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Forced convection heat transfer to turbulent flow of supercritical water in a round horizontal tube Bazargan, Majid 2001

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F O R C E D C O N V E C T I O N H E A T T R A N S F E R T O T U R B U L E N T F L O W O F S U P E R C R I T I C A L W A T E R I N A R O U N D HORIZONTAL T U B E by M A J I D B A Z A R G A N B . S c , Shar i f University of Technology, 1986 M. .Sc , The University of New Brunswick, 1995 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F M E C H A N I C A L ENGINEERING We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2001 © Majid Bazargan, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date A*VV \1 01 ^ ^ k n l NO, DE-6 (2/88) ABSTRACT Very little heat transfer data for supercritical fluids can be found in the literature over the last twenty-five years. Most of the available data are for vertical and not horizontal flows. This investigation focuses on horizontal flows. The pilot SCWO system, constructed at the Department of Mechanical Engineering, UBC in partnership with NORAM, a Vancouver based engineering company, was used to accomplish this. The current data was obtained with accurate state of the art measurement and data acquisition equipment. Measurements of heat transfer to pure supercritical water are carried out in the test section before oxygen and sample wastes are introduced to the system. This database was used to critically review and evaluate available empirical, forced convection, heat transfer correlations. For this study, a new approach leading to a more accurate empirical correlation for the flows with negligible effects of the flow acceleration and the buoyancy was offered. The conditions for the onset of buoyancy and acceleration forces were also investigated. For flows unaffected by buoyancy and acceleration, a numerical model, incorporating recent improvements in turbulence modeling, was developed. The model was found to predict the data of the current study very well. It also predicted the available data from other studies well. Overall, it can be said that this study encapsulates all the necessary knowledge required by an engineer to design a horizontal SCW system that operates within a range of safe operating conditions (i.e. no deteriorated heat transfer due to buoyancy and/or acceleration effects). ii TABLE OF CONTENTS ABSTRACT ii T A B L E OF CONTENTS iii LIST OF TABLES vii LIST OF FIGURES viii LIST OF SYMBOLS xiv A C K N O W L E D G M E N T xvi CHAPTER 1 INTRODUCTION 1 1.1 A Supercritical Fluid • 1 1.2 Heat Transfer to Supercritical Water 4 1.3 Engineering Applications 7 1.4 Objectives of Present Study 9 1.5 Method of Approach 9 CHAPTER 2 LITERATURE REVIEW 12 2.1 Empirical Correlations 13 2.2 Buoyancy Effects 19 2.3 Modeling Studies 22 CHAPTER 3 EXPERIMENTAL APPARATUS 32 3.1 Overview of SCWO Facility 33 3.2 Safety Issues 37 3.3 Temperature Measurements 38 3.4 Pressure Measurements 41 3.5 Flow Measurements 43 3.6 Power Measurements 43 3.7 Heat Flux Measurements 45 3.8 Data Acquisition 46 CHAPTER 4 RESULTS AND DISCUSSIONS 48 4.1 Experimental Procedure 49 4.1.1 Thermophysical Property Data 49 iii 4.1.2 Test Conditions and Restrictions 50 4.1.3 Calculation of Local Heat Transfer Coefficients 50 4.1.4 Typical Variations in Heat Transfer Coefficients 52 4.1.5 Construction of Graphs 53 4.1.6 Corrections to Bulk Temperature Measurements 55 4.2 Effects of Various Control Variables 57 4.2.1 Effect of Inlet Bulk Temperature 57 4.2.2 Effect of Mass Flow 59 4.2.3 Effect of Heat Flux 60 4.2.4 Effect of Pressure 61 4.3 Heat Transfer Deterioration 64 4.3.1 Deterioration along the Top Surface 64 4.3.2 Local Hot Spots on the Wall 65 4.4 Comparison with Other Studies 68 4.5 Flow Classification by Means of Dimensional Numbers 72 CHAPTER 5 EMPIRICAL CORRELATIONS 78 5.1 General Heat Transfer Correlations 78 5.2 Correlations for Supercritical Fluid Flows 79 5.3 Comparison with Experiments 82 5.4 Effect of Different Fluid Properties 88 5.5 Effect of Natural Convection 93 5.6 The Assumption of "Fully Developed Flow" 96 5.7 A New Empirical Correlation 98 CHAPTER 6 EFFECTS OF B U O Y A N C Y AND ACCELERATION . 108 6.1 Buoyancy Effects 108 6.1.1 Measurements of Buoyancy Effect 109 6.1.1.1 A Case Study with Large Buoyancy Effects 110 6.1.1.2 Experiments with Mild Buoyancy Through Increased Flow 115 6.1.1.3 Negligible Buoyancy Test 118 6.1.2 Criteria for Onset of Buoyancy Effects 120 6.1.3 Heat Transfer Calculations during Buoyancy-Free Conditions . . . 125 6.1.4 Prediction of Temp. Difference between Top and Bottom surfaces 126 6.2 The Effect of Acceleration 129 CHAPTER 7 A N A L Y T I C A L MODELING 136 7.1 Governing Equations and Simplifying Assumptions 137 7.2 Constant Property Solution 139 7.3 Variable Property Solution 142 7.3.1 Effect of Expressions for Universal Velocity Profile 146 7.3.2 Comparison with Other Data 151 7.4 Modification of the Model 153 iv 7.4.1 Modification of Expression for Eddy Diffussivity 154 7.4.2 Effect of Wall Shear Stress 155 7.4.3 Effect of Turbulent Prandtl Number 159 7.5 The Number of the Grids and the Stability of the Solutions 163 7.6 Numerical Modeling Versus Empirical Correlation 166 7.5 Significance of Buffer Zone 167 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 171 8.1 Empirical Correlations 171 8.2 Buoyancy and Acceleration Effects 173 8.3 Numerical Modeling 175 8.4 Recommendations for Future Work 176 BIBLIOGRAPHY 178 APPENDIX A CHRONOLOGICAL LITERATURE REVIEW 197 1939 The Earliest Published Study 197 1954 First Systematic Approach 198 1957 First Experimental Studies 199 1957 First Empirical Correlation 199 1960 Flow Visualization over A Cylindrical Wire 200 1961 Emergence of the Idea of Pseudo-Boiling 201 1961 Modifying the Expression for Eddy Diffusivity 202 1961 The First Reasonably Accurate Empirical Correlation 203 1962 The First Studies with Hydrogen 204 1962 Cooling Heat Transfer with Supercritical Fluids 205 1963 Systematic Study of Deteriorated Heat Transfer 205 1964 Measurements of Velocity and Temperature Profiles 206 1965 Significance of Entry Length 207 1965 Flow Oscillations 209 1966 Correlations Based on Pseudo-Boiling Proposal 211 1966 The Effect of Buoyancy 213 1967 Relation Between Deteriorated Heat Transfer and Grashof Number. . . .214 1967 First Measurements of Turbulence 215 1967 Flow Visualization in A Forced Convection Flow 216 1968 Lack of Data for Horizontal Flow 218 1969 A Comprehensive Review 219 1969 Deteriorated Heat Transfer in Downward Flow 221 1970 Large Number of Experiments with Relatively Small Success 224 1971 Confirmation of M-Shape Velocity Profile in Vertical Flow 225 1971 Qualitative Prediction of Deterioration by A Numerical Model 227 1971 Clear Distinction between Vertical and Horizontal Flows 228 1972 Advancement in Numerical Modeling 230 1972 Deteriorated Heat Transfer, The Research Priority 231 1972 Pseudo-Boiling Proposal Still Alive 233 1973 More Details on Oscillatory Nature of Flow 235 1973 Friction at Critical Region 237 1975 First Two-Dimensional Numerical Model 240 1975 Acceleration and Buoyancy effects Accounted 241 1976 Flow Similarity Conditions 244 1976 Supercritical Cryogenic Fluids, Easier To Model 245 1977 Effect of Gravity Accounted in A 1-D Model 247 1979 Contributions of Hall and Jackson 249 1979 Implementation of K-e Model 252 1981 Peak Values of Thermal Conductivity at Pseudo-Critical Temperature . .254 1982 Emerge of SCWO 255 1983 Incorporation of Density Fluctuations 258 1986 Measurements of Heat Flux and Shear Stress Profiles 261 1986 Criteria for Negligible Effects of Body Forces 262 1997 A Comprehensive Study of Entry Region 267 2001 Current Status 268 APPEND DC B TEST CONDITIONS 271 APPENDIX C COMPUTER CODES 279 APPENDIX D MIXING LENGTH, E D D Y DIFFUSIVITY AND T H E UNIVERSAL VELOCITY PROFILE 335 vi LIST OF TABLES 6.1 Summary of the test results for mixed convection flows 119 7.1 Expressions of the universal velocity profile 147 vii LIST OF FIGURES Chapter 1 1.1 Phase diagram for water 2 1.2 Variations of properties of a supercritical water with temperature at P=24 MPa .3 1.3 Typical variation of heat transfer coefficients for subcritical and supercritical water flowing in a round tube, G=800 kg/m2.s, q"=200 kW/m 2, D=6.3 m m . . . .5 1.4 Block Diagram for Supercritical Water Oxidation Process 8 Chapter 3 3.1 Schematic of UBC/NORAM Pilot Plant of SCWO 33 3.2 Electrical heating schematic for the preheaters and test section 35 3.3 Schematic of the test section 36 3.4 Spot welded attachment of thermocouples to the outside tube surface 39 3.5 Position of bulk and surface thermocouples along the test section 40 3.6 Absolute and differential pressure transducer in the test section 41 3.7 Typical variations of voltage of the secondary of the transformer under full load (top figure) and 55% power load (bottom figure) 44 Chapter 4 4.1 Outside wall and fluid bulk temperature profiles, G=662 kg/m2.s, P=24.4 MPa, q"=195 kW/m 2 52 4.2 Enhancement of heat transfer coefficients at near critical region, G=662 kg/m2.s, P=24.4 MPa, q"=195 kW/m 2 53 4.3 Variations of the wall temperature along the flow in four tests with different inlet temperatures, G=662 kg/m2.s, P=24.4 MPa, q"=195 kW/m 2 54 viii 4.4 Effect of inlet, middle and outlet temperatures on heat flux calculations 56 4.5 Effect of the flow history on the wall temperature distribution, P=24.8 MPa, G=592 kg/m2.s, q"=263 kW/m 2 58 4.6 Effect of mass flow on heat transfer coefficient, q"=300 kW/m 2, heat transfer coefficients are measured at the bottom surface of the test section 59 4.7 Effect of heat flux on temperature distribution (G=575 kg/m2.s). The wall temperatures are measured at the top surface of the test section 60 4.8 Effect of heat flux on heat transfer coefficients (G=575 kg/m2.s) . 61 4.9 Variations of the specific heat of water Cp, with pressure 62 4.10 Variations of pseudo-critical temperature of water with pressure 63 4.11 Effect of pressure on heat transfer coefficients, G=592 kg/m2.s, q"=263 kW/m 2 63 4.12 Heat transfer coefficients at different peripheral positions, P=24.3 MPa, G=432 kg/m2.s, q"=301 kW/m 2 64 4.13 Local deterioration of heat transfer, P=23.7 MPa, G=585 kg/m2.s 66 4.13 U B C experiments, P=24.3 MPa, G=432 kg/m2.s, q"=300 kW/m 2 69 4.15 Results of Shitsman (1963) for a vertical flow with D=8 mm tw: wall temperature, ts: bulk temperature, i: bulk enthalpy, P=23.3 MPa, G=430 kg/m2.s, (a) q"=233, (b) q"=291, (c) q"=326, and (d) q"=384 kW/m 2 69 4.16 UBC experiments, P=24.3 MPa, G=662 kg/m2.s, q"=303 kW/m 2 70 4.17 Results of Vikhrev and Lokshin (1964) for a horizontal flow, D=6 mm. Wall temperature is shown for (1) q"=699 , (2) q"=582 , (3) q"=465 , and (4) q"=349 kW/m 2 71 4.18 Wall temperature variations for two flows with, P=23.8 MPa, D=6.3 mm and identical q"/G of 0.3 kJ/kg 74 4.19 Heat transfer coefficients for a flow with q"/G=0.18 kg/kJ, P=24.8 MPa, G=575 kg/m2.s, q"=100 kW/m 2 74 4.20 Results of Yamagata et al. (1972) for Flows with q"/G=0.18 kg/kJ 75 ix 4.21 Variations of inside wall temperature at top surfaces of the test section for three case studies: (G=336 kg/m2.s, q"=75 kW/m 2: q".D°-2.G"° 8=0.26), (G=581 kg/m2.s, q"=265 kW/m 2: q".D° 2.G"° 8=0.59), and (G=432 kg/m2.s, q"=300 kW/m 2: q " . D ° 2 . G 0 8 = 0 . 8 5 ) 76 4.22 Results of Yamagata et al. (1972) for different values of q " D 0 2 / G 0 8 77 Chapter 5 5.1 Comparison of correlations (flow conditions identical to Swenson et al.), P=31 MPa, q"=789 kW/m 2, G=2149 kg/m2.s, D=9.42 mm 83 5.2 Comparison of correlations (flow conditions identical to Yamagata et al.), P=24.5MPa, q"=465 kW/m 2, G=1200 kg/m2.s, D=10 mm 84 5.3 Comparison of available correlations with experiments of this study, P=25.2 MPa, q"=307 kW/m 2, G=965 kg/m2.s, D=6.3 mm 87 5.4 Comparison of available correlations with data of this study for a buoyancy-affected flow, P=23.9 MPa, q"=124 kW/m 2, G=340 kg/m2.s 88 5.5 NBS and IAPWS transport properties of water at 23 MPa 90 5.6 NBS and IAPWS transport properties of water at 25 MPa 90 5.7 Effect of transport properties on empirical correlations, P=23 MPa, q"=200 kW/m 2, G=600 kg/m2.s, D=6.2 mm 91 5.8 Effect of transport properties on empirical correlations, P=25 MPa, q"=200 kW/m 2, G=600 kg/m2.s, D=6.2 mm 92 5.9 Comparison of correlations under no buoyancy conditions tried by Yamagata, P=24.5 MPa, q"=233 kW/m 2, G=1260 kg/m2.s, D=7.5 mm 95 5.10 Variation of T r ef with respect to T D and T w along the test section, P=25.2 MPa, q"=307 kW/m 2, G=965 kg/m2.s, D=6.3 mm 99 5.11 Predictions of Dittus-Boelter with use of different Tref, P=25.2 MPa, q"=307 kW/m 2, G=965 kg/m2.s, D=6.3 mm 100 5.12 Typical variation of iZone=(iPc-ib)/(ipc-iw) along the test section 101 5.13 Results of the correlation of this study compared with others and experiments, P=25.2 MPa, q"=307 kW/m 2, G=965 kg/m2.s, D=6.3 mm 103 X 5.14 Results of the correlation of this study compared with others and experiments, P=24.4 MPa, q"=103 kW/m 2, G=575 kg/m2.s, D=6.3 mm 103 5.15 Effect of the critical izone number on heat transfer coefficients, P=24.5 MPa, q"=233 kW/m 2, G=1260 kg/m2.s, D=7.5 mm 104 5.16 Comparison of the local heat transfer coefficients predicted by the correlation of this study and others. The test conditions of Swenson et al. are examined; P=31 MPa, q"=789 kW/m 2, G=2149 kg/m2.s, D=9.42 mm 105 5.17 Heat transfer coefficients measured by Yamagata et al. (1972) 106 5.18 Heat Heat transfer coefficients Predicted by the empirical correlations of this and other studies. Test conditions are the same as Fig. 17 107 Chapter 6 6.1 Wall temperature at top and bottom surfaces, P=24.4MPa, G=964 kg/m2.s, q"=0 (no heat flux) 110 6.2 Wall and bulk temperature variations along the first segment of the test section, P=24.4 MPa, G=340 kg/m2.s, q"=297 kW/m 2 I l l 6.3 Wall and bulk temperature variations along the 2 n d segment of the test section, P=24.4 MPa, G=340 kg/m2.s, q"=300 kW/m 2 112 6.4 Differences of heat transfer coefficients at top and bottom surfaces, P=24.4 MPa, G=340 kg/m2.s, q"=300 kW/m 2 113 6.5 Variation of water density with temperature at P=24.4 MPa 114 6.6 Wall and bulk temperature variations along the test section, P=24.4 MPa, G=432 kg/m2.s, q"=300 kW/m 2 115 6.7 Heat transfer coefficients corresponding to the flow specified in Fig. 6.6 . . . .116 6.8 Wall and bulk temperature variations along the test section, P=24.3 MPa, G=646 kg/m2.s, q"=303 kW/m 2 117 6.9 Heat transfer coefficients corresponding to the flow specified in Fig. 6.8 . . . .117 6.10 Wall and bulk temperature variations along the test section, P=25.2 MPa, G=964 kg/m2.s, q"=307 kW/m 2 118 6.11 Heat transfer coefficients corresponding to the flow specified in Fig. 6.10 . . . 119 xi 6.12 Variations of Gr/Re 2 with bulk enthalpy for various flows (q"=300 kW/m2). . . 120 6.13 Variations of Gr/Re 2 7 with bulk enthalpy for various flows (q"=300 kW/m2)..121 6.14 Variations of Gr q /Gr t n and wall temperature with bulk enthalpy for Case I . . . 124 6.15 Variations of Gr q /Gr t n with bulk enthalpy for cases III and IV 124 6.16 Generalized presentation of experimental data on the wall temperature difference between the top and bottom surfaces in a horizontal flow, taken from Petukhov and Polyakov (1988). (1) Data of Vikhrev et al. (1970) for water. (2) Data of Belyakov et al. (1971) for water. (3) Data of Adebiyi and Hall (1976) for carbon dioxide 127 6.17 Generalized presentation of experimental data on the wall temperature difference between the top and bottom surfaces in a horizontal flow for Cases I and III 128 6.18 Variations of acceleration parameter y with bulk enthalpy for cases 1,111 and IV 132 6.19 Comparison of buoyancy and acceleration effects for case IV (q"= 300 kW/m 2 and G=965 kg/m2.s) 134 6.20 Comparison of buoyancy and acceleration effects for case I (q"= 300 kW/m 2 and G=340 kg/m2.s) 135 Chapter 7 7.1 Comparison of the results of constant-property model with experiments, P=24.76 MPa, q=232 kW/m 2, G=571 kg/m2 and d=6.3 mm 140 7.2 Typical velocity and temperature profiles calculated by the present Variable-property model 145 7.3 Typical heat transfer coefficients predicted by the present model along with UBC data at P=24.8 MPa, q"=232 kW/m 2, G= 571 kg/m2.s 146 7.4 Effect of various expressions of universal velocity profile on the results of present numerical model for flow of a superheated steam at P=0.1 MPa, q"=788 kW/m 2, G=2150 kg/m2.s and D=9.42 mm 148 7.5 Comparison of predictions of present model utilizing various expressions for eddy diffusivity with UBC experiments, q"=307 kW/m 2, 964 kg/m2.s 149 x i i 7.6 Comparison of the results of present model utilizing various expressions of universal velocity profile with experiments, P=24.4 MPa, q"=103 kW/m 2, G=575 kg/m2.s 150 7.7 Predictions of local heat transfer coefficients by present model compared with the results of Swenson et al., P=31 MPa, q"=788 kW/m 2, G=2150 kg/m2.s, D=9.42 mm 152 7.8 Comparison of predictions of present model with results of Yamagata et al., P=24.5 MPa, q"=233 kW/m 2, G= 1260 kg/m2.s, D=7.5 mm 153 7.9 Comparison of the model predictions with use of y + and u + differently defined, P=24.8 MPa, q"=263 kW/m 2, G=592 kg/m2.s and D=6.3 mm 154 7.10 Effect of the expression used for shear stress at the wall on heat transfer, P=25.2 MPa, G=965 kg/m2.s, q"=307 kW/m 2 157 7.11 The effect of Prt on local heat transfer coefficients, P=25.2 MPa, G=965 kg/m2.s, q"=307 kW/m 2 160 7.12 Comparison of model predictions (equation of Hollingsworth et al. used for Prt) with experiments. Test conditions are the same as Fig. 7.10 161 7.13a Variations of local heat transfer coefficients with the water bulk temperature, experiments of Yamagata et al. (1972) 162 7.13b Predictions of present numerical model, empirical correlation of this study and correlation of Swenson et al., the flow conditions are as of Fig. 7.13a . . . 162 7.14 Effect of number of grids on variations of heat transfer coefficients, P=24.5 MPa, G=575 kg/m2.s, q"=103 kW/m 2 163 7.15 Effect of number and sizes of the grids on model predictions, P=24.5 MPa, G=575 kg/m2.s, q"=103 kW/m 2, Tb=380 °C 165 7.16 Radial variation of the flow region with peak values of Cp along the test section. Shaded area represents the portion of flow experiencing temperatures 168 7.17 Temperature profiles normalized by pseudo-critical temperature. Different profiles correspond to various locations along the test section 169 xiii LIST O F S Y M B O L S Bth threshold value of body forces Cp specific heat capacity, kJ/kg.K Cp integrated specific heat capacity, kJ/kg.K D tube diameter, mm f friction factor G mass flux, kg/m2.s g gravitational acceleration constant, m/s2 Gr Grashof number Gr q Grashof number (heat flux used in the definition) Gr t h threshold value of Grashof number h heat transfer coefficient, kW/m 2 .°C i specific enthalpy, kJ/kg I current, Amps Nu Nusselt number P pressure, MPa p electrical power Pr Prandtl number q" heat Flux, kW/m 2 R resistance, ohms Re Reynols number r radial distance from center of the tube, mm r 0 tube radius, mm xiv 1 T temperature, °C t time, s u fluid velocity in axial direction, m/s V -> volume, m V voltage across the pressure transducer, Volts V fluid velocity in radial direction a thermal diffusivity P volumetric expansion coefficient, 1/K e momentum and thermal eddy difussivity once Prt=1.0 eh thermal eddy diffusivity Em momentum eddy diffusivity K thermal conductivity, W/m.°C dynamic viscosity, kg/m.s V kinematic viscosity, m2/s P density, kg/m3 P resistivity, Q.m o surface tension, X shear stress, N/m 2 tt turbulent shear stress, N/m 2 Subscript b bulk rms root mean squared w wall X V ACKNOWLEDGMENT First and foremost, I need to praise the merciful God who is the origin of all the good deeds and great accomplishments. I should also thank many people whom without their support and help I could never complete this work. By name in particular, I would like to thank my supervisor Dr. Dan Fraser for his constant encouragement and kindness. His positive attitude towards my achievements compensated the dissatisfaction I always have with respect to my own work and helped everything go smoothly. His support and optimism kept me motivated throughout the whole course of this study. Although Dr. Steven Rogak was not officially my research advisor, in many occasions he was as helpful as a co-supervisor could be. I do not remember leaving his office or from a meeting with him unless I had learnt a new thing to think about. I also consider the times I spent with Dr. Phil Hill the most precious moments I had at UBC. To me, like to many other students of him, Professor Hill is certainly an exemplary scientist and teacher. The thermo-physical properties of water as well as the insight and knowledge he offered to enrich this study are very much appreciated. I wish to thank Dr. Richard Branion for his support, kindness and the time he spent to correct hundreds of grammatical error I had made in my thesis! I would like to thank Drs. Martha Salcudean, John Grace and Bruce Bowen for serving in my examination board and for their invaluable comments. I was lucky to meet and be with a number of wonderful fellow graduate students during my stay at UBC. I should thank Danijela Filipovic in particular, as well as Vladan Prodanovic, Tazim Rehmat, Wang Shuo, Paul Teshima, Xu Li, Sanja Boskovic, Ivette Vera-Perez and Mohammad Khan for their help and being so nice to me. I specially thank Alex Podut who was my office-mate as well as a very good friend. Alex was the solution to almost any engineering problem I faced in the lab that I couldn't tackle by myself. There have been a number of UBC and N O R A M technicians and engineers who were, in one way or another, involved in construction of the experimental apparatus. In particular, help from Sean Bygrave and Stuart Gairns was highly appreciated. Loving gratitude is extended to my wife, Freshteh, who left aside her own ambitions for what she thought was important for me. She took most of the responsibility, xvi including taking care of children, to provide me peace of mind to achieve my academic goal. I am also struck by unbelievable understanding and tolerance my children Alireza and Mahmoud demonstrated during the whole course of my study at UBC. I also thank my children for correcting my English whenever they had no other fun to get busy with. Here at UBC campus, thousands of kilometers away from home, my family and I never felt lonely or homesick because of sincere companionship of a number of wonderful families around us. I really cannot list them all. However, I wish to especially thank Dr. Mohammad Soleiman-Panah, Hadi Rastegar, Dr. Mohammad Movahhedi and late Dr. Rasoul Pandi. I certainly owe my success to many great men and women of the past and present who taught me the most important lessons, showed me the path and shaped my life. I sincerely appreciate the endless love and support of my late father and mentor, Morteza (Aaghaa Joon), my mother Ashraf Saadaat and my beloved sisters and brothers. Finally, the financial support provided by the Ministry of Higher Education of Iran and NSERC of Canada are gratefully acknowledged. xvii Dedicated to my magnificent family, Freshteh Alireza and Mahmoud XX *" '« ' ' C H A P T E R O N E I N T R O D U C T I O N The academic significance and wide engineering applications have made heat transfer to supercritical fluid flows an important topic of research over several decades. It is one of the most general and complicated examples of single phase forced convection. The recent development of systems (e.g., Supercritical Water Oxidation (SCWO) systems) operating at supercritical pressures necessitates revisiting the problem. A thorough understanding of heat transfer is crucial towards the optimal design of such systems. The peculiarity of forced convection heat transfer to supercritical fluid stems from the nature of fluids at supercritical pressures. 1.1. A S U P E R C R I T I C A L F L U I D The thermodynamic critical point is the state of a fluid where saturated liquid and saturated vapor are identical. The pressure and temperature at the critical point are called the critical pressure and the critical temperature. At subcritical pressures, i.e., any pressure smaller than critical pressure, there is a saturation temperature below which the fluid exists in a liquid phase. At the saturation temperature, the fluid may be in a saturated liquid or vapor state or in between, defined by a vapor mass fraction or quality, x. At temperatures greater than the saturation temperature the fluid is in a vapor phase. l At a supercritical pressure, i.e., any pressure above critical pressure, no saturation temperature exists. Nevertheless, around a certain temperature close to the critical temperature the variations in fluid density, as well as in other properties, are tremendously large. This temperature is called the pseudo-critical temperature. Raising the fluid temperature at constant pressure from below to above the pseudo-critical temperature is not associated with a change in the fluid phase. There will be a continuous variation of density and other properties and the fluid remains in a single phase state. The variations of properties, however, are so dramatic across the pseudo-critical temperature that the fluid at temperatures below and above the pseudo-critical point is referred to as liquid-like or vapor-like respectively. A typical P-T diagram of a fluid is shown in Figure 1.1. The pseudo-critical temperature is slightly greater than critical temperature and increases with pressure. The loci of pseudo-critical temperatures are shown with a dotted line. Figure 1.1. Phase diagram for water. Dotted line is the loci of pseudo-critical temperatures. 02 S u p e r c r i t i c a l R e g i o n Liquid-Like Region I Vapor-Like Region GAS Critical Point T E M P E R A T U R E 2 The term "supercritical fluid" literally defines a fluid at both a pressure and a temperature higher than the critical point. However, the fluid at a supercritical pressure is sometimes called a supercritical fluid while the temperature may be smaller or greater than the pseudo-critical temperature. The neighborhood of the pseudo-critical point, where violent variations of properties take place, is referred to as the critical region. The closer the pressure is to the critical pressure, the more drastic are the property variations. Variations of some of properties of water with temperature at a supercritical pressure of 24 MPa are shown in Figure 1.2. Note that the critical point of water is 22.1 MPa and 374 °C. The pseudo-critical temperature corresponding to a pressure of 24 MPa is 380 °C. Density as well as transport properties such as viscosity decrease dramatically across the critical region. 300 3 2 0 340 360 380 4 0 0 4 2 0 4 4 0 4 6 0 4 8 0 T e m p e r a t u r e , °C Figure 1.2. Variat ions of properties of a supercritical water with temperature at P=24 M P a Consider, for example, the variation of specific heat shown in Fig. 1.2. Of particular significance is the huge peak value occurring at the pseudo-critical 3 temperature. It is an order of magnitude higher at the peak than before or after the pseudo-critical point. This can be shown mathematically from the following relationship. Cp = Cv + dP)r (1.1) It becomes clear from the above expression, when one considers the terms in the right hand side, that there will be a maximum in values of Cp at the pseudo-critical temperature. The partial derivative (dP/9p)T will approach to an extremely small number near the pseudo-critical point due to the large variation of density over a very small change in pressure at constant temperature. That is while all the other terms in the right hand side of Eq. 1.1 have finite values at the vicinity of the pseudo-critical point. Following the anomalous variations of Cp and other properties shown above, heat transfer to supercritical fluid flows is also unusual. 1.2. HEAT TRANSFER TO SUPERCRITICAL WATER Typical variations of local heat transfer coefficients for supercritical water, as well as for water (in this case steam) at atmospheric pressure, flowing in a round tube are shown in Figure 1.3. All conditions except pressure are the same in both flows. It can be seen that heat transfer is significantly enhanced near the critical region. The variation of heat transfer coefficients with bulk fluid temperature at supercritical pressures resembles the way that specific heat varies with temperature. However, other properties are also varying dramatically (e.g., thermal conductivity decreases substantially across the critical region) and enhanced heat transfer cannot be simply explained in terms of variations of one property. Consider the following explanation for example. More discussions, including the effects of various properties on enhanced heat transfer, are presented in chapter four of this study. The thermal resistance at the wall layer is an important parameter responsible for the rate of heat transfer from the wall to the fluid. The thermal resistance at the wall 4 depends on two factors, thermal conductivity and the thickness of the layer. Thermal conductivity (apart from a very narrow peak, which does not have great significance) falls with temperature in the near-critical region. This tends to reduce the heat transfer rate. However, a decrease in viscosity and increased specific heat suppress the extent of the high resistance wall layer. In other words, the spike of Cp and drop of viscosity across the pseudo-critical temperature overcome the effects of decreased thermal conductivity by so much that heat transfer is greatly enhanced. Downstream, in the post pseudo-critical region, the specific heat decreases and so does the heat transfer rate. Figure 1.3. Typical variation of heat transfer coefficients for subcritical and supercritical water flowing in a round tube. G=800 kg/m2.s, q"=200 kW/m2, D=6.3 mm. It is important to know that the Cp variation is not always dominant over variations of other properties. The enhancement, as shown in Fig. 1.3, is not the only mode of heat transfer at supercritical pressure. In fact, within the same range of temperature, i.e., in the critical region, and under certain conditions (high heat flux and 5 low mass flow), heat transfer may deteriorate leading to hot spots in a tube. Some data on deteriorated heat transfer are presented and discussed in chapter four. An analogy has been made in the past between heat transfer at supercritical conditions and boiling at subcritical pressures. The enhancement of heat transfer was assumed to be due to a pseudo-boiling process. Instead of two distinct liquid and vapor phases in subcritical boiling, liquid-like and vapor-like fluids are involved in case of supercritical fluid flow. Hence, the extent of heat transfer enhancement in the latter is milder as compared to actual subcritical boiling. Degraded heat transfer was attributed to a vapor-like layer of fluid blanketing the wall and inhibiting the heat flow from the wall to the center. It happens when the heat flux is high and a large number of vapor-like chunks of fluid are produced near the wall. They get a chance to collect together and form a layer of low-density fluid adjacent to the wall. This, again, was a mild version of what is called the boiling crisis at subcritical pressure. The pseudo-boiling type of explanation attracted considerable attention for many years. It seemed to be able to describe both enhanced and degraded modes of heat transfer at supercritical pressure. However, since it did not lead to a successful physical model to predict heat transfer coefficients, it gradually lost its popularity. Note that our knowledge about two-phase flows is mainly based on empirical relationships whereas the theories of single-phase convection heat transfer are well established. Since it is evident that the fluid exists in one single phase at supercritical pressures, theoretically the anomalous behavior of heat transfer should be able to be modeled in terms of single-phase forced convection heat transfer. It is, in fact, a special case of the variable-property single-phase convection. The explanation follows. The flow of water under normal pressure is considered a constant-property flow. In such flows, the differences in properties of water at the wall and bulk temperatures usually are not large. Evaluation of the properties at a film temperature, i.e., the mean of bulk fluid temperature and the temperature at a wall, in many occasions describes the flow adequately. For simple flow configurations, like pipe flows, available empirical correlations or numerical models can satisfactorily predict heat transfer coefficients. In an empirical correlation, Nusselt number, and hence heat transfer coefficient, is correlated against the Reynolds and Prandtl numbers. Since the properties do not vary greatly, the 6 relationship between Nusselt, Reynolds and Prandtl numbers remains more or less the same along the course of the flow. Unless the heat transfer nature of the fluids are different, as in liquid metals and oil for example, empirical correlations developed for a constant-property flow enjoy universality to a great extent and may apply over a wide range of flow conditions. For supercritical fluid flows, the differences in fluid properties at the heating surface and the flow centerline are large. Practically, this may happen when the heat flux is extremely high and/or the fluid properties are highly sensitive to temperature variations as it occurs near the critical region. The difficulty with empirical correlations is that near the critical region none of the wall, bulk or even film temperatures can represent the flow temperature at which the properties are to be estimated. The available empirical correlations largely disagree with each other and some of them are highly case dependent. This is discussed in detail in chapter five. Similarly, since the density variation across the pseudo-critical line is large, a secondary flow is likely to develop. Buoyancy effect, in many situations is significant and cannot be neglected. In addition to buoyancy effects, large variations (drops) of density with temperature along the test section cause fluid acceleration. Thus, the axial component of convection, which is usually neglected in normal pressure flows, needs to be investigated. The effects of buoyancy and acceleration are discussed in chapter six. From the above, it becomes clear why heat transfer to supercritical fluids is considered as the most general example of the variable-property flow. It is a classic problem of convection heat transfer and an ongoing research topic. In addition to academic significance, the study of heat transfer to supercritical fluids is important from an engineering perspective. 1.3. E N G I N E E R I N G A P P L I C A T I O N S There are a number of systems in which supercritical fluids are used as coolants or propellant fluids. Supercritical boilers have been used in steam turbine cycles for decades. Supercritical boilers have an improved rate of heat transfer. The single phase flow of supercritical water in the boiler tubes eliminates the need of a steam drum to separate steam from the liquid water. More recently, the idea of supercritical heat transfer 7 has been implemented in design of the direct-cycle, supercritical water-cooled, fast breeder reactor (Oka et al., 1993). The result is a substantial improvement in thermal efficiency when compared to boiling water reactors. Supercritical helium has been used for many years in cryogenic systems. In super conducting devices, supercritical helium performs as an ideal heat transfer medium. Supercritical hydrogen also is used as both the propellant and the heat sink in chemical and nuclear propulsion systems. In some modern military aircraft, the fuel is pressurized above its critical point and used as a coolant to remove heat from the aircraft engine. Another recent application of heat transfer to supercritical fluids is in the waste management industry. The idea is to destroy toxic aqueous waste and the method is called Super-Critical Water Oxidation, SCWO. Along with many thermodynamic and transport properties, the solubility properties of water dramatically change at supercritical conditions. At normal temperature and pressure, water is a good solvent for inorganic compounds and a non-solvent for many organics. On the contrary, supercritical water is an excellent solvent for organic materials but not for inorganic salts. This leads to the idea of supercritical water oxidation. A simplified block diagram of the process is shown in Figure 1.4. Oxygen Water & Organics H Heating & Pressurizing Single Phase Reaction Cooling M Separation H co 2 H 2 0 • Ash Figure 1.4. Block Diagram for Supercritical Water Oxidation Process Destruction efficiencies of 99.99% or higher have been reported for a variety of toxic and nontoxic materials treated by SCWO. The closed operating cycle makes it controllable and assures a complete containment of all effluents. No harmful byproducts 8 such as N0X are produced and SCWO is environmentally friendly. Heat transfer data is essential to improve the design of SCWO systems. 1.4. OBJECTIVES OF PRESENT STUDY The main purpose of this study is to investigate heat transfer to horizontal flows of supercritical water towards optimizing the design of SCWO systems. The majority of the available data are for vertical flows. A lack of data for horizontal flow of supercritical fluids is evident. Most of the available data are at least twenty-five years old. Since then, improvements in knowledge of the transport properties of supercritical water has had a direct impact on the heat transfer results. Available empirical heat transfer correlations as well as numerical models are not adequate for horizontal flows. The ultimate goal is to predict heat transfer coefficients with a better precision than is available in the literature. The focus of the present study is on flows with negligible effects of buoyancy and acceleration. Whether or not such effects are important makes a considerable difference in the analysis. Thus, it is essential to accurately predict the onset of buoyancy and acceleration effects. For example, many attempts in the past to develop an empirical correlation for forced convection heat transfer in fact were based on mixed convection data. That is because natural convection becomes a factor in a supercritical fluid flow much more readily than in a constant property flow. Furthermore, to compare various correlations, it requires careful screening of the available data to exclude the flows affected by buoyancy. This has been neglected or not properly done in many previous assessments of available correlations. The current study aims to clarify any confusion existing between pure forced and mixed convection situations at supercritical conditions. Similar ambiguity exists for the effect of acceleration. The goal is to specify the conditions where the effect of acceleration is negligible. 1.5. METHOD OF APPROACH Both experimental and numerical approaches have been taken to fulfill the objectives of the current study. A pilot SCWO system has been constructed at the Department of Mechanical Engineering, U B C in partnership with NORAM, a Vancouver based engineering company. The test section in this study is horizontal. The current data 9 are obtained with accurate state of the art measurement and data acquisition equipment. A large database is produced against which empirical correlations and analytical models may be compared and evaluated. Measurements of heat transfer to pure supercritical water were carried out in the test section without oxygen or sample wastes introduced to the system. From an engineering perspective, empirical correlations are the most popular and handy tools to predict convection heat transfer rates. For supercritical fluid flows however, there is a large disagreement between available empirical correlations which effectively reduces their applicability. It is one of the main goals of the present study to develop a new empirical correlation, which predicts heat transfer coefficients more reliably than available correlations. Learning about the sources of disagreement between available correlations is essential towards a better understanding and an accurate prediction of heat transfer to the supercritical fluids. It will also lead to a thorough assessment of the available correlations. With respect to buoyancy effects, wall temperatures were measured at the top and bottom surfaces of the test section. The difference between the heat transfer coefficients at the top and bottom surfaces specifies the extent of buoyancy effects. A wide range of flow conditions, from highly affected by buoyancy to buoyancy-free flows, was tried. This provided the opportunity to study the conditions for the onset of buoyancy effects. The criteria developed in the past for distinguishing buoyancy-free regions have been well established for vertical supercritical fluid flows. However, the effects of buoyancy are different between vertical and horizontal flows. The results of studies for vertical flows are not easily applicable to horizontal flows. Very few theoretical criteria have been suggested in the literature to detect the buoyancy-free region for horizontal supercritical fluid flows. They need to be carefully tested. There is a large volume of analytical studies available in the literature. The recent numerical models focus on vertical flows under influence of buoyancy. They mostly tend to predict a deteriorated mode of heat transfer. It is difficult to apply such models to horizontal flows. Earlier models were based on one-dimensional analysis and were basically indifferent to the flow orientation. They were meant to solve flows with negligible buoyancy effects. Nevertheless, they do not predict heat transfer coefficients 10 any better than available empirical correlations. The current study aims to establish a numerical model based on the law of the wall. Recent developments in defining the turbulent Prandtl number are incorporated into the model. Variations of fluid properties are also taken into account. The model will help to understand and explain some peculiarities of supercritical fluid flows. 11 C H A P T E R T W O L I T E R A T U R E R E V I E W The concept of the thermodynamic critical point of fluids was introduced more than a century ago. It has been known, since then, that properties of a fluid beyond its critical point are quite different than those at lower pressures. Investigation about convection heat transfer to supercritical fluids, however, started about sixty years ago. In some review papers (e.g., Hendricks et al. 1969) the earliest work has been attributed to the German researcher Schmidt in 1939. He arranged an experimental setup with a supercritical environment and showed that free convection heat transfer characteristics were appreciably altered near the critical region. He drew attention to the impact that variations of property could have on heat transfer to a fluid near its critical region and ignited the idea of thinking of supercritical fluids as suitable heat carriers. Hundreds of researchers have studied the phenomenon of heat transfer to supercritical fluids so far. The most complete review of the literature, in terms of the number of studies examined, on forced convection heat transfer to supercritical fluids has been made available in the present study. About 200 documents were reviewed and are presented in a chronological order in Appendix A. Areas of interests in this field of research are wide. Relevant investigations available to the author were addressed. Nevertheless, not all such works are directly related to the present study. For example, a large volume of investigations revolved around the notion that heat transfer to 12 supercritical fluid flow was a pseudo-boiling process. The idea tried to establish an analogy between enhanced heat transfer to supercritical fluids and the two-phase boiling as well as between deteriorated heat transfer and the boiling crisis. Since it did not lead to a successful mathematical tool, it was gradually discarded. Those studies are extensively reviewed in Appendix A but are not repeated in this chapter. Another example is the local deteriorated heat transfer which has occupied a significant part of the literature, but is not of primary interest in this study. The keywords, which may best describe the main focus of the current study, are empirical correlations, enhanced mode of heat transfer, horizontal flows, onset of buoyancy effects, and numerical models. Accordingly, the highlights of Appendix A that concern these topics are reproduced in this chapter. 2.1. E M P I R I C A L C O R R E L A T I O N S Bringer and Smith (1957) were among the first to perform a set of experiments with a supercritical fluid flowing in a round tube. Carbon dioxide, with critical point of 7.4 MPa and 31 °C, was selected as the working fluid. Enhancement of heat transfer near the critical region was clearly observed in Bringer and Smith's experiments. At that time empirical heat transfer correlations in the form of Nu=aRebPrc (e.g., the correlation of Dittus and Boelter, 1930) were well developed and practiced for normal pressure flows. Bringer and Smith showed that such correlations were inadequate to predict heat transfer coefficients near the critical region. The results of Miropolski and Shitsman (1957) with forced flow of supercritical water in a round tube also confirmed the region of enhanced heat transfer near the critical region. They recommended a modified empirical correlation to predict heat transfer during turbulent forced convection flows. It was the first time an empirical relationship was offered to fit experimental data. The form of their correlation was simple and was obtained by making minimum changes to the Dittus-Boelter correlation. Weight was given to the Prandtl number, Prmin, which is the lesser of the Prandtl number at the wall and in bulk and was used instead of the typical bulk Prandtl number only. It appeared that the suggested correlation provided a better prediction of heat transfer coefficients compared to some of the more complicated correlations developed later. Miropolski and 13 Shitsman clarified, however, that their proposed correlation was good only for fluids with Prandtl numbers around unity. For cases where temperature variations, and hence changes of fluid properties were noticeable, the conventional correlations needed to be modified. This had already been practiced successfully for varying property gases and/or liquid flows. For example, Seider and Tate (1936) included a correction factor of wall-to-bulk fluid viscosity in a Dittus-Boelter type of correlation to account for property variations. Petukhov et al. (1961) expanded this idea and applied three correction factors as functions of the wall-to-bulk fluid viscosity, specific heat and thermal conductivity to model supercritical fluids. The correlation they used as the basis (before applying the correction factors) was that of Petukhov and Kirillov (1958). To obtain the most general correlation, they collected the experimental results available so far and developed the following relationship Nub = (Cf/2)RQbPvb („ Y™Y, ^ 3 / ^ - N 0 . 3 5 nj^(Cf/2)(Prb2/'-l)+L01 b Cp CPb (2.1) The subscripts w and b stand for the wall and bulk respectively. Averaged specific heat was defined as Cp=(iw-ib)/(Tw-Tb), where enthalpy was denoted by i. The heat transfer coefficients predicted by their correlation agreed satisfactorily with data for water and carbon dioxide under moderate wall to bulk temperature ratios. This correlation is still often referred to today. (It will be examined in more detail later in this study and will be shown that relatively speaking it is one of the most accurate correlations.) Hendricks et al. (1962) carried out a set of experiments with supercritical hydrogen. They offered a mathematical model based on the idea of pseudo-boiling first proposed by Goldmann (1961). Their model agreed, within 20% with their data. Only a narrow range of flow conditions was measured, hence, assessment of the model was incomplete. Their model incorporated the well-known Martinelli parameter used in two-phase flow studies. For supercritical flow, as there were no distinct phases, the fluid was considered to consist of heavy and light species. The quality x was redefined correspondingly. 14 Bishop et al. (1964) conducted an experimental heat transfer study on upward flow of water in cylindrical and concentric annulus tubes. Their data were correlated with a Dittus-Boelter type of relationship using a correction factor in terms of densities at the wall and bulk as well as a fade away term reflecting an effect of entry length. Inclusion of the wall-to-bulk ratio of the fluid density appeared to improve the data curve fitting. Swenson et al. (1965) broadened the existing database to water. Entry length effects were demonstrated. They showed that, unlike constant property flows, the effect of an entry region did not decrease monotonically with length. Thus, conventional fade away terms, similar to that used by Bishop et al. (1964), could not explain the considerably extended entry region at supercritical fluid flows. They also studied the effects of pressure and heat flux on heat transfer coefficients. Outside of an entry region, an empirical correlation was developed which matched other measurements within ±15% in 95% of the cases. Only low heat fluxes were examined. Swenson et al. also tried their correlation against carbon dioxide data and found good agreement. Since this correlation is still referred to in some recent studies, more discussion and comparison of the results will follow in later chapters. Krasnoshchekov et al. (1965) studied a wide range of conditions during pipe flow of supercritical C O 2 . For outside the entry region, an empirical correlation was suggested using correction factor of (pw/pb)°' 3(Cp/Cpb) n, where n had different values corresponding to local wall and bulk temperatures. Agreement to within ±15% with the data was claimed. Later, in a similar study, Krasnoshchekov and Protopopov (1966) compared their correlation with the data of Bringer and Smith (1957) and others. Poor agreement was reported. It will be shown in chapter five of this study that Nusselt number correlations, generally, do not attain universality and are highly case dependent. In a review paper published by Hall et al. (1967), discrepancies between correlations and theoretical results developed so far were shown to be significant. The experimental data themselves did not agree with each other even for near similar test conditions. That was partly due to uncertainties about fluid properties. Hall et al. concluded that there must be some other influencing parameters, which were not fully accounted for. One example mentioned was buoyancy effects. 15 The extensive set of measurements of Yamagata et al. (1972) confirmed that maximum heat transfer appeared at a flow cross section where the bulk temperature was slightly less than the pseudo-critical temperature. The wall temperature was always required to be greater than the pseudo-critical point. The difference between the corresponding wall temperature and the pseudo-critical temperature grew as heat flux increased. Thus in the pseudo-critical region, the smaller the heat flux, the higher the heat transfer coefficient was. At very low heat flux, measurement of temperature difference was very difficult and the heat transfer calculation could be highly erroneous. Regarding the effect of pressure they confirmed that the closer the pressure was to the critical point, the larger were the heat transfer coefficients obtained. Based on a comprehensive database collected for upward flow, Yamagata et al. developed a correlation which predicted heat transfer coefficients within a 20% range of measured values. The ratio of the wall to bulk for various properties was examined and it was discovered that none of them could reflect the effect of property variation as effectively as Cpjcpb did. Recall that the averaged specific heat, Cp, was defined as (iw-ib)/(tw-tb)- Based on the values of the Eckert number, (tpC-tb)/(tw-tb), three different correction factors were used in the suggested correlation. Dividing the heat transfer regimes into various categories raised the chance of better fitting the experimental data. On the other hand, the weak point of this method was the existence of a discontinuity in the heat transfer coefficients, as the wall temperature equaled the pseudo-critical temperature. They compared the correlation predictions against some earlier results for a sample case study. It appeared that the correlation of Yamagata et al. was in better agreement with experiments. Since that correlation is still among those currently being referred to, a detailed comparison of the results will be shown in later chapters. Like previous studies the suggested correlation was applicable only to low and moderate heat flux. Labuntsov (1972) applied a conventional Dittus-Boelter correlation with homogenized or averaged values of properties instead of the wall or bulk values. The averaging technique resulted in acceptable predictions for enhanced heat transfer. It also predicted the onset of a deteriorated regime. The method of property averaging seemed to be promising but was not referred to in later studies. 16 To exclude the buoyancy effects, Alferov et al. (1973) included only those data indifferent to direction of flow in their curve fitting analysis and developed an empirical correlation. The correlation was able to predict a number of others' results within ±30% error. Introducing a correction factor into a conventional heat transfer correlation was unlikely to yield a general solution. A particular fluid property, which is important at some cross section of the flow, may lose its significance relative to variations of another fluid property at some other position. Thus, there is no universal correlation capable of addressing the influence of property variations on heat transfer at supercritical pressure. Alternatively, a proper choice of reference temperature at which fluid properties were estimated in a Dittus-Boelter type of correlation might significantly improve the outcome. The reference temperature could be defined as a function of wall, bulk and/or film as well as the pseudo-critical temperatures. Nevertheless, the search of Hsu and Graham (1976) showed that any such temperature function was not general and, depending on test conditions, might differ from case to case. Hall and Jackson (1979a) categorized empirical correlations in two groups. In one group correction factors, in terms of some function of wall-to-bulk ratio of fluid properties, were introduced into conventional correlations. Others used a different temperature reference than wall or bulk or film at which fluid properties were estimated. Sometimes a combination of both was also tried. Neither of these methods, however, led to a satisfactory general solution. Yeroshenko and Yaskin (1981) studied the applicability of empirical correlations developed for other supercritical fluids to a vertical flow of supercritical helium. Six correlations were examined by means of three sets of experimental data collected from different sources. The data included the normal, enhanced and deteriorated regimes of heat transfer. No obvious superiority in either of the correlations was observed. The empirical correlations in many cases included term(s) dealing with friction coefficients one way or another. The friction coefficients developed for constant property flows might be modified by means of correction factors to fit variable property conditions. According to Kurganov and Ankudinov (1985), modification of friction coefficients used in a traditional correlation (e.g., the correlation of Petukhov and 17 Kirillov, 1958) may improve predictions of heat transfer coefficients more effectively than applying a correction factor directly to the heat transfer correlation. Ghajar and Asadi (1986) compared the predictions of seven different empirical correlations with available experimental data. The study was limited to the cases where no deterioration occurred and the effect of buoyancy was negligible. The reviewed correlations were different from each other in terms of the correction factor(s) they had applied to a Dittus-Boelter type of correlation. Thus, the study of Ghajar and Asadi was concerned with how to decide what type of correction might result in a better prediction of the data. They recommended the correlation of Jackson and Fewster (1975). Ghajar and Asadi used an updated set of properties for water and carbon dioxide. They adjusted the constants of the correlation of Jackson and Fewster accordingly. As a result they ended up with two sets of constants for the correlation, one valid for water and the other for carbon dioxide. The revised correlation of Jackson and Fewster will be examined later in our current study. In a review of empirical heat transfer correlations for normal regimes of heat transfer to supercritical fluids, Kakac (1987) recommended the expression of Krasnoshchekov and Protopopov (1965) as the most general and accurate one. Once again, the lack of data for horizontal flows was emphasized. Razumovskiy et al. (1990) stated that available, empirical correlations did not lead to satisfactory results in many cases because the friction factors used were not accurate. The friction coefficient had already been calculated for vertical flows of supercritical carbon dioxide by a number of investigators. Razumovskiy et al. needed to repeat similar measurements for water, as it was known that the thermo-inertia properties of water and carbon dioxide differed significantly. They came up with a new correction factor to be applied to the friction coefficient of a constant property flow to fit it into variable property conditions. Most previous correction factors were a function of wall-to-bulk ratio of fluid densities. Razumovskiy et al. included a function of viscosity ratio as well. Meanwhile they dropped the viscosity ratio from the correction factors used in the heat transfer correlation, so that the effect of variations of viscosity would not be counted twice. Thus, they recommended the classic correlation of Petukhov and Kirillov to be used for supercritical water flow with two changes. (1) A correction factor in terms of 18 values of specific heat at wall and bulk was included in the correlation. (2) The coefficient of friction was estimated as explained above. An extensive comparison of their predictions of heat transfer coefficients with experiments was not offered. Walisch et al. (1997) carried out a set of experiments with supercritical carbon dioxide in vertical, inclined and horizontal pipe flows. The wall temperature was constant. The effect of pressure variations on Nusselt number was investigated. For each pressure and flow direction a correlation, totaling more than twenty different empirical relationships was derived. The constants and coefficients in the correlations were so different that it made them impractical to use, unless the pressure and wall temperatures were exactly the same as the experimental conditions. This seems to be in contrast with the philosophy of presenting data in reduced format to generalize the case. The large number of data presented, however, could be useful for other investigators for later comparison. Unlike constant heat flux, the constant wall temperature used by Walisch et al. as a boundary condition has not been tried by researchers frequently and thus not many similar studies are available for the purpose of comparison. Bazargan and Fraser (1999) reported part of the data, presented in this study, for horizontal flow of supercritical water in a round smooth tube. The disagreements between the predictions of different studies with experiments were shown and the sources of the discrepancies were discussed. None of the available correlations are universal. It was emphasized that extra precaution needed to be taken before applying the results of vertical flows to horizontal test sections. 2.2. B U O Y A N C Y E F F E C T S Shitsman (1966) appears to be the first to investigate natural convection (buoyancy) effects during the horizontal flow of supercritical water. Such an effect manifests itself as a temperature difference between top and bottom surfaces of a tube. Shitsman measured top and bottom temperatures at various locations along the tube. Temperature differences as high as 250 °C was observed. The product of the Grashof and Prandtl numbers was taken as a measure of the significance of buoyancy. The flow was divided into a few categories. A function of (Gr.Pr) was correlated with respect to the temperature difference between top and bottom surfaces. Thus, based on flow conditions, 19 temperature differences between the top and bottom surfaces could be predicted. Since there were no similar data available in the literature at that time, Shitsman's relationship could not be examined against any data but his own. This correlation has not been referred to frequently in later works. Belyakov et al. (1971) experimentally compared the wall temperatures and heat transfer coefficients between vertical and horizontal flows. A range of variables similar to what was practiced in a typical supercritical boiler was achieved. Mass flow rate per unit cross sectional area, G, varied from 300 to 3000 kg/m2 s and heat flux q was between 232 and 1396 kW/m . At temperatures below and above the critical region, the heat transfer coefficient was not substantially affected by the magnitudes of heat flux and mass flow. Experimental results were provided for heat transfer at different peripheral positions of the flow cross sections in horizontal tubes. At 135° and 225° (with respect to the top center) the heat transfer coefficients were almost the same as that of a vertical tube. The cause of asymmetry of heat transfer around the tube cross section was explained to be due to the joint influence of forced and free convection. A similar comparison between horizontal and vertical flows of supercritical water was made by Yamagata et al. (1972). The effect of each variable was systematically studied by letting it vary over the required range and measuring heat transfer while other variables were fixed. The results of Belyakov et al. (1971) were confirmed that at low heat flux, the same heat transfer coefficients were obtained for a vertical tube and the top and bottom surfaces of horizontal flow. For higher heat flux the highest heat transfer occurred at the bottom surface of the horizontal flow. Vertical flow, with no peripheral temperature difference, appeared to attain the second highest heat transfer coefficient. The lowest heat transfer occurred through the top surface of the horizontal flow. Adebiyi and Hall (1976) studied the effects of buoyancy in a horizontal flow of supercritical carbon dioxide. They measured the wall temperature at the top and bottom surfaces. A wide range of heat flux and mass flow was tried in order to distinguish between buoyancy-free and buoyancy-dependent cases. A relatively large tube diameter of 22.1 mm in the test section facilitated free convection activities. In some of the tests the upper wall temperatures (in degree C) were three times larger than lower surface temperatures. The results were compared with wall temperatures in vertical flows where 20 no peripheral temperature asymmetries existed. In general, it was noticed that the buoyancy effect caused not only a degradation of heat transfer at the tube top surfaces but also an improvement at the bottom. The criteria already suggested in the literature for a buoyancy-free region were examined. However, the Grashof number within the buoyancy-free region was at least 30 times smaller than the threshold value in their tests. Thus, the accuracy of the proposed criteria could not be thoroughly tested by Adebiyi and Hall results. The dimensionless number Grb/ReD ' had already been used to distinguish buoyancy-affected from buoyancy-free regions in upward and downward flows of supercritical fluids. Watts and Chou (1982) arranged an experimental setup and modified this number to Grb/(Reb2 7Prb°5) to better address the matter. In a rising flow of supercritical water, they noticed that once Grb/(Reb2'7PrD0'5) was less than 10"5 the buoyancy effect was negligible. For higher values of Grb/(Reb2'7Prb°'5) they introduced some correction factors in terms of Grashof number to their empirical correlation to compensate for the degradation of heat transfer owing to buoyancy influence. For down coming flows the same dimensionless parameter appeared to be useful in classifying the flow. If Grb/(Reb"'Prb ) was below 5x10" , no effect of buoyancy was observed in the experiments. For higher values, and in contrast with upward flows, the heat transfer was improved. They developed an empirical correlation to cope with the enhanced heat transfer conditions in downward flows. Bogachev et al. (1983b) included the effects of mixed convection in the upward flow of supercritical helium in a round tube. They focused on low Reynolds number flows for which fewer data were available in the literature. For both laminar and turbulent flows they developed empirical correlations which they claimed to be accurate within ±25% and ±20% of experimental values, respectively. The Grashof number was included in their relationships to account for the buoyancy effect. Within the range of parameters studied by Bogachev et al., the effect of property variation appeared to be less important compared to the influence of mixed convection. On this basis, they proposed that the correction factors in the form of some ratios of wall to bulk properties were not necessary. No comments were presented about accuracy or applicability of their correlation to supercritical water flows. 21 Petukhov and Polyakov (1986) clarified that heat transfer to supercritical fluids, despite the large variation in their properties, was usually fully developed at x/d>50. For vertical flows, two different parameters representing the effects of buoyancy and fluid acceleration were defined. The criteria were set to specify the regions under the influence of each of them. The body of the work was based on Polyakov (1975). Similar criteria were used for horizontal flows and the contributions of buoyancy and acceleration to temperature differences between top and bottom surfaces were studied. Comparison with experiments showed satisfactory agreement. To reach a better assessment of the criteria, however, more data are needed. For horizontal flow, in particular, the criteria were almost the only available ones systematically derived and will be discussed further in chapter 6 of the current study. Kakac (1987) also pointed out that criteria established for buoyancy-free region in vertical flows could be considered as experimentally proven. Those developed for horizontal flows have not been tested enough. 2.3. MODELING STUDffiS The novel studies of Deissler (1954) and Goldman (1954) were the first classic studies performed on forced convection heat transfer to supercritical fluids. Deissler solved simultaneously the equations of shear stress and heat transfer for a pipe flow by means of a simple analytical method. He employed a law of the wall type of analysis as a turbulence model. The basic postulation in the law of the wall is that velocity variations over any cross section of a turbulent flow fall onto the same profile (universal velocity profile) once expressed in a generalized system of coordinates (y+ and u+). An analogy between momentum and heat transfer was postulated, i.e., turbulent Prandtl number was assumed to be unity. He also assumed that the radial variations of shear stress and heat flux had a negligible effect on velocity and temperature profiles. This is equivalent to neglecting the convection terms in the equations of momentum and energy. The flow was divided into near wall and core regions. In an earlier study of air flowing in a pipe under high heat flux, Deissler (1950) developed an expression for eddy diffusivity near the wall. He used the same relationship for the case of supercritical fluid flow. Away from 22 the wall, where variations of the fluid are not as severe, the expression proposed by von Karman (1934) was used. Recall that du (2.2) dy and that the eddy diffusivity can be obtained from the expression of universal velocity profile, i.e., In this document the various forms of the equation relating y+, u + and em are alternatively referred to as "universal velocity profile" and "expression for eddy diffusivity". Deissler noted that for a given heat flux and wall temperature, velocities and temperatures could be calculated at any distance from the wall. The calculated velocity and temperature profiles were compared with experimental data for air. A relatively good agreement was obtained. From there, Deissler extended his method for the case of heat transfer to supercritical water. He presented his predictions without having any experimental data to compare with. Later experiments by other investigators showed that Deissler results for supercritical water were only qualitatively acceptable. Goldmann (1954) attempted to solve the same set of equations for shear stress and heat transfer with similar boundary conditions and expressions for the eddy diffusivity as Deissler. The distinction lay in the crucial assumption Goldmann made to account for the effect of fluid property variations. He supposed that the local eddy diffusivity was a function of fluid properties (i.e., fluid temperature) and was independent of small property variations adjacent to that location. In other words, he proposed that the same expression of eddy diffusivity, used for a constant-property flow, was applicable to supercritical fluid flow provided that fluid property variations with temperature (from the wall to the core) was accounted for in the calculation of the eddy diffusivity at each point. Based on this assumption, the distance from the wall to the center was divided into many (2.3) v du+ 23 intervals. Integration of the equations of shear stress and heat transfer led to velocity and temperature profiles. Comparison with the results of Deissler showed some decrease in the heat transfer coefficient. Such a decrease was shown, through experimental results made available later, to be an improvement. To explain enhanced heat transfer in terms of single phase forced convection, Hsu and Smith (1961) took a significant step ahead. They extended the analytical model of Deissler (1954) and included a correction factor in the expression for eddy diffusivity. The correction term was derived from a simple analysis that accounted for density fluctuations. The predicted velocity profile showed that during large density changes, the maximum velocity was not at the flow center. This was different than what occurred in a constant property flow. Heat transfer predictions were compared with available data. Only in some cases was a better agreement reached. Their study was applicable to vertical pipe flows. Later comparison of the results showed that the particular modification of Hsu and Smith was not very successful. Nevertheless, the significance of their work was in the approach they introduced. The message was that the equation of the universal velocity profile, developed for constant property flow, needed to be adjusted before applying it to variable property flows. Szetela (1962) compared the predictions of Deissler's (1954) solution against measurements made for the flow of supercritical hydrogen in a round tube. Deissler's method under-predicted Szetela's data by up to 33%. Once the density variations in the form suggested by Hsu and Smith (1961) were used, the results showed a 50% to 75% overestimation. The conclusion was that neither model was adequate. It was mentioned earlier that Deissler (1954) and Goldmann (1954) took into account the influence of radial variations of fluid properties in their analysis. However, the effects on radial variations of heat flux were not yet considered. This was the major accomplishment of Petukhov and Popov (1963). They used the expression of Reichardt for the universal velocity profile. Frictional effects were also considered. They demonstrated how their formulation reduced to the well-known correlation of Petukhov and Kirillov (1958) once property variations diminished. Hess and Kunz (1965) tried a similar approach to that of Deissler (1954). They tested the model with supercritical hydrogen data. The expression of van Driest for 24 universal velocity profile was used. An acceptable agreement with low heat flux cases was found. The original van Driest expression in which k=0.4 and A+=26 is as follows. du+ 2 (2.4) dy+ ~ l + {l + 4x - 2 y + 2 [ l - e ( - y + / A + ) ] } 1 / 2 For higher heat fluxes, a modification of the dimensionless damping constant A + in the van Driest expression was suggested. For the near wall region, viscosity, and the extent to which it was damped, seemed to dominate the flow structure. Hess and Kunz recommended A + = 30.2 e"0 0 2 8 5 ( ( W instead of a constant number to account for property variations on diffusivity. Nevertheless, they admitted that their model required complex numerical calculation and they did not recommend it. An empirical correlation, which matched their results closely, was developed and suggested instead. Hall et al. (1966) applied a similar approach. They used the expression of Corcoran et al. (1956) for the universal velocity profile. The results were not satisfactory. Then, a peak thermal conductivity near the pseudo-critical temperature was assumed. The existence of such peak value was a matter of debate by thermodynamicists at that time. Heat transfer predictions by the model did not show much improvement. From there, Hall et al. concluded that molecular convection did not have a significant effect in enhancing heat transfer. In the next step, the expression for eddy diffusivity was modified. It was assumed that the eddy diffusivity around the pseudo-critical temperature had its highest value. It decreased linearly away from the wall until it passed over the buffer zone. The results were slightly improved. Finally, the expression for the universal velocity profile was further modified by using a thermal expansion factor. This led to a better agreement with experimental data. The major difference between the numerical model of Tanaka et al. (1971) and Goldman's (1954) was the use of an alternate expression for the eddy diffusivity. Tanaka et al., however, declared that the choice of equation for eddy diffusivity did not affect their predictions significantly. This observation puts a question mark on the reliability of their solution. The expressions for the universal velocity profiles were not originally developed for variable-property flows. Thus, when applied to supercritical fluid flows, 25 they are expected to lead to different results. Furthermore, it was not clear what component of the study of Tanaka et al. improved their results compared to Goldman's predictions. Inoue et al. (1972) employed the same correction factor used by Hsu and Smith (1961), em=8o (1+ 31np/3lnu), to account for the effect of property variation on eddy diffusivity. In addition, the thermal eddy diffusivity in the energy equation was also modified to account for the non-linearity of enthalpy variation. Comparison of the results with experimental data showed a significant improvement for the case of high flow rate and low heat flux. Nishikawa et al. (1972) considered the axial variation of the fluid properties in their analysis. Depending on flow conditions the assumption of fully developed flow may not be applicable due to large variation of fluid properties near the critical region. For example, the temperature gradient at the wall (dT/dy)w, which is highly dependent on the thermal conductivity of the fluid, changes with wall temperature along the tube. Note that heat flux is uniform, i.e., q"=K(dT/dy)=constant. Consequently, the term 3i/3x (axial variation of the enthalpy) is considerable and may not be eliminated from the energy equation. With the aid of some simplifying assumptions to account for axial convection, Nishikawa et al. solved the equations of momentum and energy simultaneously. Nevertheless, the results did not change noticeably. This was due to the fact that the axial variation of the fluid properties influenced only the region immediate to the wall. In this region, the amount of heat carried by axial convection was a few order of magnitudes less than the radial heat flux. Nishikawa et al. also showed that the velocity profile did not have a significant effect on heat transfer coefficient. Furthermore, by manipulating a mixing length model and accounting for density fluctuations, they developed an enhanced mixing model for eddy diffusivity of momentum, which was different than the correction factor derived by Hsu and Smith (1961). A similar procedure remained to be done in order to account for the effect of density variations on enthalpy and hence on thermal eddy diffusivity. Due to the complexity of such calculations, it was simply assumed that the eddy diffusivity of heat was equal to that of momentum. The results of the enhanced mixing model for variable property fluid flow showed a better agreement with experimental data. The 26 improvement was mainly due to a decrease in temperature drop across the buffer layer as well as the turbulent core. Yet, the results underestimated the heat transfer coefficient around the pseudo-critical temperature. Presenting a two-dimensional mixing length model was an important contribution that Sastry and Schnurr (1975) made. The buoyancy effect was neglected and flow was assumed to be axisymmetric. Axial conduction and viscous dissipation were also assumed to be negligible. In fact, compared to earlier one-dimensional solutions, the only additional term accounted for was axial convection. The expression of van Driest was used for the eddy diffusivity. The analysis started from a cross section with known velocity and enthalpy profiles. This was usually chosen to be well upstream of the flow where the temperature was far less than the pseudo-critical temperature. Simultaneous solution of equations of conservation of mass, momentum and energy provided new velocity and enthalpy profiles for the next step downstream. Having pressure and enthalpy known, the wall temperature and local heat transfer coefficient were estimated. Comparison with experimental data for water and carbon dioxide showed good agreement as long as the inlet wall temperature was less than the pseudo-critical temperature. For the range under study, neglecting free convection appeared to be a valid assumption. The results of this analytical solution also suggested a significant influence of upstream conditions on heat transfer downstream. This was in contrast with some earlier findings (e.g. Nishikawa et al., 1972). According to Jackson and Hall (1979a), there were many problems upon which the improvement of analytical models rest. The effect of property variation on turbulence structure needed to be clearly understood. The expressions employed for eddy diffusivity were generally developed for constant property flows and their use for variable property flows was questionable. Correction factors introduced to adopt such expressions did not always improve the results and evidences in this regard were contradictory. (The correction factors were usually meant to account for the effect of density fluctuations on the expression for eddy diffusivity). Finally, they questioned the assumption of the turbulent Prandtl number being unity in supercritical fluid flow. In assessment of the results, they mentioned that in many cases it was difficult to detect whether the discrepancy between the numerical models and experiments were due to inadequate 27 physical modeling or lack of precision in the numerical scheme. As the most effective way to clarify many ambiguities they suggested further careful measurements of turbulence and velocity and temperature profiles for different range of control parameters. The disagreements between available 1-D models considering normal mode of heat transfer to supercritical fluid flows (enhanced heat transfer near the critical region) are not completely resolved. However, since the mid 70's, modeling studies tended to concentrate mainly on evaluation of deteriorated heat transfer in vertical flows due to buoyancy and/or acceleration effects. Those models were mostly 2-D analyses and were not applicable to horizontal flows, which are the subject of this study. However, reviewing some of the ideas and techniques used in them might be helpful. Petukhov and Medvetskaya (1979) contributed to the advancement of analytical models by replacement of Prt=l with a relationship that allowed variation of Prt with fluid properties. Their two-dimensional model was derived for an upward flow. The thermo-gravitational (buoyancy) and acceleration terms were accounted for and the solution was considered to be general. The predicted velocity gradient verified the existence of an M -shaped profile. Need for a more compatible turbulence model was clear from previous analytical studies. A K-e model of turbulence was used by Hauptmann and Malhotra (1979) to supplant mixing length in a mathematical modeling of an upward variable density flow. The velocity and temperature profiles obtained were compared with the measurements of Wood and Smith (1964) for supercritical carbon dioxide. Good agreement for the case of low heat flux was reached. It should be noted that because of the lack of experimental data for the region very close to wall, the comparison was not complete. The changes in velocity and shear stress profiles along the tube were calculated. For high heat flux, it was shown that the maximum velocity might be shifted from the centerline to some point closer to the wall. A distance of more than twenty diameters from the flow inlet was needed to let this happen. The predicted velocity variation away from the entry region was similar to the measured profile. Quantitatively, however, they differed greatly. Bellmore and Reid (1983) developed a numerical two-dimensional model to calculate wall temperature in an upward flow of para-hydrogen near the critical region. 28 The basic improvement of the model with respect to previous studies was implementation of density fluctuations in the conservation equations. The range of applicability of their model included enhanced and deteriorated regimes of heat transfer. As for velocity distribution, the existence of an M-shaped profile was confirmed. They confirmed that variations of fluid density had a striking effect on turbulence activities and hence heat transfer at the critical region. Popov and Petrov (1985) modified the two-dimensional model of Popov (1977) to account for flow resistance due to acceleration. The variations of momentum in the axial direction were taken into account. The flow of supercritical carbon dioxide under cooling conditions was studied. The model revealed velocity and temperature profiles based on which the flow resistance owing to both wall friction and acceleration could be calculated. The expression of constant-property eddy diffusivity was modified to fit the variable property conditions. The turbulent Prandtl number was assumed to be unity. Within the examined range of heat and flow parameters, buoyancy appeared to have little effect on heat transfer in vertical flows and was neglected for horizontal flows. Kurganov et al. (1989) confirmed that a one-dimensional approach was adequate to predict the flow resistance and heat transfer near the critical region. The heat flux, however, needed to be moderate to avoid a deteriorated regime of heat transfer. To estimate the coefficient of friction, the formula used for constant property flows could be applied along with a correction factor. The earlier suggestion for a correction factor in terms of a function of ratio of the densities at the wall and bulk was shown to be satisfactory. Perhaps because of the unsatisfactory results obtained with empirical correlations, interest in them gradually diminished. The general trend of investigations was rather toward establishment of more physically meaningful models. Based on reviews of past studies, Polyakov (1991) concluded that for low heat flux cases (no heat transfer deterioration), simple, mixing-length turbulence models might adequately describe heat transfer to supercritical fluids. In fact, it had already been shown that in some cases a more complex model like a K-e model yielded relatively worse results. That was because the approximations and constants in a K-e model derived for a constant property flow required far more justifications and adjustments to be implemented in a variable property 2 9 flow. He also clarified that the assumption of fully developed flow was reasonable for low heat flux cases and x/d > 50. The lack of data for the case of horizontal flows was obvious from his report. For normal flows where no deterioration was reported and the effect of buoyancy was not substantiated, Kurganov and Kaptilnye (1993) recommended the equations of Reichardt for momentum eddy diffusivity. The eddy diffusivity of heat followed more or less the same pattern. Depending on other flow conditions a constant turbulent Prandtl between 0.5 to 1.1 led to satisfactory results. Later in chapter seven of this study the equations of Reichardt for eddy diffusivity will be examined. Zhou and Krishnan (1995) incorporated a low Reynolds number K-e model to analyze laminar and turbulent flows of supercritical carbon dioxide. Due to the application of wall functions, one major shortcoming of a standard K-e model in the case of supercritical fluid flow, they mentioned the use of coarse grids near the wall. The first node needed to be at least at y+>20 from the wall. Apparently this resolution for the situation where there are steep profiles of fluid properties at the wall was not acceptable. No such limitation existed for low Reynolds number K-e turbulence modeling. The effect of buoyancy was included in the numerical model. In laminar flows the buoyancy improved heat transfer regardless of the flow direction. In turbulent flows, the differences in upward and downward flows were clearly detected. Nevertheless, not enough comparison with experimental results was presented to quantitatively evaluate the model. The M-shaped velocity profile was obtained for upward flows under high heat flux. Deteriorated heat transfer was only qualitatively predicted. Despite the fact that the effects of property variations on the coefficients and constants of a K-e turbulence model have not been fully realized, Koshizuka et al. (1995) adopted a numerical model to analyze vertical supercritical fluid flows. The advantage of their model was that it could predict both enhanced and deteriorated regimes of heat transfer. The evaluation of the results was limited to comparison with data of Yamagata et al. (1972). Based on their predictions, Koshizuka et al. explained degraded heat transfer at high and low Reynolds numbers differently. At high mass flow rate and high heat flux, deterioration might occur due to growth in thickness of the viscous sub-layer 30 downstream of the flow, while buoyancy was claimed to be responsible for the decrease in heat transfer at low mass flows. Lee and Howell (1998) implemented the improved mixing length model of Bellmore and Reid (1983) to develop a 2-D model for vertical flow of supercritical water. They compared their results with the data of Yamagata et al. (1972) as well as predictions of a few empirical correlations and numerical models. It was interesting to see that the disagreements with experiments were more or less the same. Lee and Howell over-predicted the heat transfer coefficient peaks by 10 and 31% corresponding to Prt=1.0 and Prt=0.9 used in their model, respectively. The results of K-e modeling (Koshizuka et al., 1995) underestimated these peaks by 17%. The heat transfer coefficient peaks predicted by the empirical correlations of Dittus-Boelter and Swenson et al. (1965) occurred at higher bulk enthalpies. The magnitude of the peak was compatible with the predictions of numerical solutions or was even closer to the experimental data. Lee and Howell mentioned that the validity of the empirical correlations was limited to a narrow range of parameters. They concluded that to design process equipment operating at or near critical conditions, no traditional empirical correlation was adequate. A complete numerical analysis was recommended. For horizontal flows, a complete numerical analysis means a 3-D model, which has not yet been accomplished. The picture for flows with negligible effects of buoyancy and acceleration is not quite clear either. As mentioned earlier, the assumption of "fully developed flow" may be applied to these types of flows. Thus, theoretically, a 1-D numerical model or an empirical correlation should be able to adequately predict heat transfer coefficients. In practice, however, such a universal numerical model or empirical correlation has not yet been reached at. 31 CHAPTER THREE EXPERIMENTAL APPARATUS The specific experimental objective of this study was to measure the local heat transfer coefficients in a horizontal pipe flow of water near the critical region. Separate measurements at the top and bottom surfaces of the tube would provide additional knowledge needed to understand the effects of buoyancy. It was shown that near the critical region water properties vary dramatically with pressure and temperature. Thus, to measure local heat transfer coefficients, a specially designed setup is needed so that it can withstand high pressure and temperature while the small changes in the flow conditions (i.e., pressure and temperature and hence other fluid properties) may be detected. The heat transfer measurements were carried out in a Supercritical Water Oxidation (SCWO) facility. A pilot plant was constructed in the Department of Mechanical Engineering of the University of British Columbia (UBC) in partnership with N O R A M , a Vancouver based engineering company. Many people have been involved in completion of the plant over several years. As a member of the team, the first year of this study (1996), was mainly attributed to the construction of the apparatus. A great deal of technical knowledge on the apparatus is available in the reports produced by NORAM. Two particularly rich sources of detailed information about the plant are the Masters theses done by Teshima (1997) and Filipovich (2000). The SCWO process was introduced in chapter 1. The description of the apparatus follows. 32 experiments no reaction takes place so that the outlet from the system remains distilled water. Instead of leading it to a drain, it can be collected and pumped back to the water tank. In this sense the process may be considered as a closed cycle. The water is pumped into the system by a high pressure, triplex positive displacement metering pump (GIANT P57). A pulsation damper (Hydrodynamics Flowguard DS-10-NBR-A-1/2" NPT) was installed in the line right after the pump to control the flow fluctuations caused by the pump. Adjusting the back pressure regulator provides the required system pressure. The pump can handle outlet pressures up to 45 MPa. The operating range in this study is between 22 to 29 MPa. The pump speed may be adjusted by a Variable Frequency Drive (VFD, Reliance Electric ISU21002). The range of flow rates is between 0.6 and 2.2 £/min. At the beginning of the cycle the water gains some heat by passing through the tube side of a counter flow tube-in-shell regenerative heat exchanger. The counter flow in the shell side contains hot water leaving the reactor. The shell is 1/2" Schedule 80 seamless pipe. Approximately 30 kW of heat is recovered and transferred to the cold fluid in the tube. To allow for thermal expansion of the tube and shell arrangement, special high-pressure fittings are used at either ends of the heat exchanger. After leaving the regenerative heat exchanger, the water is next introduced to the first and second preheaters. Both preheaters are electrically heated by running A C current through the tube walls. The two preheaters are separated and the current may be different in each of them. They are used to maintain the desired flow temperature before entering the test section. The electrical power to the preheaters is supplied by a Silicon Controlled Rectifier (SCR) panel through a pair of transformers (240/24 V A C , Hammond Manufacturing). The wiring of the step-down transformers is shown in Figure 3.2. They are connected to SCR panel in series and to the heaters in parallel. Currents up to 500 Amp are available through the secondary transformers. Heavy-duty cables are required to carry such high currents. Thus, 2.5-cm thick copper cables are used for this purpose. The cables are attached to one end of stainless steel rods (SS 304) via barrel connectors. The other ends of the rods are silver-soldered to the preheater surface. The high voltage terminals (24 V A C maximum) are connected to the middle of 34 the preheater. The ground connectors are at the ends. This arrangement provides equal electrical power to each half of a preheater. It also reduces the risk of any ground loops in the system. Figure 3.2. Electrical heating schematic for the preheaters and test section The power delivered to the first preheater is manually controlled via an SCR panel. The maximum power is (24 Volts x 500 Amps = 12 kW). The second preheater is equipped with a feedback temperature controller. The SCR panel therefore adjusts the power to the second preheater automatically so that the pre-assigned temperature is maintained at the preheater outlet. Of course, the power to the second preheater can alternatively be adjusted manually by setting the feedback temperature controller off. The regenerative heat exchanger and preheaters together are capable of raising the water temperature to the pre-pseudo-critical point before it enters the test section. The part of the system where most measurements take place is the test section. The temperature measurements at other locations are for the purpose of keeping track of temperature variations throughout the cycle. The test section consists of two major parts followed and proceeded by two short sections. See Figure 3.3. Union fittings between the segments facilitate disassembly of the test section for examination of the thermocouples 35 Further cooling takes place in the process cooler so that the water temperature reduces below 50 °C. A back-pressure regulator (Tescom #54-2162T24S) regulates the supercritical pressure down to ambient pressure. The water may be led to the drain or collected in a separate container (not shown in Fig. 3.1) and pumped back to the distilled water tank. Note that the conductivity meter shown in Fig. 3.1 detects the variations of the fluid conductivity from which one may identify any changes in concentration of the solutions under study. It was used in fouling experiments. For experiments with pure water no conductivity measurements were done. 3.2. SAFETY ISSUES Over-pressurization and over-heating are two main hazards that must be avoided. Over-pressurization may occur because of plugging of the tube due to deposition. This is quite likely to happen in experiments with multi-components fluid flows. The chance of plugging of the tube in our tests with pure water was zero. Over-pressurization, however, may still occur due to fluctuations in the flow. In general, supercritical fluid flow has an oscillatory nature. Under certain flow conditions (heat flux, mass flow, pressure and temperature) the amplitude of the pressure fluctuations may escalate to a few MPa. Thus, the pressure could exceed the high-pressure set point. To avoid over-pressurization, three relief valves (as shown previously in Fig. 3.1) are placed in the system at various locations. The high-pressure relief valve in our tests is set as 29 MPa. Overheating is most likely at high heat loads and low flows. Around the critical region, the high specific heat capacity of water inhibits any large increase of temperature. If the inlet temperature to the test section is already near the pseudo-critical point and the heat flux is high and mass flow is low, there is a good chance that the critical region is reached in the first heated segment of the test section. Thus, in the second segment of the test section the temperature may grow as high as 700 °C or more. That is beyond the safe limit (650 °C) for Inconel tubing. The other scenario, under which over heating may occur, is when the flow is undesirably reduced or interrupted. In the absence of enough cooling water flow, the temperature of electrically heated tubes rises very quickly. This is dangerous and can lead 37 to tube failure. To prevent over heating, a number of temperature-sensitive alarms were placed just before and/or after all components of the system. The temperature set points of the alarms are manually assigned from the control panel. As soon as the temperature exceeds the set point, the alarm trips off and all the heaters are automatically shut off. Additional thermocouples were clamped to the tube surface at various points throughput the flow loop. The corresponding temperatures were shown by digital displays on the control panel. Thus, an overall map of temperature variations in the system is instantly available. Undesirable heating anywhere in the system may be monitored and treated manually even before the alarms trip off. There is also an emergency switch on the control panel, which shuts off both the flow (pump) and the heaters. It should be noted here that in the case of over heating, before resumption of fluid flow it is always desirable to cool down the tube. Thus, the emergency shut off is not recommended. It is designed only for situations when the whole system has to be stopped. As an example if the water in the water tank runs out in the middle of a test, the system should be shut off immediately in order to avoid damages to the pump and overheating the dry system. In the case of a mechanical failure, high pressure and temperature can be very dangerous. As a result a jet of hot water might spray out of the test facility. Eighteen gauge steel sheets are used to enclose the system. More details of safety procedures may be found in the SCWO User's Guide. 3.3. T E M P E R A T U R E M E A S U R E M E N T S Two types of temperatures were measured in our experiments. They were surface of the tube and bulk fluid temperatures. The surface thermocouples, in turn, are of two kinds. Those used in the temperature alarms are Inconel sheathed and ungrounded thermocouple probes. The rest of the thermocouples are K-type Chromel-Alumel. Twisted shielded wires are used to extend the thermocouples from the site of measurement to the data acquisition system. In various sections of the system other than the test section, the thermocouples were clamped to the tube in order to measure surface temperatures. 38 A hand pump (Omega, PCL-2HP) attached to the positive pressure port and a portable manometer (Omega, PCL-200) were used together to calibrate the DP cell. The negative port was open to atmosphere. A manometer read the pressure difference. The pump supplied the required pressures and the voltages were delivered from the carrier demodulator and read from a computer display. Zero pressure and 2.5 MPa corresponded to zero and 10 volts respectively. A calibration curve was obtained which expressed the pressure difference in terms of the voltage detected by the DP cell. In most experiments of this study the pressure drop across the test section is small and is neglected. It is important to measure absolute pressure in the test section with good precision. Note that heat transfer calculations and analysis are based on thermo-physical properties of water, which are functions of pressure. An absolute pressure transducer (Validyne Model P2) is employed to measure the pressure. The position of the transducer is shown in Fig. 3.6. The output is a signal ranging between zero and 10 volts. The uncertainty in pressure measurement was ±0.5% or ±0.12 MPa at 24 MPa. This has no significance on heat transfer calculations unless the pressure is extremely close to the critical pressure of 22.1 MPa. In this study, a deadweight tester was used to calibrate the absolute pressure transducer. The deadweight tester works by balancing the hydraulic pressure generated by means of the known weights. A screw plunger can push the oil inside against a very small surface area so that large pressures up to 27 MPa may be sustained. The force of oil is transmitted to one side (smaller arm) of a balance. The force on the other side (longer arm) is provided by the weights. Since the geometry of the device is known, the required amount of the weights to balance the oil force measures the generated pressure very accurately. From the other end, the pressurized oil is connected to the absolute pressure transducer to be calibrated. Each voltage corresponds to a pressure so generated. The curve fit of the data leads to P=-0.8698+3.4108V+0.0064V2. This equation states the relationship between the output voltage of transducer (V) in volts and the fluid pressure (P) in MPa. A second calibration of the absolute pressure transducer was performed using a digital calibrator. The results validated the first calibration. More details of the calibration 42 of both differential pressure and absolute pressure transducers are available in the UBC SCWO System Calibration Report. 3.5. FLOW MEASUREMENT Throughout this study the flow rate was adjusted between 0.6 and 2.2 £/min. The flow rate was measured by collecting the water in a graduated cylinder at the system outlet over a timed interval. The flow fluctuations are high near the critical temperature. Thus, the flow measurements should be performed once the system is at the most steady state condition that is when the system is cold (unheated). A simple correlation between the pump speed and the flow rate was derived. The uncertainty of the flow measurement was less than 1%. There are also two flow meters placed close to the inlet and outlet of the system. They are used only for a continuous check of the flow and safety purposes. The one at the outlet is connected to an alarm control system. If, due to malfunction of the pump, the flow rate goes below a pre-assigned value, the controller stops the heaters to inhibit any possible overheating of the system. 3.6. POWER MEASUREMENT The voltage difference across the test section cannot be detected by a traditional handheld voltmeter. The readings of such voltmeters for rms (root mean squared) values of A C currents can be valid only if they are referring to a perfect sinusoidal signal. This is not always the case in our system and occurs only when a transformer runs at its maximum load. As a contribution of the present study to an accurate measurement of the electrical power generated, the instantaneous magnitudes of voltage across the test section are detected by means of an oscilloscope. The results are shown in Figure 3.7 for two cases of transformers operating at 100% and 55% of maximum load. The true rms value of voltage, Vrms, was obtained by integration of the signal over a few cycles. The true rms value of electrical current, Irms, flowing through the tube also needs to be detected in order to calculate the power generated from P=V.I. A donut-shaped current transformer (CT) with 1000:5 amps ratio was used to reduce the very high electrical current (= 500 amps) down to 2.5 amps or less. A half-ohm resistor closed the CT circuit. A second channel of the oscilloscope was used to record voltage differences 43 across the resistor. The voltage difference was divided by 0.5 ohms and multiplied by 200 to obtain the corresponding electrical current in amps. Careful examination of both channel readings assured that there was no phase difference between voltage and current in the test section. Under these conditions, the product of and 1 ^ gives the true rms V a l u e O f power P^s, i.e., Prms = (V.Tjrms = Vrms . Irms Tele Run: lOOkS/s Hi Res E--T-A: 22.4 V A: 13.7ms : -7.2 V Cl Pk-Pk 60.0 V C2 Pk-Pk 3.01 V C2 RMS 1.061 V C1 RMS 21.20 V 10\6 t>' ' Ch2 ' s'o'omv ' M'S.frfrms Chi > " 'cVs'V 9 Mar 1999 15:47:52 Tek Run: I00ks /s HI Res [~-T A: 20.4 V A: 13.7ms : -1.2 V Cl Pk-Pk 56.2 V C2 Pk-Pk 2.76 V C2 RMS 783mV C1 RMS 15.86 V Hfll ' i6.6 {/' ' rih2 ' s W m v MS.ddms Chi !/ " ' '6.&V 9 M a r 1 9 9 9 15:55:13 Figure 3.7. Typical variations of voltage of the secondary of the transformer under full load (top graph) and 55% power load (bottom graph). 44 In addition to power calculations, the resistance of the Inconel tubing can be calculated from R=V/I and compared with what was obtained from the table of physical properties of Inconel 625. The variation of Vrms with Inns appeared to be linear. Thus, the slope of the straight line, R, represents the electrical resistance of the test section tubing. It suggests a value of 0.0503 Q. for electrical resistance of the test section. From the table of physical properties of Inconel, the electrical resistivity, p, changes from 1.29 uD.m at 300 °C to 1.30 ufl.m at 400 °C. (This implies that temperature variations do not affect the resistance of the tube noticeably which is consistent with our measurements). The equation R=p £/A leads to R=0.0458 Q at 300 °C. This is different from the result of the power measurements by about 10%. One potential source of discrepancy might be errors in the inner and/or outer tube diameters. To avoid measurements of true rms values of current or voltage during each experiment, the rms value of power was directly correlated against the voltage across the primary of the transformer. The following relationship is recommended for this purpose. p = 19.8807 f V ySCR v24-0.0055V 5 C T y (3.1) The voltage across the primary of the transformer is measured and shown on the SCR panel, VSCR. It is denoted by VSCR in Eq. 3.1. An average value for R and the equation P=V /R was also used to obtain the correlation. The difference between power estimated this way and P^ Vrms.Irms does not exceed 1.7%. The measurements were also done for the second heated segment of the test section. The results were assumed to be the same as for other heated segment. 3.7. HEAT FLUX MEASUREMENTS Not all electrical power generated in the test section is transferred to the fluid. There is always a portion of heat, though very small, lost via the external surface of the tube through the insulation (heat loss). To identify the net heat flux therefore the heat loss needs to be estimated and subtracted from the measured electrical power. The heat loss is 45 a function of the temperature in the test section. The procedure to calculate the heat loss is as follows. The bulk inlet temperature to the test section may be raised to the desired temperature by means of preheaters. Meanwhile the test section is kept unheated. The bulk temperature at the outlet of the test section as well as the surface temperatures at few points along the test section are measured. The fluid temperature is expected to drop across the unheated test section. From the inlet and outlet bulk temperatures the fluid enthalpies are estimated. A simple energy balance yields the heat loss corresponding to an average temperature of the tube surface. The test is repeated for various bulk inlet, and hence tube surface temperatures. A correlation between the wall temperature and heat loss may be set up. There is an alternative way of detecting heat flux, which is more accurate and is used in this study. Instead of applying an energy balance to the unheated test section to determine the heat loss, it is applied to the heated tube to obtain the heat flux directly. The excess enthalpy of the fluid at the outlet with respect to the inlet represents the energy absorbed by the fluid flow. Like heat loss calculations, a correlation between heat flux and flow conditions may be established. In practice, bulk temperatures at the inlet and outlet as well as flow rate are measured in each test. Heat flux calculations may be readily done without any extra burden. Thus, use of any relationship between heat flux and flow conditions could result in losing precision, and though very small, is not worthwhile and thus not recommended. 3.8. DATA ACQUISITION An Omega MultiScan/1200 with 24 differential channels was employed as a data acquisition board. It is an isolated instrument, as it is needed for collection of data from an electrically hot system. Four of the channels are reserved for the pressure signals and fluid conductivity. The other twenty are used for temperature signals delivered by K type thermocouples. One cold junction sensor is assigned to every eight input channels. A software-controlled calibration of the cold junctions is done based on temperatures of an ice bath and boiling water. 46 The MultiScan board is connected to a PC computer. Tempview version 4.14 is the software used for data acquisition. The sampling frequency of data collection from each channel is 1.92 kHz, equivalent to one reading per 520.83 us. To eliminate noise induced by the SCR control panel or otherwise, Tempview may be configured to average the signals at a desired rate. With averaging every 256 measurements, it takes 3.35 s to scan all 24 channels. Thus, a time of 5 s (greater than 3.35 s) was selected in the Tempview menu as the scan interval. While storing data in an appropriate file, Tempview is capable of displaying the outcome of all 24 channels on the computer display monitor. Of particular interest is the temperature gradient along the test section, which may be monitored instantly on the computer screen. It is important to start acquiring data only after the quasi-steady state is reached. 4 7 CHAPTER FOUR RESULTS AND DISCUSSIONS A systematic experimental study was performed with over 50 experiments under varying conditions. Test conditions for almost all the runs are tabulated and presented in Appendix B. The test results are usually shown as the graphs of the variations of the wall and bulk temperatures or heat transfer coefficients along the test sections. Only some of the data have been shown in the body of the thesis. The rest of the data (those not shown) have served to support the conclusions reached in this study. The main focus of the work is on enhanced heat transfer, which is the most dominant mode of heat transfer near the critical region. Some limited results with hot spots detected on the wall surface, indicating local deterioration of heat transfer, are also presented. There is a second type of degraded heat transfer as well, which is solely due to buoyancy and occurs at the top surface of the tube. This is presented briefly in this chapter and will be discussed in more detail in chapter 6. The feasibility of comparing raw experimental data with other studies is also discussed in this chapter. The presentation of the reduced data and comparison of empirical correlations, because of their importance and wide applications, are carried out separately in chapter 5. 48 4.1. EXPERIMENTAL PROCEDURE Pressure, mass flow, inlet temperature to the test section, and heat flux were independent variables measured during operation of the SCWO facility. Each of those parameters may be controlled separately and set to a desired value. The bulk and wall temperatures along the course of the flow are dependent variables of special interest from which the local heat transfer coefficients can be calculated. Wall temperature at a number of points along the test section, bulk fluid temperature at the inlet, middle and outlet of the test section, pressure and mass flow were directly measured in each test. Heat flux, variations of bulk temperature along the,test section and hence local heat transfer coefficient were calculated accordingly. Details of the calculations follow. A table of properties of water was an essential tool needed for heat transfer calculations. 4.1.1. Thermophysical Property Data A table of thermodynamic properties of water is essential in order to set an energy balance between the inlet and outlet of the test section. Accurate values of enthalpy at different temperature and pressure warrants a reliable estimation of heat flux. Other thermodynamic properties (density and specific heat) as well as transport properties (viscosity and thermal conductivity) are needed once data have been reduced and presented in a non-dimensional empirical correlation. Throughout this study the thermodynamic and transport properties of water are taken from the latest releases of the International Association of Properties of Water and Steam (IAPWS, formerly called LAPS). For thermodynamic properties a Fortran routine called iapws957.f (Pruss and Wagner, 1995) was used. Pressure and temperature are input data. For viscosity and thermal conductivity the equations recommended by LAPS, 1985a and LAPS, 1985b (see references) were used. Two Fortran subroutines were developed to evaluate the LAPS equations. The thermodynamic properties of water at the critical region have been known to a relatively high precision for quite a while. Transport properties, however, were not as well understood in many earlier studies. The differences in the water properties could cause noticeable disagreement among the results of different studies. The effect of fluid properties on heat transfer will be discussed later. 49 4.1.2. Test Conditions and Restrictions Variations of test conditions in this study are as follows. Volumetric flow rates ranged between 0.60 to 2.25 ^/min. Considering the tube diameter D=6.3 mm, the corresponding mass flux was within the range of 330 to 1210 kg/m2.s. Pressures up to 28 MPa were examined. In most experiments, however, the pressure was between 24 and 25.5 MPa. The fluid inlet temperature to the test section was controlled by means of preheaters to be below pseudo-critical temperature in the vast majority of experiments. Heat fluxes of 100 kW/m or smaller to maximum values of around 315 kW/m were achieved. Note that in the tests with very low heat flux and high flow rates, the temperature difference between wall and bulk was comparable to the error associated with the thermocouple measurements. Thus, data obtained from such cases were not relied upon to draw any important conclusions throughout this study. There are other potential variables, which were fixed in this study due to various restrictions. They may be summarized as follows. The most basic and widely used configuration, i.e., flow through a round tube, was chosen for the present investigation. The flow was always horizontal and the length and diameter of the test section was fixed. The effect of entry length was not studied in this investigation. The Reynolds number varied between 20000 and 500000. Thus, only turbulent flows were studied. The apparatus was designed for flow of water or water mixtures. Other supercritical fluids were not tried. Many findings, however, are applicable to other fluids as well. The heat flux, as generated via electrical heating of the tube, though varying from case to case, was uniform in each test. A second type of heat transfer boundary condition (constant wall temperature) was not attempted. The direction of heat flux was always from the wall to the bulk and the reverse direction (cooling) was not considered here. Note also that transient mode of heat transfer was not of interest in the current study and all measurements were taken under quasi-steady state conditions. 4.1.3. Calculation of Local Heat Transfer Coefficients To calculate local heat transfer coefficients, the first step was to estimate the bulk enthalpy at any given axial position. The pressure and the bulk temperatures at the inlet and outlet were available via measurements. As explained in chapter 3, the heat flux was 50 calculated from the difference of the inlet and outlet enthalpies. The heating surface is nd£, where (d=6.3 mm) and (.£=2946 mm) are the inside diameter and the heated length of the test section. Knowing two properties, pressure and temperature, the enthalpies at inlet and outlet were obtained from the thermodynamic properties of water (iapws957.f). The change of electrical resistance of the Inconel tube, within the range of temperature across the test section, was negligible. Variation of the tube thickness was also small so as not to violate uniformity of electrical heating of the tube. The kinetic energy of the fluid was negligibly small compared to its enthalpy. Thus, the uniform heating of the tube implies that the bulk enthalpy varied linearly along the test section. The bulk enthalpies at the inlet and outlet of the test section were used to determine the bulk enthalpy at any axial distance. The inside wall temperature, T w , was obtained from the equation of radial conduction heat transfer in a heated tube: T -T = Q"RI ( f l 2 - 2 1 n a - 1 ) (4.1) 2K (1-a ) where K is thermal conductivity of Inconel, T e is outside surface temperature, r; is internal radius and "a" is the ratio of the outside to inside radii of the test section. The uniform measured heat flux is q". In this study, the inside and outside diameters of the test section are 6.3 and 9.6 mm respectively. The average thermal conductivity of Inconel is about 17 W/m within the range of operation of the test section. The local heat transfer coefficient, h can be determined from the following equation. h = —i (4.2) (TW~TB) where the local bulk temperature, T b , is obtained from local bulk enthalpy, i b , and pressure using the table of water properties (iapws957.f). 51 4.1.4. Typical Variations in Heat Transfer Coefficients In most cases, because of the significant change of properties, there is a remarkable enhancement of heat transfer to supercritical fluids near the critical region. In a heated tube, this situation is met when the wall temperature is above and the bulk temperature is below the pseudo-critical point. Once the enhanced mode of heat transfer commences, most of the heat energy is consumed to convert the water from a liquid-like to a vapor-like state. This process is associated with only a small rise in wall temperature and hence, may be pronounced over a significant length of the test section. The axial wall temperature gradient is nearly flat in the critical region. Downstream of where the water is heated above the pseudo-critical point, heat transfer enhancement diminishes and the wall temperature increases. Consequently, the growth of the wall and bulk temperatures in the post critical region becomes similar to the pre-critical region. Typical temperature variations along the course of flow are shown in Figure 4.1. Figure 4.1. Outside wall and the fluid bulk temperature profiles, G=662 kg/m2.s, P=24.4 MPa, q"=195 kW/m 2 5 2 The critical region is roughly between i=1900 to 2400 kJ/kg. It can be seen that within the critical region the temperature differences between the wall and fluid were the lowest. This indicates an increase in values of heat transfer coefficients. Corresponding variations in heat transfer coefficients are shown in Figure 4.2. Uniformity of heat flux implies that fluid energy (enthalpy) changes linearly with distance. It makes the fluid bulk enthalpy an excellent choice for the horizontal axis in Figs. 4.1 and 4.2 as well as many others to come. D ° 60 E 2C 50 40 h c <u S 30 O 20 c *Z 10 a) I • 1600 1800 2000 2200 2400 2600 Bulk Enthalpy , kJ /kg 2800 Figure 4.2. Enhancement of heat transfer coefficients at and near the critical region, G=662 kg/m2.s, P=24.4 MPa, q"=195 kW/m2 4.1.5. Construction of Graphs For a relatively high mass flow, the test section was not long enough to accomodate all enthalpy (temperature) changes shown in the two previous figures. In fact, different symbols used in Figs. 4.1 and 4.2 indicate that they are obtained from various tests with different inlet temperatures. Use of bulk enthalpy as the horizontal axis makes it possible to have all data on one graph. To better visualize the convenience of 53 4.1.6. Corrections to Bulk Temperature Measurements Local heat transfer coefficient is a function of T D (Eq. 4.2). Furthermore, T D affects h due to the impact it has on estimating q" (obtained from an energy balance). Thus, an accurate estimate of T b is crucial. A difficulty arises when one or both inlet and outlet temperatures are close to the pseudo-critical point. Sometimes a small error of 1 or 2 °C in measuring bulk temperature can lead to noticeable miscalculations of enthalpies, and hence heat transfer coefficients, along the tube. The uncertainty in bulk temperature measurements was about ±2 °C. Consequently, error detection is even more difficult in the critical region. Away from the pseudo-critical temperature, however, this much error could be safely neglected. This simple fact was used to calculate heat flux accurately. Consequently, accurate values of heat flux were used to correct bulk temperature measurements taken near the pseudo-critical point. The procedure can be explained in more detail by means of the following example. It illustrates why heat flux calculations should be avoided where inlet and/or outlet temperatures are near the critical region. Recall from Fig. 3.5 that the test section has two back to back segments. The bulk temperatures at three points, inlet, middle and outlet were used. At the middle, the measured fluid temperature serves as the outlet temperature for the first part and as the inlet temperature to the second part of the test section. Heat flux in the first part is calculated through energy balance between inlet and middle bulk temperatures. Heat flux in the second segment is obtained from middle and outlet fluid temperatures. The variations of inlet, middle and outlet bulk temperatures with time for an arbitrary test are shown in Figure 4.4. Heat fluxes obtained from the first and second segments of the test section as well as across the whole test section (energy balance between inlet and outlet temperatures) are also shown. Since the lengths of the two segments of the test section are equal, the calculated heat flux over the whole test section was the average of the heat fluxes obtained from each segment of the test section. Note that the electrical power delivered to the test section is fixed during entire measurements. It implies that heat flux to the test section is almost constant. This is while the electrical power provided by preheaters were gradually changed during the test to produce different inlet, and hence outlet, temperatures in the test section. A small variation of heat flux in the test section is 55 however expected due to the fact that the heat loss through the insulation boards were larger at higher temperatures. 1000 2000 3000 4000 5000 6000 7000 Time, s Figure 4.4. Effect of inlet, middle and outlet temperatures on heat flux calculations It can be seen that at the beginning of the test where all temperatures (i.e., inlet, middle and outlet bulk temperatures) were below the critical region, the calculated heat flux appeared to be constant and the same in both segments. The pseudo-critical temperature is around 380 °C in this example. Note that t=0 in Fig. 4.4 denotes the start of data acquisition and not the start of the experiment. In fact, the quasi-steady state was reached before the data were collected. This fact may also be deduced from Fig 4.4. It can be seen that during 0< t <1000, although the bulk temperatures were growing, the heat flux in the test section was fixed. As the outlet temperature approached the pseudo-critical point at t=1000 s, the effect of error in the bulk temperature starts to become important. A jump in values of heat flux obtained from the second segment was observed. That was while the calculations in the first segment were still unaffected. Once the 56 middle temperature reached the critical region around t=2200 s the outcome of the first segment calculations also became unreliable. The situation remained the same until both outlet and middle temperatures grew higher than the critical region around t=5100 s. From this point on, the energy balance in the second segment resulted in a stable constant value for heat flux. The heat flux calculated during the period t > 5100s was slightly smaller than what was calculated at the beginning of the test. This occurred while the power input to the test section was constant throughout the experiment. It means that the heat loss, as expected, was slightly larger at higher temperatures. An average of heat fluxes obtained from the first segment during the beginning of the test (t<2100) and from the second segment at the end of the test (t>5100) may reliably represent heat flux throughout whole experiment. Once the heat flux was determined, it was used to correct both inlet and outlet temperatures so that an energy balance always resulted in the same value of heat flux during a test with fixed power settings, i.e., electrical heating. 4.2. EFFECTS OF VARIOUS CONTROL VARIABLES In order to more fully understand heat transfer to supercritical fluids, a thorough study was performed where all, except one, of the controlled variables were held constant. This allowed a demonstration of the effect of each of these variables independently. Such variables were inlet bulk temperature, mass flow, heat flux, and pressure. The findings for horizontal flows in this study are presented below. 4.2.1. Effect of Inlet Bulk Temperature The entry region of the flow refers to the beginning part of the test section where the wall boundary layer profile has not yet extended to the flow centerline. Potential flow was still observed at the core. For constant property flows, a distance of approximately 20 times the tube diameter was required for the flow to develop. At supercritical fluid flows, because of the variable nature of the flow, the entry length might increase to over 50D. Inlet bulk temperatures have a substantial effect on flow within the entry region. The entry region of supercritical fluid flows deserves an independent study. For a recent analysis on this subject in a vertical flow refer to Lee and Howell (1998). In the current 57 A s shown in Fig . 4.5, all graphs collapse onto more or less the same profile. This means that regardless of the axial position of surface thermocouples, i.e., regardless of inlet bulk temperature, the same wall temperatures were obtained wherever bulk temperatures were the same. It can be concluded that the inlet temperature and hence the flow history did not affect heat transfer rate noticeably. Thus, outside of the entry region, the wall temperature was decided by heat flux, mass flow and the local bulk temperature only. In other words, the assumption of fully developed flow in a sense that heat transfer coefficients can be calculated from local fluid properties was justified in this example. Small differences of the records shown in Fig . 4.5 were due to pressure fluctuations. However, the inlet temperature, as w i l l be shown later, can have a dramatic effect on deteriorated regimes of heat transfer. 4.2.2. Effect of Mass Flow The effect of mass flow on heat transfer at supercritical pressures is similar to that which occurs at constant-property flows. The higher the mass flow the lower the wall temperature w i l l be through an increase in the convection heat transfer coefficient. Heat transfer coefficients for three tests with different mass flows are shown in Figure 4.6. o o X | 1000 1200 1400 1600 1800 2000 2200 2400 2600 Bulk Enthalpy, kJ/kg Figure 4.6. Effect of mass flow on heat transfer coefficient, q"=300 kW/m2. Heat transfer coefficients are measured at the bottom surface of the test section. 59 / 4.2.3. Effect of Heat Flux The effect of heat flux is opposite to that of mass flow. With increasing heat flux, the temperature difference between wall and bulk increases. This can be seen in Figure 4.7 where the wall temperature distributions for three different heat fluxes as well as the bulk temperature are shown. The growth of the temperature difference is, however, greater than the increase of heat flux. For example, once the heat flux is increased by 1.65 times from 234 kW/m2 to 304 kW/m2, the temperature difference between the wall and bulk grows about two times larger. Thus, the heat transfer coefficients decrease with increasing heat flux. 1500 1650 1800 1950 2100 2250 2400 2550 Bu lk En tha lpy , k J / k g Figure 4.7. Effect of heat flux on temperature distribution (G=575 kg/m2.s). The wall temperatures are measured at the top surface of the test section. Heat transfer coefficients corresponding to the temperature distributions of Fig. 4.7 are shown in Figure 4.8. This behavior is one of the peculiarities of heat transfer to supercritical fluids. In constant-property flows, heat transfer enhances with an increase in heat flux. In the critical region of a supercritical fluid flow, once heat flux is low, the 60 temperature difference between the wall and bulk is small. Thus, the portion of fluid in the vicinity of the pseudo-critical temperature constitutes a noticeable part of the whole flow. By increasing heat flux, that portion, which is responsible for enhancement of heat transfer, is suppressed, i.e., less percentage of flow exists in the pseudo-critical bandwidth. Hence, heat transfer coefficients decrease. Figure 4.8. Effect of heat flux on heat transfer coefficients (G=575 kg/m 2 .s) 4.2.4. E f f e c t o f P r e s s u r e The fluid properties are sensitive to pressure variations in the critical region. The closer the pressure is to the critical point, the steeper the change of properties is near the pseudo-critical temperature. Specific heat, because of its peak value, is the property, which best represents such variations. The changes of specific heat with temperature at four different supercritical pressures are shown in Figure 4.9. 61 Pressure, MPa ) i i . 1 , 1 , 1 , i , i 600 1800 2000 2200 2400 2600 2800 Enthalpy, kJ/kg Figure 4.9. Variations of the specific heat of water Cp, with pressure At a pressure of 23 MPa, the Cp spike (not shown in Fig. 4.9) was higher than 300 kJ/kg.°C. Note that the peak values occur at slightly higher temperatures (i.e., higher enthalpies) as pressure increases. This corresponds to the variation of the pseudo-critical temperature with pressure shown in Figure 4.10. From the variation of Cp with pressure, it can be anticipated that heat transfer coefficients would also be influenced by pressure. This has been previously studied by various investigators for large pressure differences in vertical flows. Swenson et al. (1965), for example, showed large differences in the heat transfer coefficients obtained at 25 and 31 MPa. Small pressure differences, however, can also have a significant effect. Figure 4.11 shows the results of the current study for a case with a fairly small relative pressure difference. It reveals that at pressures close to the critical point, a small pressure difference (even 1 MPa as shown) can cause a considerable change in the heat transfer coefficient. Note that following the way in which 62 pseudo-critical temperature varies with pressure (Fig. 4.10), the peak in the heat transfer coefficients in Fig. 4.11 was shifted to a higher temperature with an increase in pressure. 370 1—•— 1—•— 1—•— 1—' ' •— i—•—i—•—i—•— i—i— i—i— i—^-i— 22 23 24 25 26 27 28 29 30 31 32 Pressure, MPa Figure 4.10. Variations of pseudo-critical temperature of water with pressure 1700 1850 2000 2150 2300 2450 2600 Bulk Enthalpy, kJ/kg Figure 4.11. Effect of pressure on heat transfer coefficients, G=592 kg/m 2 .s , q"=263 k W / m 2 63 4.3. HEAT TRANSFER DETERIORATION In most supercritical fluid flows, heat transfer is enhanced near the critical region. Under certain conditions, however, heat transfer may deteriorate. There are two situations in which degraded heat transfer can occur. A buoyancy effect is mainly responsible for one of them. A stratified kind of flow in horizontal test sections leads to degradation of heat transfer at the top surfaces of the tube. The other type of deterioration appears as local hot spots in the tube wall and is a more complicated phenomenon. The flow acceleration, generated as a result of large density variations, is probably a major cause in this situation. There also seems to be a correlation between this type of deteriorated heat transfer and flow oscillations. Both kinds of deteriorated heat transfer were observed in our experimental facility. The results are presented below. 4.3.1. Deterioration along the Top Surface The buoyancy effect in a horizontal flow can be detected by temperature differences between the top and bottom surfaces of the tube. High heat flux and low mass flow promote the buoyancy effect. Typical results of such cases are shown in Figure 4.12. O o c CD O 1 o O I c ro ra X 6 Bottom Surface A 45° w.r.t. top center >0 • Top Surface 1400 1600 1800 2000 2200 2400 2600 2800 Bulk Enthalpy, kJ/kg Figure 4.12. Heat transfer coefficients at different peripheral positions, P=24.3 MPa, G=432 kg/m2.s, q"=301 kW/m2 64 Heat transfer coefficients were measured at the bottom, top and at a 45° position with respect to the top centerline of the tube. It can be seen from Fig. 4.12 that the peak heat transfer coefficients at the bottom surface are more than two times larger than those at the top and slightly less than two times those at 45° locations. The degradation of heat transfer at the top surface is broad and covers a large part of the test section. Deterioration at the top surfaces was accompanied by an improvement of heat transfer at the bottom surface of the tube. The opposing effects of buoyancy on heat transfer at the top and bottom surfaces almost cancel each other. The average heat transfer coefficient is only slightly affected by buoyancy. In practice however, overheating of the tube at the top surface of the test section may sometimes become problematic. Thus, it is important to distinguish buoyancy-affected from buoyancy-free flows. Some criteria underlying the buoyancy-free situations have, been suggested in the literature. Most of those results, however, are not applicable to horizontal flows. The large amounts of buoyancy-affected data obtained in the current study deserve more analysis and are presented in chapter 6. 4.3.2. Local Hot Spots on the Wall The body forces, i.e., buoyancy and inertial forces, under certain circumstances, may lead to hot spots along the tube wall. A typical result of locally deteriorated heat transfer is shown in Figure 4.13. The results of two other non-deteriorated tests are also shown for the sake of comparison. One peculiarity of this type of deteriorated heat transfer was that it was highly irreproducible. A certain combination of control parameters, i.e., pressure, heat flux, mass flow, tube diameter, flow orientation and inlet temperature decide whether the effect of body forces can lead to local overheat of the tube wall. 65 1800 1900 2000 2100 2200 2300 2400 Bu lk En tha lpy , k J / k g Figure 4 .13. Loca l deterioration of heat transfer, P=23.7 M P a , G=585 kg/m 2 .s . The measurements were made at the top surface of the test section. An important observation was that occurrence of hot spots on the wall did not have a linear correlation with heat flux. Once the heat flux was large enough to destabilize the temperature gradient, a further increase of heat flux may even stop deterioration and return the heat transfer rate to its normal trend. This can be observed in Fig. 4.13. There was no sign of an abnormal temperature distribution along the tube for heat fluxes up to 252 kW/m . Once the heat flux was raised further to 259 kW/m the hot spot(s) occurred on the tube surface. Beyond this limit, a small increase of heat flux to 262 kW/m 2 caused the hot spot to vanish. Another observation worth mentioning is that the hot spots always appeared at the same location. It was noticed that the same conditions leading to deterioration at this certain point did not cause any anomaly elsewhere along the tube. This was examined by trying different tests with exactly the same flow but different inlet temperatures. The slight increase or decrease of inlet temperature did not shift the location of the hot spots 66 towards upstream or downstream. It could only facilitate or prevent the overheat of the wall at the same exact location. The correlation between the distance and deterioration of heat transfer suggests involvement of flow oscillations, which are highly dependent on flow geometry (the tube length). There seems to be a relationship between the nodes of flow oscillations (as they would occur in an open ended organ pipe) and the locations where the hot spots occur. Theoretically, the deteriorated mode of heat transfer under supercritical conditions is known as one of the most complicated flows. In terms of modeling, most terms in the momentum and energy equations cannot be neglected. It requires a two-dimensional (for vertical flows) or a three-dimensional (for horizontal flows) model to account for the effects of body forces. There is no universal physical model to satisfactorily account for the effects of body forces and predict the deteriorated heat transfer coefficients accurately. To give an idea about the complexity of the problem, the inadequacy of two available explanations for the cause(s) of heat transfer deterioration are presented below. It was shown in the section 4.2.3 that heat transfer decreases with an increase of heat flux in the critical region. It was stated that the larger the heat flux (and/or the lower the mass flow), the more compressed is the region of flow at the pseudo-critical temperature. Such a trend will go on until the peak in heat transfer coefficients vanishes. See Fig. 4.8. This explanation can justify disappearance of heat transfer enhancement only. It does not necessitate local deterioration of heat transfer and the occurrence of hot spots along the test section. It rather explains the situation of broad deteriorated heat transfer encountered in a vertical flow due to buoyancy effects. Furthermore, it may only describe cases of deterioration when Tb<Tpc<Tw. Our data show that the presence of local hot spots was not limited to such temperature conditions. This type of deteriorated heat transfer was also observed in cases with both bulk and wall temperatures below or above the pseudo-critical temperature. Jackson and Hall (1978) offered another explanation in terms of flow re-laminarization. With a constant-property flow, regardless of the magnitudes of heat flux and mass flow, the shear stress is highest at the wall and linearly vanishes to zero at the flow centerline. With supercritical fluid flow and under a large heat flux (and/or low mass flow), a severe density profile develops. The low-density fluid at the wall tends to 67 accelerate more than the heavier fluid at the core. Thus, at some distance from the wall a reverse shear stress profile is formed which leads to suppression of turbulence production. Heat transfer deteriorates during such a re-laminarized region of the flow resulting in a local rise of the wall temperature. This explanation also would appear not to fit horizontal flows very well and the observations mentioned in the current study may not be fully justified. The problem of deteriorated heat transfer deserves an independent study and is beyond the scope of the current investigation. 4.4. C O M P A R I S O N W I T H O T H E R STUDIES In spite of a great number of experimental studies, it is impossible to find one with exactly identical test conditions to the present study. More than 60% of the past investigations were carried out with fluids (mostly carbon dioxide) other than water. Among data for supercritical water, there are studies with a similar test section configuration, i.e., flow orientation and tube diameter, but they differ in the range of mass flow and heat flux. Consider the following examples. Shitsman (1963) studied a vertical flow with G=430 kg/m2.s and q"=233, 291, 326 and 384 kW/m 2. The tube diameter was 8 mm (the test section diameter of the current study is 6.3 mm). Mass flow and heat flux, however, closely match an experiment set of this study with G=432 kg/m2.s and q"=300 kW/m 2. The test pressure differs slightly in the two studies. The changes of inside wall temperature with the water bulk enthalpy are shown in Figures 4.14 and 4.15 for this study and that of Shitsman, respectively. The heat flux in Fig. 4.14 corresponds to a situation between cases b and c in Fig. 4.15. The different units of bulk enthalpy in the figures should not create any ambiguity. The equivalent value for an enthalpy of 500 kcal/kg is 2093 kJ/kg. Since the test conditions in Figs. 4.14 and 4.15 were not exactly identical, differences in the results were anticipated. The extent of disagreement of data, however, seems not to be proportional to the slight differences in the tube diameter and pressure in the two cases. The wall temperatures, T w , in the present study are more than 20 °C larger than those of Shitsman around the pseudo-critical point. Such a temperature difference is comparable with the difference between wall and bulk temperatures and results in a huge difference of heat transfer coefficients. Furthermore, as shown in Fig. 4.15, Shitsman reported a 68 jump in the wall temperature whereas the temperature variation in our data was quite smooth. 600 o ° 500 <D 3 1 CU Q. § 400 300 — o - Inside Wall Temperature Bulk Temperature j i i i innr exams Bulk Enthalpy, kJ /kg Figure 4.14. UBC Experiments, P=24.3 MPa, G=432 kg/m2.s, q"=300 kW/m2, D=6.3 mm 600 500 400 T w . -C a % b | 0 •-a ° ———" 4 i II "if' 700 Figure 4.15. Results of Shitsman (1963) for a vertical flow with D=8 mm, tw: wall temperature, t,: bulk temperature, i: bulk enthalpy, P=23.3 MPa, G=430 kg/m2.s, a) q"=233, b) q"=291, c) q"=326, and d) q"=384 kW/m . 69 The substantial disagreement between the results of Shitsman and the present study, both qualitatively (in terms of occurrence of the peak wall temperature) and quantitatively (in terms of magnitude of the wall temperature after the peak) suggest that there was a considerable influence of flow orientation (and hence buoyancy). The extension of the results obtained from either a vertical or horizontal flow to the other, though very similar in other aspects, may lead to significant error. In fact, the similarity conditions under which both vertical and horizontal flows remain unaffected by buoyancy, and thus behave the same, are not thoroughly understood. This matter will be revisited in greater detail in chapters five and six. The test sections in the majority of investigations are vertical, which further limits the chance of finding similar studies for the sake of comparison. The study of Vikhrev and Lokshin (1964) is one of the few works with supercritical water flowing in a horizontal test section. The results of our study with q"=300 kW/m 2 and G=642 kg/m2.s, and those of Vikhrev and Lokshin are shown in Figures 4.16 and 4.17 respectively. The graph labeled 4 in Fig. 4.17 corresponds to q"=349 kW/m 2 and G=700 kg/m2.s. It may be compared with Fig. 4.16. The inner diameter of their test section was 6 mm. 430 r • • • 1 1 1 1 1 1900 2000 2100 2200 2300 2400 2500 2600 Bulk Enthalpy, kJ/kg Figure 4.16. UBC Experiments, P=24.3 MPa, G=662 kg/m2.s, q"=303 kW/m2, D=6.3 mm 70 700 500 HOO 300 ZOO T 1 G=7i O0kg/n?s i -3 300 MQ 500 600 \ ; I x*—~~J 1000 1500 2000 2500k]/kg Figure 4.17. Results of Vikhrev and Lokshin (1964) for a horizontal flow, D=6 mm. Wall temperature is shown for 1) q"=699, 2) q"=582, 3) q"=465, and 4) q"=349 kW/m2. Despite the larger difference of heat flux and mass flow, as compared with the previous example, better agreement was observed. This further confirms the large influence of flow orientation. The quantitative differences are basically due to different values of heat flux and mass flow. Classification of flows in terms of independent variables such as mass flow and heat flux helps better evaluation of data. At least, it may give a tentative idea as to whether the heat transfer is likely to be deteriorated or enhanced for any given test conditions in the critical region. 71 4.5. FLOW CLASSIFICATION BY MEANS OF DIMENSIONAL NUMBERS The traditional non-dimensional numbers employed in empirical correlations are highly dependent on fluid properties. Since fluid properties vary drastically with pressure and temperature near the critical region, any uncertainty in the estimation of fluid properties can lead to a significant variance in the analysis. Alternatively, one may combine heat flux, mass flow and the tube diameter to derive a characteristic number through which the results of different studies can be compared effectively. Use of a well-established empirical correlation was made to derive such a dimensional number. Consider a typical Nusselt number correlation for convection heat transfer. It can be expanded to obtain Nu = — = a Rem Pr" £ = Al K V K J (4.3) is a correction factor and, as will be seen in the next chapter, is a function of fluid properties at the wall and bulk. Correction factors are usually used in correlations developed for supercritical fluid flows. "A" is the product of the constant "a" and and hence a function of fluid properties. From Eq. 4.3 the heat transfer coefficient is calculated as h = --A\ D GD V Y1 = AGmDm-lKl+npn-mCj n or, h = GmDm-lf(Tw,Tb) (4.4) where f(Tw, Tb) includes the constant A and the fluid properties and is a function of T w and T b . Equating 4.4 with h = q'7( T w - Tb) yields m r\/n—1 G D = (Tw-Tb)-f(Tw,Tb) = g(Tw,Tb) (4-5) 72 Here again "g" is a function of T w and Tb only. The above equation implies that at constant pressure, for a given bulk temperature, Tb, flows with equal q"/(G m .D m l ) will achieve the same wall temperature, T w . A question which remains to be investigated is what value of "m" is most appropriate for supercritical fluid flows. In many studies, q'7G has been used as a measure of similarity between the flows. This is equivalent to assuming "m" as unity. Then, the parameter q"/(G m .D m l ) simply reduces to q'VG. Nonetheless, the following discussion demonstrates how q"/G fails to be a universal tool by which similar supercritical fluid flows can be recognized. In Fig. 4.14, shown earlier, the temperature distribution for a test with G=432 kg/m2.s and q"=300 kW/m 2 (q'VG = 0.69) was presented. Graph number 3 of the results of Vikhrev and Lokshin shown in Fig. 4.17 corresponds to G=700 kg/m2.s and q"=465 kW/m 2 (q'VG = 0.66). The magnitudes of q'VG in two cases are very close to each other. Nevertheless, it can be seen that the discrepancy in temperature distribution is large. The differences in the wall temperatures are as large as 30 °C. In terms of heat transfer coefficients, this translates to a difference of more than 100%. Even for two flows with identical geometry, equal values of q'VG does not guarantee an identical temperature distribution. This is demonstrated by comparing the results of two tests of the present study. Figure 4.18 shows the wall temperature distributions for two cases with (q"=267 kW/m 2 and G=587 kg/m2.s) and (q"=154 kW/m 2 and G=339 kg/m2.s). The value of q'VG is 0.45 kg/kJ in both. The disagreement of the results is evident. Apart from an inadequacy of q'VG as a quantitative measure, caution should be exercised even when it is used as a qualitative tools to distinguish between various flows (e.g., buoyancy-free and buoyancy-affected flows). Consider the following example. Variations of heat transfer coefficients with fluid bulk enthalpy for both top and bottom surfaces of the test section with q"=103 kW/m 2 and G=575 kg/m2.s (q"/G=0.18) are shown in Figure 4.19. Yamagata et al. (1972) used a heat flux and mass flow about 2.3 times larger than the values in the present test, yet the ratio of q'VG was the same. Both vertical and horizontal flows were tried. Their results are shown in Figure 4.20. 73 c P C M ^ 601 C CD O o 20! c CO H TO 0 I ? P = 245 bar , G= 1260 kg/ms q" = 233 kW/m2 d =7.5 mm 300 350 Vertically upward Vertically downward Horizontal (Top) Horizontal (Bottom) 450 500 400 T b ' °C Figure 4.20. Results of Yamagata et al. (1972) for Flows with q"/G=0.18 kg/kj It can be seen that buoyancy effects are minimal in Yamagata et al. This is justified since identical results were obtained for vertical flow and both top and bottom surfaces of the horizontal tube. The difference of heat transfer coefficients at the top and bottom surfaces in the current study, however, is clearly observed in Fig. 4.19. The test section diameter in the present study (6.3 mm) was smaller than that of Yamagata et al. (7.5 mm). Note that the effect of buoyancy would be expected to be less pronounced in a test section with smaller tube diameter. The conclusion one may draw here is that q"/G is not capable of accurately categorizing supercritical fluid flow behavior. It is evident from D m l in Eq. 4.5 that any value of "m", other than unity could better account for the effect of tube diameter. An alternative value for "m", suggested by some investigators (e.g., Goldman 1961) is 0.8. Recall from Eq. 4.3 that "m" was the exponent of Reynolds number in an empirical correlation. In a number of those correlations, which were obtained from a curve fit of experimental data, Reynolds 75 number is raised to 0.8. For m=0.8, we get q 7 G m . D m l = q " D ° 2 / G 0 8 . The following examination illustrates the improved success of q " D 0 2 / G ° 8 in describing heat transfer to supercritical fluid flows. Heat flux and/or mass flow were changed in three different experiments of the present study so that the values of 0.26, 0.59, and 0.85 for the parameter q"D° 2 / G ° 8 were obtained. The units of q"D a 2 G"° 8 are (kJ/kg)°- 8(kW/m) 0- 2. The variations of inside wall temperature at the top surfaces of the test section were superimposed and are shown in Figure 4.21. Figure 4.21. Variations of inside wall temperature at top surfaces of the test section for three case studies: (G=336 kg/m2.s, q"=75 kW/m2: q'.D^.G 08=0.26), (G=581 kg/m2.s, q"=265 kW/m2: q".D02.G*8=0.59 (kj/kg)° 8 (kW/m)° 2 ), and (G=432 kg/m2.s, q"=300 kW/m2: q".D02.G08=0.85 (kJ/kg)08(kW/m)02). Figure 4.22 shows the results of Yamagata et al. (1972) for three different test conditions. The curves with q"=233, 465 and 698 kW/m2 correspond to values of 0.29, 76 0.58 and 0.87 for q"D°-2/G0 8 respectively, which are very close to those of Fig. 4.21. A careful comparison of the temperature distributions shown in Figs. 4.21 and 4.22 reveals some discrepancies. However, the extent of these is not large and the trend of temperature variations is qualitatively similar. 500 > 400 300 P=24.5 MPa G-1260 kg/irrs D=7.5 mm Bottom Top q", kW/m2 q"D 0 3/G 0 8 * A 233 0.29 0 • 465 0.58 ° • 698 0.87 1000 2000 Bulk E n t h a l p y , k J / k g 3000 Figure 4.22. Results of Yamagata et al. (1972) for different values of q"D° 2/G° 8 A great volume of previous heat transfer data for supercritical fluid flows was presented only in reduced form via empirical correlations. To evaluate such data as well as to search for a mathematical tool to predict heat transfer coefficients, Nusselt number correlations are studied in detail in the next chapter. 77 CHAPTER FIVE EMPIRICAL CORRELATIONS Representing heat transfer data in the form of a relationship between non-dimensional numbers has been general practice over decades. In this chapter an extensive review of available forced convection heat transfer correlations is presented. Comparison with experimental data is made and discrepancies between the various correlations are highlighted. Based on our assessment of these empirical correlations a new method of calculating for heat transfer coefficients is introduced. 5.1. GENERAL HEAT TRANSFER CORRELATIONS The general form of a Nusselt number correlation for forced convection heat transfer in a pipe flow is Nu=aRebPrc. The constants a, b and c are obtained via a curve fit of experimental data. Nusselt, Reynolds and Prandtl numbers are non-dimensional. They are defined as Nu=hD/K, Re=GD/p: and Pr=iiCp/K, where h is heat transfer coefficient, D is tube diameter, and G is mass flow. The thermal conductivity, viscosity and specific heat of the fluid are denoted by K, | i and Cp, respectively. This type of presentation is usually called a Dittus-Boelter correlation after Dittus and Boelter (1930). For cases where temperature variations, and hence changes of fluid properties are noticeable, the conventional correlations needed to be modified. Seider and Tate (1936) included a correction factor of wall-to-bulk fluid viscosity to account for property variation. Their 78 correlation predicted heat transfer coefficients satisfactorily for fluids having monotonic variations of properties with temperature. A more accurate correlation in a slightly different format was proposed later by Petukhov and Kirillov (1958) as follows. A , (Cf/2)RebPr ( 5 . 1 } ^ ^ ( C ^ P r ^ - l j + l . O ? where, Cf = I (5-2) f (3.641og]0Refc-3.28)2 This correlation was further modified by Gnielinski (1976) to cover a wider range of flow conditions with the same or better accuracy. For a fluid whose thermophysical properties do not vary significantly with temperature, flow similarity may be established through the use of dimensionless parameters such as Nusselt, Reynolds and Prandtl numbers. Heat transfer in such flows can be calculated via an empirical relationship between the non-dimensional numbers as explained above. The dramatic property change a supercritical fluid undergoes near the critical region, however, complicates the problem. Such property variations are too large to be addressed by correlations not specifically derived for supercritical fluid flows. 5.2. CORRELATIONS FOR SUPERCRITICAL FLUID FLOWS Many investigations have been performed towards developing a heat transfer correlation for supercritical fluid flows. As result, more than twenty different correlations were derived over the last few decades. Several popular correlations, which are more frequently referred to in the literature, are briefly reviewed and compared with data from this study. For a more complete listing of the developed correlations see Kakac (1987). Miropolski and Shitsman (1957) introduced one of the earliest correlations. Instead of implementing a correction factor, based on local flow conditions, they suggested an alternative use of wall and bulk Prandtl number in their correlation. This made their correlation, stated below, simple and easy to use. 79 Nub= 0.023 R e ; - 8 P r m i n a 8 (5.3) Pr m i n is the smaller of the Prb and Prw. Miropolski and Shitsman examined vertical flows of supercritical water in tubes with 7.8 and 8.2 mm LD. The data were given in a non-dimensional format and the raw data were not presented. The range of validity of the correlation was not discussed. The correlation of Petukhov and Kirillov (1958) for constant-property flows was modified by Petukhov et al. (1961) to correlate supercritical fluid flow data. They measured heat transfer to a horizontal flow of supercritical carbon dioxide. No difference was noted between heat transfer at the top and bottom surfaces of the tube. It assured that their data were not affected by buoyancy. However, to expand the range of applicability of their correlation, they included data of others for vertical flows of water and carbon dioxide, which were not guaranteed to be buoyancy-free flows. Their correlation is as follows. Nub = ( C ^ R e . P r 12.7^(c//2)(Pr/M)+1.07 f y O . l l U ( K A0 3 V Cpb \0.35 (5.4) where the average specific heat is given by Cp = T„,-Th (5.5) They did not specify, however, the range of conditions over which their correlation is applicable. For the sake of brevity, the correlation and the study of Petukhov et al. (1961) will simply be referred to as Petukhov from now on. In a study of upward flow of supercritical water in a vertical test section, Swenson et al. (1965) used the averaged specific heat (Eq. 5.5) and introduced an averaged Prandtl number as p7w =//M,Cp//rw • Transport properties were evaluated at the wall temperature. They also evaluated the Reynolds number at the wall to express the 80 wall Nusselt number. This correlation is one of the rare cases where the wall Nusselt number, instead of bulk Nusselt number, was used. For this reason, as will be seen later, it predicts the peak of heat transfer coefficients earlier than other correlations. Unlike many earlier studies, Swenson et al. clearly specified the test conditions. The inside diameter of the tube was 9.42 mm. Heat flux varied from 205 to 1823 kW/m 2. Mass flow was between 542 and 2149 kg/m2.s. For the sake of brevity the study and correlation of Swenson et al. (1965) will be simply referred to as Swenson in this study. The correlation of Swenson et al. is given by Nu = 0.00459 R e „ 0 9 2 3 P r / 6 1 3 \0.231 (5.6) Krasnoshchekov and Protopopov (1966) measured heat transfer to supercritical carbon dioxide flowing in a horizontal test section with an inside diameter of 4.08 mm. It was mentioned in their work that wall temperatures were measured at the top and bottom surfaces of the tube. This was not, however, addressed in their correlation. Their predictions almost exactly match the results of Petukhov in most cases. The range of variations of flow parameters on which the correlation was based was clearly stated. Their correlation is (Cf/2)RebPr 12.7^(c//2)(Pr/^)+1.07 Pb (5.7) where, n=0.4 for Tb<Tw<T, n=0.4+0.2[(Tw/Tpc)-l] n=0.4+0.2[(Tw/Tpc)-l] {l-5[(TVTpc)-l]} pc or 1.2T p c<T b<Tw for Tb<Tp c<Tw for Tp c<Tb<1.2Tpc A clear distinction between vertical and horizontal flows was made in the experimental study of Yamagata et al. (1972). They showed results for the top and bottom surfaces of a horizontal test section as well as for a vertical flow with identical test conditions. Their empirical heat transfer correlation was however developed based on 81 data for an upward vertical flow in a 10-mm tube. The correlation of Yamagata et al. is presented below. Pr p c, used to define the correction factor, F c , is the Prandtl number evaluated at the pseudo-critical temperature. Nub = 0.0135Re fc085 P r / 8 Fc (5.8) where, E = (Tp c-Tb)/(TW-Tb) Fc=1.0 for 1<E Fc = 0.61Vrpc^5(CplCpbT for 0 < E < 1 Fc=(Cp/Cpb)"2 for E < 0 and ni = -0.77 (1+1/Prpc) + 1.49 n 2 = 1.44 (1+1/Prpc)- 0.53 Based on their review of empirical correlations, Ghajar and Asadi (1986) recommended a modified version of the correlation of Jackson and Fewster (1975) as follows. Nub = 0.0064 Re 0' 9 1 P r 0 4 8 \0.6 / \n ^ ( Cp^ CPb (5.9) where, the exponent n is defined in the same manner as in correlation of Krasnoshchekov and Protopopov. Jackson and Fewster simply used a Dittus-Boelter type of correlation to apply the correction factors whereas in Krasnoshchekov and Protopopov the correlation of Petukhov-Kirillov was used as the base. 5.3. COMPARISON WITH EXPERIMENTS A Fortran program, EMPLRICAL.f, was developed to evaluate the available empirical correlations. Input data are pressure, tube diameter, heat flux, mass flow and bulk temperature. Based on a specific correlation, the Nusselt number and hence the local heat transfer coefficient is predicted. In most correlations, fluid properties at the wall are included in one way or another. Thus, the calculation starts with an initial guess for the 82 wall temperature. The actual value is obtained by iteration. The program is run repeatedly for a number of bulk temperatures so that predicted heat transfer coefficients can be presented versus bulk temperature. As explained in chapter 4, to better illustrate axial variations of heat transfer coefficients along a test section, the fluid bulk enthalpy is usually used for the horizontal axis of the graph, instead of bulk temperature. An empirical heat transfer correlation is a curve fit of experimental data. It is not surprising that the correlation matches the experimental data of the same study. The question is how well one can predict heat transfer under different test conditions. In the following example, shown in Figure 5.1, input data are set identical to a case study presented by Swenson et al. The predictions of the Swenson correlation can be considered to represent their data. Use of different fluid properties in the original work of Swenson and in the calculations by EMPIRICAL.f does not generate noticeable discrepancies. The pressure in this example (31 MPa) is reasonably far from the critical point (22.1 MPa). Note that the differences of thermophysical properties of water in various studies have been most significant at pressures close to the critical point. 1 0 1400 1600 1800 2000 2200 2400 2600 2800 Bulk Enthalpy, kJ/kg Figure 5.1. Comparison of available correlations with experiments of Swenson et al., P=31 MPa, q "=789 kW/m2, G=2149 kg/m2.s, D=9.42 mm. 83 The differing predictions of various correlations are apparent. Miropolski-Shitsman over predicts the results of Swenson. Fewster-Jackson underestimates heat transfer coefficients. Petukhov better predicts the peak value of heat transfer coefficients. However, the peak value in the results of Swenson was shifted upstream by all the other correlations. Regardless of how accurate the experiments of Swenson were, the clear message of the above comparison is the difference between the available empirical correlations. A Yamagata et al. Miropolski-Shitsman 30 340 350 360 370 380 390 400 410 420 Bulk Enthalpy, kJ/kg Figure 5.2. Compar ison of available correlations with experiments of Yamagata et al . , P=24.5MPa, q"=465 k W / m 2 , G=1200 kg/m 2 .s , D=10 m m . Let us consider another example. Shown in Figure 5.2 are the predictions of some correlations for the test conditions of Yamagata. The peak heat transfer coefficients of the Yamagata correlation, calculated by EMPIRICAL.f, are about 10% greater than their actual test results. This is attributed to the differences in properties in Yamagata and EMPLRICAL.f. The Yamagata correlation is more sensitive to changes of properties than the Swenson correlation. Furthermore, the pressure (P = 24.5 MPa) in this example is 84 closer to the critical point making calculations more sensitive to the use of different properties. It can be seen that the predictions of Miropolski-Shitsman for part of the critical and post critical regions fall on the results of Yamagata. For the pre-critical region the predictions sometimes differ noticeably from the Yamagata results. Swenson and Petukhov both under-predict significantly the peak of heat transfer coefficients. The differences between these two correlations, however, were larger than for the previous example. Again, Swenson suggested the occurrence of peak heat transfer earlier in the flow than the rest of the correlations. Comparing the Swenson and Yamagata correlations, it can be said that each of them is unsuccessful in predicting heat transfer under flow conditions for which the other correlation had been developed. This is almost always true in comparing any two correlations. It is not easy to judge which of the experiments in the literature are more reliable. The corresponding correlations claim to be more or less equally accurate within the range of experimental conditions. As such, from the apparent disagreement between correlations one can conclude that none of them are reliable over a wide range of flow conditions. By further inspection and comparison of preferred correlations, the following interim conclusions may be reached. (1) There is a pattern in the predictions of various correlations. Higher heat transfer coefficients are suggested by Miropolski-Shitsman and Yamagata. Lower values are predicted by Fewster-Jackson. Swenson and Petukhov offer intermediate results. Predictions of Krasnoshchekov and Protopopov very closely match those of Petukhov. (2) Compared to other correlations, peak heat transfer coefficients predicted by Swenson are shifted slightly upstream. That is because unlike other correlations, the Reynolds number in the Swenson correlation is estimated at the wall temperature. Since the pseudo-critical region is first experienced at the wall, the Reynolds number reaches its maximum value earlier than other correlations. Maximum Reynolds number corresponds to peak values of Nusselt number and heat transfer coefficient. One advantage of the Swenson correlation, on the other hand, is that it is easy to use and iteration to estimate wall temperature rapidly converges. 85 (3) The Yamagata correlation divides the flow into three parts, pre-pseudo-critical, near pseudo-critical, and post-pseudo-critical regions. Each region is treated separately as suggested by the differing equations shown earlier. Depending on test conditions, the interface between these three regions may or may not happen smoothly. For conditions incorporated in Fig. 5.2, the predicted heat transfer coefficients varied continuously along the flow. Under different conditions, sudden jumps were observed in the boundaries of flow regions. In fact, for this reason, and to avoid ambiguity, the predictions of Yamagata were not shown in Fig. 5.1. The difficulty in convergence of the iteration process is due to the existence of three different equations for the flow regions. Near the boundaries of each region, after an iteration, the corrected wall temperature may lie within the neighboring region. Then, a different equation will be applied for heat transfer calculations, which may not cause the predicted wall temperature to converge. This is more or less the problem with any correlation in which different equations were suggested for different regions of flow. Besides flow conditions, the use of different properties may expand these difficulties. As mentioned earlier the effect of property variations on the Yamagata correlation is significant. This will be discussed in more detail later in this chapter. (4) The Miropolski-Shitsman correlation provides results similar to Yamagata without the problem of non-converging iterations. Nevertheless, a shift from Prb to Pr w and vice versa to substitute Prmj n in their correlation can sometimes cause a jump in the estimated value of heat transfer coefficient. This can be seen in Fig. 5.2. The extent of such a discontinuity is usually less than that observed for the Yamagata correlation. (5) With respect to the experimental results of the current study, it appears that the family of correlations close to Petukhov provides better predictions. An example is shown in Figure 5.3. It should be mentioned that this conclusion is limited to tests with low heat flux where the heat transfer difference between top and bottom surfaces of the tube is negligible. 86 o o _ 60 -CM | 50 -+s c <U 40 -'o £ o 30 -O i_ 1 20 • c ra H 10 -ra a> I 0 -U B C Experiments M iropolski-Shitsmar/ Swenson et a l . / Petukhov et a l . Fewster -Jackson/ A . v . 1500 1700 1900 2100 2300 Bulk Enthalpy, kJ/kg 2500 Figure 5.3. Comparison of available correlations with experiments of this study, P=25.2 MPa, q"=307 kW/m2, G=965 kg/m2.s, D=6.3 mm. Once buoyancy is significant, depending on test conditions, one correlation may better predict the temperature at the top surface of the tube whereas another is closer to the bottom surface. This is shown in Figure 5.4. However, these correlations were developed for axisymmetric flows (mostly for a vertical test section) and were not intended to distinguish between top and bottom surfaces in a horizontal flow under buoyancy effects. Investigating the source of discrepancies between empirical correlations is helpful. It may explain the extent to which each flow parameter can influence heat transfer coefficients. It also facilitates a more realistic assessment of empirical heat transfer correlations. In the following sections potential sources of disagreement between various correlations are presented. 87 o o c a> o at o o I c ra ro X 40 30 20 10 Experiment (bottom surface) Experiment (top surface) Miropolski-Shitsman Fewster-Jackson .••*""'*•, Petukhov et al. 1400 1600 1800 2000 2200 2400 2600 2800 Bulk Enthalpy, kJ/kg Figure 5.4. Comparison of available correlations with data of this study for a buoyancy affected flow, P=23.9 MPa, q"=124 kW/m2, G=340 kg/m2.s, D=6.3 mm. 5.4. EFFECT OF DIFFERENT FLUID PROPERTIES It has been reported in the literature that the use of different fluid properties is one of the reasons that empirical correlations differ. The equation of a fitted curve to a set of experimental data would obviously vary if different tables of fluid properties were used. Nevertheless, it does not necessarily mean that if the same properties were used in all investigations, identical correlations or at least fewer discrepancies between them could be reached. This is, in fact, a difficult problem to investigate, especially since in many studies references for fluid properties were not specified. One needs to redevelop the correlations from raw data on the basis of the same table of fluid properties and see if the revised relationships look alike. Suppose Yamagata over-predicts and Swenson under-predicts an experiment for example. It is only a hypothesis to say that if they used the same properties they would end up with the same correlations. To trace back correlations 88 to raw data and redo the curve fitting with a unique property library would be an extremely tedious job and impractical. Instead, looking into the way correlations behave with a change of tabulated fluid properties may elucidate this problem. It is easy to show to what extent using different tables of fluid properties affects predictions of heat transfer coefficients. Consider, for example, the correlation of Swenson. It may be expanded to obtain ^ = 0 . 0 0 4 5 9 ^ K I P J \0.613, ^0.231 K* Tw-Tbj K P b J (5.10) Thus, ^ • 0 . 6 1 3 ^ 0 . 9 2 3 ^ . 3 1 0 ^ . 0 . 3 8 7 Changes in thermodynamic properties i and p can have larger effects on heat transfer coefficient h compared to variations of transport properties K and \i. This is true in other correlations too. Thermodynamic properties, however, have been well developed, with the exception of cases within the proximity of the pseudo-critical point, for some time. They, therefore, have little contribution towards disagreements between various correlations. The uncertainty in values of transport properties, on the other hand, has had a considerable effect. Viscosity and thermal conductivity of water from two different sources for pressures of 23 MPa and 25 MPa are shown in Figures 5.5 and 5.6 respectively. The properties denoted as IAPWS are based on IAPS (1984a) and LAPS (1984b). There exists a peak value of thermal conductivity near the pseudo-critical point. The curve-fitted equations approximate the experimental data within 10% and have been used in this study. 89 Bulk Enthalpy, kJ/kg Figure 5.5. N B S and I A P W S transport properties of water at 23 M P a Figure 5.6. N B S and I A P W S transport properties of water at 25 M P a 90 NBS values in Figs. 5.5 and 5.6 refer to tables published by Harr et al. (1984). Linear interpolation was used to obtain values not available in the tables. The temperature steps in the tables were large. Thus, interpolation creates a large error near the pseudo-critical point where properties do not vary linearly. This generates a desirable gap between the estimated values of IAPWS and NBS to better study the effect of transport properties on heat transfer coefficients. It is evident from Figs. 5.5 and 5.6 that the more supercritical the pressure, the closer to each other are the estimates of NBS and IAPWS. The uncertainty of the fluid properties, in general, is more vital at pressures closer to the critical point. It is also shown that far from the critical region, differences in values of NBS and IAPWS vanish. The way that each correlation is affected by using different tables of properties is not the same for all empirical correlations. Those correlations applying different equations or coefficients according to local flow conditions are more influenced by various properties. Shown in Figure 5.7 is a comparison of two of the previous mentioned correlations incorporating both NBS and IAPWS databases. 80 70 60 50 40 30 Miropolski-Shitsman, NBS Properties ••a Miropolski-Shitsman, IAPWS Properties - • — Swenson, NBS .m - e — Swenson, IAPWS \ 1500 1700 1900 2100 2300 2500 2700 Bulk Enthalpy, kJ/kg Figure 5.7. Effect of transport properties on empirical correlations, P=23 M P a , q"=200 k W / m 2 , G=600 kg/m 2 .s , D=6.2 m m 91 Notice the significant difference between the predictions. The Miropolski-Shitsman correlation shows the most variation due to their use of Pr^n as a deciding parameter, which can be Pr b or Prw . According to one table of transport properties Pr b can be smaller than Pr w at a certain point in the flow, while a different table of properties may suggest the opposite. In the correlation of Swenson et al., on the other hand, there is no such conditional term and hence it is less affected. In Fig. 5.7, the test pressure, 23 MPa, is close to the critical pressure. The large differences in transport properties between NBS and IAPWS did not generate much of a change in Swenson. However, a drastic change with the Miropolski-Shitsman correlation was observed. The use of NBS transport properties caused both correlations to approach each other. Figure 5.8 demonstrates the results of another example with the same flow parameters, except that the pressure was 25 MPa. Apparently, the choice of transport properties has less significance compared to that shown in Fig. 5.7. At a pressure of 30 MPa or higher, there is no difference in values of NBS and IAPWS and thus the correlations are not affected by implementing different properties. The predictions of different correlations, however, as already shown in Fig. 5.1, do not collapse onto the same curve yet. Figure 5.8. Effect of transport properties on empirical correlations, P=25 MPa, q"=200 kW/m2, G=600 kg/m2.s, D=6.2 mm 92 It is true that disagreements of correlations are greatest near the pseudo-critical temperature where various tables of properties differ most. For example at a pressure of 23 MPa and water enthalpies smaller than 1700 kJ/kg or larger than 2700 kJ/kg, the properties from both sources are about the same (Fig. 5.5). Consequently, differences in heat transfer predictions are minimal under those conditions (Fig. 5.7). However, it can be concluded from the above discussion that in contrast with what is widely stated in the literature, the use of different properties appears not to be the major cause of discrepancies between various correlations. 5.5. EFFECT OF NATURAL CONVECTION It was mentioned in chapters 2 and 4 that many early investigations did not distinguish between vertical and horizontal flows. In some earlier studies the orientation of the test section was not even reported. For horizontal test sections, some did not specify if measurements were done at the top or bottom surface of the tube. In a number of studies with vertical test sections, the differences between upward and downward flows were not accounted for. In most such studies the effect of buoyancy was already assumed to be negligible before it was investigated. Thus, many experimental results intended to represent forced convection heat transfer, in fact, likely reflect mixed convection situations. Empirical correlations are affected by such ambiguity. Neglecting buoyancy is claimed, in this study, to be the most important source of discrepancy between empirical correlations. The effect of buoyancy and related experimental results are presented in the next chapter. Here, the impact it has had on empirical correlations is studied. The mass flow and heat flux in Miropolski-Shitsman overlap the range of variables in this study. The flow orientation was not clearly stated in their study. Nevertheless, in later studies (e.g., Shitsman, 1967) it was referred to as vertical flow. The diameter of their test section, 29 mm, was relatively large which favors natural convection activities. Under similar conditions of mass flow and heat flux in our tests and theirs, a large temperature difference between top and bottom surfaces was observed in our data. It could be deduced that their data were highly affected by buoyancy. Thus, the correlation of Miropolski-Shitsman is not expected to fit buoyancy-free conditions. Use 93 of their correlation is therefore limited to vertical flows under conditions identical or close to the original work of Miropolski-Shitsman. In the study of Yamagata, a number of flows with different heat flux, mass flow and orientation were tried. The effect of natural convection in their vertical test section was to enhance heat transfer in downward flow and decrease it during upward flow. For horizontal flow they showed that bottom surfaces had higher heat transfer coefficients than top surfaces. Despite the fact that they distinguished between the buoyancy-affected and buoyancy-free flows, the data pool they used to derive a single correlation contained both types of flows. Since the correlation was based on data for upward flows, it is recommended only for upward flows. The accuracy of their prediction is expected to fall as conditions deviate from the range of parameters on which it was based. The Swenson correlation was derived for upward vertical flows. The range of flow parameters for which the correlation was derived was given. Downward flow under the same conditions was not tried, i.e., buoyancy effects were not investigated. Generalizing the range of validity of Swenson depends upon whether the buoyancy effect was significant in their study or not. Comparing Swenson results with the present study cannot answer the question. Their range of control parameters was different than this study. As will be discussed in chapter 6, it is difficult to set a precise demarcation between flows affected by natural convection and mere forced convection heat transfer at supercritical pressures. There is some evidence indicating that the effect of natural convection was not negligible in Swenson. Consider the Swenson and Yamagata correlations. Both were developed from upward flows. Tube diameters were 9.4 and 10 mm respectively. Mass-flow and heat flux in Swenson was much larger. However, the range of variations in q"D°' 2/G 0 8 in both studies was close to each other. It was shown in chapter 4 that two flows with identical values of q"D a 2/G 0- 8 may be considered similar flows. Thus, Swenson's experiments were probably influenced by buoyancy as were Yamagata's. Even if the effect of natural convection was negligible, the extension of Swenson to horizontal flows is questionable. To the best knowledge of the author, there is no investigation showing that if natural convection is not a factor in a supercritical vertical flow, it will not be important in an identical horizontal flow. Only if the mass-flow and 94 heat flux were set conservatively well beyond the available criteria for buoyancy-free conditions, would no differences in vertical and horizontal flows be expected. This includes only flows with large mass flow and small heat flux. It cannot then be concluded that buoyancy-free conditions in vertical and horizontal flows are the same. To emphasize the misleading effects of neglecting buoyancy consider Figure 5.9. The predictions of a few correlations for the test conditions for which Yamagata showed the effect of natural convection was zero, are presented. 1500 1700 1900 2100 2300 2500 2700 Bulk Enthalpy, kJ/kg Figure 5.9. Comparison of correlations under no buoyancy conditions tried by Yamagata, P=24.5 MPa, q "=233 kW/m2, G=1260 kg/m2.s, D=7.5 mm. As shown, the disagreements between predictions are not resolved. It supports the idea that the data used to develop each of the correlations were more or less affected by buoyancy so that when correlations are tried against a pure forced convection case they do not provide acceptable results. 95 The advantage of Petukhov is that they only considered cases with a small ratio of wall to bulk temperature, i.e., low heat flux regimes. The effect of natural convection, compared to other studies, was small. However, as mentioned earlier, to expand the range of applicability of their correlation they included data of others for vertical flows of water and carbon dioxide, which were not guaranteed to be buoyancy-free flows. 5.6. THE ASSUMPTION OF "FULLY DEVELOPED FLOW" Far enough from the flow inlet where growing boundary layers at opposite walls meet and there is no more potential flow at the core, the flow is considered to be fully developed. Note that since the temperature profile is extremely steep at the wall and quite flat at center, the bulk temperature almost equals the center or core temperature. The bulk and core temperatures have been taken in this study to be the same. At normal pressure and under constant heat flux, the developed velocity profile does not change along the course of flow. The developed temperature profile is also similar everywhere. At high heat flux the temperature difference between wall and bulk may become large. Consequently, the fluid properties vary from wall to the core. In such variable-property, forced convection flow, the estimated values of Reynolds and Prandtl numbers at wall and bulk are different. However, it does not essentially violate the similarity of each of the velocity and temperature profiles along the course of flow. The assumption of fully developed flow is still applicable. The impact which radial variation of properties may have on heat transfer can be satisfactorily accounted for by introducing correction factors into empirical correlations. The correction factor is usually a function of wall to bulk temperature ratio. Since the fluid properties vary dramatically near the critical region, heat transfer to supercritical fluids is of course a problem of variable-property convection heat transfer. What makes it different, however, from other examples of this category is the peculiar way in which the properties vary. It is not only the difference in properties between wall and bulk that matters. The way that properties vary from the wall to the core is important too. Near the pseudo-critical temperature changes of some properties with temperature are not monotonic. Once Tb<Tpc<Tw, specific heat capacity increases as we move away from the wall towards the flow centerline. Maximum value of specific 96 heat occurs at some distance from the wall where the pseudo-critical temperature is reached. Further away from the wall, it decreases again towards a certain value at the flow centerline. It means that the sign of the derivative of specific heat with respect to radius (i.e., temperature) changes as it crosses the critical region. That explains why implementation of correction factors in empirical correlations as applied to other flows with variable properties is inadequate in the case of supercritical fluid flows. For supercritical fluids, correction factors in the form of functions of wall-to-bulk variations of one or more properties, instead of just using (Tw/Tb)n, considerably improved the predictions. They are concerned with "at wall" and "at bulk" or even "at film" and not the way they change from the wall to the bulk. Substituting wall or bulk Cp by averaged Cp defined as (iw-ib)/(Tw-Tb) was an effective step ahead. Nevertheless, it is difficult to fully address the effect of Cp profile as well as other properties by means of correction factors so defined. On the same basis, the discrepancies between experiments and empirical correlations may be explained as follows. The Nusselt type correlation is developed through a semi-empirical approach. It stems from the "fully developed flow" assumption. Under supercritical conditions this assumption in its restrictive sense may not be helpful anymore, i.e., the velocity and temperature profiles are expected to change when the pseudo-critical region spans the flow from the wall to bulk. If the pseudo-critical temperature has not yet been reached at the wall, or when the bulk temperature has already passed the pseudo-critical point, the variation in profiles along the flow is not violent. Thus, the empirical correlations may predict the data satisfactorily. For Tb<Tpc<Tw, Reynolds and Prandtl numbers only partially characterize the flow. The question of at what temperature they should be evaluated is more difficult than any other situation to answer. Further information about the radial location of the pseudo-critical region (fast varying property zone) seems to be necessary. In other examples of variable property flow there may exist a large difference of properties between wall and bulk, but there is no region which is of particular importance. On the contrary, in supercritical fluid flow it is vital to know whether the pseudo-critical temperature is experienced in the laminar sub-layer, buffer layer or etc. Available empirical correlations do not fully reflect the physics involved. So, it is no wonder that they are case dependent. The results may not be reliable if there will be a 9 7 change in test conditions such that they are different from the ones on which they were developed. It leads us to the idea of allowing the reference temperature vary with flow conditions along the test section. 5.7. A NEW EMPIRICAL CORRELATION Calculating non-dimensional numbers at temperatures such as that of the bulk or wall, as widely practiced, cannot reflect the different regions of flow along the tube. At one point, the pseudo-critical region may be reached in the laminar sub-layer thus not promoting heat transfer substantially. Downstream at some other point where the pseudo-critical temperature is shifted to the buffer layer, turbulence activities increase and heat transfer is enhanced. To comply with the variant nature of flow along the test section, it is suggested here to evaluate Reynolds and Prandtl numbers at a variable reference temperature, T r ef. This way the assumption of fully developed flow in a specific sense has been applied. That is heat transfer may be determined from local fluid properties and is independent of the flow history. The question is what temperature can best represent the flow at each axial location, i.e., how to define T r ef. The test results shown previously in Fig. 5.3 were used to demonstrate how T r e f could be defined. The most basic heat transfer correlation, i.e., Dittus-Boelter, was employed for this purpose. Reynolds and Prandtl numbers in Dittus-Boelter are to be estimated at bulk temperature. We know from previous studies as well as this investigation that Dittus-Boelter over-predicts the peak of heat transfer coefficients for supercritical flows. Let us find out at what temperature (if any) should Reynolds and Prandtl numbers be estimated in order that the Dittus-Boelter correlation provides the best agreement with the data. For each flow cross section, the difference between wall and bulk temperature was divided into a large number of temperature points. The steps were chosen so that they were equi-distant in terms of enthalpies. This was to make the temperature intervals decrease near the pseudo-critical point. The Dittus-Boelter correlation was calculated at each of these temperatures and compared. The temperature resulting in the best agreement with experiment was called the equivalent temperature and denoted by Teq. Thus, there will be one temperature, T e q , associated with each flow cross section for which the predicted heat transfer coefficient equals the experimental 98 value, i.e., hcorTeiation ~ heXperiment- Hence, the Dittus-Boelter correlation can accurately predict the heat transfer coefficient provided that T r e f is chosen to be Teq at any point. The variation of T e q along the test section is shown in Figure 5.10. As shown, in sub-critical as well as post-critical regions T ^ T t , . In the vicinity of the pseudo-critical temperature, T e q jumps from Tb to some values close to T w . It then remains almost constant until the bulk is raised to that temperature. The small scatter of the values of T e q in the sub-critical region does not necessarily imply that choosing T r e f=Tb will not provide acceptable results. More trials confirmed that the scatter can be safely ignored and the major trend observed in variation of Teq may be relied upon. 1600 1800 2000 2200 Bulk Enthalpy, kJ/kg 2400 Figure 5.10. Variation of T r e f with respect to T b and T w along the test section, P=25.2 MPa, q "=307 kW/m2, G=965 kg/m2.s, D=6.3 mm. The predictions of Dittus-Boelter with reference temperature equal to bulk, wall and two arbitrary temperatures between wall and bulk along with experimental results are shown in Figure 5.11. The harmony between Figs. 5.10 and 5.11 helps better appreciation 99 The next step is to specify a criterion based on which T r e f should vary with temperature (along the test section). As explained earlier, the use of enthalpy is preferable over temperature. Enthalpy at T r ef is denoted as i r e f and estimated by lref ~ h +ifac(iW h) (5.11) where i w and i b are the wall and bulk enthalpies. The enthalpy factor, ifac, is to be specified for different zones of flow. In order to study the variation of i f a c along the flow, we define the dimensionless number "enthalpy zone", i z o n e , as = l_pc lb_ zone I — I pc w (5.12) where i p c is the fluid enthalpy at the pseudo-critical temperature. Wall temperature is estimated by Dittus-Boelter using T r ef=T b. Variation of i z o n e with flow enthalpy along the test section is shown in Figure 5.12. 1 o a. II C o N Bulk Enthalpy, kJ/kg Figure 5.12. Typical variation of izone=(ipC-ib)/(ipc-iw) along the test section 101 Before the pseudo-critical temperature is reached at the wall, i.e., ib<iw<iPc, izone is greater than one. It goes to positive infinity asymptotically as T w approaches T p c . For this range of values of i z o n e , Figs. 5.10 and 5.11 suggest that T r ef=Tb (i.e., iref=ib) is adequate. Where ib<ipc<iw, izone has a negative number, T r ef=Tb still leads to satisfactory results for T w values close to T p c . It was confirmed by the experimental results that this region expands to values of i z o n e from negative infinity up to about -0.9. Starting from this critical i z o n e number (« -0.9) to the point where the bulk is at pseudocritical temperature (izone=0) and even at post-critical region, ib>ipc, W varies with temperature. From Fig. 5.10 it is clear that there is a jump in T r e f from Tb to a value close to T w in the near critical region. This can be modeled using value of iZOne=-0.9 and the parameter "enthalpy factor, if a c " defined as below. lfac ~ l-L 'pc l-C-0.9) iw ^zone^pc F 9 L for -0.9 < Lone. < 1 ifac = 0 for iZone ^ -0.9 or 1 < L According to the above definitions, variations of i z o n e , ifac and T r e f may be summarized as follows. At sub-critical temperatures up to where the wall is just into critical region, if a c is zero and Tref=Tb. As iZOne crosses the critical value of -0.9, i f a c jumps from zero to (ip c/iw) which is slightly smaller than unity. Accordingly, T r e f will jump to a value close to T w . Further downstream of the flow if a c vanishes to zero again as i z o n e approaches one at the post-critical region. Heat transfer coefficients obtained from the correlation with T r e f defined as above, are shown in Figure 5.13. The predictions of a few other correlations are also shown. Good agreement of the present correlation with experiment is evident. A mild sharp curvature shown before the peak is due to a sudden jump in the values of T r e f discussed earlier. It can be smoothed out at the expense of bringing more complexity into the definition of i z o n e , which seems to be unnecessary at this stage. 102 o o _ CM " E c <u o ! t <D O o c ra ro I 60 50 40 30 20 10 r Correlation (this study) Petukhov et al. Fewster-Jackson / Miropolski-Shitsman Experiment 1500 1700 1900 2100 2300 Bulk Enthalpy, kJ/kg 2500 Figure 5.13. Results of the correlation of this study compared with others and experiment, P=25.2 MPa, q"=307 kW/m2, G=965 kg/m2.s, D=6.3 mm. 1800 2000 2200 2400 2600 Bulk Enthalpy, kJ/kg Figure 5.14. Results of the correlation of this study compared with others and experiment, P=24.4 MPa, q "=103 kW/m2, G=575 kg/m2.s, D=6.3 mm. 103 It should be noted that the same trend of T r ef variation along the test section (shown in Fig. 5.10) was observed for flows with different test conditions. Thus, similar performance is expected from the present correlation in other case studies too. The test results for lower heat fluxes and mass flows along with the correlations' predictions are shown in Figure 5.14. In this example, the Fewster-Jackson correlation also offers compatible results. Nevertheless, it is less universal compared to the present correlation. It was mentioned earlier that the critical i z o n e number specifies the demarcation between the flow regions with ifac=0 and ifac^O. In Fig. 5.14, the critical i z o n e number was selected to be -0.93, instead of -0.9 as was used in Fig. 5.13. This small change was optional and affected the results only moderately. It simply avoided a tiny discontinuity at the intersection of the two flow regions, which could occur if -0.9 was used. In fact, the interface between the two zones of the flow can be shifted back and forth by playing with this number. A simple trial and error, however, shows that there is only one choice of the critical i z o ne number at which the sub-critical and post-critical curves merge without any discontinuity. This is illustrated in Figure 5.15. Bulk Enthalpy, kJ/kg Figure 5.15. Effect of the critical imm number on heat transfer coefficients, P=24.5 MPa, q"=233 kW/m 2 , G=1260 kg/m2.s, D=7.5 mm. 104 As shown in Fig. 5.15, values greater or smaller than -0.77 for critical i z o n e number cause an abrupt jump in heat transfer coefficients. The extent of discontinuity depends on how far from -0.77 the critical i z o n e has been selected. Thus, instead of using a fixed number for i z o n e , it is recommended to find the most suitable value by trial and error for any flow conditions. This does not add much of complexity to the calculation procedure whereas the correlation generality is effectively increased. To demonstrate the competence of the present correlation in predicting the results of other investigations for horizontal flows, we need to find those measurements, which are not affected by buoyancy. Not many of such data are available in the literature. A set of measurements made by Swenson et al. at P=31 MPa may be one of those situations that because of high pressure (relatively far from the critical pressure) the effect of natural convection is minimal. The correlation of Swenson et al. match their own data and were previously shown in Fig. 1. The same results are repeated here in Figure 5.16 where the predictions of the present correlation are also shown. It can be seen that the predictions by the current empirical correlation agree the data of Swenson et al. satisfactorily. 0 70 o C N 1 601 c 50 d) o !t 0) 40 o O I c ns re 0) I 30 20 10 • Swenson et al. Miropolski-Shitsman Current Study •• Fewster-Jackson •••• 1800 2000 2200 2400 Bulk Enthalpy, kJ/kg 2600 Figure 5.16. Comparison of the local heat transfer coefficients predicted by the correlation of this study and others. The test conditions of Swenson et al. are examined: P=31 MPa, q"=789 kW/m2, G=2149 kg/m2.s, D=9.42 mm. 105 In the above example, the predictions of the correlations of Petukhov et al. and Krasnoshchekov and Protopopov (not shown in Fig. 5.16 to avoid overcrowd) were close to the predictions of the present correlation. A more convincing example follows where a set of measurements of Yamagata et al. are compared with predictions of the others and present correlation. Figure 5.17 shows the data of Yamagata et al. for four different test conditions. The graph with largest peak heat transfer coefficients corresponds to the test conditions where the buoyancy effects are negligible. This can be learnt from the fact that the heat transfer coefficients measured at the bottom and the top surfaces, as shown in Fig. 17, are identical. This is among rare data for horizontal flows in which the effects of natural convection were shown to be negligible. 80, 60r 1 40 20r P ' 245 bar G ' 1830 kg/m 2* d ' 7 5 mm Bottom Top <7. kW/m - * 233 Bi-ff M. turn a o 465 698 930 300 350 400 Tw 450 500 Figure 5.17. Heat transfer coefficients measured by Yamagata et al. (1972). 106 Eight different correlations (Gnielinski, Petukhov et al., Swenson et al., Yamagat et al., Miropolski-Shitsman, Jackson-Fewster, Razumovski et al., Krasnoshchekov-Protopopov) were tried and the predictions of the best two of them (compared to the data of Yamagata et al., Fig. 17), together with those of the present correlation were shown in Figure 18. Most of the other correlations largely over-predicted the data. The closer agreement of peak heat transfer coefficients (-75 kW/m 2 °C) predicted by this study with the data of Yamagata et al. is striking. 300 320 340 360 380 400 420 440 460 480 500 Wall Temperature, °C Figure 5.18. Heat transfer coefficients Predicted by the empirical correlations of this and other studies. Test conditions are the same as Fig. 17. It should be re-emphasized that the correlation developed in this chapter is valid for axisymmetric flows where natural convection as well as the effect of acceleration is negligible. It is therefore important to accurately predict the onset of buoyancy and acceleration effects. This has been addressed in the following chapter. 107 CHAPTER SIX EFFECTS OF BUOYANCY AND A C C E L E R A T I O N There exists a large density variation in the flow around the pseudo-critical temperature. It may have two important effects on the flow. Under certain conditions, a secondary flow may be induced so that heat transfer by natural convection becomes comparable with that of forced convection. In addition to a radial density profile, axial variations of density with temperature may also be significant. This can induce large axial variations of the fluid velocity. Thus, the momentum transferred by the term pu(du/3x) may become comparable with momentum transferred by shear stress. These two effects, buoyancy and acceleration, are sometimes referred to as body forces. Their effects are discussed in this chapter. 6.1. B U O Y A N C Y E F F E C T S It was mentioned in the previous chapter that confusion between mixed and pure forced convection is a major source of disagreement among available empirical correlations. In fact, in many studies, the effect of buoyancy was assumed to be negligible before it was even investigated. As such, most experimental results intended to represent solely forced convection heat transfer were likely mixed convection situations. It is therefore crucial to distinguish a mixed convection regime from a dominant forced convection case. 108 The vast majority of analytical and experimental studies in this respect were developed for vertical flows. Very few criteria have been suggested in the literature to detect the buoyancy-free region of a horizontal supercritical fluid flow. They were not tested extensively in the past due to a lack of data. This study extends such a limited database through careful measurements of the flows with large, moderate and minimal buoyancy effects. 6.1.1 Measurements of Buoyancy Effect In vertical flows, any difference in the hydro-thermal characteristics of two identical upward and downward flows may be attributed to the effect of buoyancy. Of course, all other controlling parameters such as mass flow, pressure and heat flux need to be the same in both flows. In a horizontal flow, the difference in surface temperature at top and bottom of the tube can be considered as a measure of buoyancy or natural convection effects. The temperature readings at the top and bottom surfaces need to be tested in an isothermal flow first. In such a flow, since there is no radial temperature profile, no density gradient exists. Fluid temperature was raised by means of preheaters up to the critical region. No heat was added to the test section. The history of temperature variations of the top and bottom surfaces at one arbitrary location is shown in Figure 6.1. As shown, no temperature difference was detected between the top and bottom surfaces. The sudden jump appearing in the wall temperatures was due to an interruption in data acquisition. Additional thermocouples were welded on the tube surface at 45° from the top center of the tube cross section. As expected they read the same temperature (not shown in Fig. 1). Once we were confident about the consistency of surface thermocouple measurements at high temperatures near the critical region, a heat flux (electrical heating) was introduced along the test section length. The Grashof number can be used as a measure of the significance of buoyancy effect. For pipe flow, it is defined as ar=S0(T„-Tt)J> 109 where bulk thermal expansion is defined as P=-(3p/dT)/p. The symbols v and g denote the kinematic viscosity of water and acceleration due to gravity respectively. At a relatively low flow rate, Reynolds numbers are small. Meanwhile, the radial temperature profile and density gradient, and hence Grashof number become greater. Thus, the dimensionless number Gr/Re , which is usually used as a measure of the buoyancy effect, increases. This justifies the simple rule that the lower the mass flow rate, the higher the influence of buoyancy. 400 380 \ 360 340 320 300 280 260 240 220 200 • Bottom Surface o Top Surface 300 600 900 1200 1500 1800 2100 2400 Time, s Figure 6.1. Wall temperature at top and bottom surfaces, P=24.4 MPa, G=964 kg/m2.s, q"=0 6.1.1.1. A Case Study with Large Buoyancy Effect To promote the effects of natural convection, the highest possible heat flux permitted by the experimental apparatus (=300 kW/m2) along with a low mass flow (=0.63 ^/min or 340 kg/m .s) was investigated. These conditions are denoted as "Case I" for later reference. The wall temperature distribution versus the water bulk enthalpy in the first segment of the test section is shown in Figure 6.2. To avoid the influence of 110 thermal entry length, data were collected only at locations were x/d > 50. The difference between the top and bottom surface temperature is about 30 °C as the bulk temperature exceeds 150 °C. It further increases up to about 70 °C at the critical region where property variations are maximum. 510 470 P 430 <u 390 « 350 0) | - 310 *- 270 230 190 150 bottom surface temperature top surface temperature fluid bulk temperature 600 900 1200 1500 1800 2100 Bulk Enthalpy, kJ/kg 2400 Figure 6.2. Wall and bulk temperature variations along the first segment of the test section, P=24.4 MPa, G=340 kg/m2.s, q "=297 kW/m2 The difference between wall and bulk temperatures reduces as a result of increasing heat capacity around the pseudo-critical temperature. Nevertheless, the difference in fluid density at wall and bulk temperatures is substantial and hence buoyancy becomes important. When temperatures were raised well above the critical region, the difference in the top and bottom surface temperatures decreased. This can be seen in Figure 6.3, where the top and bottom surface temperature at the end of the second segment of the test section are shown. Heat flux is assumed to be the same in both segments of the test section. At high temperatures up to 670 °C, the top and bottom 111 temperatures approach each other. There is a decreasing density variation moving away from the pseudo-critical temperature. Hence, the bulk density approaches the density at the wall and buoyancy diminishes. 3 5 0 2000 2200 2400 2600 2800 3000 3200 3400 Bulk Enthalpy, kJ/kg Figure 6.3. Wall and bulk temperature variations along the 2n d segment of the test section, P=24.4 MPa, G=340 kg/m2.s, q "=300 kW/m2 It is important to note the large influence that temperature differences at the top and bottom surfaces can have on the local heat transfer coefficients. Variations of local heat transfer coefficients with bulk enthalpy, corresponding to Figs. 6.2 and 6.3, are shown in Figure 6.4. Around the pseudo-critical temperature, local heat transfer coefficients at the bottom surface are up to 2.5 times larger than the top surface. No significant enhancement of heat transfer occurs at the top surface of the test section. This situation is sometimes referred to as deteriorated heat transfer due to natural convection. It is broad and covers the entire top surface of the test section. On the other hand, at the bottom surface, heat transfer is improved. This has also been confirmed through other 112 studies, e.g., Belyakov et al. (1972), Yamagata et al. (1972) and Adebiyi and Hall (1976). They showed that heat transfer coefficients in a similar vertical test section were lower than those at the bottom surface of a horizontal flow and greater than that observed at the top surfaces. 150 195 240 285 330 375 420 465 510 Bulk Temperature, °C Figure 6.4. Differences of heat transfer coefficients at top and bottom surfaces, P=24.4 MPa, G=340 kg/m2.s, q"=300 kW/m2 The accuracy of the data from the current study shows how the effect of natural convection is greater at pre-critical versus post-critical temperatures with the largest variation within the critical region. It can be seen from Figs. 6.3 and 6.4 that at high temperatures (Tb~550 °C or Tw=700 °C) the difference between top and bottom surface temperatures vanishes. It indicates that the variation of water density from the wall to the center (in spite of a temperature difference of about 140 °C between the wall and bulk) is not large enough to create a noticeable effect of natural convection. This much temperature difference at the pre-critical region, on the other hand, can still generate a 113 significant density gradient resulting in buoyancy effects (i.e., temperature difference between top and bottom surfaces) as shown in Fig. 6.4. The density variation with temperature for water is shown in Figure 6.5. Figure 6.5. Variation of water density with temperature at P=24.4 MPa It can be seen that a larger variation in density with temperature exists in the pre-critical region, hence, supporting the above observations. In Fig. 6.5, for a pressure of 24.4 MPA, the pseudo-critical temperature is 383 °C. This is where the slope of the graph, 3p/dT, is the largest. It is clear that 3p/dT is greater within the pre-critical region as compared to post-critical temperatures. The larger 3p/dT results in a greater Grashof number. The density variations and hence the Grashof number approach zero before the bulk temperature reaches 560 °C or even less. Therefore, natural convection, which is proportional to the Grashof number also, diminishes. A similar trend exists for other water properties (except specific heat capacity). Therefore, the effect of property variations at pre-critical temperatures, though not as dramatic as around the pseudo-critical point, is important and should not be neglected. 114 6.1.1.2. Experiments with Mild Buoyancy through Increased Flow Rates In the next set of experiments the volumetric flow rate was increased to 0.80 llmin (=430 kg/m2.s). This test is referred to as Case H. The other test conditions were close (within 2%) to the previous test, Case I. Wall temperature variations at top and bottom surfaces of the tube are shown in Figure 6.6. 570 550 530 510 490 470 450 430 410 390 370 350 33 o bottom surface temperature • top surface temperature • fluid bulk temperature § ftoO 1700 1900 2100 2300 2500 2700 2900 3100 Bulk Enthalpy, kJ/kg Figure 6.6. Wall and bulk temperature variations along the test section, P=24.4 MPa, G=432 kg/m2.s, q"=300 kW/m2 The peak difference between top and bottom surface temperatures occur again near the pseudo-critical point and reaches about 35 °C. Note that the corresponding temperature difference at a mass flow rate of 0.63 £/mm (Case I) was almost twice this amount. Hence, increased flow rates reduce buoyancy effects as anticipated. The wall temperature for which buoyancy effects vanished was reduced from 670 °C in Case I to about 570 °C in Case II due to the decreased buoyancy effect with increasing flow rate at a fixed heat flux. 115 The variation of heat transfer coefficient along the test section for Case II is shown in Figure 6.7. Compared to Case I, heat transfer improves at both top and bottom surfaces due to the increased flow rate. Since buoyancy effects were reduced, the top-to-bottom surface ratio of heat transfer coefficients also decreased accordingly. Figure 6.7. Heat transfer coefficients corresponding to the flow specified in Fig. 6.6. To further proceed towards conditions where buoyancy effects vanish, the flow rate was increased to 1.2 &rmn (=645 kg/m2.s). These conditions are denoted as Case III. Figure 6.8 shows the wall temperature measurements. As expected, the temperature differences between top and bottom surfaces were greatly reduced to less than 15 °C near the pseudo-critical temperature. In general, the higher the flow rate, the smaller is the difference between bulk and wall temperature. Within the post-critical region, once the bulk temperature reached about 400 °C, no significant temperature difference between top and bottom surfaces was detected. The corresponding wall temperature was about 445 °C. The density variation associated with a temperature difference of 45 °C from the wall to the center was negligible. See Fig. 6.5. 116 1200 1400 1600 1800 2000 2200 2400 2600 2800 Bulk Enthalpy, kJ/kg jure 6.8. Wall and bulk temperature variations along the test section, P=24.3 MPa, G=646 kg/m2.s, q"=303 kW/m2 O 25 r o X 1000 1300 1600 1900 2200 2500 2800 Bulk Enthalpy, kJ/kg Figure 6.9. Heat transfer coefficients corresponding to the flow specified in Fig. 6.8 117 Heat transfer coefficients associated with the temperature distribution shown previously in Fig. 6.8, are presented in Figure 6.9. The uneven results for pre-critical and post-critical regions are obvious. Note again that small temperature differences between the top and bottom surfaces resulted in a significant difference in the heat transfer coefficient between these two surfaces. 6.1.1.3. Negligible Buoyancy Test Another experiment (Case IV) was performed for which it was observed that the buoyancy effects had vanished. The volumetric flow needed was around 1.79 £/min (=965 kg/m .s). The heat flux was the same as for the previous cases. Figures 6.10 and 6.11 show the temperature distribution and heat transfer coefficients respectively. They show that the temperature differences between top and bottom surfaces and hence, the differences in heat transfer coefficients are negligible. Figure 6.10. Wall and bulk temperature variations along the test section, P=25.2 MPa, G=964 kg/m2.s, q"=307 kW/m2 118 Figure 6.11. Heat transfer coefficients corresponding to the flow specified in Fig. 6.10 q" « 300 kW/m 2 in all cases M o o t/3 a o ii > 6 3 B >< a 6 H 5 u a, ca X ! O > O •8 a 2 C3 a, o X) i3 \a a D O O U o 6 1 o 3 o 3 O OH O o o G <4-l o o G t -CASE I 340 70 670 12 2.75 CASE n 432 35 570 15 2.15 CASE m 646 12 460 24 1.5 CASE IV 965 405 37 1.2 Table 6.1. Summary of the test results for mixed convection flows 119 Table 6.1 summarizes the results of the experiments discussed in this chapter. The pressure in Case TV was around 25.3 MPa, which was slightly higher than the mean pressure of about 24.4 MPa used in the other tests. However, this did not influence the pattern of variations of wall temperature and heat transfer coefficient as listed in the table. Note that heat flux remained the same (=300 kW/m2) in all cases. 6.1.2. Criteria for Onset of Buoyancy Effects For engineering design purposes, the main question to be answered is to know where the effect of natural convection is negligible and where it is not. Grashof number can adequately measure the extent of the effect of natural convection in a normal flow where property variations are not as severe. The term Gr/Re2, which is derived through a dimensional analysis of mixed convection heat transfer, is used to distinguish the buoyancy-free region from conditions where natural convection is significant. At a normal pressure flow, once Gr/Re is in the order of magnitude of unity or higher, buoyancy effects cannot be neglected. Figure 6.12, however, shows that such a criterion is inadequate for supercritical flows. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 o •a " A •o Case IV, G=964 kg/m2.s Case III, G=646 kg/m2.s o® Case II, G=432 kg/m2.s o Case I, G=340 kg/m2.s ° 0 1400 1600 1800 2000 2200 2400 2600 2800 Bulk Enthalpy, kJ/kg Figure 6.12. Variations of Gr/Re2 with bulk enthalpy for various G (q"=300 kW/m2) 120 It can be seen from Fig. 6.12 that for Gr/Re values of less than 0.1, the buoyancy is still noticeable and needs to be accounted for. This criterion cannot set a clear demarcation between buoyancy-affected and buoyancy-free conditions. Peak values of Gr/Re vary within an order of magnitude over cases I to IV. A more fine-tuned parameter is therefore needed. The results shown in Fig. 6.12 were based on estimating the density gradient and viscosity at bulk temperature. The top surface temperature was used for wall temperature. A number of investigators have studied mixed convection heat transfer for supercritical flows. Jackson et al. (1975) recommended the use of the quantity Gr/Re 2 7 instead of Gr/Re 2 to account for the effect of buoyancy near the critical region. They showed that in a vertical flow a difference between upward and downward flows appeared once Gr/Re 2 7 > 10"5. To test this hypothesis for the current study, the quantity Gr/Re 2 7 for the experiments listed in Table 1 was plotted versus bulk enthalpy along the test section as shown in Figure 6.13. With respect to the previous criterion, this method represents a significant improvement towards predicting buoyancy effects. 18 h 1200 1400 1600 1800 2000 2200 2400 Bulk Enthalpy, kJ/kg Figure 6.13. Variations of Gr/Re2 7 with bulk enthalpy for various G (q"=300 kW/m2) 121 2 7 For Gr/Re ' less than the proposed limits, our measurements still detected differences between top and bottom surface temperatures. For example with Case III the heat transfer coefficient varied up to 45% at bulk enthalpy of 1600 kJ/kg (Fig. 6.9) when Gr/Re 2 7 < 10"5 as shown in Fig. 6.13. The peaks in Fig. 6.13 do not fall within the critical region where buoyancy becomes the most important. Thus, the criterion recommended by Jackson and Hall for vertical flows gives a sense of the order of magnitude but is not adequate for horizontal flows under moderate buoyancy effects. Petukhov et al. (1974) studied horizontal flow and derived a threshold value for Grashof number, Grth, below which buoyancy effects could be neglected. The same analysis was later refined and can be found in Petukhov and Polyakov (1988). The threshold Grashof number, Grth, was defined as Grth =3x10 Reb 2.75 Pr .0.5 2/3 l + 2.4Re " 1 / 8(Pr -1) (6.2) for which Pr = _ L - h Mb Tw Tb Kb (6.3) Considering that q"=K-3T/3x at the wall, they modified Eq. (6.1) and derived the following expression for the Grashof number where Urq~ 2 VbKb J= 1 Pb~Pw P film Tw — Tb (6.4) (6.5) In order to have negligible buoyancy, it was required that Grq<Grth. This criterion, called the Petukhov criterion in this study, has been tested in various investigations. In the experiments of Hall and Adebiyi (1976) for horizontal flow 122 of supercritical carbon dioxide, the effect of natural convection was large. The value of Gr q was much greater than Grth for all cases. Thus Petukhov's criterion was not violated. However, it could not be considered as an accurate test of the criterion since the Grashof numbers in all cases were well above the threshold value. No marginal case, i.e., a flow with weak effect of buoyancy where Gr q is just slightly greater than Gr t n was examined. Petukhov and Polyakov (1988) considered a larger number of experiments performed by different investigators and reported variations of Grq/Grth along the test section for each case. They showed that increasing the temperature difference between top and bottom surfaces increased Gr q /Gr t n . For example, for a 200 °C temperature difference between top and bottom surfaces of the tube (data of Belyakov et al., 1971), Grq/Grth increased up to 1300. The criterion of Petukhov is more focused than the earlier criteria given by Gr/Re". It is sensitive to small changes in the flow and is expected to detect even small effects of natural convection. More comprehensive data such as that obtained in the current study is needed to test its precision. The data of the present study were shown to span various flow regimes, from one highly influenced by natural convection (Case I) to one with a negligible buoyancy effect (Case IV). They therefore provide a unique opportunity to test the Petukhov criterion. Shown in Figure 6.14 are the variations of Gr q /Gr t n as a function of bulk enthalpy for Case I. Also shown are the wall temperatures. As expected, peak values of Grq/Grth (as large as 200) occurred in the pseudo-critical region where the influence of buoyancy was largest. The difference in natural convection behavior at pre and post pseudo-critical regions, as discussed earlier, was also reflected by the values of Grq/Grth. Once the effect of natural convection vanished, Grq/Grth dropped to unity, which agrees well with the Petukhov criterion. In Fig. 6.14, Grq/Grth varies between 1 and 200. In order to obtain a better assessment of the Petukhov criterion a closer look at values near unity is required. This could be met by considering the case studies which were under a lesser influence of natural convection. The results for cases III and IV are shown in Figure 6.15. 123 800 1100 1400 1700 2000 2300 2600 2900 3200 Bulk En tha lpy , kJ /kg Figure 6.14. Variations of Grq/Gru, and wall temperature with bulk enthalpy for Case I 22 h 1000 1300 1600 1900 2200 2500 2800 3100 Bulk Enthalpy, kJ/kg Figure 6.15. Variations of Gr^Gr^ with bulk enthalpy for cases IH and IV 124 For Case TV it was shown earlier in Figs. 6.10 and 6.11 that the effect of natural convection was minor along most of the test section. There was a very small difference between top and bottom surfaces of the tube around the pseudo-critical point as shown in Fig. 6.11. It is interesting to note that even such a small effect was detectable using the criterion of Petukhov et al. Such a case corresponded to a peak of only Grq/Grth= 5 as shown in Fig. 6.15. For bulk enthalpies greater than 2200 kJ/kg with no temperature difference of the top and bottom surfaces, Grq/Grth approached unity. A similar trend can also be seen for Case III in Fig. 6.15. Hence, the current study serves to verify the Petukhov criterion with very good accuracy. Such an assessment was not done before due to a lack of data on horizontal flows. Calculating Grq and Grth requires knowledge of the wall temperatures. Once buoyancy is negligible the wall temperature at the top and bottom surfaces is equal and either may be used. For cases experiencing buoyancy effects (e.g., Case III shown in Fig. 6.15) it can be shown that the choice of top or bottom surface temperature does not substantially affect the results. In fact, either of Grq and Grth alone is considerably affected by variation of T w , but their ratio, Grq/Gr,h remains about the same. This may be considered as another advantage of the Petukhov criterion that an accurate estimate of the wall temperature is not needed in order to use the criterion. 6.1.3. Heat Transfer Calculations during Buoyancy-Free Conditions The Petukhov criterion is used to assess flows where the heat transfer coefficients can be accurately determined. In summary, the procedure is as follows. Mass flow, heat flux, pressure, bulk temperature and the tube diameter are known. The correlation developed in this study is used to estimate the wall temperature. Using the estimated wall temperature, Grq and Grth are determined. There are two possibilities. If the ratio Grq/Grth is close to unity, it means that the buoyancy is negligible and there is no temperature difference between top and bottom surfaces. The calculated wall temperature, therefore, will be an acceptable estimation. Consequently, heat transfer coefficients can be evaluated accurately. The other possibility is that Grq/Grth is large, i.e., flow is influenced by natural convection. In that case, wall temperatures calculated via an empirical correlation are not 125 a reliable estimation. They represent neither top nor bottom surface temperature of the tube. The term Grq/Grth is sensitive to the wall temperature only when it is very large, i.e., when buoyancy is important. In such a situation, a more advanced method is needed to calculate both wall temperatures and heat transfer coefficients at top and bottom surfaces. In flows with a moderate effect of buoyancy (e.g., cases II and III), however, the predicted wall temperature can be used to obtain a rough estimate on the average heat transfer coefficient. 6.1.4. Prediction of the Temperature Difference between Top and Bottom Surfaces Once buoyancy effects are appreciable, the next question is how to evaluate the temperature difference between the top and bottom surfaces of a horizontal test section. The most general and detailed method available is that of Petukhov and Polyakov (1988). They employed a particular pair of coordinates that correlate the temperature difference between top and bottom surfaces with other flow parameters. They chose the quantity Gr<p/<E> for x axis in which G r (6.6) (6.7) O = 3x l0" 5 Re f c 2 7 5 Pr u m .0.5 l + 2.4Re,- 1 / 8(Prm -1) .2/3 (6.8) Pr_ = _ 2 P r i P r 2 Pr,+ Pr2 (6.9) where Pi, Pri, P2 and Pr2 are evaluated at temperatures Ti and T2, respectively, which are defined below. Enthalpy at Ti is designated as i i . ix =/ ,+43.5Re a 2 ^- (6.10) G 126 T2=Tb+(ix-ib) X". (6.11) They also defined the following quantity for y-axis Kb ' C^w(top) w(bottom ) ) Nu top Nu bottom q'.D (6.12) Figure 6.16 shows the data plotted in the x-y coordinates so defined. The figure is taken from Petukhov and Polyakov (1988) and contains the data of three different studies. Despite some scatter there seems to be a trend in Fig. 6.16. Based on that, Petukhov and Polyakov recommended use of their generalized coordinates as a quantitative tool to estimate the temperature difference between top and bottom surfaces of a horizontal tube. • Q JS o cr :—. -El Figure 6.16. Generalized presentation of experimental data on the wall temperature difference between the top and bottom surfaces in a horizontal flow, taken from Petukhov and Polyakov (1988). (1) Data of Vikhrev et al. (1970) for water. (2) Data of Belyakov et al. (1971) for water. (3) Data of Adebiyi and Hall (1976) for carbon dioxide. 1 2 7 Figure 6.17 presents the results of cases I and III of this study using the same generalized coordinates. Case III, to some extent, follows the pattern observed in Fig. 6.16. The disagreement of the data of Case I, however, is apparent. Such disagreement is most likely attributable to the effects of acceleration. Petukhov and Polyakov have recommended their method for cases with negligible effects of acceleration. As will be seen in the next section, this situation may occur in large tube diameters or during flows with only moderate buoyancy effects (e.g., Case III). 11 -10 1 9 -8 -7 -. 6 -5 -4 -3 '-2 -1 -0 '--10 1 o Case I A Case HI AAA Q O m AA A . A 11. i.Li.i i i i i 4 5 6 10L 4 5 6 101 Giyo 9 Figure 6.17. Generalized presentation of experimental data on the wall temperature difference between the top and bottom surfaces in a horizontal flow for cases I and IH The other possible cause of disagreement between the predictions of Petukhov-Polyakov for temperature difference in the top and bottom surfaces and the measurements can be the effect of peripheral conduction in the tube. Once the buoyancy effects are moderate (i.e., there is a mild temperature difference between top and bottom surfaces), 128 the heat conducted peripherally in the tube from the top to the bottom may be neglected. However, for cases with temperature differences as large as 75 °C (Case I) between top and bottom surfaces, the effect of conduction in circumference of the tube needs to be investigated. A simple calculation for "Case I", using q"=K(AT/Ax), shows that conductive heat flux in the tube from the top to the bottom surface may approximately be as high as 120 kW/m", which is comparable with heat flux to the water (300 kW/m ). The values of K=20 W/m, AT=75 °C and Ax=7tr were used in above calculations. The conduction heat transfer inside the tube was not included in the model of Petukhov and Polyakov. In general, an accurate estimate of the temperature difference between top and bottom surfaces of the tube in a horizontal flow requires a 3-D analysis with the effect of conduction in the tube taken into consideration. As yet, there is no such comprehensive method available in the literature. The complicated problem of mixed convection heat transfer in horizontal flow of supercritical fluids deserves an independent, more in depth study. 6.2. THE EFFECT OF ACCELERATION It has been shown by Kurganov and Ankudinov (1985) and others that compared to the buoyancy effect, the acceleration term may become important only in small bore tubes. For large tubes, the buoyancy is dominant. For moderate tube sizes, the acceleration is usually negligible with respect to buoyancy effects. Consequently, it can be said that once buoyancy is negligible in such flows, the acceleration term may safely be neglected too. This is further investigated in this section. In a pipe flow under normal (sub-critical) conditions, the momentum transferred by axial convection is much smaller than the contribution of the shear stress profile. At supercritical pressures, however, flow acceleration induced by dramatic variations of density with temperature may become considerable. The effect of flow acceleration, therefore, cannot be readily neglected and needs further investigation. It has been shown by various researchers (e.g., Ankudinov and Kurganov, 1981) that acceleration is the cause of heat transfer deterioration in many cases. To get a better idea about this phenomenon, consider the following analysis. 129 The acceleration term is represented by pu(du/3x). Note that the variation of the mean velocity in the x-direction is only under consideration. Thus, (3u/3x) may be reduced to (du/dx). The chain rule may be applied to obtain du _ du di ^ dx di dx Recall that heat flux was calculated from the energy balance between the inlet and outlet of the test section. Thus, the variations of enthalpy along the flow may be related to heat flux as follows. q\nDL) = q = Gfr D2/4)(iout - iin) (6.14) where (TtDL) is heat transfer surface and (7tD 2/4) is the flow cross section. Noting that di/dx=(iout-iin)/L, Eq. 6.14 leads to ^ - ^ (6.15) dx GD Under steady state conditions the mass flow G (=pu) is constant. The conservation of mass implies that p(du)=-u(dp)=-(G/p)dp. The change in velocity can be expressed in terms of density variations. Using the definition of fluid bulk expansion, P=-(l/p)(9p/dT) and di=Cp.dT as well as Eq. 6.15, Eq. 6.13 may be manipulated to obtain ^du^ApGq" ( 6 1 6 ) dx DpCp Eq. 6.16 estimates the acceleration in terms of measurable flow parameters. A similar relationship is needed for shear stress profile in order to compare it with acceleration term. Hall and Jackson (1978) postulated that acceleration becomes 130 significant as x/xw=0.95 at y+=20, where y + is dimensionless distance measured from the wall and defined as v u (6.17) Kurganov and Kaptilnye (1993) modified the above threshold to T/TW=0.95~0.97 at y+=30. Together with Eq. 6.17, it leads to _ ( l - 0 . 9 6 ) T w ^ V 2 _ 1 3 3 y l Q _ 3 ^ V 2 30 A, (6.18) Note that xw=(f/2)pu and friction factor for this analysis can be approximated by f=0.046(Re)"02. Thus Eq. 6.18 can be rewritten as 4.64xlQ- 6pJ / 2G 3 A V . R e a 3 (6.19) The required tools are available now to introduce an acceleration parameter, y, as the ratio of the acceleration term (Eq. 6.16) to the threshold value of shear stress profile (Eq. 6.19). du pU~dx _8 .62xl0 5 ygp f c / /^Re 0 3 DCppJ /2G2 (6.20) Acceleration is expected to be noticeable wherever y>l. Variations of y along the test section for cases I, III and IV (see Table 6.1) are shown in Figure 6.18. 131 Polyakov (1988) for calculation of buoyancy effects (Eqs. 6.6 to 6.12) considered only those conditions with a negligible effect of acceleration. That explains why calculation of temperature difference between top and bottom surfaces led to more unacceptable results for Case I compared to Case III (Figs. 6.16 and 6.17). The cause of both buoyancy and acceleration effects is the same, i.e., large variations of density. They are usually coupled and it is customary to study them comparatively. The effects of acceleration therefore can be evaluated in comparison to buoyancy effects, which are directly observed and detected experimentally. On the same ground that Eq. 6.2 was derived to predict the onset of buoyancy effect, Petukhov and Polyakov (1988) developed a criterion for conditions where acceleration becomes noticeable. They defined the acceleration parameter J to be Pr yTw-Tb KbPf2 2 (6.21) where p f is evaluated at (Tw+Tb)/2 and the averaged Prandtl number is defined by Eq. 6.3. For vertical flows, the following relationship was offered by Petukhov and Polyakov (1988) to satisfy the conditions of negligible body forces. ± Gr+ J < 4 x IO"4 Re 2 8 Pr (6.22) in which the positive sign of Gr q is for upward and the negative sign is for downward flows. Gr q was defined in Eq. 6.4. The right hand side of inequality 6.22 was the threshold value of body forces and was referred to as Bth-In horizontal flows, the threshold for significance of body forces was expected to be different. Petukhov and Polyakov (1988) did not specify any value for Bth-Nevertheless, it is possible to compare Gr q (representing buoyancy effect) and J (representing the effect of acceleration) against each other regardless of the flow orientation. This was particularly useful in the current study with buoyancy effects 133 CHAPTER SEVEN ANALYTICAL MODELLING Most available theoretical numerical studies have concentrated on vertical flows. In a vertical test section whether or not buoyancy is significant, the flow is axisymmetric. This reduces the complexity of the problem. A 2-D model, therefore, can account for the effects of buoyancy and acceleration. It potentially can predict enhanced and deteriorated heat transfer during a vertical flow. In a horizontal test section, however, due to flow asymmetry, any real understanding of this problem can only be accomplished through a 3-D analysis, which, at present, has yet to be developed. Acceleration and buoyancy effects are closely coupled in such flows. Thus, a 2-D model for horizontal flows is restricted to account for acceleration effects without buoyancy. This may occur in very small diameter pipes. It was shown in chapter 6 that for horizontal tube with diameters similar to or larger than that used in the present study, once buoyancy forces become unimportant, so do acceleration effects. Thus, a 1-D numerical analysis can be expected to be adequate for horizontal supercritical fluid flows in the absence of buoyancy. This is somewhat equivalent to stating that far from the entrance region, despite large property variations, the flow can be assumed to be fully developed. Such conditions occur with higher flow rates and lower heat fluxes. It means that heat transfer coefficients can be calculated from local fluid properties only, i.e., they are independent of the flow history. Various investigators have supported the idea that under certain conditions, a supercritical fluid flow may be treated as fully developed. Kurganov et al. (1986) 136 measured the velocity and temperature profiles in a supercritical carbon dioxide flow. Changes in velocity profile were shown to be small to moderate along the flow during non-deteriorated cases. The fact that a Nusselt type correlation, developed in chapter 5 of this study, predicted local heat transfer coefficients well, also verifies that heat transfer is independent of the flow history, and hence the assumption of fully developed flow is expected to be applicable. In this chapter, a 1-D mathematical model was established to calculate local heat transfer rates to the turbulent flow of water in a horizontal circular tube at supercritical pressure. The model is based on the theory of the law of the wall. Various analytical expressions for the universal velocity profile have been examined. Radial variations of heat flux and shear stress were taken into account. The effect on turbulent Prandtl number was also studied. Comparison of the results with experimental data was made and showed good agreement in region where buoyancy (and hence flow acceleration) is negligible. 7.1. GOVERNING EQUATIONS AND SIMPLIFYING ASSUMPTIONS Simplified equations of momentum and energy for a steady state, fully developed, axisymmetric, turbulent pipe flow are: -—(rr) + — = 0 r dr dx pu — = -—(rq ) dx r dr (7.1) (7.2) Additional assumptions made are as follows. a) Convection terms of momentum (acceleration) are negligible compared to the momentum transferred by shear stress. b) The gravitational term (buoyancy effect) is ignored, hence the flow is axisymmetric c) The terms for radial convection, axial conduction and viscous dissipation in the energy equation are negligible compared to the axial convection and radial conduction terms. Shear stress and heat fluxes are defined as: 137 (7.4) (7.3) E m and £h are momentum and heat eddy diffusivity and represent velocity and temperature profiles as below They account for the effect of turbulent fluctuations of velocity and temperature respectively. Turbulent Prandtl number is defined as Available data for flow of water at normal conditions shows that Prt are very close to unity. For supercritical fluid flow, £ m and £h are assumed to be equal as a first approximation, em= £ h = £. The influence of variation of Pr t will be discussed later. At supercritical pressure, the properties of water vary dramatically near the critical temperature. It makes the momentum and energy equations highly coupled. Thus, a numerical scheme is needed to solve them simultaneously. Before accomplishing this, in order to show the influence of variations in the fluid properties on the heat transfer coefficient, a primary model, neglecting such property variations, was developed. The inability of such model to predict heat transfer coefficients is demonstrated below. (7.5) (7.6) (7.7) 138 7.2. CONSTANT-PROPERTY SOLUTION As mentioned above, although the constant-property solution will no doubt be in error, it is useful to start here and see the limiting assumptions used to better understand why a variable-property solution is required. The formulations employed in this section follow the derivations presented by Kakac and Yenner (1995). Consider Eqs. 7.3 and 7.4. To establish the Reynolds analogy between shear stress and heat transfer, we assume that either v=ct or both v and a are negligible compared to e. Thus, (v-e)/(a-e)=l and a direct proportionality between shear stress and heat transfer holds, x/q is assumed not changing with r. Dividing Eqs. 7.3 by 7.4 and integrating it between the wall (u=0, T=TW) and bulk (u=U, T=Tb) leads to 4l_ = CPb(Tw-Tb) (7 8) K u The assumption x/q" = xw/qw" = constant, is justified by inspecting Eqs. 7.1 and 7.2. Note that 3i/3x = Cp(3T/3x). It requires that the terms 3P/3x and u(3T/3x) be functions of x only and do not vary with r. This is not a bad approximation for a fully developed constant-property flow. For pipe flows, x varies linearly with r. The heat flux depends on velocity and temperature profiles. Recalling that q"= h(Tw-Tb) and using xw=(l/2)CfpUb2, Eq. 7.8 leads to the Reynolds analogy: St= k = NU =Cf (7-9) pCpbU Re.Pr 2 where St and Cf are Stanton number and friction coefficient respectively. Given the mass flow, heat flux, pressure, diameter of the tube and bulk temperature as input, a Fortran routine was developed to calculate the wall temperature and heat transfer coefficient for water flowing in a tube at supercritical pressure. The results for a sample case along with data from the present study are shown in Figure 7.1. The predictions of a constant-property two-layer model, introduced later, are also shown. 139 is maintained by the mechanism of molecular conduction. For the laminar sub-layer with thickness, 5L, we have: du u, T = JU — = M^r dy SL q'=z_Kii]_ = _K(TL-TJ dy 8L UL and T L are the velocity and temperature at y=OL respectively. Thus, (7.10) Applying the Reynolds analogy (Eq. 7.8) for the core region we get (7.11) 9 _ Cpb(Tb-TL) r U-uL Assume (q7x)=(qw7Tw)=constant. Then, equating and manipulating Eqs. 7.10 and 7.11 yield S t = h = C = ( V 2 ) C / (7.12) pUbCP pUbCpb(Tw-Tb) 1 + i ^ ( p r _ n For flows under normal conditions, U i / U b may be replaced by its experimentally obtained equivalent, 5V(xw/pUb 2) or 5V(Cf/2), to get St=  ( l / 2 ) C ' (7-13) 1 +5^ /(1/2) C/Pr-1) 141 This has been applied to the same case study tried for the previous model and results were shown earlier in Fig. 7.1. A slight improvement of the predictions for the pre pseudo-critical region was observed. It is interesting to note that including the near wall region did not significantly affect the results within the pseudo-critical region. This is partially due to the fact that the empirical relationship used for ui/Ub is not applicable to a variable-property flow. No reliable data are available for supercritical fluid flow that can replace the term ui/Ub in 7.12. More importantly, the effects of property variations are too large to be simply accounted for by dividing the flow cross-section into two regions. The two-layer model was further modified to a three-layer model (von Karman analogy) by dividing the boundary layer into laminar sub-layer, buffer layer and core region. In the laminar sub-layer, attached to the wall, molecular conduction is dominant. In the core region turbulent convection plays the major role. The buffer zone is a mix of the two. The details of the three-layer model are not described here. This model did not lead to any better results. Although no discontinuity in predicted heat transfer coefficients was obtained, unrealistic peak values, up to several times larger than experiments, were predicted for the heat transfer coefficients. Hence, it can be concluded that a constant-property three-layer analysis cannot adequately predict heat transfer to the variable-property flow of supercritical fluids. Alternatively, instead of dividing the flow into just two or three layers, a much larger number of steps may be used over the entire flow cross section. This leads us to the concept of simultaneous integration of shear stress and heat transfer from the wall to the bulk. This way, the fluid property variation may be accounted for in its entirety and none of the restricting assumptions used in the Reynolds analogy apply. 7.3. V A R I A B L E - P R O P E R T Y S O L U T I O N A ID model allowing fluid properties to vary from the wall to the center was developed in this study. An extension of earlier models was performed to include the most recent advances in law of the wall modeling (e.g., newer definitions of the turbulent Prandtl Number). The general 1-D model is described as follows. 142 The model provides a solution to the boundary layer equations based on the law of the wall. The basic postulation in the law of the wall is that velocity variations over any cross section of a turbulent flow fall onto the same profile once expressed in a generalized system of coordinates. This is supported by a high volume of experimental data available in the literature for flows with negligible property variations. The non-dimensional distance y + and velocity u +, used as generalized coordinates, are defined as V + _ u The expression describing the relationship between y + and u + is called the universal velocity profile. The procedures for obtaining the actual velocity and temperature profiles are as follows. Given the flow rate, pressure, wall temperature, and heat flux, Eqs. 7.3 and 7.4 were integrated simultaneously from the wall to the center. The distance from the wall (y=0 or r=rG) to the tube center (y=r0 or r=0) was divided into a large number of steps, Ay. Starting with a known wall temperature, x w is estimated. The wall shear stress is a function of friction coefficient. There are empirical expressions stating the friction coefficient in terms of Reynolds number for constant-property flow. Either of them may be used to make an initial guess for Tw. This will be corrected as a result of the iteration. A more realistic guess for TW , though not affecting the final outcome of integration, may speed up convergence of the solution. Properties at the wall were used to estimate y + at a very small distance Ay from the wall, say yi. A number of formulations have been suggested in the literature for a universal velocity profile. Any of these equations may be used to obtain the velocity u at yi from Eq. 7.3. The effect of expression used for universal velocity profile will be discussed later. As for the boundary condition, the velocity at the wall is assumed to be zero. Note also that the eddy diffusivity e can be obtained from the expression for the universal (7.14) (7.15) 143 velocity profile. Considering T=Tw at y+=0, from Eqs. 7.3, 7.14 and 7.15 (see Appendix D) we may get 1 + £ = * 1 (7.16) v du In the present study the various forms of the equation relating y+, u + and e are referred to as "universal velocity profile". They may alternatively be referred to as the "expression for eddy diffusivity". Once the velocity was calculated at yi, knowing the heat flux at the wall qw", the heat flux Eq. 7.4 was solved to obtain the enthalpy, i. The corresponding temperature, T, was estimated via a table of the hemodynamic properties of water. To obtain velocity and temperature profiles, the same procedure was repeated and the flow cross section was marched from yi towards the tube center. It may be known how the shear stress and heat flux varies with tube radius. Then, instead of shear stress and heat flux at the wall, their corresponding values at each step can be used. Following Nishikawa et al. (1972), integrating Eqs. 7.1 and 7.2 yields T 7 w r 0 (7.17) q_=r^_2_^puLdr ( 7 1 g ) where G is the mass flow and defined as G = \)purdr (7-19) 2 r0 o In carrying out the integration to reach 7.17 and 7.18, it was assumed that the radial component of the flow velocity is relatively small so that the pressure is constant 144 over the cross section. The term 3i/3x was also assumed to be constant over the tube cross section and hence, not varying with r. Upon completion of the integration leading to the tube center, one needs to check the value of the mass flow obtained via the velocity profile and the given initial value to see if they match. If not, the first estimation of Tw is corrected and the iteration continued until the calculated mass flow converges to the initial value. Typical velocity and temperature profiles estimated by the model are shown in Figure 7.2. y + 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 0 . 0 QA 0 . 2 0 . 3 OA 0 . 5 0^6 0 . 7 0 . 8 o ' 9 1 .0 y/R Figure 7 . 2 . Typical velocity and temperature profiles calculated by the present variable-property model. The temperature profile is integrated over the flow cross section to obtain the bulk temperature. Having the bulk and wall temperatures as well as heat flux, the relation h=q7(Tw-TD) is used to evaluate local heat transfer coefficients. Predictions of the model at P=24.8 MPa, q"=232 kW/m2 and G= 571 kg/m2.s as well as the data of current study 145 are shown in Figure 7.3. Note that the test conditions are the same as those in Fig. 7.1 where predictions of the constant-property model were shown. « 10 -• 1 — 1 1 1 i i • • 1600 1800 2000 2200 2400 Bulk Enthalpy, kJ/kg Figure 7.3. Typical heat transfer coefficients predicted by the present model along with UBC data at P=24.8 MPa, q"=232 kW/m2, G= 571 kg/m2.s. 7.3.1. Effect of Expressions for Universal Velocity Profile It was mentioned earlier that the law of the wall implies that variations of u + with y + conform to almost the same profile over a wide range of flow conditions. Various investigators have measured such universal profile. The measurements were generally performed at normal pressure conditions. Different curve-fitting techniques have been used to derive them. Most well-known expressions for the universal velocity profile are listed in Table 7.1. Any of the expressions listed in Table 7.1 can be incorporated to the present model. Although the expressions for universal velocity profile look different, using any one of them, leads to more or less the same results for normal pressure flows. This is shown in Figure 7.4 where heat transfer coefficients predicted by the present model for a flow of a superheated steam at 0.1 MPa are presented. 146 The equation of universal velocity profile Range of Validity Investigator u + = y+ u + = 2.5 ln y+ + 5.5 0<y +< 11.5 11.5 <y+ Prandtl-Taylor u + = y+ u + = 5 ln y+ - 3.05 u + = 2.5 ln y+ + 5.5 0 < y+ < 5 5 < y+ < 30 30<y+ von Karrnan o du+ 2 dy" l + {l + 4K2y+2[l-e(-y+IA+)]}^ All y+ K=0.4, A=26 van Driest u + = 14.53 tanh (y+/14.53) u + = 2.5 ln y+ + 5.5 0 < y+ < 27.5 27.5 < y+ Rannie u + = 2.5 ln(l +0.4 y+)+7.8[l- e("y+/11) - (y+/ll) e ( - ° - 3 3 y + ) ] All y+ Reichardt du+ 1 dy+ l + nVy + [ l -e ( -" V / ) ] u + = 2.78 ln y+ + 3.8 n=0.124 0 < y+ < 26 26<y+ Deissler Table 7.1. Expressions of the universal velocity profile 147 Figure 7.6 the results of a case study with low heat flux are shown. As shown the best agreement with experiments was reached with the equations of Prandtl-Taylor used for eddy diffusivity. Nevertheless in some other examples not shown here, the Prandtl-Taylor model failed to even capture the peak value of heat transfer coefficient near the pseudocritical temperature. i — > — i — i — I — I — i — i — i i i i i i ' i • 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 Bulk Enthalpy [kJ/kg] Figure 7.6. Comparison of the results of present model utilizing various expressions of universal velocity profile with experiments, P=24.4 MPa, q"=103 kW/m2, G= 575 kg/m2.s. Note, however, that due to the low heat flux in the current example, the temperature difference between wall and bulk was sometimes as small as 3.5 °C near the critical region. Therefore, the experimental results shown in Fig. 7.6 are less reliable compared to previous case. There was also a minor effect of buoyancy under the flow conditions specified in Fig. 7.6. Thus, the average values of experimental heat transfer coefficients at the top and bottom surfaces of the test section were used. Despite such uncertainties in the data of Fig. 7.6, they have been presented here to show that regardless 150 of agreement with the experiment, the predictions of the model with various expressions of universal velocity profiles repeated the same pattern as was previously observed in Fig. 7.5. 7.3.2. Comparison with Other Data For later modification of the current model, it is important to know which expression for universal velocity profile is the most appropriate one for supercritical fluid flows. It is difficult to intuitively explain the differences in the predictions of the model in terms of the format of expressions employed for universal velocity profile. For example, Jackson and Hall (1978) speculated that since the eddy diffusivity suggested by van Driest contained a damping factor (e"y+) it could better cope with variable-property conditions. However, the simple relationship of von Karman was found by this study to provide similar results without such a damping factor. A comparison of the model predictions with data of this study as well as others was made over a wide range of conditions. It is concluded that the expressions of von Karman, van Driest and Rannie constituted a family of solutions resulting in a better agreement with experiment in the vast majority of the tests. An example of variations in the local heat transfer coefficients along the tube is shown in Figure 7.7. The flow parameters are the same as test conditions performed previously by Swenson et al. (1965). The predictions of an empirical correlation developed by Swenson et al., which closely match their experimental data, are also shown in Fig. 7.7. As shown, the ability to predict data with the present model is promising and warrants further improvement. The expression of van Driest for the universal velocity profile was used in this example. 151 I I I 1 1 1 I I I I I I I ' I , • . • 300 320 . 340 360 380 400 420 440 460 480 500 Film Temperature °C Figure 7.7. Predictions of local heat transfer coefficients by present model compared with the results of Swenson et al., P=31 MPa, q"=788 kW/m2, G=2150 kg/m2.s, D=9.42 mm. Another example is shown in Figure 7.8. The data of Yamagata et al. (1972) is well represented by the line shown, which is their correlation (best fit to the data). Predictions obtained with the present model using van Driest as well as Deissler expression, are also shown. It further confirms that the expression of van Driest for turbulence model provides better agreement with experiments. The accuracy of the model in predicting the peak heat transfer coefficient shown is particularly striking. 152 80 r CM a 70 h Yamagata et al. Present Model (van Driest) Present Model (Deissler) 0 330 340 350 360 370 380 390 400 41 0 420 430 440 Bulk Temperature °C Figure 7.8. Comparison of predictions of present model with results of Yamagata et al., P=24.5 MPa, q "=233 kW/m2, G= 1260 kg/m2.s, D=7.5 mm. 7.4. MODIFICATION OF THE MODEL Different techniques have been applied in the literature to modify the law of the wall type of models for supercritical fluid flows. For the current study a systematic approach on the effect of such adjustments within the model was performed. Hence, different extensions to available theories have been applied to try and fit variable-property conditions. Some of those were incorporated in the present model. One example is the integrated form of y+ and u+. An integrated form of y+ and u+ was defined by Goldman (1954) to further account for variations in fluid properties. Goldman defined the following (7.20) o V 153 density fluctuations and developed a correction factor to be included in the expression for eddy diffusivity. Their suggestion was implemented in the present model. It was found that it did not considerably affect the predictions. It was clarified in later studies (e.g., review paper of Polyakov 1991) that lack of reliable turbulence data in supercritical fluid flows is a problem common to all types of turbulence models. More recent models such as K-e, which are heavily based on empirical data, suffer equally, if not more than law of the wall models, from shortcomings of experimental data. Unless otherwise specified, the equation of van Driest together with the modification of Kays (1994) is used in this model to predict the upcoming results. According to the suggestion of Kays (1994), at small y+, em/v varies as (y+)3. The relationship developed for the region adjacent to the wall is stated below and was included into the model. Thus, the following relationships are used for eddy diffusivity in the current study. ^ = 0.001(y +) 3 for y + <5 (7.22) V ^ = 0.5[l + 4k2y+\l-e (-y+/A+))f2 for 5<y + (7.23) V where, k=0.4 and A+=26. Xu (2000) started with the expression of the mixing length and modified the universal velocity profile of van Driest (Eq. 7.23) with the property variations accounted for. The derivation of the expression of van Driest for constant property flows as well as the modified version of Xu is presented in Appendix D. However, the predictions of heat transfer coefficients using van Driest expression with modification of Xu and without it do not differ considerably. 7.4.2. Effect of Wall Shear Stress One of the two main parameters, accounted for in the current study, that was found to influence the model predictions was the equation used for the wall shear stress. 155 This will be explained below. The other important parameter (turbulent Prandtl number) will be discussed in the next section. The wall coordinates, y + and u + , are estimated based on values of the wall shear stress, xw. Furthermore, the radial variation of shear stress is usually expressed as a function of shear stress at the wall. It is therefore important to investigate how they may affect the results of the numerical solutions. The wall shear stress is defined as *w = f\pu2 (7.24) There is a number of empirical relationships expressing the friction factor , / , in terms of the Reynolds number for smooth surfaces at normal pressure. The one valid for the range of conditions of the present study is given by / = (1.58 ln(Re)- 3.28) -2 (7.25) The question is how can a supercritical environment affect the shear stress at the wall? Petukhov et al. (1983) suggested the following correction of Popov to account for property variations near the critical region. (Oo ( « \ 0.4 (7.26) (x w)o is the wall shear stress at constant property conditions obtained from Eqs. 7.24 and 7.25. A more recent correction factor, introduced by Razumovskiy et al. (1990), is (Oo Pw _ Mw Pb Pb x0.18 (7.27) 156 Such correction factors induce a double iteration loop in the model. That is because integration of the equations of shear stress and heat flux start from the wall where bulk temperatures do not apply. Applying Eqs. 7.26 or 7.27, however, requires an initial guess for the bulk temperature. Once the integration completes at the flow centerline, the bulk temperature is calculated and if it does not match the original assigned value the integration will be repeated all over from the wall with a new estimated value of T b (and hence x w ) . As the solution converges for Tb, the resultant mass flow is checked to see if it matches the original given value. If not, (Tw)0 will be corrected and a second loop of iteration applies until the mass flow condition is met. It must be noted that the final value of (x w ) 0 was within ±5% of the initial value (using Eq. 7.24) in almost all cases tried. This fact makes the whole process of implementing a correction factor for the wall shear stress meaningful. The model predictions, with and without applying correction factors for shear stress at the wall, are shown in Figure 7.10. I 50 40 Constant-property equation used to estimate shear stress at wall Correction of Razumovskiy et al . used for the wall shear stress Correction of Popov used to estimate shear stress at the wall Experiments / — 1600 1800 2000 2200 2400 Bulk Enthalpy, kJ/kg 2600 Figure 7.10. Effect of the expression used for shear stress at the wall on heat transfer, P=25.2 MPa, G=965 kg/m2.s, q"=307 kW/m2 157 The most acceptable predictions were obtained in cases where the effects of property variations on shear stress at the wall were accounted for. It should be emphasized again that the results of the numerical scheme are not dependent on initial estimate of the shear stress at the wall. However, use of an additional constraint in terms of the temperatures at the wall and bulk (e.g., Eqs. 7.26 or 7.27) acts as a fine-tune criterion for convergence of the numerical solution. Note that 1 °C difference in the bulk temperature can change the mass flux G by 15%. This justifies the use of temperature conditions in addition to mass flux to set the criteria for termination of the iterations. The expression used for the universal velocity profile in the above example is that of van Driest. Eqs. 7.26 and 7.27 were found to have a similar effect on model predictions once other universal velocity profiles listed in table 7.1 were investigated. These results are not shown here. The test conditions in Fig. 7.10 are the same as those in Fig. 7.5. The model predictions are improved considerably. Incorporating the effect of property variations on wall shear stress leads to further adjustments in the numerical technique. It was noticed that the modified numerical scheme required some values for both wall and bulk temperatures to start the calculations with. Typically, the wall temperature is given as input data and the bulk is assigned an arbitrary value. The bulk temperature is corrected later as the solution converges. Although quantitatively it appeared to make no difference, it is more reasonable to have bulk temperatures as input data and do iterations to find the wall temperatures. For a constant heat flux situation and known flow inlet conditions, fluid bulk enthalpy (and hence bulk temperature) is known at any cross section along the test section. Heat transfer directly affects the temperature distribution, particularly at the wall. It is therefore more logical to have bulk flow conditions as a priori and solve for the wall temperatures. The fact that integration starts from the wall has guided most researchers to have the wall temperature fixed and solve for the bulk temperature. It may be considered as an advantage of the present model to approach otherwise. 158 7.4.3. Effect of Turbulent Prandtl Number The turbulent Prandtl number was defined earlier to be the ratio of momentum eddy diffusivity to heat eddy diffusivity, i.e., Prt=em/eh. For fluids with a molecular Prandtl number close to one, like water at normal pressure and temperature, the turbulent Prandtl number is assumed to be unity. This is a requirement for the Reynolds analogy to hold. Measured or estimated values obtained from the literature for Prt ranges from 0.85 to 1.07. Those values were obtained for water at sub-critical pressures (i.e. near normal flows). There is no data available for Prt during supercritical fluid flow. Under such conditions measuring turbulent shear stress and velocity profiles to determine em is extremely difficult. In addition, simultaneous measurements of turbulent heat flux and temperature profile to form £h, make Prt to date, impossible to be evaluated. This is particularly important since most uncertainties exist at the near-wall region where the measurements are hardest of all. It should be noted, however, that within the region near the wall we are more concerned about the buffer zone rather than the laminar sub-layer. In most supercritical studies Prt has been assumed to be unity. This is partially because those studies were carried out before recent developments. Furthermore, in some studies, changing Prt from 1.0 to other numbers like 0.9 or 0.85 appeared to have no significant effect. Unlike such studies, the present model, as will be shown, is sensitive to values of Prt. In a recent review, Kays (1994) showed some discrepancies between the results of Direct Numerical Solutions and experiments for Prt of flow of water. For high Reynolds numbers, as in the current study, the experiments suggested that Prt approached a value of 0.85 within the core region. The core region is usually specified as y+ > -35, which covers almost the whole flow cross section except the very thin layer adjacent to the wall. Within the buffer region near the wall Prt may be greater than one. Hollingsworth et al. (1989) fitted their data to a curve, which correlated Prt with y+ as follows. Pr, = 1+ 0.855-tanh[0.2(/-7.5)] (7.28) 159 At very small y+, the equation of Hollingsworth et al. suggests values much greater than one for Prt, which is not acceptable. For this reason, Kays (1994) suggested that the Eq. 7.28 may be applied at y+ > 5 (i.e. outside the sublayer) and for smaller values of y+, Prt can be assumed to be constant and equal to unity. This modification does not influence heat transfer as it affects the sub-layer only. The effect of variations of Prt on heat transfer coefficients calculated by the present model are shown in Figure 7.11. The experimental results are not shown so as to not overcrowd the figure. However, the flow parameters were chosen to be the same as those shown earlier along with the data in Fig. 7.10. Figure 7.11. The effect of Pr t on local heat transfer coefficients, P=25.2 MPa, G=965 kg/m2.s, q"=307 kW/m2. The results shown in Fig. 7.11 at constant values of 0.85, 0.9 and 1.0 for Prt reveal that the greater the value of Pr,, the smaller the heat transfer coefficient. However, all three of them resulted in over-predicting the data. Applying the relationship of 160 Hollingsworth et al. (Eq. 7.28) yielded interesting results for the heat transfer coefficients. This is because Eq. 7.28 predicts higher values than 1 for Prt as the wall is approached (i.e. small y+). The close agreement with experiments is striking and shown separately in Figure 7.12. Compare the results with those of Fig. 7.10. One may conclude that the suggested high values of Prt at the wall by Hollingsworth et al., which sound unconventional for normal flows of water, can be realistic at supercritical fluid flows. Figure 7.12. Comparison of model predictions (equation of Hollingsworth et al. used for Prt) with experiments. Test conditions are the same as Fig. 7.10. Comparison with the results of others verified the effectiveness of the present model with the Hollingsworth modification. Figure 7.13a shows the data of Yamagata et al. (1972). Such data can be compared with the model predictions as shown in Figure 7.13b. Note that under the flow conditions shown, buoyancy effects were negligible. This can be deduced from Fig. 7.13a by noting the identical experimental results for vertical upward and downward flows and those obtained at the top and bottom surfaces for 161 As shown in Fig. 7.13a, the experiments suggested a value of about 58 kW/ m 2.°C for peak heat transfer coefficients. The correlation of Swenson et al. predicts the pick value closer to 70 kW/ m . The prediction of the present model is more accurate at about 53 kW/m 2.°C. Hence, the analytical model developed in this study predict the data well compared to others (e.g., Swenson et al. and Yamagata et al.). 7.5. THE NUMBER OF THE GRIDS AND THE STABILITY OF THE SOLUTION When integrating an equation numerically the finer the integration intervals, the more accurate are the predictions. However, the solution converges at some point and increasing the number of grids beyond that becomes futile. To illustrate this, consider the flow conditions shown earlier in Fig. 7.6. For this case the actual measured peak local heat transfer coefficient was around 18 kW/m . Figure 7.14 shows the model predictions for the same conditions as those in Fig. 7.6. It is shown that there is no chance of any reasonable predictions with N<80. Peak heat transfer coefficient approached 18 kW/m as the number of integration steps became greater than 80. P 350 |-CN ' E g 300 £ 250 [ | 200 / ° O 1 100 h-2 50 No. of Grids 1500 1800 2100 2400 Bulk Enthalpy, kJ/kg 2700 Figure 7.14. Effect of number of grids on variations of heat transfer coefficients, P=24.5 MPa, G=575 kg/m2.s, q"=103 kW/m2. 163 It can be also seen from Fig. 7.14 that the largest effect of grid numbers is around the pseudo-critical point, i.e., i D = 2100 kJ/kg. Thus, it is necessary to check the accuracy of the model only within the pseudo-critical region. If the model converges there, it no doubt converges outside of this region as well, although this was double-checked anyway. Before showing additional model predictions with increasing N, we need to introduce an additional parameter that concerns the size of the grids. In particular the grid size needs to be refined near the wall as discussed below. Recalling from Fig. 7.2, the temperature and velocity profiles are steep near the wall and flat at the core. It therefore suggests a fine grid near the wall region and coarser grids as we approach the flow centerline. Let the size of the grids be controlled by the parameter "Gratio" which is the ratio of the size of a grid to its previous one. It is assigned a number greater than, but close to, one. Starting from the wall, if the size of the first grid is yi , then the second grid y2 will be "Gratio" times larger than yi, i.e., y2 = yi*(Gratio). The size of the third and n* grid will be (y3)=(yO*(Gratio)2 and (yn)=(y1)*(Gratio)nl, respectively. The sum of the sizes of all of the grids, yi+y2+.--+yn, equals the tube radius r0. Therefore, given the number of the grids N, Gratio, and r0, the size of the first grid yi can be obtained via following relationship. _ r0(Gratio-V) ( 7 2 9 ) 1 (Gratiof-\ Thus, the two parameters, grid number N and Gratio decide how smooth the integration process is. The variation of model predictions for heat transfer coefficients versus grid number as a function of Gratio are shown in Figure 7.15. Regardless of value of Gratio, each curve asymptotically approaches a certain value for the heat transfer coefficient. This may happen rapidly or smoothly for high or low values of Gratio respectively. At small values of Gratio, like Gratio=L02, the growth of the grid sizes is very slow and at least 250 grids are needed for convergence. However, to save computing time it is also shown that at Gratio = 1.05 the same solution converges earlier at around 140 grids. Thus, it is a trade off, precision versus computing time. In the above example the two 164 layer and beginning of the core region, say -10 < y + < -50. In the laminar sub-layer the molecular diffusion is dominant and turbulence does not play an important role. For a relatively large Gratio number, like 1.4, further increase of the grid numbers beyond a certain value, only affects the distribution of the grids within the sub-layer. Thus, more number of the grids does not improve the results. In other words, while there is still room in the buffer layer for more number of grids to better model the abrupt variations of the properties, the increase of the total number of the grids were accommodated all within the sub-layer and not reached to the buffer zone. As Gratio decreases the converged value of heat transfer coefficient, as shown in Fig. 7.15, approaches asymptotically to a certain answer that can be imagined to be the answer one would obtain with Gratio=l and a very large number of grids. A variety of different examples were reviewed and a similar behavior as shown in Fig. 7.15 was found. It indicates that the numerical method employed is proper and sound. A Gratio between 1.05 and 1.1 assures the most efficient results. Hence it resulted in an average value of 105 taken as the number of grids. 7.6. Numerical Modeling Versus Empirical Correlation Upon reviewing the results of the empirical correlations discussed earlier and those of numerical models one may consider the following. It is true that in many cases the results predicted by numerical models agree with experimental data better than empirical correlations. It is also true that in some other cases the empirical correlations provide better results. Overall, it can be concluded that there is no superiority observed in numerical models over empirical correlations in term of predicting heat transfer coefficients. As such, the question is why bother to develop a numerical model with the difficulties encountered while compatible results could be obtained via empirical correlations. Especially since the analytical expressions for the universal velocity profiles, as listed in table 7.1, were obtained in the same empirical manner as the Nusselt number type of correlations were developed. In other words, both methods have their own brand of empiricism. It even seems to be a short cut to focus on the final result and directly search for a better curve fit of the Nusselt number (i.e., heat transfer coefficient) 166 rather than starting from a best fit for the universal velocity profile and then obtain the heat transfer coefficients. Nevertheless, it should be pointed out that a Nusselt number type of correlation is more limited in terms of explaining the mechanisms involved. For example, if some complexities are added to the problem, such as the occurrence of heat transfer deterioration, it could not be predicted by a Nusselt type analysis. However, a law of the wall model may be able to predict it. Such models reveal more details and enhance our understanding of the physics of the problem. To better illustrate this, consider the following where we use the numerical model to examine the buffer layer. 7.7. Significance of Buffer Zone It has been shown by many researchers that in a supercritical pressure flow and under certain circumstances, enhancement of heat transfer occurs when Tb < T p c < T w . This is related to the dramatic increase of the specific heat of the fluid within the vicinity of the pseudocritical temperature. The local decrease in thermal conductivity of the fluid near the pseudocritical temperature, though significant, is not large enough to compensate for the dominating effect of Cp towards enhancing heat transfer. Therefore, some researchers (e.g., Kakac 1987) have concluded that the peak of the heat transfer coefficient occurs at the tube cross section where the largest volume of the fluid is experiencing temperatures close to the pseudocritical. The results of the present model, however, did not support this idea. Figure 7.16 shows a region of the flow where temperatures are close to the pseudocritical. This region is arbitrarily defined as temperatures between (0.999)Tpc and (1.001)TPc and is shown by the shaded area in the figure. In other words, the upper and lower bound curves represent the radial distance for which fluid temperatures are (0.999)Tpc and (1.001)Tpc respectively. 167 Bulk Enthalpy, kJ/kg Figure 7.16. Radial variation of the flow region with peak values of Cp along the test section. Shaded area represents the portion of flow experiencing temperatures between 0.999Tpc and 1.001Tpc. Heat transfer coefficients are also shown for the sake of comparison. It can be seen that at small bulk enthalpies, ib < 1700 kJ/kW, the wall temperature has not reached the pseudo-critical point yet. Where ib > 2250 kJ/kW, the pseudo-critical temperature has already been spanned across the whole flow cross section. For bulk enthalpies between 1700 U/kW and 2250 kJ/kW the pseudo-critical region is moving from y/ro=0.0 to y/ro=1.0. An enlarged version of Fig. 7.16 (not shown here) clarified that up to ib=1900 kJ/kW, the pseudo-critical region remained attached to the wall. It was felt only within the laminar sub-layer. For 1900<ib<2050 all of the buffer layer lies within the pseudo-critical region. This corresponded to the peak heat transfer coefficients. Note however that the largest volume of the fluid was within the pseudocritical region when the bulk enthalpy varied between 2100 to 2200 kJ/kg (the shaded band was widest within 168 this region). This did not correspond to the peak heat transfer coefficient as shown. The maximum heat transfer, therefore, was decided by the buffer zone. A further clarification on the above may be offered as follows. Consider Figure 7.17 where a number of local temperature profiles are shown all at the same wall heat flux, flow rate, etc. Figure 7.17. Temperature profiles normalized by pseudo-critical temperature. Different profiles correspond to various locations along the test section. The pseudocritical region is given at T / T p c « 1. To focus on the wall region, the temperature profiles are shown only up to y+=50. Beyond y+=50 to the tube centerline, temperature changes were small and are not shown. The sub-layer of the boundary layer is usually referred to where 0 < y + < 5. The buffer zone is approximately where 5 < y + < 35 and the core is 35 < y+. The curve at the top of the Fig. 7.17 represents the temperature profile at a location where wall and bulk temperatures were higher than the pseudocritical point. The 169 profile shown at the bottom of the figure reflects the condition where the pseudocritical region was not reached. In all the profiles, the sharpest variation of temperature occurred at the wall. Note for each of the curves that the larger the heat transfer coefficient, the smaller was the temperature difference between the wall and core. The darker line in Fig. 7.17 thus denotes the largest heat transfer coefficient. The effect of the pseudocritical, or high Cp region, is to flatten the temperature profile. In the core region, however, the temperature profile is already relatively flat and occurrence of the pseudocritical in this region can only slightly increase this flatness of the profile. Having the pseudocritical region near the wall (within the laminar sub-layer) does not affect the temperature profile much as can be seen by comparing the curves. The temperature profile was sharp within the sub-layer and only a small fraction of it may coincide with pseudocritical temperature. Thus, the existence of the pseudocritical region at either the wall or core does not explain heat transfer enhancement. The buffer zone is the layer of fluid where temperature variation is large but not very sharp due to the extent of the region on the curves. Thus, the presence or absence of the pseudocritical region within the buffer zone is what makes the difference. The darker curve in Fig. 7.17 corresponds to where the pseudocritical temperature occurs within the buffer layer. It can be seen that most of the layer is at T/T p c ~ 1. The profile has the least temperature difference between its two ends, hence representing the highest local heat transfer coefficient. Other profiles in the figure are for the sake of comparison. 170 CHAPTER EIGHT CONCLUSIONS AND RECOMMENDATIONS There has been very little heat transfer data for supercritical fluids that can be found in the literature over the last twenty-five years. Most of the available data are for vertical flows. Since the effects of buoyancy are different in vertical and horizontal flows, the results of studies of vertical flows are not easily applicable to horizontal flows, unless the effect of natural convection is negligible in both flows. Yet the flow parameters leading to buoyancy-free conditions are different in vertical and horizontal orientations. This investigation is especially significant since it focuses on horizontal flows and as such, adds considerably to the worldwide database. As well, the current data was obtained with accurate state of the art measurement and data acquisition equipment. In the light of the generated data, this study encapsulates all the necessary knowledge required by a design engineer; in particular, to design a horizontal SCW system that operates within a range of safe operating conditions (i.e. no deteriorated heat transfer due to buoyancy and/or acceleration effects). Details of the contributions made by this study follow. 8.1. EMPIRICAL CORRELATIONS It was shown that none of the available empirical correlations predict heat transfer coefficients satisfactorily even within the range of negligible buoyancy conditions during 171 horizontal flows. The disagreements among available empirical correlations are large. Most of them are highly case dependent. Potential sources of discrepancies between empirical correlations were investigated. Empirical correlations developed in the past were, it is argued, in all likelihood based on data that was of mixed rather than pure forced convection cases. This is the most likely cause for disagreement between them. Some have postulated that the disagreement was due to the use of different property databases. This was shown in the current study to have a lesser effect. In other words, it is more likely that buoyancy effects, which may have occurred (and were not accounted for) in developing each correlation, were the largest cause of disagreement between their predictions. Thus, to compare various correlations, requires careful screening of the available data in order to exclude flows affected by buoyancy. This was not properly done in many previous assessments of such correlations. Various correlations have attempted to account for property variations through the use of correction factors that are functions of wall-to-bulk fluid properties. It is the opinion of the author that it would be difficult, if not impossible, to develop a correlation by this approach alone that could predict all the available data. Relatively speaking however, the correlations of Petukhov et al. (1961) and Krasnoshchekov and Popov (1966) better predicted the data of the current study. The suggested use of empirical correlations is for preliminary design purposes and even then, only cases with lower heat fluxes and higher mass flow rates. A different approach was pursued to develop a new correlation for calculation of local heat transfer coefficients in supercritical fluid flows where body forces are negligible. It was suggested to evaluate Reynolds and Prandtl numbers at variable rather than fixed reference temperatures. It was postulated that there would be one temperature for any axial position along the tube at which hcomiation^hexperiment- A relationship to express such a reference temperature in terms of the fluid properties along the course of the flow was introduced. This correlation was shown to predict the data obtained in this study as well as data obtained by other researchers with a better accuracy than previous correlations. 172 8.2. BUOYANCY AND ACCELERATION EFFECTS Natural convection can dramatically influence the flow particularly around the critical region. The vast majority of data in the literature, for both horizontal and vertical test sections, were generated under flow conditions where buoyancy was important. This was simply being ignored in early investigations and surprisingly in a number of later studies too. Buoyancy effects are more important in horizontal flows where flow asymmetry leads to a non-uniform local temperature distribution around the tube periphery. Buoyancy effects reveal themselves through the occurrence of large temperature differences between the top and bottom surfaces. It was found that natural convection effects are greater at pre pseudo-critical than post pseudo-critical temperatures. This is due to the larger variation in density with temperature within the pre pseudo-critical region. Near the critical region, peak heat transfer coefficients at the bottom were sometimes (during low flows and high heat fluxes) more than two times greater than those at the top surfaces. This type of degraded heat transfer was felt for a significant length along the top surface. It is therefore crucial to predict the onset of the buoyancy effects. The criterion developed in the past for distinguishing buoyancy-free regions for near constant property flows (e.g. water at low pressure) is not adequate for predicting supercritical fluid flows. The criteria to distinguish buoyancy-free flows have been well established for vertical flows. As such, they were found through this study to not apply for horizontal flows. Very little theoretical criteria have been suggested in the literature to detect the buoyancy-free region for horizontal supercritical fluid flows. A systematic approach was taken in the current study to span the test conditions from highly buoyancy-affected to buoyancy-free cases. This provided the opportunity to investigate the onset of buoyancy effects. The threshold Grashof number Gr th suggested by Petukhov and Polyakov (1988) was found to be very accurate in distinguishing between buoyancy-affected flows (Grq>Grth) and buoyancy-free flows (Grq<Grth) with the data of the current study. Their criterion was not thoroughly tested before due to a lack of data for horizontal flows. However, the suggested procedure of Petukhov and Polyakov (1988) for calculation of 173 the temperature difference between top and bottom tube surfaces was found inadequate with the data obtained in the current study. A new dimensionless number representing the effect of acceleration was defined. It was shown that such effects were smaller than buoyancy effects, at least for the flow configurations used in this study. This was also confirmed through the use of a previous dimensionless number introduced by Petukhov and Polyakov (1988). The main purpose of the current study was to predict heat transfer in the flows under negligible effects of body forces, i.e., buoyancy and acceleration. The interest on body forces was therefore limited to find out the onset of such effects. However, body forces have attracted a lot of attention in the past due to the fact that they may lead to deteriorated heat transfer. Evaluation of the criteria that classified the flows in terms of deteriorated and non-deteriorated regimes, and also prediction of the deteriorated heat transfer coefficients were not the focus of the current study. Some limited data however was obtained which is worthwhile to mention here. The data obtained over a wide range of heat fluxes and mass flows revealed that the values for q'VG could not adequately classify different modes of supercritical fluid flows. Such dimensional parameters have been used in the past. The term q " D 0 2 / G 0 8 suggested by Goldman (1961) and others was shown, however, to be a more effective parameter for flow classification. Tests with the same values of q " D 0 2 / G ° 8 demonstrated qualitatively similar behaviors. During deterioration, localized hot spots were not limited to tests with T D <T p c <T w as some previous studies have reported. Nor did the hot spots confine themselves within the entrance region of the test section. In this study, hot spots always appeared around the same location over the range of conditions for which they occurred. For example, a slight increase or decrease in the inlet temperature did not shift the location of the hot spots. The inlet temperature had a deciding effect on whether heat transfer was deteriorated or not, but not on the location of the hot spots. Deteriorated situations were highly irreproducible and usually accompanied by flow oscillations. As such, this is analogous to a similar behavior found in two-phase flows (i.e. dry-out associated with two-phase flow instabilities beyond the critical heat flux point). 174 Unlike what may be expected (e.g., in two-phase flows) it was found that increasing the heat flux beyond the deterioration region tended to eliminate the hot spots. There must be an upper limit to this heat flux where perhaps some other form of deterioration would occur. This was beyond the capabilities of the apparatus. Note also that before heat transfer was deteriorated (i.e., during enhanced mode of heat transfer), unlike standard single-phase forced convection cases, the heat transfer coefficients tend to increase with decreasing heat flux near the critical region. 8.3. NUMERICAL MODELING Consider flows far from the entrance region and with negligible buoyancy and acceleration effects. It was found that for a fixed mass flow, heat flux and pressure, a variation in the inlet temperature did not affect the heat transfer rate downstream. Therefore, local heat transfer coefficients are independent of upstream conditions (flow history) and hence, the assumption of a fully developed flow is reasonable. It implies that a 1-D analysis in which the radial variations of the flow from the wall to the center are •accounted for should be adequate. For flows unaffected by buoyancy and acceleration, a numerical model, incorporating recent improvements in turbulence modeling, was developed. The model was found to predict the data of the current study very well. It also predicted the available data from other studies well. The model provides a solution to the boundary layer equations based on the law of the wall. Some features of the supercritical fluid flow like velocity and temperature profiles, which are impractical or extremely difficult to measure, can be obtained from the numerical model. The model was shown to predict accurately the heat transfer during flows where buoyancy (and hence acceleration) effects are unimportant. However, to properly account for the more complicated case where buoyancy and/or acceleration occur in a horizontal flow, a 3D model is imperative. Under most conditions, both buoyancy and acceleration are closely coupled in horizontal flows. An exception would be for small diameter tubes that may have acceleration effects (due to decreasing density as the fluid passes through the pseudo-critical temperature) without buoyancy effects (due to the small diameter). The aforementioned expression derived to determine the magnitude of acceleration 175 effects is therefore important to distinguish flows wherein the effects of acceleration are important and hence to, specify the limits of the 1-D model. This is a crucial design consideration. Most of the popular expressions for universal velocity profiles were tried in the current model. Variations in the predictions were shown. It was found that using the expressions offered by Van Driest, Von Karman and Rannie provide the best prediction to the data obtained in the current study. Others were shown to dramatically over predict, or in the case of Prandtl-Taylor, under predict the data of the current study. The present model is sensitive to values of Prt. Applying the relationship of Hollingsworth et al. (1985) yields the closest agreement with the data obtained in the present study. Incorporating different expressions for the wall shear stress was shown to also affect the model predictions. The correction of Popov reported by Petukhov et al. (1982) to account for variable properties was found to predict the current data more accurately than other methods. The results of the current model show that peak heat transfer coefficients are not decided by conditions at the core, nor by the very immediate sub-layer attached to the wall. The core effect has been suggested earlier. However, in this study it was shown that temperature profiles are relatively flat in the core. On the other hand even though temperature profiles are steep within the sublayer, it is extremely thin. Instead, it was found that the buffer zone has a dominating effect on heat transfer. The buffer layer was shown to be the region of the flow across which the temperature variation is large but not very sharp. Hence it was argued that the main component of convective heat transfer is found within this region. Thus, the presence or absence of the pseudo-critical region within the buffer zone is what makes the difference. 8.4. RECOMMENDATIONS FOR FUTURE WORK Data should be obtained from test sections with different diameters. The effect of the tube diameter on horizontal flows is extremely important and has not been adequately addressed in the literature. For instance, the parameter q"D° 2 / G ° 8 or a similar term can serve to classify various flows. Such a study would further improve our knowledge of how to define the regions where buoyancy and acceleration effects dominate. 1 7 6 The current SCWO pilot plant was not capable of studying deteriorated heat -transfer over a wide range of conditions. Higher heat fluxes and lower mass flows are required for a more thorough study. Also in the current set up, heat recovery through the heat exchanger results in increasing inlet temperatures to the test section. Thus, the choice on inlet temperature, for which deterioration is highly dependent, is difficult to maintain. An independent study with additional test section designs as well as other components of the system is recommended to investigate the conditions where hot spots occur during horizontal flow of supercritical fluids. A quantitative evaluation of the buoyancy effect during horizontal flow requires further investigation and advanced modeling. For future work, data from flows with large tube diameters would be necessary to further our understanding. Even though the effects of thermal acceleration and buoyancy are usually coupled, in large tubes buoyancy effects may dominate over the effects of acceleration. The effect of acceleration is negligible whenever buoyancy effects are small. This is generally true for flows through tubes with moderate to large diameters. 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Rouse; A convenient Correlation for Heat Transfer to Constant and Variable Fluids in Turbulent Pipe Flow, Int. J. Heat Mass Transfer, Vol. 18, pp. 677-683, 1975. Stewart, E. , P. Stewart and A. Watson; Thermo-Acoustic Oscillations in Forced Convection Heat Transfer to Supercritical Pressure Water, Int. J. Heat Mass Transfer, Vol. 16, pp. 257-270, 1973. 193 Styrikovich, M.A., T.Kh. Margulova and Z.L. Miropolski; Problems in the Development of Design of Supercritical Boilers, Teploenergetika (Thermal Engineering), Vol. 14, No. 6, pp. 4-7 (5-9), 1967. Sutler, P.L. and A.F. Romanov; Experimental Dynamic Characteristics of A Supercritical Once-Through Boilers, Teploenergetika (Thermal Engineering), Vol. 19, No. 5, pp. 56-61 (79-85), 1972. Swenson, H.S., J.R. Carver and G. Szoeke; The Effects of Nucleate Boiling Versus Film Boiling on Heat Transfer in Power Boiler Tubes, Transactions of the ASME Journal of Engineering for Power, pp. 365-371, October 1962. Swenson, H.S., J.R. 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Dorfler; Heat Transfer to Supercritical Carbon Dioxide in Tubes with Mixed Convection, Proceedings of the ASME 32nd National Heat Transfer Conference, 1997. Watson, A.; The Influence of Axial Wall Conduction in Variable Property Convection - With Particular Reference to Supercritical Pressure Fluids, Int. J. Heat Mass Transfer, Vol. 20, pp. 65-71, 1977. Watts, M J . and C T . Chou; Mixed Convection Heat Transfer to Supercritical Pressure Water, Proceedings of the 7th International Heat Transfer Conference, Munchen, 1982. Wolf, H. and M.J. Zucrow; The Analytical Determination of the Local Heat Transfer Characteristics of Gases Flowing Turbulently in the Thermal Entrance Region of A Circular Duct with Large Temperature Differences Between the Gas and the Duct Wall, Technical Report PUR-35-M, Purdue Research Foundation, December, 1957. Wood, R.W. and J.M. Smith; Heat Transfer in the Critical Region - Temperature and Velocity Profiles in Turbulent Flow, A.I.Ch.E. Journal, Vol. 10, No. 2, pp. 180-186, March 1964. Xu, L.; Mass Transfer in Supercritical Water, MScE Thesis, University of British Columbia, 2000. 195 Yaskin, L.A., M.C. Jones, V.M. Yeroshenko, P.J. Giarratano and V.D. Arp; A Correlation for Heat Transfer to Super-physical Helium in Turbulent Flow in Small Channels, Cryogenics, pp. 549-552,October 1977. Yamagata, K., K. Nishikawa, S. Hasegawa, T. Fujii and S. Yoshida; Forced Convective Heat Transfer to Supercritical Water Flowing in Tubes, Int. J. Heat Mass Transfer, Vol. 15, pp. 2575-2593, 1972. Yeroshenko, V.M. and L.A. Yaskin; Applicability of Various Correlations for the Prediction of Turbulent Heat Transfer of Supercritical Helium, Cryogenics, pp. 94-96, February 1981. Yoshida, S., T. Fujii and K. Nishikawa; Analysis of Turbulent Forced Convection Heat Transfer to a Supercritical Fluid Flowing in a Tube, Heat Transfer, Japanese Research, Vol. 2, 1973. Yeroshenko, V.M. and L.A. Yaskin; Noniterative Method of Calculating Heat Transfer to Turbulent Flows of Liquids in Tubes at Supercritical Pressure, Thermal Engineering (Teploenergetika), Vol. 31, No. 8, pp. 473-475 (74-76), 1984. Zhou, N. and A. Krishnan; Laminar and Turbulent Heat Transfer in Flow of th Supercritical CO2, Proceedings of the ASME 30 National Heat Transfer Conference, Portland, Oregon, Aug. 6-8, 1995. 196 APENDIX A CHRONOLOGICAL LITERATURE REVIEW This appendix offers a comprehensive review of the literature on "forced convection heat transfer to supercritical fluids". All related investigations available to the author have been reviewed. This review therefore does not provide background for the current study only. It may also be considered as an independent extensive survey of the literature on forced convection heat transfer to supercritical fluids. Unlike most review papers, the main topic has not been divided into subtopics within the area. Instead, a mere chronological order has been used to tell the story of heat transfer to supercritical fluid flows. In addition there are sub-headings which highlight the important milestones/advances in the field. Also a brief summary is given at the end of the document which reflects the current research status on the topic. 1939 The Earliest Published Study The concept of the thermodynamic critical point of fluids was introduced more than a century ago. It has been known, since then, that properties of a fluid beyond its critical point are quite different than those at lower pressures. The study of forced convection heat transfer to supercritical fluids, however, started about sixty years ago. In some review papers (e.g., Hendricks et al. 1969) the earliest work has been attributed to the German researcher Schmidt in 1939. He arranged an experimental setup with a supercritical environment and showed that free convection heat transfer characteristics were appreciably altered near the critical region. He drew attention to the impact that 197 variations of property could have on heat transfer to a fluid near its critical region and ignited the idea of thinking of supercritical fluids as suitable heat carriers. 1954 First Systematic Approach In English literature, the earliest release goes back to the mid 50's. Two novel studies of Deissler (1954) and Goldman (1954) were the first classic studies performed on forced convection heat transfer to supercritical fluids. Both investigations were analytical studies. Deissler solved simultaneously the equations of shear stress and heat transfer for a pipe flow by means of a simple numerical method. He employed a law of the wall type of analysis for a turbulence model. An analogy between momentum and heat transfer was postulated, i.e., turbulent Prandtl number was assumed to be unity. He also assumed that radial variations of shear stress and heat flux had a negligible effect on velocity and temperature profiles. This is equivalent to neglecting the convection terms in the equations of momentum and energy. The flow was divided into near wall and core regions. In an earlier study of air flowing in a pipe under high heat flux, Deissler (1950) developed an expression for eddy diffusivity near the wall. He used the same relationship for the case of supercritical fluid flow. Away from the wall, where variations of the fluid are not as severe, the expression proposed by von Karman (1934) was used. For a given heat flux and wall temperature, velocities and temperatures could be calculated at any distance from the wall. The calculated velocity and temperature profiles were compared with experimental data for air. A relatively good agreement was obtained. From there, Deissler extended his method for the case of heat transfer to supercritical water. He presented his predictions without having any experimental data to compare with. Later experiments by other investigators showed that Deissler results for supercritical water were only qualitatively acceptable. Goldmann (1954) attempted solving the same set of equations for shear stress and heat transfer with similar boundary conditions and expressions for the eddy diffusivity as Deissler. The distinction lay in the critical assumption Goldmann made to account for the effect of fluid property variations. He supposed that the local eddy diffusivity was a function of fluid properties (i.e., fluid temperature) and was independent of small property variations adjacent to that location. In other words, he proposed that the same expression of eddy diffusivity, used for a constant-property flow, was applicable to 198 supercritical fluid flow provided that fluid property variations with temperature (from the wall to the core) would be accounted for in calculation of the eddy diffusivity at each point. Based on this assumption, the distance from the wall to the center was divided into many intervals. Integration of the equations of shear stress and heat transfer led to velocity and temperature profiles. Comparison with the results of Deissler showed some decrease in the heat transfer coefficient. Such a decrease was shown, through experimental results made available later, to be an improvement. 1957 First Experimental Studies To address the need for experimental data, Bringer and Smith (1957) were among the first to perform a set of experiment with a supercritical fluid flowing in a round tube. Carbon dioxide, with critical point of 7.4 MPa and 31 °C, was selected as the working fluid. Enhancement of heat transfer near the critical region, similar to what predicted by Deissler and Goldman, was clearly observed in Bringer and Smith's experiments. They applied the analysis of Deissler to their data. The agreement was better than what one could get out of a traditional forced convection Nusselt number correlation. Bringer and Smith concluded that empirical correlations, developed earlier for constant property flows, were inadequate to predict the heat transfer coefficient near the critical region. 1957 First Empirical Correlation The results of Miropolski and Shitsman (1957) with forced flow of supercritical water in a round tube also confirmed the region of enhanced heat transfer near the critical region. They recommended a modified empirical correlation to predict heat transfer during turbulent forced convection flows. It was the first time an empirical relationship was offered to fit experimental data. The form of their correlation was simple and was obtained by making minimum changes to the Dittus-Boelter correlation. Weight was given to the Prandtl number, Prmin, which is the lesser of the Prandtl number at the wall and in bulk and was used instead of the typical bulk Prandtl number only. For near-critical temperature, variations of Prandtl number resemble the way specific heat varies with temperature. It appeared that the suggested correlation provided a better prediction of heat transfer coefficients compared to some of the more complicated correlations developed later. Miropolski and Shitsman clarified, however, that their proposed correlation was good only for fluids with Prandtl number around unity. 199 Interest in supercritical fluids was not restricted to carbon dioxide and water. Powell (1957) measured forced convection heat transfer to supercritical oxygen. This was the first published measurement on supercritical oxygen. One main advantage of the data of Powell was the way he presented them. He defined a generalized heat transfer coefficient as a function of heat flux, tube diameter, mass flow, and bulk and wall temperatures. This way, raw experimental data was less manipulated in comparison with other data and was less influenced by different libraries of fluid properties. The different aspect of Powell's results was that he did not capture any enhancement in heat transfer near the critical region as reported for other supercritical fluids before him. The data showed only a monotonic decrease in heat transfer rate. The number of experimental studies performed with supercritical fluids was so limited that the results of Powell alone were good enough to raise doubts about previous findings. It was not clear if the phenomenon of heat transfer to a supercritical fluid was a function of the fluid chosen or not. It seemed that any generalization of the results for one supercritical fluid to another needed further investigation. Dickinson and Welch (1958) enriched the heat transfer database for supercritical water by performing a relatively comprehensive set of experiments. They covered pressures of 31.03 and 24.13 MPa; volumetric flows of 5.9 to 9.3 £/min; temperatures of 104 to 537 °C and heat fluxes of 882 and 1827 kW/m 2 in a vertical pipe flow. The enhancement of heat transfer near the critical region was reported. 1960 Flow Visualization over A Cylindrical Wire It was found interesting to experimentally span various flow pressures from subcritical to supercritical values and watch for changes. To study fluid flow, in general, it seems that no other technique is more convincing than flow visualization. Griffith and Sabersky (1960) applied flow visualization techniques to study subcritical and supercritical pressure flow of Freon (R114A) over a cylindrical wire. The wire was electrically heated to the temperatures above the critical point. Photographs showed that bubble like activities did exist within the supercritical region but certainly not as strongly nor as clearly as that which occurs during phase transition at subcritical pressures. Furthermore, unlike subcritical nucleate boiling, the formation of lower density bubbles at supercritical pressures was not accompanied by an immediate increase of heat transfer. 200 Griffith and Sabersky, therefore, concluded that such bubble-like clusters did not affect the heat transfer rate. From the study of McGarthy and Wolf (1960) it was concluded that enhanced heat transfer is peculiar to the near critical region where the pressure is slightly above the critical pressure and the temperature is close to the pseudo-critical point. They carried out a set of experiments with gaseous hydrogen at high heat flux and high supercritical pressures in a smooth, round, electrically heated tube. Pressures and temperatures were well above the critical point. They found that pressure had no effect on the heat transfer. Hence a correction factor (Tw/Tb)n, as in case of single-phase liquid or gas flow, was used in a conventional empirical heat transfer correlation. This was found to match the experimental data satisfactorily. In such cases, another example of which can be found in Barnes and Jackson, 1961, the property variations are large and cannot be neglected. The fluid properties, however, vary monotonically with temperature. Thus, unlike supercritical fluid flows, a conventional heat transfer correlation with a correction factor in form of (Tw/Tb)n can sufficiently predict heat transfer coefficients. 1961 Emergence of the Idea of Pseudo-Boiling The resemblance of enhanced heat transfer at supercritical pressure to that of subcritical boiling (with phase change) led Goldman (1961) to propose the idea of a pseudo-boiling heat transfer phenomenon at supercritical pressures. The magnitude of the increase in heat transfer at supercritical conditions is much less, and certainly not comparable to what occurs during nucleate boiling. Nevertheless, the increased capacity of a supercritical fluid in absorbing heat near its critical temperature has similarities to the role of the latent heat during the phase change boiling process at subcritical pressure. The whistle sound detected in Goldman's experiments also supported boiling-like activities. Of course, at supercritical pressure, two distinct phases could not exist in equilibrium. Thus, it was postulated that under transient conditions liquid-like fluid came in contact with the hot surface and burst into lower density vapor-like bubbles. Such bubbles departed the surface due to buoyancy. The mixing and/or turbulence created this way, in turn, excited a larger volume of cooler liquid-like fluid to contact the surface and thus heat transfer was increased. Another important contribution of Goldmann (1961) was that he detected two different modes of heat transfer. It was reported that for high heat fluxes 201 close to the pseudo-critical temperature, a deterioration of heat transfer might occur. This was indicated by local hot spot along the tube wall. No explanation was given though. The idea of pseudo-boiling was also found in the work of Bonilla and Sigel (1961). They arranged an experimental loop, probably for the first time, to measure natural convection heat transfer near the critical point. The working fluid was n-pentane. Increased natural convection heat transfer was detected near the critical region. A conventional Nusselt number correlation worked out well until higher Rayleigh numbers were reached where an enhanced heat transfer, similar to nucleate boiling, occurred. A new debate had been introduced into the field, "Is heat transfer to supercritical fluid flow a mild type of boiling process, the so called pseudo-boiling or is it an extension, though extraordinary, of single phase forced convection heat transfer?" In fact, the delay in reaching a convincing explanation along with a quantitative model from the latter approach, made the former more popular. It could at least give a qualitative explanation for the phenomenon. 1961 Modifying the Expression for Eddy Diffusivity To explain enhanced heat transfer in terms of single phase forced convection, Hsu and Smith (1961) took a significant step ahead. They extended the analytical model of Deissler (1954) and included a correction factor in the expression for eddy diffusivity. The correction term was derived from a simple analysis that accounted for density fluctuations. In fact, two effects of density variations were distinguished. One was the change of radial transfer of momentum and heat between two adjacent layers of fluid. The other was the influence of density changes on the buoyancy/gravitational term. They demonstrated the contribution of each of the two effects on the heat transfer coefficients. The predicted velocity profile showed that during large density changes, the maximum velocity was not at the flow center. This was different than what occurred in a constant property flow. Heat transfer predictions were compared with available data. Only in some cases, a better agreement was reached. Their study was applicable to vertical pipe flows. Later comparison of the results showed that the particular modification of Hsu and Smith was not very successful. Nevertheless, the significance of their work was the approach they introduced. The message was that the equation of eddy diffusivity, developed for 202 constant property flow, needed to be adjusted before applying it to variable property flow. Although supercritical carbon dioxide was not important at the time, the experimentalists preferred it to supercritical water for two reasons. Those were the lower critical point (temperature and pressure) and physical property knowledge. Koppel and Smith (1961) measured the heat transfer coefficients to supercritical carbon dioxide during pipe flow. Both normal and deteriorated heat transfer were detected. Together with experimental data for other supercritical fluids, a comparison was made against the correlation proposed by Miropolski and Shitsman (1957). Differences in test conditions were too wide to lead one to a general conclusion. 1961 The First Reasonably Accurate Empirical Correlation In a search for a generalized empirical correlation, the experimental results available so far were collected and examined by Petukhov, Krasnoshchekov and Protopopov (1961). They established a correlation, which agreed satisfactorily with data for water and carbon dioxide under moderate wall to bulk temperature ratios. This correlation is still often referred to today. (It will be examined in more detail later in this study and will be shown that it is one of the most accurate correlations.) Hines and Wolf (1962) conducted an experimental study with turbulent flow of supercritical RP-1 and diethylcyclohexane (DECH) in an electrically heated tube. Tests were intended to qualitatively study the vibration of the tube under supercritical heat transfer conditions. Such oscillations were shown to resemble the vibrations encountered in nucleate boiling along with an increase in the heat transfer rate. In fact, the existence of such oscillations further supported the idea that heat transfer to supercritical fluids might follow a pseudo-boiling pattern rather than a single phase forced convection case. Quantitatively, the enhanced heat transfer associated with the vibration was reported to be 40% whereas that of nucleate boiling was known to be as high as 100%. Likewise, tube failure in a flow of supercritical fluid did not occur as, in what is-called burnout, in nucleate boiling. A gradual tube rupture in supercritical fluid flow, however, was reported. Hines and Wolf concluded that the scattered bubbles generated due to the steep density profile at the wall could not control the heat transfer mechanism at supercritical pressure like bubble activity did in a two-phase flow. They suggested that the variation of 203 viscosity might be a major cause for anomalous heat transfer at supercritical conditions. They showed experimentally that for a large reduced pressure of P—6, where the viscosity did not alter with temperature significantly, no peak in the heat transfer coefficient was recorded. This, however, does not seem to be convincing since the other properties of the fluid also did not alter considerably with temperature at P—6. 1962 The First Studies With Hydrogen Hendricks, Graham, Hsu and Medeiros (1962) carried out a comprehensive set of experiments with hydrogen at subcritical and supercritical pressures. At supercritical pressure and near the critical temperature, the pseudo-boiling idea first proposed by Goldmann (1961), was mathematically modeled. Their model agreed, within 20%, to their data. Only a narrow range of flow conditions was measured, hence, assessment of the model was incomplete. Their model incorporated the well-known Martinelli parameter used in two-phase flow studies. For supercritical flow, as there were no distinct phases, the fluid was considered to consist of heavy and light species. The quality x was redefined correspondingly. Supercritical hydrogen was treated differently in the study of Szetela (1962). He considered heat transfer to supercritical hydrogen as a forced convection problem. He examined the method of solution by Deissler (1954) against measurements made for the flow of supercritical hydrogen in a round tube. Deissler's method under-predicted Szetela's data by up to 33%. Once the density variations in the form suggested by Hsu and Smith (1960) were used, the results showed a 50% to 75% overestimation. The conclusion was that neither model was adequate. Swenson, Carver and Szoeke (1962) studied water over a wide range of subcritical pressures. The investigation was not aimed particularly at study of supercritical conditions. It was interesting, however, that for pressures near the critical point, the film conductance was high and burnout was less likely to occur. Tests were done on two-phase boiler tubes at varying quality. Nucleate boiling, transition region and stable film boiling were noted. The Nusselt number correlation for single phase forced convection was modified for the region of film boiling. Good agreement with the data was found. The fact that the two-phase process of film boiling was modeled by a 204 modified single-phase correlation indicated that it should be even more feasible to similarly treat the supercritical fluid flow, which was already a single-phase flow. 1962 Cooling Heat Transfer with Supercritical Fluids Shitsman (1962) investigated cooling under supercritical conditions. He derived a Nusselt number correlation to predict heat transfer in a vertical pipe flow of supercritical water. An available correlation was modified by correction factors to account for varying fluid properties. It was emphasized that variation of the heat transfer coefficient resembled variations in specific heat or Prandtl number around the critical region. Natural convection to supercritical water was studied by Fritsch and Grosch (1963). They found that standard correlations were adequate in predicting the heat transfer rate to within 20%. Fritsch and Grosch believed that disagreements were due to a lack of knowledge of the transport properties of water near the critical region. The results of Fritsch and Grosch (1963) and the earlier study of Powell (1958) indicate that the flow conditions leading to an enhancement of heat transfer were not fully understood yet. It was mentioned earlier that Deissler (1954) and Goldmann (1954) took into account the influence of radial variations of fluid properties in their analysis. However, the effects on radial variations of heat flux were not yet considered. This was the major accomplishment of Petukhov and Popov (1963). They used the expression of Reichardt for eddy diffusivity. Frictional effects were also considered. They demonstrated how their formulation reduced to the well-known correlation of Petukhov and Kirillov (1958) once property variations diminished. Predictions were compared to flows of hydrogen and air under high temperatures. Good agreement was reported. 1963 Systematic Study of Deteriorated Heat Transfer Shitsman (1963) conducted a comprehensive experimental study with supercritical water flowing in a vertical tube. Pressures of 230, 240 and 250 A T M , and mass flow rates of 300, 430, 700 and 1500 kg/m2.s were studied with heat fluxes up to 1200 kw/m 2. Enhanced heat transfer data were predicted well by the Miropolski and Shitsman (1957) correlation. They studied deteriorated heat transfer around the pseudo-critical point. Local jumps in wall temperature of up to 170 °C were detected. Shitsman concluded that high heat flux, low flow rates and closeness of pressure to the critical point caused deteriorated heat transfer. At large mass flows, no impairment was noted 205 even at higher heat fluxes. Peak values of wall temperature were shown to decrease with increasing pressure. Nevertheless, in some deteriorated heat transfer cases, lowering the mass flow did not escalate heat transfer deterioration. Knowing that heat transfer may be enhanced at supercritical pressure, alternative supercritical heat transfer fluids were sought by many investigators. Aladyev, Malkina and Povarnin (1963) examined methyl alcohol at supercritical conditions. Unlike water, deteriorated heat transfer was easily noted. They explained it in terms of decomposition of the methyl alcohol and formation of a coke-like deposit on the tube wall. Enhanced heat transfer was recorded only at small reduced pressures. Aladyev et al. fitted their results into a correlation, which predicted the heat transfer rate with 20% error. Enhanced heat transfer was excluded in their correlation. Aladyev et al. (1964) considered molecular dissociation of a mixture to explain enhanced heat transfer. They supported this with data obtained for ethyl alcohol. Their explanation, however, was not applicable to a pure supercritical fluid. They also obtained data for water at 250 to 350 atm. Surprisingly, no heat transfer enhancement was reported The reason for this may be that they never studied the conditions for which enhancement can occur. Such conditions became clear in later studies. 1964 Measurements of Velocity and Temperature Profiles One important step towards understanding heat transfer to supercritical fluids was measuring velocity and temperature profiles. Wood and Smith (1964) measured, for the first time, the velocity and temperature profiles of supercritical carbon dioxide flowing in a vertical round tube. When wall and centerline temperatures were higher and lower than the pseudo-critical temperature respectively, maximum velocities were not at the centerline. The velocity profile exhibited an M shape in those cases. Pickering (1964) reviewed methods for calculating forced convection heat transfer. Nusselt correlation correction factors derived for variable property flows of gases and/or liquids failed to predict data for supercritical fluids. Correction factors were usually a function of ratios of wall to bulk temperatures. Reference temperatures for property evaluations were in the form of T x = Tb + x(Tw-Tb), where x could be any number between 0 and 1. Deissler (1954), for example, used x=0.4. It became clear later that his predictions did not match the experiments well. 206 Most experimental studies to this date considered vertical flows. One of the earliest major studies on supercritical water heat transfer during horizontal flow was that of Vikrev and Lokshin (1964). They studied steam-generating tubes at supercritical pressures with special attention given to deteriorated heat transfer. It was confirmed that, like vertical flows, degraded heat transfer occurred only at low flow rates and high heat fluxes. For example, at a mass flow rate of 400 kg/(m2 s) and a heat flux of 700 kW/m 2 deterioration occurred in a tube with 10-mm diameter. No discussion was made regarding the effect of the tube diameter. The variation of pseudo-critical temperature with pressure was empirically estimated to follow tp c = tcr + 0.348(p-pcr), where tcr and p c r were critical temperature and pressure respectively. An expression was suggested for the minimum heat transfer coefficient during deterioration. The heat transfer coefficient was expressed as a function of heat flux, mass flow and pressure. This was the first attempt ever made to quantitatively formulate the deteriorated heat transfer coefficients. Bishop et al. (1964) conducted an experimental heat transfer study on upward flow of water in cylindrical and annulus tubes. Their data were correlated with a Dittus-Boelter type of relationship using a correction factor in terms of densities at the wall and bulk as well as a fade away term reflecting an effect of entry length. Inclusion of the wall-to-bulk ratio of the fluid density appeared to improve the data curve fitting. 1965 Significance of Entry Length Swenson, Carver and Kakarla (1965) broadened the existing database to water. Entry length effects were demonstrated. They showed that, unlike constant property flows, the effect of an entry region did not decrease monotonically with length. Thus, conventional fade away terms, similar to that used by Bishop et al. (1964), could not explain the considerably extended entry region at supercritical fluid flows. They also studied the effects of pressure and heat flux on heat transfer coefficients. Outside of an entry region, an empirical correlation was developed which matched other measurements within ±15% in ±95% of the cases. Only low heat fluxes were examined. Swenson et al. also tried their correlation against carbon dioxide data and found good agreement. Since this correlation is still referred to in some recent studies, more discussion and comparison of the results were presented in the body of this thesis. 2 0 7 Krasnoshchekov et al. (1965) studied a wide range of conditions during pipe flow of supercritical CO2. They included entry length measurements. For outside the entry region, an empirical correlation was suggested using correction factor of (pw/pb)° 3(Cp/Cpb)n, where n had different values corresponding to local wall and bulk temperatures. Agreement to within ±15% with the data was claimed. Some data were plotted as (q"d a 2/G a 8) versus wall temperature for different bulk temperatures. This was an attempt to further classify supercritical fluid flows as well as minimize the influence of different property libraries. Later, in a similar study, Krasnoshchekov and Protopopov (1966) compared their correlation with the data of Bringer and Smith (1957) and others. Poor agreement was reported. It was shown in chapter five of this study in more depth that Nusselt number correlations, generally, did not attain universality and were highly case dependent. To fit data for supercritical carbon dioxide, Popov (1965) modified the analytical study of Petukhov and Popov (1963). An extra term was introduced that accounted for density changes at the wall and bulk. Comparison with limited data showed good agreement. Hess and Kunz (1965) tried a similar approach to Deissler (1954). They tested the model with supercritical hydrogen data. The expression of van Driest was used to define the eddy diffusivity. A good agreement with low heat fluxes cases was found. For higher heat fluxes, a modification of the dimensionless damping constant A + in the Van Driest expression was suggested. For the near wall region, viscosity, and the extent to which it was damped, seemed to dominate the flow structure. Hess and Kunz recommended A + = 30.2 e " 0 0 2 8 5 ( t W instead of a constant to account for property variations on diffusivity. A comparison with the experimental data of Hendricks for hydrogen found that it was not necessary to postulate any similitude of supercritical and film-boiling heat transfer as suggested by Hendricks. Nevertheless, they admitted that their model required complex numerical calculation and was not recommended. An empirical correlation, which matched their results closely, was developed. Despite the relative success of turbulent forced convection analysis to qualitatively predict heat transfer enhancement in supercritical fluid flow, the explanation of the problem in terms of a pseudo-boiling concept was still popular. Nishikawa and Miyabe (1965) studied free convection heat transfer to subcritical and supercritical 208 carbon dioxide. Photographs revealed pseudo-nucleate boiling and pseudo-film boiling associated with enhanced and degraded heat transfer modes at supercritical pressure. 1965 Flow Oscillations To better understand flow instabilities observed in previous experiments, Cornelius and Parker (1965) designed a heat transfer loop that operated under either natural or forced convection. Refrigerant 114 was selected to avoid operation at high temperature and pressure. Two types of instabilities were recorded. One with a higher frequency and acoustic nature occurred at temperatures below the pseudo-critical point. Another type, at lower frequency, was detected when wall temperatures exceeded the pseudo-critical point. This was accompanied by an improvement in heat transfer. Interaction between the near wall vapor-like layer and liquid-like fluid in the core, as well as the formation of a steep density gradient was, considered to result in this phenomenon. Whether the flow instability was the cause, and improved heat transfer the effect, or vice versa, was not clear. Thurston, Rogers and Skoglund (1966) measured the frequency and amplitude of pressure oscillations in a heated flow of subcritical and supercritical hydrogen. Extension of a two-phase model to the near critical and supercritical region was made through defining quasi two-phase flow properties. These were calculated via a graphical technique. The idea of Hendricks et al. (1962) was expanded to define the phase quality x for a pseudo-boiling process. Other properties were estimated based on x. They developed a method to predict the frequency and amplitude of oscillations. The mechanism was explained in terms of an interaction between the superheated film adjacent to the wall and heat transfer. An instantaneous pressure rise was thought to compress the film layer at the wall and hence, improve heat transfer. Better heat transfer, in turn, causes a higher rate of vapor generation and therefore thickens the film layer. This was then thought to decrease heat transfer. If the heat flux is moderate, this decrease is good enough to return the vapor layer thickness to a balanced value and thus, dampen oscillations. For higher rates of heat transfer, however, the superheated wall layer becomes too thick and thus the pressure drop decreases. The interface of the superheated film and core will be accelerated. The dense core again approaches the wall and such a repetition produces the oscillations. 209 Kafengauz and Fedorov (1966) measured the frequency and amplitude of high frequency oscillations in a flow of di-isopropyl-cyclo-hexane at subcritical and supercritical pressures. The enhancement of heat transfer and occurrence of pressure oscillation were explained as follows. At subcritical pressures, vapor bubbles were first formed at the hottest location i.e., at the wall. As the bubbles were propelled into the cooler core liquid condensed. Formation and collapse of vapor usually occurred abruptly. This created additional turbulence and hence enhanced heat transfer. This rapid formation, and condensation, was responsible for sudden local deceleration or a micro-hydraulic shock in the flow. They observed similarities in the frequency and amplitude of the oscillations in both subcritical and supercritical pressures. This observation supported the idea that a boiling like phenomenon was occurring in the supercritical region. The only difference was that the terms "vapor" and "liquid" in subcritical boiling were replaced by "vapor like" and "liquid like" concepts for the supercritical region. There was one basic question to be answered if the pseudo-boiling concept were justifiable: Was there any real bubble-like behavior within a supercritical environment? Knapp and Sabersky (1966) studied free convection and used shadowgraphs to address this question. A hot metal wire was submerged in a pool of supercritical carbon dioxide. Their pictures confirmed the presence of bubble-like flow around the wire. However, unlike nucleate boiling, the occurrence of bubble-like flow was depending on geometry, heat flux and other parameters. They concluded that a bubble forming under supercritical conditions was most likely the result of hydrodynamic instabilities due to extreme change of properties. Thus, the nature of heat transfer at supercritical environment was considered different from nucleate boiling. In addition to observations of Knapp and Sabersky, note that it is likely that decreased density around the pseudo-critical point would impart motion due to buoyancy which, of course, would enhance heat transfer in a similar manner as in nucleate boiling. However, the density differences are around two orders of magnitude less than nucleate boiling at moderate pressures. In addition, the absence of a liquid-vapor interface rules out any surface tension effects that can have a dominating effect during pool boiling. 210 1966 Correlation Based on Pseudo-Boiling Proposal Hendricks, Graham, Hsu and Friedman (1966) produced a large heat transfer database for subcritical and supercritical hydrogen. The influence of various parameters on heat transfer, including tube diameter, which was not addressed before, was investigated. Flow oscillations and entry length effects were also studied. Although flow oscillations were said to improve heat transfer, the common practice was to avoid it. That was because the vibrations could damage equipment and create havoc on control systems. It was found that the instabilities could be controlled by changing heat flux more effectively than by varying mass flow. A number of empirical correlations were examined. An extension of a correlation for gases did not effectively predict the data for supercritical hydrogen. Based on the idea of pseudo-boiling, Hendricks et al. developed two different correlations. The results were compatible with predictions obtained by other forced convection correlations. These correlations were probably the only ones based on a two-phase flow type of model. In one of them a transverse heat transport was proposed. The light species was assumed to eject from the wall. A transverse velocity was defined and its effect was included in a Dittus-Boelter type of correlation. The Martinelli two-phase parameter needed to be calculated at different temperatures within the fluid. Good agreement with experiments was reached for regions where the bulk and wall temperatures were either both below or above the pseudo-critical point. This avoided the most difficult region where Tb<Tpc<Tw. This may be why these correlations were not referred to frequently in the literature. The relative success of models based on single-phase forced convection was another reason making the model of Hendricks et al. obsolete. Hendricks et al. (1966) also demonstrated the difficulty of performing reliable back calculation of heat flux from an empirical correlation. For example, when the discrepancy between the results of two correlations in a particular case were 40%, the heat flux obtained from back calculation can differ by up to few hundred percent. That was an important point, which warned about the way data should be presented. The results of many studies cannot be used for valid comparison simply because data was manipulated and there is no way to accurately determine the original test conditions. 211 Three main differences exist between forced convection heat transfer to fluids under normal (near constant properties) and supercritical (large variable properties) conditions. These were listed by Hall, Jackson and Khan (1966). First, the equations of heat convection and diffusion were affected non-linearly with changing temperature. Consequently the heat transfer coefficient was strongly affected by slight variations in the heat flux. Second, the concept of fully developed flow was questionable due to large thermophysical and transport property variations along the length of a heated tube. Third, heat and momentum transfers were highly coupled. The temperature profile resulted in radial changes of fluid properties, which affected the velocity profile. This meant that the equations of momentum and energy should have been solved simultaneously. To better understand the heat transfer mechanism at supercritical conditions, Hall et al. designed a clever experimental setup for which axial convection was minimized. Supercritical carbon dioxide flowed between parallel plates with the upper plate heated and the lower plate cooled so that the net heat transfer was zero. A sharp increase in the heat transfer rate was observed as the hot wall approached the pseudo-critical temperature. Like Goldmann's (1954) method, they solved the equations of shear stress and heat flux along with an expression for eddy diffusivity (Corcoran et al. 1956) to obtain temperature and velocity profiles. The results were not satisfactory. Then, a peak thermal conductivity near the pseudo-critical temperature was assumed. Its existence was a matter of debate by thermodynamicists at the time. Heat transfer predictions by the model did not show much improvement. From there, Hall et al. concluded that molecular convection did not have a significant effect in enhancing heat transfer. In the next step, the expression for eddy diffusivity was modified. It was assumed that the eddy diffusivity around the pseudo-critical temperature had its highest value. It decreased linearly away from the wall until it passed over the buffer zone. The results were slightly improved. Finally, a thermal expansion factor was introduced to the expression for eddy diffusivity. This led to good agreement with experimental data. Melik-Pashaev (1966) employed a similar analytical approach. The Prandtl mixing length theory was used to model turbulence fluctuations. Density fluctuations were accounted for and turbulent shear stress and heat flux were defined as x =-(pu7v'+p'uV) and q =p7v'+p'i'v'• The fluctuating terms were all transformed to 212 expressions in terms of mixing length. Of course, because of the extra terms accounting for density fluctuations, the equations became more complicated than before. They were solved simultaneously and the temperature profile was obtained. The heat transfer coefficient was calculated from the Stanton number relationship. Only one single graph of comparison of the predictions with data for supercritical carbon dioxide was presented by Melik-Pashaev. Good agreement was shown to hold. 1966 The Effect of Buoyancy Shitsman (1966) appears to be the first investigator of natural convection (buoyancy) effects during the horizontal flow of supercritical water. Such an effect manifests itself as a temperature difference between top and bottom surfaces of a tube. Shitsman measured top and bottom temperatures at various locations along the tube. Temperature differences as high as 250 °C were observed. The product of the Grashof and Prandtl numbers was taken as a measure of the significance of buoyancy. The flow was divided into a few categories. A function of (Gr.Pr) was correlated with respect to the temperature difference between top and bottom surfaces. Thus, based on flow conditions, temperature differences between the top and bottom surfaces could be predicted. Since there were no similar data available in the literature at that time, Shitsman's relationship could not be examined against any data but his own. This correlation has not been referred to frequently in later works. It was well known that curved flows induced centripetal acceleration and affected the flow structure. Miropolski, Picus and Shitsman (1966) used a number of bent tubes with different configurations to study heat transfer to subcritical and supercritical water. The effect on the critical heat flux with curvature during boiling was studied. It was mentioned that due to thickening and thinning of the outer and inner curvature the heat flux was not held uniform. The different heat transfer coefficients at the outer and inner walls, however, were not solely because of a non-uniform heat flux. At subcritical pressure, the Nusselt correlation was modified to account for curved surfaces. The effect of centrifugal forces on a redistribution of layers of fluid with different density was used to explain the difference when compared to straight pipe flow. At supercritical pressure, however, the same line of reasoning was not followed. It was shown that there was a significant change of heat transfer at the outer and inner walls for curved tubes during 213 horizontal flows. The same conditions were repeated for vertical flow and local deterioration in heat transfer was observed. When compared with the case of subcritical water flow, the non-dimensional numbers describing free convection appeared to be two to three orders of magnitudes larger. Miropolski et al. concluded that buoyancy was most likely responsible for deteriorated heat transfer. Their correlation was modified to predict heat transfer at supercritical pressures for curved pipe flows. Further support for their findings was provided by a following, similar study by Miropolski and Picus (1967). 1967 Relation between Deteriorated Heat Transfer and Grashof Number Shitsman (1967) noticed the disparity in the literature between heat transfer to upward and downward flow of supercritical carbon dioxide in a round tube. For identical flow rates and heat fluxes, deteriorated heat transfer was only detected in upward flow. Shitsman attributed this effect to natural convection. He concluded that the direction of forces during upward flow dampened transverse turbulent activities. Consequently, the boundary layer flow became laminar in an otherwise turbulent flow. High wall temperatures occurred where the maximum thickness of the laminar boundary layer was reached. From that point on, the turbulent pulsation was probably resumed and heat transfer gradually improved along the tube. The empirical correlation of Miropolski and Shitsman (1957) was modified to include the Grashof number. This was one of the very first attempts to predict deteriorated heat transfer coefficients. The proposed correlation, which fitted Shitsman's results well, was not compared with other data. Vikhrev, Barulin and Konkov (1967) focused on deteriorated heat transfer during the vertical flow of supercritical water. Two different modes of deteriorated heat transfer were detected. One was associated with entry length effects. Temperatures were well below the pseudo-critical point. High heat flux and low flow rate, however, were required for deterioration to occur. A second type of deterioration occurred wherever pseudo-critical temperatures were reached. The latter, as reported in earlier works, was caused by dramatic property changes. Away from the deteriorated region, the data was satisfactorily predicted by the correlation of Miropolski and Shitsman (1957). A modified version of the Goldmann's (1954) model was developed by Tanaka, Nishiwaki and Hirata (1967). Cooling heat transfer was included. Predictions showed good agreement with their data during both heating and cooling. They also showed that 214 heat transfer coefficients were larger during cooling than heating. Based on the success of their model, Tanaka et al. rejected the necessity for a new type of classification (i.e., pseudo-boiling proposal) to describe heat transfer to supercritical fluids. They expressed the case simply as an extension of turbulent forced convection with very high variable fluid properties. 1967 First Measurements of Turbulence An alternative explanation was offered by Bourke and Denton (1967) in which the occurrence of maximum and minimum wall temperatures was discussed. They speculated that neither thickening of the thermal boundary layer nor reduction of conductivity across it were large enough to account for the temperature difference between wall and bulk. The core region, instead, was more likely the region where the changes took place to keep the heat flux constant from wall to the bulk. There was an interfacial, annular region of pseudo-phase change between dense, incompressible, liquid-like fluid in the core and a compressible vapor-like fluid at wall. Because of very high values of specific heat, this region acts as a heat sink. Also, the variation of properties was largest both axially and transversely within that annular region. Changes in wall temperature were explained in terms of the lifetime of eddies existing in the interface region. The lifetime and rate of expansion of eddies dictate the velocity and hence temperature fluctuations required for conservation of the angular momentum of eddies. Low and high heat flux cases corresponded to decaying and lasting eddies in the critical annular zone. The latter required more transfer of heat from the wall and hence a larger temperature differences between wall and bulk. More measurements on temperature and velocity profiles in the supercritical region, however, were needed to support Bourke and Denton proposal. Bourke et al. (1967) planned, for the first time, to measure turbulence using hot wire anemometery. This was performed during isothermal flow of carbon dioxide at supercritical pressures. Due to technical difficulties, however, measurements could not be applied to the flow of supercritical carbon dioxide with heat transfer. They did so, however, for air heated flow. Cold wire was used for the flow of turbulent air and radial velocity as well as temperature fluctuations were measured. Extension of measurements to supercritical carbon dioxide was not done. 215 In a review paper published by Hall, Jackson and Watson (1967), discrepancies between correlations and theoretical results developed so far were shown to be significant. The experimental data themselves did not agree with each other even for near similar test conditions. That was partly due to uncertainties about fluid properties. In analytical models, the major difference was the way in which turbulent diffusivity was modeled. The equations and simplifying assumptions were more or less the same. Pressure, heat flux and shear stress were considered to be constant across the flow. Hall et al. concluded that there must be some other influencing parameters, which were not fully accounted for. One example mentioned was buoyancy effects. They also presented the following explanation for heat transfer deterioration. As wall temperatures were just above the pseudo-critical point and bulk temperatures below it, the laminar boundary layer was considerably thicker, compared to the constant property situation. This was mainly due to the sudden decrease of density. The thermal resistance of the layer also increased. Thus, a larger temperature difference across the layer was expected. Once the laminar boundary layer grew further it converted to a turbulent layer and a normal heat transfer mode resumed. 1967 Flow Visualization in A Forced Convection Flow Sabersky and Hauptmann (1967) studied the flow of supercritical carbon dioxide over a flat plate. Following the bubble-like activity observed by Knapp and Sabersky (1966) in free convection heat transfer around a heated wire, Sabersky and Hauptmann were motivated to see if the same event could occur in a supercritical forced convection case. Different ranges of temperatures were tried. The case of Tb<Tp c<Tw which corresponded to enhanced heat transfer was particularly focused on. The results of flow visualization, however, did not demonstrate any significant difference between flows of enhanced and non-enhanced heat transfer. No resemblance to boiling heat transfer, i.e., bubbly flow, was detected. Furthermore, a limited number of measurements by a hot wire anemometer showed an increase in turbulence level as the wall temperature exceeded the pseudo-critical point. The observations made by Sabersky and Hauptmann were in favor of explaining the problem in terms of turbulent forced convection, rather than a pseudo-boiling proposal. 216 Water at supercritical pressure was of great interest to engineers (see for example Styrikovich, Margulova and Miropolski (1967) in which once-through boilers were discussed). However, the search for other fluids as potential supercritical heat transfer media was an ongoing task. A number of different fluids were examined by Kafengauz (1967), e.g., di-isopropyl-cyclo-hexane (DICH). He confirmed that pressure waves were detected at the critical region, which might violate thermodynamic equilibrium. Conversion of superheated liquid-like fluid to vapor-like fluid in the boundary layer and its collapse into cold nucleus was accompanied by onset of pressure waves. Phase transition (liquid-like to vapor-like) was possible when the temperature gradient was high. In two later studies with water, Kafengauz and Fedorov (1968a and b) experimentally showed that there existed a correlation between wall temperature and pressure oscillation in both turbulent subcritical and supercritical pressure flows. Pressure oscillations probably originated from shock waves due to the collapse of the steam bubbles, or vapor-like chunks in the case of supercritical pressure flow. The shock waves propagated in the flow at the speed of sound. Since the speed of sound was different in single phase and two-phase media, the reflection of waves from the tube ends formed standing waves. These standing waves enhanced bubble appearance and activities, or bubble-like activities in the case of supercritical pressure, and improved heat transfer. Shitsman (1968), in continuation of his previous study, reported that the sudden enhancement of the heat transfer coefficient seemed to be correlated with the viscosity ratio at the wall and bulk. This ratio for a temperature difference of 100 °C between wall and bulk was maximum first at a temperature range of 0-100 °C and then for 300-400 °C. Thus, near the pseudo-critical temperature was shown to be not the only region where enhancement of heat transfer was observed. He also demonstrated that for horizontal flows, the impairment of heat transfer occurred at relatively higher heat flux compared to vertical flows. The reason of deterioration in both cases was attributed to the effects of natural convection. Although heat transfer in the critical region was known to be a complex function of many parameters, Lokshin, Semenovker and Vikhrev (1968) proposed that the quantity q'VG is capable of representing different modes of heat transfer at supercritical pressure. They also introduced a correction factor, "A", to be the ratio of heat transfer in 217 the critical region to that of calculated by a conventional empirical correlation. Then a graph of "A" versus bulk enthalpy for different values of q'VG was generated. Various regimes of enhanced and deteriorated heat transfer could be distinguished from the graph. The heat transfer coefficient so obtained showed qualitative agreement with some experimental data. This study was probably among the very few attempts, which tried to systematically predict deteriorated heat transfer at supercritical pressures. 1968 Lack of Data for Horizontal Flow The unusual results of Schnurr (1968) who examined the flow of supercritical carbon dioxide in a horizontal tube raised some uncertainty about generalization of some earlier findings. He did not capture any sharp jump of heat transfer coefficient in the vicinity of the pseudo-critical temperature. He mentioned the small tube diameter (about 2.5 mm) to be the potential cause of the discrepancy. A small tube diameter conceivably reduced the chance of free convection. On the other hand, no study had shown that the presence of free convection was a requirement for enhancement of heat transfer. Even if so, the results of Schnurr detected an average of 20% better heat transfer at the bottom surface compared to the tube top surface, which was an indication of the existence of free convection. Furthermore, he showed that available correlations, such as the one derived from the work of Deissler, were adequate in predicting the heat transfer coefficient, provided that Reynolds and Prandtl numbers were estimated at appropriate temperatures. The study of Schnurr further demonstrated the lack of data for horizontal flows. More or less the same general conclusion was reached by Petukhov (1968). Heat transfer to a supercritical pressure fluid flow was mentioned as the most general problem of variable property heat transfer. The heat transfer and friction are functions of many arguments and thus generalization of experimental data is very difficult. Petukhov drew the attention to the fact that in many previous studies the results of the experiments under different conditions were plotted and compared on the same ground. This led to misleading conclusions. Inadequacy of accurate physical properties of fluids, in general, and transport properties in particular, created further discrepancy between theoretical calculations and exp