This book covers aspects of multiphase flow and heat transfer during phase change processes, focusing on boiling and con

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- Fabio Toshio Kanizawa
- Gherhardt Ribatski

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*Table of contents : PrefaceContentsNomenclatureChapter 1: Introduction 1.1 Problems ReferencesChapter 2: Fundamentals 2.1 Basic Definitions 2.2 Flow Patterns 2.2.1 Flow Patterns During Vertical Adiabatic Flow 2.2.2 Flow Patterns for Horizontal Adiabatic Flows 2.3 Void Fraction 2.3.1 Local Void Fraction 2.3.2 Line Averaged Void Fraction 2.3.3 Area Averaged Void Fraction 2.3.4 Volume Averaged Void Fraction 2.3.5 Void Fraction Predictive Methods 2.3.5.1 Slip Ratio Method 2.3.5.2 Drift Flux Model - Zuber and Findlay Method 2.3.5.3 Minimum Entropy Generation - Zivi´s Method 2.3.5.4 Minimum Kinetic Energy Method - Kanizawa and Ribatski Method 2.4 Flow Boiling Fundamentals 2.5 In-Tube Condensation Fundamentals 2.6 Transition from Macro to Microscale Conditions 2.7 Solved Example 2.8 Problems ReferencesChapter 3: Flow Patterns 3.1 Flow Pattern Identification 3.2 Flow Pattern Transition Criteria for Adiabatic Flows 3.2.1 Graphical Methods 3.2.2 Taitel and Dukler (1976) 3.2.3 Taitel, Barnea, and Dukler (1980) 3.2.4 Barnea, Shoham, and Taitel (1982a) 3.3 Predictive Methods for Convective Boiling 3.3.1 Wojtan, Ursenbacher, and Thome (2005) 3.3.2 Revellin and Thome (2007) 3.3.3 Ong and Thome (2011) 3.4 Predictive Method for Convective Condensation 3.4.1 El Hajal, Thome, and Cavallini (2003) 3.4.2 Nema, Garimella, and Fronk (2014) 3.5 Solved Examples 3.6 Problems ReferencesChapter 4: Pressure Drop 4.1 Predictive Methods for Frictional Pressure Drop Parcel 4.1.1 Homogeneous Model 4.1.2 Lockhart and Martinelli (1949) 4.1.3 Chisholm (1967) 4.1.4 Müller-Steinhagen and Heck (1986) 4.1.5 Cioncolini, Thome, and Lombardi (2009) 4.2 Solved Examples 4.3 Problems ReferencesChapter 5: Flow Boiling 5.1 Nucleate Boiling Concepts 5.2 Heat Transfer Coefficient for Convective Boiling 5.3 Predictive Methods for Convective Flow Boiling 5.3.1 Liu and Winterton (1991) 5.3.2 Saitoh et al. (2007) 5.3.3 Kandlikar and Co-workers 5.3.4 Wojtan et al. (2005a, b) 5.3.5 Thome and Co-workers 5.3.6 Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017) 5.3.7 Heat Transfer Coefficient Under Transient Heating 5.4 Solved Examples 5.5 Problems ReferencesChapter 6: Critical Heat Flux and Dryout 6.1 Introduction 6.2 Hydrodynamic Model 6.3 Macrolayer Model 6.4 Critical Heat Flux During In-Tube Flow 6.5 Solved Example 6.6 Problems ReferencesChapter 7: Condensation 7.1 Film Condensation on an Isothermal Surface 7.2 Predictive Methods for In-Tube Convective Condensation 7.2.1 Dobson and Chato (1998) 7.2.2 Cavallini et al. (2006) 7.2.3 Shah (2016) 7.2.4 Jige, Inoue, and Koyama (2016) 7.3 Solved Examples 7.4 Problems ReferencesIndex*

Mechanical Engineering Series

Fabio Toshio Kanizawa Gherhardt Ribatski

Flow boiling and condensation in microscale channels

Mechanical Engineering Series Series Editor Francis A. Kulacki, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

The Mechanical Engineering Series presents advanced level treatment of topics on the cutting edge of mechanical engineering. Designed for use by students, researchers and practicing engineers, the series presents modern developments in mechanical engineering and its innovative applications in applied mechanics, bioengineering, dynamic systems and control, energy, energy conversion and energy systems, ﬂuid mechanics and ﬂuid machinery, heat and mass transfer, manufacturing science and technology, mechanical design, mechanics of materials, micro- and nano-science technology, thermal physics, tribology, and vibration and acoustics. The series features graduate-level texts, professional books, and research monographs in key engineering science concentrations.

More information about this series at http://www.springer.com/series/1161

Fabio Toshio Kanizawa • Gherhardt Ribatski

Flow Boiling and Condensation in Microscale Channels

Fabio Toshio Kanizawa Laboratory of Thermal Sciences (LATERMO) Universidade Federal Fluminense, School of Engineering Niterói, Rio de Janeiro, Brazil

Gherhardt Ribatski Department of Mechanical Engineering University of São Paulo, São Carlos School of Engineering São Carlos, São Paulo, Brazil

ISSN 0941-5122 ISSN 2192-063X (electronic) Mechanical Engineering Series ISBN 978-3-030-68703-8 ISBN 978-3-030-68704-5 (eBook) https://doi.org/10.1007/978-3-030-68704-5 © The Editor(s) (if applicable) and The Author(s) 2021 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is an introductory textbook for multiphase ﬂow and heat transfer during convective boiling and condensation inside channels focusing on microscale systems. A proper analysis of heat transfer with phase change inside channels requires knowledge about the ﬂow characteristics, hence a signiﬁcant parcel of the book is dedicated to the analysis and modelling of hydrodynamic aspects of liquid and gas ﬂows and the parameters employed on their characterizations. Therefore, even though the ultimate objective of this book is the analysis of convective boiling and condensation, students focused on adiabatic ﬂow, such as for oil and gas industry, can also adopt this book as reference because it gathers several prominent studies of multiphase ﬂow subject into a single text with uniform nomenclature. It must be highlighted that the fundamentals of multiphase ﬂows have been formalized and studies have begun to be published in a systematic way mainly from the 1950s. From the late 1960s, the investigations in this area have experienced a search for mathematical formality and a tendency of unifying the modelling approach. Hence, multiphase ﬂows is a relatively young subject of research, and this book aims to contribute on this task by addressing classical and more recent studies of the area into a single text. The characteristics of multiphase ﬂows depend on several aspects, including ﬂuids properties, operational conditions, phases proportion, channel orientation, and geometry, and whether the ﬂow is heated, cooled, or adiabatic, among other aspects. Therefore, the complete analysis of the ﬂow requires a detailed investigation of the phenomena occurring along each phase and how the phases interact among themselves and with the ﬂow boundary conditions. Such an approach is challenging; in this context, numerical studies involving Lattice Boltzmann techniques and Molecular Dynamic and Direct Numerical Simulations are being employed to investigate the microscopic aspects of two-phase ﬂows, and based on them, complex models are being constructed. Up-to-date instrumentation such as high-speed video cameras, IR thermography, and micro PIVs, among others, are also being employed in order to investigate the microscopic aspects of heat transfer and momentum diffusion in two-phase ﬂows. However, an analysis of such studies are out of the v

vi

Preface

scope of the present book that focuses mainly on analyses assuming one-dimensional ﬂow with averaged properties along the cross section, with their variation along the channel length due to heat and/or momentum transfer. Even though modelling such a complex problem through a one-dimensional approach might seem to be over simpliﬁed, this procedure has been in use for several decades and has provided reliable results because most of the models developed based on one-dimensional approach relies on innovative modellings and adjustment of empirical constants based on broad experimental databases. Although the book’s title indicates its main focus on microscale channels, the reader shall notice that most analyses begin with descriptions of models for conventional-sized channels following a historical order and then switch to microscale channels. This sequence is adopted based on the fact that usually the models developed for microscale conditions derive from models for conventional channels. As a consequence, the basic parameters employed on the characterization of multiphase ﬂows inside conventional channels and mini and microchannels are similar. Differences arise from distinctions between the dominant forces and the main physical mechanisms, for example, stratiﬁed ﬂows in horizontal channels, characterized by the liquid ﬂowing in the lower part of the channel and vapor in its upper part, are not observed in microscale channels due to the predominance of surface tension over gravitational effects on the two-phase ﬂow distribution. The present textbook is focused on gas–liquid ﬂow of the same substance, and most of the modelling approaches consider saturated liquid and vapor in thermal equilibrium, and when not speciﬁed differently, conditions of incompressible ﬂow (low Mach number) are assumed. Additionally, most of the analyses presented here refer to the ﬂow along a single channel; however, it must be highlighted that differences related to ﬂow maldistribution effects and reverse ﬂow, among others, might result in distinct behavior for multichannel system when compared with single channel. Based on these aspects, this book starts with an introductory and motivational chapter, which aims to explain the advantages and applications concerning phase change in microscale channels. Then, Chap. 2 presents the fundamental parameters of multiphase ﬂow, including fundamental aspects of ﬂow patterns, void fraction, and heat transfer processes. Chapter 3 is dedicated to ﬂow pattern characterization and prediction, including conditions of conventional and microscale channels. Then, Chap. 4 presents the fundamental aspects of pressure drop for single- and two-phase ﬂows and derives the dominant equations for the later. Additionally, several predictive methods for estimate of the frictional parcel of the pressure drop are addressed and compared. Chapters 5 and 6 cover pool and convective boiling, critical heat ﬂux, and dry out, presenting an overview of the heat transfer trends, dominant mechanisms, and methods to predict the heat transfer coefﬁcient. Finally, Chap. 7 addresses the subject of condensation, describing the ﬁlm condensation model of Nusselt followed by an analysis of the condensation inside channels. Regarding the organization of this book, whenever possible, usual nomenclature of the ﬁeld is adopted to facilitate the transition from and to different textbooks.

Preface

vii

Additionally, when not explicitly mentioned differently, it is assumed that the working ﬂuid is a pure substance, hence it presents constant saturation temperature for a given pressure. Several enhancement techniques for heat transfer in microscale channels are available in the open literature, such as micro ﬁns, re-entrant cavities, and porous coatings. Nonetheless, since these methods are not consolidated, this book is focused on heat transfer in microscale channels without any enhancement. A complete understanding of the subjects covered in this book requires a previous knowledge of thermodynamics, ﬂuid mechanics, and convective heat transfer during single-phase ﬂow. An experienced researcher may ﬁnd the descriptions of traditional models and predictive methods a bit different from the original presentations, such as the ﬂow pattern transition criteria presented by Taitel and Dukler (1976) and the Lockhart and Martinelli (1949) developments on pressure drop, mainly related to the basic parameters adopted to describe the mixture ﬂow. Whenever possible and convenient, the present authors address the models based on the parameters presented in Chap. 2, which include void fraction, superﬁcial velocities, and so on, rather than liquid height, which are not uncommon in classical literature. By completing the study of the subject addressed in this book, the reader shall be able to discuss the main aspects of multiphase ﬂow and heat transfer processes. Additionally, it is expected that the students should be able to infer the dominant mechanisms for heat and momentum transfer, design equipment and systems operating with phase change, and continue their studies with speciﬁc literature for the subject of investigation. São Paulo, Rio de Janeiro, Brazil Niteròi, Rio de Janeiro, Brazil São Carlos, São Paulo, Brazil

Fabio Toshio Kanizawa Gherhardt Ribatski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 10 10

2

Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Flow Patterns During Vertical Adiabatic Flow . . . . . . . . . 2.2.2 Flow Patterns for Horizontal Adiabatic Flows . . . . . . . . . 2.3 Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Local Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Line Averaged Void Fraction . . . . . . . . . . . . . . . . . . . . . 2.3.3 Area Averaged Void Fraction . . . . . . . . . . . . . . . . . . . . . 2.3.4 Volume Averaged Void Fraction . . . . . . . . . . . . . . . . . . 2.3.5 Void Fraction Predictive Methods . . . . . . . . . . . . . . . . . . 2.4 Flow Boiling Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 In-Tube Condensation Fundamentals . . . . . . . . . . . . . . . . . . . . . 2.6 Transition from Macro to Microscale Conditions . . . . . . . . . . . . 2.7 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 20 21 23 28 28 29 29 31 33 43 49 51 56 60 61

3

Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1 Flow Pattern Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Flow Pattern Transition Criteria for Adiabatic Flows . . . . . . . . . . . 68 3.2.1 Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Taitel and Dukler (1976) . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.3 Taitel, Barnea, and Dukler (1980) . . . . . . . . . . . . . . . . . . . 85 3.2.4 Barnea, Shoham, and Taitel (1982a) . . . . . . . . . . . . . . . . . 96 3.3 Predictive Methods for Convective Boiling . . . . . . . . . . . . . . . . . 103 3.3.1 Wojtan, Ursenbacher, and Thome (2005) . . . . . . . . . . . . . . 103

. . . . . . . . . . . . . . . . .

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Contents

3.3.2 Revellin and Thome (2007) . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Ong and Thome (2011) . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Predictive Method for Convective Condensation . . . . . . . . . . . . . 3.4.1 El Hajal, Thome, and Cavallini (2003) . . . . . . . . . . . . . . 3.4.2 Nema, Garimella, and Fronk (2014) . . . . . . . . . . . . . . . . 3.5 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

107 110 112 114 117 120 121 122

4

Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Predictive Methods for Frictional Pressure Drop Parcel . . . . . . . . 4.1.1 Homogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Lockhart and Martinelli (1949) . . . . . . . . . . . . . . . . . . . . 4.1.3 Chisholm (1967) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Müller-Steinhagen and Heck (1986) . . . . . . . . . . . . . . . . 4.1.5 Cioncolini, Thome, and Lombardi (2009) . . . . . . . . . . . . 4.2 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

125 137 137 140 146 149 151 156 158 159

5

Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Nucleate Boiling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Heat Transfer Coefﬁcient for Convective Boiling . . . . . . . . . . . . 5.3 Predictive Methods for Convective Flow Boiling . . . . . . . . . . . . 5.3.1 Liu and Winterton (1991) . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Saitoh et al. (2007) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Kandlikar and Co-workers . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Wojtan et al. (2005a, b) . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Thome and Co-workers . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017) . . . . . . . . . . . . . . 5.3.7 Heat Transfer Coefﬁcient Under Transient Heating . . . . . 5.4 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

161 163 179 189 190 191 194 196 199

. . . . .

205 208 210 213 214

Critical Heat Flux and Dryout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Hydrodynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Macrolayer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Critical Heat Flux During In-Tube Flow . . . . . . . . . . . . . . . . . . . 6.5 Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

217 217 219 224 228 235 238 239

6

Contents

7

Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Film Condensation on an Isothermal Surface . . . . . . . . . . . . . . . 7.2 Predictive Methods for In-Tube Convective Condensation . . . . . . 7.2.1 Dobson and Chato (1998) . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Cavallini et al. (2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Shah (2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Jige, Inoue, and Koyama (2016) . . . . . . . . . . . . . . . . . . . 7.3 Solved Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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241 242 250 251 254 256 257 260 262 263

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Nomenclature

A Cf d ds D E Fc f g G h H î j k K L m ṁ M p P q Q ! r r Rp Ra s

transversal area, m2 Darcy friction factor, non-dimensional diameter, m line segment, m domain energy, J enhancement factor of convective effects, non-dimensional Fanning friction factor, non-dimensional gravitational acceleration, m/s2 mass ﬂux, kg/m2s heat transfer coefﬁcient, W/m2K height, m speciﬁc enthalpy, J/kg superﬁcial velocity, m/s thermal conductivity, W/mK momentum coefﬁcient, non-dimensional length, m mass, kg mass ﬂow rate, kg/s molar mass, kg/kmol pressure, Pa perimeter, m heat transfer rate, W volumetric ﬂow rate, m3/s position vector, m radius, m surface peak roughness, μm surface averaged roughness, μm entropy, J/kgK xiii

xiv

ŝ S Snb T Tˆ t u V V w x X x y Ŷ z Δp Δt

Nomenclature

Laplace variable, 1/s surface area, m2 suppression factor of nucleate boiling effects, non-dimensional temperature, C time constant, s time instant, s local velocity, m/s velocity, m/s volume, m3 in situ axial velocity, m/s axis perpendicular to z and y density function, non-dimensional vapor quality, non-dimensional horizontal axis perpendicular to the duct axis gravitational parameter of Taitel and Dukler model, non-dimensional axial direction pressure difference, Pa time interval, s

Greek Symbols α β βc ε Γ γ δ η θ λ Λ μ ρ σ τ ϕ ω ξ

void fraction, non-dimensional volumetric fraction, non-dimensional contact angle, entrainment factor, non-dimensional propagation velocity of interfacial perturbations, m/s stratiﬁcation angle, rad liquid ﬁlm thickness, m generic parameter inclination relative to horizontal plane, rad mean free path, m wavelength, m viscosity, kg/m.s density, kg/m3 surface tension, N/m shear stress, Pa heat ﬂux, W/m2 frequency, Hz liquid holdup of entrained droplets, non-dimensional

Nomenclature

Subscripts 0 1 2 3 b c cs e f ﬂuid g h i I k l l0 lv m n nb r sat v v0 z w cap

relative to a point relative to a line segment relative to an area relative to a volume bubble relative to convective effects relative to the channel wall section equivalent frictional parcel evaluated for bulk conditions gravitational parcel hydraulic relative to phase i interface relative to kinetic energy component relative to liquid phase assuming the mixture ﬂowing as liquid vaporization momentum parcel domain dimension relative to nucleate boiling reduced property saturation condition relative to vapor phase assuming the mixture ﬂowing as vapor axial direction evaluated at the wall temperature relative to capillar effects

Nondimensional Parameters Bd Bo Ca Cn Co Eo Fr Ga Kn La*

Bond number Boiling number Capillary number Convection number Conﬁnement number Eötvos number Froude number Galileo number Knudsen number Laplace number

xv

xvi

Pr Re We X^lv

Nomenclature

Prandtl number Reynolds number Weber number Lockhart and Martinelli parameter

Operators n

Space average operator Time average operator

Chapter 1

Introduction

Multiphase ﬂow corresponds to simultaneous ﬂow of two or more immiscible phases, which can be gas-liquid, liquid-liquid, gas-solid, liquid-solid, or a combination of these pairs, and its characteristic is strongly dependent on the interface between the phases. The simplest multiphase ﬂow corresponds to two-phase ﬂow that consists in simultaneous ﬂow of two immiscible phases. For liquid-liquid ﬂow to be treated as a multiphase ﬂow, it is necessary that they are composed of immiscible liquids, such as oil and water. Similarly, the tendency of gases to get mixed when put in contact also precludes mixtures of different gases to be characterized as multiphase ﬂows. Multiphase ﬂow is present in several natural and industrial processes. The morning shower can be treated as a two-phase ﬂow, with liquid water droplets ﬂowing downward due to gravitational forces and interacting with air and vapor mixture. The water boiling in the kettle for the morning coffee or tea is also a condition of multiphase ﬂow, with water vapor being formed in the kettle bottom surface, detaching and rising due to buoyance forces. The operation of internal combustion engines of the commute involves multiphase ﬂow in several processes, such as the fuel injection that consists of a mixture of air and dispersed droplets of liquid fuel, and the lubricating oil in contact with air and engine parts in the oil pan, among others. In recreational aspects, the multiphase ﬂow is also present, such as when pouring beer from the bottle into a pint, the condition in the bottle neck corresponds to counter current the ﬂow of air entering the recipient and liquid exiting it. With the pressure reduction due to the opening of the bottle, the dissolved gas tends to be released in the form of bubbles forming the beer foam, and when in contact with hotter surface of the glass, additional bubbles are formed and rise due to buoyance forces. Several examples of multiphase ﬂow can be found also in natural processes, such as rain, fog, clouds, wave formation, dust, sandstorm, etc. The cases of multiphase ﬂow with a solid phase, such as in dust transport by the wind or slurry ﬂow, the mixture behaves as ﬂuid and then can be treated as multiphase ﬂow.

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_1

1

2

1 Introduction

The subject of multiphase ﬂow has been investigated by several industrial areas, including power generation from fossil fuels, nuclear source, or even based on renewable energy source more recently, when considering the application of organic Rankine cycles (ORC) for solar or geothermal energy harvesting. Additionally, and according to that described above, the multiphase ﬂow is also present in refrigeration and heat management applications, oil and gas industry, and material transport, among others. The pioneer studies on multiphase problems have been addressed at the beginning of the twentieth century, by investigations of heat transfer problems with phase change. It should be highlighted that in this period, even single-phase ﬂow was a challenging and unknown world, and Prandtl solution of the boundary layer problem was just coming up in 1904. Just a few years later, Nusselt (1916) described the modelling of condensation on a vertical isothermal plate which can be considered one of the ﬁrst theoretical studies for heat transfer of phase change processes. From the 1940s, several studies focusing on multiphase ﬂows have been published, with successive increment of complexity and agreement with empirical observations. Subsequently, from the 1970s the number of publications focused on nuclear industry has increased, and since then the overall publications related to multiphase ﬂow have increased sharply, emphasizing the growing interest and relevance of the theme. Figure 1.1 depicts the number of publications along the decades according to different areas of applications and segregating two-phase ﬂow multiphase ﬂow studies. Boiling, condensation, freezing, melting, and sublimation are heat transfer processes associated to multiphase ﬂows that provide reasonable high heat transfer

Number of publications 100000 10000 1000 100 10 1

Refrigeration

Two-phase flow

Multiphase flow

Steam generator

Nuclear multiphase Oil two-phase flow

Fig. 1.1 Number of publications along the time according to search term for abstract, title, or keyword in July of 2019, according to Science Direct database in June of 2019

1 Introduction

3

coefﬁcients. For example, R134a single-phase liquid ﬂow in a 1.5 mm ID channel the heat transfer coefﬁcient is approximately 1300 W/m2K for a mass velocity of 500 kg/m2s and a ﬂuid temperature of 30 C, while, under conditions of convective boiling, values in the order of 5000 to 10,000 W/m2K are achieved for the same mass ﬂow rate and temperature. Recall that the heat transfer coefﬁcient for natural convection of a surface exposed to gases is of the order of 5 W/m2K, and for forced convection of gases is of the order of 80 W/m2K. The high heat transfer coefﬁcients typical of phase change processes allows the minimization of the equipment size compared to similar equipment working under single-phase conditions. Nonetheless, it is important to highlight that it is also possible to obtain high heat transfer coefﬁcients under single-phase ﬂow conditions, such as for forced convection of liquid metals and forced convection of conventional ﬂuids at high or extremely high velocities. However, their use is restricted to special applications, such as nuclear power generation. In this context, the reader should recall from the thermodynamic course that one example of irreversibility is related to heat transfer with a ﬁnite temperature difference (ΔT ), whereas the reversible heat transfer process requires inﬁnitesimal temperature difference (dT). Hence, recalling the Newton’s Cooling Law from the heat transfer course, the heat ﬂux for convective heat transfer is given as follows: ϕ ¼ hΔT

ð1:1Þ

where ϕ is the heat ﬂux in W/m2 from the surface to the ﬂuid, ΔT is the temperature difference between the surface and the ﬂuid, and h is the heat transfer coefﬁcient in W/m2K in SI units. Hence, for a given heat ﬂux, the lower the heat transfer coefﬁcients, the higher the temperature difference and consequently the irreversibility. The condition of inﬁnitesimal temperature difference would be achieved only for very high heat transfer coefﬁcient, as discussed before (ϕ ¼ hΔT hdT for very high h). Additionally, the reader shall remember that the Carnot cycle is the most efﬁcient thermodynamic cycle, either for power generation or refrigeration, composed by two isentropic and adiabatic processes, corresponding to compression, or pumping, and expansion processes and two isothermal processes, corresponding to heat transfer processes with inﬁnitesimal temperature difference. Hence, a single-phase liquid ﬂowing along a channel that receives or rejects energy will present temperature variation which deviates from the isothermal process, unless conditions of very highspeciﬁc heat, density, and/or mass ﬂow rate are imposed. The ﬂuid temperature variation could be overcome using phase change processes for pure substance, whereas for vapor-liquid in equilibrium of pure substances the temperature variation is related only to pressure variation. Therefore, by properly designing the heat exchangers to obtain low pressure drop, the heat transfer process in the cycle can be considered as almost isothermal. Figure 1.2 schematically depicts the thermodynamic diagram of theoretical cycles for refrigeration based on vapor compression and power generation, assuming pure substance and thermal reservoirs with uniform temperature. Considering the case of refrigeration, depicted in Fig. 1.2a, the refrigerated space temperature Tref must be higher than evaporation temperature, and the

4

1 Introduction

=

cte

c

p

p=

cte

b

p=

cte

a

Tsource T

Text

T

Tref

Carnot cycle Text

Carnot cycle

Theoretical cycle s Refrigeration - Vapor compression

Theoretical cycle s Power generation - Rankine

T

Text

Carnot cycle

Tref Theoretical cycle s Refrigeration - Gas cycle

Fig. 1.2 Schematics of refrigeration and power generation cycles in comparison with Carnot cycle – temperature vs. entropy diagram (a) Refrigeration - Vapor compression cycle (b) Power generation - Rankine (c) Refrigeration - Gas cycle

condensation temperature must be higher than the temperature of the ambient to which heat is rejected Text. Similarly, in the case of power generation depicted in Fig. 1.2b, the temperature of the thermal reservoir that is the energy source to generate vapor Tsource must be higher than the ﬂuid temperature along the heat exchanger, and the condensing temperature must be higher than the temperature of the ambient to which energy is rejected Text. In a practical case, the Tsource could be deﬁned in two stages, with a lower temperature in the phase change region, and higher in the superheating region to reduce requirements of the hot thermal reservoir. In any case, either for refrigeration or power generation, Fig. 1.2 also depicts the Carnot cycle, which is closer to the theoretical cycle with phase change in comparison with the single-phase cycle, depicted in Fig. 1.2c for refrigeration with gas cycle. It can be added to this discussion that the high vaporization latent heat of conventional ﬂuids, for example, for water at atmospheric pressure, the vaporization enthalpy is approximately 2.257 MJ/kg, and the speciﬁc heat at constant pressure of the saturated liquid at the same pressure is 4.217 kJ/kgK. Hence, to sustain the same heat transfer of vaporization, the same mass of liquid water would have to experience a temperature variation higher than 500 K, and would not be liquid anymore. Even considering ﬂuid and operational conditions that support such temperature variation, the temperature of the hot source needs to be higher than the highest temperature of the working ﬂuid, such as depicted in Fig. 1.2, which impacts the costs of the heat source and of the materials to sustain such temperatures. Conversely, when using liquid-vapor ﬂow at almost uniform temperature, the heat source can be at a temperature slightly above the saturation temperature. Still considering refrigeration processes, the assumption of isobaric heat transfer processes in the heat exchangers is usually adopted for didactical and modelling purposes, which relies on the condition that the pressure drop in heat exchangers is null. However, in real systems the pressure reduces along the ﬂow path due to ﬂuid friction with the channel wall, and due to phase interactions in the case of multiphase ﬂow. Figure 1.3 depicts schematically the pressure vs. enthalpy diagram for vapor compression cycle, similar to the one presented in Fig. 1.2a, however also with the

p

Theoretical cycle Text

s=

Fig. 1.3 Vapor compression cycle with effect of pressure drop along the heat exchangers – pressure vs. enthalpy diagram

5

cte

1 Introduction

Carnot cycle

s=c te

Tref

Δic,1

Cycle with Δic,2 pressure drop i Refrigeration - Vapor compression

effect of pressure drop along the heat exchangers. According to Fig. 1.3, the pressure drop in heat exchangers implies on variation of the ﬂuid temperature, increasing the deviation from the Carnot cycle. Additionally, according to Fig. 1.3 the pressure in the heat exchangers implies on increment of the compressing power. In the case of theoretical cycle with no pressure drop, the compressing power is given as mass ﬂow rate multiplied by the enthalpy variation in the compressor, denoted by Δic,1, while in the case that pressure drop is accounted for, the enthalpy difference increases to Δic,2. The reader might have noticed that even the cooling capacity has been impacted, and an increase in the mass ﬂow rate, or superheating of the vapor, would be necessary to keep the same capacity. Based on this discussion, it can be concluded that it is impossible to treat only the heat transfer processes and not consider the momentum transfer processes, associated with pressure drop of internal ﬂows. Therefore, this book dedicates Chap. 4 to this subject. In this context, it could be mentioned that systems with large capacity, such as air-conditioning in shopping malls or refrigeration in meat industry, usually counts with shell and tube heat exchangers as evaporator and condenser, with refrigerant operating in the shell side, hence with lower pressure drop. However, considering systems with low or intermediate capacity, such as domestic and commercial refrigerator and air-conditioning, usually the evaporator and condenser consist in coiled and ﬁnned heat exchangers with refrigerant in the tube side, which consists in a condition of relevant ﬂow resistance. Still in the subject of refrigeration and air-conditioning, a recent report published by International Energy Agency (IEA 2018) pointed out that space cooling consumed more than 25% of USA’s total electrical energy demand during a condition of peak consumption in 2016, and the world averaged parcel is of the order of 10%. The same report indicated that the expected increment of energy demand for space cooling in the world from 2016 to 2050 is approximately 2500 GW, which is higher than the combined generation capacity of the USA, Europe, and India in 2016,

6

1 Introduction

Fig. 1.4 Share of renewable energy in global consumption in 2010. The World Bank (2018)

implying in higher demand for natural resources and impacting the environment. A possible way to mitigate this effect is the development of more efﬁcient refrigeration systems, which rely on improvement of compressor, heat exchangers, and development of new materials and ﬂuids. In this context, The World Bank (2018) indicated that more than 75% of the electrical energy was generated from non-renewable sources in 2010, such as depicted in Fig. 1.4, which impacts the environment and motivates additional research to develop and improve renewable energy sources to sufﬁce the inevitable increment of demand. Hence, photovoltaic solar cells have been in development for several decades, and nowadays their use has become more common due to reduction of price. Nonetheless, the current efﬁciency in converting solar energy into electrical energy is between 12 to approximately 17% for single-junction and in laboratorial conditions according to the report of Fraunhofer Institute for Solar Energy Systems (Fraunhofer Institute for Solar Energy Systems, ISE 2019), and the non-converted parcel is reﬂected or absorbed by the material as heat, which must be dissipated to avoid superheating that would cause severe efﬁciency deterioration or even permanent damage of the component. Singh and Ravindra (2012) evaluated the effect of operating temperature on the open circuit tension, closed circuit current, and converting efﬁciency and concluded that the three parameters reduced with increment of temperature. The heat can be properly dissipated by natural convection in the case of conventional solar cells, however in the case of concentrated solar cells, the energy input ﬂux can be of the order of 300 kW/m2 according to Sinton et al. (1986) and Green et al. (2015). In these cases, air cooled heat spreaders do not sufﬁce the heat dissipation, and alternative approaches are needed, such as using microchannel heat sinks with phase change process, which has potential of absorbing high heat ﬂuxes with low temperature variation. In the same context, another approach for solar energy harvesting that has gained interest is the organic Rankine cycle (ORC), which is fundamentally similar to the steam Rankine cycle (SRC), however, it operates with organic ﬂuids rather than water. Hence, due to the characteristics of the organic ﬂuids, the energy source for vapor generation in the ORC can be of lower temperature in comparison with SRC, which allows the operation with low grade thermal energy sources, such as from solar collectors, geothermal reservoirs, and industrial waste heat. In this context,

1 Introduction

7

Fig. 1.5 Variation of transistors count in processors between 1971 and 2018 – Moore’s Law

Tocci et al. (2017) recently presented a review about this concept and listed the main advantages and disadvantages of ORC in comparison with SRC systems. Nonetheless, even though ORC-based systems are already commercially available, several aspects of the heat and momentum transfer for the characteristic operational conditions are still unclear, because they are way distinct from those for refrigeration systems. This aspect emphasizes the need for additional studies focusing on the understanding of the main mechanisms occurring during heat transfer process in the heat exchangers, aiming to improve the design, reliability, and efﬁciency of systems based on ORC. Another application that heat transfer with phase change has gained attention in the recent decades is related to heat management of electrical and electronic equipment, mostly focused on semiconductor components, such as processors, high power transistors, and laser sources. In this context, in 1965 Gordon E. Moore foresaw that the number of transistors in processors would double every year, which was revised in 1975 to doubling every 2 years, and this prediction is nowadays known as Moore’s Law, which has shown to be in agreement with the evolution of computer processors as shown in Fig. 1.5. With the increment of transistor numbers and clocks, the heat dissipation per unit volume has increased accordingly, and the heat dissipation to keep the semiconductor temperature within the allowed operational range, usually below 65 C, can be accomplished by air-cooled heat sinks until recently. However, considering the trend of continuous increment of power dissipation in reduced spaces/areas, air cooled systems will not sufﬁce and use of other working ﬂuid is an alternative, which in turns bring the use of microchannel due to space restrictions.

8

1 Introduction

The appealing aspect of increment of efﬁciency of heat management of processes can also be seen on basis of the energy consumption of data centers. According to Zhang et al. (2017) and Song et al. (2015), approximately 40% of the electricity consumption of data centers is related to air-conditioning to keep the processor in the optimal temperature range. And according to Jones (2018), the share of datacenter in 2018 on the electricity consumption is approximately 1% worldwide; however, it is expected that by 2030 this parcel would increase to approximately 8%, also increasing the demand for more energy production. Again, a way of reducing this impact would be the development of more efﬁcient processors and heat management systems, such as direct cooling of the chip instead of the entire system and building. In this context, the reduction of channel diameter itself already implies the increment of the heat transfer coefﬁcient. For example, consider a condition of laminar ﬂow with imposed heat ﬂux, which is quite common since the ﬂow rate required to reach turbulent ﬂow regime is usually very high in microscale channel due to its reduced diameter. Therefore, the heat transfer problem in developed singlephase ﬂow consists in a condition of constant Nusselt number of 4.364 for uniform heat ﬂux condition, and the diameter reduction implies on the increment of the heat transfer coefﬁcient as follows: Nu ¼ 4:364 ¼

hd k ) h ¼ 4:364 k d

ð1:2Þ

where h is the heat transfer coefﬁcient in W/m2K, d is the channel internal diameter in meters, and k is the ﬂuid thermal conductivity in W/m.K. Hence, the reduction of channel size implies on more efﬁcient heat exchangers. On the other hand, the heat transfer rate for this speciﬁc condition is independent of the channel diameter, as follows: k q_ ¼ hAΔT lm ¼ 4:364 πdLΔT lm ¼ 4:364πkLΔT lm d

ð1:3Þ

where L is the length of the considered channel and ΔTlm is the log mean temperature difference between the duct wall and ﬂuid bulk temperature. Therefore, reduction of the channel size implies on reduction of the material cost and refrigerant charge for the same heat transfer rate. It must be highlighted that, according to the analysis of this simple case, the mass ﬂow rate presents no inﬂuence on the total amount of heat removed by the ﬂuid, as long as the ﬂow is kept under laminar regime. The inﬂuence of the ﬂow rate could be perceived in the enthalpy variation, which can lead to the saturation condition and consequent two-phase ﬂow boiling in the case of positive heat ﬂux.

1 Introduction

9

On the other hand, the friction factor for laminar developed ﬂow is given as follows: 16 f ¼ Gd ð1:4Þ μ

where G is the mass ﬂux in kg/m2s and μ is the ﬂuid viscosity in kg/ms. And the pressure drop is given as follows: Δp ¼ 2f

G2 L GL ¼ 32μ 2 ρd ρd

ð1:5Þ

Therefore, the pressure drop increases according to a square power with the reduction of the channel diameter for round duct and laminar ﬂow. Continuing the analysis, the pumping power parcel due to the pressure drop in the considered channel can be given by the product of the pressure drop and the volumetric ﬂow rate, as follows: Pumping power ¼ ΔpQ ¼ Δp

G πd2 GL G πd2 G2 L ¼ 32μ 2 ¼ 8πμ 2 ρ 4 ρ ρd ρ 4

ð1:6Þ

Therefore, for the same mass velocity, or mass ﬂux, the pumping power is independent of the duct diameter. It must be emphasized that this analysis is restricted to the condition of laminar ﬂow in round duct, and its validity depends on limitation of heat ﬂux to avoid phase change by vaporization. Nonetheless, this analysis gives us an idea about the advantages of the use of microscale channels related to reduction of channel size and refrigerant charge for a similar heat transfer rate and pumping power. In the case of multiphase ﬂow of pure substance, as above mentioned, the temperature variation depends basically on pressure variation, and this characteristic provides more uniform temperature of the surface to be cooled. Considering the applications for heat management of semiconductors, this aspect is also very important because the presence of hot spots along the component can deteriorate its capacity, as pointed out by Royne et al. (2005) and Baig et al. (2012) for photovoltaic cells. These studies concluded that hot spots could deteriorate the system efﬁciency, or even damage it permanently. Nonetheless, a similar analysis as the one presented above for phase change problems would be way more complex, as the reader will see along the book. Therefore, this textbook aims to present the fundamental concepts of multiphase ﬂow, with successive increment of complexity in the analysis for each topic. Hence, the foregoing chapters deal with distinct aspects related to multiphase ﬂow, and for heat transfer during boiling and condensation.

10

1 Introduction

1.1

Problems

1. List examples of multiphase ﬂow that you faced since you woke up. 2. List examples of phase change processes that you faced since you woke up. 3. Consider a refrigeration system operating with R134a with evaporating temperature of 20 C and condensing temperature of 40 C. Evaluate the coefﬁcient of performance COP considering Carnot cycle, theoretical cycle, and a cycle with pressure drop of 10% of the inlet pressure in each heat exchanger assuming isentropic compression. 4. Repeat Exercise 3 assuming isentropic efﬁciency of 75%. 5. Evaluate the required heat transfer area for a tube-in-tube counter-current heat exchanger for 1 kW operating with the following (tip: use effectiveness approach, and assume the thermal resistance related to conduction as negligible for items a, b, and c): (a) Single-phase water on both sides, with inlet temperatures of 90 and 20 C, and respective mass ﬂow rates of 0.025 and 0.030 kg/s, and heat transfer coefﬁcient for each side as approximately 1000 W/m2K. (b) Repeat previous item assuming isothermal ﬂuids. (c) Single-phase water ﬂow in the shell at 90 C and mass ﬂow rate of 0.025 kg/s, and R134a boils inside at evaporating temperature of 20 C and averaged heat transfer coefﬁcient of 5000 W/m2K. Estimate the vaporization rate in kg/s. (d) Condensation of water at 100 C in the shell, and convective boiling of R134a on the tube side, both with averaged heat transfer coefﬁcient of 5000 W/m2K. 6. Consider a hot thermal reservoir with 300 C and a cold thermal reservoir with 30 C, estimate the efﬁciency of theoretical thermal machines operating with: (a) Rankine cycle operating with water, evaporating temperature (Tsat) of 200 C, outlet of steam generator at 300 C, condensing at 30 C. Assume isentropic pumping and expansion. (b) ORC operating with R245fa, without superheating, and ﬂuid operating at 300 C in the vapor generator, and 30 C. Assume isentropic pumping and expansion.

References Baig, H., Heasman, K. C., Sarmah, N., & Mallick, T. (2012, October). Solar cells design for low and medium concentrating photovoltaic systems. In AIP conference proceedings (Vol. 1477, 1, pp. 98–101). AIP. Fraunhofer Institute for Solar Energy Systems, ISE. (2019). Photovoltaics report - with support of PSE GmbH. (report). November, 14th of 2019. Accessed in April, 9th of 2020, available at: https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/Photovol taics-Report.pdf.

References

11

Green, M. A., Emery, K., Hishikawa, Y., Warta, W., & Dunlop, E. D. (2015). Solar cell efﬁciency tables (version 45). Progress in Photovoltaics: Research and Applications, 23(1), 1–9. IEA. (2018). The future of cooling. Opportunities for energy-efﬁcient air conditioning. Paris: International Energy Agency. Jones, N. (2018). The information factories. Nature, 561(7722), 163–166. Nusselt, W. (1916). The condensation of steam on cooled surfaces. Zeitschrift des Vereins Deutscher Ingenieure, 60, 541–546. Prandtl, L. (1904). On ﬂuid motions with very small friction (in German). Third International Mathematical Congress (pp. 484–491). Heidelberg. Royne, A., Dey, C. J., & Mills, D. R. (2005). Cooling of photovoltaic cells under concentrated illumination: A critical review. Solar Energy Materials and Solar Cells, 86(4), 451–483. Singh, P., & Ravindra, N. M. (2012). Temperature dependence of solar cell performance—An analysis. Solar Energy Materials and Solar Cells, 101, 36–45. Sinton, R. A., Kwark, Y., Gan, J. Y., & Swanson, R. M. (1986). 27.5-percent silicon concentrator solar cells. IEEE Electron Device Letters, 7(10), 567–569. Song, Z., Zhang, X., & Eriksson, C. (2015). Data center energy and cost saving evaluation. Energy Procedia, 75, 1255–1260. The World Bank. (2018). Global tracking framework. Sustainable energy for all. 77889 v.3. Tocci, L., Pal, T., Pesmazoglou, I., & Franchetti, B. (2017). Small scale Organic Rankine Cycle (ORC): A techno-economic review. Energies, 10(4), 413. Zhang, X., Lindberg, T., Xiong, N., Vyatkin, V., & Mousavi, A. (2017). Cooling energy consumption investigation of data Center IT room with vertical placed server. Energy Procedia, 105, 2047–2052.

Chapter 2

Fundamentals

Two-phase ﬂow conditions are characterized by the simultaneous ﬂow of two identiﬁable phases, which can be of the same substance ﬂowing as different phases and of distinct and immiscible substances, such as liquid and non-condensable gas. Single-phase ﬂow in conventional ducts, such as straight pipes with circular crosssection and large diameters, are satisfactorily well characterized by the ﬂow rate, ﬂuid properties, and duct geometry. On the other hand, in the case of two-phase ﬂow, the channel orientation, phase geometrical distributions characterized by the ﬂow patterns, and their parcels along the duct also play an important role and preclude the unambiguous characterization of the ﬂow conditions. Therefore, additional deﬁnitions are required to characterize the two-phase ﬂow. These parameters ultimately inﬂuence the heat transfer coefﬁcient and pressure drop.

2.1

Basic Deﬁnitions

Two-phase ﬂow parameters are deﬁned in this chapter. The characterization of the operational conditions of two-phase ﬂows requires the deﬁnition of several parameters such as velocity, saturation temperature, phases distributions, and vapor fraction, among others; therefore, it is essential to deﬁne these parameters precisely and unambiguously. Most of the deﬁnitions presented here are similar to those adopted by Delhaye (1981a), Wallis (1969), and Collier and Thome (1994), and they are valid for two-phase ﬂow in microscale channels. Moreover, most of the deﬁnitions are also valid for adiabatic conditions, commonly seen in oil and gas industry. Even though the gas-liquid two-phase ﬂow rarely can be considered strictly developed due to variation of phase velocities related to phase change and/or pressure drop that causes variation of vapor density and fraction, it is possible to deﬁne its characteristics for a given cross-section, or for a “short” region of the channel. Therefore, based on the deﬁnition for an arbitrary cross-section, it is possible to extend the deﬁnitions and analysis for any cross-section along the ﬂow © The Author(s) 2021 G. Ribatski, F. Kanizawa, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_2

13

14

2 Fundamentals

path. Moreover, even though the schematics presented here are generally for circular cross-sections, the vast majority of deﬁnitions are valid for any geometry. Due to the intrinsic ﬂuctuation characteristic of two-phase ﬂow, the deﬁnition of two-phase ﬂow parameters require presentation of average operators, which can be deﬁned for time or space domains. These operators are adopted for the deﬁnition of several parameters such as void fraction and in situ velocities. The deﬁnitions presented here are similar to those addressed by Delhaye (1981b). Phase Density Function The phase density function corresponds to a binary function that indicates the ! presence of a given phase i in a position r of the domain at instant t, and is deﬁned as follows:

!

Xi r , t ¼

8

n ¼

Dn η

R

r , t dDn

!

Dn dDn

ð2:3Þ

where the subscript n denotes the domain dimension, and n denotes spatial average in the domain Dn, which can be a line segment s, an area A, or a volume V. Alternatively, the average propertie can be evaluated only in the region occupied by one of the phases as follows: ! ! η r , t X i r , t dDn Dn ¼ R ! Dn X i r , t dDn R

< ηðt Þ>n,i

ð2:4Þ

2.1 Basic Deﬁnitions

15

where the subscript i refers to the phase vapor or liquid, and the term Xi corresponds to the density function deﬁned by Eq. (2.1). Time Average Operator Two-phase ﬂow is also characterized by the temporal variation of the properties and parameters along the time. Thus, it is interesting and useful to deﬁne the time averaged operator of a parameter η that can be written as follows: Z 1 t0 þΔt ! ! η r ¼ η r , t dt Δt t0

ð2:5Þ

where over-bar denotes a time-averaged value. Additionally, it is possible to deﬁne the time average operator for a parameter η only in phase i as follows: ! ηi r ¼

R t0 þΔt ! ! R t0 þΔt ! ! η r , t X r , t dt η r , t X i r , t dt i t0 t0 ¼ R t0 þΔt ! ! X r , t dt r Δt i i t0

ð2:6Þ

!

where the term Δti corresponds to the time interval that point r is occupied by phase i. It must be highlighted that both average operators can be applied for a given parameter. For example, the proportion of phases along the cross-section, which consists in the spatial average of the density function along the cross-section, can be averaged in time. Moreover, based on the average operator properties it is possible to infer some commutative properties of the time and space average operators as follows: n < η>n ¼ < η>

ð2:7Þ

With the deﬁnition of the average operators, it is possible to introduce basic parameters for two-phase ﬂows. Delhaye (1981b) is indicated as supplementary material for further analyses of average operators. Two-Phase Flow Basic Parameters Figure 2.1 schematically depicts two-phase ﬂow in a duct with cross-sectional area A and with inclination θ relative to the horizontal plane. It is assumed that the transversal areas occupied by the vapor and liquid phases are, respectively, Av and Al, and the mass ﬂow rates of each phase through the cross-section are ṁv and ṁl, respectively. In this ﬁgure, and along the entire text, it will be assumed that the axial direction is z, the horizontal axis is y, and x is perpendicular to these axes, all denoted by bold, italic, and lower case letters.

16

2 Fundamentals

A Av Av

x

mv

y

g

g x

m z y

ml

Al

Al

0

Fig. 2.1 Schematics of two-phase ﬂow in a duct

The mixture instantaneous mass ﬂow rate ṁ is given by the sum of both phases mass ﬂow rates, as follows: m_ ¼ m_ l þ m_ v

ð2:8Þ

where ṁl and ṁv correspond to the liquid and vapor mass ﬂow rates, respectively. The instantaneous mass ﬂow rate can be determined as follows: Z

!

m_ i ðt Þ ¼

!

Z

X i ρV dA ¼ A

!

!

ρV dA

ð2:9Þ

Ai

where ρ corresponds to the phase density in kg/m3 and the integral domains correspond to the cross-sectional areas. It must be highlighted that even though the sub index i was omitted for the ﬂuid density and velocity, the integration was restricted to the cross-sectional area occupied by the phase i. This restriction is imposed by the phase density function in the ﬁrst equality and by the integration domain in the second. It is usually not feasible to determine the instantaneous mass ﬂow rate of each phase, therefore the time-averaged values are adopted and the overbar symbol is omitted. Moreover, most of the foregoing deﬁnitions are based on time-averaged parameters, and the time-averaged operator will be omitted for simplicity and to avoid pollution of the text. Additionally, the variation of the ﬂuid properties along the cross-section, such as density and viscosity, is usually considered negligible; therefore, whenever it is not speciﬁed differently, the properties addressed correspond to the mean value over the cross-section, which are determined based on the saturation condition for phase change processes. According to the thermodynamic deﬁnition, the vapor quality x is deﬁned as the vapor mass fraction in a vapor-liquid mixture. In two-phase ﬂow conditions, even considering conditions of non-equilibrium that occur during phase change processes,

2.1 Basic Deﬁnitions

17

it is assumed that the vapor quality corresponds to the relative fraction of the vapor mass ﬂow rate, and is deﬁned as follows: x¼

m_ v m_

ð2:10Þ

In conditions of two-phase ﬂow with phase change, it is quite common to deﬁne the vapor quality based on the energy balance, which is recurrently referred in the literature as the equilibrium thermodynamic vapor quality. This approach is adopted because it is usually not feasible to directly measure the mass ﬂow rate of each phase, as discussed above. Thus, the equilibrium thermodynamic vapor quality, or simply vapor quality, is determined based on the local enthalpy i assuming thermodynamic equilibrium as follows: x¼

bi bil bilv

ð2:11Þ

where the ﬂuid local enthalpy î is usually evaluated based on energy balance. The terms îl and îlv correspond, respectively, to the liquid and vaporization enthalpies evaluated at the local saturation pressure. Based on Eqs. (2.11) and (2.10), it is possible to evaluate the mass ﬂow rate of the vapor phase, and from Eq. (2.8) it is possible to evaluate the liquid mass ﬂow rate. The mass velocity G, also referred as mass ﬂux, is deﬁned as the mixture mass ﬂow rate per unit of cross-sectional area, and is given as follows: G¼

m_ m_ þ m_ v m_ l m_ v ¼ l ¼ þ ¼ Gl þ Gv A A A A

ð2:12Þ

The mass velocities of each phase, Gl and Gv, are already deﬁned in Eq. (2.12), which correspond to the mass ﬂow rate of each phase per unit of the total crosssectional area. Based on the deﬁnitions of mass velocity and vapor quality, given by Eqs. (2.10) and (2.12), respectively, it is possible to write the mass velocities of the vapor and liquid phases as follows: m_ v m_ ¼ x¼Gx A A _ m_ m Gl ¼ l ¼ ð1 xÞ ¼ G ð1 xÞ A A Gv ¼

ð2:13Þ ð2:14Þ

In the foregoing discussion, the overbar in the mass ﬂow rate term will be suppressed because it is usually considered the time averaged value, hence, ṁ ¼ ṁ.

18

2 Fundamentals

The superﬁcial velocity j, or volumetric ﬂux, is commonly deﬁned and adopted in the related literature to characterize the operational condition, as well as coordinated axis for some ﬂow pattern maps. The superﬁcial velocities correspond to the volumetric ﬂow rate per unit of cross-sectional area, and are deﬁned for the mixture and each phase as follows: j ¼ jl þ jv 1 jv ¼ hX vV z i2 ¼ Δt zi ¼ 1 jl ¼ hX l V 2 Δt

Z

Z t0

t 0 þΔt t0

t 0 þΔt

1 A

Z

1 A

Z

!

!

X v V dA dt ¼ A

!

ð2:15Þ

h i

h i !

X l V dA dt ¼ A

m_ l ρl

A

¼

m_ v ρv

A

¼

G x ρv

G ð 1 xÞ ρl

ð2:16Þ

ð2:17Þ

where the term ρ corresponds to the ﬂuid density and the subscripts l and v correspond to the liquid and vapor phases, respectively. The terms inside square brackets in Eqs. (2.16) and (2.17) correspond to the volumetric ﬂow rate of vapor and liquid phases, respectively. It must be highlighted that, even though the mixture superﬁcial velocity given by Eq. (2.15) corresponds to the mean ﬂow velocity, this parameter is different from the actual ﬂow velocity, and the difference is mainly related to the slip between the phases. It will be shown below that the actual ﬂow velocity is related to the superﬁcial velocity through the deﬁnition of the void fraction α. In an alternative deﬁnition, the superﬁcial velocities can be considered as the mean velocity of each phase if they were ﬂowing alone in the same duct, which is derived using Eq. (2.3). The instantaneous velocity u, which is closer to the actual ﬂow velocity when compared to the superﬁcial velocity, corresponds to the mean velocity evaluated along the parcel of the cross-sectional area occupied by the respective phase. It is calculated according to Eq. (2.4), and is deﬁned as follows: R V X dA z v A V z dA uv ¼< V z >2,v ¼ R ¼ AR ¼ v Av D2 X v dD2 A X v dA R R R V z X l dD2 V z X l dA A V z dA ¼ l ¼ AR ul ¼< V z >2,l ¼ DR2 Al D2 X l dD2 A X l dA R

D2 V z X v dD2

R

ð2:18Þ ð2:19Þ

where the term Vz corresponds to the axial component of the velocity vector, and Av and Al correspond to the transversal area occupied by vapor and liquid phases, respectively. The in situ velocity is deﬁned as the time averaged value of the instantaneous velocity adopting the deﬁnition of area averaged void fraction α2, which will be presented in item 2.3. Recognizing that the numerators of the last members of Eqs. (2.18) and (2.19) correspond to the volumetric ﬂow rate of vapor and liquid

2.1 Basic Deﬁnitions

19

phases, respectively, and assuming steady state condition and developed ﬂow, it is possible to write the in situ velocity as follows: uv ¼

ul ¼

R R 1 t 0 þΔt Av V z X v dAv dt Δt t 0 R R t þΔt 0 1 Av X v dAv dt Δt t 0

R R 1 t 0 þΔt Al V z X l dAl dt Δt t 0 R R t þΔt 0 1 Al X l dAl dt Δt t 0

¼

¼

j Qv Qv A Gx ¼ ¼ v¼ Av AAv α2 ρv α2

Gð1 xÞ jl Ql Ql A ¼ ¼ ¼ Al AAl 1 α2 ρl ð1 α2 Þ

ð2:20Þ

ð2:21Þ

where the term Q refers to the time averaged volumetric ﬂow rate, and Av and Al refer to time averaged cross-sectional area occupied by vapor and liquid phases, respectively. The overbar for in situ velocity u will be kept along this textbook to ensure distinction from instantaneous velocity. Additionally, based on Eqs. (2.20) and (2.21), it is possible to conclude that the void fraction α2 is equal to Av/A, and this parameter will be more deeply described and analyzed in Sect. 2.3. Useful relationships between the in situ velocity, superﬁcial velocity, mass ﬂux, and void fraction can be inferred from Eqs. (2.21) and (2.22). These equalities are recurrently used along the text as well as for derivation of models and predictive methods. The drift velocities are also recurrently presented in two-phase ﬂow textbooks, and correspond to the difference between the phases and mixture mean velocities, as follows: uvj ¼ uv j

ð2:22Þ

ulj ¼ ul j

ð2:23Þ

The volumetric fraction of the gas phase β, or simply volumetric fraction, is also a common parameter cited in the literature, and corresponds to the fraction of volumetric vapor ﬂow rate, as follows: h i m_ v

ρv jv Qv ¼h i h i¼ β¼ Qv þ Ql j þ jl _mv _ml v þ ρv

ð2:24Þ

ρl

It will be shown in Sect. 2.3 that the volumetric fraction given by Eq. (2.24) corresponds to the void fraction according to the homogeneous model. The heat transfer coefﬁcient h relates the heat ﬂux ϕ and the temperature difference between a reference solid wall Tw and the ﬂuid bulk temperature T. The heat transfer coefﬁcient is given according to Newton’s cooling law as follows: h¼

ϕ Tw T

ð2:25Þ

20

2 Fundamentals

In the following subsections, speciﬁc aspects of two-phase ﬂow focused on phase change processes in microscale channels will be discussed.

2.2

Flow Patterns

Deﬁnition and characterization of ﬂow patterns occurring during two-phase ﬂow in microscale channels. The differences between two-phase ﬂow in macro and microscale are introduced, as well as the difference between ﬂow patterns during convective condensation and boiling. Due to the two-phase ﬂow complexities, it is convenient to segregate the solutions of the two-phase ﬂow problems according to intrinsic ﬂow characteristics associated to the dominant geometrical and dynamic parameters. In two-phase ﬂows, depending on the operational conditions, the phases are distributed according to distinct geometrical conﬁgurations. Flow topologies with similar characteristics are deﬁned as ﬂow patterns. The correct prediction of the ﬂow pattern is a key aspect for the development of accurate predictive methods for heat transfer coefﬁcient, pressure drop, ﬂow-induced vibration and noise, etc., since they are intrinsically correlated to the ﬂow pattern. In this context, several heat transfer coefﬁcient and pressure drop predictive methods that account for the ﬂow pattern are developed based on a mechanistic approach. According to Ishii and Hibiki (2011), the ﬂow patterns can be categorized based on the continuity of the interface as follows: • Separated ﬂows: the phases are continuum and segregated by a single interface between them. These ﬂows can be characterized as liquid in contact with the duct walls and vapor ﬂowing in the section core, and as a liquid ﬁlm in the bottom region of the cross-section and vapor in the upper part. • Dispersed ﬂows: one of the phases is dispersed in the other. Liquid droplets ﬂow dispersed within the vapor, and vapor bubbles ﬂow within the liquid phase. The interface is not continuous, but it is formed of several segments. • Mixed ﬂows: the ﬂow patterns pertaining this group are composed of a combination of characteristics of separated and dispersed ﬂows, that is, large vapor bubbles (large and continuum portion of vapor) ﬂowing intermittently separated by liquid pistons containing dispersed vapor bubbles, and liquid ﬂowing on the duct wall along its entire perimeter with vapor ﬂow in the test section core containing entrained liquid droplets. Detailed classiﬁcations for the ﬂow patterns are frequently found based on the structure of the phases interface. Most of them are classiﬁed based on adiabatic ﬂows, even though the heat transfer process itself can affect the phases distribution, such as during condensation that precludes dry regions on the duct surface, and pure stratiﬁed ﬂow is not expected. Nonetheless, the discussion presented in this section corresponds only to description of ﬂow patterns in conventional and microscale channels. Flow pattern prediction methods are addressed in Chap. 3. It should be

2.2 Flow Patterns

21

highlighted that different classiﬁcations are available in the open literature, adopting distinct nomenclature for the same ﬂow pattern and the same nomenclature for different ﬂow patterns.

2.2.1

Flow Patterns During Vertical Adiabatic Flow

Figure 2.2 depicts the ﬂow patterns observed for vertical upward ﬂow, which are classiﬁed based on subjective approach (visual observations) as follows: • Bubbles: characterized by the vapor phase distributed according to discrete bubbles smaller than the duct diameter within the continuum liquid ﬂow. This ﬂow pattern is characteristic by reduced vapor fraction and high liquid velocities. • Slug ﬂow: as the vapor fraction increases, bubbles coalescence takes place causing the increase of bubble size. This implies on the transition to slug ﬂow pattern, characterized by successive passage of large vapor bubbles with transversal dimensions of the same order of the duct diameter. These bubbles are denominated as Taylor bubbles with a hemispherical nose and an amorphous trailing region. During two-phase ﬂow under certain conditions, the liquid ﬁlm formed between the large bubble and the duct wall may ﬂow downwards,

Bubbles

Slug

Churn

Annular

Fig. 2.2 Schematics of ﬂow patterns for vertical upward ﬂow

Wispy annular

Wispy

22

•

•

•

•

2 Fundamentals

counter-current to the rising vapor. Small vapor bubbles are observed within the liquid slug between successive Taylor bubbles. Churn ﬂow: with additional increment of vapor fraction, the increment of inertial effects and reduction of liquid fraction disrupt liquid bridges between successive large bubbles, and the ﬂow becomes chaotic. This ﬂow pattern is characterized by chaotic movement of liquid and vapor phases and is observed in channels of conventional dimensions. Churn ﬂow is frequently assumed as a transitional ﬂow pattern. Annular ﬂow: under conditions of high vapor velocities, the dominant inertial effects of the vapor ﬂow moves the liquid phase to the duct walls, forming a continuous liquid ﬁlm on the duct surface. The interface between the phases is perturbed and usually contains interfacial waves. Depending on the operational conditions, small bubbles are wrapped by the liquid ﬁlm. Wispy annular: the high ﬂow velocity of the vapor ﬂow typical of annular ﬂows might detach liquid droplets from the interfacial waves. This behavior implies on a liquid ﬁlm along the tube perimeter with the vapor ﬂowing in the center of the section containing entrained liquid droplets. Wispy ﬂow: this ﬂow pattern is also referred in the literature as mist ﬂow. Under conditions of very high vapor velocities and reduced liquid fraction, the liquid ﬁlm thins and becomes unstable, being ceased by the vapor ﬂow. Then, the ﬂow is characterized by a dry wall and a continuum vapor phase with entrained liquid droplets.

Studies focused on downward ﬂow patterns are also found in literature, such as the one presented by Barnea et al. (1982). According to them, only annular ﬂow is naturally occurring in downward ﬂow because buoyancy effects preclude concurrent natural downward ﬂow of vapor and liquid. The bubbles and slug ﬂow patterns are reported only for signiﬁcantly high liquid velocities. Figure 2.3 depicts images of downward air-water ﬂows in square microchannel and Fig. 2.4 depicts them schematically. The annular ﬂow corresponds to a falling ﬁlm inside tubes, and it is likely to occur during condensation inside vertical tubes. In the case of conditions with higher liquid content, such as slug and bubbles ﬂow, the liquid velocity must be high enough to overcome the slip velocity of the vapor phase, in order of resulting in co-current downward ﬂow. Moreover, Lavin et al. (2019) observed coring effect during bubble ﬂow, as shown in Fig. 2.3, with higher concentration of bubbles close to the center of the cross-section, and the main quoted mechanism for this phenomena are related to pressure gradient due to shear stress gradient, and transverse lift force induced by velocity gradient. In addition to concurrent upward and downward ﬂow patterns in vertical ducts, there is also the possibility of counter-current gas-liquid ﬂow with downward liquid ﬂow and upward gas ﬂow. In this case, annular or slug ﬂow patterns are likely to occur. Depending on the superﬁcial velocities of both phases, the counter current ﬂow might not exist, whereas one of the phases is dragged by the other, such as discussed by Wallis (1969) for conventional sized channels.

2.2 Flow Patterns

Annular

23

Bubbles

Slug

Fig. 2.3 Images of air-water downward ﬂow in microchannel, Lavin et al. (2019)

The condition of counter-current ﬂow in microscale channels is unlikely to occur due to effects of bubble conﬁnement, hence, it will not be discussed in this book, but the interested readers are encouraged to check Wallis (1969).

2.2.2

Flow Patterns for Horizontal Adiabatic Flows

During vertical ﬂows, even for concurrent upward and downward ﬂows, the buoyancy effects inﬂuence the slip between the phases, whereas the vapor phase presents higher vertical upward velocity component. Nonetheless, these effects do not

24

2 Fundamentals

Fig. 2.4 Schematics of vertical downward ﬂow

Annular

Bubbles

Slug

necessarily imply on axi-asymmetric phase distribution for vertical channels. On the other hand, for horizontal ﬂow, buoyancy effects imply on predominantly higher concentration of gas phase in the upper region of the duct, with consequent asymmetry in the phase distributions. Figure 2.5 schematically depicts the horizontal ﬂow patterns for conventional channels that are described as follows: • Bubbles: as for vertical ﬂows, this ﬂow pattern is characterized by bubbles dispersed in a continuum liquid phase. Buoyancy effects are responsible for a higher concentration of bubbles in the upper region of the tube. This ﬂow pattern is veriﬁed for reduced gas fraction and high liquid velocities. • Stratiﬁed: characterized by the liquid as a continuum ﬁlm in the bottom region of the section and gas ﬂowing in the upper region with the phases separated by a smooth interface. This ﬂow pattern is veriﬁed for reduced ﬂow velocities and under predominance of gravitational effects on the two-phase ﬂow topology. The liquid-vapor interface is curved in the azimuthal direction due to inertial and capillary effects. The stratiﬁed ﬂow is absent in microscale channels because the capillary effects tend to ﬁll the surface with liquid ﬁlm.

2.2 Flow Patterns Fig. 2.5 Schematics of horizontal ﬂow patterns, ﬂowing from left to the right

25

Bubbles

Stratified

Stratified wavy

Plug

Slug

Annular

Mist

• Stratiﬁed wavy: with the increment of vapor phase velocity, interfacial waves with considerable amplitude are formed, moving according to the ﬂow main direction. This ﬂow pattern is also veriﬁed only for channels of conventional size. • Plug: this ﬂow pattern corresponds to a type of intermittent ﬂow pattern, where the vapor is distributed according to discrete large bubbles, but with transversal dimensions smaller than the channel diameter. The liquid slugs between successive bubbles contain small vapor bubbles dispersed within the liquid phase. This ﬂow pattern is veriﬁed for intermediate ﬂow velocities. • Slug: this is another sub-classiﬁcation of intermittent ﬂows and is characterized by vapor bubbles larger than those observed for plug ﬂow and is also referred in literature as elongated bubbles. A thin liquid ﬁlm separates the upper part of the tube from the bubbles, which correspond to parcel of the liquid that remains from the liquid slug passage. Small vapor bubbles might be observed dispersed within the liquid slugs. • Annular: under conditions of high vapor velocities, the inertial effects of the vapor phase move the liquid to the duct periphery; thus, the liquid ﬂows as a continuum liquid ﬁlm along the entire tube perimeter. Distinctly than vertical ﬂow, the liquid ﬁlm thickness may present signiﬁcant non-uniformity along the channel perimeter due to gravitational effects. The ﬁlm is thicker in the bottom region of the channel.

26

2 Fundamentals

Fig. 2.6 Horizontal ﬂow patterns for vapor-liquid ﬂow in microchannel. R245fa in 1.1 mm ID tube, Tibiriçá and Ribatski (2013)

• Mist: the characteristics of this ﬂow pattern are similar for vertical and horizontal ﬂows. The inertial effects of the vapor phase dominate the phase distribution. It must be highlighted that these classiﬁcations and deﬁnitions are commonly found in textbooks concerning the subject of two-phase ﬂows and are valid mostly for channels of conventional size. Moreover, the occurrence of some ﬂow patterns is unlikely to occur during phase change phenomena; for example, the stratiﬁed and wavy-stratiﬁed ﬂow patterns as described in this item are unlikely to occur during condensation, since the phase change process implies on the formation of a liquid ﬁlm along the entire tube perimeter. In the case of two-phase ﬂow in microchannels, the surface tension effects have signiﬁcant contribution on the phase distribution, and ﬂow patterns dominated by buoyance effects are not veriﬁed, as shown in Fig. 2.6, and schematically depicted in Fig. 2.7. According to Tibiriçá and Ribatski (2013), and similar to Ishii and Hibiki (2010), the ﬂow patterns during horizontal ﬂow in microchannels were classiﬁed as separated, dispersed, and mixed. The following sub-classiﬁcation was proposed by Tibiriçá and Ribatski (2013): • Dispersed: – Bubble ﬂow pattern: similar to the case of two-phase ﬂow in conventional channels, with continuum liquid phase with dispersed vapor bubbles slightly smaller than channel diameter. It is veriﬁed only for reduced vapor content. – Mist ﬂow: similar to the description for mist ﬂow in conventional channels, this ﬂow pattern is characterized by continuum vapor phase with dispersed liquid droplets, even though it was not reported by Tibiriçá and Ribatski (2013) because it is close to the condition of wall dryout. It is veriﬁed for conditions of low liquid content.

2.2 Flow Patterns

27

Dispersed (Bubbles)

Intermittent (Slug)

Intermittent (Elongated bubbles)

Intermittent (Churn)

Annular Fig. 2.7 Schematics of horizontal ﬂow patterns in microscale channels

• Separated: – Annular ﬂow: characterized by continuum liquid ﬁlm along the channel wall, with vapor ﬂowing in the channel core. • Mixed (intermittent): – Intermittent (slug): it is characterized by successive passage of vapor bubbles with characteristic dimensions of the same order of magnitude of the channel diameter. – Intermittent (elongated bubbles): it is similar to the slug ﬂow, but the vapor bubble length corresponds to several diameters. – Intermittent (churn): it can be considered as transitional ﬂow pattern between elongated bubbles and annular ﬂow patterns, with the liquid pistons being disrupted by the vapor ﬂow. It must be highlighted that stratiﬁed ﬂow is not observed for microscale channels due to suppression of buoyance effects, and that dispersed bubbles usually count with aligned single bubbles.

28

2.3

2 Fundamentals

Void Fraction

Deﬁnition of area averaged void fraction, as well as volume averaged void fraction and local void fraction. A short discussion about the importance of these parameters on two-phase ﬂow problems is presented. The void fraction α is deﬁned as the time average fraction of vapor phase in a region of the space occupied by the two-phase mixture and can be deﬁned according to distinct domains D including a single point (local), line segment, area and volume, denoted in this text by the subscripts 0, 1, 2, and 3, respectively. Void fraction is one of the most important parameters of two-phase ﬂow because it is directly related to the accelerational and gravitational pressure drops. Moreover, several parameters of two-phase ﬂow, such as local ﬂow pattern and consequently frictional pressure drop and heat transfer coefﬁcient, are interrelated with the void fraction. Additionally, the void fraction is a critical parameter for the evaluation of dryout occurrence, which can lead to the heat transfer surface burnout and consequent damage of the heat exchanger under conditions of imposed heat ﬂux, which must be avoided specially in systems such as vapor generators of nuclear power plants. Even though the vapor quality is a measurement of the vapor fraction in a mixture, this parameter does not represent the actual amount of vapor in a given region of the ﬂowing mixture. From a practical point of view, it is possible to infer this aspect when we evaluate the volume fraction that would be occupied by a mixture at a given vapor quality x. Let us assume saturated R134a at 5 C. For this condition, the liquid and vapor densities are 1311 and 12.1 kg/m3, respectively. For a motionless (non-ﬂow) condition, a vapor quality value of 0.009 corresponds to similar volumes occupied by both phases, that is, 50% in volume for each phase. Therefore, marginal vapor quality values would correspond to the cross-section almost entirely occupied by vapor phase, which is not observed experimentally, and the slip between the phases justiﬁes the higher liquid content. Thus, several approaches for estimation of void fraction are based on modelling of the slip between the phases.

2.3.1

Local Void Fraction !

The local void fraction corresponds to the parcel of time that a location (point r ) is occupied by the vapor phase, and it is deﬁned based on the density function given by Eq. (2.1). The local void fraction is given as follows: α0 ¼

1 Δt

Z

t 0 þΔt

X v dt t0

ð2:26Þ

2.3 Void Fraction

29

The local void fraction can be experimentally determined using intrusive approaches, such as resistive and optical probes, as well as non-intrusive approaches such as some based on optical method. It should be highlighted that it is nonsense to evaluate the integral in the space domain deﬁned by a point, because, theoretically, a point has no dimensions, even though the measuring probe must have a ﬁnite size.

2.3.2

Line Averaged Void Fraction

The line averaged void fraction corresponds to the time average of the space parcel occupied by the vapor phase along a line segment D1. It is deﬁned as follows: 1 α1 ¼ Δt

t 0 þΔt Z

Z

X v dD1 =

dD1

D1

t0

Z D1

1 dt ¼ Δt

t 0 þΔt Z

Z

Z X v ds=

t0

L

ds dt ð2:27Þ

L

Assuming a rectilinear segment, Eq. (2.27) can be rewritten as a function of parcel of the line corresponding to the vapor phase (Lv/L ) as follows: 1 α1 ¼ Δt

Z

t 0 þΔt t0

Lv dt L

ð2:28Þ

Certain experimental approaches provide the mixture density along a segment of line, such as gamma and x-ray attenuation techniques. Based on this approach and knowing the index of attenuation of each phase, it is possible also to experimentally determine the phase parcels and consequently the instantaneous vapor fraction and void fraction along the segment of line. Falcone et al. (2009) provide a careful discussion about these methods.

2.3.3

Area Averaged Void Fraction

Similar to the line averaged void fraction, the area averaged void fraction, also referred as superﬁcial void fraction, corresponds to the time averaged parcel of area occupied by the vapor phase in a given two-dimensional domain D2, which usually refers to the cross-sectional area, and is deﬁned as follows: 1 α2 ¼ Δt

Z Z Z 1 t0 þΔt X v dD2 = dD2 dt ¼ X v dA= dA dt ð2:29Þ Δt t0 D2 D2 A A

t 0 þΔt Z

Z t0

Z

30

2 Fundamentals

Assuming that the instantaneous area occupied by the vapor phase along the duct cross-section is Av and the total area in analysis is A, Eq. (2.29) can be rewritten as follows: 1 α2 ¼ Δt

Z

t 0 þΔt t0

Av dt A

ð2:30Þ

In this text, when not speciﬁed differently, the term void fraction refers to the area averaged void fraction and will be denoted simply by α. There are several experimental methods for determining the area averaged void fraction, such as some optical method, capacitive sensing system, wire mesh sensor, and array of radioactive emitter-receptor pairs, such as discussed by Falcone et al. (2009). Nonetheless, the reader must be aware that experimental investigation of area averaged void fraction is challenging, and most of the studies have performed experimental determination of local, line averaged, or volume averaged void fraction. Figure 2.8 from Marchetto (2019) illustrates the evolution of the instantaneous local void fraction for intermittent ﬂow in a small diameter channel during the passage of three consecutive elongated vapor bubbles. In this ﬁgure, the evolution of the time-averaged superﬁcial void fraction estimated according to Eq. (2.30) is also shown. From a comparison of both curves displayed in Fig. 2.8, it can be noted that the time-averaged void fraction tends to a value of approximately 0.6 as the upper integer limit in Eq. (2.30) increases. Therefore, it can be concluded that despite of the ﬂow intermittence, a constant time-averaged superﬁcial void fraction can be

Fig. 2.8 Variation of the instantaneous and time-averaged void fraction over time for an intermittent ﬂow, Marchetto (2019)

2.3 Void Fraction

31

measured under developed ﬂow conditions if the integer upper limit in Eq. (2.30) is long enough to characterize the ﬂow.

2.3.4

Volume Averaged Void Fraction

The volume averaged void fraction, also referred as volumetric void fraction, is frequently mentioned in literature. The term gas holdup is also found in literature mainly by engineers working in the sector of oil and gas. The volumetric void fraction is deﬁned as the time averaged volume parcel of the gas phase, and is given as follows: α3 ¼ ¼

1 Δt 1 Δt

Z Z

t 0 þΔt t0 t 0 þΔt

Z ð Z ð

Z X v dD3 = D3

Z

X v dV= V

t0

dD3 Þdt D3

dVÞdt

ð2:31Þ

V

Supposing that the instantaneous volume occupied by the vapor phase is V v and the total volume is V, the volume averaged void fraction is given as follows: α3 ¼

1 Δt

Z

t 0 þΔt t0

Vv dt V

ð2:32Þ

The quick closing valve technique is recurrently mentioned in the literature as an experimental approach for volumetric void fraction measurements. This technique consists on trapping the two-phase mixture in a given volume of the test section by closing simultaneously valves located upstream and downstream of the measurement target region. Subsequently, the phase volumes are measured, and the fractions of liquid and gas phases are calculated. This approach is used for two-phase ﬂow of liquid and immiscible gas, as well as for two-phase ﬂow of saturated substance. The volumes of each phase can be measured directly in the ﬁrst case because the phases are separable; on the other hand, in the case of two-phase ﬂow of a single substance (or solution), the heat transfer between the ﬂuid and its neighborhood might imply on variation of phase proportion between the trapping and measuring instants. Thus, researchers usually measure the trapped mass, and based on the phase densities for the experimental condition, they evaluate the phase volumetric fraction. The distance between the upstream and downstream valves must be long enough to avoid effects of intermittency, characteristic of intermittent ﬂow patterns. Moreover, when applied to two-phase developing ﬂow, the results obtained through the quick closing valve method become typical of the measurement length and the distance between the valves and the test section inlet, therefore, they cannot be considered as general.

32

2 Fundamentals

Ac

La

Pa

Lb

Aa

Ab

Flow direction Lc

Ld

Pc Pb

Fig. 2.9 Schematics of a vapor bubble passage in a duct

Falcone et al. (2009) present several experimental techniques for evaluation of two-phase ﬂow parameters, including void fraction, and is here indicated as supplementary material. It covers details about installation, application, and range of validity mostly focused on applications in the oil and gas industry. It can be recognized at this point that, similarly to the density function deﬁned by Eq. (2.1), the time averaged fraction of the liquid phase in a given domain is equal to the unity minus the void fraction (1 – αn). In the same context, specialists of the oil and gas industry refer to the volumetric liquid content as liquid holdup, or simply holdup. It must be mentioned that the point, line, area, and volume averaged void fractions are not necessarily similar due to variation of ﬂow parameters in space and time. These four parameters would correspond to the same value only in conditions of homogenous ﬂow, corresponding to uniform distribution of the phases along the entire domain region. Figure 2.9 schematically depicts these differences in the case of elongated bubble ﬂow patterns. According to this ﬁgure, it can be noticed that the density function for points Pa, Pb, and Pc are distinct from each other, since the upper and central points are in contact with the vapor phase while point Pc is in contact with the liquid phase. Therefore, the local void fraction, which consists in the time averaged density function of the vapor phase given by Eq. (2.26), is distinct for each point. Similarly, the instantaneous vapor fraction along the lines La, Lb, Lc, and Ld are distinct from each other, and the differences get even higher due to buoyancy effects that make the ﬂow non-axisymmetric. Therefore, the void fraction for distinct line segments would be different as well. Moreover, the determination of the area averaged void fraction based on the line averaged void fraction requires a previous knowledge about the ﬂow pattern and phases distributions, and this information is as difﬁcult to obtain as the area averaged void fraction. Some authors present experimental studies, such as described by Falcone et al. (2009), in which they determine the area averaged void fraction based on measurement of three lines’ averaged void fractions. Based on empirical relationships between line and area average void fractions for a given phase distribution, the area averaged void fraction is estimated.

2.3 Void Fraction

33

Therefore, based on this discussion, it can be concluded that the point, line, and area averaged void fraction are distinct from each other, and some of these parameters are strongly dependent on the measurement location. The volume averaged void fraction is also a function of the region of analysis. According to Fig. 2.9, distinct instantaneous vapor volume fractions would be obtained if we consider the volumes formed between section pairs Ac - Aa, Ac - Ab, and Aa - Ac. Thus, even considering that the ﬂow is completely developed, a high number of samples of these volumes would be required to obtain a reasonably low deviation from the area averaged void fraction, because the instant that the sample is trapped affects the result. Alternatively, a larger volume would reduce the inﬂuence of the instant of time of the sample for a completely developed ﬂow. However, for two-phase ﬂow in microscale channels the high pressure drop might result in signiﬁcant variation of vapor-speciﬁc volume, with consequent variation of phase fractions and velocities along the channel length. In conclusion, the integral in time of the vapor area parcel is similar to the volume averaged void fraction only if the ﬂow is fully developed, under steady state condition, non-intermittent, for a uniform cross-section, adiabatic, and under conditions of negligible compressibility effects. In conclusion, the reader might have noticed that experimental determination of void fraction is challenging, and even today several research groups are engaged in developing instrumentation and methods for precise determination of void fraction. Additionally, several of the measurement techniques are intrusive, that is, they correspond to instrumentation and sensor inserted into the ﬂow stream, disturbing the velocity and temperature proﬁles, and consequently altering the parameter that was intended to be measured. In this context, it should be mentioned that when analyzing a real thermal cycle, such as refrigeration or power generation, the experimental determination of void fraction would help to obtain the thermodynamic state along the cycle. For instance, consider the outlet of expansion device in refrigeration cycle, whereas the conventional instrumentation would correspond to pressure and/or temperature transducers. But considering that it is two-phase ﬂow in this point, the pressure and temperature data does not allow determination of thermodynamic state. Alternatively, the expansion device could be considered as adiabatic, and by evaluating the inlet condition, which is probably subcooled liquid, it is possible to infer the outlet condition. However, this approach requires the hypothesis of adiabatic ﬂow that is not completely attained and knowing the phase fraction can help in determining the local vapor quality. Moreover, in some industries, such as in oil and gas, the correct determination of each phase fraction is needed to calculate amount, taxes, prices, etc.

2.3.5

Void Fraction Predictive Methods

Several approaches for void fraction prediction are available in the open literature. Woldesemayat and Ghajar (2007) presented an extensive review of literature

34

2 Fundamentals

concerning void fraction predictive methods, and classiﬁed them as slip correlation, drift ﬂux models, general correlations, and as correction factors for the void fraction estimated according to homogeneous model. In the present textbook, an alternative classiﬁcation is adopted as follows: • Kinematic approaches: these methods consist in correlations that assume some characteristic of the velocity proﬁles and based on average operators and continuity it is possible to estimate the void fraction. This category includes, for example, the slip ratio methods and drift ﬂux models. • General methods: these methods consist in purely empirical correlations and methods based on physical principles, other than the assumption of velocity characteristics. These approaches include those developed based on Lockhart and Martinelli (1949) parameter, minimum entropy generation, minimum kinetic energy, and purely empirical methods.

2.3.5.1

Slip Ratio Method

The simplest modeling approach for two-phase ﬂow consists in the homogeneous model that is based on the assumption of the liquid and gas phases ﬂowing as a single pseudo-ﬂuid with averaged properties. This is a kinematic model and considers absence of slip between the phases and uniform velocity proﬁles, thus the in situ velocities of the liquid and gas phases are similar. Hence, from Eqs. (2.20) and (2.21), eliminating the mass ﬂux term, we obtain the following expression: ð 1 xÞ x ¼ ρl ð1 α2 Þul ρv α2 uv

ð2:33Þ

Solving Eq. (2.33) for α2, we obtain the void fraction given as a function of the slip ratio as follows: α2 ¼

1

1þ

uv ρv 1x ul ρl x

ð2:34Þ

where the term uv / ul corresponds to the slip ratio, and according to the homogeneous model assumption is equal to the unity (uv / ul ¼ 1). It is common sense in literature that the void fraction is a function of vapor quality x and ﬂuids properties, as well as of the ﬂow velocity and channel geometry, which effects on the void fraction are not totally captured by the homogenous model. Moreover, in the case of vertical upward ﬂow, buoyancy effects might imply on higher velocity of vapor phase, contributing to deviation from the homogenous assumption. Even during horizontal ﬂow, the vapor phase usually possesses higher velocity than the liquid phase, which is related to the friction between the

2.3 Void Fraction

35

phase and duct wall and/or the bubble nose shape. Therefore, the assumption of no-slip is not representative of the real phenomena for these conditions, and the homogenous model provides the highest void fraction value possible, otherwise the results would be physically incorrect because it would correspond to liquid ﬂowing faster than vapor. An in situ liquid velocity higher than the vapor velocity would be possible in the case of liquid ﬂowing downward along the duct wall, due to gravitational effects, and the vapor ﬂowing concurrently dragged by the liquid phase. Some researchers adjust correlations for the slip ratio in order of developing a void fraction predictive method that is dependent on ﬂow conditions and duct geometry, such as proposed by Tibiriçá et al. (2017) for microchannels, according to which the slip ratio is given as follows: 0:31 uv 1 x 0:267 0:1082 ρv ¼ 1:2364 Fr x ul ρl

ð2:35Þ

where the Froude number is evaluated for the mixture as follows: Fr ¼

2.3.5.2

G2 ðρl ρv Þ2 gd

ð2:36Þ

Drift Flux Model – Zuber and Findlay Method

Several approaches have been proposed to account for the effects of slip between the phases and non-uniform distribution and velocity proﬁle of the phases along the cross-section, and among them, the drift ﬂux model is one of the most used methods for prediction of void fraction and was ﬁrstly presented by Zuber and Findlay (1965). The derivation of this method assumes non-uniform velocity proﬁles and phase distributions and takes into account the slip between the phases. The method is derived based on spatial average operator, as given by Eq. (2.4), for steady state condition. Based on the deﬁnition of the drift velocity of vapor phase, given by Eq. (2.22), the instantaneous and local vapor drift velocity are written as follows: V z,vj ¼ V z,v j

ð2:37Þ

which can vary along the cross-section. Thus, rearranging this equation as follows and multiplying it by the vapor density function, we get: V z,v X v ¼ V z,vj X v þ jX v

ð2:38Þ

Therefore, by evaluating the mean value of this function along the entire crosssection, we obtain:

36

2 Fundamentals

R

A VRz,v X v dA A dA

R ¼

A VRz,vj X v dA A dA

R

jX v dA þ AR A dA

ð2:39Þ

We can recognize that the numerator of the ﬁrst member of Eq. (2.39) corresponds to the vapor volumetric ﬂow rate, which divided by the cross-sectional area corresponds to the vapor superﬁcial velocity jv. Multiplying this equation by the inverse of the area averaged void fraction, and adopting the relationship between the void fraction, vapor superﬁcial velocity and void fraction, we can re-write Eq. (2.39) as follows1: R

A V z,v X v dA=A

α2

j ¼ v¼ α2

R

A V z,vj X v dA=A

α2

R þ

A jX v dA=A

α2 j

j

ð2:40Þ

where a term j / j was multiplied by the last term of the second member. Zuber and Findlay (1965) rewrote Eq. (2.40) as follows: jv e ¼ V vj þ C 0 j α2

ð2:41Þ

where the term C0 corresponds to the distribution parameter that takes into account e vj correthe non-uniform distribution of the phases along the cross-section, and V sponds to the drift parameter that takes into account the slip between the phases. These parameters are deﬁned empirically, and, according to the original authors, are dependent on the local ﬂow pattern. The distribution and drift parameters are written explicitly as follows: R C0 ¼ R e vj ¼ V

A jX v dA=A

α2 j

A V z,vj X v dA=A

α2

ð2:42Þ ð2:43Þ

Recall that the numerator in Eq. (2.43) does not correspond to the superﬁcial velocity, because it consists in the integral of the slip velocity. By multiplying Eq. (2.41) by α2/j, the following relationship is obtained: jv ¼ j

! e vj V þ C 0 α2 j

ð2:44Þ

1 This presentation is slightly different than the original description, which is based on weighted averages of the two-phase parameters, but the ﬁnal relationship for void fraction is exactly the same.

2.3 Void Fraction

37

where the ﬁrst member corresponds to the volumetric ratio β, deﬁned by Eq. (2.24). Thus, rearranging the terms, it is possible to obtain a relationship for void fraction given as follows: α2 ¼

β C0 þ

e V vj

ð2:45Þ

j

In order to check the validity of this method for the case of homogeneous ﬂow, it is expected that the phase distribution is uniform, thus C0 is equal to the unity. e vj is null. Therefore, Additionally, the drift between the phases is negligible, thus V the void fraction is equal to the volumetric fraction, which is similar to the void fraction according to the homogenous model. According to the original authors (Zuber and Findlay, 1965), the distribution and drift parameters depend on the ﬂow pattern, and consequently, regressions for e vj are required for each ﬂow pattern. Collier determination of values for C0 and V and Thome (1994) present some values and relationships from literature for these parameters depending on the local ﬂow pattern. The rising velocity of a bubble in stagnant media is recurrently adopted as the drift parameter, even for horizontal ﬂow. The drift term is generally related to gravitational effects on the phases, and in the case of ﬂow in horizontal channels the bubble nose geometry is the main parameter deﬁning the drift between the phases. As the channel size decreases, the importance of gravitational effects on two-phase ﬂow topology reduces, the ﬂow tends to be axisymmetric, and the drift between the phases tends to zero, as shown by Sempertegui-Tapia et al. (2013). Subsequently to the Zuber and Findlay (1965) study, several authors proposed adjustments for the distribution and drift parameters. Constants were proposed for narrow ranges of operational conditions and correlations for wider ranges. In this context, Rouhani (1969) and Rouhani and Axelsson (1970) proposed correlations for the distribution and drift parameters for horizontal and vertical upward ﬂows, respectively. Considering conditions of phase change, j and β vary along the channel length as function of vapor quality, therefore it is more convenient to present Eq. (2.45) as a function of mass velocity and vapor quality, as follows: ( α2 ¼

" #)1 e vj V ρv x 1x þ C0 þ ρv ρl x G

ð2:46Þ

The mass velocity and vapor quality were introduced because in applications involving phase change processes, the ﬁrst is constant in case of ﬁxed cross-section and the second can be obtained directly from an energy balance. Rouhani (1969) and Rouhani and Axelsson (1970) proposed the rising velocity of a bubble in a stagnant media as the drift parameter, as proposed by Zuber and Findlay (1965), given as follows:

38

2 Fundamentals

14 e vj ¼ 1:18 gσ ðρl ρv Þ V ρ2l

ð2:47Þ

For horizontal ﬂow, Rouhani (1969) proposed the following relationship for the distribution parameter: 1 gdρ2l 4 C 0 ¼ 1 þ 0:2ð1 xÞ G2

ð2:48Þ

where the term inside the brackets can be considered as the inverse of the Froude number. For vertical upward ﬂow, Rouhani and Axelsson (1970) proposed the following values for the distribution parameter:

C0 ¼

1:10

for G 200 kg=m2s

1:54

for G < 200 kg=m2s

ð2:49Þ

The methods proposed by Rouhani (1969) and Rouhani and Axelsson (1970) were developed originally focused on nuclear applications, thus based on experimental results for liquid water and steam. Nonetheless, several studies, such as Wojtan et al. (2005a, b), indicate their validity for refrigerant ﬂow during phase change processes.

2.3.5.3

Minimum Entropy Generation – Zivi’s Method

Differently than the kinematic approaches, general approaches are also available in the open literature, which are not necessarily based on the characteristics of velocity proﬁle. Zivi (1964) proposed a method based on the minimization of the entropy generation of the two-phase ﬂow. Zivi assumed that the minimum entropy generation condition is attained for minimum kinetic energy ﬂux integrated along the crosssection, assuming uniform velocity proﬁles. The kinetic energy of a ﬂuid particle is equal to u2/2, thus the kinetic energy ﬂux along the cross-section is given as follows: Z Ek ¼

u2 ! ! ρV dA ¼ 2A

A

2

ul 2A

Z

ul 2 ! ! ρV dA þ 2A

Al

!

!

ρV dA þ Al

Z

2

uv 2A

Z

Av

uv 2 ! ! ρV dA ¼ 2A

Av !

!

ρV dA ¼

G 2 ul ð1 xÞ þ uv 2 x 2

Z

2

ul m_ l uv 2 m_ v þ ¼ 2 A 2 A

ð2:50Þ

2.3 Void Fraction

39

Writing the in situ velocities as a function of the mass ﬂux, vapor quality, and ﬂuid densities, given by Eqs. (2.20) and (2.21), and then minimizing the kinetic energy ﬂux through derivation in relation to α2, the following is obtained: ( " #) 3 ∂Ek ∂ G 3 ð 1 xÞ 1 x3 1 ¼ þ ¼ ∂α2 ∂α2 2 ρ2l ð1 α2 Þ2 ρ2v α22 " # 3 3 ð 1 x Þ x þ ¼0 G3 ð1 α2 Þ3 ρ2l α32 ρ2v

ð2:51Þ

Solving for α2, a relationship for void fraction is obtained, given as follows: α¼ 1þ

1 23 ρv 1x ρl

ð2:52Þ

x

Similar to the homogenous model, the Zivi (1964) method does not take into account mass ﬂow rate effects. Nonetheless, this method has shown reasonably good agreement with experimental results for distinct operational conditions.

2.3.5.4

Minimum Kinetic Energy Method – Kanizawa and Ribatski Method

Considering a similar approach of Zivi (1964), Kanizawa and Ribatski (2016) proposed a void fraction predictive method based on the minimization of the kinetic energy along the cross-section assuming non-uniform velocity proﬁle of the phases along the cross-section. They assumed that the void fraction is associated to a condition of minimum momentum ﬂux along the cross-section of the pipe (instead of kinetic energy ﬂux as adopted by Zivi, 1964). In their method, the momentum ﬂux of each phase is averaged in relation to its respective area, given by (2.4), as follows: R 2 ! ! V z,l dA X v ÞρV dA K l u2l Al ¼ ρl ARl ¼ ρl ¼ K l ρl u2l A ð 1 X ÞdA dA l v A Al

R < E k >2,l ¼

ð1 A V z,l R R

< E k >2,v ¼

!

X ρV A V z,v R v

A X v dA

!

dA

ð2:53Þ

R

2 K v u2v Av A V z,v dA ¼ ρv Rv ¼ ρv ¼ K v ρv u2v Av Av dA

ð2:54Þ

where the term K corresponds to the momentum coefﬁcient (form factor of velocity proﬁle) and corresponds to the contribution R 2 of the non-uniformities of the velocity V z dAÞ proﬁle and phases distribution (K ¼ A u2 A ). In conditions of uniform velocity

40

2 Fundamentals

proﬁles, even considering slip between the phases, the form factor of each phase becomes equal to the unity. The kinetic energy of the two-phase mixture is given as the sum of the parcels of both phases. Similar to the Zivi (1964) method, writing the in situ velocities as a function of the mass ﬂux, vapor quality, and ﬂuid density, and minimizing the function in α2, the following relationship is obtained: ∂ < E k >2 ∂ ¼ K l ρl ul 2 þ K v ρv uv 2 ¼ ∂α2 ∂α2 ( 2 ) 2 G ð 1 xÞ ∂ Gx K l ρl þ K v ρv ¼ ρv α 2 ρl ð1 α2 Þ ∂α2 ( " # 2 ) 2 1 K v x2 ∂ 1 2 K l ð 1 xÞ ∂ þ ¼ G ρl ρv ∂α2 α2 ∂α2 ð1 α2 Þ2 ( ) 2 K v x2 2 K l ð1 xÞ ¼0 2G ð1 α2 Þ3 ρl α32 ρv

ð2:55Þ

Assuming G different from zero (ﬂow condition) and solving for α2, the following relationship is obtained: α2 ¼ 1þ

1 13 13 2 ρv Kl 1x 3 Kv

ρl

ð2:56Þ

x

As above discussed, the momentum factor of each phase is related to the non-uniformities of the velocity proﬁles, therefore it is expected that their values depend on phase distribution and, consequently, on the ﬂow pattern. Thus, the momentum coefﬁcient ratio was correlated by Kanizawa and Ribatski (2016) as a function of non-dimensional parameters for horizontal and vertical upward ﬂows. It should be highlighted that this method is valid only for conditions under which the estimated void fraction is smaller than the estimative according to the homogenous model, and this condition is given as follows:

Kl Kv

13

23 1 ρv 1x 3 x ρl

ð2:57Þ

hence, in the case that Eq. (2.57) is not satisﬁed, the homogeneous model must be considered. For horizontal ﬂows, the two-phase ﬂow topology is dominated mainly by the balance between buoyancy and inertial effects. Thus, the momentum coefﬁcient ratio was correlated as a function of the Froude number and the viscosity ratio as follows:

2.3 Void Fraction

41

Kl Kv

13

¼ 1:021 Fr 0:092 m

0:368 μl μv

ð2:58Þ

where the Froude number of the mixture Frm corresponds to the balance between gravitational and inertial effects, and is given as follows: Fr m ¼

G2 gðρl ρv Þ2 d

ð2:59Þ

The correlation given by (2.58) was adjusted for a database with wide experimental conditions comprising results for R12, R22, R134a, and R410A, for ﬂow in ducts with diameters ranging from 0.5 to 13.8 mm, saturation temperatures between 5 and 50 C, and mass ﬂux between 70 and 800 kg/m2s. For vertical ﬂows, the two-phase ﬂow is dominated by a balance between inertial and interfacial effects, thus the momentum coefﬁcient ratio is correlated as a function of the liquid-vapor viscosity ratio and the Weber number of the mixture, as follows:

Kl Kv

13

¼ 14:549 We0:222 m

1:334 μl μv

ð2:60Þ

where the Weber number of the mixture corresponds to the balance between inertial and surface tension effects, and is given as follows: Wem ¼

G2 d ðρl ρv Þσ

ð2:61Þ

Equation (2.60) was adjusted for a database comprising basically in-tube ﬂows of water and heavy water, hydraulic diameters ranging from 6 to 13.4 mm, saturation temperatures between 164 and 275 C, and mass velocities between 52 and 2030 kg/m2s. Unfortunately, the number of studies for refrigerant ﬂow in vertical channels is quite limited, and no data was capable of validating this model for other ﬂuids. In the case that the estimative of the momentum coefﬁcient ratio according to Eqs. (2.58) and (2.60) does not satisfy the restriction given by Eq. (2.57), the homogenous model should be considered for void fraction estimative. Figures 2.10 and 2.11 depict the void fraction predicted values according to the described methods in this section. It can be noticed according to these ﬁgures that the homogeneous void fraction model predicts the highest values among all the methods, which is coherent because this method is based on the assumption of equal ﬂow velocity for both phases. Void fraction values higher than the homogeneous model correspond to liquid in situ velocities higher than the vapor velocity for horizontal and vertical upward ﬂows. In most applications, the liquid velocity should be lower due to gravitational and frictional forces.

2 Fundamentals 1.0

1.0

0.8

0.8

0.6

0.6

Homogeneous model Kanizawa and Ribatski (2016) Rouhani (1969) Zivi (1964)

0.4 0.2

a [-]

a [-]

42

0.0 0.0

0.0 0

0.2

0.4

x [-]

0.6

0.8

1

1.0

1.0

0.8

0.8

0.0 0.0

a [-]

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4 0.2

Horizontal flow R134a, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

0.6

0.6

a [-]

0.4 0.2

Horizontal flow R134a, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

Homogeneous model Kanizawa and Ribatski (2016) Rouhani (1969) Zivi (1964)

0.4

x [-]

0.6

0.8

0.6

0.8

1.0

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4 0.2

Vertical flow R134a, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

x [-]

Vertical flow R134a, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.0 0.0

1.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 2.10 Predicted void fraction values for R134a in 1.0 mm ID tube 1.0

1.0

0.8

0.8 Homogeneous model Homogeneous model Kanizawa andRibatski Ribatski (2016) Kanizawa and (2016) Rouhani (1969) Rouhani (1969) Zivi (1964) (1964)

0.4 0.2 0.0 0.0

Horizontal flow Isobutane, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

0.2 0.0 0.0

1.0

1.0

0.8

0.8

α [-]

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4

0.0 0.0

Horizontal flow Isobutane, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

1.0

0.6

0.6

α [-]

0.4

1.0

0.2

Homogeneous model Homogeneous model Kanizawa andRibatski Ribatski (2016) Kanizawa and (2016) Rouhani (1969) Rouhani (1969) Zivi (1964) Zivi (1964)

0.6

α [-]

α [-]

0.6

0.2

Vertical flow Isobutane, d = 1 mm, G = 100 kg/m²s, Tsat = 20 °C

0.2

0.4

x [-]

0.6

0.8

1.0

Homogeneous model Kanizawa and Ribatski (2016) Rouhani and Axelsson (1970) Zivi (1964)

0.4

0.0 0.0

Vertical flow Isobutane, d = 1 mm, G = 500 kg/m²s, Tsat = 20 °C

0.2

0.4

Fig. 2.11 Predicted void fraction values for isobutane in 1.0 mm ID tube

x [-]

0.6

0.8

1.0

2.4 Flow Boiling Fundamentals

43

Up to date, there is no reliable predictive method for downward ﬂows, even though the condition of condensation inside channels during downward ﬂow is not uncommon (Fig. 2.11).

2.4

Flow Boiling Fundamentals

The fundamentals of convective boiling (also named as ﬂow boiling) are presented in this section. Nucleate boiling and convective boiling mechanisms are addressed. Heat transfer during convective boiling is dominated by a combination of convective and nucleate boiling effects with both schematically depicted in Fig. 2.12. The convective effects are strictly associated to the ﬂuid ﬂow effects, while the nucleate boiling effects are related to the heat transfer due phase change process, characterized by bubble formations and detachment. Conditions dominated by convective effects are characterized by signiﬁcant inﬂuence of ﬂow velocity associated to mass velocity and vapor fraction, with phase change occurring mainly on the vapor–liquid interface. On the other hand, under conditions dominated by nucleate boiling mechanism, the inﬂuence of ﬂow velocity on heat transfer is minimized and the phase change process is associated with the bubble nucleation, growth, and departure from the channel wall. Figure 2.13 schematically depicts the evolution of the two-phase ﬂow characteristics, wall and bulk temperatures, and main heat transfer mechanisms for convective boiling in vertical upward ﬂow with imposed heat ﬂux, assuming subcooled condition (mean cross-sectional ﬂuid enthalpy lower than the saturated liquid enthalpy at the local pressure) at the inlet section. Under single-phase ﬂow conditions, the ﬂuid temperature rises along the duct length due to the heat transferred from the wall through forced convection. Moreover, the temperature difference between the bulk and the wall is almost constant due to a heat transfer coefﬁcient that presents only Convective effects

Nucleate boiling effects

Hydrodynamic boundary layer

Thermal boundary layer

Velocity profile

+

Temperature profile

Fig. 2.12 Schematics of convective and nucleate boiling effects for convective ﬂow boiling inside ducts

44

2 Fundamentals

Averaged fluid temperature

Vapor single-phase flow

Forced convection to the vapor flow

Mist flow

Drywall + droplets deposition

x=1 Fluid temperature in the center of the flow

Wall temperature Wall dryout

Annular wispy flow

Fluid temperature

Annular flow

Wall temperature

Fluid temperature in the center of the flow

x=0

Saturation temperature T

Forced convection heat transfer through liquid film

Plug flow

Nucleate saturated boiling

Bubbles

Averaged fluid temperature

Subcooled boiling Liquid single-phase flow

z

Forced convection to the liquid flow

Fig. 2.13 Schematics of convective ﬂow boiling during vertical upward ﬂow, modiﬁed from Collier and Thome (1994)

minor variations associated to the changes of the ﬂuid transport properties with increasing temperature. The bubble nucleation process is triggered for a wall temperature higher than the saturation temperature at the local pressure. This characterizes a phenomenon known in literature as the onset of nucleate boiling (ONB) and may occur even for bulk ﬂuid temperature inferior than the saturation temperature because the ﬂuid closer to the wall is hotter than the temperature at the channel core. Bergles and Rohsenow (1964), based on the approach of Hsu and Graham (1961), proposed a model for the prediction of the ONB, which assumes vapor contained into cavities along the surface, such as schematically depicted in Fig. 2.14, and estimates the superheating or heat ﬂux necessary to promote the bubble growth. According to this approach, the pressure difference between the vapor trapped into cavities with radius r and the external liquid is evaluated based on the force balance as follows: pv p l ¼

2σ r

ð2:62Þ

2.4 Flow Boiling Fundamentals

45

Tv (r) - Eq. (2.65)

Liquid

y=r Tl (y) = Tv (r) dTl (y) = dTv (r) dy dr Eq. (2.67)

↑ϕ

y Tl (y) - Eq. (2.66)

r

r

r

↑ Tw Temperature Surface with cavities Fig. 2.14 Schematics of vapor bubble into cavity for ﬁxed Tw, adapted from Bergles and Rohsenow (1964)

where pv and pl correspond to the pressure inside and just outside the bubble, respectively, σ corresponds to the surface tension and r to the cavity radius. Therefore, the pressure inside the bubble is higher than the liquid pressure, and consequently the vapor is superheated in relation to local liquid pressure (Tv > Tsat ( pl)). It is possible to relate the corresponding superheating temperature based on the pressure difference using the Clausius–Clapeyron equation for ideal gas, given as follows for vaporization: bilv bi dpv p bi ¼ lv ¼ v lv2 dT T v T v ð vv vl Þ Rv T v v v

ð2:63Þ

where îlv corresponds to the latent heat of vaporization, Rv to the gas constant of the substance, and Tv to the vapor temperature. Hence, by integrating Eq. (2.63) from the liquid to the vapor condition, the following relationship is obtained: bi ðT T sat Þ pv pv pl ln þ 1 ¼ lv v ¼ ln pl pl Rv T v T sat

ð2:64Þ

which combined with Eq. (2.62) results in the following relationship: T v T sat ¼

Rv T v T sat 2σ ln þ1 rpl bilv

ð2:65Þ

which represents the superheating temperature of the vapor in relation to the saturation condition at pl to maintain the bubble stable.

46

2 Fundamentals

Bergles and Rohsenow (1964) assumed pure heat conduction along the liquid to estimate the temperature proﬁle, which results in the following relationship: Tl ¼ Tw

ϕy kl

ð2:66Þ

where ϕ corresponds to the heat ﬂux, kl to the liquid thermal conductivity, and Tw to the surface temperature. Therefore, considering that at the bubble tip the liquid and vapor must be in equilibrium, whereas: T l ¼T v dT l dT v ¼ dy dr

ð2:67Þ

for y ¼ r, Bergles and Rohsenow (1964) assumed Eq. (2.67) as the condition for the onset of nucleation boiling. It must be mentioned that Eqs. (2.65), (2.66) and (2.67) cannot be solved analytically and an interactive method is required for calculating Tv (or Tl), ϕ (or Tw), and r (or y). Figure 2.15 depicts the predictions of parameters for ONB according to the model proposed by Bergles and Rohsenow (1964). According to Fig. 2.15a, the wall superheating increases asymptotically with heat ﬂux for ONB, and according to Fig. 2.15b, the cavity radius that would be activated as a vapor formation nuclei decreases with increasing wall superheating and/or heat ﬂux. It must be mentioned that by solving Eqs. (2.65), (2.66) and (2.67), a single cavity radius value is obtained, which corresponds to the critical value. However, according to the analysis of Bergles and Rosehnow (1964) the bubble would form and grow as long as the liquid temperature, given by Eq. (2.66), is higher than the vapor temperature of equilibrium, given by Eq. (2.65). Hence, a range of active cavities can be estimated by imposing Tl ¼ Tv, as shown schematically by the dotted line on b) 104

100

R134a R1234ze(Z) R600a Water

103

r = y [m m]

T w - T sat [K]

a) 101

10-1

R134a R1234ze(Z) R600a Water

Tsat = 20 °C

102

101 T sat = 20 °C

10-2 10

100

1000 2 f [W/m ]

10000

100 10-2

10-1

100

101

Tw - Tsat [K]

Fig. 2.15 Onset of nucleate boiling according to the approach proposed by Bergles and Rohsenow (1964), depicting variation of (a) wall superheating with heat ﬂux, and (b) cavity sizes with wall superheating

2.4 Flow Boiling Fundamentals

47

the right in Fig. 2.14. Nonetheless, keep in mind that increasing the heat ﬂux results in an increment of wall temperature and vice versa. Based on this analysis, it can be concluded that high wall superheating and heat ﬂux are needed to activate small cavities, as shown in Fig. 2.15b. Therefore, by combining an energy balance equation with a correlation for forced convection, such as Gnielinski (1976) valid for Reynolds higher than 2500, and the above described method, the ONB position along the ﬂow path can be estimated. Recently, Kandlikar (2006) described a predictive method developed by his research group, which consists in a modiﬁcation of the model proposed by Hsu and Graham (1961) and Bergles and Rohsenow (1964) for microchannels during water ﬂow. According to his proposal, the wall superheating is given as follows: T w T sat ¼ 1:1

2σ sin ðβr Þ T sat rϕ þ r k l sin ðβr Þ ρvbilv

ð2:68Þ

where r stands for the cavity radius and βr to the receding contact angle, which depends on the pair ﬂuid-surface, as well as on the thermodynamic state of the ﬂuid. In Fig. 2.13, the bubble nucleation is established upstream the state of saturated liquid, characterizing subcooled ﬂow boiling. Downstream the ONB, as the ﬂuid enthalpy increases, the number of nucleation sites rises, and nucleate boiling becomes the main heat transfer mechanism. Eventually, the high vapor fraction causes bubbles coalescence with consequent formation of large bubbles, characterizing plug ﬂow. With additional increments of vapor fraction, the liquid plugs between consecutive vapor bubbles vanish and the ﬂow becomes characterized by a continuum liquid ﬁlm on the duct wall and vapor ﬂowing in the core of the channel, characterizing annular ﬂow. Due to the higher speciﬁc volume of vapor phase compared to liquid, the vapor velocity and, consequently, the two-phase velocity are increased as the liquid is evaporated. Under this circumstance, the shear of the vapor at high velocity on the liquid ﬁlm might cause detachment of liquid droplets. Thus, the ﬂow transitions from pure annular to wispy ﬂow characterized by a liquid ﬁlm along the tube perimeter with vapor ﬂow in the center of the section and entrained liquid droplets. For annular and wispy ﬂows, the heat transfer rate is given as a result of convective effects and the contribution of nucleate boiling effects is suppressed. As shown in Fig. 2.13 for the region comprising bubbles, plug, annular, and annular wispy ﬂow patterns, assuming negligible pressure drop, the ﬂuid temperature is almost constant and corresponds to the saturation temperature. On the other hand, the heat transfer coefﬁcient is high and as a consequence the difference between the ﬂuid and wall temperature is low. Eventually the liquid ﬁlm will vanish due to a combination of droplet detachment and their vaporization, and the liquid phase becomes distributed as small dispersed droplets within the vapor phase, characterizing mist ﬂow. As result of the surface dryout, the heat transfer coefﬁcient decreases signiﬁcantly, and the wall temperature presents a sudden increase. The heat transfer from the wall to ﬂuid under mist ﬂow conditions occurs ﬁrstly from the wall to the vapor contacting the duct, and then,

48

2 Fundamentals

Intermittently dry

Singlephase liquid

Bubbles

Forced Subcooled convection to boiling the liquid flow T

Plug

Slug

Nucleate saturated boiling

Annular

Forced heat transfer though liquid film

Singlephase vapor

Stratified

Forced convection to the vapor flow

Dry wall + droplets deposition

Wall temperature

x=0

Average fluid temperature

Fluid temperature in the center of the section

Saturation temperature x=1

Fig. 2.16 Schematics of convective ﬂow boiling during horizontal ﬂow

from the vapor to the liquid. Droplet deposition on the tube wall also contributes to the overall heat transfer process, promoting a slight increase in the heat transfer coefﬁcient, with a proportional reduction on the wall temperature. Downstream the surface dryout, the droplets will eventually vanish, and single-phase forced convection to the vapor ﬂow becomes the heat transfer mechanism. Based on the above discussion, it should be emphasized that the main heat transfer mechanisms during ﬂow boiling depend on the phases’ distribution, characterized as ﬂow patterns. The ﬂow patterns, in turns, depend on the vapor fraction. Similarly, Fig. 2.16 depicts the evolution of the ﬂow characteristics along the duct length for horizontal ﬂow in channels of conventional size under condition of imposed heat ﬂux. This ﬁgure also depicts the draft of wall and ﬂuid temperatures, as well as main heat transfer mechanisms in each region. Differently than vertical ﬂow, buoyancy effects exert a crucial effect on the phase distribution along the cross-section making it non-axisymmetric. Additionally, even though the pressure variation along the cross-section is small, the onset of bubble nucleation is more likely to occur in the upper region of the channel due to local slightly smaller pressure and predominance of hotter liquid in this region due to buoyance effects. The non-equilibrium and higher temperature of the duct wall cause bubble nucleation even under conditions of subcooled liquid ﬂow, which is depicted on the left side in Fig. 2.16 as subcooled ﬂow boiling. With additional increment of the ﬂuid enthalpy, the ﬂow will eventually become saturated, which is denominated as nucleate saturated boiling. For reduced vapor quality, the ﬂow is characterized by the ﬂow of vapor as small dispersed bubbles, hence corresponding to the bubble ﬂow pattern. With subsequent increment of vapor fraction, the bubble size and number increase, implying on their coalescence and ﬂow pattern transition from plug to slug ﬂow. Under this condition, the contribution of convective effects becomes more signiﬁcant, even though the heat transfer process is dominated by nucleate boiling effects.

2.5 In-Tube Condensation Fundamentals

49

As the liquid slugs between successive vapor bubbles are disrupted, the ﬂow shifts either to stratiﬁed ﬂow pattern under conditions of reduced ﬂow velocity or annular ﬂow pattern under conditions of intermediate and high ﬂow velocities. For stratiﬁed ﬂow, the upper portion of the duct wall is in contact with the vapor phase, implying on reduced heat transfer coefﬁcient and higher wall temperature along this region. The occurrence of stratiﬁed ﬂow pattern is restricted to conventional channels and low mass velocity ﬂows; therefore, it is expected that the contribution of the convective effects to heat transfer plays a small role while the nucleate boiling is the major heat transfer mechanism. This aspect is observed empirically, because in general, the heat transfer coefﬁcient during stratiﬁed ﬂow presents negligible variation with the increment of vapor quality; even a slight reduction of heat transfer coefﬁcient with vapor quality is reported in some studies for stratiﬁed ﬂow, which is attributed to reduction of the liquid portion in contact with the channel wall, and consequently increment of the surface exposed to the vapor phase. Annular ﬂow pattern is characterized by high heat transfer coefﬁcients along the entire duct perimeter and, consequently, a perimeter-averaged heat transfer coefﬁcient higher than the value for stratiﬁed ﬂow. Similar to the case of vertical ﬂows, the heat transfer process during annular ﬂow pattern in horizontal channel presents a signiﬁcant contribution of convective effects. Nonetheless, for annular ﬂow in horizontal ducts, the liquid ﬁlm in the upper region is thinner due to gravitational effects; therefore, additional increment of vapor fraction progressively dries the liquid ﬁlm from the upper region to the tube bottom. Some authors, such as Wojtan et al. (2005a), deﬁne the condition of gradual wall drying as a dryout ﬂow pattern, assumed as a transitional condition between annular and mist ﬂow (observed under high mass velocity conditions).

2.5

In-Tube Condensation Fundamentals

The fundamentals of in-tube condensation are presented in this section. The main heat transfer mechanisms during condensation are presented and discussed. Condensation during forced convection inside ducts is dominated by convective and gravitational effects, as schematically depicted in Fig. 2.17. Convective effects are dominant for high ﬂow velocities. Under this condition, the heat transfer coefﬁcient is mainly a function of the two-phase ﬂow velocity and its value increases with raising vapor quality and mass velocity, and effects on the heat transfer coefﬁcient of wall subcooling are negligible. On the other hand, condensation inside horizontal tubes dominated by gravitational effects are typical of low ﬂow velocities. The heat transfer process is basically controlled by the vapor condensation on the upper part of the duct and the resulting liquid is driven by gravity to the bottom region of the tube and is simultaneously propelled according to the main ﬂow direction. The heat transfer process in the upper part of the duct is somewhat similar to the condensation in vertical walls as addressed by Nusselt (1916). Under this condition, the heat transfer coefﬁcient is mainly a function of the wall subcooling.

50

2 Fundamentals

Convective effects

Gravitational effects

Hydrodynamic boundary layer

+

Thermal boundary layer

Velocity profile

Temperature profile

Fig. 2.17 Schematics of convective and gravitational effects for convective ﬂow condensation inside ducts

For small diameter tubes, surface tension effects are also indicated as relevant to the heat transfer process, as pointed out by Rossato et al. (2017). Figure 2.18 schematically depicts the two-phase ﬂow evolution, ﬂuid temperature, and heat transfer coefﬁcient variations for condensation along horizontal channels. This ﬁgure also depicts the dominant mechanisms affecting the heat transfer coefﬁcient along the condensation process. This description is similar to the one addressed by Ribatski and Da Silva (2016). The onset of liquid condensation requires a duct wall temperature lower than the saturation temperature at the local pressure, hence, liquid is condensed even for conditions corresponding to thermodynamic vapor qualities higher than the unity. Initially, the condensate forms a liquid ﬁlm along the duct internal perimeter, and with additional vapor quality reduction and high ﬂow velocities waves are present on the liquid–vapor interface. This ﬂow structure corresponds to a wavy-annular ﬂow pattern. Additional vapor quality reductions imply lower vapor shear effects on the liquid ﬁlm with the interface becoming smoother and the ﬂow transitioning to annular ﬂow. The heat transfer coefﬁcient for wavy-annular ﬂow is higher than for annular ﬂow due to both a thinner perimeter-averaged liquid ﬁlm and high two-phase ﬂow velocity. Under conditions of high ﬂow velocities, the vapor ﬂow may detach liquid droplets from the wavy ﬁlm, and the droplets ﬂow entrained within the vapor phase. As the vapor quality decreases, a signiﬁcant parcel of the liquid accumulates in the bottom region of the tube due to buoyancy effects, and the ﬂow becomes stratiﬁed with liquid segregated in the lower part of the tube and as a ﬁlm on the tube wall and vapor ﬂowing in the upper part of the tube. Depending on the vapor velocity, the vapor–liquid interface may be either smooth or wavy, corresponding to stratiﬁed and wavy-stratiﬁed ﬂow patterns, respectively. The heat transfer for these ﬂow patterns is dominated by gravitational effects. Convective effects are dominant for annular and wavy-annular ﬂow patterns while gravitational effects dominate under stratiﬁed ﬂow conditions.

2.6 Transition from Macro to Microscale Conditions

Superheated vapor

Wavyannular

Annular

Wavy stratified

51

Stratified

Slug

x = 1.0

Subcooled liquid x = 0.0

Convective effects

Gravitational effects

Convective effects

T Fluid temperature in the center of the flow

Fluid temperature

Wall temperature

Heat transfer coefficient

Fig. 2.18 Schematics of convective ﬂow condensation inside horizontal channel, adapted from Ribatski and da Silva (2016)

Successive reductions of vapor quality will eventually result in liquid slugs between elongated bubbles, as depicted in Fig. 2.18. According to Ribatski and Da Silva (2016), the heat transfer process for intermittent ﬂow patterns is dominated by convective effects, since the heat transfer coefﬁcient is mainly associated to the ﬂow characteristics close to the tube wall. Finally, further vapor quality reductions promote bubbles to shrink and, ultimately, their collapse when liquid single-phase ﬂow is established. However, as shown in Fig. 2.12, some vapor bubbles may remain within the ﬂow even for subcooled ﬂow, characterizing a non-equilibrium condition. For slug ﬂow, as the vapor quality decreases the two-phase ﬂow velocity and, consequently, the heat transfer coefﬁcient are reduced.

2.6

Transition from Macro to Microscale Conditions

Several approaches and criteria that deﬁne the transition between micro and macroscale available in the open literature are presented and discussed. Approaches based on manufacturing techniques, channel geometry, bubble conﬁnement, and two-phase ﬂow characteristics are addressed. In small diameter channels, the duct dimension becomes of the same order of magnitude of the bubble departure diameter, hence it is expected that surface tension forces present signiﬁcant inﬂuence on the ﬂow behavior and heat transfer process, which justiﬁes the need for different modelling and analysis approaches for micro and conventional channels. Moreover, it is expected that the ﬂow restriction caused

52

2 Fundamentals

by the bubble growth itself induces an earlier bubble detachment than observed for conventional channels and pool boiling conditions in a stagnant media as pointed out by Jacobi and Thome (2002). Therefore, the characterization of the channel size scales might be important based on the fact that the predominant forces and mechanisms change as the channel dimension reduces. In this context, Thome (2004) pointed out several aspects that justify the classiﬁcation of channels according to the dimensions. In his review, Thome gathered experimental results from the literature and indicated axisymmetric phase distribution in the case of microscale channels, as well as differences of trends and values for heat transfer coefﬁcient, critical heat ﬂux, void fraction, and pressure drop between micro and macroscale conditions. Considering the hydrodynamic aspect, it is common sense that two-phase ﬂow in conventional, also named as macroscale channels, is dominated by inertial and gravitational forces with transport properties such as phase densities and viscosity playing the dominant role. Nonetheless, although surface tension also inﬂuences the phase distribution and interface geometry for two-phase ﬂow in conventional channels, such as pointed out by Rodriguez and Castro (2014) setting the curvature of interface for stratiﬁed ﬂow and affecting two-phase ﬂow transitions, its effects are not dominant. On the other hand, for two-phase ﬂows in microscale channels, the reduced curvature radius of the interface associated to the two-phase ﬂow conﬁnement implies on the predominance of surface tension effects. In the state-of-the-art review of Mehendal et al. (2000), the authors pointed out distinguished behaviors between micro and mesoscale channels for Nusselt number and friction factor, even for single-phase ﬂow, compared to conventional channels. Based on the literature review, these authors also speculated that the transition from laminar to turbulent ﬂow regimes occurs for Reynolds number within the range of 200–900 for microscale channels, well below the usual values observed for conventional channels quoted around 2300. In fact, currently it is well established for single-phase ﬂow that the hydrodynamics and heat transfer behaviors are similar for channels with dimensions larger than the condition for which rarefaction effects become relevant. As pointed out by Morini (2006), these differences observed in the earlier studies for single-phase ﬂows were linked mainly to experimental uncertainties associated to surface roughness and channel dimension evaluation. Based on this discussion and on contrary to single-phase ﬂow, it can be concluded that the classiﬁcation of micro and conventional scale channels for two-phase ﬂows is not just a matter of channel size, but it is related to the dominant forces that govern the ﬂow and dominant heat transfer mechanisms. Researchers have proposed criteria for the classiﬁcation of channels according to size scales, and Mehendal et al. (2000) proposed their classiﬁcation according to an order of increasing their size as: nano, micro, mini (or meso), and macrochannels (or conventional). Classiﬁcations based on manufacturing processes with the transitional diameters deﬁned arbitrarily were proposed by Mehendal et al. (2000) and Kandlikar and Grande (2003). Mehendal et al. (2000) proposed the following

2.6 Transition from Macro to Microscale Conditions

53

classiﬁcation based on the hydraulic diameter: micro 1 ~ 100μm; meso 100 ~ 1000μm; compact 1000 ~ 6000μm; conventional >6000μm. Kandlikar and Grande (2003) introduced in this discussion aspects related to the range of dimensions for which the random motion of the particles affects the ﬂow conditions, and classiﬁed the channel scales as follows: molecular nanochannel 3000μm. According to Thome (2008), Bretherton (1961) was the ﬁrst author to propose a channel scale classiﬁcation based on bubble conﬁnement, even though not intentionally focused on microscale ﬂows, who indicated the pipe diameter below which a Taylor bubble would not rise due to buoyancy effects. Bretherton (1961) concluded that a Taylor bubble would not rise for Eötvos numbers smaller than 6.736. Kew and Cornwell (1997) proposed also a classiﬁcation based on mechanistic aspects of two-phase ﬂow and convective boiling. According to them, prediction methods for the heat transfer coefﬁcient developed for conventional channels are not valid for conﬁnement numbers higher than 0.5; hence, the ﬂow boiling process is considered to occur under microscale conditions for Co higher than 0.5, with the conﬁnement number deﬁned as follows: Co ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gðρl ρv Þd2

ð2:69Þ

which is associated to the ratio between surface tension and buoyance forces. It must be emphasized that even though this method takes into account surface tension and buoyance effects, it does not include inertial effects associated to two-phase ﬂow velocity that may favor the detachment of smaller bubbles compared to quiescent conditions. Moreover, the evaluation of the degree of ﬂow conﬁnement for diabatic conditions based on two-phase topology may depend on heat ﬂux intensity and if the ﬂuid is being evaporated or condensed. Based on the same approach of Kew and Cornwell (1997), Ullman and Brauner (2007) proposed a transition criterion based on the Eötvos number, given as follows: Eo ¼

gðρl ρv Þd 2 8σ

ð2:70Þ

which corresponds to the inverse of squared conﬁnement number divided by eight. According to Ullman and Brauner (2007), a channel can be considered a microscale channel for conditions with Eötvos number smaller than 0.2. It must be emphasized that this approach is focused on two-phase ﬂow inside channels, taking into account characteristics of both phases, and assumed that the condition to characterize a microchannel corresponds to the impossibility of occurring stratiﬁed ﬂows. Kawaji and Chung (2003) proposed a transition criterion based on six dimensionless numbers that takes into account the inﬂuence of ﬂow velocity. The transition criteria proposed by them is given as follows:

54

2 Fundamentals

ðρl ρv Þgd 2 1 4σ j2 dρ G 2 ð 1 xÞ 2 d ¼ l l¼ 1 ρl σ σ j2 dρ G2 x2 d ¼ v v¼ 1 ρv σ σ ρ j d Gð1 xÞd ¼ l l ¼ < 2000 μl μl ρ j d Gd ¼ v v ¼ < 2000 μv μv μ G ð 1 xÞ μj 1 ¼ l l¼ l ρl σ σ

Bd ¼ Wel Wev Re l Re v Cal

ð2:71Þ

where Cal is the capillary number based on the liquid velocity and viscosity, and Bd is the Bond number, which consists of two times the Eötvos number. In their method, all the criteria must be satisﬁed, and the smallest diameter should be considered as the threshold value between micro and macroscale conditions. Even though the capillary number does not take into account the tube hydraulic diameter, it is used to evaluate the balance between viscous dissipation and surface tension, and must be satisﬁed to characterize the two-phase ﬂow in a pipe as under microscale conditions. Similarly, Ong and Thome (2011) proposed a transition criterion to classify micro, meso, and macroscale channels based on their investigation of channel effect on ﬂow pattern transitions. The criteria to classify a certain channel as micro or macro was based on whether ﬂow patterns characteristic of microscale occurred. The mesoscale channels consisted of dimensions with thermohydraulic characteristics of micro and macroscales. Hence, according to Ong and Thome (2011), a channel can be considered microchannel for conﬁnement number higher than 1.0, macrochannels correspond to conﬁnement number smaller than 0.34, and mesoscale for the intermediate range, where the conﬁnement number Co is given by Eq. (2.69). Macro-to-microscale segregation methods based on the bubble departure diameter are suitable only for ﬂow boiling, whereas the transition for adiabatic ﬂows and convective condensation cannot be predicted based on these approaches. Therefore, Li and Wang (2003) proposed a method to characterize micro-to-macroscale transitions based on the Young–Laplace equation accounting for the pressure difference between both sides of the interface imposed by the surface tension. In their method, the transition is given as a function of capillary length, deﬁned as follows: Lcap ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gð ρl ρv Þ

ð2:72Þ

Li and Wang (2003) evaluated the symmetry of the phase distribution during condensation inside round channels, and proposed criteria for identiﬁcation of channel dimensions for which the occurrence of stratiﬁed diameter is possible, deﬁned as dcritical, and the channel dimension for which the ﬂow would be minimally axisymmetric, deﬁned as dtransition. Their transitional dimeters are given as follows:

2.6 Transition from Macro to Microscale Conditions

d [m]

5 x10

R134a, G = 100 kg/m²s, ∆ T = 10 °C, x = 0.10

-3

10

10

Ong and T Brethert home (2011), c on (196 1) - c , macro Kew an d Cornw ell (199 Li and W 7) - c ang (20 03), criti cal Ullman and Bra uner (20 Ong and 07) - c Thome (2011), c, micro

-3

-4

Kutate Nishikawa et al. (1976) Li and W ladze and Gog o ang (20 03), tran nin (1979) sition Kandlikar and Grande (2003) 003) - c (2 ng hu C d Fritz (1 Kawaji an 935) Mehendal et al. (2000)

0,0

4 x10

0,2

-3

-4

p r [-]

0,6

0,8

1,0

Ong an d Thom Brethe e (2011), c, macro rton (1 961) Kew a nd Li and Cornwell ( 19 Wang (2003) 97) Ullma , critic n and B al rauner Ong an (2007) d Thom e (201 1), c, m icro

d [m] 10

0,4

Isobutane, G = 100 kg/m²s, ∆ T = 10 °C, x = 0.10

-3

10

55

Kuta Nishikawa et al. (1976) Li and teladze and G Wang (2003) ogonin (197 9) , transit ion Kandlikar and Grande (2003) Fritz ( Kawaji and Chung (2003) 1935) Mehendal et al. (2000)

0,0

0,2

0,4

p r [-]

0,6

0,8

1,0

Fig. 2.19 Micro and macroscale transition diameters for R134a and isobutene at wall superheat of 10 C, x of 0.10, and mass velocity of 100 kg/m2s

dtransition ¼ 0:255 Lcap

ð2:73Þ

d critical ¼ 1:749 Lcap

ð2:74Þ

Figure 2.19 depicts the variation of the transition diameter according to distinct criteria discussed in the present section, for R134a and isobutane for reduced

56

2 Fundamentals

pressure pr varying from values close to zero to values close to unity. According to this ﬁgure, as the working pressure increases, the transition diameter that characterizes the microscale channels is reduced according to most methods proposed based on ﬂuid properties, on the contrary to the criteria of Mehendal et al. (2000) and Kandlikar and Grande (2003), which consist on ﬁxed transitional values. In conclusion, a criterion to characterize macro to microscale conditions is suitable only if different behaviors for heat transfer, pressure drop, ﬂow induced vibration, and noise are observed, since those are the parameters that a designer of a heat transfer device should care. Moreover, if this is the case, different criteria should be considered according to the two-phase ﬂow conditions such as adiabatic ﬂows, ﬂow boiling, and convective condensation. Aspects such as ﬂow patterns should also be considered since a bubble ﬂowing under narrowed conditions does not represent the physics of conﬁned annular ﬂow.

2.7

Solved Example

1. Consider two-phase ﬂow of R134a inside a 1 mm ID channel, with saturation temperature of 15 C, mass ﬂux of 250 kg/m2s, and vapor quality of 0.15. Present the mass ﬂow rate. Superﬁcial velocities of each phase and the mixture velocity. Volumetric fraction. Void fraction according to homogeneous model, Zivi approach, minimum kinetic energy, and Rouhani approaches for horizontal and vertical upward ﬂow. (e) The in situ velocities assuming the four void fraction predictions. (f) Does this condition correspond to macro or microscale channel according to Kew and Cornwell criteria?

(a) (b) (c) (d)

Firstly, let us list the input data: Fluid ¼ R134a; d ¼ 1 mm ¼ 0.001 m; Tsat ¼ 15 C; G ¼ 250 kg/m2s; x ¼ 0.15. Considering that this problem comprises simultaneous ﬂow of liquid and vapor phases, it is interesting to list the thermophysical properties of both phases for Tsat ¼ 15 C, as follows: ρl ¼ 1243 kg/m3; ρv ¼ 27.78 kg/m3; σ ¼ 0.009359 N/m; μl ¼ 0.00022 kg/ms; μv ¼ 1.153105 kg/ms. (a) The mass ﬂux consists to the ratio between the mass ﬂow rate and the crosssectional area, as follows: G¼

m_ m_ ¼ A πd2 4

Then, solving for ṁ, the mass ﬂow rate is equal to 7.854107 kg/s.

2.7 Solved Example

57

(b) The superﬁcial velocity, or volumetric ﬂux, consists in the ratio between the volumetric ﬂow rate and the cross-sectional area. Therefore, for each phase, the volumetric ﬂux can be given as follows: jv ¼

Qv Q and jl ¼ l A A

Considering the available input parameters, it is easier to represent the volumetric ﬂow rate as a function of the mass ﬂux, as follows: jv ¼

Qv _ Gx m m_ mx ¼ v ¼ ¼ ¼ 1:577 s A ρv A ρv A ρv

Similarly, for the liquid phase: jl ¼

Ql Gð1 xÞ m ¼ 0:171 ¼ ρl s A

And the mixture superﬁcial velocity is given as the sum of the volumetric ﬂux of each phase, as follows: j ¼ jv þ jl ¼ 1:748

m s

Notice that even for a relatively low vapor quality value, the liquid velocity corresponds to only approximately 10% of the mixture velocity. (c) The volumetric fraction corresponds to the ratio between the volumetric ﬂow rate of the vapor phase and the mixture volumetric ﬂow rate, as follows: β¼

Qv Qv þ Ql

Dividing the numerator and denominator by the cross-sectional area A, it is possible to use directly the results for superﬁcial velocities obtained in the previous item, as follows: β¼Q

v

A

Qv A

þ QAl

¼

jv j ¼ v ¼ 0:902 jv þ jl j

(d) According to the homogenous model, it is assumed that the slip velocity ratio is equal to the unity, hence, the void fraction is given as follows: αH ¼

1 1 þ uuvl ρρv l

1x x

58

2 Fundamentals

where uv/ul ¼ 1.0. Hence, substituting the values, the void fraction according to the homogeneous model is 0.902, which is identical to the volumetric fraction, as discussed along the chapter. According to the Zivi method, the void fraction is given as follows: αZivi ¼ 1þ

1 23 ρv ρl

1x x

and substituting the corresponding values, the resulting value is 0.712, which is considerably lower than αH. According to the minimum kinetic energy approach, the void fraction depends on the momentum coefﬁcient ratio, and is given as follows: α min ,kinetic ¼ 1þ

1 13 13 2 ρv Kl 1x 3 Kv

ρl

x

whereas for horizontal ﬂow, the momentum coefﬁcient ratio is given as:

Kl Kv

13

¼ 1:021 Fr 0:092 m

0:368 μl μv

with Froude number given as: Fr m ¼

G2 ¼ 4:284 gðρl ρv Þ2 d

resulting in momentum coefﬁcient ratio of 0.3018. And for vertical upward ﬂow, the momentum coefﬁcient ratio is given as follows:

Kl Kv

13

¼ 14:549 We0:222 m

1:334 μl μv

with Weber number given as: Wem ¼

G2 d ¼ 5:476 ðρl ρv Þσ

which results in momentum coefﬁcient ratio of 0.1954. Both values must be evaluated to verify whether they satisfy the condition that the maximum void fraction possible corresponds to the homogenous model. Hence, considering the inequality:

2.7 Solved Example

59

Kl Kv

13

23 1 ρv 1x 3 x ρl

the right-hand side is equal to 0.1275. Hence, this restriction is satisﬁed for horizontal and vertical ﬂows. Therefore, by substituting both values for void fraction, we obtain void fraction values of 0.796 for horizontal ﬂow and 0.858 for vertical ﬂow. Finally, the predictions according to the drift ﬂux model adjusted by the Rouhani research group are given as follows: ( α2 ¼

" #)1 e vj V ρv x 1x C0 þ þ ρv ρl x G

where the slip parameter is estimated for gas bubbles rising in stagnant liquid, given as follows: 1 gσ ðρl ρv Þ 4 m e V vj ¼ 1:18 ¼ 0:1088 s ρ2l For horizontal ﬂow, the distribution parameter is given as follows: C 0 ¼ 1 þ 0:2ð1 xÞ

1 gdρ2l 4 ¼ 1:119 G2

and for vertical ﬂow, it is equal to 1.10 because the mass ﬂux is higher than 200 kg/m2s, otherwise it would be 1.54. Therefore, substituting in the relationship for void fraction, the obtained values are, respectively, 0.764 and 0.776 for horizontal and vertical ﬂows. (e) The in situ velocity depends on the local void fraction, as follows: uv ¼

jv jl u ¼ α l 1α

Hence, considering the results from the previous item, the following in situ velocity values are obtained: Void fraction Homogenous model Zivi (1964) model Kanizawa and Ribatski (2016) horizontal Kanizawa and Ribatski (2016) vertical Rouhani (1969) horizontal Rouhani and Axelsson (1970) vertical

α [] 0.902 0.712 0.796 0.858 0.764 0.776

jv [m/s] 1.577

jl [m/s] 0.171

uv [m/s] 1.748 2.032 1.982 1.839 2.065 2.032

ul [m/s] 1.748 0.593 0.837 1.200 0.723 0.764

60

2 Fundamentals

Notice the difference between superﬁcial and in situ velocities. Additionally, the void fraction value estimated according to the homogenous model is considerably higher than according to other methods. (f) The Kew and Cornwell criterion is based on the conﬁnement number, whereas a channel is considered as micro for Co higher than 0.5. Hence, the conﬁnement number is given as follows: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ Co ¼ ¼ 0:885 > 0:5 gðρl ρv Þd2 Hence, it can be considered as a microchannel according to this criterion.

2.8

Problems

1. Derive the relationships given by Eqs. (2.20) and (2.21). 2. Derive the relationships for superﬁcial velocities based on average operators. 3. Show that the volumetric fraction β is similar to the void fraction estimated according to the homogenous model. 4. Assume an axisymmetric annular ﬂow absent of liquid droplets in the vapor stream. What is the area averaged void fraction if the linear void fraction, along a rectilinear segment passing by the center of the cross-section, is (a) 85, (b) 90, and (c) 95%? Are they different from the volume averaged void fraction in this case? 5. Redo exercise 4 by assuming stratiﬁed ﬂow, with linear void fraction measurement along vertical direction in the central plane of the channel. 6. Redo exercise 5 by assuming the same conditions, but consider that the surface tension is equal to 0,01073 N/m and that the contact angle is equal to 45 . For simplicity, consider vertical wall for the estimation of liquid portion due to surface tension. 7. Consider R134a ﬂowing along a horizontal microscale channel with 0.5 mm of internal diameter. Assuming that a uniform heat ﬂux of 10 kW/m2 is imposed in the internal wall, the ﬂuid in the duct inlet is at 5 C subcooling temperature for a pressure of 293 kPa at mass ﬂux of 300 kg/m2s, determine: (a) the vapor quality 0.1 m away from the inlet? (b) the void fraction according to the homogenous model, Kanizawa and Ribatski method, and drift ﬂux model. Assume that the pressure drop parcel is negligible. 8. Repeat exercise 7 assuming vertical upward ﬂow. 9. Repeat exercise 7, and calculate the phases and mixture superﬁcial velocities for vapor qualities of 0.1, 0.5, and 0.8. 10. Compare the void fraction predicted values for vapor quality ranging from 0 to 1 (increment of 0.1) according to the predictive methods described in this study

References

11. 12.

13.

14.

61

for vertical and horizontal ﬂows. Assume internal diameter of 1 mm, and two-phase ﬂow of R22, R134a, CO2, and R1234ze at 0 and 40 C. Discuss the differences between the estimated values. Based on the deﬁnitions presented in this chapter, derive Eq. (2.46) starting from Eq. (2.45). Discuss if the condition of α higher than the homogenous model for horizontal or vertical upward ﬂow is feasible. Why, or why not? What about the case of downward ﬂow? The line averaged void fraction determined with a radioactive densitometer indicated a value of α1 for annular ﬂow in a round channel, measured transversal to the tube axis and passing by its center line. What is the corresponding area averaged void fraction? Estimate for α1 ¼ 95%. Now, assume a stratiﬁed ﬂow whereas the line averaged void fraction measured vertically and transversally to the channel axis, passing by its centerline, is equal to α1. Determine the corresponding area averaged void fraction. Estimate for α1 of 95%, 80%, 50%, and 25%. Compare with previous problem.

References Barnea, D., Shoham, O., & Taitel, Y. (1982). Flow pattern transition for vertical downward two phase ﬂow. Chemical Engineering Science, 37(5), 741–744. Bergles, A. E., & Rohsenow, W. M. (1964). The determination of forced-convection surfaceboiling heat transfer. Journal of Heat Transfer, 86, 365–372. https://doi.org/10.1115/1.3688697 Bretherton, F. P. (1961). The motion of long bubbles in tubes. Journal of Fluid Mechanics, 10(02), 166–188. Collier, J. G., & Thome, J. R. (1994). Convective boiling and condensation. Oxford: Clarendon Press. Delhaye, J. M. (1981a). Two-phase ﬂow patterns. In A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, & F. Mayinger (Eds.), Two-phase ﬂow and heat transfer in the power and process industries (pp. 234–235). Washington: Hemisphere publishing corporation. Delhaye, J. M. (1981b). Equations for two-phase ﬂow modeling. In A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, & F. Mayinger (Eds.), Two-phase ﬂow and heat transfer in the power and process industries (pp. 234–235). Washington: Hemisphere publishing corporation. Falcone, G., Hewitt, G., & Alimonti, C. (2009). Multiphase ﬂow metering: Principles and applications (Vol. 54). Amsterdam: Elsevier. Gnielinski, V. (1976). New equations for heat and mass transfer in turbulent pipe and channel ﬂow. International Chemical Engineering, 16(2), 359–368. Hsu, Y. Y., & Graham, R. W. (1961). An analytical and experimental study of the thermal boundary layer and ebullition cycle in nucleate boiling (Vol. 594). Washington, DC: National Aeronautics and Space Administration. Ishii, M., & Hibiki, T. (2010). Thermo-ﬂuid dynamics of two-phase ﬂow. New York: Springer Science & Business Media. Ishii, M., & Hibiki, T. (2011). Thermo-ﬂuid dynamics of two-phase ﬂow. Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-7985-8 Jacobi, A. M., & Thome, J. R. (2002). Heat transfer model for evaporation of elongated bubble ﬂows in microchannels. Journal of Heat Transfer, 124(6), 1131–1136. https://doi.org/10.1115/ 1.1517274

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Kandlikar, S. G. (2006). Nucleation characteristics and stability considerations during ﬂow boiling in microchannels. Experimental Thermal and Fluid Science, 30(5), 441–447. Kandlikar, S. G., & Grande, W. J. (2003). Evolution of microchannel ﬂow passages– Thermohydraulic performance and fabrication technology. Heat Transfer Engineering, 24(1), 3–17. Kanizawa, F. T., & Ribatski, G. (2016). Void fraction predictive method based on the minimum kinetic energy. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 209–225. Kawaji, M., & Chung, P. Y. (2003, January). Unique characteristics of adiabatic gas-liquid ﬂows in microchannels: Diameter and shape effects on ﬂow patterns, void fraction and pressure drop. In ASME 2003 1st international conference on microchannels and minichannels (pp. 115–127). American Society of Mechanical Engineers, Rochester, New York, USA. Kew, P. A., & Cornwell, K. (1997). Correlations for the prediction of boiling heat transfer in smalldiameter channels. Applied Thermal Engineering, 17(8), 705–715. Lavin, F. L., Kanizawa, F. T., & Ribatski, G. (2019). Analyses of the effects of channel inclination and rotation on two-phase ﬂow characteristics and pressure drop in a rectangular channel. Experimental Thermal and Fluid Science, 109, 109850. Li, J. M., & Wang, B. X. (2003). Size effect on two-phase regime for condensation in micro/mini tubes. Heat Transfer—Asian Research, 32(1), 65–71. Marchetto, D. B. (2019). Analysis of thermohydraulic performance of a polymeric heat sink based on convective boiling in microchannels. Dissertation (Master) – Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos, Brazil, 173 p. Mehendale, S. S., Jacobi, A. M., & Shah, R. K. (2000). Fluid ﬂow and heat transfer at micro-and meso-scales with application to heat exchanger design. https://doi.org/10.1115/1.3097347 Morini, G. L. (2006). Scaling effects for liquid ﬂows in microchannels. Heat Transfer Engineering, 27(4), 64–73. Nusselt, W. (1916). The surface condensation of water vapour. VDI Z, 60, 541–546. Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase ﬂow: Part 1– Two-phase ﬂow patterns and ﬁlm thickness measurements. Experimental Thermal and Fluid Science, 35(1), 37–47. Ribatski, G., & Da Silva, J. D. (2016). Condensation in microchannels. In S. K. Saha (Ed.), Microchannel phase change transport phenomena (pp. 287–324). Amsterdam/Boston: Elsevier. ISBN: 9780128043189. Rodriguez, O. M., & Castro, M. S. (2014). Interfacial-tension-force model for the wavy-stratiﬁed liquid–liquid ﬂow pattern transition. International Journal of Multiphase Flow, 58, 114–126. Rossato, M., Da Silva, J. D., Ribatski, G., & Del Col, D. (2017). Heat transfer and pressure drop during condensation of low-GWP refrigerants inside bar-and-plate heat exchangers. International Journal of Heat and Mass Transfer, 114, 363–379. Rouhani, S. Z. (1969). Modiﬁed correlations for void-fraction and pressure drop. AB Atomenergi Sweden, AE-RTV-841, pp. 1–10. Rouhani, S. Z., & Axelsson, E. (1970). Calculation of void volume fraction in the subcooled and quality boiling regions. International Journal of Heat and Mass Transfer, 13(2), 383–393. Sempértegui-Tapia, D., De Oliveira Alves, J., & Ribatski, G. (2013). Two-phase ﬂow characteristics during convective boiling of halocarbon refrigerants inside horizontal small-diameter tubes. Heat Transfer Engineering, 34(13), 1073–1087. Thome, J. R. (2004). Engineering data book III. Wolverine Tube Inc, 2010. Thome, J. R. (2008). Engineering data book III. Wolverine Tube Inc, 2010. Tibiriçá, C. B., & Ribatski, G. (2013). Flow boiling in micro-scale channels–Synthesized literature review. International Journal of Refrigeration, 36(2), 301–324. Tibiriçá, C. B., Rocha, D. M., Sueth Jr, I. L. S., Bochio, G., Shimizu, G. K. K., Barbosa, M. C., & dos Santos Ferreira, S. (2017). A complete set of simple and optimized correlations for microchannel ﬂow boiling and two-phase ﬂow applications. Applied Thermal Engineering, 126, 774–795. https://doi.org/10.1016/j.applthermaleng.2017.07.161

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Ullmann, A., & Brauner, N. (2007). The prediction of ﬂow pattern maps in minichannels. Multiphase Science and Technology, 19(1), 49–73. https://doi.org/10.1615/MultScienTechn. v19.i1.20 Wallis, G. B. (1969). One-dimensional two-phase ﬂow. New York: McGraw Hill Book Company. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005a). Investigation of ﬂow boiling in horizontal tubes: Part I—A new diabatic two-phase ﬂow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005b). Measurement of dynamic void fractions in stratiﬁed types of ﬂow. Experimental Thermal and Fluid Science, 29(3), 383–392. Woldesemayat, M. A., & Ghajar, A. J. (2007). Comparison of void fraction correlations for different ﬂow patterns in horizontal and upward inclined pipes. International Journal of Multiphase Flow, 33(4), 347–370. Zivi, S. M. (1964, May). Estimation of steady state steam void fraction by means of the principle of minimum entropy production. Journal of Heat Transfer, 86, 247–252. Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase ﬂow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 3

Flow Patterns

This chapter is focused on the analysis of ﬂow patterns during gas-liquid ﬂows inside channels. The ﬂow patterns commonly found in literature for adiabatic two-phase ﬂow in conventional channels are described in Sect. 2.2. These ﬂow patterns are schematically illustrated in Figs. 2.2, 2.3, and 2.4 for vertical upward, vertical downward, and horizontal ﬂows, respectively. According to the discussion presented in Sect. 2.6, the deﬁnition of microscale channel itself is usually based on the phase distribution within the ﬂow and the dominant forces, as well as the dominant heat transfer mechanisms. Therefore, it is expected that the ﬂow patterns during two-phase ﬂows in microscale channels present different characteristics from those for conventional channels, or at least, the transition between ﬂow patterns are not the same. Despite these aspects, this chapter also comprises descriptions of the predictive methods for ﬂow pattern transitions in conventional channels, because they are used as the basis for the development of predictive methods for microscale channels.

3.1

Flow Pattern Identiﬁcation

The ﬂow patterns can be identiﬁed subjectively, based on visual observation and the subjective judgement by an observer, and objectively, based on an analysis of time response of some ﬂow parameters, such as void fraction and pressure drop. Subjective Approach Generally speaking, ﬂow patterns are classiﬁed based on the geometrical characteristics of the gas–liquid interface, as discussed in Chap. 2, and named through intuitive denominations such as bubbles, annular, and so on. This procedure allows to identify them based on the judgement of an observer considering visual observations of the ﬂow pattern at naked eye as well as helped by ﬂow images and videos captured along a translucid section. Considering that different ﬂow pattern © The Author(s) 2021 G. Ribatski, F. Kanizawa, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_3

65

66

3

Flow Patterns

judgments may be provided by distinct observers, this approach is nominated as subjective. Flow pattern segregations based only on visual observations are appropriate for conditions of low mass velocities and separated ﬂows, while under conditions of high ﬂow velocities and/or non-continuous ﬂow patterns, ﬂow pictures and videos are necessary. The subjective approach provides reasonable results and, generally, the ﬂow patterns segregated according to this method are associated with different heat transfer and pressure drop behaviors. In this context, predictive methods for heat transfer coefﬁcient and frictional pressure drop are frequently developed based on the local ﬂow pattern characterized through the subjective approach. Objective Approach Under certain conditions, the visual access to the ﬂow is not possible, such as for test sections with metallic walls, opaque ﬂuids, and applications involving extremely high pressure and/or temperature. Consequently, under these circumstances, ﬂow patterns are inferred from time varying properties of the ﬂow. Jones and Zuber (1975) were among the pioneers to implement an objective approach to segregate ﬂow patterns. They identiﬁed ﬂow patterns for vertical gas-liquid ﬂow in a rectangular channel based on the analysis of chordal void fraction measurements performed with X-ray densitometry with the presence of liquid phase corresponding to higher attenuation of the X-ray. The probability density function (PDF) of the void fraction signal was used to identify the following ﬂow patterns: bubbles, intermittent, and annular. Figure 3.1 schematically depicts the void fraction signature with time of the normalized X-ray signal attenuated by the ﬂow and the corresponding power spectral density of the chordal void fraction α for bubbles, intermittent, and annular ﬂow patterns. Two-phase ﬂow under bubble ﬂow patterns corresponds predominantly to a continuum liquid with dispersed bubbles, therefore, the PDF peak corresponds to reduced void fraction values, as shown in Fig. 3.1. On the other hand, the annular ﬂow pattern corresponds to a thin liquid ﬁlm ﬂowing on the duct wall, and gas a)

b) Bubbles Intermittent Annular

Probability Density Function

Vapor 1.0

α [-]

0.8 0.6 0.4 0.2 Liquid 0.0

Time

0.0 Liquid

0.2

0.4

0.6 α [-]

Fig. 3.1 Schematics of power spectral density analysis for different ﬂow patterns

0.8

1.0 Vapor

3.1 Flow Pattern Identiﬁcation

67

ﬂowing in the core, therefore, corresponds to conditions of predominantly high void fraction values, as also shown in Fig. 3.1. The intermittent ﬂow pattern corresponds to the intermittent passage of elongated bubbles separated by liquid slugs containing small dispersed bubbles (refer to the slug ﬂow schematically depicted in Fig. 2.2). Hence, during the elongated bubble passage across the sensor signal, the void fraction value is similar to that of the annular ﬂow, and during the liquid passage, the void fraction signal is similar to the condition of bubble ﬂow pattern. As shown in Fig. 3.1, this condition corresponds to two peaks, at low and high void fraction values. Even though the approach based on the power spectral density is less dependent on the judgement of an observer, the classiﬁcation still relies on the deﬁnition of threshold values based on arbitrary judgements. In this context, methods to detect and segregate the ﬂow patterns based on artiﬁcial intelligence such as clustering and neural network algorithms have become a promising approach, such as performed by Mi et al. (1998, 2001). The clustering methods are independent of subjective judgments and their principle is grouping the data points according to similarities. One example of clustering method is the k-means that groups the data points to form clusters of data based on the Euclidean distance among them and the centroid of each cluster. The k-means method consists in an iterative method, and as a result, each experimental data is attributed to a data cluster that can be considered as a ﬂow pattern. The correspondence between the obtained groups and the ﬂow patterns visually identiﬁed is not necessarily attained. Moreover, according to the literature, the ﬁnal groups might depend on the initial guess for the centroids. Routines to implement such methods are not a major problem since they are already available in most engineering software, such as Matlab, Scilab, and Canopy, which provide libraries with these functions. The implementation of the clustering methods to ﬂow pattern identiﬁcation initially requires the deﬁnition of the typical ﬂow parameters to be used as data input in the algorithm. The mean value, standard deviation, and peak to valley value of the instantaneous void fraction along a period, among others, can be used as input parameters for the clustering method. Other parameters, such as local pressure, pressure drop along a channel length, and temperature, could also be used as input parameters for the clustering methods, as well as combinations of them. However, it is desirable to use non-dimensional parameters; therefore, the standard deviation, peak to valley values, etc., can be normalized by the mean value of each experimental/operational condition. It must be mentioned that ﬂow pattern identiﬁcation should mainly consider the effects of two-phase ﬂow topology on heat exchanger design parameters, such as heat transfer coefﬁcient, pressure drop, ﬂow-induced vibration, ﬂow-induced noise, and critical heat ﬂux, since these are the aspects considered by the designer. The next sections are dedicated to the presentation of ﬂow pattern predictive methods.

68

3.2

3

Flow Patterns

Flow Pattern Transition Criteria for Adiabatic Flows

This section addresses ﬂow pattern predictive methods that were developed focused on conventional channels. Usually, these methods are based on experimental results for channels with dimensions typical of applications in nuclear and oil and gas industries, and quite commonly air and water mixtures close to atmospheric pressure and ambient temperature are used as working ﬂuids. The methods nominated as ﬂow pattern maps are simply graphical representations characterizing regions associated to the ﬂow patterns, that is, based on the operational conditions, two coordinates are deﬁned that characterize a data point contained within a region of the graph that corresponds to a speciﬁc ﬂow pattern. Prediction methods based on phenomenological approaches are also available. They consider the physical mechanisms responsible for the transitions. These methods are detailed in the present text, since they are used as the basis for the development of ﬂow pattern predictive methods for two-phase ﬂow in microscale channels. The graphical methods, even though are relevant from a historical point of view, are usually developed based on restrict experimental databases, and therefore, because they do not include a mechanistic approach, are recommended only to the range of experimental conditions considered in their development. On the other hand, predictive methods based on physical mechanisms can be considered as general, as long as the dominant mechanisms are correctly chosen. In this context, this chapter describes the methods developed by Taitel and coworkers, which states the main ﬂow pattern transitional mechanism for adiabatic two-phase ﬂows adopted in most studies after the classical paper published by Taitel and Ducker (1976). Due to the differences that the heat transfer process may impose on ﬂow pattern transitions, this chapter also addresses predictive methods for ﬂow patterns during evaporation and condensation.

3.2.1

Graphical Methods

Baker (1954) was pioneer to propose one of the ﬁrst ﬂow pattern maps for oil and gas horizontal ﬂows focused on applications in the oil and gas industry. His map shown in Fig. 3.2 comprises in its coordinated axes ﬂow conditions and dimensionless transport properties referred to the corresponding properties of air and water, aiming to give a universal characteristic to the map. This method does not account for the effect of tube diameter on the ﬂow pattern transitions, even though it is expected that this parameter inﬂuences signiﬁcantly the transitions. In the subsequent years, several ﬂow pattern maps were proposed based on different experimental databases. Even though the maps are useful and the ﬂow pattern information is easily obtained from them, these methods are usually valid for a narrow range of operational conditions and working ﬂuids as mentioned above.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

69

105

Dispersed flow

Annular

Wavy

4

Bubbles

G x ρair,STP ρwater,STP ρv ρl

1/2

10

Slug 10³

Stratified Plug 10²

10-1

100

101

1 - x ρv x ρair,STP

1/2

ρl ρwater,STP

1/6

102

σwater,STP μ σ μwater,STP

1/3

103

104

Fig. 3.2 Flow pattern map for horizontal gas and oil ﬂows, Baker (1954)

3.2.2

Taitel and Dukler (1976)

In 1976, Taitel and Dukler proposed an analytical model to predict ﬂow pattern transitions for gas and liquid ﬂows in horizontal and near horizontal round channels, which has been used as the basis for several foregoing predictive methods. This method accounts for the effects of ﬂuids properties, channel dimension, and duct inclination on the ﬂow pattern transitions. The original development presented by Taitel and Dukler (1976) is based on the liquid height Hl, as schematically shown in Fig. 3.3, associated to stratiﬁed ﬂows. Additionally, the original authors pointed out that their method is valid for adiabatic ﬂows, implying on neglecting aspects such as possible wall dryout due to vaporization, or ﬁlm formation due to condensation. The focus of the method is on the oil and gas industries, under conditions that heat transfer processes do take place, but slightly affect the phase distribution. Nonetheless, the transition criteria proposed by Taitel and Dukler (1976) are extended to applications involving two-phase vapor-liquid ﬂow with phase change by several authors, for example, Kind et al. (2010) and Wojtan et al. (2005). The basic assumptions of their model are the following: steady state ﬂow, hence, constant time averaged properties and parameters, completely developed ﬂow, and small duct inclinations. Taitel and Dukler (1976) proposed transition criteria considering the following ﬂow patterns: smooth stratiﬁed, wavy stratiﬁed, intermittent (slug and plug), annular with dispersed liquid, and dispersed bubble. In their method, the ﬂow parameters are non-dimensionalized adopting as the scaling parameters the pipe diameter d and the

70

3

d

uv τwv

Sv

g

τi

Aα

τi ul

τwl

Flow Patterns

Si θ

Hl

A(1 - α)

Sl

z Fig. 3.3 Schematics of stratiﬁed ﬂow in inclined duct

superﬁcial ﬂow velocities, jl and jv. The prediction of stratiﬁed wave emphasizes the fact that this method might not be valid for microscale channels, in which the stratiﬁcation effects are suppressed. Hence, subsequent to the analysis of momentum balance for both phases, the transition criteria are described according to physical aspects. Based on the schematics presented in Fig. 3.3 and by neglecting pressure gradients along the channel cross-section, the force balance in the axial direction for an element dz long for the liquid and vapor phases are given as follows, respectively: dp τwl Sl þ τi Si ρl Að1 αÞgsinðθÞ ¼ 0 dz

ð3:1Þ

dp τwv Sv τi Si ρv AαgsinðθÞ ¼ 0 dz

ð3:2Þ

Að1 αÞ Aα

where the term τwl and τwv refer to the shear stress between the liquid and vapor phases and the duct wall, respectively, Sl and Sv correspond to the contact area between the phases and the wall, τi corresponds to the interfacial shear stress, and Si corresponds to the interfacial area between the phases. Eliminating the pressure gradient term of both equations, we obtain the following relationship: τwl Sl τi Si τ S þ τ i Si þ ρl Agsinθ ¼ wv v þ ρv Agsinθ 1α α

ð3:3Þ

and rearranging the terms, we obtain:

τwl Sl τwv Sv τ i Si þ þ ðρl ρv ÞAgsinθ ¼ 0 1α α α ð1 α Þ

ð3:4Þ

The shear stress among the phases and the duct wall can be estimated based on the Fanning friction factor f as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

71

τwl ¼ f l

ρl u2l 2

ð3:5Þ

τwv ¼ f v

ρv u2v 2

ð3:6Þ

τi ¼ f i

ρv ð uv ui Þ 2 2

ð3:7Þ

The friction factor can be evaluated via Blasius correlation for turbulent ﬂow, which in turn is a function of the Reynolds number given as a function of the in situ velocity. Taitel and Dukler (1976) considered that the friction factor for the interfacial shear stress is similar to the one of the vapor phase ( fi ¼ fv). Moreover, it is assumed that the vapor velocity is considerably higher than the interface velocity, u̅v u̅i, and the interfacial tension can be evaluated as a function of the velocity only of vapor phase. Hence, it is necessary to deﬁne characteristic lengths for both phases and Taitel and Dukler adopted the hydraulic diameters for both phases deﬁned as follows: d hl ¼ 4

Að1 αÞ Sl

ð3:8Þ

Aα Sv þ Si

ð3:9Þ

d hv ¼ 4

where it is considered that the wall imposes the main friction to the liquid phase; conversely, the wall and gas–liquid interface imposes restriction to the vapor phase ﬂow. Hence, the Reynolds number and friction factor are given as follows: Re hl ¼

ρl ul d hl μl

ð3:10Þ

Re hv ¼

ρv uv dhv μv

ð3:11Þ

hl f l ¼ C hl Re n hl

ð3:12Þ

fv ¼

hv C hv Re n hv

ð3:13Þ

where the constant Ch and exponent nh are, respectively, 16 and 1 for laminar ﬂow (Reh < 2300), and 0.046 and 0.2 for turbulent ﬂow. Hence, Eq. (3.4) can be rewritten as follows:

72

3

Flow Patterns

ρl u2l Sl ρv u2v Sv Chl C hv Si n n þ ðρl ρv ÞAgsinθ ¼ 0 þ þ ρl ul d hl hl 2 1 α ρv uv d hv hv 2 α αð1 αÞ μl

μv

ð3:14Þ Developing the above equation, we ﬁnd the following relationships:

Chl Þ 4 ρl ul AμðS1α l

nhl

l

2

ρl u2l Sl þ 2 1α 4

C hv

nhv

ρv uv Aα μv ðSv þSi Þ

3

ρv u2v Sv Si þ ðρl ρv ÞAgsinθ ¼ 0 2 α α ð1 α Þ

2

3

ð3:15Þ

ρ j2 7 S nhl ρ j2 7S þ Si nhv 1 Sl 6 C 6 C 4 hlnhl l l 5 l þ 4 hvnhv v v 5 v 3 ρv jv d ρl jl d 2 πd 2 πd α2 ð1 α Þ μl

S Si vþ ðρl ρv ÞAgsinθ α αð1 αÞ

μv

¼0

ð3:16Þ

We can recognize that the terms inside the square brackets of the ﬁrst and second terms of Eq. (3.16) correspond, respectively, to the frictional pressure drop for only the liquid and vapor phases ﬂowing in the channel. Therefore, dividing all the terms by the vapor single-phase pressure drop, we obtain the following non-dimensional relationship: b X

2

Sl πd

nhl

Sl Sv þ Si nhv 1 Sv Si b ¼0 þ þ Y d α2 αd αdð1 αÞ d ð1 α Þ3

ð3:17Þ

where the term X^ is the Lockhart and Martinelli parameter, named after the authors for their study presented in 1949, which corresponds to the ratio between the pressure gradients for only liquid and gas ﬂowing in the channel, given as follows: nhl

ρl j2l 2d

ρ Cj dhv nhv

ρv j2v 2d

2

b ¼ X

C hl

ρl j l d μl

vv μv

¼

dp dz l dp dz v

ð3:18Þ

and in terms of mass ﬂux and vapor quality, the Lockhart and Martinelli parameter is given as follows: n

2nh,l h,l b 2 ¼ C hl ðGd Þnh,v nh,l μnl ρv ð1 xÞ X h,v 2n Chv x h,v μv ρl

ð3:19Þ

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

73

Fig. 3.4 Schematics of the geometric distribution of phases during stratiﬁed ﬂow

d Sv Aα γ Hl

Si Sl

A(1 - α) Stratiﬁed ﬂow

This parameter is presented again in Chap. 4, and is recurrently mentioned in two-phase ﬂow studies, as well as along this book. And the term Ŷ in Eq. (3.17) takes into account the channel inclination and is given as follows: b ¼ ðρl ρv ÞAgsinθ Y ¼ ρ j2 d ρv Cjv dhv nhv v2 v μv

ðρl ρv ÞAgsinθ d dp dz v

ð3:20Þ

Despite the fact that transitions from stratiﬁed to annular or intermittent ﬂow patterns, between intermittent and annular, and between annular and dispersed bubbles are expected, the method proposed by Taitel and Dukler considers only the force balance for stratiﬁed ﬂow, hence using the liquid height Hl, which is schematically depicted in Fig. 3.4. Assuming the phases are distributed according to stratiﬁed conﬁguration, all parameters can be written as a function of the liquid height Hl as follows1: 2H l 1 d Sv 2H l ¼ cos 1 1 d d Sl 2H l ¼ π cos 1 1 d d rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Si 2H l ¼ 1 1 d d γ ¼ cos 1

1

ð3:21Þ ð3:22Þ ð3:23Þ ð3:24Þ

The cross-sectional area occupied by the liquid phase Al can be easily evaluated based on the circle section (0!π/2-γ and π/2 + γ!0) plus the isosceles triangle area (π/2-γ!π/2 + γ). Similarly, the interfacial perimeter Si can be evaluated based on the Pythagoras theorem for the same isosceles triangle.

74

3

1α¼

Flow Patterns

Al πd2 =4

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2H l 1 1 d rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 A cos 1 2H l 1 2H l 2H l 1 α ¼ 2v ¼ 1 1 1 π d π d d πd =4 cos 1 2H l 1 2H l ¼1 1 þ 1 π d π d

ð3:25Þ ð3:26Þ

Therefore, all the terms of Eq. (3.17) are a function of the liquid height to diameter ratio Hl / d, or 2Hl/d-1, Lockhart and Martinelli parameter X^, and other known operational parameters, such as ﬂow velocities and ﬂuids properties. Hence, the variation of non-dimensional liquid height with Lockhart and Martinelli parameter can be represented graphically, as shown in Fig. 3.5. The reader is encouraged to implement computationally Eqs. (3.21) to (3.26) and Eq. (3.17), varying Hl / d from unity to zero, to construct a graph similar to the one shown below. In the case that the reader would like to study the original paper of Taitel and Dukler, please be advised that the channel inclination angle is deﬁned differently than in this study, which implies in negated values of Ŷ, hence, Fig. 3.5 is not identical to the one available in the original paper. We might be interested in representing Eq. (3.17) as a function of the area averaged void fraction, rather than the liquid height. Hence, the geometrical terms, more speciﬁcally Sl, Sv, and Si, can be written as a function of the area averaged void

1.0

0.8

100

0.6

Y = 10.0

0.0 10 -3

10 -2

10 -1

0

Turbulent - Turbulent

-10

0.2

-10

0.4 0.0

(Hl / d ) [-]

1000

10 0

Laminar - Turbulent

10 1

10 2

10 3

10 4

X [-] Fig. 3.5 Variation of non-dimensional liquid height with Lockhart and Martinelli parameter for stratiﬁed ﬂow

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

75

1.0 Turbulent - Turbulent Laminar - Turbulent

-10

0.0

-100

0.8

Stratified flow

α [-] a

0.6 Y = 10

0.4

0.2

100 1000

0.0 10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

X [-] Fig. 3.6 Variation of void fraction with Lockhart and Martinelli parameter for stratiﬁed ﬂow pattern

fraction α, and Fig. 3.4 depicts the schematics for stratiﬁed ﬂow. The stratiﬁcation angle γ can be related to the void fraction α in a transcendental formulation as follows: α¼

γ sin 2γ π 2π

ð3:27Þ

And the perimeters Sl, Sv, and Si can be written, respectively, as a function of γ as follows: Sl ¼πγ d Sv ¼γ d Si ¼ sin γ d

ð3:28Þ ð3:29Þ ð3:30Þ

Hence, by knowing the Lockhart and Martinelli parameter X^, it is possible to infer the void fraction α based on Eq. (3.17), and analogous to Fig. 3.5, Fig. 3.6 depicts the variation of void fraction with Lockhart and Matinelli parameter for stratiﬁed ﬂow pattern, which again is based on Eq. (3.17) obtained from the momentum balance of liquid and vapor phases.

76

3

Flow Patterns

Hence, by knowing the operational conditions, it is possible to estimate the Lockhart and Martinelli parameter, and based on Eq. (3.17), it is possible to estimate the liquid height or void fraction, which could also be performed through Figs. 3.5 and 3.6, respectively. Based on these parameters, it is possible to determine the in situ velocities of liquid and vapor phases, which are used for determination of ﬂow pattern transitions described in the following paragraphs. The stratiﬁed ﬂow pattern is characterized by the dominance of gravitational forces over inertial and/or surface tension forces, hence for very low ﬂow velocities and for channels with conventional dimensions. In this context, Taitel and Dukler (1976) considered smooth and wavy stratiﬁed ﬂows as different ﬂow patterns. The ﬁrst one corresponds to the ﬂow of liquid in the bottom region and vapor in the upper region of the cross-section separated by a smooth interface. In this condition, the gas/vapor velocity is not high enough to disturb the interface and to generate waves. Taitel and Dukler adopted the method addressed by Stewart (1967), which in turn is based on studies by Jeffreys (1925, 1926). According to this approach, interfacial waves for gas and liquid ﬂow are formed when: ð uv uw Þ 2 u w >

4μl gðρl ρv Þ sρl ρv

ð3:31Þ

where uw is the wave velocity, which is considerably lower than the vapor velocity (u̅v uw), and s is a sheltering coefﬁcient, equal to 0.3. The wave velocity is approximated by the liquid ﬁlm velocity, u̅l; therefore, based on these simpliﬁcations the transition between smooth and stratiﬁed ﬂow patterns is given as follows: 1=2 4μl gðρl ρv Þ cos θ uv > ul sρl ρv

ð3:32Þ

where the phase in situ velocities are determined based on the void fraction determined through Eqs. (3.17), (2.18), and (2.19). The wavy stratiﬁed is similar to the smooth stratiﬁed ﬂow pattern, however, the interface is rough with interfacial waves. Equation (3.32) can be made non-dimensional for a modiﬁed Froude number and liquid Reynolds number as follows: 1=2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2αð1 αÞ1=2 ρv ρl j l d jv > d ðρl ρv Þgcosθ μl s1=2 Fr v Re l

1=2

>

2αð1 αÞ1=2 s1=2

ð3:33Þ ð3:34Þ

where α is determined based on Eq. (3.17) and Eqs. (3.27) to (3.30) as a function of X^ and Ŷ. The modiﬁed Froude number and liquid Reynolds number are given, respectively, as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.7 Schematics of a solitary wave ﬂowing in a closed channel

77

p' uv'

uv p ul

Fr v ¼ jv

Hv Hl

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρv d ðρl ρv Þgcosθ

Re l ¼

ρl jl d μl

Hv' Hl'

ð3:35Þ ð3:36Þ

With the increment of vapor velocity, the formed waves tend to grow and can occur transitional from stratiﬁed wavy to annular or intermittent ﬂow patterns, which is given based on the balance between liquid wave weight and pressure reduction in the wave tip, caused by vapor ﬂow acceleration imposed by the restriction generated by the wave, as schematically depicted in Fig. 3.7. Assuming that interfacial waves already exist, whose occurrence is given by the condition of Eq. (3.32), the transition from wavy stratiﬁed to annular or intermittent ﬂow patterns occurs for the condition that the wave will grow and block the channel section. Therefore, a force balance is performed in the channel cross-section comprising the wave tip, assuming that the wave is subjected to gravitational forces, depicted by the darker region in Fig. 3.7, and the pressure reduction in the vapor region, which is determined based on Bernoulli’s equation. Hence, the pressure in the wave tip, denoted by p’ is estimated based on the gas velocity variation from a ﬂat region to a region with wave, neglecting variation of potential energy, resulting in the following relationship: p p0 ¼ ρ v

u0 2v u2v 2

ð3:37Þ

Assuming that the pressure along the liquid ﬁlm is uniform and equal to p, and that the wave and liquid ﬁlm velocities are similar, it is possible to perform a force balance for the dark region of Fig. 3.7 in the “vertical” direction, and the criterion that dictates when the wave will grow is related to the condition that the force due to pressure reduction is higher than the buoyance forces as follows: ðp p0 ÞdA ðρl ρv Þ H 0l H l gcosθdA

ð3:38Þ

Hence, combining Eq. (3.37) in Eq. (3.38) we obtain: ρv

u0 2v u2v 2

dA ðρl ρv Þ H 0l H l gcosθdA

ð3:39Þ

78

3

Flow Patterns

or: ρ ρv 0 2 u0 v u2v 2 l H l H l gcosθ ρv

ð3:40Þ

where the in situ velocity u̅v and liquid height Hl are directly related to the void fraction α, hence, determined based on Eq. (3.17); in the same token, the restriction vapor velocity u̅v’ and liquid height Hl’ can be given as a function of a local void fraction α’. Moreover, according to the schematics depicted in Fig. 3.7, it is possible to evaluate the velocity u̅v’ based on the continuity equation assuming incompressible ﬂow, and a local void fraction α’; therefore, Eq. (3.40) can be rewritten as follows: ρ ρv u2v 2 l ρv

H 0l H l ðα=α0 Þ2 1

! gcosθ

ð3:41Þ

For small disturbance, the term inside the brackets can be approximated by a Taylor series, namely, the term H’l, resulting in the following relationships: u2v

02 ρl ρv α ∂H l gcosθ ρv α ∂α

ð3:42Þ

where, again for small disturbance α’ α, and the liquid height can be given as a function of stratiﬁcation angle γ as follows: Hl ¼ 1 þ cos γ d

ð3:43Þ

which is ultimately a function of void fraction given by Eq. (3.20). Therefore: u2v

ρl ρv sin γ dgcosθ α 1 cos 2γ ρv

j2v ρl ρv α π dgcosθ sin γ 2 ρv α2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρv πα3 jv 2 sin γ ðρl ρv Þdgcosθ

ð3:44Þ ð3:45Þ ð3:46Þ

where the term on the left side is recognized as a modiﬁed vapor Froude number Frv*, and Eq. (3.46) can be written as follows: Fr v

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ πα3 2 sin γ

ð3:47Þ

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

79

a) Hl/d > 0.5

Hl/d > 0.5

b) Hl/d < 0.5

Hl/d < 0.5 Fig. 3.8 Schematics of sinusoidal wave growth

The term on the right side of (3.47) is basically related to the void fraction α, which is determined based on Eqs. (3.17) and (3.27) as a function of X^ and Ŷ. Therefore, for values of modiﬁed Froude number smaller than that deﬁned by Eq. (3.47), the wave growth would imply in transition to annular or intermittent ﬂow patterns, depending on the amount of liquid. In conditions of high liquid level, the amount of liquid is enough to form a stable liquid slug, blocking the channel section, and then the vapor ﬂows as discrete bubbles. On the other hand, for reduced liquid heights, the grown wave tends to block the vapor path, but this slug is unstable and the liquid is pushed to the channel periphery, which corresponds to an annularlike ﬂow. Therefore, the transition between intermittent and annular-like ﬂows depends basically on the liquid height, or void fraction, and Taitel and Dukler proposed and justiﬁed a value of 50% for Hl / d and α as transition criteria. To reinforce the assumption of Hl / d ¼ 0.5 (α ¼ 0.5) for the transition between intermittent and annular-like ﬂow patterns, the original authors claim the reader to imagine a sinusoidal wave, such as depicted schematically in Fig. 3.8. The growth of the wave requires liquid from the vicinity, which results in valleys in the interface; therefore, if the interface mean line is below the channel center (Hl / d < 0.5), in conditions of growing wave amplitude, the interface would reach the bottom channel wall before the wave tip reaches the top channel wall, with consequent unstable slug. On the other hand, for liquid height above the channel center for smooth interface, when increasing the wave amplitude, the wave tip would reach the top channel wall before the interface valley reaches the channel bottom. Therefore, by deﬁning Hl / d ¼ 0.5 (or α ¼ 0.5), it is possible to determine the corresponding X^ from Eq. (3.17), or from Figs. 3.5 or 3.6.

80

3

Flow Patterns

Nonetheless, it is intuitive that for Hl / d ¼ 0.5, the stratiﬁcation angle γ is equal to π/2, Si / d ¼ 1.0, and Sl / d ¼ Sv / d ¼ π/2. Therefore, by substituting these values in Eq. (3.17), the Lockhart and Martinelli parameter is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2nkl 1 1 nkv b b X¼ Y þ 5 þ 2 π π

ð3:48Þ

which, for turbulent ﬂow regimes for both phases (nkl ¼ nkv ¼ 0.2) and horizontal ﬂow (Ŷ ¼ 0.0), results in a constant value of 1.325. Therefore, according to the deﬁnition of the parameter X^, given by Eq. (3.19) for turbulent ﬂow regimes, we ﬁnd: 0:1 ρ 0:5 1 x 0:9 μl v b X tt ¼ x μv ρl

ð3:49Þ

equal to 1.325, where the subscripts tt refers to turbulent ﬂow regimes for liquid and vapor phases. Therefore, for a ﬂuid at a given saturation temperature the transition between intermittent and annular-like ﬂow patterns occurs for constant vapor quality values. Subsequently, Barnea et al. (1982b) proposed that his transition occurs for Hl/d of 0.35, and the difference is attributed to the vapor fraction within the liquid slugs during the transition, which would require less liquid to close the cross-section. Nonetheless, in this text we will keep the original proposal. It will be seen later, in Sect. 3.3.1, that assuming constant α of 0.5 for the transition between intermittent and annular ﬂow patterns is not conﬁrmed experimentally, and alternative proposals for this transition are available in the open literature. In conditions of intermittent ﬂow pattern with high liquid content, the vapor bubbles can be broken due to high ﬂow turbulence. Hence, the intermittent ﬂow pattern transits to dispersed bubbles in conditions that the liquid ﬂow turbulence overcomes the buoyance forces of the vapor phase. In this context, Taitel and Dukler addresses relationships for the buoyancy force as follows: F b ¼ gcosθðρl ρv ÞαA

ð3:50Þ

The turbulence forces are given according to the method proposed by Levich (1962) as follows: Ft ¼

ρl u0l 2 Si 2

ð3:51Þ

where ul’ is the radial velocity ﬂuctuation of the liquid phase due to turbulence, whose root mean square is evaluated as a function of the friction factor and liquid velocity as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

u0l 2

1=2

¼ ul

fl 2

81

1=2 ð3:52Þ

Therefore, by deﬁning the condition of Ft > Fb, the following relationship is obtained: ρl u2l f l Si > gcosθðρl ρv ÞαA 4

ð3:53Þ

The equation above can be rewritten as a function of non-dimensional parameters as follows: fl

ρl j2l Si πd > gcosθðρl ρv Þα 4 2d 4ð1 αÞ2

ð3:54Þ

ρl j2l gcosθðρl ρv Þαð1 αÞ2 π > 2d Si =d

ð3:55Þ

ρl j2l 4πgcosθðρl ρv Þαð1 αÞ2 > 2d sin γ

ð3:56Þ

fl 4fl

dp dz

¼ Cl l

ρl j2l 4πgcosθðρl ρv Þαð1 αÞ2 > 2d sin γ

ðdp=dzÞl 4παð1 αÞ2 > sin γ gcosθðρl ρv Þ T>

4παð1 αÞ2 sin γ

ð3:57Þ ð3:58Þ ð3:59Þ

where the term T on the left side can be considered as the ratio between turbulence and buoyance forces acting on the vapor phase and is evaluated according to the ﬂow conditions and ﬂuid properties, and the term on the right side is a function of α. Therefore, the non-dimensional transitions given by Eqs. (3.34), (3.47), (3.48), and (3.59) can be presented graphically, as shown in Fig. 3.9. It must be emphasized that all the listed transitions can be given as a function of the non-dimensional liquid height or void fraction, which are a function of the Lockhart and Martinelli parameter, as given by Eq. (3.17) and shown by Figs. 3.5 or 3.6. Hence, by knowing the operational conditions, namely, ﬂuid properties, mass velocity, vapor quality, and channel diameter, it is possible to infer the Lockhart and Martinelli parameter, given by Eq. (3.19), and from Eq. (3.17) or from Figs. 3.5 and 3.6, it is possible to infer the non-dimensional liquid height or void fraction. From these parameters, it is possible to estimate all the transitions, depicted in Fig. 3.9, which is convenient due to the validity for several ﬂuids and channel sizes.

82

3 3

1 x10

Frv* . Rel0.5 [-]

Annular

10

2

10

1

Dispersed bubbles

1

Wavy stratified

1 x10

-1

1 x10

-2

1 x10 10 4

-3

Intermittent

10

0

Frv* or T [-]

10

Flow Patterns

Smooth stratified

10

-1

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 3

X [-] Fig. 3.9 Non-dimensional ﬂow pattern transitions for horizontal ﬂow (θ ¼ 0 ) according to Taitel and Dukler (1976)

Alternatively, the ﬂow pattern transitions can be given as a function of mass velocity and vapor quality for a given ﬂuid, saturation temperature, and channel diameter as shown in Fig. 3.10. This form of presentation is more appropriate for heat transfer problems because usually the ﬂuid ﬂows along a channel of uniform cross-section, and due to the heat transfer process, there is variation of its enthalpy, and consequently of its vapor quality. In this presentation form, it is possible to infer the ﬂow pattern evolution along the channel as the ﬂuid enthalpy increases or decreases. The reader can observe in Fig. 3.10 that smooth stratiﬁed ﬂow occurs only for very low ﬂow velocities, which is coherent with the intuition and experimental observations. With the increment of ﬂow velocity, the ﬂow pattern can suffer a transition to wavy ﬂow, and additional increments of ﬂow velocity can result in a transition to annular, or intermittent/dispersed bubble ﬂow patterns. Conversely, assuming a tube of an evaporator and saturated liquid at its inlet for G ﬁxed at 600 kg/m2s, at very low x values, the local ﬂow pattern corresponds to wavy stratiﬁed, and with sensible increment of vapor quality a transition to intermittent ﬂow patterns occurs. With additional increments of x, the ﬂow pattern will transit to annular ﬂow, and this transition occurs at a ﬁxed vapor quality of approximately 0.17 for this ﬂuid and saturation temperature, with occurrence of annular ﬂow pattern until total liquid dryout at a vapor quality equal to 1. On the other hand, for a mass velocity of 10 kg/m2s, the same ﬂuid, channel, and saturation temperature, at the beginning of the channel, corresponding to a vapor quality close to 0, the ﬂow is smooth stratiﬁed due to buoyancy forces and negligible interfacial tension and

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

10

83

5

Dispersed bubbles

10

4

10

3

10

2

R134a, d = 10 mm, Tsat = 50 °C

G [kg/m²s]

Intermittent Annular

Wavy stratified

10

1

Smooth Stratified 0

10 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.10 Flow pattern transitions for horizontal ﬂow of R134a at 50 C in 10 mm ID channel

turbulence. This condition is kept for successive increments of vapor quality until achieving a value of approximately 0.5, which corresponds to a transition to wavystratiﬁed ﬂow maintaining this ﬂow pattern until the total liquid dryout. The above mind exercise is not entirely correct, because practical applications of evaporation at 50 C in 10 mm ID tube is uncommon in real applications, and during the heat transfer process the local ﬂow pattern can be affected by the heat transfer rate and direction (evaporation or condensation), as will be seen in Sects. 3.3 and 3.4. Additionally, even though this method is valid for adiabatic ﬂow, subsequent developments in this area, such as the study of Wojtan et al. (2005) presented in Sect. 3.3.1, used it as a basis for the derivation of transition criteria for two-phase ﬂow during evaporation. Additionally, with the reduction of the channel diameter, the surface tension effects present more signiﬁcant contribution to the phases distribution, and in this context, Barnea et al. (1983) proposed in their pioneer study a ﬂow pattern predictive method for microscale channels. The experimental database used by the authors to develop their predictive method comprises results for air-water ﬂow, close to atmospheric pressure, for channels with internal diameter ranging between 4.0 and 12.3 mm. Based on the evaluated experimental results, the main difference was veriﬁed by the authors for the transition between stratiﬁed and non-stratiﬁed ﬂows, whereas for conventional channels it is given by the conditions of wave growth due to vapor acceleration. In the case of microscale channel, the surface tension starts to play an important contribution pushing the liquid upward and promoting an earlier transition from stratiﬁed ﬂow. Figure 3.11 schematically depicts the liquid slug

84

3

Fig. 3.11 Schematics of liquid slug formation due to surface tension forces

R

Hv

Flow Patterns

σ g

V2

V1

formation according to the model proposed by Barnea et al. (1983), who considered a two-dimensional wave, whereas the wave formation requires liquid from the vicinity and results in a valley. Moreover, by assuming a wave with circular format and radius R, centered in the upper wall, it is possible to infer the radius R by equating the volumes V 1 and V 2 , which results in the following relationship: 4 R ¼ Hv π

ð3:60Þ

Therefore, by equating the weight of the wave (darker region in Fig. 3.11) with the surface tension force σ, we ﬁnd: π Hv ¼ 2

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gð ρl ρv Þ ð 4 π Þ

ð3:61Þ

And, based on this relationship it is possible to determine the non-dimensional liquid height for the transition: Hl H π ¼1 v ¼1 2d d d

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gð ρl ρv Þ ð 4 π Þ

ð3:62Þ

which, substituting in Eq. (3.17), provides a constant value for the Lockhart and Martinelli parameter X^, and consequently a constant vapor quality for a given ﬂuid and saturation pressure. Additional to the transition from stratiﬁed to non-stratiﬁed ﬂows, Barnea et al. (1983) also adopted the modiﬁed value for the transition between annular and intermittent-like ﬂows, as suggested by Barnea et al. (1982b), according to which Hl / d ¼ 0.35 and also results in a constant X^ and vapor quality. Figure 3.12 depicts the ﬂow pattern transitions for R134a in a 4 mm ID channel, and it can be noticed according to this ﬁgure that the transitions are considerably different from those for conventional channels, depicted in Fig. 3.10, and the main differences are due to the surface tension effects, neglected by Taitel and Dukler (1976). Based on the analysis of the ﬂow pattern transitions for small channels depicted in Fig. 3.12b, it can be speculated that the absence of transition from wavy stratiﬁed to dispersed bubbles for high mass velocities is incoherent, since the stratiﬁed ﬂow patterns for very high mass velocities is unexpected. The reader should be aware that the dominant mechanism for ﬂow pattern transitions during horizontal ﬂow are distinct from those for vertical ﬂow. Therefore, below we present the classical predictive method also proposed by the research group of Taitel and Dukler, concerning vertical ﬂows.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows 1

1 x10

5

10

4

10

3

10

2

10

1

Dispersed bubbles

Dispersed bubbles

R134a, d = 4 mm, Tsat = 35 °C

1

0

T [-]

10

b) 10

Intermittent

1 x10

-1

Smooth stratified

G [kg/m²s]

Frv* . Rel0.5 [-]

Annular

1

R134a, d = 4 mm, Tsat = 35 °C

Wavy stratified

a) 10

85

Wavy stratified Intermittent

Annular

Smooth Stratified

10

-1

10

-3

10

-2

10

-1

10

0

10

X [-]

1

10

2

10

3

1 x10 10 4

-2

0

10 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.12 Flow patterns transition for microscale channels according to Barnea et al. (1983)

3.2.3

Taitel, Barnea, and Dukler (1980)

Subsequent to the proposal of the method for ﬂow pattern transitions in horizontal and near horizontal ﬂows, Dukler and coworkers proposed criteria for ﬂow pattern transitions during vertical upward adiabatic ﬂow that considers transitions among slug, churn, bubble, ﬁnely dispersed bubbles, and annular ﬂow patterns. Analogous to horizontal ﬂow, the authors deﬁned the dominant mechanisms responsible for the transitions, and then described the governing equations associated with these mechanisms. On the other hand, different from the case of horizontal ﬂow, where the liquid height was considered as the main characteristic length of the ﬂow, in the case of vertical ﬂow these authors took the void fraction α as the main geometrical parameter. Additionally, in this method the transition criteria do not result in non-dimensional relationships. According to Taitel et al. (1980), the bubble and dispersed bubble ﬂow patterns are expected to occur in conditions of reduced void fraction, and the limiting conditions for their occurrence are related to the turbulence intensity and void fraction. The turbulence intensity should be high enough for the occurrence of bubble ﬂow, in order of breaking large bubbles into small ones, or to inhibit their coalescence into larger bubbles. Conversely, the void fraction should be small enough to inhibit the collisions among the bubbles, favoring their coalescence. Therefore, assuming a cubic lattice distribution for the bubbles, such as depicted in Fig. 3.13a, the corresponding maximum void fraction is 0.52. Nonetheless, any random movement of the bubbles for this packaging results in collision among them, favoring their coalescence and the transition to slug ﬂow. Therefore, at least some space between the bubbles is required to avoid collision and coalescence, and in this context the authors proposed to adopt a separation of half of the bubble radius, such as depicted in Fig. 3.13b, for which the void fraction is approximately 0.25, as the transitional criterion. Taitel, Barnea, and Dukler justify this consideration based on experimental results, which show that slug ﬂow is not observed for void fraction lower than 0.20 and that bubble ﬂow rarely occurs for void fraction higher than 0.30. Additionally, this hypothesis is coherent with previous studies available in the literature at that time. Therefore, assuming a constant void fraction α of 0.25 it is

86

3

Flow Patterns

a)

d=a

α = 0.52

a a

b)

d=(4/5)a

d/4

α = 0.268 ≈ 0.25

a a Fig. 3.13 Schematics of bubbles distribution according to cubic lattice

possible to determine a relationship between the liquid and vapor superﬁcial velocities. The liquid in situ velocity is given according to Eq. (2.21) as follows: ul ¼

jl 1α

ð3:63Þ

In the case of vapor ﬂow, the in situ velocity is given as the sum of u̅l and the slip velocity between the bubble and liquid phase U0, which is mainly related to buoyancy forces as follows: uv ¼

jl jv þ U0 ¼ ul þ U 0 ¼ 1α α

ð3:64Þ

The slip velocity is given according to the correlation proposed by Harmathy (1960) for relatively large bubble rising velocity in stagnant liquid media, which is insensitive to the bubble characteristic diameter, and is given as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

U 0 ¼ 1:53

gðρl ρv Þσ ρ2l

87

1=4 ð3:65Þ

Based on the above equations, it is possible to determine a relationship between the liquid and vapor superﬁcial velocities as follows: jl ¼

1=4 gðρl ρv Þσ 1α jv 1:53 ð1 αÞ α ρ2l

ð3:66Þ

From which for α ¼ 0.25, we obtain the following relationship: 1=4 gðρl ρv Þσ jl ¼ 3 jv 1:15 ρ2l

ð3:67Þ

The above equation provides the transition criterion between bubble and slug ﬂow for conditions under which dispersion forces are not dominant. The proposers have reinforced the validity of this criterion by comparing its predictions with ﬂow pattern maps available in the open literature. The bubble ﬂow pattern delimited by this restriction corresponds to the vapor phase ﬂowing as bubbles of intermediate diameter in a zig-zag manner due to the bubble non-spherical form with higher velocity than the liquid phase due to buoyance forces. In conditions of high liquid velocity, the ﬂow turbulence can break large bubbles into small ones even for conditions of intermediate void fractions, including α higher than 0.25, and the bubbles tend to be more spherical due to their reduced size. Based on the studies presented by Hinze (1955), who theoretically evaluated the characteristic dimension of a dispersion caused by turbulence, and by Brodkey (1967), who presented a criterion to deﬁne the critical diameter of a bubble to be spherically stable, Taitel, Barnea, and Dukler proposed the transition criterion between bubble and dispersed bubble ﬂow patterns. According to their criterion, the maximum diameter of a bubble generated due to ﬂow turbulence is given as the balance between surface tension and turbulence forces as follows: dmax

3=5 σ ¼k E 2=5 ρl

ð3:68Þ

where the term k is an empirical constant equal to 1.14 for gas dispersed in liquid, and E is the rate of energy dissipation per unit mass, which is evaluated based on the pressure drop, and is given as follows: dp j Ε ¼ dz m ρH

ð3:69Þ

88

3

ρ j2 dp ¼ 2f H dz m d 0:2 ρ jd f ¼ 0:046 l μl

Flow Patterns

ð3:70Þ ð3:71Þ

Therefore, rewriting Eqs. (3.68) to (3.71), we obtain the following relationship:

dmax

3=5 0:2 3 !2=5 ρl jd σ j ¼k 0:092 ρl d μl

ð3:72Þ

In contrast, the critical diameter of the bubbles to maintain their spherical shape is given as follows: dcrit ¼

0:4σ ðρl ρv Þg

1=2 ð3:73Þ

For dmax > dcrit the bubble rising velocity in stagnant liquid is almost independent of the bubble size and is given by Eq. (3.65), which corresponds to the bubble ﬂow pattern. Conversely, for dmax < dcrit the rising velocity sharply decreases with bubble diameter. Under conditions that the turbulence intensity is high enough, the coalescence is suppressed and there is no transition to slug ﬂow even for void fraction values higher than 0.25. Therefore, by equating dmax and dcrit in Eqs. (3.72) and (3.73), it is possible to infer the transition between bubble and dispersed bubble ﬂow patterns, which is given as follows: 3=5 0:2 3 !2=5 1=2 ρl jd σ j 0:4σ k 0:092 ¼ ρl d μl ðρl ρv Þg 1=1:12 σ 0:1 d0:48 1=2 j ¼ 4:0 0:52 0:08 ððρl ρv ÞgÞ ρl μ l 0:1 0:48 1=1:12 σ d 1=2 jl ¼ jv þ 4:0 0:52 0:08 ððρl ρv ÞgÞ ρl μ l

ð3:74Þ

ð3:75Þ ð3:76Þ

We should recall the geometric analysis previously performed, such as depicted in Fig. 3.13, according to which it is impossible for a bubble ﬂow pattern to exist for void fraction higher than 0.52. Therefore, the transition given by Eq. (3.76) is limited by this condition. Additionally, for high ﬂow velocity and turbulence intensity and reduced bubble diameter, the slip between the phases is almost negligible, hence u̅l u̅v. Then, by considering α ¼ 0.52 it is possible to determine the complementary transition condition as follows:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

89

jl ¼ 0:92 jv

ð3:77Þ

Therefore, the conditions of liquid superﬁcial velocity higher than those given by Eqs. (3.76) and (3.77) the vapor-liquid ﬂow corresponds to dispersed ﬂow of the vapor phase as small bubbles, moving within the liquid phase with negligible slip. In the case of two-phase vertical ﬂow in small dimension channels, the possibility of existing a sporadic vapor slug existing must be evaluated. The rising velocity of a Taylor bubble inside a round channel is proportional to the square root of the channel diameter, and can be evaluated according to the Nicklin (1962) method as follows: U T ¼ 0:35

pﬃﬃﬃﬃﬃ dg

ð3:78Þ

where d is the channel internal diameter and g is the gravitational acceleration. Therefore, assuming simultaneous ﬂow of bubbles with intermediate dimensions and Taylor bubbles, whenever UT > U0, the smaller bubbles would be swept around the Taylor bubble nose to the channel periphery, such as schematically depicted in Fig. 3.14a, and then directed to the wake region without coalescence. Therefore, there is no transition to slug ﬂow. On the other hand, assuming that the smaller bubbles present higher velocity than the Taylor bubble, UT < U0, veriﬁed for reduced diameter channel and schematically depicted in Fig. 3.14b, the migration of the bubbles toward the Taylor bubble tail favors coalescence, and consequent transition to slug ﬂow. Therefore, depending on the channel diameter and two-phase mixture velocity, the bubble ﬂow pattern will not occur. For instance, for air-water ﬂow at 20 C and atmospheric pressure, the limiting channel diameter is 52 mm; while for R134a at 20 and 50 C, the limiting channel diameters are, respectively, 19.8 and 12.5 mm, whereas no bubble ﬂow occurs in channels with smaller dimensions than these.

Fig. 3.14 Schematics of simultaneous ﬂows of Taylor bubbles and bubbles with intermediate dimensions relative to liquid ﬂow

a)

b)

U0

UT

UT

g

U0

UT > U0

UT < U 0

90

3

Flow Patterns

Before analyzing the transition between slug and churn ﬂow patterns, Taitel et al. (1980) expatiated about the deﬁnition of the churn ﬂow itself. In fact, since several classiﬁcations are based on visual observations and subjective method, therefore sensible to personal judgement of the researcher, distinct experimental results can be obtained for a certain set of results depending on who classiﬁes the local pattern. Therefore, according to Taitel et al. (1980) the churn ﬂow pattern is characterized by the oscillatory ﬂow of liquid slugs and vapor Taylor bubbles, with successive destruction and construction of these entities along the channel. Complimentary to this classiﬁcation, Taitel et al. (1980) stated that churn ﬂow corresponds to a transitionary ﬂow pattern, occurring only in the inlet region of the section. In the entrance region, short lengthened liquid slugs and Taylor bubbles are formed and ﬂow in the main ﬂow direction, and since these liquid slugs are unstable due to their thickness, they break up, fall, and combine with the liquid slug in the lower region increasing the size of the liquid slugs. Then the liquid slug is propelled again by the stream and a net ﬂow is veriﬁed. Simultaneously, the successive Taylor bubbles combine creating longer bubbles. This process occurs along the channel length until a region where the liquid slugs are stable enough to sustain the slug ﬂow pattern. Therefore, the transition criterion is actually based on the channel length required to develop the slug ﬂow, whereas churn ﬂow occurs upstream to this point, and the derivation is performed based on the required conditions to obtain stable liquid slugs and Taylor bubbles. The Taylor bubble velocity in ﬂowing liquid is given according to the proposal of Nicklin (1962) as follows: uv,T ¼ 1:2ul þ 0:35

pﬃﬃﬃﬃﬃ gd

ð3:79Þ

where the second term on the right side corresponds to the rising velocity of a Taylor bubble inside a circular duct with stagnant liquid, and the ﬁrst term corresponds approximately to the liquid velocity at the center of the tube, which is 20% higher than the average velocity for turbulent ﬂow. In this development, it is also assumed that the liquid slugs separating successive Taylor bubbles contain dispersed bubbles with intermediate dimensions, such as schematically depicted in Fig. 3.15a. In these regions, it is assumed that the void fraction is equal to 0.25, in a similar way to the characterization of the bubble ﬂow pattern (refer to Fig. 3.13). Moreover, it is considered that the smaller bubbles move with the same velocity of the Taylor bubbles; therefore, the vapor in situ velocity u̅v is given according to Eq. (3.79). The mixture superﬁcial velocity is a known parameter, and can be related to the velocities of each phase as follows: j ¼ jv þ jl ¼ uv α þ ul ð1 αÞ

ð3:80Þ

By solving Eqs. (3.79) and (3.80) for the in situ velocities, we obtain the following expressions:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.15 Schematics of the ﬂow of stable slugs

91

a)

b)

ul r

c)

ul u̅v

r z*

ls

d) u l

1.2 u̅l

r α = 0.25 lT g r

pﬃﬃﬃﬃﬃ j 1:2 1α þ 0:35 gd α 1:2 1α þ1 pﬃﬃﬃﬃ 0:35α gd j ul ¼ 1α α 1α 1:2 1α þ 1

uv ¼

ð3:81Þ

ð3:82Þ

During the passage of a large bubble, the liquid ﬁlm in the channel periphery ﬂows downstream in order to obey the mass conservation, such as depicted schematically in Fig. 3.15b. Therefore, a jet is formed in the liquid slug just upstream the bubble tail, Fig. 3.15c, which can make the liquid slug unstable due to reverse ﬂow. Hence, the occurrence of slug ﬂow requires stable and consequently quite thick

92

3

Flow Patterns

liquid slugs, to allow the reestablishment of the liquid velocity proﬁle, such as schematically depicted in Fig. 3.15d. In the case of short liquid slugs, the slug length would not be enough to allow reestablishment of the ﬂow, therefore the liquid bridge would collapse and fall to the lower liquid portion, implying on the transition to churn ﬂow. Based on experimental results obtained in their laboratory for air-water ﬂows, Taitel et al. (1980) state that the liquid slug length observed for stable slug ﬂow is approximately 16 times the channel diameter, ls / d ¼ 16 (Fig. 3.15a), independently of the channel diameter. Hence, just downstream the inlet section of the vapor-liquid mixture, liquid slugs, as gas bubbles, of short length are formed and collapse forming larger slugs and bubbles. This process continues and evolves downstream until a point that stable liquid slug is formed; recalling that the stable liquid slug length is approximately 16 times the diameter, two slugs of 8 diameters in length are required to form a stable slug. Therefore, referring to the z* coordinate depicted in Fig. 3.15a, the liquid velocity in the channel center (r ¼ 0) must vary from u̅v in z* ¼ 0 to 1.2 u̅l in z* ¼ ls, and this velocity variation can be modelled as an exponential function as follows: ul,r¼0 ¼ uv eϑz=ls þ 1:2ul 1 eϑz=ls

ð3:83Þ

where the term ϑ corresponds to the decay constant, and it is assumed as equal to ln (100) ¼ 4.6. Hence, by considering the in situ velocities of both phases given by Eqs. (3.81) and (3.82), with the void fraction of the slug region equal to 0.25 (recall the modelling approach for the bubble ﬂow pattern), the following relationship is obtained: ul,r¼0 ¼

pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 1:6j þ 0:35 gd ϑz=ls 1:6j 0:35 gd=3 e þ 1 eϑz=ls 1:4 1:4 pﬃﬃﬃﬃﬃ 0:35 gd 4 ϑz=ls 1 1:6 jþ e ul,r¼0 ¼ 1:4 3 3 1:4

ð3:84Þ ð3:85Þ

Therefore, by assuming the entrance region, the approaching velocity between two consecutive bubbles is given as follows: ul,r¼0 jz¼0 z¼ls ¼

pﬃﬃﬃﬃﬃ 4 0:35 gd 1 eϑ 3 1:4

ð3:86Þ

By integrating Eq. (3.86), it is possible to infer the time interval required for collision between successive bubbles in the developing region as follows: 48d t i ¼ pﬃﬃﬃﬃﬃ gd ð1 eϑ Þ

ð3:87Þ

The product of the period required for the collision of two consecutive bubbles and the vapor phase in situ velocity, estimated through Eq. (3.81), gives the length

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

93

along which slug collapses occur, which corresponds to the entrance length le, where the churn ﬂow pattern is present2: pﬃﬃﬃﬃﬃ 1:6j þ 0:35 gd 48d t i uv ¼ le ¼ pﬃﬃﬃﬃﬃ 1:4 gd ð1 eϑ Þ le j ¼ 55:41 pﬃﬃﬃﬃﬃ þ 0:22 d gd

ð3:88Þ ð3:89Þ

Hence, in a set of ﬂow conditions that would correspond to slug ﬂow, there can be either slug or churn ﬂow pattern depending on the position along the channel; whereas, in the entrance region, z < le, churn ﬂow is present and for developed region, z > le, slug ﬂow is present. Subsequently in this section, the ﬂow pattern transitions will be plotted in charts according to the ﬂow conditions; therefore, it is interesting to present Eq. (3.89) in the form of superﬁcial velocities in a similar way to the previously described transitions as follows:

pﬃﬃﬃﬃﬃ le gd jl ¼ jv þ 0:22 55:41d

ð3:90Þ

Hence, upstream a given position deﬁned by le / d the ﬂow corresponds to churn, otherwise it corresponds to slug ﬂow, as long as the conditions do not correspond to bubble or dispersed bubble ﬂow. During annular ﬂow in vertical channels, the vapor ﬂows in the central region of the channel, and the liquid ﬂows predominantly as a ﬁlm adjoined to the channel wall. The higher density of the liquid phase, summed to the friction imposed by the wall, makes the liquid ﬁlm velocity considerably inferior to the vapor ﬂow velocity, with consequent high interfacial shear stress between the phases. Therefore, interfacial waves tend to be formed, which favor detachment of portions of liquid from the interfacial disturbances and consequently part of the liquid phase ﬂows as dispersed droplets within the vapor phase. Hence, the transition to annular ﬂow is given as a function of the balance of forces in a liquid droplet ﬂowing within the vapor stream, whereas the annular ﬂow pattern occurs when the drag forces overcome the gravity forces as follows: Cd

πd2d ρv u2v πd3d ¼ gðρl ρv Þ 4 2 6

ð3:91Þ

where Cd is the drag coefﬁcient that can be assumed as constant and equal to 0.44, and dd is the droplet diameter, which is estimated according to the method proposed by Hinze (1955) for the maximum stable droplet size as follows:

2 The coefﬁcient of this equation is 36% higher than the original proposal due to few simpliﬁcations adopted in this book to be more didactical.

94

3

dd ¼

Wecrit σ ρv u2v

Flow Patterns

ð3:92Þ

where the critical Weber number Wecrit is assumed to be 30 as suggested by Hinze (1955). Hence, by substituting the droplet diameter into Eq. (3.91) we obtain the corresponding vapor velocity for transition to annular ﬂow. Moreover, under conditions of annular ﬂow the liquid ﬁlm is considerably thin, and the assumption of void fraction of approximately unity leads to negligible difference in the superﬁcial velocities. Therefore, the transition to annular ﬂow is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 4Wecrit σgðρl ρv Þ 4 σgðρl ρv Þ ¼ 3:09 jv ¼ 3C d ρ2v ρ2v

ð3:93Þ

Therefore, the transition to annular ﬂow is independent of the liquid ﬂow velocity. Figure 3.16 depicts the ﬂow pattern transitions for air-water ﬂow in 26 mm ID tube, where the coordinate axis is given as a function of the superﬁcial velocities of each phase. Again, under phase-change conditions, it is common to face a situation with constant mass ﬂow rate, and variation of vapor quality along the evaporator or condenser, which are more properly represented by pairs of G and x for the transition 10

1

Dispersed bubble

10

0

500 200

jl [m/s]

Bubble

10

Churn

100

-1

Air-water T = 20 °C, p = p atm ,

le / d = 50

10

-2

10

-3

d = 26 mm, d crit = 52 mm

Slug

10 -2

Annular

10 -1

10 0

10 1

10 2

jv [m/s] Fig. 3.16 Flow pattern transitions for vertical upward ﬂow of air-water in 26 mm ID channel at 20 C and patm

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

95

instead of jl and jv. Therefore, Eqs. (3.67), (3.76), (3.77), (3.90), and (3.93) can be rewritten, respectively, as follows: 1=4 1:15 gðρl ρv Þσ G ¼ 1x x ρ2l ρ 3ρ

ð3:94Þ

0:1 0:48 1=1:12 4:0 σ d G ¼ 1x x 0:52 0:08 ððρl ρv ÞgÞ1=2 μl ρ þ ρ ρl

ð3:95Þ

l

l

v

v

1 ρl x ¼ 1 þ 0:92 ρv pﬃﬃﬃﬃﬃ gd le G¼ 0:22 1x x 55:41d ρl þ ρv qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3:09 4 G¼ σgðρl ρv Þρ2v x

ð3:96Þ ð3:97Þ ð3:98Þ

Figure 3.17 depicts the transitions among ﬂow patterns for vertical ﬂow of R134a in a 20 mm ID tube. This diameter was selected based on the critical diameter, which is considerably large for this ﬂuid, otherwise the bubble ﬂow pattern would not occur. Independently, the dispersed bubble ﬂow is not observed in the present analysis, which is mainly related to the restriction of α 0.52 and no-slip between 100 R134a, Tsat = -20 °C, d = 20 mm, d crit = 19.8 mm

Annular

G [kg/m²s]

75

Slug

50 500 Churn

25

200 Bubble

0 0.0

le / d = 100

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.17 Flow pattern transitions for R134a vertical upward ﬂow in 20 mm ID tube at 20 C

96

3

Flow Patterns

the phases. In fact, the prediction of bubble ﬂow pattern is only possible due to the slip velocity between the phases, otherwise the void fraction would be higher than 0.52. From Fig. 3.17 and considering a mass velocity of 75 kg/m2s and vertical evaporation of R134a at 20 C as saturated liquid at the channel inlet, it can be ﬁgured out that as the ﬂuid enthalpy increases, bubbles are formed on the channel wall and, then, they ﬂow dispersed within the liquid. With the increment of the vapor quality due to heat addition along the channel length, the coalescence of the bubbles takes place, implying on a transition to churn or slug ﬂow, depending on the ﬂow length, at a vapor quality of approximately 5%. Further addition of heat to the ﬂuid promotes the transition to annular ﬂow at a vapor quality of approximately 40%. It is important to highlight that this analysis neglects the effect of heat ﬂux intensity and its direction on the ﬂow pattern transitions. The ﬂow pattern transition method proposed by Mishima and Ishii in 1984 for vertical upward ﬂow of vapor-liquid mixtures can also be cited. Analogous to the method of Taitel et al. (1980), the method proposed by Mishima and Ishii (1984) is based on the physical mechanisms mostly related to the geometric distribution of the phases. This method provides values of void fraction associated to the transitions between the ﬂow patterns. The method is mainly based on the drift-ﬂux model proposed by Zuber and Findlay (1965) and described in Sect. 2.3, to determine the superﬁcial velocities associated to the ﬂow pattern transitions.

3.2.4

Barnea, Shoham, and Taitel (1982a)

This subsection addresses the predictive method for vertical downward ﬂow proposed by Taitel and his coworkers. Even though vertical downward ﬂow is not commonly veriﬁed in practice for convective ﬂow boiling, this ﬂow direction is commonly observed for condensation applications. As described in Chap. 2, basically three ﬂow patterns are expected to occur during vertical downward co-current vapor-liquid ﬂow, namely, annular, bubble, and slug (intermittent) ﬂows. Different from the condition of upward ﬂow, where the vapor in situ velocity is higher or equal to the liquid phase velocity, due to the gravity and density differences, the same does not occur for downward ﬂow, when a liquid velocity higher than the vapor velocity is possible. Analogous to the case of horizontal ﬂow method proposed by Taitel and Dukler (1976), the modelling and derivation of the ﬂow pattern transitions for vertical downward ﬂow begin by the analyzes of the momentum balance of vapor and liquid phases, considering the schematics depicted in Fig. 3.18. The most natural ﬂow pattern during vertical downward ﬂow inside a duct of co-current vapor and liquid phases is the annular ﬂow, which would occur even without a propelling system for the ﬂow.

3.2 Flow Pattern Transition Criteria for Adiabatic Flows Fig. 3.18 Schematics of downward ﬂow inside a channel

97

z ul τw,l

g

τi

uv

τi

δ

τw,l

τi τi

τw,v = 0

Assuming a coordinate axis in the main ﬂow direction, such as depicted in Fig. 3.18, the force balance for liquid and vapor phases in the z direction are given, respectively, as follows: τw,l Sl τi Si dp Al þ ρl Al g dz ¼ 0

ð3:99Þ

þτi Si dp Av þ ρv Av g dz ¼ 0

ð3:100Þ

where, analogous to horizontal ﬂow, the term S corresponds to the contact areas and τ to the shear stresses, and the subscripts w and i correspond to the wall and interface, respectively. By eliminating the pressure difference in both equations, the following relationship is obtained: ðρl ρv ÞgAdz ¼

τ S τ i Si þ w,l l α ð1 α Þ 1 α

ð3:101Þ

Whereas, the terms involving dimensions such as the liquid perimeter can be rewritten as functions of the channel diameter and void fraction. Recalling the deﬁnition of the interfacial perimeter between liquid and channel wall, and vapor– liquid interface, are given as follows: Sl ¼ πdz d pﬃﬃﬃ Si ¼ π αdz d

ð3:102Þ ð3:103Þ

Substituting these relationships in Eq. (3.101), the following equation is derived: ðρl ρv Þgd τ τ ¼ pﬃﬃﬃ i þ w,l 4 α ð1 α Þ 1 α

ð3:104Þ

where the shear stress terms are given as functions of the friction factor as follows:

98

3

τw,l ¼ f w,l τi ¼ f i

ρl u2l 2

Flow Patterns

ð3:105Þ

ρv ð uv ul Þ 2 2

ð3:106Þ

The friction factors are estimated based on their respective Reynolds numbers based on the hydraulic diameters as follows: nl ρl ul dh,l f w,l ¼ C l μl nv ρv uv dh,v f i ﬃ f v ¼ Cv μv

ð3:107Þ ð3:108Þ

where the constants C and exponent n are, respectively, equal to 0.046 and 0.2 for turbulent ﬂow, and 16 and 1.0 for laminar ﬂow. The hydraulic diameters are given as follows: dh,l ¼

4Al 4πd2 ð1 αÞ ¼ dð1 αÞ ¼ 4πd Sl

ð3:109Þ

pﬃﬃﬃ 4Av 4πd2 α pﬃﬃﬃ ¼ d α ¼ Si 4πd α

ð3:110Þ

dh,v ¼

where it is considered that for momentum transfer issues, only the channel wall imposes resistance to the liquid ﬂow. And the in situ velocities are given as functions of the superﬁcial velocities and void fraction, as described in Chap. 2. Therefore, substituting Eqs. (3.102) and (3.103), and (3.105) to (3.110) into Eq. (3.104), the relationship for the momentum balance of both phases is given as follows: ðρl ρv Þgd 1 ¼ 4 ð1 αÞ3 nv nl 2 ρv j2v ðð1 αÞ=α jl = jv Þ2 ρl j l ρ jd ρl j l d Cv v v þ C l μv μl 2 2 αð1nv Þ=2 ð3:111Þ Then, dividing the equation above by the frictional pressure drop gradient of vapor single-phase ﬂow, the following relationship is derived: 2

nl 2 3 ρl jl ρl jl d C l μl 2d ðρl ρv Þg=4 ðð1 αÞ=α jl = jv Þ 1 6 nv 2 ¼ nv 2 7 þ 5 ð3:112Þ 34 ð1nv Þ=2 ρv jv ρ ρv jv d ρ j d α v jv ð1 αÞ C v vμ v Cv μ 2d 2d 2

v

v

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

99

where the second term inside the square brackets on the right side can be recognized as the square of Lockhart and Martinelli parameter X^. The term on the left side can be considered as the ratio between buoyancy forces and the vapor friction forces, and is represented by Ŷ. Hence, the previous equation is re-written as follows: b¼ Y

ðð1 αÞ=α jl = jv Þ2 b 2 1 þX αð1nv Þ=2 ð1 αÞ3

ð3:113Þ

from which it is possible to determine the void fraction, or liquid ﬁlm thickness, as a function of the superﬁcial velocities and ﬂuids properties. Different from that performed for horizontal and vertical upward ﬂow, a mental exercise is performed assuming initially reduced liquid ﬂow rate with successive increment, rather than increment of the vapor phase fraction. This analysis is more coherent with the process occurring inside a vertical condenser. Analogous to the case of horizontal ﬂow, the transition between annular and slug ﬂow for vertical downward ﬂow depends basically on the liquid holdup. Therefore, Barnea et al. (1982a) considered that the transition from annular to slug ﬂow occurs when the liquid holdup (1 – α) during the slug ﬂow is approximately twice the liquid holdup of the annular ﬂow immediately before the transition. Hence, by assuming the same void fraction value taken for transition between bubbles and slug ﬂow for vertical upward ﬂow, which was of approximately 25% ((1 – α)slugﬂow ¼ 0.75 0.7), it is possible to determine the annular to slug transition condition as follows: 2 ð1 αÞtransition ¼ ð1 αÞslugflow

ð3:114Þ

αtransition ¼ 0:65

ð3:115Þ

The reason for deﬁning factor 2 as a transition criteria is not clear, but this assumption results in a transition condition that agrees with the experimental results. By substituting this value in Eq. (3.113), we can obtain the transition condition as follows: " jl ¼ jv 0:538 0:65

ð1nv Þ=4

b Y b2 X 23:32

0:5 # ð3:116Þ

With successive increment of the liquid ﬂow rate, eventually the slug ﬂow pattern will transition to dispersed bubbles, and analogous to the cases of vertical upward ﬂow and horizontal ﬂow, this transition is characterized by the condition that turbulence forces overcome the interfacial tension forces. Barnea, Shoham, and Taitel stated the transition given by Eq. (3.76) between slug and dispersed bubbles for vertical upward ﬂow is also reasonable for downward ﬂows, but the prediction can be improved by considering the effect of void fraction on the maximum possible diameter of a bubble generated by turbulence, which is given by Calderbank (1958) as follows:

100

3

dmax ¼ 4:15α1=2 þ 0:725 3=5 2=3 ðσ=ρl Þ E

Flow Patterns

ð3:117Þ

where again σ is the surface tension, and E is the rate of energy dissipation per mass unit, given by Eqs. (3.69) to (3.71), that results in the following relationship: 0:2 3 ρl jd j Ε ¼ 0:092 d μl

ð3:118Þ

and the homogeneous model is assumed for the estimative of void fraction α as follows: α¼

jv j

ð3:119Þ

Therefore, the transition criterion can be determined by equating the maximum bubble diameter generated by turbulence during downward ﬂow with the critical diameter for a bubble to keep its spherical shape. Different from the upward ﬂow, for downward ﬂows Barnea, Shoham, and Taitel indicated that for bubbles with diameter twice the one given by Hinze (1955) and Eq. (3.73) the shape is still spherical. Therefore, by considering these conditions, the transition criterion is given as follows: 0:487

μl 0:08 ρl 0:52 j1:62 ððρl ρv ÞgÞ1=2 σ 0:1 d 0:48

0:725 j0:5 ¼ 4:15 jv 0:5

ð3:120Þ

which is transcendental on j, hence cannot be explicitly solved on j and a numerical method is needed for its solution. In a similar way to the case of upward ﬂow, the downward ﬂow cannot sustain dispersed bubble ﬂow pattern for void fraction higher than 0.52, and this restriction is related to the maximum packaging of a cubic lattice distribution, schematically depicted in Fig. 3.13. Analogous to vertical upward ﬂow, the transition is determined based on the deﬁnition of the in situ velocities of the vapor phase as follows: uv ¼ ul U 0

ð3:121Þ

where the buoyance forces tend to decrease the vapor in situ velocity in relation to the liquid velocity, and the difference is given by Eq. (3.65), which corresponds to the rising velocity of bubbles in quiescent liquid. Therefore, by rewriting the in situ velocities as a function of the superﬁcial velocities and void fraction, the following relationship can be obtained:

3.2 Flow Pattern Transition Criteria for Adiabatic Flows

jl ¼ 0:923 jv þ 0:734

101

1=4 gðρl ρv Þσ ρ2l

ð3:122Þ

In a similar way to the case of vertical upward ﬂow, the existence of the bubble ﬂow pattern during downward ﬂow is also restricted to large channels, whereas the transition value is given by comparing the velocities of a Taylor bubble, given by Eq. (3.78), and of regular amorphous bubbles, given by Eq. (3.65). Anyway, it should be remembered that during downward ﬂow the vapor phase tends to be slower than the liquid due to buoyance forces, and under conditions that the amorphous bubbles have higher velocity in absolute value than the Taylor bubble, the bubble ﬂow pattern can occur. Otherwise, it is impossible for such a ﬂow pattern to exist. This restriction is given as follows: d ¼ 4:372

ðρl ρv Þσ gρ2l

1=2 ð3:123Þ

which for air-water mixture at patm and 20 C corresponds to 52 mm, and R134a corresponds to 16 mm. Finally, under conditions of very low vapor velocities and high liquid holdup, the amount of liquid might be insufﬁcient to form a Taylor bubble or the continuum core vapor stream of annular ﬂow. Hence, bubble ﬂow occurs for these conditions, and analogous to the analysis performed for upward ﬂow, the transition to slug ﬂow would occur for void fraction of approximately 25%, which is restricted by the maximum packaging of bubbles in a cubic lattice distribution with half radius of spacing between the bubbles. Therefore, by assuming the slip given by Eq. (3.65), this transition is given as follows: jl ¼ 3 jv þ 1:148

1=4 gðρl ρv Þσ ρ2l

ð3:124Þ

Figures 3.19 and 3.20 depict the ﬂow pattern transitions for air-water and R134a, respectively, for two diameters. In Fig. 3.19 for air-water ﬂow, the transition between slug and dispersed bubbles is not represented for the channel with 25 mm of internal diameter because it is smaller than the critical diameter, given by Eq. (3.123). The ﬂow pattern transitions for R134a are presented as a function of the mass velocity and vapor quality, since for heat transfer problems it is common to face constant mass ﬂow rate and ﬂuid enthalpy variation, rather than ﬁxed superﬁcial ﬂow velocity of one of the phases. In any case, it can be speculated that the transition from annular to slug ﬂow for high jv or x is incoherent because both are associated to the increment of the vapor fraction. These conditions would be favorable for annular ﬂow condition; therefore, it is reasonable to assume an extrapolation of the transition curve with positive or null derivative rather than attaining to the transition given by Eq. (3.116) for the entire range of vapor quality and superﬁcial vapor velocity.

102

3

10

Flow Patterns

1

Dispersed bubbles

jl [m/s]

Slug

10

0

Annular Air - water flow at p atm and T = 20 °C, d crit = 52 mm d = 60 mm d = 25 mm

10

-1

10 -2

10 -1

10 1

10 0

10 2

jv [m/s] Fig. 3.19 Flow pattern transitions for vertical downward ﬂow of air water ﬂows at patm and 20 C

3 x10

4

10

R134a, Tsat = 35 °C, dcrit = 13.7 mm

4

G [kg/m²s]

Dispersed bubbles

Slug

10

3

d = 50 mm

Annular

d = 25 mm 2

3 x10 10 -4

10 -3

10 -2

10 -1

x [-] Fig. 3.20 Flow pattern transitions for vertical downward ﬂow of R134a at 35 C

10 0

3.3 Predictive Methods for Convective Boiling

3.3

103

Predictive Methods for Convective Boiling

The transitions for ﬂow patterns during adiabatic ﬂows were addressed in the previous subsection for conventional and minichannels. During the heat transfer process, the ﬂow pattern transitions, and the ﬂow patterns themselves can be different. For example, the pure stratiﬁed ﬂow is not expected during the condensation process because the cooler surface implies on a liquid ﬁlm being formed along the wall. Conversely, during convective ﬂow boiling in horizontal channel, the gravitational effects during annular ﬂow act in order to reduce the ﬁlm thickness in the upper region of the tube; therefore, the ﬁlm in the upper region tends to dry ﬁrst, creating a transitional ﬂow pattern, here deﬁned as dryout. It is recurrently mentioned in the open literature that the gravitational effects are less pronounced during two-phase ﬂow in microchannels, hence the ﬂow patterns might be different from those for conventional channels. Therefore, this subsection addresses some predictive methods for ﬂow patterns during convective boiling, and the next for convective condensation. Several ﬂow pattern predictive methods for convective boiling are available in the open literature, such as Kattan et al. (1998) and subsequently Wojtan et al. (2005), who included the effects of heat transfer process on the ﬂow pattern transitions. Most of them take the physical mechanisms of the Taitel and Dukler (1976) method to predict the transitions, and adjusted new empirical constants and exponents according to their experimental database. Moreover, since these methods are focused on heat transfer problems, the transitions are presented as a function of the mass velocity and vapor quality, which is more elucidating for phase change processes than superﬁcial velocities.

3.3.1

Wojtan, Ursenbacher, and Thome (2005)

The method of Wojtan et al. (2005) proposed to predict ﬂow pattern transitions during convective boiling in conventional channels is an updated version of the method proposed by Kattan et al. (1998). The later method, in turn, is a modiﬁcation of the method proposed by Steiner (Kind et al. 2010, which was available in 1993 in the ﬁrst edition of VDI Heat Atlas) and ultimately is based on adjustments of the Taitel and Dukler (1976) predictive method. Even though the experimental results used to validate the method were obtained for channels with internal diameter of 8.0 and 13.8 mm, this condition still corresponds to conventional scale channel according to the threshold values indicated by Kew and Cornwell (1995), described in Chap. 2, for refrigerants R410A and R22. The main difference between the Wojtan et al. (2005) and the Taitel and Dukler (1976) methods is related to the prediction of a dryout ﬂow pattern, which corresponds to the annular ﬂow being gradually drought from the top to the tube bottom region. Additionally, the method also predicts some transitional ﬂow patterns,

104

3

Flow Patterns

such as Slug + Stratiﬁed wavy ﬂows, which possesses characteristics of both patterns. Moreover, the authors expatiated about the ﬂow pattern identiﬁcation method that based on void fraction ﬂuctuations during time, in a similar way to the description presented by Jones and Zuber (1975); the void fraction is determined based on an optical method with the use of ﬂuorescent dye dispersed in the liquid. In this textbook, basically the implementation of the Wojtan et al. (2005) is presented, and those interested are encouraged to check the original paper to get the steps of its development. In a similar way to Taitel and Dukler, it is considered the geometrical distribution of stratiﬁed ﬂow as a basis for the remaining parameters, however the authors considered the stratiﬁcation angle γ, shown in Fig. 3.4. Moreover, instead of determining the void fraction, or liquid height, based on the momentum balance of both phases, α is determined based on the Rouhani (1969) method, such as described in Sect. 2.3. This approach eliminates the iterative method required for the liquid height evaluation, that is, a function of the Lockhart and Martinelli parameter X^, as performed by Taitel and Dukler (1976). The transition from smooth stratiﬁed and wavy stratiﬁed is given according to the proposal of Kattan et al. (1998) as follows: 1=3 800:18ð1 αÞα2 ρv ðρl ρv Þμl g G¼ x2 ð1 xÞ

ð3:125Þ

The transition from stratiﬁed wavy to non-stratiﬁed ﬂows is given by the following relationship, which comes from successive adjustments of the Taitel and Dukler (1976) proposal: 2

!

31=2

πα3 gdρv ρl π3 Fr l0 6 7 ﬃ G ¼ 4 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ1 5 2 2 25ðH l =d Þ Wel0 2 4x 1 ð2H l =d 1Þ

þ 50 kg=m2 s

ð3:126Þ

where the Froude and Weber numbers for the mixture ﬂowing as liquid phase are given, respectively, as follows: Fr l0 ¼ G2 =ρ2l gd

ð3:127Þ

Wel0 ¼ G2 d=ρl σ

ð3:128Þ

hence, the ratio Frl0 / Wel0 is independent of the mass velocity. The vapor quality corresponding to the transition between intermittent and annular-like ﬂow patterns is given as follows, which comes from the assumption of constant Lockhart and Martinelli parameter:

3.3 Predictive Methods for Convective Boiling

" x ¼ 0:34

1=0:875

105

#1 1=1:75 1=7 ρl μv þ1 ρv μl

ð3:129Þ

The condition corresponding to the dryout inception is given by the following relationship, which comes from the deﬁnition of the vapor quality for dryout inception, proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005): ( G¼

0:25 0:7 )1=1:08 ½gdρv ðρl ρv Þ 0:37 ρv σ 0:17 ρl ϕcrit 0:58 þ 0:52 ln x 0:235 ρv ϕ d ð3:130Þ

where the critical heat ﬂux ϕcrit is given according to Kutateladze (1948) as follows: 0:25 ϕcrit ¼ 0:131ρ0:5 v ilv ½gσ ðρl ρv Þ

ð3:131Þ

Similarly, the end of the dryout region is given as a function of the vapor quality for dryout completion, as proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005), as follows: ( G¼

0:09 0:27 )1=1:06 ½gdρv ðρl ρv Þ 0:15 ρv σ 0:38 ρv ϕcrit 0:61 þ 0:57 ln x 0:0058 ρl ϕ d ð3:132Þ

Figure 3.21 depicts the transitions according to the aforementioned conditions for R134a ﬂow at 10 C in an 8 mm ID tube with heat ﬂux of 10 kW/m2. This ﬁgure also depicts the limits for the transitional ﬂow patterns proposed by Wojtan et al. (2005). It must be highlighted that, analogous to Taitel and Dukler’s (1976) original method, the transition between intermittent and annular ﬂow patterns is predicted to occur at constant vapor quality. However, according to Barbieri et al. (2008) and Felcar et al. (2007), the vapor quality corresponding to this transition reduces with the increment of mass velocity, hence the curve presents a negative derivative on x. According to Barbieri et al. (2008), this transition is given as a function of the liquid Froude number and Lockhart and Martinelli parameters as follows: 2:40

b tt Fr l ¼ 3:75X pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " #1:2 3:75ρ2l gd 1 x0:9 ρv 0:5 μl 0:1 G¼ x ρl μv ð 1 xÞ

ð3:133Þ ð3:134Þ

Figure 3.22 depicts the ﬂow pattern transitions depicted in Fig. 3.21, as well as the transition between intermittent and annular ﬂow patterns described above,

106

3

Flow Patterns

500 Mist

R134a, Tsat = 10 °C, d = 8 mm, φ = 10 kW/m²

Intermittent

300

ut Dryo

G [kg/m²s]

400

200

Annular

Slug

100

Slug + Stratified wavy

Stratified wavy x ia

0 0.0

0.2

Smooth stratified

0.4

x [-]

0.6

0.8

1.0

Fig. 3.21 Flow pattern transition for R134a ﬂow at 10 C in an 8 mm ID tube according to Wojtan et al. (2005)

500 Barbieri, Jabardo and Bandarra Filho (2008)

Mist

400

R134a, Tsat = 10 °C, d = 8 mm, φ = 10 kW/m²

200

Annular

Slug

100

Slug + Stratified wavy

Stratified wavy x ia

0 0.0

ut Dryo

G [kg/m²s]

Intermittent

300

0.2

Smooth stratified

0.4

x [-]

0.6

0.8

1.0

Fig. 3.22 Flow pattern transition according to Wojtan et al. (2005) and the intermittent to annular ﬂow patterns according to Barbieri et al. (2008)

3.3 Predictive Methods for Convective Boiling

107

highlighting the differences in this transition. Therefore, for ﬂow with heat transfer process, the consideration of constant void fraction or liquid height might not be coherent with reality. It must be advised that the predictive method proposed by Wojtan et al. (2005) was developed based on experimental results for conventional channels (d 8 mm); nonetheless, it is presented in this book for didactical purposes. Moreover, this aspect can be recognized by the fact that even this method and the Taitel and Dukler (1976) method do not consider surface tension forces on ﬂow pattern transitions. Conversely, Barnea et al. (1983) consider this mechanism for the transition between stratiﬁed wavy and non-stratiﬁed ﬂow patterns, as discussed in Sect. 3.2.2. Most recently, Cheng et al. (2008) modiﬁed the method of Wojtan et al. (2005) to predict the ﬂow pattern transitions of CO2 based on results from literature for ﬂow patterns and heat transfer coefﬁcient of carbon dioxide. Their updated prediction method is capable of providing accurate ﬂow pattern predictions for tube diameters from 0.6 to 10 mm, therefore being suitable to micro and macroscale conditions. The database used for its formulation covers mass velocities from 50 to 1500 kg/m2s, heat ﬂuxes from 1.8 to 46 kW/m2, and saturation temperatures from 28 to +25 C (reduced pressures from 0.21 to 0.87).

3.3.2

Revellin and Thome (2007)

Revellin and Thome (2007) proposed a ﬂow pattern predictive method for convective ﬂow boiling in small-scale channels. As previously discussed, during two-phase ﬂow in reduced diameter channels, the surface tension forces play a signiﬁcant role on the occurring phenomena and can overcome the buoyance forces under some conditions. Therefore, the gravity-dominated ﬂow patterns, namely, the stratiﬁedlike ﬂow patterns, are unlikely to occur during two-phase ﬂow in microchannels, and basically bubbles, slug, and annular ﬂow patterns are expected. Revellin and Thome (2007) also characterized the transitional ﬂow patterns that correspond to ﬂow patterns combining the characteristic of two ﬂow patterns, but did not address the transition conditions among them. The ﬂow patterns identiﬁed by them are schematically depicted in Fig. 3.23. The transitions predicted by their method were compared with experimental results for R134a and R245fa in 0.509 and 0.790 mm ID tubes, and reasonable agreement was found. These authors performed an objective method for ﬂow pattern identiﬁcation based on bubble length and their frequency. The transition between bubbles (nominated by them as isolated bubbles) and slug ﬂow patterns was characterized by the vapor quality corresponding to the maximum bubbles’ frequency. For vapor qualities higher than the one corresponding to the vapor quality associaty to the bubbles frequency peak, further increase in vapor quality implies on the bubble coalescence and reduction of their frequency. It was found by them that this transition depends on the mass velocity, heat ﬂux, ﬂuids properties including the surface tension, and geometrical parameters. Therefore,

108

3

Flow Patterns

Bubbles

Bubbles to Slug transition

Slug

Slug to annular transition

Wavy annular

Annular

Flow direction Fig. 3.23 Schematics of ﬂow patterns in mini and microchannels

Revellin and Thome proposed that the vapor quality corresponding to the transition can be correlated as a function of the liquid only Reynolds number, Boiling number, and vapor only Weber number, given as follows: x ¼ 0:763

Re l0 Bo Wev0

Re l0 ¼ Bo ¼

Gd μl

ϕ Gilv

Wev0 ¼

G2 d ρv σ

0:41 ð3:135Þ ð3:164Þ ð3:137Þ ð3:138Þ

Equation 3.135 can be rewritten as a function of the mass velocity, given as follows:

3.3 Predictive Methods for Convective Boiling

G¼

109

1=2 0:719 ϕρv σ x1=0:82 μl ilv

ð3:139Þ

Analogous to the transition between bubbles and intermittent ﬂow patterns, the transition from slug to annular ﬂow is given as a function of the bubble frequency when its value tends to zero, annular ﬂow is achieved, corresponding to the absence of bubbles. Hence, based on their experimental results, Revellin and Thome identiﬁed that this transition depends mainly on the liquid Reynolds and Weber number, and is given as follows: 1:23 x ¼ 1:4 104 Re 1:47 l0 Wel0

ð3:140Þ

where the liquid only Weber number Wel0 is given as follows: Wel0 ¼

G2 d ρl σ

ð3:141Þ

The transition from annular to dryout conditions is estimated based on the critical heat ﬂux estimated according to the method of Katto and Ohno (1984) modiﬁed by Wojtan et al. (2006), based on data for ﬂow boiling of halocarbon refrigerants in microscale channels and given as follows: ϕcrit

0:073 0:72 ρ d Gilv ¼ 0:437 v L ρl We0:24 l0

ð3:142Þ

Then, based on energy balance by knowing the inlet enthalpy iin, it is possible to determine the vapor quality for the transition as follows: x¼

ϕ L iin il L þ 4 crit ¼ xin þ 4Bocrit d ilv Gilv d

ð3:143Þ

Figure 3.24 depicts the ﬂow pattern transitions according to the Revellin and Thome (2007) predictive method for R134a. According to this ﬁgure, the dryout condition would only occur for very high mass velocities. Conversely, the bubble ﬂow pattern is observed for the entire mass velocity range, however, only for very low vapor quality values. Again, the stratiﬁed ﬂow pattern is absent during two-phase ﬂow in minichannels because the surface tension forces overcome the buoyance forces. Nonetheless, the experimental database for ﬂow patterns during convective boiling of refrigerants in vertical microchannels is limited, and consequently, as far as the present authors know, a ﬂow pattern prediction method for ﬂow boiling in vertical small diameter channels is still not available in the open literature. However, due to the predominance of surface tension over gravitational effects for small diameter channels, it can be speculated that prediction methods developed for horizontal microchannels are also suitable for ﬂows boiling in vertical small diameter channels.

110

3

2000

Dryout R134a, T = 20 °C, φ = 10 kW/m², φcrit = 3.4 MW/m²

1600

G [kg/m²s]

Flow Patterns

d = 0.509 mm

1200

d = 0.790 mm

800 Annular

400 Slug Bubbles

0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.24 Flow pattern transitions for microchannels during convective ﬂow boiling of R134a according to Revellin and Thome (2007)

3.3.3

Ong and Thome (2011)

Ong and Thome (2011) proposed ﬂow pattern transitions for two-phase ﬂow in microchannels. The experimental database obtained by the authors to adjust the empirical coefﬁcients and exponents in their method includes results for R134a, R236fa, and R245fa ﬂowing inside channels with 1.03, 2.20, and 3.04 mm, for mass ﬂuxes ranging between 100 and 1500 kg/m2s, and saturation temperature between 25 and 35 C. They also add to this database the data previously obtained by Revellin and Thome (2007). The ﬂow patterns were identiﬁed by an objective method using two sets of laser and photodiode installed along the visualization section of their experimental facility, and the signal from this instrumentation was evaluated through a PDF analysis. Based on the experimental results for ﬂow patterns, the authors proposed transitions criteria between ﬂow patterns by data ﬁtting with governing non-dimensional parameters. The authors adopted non-dimensional parameters, such as Reynolds and Weber numbers, but also included conﬁnement number Co, to account for the effect of a conﬁned bubble, and the Froude number Fr to account for the relative effect of inertial and gravitational effects. Based on all parameters, it is claimed that the predictive methods capture the transition from conditions governed by conﬁnement effects, such as for bubble ﬂow, and conditions governed by shear effects, such as for annular ﬂow. In addition to bubble (isolated bubbles), slug (coalescing bubbles), and annular ﬂow patterns, Ong and Thome (2011) also deﬁned the slug-plug ﬂow that is constituted of a long

3.3 Predictive Methods for Convective Boiling

111

vapor bubble separated by liquid plugs that exhibit strong buoyancy effects and a thick stratiﬁed layer of liquid at the bottom of the elongated bubbles. The vapor quality for the transition between bubbles (isolated bubbles) and slug (coalescing bubbles) (from the top of Fig. 3.23, transition from second to fourth ﬂow patterns) is given as follows: x ¼ 0:36Co

0:20

0:65 0:90 μv ρv 0:25 Re 0:75 We0:91 v0 Bo l0 μl ρv

ð3:144Þ

where the Reynolds number is evaluated assuming the mixture ﬂowing as vapor, the Boiling number Bo is evaluated according to Eq. (3.137), the conﬁnement number Co is given as follows: Co ¼

1 d

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gðρl ρv Þ

ð3:145Þ

and the Weber number for the mixture ﬂowing as liquid is given as follows: Wel0 ¼

G2 d σρl

ð3:146Þ

The transition from slug (coalescing bubbles)-to-annular and annular ﬂow patterns is given as follows: x ¼ 0:047Co

0:05

0:70 0:6 μv ρv 0:91 Re 0:8 v0 Wel0 μl ρl

ð3:147Þ

The transition between slug to slug-plug ﬂow patterns is given as follows: x ¼ 9Co0:20

0:9 ρv Fr 1:2 Re 0:1 l0 l0 ρl

ð3:148Þ

where the Reynolds number is evaluated assuming the mixture ﬂowing as liquid, and the Froude number for the mixture ﬂowing as liquid is given as follows: Frl0 ¼

G2 ρ2l gd

ð3:149Þ

According to the proposal, the value estimated according to Eq. (3.148) should be considered only when it is lower than the vapor quality estimated according to Eq. (3.147). Figure 3.25 depicts the ﬂow pattern transitions according to the proposal of Ong and Thome (2011) for R134a and R245fa in 1 and 2 mm ID channels.

112

3

a) 1200

b) 1200

1000

1000

R134a, T sat = 20 °C, φ = 10 kW/m²

Annular

G [kg/m²s]

100

Slug

R245fa, T sat = 20 °C, φ = 10 kW/m²

ar ul nn -a -to ug Sl

Slu g-t o-a nnu lar

G [kg/m²s]

Flow Patterns

d = 1 mm d = 2 mm

Annular

100

Slug Bubbles

12 0.0

0.2

Bubbles

d = 1 mm d = 2 mm

0.4

x [-]

0.6

0.8

1.0

12 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 3.25 Flow patterns transition during convective boiling inside microchannels according to the Ong and Thome (2011) proposal

3.4

Predictive Method for Convective Condensation

At the present, the number of studies focused on the ﬂuid mechanics and heat transfer problems during convective condensation are considerably inferior to those for convective boiling. Nonetheless, in this textbook we address the predictive method proposed by El Hajal et al. (2003) for conventional sized channels. For practical applications, sometimes it is satisfactory deﬁning the ﬂow as gravity dominated or convection dominated, such as performed by Cavallini et al. (2006), who concluded that this transition can be predicted based on the non-dimensional vapor velocity jv*, such as deﬁned by Wallis (1969), as follows: jv ¼

h

1=3 i3 3 b 1:111 7:5= 4:3X þ 1 þ C tt t

ð3:150Þ

where the constant Ct depends on the ﬂuid type and is equal to 1.6 for hydrocarbons and 2.6 for other refrigerants. The non-dimensional vapor velocity is given as follows: xG jv ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ gdρv ðρl ρv Þ

ð3:151Þ

which is somewhat similar to the Froude number, corresponding to the ratio between inertial and gravitational forces. This transition is depicted in Fig. 3.26 for hydrocarbons and other refrigerants, and according to this ﬁgure the difference between types of ﬂuids vanishes for high values of the Lockhart and Martinelli parameter, which corresponds to low vapor quality conditions. It should be mentioned that Cavallini et al. (2006) indicated conditions dominated by wall subcooling, which will be seen in Chap. 7 to correspond to gravity-dominated conditions; however, the interpretation is similar to the one addressed in this book.

3.4 Predictive Method for Convective Condensation

10

113

1

jv* [-]

Predominance of inertial effects

10

0

Predominance of gravitational effects

Refrigerants in general Hydrocarbons

10

-1

0,01

0,1

Xtt [-]

1

10

Fig. 3.26 Transition of predominant effects during condensation, Cavallini et al. (2006)

Hence, in conditions that the actual non-dimensional vapor velocity, given by Eq. (3.151), is higher than the transitional value, given by Eq. (3.150), the ﬂow condition is dominated by inertial effects and the heat transfer is dominated by convective effects, with a corresponding speciﬁc predictive method for the heat transfer coefﬁcient, such as depicted in Fig. 3.27. Conversely, if the actual jv* is smaller than the transitional value, the heat transfer process is governed by gravitational effects, and the heat transfer coefﬁcient predictive method is estimated as a function of the temperature difference, and terms that represent the buoyancy forces, such as thermal expansion coefﬁcient, schematically depicted in Fig. 2.17. These aspects are well addressed in Chap. 7. Despite the fact that the correlation given by Eq. (3.150) had been developed for conventional sized channels, Matkovic et al. (2009), Del Col et al. (2015), and Lopez-Belchi et al. (2016) concluded that this method is also appropriate for microscale channels. In a similar analysis, Jige et al. (2016) proposed a predictive method for heat transfer coefﬁcient during condensation inside rectangular microchannels segregating into intermittent and annular ﬂow patterns. According to these authors, the annular ﬂow pattern in turn can be dominated by vapor shear stress or surface tension effects, while the intermittent ﬂow pattern is given by combination of annular and liquid single-phase ﬂow conditions. Nonetheless, the prediction of heat transfer coefﬁcient is given by an asymptotic combination of all parcels, as described in Chap. 7, and no transition of predominant effects can be inferred.

114

3

3.4.1

Flow Patterns

El Hajal, Thome, and Cavallini (2003)

El Hajal, Thome, and Cavallini (2003) proposed a ﬂow pattern predictive method for condensation inside horizontal channels of conventional dimensions. This method is similar to the Wojtan et al. (2005) method for ﬂow patterns during convective boiling, in the sense that it is based on the method of Kattan et al. (1998), which in turn is based on the method of Steiner, described in Kind et al. (2010), and ultimately is based on the method of Taitel and Dukler (1976) for horizontal ﬂows. As expected, some ﬂow patterns related to the heat transfer process predicted by the Wojtan et al. (2005) method are absent, such as dryout. This is reasonable because this ﬂow pattern depends on the heat transfer process. The physical mechanisms that govern the transitions are analogous to the method of Taitel and Dukler (1976), and the differences are related to the transition values, which are adjusted by means of the empirical constants to experimental results for condensation. Therefore, the complete description and discussion about the transitions are not presented in this chapter, and the reader is encouraged to turn to the corresponding section to remember the physical aspects of the transitions. It must be mentioned that a genuine stratiﬁed ﬂow is not expected during the condensation process, because the contact between the saturated vapor with the cooler wall would result in the formation of a liquid ﬁlm or droplets along all the channel perimeter. Nonetheless, the authors keep the prediction of this transition 600

G [kg/m²s]

Predominance of inertial effects

d = 10 mm d = 5 mm d = 2 mm

400

200 Predominance of gravitational effects R134a, Tsat = 20 °C

0 0,0

0,2

0,4

x [-]

0,6

0,8

1,0

Fig. 3.27 Transition between conditions dominated by inertial or gravitational effects, Cavallini et al. (2006)

3.4 Predictive Method for Convective Condensation

115

considering that the liquid in the upper region does not signiﬁcantly disturb the vapor phase ﬂow, and, therefore, stratiﬁed ﬂow becomes a reasonable deﬁnition. Analogous to the Wojtan, Ursenbacher, and Thome (2005) method for prediction of ﬂow pattern during convective boiling, the El Hajal et al. (2003) method also takes into account a predictive method for void fraction α to avoid the iterative method required by the Taitel and Dukler (1976) for determination of the liquid height. In the present method, the authors adopted the log mean value of the void fractions estimated according to the homogeneous model, given by Eq. (2.34), and the Rouhani (1969) method, based on the Zuber and Findlay (1965) predictive method, given by Eqs. (2.46), (2.47), and (2.48). Therefore, the log mean void fraction is given as follows: α¼

αH αR ln ðαH =αR Þ

ð3:152Þ

where the subscripts H and R attain to the homogeneous model and Rouhani (1969) methods, respectively. Based on the void fraction given by Eq. (3.152), it is possible to infer the non-dimensional liquid height and other geometrical parameters according to Eq. (3.21). The transition from smooth to wavy stratiﬁed ﬂow is determined for mass velocities given as follows: G¼

800:18ð1 αÞα2 ρv ðρl ρv Þμl g x2 ð 1 xÞ

1=3

þ 20 kg=m2 s x

ð3:153Þ

And the transition between stratiﬁed wavy and non-stratiﬁed ﬂows is given for the following mass velocity:

G¼

8 >

:4x2

3

"

dπ 5H l

2

Fr l0 Wel0

1:023 þ1

9 #>0:5 = > ; ð3:154Þ

where the inﬂuence of the heat ﬂux was neglected. The transition between intermittent and annular ﬂows is given in a similar way to Taitel and Dukler’s (1976) original method, deﬁned for a constant Lockhart and Martinelli value of 0.34. Therefore, the vapor quality for this transition is given according to Eq. (3.129).

116

3

5000

Bubbly

Flow Patterns

Mist flow

1000

G [kg/m²s]

Intermittent Annular

100

Stratified wavy

T = 40 °C

Smooth stratified

10 0.0

T = 60 °C

0.4

0.2

x [-]

0.8

0.6

1.0

Fig. 3.28 Flow pattern transitions according to El Hajal et al. (2003) for condensation of R134a in 8 mm ID tub, G ¼ 300 kg/m2s

The transition to mist ﬂow occurs for conditions of high mass velocities, and intermediate and high vapor qualities, and is given as follows: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ" 2 # 480α2 gdρl ρv Fr l0 8 1:138 þ ln G¼ Wel0 3ð1 αÞ x2

ð3:155Þ

where the Froude and Weber for the mixture ﬂowing as liquid are given, respectively, by Eqs. (3.127) and (3.128). The transition to bubbly ﬂow is given as follows: " G¼

4παð1 αÞ2 d 5=4 ρl ðρl ρv Þg 0:3164ð1 xÞ7=4 Pid μ0:25 l

#4=7 ð3:156Þ

For all the calculations in this method, it is considered that the void fraction is evaluated for a given mass ﬂux, that is, it is not evaluated for the mass velocity of the corresponding transition. Figure 3.28 depicts the transitions according to this method for R134a in a channel with an internal diameter of 8 mm.

3.4 Predictive Method for Convective Condensation

3.4.2

117

Nema, Garimella, and Fronk (2014)

Nema et al. (2014) proposed predictive methods for ﬂow pattern transitions during condensation inside mini and microchannels. In their map, the ﬂow patterns were classiﬁed as wavy, intermittent, annular, and dispersed bubbles. The database used by the authors to adjust and compare the proposed transition criteria includes results for condensation of R134a for mass ﬂuxes between 150 and 750 kg/m2s, circular, square, and rectangular horizontal channels with hydraulic diameters ranging between 1.00 and 4.91 mm. Given the range of the channel size, part of the results comprise a stratiﬁed-like ﬂow pattern, hence justifying the inclusion of transition between discrete and dispersed wave ﬂow patterns. Additionally, the authors pointed out that the dispersed bubble ﬂow pattern is unlikely to occur in refrigeration applications because it requires very high mass ﬂuxes. The intermittent ﬂow pattern is expected to occur for reduced vapor content, and Nema et al. (2014) adopted a transition criterion similar to Barnea et al. (1983), schematically represented in Fig. 3.11. By solving Eq. (3.61) for σ, and using the result of Eq. (3.60), the following relationship was obtained: π σ ¼ R2 g 1 ðρl ρv Þ 4

ð3:157Þ

where R is the radius of the bubble represented in Fig. 3.11. According to Nema et al. (2014), the criterion for transition from wavy or annular to intermittent ﬂow pattern depends basically on the liquid content. For mini and microchannels, this condition corresponds to the minimum liquid content required to block the cross-section, which is obtained for R ¼ dh. Hence, by equating R to the channel hydraulic diameter (R ¼ dh), it is possible to obtain a critical Bond number based on the surface tension evaluated according to Eq. (3.157) as follows: Bd crit =

ðρl 2 ρv Þgd 2h 4 ¼ 4:66 ¼ σ crit ð 4 πÞ

ð3:158Þ

The critical Bond number given by Eq. (3.158) corresponds to the transition criterion between micro and conventional scale, whereas for Bond numbers smaller than Bdcrit it corresponds to microscale. The two-phase mixture Bond number is given as follows: Bd ¼

ðρl ρv Þgd 2h σ

ð3:159Þ

Hence, for mini and microchannels (Bd Bdcrit) the transition to intermittent ﬂow pattern is similar to the criterion proposed by Barnea et al. (1983) for conventional channels, given by a constant Lockhart and Martinelli parameter (X^tt ¼ 0.3521) assuming turbulent regime for both phases. The Lockhart and

118

3

Flow Patterns

Martinelli parameter for assuming both phases as turbulent X^tt is given by Eq. (3.49). In the case that the Bond number is higher than the critical value (Bd > Bdcrit), Nema et al. (2014) performed a regression with their data and proposed the following transition criteria: b tt ¼ 0:3521 þ X

1:2479 1 þ 5:5=ðBd Bd crit Þ

ð3:160Þ

which tends asymptotically to 1.6 as Bd increases, and according to Nema et al. (2014) this corresponds to the condition of intermittent ﬂow pattern transition proposed by Taitel and Dukler (1976) for conventional size channels. However, as we have seen in Eqs. (3.48) and (3.49) of Sect. 3.2.2, the condition of α equal to 0.5 corresponds to X^tt ¼ 1.325. Nonetheless, the original proposal presented by Nema et al. (2014) will be kept in this textbook. For sufﬁciently large Bond numbers (Bd Bdcrit), corresponding to large channels, Nema et al. (2014) proposed a transition between discrete and dispersed wave, both corresponding to a stratiﬁed-like ﬂow pattern; however, the discrete wave pattern corresponds to a condition of identiﬁable waves with a long amplitude and wavelength, which is more common for low velocities. On the other hand, a dispersed wave ﬂow pattern corresponds to a condition that the waves have small amplitude and wavelength, and the liquid–vapor interface is almost indistinguishable, being more common for intermediate ﬂow velocities. The transition between discrete and dispersed wave ﬂow patterns is given by a constant value of modiﬁed vapor Froude number of 2.75, and the modiﬁed vapor Froude number is deﬁned as follows: Gx Fr v ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dh gρv ðρl ρv Þ

ð3:161Þ

Recall that the transition between or to/from stratiﬁed ﬂow pattern is not observed for microscale channels (Bd Bdcrit) according to the literature. Nema et al. (2014) also proposed distinct transitions to annular ﬂow pattern depending on the channel dimensions. According to the authors, the annular ﬂow pattern is characterized by the dominance of vapor inertial effects over surface tension effects. In the case of microscale channels (Bd Bdcrit), the occurrence of annular ﬂow is given by: Wev 35

ð3:162Þ

Wev < 35 and bX tt 0:3521

ð3:163Þ

or

where the Weber number of the vapor phase is given as follows:

3.4 Predictive Method for Convective Condensation

Wev ¼

119

dh G2 x2 ρv σ

ð3:164Þ

and the Lockhart and Martinelli parameter for both phases as turbulent X^tt is given by Eq. (3.49). In the case of ﬂow in channels with macroscale characteristics (Bd > Bdcrit), the transition to annular ﬂow is also given as a function of the vapor Weber number as follows: Wev ¼ 6 þ 7ðBd Bdcrit Þ1:5

ð3:165Þ

The authors identiﬁed the occurrence of the mist ﬂow pattern for Wev higher than 700 and X^tt lower than 0.175, independent of the channel diameter. Additionally, Nema et al. (2014) indicated that the number of experimental results for dispersed bubbles in their database was limited, and consequently they suggested to adopt for this transition the criterion proposed by Taitel and Dukler (1976), described in Sect. 3.2.2. Figure 3.29 depicts the ﬂow pattern transitions according to the predictive method proposed by Nema et al. (2014) for condensation inside 1 mm ID channel. In this ﬁgure, the transition to dispersed bubbles was not included because it occurs for G much higher than the values expected in most applications, and the transitions to and between stratiﬁed-like ﬂow patterns were not included because the displayed conditions correspond to a microscale case (Bd < Bdcrit). 1500 dh = 1 mm,Tsat = 40 °C

1200

G [kg/m²s]

Mist

900

600

300

Annular

Intermittent

R134a (Bd = 1.76) R410A (Bd = 2.58)

0 0.0

0.2

0.4

x [-]

0.6

Fig. 3.29 Flow pattern transitions according to Nema et al. (2014)

0.8

1.0

120

3.5

3

Flow Patterns

Solved Examples

Suppose that, similar to the study presented by Felcar et al. (2007), you have performed a series of experiments to infer the transition between intermittent and annular ﬂow patterns for R134a ﬂowing inside a 10 mm ID tube at 5 C. By performing a regression analysis by minimum square, you have found that the relationship between the mass ﬂux and vapor quality for the transitional condition can be given according to the following dimensional relationship: G ¼ 57:531 x1 where G is given in kg/m2s and x as non-dimensional (0–1). Assuming that this transition can be given by the vapor inertial forces overcoming buoyance forces of the mixture, it is reasonable to consider the following modiﬁed Froude number as a ﬁrst attempt to correlate: Fr ¼

ðGxÞ2 ðρl ρv Þgd

Find the corresponding non-dimensional correlation for this transition, and evaluate the root mean square deviation between it and the assumption of constant vapor quality for mass velocities between 100 and 400 kg/m2s. Solution: For saturation temperature of 5 C, the following properties can be found for R134a: ρl ¼ 1311 kg/m3 ρv ¼ 12.09 kg/m3 μl ¼ 2.832 104 kg/m s μv ¼ 1.071 105 kg/m s Based on the given information, d ¼ 0.010 m and g ¼ 9.806 m/s2. Hence, by substituting the obtained dimensional correlation into the deﬁnition of the modiﬁed Froude number, we can notice that the numerator term results in a constant. Additionally, since there is mention about variation of operation conditions, the denominator is also a constant. Therefore, the transition can be estimated based on a constant value of Froude number as follows: Fr trans ¼ 0:02 where it is an annular ﬂow pattern for Fr* > Fr*trans, and intermittent ﬂow pattern for lower values. We can recall Eq. (3.129) to evaluate the transition based on the assumption of constant liquid height as follows:

3.6 Problems

121

" xTaitel ¼ 0:34

1=0:875

#1 1=1:75 1=7 ρl μv þ1 ρv μl

which is a function of the ﬂuid properties only and is equal to 0.2734 for the informed values. Therefore, it is possible to infer the difference in the root mean square deviation between both approaches as follows: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u G¼400kg=m2 s u Z u ðxtrans xTaitel Þ2 Dev ¼ u dG t G G¼100kg=m2 s

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u h i2 2s u G¼400kg=m Z ð0:02ðρl ρv ÞgdÞ0:5 =G xTaitel u ¼u dG t G G¼100kg=m2 s

which results in a deviation of 0.15.

3.6

Problems

1. Analyze the momentum balance equation obtained based on the Taitel and Dukler (1976) approach for annular ﬂow, given by Eq. (3.17). (a) Is it valid for homogeneous ﬂow condition, speciﬁcally for non-slip condition? (b) What can you conclude about annular ﬂow? (c) Do the same analysis for stratiﬁed ﬂow, Eq. (3.17). 2. Verify whether Eq. (3.76) is dimensionally correct. 3. Derive Eq. (3.77). 4. Assume that the transition between bubbles and intermittent ﬂow patterns proposed by Taitel et al. (1980) is given by the bubbles distant one diameter from each other, rather than half diameter. Derive the corresponding relationship between jl and jv. 5. Demonstrate the derivation of Eqs. (3.94) to (3.98). 6. Present the transitions for horizontal ﬂow according to Taitel and Dukler (1976) as a function of G and x. 7. Rewrite all the ﬂow pattern transitions according to the Revellin and Thome (2007) method in terms of mass velocity.

122

3

Flow Patterns

References Baker, O. (1954). Simultaneous ﬂow of oil and gas. New pipeline techniques, July 26, 1954, 185–194. Barbieri, P., Jabardo, J., & Bandarra Filho, E. (2008). Flow patterns in convective boiling of refrigerant R-134a in smooth tubes of several diameters. In Proceedings of the 5th European Thermal-Sciences Conference, Eindhoven, The Netherlands. Barnea, D., Luninski, Y., & Taitel, Y. (1983). Flow pattern in horizontal and vertical two phase ﬂow in small diameter pipes. The Canadian Journal of Chemical Engineering, 61(5), 617–620. Barnea, D., Shoham, O., & Taitel, Y. (1982a). Flow pattern transition for vertical downward two phase ﬂow. Chemical Engineering Science, 37(5), 741–744. Barnea, D., Shoham, O., & Taitel, Y. (1982b). Flow pattern transition for downward inclined two phase ﬂow; horizontal to vertical. Chemical Engineering Science, 37(5), 735–740. Brodkey, R. S. (1967). The phenomena of ﬂuid motions. Reading: Addison-Wesley Press. Calderbank, P. H. (1958). Physical rate processes in industrial fermentation, Part I: The interfacial area in gas-liquid contacting with mechanical agitation. Transactions of the Institution of Chemical Engineers, 36, 443–463. Cavallini, A., Col, D. D., Doretti, L., Matkovic, M., Rossetto, L., Zilio, C., & Censi, G. (2006). Condensation in horizontal smooth tubes: A new heat transfer model for heat exchanger design. Heat Transfer Engineering, 27(8), 31–38. Cheng, L., Ribatski, G., Moreno-Quibén, J., & Thome, J. R. (2008). New prediction methods for CO2 evaporation inside tubes: Part I – A two-phase ﬂow pattern map and a ﬂow pattern based phenomenological model for two-phase ﬂow frictional pressure drops. International Journal of Heat and Mass Transfer, 51, 111–124. Del Col, D., Bortolato, M., Azzolin, M., & Bortolin, S. (2015). Condensation heat transfer and two-phase frictional pressure drop in a single minichannel with R1234ze (E) and other refrigerants. International Journal of Refrigeration, 50, 87–103. El Hajal, J., Thome, J. R., & Cavallini, A. (2003). Condensation in horizontal tubes, Part 1: Two-phase ﬂow pattern map. International Journal of Heat and Mass Transfer, 46(18), 3349–3363. Felcar, H. O. M., Ribatski, G., & Jabardo, J. S. (2007, September). A gas-liquid ﬂow pattern predictive method for macro-and-mini-scale round channels. Proceedings of the 10th UK heat transfer conference, Edinburgh, Scotland. Harmathy, T. Z. (1960). Velocity of large drops and bubbles in media of inﬁnite or restricted extent. AICHE Journal, 6(2), 281–288. Hinze, J. O. (1955). Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AICHE Journal, 1(3), 289–295. Jeffreys, H. (1925). On the formation of water waves by wind. Proceedings of the Royal Society A, 107(742), 189–206. Jeffreys, H. (1926). On the formation of water waves by wind (second paper). Proceedings of the Royal Society A, 110(754), 241–247. Jige, D., Inoue, N., & Koyama, S. (2016). Condensation of refrigerants in a multiport tube with rectangular minichannels. International Journal of Refrigeration, 67, 202–213. Jones, O. C., & Zuber, N. (1975). The interrelation between void fraction ﬂuctuations and ﬂow patterns in two-phase ﬂow. International Journal of Multiphase Flow, 2(3), 273–306. Kattan, N., Thome, J. R., & Favrat, D. (1998). Flow boiling in horizontal tubes: Part 1 – Development of a diabatic two-phase ﬂow pattern map. Journal of Heat Transfer, 120(1), 140–147. Katto, Y., & Ohno, H. (1984). An improved version of the generalized correlation of critical heat ﬂux for the forced convective boiling in uniformly heated vertical tubes. International Journal of Heat and Mass Transfer, 27(9), 1641–1648. Kew, P. A., & Cornwell, K. (1995). Conﬁned Bubble Flow and Boiling in Narrow Spaces, 10th IhTC. Brighton, UK.

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Kind M. et al. (2010) H3 Flow Boiling. In: VDI e. V. (eds) VDI Heat Atlas. VDI-Buch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77877-6_124 Kutateladze, S. S. (1948). On the transition to ﬁlm boiling under natural convection. Kotloturbostroenie, 3, 10–12. Levich, V. G. (1962). Physicochemical hydrodynamics. Englewood Cliffs: Prentice Hall. López-Belchí, A., Illán-Gómez, F., Cascales, J. R. G., & García, F. V. (2016). R32 and R410A condensation heat transfer coefﬁcient and pressure drop within minichannel multiport tube. Experimental technique and measurements. Applied Thermal Engineering, 105, 118–131. Matkovic, M., Cavallini, A., Del Col, D., & Rossetto, L. (2009). Experimental study on condensation heat transfer inside a single circular minichannel. International Journal of Heat and Mass Transfer, 52(9–10), 2311–2323. Mi, Y., Ishii, M., & Tsoukalas, L. H. (1998). Vertical two-phase ﬂow identiﬁcation using advanced instrumentation and neural networks. Nuclear Engineering and Design, 184(2–3), 409–420. Mi, Y., Ishii, M., & Tsoukalas, L. H. (2001). Flow regime identiﬁcation methodology with neural networks and two-phase ﬂow models. Nuclear Engineering and Design, 204(1–3), 87–100. Mishima, K., & Ishii, M. (1984). Two-ﬂuid model and hydrodynamic constitutive relations. Nuclear Engineering and Design, 82(2–3), 107–126. Mori, H., Yoshida, S., Ohishi, K., & Kakimoto, Y. (2000). Dryout quality and post-dryout heat transfer coefﬁcient in horizontal evaporator tubes. In European thermal sciences conference, pp. 839–844. Nema, G., Garimella, S., & Fronk, B. M. (2014). Flow regime transitions during condensation in microchannels. International Journal of Refrigeration, 40, 227–240. Nicklin, D. J. (1962). Two-phase bubble ﬂow. Chemical Engineering Science, 17(9), 693–702. Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase ﬂow: Part 1– Twophase ﬂow patterns and ﬁlm thickness measurements. Experimental Thermal and Fluid Science, 35(1), 37–47. Revellin, R., & Thome, J. R. (2007). A new type of diabatic ﬂow pattern map for boiling heat transfer in microchannels. Journal of Micromechanics and Microengineering, 17(4), 788. https://doi.org/10.1088/0960-1317/17/4/016 Rouhani, S. Z. (1969). Modiﬁed Correlations for Void-Fraction and Pressure Drop. AB Atomenergi Sweden, AE-RTV-841, 1–10. Stewart, R. W. (1967). Mechanics of the Air—Sea Interface. The Physics of Fluids, 10(9), S47–S55. Taitel, Y., & Dukler, A. E. (1976). A model for predicting ﬂow regime transitions in horizontal and near horizontal gas-liquid ﬂow. AICHE Journal, 22(1), 47–55. Taitel, Y., Barnea, D., & Dukler, A. E. (1980). Modelling ﬂow pattern transitions for steady upward gas-liquid ﬂow in vertical tubes. AICHE Journal, 26(3), 345–354. Wallis, G. B. (1969). One-dimensional two-phase ﬂow. McGraw Hill Book Company. Wojtan, L., Revellin, R., & Thome, J. R. (2006). Investigation of saturated critical heat ﬂux in a single, uniformly heated microchannel. Experimental Thermal and Fluid Science, 30(8), 765–774. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005). Investigation of ﬂow boiling in horizontal tubes: Part I – A new diabatic two-phase ﬂow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase ﬂow systems. Journal of Heat Transfer, 87(4), 453–468. https://doi.org/10.1115/1.3689137

Chapter 4

Pressure Drop

The pressure drop corresponds to the pressure reduction during the ﬂuid passage through a channel segment. It is an important parameter in the design of heat exchanger since it is directly related to the pumping power, which impacts the overall efﬁciency of the thermal system. Additionally, and as expected especially for two-phase ﬂows, the variation of ﬂuid pressure along the ﬂow path affects ﬂuid properties, such as vapor-speciﬁc volume and vapor quality, which might cause increment of ﬂow velocity. The pressure reduction during gas-liquid ﬂow of a substance under saturation conditions cause reduction of ﬂuid temperature, which may impact the heat transfer performance and cause non-uniformity of the solid surface temperature. The pressure drop imposed to the ﬂow can be related to the friction between the phases, friction between the ﬂuid and the wall, variation of the ﬂow height, and of ﬂow velocity. These contributions are classiﬁed as frictional, gravitational, and accelerational parcels, respectively. The ﬁrst parcel is dissipative and hence cannot be recovered, while the gravitational and accelerational parcels are conservative and can be recovered. In this textbook, a mathematical relationship for these parcels is derived based on the Reynolds transport theorem for the conservation of mass and momentum quantity for a short duct segment for each phase. The Reynolds transport theorem for conservation of mass and momentum quantity, which can be found in undergraduate textbooks, for one of the phases is given, respectively, as follows: 0¼

∂ ∂t

Z

Z

!

ρdV þ CV

!

ρV dA

ð4:1Þ

CS

Z Z X! X! ! ! ! ! ∂ F surface ¼ V ρdV þ V ρV dA F body þ ∂t CV CS

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_4

ð4:2Þ

125

126

4 Pressure Drop

dz τwv

AI

Awv τI

τI τwl

m Awl

θ z

A x α p ρl ρv ul uv

g

A + dA x + dx α + dα p + dp ρl + dρl ρv + dρv ul + duv uv + duv

Fig. 4.1 Schematics of two-phase ﬂow in a duct

Figure 4.1 schematically depicts the parallel and concurrent ﬂow of vapor and liquid in a duct dz long with a not necessarily uniform cross-section. In the point z, the duct has area A and the ﬂow is characterized by a vapor quality x, void fraction α, pressure p, liquid ρl and vapor ρv densities, and liquid ul and vapor uv axial velocities. Assuming that all the parameters might present variations along a duct length dz, these variables are represented by their original values plus an inﬁnitesimal variation. Figure 4.1 also depicts the shear stress between the liquid and vapor phases with the duct wall, denoted, respectively, by τwv and τwv, and between the phases along the interface, denoted by τI. Since under conditions of positive inclination θ, it is expected that the vapor phase presents higher velocity than the liquid phase, the interfacial shear stress tends to reduce the vapor velocity and increase the liquid velocity. On the other hand, the shear stress with duct wall always tends to reduce the velocity of both phases. Based on the schematics presented in Fig. 4.1, it is convenient deﬁning the mean values of the parameters for the length dz. In the

4 Pressure Drop

127

present development, the domain dimension for the evaluation of the space average operator will be suppressed: A þ A þ dA dA ¼Aþ 2 2 α þ α þ dα dα ¼αþ < α >¼ 2 2 p þ p þ dp dp < p >¼ ¼pþ 2 2 ρ þ ρl þ dρl dρ < ρl >¼ l ¼ ρl þ l 2 2 ρv þ ρv þ dρv dρv < ρv >¼ ¼ ρv þ 2 2 < A >¼

ð4:3Þ ð4:4Þ ð4:5Þ ð4:6Þ ð4:7Þ

Moreover, the deﬁnition of the volume occupied by each phase is useful in the foregoing development and is given as follows: dA dα Aþ αþ dz 2 2 dA dα 1α dz V l ¼< A > ð1 < α >Þdz ¼ A þ 2 2 d ð1 αÞ dA ¼ Aþ 1αþ dz 2 2 V v ¼< A >< α > dz ¼

ð4:8Þ

ð4:9Þ

In this development, it is assumed that the system is under steady state condition (∂/∂t ¼ 0) and that the velocity proﬁles are uniform along the cross-sections; therefore, even though the over bar sign was not introduced in this development to avoid excessive pollution of the equations that are already long, the ﬂow velocities depicted in Fig. 4.1 consist on the mean values evaluated for each phase along the cross-section. Additionally, the following simplifying hypotheses are assumed along a cross-section: uniform transport properties for liquid and gas, and the pressure is uniform. Mass Conservation The mass conservation given by Eq. (4.1) assuming the above-mentioned hypothesis for the gas phase is given as follows:

Z

!

!

ρV dA ¼ ρv uv Aα þ ðρv þ dρv Þðuv þ duv ÞðA þ dAÞðα þ dαÞ m_ I

0¼ CS,v

ð4:10Þ where ṁI corresponds to the mass ﬂow rate from the liquid to the vapor phase across the interface, and its occurrence is associated to the phase change process. It is

128

4 Pressure Drop

assumed that the sign of ṁI is negative for condensation; therefore, the present development is valid for both cases. This term can be rewritten as ṁdx, where dx is the variation of vapor quality along the length dz. Simplifying Eq. (4.10), neglecting multiplication of inﬁnitesimal terms, the following relationship is obtained: _ 0 ¼ dðρv uv AαÞ mdx

ð4:11Þ

The ﬁrst term of the right side in Eq. (4.11) corresponds to the variation of vapor mass ﬂow rate along the inﬁnitesimal length dz, and the last one corresponds to the mass transfer due to phase change, which must be equal for non-penetrating walls. Similarly, the same approach can be applied for the liquid phase as follows: Z 0¼

!

!

ρV dA CS,l

¼ ρl ul Að1 αÞ þ ðρv þ dρv Þðuv þ duv ÞðA þ dAÞð1 α dαÞ þ m_ I

ð4:12Þ

and after the simpliﬁcations we obtain the following relationship: _ 0 ¼ dðρl ul Að1 αÞÞ þ mdx

ð4:13Þ

where the ﬁrst term of the right side corresponds to the variation of liquid mass ﬂow rate along the inﬁnitesimal length dz, and the last term corresponds to the mass transferred due to phase change, and can be considered as the linking term between the phases. Momentum Quantity Conservation In a similar way to the development performed for mass conservation, the momentum conservation is evaluated for each phase separately, using an interfacial term to link the phases. The momentum balance for the control volume deﬁned by the vapor phase in the axial direction is given by Eq. (4.2) for the z direction as follows: τwv Awv τi Ai þ pAα ðp þ dpÞðA þ dAÞðα þ dαÞþ < p > dA < α > ρv V v gsinθ ¼ 0 uv ρv uv Aα þ ðuv þ duv Þ½ðρv þ dρv Þðuv þ duv ÞðA þ dAÞðα þ dαÞ _ mdxV ð4:14Þ I where VI corresponds to the axial component of the velocity of mass ﬂowing across the interface. Notice that the ﬁfth term on the left-hand side refers to the force that pressure imposes to the vapor due to variation of the cross-sectional area, and the mean value of pressure and void fraction were considered for its estimative. The term in square brackets of Eq. (4.14) is equal to the second term of the last member in Eq. (4.10). Additionally, the term of mean values can be rewritten based

4 Pressure Drop

129

on Eqs. (4.3), (4.4), (4.5), (4.6), and (4.7). Therefore, the above equation can be simpliﬁed as follows, where the terms corresponding to product of inﬁnitesimal terms are neglected: τwv Awv τi Ai Ad ðpαÞ Aαρv gsinθdz ¼ ρv Aα

du2v _ V I mdx _ ð4:15Þ þ uv mdx 2

Similarly, for the liquid phase the momentum conservation equation can be derived for the liquid phase, and the resulting relationship is given as follows: τwl Awl þ τi Ai Adp þ Ad ðpαÞ Að1 αÞρl gsinθdz ¼ ρl Að1 αÞ

du2l _ þ V I mdx _ ul mdx 2

ð4:16Þ

We can evaluate the momentum conservation for the mixture in the element based on the sum of Eqs. (4.15) and (4.16). In order to simplify the relationship: ðτwv Awv þ τwl Awl Þ Adp A½αρv þ ð1 αÞρl gsinθdz ¼ ρv Aα

du2v du2 _ ul mdx _ þ ρl Að1 αÞ l þ uv mdx 2 2

ð4:17Þ

Dividing Eq. (4.17) by A.dz and simplifying, we obtain the following relationship:

ðτwv Awv þ τwl Awl Þ dp ½αρv þ ð1 αÞρl gsinθ Adz dz ¼

ρv α du2v ρl ð1 αÞ du2l m_ dx þ þ ð uv ul Þ A dz dz 2 dz 2

ð4:18Þ

Hence, rearranging: ðτwv Awv þ τwl Awl Þ dp f½αρv þ ð1 αÞρl gsinθg ¼ Adz dz ρv α du2v ρl ð1 αÞ du2l m_ dx þ þ ð uv ul Þ 2 A dz dz 2 dz

ð4:19Þ

Therefore, according to Eq. (4.19) the pressure drop gradient can be considered as a combination of three main parcels. The ﬁrst term on the right-hand side of this equation corresponds to the friction between both phases and the duct wall, and can therefore be considered as a frictional parcel. The second term corresponds to the pressure variation due to gravity, where the ﬂuid density is averaged based on the void fraction. The last term corresponds to pressure variation due to the change of ﬂuid momentum. Thus, Eq. (4.19) can be rewritten as follows:

130

4 Pressure Drop

dp dp dp dp ¼ dz dz frictional dz gravitational dz accelerational

ð4:20Þ

where each parcel is given as follows: ðτwv Awv þ τwl Awl Þ dp ¼ Adz dz frictional dp ¼ f½αρv þ ð1 αÞρl gsinθg dz gravitational ρv α du2v ρl ð1 αÞ du2l dp m_ dx þ þ ðuv ul Þ ¼ 2 dz accelerational A dz dz 2 dz

ð4:21Þ ð4:22Þ ð4:23Þ

The shear stress between each phase with the duct wall, as well as the contact area between each phase and the duct wall, are difﬁcult to estimate during two-phase ﬂows by analytical and numerical methods, because these parameters depend on the phase distribution and velocity proﬁle. Therefore, the frictional parcel of the pressure drop is usually evaluated based on prediction methods developed considering the adjustment of empirical constants. This parcel depends on the ﬂow velocity, ﬂuids properties, and duct geometry, among other parameters. The gravitational parcel of the pressure drop given by Eq. (4.22) depends on the area averaged void fraction. When integrating along a length of interest, the area averaged void fraction can be similar to the volume averaged void fraction. The correct estimation of this parameter is critical under conditions of reduced ﬂow velocity, or for conditions of reduced friction, because this parcel can correspond to more than 90% of the total pressure drop. In fact, under conditions of vertical and inclined ﬂows, the estimative of this pressure drop parcel is strongly dependent of the prediction method adopted for the estimative of the area averaged void fraction. On the other hand, under conditions of horizontal ﬂow the duct inclination in relation to the horizontal plane is null, θ ¼ 0, therefore this parcel is null as well. The accelerational parcel of the pressure drop, given by Eq. (4.23), can be simpliﬁed adopting the relationship between the in situ velocity, mass ﬂux, vapor quality, and ﬂuid densities, given by Eqs. (2.20) and (2.21). Therefore, Eq. (4.23) can be rewritten as follows:

( ) 2 2 ρl ð1 αÞ d Gð1 xÞ ð1 xÞ ρv α d Gx dp x 2 dx G ¼ þ þ dz accelerational dz ρl ð1 αÞ ρ v α ρ l ð 1 αÞ dz 2 dz ρv α 2

2 ð 1 xÞ 2 ð1 xÞ2 dG2 d x 1 x2 2 ¼ þ þ G dz 2 αρv ð1 αÞρl dz αρv ð1 αÞρl

ð4:24Þ

4 Pressure Drop

131

Notice that Eq. (4.24) can be derived from Eq. (4.23) only under conditions of two-phase ﬂow, otherwise either α or (1-α) would be null, and the relationships between in situ and mass velocities would be invalid. Hence, imposing the limiting values for x (or α) tending to unity or zero directly in Eq. (4.24) is incoherent. For steady-state two-phase ﬂow along a uniform cross-section duct and with null mass transfer across the walls, the mass ﬂow rate and mass velocity are constant and hence the second term of the last member of Eq. (4.24) is null. In this case, the term G2 of the ﬁrst term can be taken out of the derivative and the integration of pressure gradient can be evaluated based on the Fundamental Calculus Theorem, which is basically related to variation of phase fraction and densities. The contribution of this parcel can be considered negligible under conditions of adiabatic horizontal two-phase ﬂow with low or moderate ﬂow velocities in conventional channels; however, under conditions that the frictional or gravitational pressure drop parcels are signiﬁcant, and during phase change along the channel, the variation of ﬂow momentum cannot be neglected and this parcel must be taken into account. In this context, under conditions of high-pressure gradient, even during adiabatic ﬂow, the variation of saturation pressure may impact the heat transfer process. Figure 4.2 depicts the pressure-enthalpy diagrams for R134a and R600a, and assuming a condition of isenthalpic two-phase ﬂow, it can be noted in these plots that the reduction of pressure causes increment of vapor quality, which in turn results in increment of ﬂow velocity, enhancing the frictional pressure drop parcel. According to Fig. 4.2, this effect is more prominent for R600a than for R134a for high speciﬁc enthalpy values, while for reduced enthalpy, such as exempliﬁed in this ﬁgure, the variation of vapor quality for isenthalpic processes are of the same order for both ﬂuids. Thus, the contribution of the vapor quality increment due to pressure reduction must be evaluated during the analysis of heat exchangers. The process of increasing the vapor content due to pressure drop is known as ﬂashing effect and is important for conditions of high pressure drop such as occurring in reduced diameter channels, for example, in capillary tubes of refrigeration systems. Based on these derivations, it is possible to infer the importance of a precise estimation of the phases fraction, namely, void fraction and vapor quality. Chapter 2 introduced approaches for evaluation of vapor quality x based on energy balance and inlet properties. Moreover, Sect. 2.3 introduced predictive methods for estimating void fraction α. Some predictive methods for the frictional parcel of the pressure drop are addressed below. Nonetheless, some general aspects relating frictional pressure drop during two-phase ﬂow can be discussed. Due to the fact that this textbook is focused on phase change processes, corresponding to evaporation and condensation, it is reasonable to analyze the parameters for a given mass ﬂow rate. The reader might ﬁnd some studies that deal mainly with superﬁcial velocities, which is reasonable for adiabatic (or almost adiabatic) ﬂows, such as veriﬁed for oil and gas industry.

132

4 Pressure Drop

Fig. 4.2 Pressure and enthalpy diagrams for R134a and R600a

Therefore, assuming a ﬁxed mass velocity and saturation pressure, it is possible to estimate the frictional pressure drop parcels for liquid and vapor single-phase ﬂows as follows:

dp dz

¼ 2fi f ,i

G2i ρi d

ð4:25Þ

4 Pressure Drop

133

20000 R134a, d = 5 mm, G = 300 kg/m²s

Linear interpolation (dp/dz)f,v

dp/dz [Pa/m]

15000

Tsat = -10 °C

10000

Tsat = -5 °C

5000

0 0.0

(dp/dz)f,l

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 4.3 Pressure drop for single and two-phase ﬂows (linear interpolation)

where the friction factor for round channel can be determined based on the Hagen– Poiseuille velocity proﬁle for laminar ﬂow, or Blasius correlation for turbulent ﬂow regime as follows: f i ¼ c1 = Re m i

ð4:26Þ

with c1 and m, respectively, equal to 16 and 1 for laminar ﬂow regime, usually assumed for Reynolds numbers below 2300 for round channels, and c1 ¼ 0.079 and m ¼ 0.25 for turbulent ﬂow regime, assumed to occur for Reynolds numbers higher than 10,000, and the region between these limits is considered as a transitional condition. Again, c1 ¼ 0.046 with m ¼ 0.2 are also commonly seen in the literature for turbulent ﬂow regime. The frictional pressure drop gradient for liquid and vapor single-phase ﬂows are estimated according to Eq. (4.25), and are depicted in Fig. 4.3 for R134a ﬂowing in a 5 mm ID channel at 10 C, and they must be the limits for two-phase ﬂow conditions, corresponding to x!0 and x!1, respectively. Even though the viscosity of the liquid phase is higher than that of the vapor phase, the pressure drop of the vapor phase is signiﬁcantly higher due to the respective phases’ velocities, which are related to the speciﬁc volumes. For example, for R134a at 10 C the liquid and vapor densities are, respectively, 1327 and 10.05 kg/m3; therefore, for the same mass velocity, the vapor single-phase ﬂow velocity for a given mass ﬂow rate would be more than 130 times higher than the liquid velocity. In comparison, for the same saturation temperature the liquid and vapor viscosities are approximately,

134

4 Pressure Drop

respectively, 3.02104 and 1.05105 kg/m.s, a ratio of approximately 30 times. Similar velocity ratios are also found for different ﬂuids and saturation pressures. Another important point is related to the effect of saturation temperature and pressure. The property that has higher impact on the pressure drop is the speciﬁc volume of the vapor phase. For instance, considering the conditions depicted in Fig. 4.3 for R134a, namely, saturation temperatures of 10 and 5 C, the corresponding liquid viscosities are 3.02104 and 2.83104 kg/m.s (variation of 6.3%), and the vapor viscosities are, respectively, 1.05105 and 1.07105 kg/m.s (variation of 1.9%); while the vapor-speciﬁc volume varied from 0.09952 m3/kg at 10 C to 0.08274 m3/kg (variation of 16.9%). The frictional pressure drop parcel for vapor single-phase ﬂow reduced by 16.5% for the same temperature variation (10 to 5 C), which indicates that the variation of ﬂow velocity is the dominant effect on the pressure drop variation. Again, similar results are veriﬁed for other ﬂuids and saturation pressures/temperatures. Returning to the discussion about the frictional pressure drop parcel during liquid and vapor simultaneous ﬂow, the reader might be wondering whether considering a linear interpolation between the pressure drop for liquid and vapor ﬂows to estimate the frictional pressure drop during liquid and vapor simultaneous ﬂow as a function of the vapor quality is reasonable, such as depicted in Fig. 4.3. In fact, this hypothesis makes sense, however due to the ﬂow complexity and interaction between the phases, the mechanical energy dissipation is higher than a combination of the pressure drop gradient for both phases. Therefore, the frictional pressure drop parcel for two-phase ﬂow should be somewhat like the curve depicted for homogeneous ﬂow in Fig. 4.4. 20000 R134a, d = 5 mm, G = 300 kg/m²s

dp/dz [Pa/m]

15000

(dp/dz)f,v

Homogeneous model

Tsat = -5 °C Tsat = -10 °C

10000

5000

0 0.0

(dp/dz)f,l

0.2

0.4

x [-]

0.6

0.8

Fig. 4.4 Pressure drop for single and two-phase ﬂows (homogeneous model)

1.0

4 Pressure Drop

135

Alternatively, the increment of pressure drop beyond a linear interpolation can be interpreted as partial restriction of the channel cross-section by each phase, that is, the liquid portion constrains the vapor ﬂow and vice versa. Hence, the pressure gradient increases for intermediate vapor qualities, and for reduced or high vapor qualities the ﬂow tends to present characteristics more similar to single-phase ﬂow. As shown in Fig. 4.4 and observed in most of the experimental studies from literature, the frictional pressure gradient increases with quality until it reaches a peak. Further increments in the vapor quality promote the reduction of the pressure drop gradient. Based on experimental results for the liquid ﬁlm behavior during annular ﬂow of R245fa, Moreira et al. (2020) pointed out that at high vapor quality, when the liquid ﬁlm is very thin, the liquid–vapor interface becomes smoother and disturbance waves slow down and vanish. They linked this shift in the trend of the pressure gradient with vapor quality to the disappearance of disturbance waves. It was mentioned in Chap. 3 that the local ﬂow pattern should impact several twophase ﬂow parameters, including the pressure drop, which is veriﬁed in practice. Therefore, few predictive methods for the frictional pressure drop parcel account for the local ﬂow pattern. Another important aspect when selecting a predictive method for the frictional pressure drop parcel is related to the attainment of the limiting conditions of saturated vapor and liquid. Some predictive methods are developed based on adjustment of empirical parameters with experimental results for a limited range, for example, comprising only low vapor quality values. Therefore, it is not uncommon to ﬁnd predictive methods whose estimative for x equal unity is different than that for vapor single-phase ﬂow. Hence, the reader must check the range of validity of the proposed methods. It has been veriﬁed experimentally by several researchers that the vapor quality value corresponding to the maximum pressure drop reduces with the increment of mass velocity, as shown in Fig. 4.5. Therefore, it is desirable that the predictive method satisfactorily captures this trend. In the case of ﬂow in vertical or inclined channels, the gravitational pressure drop parcel might correspond to a signiﬁcant parcel of the total pressure drop, speciﬁcally for reduced or intermediate ﬂow velocities and high liquid content, which emphasizes the need for reliable and precise void fraction predictive methods, as indicated by Eq. (4.22). In this context, Lavin, Kanizawa, and Ribatski (2019) presented an analysis of the effect of channel inclination on ﬂow patterns and pressure drop, and based on their analyses, the authors pointed out that the experimental determination of the frictional pressure drop parcel for non-horizontal channels relies on the selection of appropriate void fraction predictive method, and depending on the chosen method, even negative values for Δpf might be obtained, such as illustrated in Fig. 4.6 from their study, which is incoherent with the physical problem. Hence, when evaluating frictional pressure drop parcel for vertical or inclined channels by using a method developed based on experimental results, it is necessary to check which void fraction predictive method was adopted for data regression, and adopt it to estimate the total pressure drop. Otherwise, the error on the estimation of the total pressure drop might be signiﬁcant.

136

4 Pressure Drop

G1

dp/dz [Pa/m]

G2 G3 G1 > G 2 > G 3

0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 4.5 Schematics of the effect of mass velocity on the pressure drop

1.0

5 4 Downward flow θ =-90°

0.8

2 1

0.6

0

α [-]

∆ pf / L [kPa/m]

3

-1 Channel rotation =0°

-2 -3

j l = 0.195 m/s G È 195 - 200 kg/m²s

-4 -5 0.1

0.4

Channel rotation = 30° Channel rotation = 60° Channel rotation = 90°

1

jv [m/s]

10

0.2 20

Fig. 4.6 Variation of pressure drop with gas superﬁcial velocity for air-water ﬂow in rectangular channel with hydraulic diameter of 6.24 mm (Lavin et al. 2019)

Finally, we try to attain to the original author when describing the predictive methods, hence, depending on the school and habit, some researchers prefer to use the Darcy friction factor Cf while others prefer the Fanning friction factor f. The reader should remember their deﬁnition from the ﬂuid mechanics course.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

137

Nonetheless, their equivalence is simply given by a product Cf ¼ 4f (the Darcy type friction factor is equivalent to 4 times the Fanning type friction factor), and the frictional pressure drop relationship corresponds to each accordingly.

4.1

Predictive Methods for Frictional Pressure Drop Parcel

This section addresses predictive methods for the frictional pressure drop parcel, given by Eq. (4.21). Due to the complexity of the two-phase ﬂow, it is not feasible to determine the velocity proﬁles and precise geometrical distribution of each phase, therefore it is a difﬁcult task to determine the frictional pressure drop parcel analytically. Thus, the pressure drop predictive methods are usually based on simplifying assumptions followed by adjustment of empirical parameters. One of the simplest approaches, denominated as homogeneous model, is based on the assumption that the mixture behaves as a pseudo ﬂuid with averaged properties, and for this approach the main difﬁculty is the determination of the mixture equivalent viscosity for the estimative of the Reynolds number and friction factor. One of the most common approaches for frictional pressure drop estimative is based on the two-phase multipliers, which consists in multiplying the single-phase pressure drop, evaluated based on the friction factor for the corresponding ﬂow regime, by an estimated factor. For laminar ﬂow, the friction factor is determined based on the Hagen–Poiseuille velocity proﬁle, and for turbulent ﬂow regime it can be estimated based on Blasius, Colebrook (1939), Churchill (1977), or any other method. The ﬁrst proposal for the two-phase ﬂow multiplier was developed by Lockhart and Martinelli (1949), and is described below. Subsequent to these authors, several adjustments and methods were proposed based on this method, such as Friedel (1979) and Grönnerud (1979). Finally, there are also predictive methods that are purely empirical, and one of the most known and reliable among them is the method proposed by Müller-Steinhagen and Heck (1986). Additionally, there are methods based on the pre-deﬁnition of the local ﬂow pattern, which adopts distinct predictive methods for each ﬂow pattern, such as the methods proposed by Moreno-Quibén and Thome (2007). Cheng et al. (2008) for CO2, and Cioncolini et al. (2009) for annular ﬂow in micro and conventional channels. In general, reasonable pressure drop predictions are obtained through methods that do not take into account the ﬂow patterns, and, therefore, are easier to implement.

4.1.1

Homogeneous Model

The homogeneous model is the simplest frictional pressure drop predictive method, and assumes that the mixture is a pseudo-ﬂuid with averaged properties, which is

138

4 Pressure Drop

A

A

Av mv x

x

z y

m

Al

g

z y

m

g

ml

Fig. 4.7 Assumptions of the homogeneous model, from two-phase ﬂow to a pseudo ﬂuid

schematically depicted in Fig. 4.7, and the pressure drop is evaluated assuming single-phase correlations. Therefore, this approach is more coherent with conditions of dispersed ﬂow such as dispersed bubbles or mist ﬂow. In this model, the frictional pressure drop parcel is evaluated as follows: dp G2 ¼ 2fm dz tp ρm d

ð4:27Þ

where the mixture density ρm is evaluated based on the speciﬁc volume of each phase assuming equilibrium condition as follows: ρm ¼

x 1x þ ρv ρl

1 ð4:28Þ

In Eq. (4.27), the sub index tp refers to the frictional pressure drop parcel during two-phase ﬂow, for cleanliness of the text. Since we are considering that the mixture behaves as a pseudo-ﬂuid, the friction factor can be evaluated based on the mixture Reynolds number as follows: fm ¼

C Re nm

ð4:29Þ

where the constant C and exponent n are, respectively, 16 and 1 for laminar ﬂow and, respectively, 0.046 and 0.2 for turbulent ﬂow (0.079 and 0.25 are also commonly used). The mixture Reynolds number is given as follows: Re m ¼

Gd μm

ð4:30Þ

The main challenge here is determining the mixture viscosity, which cannot even be determined for a mixture in equilibrium as performed for other parameters, such

4.1 Predictive Methods for Frictional Pressure Drop Parcel

139

as mixture-speciﬁc volume and enthalpy that can be estimated based on the vapor quality. In this token, Wallis (1969) addresses several models to estimate the mixture viscosity. For dispersed ﬂow with low concentration of the dispersed phase, especially for a condition of solid aspherical particles, the Einstein model provides reasonable predictions and is given by the following relationship: μm ¼ μc ð1 þ 2:5αd Þ

ð4:31Þ

where μc stands for the continuous phase viscosity and αd refers to the volumetric fraction of the dispersed phase. In the case of dispersed gas bubbles in continuum liquid, where the gas has low viscosity, the mixture viscosity is given as follows: μm ¼ μc ð1 þ αd Þ

ð4:32Þ

For conditions distinct than dispersed ﬂow, rheological models are usually not capable of correctly capturing the mixture viscosity behavior. In these cases, several methods have been proposed, accounting for the constraint of single-phase ﬂows. The method proposed by McAdams et al. (1942) follows the same approach adopted for the estimative of mixture density, and is given as follows: μm ¼

x 1x þ μv μl

1 ð4:33Þ

On the other hand, the method proposed by Cicchitti et al. (1959) is similar to the evaluation of the mixture-speciﬁc volume, and is given as follows: μm ¼ xμv þ ð1 xÞμl

ð4:34Þ

Finally, the method proposed by Dukler et al. (1964) considers the volumetric fraction of each phase as pondering parameter, and the mixture viscosity is estimated as follows: μm ¼ βμv þ ð1 βÞμl

ð4:35Þ

where the volumetric fraction β is given by Eq. (2.24). Several studies available in the open literature compared the experimental results for two-phase ﬂow of liquid and condensable vapor with the homogeneous model, and the adoption of Cicchitti et al. (1959) approximation of the mixture viscosity provided reasonable agreement with experimental results for microchannels. This result seems to be associated with the fact that the phase slip is deteriorated in microchannels in comparison with conventional channels. Recall the homogeneous model for void fraction, which explicitly assumes that the phases ﬂow at the same velocity.

140

4 Pressure Drop

(dp dz )

Av A

x

mv

z y

Av

+

mv x

z y

v

m

Al

dp dp (dp dz ) = (dz ) = (dz ) v

l

tp

g

ml

(dp dz )

l

x

Al

z

g

ml y

Fig. 4.8 Assumptions of the Lockhart and Martinelli model, from two-phase ﬂow in a channel to single-phase ﬂows in two channels

4.1.2

Lockhart and Martinelli (1949)

As above described, and according to the present authors’ knowledge, Lockhart and Martinelli (1949) were the ﬁrst to propose the prediction of the frictional pressure drop parcel during two-phase ﬂow based on two-phase multipliers. Even though the method does not provide satisfactory predictions of experimental results for mini and microchannels, it is presented here for historical reasons and because it is didactical for two-phase ﬂow students. Nonetheless, subsequent improvements and adjustments, as performed by Chisholm (1967) among others, have turned this method appropriate for a variety of applications, including phase-change ﬂows in minichannel, and across tube bundles. This method can be considered as a two-ﬂuid model because it considers the existence of two parallel ﬂow streams, one corresponding to the liquid phase and another to the vapor phase, each one ﬂowing along an imaginary channel with a characteristic hydraulic diameter, as schematically depicted in Fig. 4.8. Additionally, the model is based on the following simplifying assumptions: • The static pressure in each section of the liquid ﬂow is equal to the static pressure in the vapor ﬂow. Therefore, the pressure gradient is similar for both phases. • The sum of the volume (area) occupied by liquid and vapor phase must equal to the channel volume (area).

4.1 Predictive Methods for Frictional Pressure Drop Parcel

141

Table 4.1 Possible combinations of ﬂow regimes ll lt tl tt

Liquid phase Laminar Laminar Turbulent Turbulent

Vapor phase Laminar Turbulent Laminar Turbulent

Condition Rehl < 1000 Rehl < 1000 Rehl > 2000 Rehl > 2000

Rehv < 1000 Rehv > 2000 Rehv < 1000 Rehv > 2000

c1,l 16 16 0.046 0.046

c1,v 16 0.046 16 0.046

ml 1.0 1.0 0.2 0.2

mv 1.0 0.2 1.0 0.2

Based on the above-mentioned hypothesis, the following relationships can be addressed:

ρ u2 dp ¼ 2 f hl l l dz tp dhl ρ u2 dp ¼ 2 f hv v v dz tp dhv

ð4:36Þ ð4:37Þ

where dhl and dhv are the hydraulic diameters of the liquid and vapor phases, respectively, and fh corresponds to the friction factor which is a function of the Reynolds number based on the hydraulic diameter of each phase, as schematically depicted in Fig. 4.8. At this point the reader might be wondering which ﬂow regime, laminar or turbulent, should be considered to estimate the friction factor, and in fact, it is possible to have four combinations of ﬂow regimes, as shown in Table 4.1, which also addresses the ﬂow regime transition criteria that will be discussed subsequently. The pressure drop during two-phase ﬂow is higher than that for the case of only one of the phases ﬂowing in the channel, which is a consequence of the cross-section reduction of each phase and the interfacial forces that results in non-reversible work. Therefore, it is reasonable to expect that the hydraulic diameters dhl and dhv are smaller than the channel diameter. Moreover, Lockhart and Martinelli (1949) deﬁned the hydraulic diameters based on classical deﬁnition as follows: dhl ¼ 4Að1 αÞ=dπψ l ¼ dð1 αÞ=ψ l

ð4:38Þ

dhv ¼ 4Aα=dπψ v ¼ dα=ψ v

ð4:39Þ

where the terms ψ l and ψ v correspond to the correction factor for the non-circularity of the phases’ geometries. These parameters are in principle determined from experiments; therefore, they consider the slip between the phases. The friction factor is given as a function of the Reynolds number, assuming the Blasius form as follows: f hi ¼

c1i Re mi h,i

ð4:40Þ

142

4 Pressure Drop

where the coefﬁcient and exponent are presented in Table 4.1, for each phase i. The Reynolds number in this equation is given as a function of the in situ velocity and of the hydraulic diameter, given by Eqs. (4.38) or (4.39), as follows: Re h,i ¼

ρi ui d h,i μi

ð4:41Þ

By equating Eqs. (4.36) and (4.37), which is one of the main assumptions of the model proposed by Lockhart and Martinelli, the following equation is obtained: ρ u2 ρ u2 dp ¼ 2 f hl l l ¼ 2 f hv v v dz tp dhl dhv

ð4:42Þ

2v c1,l ρl u2l c1,v ρv u ¼ m,v dhl Re h,v dhv Re m,l h,l

ð4:43Þ

2v ρl u2l ρv u c1,l c1,v m,l d ¼ ρ u d m,v d ρl ul d h,l hl v v h,v hv

ð4:44Þ

ρl j2l ρv j2v c1,l c1,v m,l dð1αÞ3 ¼ ρ j d m,v dα3

ð4:45Þ

μv

μl

ρl jl d μl ψ l

ψl

v v

ψv

μv ψ v

m,l

c1,l

1 α 3 ψ 1þm,v v ¼ l α ψ 1þm

ρ l

ρl j l d μl

c1,v

v jv d μv

m,v

ρl j2l d ρv j2v d

ð4:46Þ

We can recognize that the term on the right side of Eq. (4.46) corresponds to the pressure drop gradient ratio, considering only the liquid and vapor phases ﬂowing in the channel, even though the friction factor coefﬁcients and exponents are estimated based on the actual velocities and ﬂow area. This ratio is deﬁned as the square of Lockhart and Martinelli parameter, given as follows: # )1=2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðdp=dzÞl ρl j2l ρv j2v c1l c1v ¼ = ðdp=dzÞv ðρv jv d=μv Þmv d ðρl jl d=μl Þml d

(" b¼ X

ð4:47Þ

where X^ is the Lockhart and Martinelli parameter, also used by Taitel and Dukler for the development of their ﬂow pattern transition criteria as previously shown in Chap. 3. Conversely, the left-hand side of Eq. (4.46) represents the unknown parameters that we want to estimate.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

143

Returning to the pressure drop relationship, given by Eq. (4.36): ρ u2 ρl j2l dp c1l ¼ 2 f hl l l ¼ 2 dz tp d hl ðρl jl d hl =μl ð1 αÞÞml ð1 αÞ2 dhl ρl j2l dp c1l ¼2 ml dz tp ðρl jl d=μl ψ l Þ ð1 αÞ3 d=ψ l ψ 1þml ρl j2l dp c1l l ¼ 2 dz tp ð1 αÞ3 ðρl jl d=μl Þml d

ð4:48Þ ð4:49Þ ð4:50Þ

that can be rewritten as follows:

dp dz

¼ tp

Φ2l

dp dz l

ð4:51Þ

where the term Φ2l is the liquid two-phase multiplied, and (dp/dz)l is the pressure drop evaluated assuming only the liquid phase ﬂowing in the channel with diameter d. A similar result is obtained for the vapor phase, from Eq. (4.37), which is given by:

dp dz

tp

ψ 1þmv dp 2 dp v ¼ ¼ Φv dz v dz v α3

ð4:52Þ

Both two-phase multipliers, Φ2l and Φ2v , are experimentally determined, and according to Eqs. (4.46), (4.47), (4.51), and (4.52), they seem to be related to the Lockhart and Martinelli parameter X^. Hence, using two-phase experimental results for air-water, air-benzene, and air-kerosene for diameters ranging from 1.5 to 25.8 mm, the original authors veriﬁed whether these parameters are in fact a function of X^, which was conﬁrmed. The authors plotted a variation of two-phase multipliers as a function of the Lockhart and Martinelli parameter for distinct combinations of ﬂow regimes and conﬁrmed that X^ is the main variable for representation of the two-phase multipliers. It should be mentioned that this conﬁrmation was based on analysis of di-log graphs, which hide signiﬁcant data dispersion, therefore it might not be so convincing and concluding. Nonetheless, deviations higher than 30% for studies in two-phase area, even among experimental results for similar conditions and between experimental and predicted results, are reasonable, common, and marginally acceptable; hence, the conclusions presented by Lockhart and Martinelli are reliable enough for applications. Moreover, this study was the ﬁrst to address the concept of two-phase multiplier, which is adopted until nowadays. Lockhart and Martinelli (1949) addressed tabular data for the relationship between two-phase multiplier and the parameter X^, which are depicted graphically in Fig. 4.9.

4 Pressure Drop 2

1.0 x10

2

Φ l [-]

2.0 x10

1.0 x10

Turbulent - Turbulent Laminar - Turbulent Turbulent - Laminar Laminar - Laminar

1

0

1.0 x10 10 -2

10 -1

2

1.0 x10

2

1.0 x10

1

1.0 10 2

10 1

10 0

2.0 x10

Φ v [-]

144

X [-] Fig. 4.9 Two-phase multiplier according to Lockhart and Martinelli (1949)

10

-1

a [-]

5 x10

0

2 x10

-1

10

-1

10 -2

10 -1

10 0

10 1

10 2

X [-] Fig. 4.10 Void fraction value according to Lockhart and Martinelli (1949)

According to Eq. (4.46), it can be speculated that the void fraction should also be a function of the Lockhart and Martinelli parameter, which was also proofed from analysis of experimental results by the original authors, and the relationship is depicted graphically in Fig. 4.10.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

145

The term X^ can be presented in a more rapidly useful form, as a function of phase properties, and in the case of phase change problems, as a function of the vapor quality as follows: b ¼ X 2

ρl j2l c1l ðρl jl d=μl Þml d ρv j2v c1v ðρv jv d=μv Þmv d

b¼ X

c1l c1v

¼

1=2

1ml ðGð1 xÞ=ρl Þ2ml c1l mvml μml l ρl d mv 1mv c1v μ v ρv ðGx=ρv Þ2mv

ðdGÞðmvmlÞ=2

μml l ρv μmv v ρl

1=2

ð1 xÞ1ml=2 x1mv=2

ð4:53Þ

ð4:54Þ

from which, the most seen is the Lockhart and Martinelli parameter for turbulent ﬂow regimes for both phases (c1l ¼ c1v, and ml ¼ mv ¼ 0.2): 0:9 μ 0:1 ρ 0:5 l v b tt ¼ 1 x X x μv ρl

ð4:55Þ

which takes the sub index tt, which was suppressed until this point. Then, along this textbook, whenever the Lockhart and Martinelli parameter is mentioned with distinct combinations of sub-indexes (ll, lt, tl or tt), the ﬁrst one is relative to the liquid phase and the second to the gas phase. Another point is related to the ﬂow regime transition criteria presented in Table 4.1. Lockhart and Martinelli performed a qualitative analysis of the conditions that would contribute to the transition from laminar to turbulent ﬂow regime, and concluded that, different from that veriﬁed for single-phase ﬂows in round channels, the laminar ﬂow regime occurs for Reynolds numbers lower than 1000, and turbulent ﬂow regime occurs for Reh > 2000. These limits are distinct from the ones commonly assumed for single-phase ﬂow in round channels, which correspond to laminar ﬂow for Re 2300 and turbulent for Re > 10,000, and the difference is caused by the interaction between the phases. Nonetheless, when determining which group of parameters c1 and m would be selected, the reader does not necessarily have the in situ velocity; therefore, the present authors suggests to evaluate the Reynolds number based on the actual diameter and superﬁcial velocities, and then correct the estimations based on void fraction values if the obtained values are close to the regime transition. The Lockhart and Martinelli method is graphically based, therefore it is not promptly implemented computationally; hence, subsequent improvements were proposed by other researchers, such as the Chisholm (1967) approach, which is more widely used than Lockhart and Martinelli’s original method. In this context, it is appropriate to introduce alternate forms of the two-phase multipliers, given by Φ2l0 and Φ2v0 that are related to the factor to be multiplied by the pressure drop of the mixture ﬂowing as liquid and vapor, respectively.

146

4 Pressure Drop

Table 4.2 Coefﬁcients C for Chisholm (1967) correlation of two-phase multiplier

4.1.3

Liquid Turbulent Laminar Turbulent Laminar

Vapor Turbulent Turbulent Laminar Laminar

C 20 12 10 5

Chisholm (1967)

Subsequently to the proposal presented by Lockhart and Martinelli (1949), several authors have contributed to the development and improvement of predictive methods for frictional pressure drop parcel during vapor-liquid two-phase ﬂow, and several of them are based on the concept of two-phase multiplier, such as Chisholm (1967). In his study, Chisholm presented an extended theoretical analysis of the Lockhart and Martinelli model including several aspects in the analysis, such as shear force. However, in conclusion the author proposed a simpliﬁed correlation to estimate the two-phase multiplier, which is given as follows: Φ2l ¼ 1 þ

C 1 þ 2 b b X X

ð4:56Þ

where the term X^ is the Lockhart and Martinelli parameter, deﬁned by Eq. (4.47) or (4.54). The parameter C depends on the ﬂow regimes, presented in Table 4.2, which are valid for two-phase ﬂow mixtures with density ratio similar to that of air-water close to atmospheric pressure. Different from Lockhart and Martinelli (1949), Chisholm adopted a value of 2000 for Reynolds number as transition between laminar and turbulent ﬂow regimes for each phase. Alternatively, Eq. (4.56) can be rewritten as a two-phase multiplier for the vapor phase as follows: bþX b2 Φ2v ¼ 1 þ C X

ð4:57Þ

with the C parameter also given by Table 4.2. Subsequently to these studies, several investigators have tried to correlate the parameter C of Eq. (4.56) for distinct ﬂow condition and duct geometry. This approach is adopted even for external ﬂow across tube bundles, such as in the study presented by Ishihara et al. (1980), who assumed the condition of turbulent– turbulent regimes for both streams and assumed the friction factor for internal ﬂow in round smooth pipes. Speciﬁcally for the case of internal ﬂow inside channels with reduced diameter, the correlation for the parameter C proposed by Mishima and Hibiki (1996) was developed for reduced scale channels, which can be considered as an adjustment of the constant value 20 of Table 4.2 as a function of the channel diameter. The

4.1 Predictive Methods for Frictional Pressure Drop Parcel

147

experimental database used for its development comprises results for air-water ﬂows in channels with internal diameter ranging from 1.05 to 4.08 mm, made of glass and aluminum, and the resulting correlation to be used with Eq. (4.56) is given as follows1: C ¼ 21 1 ed=d0

ð4:58Þ

where d0 is a reference diameter equal to π mm. Equation (4.58) is indicated for conditions comprising density ratios similar to that of air-water ﬂows (ρl / ρv 800) for horizontal and vertical channels, in round and rectangular channels assuming the hydraulic diameter as the characteristic dimension. This relationship seems to be adequate for turbulent–turbulent ﬂow regimes. Adopting the same approach, Zhang et al. (2010) proposed alternate correlations for the parameter C of Chisholm’s method, which were adjusted based on experimental results for air-water, water-N2, R113, R134a, R22 among others ﬂuids, in horizontal and vertical channels with internal diameter ranging from 0.014 to 6.25 mm for diabatic and adiabatic conditions, and for liquid and vapor Reynolds numbers below 2000. The parameter C is given as follows: C ¼ 21 1 e0:358=La

ð4:59Þ

where La* is the modiﬁed Laplace number, which is given as follows: La ¼

1 d

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ gðρl ρv Þ

ð4:60Þ

where the internal diameter d is substituted by the hydraulic diameter for non-circular channels. The reader shall ﬁnd several alternate forms for the parameter C and/or for the two-phase multipliers, Φ2l and Φ2v, as a function of Froude, Weber, and Reynolds numbers, among others non-dimensional parameters. Alternatively, the reader might also ﬁnd predictive methods for two-phase multipliers that assume the pressure drop of the mixture ﬂowing as liquid or vapor, which are given, respectively, as Φ2l0 and Φ2v0. In this context, the predictive method proposed by Friedel (1979) can be addressed, according to which the frictional pressure drop parcel during two-phase ﬂow is given as follows:

The original correlation is given as C ¼ 21(1 e0.319d), but to be coherent with the units, the present authors presented the exponential as a diameter ratio.

1

148

C

f ,i0

4 Pressure Drop

¼

dp 2 dp ¼ Φl0 ð4:61Þ dz tp dz l0 ρ C f ,v0 Φ2l0 ¼ ð1 xÞ2 þ x2 l ρv C f ,l0 0:91 0:19 0:7 0:78 3:21x ð1 xÞ0:224 ρl μv μv 1 ð4:62Þ þ ρv μl μl Fr 0:0454 We0:035 tp tp 64= Re i0 for Re i0 1055 f0:86859 ln ½ Re i0 =ð1:964 ln ð Re i0 Þ 3:8215Þg2

ð4:64Þ

G2 gdρ2H

ð4:65Þ

G2 d ρH σ

ð4:66Þ

Wetp ¼ ρH ¼

dp dz

x 1x þ ρl ρv ¼C

l0

Re i0 > 1055 ð4:63Þ

Gd μi

Re i0 ¼ Fr tp ¼

for

f ,l0

1

G2 2dρl

ð4:67Þ ð4:68Þ

Notice that the Reynolds number and the single-phase pressure drop are evaluated for the two-phase mixture ﬂow rate (G) rather than the phase ﬂow rate (G(1-x) for liquid, or Gx for vapor), which characterizes the difference from the subscripts l0 and l. This method was developed based on an experimental database comprising more than 25,000 experimental results for frictional pressure drop during single and two-component mixtures ﬂowing in channels with internal diameter ranging from 1 to 260 mm. At this point, the signiﬁcance of the non-dimensional parameters adopted in this method are discussed. The Froude number is associated to the balance between inertial and buoyancy effects, whereas high values indicate predominance of inertial forces. It is commonly observed in literature that gravitational effects on the two-phase topology become negligible during two-phase ﬂows in minichannels; therefore, this method might not be appropriate for two-phase ﬂow in miniscale channels. For instance, the Weber number corresponds to the balance between inertial and surface tension effects, where low values represent predominance of surface tension effects, and in the case of two-phase ﬂow inside minichannels the contribution of surface tension effects is signiﬁcant. Considering the fact that the two-phase multiplier reduces with the increment of Froude and Weber numbers, which possess exponents of similar magnitude, the effect of surface

4.1 Predictive Methods for Frictional Pressure Drop Parcel

149

tension effects compensates the negligible effect of buoyance forces during two-phase ﬂow in microchannels.

4.1.4

Müller-Steinhagen and Heck (1986)

Müller-Steinhagen and Heck proposed an empirical method for prediction of frictional pressure drop parcel during two-phase ﬂow, which is given as a combination of the pressure drop during single-phase ﬂows of vapor and liquid podered as a function of the vapor quality. This method has been elaborated based on the qualitative variation of pressure drop with vapor quality, as discussed before and depicted in Figs. 4.3 and 4.4, that is, the frictional pressure drop parcel should present positive ﬁrst-order derivative, for low and intermediate vapor quality values, and negative second-order derivative, attaining to the single-phase pressure drops. The coefﬁcients and exponents of this method have been adjusted based on an experimental database comprising more than 9300 experimental data points for wide ranges of channel diameter (down to 5 mm), reduced pressure, and working ﬂuids. This method is given as follows:

dp dz

¼ tp

dp dp dp dp þ2 x3 x ð1 xÞ1=3 þ dz l0 dz v0 dz l0 dz v0

ð4:69Þ

where:

dp dz

¼C i0

f ,i0

G2 2dρi

ð4:70Þ

and: C

f ,i0

¼

64= Re i0 0:3164= Re 0:25 i0

for for

Re i0 1187 Re i0 > 1187

ð4:71Þ

By assuming the limits of x!0 and x!1, it can be observed that this method obeys the limits of single-phase ﬂows. Moreover, as mentioned above, it has been reported in several studies available in the open literature that the frictional pressure drop estimated according to Müller-Steinhagen and Heck (1986) agrees reasonably well with experimental results for two-phase ﬂow inside conventional and miniscale channels. Despite the reasonably good predictions, some researchers have proposed adjustments for the Müller-Steinhagen and Heck (1986) predictive method in order of turning it more adequate for miniscale channels, such as performed by SemperteguiTapia and Ribatski (2017). These authors have proposed modiﬁcations of constants

150

4 Pressure Drop

and exponents of the Müller-Steinhagen and Heck method, which were adjusted based on experimental database obtained with low global warming potential (GWP) refrigerants, in circular, triangular, and square channels. The resulting method is given as follows:

dp dz

¼ tp

dp dp þ 3:01 exp 4:64 106 Re v0,e dz l0 dz v0 dp dp Þxð1 xÞ1=2:31 þ x2:31 dz l0 dz v0

ð4:72Þ

where the Reynolds number for the mixture ﬂowing as vapor is evaluated based on the equivalent diameter de,2 hence, the subscript e, which is given as follows: Re v0,e ¼

Gd e μv

ð4:73Þ

The frictional pressure drop parcel for the mixture ﬂowing as liquid and vapor are evaluated according to the ﬂow regime, hence differentiating between laminar and turbulent ﬂow regimes with transition deﬁned for Reynolds number of 2300. In the case of circular channels, the friction factor is given by the already familiar Eq. (4.40), with constant of 16 and exponent of 1.0 for laminar, and constant of 0.0791 and exponent of 0.25 for turbulent ﬂow regime. Due to the fact that the database of Sempertegui-Tapia and Ribatski (2017) included results for non-circular channels, with equivalent diameter ranging from 0.835 to 1.1 mm (hydraulic diameter between 0.634 and 1.1 mm), the friction factor also depends on the geometry for laminar ﬂow regimes. On the other hand, for turbulent ﬂow regimes, the friction factor can be considered as independent of the channel geometry. For laminar ﬂow inside rectangular channel, the friction factor is given according to Shah and London’s (1978) proposal as follows:

24 1 1:3553ζ þ 1:9467ζ 2 1:7012ζ 3 þ 0:9564ζ 4 0:2537ζ 5 de f ¼ ð4:74Þ Re i0,e dh where ζ is the aspect ratio ( 1.0), and the Reynolds number Rei0,e is evaluated based on the hydraulic diameter and total mass ﬂow rate. The terms de and dh refer, respectively, for equivalent and hydraulic diameters. For ﬂows inside triangular channels with sharp and rounded corner, the singlephase friction factor is given, respectively, by:

Equivalent diameter is deﬁned as de ¼ (4A/π)1/2 and corresponds to the diameter of a circular channel with same cross-sectional area A.

2

4.1 Predictive Methods for Frictional Pressure Drop Parcel

151

f ¼

13:333 d e Re i0,e dh

ð4:75Þ

f ¼

15:993 d e Re i0,e dh

ð4:76Þ

For non-circular channels in turbulent ﬂow regime (Rei0,e > 2300), the friction factor is evaluated according to the relationship that is used for circular channels, with the Reynolds number evaluated with the actual velocity and equivalent diameter.

4.1.5

Cioncolini, Thome, and Lombardi (2009)

Cioncolini et al. (2009) proposed a predictive method for pressure drop during annular ﬂow in micro and macrochannels accounting for the effect of liquid entrainment in the vapor core ﬂow. According to the proposal presented by the authors, the frictional pressure drop parcel is similar to the homogenous model, however, predictive methods for the friction factor were proposed by the authors mainly as a function of a modiﬁed Weber number, rather than only Reynolds number conventionally used for estimation of f for single-phase ﬂow. According to the proposal, the total pressure gradient for two-phase developed ﬂow along channels with uniform cross-section is given as follows:

dp dz

¼ 2 f tp

G2c εð 1 xÞ 2 ð1 εÞ2 ð1 xÞ2 d x2 þ G2 þ þ dz ρv α ρl εξð1 αÞ ρl ð1 ξÞð1 αÞ ρc d

þ ½ρl ð1 αÞ þ ρv αg sin θ

ð4:77Þ

where the terms on the right-hand side correspond, respectively, to frictional, accelerational, and gravitational parcels. In this textbook, Eq. (4.77) will be kept for this development rather than Eqs. (4.20), (4.21), (4.22), and (4.23) to emphasize the fact that Cioncolini, Thome, and Lombardi (2009) accounted for the entrainment fraction ε, which corresponds to the fraction of liquid that ﬂows as dispersed droplets in the vapor stream, and can be derived by a similar approach adopted for the derivation of Eq. (4.20). In Eq. (4.77), the term ξ corresponds to the liquid holdup of the entrained droplets and is given as follows: ξ¼ε

α 1 x ρv 1 α x ρl

ð4:78Þ

The entrainment fraction ε is evaluated according to Oliemans, Pots, and Trompé (1986) as follows:

152

4 Pressure Drop

Table 4.3 Exponents for entrainment factor according to Oliemans, Pots, and Trompé (1986)

Exp. C0 C1 C2 C3 C4 C5 C6 C7 C8 C9

Par. 10 ρl [kg/m3] ρv [kg/m3] μl [kg/ms] μv [kg/ms] σ [N/m] d [m] jl [m/s] jv [m/s] g [m/s2]

All data points Standard Value error 2.52 0.40 1.08 0.05

Film Reynolds number, Relf 103– 3 100–300 300–10 3103 0.69 1.73 3.31 0.63 0.94 1.15

3103– 104 8.27 0.77

104– 3104 6.38 0.89

3104– 105 0.12 0.45

0.18

0.06

0.96

0.62

0.40

0.71

0.70

0.25

0.27

0.04

0.80

0.63

1.02

0.13

0.17

0.86

0.28

0.11

0.09

0.50

0.46

1.18

0.55

0.05

1.80 1.72 0.70 1.44 0.46

0.08 0.05 0.03 0.05 0.03

0.88 2.45 0.91 0.16 0.86

1.42 2.04 1.05 0.96 0.48

1.00 1.97 0.95 0.78 0.41

0.17 1.16 0.83 1.45 0.32

0.87 1.67 1.04 1.27 0.07

1.51 0.91 1.08 0.71 0.21

ε¼

1þ

1

1 10C0 ρCl 1 ρCv 2 μCl 3 μCv 4 σ C5 d C6 jCl 7 jCv 8 gC9

ð4:79Þ

which is a dimensional expression, with all parameters in SI units. The exponents of Eq. (4.79) are presented in Table 4.3, where the reference Reynolds number Relf is estimated for the liquid ﬁlm, given as follows: Relf ¼ ð1 εÞð1 xÞ

Gd μl

ð4:80Þ

Notice that the determination of entrainment factor depends on the liquid ﬁlm Reynolds number, which in turn depends on ε, hence, an iterative method is needed. Cioncolini et al. (2009) adopted the void fraction predictive method proposed by Woldesemayat and Ghajar (2007), which is an adjustment of the Dix (1971) proposal, that is based on the drift ﬂux model proposed by Zuber and Findlay (1965), given as follows3:

3 The original proposal of Woldesemayat and Ghajar (2007) includes a diameter d inside the second square brackets of the denominator, which makes the equation dimensionally incorrect. Additionally, the original proposal of Dix (1971) does not count with d in the drift parameter. Hence, in this textbook we will keep the dimensionally correct representation. Additionally, beware that for some programming languages, such as F-Chart EES, setting the exponent of an exponent requires additional brackets, such as x^(y^z) instead of simply x^y^z.

4.1 Predictive Methods for Frictional Pressure Drop Parcel

α¼

153

jv ð4:81Þ 0:1 3 ρ ρv h i0:25 patm l 5 þ 2:9 gσ ð1þ cos 2θÞðρl ρv Þ jv 41 þ jjl ð1:22 þ 1:22 sin θÞ p ρ 2

v

l

where θ is the channel inclination in relation to horizontal plane, and patm corresponds to 1 atm (101.325 kPa). The two-phase friction factor ftp for macrochannels is given as follows: f tp ¼ 0:172We0:372 c

ð4:82Þ

where the modiﬁed Weber number Wec, corresponding to core Weber number, is given as follows: Wec ¼

G2c d c ρc σ

ð4:83Þ

and the core mass ﬂux Gc is given as follows: Gc ¼

4m_ ½ x þ εð 1 xÞ πd 2c

ð4:84Þ

and the core diameter is given as follows: dc d

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ α þ ξ αξ

ð4:85Þ

The core density ρc is given as follows: ρc ¼ ð1 αc Þρl þ αv ρv

ð4:86Þ

where αc is the droplet laden core void fraction, given as follows: αc ¼

α α þ ξð1 αÞ

ð4:87Þ

In the case of microchannels, the two-phase friction factor is given as follows: Re 0:318 f tp ¼ 0:0196We0:372 c lf

ð4:88Þ

154

4 Pressure Drop

Cioncolini et al. (2009) used the criterion proposed by Kew and Cornwell (1997) to characterize whether a channel is micro or macro, which is based on the Bond number, given as follows: Bd ¼

gðρl ρv Þd2 σ

ð4:89Þ

whereas if Bd is lower than 4, it corresponds to microchannel, and if Bd > 4, it corresponds to macrochannel. The reader now counts with a myriad of predictive methods. Despite the fact that all the methods should predict similar values, since they intend to represent the real phenomena, this aspect is not necessarily true. Figure 4.11 depicts predicted values for frictional pressure drop parcel during two-phase ﬂow according to the methods described in this chapter, where the homogeneous model assumes the mixture viscosity given by the Cichitti model, and the method of Lockhart and Martinelli (1949) was implemented based on cubic interpolation of the tabular data. Assuming two-phase ﬂow of R134a in channel with 3 mm of internal diameter at saturation temperature of 20 C and mass velocity of 100 kg/m2s as baseline for the analysis (Fig. 4.11.a), we can evaluate the effects of changing some operational conditions on the pressure drop gradient. The increment of mass velocity, depicted by Fig. 4.11.b, causes increment of the pressure drop, which is properly captured by all methods; similarly, reduction of channel diameter implies on the increment of pressure drop, which can be concluded by comparing Fig. 4.10a, c, and both responses are already expected and common sense. Finally, the increment of saturation temperature (and pressure), results in reduction of vapor-speciﬁc volume and consequently of the pressure drop, which can be inferred based on the comparison between Fig. 4.10a, d. As discussed previously in this chapter, the reduction of pressure drop in this case is mainly related to the reduction of the speciﬁc volume of vapor phase, rather than on the reduction of liquid viscosity, which also presents some contribution but it is not dominant. Now we shall discuss the differences between different predictive methods, which are signiﬁcant as can be observed for all conditions depicted in Fig. 4.11. One important aspect that must be mentioned is related to discontinuities of the predicted values, which are present for some of the predictive methods. In the case of homogeneous model, the discontinuities are related to the transition between laminar and turbulent ﬂow regimes. These discontinuities also occur for the Lockhart and Martinelli (1949), Chisholm (1967), Friedel (1979), and Cioncolini, Thome, and Lombardi (2009) methods, which are mostly related to transition between laminar and turbulent ﬂow regimes. One important aspect to be mentioned is related to the quite weird behavior of the predictions according to Lockhart and Martinelli, which include also the effects of ﬂow regime transition and a numerical issue related to the interpolation of tabular data. Regarding the differences among them, we can notice that in any operational conditions the differences can be as high as 100%, therefore, the reader must check the experimental database used in the development of the method to select a method

Fig. 4.11 Predicted values for frictional pressure drop parcel during two-phase ﬂow inside channels for R134a and a) d ¼ 3 mm, G ¼ 100 kg/m2s, Tsat ¼ 20 C; b) d ¼ 3 mm, G ¼ 250 kg/m2s, Tsat ¼ 20 C; c) d ¼ 1 mm, G ¼ 100 kg/m2s, Tsat ¼ 20 C; d) d ¼ 3 mm, G ¼ 100 kg/m2s, Tsat ¼ 0 C.

4.1 Predictive Methods for Frictional Pressure Drop Parcel 155

156

4 Pressure Drop

developed for similar experimental conditions, which includes working ﬂuids and pressure/temperature, channel geometry, and ﬂow velocities. The maximum pressure drop predicted by several methods is independent of the ﬂow velocity, therefore, do not properly capture this experimental behavior.

4.2

Solved Examples

Consider R600a ﬂowing inside a 2 mm ID tube according to mass ﬂuxes of G1 ¼ 100 kg/m2s and G2 ¼ 400 kg/m2s, and vapor quality of 0.5 for saturation temperature of 50 C. Assuming horizontal and vertical upward ﬂows, evaluate the frictional and gravitational pressure drop parcels and the respective fraction of the total pressure drop. Neglect the accelerational pressure drop parcel and assume the Müller-Steinhagen and Heck (1986) predictive method for frictional pressure drop parcel, and minimum kinetic energy model for void fraction. Solution For Tsat ¼ 50 C, the following properties are obtained: ρv ¼ 17.59 kg/m3 ρl ¼ 516 kg/m3 μv ¼ 8.642106 kg/ms μl ¼ 1.144104 kg/ms σ ¼ 7.112 N/m Based on the Müller-Steinhagen and Heck (1986) approach, given by Eqs. (4.69), (4.70), and (4.71), it is possible to estimate the frictional pressure drop parcel, which, respectively for G ¼ 100 and 400 kg/m2s, results in: dp ¼ 3349 Pa dz f and dp ¼ 337893 Pa dz f In the case of horizontal ﬂow, the void fraction values according to the Kanizawa and Ribatski (2016) model is estimated as 0.893 and 0.915 for G of 100 and 400 kg/m2s, respectively. Nonetheless, for horizontal ﬂow the gravitational pressure drop parcel is null, and the frictional parcel corresponds to 100% of the total pressure drop. In the case of vertical ﬂow, the void fraction values according to the same approach, accounting for the difference in the formulation, is 0.907 and 0.948, which results in the gravitational pressure drop parcel of 626.8 and 428.9 Pa/m

4.2 Solved Examples

157

(Eq. (4.22)), for G of 100 and 400 kg/m2s, respectively. Hence, for G1 ¼ 100 kg/m2s, the frictional pressure drop parcel corresponds to 15.8% of the total pressure drop, and for G2 ¼ 400 kg/m2s, this contribution reduces to 1.1%. Further Discussion This example illustrates the importance of the correct estimative of void fraction by selecting reliable predictive methods, since the gravitational pressure is directly a function of α, as shown in Eq. (4.22). This aspect is more pronounced for two-phase ﬂow in macroscale channels, where the frictional parcel is naturally smaller due to larger diameter, and gravitational parcel corresponding to more than 90% of the total pressure drop is not uncommon. Additionally, the reader might be wondering why the accelerational pressure drop parcel was not evaluated, and this aspect is related to the fact that the channel length was not informed. Recall that this parcel corresponds to the variation of ﬂow kinetic energy between two points, and in this case only a point was informed, which corresponds to the location with vapor quality of 50%. If at least a channel length was provided, for example, 1 m, even for adiabatic ﬂow it is possible to estimate iteratively the accelerational pressure drop parcel. Assuming uniform frictional pressure drop parcel for horizontal ﬂow, by performing an iterative method, it is possible to estimate the accelerational pressure drop parcel according to the following simple algorithm for adiabatic ﬂow: Evaluate (dp/dz)f according to Müller-Steinhagen and Heck (1986). Outlet-speciﬁc enthalpy ¼ inlet-speciﬁc enthalpy (îout ¼ îin) Initialize Δpa ¼ Δpa,aux ¼ 0 Pa Error ¼ high value Tolerance ¼ low value While (Error > Tolerance) pout ¼ pin – (dp/dz)f L – Δpa Evaluate Δpa as function of îout, pout, îin, pin according to Eq. (4.24) for constant G Error ¼ |Δpa – Δpa,aux| Δpa,aux ¼ Δpa End while Based on this approach, the accelerational pressure drop parcel for 1 m in channel length assuming adiabatic and uniform frictional pressure drop parcel for horizontal ﬂow is 1.5 and 286.3 Pa for G of 100 and 400 kg/m2s, respectively. The corresponding vapor quality variation is 0.13% and 1.5% for both mass ﬂuxes. In the case of diabatic ﬂow, with uniform heat ﬂux, for example, it is important to account for vapor quality variation along the length, because the frictional and gravitational pressure drop parcels are signiﬁcantly affected by vapor quality variation, as discussed in this chapter, as well as accelerational pressure drop parcel. Moreover, these parameters are not linearly dependent on the vapor quality, hence getting the mean value between inlet and outlet is incorrect. Therefore, it is interesting to discretize the domain in short segments and sum all the contributions.

158

4.3

4 Pressure Drop

Problems

1. Derive Eq. (4.11) starting from Eq. (4.10), adopting the necessary hypothesis. Do the same for Eq. (4.13). 2. Derive step by step Eq. (4.19). 3. Check whether Eq. (4.54) is non-dimensional. 4. Derive Eq. (4.57). 5. Assume homogenous model for void fraction and frictional pressure drop estimative during water downward ﬂow inside vertical pipe of 6 mm of internal diameter, at 40 C. Estimate the pressure drop gradient due to frictional and gravity for mass ﬂux of 250 kg/m2s and vapor quality of 50%. 6. Now, assume vertical upward ﬂow for the conditions described in exercise 5. 7. Now, assume horizontal ﬂow for the conditions described in exercise 5. 8. Check which predictive methods described in this chapter satisfy the conditions of single-phase ﬂow. 9. Consider adiabatic ﬂow of R134a inside a 1 mm ID tube, with inlet condition corresponding to saturated liquid at 40 C, and mass ﬂux of 600 kg/m2s. Evaluate the required tube length to reduce the pressure until saturation temperature of 5 C. What is the outlet vapor quality? (In this exercise, practice the ﬂashing effect, even though in real cases it is possible to obtain blocked ﬂow due to the high ﬂow velocity). Assume horizontal ﬂow. 10. Assume an annular ﬂow in horizontal round channel with liquid ﬁlm along the channel surface, and vapor ﬂow in the core region, both in laminar regime. Present the velocity proﬁles and wall shear stress, as well as pressure gradient. Consider α varying from 0.80 to 0.99 and present the variation of pressure drop for R134a ﬂowing at 5 C in a channel of d ¼ 3 mm, at mass ﬂux of 50 kg/m2s, and plot the variation of pressure drop with void fraction. 11. Consider R134a ﬂow in a 3 mm ID tube at 0 C, G of 100 kg/m2s, and vapor quality of 25%. Based on the Lockhart and Martinelli (1949) approach, and assuming homogeneous void fraction model with both phases distributed according to circular geometry (which is not feasible, but it is a good exercise), evaluate: (a) (b) (c) (d)

The hydraulic and equivalent diameter of both phases. The parameters Ψ l and Ψ v based on Eqs. (4.38) and (4.39). Conﬁrm the validity of Eq. (4.46) with the obtained values. Evaluate the two-phase multiplier values, and the two-phase pressure drop gradient.

12. Repeat exercise 11 assuming the minimum kinetic energy model for horizontal ﬂow.

References

159

References Cheng, L., Ribatski, G., Moreno-Quibén, J., & Thome, J. R. (2008). New prediction methods for CO2 evaporation inside tubes: Part I – A two-phase ﬂow pattern map and a ﬂow pattern based phenomenological model for two-phase ﬂow frictional pressure drops. International Journal of Heat and Mass Transfer, 51, 111–124. Chisholm, D. (1967). A theoretical basis for the Lockhart-Martinelli correlation for two-phase ﬂow. International Journal of Heat and Mass Transfer, 10(12), 1767–1778. Churchill, S. W. (1977). Friction-factor equation spans all ﬂuid-ﬂow regimes. Chemical Engineering, 84(24), 91–92. Cicchitti, A., Lombardi, C., Silvestri, M., Soldaini, G., & Zavattarelli, R. (1959). Two-phase cooling experiments: pressure drop, heat transfer and burnout measurements (No. CISE-71). Milan: Centro Informazioni Studi Esperienze. Cioncolini, A., Thome, J. R., & Lombardi, C. (2009). Uniﬁed macro-to-microscale method to predict two-phase frictional pressure drops of annular ﬂows. International Journal of Multiphase Flow, 35(12), 21138–21148. Colebrook, C. F. (1939). Turbulent ﬂow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156. Dix, G. E. (1971). Vapor void fractions for forced convection with subcooled boiling at low ﬂow rates. GE Report, Berkeley: University of California. Dukler, A. E., Wicks, M., & Cleveland, R. G. (1964). Frictional pressure drop in two-phase ﬂow: A. A comparison of existing correlations for pressure loss and holdup. AICHE Journal, 10(1), 38–43. Friedel, L. (1979). Improved friction pressure drop correlations for horizontal and vertical two phase pipe ﬂow. 3R International, 485–491, July 1979. Grönnerud, R. (1979) Investigation of liquid hold-up, ﬂow resistance and heat transfer in circulation type evaporators. Part IV. Two-phase ﬂow resistance in boiling refrigerants. Bulletin de L’Institut International Du Froid, 1972-1. Ishihara, K., Palen, J. W., & Taborek, J. (1980). Critical review of correlations for predicting two-phase ﬂow pressure drop across tube banks. Heat Transfer Engineering, 1(3), 23–32. Kanizawa, F. T., & Ribatski, G. (2016). Void fraction predictive method based on the minimum kinetic energy. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 209–225. https://doi.org/10.1007/s40430-015-0446-x Kew, P. A., & Cornwell, K. (1997). Correlations for the prediction of boiling heat transfer in smalldiameter channels. Applied thermal engineering, 17(8–10), 705–715. https://doi.org/10.1016/ S1359-4311(96)00071-3 Lavin, F. L., Kanizawa, F. T., & Ribatski, G. (2019). Analyses of the effects of channel inclination and rotation on two-phase ﬂow characteristics and pressure drop in a rectangular channel. Experimental Thermal and Fluid Science, 109, 109850. Lockhart, R. W., & Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase, two-component ﬂow in pipes. Chemical Engineering Progress, 45(1), 39–48. McAdams, W. H., Woods, W. K., & Heroman, L. C. (1942). Vaporization inside horizontal tubesII-benzene-oil mixtures. Transactions of the ASME, 64(3), 193–200. Mishima, K., & Hibiki, T. (1996). Some characteristics of air-water two-phase ﬂow in small diameter vertical tubes. International Journal of Multiphase Flow, 22(4), 703–712. Moreira, T. A., Morse, R. W., Dressler, K. M., Ribatski, G., & Berson, A. (2020). Liquid-ﬁlm thickness and disturbance-wave characterization in a vertical, upward, two-phase annular ﬂow of saturated R245fa inside a rectangular channel. International Journal of Multiphase Flow, 132, 103412. Moreno-Quibén, J., & Thome, J. R. (2007). Flow pattern based two-phase frictional pressure drop model for horizontal tubes, Part II: New phenomenological model. International Journal of Heat and Fluid Flow, 28(5), 1060–1072.

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Müller-Steinhagen, H., & Heck, K. (1986). A simple friction pressure drop correlation for two-phase ﬂow in pipes. Chemical Engineering and Processing: Process Intensiﬁcation, 20 (6), 297–308. Oliemans, R. V. A., Pots, B. F. M., & Trompe, N. (1986). Modelling of annular dispersed two-phase ﬂow in vertical pipes. International Journal of Multiphase Flow, 12(5), 711–732. Sempértegui-Tapia, D. F., & Ribatski, G. (2017). Two-phase frictional pressure drop in horizontal micro-scale channels: Experimental data analysis and prediction method development. International Journal of Refrigeration, 79, 143–163. Wallis, G. B. (1969). One-dimensional two-phase ﬂow. McGraw Hill. Woldesemayat, M. A., & Ghajar, A. J. (2007). Comparison of void fraction correlations for different ﬂow patterns in horizontal and upward inclined pipes. International Journal of Multiphase Flow, 33(4), 347–370. Zhang, W., Hibiki, T., & Mishima, K. (2010). Correlations of two-phase frictional pressure drop and void fraction in mini-channel. International Journal of Heat and Mass Transfer, 53(1), 453–465. Zuber, N., & Findlay, J. (1965, November). Average volumetric concentration in two-phase ﬂow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 5

Flow Boiling

This chapter concerns an analysis of heat transfer during in-tube convective boiling (also named in literature as convective boiling), focusing on small-scale channels. The heat transfer process during ﬂow boiling is composed of two main mechanisms, namely, nucleate boiling and forced convection, such as schematically depicted in Fig. 2.12. Based on this fact, the majority of the predictive methods for heat transfer coefﬁcient during ﬂow boiling are based on the superposition of the contribution of these effects. As proposed by Kutateladze (1961), this approach is given as follows: h¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n ðSnb hnb Þn þ ðF c hc Þn

ð5:1Þ

where, usually, hnb and hc are the heat transfer coefﬁcients estimated based on methods for pool boiling and single-phase forced convection, respectively; Snb and Fc are the factors associated to the suppression of nucleate boiling effects and enhancement of convective effects, respectively, as the vapor quality increases; and n corresponds to the asymptotic exponent that is usually 1 or 2, depending on the predictive method. The approach given by Eq. (5.1) for heat transfer coefﬁcient estimation is recurrently referred in the open literature as the Chen (1966) approach, who adopted unitary asymptotic exponent (n ¼ 1). In general, the value of the nucleate boiling suppression factor varies from zero to the unity (0 Snb 1) and reduces with the increment of ﬂow inertial effects. Conversely, the convective enhancement factor is usually higher than one (Fc 1), and its value increases with increasing ﬂow inertial effects, as follows: Inertial effects "!Snb # Inertial effects "!Fc "

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_5

161

162

5 Flow Boiling

Additionally, under typical operational conditions corresponding to conventional channels, the heat transfer coefﬁcient associated to nucleate boiling effects is usually much higher than the value associated to the forced convection parcel (hnb hc); on the contrary, for convective boiling in microchannels both parcels tend to present a similar order of magnitude. Under conditions such as reduced ﬂow velocities and/or high heat ﬂuxes, the nucleate boiling mechanism is dominant, and the convective effects play a reduced role on the heat transfer process, implying on negligible variations of the heat transfer coefﬁcient with mass velocity and vapor quality prior to the surface dryout. Conversely, for conditions of high mass velocities and/or low heat ﬂuxes, convective effects prevail over nucleate boiling, hence the heat transfer coefﬁcient increases with increasing two-phase ﬂow velocity associated with the increment of mass ﬂux and/or vapor quality. The required wall superheating for the onset of nucleate boiling (ONB) was already introduced in Chap. 2 and is deeply discussed in this chapter. Nucleate boiling consists of the vapor formation as discrete bubbles in the liquid phase and can be characterized as homogeneous when it occurs away from a surface and as heterogeneous when it occurs on a surface, usually solid. Under both conditions, beginning from a state of subcooled liquid, heating the liquid may eventually lead the ﬂuid to a saturated state, with no phase change because a temperature higher than Tsat is necessary to trigger the vaporization process. This additional energy parcel is related to several aspects, including the interfacial energy, the surface tension effects, the inertia due to displacement of the liquid in the neighborhood, and the thermodynamic non-equilibrium between inside and outside of the bubble, among other effects. In this context, consider the force balance of a stable vapor bubble with diameter db in a liquid media, such as schematically depicted in Fig. 5.1. Based on Newton’s

pe

σ db

pi

Fig. 5.1 Schematics of vapor bubble in mechanical equilibrium with liquid media.

5.1 Nucleate Boiling Concepts

163

Second Law, and assuming only pressure and surface tension forces, we can infer the pressure difference between inside pi and outside pe of the bubble as follows: Forcesurface tension Forcepressure ¼ πdb σ ðpi pe Þ ð pi pe Þ ¼

4σ db

πd2b ¼0 4

ð5:2Þ ð5:3Þ

Hence, if we consider that the vapor would be in thermal equilibrium with a saturated liquid, the pressure inside the bubble would be higher than the saturation pressure, then the bubble would collapse. Therefore, additional energy is needed to overcome the listed effects related to bubble formation, where the force balance described above is only one of them. In the subsequent sections, these effects are discussed in more detail.

5.1

Nucleate Boiling Concepts

Homogeneous nucleation is characterized by vapor formation away from a wall within the liquid media under the addition of uniform energy to the ﬂuid along its volume. The heating of a cup of water in a microwave until the establishment of phase change condition can be presented as an approximate experiment, even though this does not really correspond to homogeneous nucleation. You are advised in NOT doing this experiment because under such a condition the phase change from liquid to vapor may start suddenly, spilling the hot water over you. Different from the condition of a conventional pan, where the energy is supplied from a hot surface, the heating in the microwave is directly into the water molecules, with higher heat transfer intensity in the periphery. Hence, the liquid will receive energy and the temperature in the peripheric region will increase to be even higher than the saturation temperature without previous phase change, and a perturbation will eventually promote phase change at some point. Remembering that the speciﬁc volume of the vapor phase at atmospheric condition is considerably higher than that of the liquid, the initial vapor formation will disturb the entire volume in the cup due to ﬂuid displacement, which can promote phase change from liquid to vapor phase at several points along the ﬂuid in an explosive way. Hence, even though the liquid was superheated, it did not vaporize at saturation temperature, remaining in a metastable condition. Another day-to-day example of metastable state is the subcooled liquid, such as beer in a refrigerator, that is below the freezing point but turns into a solid-liquid mixture only when it is disturbed, which can be by abrupt opening or by holding it with bare hands. Even though the ﬂuid was at a temperature below the freezing point, it was in liquid state.

164

5 Flow Boiling

Heating

Liquid

Liquid

T < Tsat

T ≥ Tsat

Fluid molecules with Brownian motion

Motion intensiﬁes and energy of parcel of the molecules increases

Heating

Nucleation Vapor

Liquid

Liquid

T ≥ Tsat

T ≥ Tsat

That promotes transition to vapor phase, displacing the ﬂuid in the vicinity and disturbing the media

Eventually, molecules with high energy form a cluster (region) of high energy

Fig. 5.2 Schematic representation of homogeneous nucleate boiling

The vapor formation requires liquid superheating that under a metastable condition changes to the vapor phase. There is no consensus about the mechanisms that favor the bubble nucleation, but one of the proposals argues that the following steps should occur to promote bubble formation, as schematically represented in Fig. 5.2, assuming the absence of a surface: the ﬂuid molecule energy increases due to the ﬂuid heating; the random movement of the molecules promotes diffusion of energy by net displacement of molecules with high energy, and/or by energy transfer to other molecules that get in contact with them; hence, portions of molecules with higher energy will eventually form a cluster (therefore some statistical parameter to

5.1 Nucleate Boiling Concepts

165

Liquid βc = 0 Perfect Hydrophilic

0° < βc < 90° Hydrophilic

Vapor

Vapor

90° < βc < 180° Hydrophobic

βc

βc

Vapor

βc

Solid Vapor 0° < βc < 90° Hydrophilic

90° < βc < 180° Hydrophobic Liquid βc

βc Liquid

βc = 180° Perfect Hydrophobic Liquid βc

Solid Fig. 5.3 Schematics of contact angle

consider this aspect is required), which corresponds to a region of higher energy and with capacity to promote vapor formation. It must be emphasized that the vapor bubble formation requires displacement of the neighboring liquid, which corresponds to additional work of displacement against the pressure and to increase the ﬂuid kinetic energy of the high energetic cluster. Additionally, an energy parcel related to surface tension forces is needed for bubble formation due to the liquid–vapor interface. Methods available in literature for homogenous nucleation such as Blander and Katz (1975) predict water superheating higher than 200 C for the onset of vapor formation under atmospheric conditions. This value is considerably higher than those veriﬁed when water is boiled to prepare the morning coffee. Different from the homogeneous vapor nucleation that requires a uniform heating of a liquid, the most common approach to increase the liquid energy/temperature in practical usage is through a hot surface. Under such a condition, the liquid in contact with the surface tends to be at the same temperature as the surface, with a temperature gradient along the ﬂuid. In addition to the deﬁned region with higher temperature, the presence of a surface favors the bubble formation whose behavior is affected by the surface wettability and roughness. Under conditions that the surface-liquid pair presents low afﬁnity deﬁned as hydrophobicity, characterized by a contact angle βc higher than 90 , as schematically depicted in Fig. 5.3, the required superheating for the formation of bubble is inferior than for the condition of high wettability (hydrophilic surface), characterized by βc smaller than 90 . The reader can experience qualitatively the wettability contribution on the bubble formation in the kitchen by the conditions of water boiling in a nonstick (Teﬂon® covered) and

166

5 Flow Boiling

Liquid Vapor

βc=βc’

βc

βc’

Cavity angle

Solid

βc

βc’

Cavity angle

βc

βc’

Cavity angle

Fig. 5.4 Schematics of the effect of surface roughness on the effective contact angle

conventional pan. In this experiment, the reader shall notice that for similar conditions, the number of bubbles in the nonstick surface is considerably higher than that in the conventional pan. Usually, metallic surfaces and water present a contact angle close to 80 , and other ﬂuids, such as oils and halocarbon refrigerants, and metallic surfaces usually present angles ranging from 7 to 30 , being hydrophilic. Conversely, non-stick surfaces present a contact angle of the order of 110 , thus with hydrophobic characteristics. Even though contact angles higher than 90 are desirable when regarding the bubble formation, industrially it is difﬁcult to maintain a non-stick surface for long-term service, due to surface degradation and fouling. Additionally, as will be discussed in Chap. 6, surfaces with low wettability (high contact angle), usually result in low critical heat ﬂux (CHF), which can also limit the operational range of the equipment. In fact, it can be shown that the energy necessary for the formation of a vapor nucleus on a surface is lower than the energy necessary for the formation within a liquid medium; moreover, this amount of energy is inversely proportional to the contact angle. Nonetheless, even considering the wettability properties of the surfaces and assuming smooth surfaces, the resulting superheating would be considerably higher than the value experimentally observed, and this difference can be partially justiﬁed on the surface roughness, which acts to increase the effective contact angle and to reduce the required superheating. Figure 5.4, adapted from Collier and Thome (1994), illustrates the inﬂuence of the cavity angle on an effective contact angle βc0 , whereas for a ﬁxed βc the reduction of cavity angle implies on the increment of the angle between the surface and the bubble surface, βc0 , and hence reduces the required superheating of the surface for bubble formation. In this context, several research groups have been investigating surface modiﬁcations, such as roughness modiﬁcation, covering with porous material, deposition of nanoparticles, among others, to enhance the heat transfer coefﬁcient during nucleate boiling. In order to reinforce this discussion, the example of glass recipients for laboratory applications can be addressed, such as Becker recipients. These accessory surfaces present very low surface roughness due to the material characteristics and the manufacturing processes, which makes bubble formation more difﬁcult and require higher superheating to initiate bubble formation along the surface. Hence, lab users

5.1 Nucleate Boiling Concepts

Liquid

Noncondensable gas

Solid

Cavity t1

167

Liquid

Liquid

Liquid

Solid

Solid t2

t3

Trapped non-condensable gas t4

t1 < t2 < t3 < t4

Fig. 5.5 Schematics of liquid front trapping non-condensable gas into surface cavity along time

usually intentionally introduce surface roughness mechanically or by chemical attack to facilitate bubble formation. In the case of a non-smooth surface initially dry and exposed to non-condensable gas, such as air, that is wetted with the working ﬂuid in liquid phase, such as schematically depicted in Figure 5.5, the non-condensable gas can be trapped in the cavity, which reduces the required surface superheating. Recall Eq. (5.3), which would also be valid for a hemispherical bubble in a cavity, such as schematically depicted in Fig. 5.4, where db can be read as cavity diameter; however, the internal pressure corresponds to the sum of the vapor and non-condensable gas partial pressures, and the external pressure corresponds to the liquid pressure, as follows: pv þ pa pl ¼

4σ db

ð5:4Þ

where pv and pa correspond to the vapor and non-condensable-gas partial pressures, respectively. Again, similar to the approach proposed by Bergles and Rohsenow (1964) and described in Sect. 2.4, the Clapeyron equation for ideal gas can be used, which is given as follows: dpv ^ilv ρv p ^i ¼ v lv2 dT v Tv Rv T v

ð5:5Þ

where Rv stands for the speciﬁc gas constant. Equation. (5.5) is a differential equation that can be solved for pv and Tv by separation of variables, as follows: ^i 1 dp lv dT pv v Rv T 2v v ^i ðT T l Þ ^i 1 1 lnðpv Þ lnðpl Þ lv ¼ lv v Rv T l T v Rv T l T v

ð5:6Þ ð5:7Þ

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5 Flow Boiling

And by using the result of Eq. (5.4), assuming that the liquid is at saturation temperature, the following relationship can be obtained for the required vapor superheating for equilibrium: T v T sat ¼

p Rv T sat T v 4σ lnð þ 1 aÞ ^ilv d b pl pl

ð5:8Þ

Then, recalling the vapor superheating for equilibrium without non-condensable gas given in Sect. 2.4 by Eq. (2.65), the vapor superheating is given as follows: T v T sat ¼

Rv T v T sat 2σ lnð þ 1Þ ^ilv rpl

ð5:9Þ

and by comparing it with Eq. (5.8), it can be concluded that the presence of non-condensable gas trapped into the cavities reduces the required superheating for bubble equilibrium, and consequently to bubble nucleation and growth. Recall that the determination of the condition that the bubble will grow requires to estimate the liquid temperature as well, as given by Eqs. (2.66) and (2.67). Hence, bubbles shall be formed preferentially in cavities along the heated surface, either for the cases with or without non-condensable gases, and successive energy transfer from the surface to the ﬂuid result in the bubble growth due to vaporization. The bubble size will grow until eventually buoyance forces and drag forces (relevant in case of forced ﬂow) overcome surface tension and inertial forces, implying on the bubble detachment from the surface. In the active cavities, deﬁned as the cavities that propitiated the bubble formation, a small portion of vapor tends to remain; hence, the formation of the next bubble in the same cavity does not require the same superheating, and successive bubbles will be formed in the same region until other cavities also become active due to the increase of the surface heating. In the case that the bubble nucleation was facilitated by the presence of non-condensable gases, fractions of the gas will be displaced with the vapor when the bubble departs, until the obtainment of only vapor into the cavity. The reader can notice this aspect by boiling water in a kettle at your kitchen, when in the beginning of the boiling process, the bubbles are successively formed in speciﬁc locations. Recall that in the beginning of the boiling process, the non-equilibrium temperature non-equilibrium implies on higher temperature close to the pan solid surface, hence away from the surface the water might be below the saturation temperature. Consequently, the reader will observe that the bubble collapses just after its detachment, because it transfers energy to the subcooled liquid and is not self-sustained anymore, despite the reduction of pressure due to the liquid column. Hence, up to this point, the requirements for bubble formation in a heated surface have been brieﬂy discussed, and the reader is encouraged to review the discussion presented in Sect. 2.4 regarding ONB. Focused on historical and didactical aspects, it is interesting to present the boiling curve as shown in Fig. 5.6, which can be obtained either by controlling the heat ﬂux

5.1 Nucleate Boiling Concepts

169

7

a

b 6-7 6' - 7

6'

5

5-6

4 5 CHF 3'

3-4

2-3 ONB 3'' 1

2

3

6

1-2

1'

ΔT = Tw - Tsat Fig. 5.6 Schematic boiling curve for imposed heat ﬂux and dominant heat transfer mechanisms. (a) Representation of boiling curve, (b) schematics of heat transfer processes in each condition

or the temperature difference between the surface and the ﬂuid, whereas the ﬂuid away from the heater is stagnant, which is referred in the literature as pool boiling condition. Let us assume a cylindrical heater with small diameter, disposed horizontally in the subcooled liquid media. The heating can be performed by Joule effect via electrical current applied directly to the heater surface, which provides approximately uniform and imposed heat ﬂux condition along the external surface, or by considering vapor condensing internally to a microtube that can be approximated as a condition of imposed surface temperature. The heat transfer process from the heater to the neighbor liquid results in temperature gradient along the liquid, whereas the hotter regions are closer to the heater wall. This experiment was ﬁrstly reported by Nukiyama (1934) and even though it seems to be simple, it gives important insights about the heat transfer processes associted to the nucleate boiling phenomenon. The condition of imposed heat ﬂux as displayed in Fig. 5.6 may correspond to the thermal management of electronic components, concentrated solar collectors, and nuclear reactors. For reduced heat ﬂux, region 1–2 in Fig. 5.6a, the heat transfer process corresponds to natural convection from a horizontal cylinder, and well established predictive methods for the heat transfer coefﬁcient are available in the

170

5 Flow Boiling

open literature for distinct geometries, such as in Lienhard and Lienhard (2020). The heat transfer coefﬁcient increases with the heat ﬂux due to the increment of density difference and induced ﬂow around the heater. Under this condition, the liquid adjoined to the heater can be hotter than the saturation temperature without bubble formation, as previously pointed out in the discussion about the conditions necessary for bubble formation and its growth. When the heater surface achieves the temperature corresponding to the required superheating, bubble formation will eventually occur and propagate over all the surface, corresponding to point 2 in Fig. 5.6, denominated as ONB. In the imminence of bubble formation, the heat transfer coefﬁcient presents a sharp increment, with consequent reduction of the heater wall temperature, such as depicted by the gap from point 2 to 3 in Fig. 5.6a. The reader should keep in mind the Newton’s Cooling Law, which states that the heat ﬂux during convective heat transfer is proportional to the temperature difference between the surface and the ﬂuid, whereas the heat transfer coefﬁcient is the proportionality parameter and is given as follows: ϕ ¼ hðT w T fluid Þ

ð5:10Þ

Hence, a subtle increment of h for a ﬁxed ϕ implies on a sharp reduction of ΔT ¼ Tw – Tﬂuid. The reduction of the wall temperature does not necessarily eliminate the bubble generation, and this aspect is related to the presence of active cavities, such as previously discussed, which corresponds to regions with vapor nuclei that eliminate the need of high superheating, and vapor bubbles just grow and detach. The difference between the pool boiling curves obtained for increasing and decreasing heat ﬂux is nominated as hysteresis. This phenomenon is illustrated in Fig. 5.7 that 120 Increasing f

100

Reducing f

f [kW/m2]

80

Nucleate boiling

60 40 Natural convection

20 0 0

20

Tw - Tsat [K]

40

60

Fig. 5.7 Variation of wall superheating with heat ﬂux for pool boiling of R11 at Tsat ¼ 5.5 C in horizontal tube with 19 mm of external diameter. (Ribatski 2002)

5.1 Nucleate Boiling Concepts

171

depicts experimental results for R11 pool boiling on a horizontal tube with 19 mm of external diameter, under conditions of increasing and decreasing heat ﬂux. When increasing the heat ﬂux, depicted by empty symbols, the wall temperature tends to increase with heat ﬂux until a point where the nucleate boiling begins, and from this point on the temperature difference is inferior to the condition of single-phase ﬂow. Successive increment of ϕ results in sensible increment of the wall temperature, indicating high heat transfer coefﬁcient. Successive increments of the heat ﬂux beyond the ONB implies on the achievement of the fully developed boiling regime, as nominated by Jabardo et al. (2004), with active cavities propitiating successive generation, growth, and detachment of bubbles. This condition corresponds to considerably high heat transfer coefﬁcients, which usually are higher than 1 kW/m2K and can reach values of the order of 10 kW/ m2K (or even 100 kW/m2K for microchannels depending on the ﬂuid and operational conditions), depicted by a variation of ϕ with ΔT according to a power law with a constant exponent usually between 2 and 4 for the wall superheating as schematically depicted in Figs. 5.6 and 5.7. There is still no consensus about the dominant mechanisms that result the high heat transfer coefﬁcients observed during the nucleate boiling process. In fact, it is expected that more than one mechanism contributes to the overall heat transfer with their relative inﬂuence varying according to the reduced pressure, gravitational acceleration, heat ﬂux, surface roughness, wettability, ﬂuid type, and degree of ﬂuid subcooling. Earlier models such as Rosenhow (1952) and Forster and Zuber (1955) consider bubble agitation and microconvection effects as the main mechanisms responsible for the heat transfer in pool boiling and, then, based on an analogy between nucleate boiling and forced convection a correlation for the heat transfer coefﬁcient is proposed. Kim (2009) pointed out the following three main mechanisms as the most important to explain the high heat transfer coefﬁcients observed under nucleate boiling conditions: • Liquid replacement: the bubble detachment causes disruption of the thermal boundary layer close to the hot surface, and part of the hot liquid is dragged by the bubble movement. Hence, colder liquid away from the surface ﬂows toward the surface, as described by Rohsenow (1971), and schematically represented in Fig. 5.8a. In general, the models based on such mechanism neglect wall heat transfer during the bubble growth process and assume transient conduction during the bubble waiting time as proposed by Han and Grifﬁth (1965) and Mikic and Rohsenow (1969). Most recently, Gerardi et al. (2010), based on high-speed video of the boiling hydrodynamics and infrared thermometry of the heated wall, pointed out transient conduction following bubble departure as the dominant contribution to nucleate-boiling heat transfer. • Microlayer evaporation: Cooper (1969) and Cooper and Lloyd (1969) based on their experimental results have proposed a model considering that a thin layer of liquid (the microlayer) is trapped between the liquid–vapor interface and the solid wall as the bubble grows, as schematically depicted in Fig. 5.8b. This liquid ﬁlm

172

5 Flow Boiling

Condensation (evaporation) into bulk of ﬂuid

Bubble Microlayer evaporation Evaporation from thermal boundary layer

Hot surface

Vapor Li qu id

Tra ns T ﬂowverse i

Micro region Adsorbed ﬁlm

Hot surface

Warmer liquid

Cold liquid

Tw,mic δ

Triple point

Local heat ﬂux

Detail¹

Detail²

Microconvection

Hot surface

Hot surface

Fig. 5.8 Schematics of heat transfer enhancement mechanisms for nucleate boiling. (a) liquid replacement, (b) microlayer evaporation, adapted from Cooper and Lloyd (1969), (c) evaporation at the triple contact line, adapted from Stephan and Hammer (1994), (d) microconvection due to Marangoni effect

is denominated as a microlayer that rapidly evaporates with subsequent heating implying on high heat transfer rates. As the liquid ﬁlm evaporates, the bubble grows and eventually detaches, promoting rewetting of the surface. Near the dry patch, where the microlayer is thinnest, the heat transfer rate is maximum. In the models based on this mechanism, the higher heat transfer rates are observed during the bubble growth period. Judd and Hwang (1976) proposed a model considering the following mechanisms during the bubble life span: (i) transient conduction during the bubble waiting time, as proposed by Mikic and Rohsenow (1969); (ii) microlayer evaporation during the bubble growing time; and (iii) natural convection during the bubble life span in the area not inﬂuenced by the bubble. According to their model, microlayer evaporation is responsible for approximately one-third of the total heat transfer. This mechanism seems to prevail for hemispherical bubbles of non-metallic ﬂuids boiling under conditions of low reduced pressures and microgravity. For bubbles growing in a subcooled

5.1 Nucleate Boiling Concepts

173

liquid, a signiﬁcant amount of condensation over the curved surface of the bubble, as shown in Fig. 5.8b, may happen, leading to a slower bubble growth. • Evaporation at the triple contact line: In this mechanism, as proposed by Stephan and Hammer (1994), the main mode of heat transfer is the evaporation close to the triple (or three-phase) contact line (schematically depicted in Fig. 5.8c). According to their model, in this region, there is a liquid ﬁlm adsorbed between the wall and the bubble, which consists only of a few molecular layers and cannot be evaporated due to adhesion pressure, also known as disjoining pressure. This adhesion force is inversely proportional to the fourth power of the distance from the wall and causes the liquid to spread out over the wall. In the thicker part of this region, capillary forces due to an increase of interfacial curvature generates a driving pressure gradient that drives liquid against the triple contact line direction. Assuming one-dimensional conduction through the liquid ﬁlm due to the difference between the wall and the saturation temperatures, the maximum heat transfer occurs when the liquid ﬁlm is sufﬁciently thick, such that the disjoining forces become small, and thin enough so that its thermal resistance is low. Stephan and Hammer (1994) suggested heat ﬂuxes in the region close to the triple contact line 100 times larger than the CHF. Although thermocapillary convection, also known as Marangoni Convection, was not proved to occur for saturated pool boiling because most of bubble interface is at saturation temperature, this mechanism can be pointed as relevant for subcooled boiling mainly under microgravity conditions. Surface tension gradients are induced in the bubble interface due to temperature gradients along its curvature (whereas usually the higher the temperature the lower the surface tension), therefore, at the region of the bubble surface away from the heating wall, the temperature is inferior than the temperature close to the hot surface, hence the surface tension increases from the triple contact line to the bubble top, such as schematically depicted in Fig. 5.8d. Therefore, the tangential stress caused by the gradients in surface tension are balanced by viscous stresses associated with liquid motion since shear force exerts by vapor phase can be neglected due to the low vapor viscosity. Therefore, the Marangoni Convection promotes the circulation of warmer liquid from the wall to the colder region of the pool. Jets of warmer liquid looking as mushroom-like clouds originating from the top of the bubble are also pointed in the literature as promoted by Marangoni effects (Hetsroni et al. 2015). It is important to highlight that chemical composition variations along the bubble interface can also generate surface tension gradients and, consequently, Marangoni Convection. The region between points 3 and 4 depicted in Fig. 5.6 corresponds to the condition desired in most of engineering applications, characterized by high heat transfer coefﬁcient, with vapor formation under stable and safe condition. The increment of heat ﬂux will eventually lead the system to the CHF, deﬁned by point 5 in Fig. 5.6, which corresponds to a condition that the vapor generation in the surface is high enough to inhibit the rewetting of the surface, hence the heat transfer process to the liquid is deteriorated and occurs partially from the surface to a vapor layer adjoined to the surface, and the vaporization occurs at the liquid–vapor

174

5 Flow Boiling

interface. Additionally, the heat transfer by radiation starts to have a signiﬁcant contribution to the total heat transfer process due to subtle increment of the surface temperature. Assuming that the heat ﬂux is the controlled parameter, subsequent increment of heat ﬂux beyond the CHF condition implies on the formation of a vapor cushion adjoined to the surface, and the heat transfer coefﬁcient presents a sharp reduction with consequent sharp increment of the surface temperature, as indicated by the gap from the point 5 to 60 in Fig. 5.6a. It must be mentioned that this condition is usually undesired for practical applications because it can result in failure of the heat transfer surface by excessive temperature, referred in literature as burnout. Starting the reduction of heat ﬂux from point 7, the curve would follow the same path from 7 to 60 , and then to 6, which corresponds to the minimum heat ﬂux that is able to maintain stable ﬁlm boiling. Further reduction of the heat ﬂux implies on the surface rewetting and the establishment of nucleate boiling with the boiling curve jumping to point 300 . From point 300 to 3 the expected curve is similar to the one for the condition of increasing ϕ, but since the surface cavities have already been activated, the wall temperature does not change abruptly to point 2 that corresponds to a sudden reduction of the heat transfer coefﬁcient, but keeps a relatively high heat transfer coefﬁcient and curve reach point 10 , where the bubble nucleation is not sustained anymore and the main heat transfer mechanism is natural convection. Subsequent heating processes of the surface do require lower superheating, since parcel of cavities are already active with bubble nuclei. Conversely, under conditions that the experiment is performed through the control of the surface temperature, the heat transfer behavior for single-phase natural convection is similar to that previously discussed for controlled heat ﬂux. When the ONB condition is achieved, the heat transfer coefﬁcient h presents a sharp increment, and since the surface temperature is the controlled parameter, there is a sharp increment of the heat ﬂux (recall Eq. (5.10)), and the operational condition jumps from point 2 to 30 of Fig. 5.6, corresponding to a higher heat ﬂux for the same wall superheating. Subsequent increments of the surface temperature provide the same behavior of the relationship between heat ﬂux and ΔT as observed for conditions of controlled heat ﬂux until the condition of CHF. For wall superheating higher than the one corresponding to the CHF condition, the portion of the surface that is occupied by vapor increases and the heat transfer coefﬁcient sharply decreases. Hence, successive increments of surface temperature result in a reduction of heat ﬂux values, depicted by the line region 5–6 in Fig. 5.6, and commonly referred in literature as transition boiling (Collier and Thome 1994), which is characterized by a combination of conditions of nucleate boiling, and regions partially occupied by vapor cushions. Continuing the increment of surface temperature, it will eventually become dry corresponding to point 6 in Fig. 5.6a, and the heat transfer from the surface to the liquid will be performed partially by convection through the vapor cushion formed around the heater, and a signiﬁcant parcel will be transferred by thermal radiation. Hence, subsequent increments of the surface temperature will increase the heat ﬂux due to the increment of radiative parcel, even though the convective parcel itself has been compromised, up to point 7.

5.1 Nucleate Boiling Concepts

175

Analogous to the case of controlled heat ﬂux, by reducing ΔT from point 7 to point 1, the curve would present the same shape in the regions 7–6 and 6–5 as for increasing ΔT. However, the path of the curves of increasing and decreasing ΔT may differ in the region 60 –5 corresponding to the transition from ﬁlm to nucleate boiling due to the wetting characteristics of the liquid on the solid surface. After achieving the condition of CHF, point 5 in Figure 5.6, subsequent reductions of the wall temperature result in a condition of stable nucleate boiling condition. In a similar way to the condition of controlled heat ﬂux, the active cavities for nucleation keep the vapor generation, and there is no sudden reduction of heat transfer coefﬁcient from point 30 to 2, but the heat transfer process occurs according to curve 5-4-3. Additionally, and similar to the condition of controlled heat ﬂux, when reaching point 3 the cavities should still be active, and the heat transfer process should follow steps 3-10 -1. Again, when operating a system with a surface cooled by convective and/or nucleate boiling, it is desirable to achieve the conditions in the region of stable bubble formation, represented by the region 3–4 in Fig. 5.6, which provides very high heat transfer coefﬁcients and relatively low difference between the surface and ﬂuid temperatures. The achievement of the CHF is usually catastrophic for conditions of imposed heat ﬂux, due to sudden increment of the surface temperature, and leads to system failure due to burnout, ﬂuid leakage, and other related problems. This text does not aim to provide a broad discussion concerning nucleate boiling mechanisms, but it is intended to discuss the general aspects without further details and to provide the fundamental aspects needed for introducing the heat transfer mechanisms concerning convective boiling and how they are affected by the operational conditions. The reader is encouraged to check dedicated publications concerning this subject for a deep knowledge on nucleate boiling. Several predictive methods for nucleate boiling in stagnant ﬂuids (named in literature as pool boiling) are available in the open literature, such as Cooper (1984), Gorenﬂo and Kenning (2010), and Ribatski and Saiz-Jabardo (2003), for the region 3–4 of Fig. 5.6. Most of the predictive methods for prediction of the heat transfer coefﬁcient under pool boiling conditions accounts for heat ﬂux and ﬂuid properties, including thermal conductivity, molar mass, and reduced pressure. It has been veriﬁed experimentally that the heat transfer coefﬁcient for pool boiling is approximately proportional to the heat ﬂux to approximately 0.7, as follows: h / ϕ0:7

ð5:11Þ

Additionally, some of the methods such as Rohsenow (1952) takes into account the surface roughness and surface-ﬂuid characteristics, such as contact angle βc and wettability, through an empirical parameter. However, it is important to highlight the difﬁculties associated with the quantitative characterization of the effects related to the interactions between surface and ﬂuid and the possibility of surface contamination and boiling aging effects. Hence, contact angle values available in the open literature might not be reliable for design purposes due to oxidation of the heating surface and fouling effects.

176

5 Flow Boiling

The predictive method proposed by Cooper (1984) is one of the most employed to evaluate the heat transfer coefﬁcient during pool boiling due to its simplicity and reasonable predictions. This method was developed based on an extensive analysis of inﬂuencing parameters during pool boiling, including ﬂuids and surface properties, considering a database gathered from several independent laboratories containing more than 5800 data points, comprising experiments with water, synthetic refrigerants, and cryogens, among other ﬂuids, on surfaces made of copper and stainless steel, among other materials. In his analysis, Cooper evaluated the effects of several ﬂuid properties on the heat transfer coefﬁcient for pool boiling, and concluded that the most important parameters deﬁning the heat transfer coefﬁcient are the heat ﬂux, reduced pressure, molar mass, and surface roughness. The resulting dimensional correlation proposed by Cooper (1984) is given as follows: h ¼ h0

ϕ ϕ0

0:67 pr

0:120:2log10

Rp Rp0

ðlog10 pr Þ0:55

M M0

0:5

ð5:12Þ

where the heat ﬂux ϕ is given in W/m (Bergles and Rohsenow 1964), the surface peak roughness Rp in μm, the molar mass M in kg/kmol, and the resulting heat transfer coefﬁcient is given in W/m2K. The reference values are h0 ¼ 55 W/m2K, ϕ0 ¼ 1 W/m (Bergles & Rohsenow, 1964), Rp0 ¼ 1 μm, and M0 ¼ 1 kg/kmol. Similarly, Gorenﬂo and Kenning (2010) presented a correlation for prediction of heat transfer coefﬁcient under pool boiling conditions given as a function of the ﬂuid and surface parameters, as well as heat ﬂux. The correlation is given as follows1: h ¼ 3580

h

i W F F F F 2 m K ϕ pr w f

ð5:13Þ

where the terms Fϕ, Fpr Fw, and Ff stand, respectively, for the effects of heat ﬂux, reduced pressure, surface, and ﬂuid property effects. These parameters are given as follows: Fϕ ¼

ϕ 20, 000ϕ0

0:950:3p0:3 r

1:4pr 1 pr 152 ðkρcp Þw 0:25 Ra Fw ¼ 0:4Rp0 ðkρcp ÞCu F pr ¼ 0:7p0:2 r þ 4pr þ

1

Check the original reference for water as working ﬂuid.

ð5:14Þ ð5:15Þ ð5:16Þ

5.1 Nucleate Boiling Concepts

177

" Ff ¼

dp ðdT Þsat

#0:6 ð5:17Þ

σ 1½μmK 1

where pr is the reduced pressure (p/pcrit), ϕ is the heat ﬂux in W/m2, Ra is the surface average roughness in μm, and (kρcp)0.5 is the thermal effusivity of the material, whereas w stands for the wall material and Cu stands for copper material, (dp/dT)sat is the derivative of pressure with temperature for saturation condition, σ is the surface tension in N/μm, and ϕ0 and Rp0 stand for unitary heat ﬂux in W/m2 and roughness in μm, respectively. Ribatski and Saiz-Jabardo (2013) also proposed a predictive method for heat transfer coefﬁcient during pool boiling of synthetic refrigerants, including R11, R123, R12, R134a, and R22, for surfaces made of copper, brass, and stainless steel. The resulting predictive method consists in a correlation given as follows: h ¼ Fw

ϕ ϕ0

0:90:3p0:2 r

0:8 p0:45 r ðlogpr Þ

Ra Rp0

0:2

M M0

0:5

ð5:18Þ

where the term Fw depends on the surface material and is equal to 100, 110, and 85 W/m2K, respectively, for copper, brass, and stainless steel. The heat ﬂux is given in W/m2, molar mass M in kg/kmol, and surface averaged roughness Ra in μm, and the resulting heat transfer coefﬁcient in W/m2K. The terms ϕ0, Rp0, and M0 stand for unitary heat ﬂux in W/m2, roughness in μm, and molar mass in kg/kmol, respectively. Finally, it worth describing the heat transfer coefﬁcient predictive method for pool boiling conditions proposed by Stephan and Abdelsalam (1980), who also gathered more than 5000 data points from independent laboratories. These authors presented correlations for each speciﬁc group of ﬂuids. Their correlation for water is given as follows (recommended for 104 pr 0.886, βc 45 ): !1:58 !1:26 0:673 5:22 ^ilv d 2b cpl T sat d2b ρl ρv h db ϕ db 6 ¼ 2:46 10 kl kl T sat ρl ðk l = ρl cpl Þ2 ðk l = ρl cpl Þ2 ð5:19Þ For hydrocarbons, Stephan and Abdelsalam (1980) presented the following relationship (5103 pr 0.9, βc 35 ): !0:248 0:670 0:335 4:33 ^ilv d2b ρv ρl ρv h db ϕ db ¼ 0:0546 ð5:20Þ kl k l T sat ρl ρl ðkl = ρl cpl Þ2 For cryogenic ﬂuids, the correlation is given as follows (4103 pr 0.97, βc 1 ):

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5 Flow Boiling

!0,374 !0:329 0:624 0:257 bilv d2b ρs cps k s 0:117 cpl T sat d 2b ρv hd b ϕdb ¼ 4:82 2 2 kl k l T sat ρl ρl cpl k l k l =ρl cpl k l =ρl cpl

ð5:21Þ where subscript s attains for the surface properties. And for halocarbon refrigerants, the correlation is given as follows (3103 pr 0.78, βc 35 ): 0:745 0:581 ρv h db ϕ db ¼ 207 Pr0:533 l kl kl T sat ρl

ð5:22Þ

where the characteristic length db corresponds to the equilibrium break-off diameter of a bubble detaching from a heated surface, given as follows: d b ¼ 0:146 βc

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2σ gðρl ρv Þ

ð5:23Þ

where βc corresponds to the contact angle, and in the absence of reliable data, it can be adopted 45 for water, 35 for refrigerants and hydrocarbons, and 1 for cryogenic ﬂuids. It should be noticed that the heat transfer coefﬁcient predictive methods proposed by Cooper (1984) (Eq. (5.12)), Gorenﬂo and Kenning (2010) (Eqs. (5.13) to (5.17)), Ribatski and Saiz-Jabardo (2013) (Eq. (5.18)), and Stephan and Abdelsalam (1980) (Eqs. (5.19) to (5.23)) present several similarities among them, namely, the exponent of the heat ﬂux is close to 0.7 for all methods, and the heat transfer coefﬁcient is proportional to a function of the reduced pressure, and in most of them is correlated as a function of the surface roughness and molar mass. Figure 5.9 depicts the variation of heat transfer coefﬁcient with reduced pressure and heat ﬂux for pool boiling of R134a on a copper surface. According to this ﬁgure, the predictive method proposed by Gorenﬂo and Kenning (2010) presents steeper 5 a ) 10

b ) 2000

10

4

1000

h [W/m²K]

h [W/m²K]

Cooper (1984) Gorenflo and Kenning (2010) Ribatski and Saiz-Jabardo (2013) Stephan and Abdelsalam (1980) R134a, copper, f = 50,000 W/m², Ra = 0.4 µm, Rp = 1.0 µm

Cooper (1984) Gorenflo and Kenning (2010) Ribatski and Saiz-Jabardo (2013) Stephan and Abdelsalam (1980)

500 R134a, copper, pr = 0.08, Ra = 0.4 µ m, Rp = 1.0 µm

3

10 0.01

0.1

p r [-]

0.5

200 1000

2000

5000

10000

2 f [W/m ]

Fig. 5.9 Variation of heat transfer coefﬁcient for pool boiling with (a) reduced pressure pr and (b) heat ﬂux

5.2 Heat Transfer Coefﬁcient for Convective Boiling

179

variation of heat transfer coefﬁcient with reduced pressure than the other methods, while all of them present similar trends with variation of heat ﬂux for pr of 0.08.

5.2

Heat Transfer Coefﬁcient for Convective Boiling

Firstly, it is important to highlight that a predictive method should not be only statistically accurate, but should also capture the trends observed experimentally. Figure 5.10 illustrates the expected behavior of the heat transfer coefﬁcient with vapor qualities under conditions of low heat ﬂuxes and mass velocities. For subcooled liquid, the heat transfer coefﬁcient depends basically on the channel geometry and Reynolds and Prandtl numbers, which in turn are functions of the ﬂuid properties and present reduced variation with ﬂuid enthalpy, which is mainly related with variations of liquid viscosity and thermal conductivity. It has been pointed out in this chapter that the ONB for stagnant ﬂuid is characterized by a sudden increment of the heat transfer coefﬁcient, and a similar behavior is observed under conditions of forced convection. For vapor qualities just higher than the one corresponding to the ONB, the heat transfer coefﬁcient presents a sudden increment, as depicted in Fig. 5.10, which is related to phase change due to nucleate boiling. In this context, Chen (1966) pointed out that the heat transfer during convective boiling is given as the superposition of macroconvection, associated to the ﬂuid ﬂow, and microconvection related to bubble nucleation, growth, and detachment. Based on this, he proposed the heat transfer coefﬁcient given as the sum of both parcels, with proper weighting factors.

rve

ted cu

expec

M

you t

Onset of Nucleate Boiling (ONB)

Dr

h [W/m²K]

less ore or

hONB Linear interp olation

hl Liquid single-phase f low

hv Vapor single-phase f low 0.0

0.2

0.4

Fig. 5.10 Expected heat transfer coefﬁcient curve

x [-]

0.6

0.8

1.0

180

5 Flow Boiling

Under pool boiling conditions, characterized by nucleate boiling in stagnant liquid, a temperature gradient is formed along the ﬂuid, whereas the liquid is hotter close to the heater surface and cooler away from it, and the ﬂuid movement is mainly associated with buoyancy effects associated with the bubbles’ growth and their detachment. Therefore, as discussed before, the higher liquid temperature close to the heater surface favors bubble formation due to liquid superheating. Conversely, in the case of forced convection, the temperature proﬁle is dependent of the velocity proﬁle and the temperature gradient close to the hot surface is affected by the boundary layer characteristics, what implies on higher temperature gradients close to the wall compared to nucleate boiling under quiescent conditions. Therefore, the achievement of conditions for bubble nucleation and growth is mitigated by the ﬂow, and, therefore, the contribution of nucleate boiling effects to the overall heat transfer under convective boiling conditions is reduced when compared to pool boiling. Such an effect is taken into account in predictive methods based on the superposition of effects for the heat transfer coefﬁcient under ﬂow boiling conditions through a nucleate boiling suppression factor, Snb, that usually, in these methods, multiplies a heat transfer coefﬁcient given by a pool boiling correlation. Therefore, the contribution of nucleate boiling to the heat transfer coefﬁcient for conditions close to the ONB during forced convection tends to be lower than during pool boiling. In this context, the suppression factor of nucleate boiling effects should tend to unite when the ﬂow velocity tends to zero (G ! 0 ) Snb ! 1). Therefore, it is possible to conclude that, different from the two-phase pressure drop curve, the heat transfer coefﬁcient for convective boiling with x tending to 0.0 (x ! 0+) does not necessarily tend to the heat transfer coefﬁcient of liquid singlephase ﬂow. Moreover, since subcooled liquid boiling is possible depending on heat ﬂux, ﬂuid properties, and surface characteristics, the condition of heat transfer coefﬁcient of single-phase liquid ﬂow is unattainable even for slightly negative thermodynamic vapor qualities (Eq. (2.11)). It must be emphasized that the occurrence of subcooled nucleate boiling is common, and is characterized by the formation of vapor bubbles with ﬂuid averaged enthalpy lower than that of the saturated liquid (î < îl), which is a consequence of the temperature proﬁle with hotter ﬂuid close to the wall and cooler ﬂuid in the core region of the channel; therefore, the point of ONB occurs for negative vapor quality values (x < 0.0), such as represented in Fig. 5.10. Hence, bubbles are formed and grow in the channel surface, and the drag force facilitates their detachment. Just after detachment, the bubbles tend to collapse under conditions of subcooled boiling because the liquid away from the solid surface is colder. With successive increment of ﬂuid enthalpy along the channel, the liquid temperature away from the surface eventually will be high enough to allow the migration of the bubble without collapsing, and the simultaneous ﬂow of vapor and liquid is possible. In this context, Tibiriçá and Ribatski (2014) presented an experimental investigation based on ﬂow visualizations of bubble formation and departure during R134a and R245fa boiling inside microchannels, inferring the size of the bubbles and departure frequency. Based on their results, as well as based on the literature review addressed by them, it was concluded that the bubble diameter reduces with the increment of ﬂow velocity,

5.2 Heat Transfer Coefﬁcient for Convective Boiling

181

and Tibiriçá and Ribatski (2014) proposed a correlation for bubble departure diameter during convective boiling in microchannels, given as follows: db 130 ¼ Rel0 d

ð5:24Þ

where d is the channel diameter, and Rel0 corresponds to the Reynolds number estimated for the mixture ﬂowing as liquid. Tibiriçá and Ribatski (2014) also analyzed the bubble growth and detachment frequency, and in a similar way to Mikic and Rohsenow (1969), and is given as follows: f reqb ¼

1:195085 ½m2 =s d2b

ð5:25Þ

where db is given by Eq. (5.24), and the resulting frequency is in Hz. Consequently, after nucleation, successive increment of the ﬂuid enthalpy, implies on the increment of the ﬂow velocity. In this context, the mixture superﬁcial velocity that corresponds to the mean velocity can be recalled: Gð1 xÞ G x 1x x Gx j ¼ jl þ jv ¼ þ ¼G þ ρl ρv ρl ρv ρv

ð5:26Þ

Considering that the liquid density is usually in the order of 103 kg/m3 while the vapor density is usually in the order of 10 to 101 kg/m3, the parcel of Eq. (5.26) related to liquid phase can be neglected, as shown in the last term. Hence, the ﬂow velocity increases along the channel for convective boiling conditions (increasing x) for a ﬁxed mass velocity, and the convective parcel tends to present higher contribution with the increment of vapor quality, denoted by the positive slope of the curve for conditions with enthalpy higher than the ONB in Fig. 5.10. The velocity of vapor phase tends to be higher than that of the liquid phase because the liquid phase tends to adhere to the channel wall due to wettability properties. Additionally, as described in Chaps. 2 and 3, the two-phase ﬂow tends to be annular for intermediary and high vapor content or for conditions dominated by vapor inertial effects. Hence, with successive evaporation of the liquid phase, the wall will eventually become dry and partially exposed to the vapor phase, which is characteristic of smaller heat transfer coefﬁcient. Therefore, the averaged heat transfer coefﬁcient in a given cross-section tends to reduce after the dryout initiation, which is schematically depicted in Fig. 5.10 by the region with negative slope. Additionally, since the transition from two-phase ﬂow to vapor single-phase ﬂow is smooth with successive reduction of the heat transfer coefﬁcient, rather than abrupt variation such as veriﬁed for the ONB, it is expected that the heat transfer coefﬁcient must tend to the values estimated for vapor single-phase ﬂow.

182

5 Flow Boiling

i.a

G1

i.b

i.c

G3

h

i.d

G1,G2,G3

G2 h

G1 G2

h

G1,G2,G3

h G3

x G1>G2>G3

x G1>G2>G3 ii.b

ii.a ϕ2 ϕ3

ϕ2 ϕ3

h

x ϕ1 > ϕ2 > ϕ3

x ϕ1 > ϕ2 > ϕ3

ii.d ϕ1

h

ϕ2

ϕ1

Tsat,1 Tsat,2 Tsat,3 x Tsat,1 >Tsat,2 >Tsat,3

ϕ2

h

ϕ3

ϕ3

x ϕ1 > ϕ2 > ϕ3

x ϕ1 > ϕ2 > ϕ3

iii.b

iii.a h

x G1 >G2 >G3

ii.c ϕ1

ϕ1 h

x G1>G2>G3

iv.a

d1 d2 d 3

Tsat,1 h Tsat,2

h Tsat,3

x Tsat,1 >Tsat,2 >Tsat,3

x d1 G 2

ϕ1, G1 ϕ2, G2 xdi,1

xdi,2 x

Table 5.1 Predictive method for dryout inception (xdi) and completion (xdc) according to Mori et al. (2000) Dryout inception

0:86 xdi,1 ¼ 0:94 1:75 106 ðRev0 BoÞ1:75 ρρv l 0:08 ρv 5 0:96 xdi,2 ¼ 0:58 exp 0:52 2:1 10 Wev0 Fr 0:02 v0 ρl 0 0:16 1 0:05 ρv ;C 0:98Fr 0:05 B v0 Bo ρl C B C B xdi,3 ¼ min B 0:21 C A @ 0:04 0:40 0:09 ρv 0:172Fr v0 Bo Wev0 ρl 8 xdi,2 xdi,3 > < xdi,3 if xdi ¼ xdi,2 if xdi,2 < xdi,3 , and xdi,2 xdi,1 > : xdi,1 if xdi,2 < xdi,3 , and xdi,2 < xdi,1 Dryout completion 0:86 xdc,1 ¼ 1:02 1:75 106 ðRev0 BoÞ1:75 ρρv l 0:08 ρv 5 0:94 xdc,2 ¼ 0:61 exp 0:57 2:65 10 Wev0 Fr 0:02 v0 ρ 0 B xdc,3 ¼ min @ 8 > < xdc,3 xdc ¼ xdc,2 > : xdc,1

1

1:01; 0:16 C A 0:22 0:09 ρv Wev0 0:690Fr 0:02 v0 Bo ρl if if if

xdc,2 xdc,3 xdc,2 < xdc,3 , and xdc,2 xdc,1 xdc,2 < xdc,3 , and xdc,2 < xdc,1

l

186

5 Flow Boiling

The non-dimensional parameters are the Boiling number Bo, Reynolds, Froude, and Weber number for the mixture ﬂowing as vapor, and are given, respectively, as follows: ϕ G ^ilv

ð5:27Þ

Gd μv

ð5:28Þ

G2 g d ρv ðρl ρv Þ

ð5:29Þ

G2 d ρv σ

ð5:30Þ

Bo ¼

Rev0 ¼ Fr v0 ¼

Wev0 ¼

Subsequent to the study of Mori et al. (2000), Wojtan et al. (2005a) adapted the second conditions (xdi,2 and xdc,2), which provided the best agreement their results, according to their experimental results. The correlations proposed by Wojtan et al. (2005a) are given as follows: " xdi ¼ 0:58 exp 0:52

0:37 0:235We0:17 v0 Fr v0

0:25 0:70 # ρv ϕ ϕcrit ρl

" xdc ¼ 0:61 exp 0:57 5:8 10

3

0:15 We0:38 v0 Fr v0

0:09 0:27 # ρv ϕ ϕcrit ρl

ð5:31Þ ð5:32Þ

where the Froude number of the mixture ﬂowing as vapor is given as Frv0 ¼ G2/ (ρv2gd), and ϕcrit is given according to Kutateladze’s (1948) proposal, as follows: 0:25 ^ ϕcrit ¼ 0:131 ρ0:5 v ilv ½g σðρl ρv Þ

ð5:33Þ

An alternative approach is based on the fact that subsequent to the dryout inception, additional increments of the ﬂuid enthalpy (vapor quality) imply on the reduction of the heat transfer coefﬁcient, which is similar to the occurrence of CHF, even though the ﬁrst phenomenon is not as critical for the heat transfer problem as the CHF. Additionally, the precise experimental identiﬁcation of dryout inception is complicated, while the identiﬁcation of CHF is more unambiguous, and several reliable predictive methods for CHF are available in the open literature, such as Zhang et al. (2006) and Kutateladze (1948) above presented, in this last case for pool boiling conditions. Therefore, Kanizawa et al. (2016) proposed the adoption of the CHF predictive method equaling the CHF to the actual heat ﬂux (ϕcrit ¼ ϕ) for conditions with known and/or imposed heat ﬂux, and by an inverse problem it is possible to infer the conditions for dryout inception.

5.2 Heat Transfer Coefﬁcient for Convective Boiling

187

This subject will be discussed with more detail in the next chapter, but for instance consider the predictive method for CHF proposed by Zhang et al. (2006), which is originally given as follows: "

Bocrit

# 0:361 0:295 Lcrit 2:31 ρv ¼ 0:0352 Wel0 þ 0:0119 d ρl " # 0:170 ρ Lcrit 0:311 2:05 v xin d ρl

ð5:34Þ

where L stands for the heat length of the section, and xin corresponds to the inlet vapor quality. The Weber number for the mixture ﬂowing as liquid is given by Wel0 ¼ G2d/ (σ ρl) and the boiling number is a function of the actual heat ﬂux Bocrit ¼ ϕcrit/Gilv. Therefore, by assuming that the CHF is equal to the actual heat ﬂux (ϕcrit ¼ ϕ), the corresponding heated length (Lcrit) is determined, and based on energy balance, it is possible to determine the vapor quality for dryout inception.2 Therefore, the following steps are taken for determination of vapor quality for dryout inception adopting a predictive method for CHF: • Evaluate Bocrit, Wel0, and other properties for the saturation temperature assuming ϕcrit ¼ ϕ. • Evaluate the inlet vapor quality or assume as null (xin ¼ 0.0). • Determine Lcrit/d. • Evaluate the vapor quality for dryout inception based on energy balance along a test section Lcrit long. In this context, Tibiriçá et al. (2015) experimentally investigated CHF during convective boiling in microchannels and found that the CHF occurrence depends on the system stability, whereas ﬂuid instabilities lead to lower CHF values, and consequently lower corresponding vapor quality values. It has been shown that imposing ﬂow restrictions upstream the evaporator results in higher ﬂow stability, and consequently higher CHF and corresponding vapor quality. Additionally, these authors pointed out that the vapor quality corresponding to CHF reduces with increment of mass velocity, therefore, the higher the mass ﬂux, the higher the instability of the system. Even though this study was focused on conditions for CHF, which is distinct from dryout, it provides insight about dryout inception in microchannels. It is worth mentioning that the condition for CHF occurrence in the evaporator depends on the other components of the system, such as pumping system and ﬂow restrictions.

2

For this case, it is impossible to directly determine the Lcrit, hence, an iterative method is required for the solution. In the case of Zhang et al.’s (2006) method, it is possible to isolate the term (Lcrit/ d )0.311, and then iteratively ﬁnd new values for the critical length.

188

5 Flow Boiling

a ) 1.2

R134a, d = 1.5 mm, Tsat = 30 °C, f = 10,000 W/m²

1.0

x [-]

0.8 0.6 0.4 0.2 0.0 10

Dryout inception: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a) Saitoh, Daiguji, Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Dryout completion: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a)

100

G [kg/m²s]

1000

10000

b ) 1.2 R134a, d = 1.5 mm, Tsat = 30 °C, G = 300 kg/m²s 1.0

x [-]

0.8 0.6 0.4 0.2 0.0 100

Dryout inception: Mori etal. (2000) Wojtan, Ursenbacher and Thome (2005a) Saitoh, Daiguji, Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Dryout completion: Mori et al. (2000) Wojtan, Ursenbacher and Thome (2005a)

1000

f [W/m²]

10000

100000

Fig. 5.13 Variation of vapor quality for dryout inception and completion with (a) mass ﬂux and (b) heat ﬂux for R134a in a 1.5 mm channel

Figure 5.13 depicts the variation of the dryout inception and completion with mass and heat ﬂux for R134a at 30 C in a 1.5 mm channel. This ﬁgure also includes the prediction according to the approach presented by Saitoh et al. (2007) that is described in Sect. 5.3.2. According to Fig. 5.13a, it can be observed that in general

5.3 Predictive Methods for Convective Flow Boiling

189

the vapor quality for dryout inception and completion tends to reduce with the increment of mass ﬂux, and a similar relationship is veriﬁed for the effect of heat ﬂux in Fig. 5.13b. Only the predictions for dryout inception xdi according to Mori et al. (2000) predicted an increment of vapor quality for dryout inception with mass velocity, which is not expected because the higher the ﬂow velocity, the higher is the probability of occurrence of mist ﬂow, characterized by continuum vapor ﬂow with entrained liquid droplets. It should also be mentioned that some of the methods predict vapor qualities higher than unity, which is related to thermodynamic non-equilibrium between the liquid and vapor phases, whereas the vapor is superheated and the liquid ﬂows as entrained droplets within the vapor stream.

5.3

Predictive Methods for Convective Flow Boiling

Up to date predictive methods for heat transfer coefﬁcient during convective ﬂow boiling available in the open literature are presented and discussed. Several predictive methods for heat transfer coefﬁcient during convective boiling in microchannels have been recently proposed, and most of them are based on the approach ﬁrstly proposed by Chen (1966), which combines the contributions of nucleate boiling and convective effects. Conversely, other more complicated methods were also proposed, such as the three zones model proposed by Thome et al. (2004) for convective boiling in microchannels. Considering convective boiling in conventional channels, other approaches have also been proposed, such as the phenomenological method proposed by Wojtan et al. (2005a, b), according to which the heat transfer coefﬁcient is estimated according to the corresponding ﬂow pattern, which in turn is predicted assuming the method for ﬂow pattern prediction described in Chap. 3. Additionally, regarding historical aspects, methods based on the approach of twophase multipliers, as usually adopted for the prediction of pressure drop (see item 4.1), were also proposed for the heat transfer coefﬁcient, where a factor given by a function multiplies the corresponding value of the heat transfer coefﬁcient for singlephase ﬂow, such as presented by Bergles and Rohsenow (1964) and Gungor and Winterton (1987). However, these methods capture only behaviors similar to the curves presented in Fig. 5.11i.a, corresponding to conditions dominated by convective effects. It must be remembered that it is expected that the heat transfer coefﬁcient trends should be similar to a combination of curves, Fig. 5.11i.a and ii.c, hence with more pronounced contribution of the nucleate boiling mechanisms at least for speciﬁc ranges of operational conditions, characterized by low vapor qualities, high heat ﬂuxes, and high saturation temperatures. Therefore, the approach presented by Chen (1966) is more appropriate to capture the contribution of nucleate boiling mechanisms; nonetheless, the original method is

190

5 Flow Boiling

graphic based, since it requires plots to infer some parameters such as enhancement factor and will not be presented in this book since it is complicated to implement it computationally. Nonetheless, several methods have been subsequently proposed based on the combination of both parcels and some of them are presented in the following items.

5.3.1

Liu and Winterton (1991)

Liu and Winterton (1991) proposed a predictive method for the heat transfer coefﬁcient that proved to be slightly better than the previous Gungor and Winterton (1986, 1987) methods regarding the agreement with experimental results. The method proposed by Liu and Winterton (1991) is based on the Chen (1966) approach, considering the superposition of convective and nucleate boiling effects, and is given as follows: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h ¼ ðec F c hl0 Þ2 þ ðenb Snb hnb Þ2

ð5:35Þ

where Fc and Snb are, respectively, the factor of enhancement of convective effects and the factor of suppression of nucleate boiling effects, hl0 and hnb are the heat transfer coefﬁcients estimated for the convective and nucleate boiling parcels, respectively. The terms ec and enb were included by Liu and Winterton (1991) into their method to account for stratiﬁcation effects for horizontal ﬂow that tends to reduce the perimeter-averaged heat transfer coefﬁcient; then, for vertical ﬂow or for horizontal ﬂow with Froude number higher than 0.05 (Frl0 ¼ G2/(ρl2gd) > 0.05) these parameters are equal to unity, and horizontal ﬂows with Froude number lower than 0.05 (Frl0 0.05) are given as follows: l0 ec ¼ Fr 0:12Fr l0 pﬃﬃﬃﬃﬃﬃﬃﬃ enb ¼ Fr l0

ð5:36Þ ð5:37Þ

The heat transfer coefﬁcient for convective parcel hl0 is given according to the Dittus and Boelter (1985) correlation for turbulent ﬂow assuming the mixture ﬂowing as liquid, as follows: hl0 ¼ 0:023

kl 0:8 0:4 Re Pr d l0 l

ð5:38Þ

with the Reynolds number for the mixture ﬂowing as liquid given by Rel0 ¼ Gd/ μl. The heat transfer coefﬁcient for nucleate boiling parcel is given according to the Cooper (1984) predictive method given by Eq. (5.12) assuming the peak roughness equal to 1μm.

5.3 Predictive Methods for Convective Flow Boiling

191

The intensiﬁcation factor of convective effects is given as a function of the vapor quality (the higher x, the higher F), density ratio, and liquid Prandtl number that accounts for the effect of convective heat transfer along the liquid in contact with the wall and is given as follows: 0:35 ρl F c ¼ 1 þ xPr l 1 ρv

ð5:39Þ

The suppression factor of nucleate boiling effects must be smaller than unity, because it is intended to suppress and tends to reduce its value with the increment of convective effects. Therefore, it is given as follows: 0:16 1 Snb ¼ 1 þ 0:055F 0:1 c Rel0

ð5:40Þ

The condition of dryout occurrence is not predicted by this method, therefore, the heat transfer coefﬁcient tends to always increase with vapor quality. This method was compared by the original authors with an extensive database of experimental results for saturated and subcooled convective boiling, comprising results for mass ﬂux between 12 and 8150 kg/m2s, heat ﬂux between 348 and 2.62106 W/m2, and channels with internal diameter ranging from 2.95 to 32 mm. It could be mentioned that the adoption of an asymptotic exponent equal 2 in Eq. (5.35) emphasizes the dominant heat transfer mechanism, whereas the resulting value tends to the highest value of the sum, and the smallest one present reduced contribution, which is even negligible depending on the orders of magnitude and exponents (1002 + 12 100 100 + 1 ¼ 101). For additional details on asymptotic correlations, it is suggested to check the study presented by Churchill (2000).

5.3.2

Saitoh et al. (2007)

The predictive method for heat transfer coefﬁcient during convective boiling proposed by Saitoh et al. (2007), focused on microchannels, is based on the Chen (1966) approach by combining according to an additive manner the nucleate boiling and convective effects under pre-dryout conditions, while another approach is adopted for post-dryout conditions. It is important to clarify that these authors consider that dryout corresponds to the onset of wall drying. This method has been evaluated by several independent laboratories and showed good agreement with experimental results for heat transfer coefﬁcient during convective boiling in small-scale channels. According to the authors, during liquid-vapor ﬂow in microchannels the surface tension effects are predominant over gravitational effects. Hence, they used the Weber number to account for this effect instead of using only the vapor quality, density ratio, and Prandtl number. The dryout occurrence is modeled based on the vanishing process of a liquid ﬁlm during annular ﬂow, which is quoted to be the ﬂow pattern that precedes the dryout.

192

5 Flow Boiling

Then, based on experimental results for the heat transfer coefﬁcient during convective boiling in microchannels, the authors identiﬁed the vapor quality corresponding to the maximum heat transfer coefﬁcient, whereas successive increments of vapor quality after the peak imply on reduction of h (recall Fig. 5.11ii.c). Additionally, they considering that the heat transfer coefﬁcient during annular ﬂow can be given as a function of the liquid ﬁlm thickness δ as follows: h¼

kl δ

ð5:41Þ

which assumes liquid ﬁlm thickness uniform along the channel perimeter, absence of entrainment of liquid within the vapor ﬂow, and solely conduction along the liquid ﬁlm. Therefore, based on experimental results for the dryout inception, the liquid ﬁlm thickness for this condition was evaluated as 15μm independent of the channel diameter, mass ﬂux, and heat ﬂux. Hence, the authors adopted the relationship for void fraction based on the slip ratio, given by Eq. (2.34), where the slip ratio s ¼ uv/ul is given based on the minimum momentum of two-phase ﬂow as follows: u s¼ v¼ ul

0:5 ρl ρv

ð5:42Þ

Based on the geometry of annular ﬂow, and by assuming the uniform liquid ﬁlm thickness, the void fraction is determined as follows: α¼

π ðd2δÞ2 4 πd2 4

2δ 2 1 ¼ 1 ¼ ρv ð1xdi Þ d 1 þ s ρ xdi

ð5:43Þ

l

and solving for the vapor quality: 1 1¼ xdi xdi ¼

1þ

2δ 1 d

2

0:5 ρl ρv 1 ρv ρl

1

1

2δ 2 d

1

0:5 ρl ρv

ð5:44Þ ð5:45Þ

with the liquid ﬁlm thickness δ as constant and equal to 15μm. Then, for conditions with x xdryout, the heat transfer coefﬁcient is given as a combination of convective and nucleate boiling effects, as follows: h ¼ F c hl þ Snb hnb

ð5:46Þ

where Fc and Snb are, respectively, the convective enhancement and nucleate boiling suppression factors, and hl and hnb refer to the heat transfer coefﬁcients estimated for forced convection and pool boiling. These enhancement and suppression factors are given as follows:

5.3 Predictive Methods for Convective Flow Boiling

193

b 1:05 X tt Fc ¼ 1 þ 1 þ We0:4 v Snb ¼

1 þ 0:4

ð5:47Þ

1 F 1:25 c Rel

104

1:4

ð5:48Þ

where the Weber number (Wev ¼ (Gx)2d/σρv) assumes only vapor ﬂow along the channel and accounts for the surface tension effects. The heat transfer coefﬁcient associated to purely convective effects is given according to Dittus and Boelter (1985) for turbulent ﬂow and by a constant Nusselt number for laminar ﬂow as follows:

hl ¼

8 >

: 0:023 k l Re0:8 Pr3 l d l

for

Rel < 1000

for

Rel > 1000

ð5:49Þ

where the liquid Reynolds number is evaluated based on the liquid ﬂow rate (Rel ¼ G(1 x)/μl). Conversely, the heat transfer coefﬁcient for nucleate boiling parcel is estimated according to Stephan and Abdelsalam (1980) correlation for synthetic refrigerants, given by Eqs. (5.22) and (5.23). The Lockhart and Martinelli parameter X^ in Eq. (5.47) is given assuming turbulent regime for both phases given by Eq. (4.55), as follows: 0:5 μ 0:1 1 x 0:9 ρv l b X tt ¼ x ρl μv

ð5:50Þ

Therefore, for pre-dryout conditions the heat transfer coefﬁcient is evaluated based on Eqs. (5.46) to (5.50). For vapor qualities higher than the one corresponding to the dryout, the heat transfer coefﬁcient tends to reduce with additional increments of the vapor quality, until the condition of dryout completion, which was assumed by Saitoh et al. (2007) as happening for vapor quality equal to unity. Nonetheless, instead of just interpolating the heat transfer coefﬁcient between the imminence of dryout until saturated vapor single-phase ﬂow, the authors modelled the fraction of dry perimeter Fd as a function of the normalized vapor quality xnor as follows: F d ¼ x3nor þ x2nor þ xnor 0:03

ð5:51Þ

where the vapor quality is normalized between the dryout inception and completion, assumed as unity (xdc¼1.0), as follows: xnor ¼

x xdi 1 xdi

ð5:52Þ

Then, the resulting heat transfer coefﬁcient for this region is given as follows:

194

5 Flow Boiling

h ¼ ð1 F d Þhpre dryout þ F d hv

ð5:53Þ

where the pre-dryout heat transfer coefﬁcient hpre dryout is estimated according to Eqs. (5.46) to (5.50) for x ¼ xdi and hv is estimated for vapor turbulent ﬂow as follows: hv ¼ 0:023

kv 0:8 13 Re Pr d v0 v

ð5:54Þ

with the Reynolds number estimated for only vapor ﬂow (Rev0 ¼ Gd/μv). The correlations of this method were adjusted based on experimental results of the original authors for R134a saturated boiling in channels with diameters ranging from 0.51 to 10.92 mm.

5.3.3

Kandlikar and Co-workers

Kandlikar and co-workers have proposed several predictive methods for heat transfer coefﬁcient for micro and conventional-sized channels. Considering that channel reduction results in eventual transition from turbulent to laminar ﬂow, Kandlikar and co-workers proposed a distinct predictive method accounting for this aspect. Hence, Kandlikar (1990, 1991) proposed a predictive method for heat transfer coefﬁcient during convective boiling inside macroscale channels, which is given as follows: h ¼ max ðhnbd , hcd Þ

ð5:55Þ

where the nucleate boiling dominated heat transfer hnbd coefﬁcient is estimated assuming an approach similar to two-phase multipliers, however accounting for nucleate boiling and convective parcels, it is given as follows: 0:8 0:7 hnbd ¼ 0:6683Cn0:2 Fr0:3 l0 þ 1058:0Bo F fl ð1 xÞ hl0

ð5:56Þ

where the ﬁrst term inside the square brackets corresponds to the parcel relative to convective effects and the second one is related to nucleate boiling effects. The boiling number is given according to Eq. (5.27), the convection number Cn is given as follows: Cn ¼

1x x

0:8 ρ 0:5 v

ρl

ð5:57Þ

and the Froude number for the mixture ﬂowing as liquid is given as follows: Frl0 ¼

G2 ρ2l gd

ð5:58Þ

5.3 Predictive Methods for Convective Flow Boiling Table 5.2 Fluid-dependent parameter of the Kandlikar model (Kandlikar 1990, 1991)

Fluid Water R11 R12 R13B1 R22 R113 R114 R134a R152a Nitrogen Neon

195 Fﬂ 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 4.70 3.50

The term Fﬂ depends on the type of ﬂuid and is given in Table 5.2. The heat transfer coefﬁcient for the mixture ﬂowing as liquid hl0 is given according to Gnielinski (1976), given as follows: hl0 ¼

ðf l0 =8Þð Re l0 1000ÞPrl kl =d pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2=3 1 þ 12:7 f l0 =8 Pr l 1

ð5:59Þ

or Petukhov (1970), given as follows: hl0 ¼

ðf l0 =8Þ Re l0 Prl kl =d pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2=3 1:07 þ 12:7 f l0 =8 Pr l 1

ð5:60Þ

where the friction factor f is given as follows: f l0 ¼ ð1:82 log 10 Re l0 1:64Þ2

ð5:61Þ

The heat transfer coefﬁcient for a condition dominated by convective effects hcd is given as follows: 0:8 0:7 hcd ¼ 1:136Cn0:9 Fr0:3 l0 þ 667:2Bo F fl ð1 xÞ hl0

ð5:62Þ

According to Kandlikar and Balasubramanian (2004), the heat transfer coefﬁcient given by Eqs. (5.55) to (5.62) is valid for macroscale channels, which were characterized for Rel0 higher than 3000 (Rel0 3000). For minichannels, which were characterized for 1600 < Rel0 < 3000, corresponding to a transitional condition, the heat transfer coefﬁcient is estimated based on linear interpolation based on the Reynolds number for heat transfer coefﬁcient evaluated at Rel0 ¼ 3000, given by Eq. (5.55), and the heat transfer

196

5 Flow Boiling

coefﬁcient estimated for Rel0 ¼ 1600, also given by Eq. (5.55); however, the nucleated boiling and convective dominated heat transfer coefﬁcients are given, respectively, by Eqs. (5.56) and (5.62) assuming Froude number as unitary (Frl0 ¼ 1.0), because the gravitational effects vanish with reduction of channel diameter. For laminar ﬂow (Rel0 < 1600), the heat transfer coefﬁcient for the mixture ﬂowing as liquid in circular channel is given as follows:

hl0 ¼

4:364kl =d

for

ϕ ¼ constant

3:657kl =d

for

T s ¼ constant

ð5:63Þ

Notice that for non-circular channels, the constant Nusselt number must be adjusted, such as described by Shah and London (1978). In the case of very small channels, characterized by Kandlikar and Balasubramanian (2004) as operational conditions corresponding to Rel0 100, the nucleate boiling mechanism is dominant; hence, the heat transfer coefﬁcient is given by Eq. (5.56) with unitary Froude number, and the heat transfer coefﬁcient for liquid single-phase ﬂow is estimated based on the constant Nusselt number. The predictive methods proposed by Kandlikar and co-workers were compared and validated with more than 10,000 independent experimental results, comprising results for ﬂow boiling of refrigerants in circular and non-circular channels with hydraulic diameter as small as 0.19 mm.

5.3.4

Wojtan et al. (2005a, b)

We have seen in Sect. 3.3.1 a ﬂow pattern map proposed by Wojtan et al. (2005a), and in the same study (Wojtan et al. 2005b), a predictive method was proposed for heat transfer coefﬁcient for each ﬂow pattern. Considering that the ﬂow pattern predictive method is valid for channels of conventional size, the heat transfer predictive method is also valid for macrochannels. The proposed method is based on the proposal of Kattan et al. (1998), which accounts for the stratiﬁcation angle γ, such as schematically depicted in Fig. 3.4, as a main geometrical parameter. Based on this parameter, the heat transfer coefﬁcient for stratiﬁed, intermittent, and annular ﬂows are given as weighted averaged values as follows: h¼

γhv þ ðπ γ Þhliq π

ð5:64Þ

where the stratiﬁcation angle is given as a function of the void fraction estimated value, rather than by solving the momentum balance as performed by Taitel and Dukler (1976). Wojtan et al. (2005b) adopted the void fraction predictive method proposed by Rouhani (1969), given by Eqs. (2.46) to (2.48) and based on the drift

5.3 Predictive Methods for Convective Flow Boiling

197

ﬂux model proposed by Zuber and Findlay (1965), and repeated here to facilitate the implementation of the method: ( α2 ¼

ρv x

("

1 #

1 ))1 gdρ2l 4 x 1x 1:18 gσ ðρl ρv Þ 4 þ 1 þ 0:2ð1 xÞ þ ρv ρl G ρ2l G2 ð5:65Þ

The stratiﬁcation angle for the smooth stratiﬁed ﬂow is evaluated based on the area-averaged void fraction as proposed by Biberg (1999) as follows: 1 h i 1 1 3π 3 1 1 2α2 þ α32 ð1 α2 Þ3 þ α ð1 α2 Þ 2 200 2 h i

ð1 2α2 Þ 1 þ 4 α22 þ ð1 α2 Þ2 þ α2

γ ss ¼ π πα2

ð5:66Þ

The heat transfer coefﬁcient for the dry region of the channel wall, represented by hv in Eq. (5.64), is given as follows: 0:8 kv hv ¼ 0:023 Re v Pr 0:4 v d

ð5:67Þ

where the modiﬁed Reynolds number of the vapor ﬂow is given as follows: Re v ¼

Gxd μv α2

ð5:68Þ

The heat transfer coefﬁcient corresponding to the wetted region of the channel wall, given by hliq, is given as an asymptotic combination of convective and nucleate boiling effects as follows: hliq ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 h3cb þ ð0:8hnb Þ3

ð5:69Þ

The heat transfer coefﬁcient for nucleate boiling parcel hnb is estimated according to Cooper (1984) correlation, given by Eq. (5.12), and the heat transfer coefﬁcient for convective boiling parcel hcb is given as a function of the liquid ﬁlm thickness δ as follows: 0:4 hcb ¼ 0:0133 Re 0:69 δ Pr l

kl δ

ð5:70Þ

and the liquid ﬁlm thickness is given as a function of void fraction and stratiﬁcation angle as follows:

198

5 Flow Boiling

δ¼

πd ð1 α2 Þ 4ð π γ Þ

ð5:71Þ

The Reynolds number based on the liquid ﬁlm thickness of Eq. (5.70) is given as follows: Re δ ¼

4Gδð1 xÞ μl ð1 α2 Þ

ð5:72Þ

Considering that the stratiﬁed ﬂow pattern was divided into three ﬂow patterns, namely, smooth stratiﬁed, stratiﬁed wavy, and slug+stratiﬁed ﬂow, and the stratiﬁcation angle given by Eq. (5.66), Wojtan et al. (2005b) proposed corrections of stratiﬁcation angle as a function of the mass velocity. Hence, considering the ﬂow patterns classiﬁed according to the methods described in Sect. 3.3.1, during smooth stratiﬁed ﬂow, the stratiﬁcation angle is given by Eq. (5.66) (γ ¼ γ ss). On the other hand, during intermittent and annular ﬂow patterns the stratiﬁcation angle should be null (γ ¼ 0). During stratiﬁed wavy ﬂow, the stratiﬁcation angle is corrected as follows: γ¼

Gwavy G Gwavy Gss

0:61 γ ss

ð5:73Þ

where Gwavy and Gss correspond to the mass ﬂuxes for transition to stratiﬁed wavy and smooth stratiﬁed, respectively, given by Eqs. (3.126) and (3.125), respectively. and G is the current mass ﬂux. Similarly, γ is also corrected for the condition of slug+stratiﬁed wavy ﬂow, however, also accounting for the effect of vapor quality as follows: γ¼

0:61 x Gwavy G γ ss xia Gwavy Gss

ð5:74Þ

where xia corresponds to the vapor quality for transition between intermittent and annular ﬂow, which is assumed as a constant as proposed by Taitel and Dukler (1976) and is given by Eq. (3.129). Therefore, by estimating the stratiﬁcation angle γ, it is possible to estimate the heat transfer coefﬁcient by Eq. (5.64) for smooth stratiﬁed, stratiﬁed wavy, slug +stratiﬁed wavy and slug ﬂow. For mist ﬂow, the authors adjusted the correlation proposed by Groeneveld (1973), resulting in the following relationship: ( h¼

1:06 0:0117 Re 0:79 h Pr v

0:4 )1:83 ρl kv 1 ð 1 xÞ ρv d

1 0:1

ð5:75Þ

5.3 Predictive Methods for Convective Flow Boiling

199

where the term inside the brackets corresponds to a correction for the Reynolds number for homogeneous ﬂow proposed by Groeneveld (1973), which is given as follows: Re h ¼

ρv Gd xþ μv ρ l ð 1 xÞ

ð5:76Þ

In the condition of dryout ﬂow, limited by the transitions given by Eqs. (3.130) to (3.132), the heat transfer coefﬁcient reduces sharply with the increment of vapor quality. In this condition, Wojtan et al. (2005b) proposed to interpolate the heat transfer coefﬁcient conditions of dryout inception and completion as follows: h¼

hðxdc Þ hðxdi Þ ðx xdi Þ þ hðxdi Þ xdc xdi

ð5:77Þ

where xdi and xdc correspond to the vapor quality for dryout inception and completion, respectively, proposed by Mori et al. (2000) and adjusted by Wojtan et al. (2005a), given by Eqs. (5.31) and (5.32), respectively. The implementation of this method, compared to the correlation based on the superposition of effects, requires a more complex programming and the resulting predictions are not signiﬁcantly more accurate than the prediction provided by simpler methods when compared with experimental results, such as the one proposed by Saitoh et al. (2007). Nonetheless, the method cares with the involved phenomena and, therefore, is suitable because it tries to capture different dominant heat transfer mechanisms. Moreover, the method considers the conditions of dryout and mist ﬂow patterns, which were addressed by few investigations.

5.3.5

Thome and Co-workers

Thome et al. (2004) were pioneers to propose a phenomenological model that takes into account the two-phase ﬂow topology to predict the heat transfer coefﬁcient during convective boiling in microchannels. This model is referred in literature as the 3-zone model, since it is based on the assumption that two-phase ﬂow inside a microchannel can be represented by weighing the contributions of liquid slug, elongated bubble (or annular), and a dry region, as depicted schematically in Fig. 5.14. Recall that during two-phase ﬂows inside microchannels, stratiﬁed-like ﬂow patterns are not expected. The model was developed for conditions of uniform heat ﬂux and assumes vapor and liquid ﬂow at the same velocities (no-slip), no superheating; hence, the liquid and vapor are kept at Tsat and the energy supplied to the ﬂuid promotes its vaporization. Moreover, it is also assumed that the ONB occurs for null vapor

5 Flow Boiling

Dry region

δmax

Elongated bubble

δmin

Flow direction

Ldry

Lﬁlm

Liquid slug

d

200

Ll

Fig. 5.14 Schematics of the three-zone model for heat transfer during convective boiling inside microchannels. (Adapted from Thome et al. 2004)

quality (x ¼ 0.0). It must be mentioned that this model aims to capture the evolution of the ﬂow along the channel, rather than just estimating the heat transfer coefﬁcient at a given position. Hence, considering the point that the ﬂuid is under saturated liquid condition, corresponding to the assumed ONB, bubbles start to form and grow until the conditions that drag and buoyance forces overcome the surface tension forces. The authors adopted the bubble growth and detachment period as proposed by Plesset and Zwick (1954), given as follows: Δt b ¼

2 πρl cpl ρvbilv d 48k l ρl cpl ðT w T sat Þ

ð5:78Þ

where cpl is the speciﬁc heat for constant pressure of the saturated liquid, and (Tw Tsat) corresponds to the wall superheating. Notice that the term inside the brackets corresponds to the inverse of the Jakob number multiplied by the density ratio. Therefore, during the bubble growing period Δtb, the liquid velocity is kept constant, and based on this assumption, it is possible to estimate the initial liquid slug length Ll, shown in Figure 5.14, assuming that the mass transfer to the bubble is negligible (ρl ρv), as follows: Ll ¼

G Δt ρl b

ð5:79Þ

Similarly, assuming that the bubble will detach when it blocks the channel crosssection, and that the liquid ﬁlm thickness is much smaller than the channel diameter (δ d ), it is possible to estimate the initial bubble length as follows: 2 Lfilm ¼ d 3

ð5:80Þ

In the initial condition, the dry region would be null (Ldry ¼ 0), and the vapor quality just after the bubble detachment (t ¼ 0 + Δtb) is estimated based on the mass of each phase as follows:

5.3 Predictive Methods for Convective Flow Boiling

x¼

201

mv 1 1 ¼ ¼ 3GΔt b mv þ ml 1 þ Ll ρl 1 þ 2ρv d Lfilm ρ

ð5:81Þ

v

Between the phases, no-slip has been assumed, and, consequently, the homogeneous model for void fraction was adopted. Considering the boundary condition of uniform heat ﬂux and that all the energy supplied to the ﬂuid is used for its vaporization, the vapor quality downstream the ONB can be estimated based on the energy balance. Hence, recognizing that the frequency of bubble passage does vary along the bubble length, the total length of the group of liquid slug, ﬁlm region, and dry region can be evaluated as follows:

L ¼ Ll þ Lfilm þ Ldry

x 1x ¼ Δt b G þ ρv ρl

ð5:82Þ

where the vapor quality depends on the axial position. Based on this parameter, it is possible to estimate the overall vapor length based on the void fraction as follows: Lv ¼ Lfilm þ Ldry ¼ Lαh ¼

Δt b Gx ρv

ð5:83Þ

and, therefore, the corresponding period is estimated as follows: L Δt b Δt v ¼ v ¼ ρv 1x x 1x 1 þ G ρ þ ρ ρl x v

ð5:84Þ

l

Similarly, the period of the liquid slug passage can be evaluated as follows: Lð1 αh Þ L Δt ¼ ¼ ρl x b Δt l ¼ l x 1x x 1x ρv 1x þ 1 G ρ þ ρ G ρ þ ρ v

l

v

ð5:85Þ

l

The period of time corresponding to vapor ﬂow, given by Eq. (5.84), comprises the period of ﬁlm ﬂow and dry wall, and to infer both parcels it is necessary to access the liquid ﬁlm thickness proﬁle. As shown in Fig. 5.14, the liquid ﬁlm is thickest close to the liquid slug, and thinnest close to the dry region, and for the maximum value, Thome et al. (2004) adjusted a correlation initially proposed by Moriyama and Inoue (1996), which resulted in the following relationship: 0

10:84 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i18 8 μ δmax B l C 0:07Bd 0:41 þ 0:18 ¼ Cδ, max @3 A d ρl dG ρx þ 1x ρ v

l

ð5:86Þ

202

5 Flow Boiling

where Cδ,max corresponds to an empirical parameter obtained based on the authors’ experimental results (Thome et al. 2004) given by a constant value of 0.29, and the modiﬁed Bond number Bd* is given as follows: 2 ρl d x 1x Bd ¼ þ G ρv ρl σ

ð5:87Þ

Considering that the energy supplied to the ﬂuid results only in its vaporization, it is possible to estimate the period of time required for the liquid ﬁlm to vaporize from δmax to δmin. For an inﬁnitesimal channel length dz, the heat transfer rate can be related to the ﬁlm thickness variation and with the imposed heat ﬂux as follows: q_ ¼ ρl 2π

d dδ b δ dz ilv ¼ ϕπd 2 dt

ð5:88Þ

Therefore, solving for the liquid ﬁlm thickness, assuming that initial thickness is δmax, that the minimum thickness is δmin, and that d/2 δ, the following relationship is obtained: δmin ¼ δmax

ϕ Δt δ, min ρlbilv

ð5:89Þ

where the minimum liquid ﬁlm thickness was adjusted by Thome et al. (2004) based on their experimental database as equal to 0.3106 m. Solving for the period of time for the liquid ﬁlm to reduce from δmax to δmin, the following relationship is obtained: Δt δ, min ¼ ðδmax δmin Þ

ρlbilv ϕ

ð5:90Þ

Hence, if the required time to vaporize the liquid ﬁlm until the minimum value is lower than the total time of the vapor bubble passage in a given position (Δtδ,min Δtv), there should be no dry region in the bubble tale (Ldry ¼ 0). Otherwise, if Δtδ,min > Δtv there should be a dry region, which length is given as follows: x 1x þ ðΔt v Δt δ, min Þ Ldry ¼ G ρv ρl

ð5:91Þ

Therefore, up to this point the corresponding geometrical characteristics of the set of liquid slug, annular-like ﬁlm ﬂow, and vapor bubble have been addressed, and based on this characterization it is possible to estimate the time-averaged heat transfer coefﬁcient. The reader should notice that the objective of this model is inferring the geometrical parameters for a given axial location of the channel, for which it is possible to estimate the vapor quality based on energy balance and all the other parameters.

5.3 Predictive Methods for Convective Flow Boiling

203

The heat transfer coefﬁcient for liquid slug and dry region are evaluated based on predictive methods for single-phase ﬂow, assuming ﬂow developing conditions. Thome et al. (2004) adopted the method proposed by Churchill and Usagi (1972) that combines the heat transfer coefﬁcient for laminar and turbulent regimes under ﬂow-developing conditions, which is given as follows: hs ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 h4s,lam þ h4s,turb

ð5:92Þ

where the averaged heat transfer coefﬁcient for the laminar parcel is given by the Shah and London (1978) correlation as follows: hs,lam

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Res ks ¼ 2 0:455 Pr Ls =d d 1 3

ð5:93Þ

and the averaged heat transfer coefﬁcient for the turbulent regime is given based on the proposal of Gnielinski (1976) for developing ﬂow as follows: hs,turb

" 2=3 # ðf s =8ÞðRes 1000ÞPrs d ks pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2=3 ¼ 1þ L d s 1 þ 12:7 f s =8 Prs 1

ð5:94Þ

where the friction factor f is given by Eq. (5.61), and the subscript s corresponds to each single-phase ﬂow, either liquid slug l or vapor bubble ﬂow dry. The Reynolds number is evaluated based on the classic deﬁnition (Res ¼ G d/μs). The heat transfer coefﬁcient during ﬁlm liquid ﬂow is estimated assuming stagnant liquid ﬁlm, hence corresponding to a local thermal resistance of conduction. Considering that the thickness reduces from δmax to δmin, Thome et al. (2004) proposed to adopt the averaged thickness value, and the resulting heat transfer coefﬁcient is given as follows: hfilm ¼

2kl δmax þ δmin

ð5:95Þ

The resulting heat transfer coefﬁcient for convective boiling is given as average heat transfer coefﬁcient, weighted according to the periods of time cooresponding to the different heat transfer mechanisms as follows: h¼

hl Δt l þ hfilm Δt film þ hdry Δt dry Δt b

ð5:96Þ

where Δtdry corresponds to max(0, Δtv – Δtδ,min) and Δtﬁlm corresponds to min (Δtmin, Δtv).

204

5 Flow Boiling

As the reader might have noticed, the evaluation of the wall temperature in Eq. (5.78) depends on the heat transfer coefﬁcient, which in turn depends on Δtb, hence an iterative method is needed for estimation of h. It is suggested to use a guess value for wall temperature, and after evaluating the heat transfer coefﬁcient, estimate the new wall temperature value as Tw,new ¼ Tsat + ϕ/h, and then update the corresponding Tw as Tw ¼ ((factor-1)Tw + Tw,new)/factor, where factor > 1, to reduce the number of required iterations, otherwise the wall temperature and heat transfer coefﬁcient might vary signiﬁcantly. It should be highlighted that annular ﬂow is dominant in microchannels, and, therefore, this model should be restricted to a narrow range of vapor quality. However, the authors neglected the ﬂow pattern transitions when adjusting the empirical constants of their model by using a database involving vapor qualities from 0 to 1. Later, Cioncolini and Thome (2011) proposed a method to predict the heat transfer coefﬁcient for annular ﬂows. Their method is based on an algebraic model of turbulence applied to the liquid ﬁlm. Most recently, Costa-Patry et al. (2012) combined the 3-zone model proposed by Thome et al. (2004), which is valid for elongated bubbles, and the model of Cioncolini and Thome (2011), which is valid for annular ﬂow, into a single model for microscale ﬂow boiling, where the ﬂow pattern transition is predicted according to the method proposed by Ong and Thome (2011). Their new method provided reasonable predictions of the independent results of Costa-Patry et al. (2012) for ﬂow boiling in a multichannel conﬁguration. They assumed the minimum ﬁlm thickness in the 3-zone model equal to the surface roughness. Schweizer et al. (2010) obtained simultaneous images of the ﬂow for a single microchannel of an elongated bubble and the associated temperature distribution on the heated surface. Flow images were obtained in the upper region of the channel with a high-speed camera and the temperature ﬁeld in the lower region using infrared thermography. From the temperature ﬁeld along the thin metal surface used as heater, local heat ﬂuxes were estimated. On contrary to the 3-zone model that predicts high heat ﬂuxes at the terminal region of an elongated bubble, Schweizer et al. (2010) observed two regions with high heat transfer rates corresponding to the frontal and end regions of the elongated bubble. The authors suggest that the observed behavior is related to intense evaporation in the triple contact line established in the frontal and terminal regions of the elongated bubble. Such a hypothesis would imply on a dry region between the triple contact lines associated with an elongated bubble. This fact was not corroborated by the results provided by the authors because according to them the heat ﬂux in the regions comprised by the bubble and in the presence of the liquid piston are similar. It seems more feasible that the high heat transfer coefﬁcient close to the bubble front is promoted by the displacement of liquid from the central region of the channel toward the heating surface. This hypothesis is corroborated by the three-dimensional numerical simulations performed by Liu et al. (2012), according to which lower wall temperatures are observed just after the bubble passage due to micro-convection effects resulting from its displacement. They also reported regions of lower wall temperatures at the intermediary region of the bubble, just downstream its front nose.

5.3 Predictive Methods for Convective Flow Boiling

5.3.6

205

Ribatski and Co-workers (Kanizawa et al. 2016; Sempertegui-Tapia and Ribatski 2017)

Kanizawa et al. (2016) presented an extensive analysis of a database comprising experimental results for convective boiling of R134a, R245fa, and R600a in microchannels with internal diameters from 0.38 to 2.6 mm. Based on this analysis, they concluded that the method of Saitoh et al. (2007) provided the best predictions of their experimental results. Nonetheless, this method did not adequately capture most of the identiﬁed trends identiﬁed displayed by the experimental results. Therefore, these authors proposed an adjustment of the method of Saitoh and co-workers according to their database. For conditions prior to dryout occurrence, the heat transfer coefﬁcient is evaluated according to Kutateladze (1961)–Chen (1966) approach, consisting in combining the convective and nucleate boiling contributions with corresponding enhancement and suppression factors. In the method of Kanizawa et al. (2016), the heat transfer coefﬁcient is given as follows: h ¼ F c hl þ Snb hnb

ð5:97Þ

where the heat transfer coefﬁcient associated to convective effects hl is evaluated by assuming only liquid ﬂow according to Eq. (5.49), as per the procedure adopted by Saitoh et al. (2007), and the heat transfer coefﬁcient for nucleate boiling effects is evaluated according to Stephan and Abdelsalam (1980) that for synthetic refrigerants is given by Eqs. (5.22) and (5.23). Considering that the database used in this study includes hydrocarbons, the predictive method for nucleate boiling parcel for these ﬂuids is given by Eq. (5.20) and (5.23). The enhancement factor of convective parcel is given as follows: F c ¼ 1 þ 2:50

b 1:32 X lv 1 þ We0:24 uv

ð5:98Þ

where the Weber number is based on the vapor in situ velocity, given as follows: Weuv ¼

ρv u2v d σ

ð5:99Þ

and the in situ velocity of the vapor phase uv is given as function of the void fraction α, which in turn is evaluated based on the Kanizawa and Ribatski (2016) method, given by Eqs. (2.56) and (2.58). The Lockhart and Martinelli (1949) parameter X^lv is evaluated assuming turbulent ﬂow for liquid phase, as recommended by Da Riva et al. (2012), as follows:

206

b tv ¼ X

5 Flow Boiling

8 > >

1 x 0:9 ρv l > b tl ¼ 1 Re0:4 :X 18:7 v x ρl μv

for

Rev > 1000

for

Rev 1000

ð5:100Þ

with the vapor Reynolds number equal to Rev ¼ G(1 x)d/μv. The suppression factor of nucleate boiling effects is given as follows: 3

Snb ¼ 1:06

Bd 810 0:86 Rel0 F 1:25 c 1 þ 0:12 10000

ð5:101Þ

where the Bond number Bd is introduced in order to capture bubble conﬁnement effects and is deﬁned as follows: Bd ¼

ðρl ρv Þd2 g σ

ð5:102Þ

The dryout inception is estimated based on the method of Zhang et al. (2006) for CHF, from which, based on an energy balance, it is possible to obtain the vapor quality for dryout inception xdi. In a similar way to Saitoh et al. (2007), it is assumed that the dryout completion occurs for vapor quality equal to unity, and for x ¼ 1 the heat transfer coefﬁcient should correspond to the saturated vapor single-phase ﬂow, which is given by the Dittus and Boelter (1985) correlation as follows: hv0 ¼ 0:023

kv 0:8 13 Re Pr d v0 v

ð5:103Þ

where the Reynolds number is evaluated assuming the mixture ﬂowing as vapor (Rev0 ¼ Gd/μv). Therefore, for the dryout condition (xdi x 1) the heat transfer coefﬁcient is given as a linear interpolation between the imminence of dryout inception and completion as follows: h ¼ hpre ðxdi Þ

1x x xdi þ hv0 1 xdi 1 xdi

ð5:104Þ

Subsequently, Sempertégui-Tapia and Ribatski (2017) complemented the database with results for R1234ze(E) and R1234yf for channels with 1.1 mm of internal diameter. The evaluation of dryout inception and completion are like the approach presented by Kanizawa et al. (2016). Additionally, the Chen (1966) approach is also adopted for evaluation of the heat transfer coefﬁcient before dryout, however it adopted an asymptotic exponent equal to 2 as adopted by Liu and Winterton (1991). Therefore, the heat transfer coefﬁcient in their method is given as follows:

5.3 Predictive Methods for Convective Flow Boiling

h¼

207

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðF c hl Þ2 þ ðSnb hnb Þ2

ð5:105Þ

where the heat transfer coefﬁcients for convective and nucleate boiling parcels are given by the same methods adopted by Kanizawa et al. (2017), and the enhancement and suppression factors for convective and nucleate boiling parcels are given as follows: F c ¼ 1 þ 2:55 Snb ¼ 1:427

b 1:04 X tv 1 þ We0:194 uv

ð5:106Þ

Bd 0:032 4 0:981 1 þ 0:1086 Rel F 1:25 c 10

ð5:107Þ

Figure 5.15 depicts the variation of heat transfer coefﬁcient with vapor quality according to the predictive methods presented in this chapter for small diameter channels, except for the methods developed by Wojtan et al. (2005a, b) and Liu and Winterton (1991) which were mainly developed for macroscale channels. It can be observed from this ﬁgure that, as expected, the heat transfer coefﬁcient tends to increase with x for reduced and intermediate vapor qualities, and after dryout inception tends to reduce, except for the Liu and Winterton (1991) method that does not predict dryout occurrence. Additionally, it can be observed in this ﬁgure that the predictive method of Saitoh et al. (2007) presents a discontinuity in the variation of h with x for vapor quality lower than the one corresponding to the dryout inception. This behavior is associated with the transition from laminar to turbulent ﬂow (and vice versa) of the phases. With the increment of vapor quality, the liquid ﬂow rate reduces with consequent reduction of inertial effects of liquid phase; conversely, the vapor ﬂow rate increases with vapor quality with consequent increment of inertial effects. It can also be observed according to this ﬁgure that for reduced and intermediate vapor qualities, the methods predict similar trends, indicating a certain degree of

a) 12000

h [W/m²K]

8000

b) 25000 Liu and Winterton (1991) Saitoh, Daiguji and Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Sempertegui-Tápia and Ribatski (2017) Kandlikar and Balasubramanian (2004) Thome, Dupont and Jacobi (2004a,b) (2004a,b)

6000 4000 2000 0 0.0

Liu and Winterton (1991) Saitoh, Daiguji and Hihara (2007) Kanizawa, Tibiriçá and Ribatski (2016) Sempertegui-Tápia and Ribatski (2017) Kandlikar and Balasubramanian (2004) Thome, Dupont and Jacobi (2004a,b) (2004a,b)

20000

h [W/m²K]

10000

15000

10000

5000 R134a, d = 1.5 mm, G = 300 kg/m²s, φ = 10 kW/m², T sat = 30 °C

0.2

0.4

x [-]

0.6

0.8

1.0

0 0.0

R134a, d = 1.5 mm, G = 500 kg/m²s, φ = 20 kW/m², T sat = 30 °C

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 5.15 Variation of the heat transfer coefﬁcient with vapor quality for R134a convective boiling in 1.5 mm channel

208

5 Flow Boiling

coherence among them, which in turn indicates that the databases used in the adjustment of their empirical constants displays similar heat transfer trends. However, with the increment of the ﬂuid enthalpy (or vapor quality), the deviations among the predictions can be as high as 100%, indicating that previous to the adoption of a predictive method, the reader must check the reference data used for the development of the method, otherwise the errors during the design stage of heat exchangers can lead to over or under sizing. It must be mentioned that the signiﬁcant deviation among the distinct predictive method emphasizes the need for additional experimental and theoretical studies with the objective of providing higher agreement between predicted and actual heat transfer coefﬁcient values.

5.3.7

Heat Transfer Coefﬁcient Under Transient Heating

Thermal management applications such as the cooling of electronic components and photovoltaic panels deals with transient heating and hot spots. In this context, Aguiar and Ribatski (2019) were pioneers to develop a heat transfer prediction method for ﬂow boiling under conditions of transient heat ﬂuxes and hot spots. In their method, the temperature difference between the channel wall and the ﬂuid ΔTw is estimated along time as a function of the variable heat ﬂux. They assumed that the instantaneous wall superheating and heat ﬂux are given as the sum of their mean values ΔT w and ϕ , and the respective ﬂuctuations as ΔT 0 w(t) and ϕ0 (t) as follows: ΔT w ðt Þ ¼ ΔT w þ ΔT 0w ðt Þ

ð5:108Þ

ϕðt Þ ¼ ϕ þ ϕ0 ðt Þ

ð5:109Þ

The averaged heat ﬂux and temperature difference are related through Newton’s Cooling Law, given by Eq. (5.10), by adopting a proper predictive method for heat transfer coefﬁcient developed for steady-state conditions such those described in this chapter, in Sects. 5.3.1 to 5.3.5. Hence, for a given ϕ it is possible to estimate the corresponding ΔT w , and the steps depicted in Fig. 5.16 are performed for determination of the time varying ΔTw(t). The second step of the procedure described in Fig. 5.16 corresponds to the estimation of the effect of variation of heat ﬂux on the wall superheating and, according to the original authors, can be estimated by a central ﬁnite difference scheme, as follows: 0 0 ∂ΔT w ΔT w T sat , G, d, x, ϕ þ Δϕ ΔT w T sat , G, d, x, ϕ Δϕ 0 ¼ 2Δϕ0 ∂ϕ

ð5:110Þ

5.3 Predictive Methods for Convective Flow Boiling

209

Fig. 5.16 Summary of the methodology for estimation of wall superheating during transient heat ﬂux conditions. (Adapted from Aguiar and Ribatski 2019)

Start

Evaluate ΔTw based on ϕ, Tsat, G, d and x

Estimate ∂ΔTw/∂ϕ' at ϕ

Based on ϕ'(t), estimate ΔTw': ΔTw'(t)=TF ϕ'(t)

Compose the wall superheating ﬂuctuation with the averaged value: ΔTw(t)= ΔTw+ΔTw'(t)

End

where the wall superheatings are estimated based on a predictive method proposed for steady state conditions, and Δϕ0 consists of an inﬁnitesimal increment of the heat ﬂux. The transfer function TF of the third step presented in the diagram of Fig. 5.16, which gives ΔT 0w ðt Þ as a function of ϕ0 (t), is a ﬁrst-order lag, given as follows: TF ¼

∂ΔTw =∂ϕ bs þ 1 Τb

ð5:111Þ

^ where ŝ is the Laplace variable, and the time constant T can be determined experimentally, or by assuming a lumped capacitance model for the channel wall, as follows: b ¼ ðρcÞw Acs Τ P h

ð5:112Þ

where (ρc)w is the product of density and speciﬁc heat of the wall material, h corresponds to the mean heat transfer coefﬁcient estimated for ϕ, Acs to the crosssection of the channel wall, and P to the wetted perimeter. It is important to highlight that knowing the transient wall superheating ΔTw(t) is extremely important to evaluate the possibility of transient wall temperature

210

5 Flow Boiling

incursion that could damage the cooled device. Based on the time varying wall superheating ΔTw(t), and by knowing the heat ﬂux ϕ(t), it is possible to estimate the instantaneous heat transfer coefﬁcient h(t) as follows: hð t Þ ¼

ϕð t Þ ΔT w ðt Þ

ð5:113Þ

Aguiar and Ribatski (2019) validated their method based on experimental results for transient heat transfer coefﬁcient during convective boiling of R134a inside round channels with internal diameter between 0.5 and 1.1 mm. The test sections used in their study were 90 mm long channels with a 10 mm central region subjected to variable heat ﬂux with sinusoidal, square, and sawtooth waveforms, with frequencies ranging between 0.5 and 2.0 Hz, mean values between 80 and 120 kW/m2, and half amplitude of 20 and 40 kW/m2. In addition to the predictive methods presented in this chapter, several other methods are being developed and presented in the open literature, and the reader is encouraged to keep up to date with the recent publications. It must be emphasized that most of the predictive methods for heat transfer coefﬁcient during convective boiling are developed based on the adjustment of empirical constants based on experimental databases, which are ﬁnite and, generally, comprise a limited number of ﬂuids, range of channel diameter, heat ﬂux, etc. Hence, previous to the adoption of a predictive method it is advisable to check its original publication and conﬁrm whether the dataset used for its development comprises the conditions that will be simulated, since deviations as high as 100% can be veriﬁed among methods even for similar operational conditions.

5.4

Solved Examples

Consider R134a ﬂowing at a temperature of 20 C and mass ﬂux of 200 kg/m2s, for uniform heat ﬂux of 5000 W/m2. Evaluate the local heat transfer coefﬁcient for: (a) Subcooled liquid and saturated vapor single-phase ﬂows inside a channel of 10 mm of internal diameter. (b) Subcooled liquid and saturated vapor single-phase ﬂows inside a channel of 1 mm of internal diameter. (c) Convective boiling inside a channel with 10 mm of internal diameter for vapor quality of 0.1. (d) Convective boiling inside a channel with 1 mm of internal diameter for vapor quality of 0.1.

5.4 Solved Examples

211

Solution: Based on the deﬁned Tsat ¼ 20 C, and considering that liquid properties are mainly a function of the temperature, it is possible to determine the following properties: ρl ¼ 1225 kg/m3 ρv ¼ 27.8 kg/m3 μl ¼ 2.068104 kg/ms μv ¼ 1.175105 kg/ms σ ¼ 8.688103 N/m kl ¼ 8.562102 W/mK kv ¼ 1.406102 W/mK Prl ¼ 3.393 Prv ¼ 0.836 In any of the above items, it is interesting to evaluate the Reynolds number for liquid and vapor single-phase ﬂows as follows: Re l0 ¼

Gd μl

Re v0 ¼

Gd μv

where for liquid it is 967 and 9673, and for vapor it is 17,026 and 170,258 for channel diameters of 1 and 10 mm, respectively. Hence, the regime is laminar only for liquid ﬂowing in 1 mm channel, and turbulent, or in transitional regime, for other conditions. (a) As discussed before, the ﬂow regime for both phases ﬂowing in 10 mm ID tube is turbulent. The Dittus and Boelter (1985) correlation is valid for Reynolds numbers higher than approximately 20,000. Therefore, let us adopt the Gnielinski (1976) predictive method, which is valid for Reynolds number in the range of 2500 to 510 (Churchill, 2000), and is given as follows: Nud ¼

ð f =8ÞðRed 1000ÞPr hd pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ k 1 þ 12:7 f =8 Pr2=3 1

where the friction factor is estimated as follows: f ¼ ð1:82 log 10 Red 1:64Þ2 Hence, the heat transfer coefﬁcient for subcooled liquid and saturated vapor ﬂowing in a 10 mm ID tube is 498.1 and 428.0 W/m2K, respectively.

212

5 Flow Boiling

(b) In the case of single-phase ﬂow in channel with 1 mm of internal diameter, the procedure of an item can be repeated for the vapor phase, which results in heat transfer coefﬁcient of 699.5 W/m2K. Considering that the subcooled liquid single-phase ﬂow corresponds to laminar regime, and the boundary condition for the heat transfer problem corresponds to uniform heat ﬂux, the Nusselt number is constant and equal to 4.364; hence: Nud ¼

hd ¼ 4:364 k

resulting in 373.6 W/m2K. Notice that, even considering the difference in ﬂow regime, the heat transfer coefﬁcient for laminar ﬂow is approximately 25% lower than the condition for turbulent ﬂow only because of the channel reduction. (c) In the case of convective boiling, it is necessary to check which predictive method is more appropriate for these conditions. The predictive method proposed by Saitoh et al. (2007) was validated for refrigerant ﬂow inside channels with a diameter between 0.5 and 10.9 mm, hence it is within both diameters evaluated in this exercise. Therefore, by implementing Eqs. (5.41) to (5.54), recalling that the nucleate boiling parcel is estimated according to Stephan and Abdelsalam (1980) correlations (Eq. (5.22)), the following parameters are obtained: • Heat transfer coefﬁcient for convective parcel: hl ¼ 419.7 W/m2K • Heat transfer coefﬁcient for nucleate boiling parcel: hnb ¼ 1279 W/m2K • Vapor quality for dryout inception: xdi ¼ 0.9615, hence, the current condition is previous to dryout inception. • The suppression factor of nucleate boiling mechanism is Snb ¼ 0.596, and the enhancement factor of convective effects is Fc ¼ 1.511. • Therefore, the resulting heat transfer coefﬁcient is h ¼ 1396 W/m2K. Notice that the heat transfer coefﬁcient, even for small vapor quality value, is approximately three times higher than the value for single-phase ﬂow. Additionally, notice that the heat transfer coefﬁcient for convective parcel is different from that evaluated for single-phase ﬂow, because it is assumed only for the liquid parcel, corresponding to 90% of the mass ﬂow rate. (d) Repeating the procedure of the previous item, but now assuming internal diameter of 1 mm, the following results are obtained: • Heat transfer coefﬁcient for convective parcel: hl ¼ 665.2 W/m2K • Heat transfer coefﬁcient for nucleate boiling parcel: hnb ¼ 1279 W/m2K • Vapor quality for dryout inception: xdi ¼ 0.7057, hence, the current condition is also previous to dryout inception. • The suppression factor of nucleate boiling mechanism is Snb ¼ 0.977, and the enhancement factor of convective effects is Fc ¼ 1.373. • Therefore, the resulting heat transfer coefﬁcient is h ¼ 2164 W/m2K.

5.5 Problems

213

Notice that the heat transfer coefﬁcient for nucleate boiling parcel is independent of the channel diameter, however, the suppression factor increases with diameter reduction. Conversely, even though the heat transfer coefﬁcient for the convective parcel increases substantially with channel reduction, the enhancement factor presents marginal reduction due to increment of viscous effects. Additionally, notice the effect of channel diameter on the vapor quality for dryout inception, which reduced from 96% for d ¼ 10 mm to 71% for d ¼ 1 mm.

5.5

Problems

A list of problems comprising the concepts of the present and previous chapters is proposed. 1. Derive Eq. (5.41). 2. Discuss the relationship between the variation of heat ﬂux and wall superheating for pool boiling conditions. Start with Eq. (5.11) to obtain a relationship between ΔT and heat ﬂux. 3. Consider annular ﬂow of R134a inside a channel with diameter d, with liquid ﬁlm thickness δ, at given vapor quality x, mass ﬂux G, and imposed heat ﬂux from the channel surface. Assuming laminar regime for both phases, and properties of saturated liquid and vapor, evaluate the heat transfer coefﬁcient. It is suggested to derive the velocity proﬁles of both phases by imposing non-slip in the interface as well as force balance. Compare the result for liquid single-phase ﬂow for a similar mass ﬂow rate. 4. Assume R600a two-phase ﬂow in a horizontal channel of 10 mm ID at mass ﬂux of 50 kg/m2s, heat ﬂux of 5000 W/m2, and saturation temperature of 0 C. In a given position, the ﬂow is stratiﬁed with void fraction of 50% (vapor quality of approximately 1.94%). Adopting an approach similar to that of Lockhart and Martinelli (1949) to estimate the heat transfer coefﬁcient (evaluate the heat transfer coefﬁcient for each phase assuming single-phase ﬂow in a smaller channel, with the corresponding heat transfer coefﬁcient), estimate the mean heat transfer coefﬁcient in this cross-section. Assume that the perimeter for liquid phase comprises only the contact with channel wall, while for the vapor phase include the contribution of the interface. Compare with the predictions according to Liu and Winterton’s (1991) and Saitoh et al.’s (2007) predictive methods. 5. Now, consider R134a ﬂowing in a 5 mm ID tube at saturation temperature of 5 C, heat ﬂux of 10,000 W/m (Bergles & Rohsenow, 1964), and mass ﬂux of 500 kg/m2s. Assuming annular ﬂow with void fraction of 95% (vapor quality of approximately 60.2%), and uniform liquid ﬁlm thickness, estimate the corresponding heat transfer coefﬁcient based on the thermal resistance imposed by the liquid ﬁlm, as given by Eq. (5.41). Compare with the predictions according to Liu and Winterton (1991) and Saitoh et al. (2007).

214

5 Flow Boiling

References Aguiar, G. M., & Ribatski, G. (2019). An experimental study on ﬂow boiling in microchannels under heating pulses and a methodology for predicting the wall temperature ﬂuctuations. Applied Thermal Engineering, 159, 113851. Bergles, A. E., & Rohsenow, W. M. (1964). The determination of forced-convection surfaceboiling heat transfer. Biberg, D. (1999). An explicit approximation for the wetted angle in two-Phase stratiﬁed pipe ﬂow. The Canadian Journal of Chemical Engineering, 77(6), 1221–1224. Blander, M., & Katz, J. L. (1975). Bubble nucleation in liquids. AIChE Journal, 21(5), 833–848. Chen, J. C. (1966). Correlation for boiling heat transfer to saturated ﬂuids in convective ﬂow. Industrial & engineering chemistry process design and development, 5(3), 322–329. Churchill, S. W. (2000). The art of correlation. Industrial & Engineering Chemistry Research, 39 (6), 1850–1877. Churchill, S. W., & Usagi, R. (1972). A general expression for the correlation of rates of transfer and other phenomena. AIChE Journal, 18(6), 1121–1128. Cioncolini, A., & Thome, J. R. (2011). Algebraic turbulence modeling in adiabatic and evaporating annular two-phase ﬂow. International Journal of Heat and Fluid Flow, 32(4), 805–817. Collier, J. G., & Thome, J. R. (1994). Convective boiling and condensation. Clarendon Press. Cooper, M. G. (1969). The microlayer and bubble growth in nucleate pool boiling. International Journal of Heat and Mass Transfer, 12(8), 915–933. Cooper, M. G. (1984). Heat ﬂow rates in saturated nucleate pool boiling-a wide-ranging examination using reduced properties. Advances in heat transfer, 16, 157–239. Cooper, M. G., & Lloyd, A. J. P. (1969). The microlayer in nucleate pool boiling. International Journal of Heat and Mass Transfer, 12(8), 895–913. Costa-Patry, E., Olivier, J., & Thome, J. (2012). Heat transfer charcacteristics in a copper microevaporator and ﬂow pattern-based prediction method for ﬂow boiling in microchannels. Frontiers in Heat and Mass Transfer (FHMT), 3(1). Da Riva, E., Del Col, D., Garimella, S. V., & Cavallini, A. (2012). The importance of turbulence during condensation in a horizontal circular minichannel. International Journal of Heat and Mass Transfer, 55(13–14), 3470–3481. Dittus, F. W., & Boelter, L. M. K. (1985). Heat transfer in automobile radiators of the tubular type. International Communications in Heat and Mass Transfer, 12(1), 3–22. Forster, H. K., & Zuber, N. (1955). Dynamics of vapor bubbles and boiling heat transfer. AIChE Journal, 1(4), 531–535. Gerardi, C., Buongiorno, J., Hu, L. W., & McKrell, T. (2010). Study of bubble growth in water pool boiling through synchronized, infrared thermometry and high-speed video. International Journal of Heat and Mass Transfer, 53(19–20), 4185–4192. Gnielinski, V. (1976). New equations for heat and mass transfer in turbulent pipe and channel ﬂow. Int. Chem. Eng., 16(2), 359–368. Gorenﬂo, D., & Kenning, D. B. R. (2010). Pool boiling (Chapter H2). Berlin/Heidelberg: VDI Heat Atlas. Springer-Verlag. Groeneveld, D. C. (1973). Post-dryout heat transfer at reactor operating conditions (No. AECL-4513). Atomic Energy of Canada Ltd. Gungor, K. E., & Winterton, R. H. S. (1986). A general correlation for ﬂow boiling in tubes and annuli. International Journal of Heat and Mass Transfer, 29(3), 351–358. Gungor, K. E., & Winterton, R. H. S. (1987). Simpliﬁed general correlation for saturated ﬂow boiling and comparisons of correlations with data. Chemical Engineering Research and Design, 65(2), 148–156. Han, C. H., & Grifﬁth, P. (1965). The mechanism of heat transfer in nucleate pool boiling—part II: the heat ﬂux-temperature difference relation. International Journal of Heat and Mass Transfer, 8(6), 905–914.

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Hetsroni, G., Mosyak, A., & Pogrebnyak, E. (2015). Effect of Marangoni ﬂow on subcooled pool boiling on micro-scale and macro-scale heaters in water and surfactant solutions. International Journal of Heat and Mass Transfer, 89, 425–432. Jabardo, J. M., Silva, E., Ribatski, G., & de Barros, S. F. (2004). Evaluation of the Rohsenow correlation through experimental pool boiling of halocarbon refrigerants on cylindrical surfaces. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 26(2), 218–230. Judd, R. L., & Hwang, K. S. (1976, November). A comprehensive model for nucleate pool boiling heat transfer including microlayer evaporation. Journal of Heat Transfer, 98(4), 623–629. Kandlikar, S. G. (1990). A general correlation for saturated two-phase ﬂow boiling heat transfer inside horizontal and vertical tubes. ASME Journal of Heat Transfer, 112, 219–228. Kandlikar, S. G. (1991). A model for predicting the two-phase ﬂow boiling heat transfer coefﬁcient in augmented tube and compact heat exchanger geometries. ASME Journal of Heat Transfer, 113, 966–972. Kandlikar, S. G., & Balasubramanian, P. (2004). An extension of the ﬂow boiling correlation to transition, laminar, and deep laminar ﬂows in minichannels and microchannels. Heat Transfer Engineering, 25(3), 86–93. Kanizawa, F. T., & Ribatski, G. (2016). Void fraction predictive method based on the minimum kinetic energy. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(1), 209–225. Kanizawa, F. T., Tibiriçá, C. B., & Ribatski, G. (2016). Heat transfer during convective boiling inside microchannels. International Journal of Heat and Mass Transfer, 93, 566–583. https:// doi.org/10.1016/j.ijheatmasstransfer.2015.09.083 Kattan, N., Thome, J. R., & Favrat, D. (1998). Flow boiling in horizontal tubes: Part 1—Development of a diabatic two-phase ﬂow pattern map. Journal of Heat Transfer, 120(1), 140–147. Kim, J. (2009). Review of nucleate pool boiling bubble heat transfer mechanisms. International Journal of Multiphase Flow, 35(12), 1067–1076. Kutateladze, S. S. (1961). Boiling heat transfer. International Journal of Heat and Mass Transfer, 4, 31–45. Kutateladze, S. S. (1948). On the transition to ﬁlm boiling under natural convection. Kotloturbostroenie, 3, 10–12. Lienhard IV, J. H., & Lienhard V, J. H. (2020). A heat transfer textbook. 5th Edition. Phlogiston press. Liu, Q., Palm, B., & Anglart, H. (2012, July). Simulation on the ﬂow and heat transfer characteristics of conﬁned bubbles in micro-channels. In International conference on nanochannels, microchannels, and minichannels (Vol. 44793, pp. 63-70). American Society of Mechanical Engineers. Liu, Z., & Winterton, R. H. S. (1991). A general correlation for saturated and subcooled ﬂow boiling in tubes and annuli, based on a nucleate pool boiling equation. International Journal of Heat and Mass Transfer, 34(11), 2759–2766. Lockhart, R. W. & Martinelli, R. C. (1949). Proposed correlation of data for isothermal two-phase, two-component ﬂow in pipes. Chemical Engineering Progress, Vol. 45–1, 39–48. Mikic, B. B., & Rohsenow, W. M. (1969, May). A new correlation of pool-boiling data including the effect of heating surface characteristics. Journal of Heat Transfer, 91(2), 245–250. Mori, H., Yoshida, S., Ohishi, K., & Kakimoto, Y. (2000). Dryout quality and post-dryout heat transfer coefﬁcient in horizontal evaporator tubes. In 3rd European thermal sciences conference (Heidelberg, 10–13 September 2000) (pp. 839–844). Moriyama, K., & Inoue, A. (1996). Thickness of the liquid ﬁlm formed by a growing bubble in a narrow gap between two horizontal plates. Journal of Heat Transfer, 118(1), 132–139. Nukiyama, S. (1934). Film boiling water on thin wires. Society of Mechanical Engineering, 37. Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase ﬂow: Part 1– Two-phase ﬂow patterns and ﬁlm thickness measurements. Experimental Thermal and Fluid Science, 35(1), 37–47.

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Petukhov, B. S. (1970). Heat transfer and friction in turbulent pipe ﬂow with variable physical properties. Advances in heat transfer, 6(503), i565. Plesset, M. S. & Zwick, S. A. (1954). The growth of vapor bubbles in superheated liquids. Journal of applied physics, 25(4), 493–500. Ribatski, G. (2002). Theoretical and experimental analysis of pool boiling of halocarbon refrigerants (Doctoral Thesis, University of São Paulo). Ribatski, G. (2013). A critical overview on the recent literature concerning ﬂow boiling and two-phase ﬂows inside micro-scale channels. Experimental Heat Transfer, 26(2-3), 198–246. Ribatski, G., & Saiz-Jabardo, J. M. (2003). Experimental study of nucleate boiling of halocarbon refrigerants on cylindrical surfaces. International Journal of Heat and Mass Transfer, 46(23), 4439–4451. https://doi.org/10.1016/S0017-9310(03)00252-7 Rohsenow, W. M. (1971). Boiling. Annual Review of Fluid Mechanics, 3(1), 211–236. Rosenhow, W. M. (1952). A method of correlating heat transfer data for surfaceboiling of liquids. Transactions of ASME, 4, 969–975. Rouhani, Z. (1969). Modiﬁed correlations for void and two-phase presssure drop. Nyköping: Aktiebolaget Atomenergi. Saitoh, S., Daiguji, H., & Hihara, E. (2007). Correlation for boiling heat transfer of R-134a in horizontal tubes including effect of tube diameter. International Journal of Heat and Mass Transfer, 50(25–26), 5215–5225. https://doi.org/10.1016/j.ijheatmasstransfer.2007.06.019 Schweizer, N., Freystein, M., & Stephan, P. (2010, January). High resolution measurement of wall temperature distribution during forced convective boiling in a single minichannel. In International conference on nanochannels, microchannels, and minichannels (Vol. 54501, pp. 101–108). Sempértegui-Tapia, D. F., & Ribatski, G. (2017). Flow boiling heat transfer of R134a and low GWP refrigerants in a horizontal micro-scale channel. International Journal of Heat and Mass Transfer, 108, 2417–2432. Shah, R. K., & London, A. L. (1978). Laminar ﬂow forced convection in ducts (Advances in Heat Transfer, supplement 1). Stephan, K., & Abdelsalam, M. (1980). Heat-transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer, 23(1), 73–87. Stephan, P., & Hammer, J. (1994). A new model for nucleate boiling heat transfer. Heat and Mass Transfer, 30(2), 119–125. Taitel, Y., & Dukler, A. E. (1976). A model for predicting ﬂow regime transitions in horizontal and near horizontal gas-liquid ﬂow. AIChE Journal, 22(1), 47–55. Thome, J. R., Dupont, V., & Jacobi, A. M. (2004). Heat transfer model for evaporation in microchannels. Part I: Presentation of the model. International Journal of Heat and Mass Transfer, 47(14-16), 3375–3385. Tibirica, C. B., & Ribatski, G. (2014). Flow patterns and bubble departure fundamental characteristics during ﬂow boiling in microscale channels. Experimental Thermal and Fluid Science, 59, 152–165. Tibirica, C. B., Czelusniak, L. E., & Ribatski, G. (2015). Critical heat ﬂux in a 0.38 mm microchannel and actions for suppression of ﬂow boiling instabilities. Experimental Thermal and Fluid Science, 67, 48–56. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005a). Investigation of ﬂow boiling in horizontal tubes: Part I—A new diabatic two-phase ﬂow pattern map. International Journal of Heat and Mass Transfer, 48(14), 2955–2969. Wojtan, L., Ursenbacher, T., & Thome, J. R. (2005b). Investigation of ﬂow boiling in horizontal tubes: Part II—Development of a new heat transfer model for stratiﬁed-wavy, dryout and mist ﬂow regimes. International Journal of Heat and Mass Transfer, 48(14), 2970–2985. Zhang, W., Hibiki, T., Mishima, K., & Mi, Y. (2006). Correlation of critical heat ﬂux for ﬂow boiling of water in mini-channels. International Journal of Heat and Mass Transfer, 49(5–6), 1058–1072. https://doi.org/10.1016/j.ijheatmasstransfer.2005.09.004 Zuber, N., & Findlay, J. (1965). Average volumetric concentration in two-phase ﬂow systems. Journal of Heat Transfer, 87(4), 453–468.

Chapter 6

Critical Heat Flux and Dryout

6.1

Introduction

The critical heat ﬂux (CHF) is an undesired operational condition, resulting from a deﬁcient rewetting of the heating surface accompanied by a sharp increase of the wall superheating as a consequence of the heat transfer coefﬁcient h reduction under a condition of imposed heat ﬂux. In the 1950s, the expansion of the nuclear industry implied on the need of a better understanding of the CHF and an enormous growth in the number of studies concerning this subject, as pointed out by Groeneveld et al. (2007). In the same paper, these authors highlighted the proposal of more than thousand correlations for the prediction of CHF of water ﬂowing inside ducts. Groeneveld et al. (2007) also indicated the complexity of the mechanisms associated to the CHF, concluding that a theory for CHF capable of being generalized to all applications was not available until that time. It can be certainly mentioned that, despite the recent advances, the same status quo remains nowadays and a unique theory is still not available. Figure 5.6 schematically depicts the boiling curve, whereas the CHF condition (point 5) corresponds to an abrupt increment of the wall temperature Tw in case of imposed heat ﬂux, or by a sharp reduction of heat ﬂux ϕ in case of controlled wall temperature. Under conditions of controlled heat ﬂux, such as electronics and nuclear applications, the abrupt increment of surface temperature is usually catastrophic for the heated surface and may cause irreversible damages to the equipment. Therefore, CHF is also referred in literature as burnout and boiling crisis (Hewitt 1981). The term departure from nucleate boiling (DNB) is also common in literature, such as in Collier and Thome (1994). This last nomenclature derives from the process of ceasing of vapor generation on the heated surface by nucleate boiling mechanism (getting away from the nucleate boiling condition of Fig. 5.6). The CHF condition corresponds to the formation of vapor blankets on the heater surface, implying on a much lower heat transfer coefﬁcient compared to a condition under which the liquid phase is in almost permanent contact with the heated surface. © The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_6

217

218

6

Critical Heat Flux and Dryout

Consequently, the heat transfer coefﬁcient presents a drastic reduction, and energy is transferred by convection through the vapor ﬁlm. In the case of imposed heat ﬂux, subsequent heat ﬂux increments beyond the occurrence of CHF lead to a continuous vapor blanket along a large portion of the surface. Under this condition, ﬁlm boiling is established, and thermal energy is transferred from the wall to the liquid phase by convection and conduction through the vapor ﬁlm, with radiation becoming relevant as the wall temperature increases. The term ﬁlm boiling comes from the fact that vaporization occurs in the interface between the vapor ﬁlm and neighboring liquid. Some studies, such as Yagov et al. (2016), investigate the ﬁlm boiling condition focused on the required surface temperature for its occurrence, which is related to the Leidenfrost phenomenon which corresponds to the condition that the liquid hovers over the heated surface without touching it. Conversely, under conditions of controlled surface temperature, successive increments of temperature beyond the CHF results in increment of surface area occupied by vapor blankets, with consequent reduction of heat transfer coefﬁcient and heat ﬂux. It is speculated that the vapor blanket is unstable and large portions of vapor are released periodically from the ﬁlm, and this condition is denominated as transition boiling, such as detailed by Collier and Thome (1994). Heat transfer predictive methods for ﬁlm boiling and transition boiling are available in the open literature, such as presented by Collier and Thome (1994) and Hewitt (1981). These methods are not described in the present text because the occurrence of CHF is undesirable for the operation of heat dissipation systems due to low heat transfer coefﬁcient, and high probability of damaging the cooling device. Therefore, the above-mentioned literature is recommended to those interested on further details. On the other hand, for several applications it is desirable to predict the conditions of occurrence of CHF, so the engineer can design the equipment to avoid its achievement. In this context, several aspects should be taken into account when dealing with CHF, such as boiling under quiescent or forced convection condition, heater geometry, gravity level, among others (Katto 1994), with speciﬁc complication for each case. Therefore, it is not surprising that most of the CHF predictive methods are developed based either on an adjustment of a simple model or by direct correlation of the relevant non-dimensional parameters based on a broad database. Even considering the phenomenological models, an adjustment of empirical parameters due to the phenomenon complexity is also performed to provide good agreement with experimental results. According to Katto (1994), in general, the predictive models can be categorized as hydrodynamic and microlayer models. Both approaches were originally developed for pool boiling conditions, and subsequently adjusted for convective ﬂow. In this context, it is interesting to discuss the effect of surface wettability, such as introduced in Chap. 5. Hydrophobic surfaces favor bubble nucleation due to the repelling characteristic of the liquid phase, and consequently implies on lower surface superheating for nucleate boiling, and provides higher heat transfer

6.2 Hydrodynamic Model

219

coefﬁcient during nucleate boiling conditions because of the higher number of active cavities. However, the advantage of vapor formation can anticipate the CHF exactly due to excessive vapor formation. On the other hand, hydrophilic surfaces require higher surface superheating for the onset of nucleate boiling and tend to present lower heat transfer coefﬁcient during nucleate boiling conditions in comparison with hydrophobic surfaces, related to higher difﬁculty of vapor formation and tendency to maintain contact with liquid phase. However, after the proper establishment of active nuclei, the CHF tends to be delayed exactly due to tendency of the surface to be wetted. Hence, even though these aspects are not always accounted for in predictive methods, the reader must be aware of the possible impact on the deﬁnition of operational range of heat transfer devices, and surface wettability characteristics. Recently, by assuming that the CHF is a near wall phenomenon associated to ﬂuid–solid interactions on the heating surface instead of a macroscale hydrodynamic instability, Zhang et al. (2019) proposed a Monte Carlo model based on continuum percolation that describes the near-wall stochastic interaction of bubbles. According to the authors, this model is capable of capturing the effect of the surface characteristics on the CHF because it follows the evolution of the bubble footprint area as the heat ﬂux increases with the CHF coinciding with a critical condition in the percolation process. In the foregoing subsections, the most widely accepted models for CHF are addressed.

6.2

Hydrodynamic Model

The hydrodynamic model has been under development since the 1950s and was ﬁrtly introduced by the Kutateladze’s research group, and was subsequently considered and adjusted by other groups, such as Zuber and Tribus (1958) that provided one of the most classical presentations. This approach considers that during nucleate boiling under pool boiling conditions, vapor is generated along the heater surface and ﬂows away from the surface, with simultaneous liquid ﬂow toward the surface. With successive increment of vapor ﬂow rate by the increment of heat ﬂux, eventually the counter current ﬂow will not be sustained, and the liquid refeeding to the surface becomes insufﬁcient to maintain the surface wet. The impossibility of simultaneous ﬂow of vapor away from the surface and liquid toward the surface, such as depicted in Fig. 6.1, is limited by interfacial instabilities, which can cause increment of interfacial waves with consequent partial blocking of the available cross-section for liquid ﬂow. Zuber and Tribus (1958) reproduced pictures obtained during pool boiling experiments, depicting characteristics of vapor and liquid ﬂow during CHF, showing that vapor detaches from the surface as columns with approximately uniform spacing among each of them, such as schematically depicted by Λcrit in Fig. 6.1.

220

6

Critical Heat Flux and Dryout

Λcrit

Liquid

Λcrit

Vapor

Heater

Λcrit

Vapor

Λcrit

Heater

Front view Λcrit

Λcrit

Top view Fig. 6.1 Schematics of the hydrodynamic model for critical heat ﬂux

Additionally, and according to the visual observations and measurements, the authors found out that the spacings among the vapor columns are approximately similar to the critical wavelength proposed by Bellman and Pennington (1954), which characterizes the interfacial perturbation length that causes exponential growth or suppression of waves, and is dominated by surface tension and buoyancy effects. The critical wavelength is given according to the Kelvin equation, which provides two values as follows:

Λcrit,1 Λcrit,2

12 σ ¼ 2π gð ρl ρv Þ 12 3σ ¼ 2π gð ρl ρv Þ

ð6:1Þ ð6:2Þ

Additionally, these authors observed the formation of vapor bubbles/slugs shortly downstream the surface and concluded that the spacing between the vapor columns is independent of the heat ﬂux. Therefore, the increment of heat ﬂux causes increment of the frequency of bubble detachment, keeping the spacings among their departure points ﬁxed. Zuber and Tribus (1958) modelled the condition of CHF assuming that the formed vapor slugs are spherical and present a diameter of Λcrit/2. Therefore, the vapor mass per bubble is given as follows: msphere ¼

1 πΛ3 ρ 48 crit v

ð6:3Þ

6.2 Hydrodynamic Model

221

Additionally, the mass ﬂux of vapor Gv can be given as a function of the heat ﬂux ϕ and enthalpy of vaporization ilv as follows: ϕ ¼ Gvbilv

ð6:4Þ

Therefore, based on these aspects and considering that one bubble is generated per each area of Λcrit Λcrit, and that an oscillating interface presents a double critical frequency, as pointed out by Rayleigh (1896), it is possible to relate the bubble frequency ωb and heat ﬂux by equating mass ﬂuxes Gv according to both approaches as follows: πΛ3crit ρv ϕ 2ωb ¼ bilv 48Λ2crit π ϕ ¼ Λcrit ρv^ilv ωb 24

ð6:5Þ ð6:6Þ

where the critical wavelength Λcrit is given either by Eq. (6.1) or (6.2). According to the discussion presented by Zuber and Tribus (1958), as well as in previous studies, the bubble frequency for CHF is evaluated based on the Kelvin and Helmholtz instability theory as a function of phase properties and critical wavelength as follows: 1 ωb ¼ Λcrit

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ σ 2π ρl þ ρv Λcrit

ð6:7Þ

And combining with Eqs. (6.1) and (6.6), we obtain the following relationship for the CHF: ϕcrit

π ¼ ρvbilv 24

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 gσ ðρl ρv Þ ð ρl þ ρv Þ 2

ð6:8Þ

and by considering that ρl ρv, the above equation can be simpliﬁed to: ϕcrit

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 gσ ðρl ρv Þ ¼ 0:131 ρvbilv ρ2l

ð6:9Þ

which is valid for low-pressure systems. For higher pressure systems, the dominant phenomenon that leads to instabilities of the countercurrent ﬂow are related to Taylor instabilities rather than Kelvin instabilities. According to Zuber and Tribus (1958), the square of propagation velocity of small interfacial disturbances is given as follows:

222

6

Γ2 ¼

Critical Heat Flux and Dryout

ρl ρv 4σ ð j þ j l Þ2 Λðρl þ ρv Þ ðρl þ ρv Þ2 v

ð6:10Þ

where Λ is the perturbation wavelength, and jv and jl are, respectively, the liquid and vapor superﬁcial velocities. It is required that Γ should be real for stability, therefore in the limit the right-hand side (RHS) of Eq. (6.10) should be null. The vapor and liquid velocities can be related by continuity equation, because for steady-state condition, the mass ﬂow rate of vapor leaving the surface should equate the liquid ﬂow rate toward the surface as follows: jv ρv ¼ jl ρl

ð6:11Þ

Therefore, by equating the RHS of Eq. (6.10) to zero, and substituting the liquid superﬁcial velocity according to Eq. (6.11), it is possible to infer the critical velocity of vapor phase as follows: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρl 4σ jv ¼ Λρv ρl þ ρv

ð6:12Þ

Now, returning to the relationship between bubble volume and frequency ωb, and the ﬂow rate by assuming that the region of reference for the vapor ﬂow rate is circular with the radius of Λ /4, it is possible to relate the estimated superﬁcial velocity according to Eq. (6.4), assuming a circular area of inﬂuence with a diameter of Λ/2 as follows: 2 4 Λ 3 Λ ρv π 2ωb ¼ ρv π jv 3 4 4

ð6:13Þ

from which, it is possible to determine a relationship for the bubble frequency as follows: ωb ¼

3 2Λ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃrﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρl 4σ Λρv ρl þ ρv

ð6:14Þ

Again, the mass ﬂux of vapor can be related to the heat ﬂux and enthalpy of vaporization as follows: ϕ 4 Λ 3 2ωb ¼ ρv π 3 4 bilv Λ2 π ϕ ¼ ^ilv ρv Λωb 24

Gv ¼

ð6:15Þ ð6:16Þ

6.2 Hydrodynamic Model

223

π 3^ i ρ ϕ¼ 24 2 lv v

rﬃﬃﬃﬃﬃﬃﬃﬃﬃrﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρl 4σ Λρv ρl þ ρv

ð6:17Þ

where, under critical conditions the wavelength Λ is given either by Eqs. (6.1) and (6.2). Hence, the CHF is given as follows: ﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃrﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρl π 3 ^ pﬃﬃﬃﬃﬃp pﬃﬃﬃﬃﬃ ilv ρv 4 σgðρl ρv Þ 24 2π ρl þ ρv r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρl π 3 ^ pﬃﬃﬃﬃﬃp 4 p ﬃﬃﬃﬃﬃ p ﬃﬃ ﬃ ilv ρv σgðρl ρv Þ ¼ 4 ρ þ ρv 24 3 2π l

ϕcrit,1 ¼ ϕcrit,2

ð6:18Þ ð6:19Þ

which corresponds to the limits for the occurrence of CHF. Zuber and Tribus (1958) suggested to simplify the above equations by assuming that the liquid phase density is much higher than the vapor phase density (ρl ρv). Additionally, they 3 ﬃﬃﬃﬃ recommended to adopt the maximum estimated value, considering p4 ﬃﬃ3p 1, 2π given as follows: ϕcrit ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ π ^ 12 p ilv ρv 4 gðρl ρv Þσ ¼ 0:131 ^ilv ρ2v 4 gðρl ρv Þσ 24

ð6:20Þ

This relationship was compared with experimental data by Zuber and Tribus (1958) and provided satisfactory agreement. Subsequently, Lienhard and Dhir (1973) argued that Taylor instabilities limit the countercurrent ﬂow rather than Kelvin instabilities. Therefore, Lienhard and Tribus (1973) concluded that the constant of Eq. (6.20) should be increased by 14%, resulting in a coefﬁcient of 0.149 rather than 0.131. It is important to highlight that Kutateladze (1951), based on experimental results and on an analogy with the liquid-vapor hydrodynamics in a distillation column and the two-phase ﬂow at the CHF, has correlated experimental results for CHF with an expression similar to Eq. (6.20), however, with a multiplier constant of 0.16, also corroborating the validity of the modelling approach proposed by Zuber and Tribus (1958). It must be remarked that the predictions of CHF according to Eq. (6.20) is valid for horizontal inﬁnite plate, and Lienhard and Dhir (1973) proposed correction factors for ﬁnite geometries, and the reader is encouraged to check this paper for further and speciﬁc methods. Carey (1992), supported by evidences in literature, pointed out the following main drawbacks of the hydrodynamic models: (i) a logical reason that justiﬁes the radius of the vapor columns as a fraction of Λcrit and independent of heat ﬂux was not given; (ii) vapor columns apart of ﬁxed spacings are typical of ﬁlm boiling conditions, however, it is not clear if this behavior persists throughout the transition boiling regime until the CHF; (iii) as mentioned above, results from literature revealed that the CHF is affected by the surface wettability whose effect is not

224

6

Critical Heat Flux and Dryout

taken into account by the hydrodynamic models; and (iv) results for boiling of metal liquids revealed CHF values from 2 to 4 times higher than the predictions given by the hydrodynamic models. Such a behavior may be associated with stronger effects of conduction and convection occurring for these ﬂuids which are not taken into account in the models that consider only vapor transport mechanisms.

6.3

Macrolayer Model

Another classical approach reported in literature to model the CHF was proposed by Haramura and Katto (1983), which is referred in the literature as macrolayer model. This term derives from the fact that the model assumes a thin liquid ﬁlm laying on the heating surface under the CHF condition, referred to as macrolayer, and the vapor bubble (or blanket) hovers over the liquid ﬁlm in the condition of CHF. It must be mentioned that reported a thin liquid ﬁlm between the bubble and heater surface close to the CHF condition was previously reported in literature. Therefore, vapor is generated by the liquid ﬁlm to the bubble vapor, which grows attached to the macrolayer due to the balance of buoyancy and inertial forces, where the last parcel is related to the displacement of liquid portion around the bubble. It is considered that the CHF occurs if the liquid ﬁlm beneath the bubble evaporates away before the bubble detachment from the surface. Figure 6.2 presents

Vapor bubble

δc

δc

uv

Liquid ﬁlm

Λd

Λd

ul

Λd

Fig. 6.2 Schematics of the vapor boiling during CHF according to macrolayer model, adapted from Haramura and Katto (1983)

6.3 Macrolayer Model

225

schematically the model adopted by Haramura and Katto (1983) to develop the predictive method for CHF, where the most important geometrical parameters are the spacing between bubbles Λd and the liquid ﬁlm thickness δc. Initially, it is considered the propagation velocity of interfacial wave, given by: Γ2 ¼

ρl ρv 2πσ ð u þ ul Þ 2 Λ ð ρl þ ρv Þ ð ρl þ ρv Þ 2 v

ð6:21Þ

which should be a real value to be stable. Based on mass conservation in the interface between solid and ﬂuid, it is possible to relate the heat ﬂux and vapor velocity as follows: ϕAw ¼ uv ρv Av bilv

ð6:22Þ

where the vapor velocity is considered to be much higher than liquid velocity due to continuity and density difference. Therefore, by assuming the limiting value for the wave propagation velocity to be real, that is, stable, and substituting the vapor velocity in Eq. (6.21), the following relationship for the wavelength in the imminence of instability can be found: Λinst

ðρ þ ρv Þρv ¼ 2πσ l ρl

2 2 bilv Av ϕ Aw

ð6:23Þ

where Av is the cross-sectional area of the bubble stem, and Aw is the corresponding area of inﬂuence of each bubble/stem along the heater surface. Previous to the instability/dryout, the liquid ﬁlm thickness δc depends on the wavelength and should be between 0 and Λinst / 2, therefore Haramura and Katto (1983) adopted the mean arithmetic value between these limits as follows: 0 þ Λ2inst π ðρl þ ρv Þρv δc ¼ ¼ σ 2 2 ρl

2 2 bilv Av ϕ Aw

ð6:24Þ

which relates the heat ﬂux and liquid ﬁlm thickness, or interfacial wavelength. Therefore, Haramura and Katto (1983) suggested to adopt the most critical (most dangerous) wavelength for the spacing between bubbles Λd, which is given as a function of Taylor instabilities, as follows: Λd ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ σ 3 2π gð ρl ρv Þ

ð6:25Þ

226

6

Critical Heat Flux and Dryout

Therefore, by assuming that the inﬂuence region of one bubble as Λd (Berenson 1961), it is possible to estimate the bubble volumetric growing ratio as follows: Qb ¼

ϕΛ2d bilv ρv

ð6:26Þ

As mentioned before, the stability of the bubble over the liquid ﬁlm depends on the balance between buoyance forces and inertial forces of the liquid displaced by bubble growth, which is inherently a transient process. Therefore, based on the modelling of the bubble growth process performed by Haramura and Katto (1983), the period of bubble growth during its cycle process is given as follows:

3 Δt b ¼ 4π

15 4 11 ρ þ ρ 35 1 v 16 l Q5b gð ρl ρv Þ

ð6:27Þ

Now, by assuming that the CHF occurs when the liquid refeeding is deﬁcient during the hovering period, and that the liquid ﬁlm evaporates during this period, it is possible to infer a relationship between the CHF as follows: Δt b ϕAw ¼ ρl δc ðAw Av Þbilv

ð6:28Þ

and substituting the previously derived parameters and solving for the heat ﬂux as follows:

3 4π

15

!35 ϕΛ2 15 ρl þ ρv ρv ^i 2 A 2 4 11 16 ρl þ ρv π v lv d Aw Av ^ilv ϕAw ¼ ρl σ ilv ρv ρl ϕ Aw 2 g ρl ρv ð6:29Þ

16

ϕ5 ¼

1 1 6 π 4π 5 b165 1 5 σ ilv 2 ðρl þ ρv Þρ5v 2 3 Λd 0

gð ρl ρv Þ

4 11 16 ρl þ ρv

Av Aw

2 A 1 v ð6:30Þ Aw

115

C 1 B 6 π 4π 5 b165 B 1 C ϕ ¼ σ ilv B ðρ þ ρv Þρ5v 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C 2 3 A l @ pﬃﬃﬃ 3 2π gðρlσρv Þ 16 5

!35

gðρl ρv Þ

4 11 16 ρl þ ρv

!35

Av Aw

2 1

Av Aw

ð6:31Þ

6.3 Macrolayer Model

227

1 1 6 π 4π 5 b165 gðρl ρv Þ 5 ϕ ¼ σ ilv ðρl þ ρv Þρ5v 2 2 3 12π σ 16 5

gðρl ρv Þ

4 11 16 ρl þ ρv

!35

Av Aw

2 1

Av Aw

ð6:32Þ 1 6 4 π 1 5 45b165 ϕ ¼ σ ilv ðgðρl ρv ÞÞ5 ðρl þ ρv Þρ5v 2 9π 16 5

1

ρ 4 11 16 l þ ρv

!35

Av Aw

2 A 1 v Aw ð6:33Þ

1 6 4 π 1 5 45b165 ϕ ¼ σ ilv ðgðρl ρv ÞÞ5 ðρl þ ρv Þρ5v 2 9π 16 5

ϕ

16

5 π 1 4b16 ¼ σ i ðgðρl ρv ÞÞ4 ðρl þ ρv Þ5 ρ6v 2 9π lv

1

ρ 4 11 16 l þ ρv

!35

1 11

4 16 ρl þ ρv

Av Aw

!3

2 A 1 v Aw ð6:34Þ

Av Aw

10

A 1 v Aw

5

ð6:35Þ ϕcrit ¼

π

4

211 32

161

0 B @

ρl ρv

þ1

11 ρl 16 ρv

5 1

þ1

C 3 A

1 16

Av Aw

2

Av Aw

3 !165 1 1 bilv ρ2v ðσgðρl ρv ÞÞ4 ð6:36Þ

Now, the only unknown variable in Eq. (6.36) is the vapor and heater area ratio Av / Aw. The reader should notice that Eq. (6.36) obtained through the macrolayer model is somewhat similar to Eq. (6.20) obtained by Zuber and Tribus (1958) based on the hydrodynamic model. Therefore, Haramura and Katto (1983) equated the CHF given by Eq. (6.36) with the estimation according to Eq. (6.20), to obtain the relationship for the area ratio. Additionally, these authors assumed that Av / Aw 1 and that ρl ρv, obtaining the following relationship for the area ratio: Av ¼ Aw

3

3

π 4 1116 7 49 38 216

85 15 15 ρv ρ ¼ 0:0584 v ρl ρl

ð6:37Þ

Recall that the prediction according to the hydrodynamic model, Eq. (6.20), provided reasonable predictions of the experimental results for large horizontal ﬂat plates and high pressure ﬂuids, therefore, it is also expected that the prediction according to Eq. (6.36) are also satisfactory for such conditions. Moreover, Kutateladze (1951) correlated the CHF with a relationship similar to Eq. (6.36). Haramura and Katto (1983) also presented predictive methods for other geometries, such as ﬁnite plates and cylindrical surfaces.

228

6

Critical Heat Flux and Dryout

Haramura and Katto (1983) also proposed an analytical approach for the prediction of CHF during forced convection over ﬂat plates. In their method, it is considered that the thermal energy supplied along a length L to the ﬂuid is responsible for the vaporization of the liquid of the macrolayer with velocity u as follows: ϕL ¼ ρlbilv uδc

ð6:38Þ

Therefore, by substituting the liquid ﬁlm thickness δc given by Eq. (6.24) in the above equation, adopting the area ratio simpliﬁcation of Eq. (6.37), and assuming mass ﬂux G ¼ uρl, it is possible to estimate the CHF as follows: ϕcrit

157 1 1 ρv ρv 3 ρl σ 3 b ¼ 0:1749 ilv G 1þ ρl ρl G2 L

ð6:39Þ

where the last term of the RHS can be recognized as inverse of the Weber number, which relates the ratio between inertial and surface tension forces. Haramura and Katto (1983) also proposed predictive methods for CHF during cross ﬂow through cylinders. Remark that CHF might occur even for saturated or subcooled condition, but the above described methods should be evaluated for properties estimated at saturated conditions for the operational pressure.

6.4

Critical Heat Flux During In-Tube Flow

Katto (1994), in his wide review, indicated that the CHF during internal ﬂow in channels is more probable to occur close to the outlet of the channel, even though some researches have reported the occurrence of hot spots upstream the outlet. Katto (1994) argued that downstream the point of CHF incipience, the heat exchanger probably operates also with CHF due to increment of vapor fraction, and considering that CHF is usually accompanied by sharp increment of heater temperature with possible damage of the heat transfer surface, it is unlikely that CHF occurs upstream the outlet. It is also important to characterize the difference between saturated and subcooled CHF, because this distinction has been considered by the authors when developing their prediction methods. The last occurs under vapor quality at the test section outlet lower than 0 while the ﬁrst is characterized by vapor qualities higher than 0. Recall that usually CHF is considered as different from dryout, because dryout during convective boiling results in reduction of heat transfer coefﬁcient but within safe limits, due to partial and progressive drying of the channel wall. In the case of convective boiling, the ﬂow inertia also tends to push liquid droplets periodically to the surface, settling the reduction of heat transfer coefﬁcient. On the other hand,

6.4 Critical Heat Flux During In-Tube Flow

229

Onset of annular ﬂow

b)

d)

G Droplets deposition Dryout a) Evaporation Liquid droplets detachment

Subcooling

Saturated

0

x

c)

Fig. 6.3 Critical heat ﬂux mechanisms during ﬂow boiling, adapted from Semeria and Hewitt (1972)

during the CHF the rewetting of the surface is deteriorated due to inertial forces resulting from the vapor ﬂow away from the surface or from the liquid–vapor interface. Nonetheless, some authors also refer to CHF and dryout as synonyms, such as Hewitt (1981), but in this book we keep them as distinct phenomena. Figure 6.3 modiﬁed from Semeria and Hewitt (1972) illustrates the different mechanisms associated to the CHF during ﬂow boiling inside vertical tubes as follows: (a) dryout under a vapor clot; (b) bubble crowding and vapor blanketing; (c) evaporation of liquid ﬁlm surrounding a slug bubble; and (d) lack of enough liquid to sustain and stable liquid ﬁlm on the heating surface. As shown in these ﬁgures, under subcooled conditions and high mass velocities, turbulence effects, responsible for breaking the bubbles in smaller ones, plus drag effects, responsible for detaching the bubbles from the wall, both act to avoid the formation of vapor clots close to the tube wall. Therefore, intense nucleation and detachment of small bubbles become necessary to avoid the tube wall rewetting and the establishment of CHF. On the other hand, under saturated conditions the amount of liquid entrained within the vapor phase in the channel core increases with increasing the mass velocity due to the increment of shear effects of the vapor on the liquid–vapor interface, intensifying the detachment of liquid droplets from the crest region of

230

6

Critical Heat Flux and Dryout

the disturbance waves. This behavior implies on the reduction of the vapor quality dryout as the mass velocity increases under annular ﬂow conditions. Under conditions of extremely high heat ﬂux and annular ﬂows, bubble nucleation followed by its growth and detachment may cause the liquid ﬁlm rupture promoting also a lower dryout vapor quality. Under low mass velocities and vapor qualities close to zero, as illustrated by Semeria and Hewitt (1972), the presence of elongated bubbles with a dry region at the bubble tale may result on the establishment of CHF if the transient wall temperature evolves up to such a value that its rewetting is not possible. For high liquid subcooling and low and intermediary mass velocities, the mechanisms described in Sect. 6.3 are expected as being associated to the CHF under ﬂow boiling conditions. Katto and Ohno (1984) proposed a predictive method for CHF during convective boiling inside vertical and uniformly heated tube, resulting in a relationship similar to Eq. (6.39), but also including the effect of the ratio between the channel diameter and the heated length. Additionally, the authors included the possible effect of subcooling of the ﬂuid at the channel inlet, by considering the subcooling enthalpy. Although, their method is not recent, it provides reasonable predictions of independent CHF data obtained for ﬂow boiling under micro and macroscale conditions and is one of the most used prediction methods for CHF, as pointed out by Shah (1987). Their method was adjusted based on their own data and on experimental data from literature for water, halocarbon refrigerants, and helium and tube diameters down to 1 mm. In this method, the CHF is given as follows: Δ^i ϕcrit ¼ ϕco 1 þ K i ^ilv

ð6:40Þ

where Δii is the inlet subcooling enthalpy, bilv is the latent heat of evaporation, and ϕco is the basic heat ﬂux as nominated by them, which can be calculated according to the following ﬁve equations: ϕco ð1Þ 1 ¼ CðWedh Þ0:043 b L=d h Gilv 0:133 ϕco ð2Þ ρ 1 ¼ 0:10 v ðWedh Þ1=3 ρl 1 þ 0:0031ðL=dh Þ Gbilv 0:133 ϕco ð3Þ ðL=dh Þ0:27 ρ ¼ 0:098 v ðWedh Þ0:433 ρl 1 þ 0:0031ðL=d h Þ Gbilv 0:60 ϕco ð4Þ ρ 1 ¼ 0:0384 v ðWedh Þ0:173 ρ 1 þ 0:280ðWedh Þ0:233 ðL=dh Þ l Gbilv

ð6:41Þ ð6:42Þ ð6:43Þ

ð6:44Þ

6.4 Critical Heat Flux During In-Tube Flow

231

0:513 ϕco ð5Þ ðL=dh Þ0:27 ρ ¼ 0:234 v ðWedh Þ0:433 ρl 1 þ 0:0031ðL=d h Þ Gbilv

ð6:45Þ

where the Weber number is given as follows: Wedh ¼

G2 dh σρl

ð6:46Þ

and the dimensionless parameter C of Eq. (6.41) is obtained as follows:

C¼

8 > < > :

0:25

for

L=d h < 50

0:25 þ 0:0009½ðL=d h Þ 50 0:34

for for

50 L=dh 150 L=d h > 150

ð6:47Þ

The inlet subcooling parameter K in Eq. (6.40) is giving by the following three equations: 1:043 4C ðWedh Þ0:043

ð6:48Þ

0:0124 þ ðdh =LÞ 5 6 ðρ =ρ Þ0:133 ðWed Þ1=3

ð6:49Þ

Kð1Þ ¼ Kð2Þ ¼

v

Kð3Þ ¼ 1:12

l

h

1:52ðWedh Þ0:233 þ ðd h =LÞ ðρv =ρl Þ0:60 ðWedh Þ0:173

ð6:50Þ

The values of ϕco and K in Eq. (6.40) are determined according to the criteria presented in Fig. 6.4. Shah (1987) developed a correlation based on a broad database gathered from 16 independent studies involving 23 ﬂuids, tube diameters from 0.315 to 37.5 mm, Start

ρv / ρl < 0.15

Yes

No ϕco(1) < ϕco(5)

No

K(1) > K(2)

Yes ϕco = ϕco(1)

No

ϕco = min{ϕco(1), ϕco(2), ϕco(3)} K = max{K(1), K(2)}

Yes ϕco = max{ϕco(4), ϕco(5)}

K = K(1)

K = min{K(2), K(3)}

End

Fig. 6.4 Algorithm for determination of parameters ϕco and K of Eq. (6.40)

232

6

Critical Heat Flux and Dryout

reduced pressures from 0.0014 to 0.96, inlet vapor quality from 4.00 to 0.85, mass velocities from 3.9 to 29.051 kg/m2s, and critical vapor quality (corresponding to the CHF) from 2.6 to 1. In his method, two different calculation procedures are proposed, considering the following parameter as the transitional criterion: Y ¼ Pe Fe0:4

μ 0:6 l

μv

ð6:51Þ

where Pe and Fe are, respectively, the Péclet number and the entrance effect factor given by Pe ¼

Gd h cp,l kl

ð6:52Þ

L Fe ¼ 1:54 0:032 dh

ð6:53Þ

where L is the axial distance between the tube entrance and the CHF location, comprising the total tube length. If Eq. (6.53) gives Fe less than 1, Fe ¼ 1 should be adopted. Besides the length L, the method of Shah (1987) also deﬁnes the effective length Le equal to L when xeq,in 0 and equal to the boiling length when xin > 0 given as follows for uniform heated channels1: 8 > < L þ Gxeq,inbilv d =ð4ϕcrit Þ Lb ¼ > : L þ Gxeq,inbilv At =ðϕcrit PÞ

for

round channels

for

other geometries

ð6:54Þ

where At is the channel cross-sectional area and P its perimeter. For Helium or when Y 106 for other ﬂuids or when the effective tube length is higher than 160=p1:14 r , the following procedure should be adopted: Bocrit ¼ 0:124

0:89 4 n dh 10 ð1 xIE Þ Le Y

ð6:55Þ

where xIE ¼ xeq,in when the ﬂuid is subcooled (xeq,in 0) at the test section inlet, otherwise xIE ¼ 0. The critical Boiling number in Eq. (6.55) is given as follows:

1 In the present study the method of Shah (1987) was extended also to non-circular channels since Tibiriçá et al. (2008) observed reasonable prediction by this method for non-circular channels.

6.4 Critical Heat Flux During In-Tube Flow

233

Bocrit ¼

ϕcrit bilv G

ð6:56Þ

The value of n in Eq. (6.55) is estimated as follows:

n¼

8 > < > :

ðd h =Le Þ0:33 ðd h =Le Þ

0:54 0:5

0:12=ð1 xIE Þ

for

Helium

for

Y 106

for

Y > 106

ð6:57Þ

Otherwise the critical Boiling number is calculated as follows: Bocrit ¼ Fe Fx Bo0

ð6:58Þ

and the lowest value for the critical Boiling number given by Eqs. (6.55) and (6.58) is adopted for the estimative of the CHF through Eq. (6.56). In Eq. (6.58), Bo0 is the highest of the values given by the following three equations: Bo0 ð1Þ ¼ 15ðY Þ0:612

Bo0 ð2Þ ¼ 0:082ðY Þ0:3 1 þ 1:45p4:03 r

Bo0 ð3Þ ¼ 0:0024ðY Þ0:105 1 þ 1:15p3:39 r

ð6:59Þ ð6:60Þ ð6:61Þ

and Fx is estimated as follows: For xcrit > 0

Fx ¼

8 2 3n

0:24157xcrit > 1:25105 > 1 ð p 0:6 Þ 0:833x r crit Y > 1:25105 > 41 þ 5 < Y 0:35

for

xcrit > 0

> > > > :

for

xcrit < 0

h in Þðpr 0:6Þ F 1 1 ð1F 20:35

ð6:62Þ where n¼ and

0 1

for for

pr 0:6 pr > 0:6

ð6:63Þ

234

6

( F1 ¼

1 þ 0:0052ðxcrit Þ0:88 Y 0:41 1 þ 4:452ðxcrit Þ

0:88

Critical Heat Flux and Dryout

for

Y 1:4 107

for

Y > 1:4 107

ð6:64Þ

and F1 ¼

F 0:42 1

for

F1 4

0:55

for

F1 > 4

ð6:65Þ

Most recently, Zhang et al. (2006) proposed a predictive method for the CHF inside microchannels. These authors gathered a wide experimental database from the open literature and performed an analysis using artiﬁcial neural network to identify the dominant non-dimensional parameters. Based on this analysis, these authors found that the boiling number for CHF, Bocrit, can be presented as a function of Weber number, density ratio, inlet vapor quality, and channel length to diameter ratio. Subsequently, Zhang et al. (2006) performed the regression based on artiﬁcial neural network and on the mentioned database to determine the constants and exponents. The resulting predictive method is given as follows: "

Bocrit

# 2:31 0:361 #0:295 0:311 " 0:170 ρv ρv L L ¼ 0:0352 Wedh þ 0:0119 2:05 xeq,in dh dh ρl ρl

ð6:66Þ The sub index h in the diameter term for the above equations refers to the hydraulic diameter because the predictive method is also suitable to non-circular channels. Additionally, the vapor quality at the section inlet xeq,in in Eq. (6.66) is 108

10-1

L/d = 10 L/d = 50

107

L/d = 10 L/d = 50 L/d = 100

L/d = 100

Bocrit [-]

φcrit [W/m²]

10-2

10-3

Zuber and Tribus (1958)

106

105

104

Zuber and Tribus (1958) 10-4 10-3

10-2

10-1

100

101

Wed [-]

102

103

104

103 100

101

102

103

104

G [kg/m²s]

Fig. 6.5 Variation of (a) boiling number with Weber number, and (b) heat ﬂux with mass velocity, for CHF conditions for water convective boiling, d ¼ 3 mm, pin ¼ 1 MPa, xeq,in ¼ 0.20 according to the model of Zhang et al. (2006)

6.5 Solved Example

235

evaluated assuming thermal equilibrium and is given as a function of the inlet enthalpy. Figure 6.5 depicts the variation of the critical heat ﬂux with mass ﬂow rate in non-dimensional and dimensional form, showing that the CHF increases with mass ﬂow rate, as expected. These pictures also depict the estimation according to the Zuber and Tribus (1958) model, which was developed for stagnant conditions. The increment of working pressure and reduction of inlet enthalpy increase marginally the CHF, but as not as pronounced as the reduction of the length to diameter ratio. A comparison performed by Tibiriça et al. (2008) of a microscale CHF database from literature against nine prediction methods revealed that the best predictions were provided by Shah (1987) as a general method (saturated and subcooled) and Zhang et al. (2006) for saturated conditions. In the case of the database containing only subcooled CHF data, the method of Hall and Mudawar (2000) provided the most accurate predictions. Their database comprised more than 1000 data points and includes both subcooled and saturated CHF results, covering 7 ﬂuids (water, R12, R113, R134a, R123, CO2, and Helium), mass velocities from 10.5 to 134,000 kg/m2s, and experimental CHFs up to 276 MW/m2 for both multi- and single-microchannel conﬁgurations. Most recently, predictive methods for CHF inside microchannels were also proposed in the literature, such as Ong and Thome (2011) and Tibiriçá et al. (2012), which were also developed for convective boiling subjected to uniform heat ﬂuxes. Based on the previous discussion, it is expected that CHF might occur in the channel outlet, therefore at the point L downstream the channel inlet. Therefore, Eq. (6.66) can be seen as an energy balance for determination of vapor quality at the CHF. On the other hand, some researchers, such as Kanizawa et al. (2016), adopted the predictive method for CHF to model the dryout occurrence, which corresponds to the maximum heat transfer coefﬁcient possible for ﬁxed mass and heat ﬂuxes, as well as channel diameter and ﬂuid properties. According to this approach, it is considered that the operational heat ﬂux ϕ is equal to the CHF given by Eq. (6.66), and the channel point of CHF (L/d) is determined by an iterative method. Then, based on energy balance it is possible to estimate the vapor quality for dryout.

6.5

Solved Example

Consider a horizontal surface facing upward submersed in stagnant water at 1 atm of absolute pressure. For this condition, evaluate the CHF value according to the hydrodynamic model proposed by Zuber and Tribus (1958) and evaluate the heat transfer coefﬁcient and wall superheating in the imminence of CHF assuming the predictive method proposed by Stephan and Abdelsalam (1980). Then, evaluate the heat transfer coefﬁcient during ﬁlm boiling assuming the heat ﬂux equal to CHF based on Berenson (1961), which is given as follows:

236

6

2

hfb ¼

Critical Heat Flux and Dryout

31=4

3 0 6 gðρl ρv Þρv kvbilv 7 0:4254 h i1=2 5 μv ΔT w,fb gðρ σρ Þ l v

where the modiﬁed enthalpy of vaporization is given as follows: h i bi0lv ¼ bilv 1 þ 0:5 cpv ΔT w,fb =bilv and is combined with radiation by assuming the equivalent heat transfer coefﬁcient for radiation for inﬁnite parallel black plates:

hrad ¼

σ SB T 4w,fb T 4sat T w,fb T sat

where σ SB corresponds to the Stephan and Boltzmann constant equal to 5.67108 W/m2K4, and the corresponding wall superheating. Finally, assuming a stagnant vapor ﬁlm with thickness δ of 0.2 mm, and heat transfer by conduction combined with thermal radiation from the surface to the liquid at Tsat, evaluate the overall heat transfer coefﬁcient. Assume vapor properties evaluated at saturation conditions and black surfaces. Solution: For water at 1 atm, the following properties can be found in thermodynamic tables: Tsat ¼ 100 C ¼ 373.15 K. ρl ¼ 958.4 kg/m3, ρv ¼ 0.5975 kg/m3, μl ¼ 2.819104 kg/ms, μv ¼ 1.227105 kg/ms, σ ¼ 0.05891 N/m, îlv ¼ 2.257106 J/kg, cpl ¼ 4217 J/kgK, cpv ¼ 2044 J/kgK, kl ¼ 0.6651 W/mK, kv ¼ 0.02508 W/mK. Hence, by solving Eq. (6.20), it is possible to estimate the CHF as follows: 1

ϕcrit ¼ 0:131 ilv ρ2v

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 gðρl ρv Þσ ¼ 1:108 106 W=m2

Therefore, considering the imminence of the CHF, let us consider the Stephan and Abdelsalam (1980) predictive method for heat transfer coefﬁcient during pool boiling, given by Eq. (5.19) as follows:

6.5 Solved Example

237

!1:58 !1:26 0:673 5:22 bilv d2b cpl T sat d2b ρl ρv hnb d b ϕd b 6 ¼ 2:46 10

2

2 kl k l T sat ρl kl =ρl cpl k l =ρl cpl where db is the bubble departure diameter given by Eq. (5.23) as follows: d b ¼ 0:146

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2σ ¼ 0:5171 mm gð ρl ρv Þ

and the corresponding hnb for ϕ ¼ ϕcrit is equal to 188,285 W/m2K. The corresponding wall superheating can be evaluated based on the Newton cooling law as follows: ϕ ¼ hnb ΔT w,nb ¼ hnb ðT w,nb T sat Þ which results in superheating of 5.89 K, that can be considered relatively low for the heat ﬂux value. But the corresponding heat transfer coefﬁcient is of the order of hundreds of kW/m2K, hence, the result is coherent. Subsequently, let us evaluate the system performance after the occurrence of the CHF, initially accounting for the heat transfer coefﬁcient predictive method given in this exercise. Considering that the predictive method for heat transfer coefﬁcient and for the thermal radiation parcel depend on the wall superheating, which in turn depends on the heat transfer coefﬁcient, an iterative method is required for the solution. Moreover, considering the combination of heat transfer by convection during ﬁlm boiling and radiation, the problem is non-linear and the use of at least a spreadsheet is advisable. The following scheme can be adopted: • Guess the surface temperature, for example, assuming overall heat transfer coefﬁcient as 1000 W/m2K, the resulting surface superheating ΔTw,fb would be 1108 K for ϕcrit. • Tolerance ¼ small value. • Error ¼ large value. • ΔTw,fb,aux ¼ ΔTw,fb. • Repeat: – – – – – –

Evaluate hfb and hrad based on ΔTw,fb. Evaluate the overall heat transfer coefﬁcient h ¼ hfb + hrad. Update the wall superheating ΔTw,fb ¼ ϕcrit / h. Error ¼ | ΔTw,fb.- ΔTw,fb,aux|. ΔTw,fb,aux ¼ ΔTw,fb. While (Error > Tolerance).

238

6

Critical Heat Flux and Dryout

By performing the above approach, the ΔTw,fb is obtained as 1628 K, and the corresponding heat transfer coefﬁcient as hfb ¼ 122.6 W/m2K, hrad ¼ 558 W/m2K, and h ¼ 680.7 W/m2K, which is 276 times lower than the estimation for nucleate boiling condition. It must be emphasized that 82% of the heat transfer rate occurs by thermal radiation due to the high temperature difference and relatively low heat transfer coefﬁcient for ﬁlm boiling. Nonetheless, considering that melting temperature of 1010 carbon steel is approximately 1516 C, a heat exchanger made of this material would fail during CHF with imposed heat ﬂux and appropriate material must be selected. This example justiﬁes the usage of burnout as a synonym for CHF since the surface would be damaged. Additionally, if the radiation parcel was neglected, the temperature difference would rise to 10,682 K, and if only radiation was considered, neglecting the ﬁlm boiling parcel, the temperature difference would be 1730 K, emphasizing the importance of accounting for thermal radiation. Finally, let us consider the simpliﬁcation of a vapor ﬁlm formed in contact with the surface that corresponds to a thermal resistance of conduction in parallel with heat transfer by radiation. The thermal radiation parcel can be estimated based on the same approach of the previous case. In the case of the thermal resistance due to conduction through the vapor ﬁlm, the following equivalent heat transfer coefﬁcient can be obtained: hcond ¼

kv δ

Hence, by repeating the algorithm described in the previous case, it is possible to estimate the overall heat transfer coefﬁcient and solve the problem. In this case, the temperature difference is 1626 K, with hcond ¼ 125.4 W/m2K and hrad ¼ 556.3 W/ m2K, with overall heat transfer coefﬁcient of 681.7 W/m2K. This result is pretty similar with the analysis of ﬁlm boiling due to the deﬁnition of the vapor ﬁlm thickness, with 82% of the heat transfer rate being transferred from the surface to the liquid via radiation. Considering a vapor ﬁlm thickness of 1.0 mm instead of 0.2 mm, the temperature difference raises to 1709 K, and the radiative parcel to 96% of the total heat transfer rate, which again emphasizes the importance of accounting for radiative heat transfer. In conclusion, the reader must bear in mind that, in general, CHF should be avoided otherwise the heat transfer surface might be damaged in case of imposed heat ﬂux, and evaluating the heat transfer coefﬁcient value after CHF might be pointless.

References

6.6

239

Problems

1. Consider a vapor layer with thickness δ in contact with a ﬂat horizontal surface with imposed heat ﬂux ϕ, with liquid above it, which can be considered at Tsat in the interface. Assuming the vapor as stagnant: (a) Derive a relationship for the heat transfer coefﬁcient assuming ﬂuid reference temperature as Tsat. (b) Considering water at atmospheric pressure, evaluate the variation of heat transfer coefﬁcient with δ between 0.1 to 5.0 mm. (c) Compare the estimated heat transfer coefﬁcient with conditions of liquid single-phase ﬂow, and nucleate boiling according to Stephan and Abdelsalam (1980). (d) Include the effect of thermal radiation by assuming inﬁnite black bodies

4 4 (ϕrad ¼ σ SB T surf T sat , σ SB as the Stephan–Boltzmann constant, and equal to 5.607108 W/m2K4), and evaluate the overall equivalent heat transfer coefﬁcient. 2. Estimate the CHF for R134a ﬂow boiling inside a 2 mm ID channel, 50 mm long, at 200 kg/m2s, and saturation temperature of 30 C. Assume saturated liquid in the inlet. 3. Repeat the previous exercise assuming subcooling degree of 2 C in the inlet. 4. Compare the heat transfer coefﬁcient in the imminence of CHF during pool boiling conditions for water at 101.3 C, R134a at 770.6 kPa, and R600a at 1683 kPa. Assume the predictive method proposed by Stephan and Abdelsalam (1980) to estimate the heat transfer coefﬁcient, paying attention that the corresponding correlation depends on the ﬂuid type, and consider either the hydrodynamic model or the macrolayer model for CHF. 5. Determine the variation of the CHF with reduced pressure for ﬂow boiling of water and R134a inside a 5 mm ID tube, 150 mm long, at 500 kg/m2s, and subcooling degree at the channel inlet of 20 C according to Katto and Ohno (1984). 6. Repeat the previous exercise considering the method of Shah (1987).

References Bellman, R., & Pennington, R. H. (1954). Effects of surface tension and viscosity on Taylor instability. Quarterly of Applied Mathematics, 12(2), 151–162. Berenson, P. J. (1961). Film-boiling heat transfer from a horizontal surface. Journal of Heat Transfer, 83, 351–358. Carey, V. P. (1992). Liquid-vapor phase-change phenomena: An introduction to the thermophysics of vaporization and condensation processes in heat transfer equipment. New York: Taylor & Francis. Collier, J. G., & Thome, J. R. (1994). Convective boiling and condensation. Oxford: Clarendon Press.

240

6

Critical Heat Flux and Dryout

Groeneveld, D. C., Shan, J. Q., Vasić, A. Z., Leung, L. K. H., Durmayaz, A., Yang, J., et al. (2007). The 2006 CHF look-up table. Nuclear Engineering and Design, 237(15–17), 1909–1922. Hall, D. D., & Mudawar, I. (2000). Critical heat ﬂux (CHF) for water ﬂow in tubes—II: Subcooled CHF correlations. International Journal of Heat and Mass Transfer, 43(14), 2605–2640. https:// doi.org/10.1016/S0017-9310(99)00192-1 Haramura, Y., & Katto, Y. (1983). A new hydrodynamic model of critical heat ﬂux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids. International Journal of Heat and Mass Transfer, 26(3), 389–399. Hewitt, G. F. (1981). Chapter 9: Burnout. In A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, & F. Mayinger (Eds.), Two-phase ﬂow and heat transfer in the power and process industries (pp. 545–560). New York: Hemisphere. Kanizawa, F. T., Tibiriçá, C. B., & Ribatski, G. (2016). Heat transfer during convective boiling inside microchannels. International Journal of Heat and Mass Transfer, 93, 566–583. Katto, Y. (1994). Critical heat ﬂux. International Journal of Multiphase Flow, 20, 53–90. Katto, Y., & Ohno, H. (1984). An improved version of the generalized correlation of critical heat ﬂux for the forced convective boiling in uniformly heated vertical tubes. International Journal of Heat and Mass Transfer, 27(9), 1641–1648. Kutateladze, S. S. (1951). A hydrodynamic theory of changes in a boiling process under free convection. Izvestia Akademia Nauk Otdelenie Tekhnicheski Nauk, 4, 529–536. Linehard, J. H., & Dhir, V. K. (1973). Extended hydrodynamic theory of the peak and minimum pool boiling heat ﬂuxes. Washington, DC: National Aeronautics and Space Administration. Lienhard, J. H., & Dhir, V. K. (1973). Hydrodynamic prediction of peak pool-boiling heat ﬂuxes from ﬁnite bodies. https://doi.org/10.1115/1.3450013 Ong, C. L., & Thome, J. R. (2011). Macro-to-microchannel transition in two-phase ﬂow: Part 2– ﬂow boiling heat transfer and critical heat ﬂux. Experimental Thermal and Fluid Science, 35(6), 873–886. Rayleigh, J. W. S. B. (1896). The theory of sound (Vol. 2). London: Macmillan. Semeria, R., & Hewitt, G. F. (1972). Aspects of gas–liquid ﬂow. In Seminar on recent developments in Heat Exchangers, International Centre for Heat and Mass Transfer, Trogir, Croatia. Shah, M. M. (1987). Improved general correlation for critical heat ﬂux during upﬂow in uniformly heated vertical tubes. International Journal of Heat and Fluid Flow, 8(4), 326–335. Stephan, K., & Abdelsalam, M. (1980). Heat-transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer, 23(1), 73–87. Tibiriçá, C. B., Felcar, H. O. M., & Ribatski, G. (2008, May). An analysis of experimental data and prediction methods for critical heat ﬂuxes in micro-scale channels. In 5th European ThermalSciences Conference, Eindhoven. Tibiriçá, C. B., Ribatski, G., & Thome, J. R. (2012). Saturated ﬂow boiling heat transfer and critical heat ﬂux in small horizontal ﬂattened tubes. International Journal of Heat and Mass Transfer, 55(25–26), 7873–7883. Yagov, V. V., Lexin, M. A., Zabirov, A. R., & Kaban’kov, O. N. (2016). Film boiling of subcooled liquids. Part I: Leidenfrost phenomenon and experimental results for subcooled water. International Journal of Heat and Mass Transfer, 100, 908–917. Zhang, W., Hibiki, T., Mishima, K., & Mi, Y. (2006). Correlation of critical heat ﬂux for ﬂow boiling of water in mini-channels. International Journal of Heat and Mass Transfer, 49(5–6), 1058–1072. Zhang, L., Seong, J. H., & Bucci, M. (2019). Percolative scale-free behavior in the boiling crisis. Physical Review Letters, 122(13), 134501. Zuber, N., & Tribus, M. (1958). Further remarks on the stability of boiling heat transfer. Report 58-5 (No. AECU-3631). California of University, Los Angeles. Department of Engineering.

Chapter 7

Condensation

In Chap. 5, a discussion concerning homogeneous nucleate boiling is presented, which is modelled by accounting for the required energy to generate a vapor bubble within a liquid media that in turn depends on the probability of an amount of highly energetic particles to collide forming a high energetic region, which propitiates the formation of a vapor nuclei, with subsequent bubble growth. Considering the case of condensation, homogeneous transition from vapor to liquid phase can also occur and can be modelled by considering a similar approach as in the case of homogeneous vapor nucleation. However, this subject is not the focus of the present study because the phenomenon of homogeneous condensation is rarely observed in industrial applications. For those interested, the books of Collier and Thome (1994) and Carey (2007) are suggested as complementary material. In the case of heterogeneous nucleation of liquid droplets, it is worth mentioning that certain degrees of subcooling of the surface and of the vapor contacting the surface are required to form the initial liquid portion in a similar way to bubble formation phenomenon, and several approaches are found in literature to estimate the required subcooling, such as the one proposed by Hill et al. (1963). As discussed in Chap. 5, the boiling phenomenon might present critical conditions, such as critical heat ﬂux or dryout. However, under condensation conditions no such thing is present and consequently the analysis is a bit simpler, unless the problem is extended until solidiﬁcation with consequent formation of a solid thermal resistance. Additionally, and as discussed in Chap. 2, the heat transfer coefﬁcient during convective condensation can be modelled by combination of gravitational and convective effects, in a similar way of the approach presented by Chen (1966) for convective boiling, which considers a combination of nucleate boiling and convective effects, such as schematically depicted in Fig. 2.17. Therefore, in this chapter, initially the modelling approach for ﬁlm-wise condensation is described in a similar way to the classical study presented by Nusselt (1916)

© The Author(s) 2021 F. Kanizawa, G. Ribatski, Flow boiling and condensation in microscale channels, Mechanical Engineering Series, https://doi.org/10.1007/978-3-030-68704-5_7

241

242

7 Condensation

for condensation in a stagnant vapor medium contacting a vertical ﬂat surface, and then this mechanism is combined with convection to model the heat transfer process inside channels. Finally, the sublimation might also be present with direct transition from vapor to solid phase in some engineering and natural conditions. Hence, the frost formation, which is common on ﬁnned evaporators for refrigeration and air-conditioning application, for example, is also a subject of studies and the interested reader is encouraged to check the studies of Hermes and coworkers (Hermes et al. 2009) for further information.

7.1

Film Condensation on an Isothermal Surface

The modelling of ﬁlm condensation on a ﬂat isothermal surface was originally presented by Nusselt (1916), focusing on conditions of a surface exposed to saturated vapor of a pure substance. Figure 7.1 depicts schematically the liquid ﬁlm ﬂowing along the surface, which can be inclined or vertical. The heat transfer process is based on the following simplifying assumptions: (i) (ii) (iii) (iv) (v)

Steady state condition. Two-dimensional problem. Transport properties variations in each phase are negligible. Newtonian ﬂuid. Isothermal surface at temperature Ts. y

z

y δ δ

y

y

Tsat

u Ts

dṁ

Tsat Ts

δ

T

δ u

Ts

z

dz z+dz

g

θ

Fig. 7.1 Schematics of ﬁlm condensation on isothermal ﬂat surface

u

Control volume

7.1 Film Condensation on an Isothermal Surface

243

(vi) Vapor at saturation temperature at Tsat. (vii) The liquid at vapor–liquid interface is at Tsat. (viii) The shear stress in the liquid vapor interface is negligible because of the relatively low viscosity of vapor phase, and because of the low ﬂow velocity of the liquid ﬁlm. (ix) Convective parcels of momentum and energy transfer are negligible because the ﬁlm ﬂow velocity is considerably low. Hence, the momentum and heat transfer are governed by diffusive and buoyancy forces. (x) Variation of velocity and temperature in the direction perpendicular to the surface is considerably higher than parallel to it, similarly to the analysis of the boundary layer during forced convection parallel to a ﬂat plate. (xi) Uniform pressure along the horizontal position. (xii) Based on the previous assumption, it is possible to estimate the pressure variation based on the hydrostatic problem along the vapor side. (xiii) No viscous dissipation along the liquid ﬂow. (xiv) The sensible heat for liquid subcooling is negligible in comparison with the latent heat. (xv) The ﬂuid properties are evaluated at the mean liquid ﬁlm temperature corresponding to the average temperature between the wall and the saturation temperatures. Before starting with the development of velocity and temperature proﬁles, it is interesting to perform a dimensional analysis of the relevant parameters in this problem. The heat transfer coefﬁcient should be given as a function of liquid speciﬁc heat, liquid density, ﬂuid vaporization enthalpy, buoyance force, liquid thermal conductivity, liquid viscosity, temperature difference between the surface and vapor, and axial position or length, therefore: h ¼ f nðρl , ^ilv , gz ðρl ρv Þ, kl , μl , cp,l , ðT sat T s Þ, L or zÞ

ð7:1Þ

where units can be written as a combination of kg, J, m, s, and K. Therefore, the problem is governed by 9 variables that can be written according to 5 dimensional groups, resulting in 4 non-dimensional parameters, as follows: μ cp,l cp,l ðT sat T s Þ ρl gz ðρl ρv Þ^ilv ðL or zÞ3 hðL or zÞ ¼ fn l , , ^ilv kl kl μl k l ðT sat T s Þ

! ð7:2Þ

or: NuL or z ¼ f nðPrl , Ja, ΠÞ with:

ð7:3Þ

244

7 Condensation

hðL or zÞ kl μ cp,l Prl ¼ l kl

NuL or z ¼

ð7:5Þ

cp,l ðT sat T s Þ bilv

ð7:6Þ

ρl gz ðρl ρv Þ^ilv ðL or zÞ3 μl kl ðT sat T s Þ

ð7:7Þ

Ja ¼ Π¼

ð7:4Þ

where the ﬁrst two non-dimensional parameters are Nusselt and liquid Prandtl numbers, respectively. The parameter Ja corresponds to the Jakob number and represents the ratio between sensible and latent heat of a ﬂuid for a given temperature difference, and its value is usually small. The fourth term has no speciﬁc denomination and will be derived in the foregoing analysis. In this analysis, the contribution of the gravitational acceleration gz is taken according to the ﬂow direction, therefore given by gsin(θ), where θ is the inclination of the plate in relation to the horizontal plane. Therefore, starting with the z component of Navier–Stokes equation for liquid phase as follows: 2 2 ∂u ∂u ∂u ∂p ∂ u ∂ u ρl þ þu þv þ ρl gz þ μ l ¼ ∂t ∂z ∂y ∂z ∂z2 ∂y2

ð7:8Þ

where the parcel relative to the perpendicular direction of the paper was already simpliﬁed, and by accounting for the aforementioned assumptions, the left-hand side (LHS) is null by hypothesis i and ix. Similarly, the ﬁrst term inside the bracket of the right-hand side (RHS) is simpliﬁed by hypothesis x (∂2u/∂y2 ∂2u/∂z2), resulting in the following relationship: 2

0¼

∂p ∂ u þ ρl gz þ μ l 2 ∂z ∂y

ð7:9Þ

Based on hypotheses xi and xii, the pressure variation along z direction can be estimated as follows: ∂p ¼ ρv gz ∂z

ð7:10Þ

and substituting into Eq. (7.9), we obtain the following relationship: ðρ ρv Þgz ∂ u ¼ l μl ∂y2 2

ð7:11Þ

7.1 Film Condensation on an Isothermal Surface

245

We may try to ﬁnd a velocity proﬁle as a function of y coordinate only, whereas the inﬂuence of the z coordinate will appear in the liquid ﬁlm thickness δ. Therefore, by integrating the previous equation twice along the y direction, we obtain the following relationship for the velocity proﬁle: u¼

ðρl ρv Þgz y2 þ C1 y þ C2 μl 2

ð7:12Þ

The integrating constants can be determined from the boundary conditions, where the ﬁrst one is related to non-slip condition in the solid–liquid interface, therefore: uðy ¼ 0Þ ¼ 0

ð7:13Þ

then C2 is null. The second boundary condition can be determined from hypothesis viii, related to null shear stress between the liquid and vapor. Therefore, based on Newton’s Viscosity Law for a Newtonian ﬂuid, this boundary condition is given as follows: μl

∂u ¼0 ∂y y¼δ

ð7:14Þ

where δ is the liquid ﬁlm thickness. Hence, the constant C1 is given as follows:

ρg ρv gz C1 ¼ δ μl

ð7:15Þ

and the velocity proﬁle is given as follows: u¼

ðρl ρv Þgz y2 þ δy μl 2

ð7:16Þ

where the unknown parameter is the liquid ﬁlm thickness δ, which will be determined based on the mass and energy balance along the ﬂat surface. The mass ﬂow rate of the liquid in a given position can be evaluated based on the integration of velocity proﬁle as follows: Z m_ l ¼

δ

ρl udA ¼

0

m_ l ¼ where W is the surface width.

ρl ðρl ρv Þgz y3 δy2 δ W þ μl 6 2 0

ð7:17Þ

ρl ðρl ρv Þgz δ3 W μl 3

ð7:18Þ

246

7 Condensation

By evaluating the mass balance for an inﬁnitesimal control volume with length dz, such as depicted in Fig. 7.1, the mass variation between two positions is related to condensation rate ṁcond, as follows: d m_ ¼ m_ cond

ð7:19Þ

Additionally, by considering the energy balance over this control volume accounting for hypothesis xiv, we obtain the following relationship for steadystate condition, null net work, null viscous energy dissipation (hypothesis xiii), and negligible contribution of kinetic and potential energy variations: Z q_ cv ¼

!

!

biρV dA ¼ bil d m_ biv m_ cond ¼ bilv dm_

ð7:20Þ

cs

The mass ﬂow rate is given by Eq. (7.18). In this equation, the only variable term is the liquid ﬁlm thickness. Therefore, by differentiating Eq. (7.18) and combining with Eq. (7.20) we obtain the following relationship: q_ cv ¼ bilv

ρl ðρl ρv Þgz 1 3 W dδ 3 μl

ð7:21Þ

According to hypothesis vi and vii, the vapor and the liquid–vapor interface are under saturated conditions, and the heat transferred associated to the liquid subcooling is neglected according to hypothesis xiv; therefore, the control volume of Fig. 7.1 exchanges heat only with the solid surface, which can be evaluated based on the temperature proﬁle of the liquid ﬁlm, through the Fourier law. Therefore, starting with the energy equation in differential form for an incompressible and Newtonian ﬂuid, we have: ∇ ðkl ∇T Þ þ g_ ¼ ρcp l

∂T ! þ V ∇T ∂t

ð7:22Þ

which can be simpliﬁed to liquid ﬁlm based on the listed hypothesis (i, ii, iii, ix, x, xiii) to the following relationship: 2

∇2 T ¼

∂ T ¼0 ∂y2

ð7:23Þ

Integrating the above equation twice, and considering the boundary conditions given by assumptions v and vi, we obtain the temperature proﬁle along the liquid ﬁlm as follows: y T ¼ ðT sat T s Þ þ T s δ

ð7:24Þ

7.1 Film Condensation on an Isothermal Surface

247

Then, by accounting for no-slip condition in the solid–liquid interface, it is possible to evaluate the heat ﬂux based on the Fourier law for heat conduction as follows: ϕs ¼ kl

k ðT T s Þ ∂T ¼ l sat δ ∂y y¼0

ð7:25Þ

Therefore, by evaluating the heat transfer rate in the control volume with the heat ﬂux (qcv ¼ ϕs W dz), we obtain the following ordinary differential equation for the liquid ﬁlm thickness from Eq. (7.21):

ρ ðρ ρv Þgz 1 3 k l ðT sat T s Þ Wdz ¼ ilv l l W dδ δ 3 μl dz ¼

ρl gz ðρl ρv Þilv 3 δ dδ μl kl ðT sat T s Þ

ð7:26Þ ð7:27Þ

Therefore, integrating the above equation from the starting point of the liquid ﬁlm thickness, assuming δ (z ¼ 0) ¼ 0, we obtain the following relationship: μ k ðT T s Þ δ4 ¼ 4 l l sat z4 ρl gz ðρl ρv Þilv z3

ð7:28Þ

The liquid ﬁlm thickness increases along the ﬂow direction proportional to the distance of the starting point at ¼. Since we are interested in a relationship for the heat transfer coefﬁcient, we can equate the heat ﬂux given by Fourier law, Eq. (7.25), with the heat ﬂux estimated through the Newton’s Cooling Law, as follows: ϕs ¼

kl ðT sat T s Þ ¼ hðT s T sat Þ δ

ð7:29Þ

then, by solving for h and considering the liquid ﬁlm thickness of Eq. (7.28), we determine a relationship for the heat transfer coefﬁcient as follows: k k 1 ﬃﬃﬃ h¼ l¼ l p δ z 44

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 4 ρ g ðρ ρ Þ^ l z l v ilv z μl kl ðT sat T s Þ

ð7:30Þ

and in non-dimensional form as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 4 ρ g ðρ ρ Þ^ hz l z l v ilv z Nuz ¼ ¼ 0:707 kl μl k l ðT sat T s Þ

ð7:31Þ

248

7 Condensation

Recall that the term inside the root is the non-dimensional parameter determined from the dimensional analysis, and given by Eq. (7.7). Additionally, neither Jakob and liquid Prandtl numbers appear in the resulting equation. In fact, both parameters account for the sensible parcel of the liquid energy variation, which was neglected during the modelling. Therefore, Sadasivan and Lienhard (1987) proposed a correction for the latent heat accounting for Prl and Ja based on experimental results for ﬁlm condensation, which is valid for Prl 0.6 and is given as follows:

bilv ¼ bilv 1 þ Ja 0:683 0:228 Prl

ð7:32Þ

which should substitute îlv in Eqs. (7.28), (7.30), and (7.31). Notice that the Jakob number shall present small values, since the temperature difference is usually not high enough to compensate for the small speciﬁc heat in comparison to the latent heat. Therefore, the correction for the latent heat given by Eq. (7.32) is close to unity. Similarly, Bejan (2013) presented a modelling approach that accounted for the sensible heat of cooling, and, therefore, without taking into account the hypothesis xiv, he obtained an equivalent latent heat of vaporization given as follows: h i bilv ¼ bilv 1 þ 3 Ja 8

ð7:33Þ

However, the expression (7.32) is preferable because it was validated with experimental results. Additionally, the reader might have noticed that the heat transfer coefﬁcient reduces along the ﬂow direction, being proportional to the inverse of the distance at ¼, and this aspect is related to the increase of thermal resistance imposed by the liquid ﬁlm. In this context, it is interesting to evaluate the conditions that correspond to laminar ﬂow, which is given based on the Reynolds number. For this application, the Reynolds number is deﬁned based on the liquid ﬁlm thickness, rather than on the distance from the upper board of the plate, as follows: Reδ ¼

ρl uδ m_ ¼ Wμl μl

ð7:34Þ

and by combining this equation with Eq. (7.18), it is possible to estimate the Reynolds number with more readily known parameters, as follows: Reδ ¼

ρl gz ð ρl ρv Þ δ 3 3 μ2l

ð7:35Þ

where the liquid ﬁlm thickness is given by Eqs. (7.28) and (7.32). According to Gregorig et al. (1974), the modelled approach presents negligible deviation from experimental results for Reynolds numbers up to approximately 7, and acceptable

7.1 Film Condensation on an Isothermal Surface

249

predictions for Reynolds numbers up to 400. According to these authors, for Reδ of approximately 8, ripples are noted along the liquid ﬁlm, indicating the incipience of instabilities, and for Reδ higher than approximately 400, the ﬂow is completely turbulent. For design purposes, or for simpliﬁed analysis, it is interesting to evaluate the mean heat transfer coefﬁcient, which can be determined by its deﬁnition as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 0 1 L kl 1 4 ρl gz ðρl ρv Þ^ilv z3 ﬃﬃ ﬃ p hdz ¼ dz L 0z 44 μl kl ðT sat T s Þ 0 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 4 kl 3 4 ρl gz ðρl ρv Þ^ilv L3 ﬃﬃ ﬃ ¼ p L 44 μl kl ðT sat T s Þ

h ¼ 1 L

Z

L

ð7:36Þ

and an equivalent Nusselt number is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 3 4 ρ g ðρ ρ Þb hL l z l v ilv L NuL ¼ ¼ 0:943 kl μl kl ðT sat T s Þ

ð7:37Þ

The average heat transfer coefﬁcient for condensation on a horizontal tube with uniform wall temperature is obtained following an almost similar procedure, except by the fact that the angle θ varies from 0 at the upper region of the tube down to 180 at its bottom. Therefore, the perimeter-averaged heat transfer coefﬁcient on a horizontal tube is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 3 4 ρ gðρ ρ Þb hD l l v ilv D Nu ¼ ¼ 0:728 kl μl kl ðT sat T s Þ

ð7:38Þ

In the above equation, as well as in Eq. (7.37), the ﬂuid properties are evaluated based on the ﬁlm temperature that is the arithmetic average between the wall and saturation temperature. In this context, it is interesting to mention that drop-wise condensation might also occur in conditions of stagnant vapor. This process corresponds to the nucleation of liquid droplet in a cooled surface exposed to vapor, which grows statically due to successive condensation or coalescence with neighboring droplets, and then falls on the surface due to gravity or drag forces when it reaches a critical size without the formation of laminar ﬁlm ﬂow. This process is more likely to occur in systems where the contact angle is high (hydrophobic surfaces), and the adhesion forces of the liquid to the solid surface is not high enough to propitiate a liquid formation, and it is of difﬁcult modelling and experimental investigation because the droplet nucleation site is usually unknown. Jakob (1936) proposed a modelling approach for the dropwise condensation, which considers that the condensation actually starts as a liquid ﬁlm, and due to surface tension forces the ﬁlm contracts forming a droplet.

250

7 Condensation

Subsequently, several authors have reinforced this model by adding details or by comparing with experimental results. However, even today this mechanism is not completely understood, and will not be detailed in this book.

7.2

Predictive Methods for In-Tube Convective Condensation

This section describes predictive methods for heat transfer coefﬁcient during convective condensation inside channels of conventional and reduced sizes. As a general comment, we might expect that the heat transfer coefﬁcient during condensation increases with mass velocity and vapor quality and reduces with increasing channel diameter and temperature difference between the surface and ﬂuid. The last aspect is more pronounced under conditions dominated by gravitational effects, as can be inferred from Eq. (7.36), and is related to the increase of the thermal resistance due to liquid ﬁlm thickening. Additionally, during condensation the vapor quality reduces along the ﬂow path, therefore the starting point of condensation presents the highest heat transfer coefﬁcient with successive reduction until liquid single-phase ﬂow. In general, the methods for prediction of the heat transfer coefﬁcient during condensation can be classiﬁed as follows: (i) purely empirical; (ii) based on the Colburn analogy; and (iii) ﬂow-pattern based. The ﬁrst group is based on the curve ﬁtting of an empirical correlation to a database. The purely empirical predictive methods are based on curve ﬁtting of a correlation to a database, and most of the proposals are focused on conditions of low mass velocities, when gravitational effects are dominant. In these cases, Eq. (7.38), derived based on the Nusselt model for ﬁlm condensation on a round tube, is modiﬁed to account for the region of the tube internal wall along which the liquid ﬁlm ﬂows due to gravity. In this group, the methods developed for conditions of low mass velocities when gravitational effects are dominant. Equation () based on the model for ﬁlm condensation on a round tube is modiﬁed in order to take into account the stratiﬁed perimeter, corresponding to the internal region of the tube along which the liquid ﬁlm ﬂows due to gravity. Among the purely empirical methods the ones proposed by Chato (1962) and Jaster and Kosky (1976) are the most commonly cited in the literature. In these methods, the heat transfer is neglected along the region over which the liquid ﬂows according to the main ﬂow direction, corresponding to the ﬂooded part of the tube. Chato (1962) based on his experimental results proposed that the heat transfer coefﬁcient is given simply through the product between the heat transfer coefﬁcient for condensation on horizontal tubes, given by Eq. (7.38) and 0.76, where this multiplier is associated to the parcel of the tube perimeter corresponding to the mechanism of ﬁlm-wise condensation. A different approach is adopted when inertial effects are dominant corresponding to conditions of high two-phase ﬂow velocities. In this case, a combination of a

7.2 Predictive Methods for In-Tube Convective Condensation

251

single-phase correlation for heat transfer coefﬁcient for in-tube turbulent ﬂow and either a two-phase multiplier as proposed by Shah (1979) or an equivalent Reynolds Number is adopted (Akers et al. 1959; Cavallini and Zechin 1974). In the methods classiﬁed within group ii, generally developed for annular ﬂows, as the model proposed by Moser et al. (1998), the momentum-heat analogy is considered and the Nusselt number is equationed based on the velocity and temperature proﬁles along the liquid ﬁlm thickness. Analogous to ﬂow-pattern-based methods for the heat transfer coefﬁcient during ﬂow boiling, as the one proposed by Wojtan et al. (2005a, b), the methods for in-tube condensation of group iii are also developed based on ﬂow pattern predictive methods, and then the heat transfer process is modelled according to the main heat transfer mechanism corresponding to each ﬂow pattern. Among the prediction methods belonging to this group, Dobson and Chato (1998) were the pioneers to propose a ﬂow-pattern-based method for the heat transfer coefﬁcient during in-tube condensation. A similar approach was employed by Thome et al. (2003) and Cavallini et al. (2006). Mostly recently, artiﬁcial intelligence tools have been successfully employed as methods for prediction of the heat transfer coefﬁcient for in-tube condensation. Furlan and Ribatski (2020), based on a broad database from literature and using dimensionless numbers as input parameters, found that machine learning methods (multi-layer perceptron with backpropagation and gradient boosted decision tree) provided satisfactory predictions of independent databases and they were also able to accurately predict the effects on the heat transfer coefﬁcient of the vapor quality, mass velocity, saturation temperature, channel diameter, and ﬂow pattern when compared to the experimental results, overperforming the most accurate end up-todate correlations and models from literature. Moreover, the machine learning methods presented 90% lower processing time than the model of Cavallini et al. (2006) when applied to a simulation of a tube-in-tube condenser.

7.2.1

Dobson and Chato (1998)

Dobson and Chato (1998) proposed their method segregating the ﬂow as gravitydominated and shear-dominated corresponding approximately to stratiﬁedwavy and annular ﬂow patterns, respectively. In their method, the ﬂow pattern transitions are deﬁned as the criteria proposed by Soliman (1982) modiﬁed by them according to their database that comprises heat transfer and ﬂow pattern results for horizontal ﬂows in smooth tubes with diameters ranging from 3.14 mm and 7.04 mm for the refrigerants R12, R22, R134a, and near-azeotropic blends of R32/R125 in 50 percent/50 percent and 60 percent/40 percent. According to their method, annular ﬂow is observed for either G 500 kg/m2s or FrSo 20 while stratiﬁed ﬂow occurs for G < 500 kg/m2s and FrSo < 20. The Froud number based on the deﬁnitions of Soliman (1982) is given as follows:

252

7 Condensation

For Rel 1250: 0:025 Re 1:59 l

b tt 1 þ 1:09X b X tt

1:26 Re 1:04 l

b tt 1 þ 1:09X b tt X

0:039

Fr So ¼

!1:5 1 Ga0:5

ð7:39Þ

1 Ga0:5

ð7:40Þ

For Rel > 1250: 0:039

Fr So ¼

!1:5

where Rel is the Reynolds number considering only the ﬂow of the liquid parcel and Ga is the Galileo number relating buoyance and viscous effects given as follows: Ga ¼

ρl ðρl ρv Þd 3 μ2l

ð7:41Þ

The heat transfer for the annular ﬂow pattern was modelled based on the twophase multiplier approach, with the liquid Nusselt number given as follows: " # hd 2:22 0:4 ¼ 0:023 Re 0:8 1 þ 0:89 Nul ¼ l Pr l kl b X

ð7:42Þ

tt

In contrast to the hypothesis adopted by Chato (1962), in this study the authors observed that as the vapor velocity is increased, the heat transfer contribution at the bottom of the tube becomes relevant, and assuming only ﬁlm-wise condensation along a parcel of the tube perimeter, neglecting forced convection in the bottom region, implies on underestimating the perimeter-averaged heat transfer coefﬁcient. Moreover, as the vapor velocity increases, interfacial shear effects on the liquid ﬁlm are also increased implying on an axial ﬁlm velocity component. As the relative importance of gravitational force decreases with increasing vapor velocity, the ﬁlm-wise solution becomes inappropriate and at a certain limit only the annular ﬂow solution becomes satisfactory. Therefore, the method proposed by Dobson and Chato (1998) considers the addition of the contributions of the forced convection heat transfer in the lower part of the horizontal tube and the ﬁlm-wise condensation in its upper part weighted according to the stratiﬁed angle γ, deﬁned as illustrate in Fig. 3.4. They suggested the approximation proposed by Jaster and Kosky (1976) to estimate the stratiﬁcation angle as a function of the superﬁcial void fraction, given as follows: γ ﬃ1

cos 1 ð2α 1Þ π

ð7:43Þ

with the void fraction, α, given according to the Zivi (1964) correlation, Eq. (2.52).

7.2 Predictive Methods for In-Tube Convective Condensation

253

The ﬁnal correlation proposed by Dobson and Chato (1998) is given as follows:

0:25 0:23 Re 0:12 hd GaPr l v0 Nul ¼ ¼ þ ð1 γ=π ÞNuFC kl Jal b 0:58 1 þ 1:11X tt

ð7:44Þ

where 0:4 NuFC ¼ 0:0195 Re 0:8 l Pr l

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c 1 þ c12 b X tt

ð7:45Þ

and the constants c1 and c2 for 0 Frl0 0.7 are given as follows: c1 ¼4:172 þ 5:48Frl0 1:564Fr2l0 c2 ¼1:773 0:169Frl0 and for Frl0 > 0.7: c1 ¼7:242 c2 ¼1:655 Frl0 is the Froude number for the two-phase mixture ﬂowing as liquid, Eq. (3.127), Rev0 is the Reynolds number for the two-phase mixture ﬂowing as vapor (Rev0 ¼ Gd/μv), Ga is the Galileo number, Eq. (7.41), Jal is the liquid Jakob number, Eq. (7.6), and Rel is the Reynolds number considering only the liquid parcel (Rel ¼ G(1-x)d/μl). Dobson and Chato (1998) were pioneers in proposing a ﬂow-pattern-based method for the prediction of the heat transfer coefﬁcient for in-tube horizontal condensation. However, some aspects were not properly addressed in their method as follows: (i) the correlation proposed by Zivi (1964) adopted by Dobson and Chato (1998) to estimate the void fraction neglects the effect of the mass velocity on the void fraction, which can be relevant in certain conditions; (ii) the heat transfer coefﬁcient variation is not continuous as the ﬂow pattern changes from stratiﬁed to annular. Such a behavior is not realistic according to the experimental studies and may result in convergence problems when this method is incorporated in a software for heat exchanger simulation and optimization; (iii) shear effects and secondary ﬂows close to the tube wall promote an increase in the liquid height near the wall for stratiﬁed ﬂows, reducing the effective stratiﬁed angle, γ. Later, these aspects were properly addressed by Thome and coworkers (El Hajal et al. (2003) and Thome et al. (2003)) in their ﬂow-pattern-based method for condensation inside horizontal tubes.

254

7 Condensation

7.2.2

Cavallini et al. (2006)

As mentioned before, the heat transfer during condensation in conventional channels is governed mainly by gravitational and convective effects, and predictive methods were published based on the combination of these effects. Therefore, in order to simplify the approach, Cavallini et al. (2006) proposed a predictive method for the heat transfer coefﬁcient that identiﬁes conditions dependent and independent of ΔT, with speciﬁc corresponding governing mechanisms. Based on this characterization, it is possible to infer the local ﬂow patterns that can be annular, wavy stratiﬁed, or intermittent ﬂow. The annular ﬂow pattern is characterized by conditions of high ﬂow velocities, hence, the heat transfer process is dominated by convective effects, and presents small contribution of gravitational effects. Conversely, the heat transfer during stratiﬁed wavy ﬂow is dominated by gravitational effects, and during intermittent ﬂows by the combination of gravitational and convective effects, with dominance of convective effects. The conditions dominated by either mechanism is determined based on inertial effects of vapor phase, represented by the non-dimensional gas velocity, described by Wallis (1969), as follows: Gx jv ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ g d ρv ð ρl ρv Þ

ð7:46Þ

The transition value of the non-dimensional gas velocity was proposed by Cavallini et al. (2006) based on their experimental results as follows:

jv,trans ¼

8"

jv,trans ): " hann ¼ hl0 1 þ 1:128 x

0:8170

# 0:3685 0:2363 2:144 ρl μl μv 0:100 1 Prl ρv μv μl ð7:48Þ

7.2 Predictive Methods for In-Tube Convective Condensation

255

where hl0 is estimated according to Dittus and Boelter (1985) for the mixture ﬂowing as liquid as follows: 0:8 Gd kl hl0 ¼ 0:023 Pr 0:4 l d μl

ð7:49Þ

For conditions that gravitational effects contribute signiﬁcantly to the heat transfer process ( jv jv,trans), the ﬂow might correspond to stratiﬁed wavy or intermittent ﬂows. Therefore, to account for these conditions with no discontinuity of the heat transfer coefﬁcient estimative due to ﬂow pattern transition, Cavallini et al. (2006) proposed the following relationship to predict the heat transfer coefﬁcient: "

hsw,slug ¼ hann

jv,trans jv

#

0:8 hsw

jv

jv,trans

ð7:50Þ

þ hsw

where hsw corresponds to the heat transfer coefﬁcient estimated for stratiﬁed wavy ﬂow, adjusted by the researchers accordding to their experimental results, and given as follows:

hsw

0:725 ¼ 0:3321 1 þ 0:741 1x x

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 4 k ρ ðρ ρ Þgb v ilv l l l þ 1 x0:087 hl0 μl d ðT sat T s Þ

ð7:51Þ

where hl0 is given by Eq. (7.49). Notice that the ﬁrst term of the RHS is similar to the derived relationship for the heat transfer coefﬁcient during ﬁlm condensation, and given by Eq. (7.36), emphasizing that this term accounts for gravitational parcel and is dependent on the temperature difference. Figure 7.2 depicts the variation of heat transfer coefﬁcient with vapor quality predicted according to the Cavallini et al. (2006) method for R600a and R1234ze (Z) in 12.7 mm ID tube, showing also the transition between conditions dominated or not by gravitational effects. It can be observed according to this ﬁgure that the heat

20000

b) 14000 G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

R600a, d = 12.7 mm, Tsat = 40 °C, Ts = 36 °C

12000 10000

h [W/m²K]

h [W/m²K]

a) 30000

10000

8000

G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

R1234ze(Z), d = 12.7 mm, Tsat = 40 °C, Ts = 36 °C

6000 4000

j*v,trans

j*v,trans

2000 0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

0 0.0

0.2

0.4

x [-]

0.6

0.8

1.0

Fig. 7.2 Variation of the heat transfer coefﬁcient with vapor quality according to Cavallini et al.’s (2006) predictive method for (a) R600a and (b) R1234ze(Z) in a 12.7 mm ID tube

256

7 Condensation

transfer coefﬁcient is signiﬁcantly impacted by the mass velocity. Moreover, it is also shown in this ﬁgure that R600a presents considerably higher heat transfer coefﬁcient than R1234ze(Z). This behavior is associated to the higher vapor-speciﬁc volume of R245fa compared to R134a. Furlan (2020) suggests the method of Cavallini et al. (2006) for the prediction of the heat transfer coefﬁcient during condensation in small diameter channels of hydrocarbons based on a comparison of 14 prediction methods and a broad database from the literature.

7.2.3

Shah (2016)

Shah (2016) proposed a new method for prediction of the heat transfer coefﬁcient during condensation in horizontal small diameter tubes (deq 3 mm) using a broad database gathered from literature including experimental results for 13 ﬂuids (water, CO2, halocarbon refrigerants, and hydrocarbons), hydraulic diameters from 0.10 to 2.8 mm; reduced pressures ranging from 0.0055 to 0.94; mass velocities from 20 to 1400 kg/m2s; and channels of rectangular, circular, semi-circular, triangular, and square cross-sectional shapes. The original Shah correlation (Shah 1979) for conventional channels and its updated versions (Shah 2009 and Shah 2013) were considered as the starting point for this new method. The ease of its implementation and the reasonable predictions when compared to independent databases are the main features of the method of Shah (2016). In this method, the heat transfer coefﬁcient is estimated according to heat transfer regimes deﬁned by the author based on heat transfer behaviors instead of different two-phase ﬂow topologies. The regimes are given as follows: Regime I: When Wev0 100 and the dimensionless vapor velocity deﬁned in Eq. (7.46) is: jv 0:98ðZ þ 0:63Þ0,62

ð7:52Þ

where Z was deﬁned by Shah (1979) as follows: Z ¼ ð1=x 1Þ0:8 p0:4 r

ð7:53Þ

the condensation heat transfer coefﬁcient is given by Eq. (7.48) with hl0 estimated through Eq. (7.49). For non-circular channels, the all-vapor Weber number is calculated using the hydraulic diameter. In the method of Shah (2016), all the other equations are based on an equivalent diameter, deﬁned by him as the ratio of four times the cross-sectional ﬂow area and the heat transfer perimeter.

7.2 Predictive Methods for In-Tube Convective Condensation

257

Regime II: If the regime is neither I nor III according to the respective criteria, the heat transfer process corresponds to regime II and the condensation heat transfer coefﬁcient is given as follows: h ¼ hann þ hNu

ð7:54Þ

where hann is given by Eq. (7.48) and hNu is given by the following equation: 1=3

hNu ¼ 1:32 Re l

ρl ðρl ρv Þgk 3l μ2l

1=3 ð7:55Þ

Regime III: If 1 jv 0:95 1:254 þ 2:27Z 1:249

ð7:56Þ

the heat transfer coefﬁcient is equal to hNu given by Eq. (7.38).

7.2.4

Jige, Inoue, and Koyama (2016)

Jige, Inoue, and Koyama (2016) performed a series of experiments for determination of pressure drop and heat transfer coefﬁcient during condensation inside multiport microchannel heat sink, operating with synthetic refrigerant ﬂuids. During convective condensation inside small-scale channels, the surface tension forces also have signiﬁcant contribution on ﬂow development and heat transfer process, and this contribution increases as the channel diameter decreases. Conversely, the gravitational effects are suppressed due to reduction of channel size, as discussed in Chap. 3. Based on these aspects, Jige and coworkers proposed a heat transfer coefﬁcient predictive method that accounts for these mechanisms and basically comprises intermittent and annular ﬂow patterns. The heat transfer for annular ﬂow pattern was modelled by the authors by solving a simpliﬁed model of the liquid ﬁlm ﬂow in a rectangular channel accounting for the shear stress and surface tension effects, obtaining different correlations according to the dominant mechanism. For conditions dominated by friction force, which is related mainly to convective effects, the heat transfer coefﬁcient is given as follows: hf ¼

kl Φv0 dh 1 x

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ρ f v0 l Rel 0:6 þ 0:06Re0:4 Pr 0:3 l l ρv

ð7:57Þ

258

7 Condensation

where the term inside the brackets can be considered as intensiﬁcation factors of convective effects, in a similar way to the proposition of Chen (1966) for convective boiling and discussed in Chap. 5. The term dh corresponds to the hydraulic diameter, and Rel corresponds to the Reynolds number only for the liquid phase based on Þdh hydraulic diameter (Rel ¼ Gð1x ). The two-phase multiplier of the vapor-phase μl Φv0 was proposed by Jige et al. (2016) based on their experimental results and is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1:25 0:75ﬃ ρ f μl ρv 1:8 1:43 v l0 Φv0 ¼ x1:8 þ ð1 xÞ þ 0:65x0:68 ð1 xÞ ρl f v0 μv ρl

ð7:58Þ

The friction factor for the mixture ﬂowing as each phase for circular channel are given as follows:

f v0

f l0

8 16 > > < Re for v0 ¼ 0:046 > > for : Re 0:2 v0 8 16 > > < Re for l0 ¼ 0:046 > > for : Re 0:2 l0

Re v0 1500 Re v0 > 1500

ð7:59Þ

Re l0 1500 Re l0 > 1500

ð7:60Þ

with the Reynolds numbers for the mixture ﬂowing as each phase is also evaluated Gd h h based on hydraulic diameter ( Re l0 ¼ Gd μl and Re v0 ¼ μv ). For non-circular channels, it is recommended to check the original publication, or Kays and London (1998), for the corresponding friction factor or Nusselt number relationship. In the same token, the heat transfer coefﬁcient for conditions dominated by surface tension effects is given as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 ρlbilv σdh kl hs ¼ 0:51 μl kl ðT sat T s Þ dh

ð7:61Þ

where no suppression factor for the surface tension effects was indicated by the original authors. Based on these aspects, Jige et al. (2016) combined both parcels for estimative of the heat transfer coefﬁcient during annular ﬂow in a canonical way as proposed by Churchill (2000), given as follows: hann ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 h3f þ h3s

ð7:62Þ

7.2 Predictive Methods for In-Tube Convective Condensation

259

Flow direction

Liquid slug

Vapor bubble

Liquid single-phase flow

Annular flow

Liquid single-phase flow

Annular flow

Fig. 7.3 Schematics of intermittent ﬂow, adapted from Jige et al. (2016)

The authors modelled the intermittent ﬂow as a successive passage of liquid slugs and vapor bubble, and the heat transfer coefﬁcient is evaluated assuming liquid single-phase ﬂow, and annular ﬂow, respectively. This approach is somewhat similarly to the three-zone model proposed by Thome et al. (2004) for convective boiling inside microchannels. The model proposed by Jige et al. (2016) is schematically depicted in Fig. 7.3, and the resulting heat transfer coefﬁcient is given as follows: h ¼ ð1 αÞhl þ αhann

ð7:63Þ

where the weighting factor α is the void fraction evaluated according to a homogeh i1 Þ neous model (α ¼ 1 þ ρρv ð1x ). The heat transfer coefﬁcient for liquid singlex l

phase ﬂow hl is evaluated according to the Gnielinski (1976) correlation for turbulent ﬂow, and the equivalent for laminar ﬂow as follows:

hl ¼

8 > > > > > >

rﬃﬃﬃﬃﬃ > > d h , Rel > 2000 > 2 > f l > : 1 þ 12:7 Pr 3l 1 2

ð7:64Þ

with the liquid Reynolds evaluated assuming hydraulic diameter and only the Þdh liquid parcel of the two-phase ﬂow (Rel ¼ Gð1x ), and the liquid friction factor μl evaluated according to Eq. (7.60) however with Rel instead of Rel0. In summary, the heat transfer coefﬁcient is estimated according to Eq. (7.63), and the annular ﬂow condition consists of long vapor bubbles, without liquid slugs. Figure 7.4 depicts the variation of the heat transfer coefﬁcient with vapor quality for

260

7 Condensation

14000

20000 G = 50 kg/m²s

h [W/m²K]

10000 8000

G = 100 kg/m²s G = 200 kg/m²s

16000

G = 400 kg/m²s G = 800 kg/m²s

h [W/m²K]

12000

12000

G = 1600 kg/m²s

6000

G = 50 kg/m²s G = 100 kg/m²s G = 200 kg/m²s G = 400 kg/m²s G = 800 kg/m²s G = 1600 kg/m²s

8000

4000 4000 2000 R134a, d = 1 mm, Tsat = 40 °C, Ts = 36 °C

0 0,0

0,2

0,4

0,6

x [-]

0,8

1,0

0 0,0

R245fa, d = 1 mm, Tsat = 40 °C, Ts = 36 °C

0,2

0,4

0,6

0,8

1,0

x [-]

Fig. 7.4 Heat transfer coefﬁcient variation with vapor quality during convective condensation inside microchannels, according to the method of Jige et al. (2016)

R134a and R245fa inside 1 mm ID microchannel. In a similar way to the predictions for macrochannels, the heat transfer coefﬁcient increases with vapor quality and mass velocity, which is related to intensiﬁcation of convective parcel. Rossato et al. (2017) pointed out that the predictive method proposed by Jige et al. (2016) provided the best agreement of their experimental results for heat transfer coefﬁcient during convective condensation inside multichannel heat exchanger.

7.3

Solved Examples

Examples comprising concepts discussed in this chapter are presented and discussed. Considering R134a at a temperature of 40 C ﬂowing at 250 kg/m2s in horizontal channel, which has uniform wall temperature of 35 C, estimate the heat transfer coefﬁcient for the following: (a) For internal diameter of 10 mm, evaluate the heat transfer coefﬁcient of saturated liquid and superheated vapor, with superheating degree of 5 C. (b) Similar to the previous exercise, however for channel with internal diameter of 1 mm. (c) Now, consider two-phase ﬂow at Tsat ¼ 40 C with vapor quality of 90% in a 10 mm ID channel. (d) Reevaluate for convective condensation with vapor quality of 90% in a 1 mm ID channel. Solution: Based on the saturation temperature, it is possible to obtain the thermophysical properties of the ﬂuid for saturation conditions, as follows: ρl ¼ 1147 kg/m3 ρv ¼ 50.12 kg/m3 μl ¼ 1.612104 kg/ms μv ¼ 1.268105 kg/ms

7.3 Solved Examples

261

Prl ¼ 3.19 Prv ¼ 0.9017 îlv ¼ 163,003 J/kg cpl ¼ 1498 J/kgK cpv ¼ 1145 J/kgK kl ¼ 7.572102 W/mK kv ¼ 1.61102 W/mK In the case of liquid ﬂow, the properties are evaluated for saturation conditions. In the case of vapor, for superheating of 5 C at T ¼ 40 C, the corresponding pressure is 887.5 kPa, with resulting relevant properties: ρv ¼ 42.03 kg/m3 μv ¼ 1.261105 kg/ms kv ¼ 1.592102 W/mK Prv ¼ 0.8506 Then, it is possible to solve the example items. (a) Single-phase ﬂow inside channel with 10 mm of internal diameter, and wall temperature of 35 C. Based on the input data and thermophysical properties, it is possible to estimate the Reynolds number for both phases (Re ¼ Gd/μ), resulting in 15,511 and 198,187 for saturated liquid and superheated vapor, respectively, hence, turbulent regime for both phases. Then, by selecting the predictive method proposed by Gnielinski (1976), which is valid for Reynolds number higher than 2500, it obtained heat transfer coefﬁcient values of 651.8 and 553.2 W/m2K for liquid and vapor, respectively. (b) Now, repeating the previous item, the Reynolds number for liquid and vapor phases are, respectively, 1551 and 19,819. Hence, it corresponds to laminar regime for saturated liquid ﬂow, and turbulent for vapor ﬂow. Then, adopting the Gnielinski (1976) predictive method for superheated vapor for heat transfer coefﬁcient, the obtained value is 900.8 W/m2K, which is 63% higher than the case of ﬂow inside a larger channel despite the Reynolds number reduction, and can be attributed only to channel dimension reduction. In the case of saturated liquid ﬂow, considering that the boundary condition corresponds to uniform wall temperature, the Nusselt number is constant and equal to 3.657, resulting in heat transfer coefﬁcient of 276.9 W/m2K, which is 58% lower than the case of ﬂow inside a larger channel due to ﬂow regime difference. (c) In the case of two-phase ﬂow, since the surface temperature is smaller than the ﬂuid temperature (Tw < Tsat), the heat transfer process occurs by condensation. Hence, a proper predictive method must be selected, and considering the channel size, which is 10 mm, it is possible to infer whether it corresponds to micro or macrochannel. Based on Kew and Cornwell (1997), given by Eq. (2.69), the

262

7 Condensation

transitional channel diameter is 1.508 mm, hence for d ¼ 10 mm the channel can be considered a macroscale case. Therefore, let us consider the predictive method proposed by Cavallini et al. (2006) for heat transfer coefﬁcient during condensation inside macroscale channels. By solving Eqs. (7.46), (7.47), (7.48), (7.49), (7.50), and (7.51), the following main parameters are found: jv* ¼ 3.065 jv,trans* ¼ 2.552 (