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Analysis of fluid circulation in a spherical cryogenic storage tank and conjugate heat transfer in a circular microtube

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Title:
Analysis of fluid circulation in a spherical cryogenic storage tank and conjugate heat transfer in a circular microtube
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English
Creator:
Rao, P. Sharath Chandra
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University of South Florida
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Tampa, Fla.
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Subjects / Keywords:
zero-boiloff
numerical simulation transient thermal management
cryocooler
electronic cooling
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Summary:
ABSTRACT: The study considered development of a finite-element numerical simulation model for the analysis of fluid flow and conjugate heat transfer in a zero boil-off (ZBO) cryogenic storage system. A spherical tank was considered for the investigation. The tank wall is made of aluminum and a multi-layered blanket of cryogenic insulation (MLI) has been attached on the top of the aluminum. The tank is connected to a cryocooler to dissipate the heat leak through the insulation and tank wall into the fluid within the tank. The cryocooler has not been modeled; only the flow in and out of the tank to the cryocooler system has been included. The primary emphasis of this research has been the fluid circulation within the tank for different fluid distribution scenario and for different level of gravity to simulate all the potential earth and space based applications. The steady-state velocity, temperature, and pressure distributions were calculated for different inlet positions, inlet velocities, and for different gravity values. The simulations were carried out for constant heat flux and constant wall temperature cases. It was observed that a good flow circulation could be obtained when the cold entering fluid was made to flow in radial direction and the inlet opening was placed close to the tank wall. The transient and steady state heat transfer for laminar flow inside a circular microtube within a rectangular substrate during start up of power has also been investigated. Silicon, Silicon Carbide and Stainless Steel were the substrates used and Water and FC-72 were the coolants employed. Equations governing the conservation of mass, momentum, and energy were solved in the fluid region. Within the solid wafer, the heat conduction was solved. The Reynolds number, Prandtl number, thermal conductivity ratio, and diameter ranges were: 1000--1900, 6.78--12.68, 27--2658, and 300 &micro m--1000 &micro m respectively. It was found that a higher aspect ratio or larger diameter tube and higher thermal conductivity ratio combination of substrate and coolant requires lesser amount of time to attain steady state. It was seen that enlarging the tube from 300 &micro m to 1000 &micro m results in lowering of the fluid mean temperature at the exit. Nusselt number decreased with time and finally reached the steady state condition. It was also found that a higher Prandtl number fluid attains higher maximum substrate temperature and Nusselt number. A correlation for peripheral average Nusselt number was developed by curve-fitting the computed results with an average error of 6.5%. This correlation will be very useful for the design of circular microtube heat exchangers.
Thesis:
Thesis (M.S.M.E.)--University of South Florida, 2004.
Bibliography:
Includes bibliographical references.
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by P. Sharath Chandra Rao.
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Title from PDF of title page.
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Document formatted into pages; contains 121 pages.

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University of South Florida Library
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University of South Florida
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aleph - 001478814
oclc - 56564328
notis - AJS2505
usfldc doi - E14-SFE0000461
usfldc handle - e14.461
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Analysis of Fluid Circulation in a Spherical Cryogenic Storage Tank and Conjugate Heat Transfer in a Circular Microtube by P. Sharath Chandra Rao A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Autar K. Kaw, Ph.D. Roger A. Crane, Ph.D. Date of Approval: July 8, 2004 Keywords: Cryocooler, Electronic cooling, Numerical simulation Transient thermal management, Zero-Boiloff Copyright 2004, P. Sharath Chandra Rao

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TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv LIST OF SYMBOLS x ABSTRACT xiii CHAPTER ONE: INTRODUCTION AND LITERATURE REVIEW 1.1 Introduction 1.2 ZBO Storage of Cryogens 1.3 Circular Microtubes 1.4 Objectives 1 1 1 7 13 CHAPTER TWO: ANALYSIS OF LIQUID NITROGEN FLOW IN A SPHERICAL TANK 2.1 Mathematical Model 2.2 Numerical Simulation 2.3 Results and Discussion 14 14 17 17 CHAPTER THREE: STEADY STATE CONJUGATE HEAT TRANSFER IN A CIRCULAR MICROTUBE INSIDE A RECTANGULAR SUBSTRATE 3.1 Mathematical Model 3.2 Numerical Simulation 3.3 Results and Discussion 35 35 38 39 CHAPTER FOUR: TRANSIENT CONJUGATE HEAT TRANSFER IN A CIRCULAR MICROTUBE INSIDE A RECTANGULAR SUBSTRATE 4.1 Mathematical Model 4.2 Results and Discussion 58 58 60 CHAPTER FIVE: CONCLUSIONS 5.1 Analysis of Cryogenic Storage 5.2 Steady State Analysis of Circular Microtube 5.3 Transient Analysis of Circular Microtube 80 80 81 82 i

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ii REFERENCES 83 APPENDICES Appendix A: Analysis of Liquid Nitroge n Flow in a Spherical Tank Appendix B: Steady State Conjugate H eat Transfer in a Circular Microtube in side a Rectangular Substrate Appendix C: Transient Conjugate Heat Transfer in a Circular Microtube Inside a Rectangular Substrate Appendix D: Thermodynamic Properties of Different Solids and Fluids Used in the Analysis 87 88 98 102 106

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LIST OF TABLES Table 2.1 Average outlet temperature of the fluid and maximum fluid temperature obtained for different positions of the inlet pipe (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 34 Table 3.1 Maximum temperature in the substrate, average heat transfer coefficient, and average Nusselt number for different tube diameters (Substrate = Silicon, Coolant = Water, q=300 kW/m 2 ) 57 Table 3.2 Maximum temperature in the substrate, average heat transfer coefficient, and average Nusselt number for different combinations of substrates and coolants (D=500m, Re=1500, q=40 kW/m 2 ) 57 Table A1 Thermodynamic properties of different solids 106 Table A2 Thermodynamic and transport properties of different fluids 106 iii

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LIST OF FIGURES Figure 2.1 Schematic of liquid nitrogen storage tank 15 Figure 2.2 Velocity vector plot for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 18 Figure 2.3 Stream line contour plot for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 19 Figure 2.4 Temperature contour plot for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 19 Figure 2.5 Temperature contour plot (within the tank) for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 20 Figure 2.6 Streamline contour plot for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 21 Figure 2.7 Temperature contour plot for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 21 Figure 2.8 Temperature contour plot (within the tank) for the tank with the inlet at the bottom (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 22 Figure 2.9 Streamline contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 23 iv

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Figure 2.10 Streamline contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0059 kg/s, g=0, q =3.75 W/m 2 ) 24 Figure 2.11 Temperature contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 25 Figure 2.12 Temperature contour plot (within the tank) of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 25 Figure 2.13 Streamline contour plot of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of the three openings = 0.005m, 0.0075m and 0.02m, Flow rate=0.0059 kg/s, g=0, q =3.75 W/m 2 ) 26 Figure 2.14 Temperature contour plot (within the tank) of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of the three openings = 0.005m, 0.0075m and 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 27 Figure 2.15 Streamline contour plot of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of all three openings = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 28 Figure 2.16 Temperature contour plot (within the tank) of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of all three openings = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 29 Figure 2.17 Streamline contour plot for the tank with the inlet extended 40% into the tank and radial discharge at 45 o from the axis (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 30 Figure 2.18 Temperature contour plot for the tank with the inlet extended 40% into the tank and radial discharge at 45 o from the axis (Diameter of the inlet = 0.02m, Width of the opening = 0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 30 v

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Figure 2.19 Streamline contour plot for the tank with the inlet extended 35% into the tank and radial discharge at 60 o from the axis (Diameter of the inlet = 0.02m, Width of the opening = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 32 Figure 2.20 Temperature contour plot (within the tank) for the tank with the inlet extended 35% into the tank and radial discharge at 60 o from the axis (Diameter of the inlet = 0.02m, Width of the opening = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 32 Figure 2.21 Streamline contour plot for the tank with radial flow in a Cchannel (Diameter of the inlet = 0.02m, Flow rate=0.0138 kg/s, g=0, q =3.75 W/m 2 ) 33 Figure 2.22 Temperature contour plot (within the tank) for the tank with radial flow in a C-channel (Diameter of the inlet = 0.02m, Flow rate=0.0138 kg/s, g=0, q =3.75 W/m 2 ) 33 Figure 3.1 Three dimensional view of a section of microtube heat sink 35 Figure 3.2 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different grid sizes (Substrate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 39 Figure 3.3 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.15, Re=1500) 41 Figure 3.4 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 42 Figure 3.5 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.5, Re=1500) 43 Figure 3.6 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 44 Figure 3.7 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=FC-72, =2658, =0.25, Re=1500) 45 vi

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vii Figure 3.8 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon Carbide, Coolant= FC-72, =2020, =0.25, Re=1500) 46 Figure 3.9 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water, =248) 47 Figure 3.10 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) 48 Figure 3.11 Variation of dimensionless local peripheral average interface heat flux along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water=248) 49 Figure 3.12 Variation of dimensionless local peripheral average interface heat flux along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) 50 Figure 3.13 Variation of Nusselt number along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water, =248) 51 Figure 3.14 Variation of Nusselt number along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) 51 Figure 3.15 Variation of Nusselt number along the length of the tube for different Graetz numbers (Substrate=Silicon, Coolant=Water, 0.15< < 0.5, =248) 52 Figure 3.16 Variation of Nusselt number along the length of the tube for different Graetz numbers (=0.25, Re=1500, 6.78 Pr 12.68, 27 2658) 53 Figure 3.17 Comparison of numerical to predicted Nusselt number based on equation (12) (1000 Re 1900, 6.78 Pr 12.68, 27 2658, 0 L 0.025 m, and 300 m D 1000 m) 54 Figure 3.18 Comparison of average Nusselt number with experimental and macro-scale correlations (Substrate = Stainless Steel, Coolant = Water, D=290 m, L=0.026 m, q=150 kW/m 2 ) 55

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Figure 4.1 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water=248, =0.15, =0.4, Re=1500) 62 Figure 4.2 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water=248, =0.5, =0.4, Re=1500) 62 Figure 4.3 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=FC-72=2658, =0.25, =0.4, Re=1500) 64 Figure 4.4 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water, =248, =0.15, =0.4, Re=1500) 65 Figure 4.5 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water, =248, =0.25, =0.4, Re=1500) 66 Figure 4.6 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=FC-72, =2658, =0.25, =0.4, Re=1500) 67 Figure 4.7 Variation of dimensionless local peripheral average interface temperature along the length of the tube at different time intervals (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 68 Figure 4.8 Variation of dimensionless local peripheral average interface temperature along the length of the tube at different time intervals (Substrate=Silicon Carbide, Coolant=FC-72, =2020, =0.25, Re=1500) 69 Figure 4.9 Variation of dimensionless transient fluid mean temperature at the exit for different inlet diameters (Substrate=Silicon, Coolant=Water, =248, Re=1500) 70 Figure 4.10 Variation of dimensionless transient fluid mean temperature at the exit for different combinations of substrates and coolants (=0.25, Re=1500) 71 Figure 4.11 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon, Coolant=Water, =248, =0.15, Re=1500) 72 viii

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ix Figure 4.12 Variation of Nusselt number alon g the length of the tube at different time intervals. (Subs trate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 73 Figure 4.13 Variation of Nusselt number alon g the length of the tube at different time intervals. (Subs trate=Silicon, Coolant=Water, =248, =0.5, Re=1500) 74 Figure 4.14 Variation of Nusselt number alon g the length of the tube at different time intervals. (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 75 Figure 4.15 Variation of Nusselt number alon g the length of the tube at different time intervals. (Substrate=Silicon Carbide, Coolant=FC72, =2020, =0.25, Re=1500) 75 Figure 4.16 Variation of average Nusselt number for different inlet diameters at different time intervals (Subs trate=Silicon, Coolant=Water, =248, Re=1500) 76 Figure 4.17 Variation of average Nusselt number for different combinations of substrates and coolants at different time intervals ( =0.25, Re=1500) 77 Figure 4.18 Variation of maximum substrate temperature for different inlet diameters at different time intervals (Substrate=Silicon, Coolant=Water, =248, Re=1500) 78 Figure 4.19 Variation of maximum substrate temperature for different combinations of substrates and coolants at different time intervals ( =0.25, Re=1500) 79

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LIST OF SYMBOLS B Half of tube spacing, m C Specific heat, J/kg-K D Tube diameter, m Fo Fourier number, f t/(D/2) 2 Gz Graetz number, RePr(D/L) H Height of the substrate, m k Thermal conductivity, W/m-K k Turbulent kinetic energy L Tube length, m Nu Local peripheral average Nusselt number, (q intf D)/(k f (T intf -T b )) Nu Local Nusselt number Nu avg verage Nusselt number for the entire tube p Pressure, N/m 2 P Dimensionless Pressure, (p-p o )/(v in 2 ) Pr Prandtl number q Heat flux, W/m 2 Q Dimensionless local peripheral average interface heat flux, q intf /q w r Radial coordinate, m x

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R Dimensionless radial coordinate, r/H Re Reynolds number, (v in D)/ T Temperature, o C v Velocity, m/s V Dimensionless velocity, v/v in W Water x Horizontal coordinate, m X Dimensionless horizontal coordinate, x/H y Vertical coordinate, m Y Dimensionless vertical coordinate, y/H z Axial coordinate, m Z Dimensionless axial coordinate, z/H Greek symbols Thermal diffusivity, k/( C p ), m 2 /s Aspect ratio, D/H Viscous dissipation rate of turbulent kinetic energy Thermal diffusivity ratio, s / f Angular direction max Maximum angular location from bottom to the top of the tube, 180 o Thermal conductivity ratio, k s /k f Kinematic viscosity, m 2 /s xi

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xii Dimensionless axial coordinate, z/L Density, kg/m 3 Dimensionless time, (t v in )/D Dimensionl ess temperature, ((T-T in ) k f )/(q w D) Dimensionless angular coordinate, max Subscripts b Bulk f Fluid in Inlet intf Interface k Turbulent kinetic energy m Mean max Maximum o Outlet r Radial direction s Solid t Turbulence w Bottom wall of wafer z Axial direction Viscous dissipation rate of turbulent kinetic energy Angular direction

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ANALYSIS OF FLUID CIRCULATION IN A SPHERICAL CRYOGENIC STORAGE TANK AND CONJUGATE HEAT TRANSFER IN A CIRCULAR MICROTUBE P. Sharath Chandra Rao ABSTRACT The study considered development of a finite-element numerical simulation model for the analysis of fluid flow and conjugate heat transfer in a zero boil-off (ZBO) cryogenic storage system. A spherical tank was considered for the investigation. The tank wall is made of aluminum and a multi-layered blanket of cryogenic insulation (MLI) has been attached on the top of the aluminum. The tank is connected to a cryocooler to dissipate the heat leak through the insulation and tank wall into the fluid within the tank. The cryocooler has not been modeled; only the flow in and out of the tank to the cryocooler system has been included. The primary emphasis of this research has been the fluid circulation within the tank for different fluid distribution scenario and for different level of gravity to simulate all the potential earth and space based applications. The steady-state velocity, temperature, and pressure distributions were calculated for different inlet positions, inlet velocities, and for different gravity values. The simulations were carried out for constant heat flux and constant wall temperature cases. It was observed that a good flow circulation could be obtained when the cold entering fluid was made to flow in radial direction and the inlet opening was placed close to the tank wall. xiii

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xiv The transient and steady state heat transf er for laminar flow inside a circular microtube within a rectangular substrate during start up of power has also been investigated. Silicon, Silicon Carbide and Stai nless Steel were the substrates used and Water and FC-72 were the coolants employed. Equations governing the conservation of mass, momentum, and energy were solved in th e fluid region. Within the solid wafer, the heat conduction was solved. The Reynolds number, Prandtl number, thermal conductivity ratio, and diameter ranges were: 1000, 6.78.68, 27, and 300 m m respectively. It was found that a higher aspect ratio or larger diameter tube and higher thermal conductivity ratio comb ination of substrate and cool ant requires lesser amount of time to attain steady state. It was seen th at enlarging the tube from 300 m to 1000 m results in lowering of the fluid mean temper ature at the exit. Nusselt number decreased with time and finally reached the steady state condition. It was also found that a higher Prandtl number fluid attains higher maximum substrate temperature and Nusselt number. A correlation for peripheral average Nusselt nu mber was developed by curve-fitting the computed results with an average error of 6.5%. This correlation will be very useful for the design of circular microtube heat exchangers.

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CHAPTER ONE INTRODUCTION AND LITERATURE REVIEW 1.1 Introduction The present study analyses fluid flow and heat transfer in two different fields: namely Cryogenic storage and Circular microtubes. Numerical simulations were performed to investigate the velocity and temperature distributions in the above mentioned problems by varying geometrical dimensions and flow parameters. Several interesting observations were made. A brief report on the type of work carried out and the corresponding results that were obtained is highlighted in the coming chapters. 1.2 ZBO Storage of Cryogens Liquid nitrogen finds its applications in super conductivity research, food refrigeration, genetic engineering, and space exploration. It is preferable to store nitrogen in liquid state because we gain very large amount of volume savings for the same mass of material stored. The development of a finite-element numerical simulation model for the analysis of fluid flow and conjugate heat transfer in a zero boil-off (ZBO) cryogenic storage system for liquid Nitrogen is the objective of the present investigation. An effective, affordable, and reliable storage of cryogenic fluid is essential for propellant and life support systems in space vehicles. The extension of the human exploration of space from low earth orbit (LEO) into the solar system is one of the 1

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NASAs challenges in the future. Without safe and efficient cryogenic storage, economically feasible long duration space missions will not be possible. The ZBO concept has recently evolved as an innovative means of storage tank pressure control, which reduces mass through a synergistic application of passive insulation, active heat removal, and forced liquid mixing. ZBO involves the use of a cryocooler/radiator system to intercept and reject cryogenic storage system heat leak such that boil-off and the necessity for venting are eliminated. A cryocooler (with a power supply, radiator, and controls) is integrated into a traditional orbital cryogenic storage subsystem to reject the storage system heat leak. With passive storage, the storage tank size and insulation weight increase with days in orbit, whereas in the ZBO storage system, mass remains constant. Literature Review on Cryogenic Storage Hastings et al. [1] made an effort to develop ZBO concepts for in-space storage of cryogenic propellants. Analytical modeling for the storage of 670 kgs of liquid hydrogen and 4000kg of LO 2 in low-earth orbit (LEO) was performed and it was observed that the ZBO system mass advantage, compared with passive storage begins at 60 days and 10 days for the LH 2 and LO 2 storage. Another important observation was that ZBO substantially adds operational flexibility as mission timelines can be extended in real time with no propellant losses. Haberbusch et al. [2] developed a thermally optimized in-space zero boil-off densified cryogen storage system model. The spherical liquid hydrogen tank model was used to investigate the effects of fluid storage temperature, multilayer insulation (MLI) 2

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thickness, and actively cooled shields on the overall storage system mass, cryocooler input power, and system volume. It was found that the storage of liquid hydrogen in a densified (subcooled) state resulted in significant system mass and volume advantages. Levenduski and Scarlotti [3] conducted a scalability study on Joule-Thomson cryocooler for space applications. The objective of their study was to (1) create a preliminary design for a J-T cryocooler that met an extreme set of cooling requirements, (2) determine any basic limitations of the J-T technology that would require enabling technologies to meet the new requirements, and (3) identify enhancing technologies that would improve system performance. Their study confirmed that the existing design was robust and could accommodate a wide range of heat loads. Aceves et al. [4] conducted analytical and experimental evaluations of commercially available aluminum-fiber insulated pressure vessels for cryogenic hydrogen storage. They found that: though the commercially available pressure vessels were not designed for operation at cryogenic temperature, no performance losses or significant damages occurred when these vessels were subjected to cryogenic temperatures and high pressures. Mueller and Durrant [5] presented an analysis of cryogenic liquefaction and storage methods for in-situ produced propellants (O 2 and CH 4 ) on Mars. They varied the insulation thickness and the cryocooler capacity to find optimum combinations for various insulation configurations, including multilayer insulation and microspheres. Their investigation showed that microsphere insulation is preferred for a human mission. Russo and Sugimura [6] validated a 65 K cryogenic system in zero-g space for focal planes, optics, instruments/other equipments viz. gamma-ray spectrometers and 3

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infrared imaging instruments that require continuous cryogenic cooling. These experiments were conducted in flight. The main flight experiment consisted of the following two on-orbit test sequences: (1) test of the cryogenic diode heat pipe and (2) test of Stirling-cycle, 2W, 65 K Improved Standard Spacecraft Cryocooler (ISSC). The results of the first test showed that the heat pipe can transport the cryocooler heat load with the overall temperature drop from condenser to evaporator limited to 3.08 K which was in agreement with the ground test results. The second test revealed that the ISSCs performance would not be affected throughout the flight experiment. No significant change in ISSCs performance was observed following its re-entry and Orbiter deintegration. Marquardt [7] analyzed cryocooler reliability issues for space applications. He demonstrated that the classical reliability analyses like statistical sampling and comparing failure modes couldnt be applied to cryocoolers. The statistical results for cryocoolers were not available as industries hadnt built many cryocoolers. It was also found that the comparison of failure modes of similar systems to that of the cryocooler was not possible as aerospace cryocooler was designed to have no failure modes. He concluded that the no-failure theory could not be guaranteed. Jun et al. [8] numerically investigated characteristics of boiling two-phase flow of liquid nitrogen inside a duct. They found that the phase change of liquid nitrogen occurs in quite a short time interval compared to two-phase pressurized water at high temperature. They also found that the boiling two-phase flow of liquid nitrogen showed a different flow structure when compared to the two-phase pressurized water at high temperature. This difference was attributed to the characteristic properties of two-phase 4

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cryogenic fluid flow, namely, rapid phase change velocity, large coefficient of compressibility, and low velocity of sound. Akyuzlu and Malipeddi [9] investigated laminar film boiling heat transfer from vertically suspended smooth surfaces in cryogenic fluids subjected to constant wall heat flux. They made comparisons between the numerical and experimental results. The mathematical model was described by conservation equations. The physical model comprised of a vertical plate suspended in liquid nitrogen with electric current as constant wall heat flux boundary condition. It was found that the mathematical model overestimated the velocities inside the vapor film and underestimated the vapor film thickness compared to the physical model. Boukeffa et al. [10] compared the experimental results concerning heat transfer between the vapor and the cryostat necks obtained for liquid nitrogen cryostat with numerical and theoretical results. They found a good agreement between the experimental and numerical results. The results also indicated that the theoretical model with an assumed perfect heat transfer between gas and solid was unable to describe the heat losses within the cryostat. Kamiya et al. [11] developed a large experimental apparatus to measure the thermal conductance of various insulations. Various specimens with allowable dimensions: diameter 1.2m and thickness up to 0.3m could be tested. The structural analysis of experimental apparatus was performed. The results of the deflection and stress of the vessel at room and the liquid nitrogen temperature were verified by the analytical models. In a later study, Kamiya et al. [12] measured the thermal conductance of different insulation structures for large mass LH 2 storage systems. The actual insulation 5

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structures comprised not only the insulation material but also reinforced members and joints. They tested two specimens, a vacuum multi-layer insulation with a glass fiber reinforced plastic (GFRP) and a vacuum solid insulation. Li et al. [13] investigated performances of non-loss storage for cryogenic liquefied gas. They found that the insulation performance and fraction of liquid volume were the main factors that affected the non-loss storage performance under the given pressure. They suggested that mechanical mix, thermal mix, insulation short, condensation of the vapor to transfer the heat and adding fin could be used to reduce or eliminate the temperature stratification of the liquid and increase the non-loss storage time. Kittel [14] made a study on the parasitic heat loads on the propellant and he proposed an alternative approach of using a re-liquefier to carry away the heat from the storage tank. He compared two schemes to remove the heat from the propellant. One scheme uses a sealed closed cycle cooler with a mixer. The mixer circulates propellant cooled by the refrigerator, isothermalizing the tank. The other scheme uses a cooler that uses the propellant vapor as its working fluid. He concluded that the first scheme offers advantages in efficiency and the ability to test the cooler before integration while the second scheme is simpler to integrate and provides an emergency vent route that intercepts the parasitic heat of the cooler. From the above literature review, it may be noted that storage of liquid nitrogen as well as other cryogenic fluids is needed for long-term space missions. Even though quite a few proof of concept studies have been done, a detailed simulation of fluid flow and heat transfer in cryogenic storage vessel has not been reported. 6

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1.3 Circular Microtubes The advent of microchannels has remarkably changed the outlook of Micro Electro Mechanical Systems (MEMS). Over the last twenty years several successful experiments and numerical investigations has led to exponential growth in this technology. As the need for chip reliability at elevated temperature increases so will the importance of microchannels be realized and utilized. As a matter of fact, the increase in power dissipation of electronic circuit has led to the usage of different geometries, different materials and different coolants as substrates and working fluids to effectively remove the heat. In this study we explore the steady state and transient analysis of fluid flow and heat transfer processes in circular microtubes embedded in a rectangular substrate. Literature Review (Steady State Analysis) Harms et al. [15] carried out experiments on single phase forced convection in deep rectangular micro channels. Two configurations were tested, a single channel system and a multiple channel system. The results showed that decreasing the channel width and increasing the channel depth provide better flow and heat transfer performance. The experimentally obtained local Nusselt number agreed reasonably well with classical developing channel flow theory. Ambatipudi and Rahman [16] studied heat transfer in a silicon substrate containing rectangular microchannels numerically. They found that a higher Nusselt number is obtained for a system with larger number of channels and higher Reynolds number. They demonstrated that for a given Reynolds number and channel width, the pressure drop is inversely proportional to the depth of the 7

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channel. They also observed that Nusselt number increased with channel depth, attained a peak, and then decreased with further increase of channel depth. Qu et al. [17] experimentally investigated heat transfer characteristics of water flowing through trapezoidal silicon microchannels with a hydraulic diameter from 62m to 169m. They also carried out numerical analysis. The results indicated that the experimentally determined Nu is much lower than that calculated from the numerical analysis. They attributed this to the effects of surface roughness of the microchannel walls. They also developed a relation which accounted for the roughness-viscosity effects and was used to interpret the experimental results. Federov and Viskanta [18] numerically studied the steady state three-dimensional heat transfer in an asymmetric rectangular channel having a laminar flow. Silicon was used as the substrate and water was the working fluid. A uniform heat flux of 90 W/cm 2 was imposed on one of the walls. They pointed out that extremely large temperature gradients occur within the solid walls in the immediate vicinity of the channel inlet, which has a potential for significant thermal stresses and structural failure of the heat sink. Lelea et al. [19] conducted experimental and numerical research on microtubes. The diameters were 0.1, 0.3, and 0.5 mm and the flow regime was laminar (Re = 95-774). The working fluid was distilled water and the tube material was stainless steel. The experimental results of flow and heat transfer characteristics confirmed that conventional/classical theories are applicable for water flow through microtubes of above size/range. Yu et al. [20] investigated fluid flow and heat transfer characteristics of dry nitrogen gas and water in microtubes, with diameters of 19, 52, and 102 micrometers for 8

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Re ranging from 250 to over 20000 and Pr ranging from 0.5 to 5. The range of laminar-turbulent transition zone in micro flow was found to lie between 2000
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Literature Review (Transient Analysis) Quadir et al. [27] used Galerkin finite element formulation to study the performance of a microchannel heat exchanger. The analysis was compared with the available experimental, analytical, and CFD results for the same channel geometry and fluid flow conditions. Their method predicted the surface temperature, fluid temperature, and the total thermal resistance of the heat sink satisfactorily. The method had an additional advantage of considering the non-uniform heat flux distribution as well. Toh et al. [28] investigated three-dimensional fluid flow and heat transfer in a microchannel. The effects of various parameters on the local thermal resistance, liquid mass flow-rate in the microchannels, heat flux through the heat sink and size of the heat sink were examined. The results obtained were compared with the experimental data of Tuckerman [29]. Ameel et al. [30] found an analytical solution to the laminar gas flow in microtubes with a constant heat flux boundary condition at the wall. The fluid was assumed to be hydrodynamically developed at the tube entrance. The Nusselt number was found to decrease with increasing Knudsen number. This was attributed to increase in the temperature jump at the wall with increasing Knudsen number. Also, the entrance length was found to vary with Knudsen number, with an increase in slip flow resulting in a longer entrance length. Shevade and Rahman [31] performed a transient analysis of fluid flow and heat transfer process in a rectangular channel, during the magnetic heating of the substrate material. Gadolinium was used as the substrate and water was the working fluid. They found that the peripheral average heat transfer coefficient and Nusselt number is large 10

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near the channel entrance and decreases as the flow proceeds towards exit. The results also showed that as the Reynolds number was increased, the outlet temperature decreased which in turn increased the average heat transfer coefficient. Rujano and Rahman [32] investigated the transient heat transfer for hydrodynamically and thermally developing laminar flow inside a trapezoidal microchannel heat sink. They conducted a systematic study to understand the effects of channel depth and width, Reynolds number, spacing between channels, and solid to fluid thermal conductivity ratio. The results showed that the time required for the heat transfer to reach steady state condition is longer for the system with larger channel depth or spacing and smaller channel width or Reynolds number. Quadir et al. [33] performed transient finite element analysis of microchannel heat exchangers in a generalized manner so that microchannel design wasnt restricted to a particular set of geometry and/or any specific operating conditions. The dimensions of Tuckerman and Pease [34] were employed and the analysis used water as the working fluid. The performance of the microchannel was obtained in terms of maximum temperature which was a function of several non-dimensional parameters chiefly Biot number, conductivity ratio, length to width, and length to height ratios. This was essentially done so that one could calculate the total thermal resistance. Jiang et al. [35] fabricated a microsystem consisting of a heater, microchannels, and temperature sensors. The transient temperature behavior of the device was experimentally studied for a variety of power dissipation levels and forced convection flow rates of de-ionized water. They found that the dry device heat-up time constant is 11

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longer than the cool-down time constant. It was observed that the forced convection leads to significantly lower operational temperature compared to dry device. Karimi and Culham [36] numerically studied the transient electro-osmotic pumping in rectangular microchannels. The numerical solutions showed significant influences of channel hydraulic diameter, aspect ratio, and applied voltage on the volumetric flow rates under transient and steady state conditions. They found that as the channel hydraulic diameter was increased, it took longer period for the flow to attain steady state. Brutin et al. [37] performed experiments to determine friction factor of laminar flow in microtubes using transient and steady-state methods. The friction factor obtained was slightly higher compared to the classical Poiseuille law. From the above literature review, it appears that past studies performed on circular microchannels were primarily experimental in nature. The studies focused mainly on comparisons with classical theories. In addition, the number of studies on circular microchannel has been very small compared to rectangular or trapezoidal microchannels. A comprehensive study dealing with the effects of all relevant geometric and flow parameters for conjugate heat transfer is currently not available in the literature. The present research explores numerical simulation model for fluid flow and heat transfer in a circular microtube. The tubes have been drilled in a rectangular block of wafer commonly used in the fabrication of microelectronics or bio-medical devices. A constant heat flux has been applied to one side of wafer to simulate heat generation due to microelectronics. The wafer is modeled taking into account heat generation in circuit components, conduction within the solid and convection of heat to the working fluid. 12

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13 Circular microtubes using Silicon, Silicon Carbid e, and Stainless Steel as wafer materials are considered in this study. Water and FC-72 are used as coolants. 1.4 Objectives The main objectives of the present investigation are: To develop a numerical model for fluid flow and heat transfer in cryogenic storage tank with constant heat flux applied on the tank wall. To investigate the geometric and flow parameters, optimizing the tank design for good fluid circulation and temperature uniformity within the tank. To develop a numerical model to highlight the steady state and transient responses of fluid flow and heat tran sfer processes in a wafer c ontaining integrated circuit devices and circular microtubes. To explore the effects of channel diam eter, solid and fluid properties, and Reynolds number on the fluid flow and heat transfer characteristics within the microtube.

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CHAPTER TWO ANALYSIS OF LIQUID NITROGEN FLOW IN A SPHERICAL TANK 2.1 Mathematical Model The storage tank for which the simulations are performed is represented schematically in Figure 2.1. The tank is similar to the one used for ZBO concept evaluation at NASA Glen Research Center (Hastings et al. [1]). The physical structure of the model comprises of a spherical body with openings at the top and bottom. A two-dimensional axi-symmetric jet enters the tank from the bottom and exits from the top. The diameter of the both inlet and outlet are: D = 0.20m. The diameter of the tank is: A = 1.4m. The tank wall is made of aluminum and is: C = 0.0127m thick. The tank is surrounded by an insulation of 0.1m thickness. Heat flux or temperature was applied at the outer wall. The working fluid in this problem is liquid nitrogen. Different ideas for channeling the flow in the tank were implemented. A steady fluid flow heat transfer model has been used to carry out the analysis. Assuming the fluid to be incompressible and Newtonian, the equations describing the conservation of mass, momentum, and energy in axi-symmetric cylindrical coordinates can be written as: [38] 14

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Figure 2.1 Schematic of liquid nitrogen storage tank 0zvrv r r 1zr (1) zvrvzrvrv2rrp1zvvrvrzrrtfrzrr v (2) 2z2zrtfzzzrzv2rvzvrrr1zp1gzvvrvv (3) 2f2f2f2ttfzfrzTrTr1rTPrzTvrT v (4) The effects of turbulence in the flow field were determined by using the kmodel. In this model, the turbulent kinetic energy and its dissipation rate were calculated by using the following equations. 15

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2zr2z2r2rt22tzrrrvzvzvrvrv2zrrrr1zvrvrvkkkkkk (5) kk222zr2z2r2rt122tzrrCrvzvzvrvrv2Czrrrr1zvrvrv (6) 2tCk (7) The empirical constants appearing in equations (5-7) are given the following values (Kays and Crawford [39]): C=0.09, C 1 =1.44, C 2 =1.92, k =1, =1.3, Pr t =1. The above values hold good for isothermal flows with no mass transfer. The present study involves both uniform heat flux and uniform temperature cases. The empirical constants which exclusively takes into consideration the above two cases have not been found. As the above-mentioned values have been optimized for adequate prediction of wide range of flows all the numerical simulations used these values. The equation used for the conservation of energy within the solid can be written as follows: [40] 0zT r T r 1 r T2s2s2s2 (8) The boundary conditions needed to solve the above equations included uniform axial velocity at the inlet, no slip condition at the solid-fluid interface and constant heat flux or constant temperature at the outer surface of the tank. 16

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2.2 Numerical Simulation The above governing equations along with the boundary conditions were solved using the finite-element method. The solid and fluid regions were both divided into a number of quadrilateral elements. After the Galerkin formulation was used to discretize the governing equations, the Newton-Raphson method was used to solve the ensuing algebraic equations. The finite element program called FIDAP was used for this computation. Convergence is based on two criteria being satisfied simultaneously. One criterion is the relative change in field values from one iteration to the next; the other is the residual for each conservation equation. In this problem a tolerance of 0.1 percent (or 0.001) for both convergence criteria was applied. 2.3 Results and Discussion The steady-state velocity, temperature, and pressure distributions were calculated for different inlet positions, inlet opening sizes, inlet velocities, and for different gravity values. The above simulations were carried out for constant heat flux and constant wall temperature cases. The simulations were performed so that a good flow circulation could be obtained. As the model was primarily built to study the heat transfer fluid flow characteristics in space, the steady state simulations were performed at g = 0 and g = 9.81 m/s 2 Figure 2.2 and Figure 2.3 show the velocity vector plot and streamline contour plots for the tank, which has inlet at the bottom. Figure 2.4 shows the temperature contour plot for the tank, which has inlet at the bottom. Different velocities of fluid flow have been simulated. As the fluid enters the tank it moves upward as a submerged jet and 17

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expands. Due to heat transfer, the temperature of the fluid near the wall increases and it rises upward as a wall plume due to buoyancy and this causes circulation in the tank. Finally, the fluid streams moving upward due to the buoyancy and that due to forced convection mixes and exits from the outlet at the top. It was observed that as the inlet velocity increases, the momentum of the incoming jet surpasses the buoyant force and that reduces circulation within the tank and more direct flow from inlet to outlet is seen. The temperature of the fluid decreases rapidly from the tank wall to the center of the tank which can be clearly made out from Figure 2.5. Figure 2.5 shows the temperature contour plot within the tank. Large amount of temperature reduction is seen in the insulation and this is because of much lower thermal conductivity of the insulation compared to the fluid or tank wall. Figure 2.2 Velocity vector plot for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 18

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Figure 2.3 Stream line contour plot for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) Figure 2.4 Temperature contour plot for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) 19

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Figure 2.5 Temperature contour plot (within the tank) for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=9.81 m/s 2 q =3.75 W/m 2 ) Simulations were also carried out at zero gravity condition. Figure 2.6 and Figure 2.7 show the streamline contour plot and the temperature contour plot for the tank, which has inlet at the bottom. There is no buoyancy force in this case as the gravity is zero. The circulation that is taking place in this situation is only because of the momentum, which is carried by the incoming fluid. The incoming fluid jet expands and impinges at the top wall of the tank. Then the fluid moves downward along the wall carrying heat with it. Figure 2.8 shows the temperature contour plot within the tank excluding tank wall and insulation regions. The hot and cold fluids mix at the bottom portion of the tank where more changes of temperature is seen in the temperature contour plot. The fluid circulates within the tank and exits from the outlet at the top. An almost linear variation in the pressure within the tank was observed from the inlet to the outlet. 20

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Figure 2.6 Streamline contour plot for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.7 Temperature contour plot for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 21

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Figure 2.8 Temperature contour plot (within the tank) for the tank with the inlet at the bottom (Diameter of the inlet=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figures 2.9 and Figure 2.10 shows the streamline contour in the tank when the inlet is extended axially to the center of the tank and the fluid is discharged radially from a single opening. The opening width is 0.01m and the flow rates are 0.0033kg/s and 0.0059 kg/s respectively. In both the cases the fluid moves towards the tank wall because of the momentum. When the fluid impinges the tank wall, some fluid moves down towards the bottom of the tank along the wall and some fluid moves toward the exit. The above phenomenon can be clearly observed in Figure 2.9. The fluid that has moved down towards the bottom makes a circulation in the lower portion of the tank. The fluid in the upper portion also makes a circulation and then mixes with the fluid coming from the lower portion and then exits from the outlet. The larger diameter opening allows more fluid to exit without proper mixing. It can also be observed that circulation is improved within the tank when compared to the previous design of inlet at the bottom because the 22

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fluid is first made to divide into parts and then circulate in each part before leaving the tank. The average temperature of the fluid within the tank was found to be 49.52 o C where as for the tank with the inlet at the bottom it was 46.29 o C. Thus it can be said that this approach reduces temperature non-uniformity in the fluid and attains better uniformity compared to the earlier case. Analogous to inlet at the bottom, an almost linear pressure variation was observed within the tank from the inlet to the outlet. Figure 2.9 Streamline contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet = 0.02m, Width of the opening=0.01 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.11 and Figure 2.12 respectively show temperature contour plots in the tank including insulation and tank wall and just for the fluid within the tank, when the inlet is extended axially to the center of the tank and the fluid is discharged radially from a single opening. The fluid takes the heat from the walls and exits via the outlet at a mean temperature of 44.8 o C. The plot also shows a large temperature drop within the 23

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insulation. It can also be observed from Figure 2.12 that a better circulation and mixing reduces temperature non-uniformity within the fluid. In the present case, an overall temperature difference of 20 o C was observed within the tank where as the tank with the inlet at the bottom had a temperature difference of 10 o C. Figure 2.10 Streamline contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet=0.02 m, Width of the opening=0.01 m, Flow rate=0.0059 kg/s, g=0, q =3.75 W/m 2 ) 24

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Figure 2.11 Temperature contour plot of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet=0.02 m, Width of the opening=0.01m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.12 Temperature contour plot (within the tank) of radial flow from a single opening for the tank with inlet pipe extended 50% into the tank (Diameter of the inlet=0.02 m, Width of the opening=0.01 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 25

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Figure 2.13 and Figure 2.14 show the streamline contour and temperature contour in the tank when the inlet pipe is extended axially into the tank and the fluid is discharged radially from three openings. The openings are placed at a distance of one-fourth, half and three-fourth the tank size. The widths of the openings are 0.005m, 0.0075m and 0.02m respectively. The smaller widths allow a constant fluid passage through all the openings. This allows the fluid to cover larger area. From the temperature contour it can be seen that the fluid temperature doesnt change in larger parts of the tank staying close to the fluid inlet temperature which highlights the fact that this kind of opening leads to less mixing of the fluid. The fluid closer to the wall attains higher temperature and leaves the tank. The Temperature distribution from the insulated wall to the center of the tank can be clearly observed in Figure 2.14. The maximum temperature difference within the tank was found to be 10 o C. Figure 2.13 Streamline contour plot of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of the three openings=0.005 m, 0.0075 m and 0.02 m, Flow rate=0.0059 kg/s, g=0, q =3.75 W/m 2 ) 26

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Figure 2.14 Temperature contour plot (within the tank) of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet = 0.02m, Width of the three openings=0.005 m, 0.0075 m and 0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.15 shows the streamline contour plot in the tank when the inlet pipe is extended axially into the tank and the fluid is discharged radially from three openings, each measuring 0.02m in width. The openings are placed at a distance of one-fourth, half and three-fourth the tank size. It was observed that most of the flow entered the tank from the first opening. The bigger opening allowed more fluid passage through it. Thus the second and third openings were not utilized effectively. A good circulation and mixing occurs in the bottom portion of the tank. Figure 2.16 shows the temperature contour plot within the tank for the above mentioned scenario. It can be seen that a large portion of the tank contains fluid at 45 o C. Thus the case with three openings of same size attains better temperature uniformity compared to three openings of different sizes. The average fluid 27

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temperature within the tank was found to be 43.85 o C where as the three openings of different sizes recorded 41.78 o C. Figure 2.15 Streamline contour plot of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet=0.02 m, Width of all three openings=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.17 and Figure 2.18 show the streamline contour plot and temperature contour plot for the tank which has the inlet extended axially about 40% into the tank and the fluid is discharged at an angle of 45 o to the axis. Better overall circulation was observed in this case. Various lengths of inclined pipe were tried and it was observed that as the pipe length decreased the fluid is discharged at an earlier stage in the tank thereby efficiently utilizing the tank volume. It can be seen that the bottom portion of the tank along the inclined pipe shows no considerable circulation. This can be avoided by using a smaller inclined pipe. The temperature contour shows a large drop within the insulation. The hottest region within the fluid is the layer which lies adjacent to the Aluminum-liquid 28

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nitrogen interface region. An almost linear pressure variation was observed within the tank from the inlet to the outlet. Figure 2.16 Temperature contour plot (within the tank) of radial flow from three openings for the tank with inlet pipe extended into the tank (Diameter of the inlet=0.02 m, Width of all three openings =0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.19 and Figure 2.20 show the streamline contour plot and temperature contour plot for the tank which has the inlet extended axially about 35% into the tank and the fluid is discharged at an angle of 60 o to the axis. As the jet of fluid was forced along the periphery of the tank wall good circulation was observed. Good mixing of hot fluid with the cold fluid can be observed in both the lower as well as upper portion of the tank. A uniform temperature distribution from the tank wall to the center of the tank was recorded. The 60 o discharge attained better heat transfer and fluid flow performance compared to the 45 o discharge. 29

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Figure 2.17 Streamline contour plot for the tank with the inlet extended 40% into the tank and radial discharge at 45 o from the axis (Diameter of the inlet=0.02 m, Width of the opening=0.01 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.18 Temperature contour plot for the tank with the inlet extended 40% into the tank and radial discharge at 45 o from the axis (Diameter of the inlet=0.02 m, Width of the opening=0.01 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 30

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A developed stage of the above mentioned channeling is the C-channel. In this case, the inlet extended along the circumference of the circular wall to a certain length. A very good amount of circulation is observed in this design. There are two circulations formed one right at the C-channel opening and the other at the exit. An efficient way to utilize the C-channel would be to increase the length of the channel along the elliptical wall; this forces more fluid to flow and circulate along the tank boundary all the way to the exit. Figure 2.21 and Figure 2.22 show the streamline contour and temperature distribution within the tank. The fluid that comes in contact with the tank wall gets heated up as it rises upward. Since the fluid is forced to flow along the tank wall large amount of fluid is heated in relatively small time unlike the other channeling designs. The temperature of the fluid decreases from the tank wall to the tank axis. A linear variation in the pressure distribution was observed within the tank from the inlet to the outlet. Table 2.1 shows the average outlet temperature of the fluid and the maximum temperature obtained for different positions of the inlet pipe. All the cases were subjected to the following conditions: diameter of the inlet = 0.02m, flow rate=0.0033 kg/s, g=0 and q =3.75 W/m 2 The maximum temperature was obtained adjacent to aluminum layer. When the fluid is discharged radially from an opening of diameter 0.01m it results in the attainment of the highest temperature. The lowest temperature is obtained in the case when the fluid is discharged radially from three openings of diameters 0.02m each. It can also be observed that the highest temperature case results in higher temperature non-uniformity in the fluid. 31

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Figure 2.19 Streamline contour plot for the tank with the inlet extended 35% into the tank and radial discharge at 60 o from the axis (Diameter of the inlet=0.02 m, Width of the opening=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.20 Temperature contour plot (within the tank) for the tank with the inlet extended 35% into the tank and radial discharge at 60 o from the axis (Diameter of the inlet=0.02 m, Width of the opening=0.02 m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) 32

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Figure 2.21 Streamline contour plot for the tank with radial flow in a C-channel (Diameter of the inlet=0.02 m, Flow rate=0.0138 kg/s, g=0, q =3.75 W/m 2 ) Figure 2.22 Temperature contour plot (within the tank) for the tank with radial flow in a C-channel (Diameter of the inlet=0.02 m, Flow rate=0.0138 kg/s, g=0, q =3.75 W/m 2 ) 33

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34 No clear trend was observed in the case of average outlet temper ature though inclination of the inlet at an angle of 45 o to the axis yielded the highest temperature. The average outlet temperature of the fluid flow ing through different inlets was 44 o C. Thus flow in a C-Channel and flow through openings of same diameters provides a better heat transfer from the tank wall to the cold fluid. Table 2.1 Average outlet temperature of th e fluid and maximum fluid temperature obtained for different positions of the inle t pipe (Diameter of the inlet = 0.02m, Flow rate=0.0033 kg/s, g=0, q =3.75 W/m 2 ) Sl. No Type of Opening (T avg ) out (T f ) max ( o C) 1 Inlet pipe extended axially a nd the fluid is discharged radially from an opening of diameter 0.01m 44.79 55.68 2 Inlet pipe extended axially about 40% into the tank and the fluid is discharged at an angle 45 o to the axis. 44.89 52.25 3 Inlet pipe extended axially about 35% into the tank and the fluid is discharged at an angle 60 o to the axis. 44.23 51.61 4 Inlet at the bottom of the tank. 44.12 49.30 5 Inlet pipe extended axially a nd the fluid is discharged radially from three openings of diameters 0.005m, 0.0075m, and 0.02m respectively and placed equi-distant from one another. 43.21 49.28 7 Radial flow of fluid in a C-Channel 44.04 46.3 8 Inlet pipe extended axially a nd the fluid is discharged radially from three openings of diameters 0.02m each placed equi-distant from one another 43.98 45.04

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CHAPTER THREE STEADY STATE CONJUGATE HEAT TRANSFER IN A CIRCULAR MICROTUBE INSIDE A RECTANGULAR SUBSTRATE 3.1 Mathematical Model The physical configuration of the system used in the present investigation is schematically shown in Figure 3.1. Because of the symmetry of the adjacent channels and uniform heat flux at the bottom, the analysis is performed by considering a cross-section of the heat sink containing half of distance between tubes in horizontal direction. It is assumed that the fluid enters the tube at a uniform velocity and temperature and hence the effects of inlet and outlet plenums are neglected. Figure 3.1 Three dimensional view of a section of microtube heat sink 35

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The differential equations were solved using dual coordinate systems. In the solid substrate a Cartesian coordinate system is used. In the case of fluid region, differential equations in cylindrical coordinate system were solved. The applicable differential equations in cylindrical coordinate system for the conservation of mass, momentum, and energy in the fluid region for incompressible flow are [38], 0ZzVVR1rVRrVR (1) 2Zr22R22r22R1RRVR1RVReRPZrzR2rVRRrVrVVVVVVVVV22rr2r2 (2) 2ZV2V2R22V22R212RVRVR12RV2PR1ZzVRVrVVRVRVrVReV (3) 2ZzV22zV22R21RzVR12RzV2ZPZzVzVzVRVRzVrVRe (4) 2Zf22f22R21RfR12Rf2ZfzVfRVRfrVPrRe (5) The equation for steady state heat conduction in solid region is [40], 02Zs22Ys22Xs2 (6) The following boundary conditions have been employed, ,2R0 0,At Z V r = 0, V = 0, V z = 1, f = 0 (7) ,2R0 ,HLAt Z 0Z 0,ZV ,0 Pfz (8) 36

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,HLZ0 ,21R2-1 ,HBXAt 0Xf,0XzV,0XrV0,V (9) ,HLZ0 ,HBX0 0,YAt 1Ys (10) ,HLZ0 ,2RAt RR ,fsfs (11) The remaining sides comprising the solid substrate were symmetric or insulated where the temperature gradient normal to the surface is zero. It can be observed that the non-dimensionalization of governing transport equations and boundary conditions were carried out using height of the substrate as the length scale and the inlet velocity as the velocity scale. All dimensionless groups have been defined in the Nomenclature section. The Reynolds number is the most important flow parameter in the governing equations. The transport properties give rise to two important dimensionless groups, namely, Prandtl number Pr and solid to fluid thermal conductivity ratio The important geometrical parameters are: L/H, B/H, channel aspect ratio and dimensionless axial coordinate The dependent variables selected to specify the results are the dimensionless temperature the dimensionless interfacial heat flux Q, and the Nusselt number Nu. 37

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3.2 Numerical Simulation The governing equations along with the boundary conditions (7-11) were solved using the Galerkin finite element method. Equations for solid and fluid phases were solved simultaneously as a single domain conjugate problem. Four-node quadrilateral elements were used. In each element, the velocity, pressure, and temperature fields were approximated which led to a set of equations that defined the continuum. The Newton-Raphson algorithm was used to solve the nonlinear system of discretized equations. An iterative procedure was used to arrive at the solution for the velocity and temperature fields. The solution was considered converged when the field values became constant and did not change from one iteration to the next. The distribution of cells in the computational domain was determined from a series of tests with different number of elements in the x, y, and z directions. The results obtained by using 8x48x40 (in the radial direction, number of cells, nr = 24) and 10x64x40 (nr = 32) captured most of the changes occurring in the system. The dimensionless local peripheral average interface temperature distribution as seen in Figure 3.2 was within 0.75% for the above two cases. Therefore, 8x48x40 elements in the x-, y-, and zcoordinate directions along with 24 cells in the radial direction (within the tube) was chosen for all numerical computations. 38

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00.020.040.060.080.10.120.140.160.180.200.20.40.60.81Dimensionless axial coordinate Dimensionless local peripheral average interface temperature, intf nx=4,ny=24,nz=30,nr=12 nx=6,ny=40,nz=40,nr=20 nx=8,ny=48,nz=40,nr=24 nx=10,ny=64,nz=40,nr=32 Figure 3.2 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different grid sizes (Substrate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 3.3 Results and Discussion A thorough investigation for velocity and temperature distribution was performed by varying the tube diameter and Reynolds number. Silicon (Si), Silicon Carbide (SiC), and Stainless Steel (SS) were the substrates and water and FC-72 were the working fluids. The length of the microtube was kept constant for all the configurations viz. 0.025 m. When water was used as the working fluid a constant heat flux of 300 kW/m 2 was applied to the bottom of the wafer. A constant heat flux measuring 40 kW/m 2 was applied when FC-72 was used. The fluid entered the tube at a uniform velocity and constant inlet temperature, T in = 20 o C. Interfacial temperature, interfacial heat flux, heat transfer 39

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coefficient, and Nusselt number were calculated at different sections along the length of the tube. The configuration was tested for diameters D: 300 m, 500 m, 1000 m and heat flux q: 40 kW/m 2 300 kW/m 2 The dimensions in Figure 3.1 are: B = 1000 m, H = 2000 m and L = 0.025 m. The local Nusselt number was calculated at locations = 0.1, 0.2, 0.4, 0.6, 0.9 and 1. Figures 3.3, 3.4, and 3.5 show the variation of local Nusselt number along the periphery of the tube diameter for the afore-mentioned locations for Silicon and water combination (= 248) for different aspect ratios: = 0.15, 0.25, and 0.5 respectively. The Reynolds number of the flow is 1500. At the inlet, as one moves along the periphery of the tube in the -direction a sinusoidal trend in the Nusselt number values is observed. As the fluid nears the exit the values vary over a much smaller range around the periphery of the tube. As the fluid moves from the inlet to the outlet the Nusselt number decreases along the tube length. During the transit the fluid absorbs heat all along its path. But the amount of heat absorbed decreases as the fluid moves downstream. This can be attributed to the development of thermal boundary layer along the tube wall. As the thickness of the boundary layer increases, the resistance to heat transfer from the wall to the fluid increases. Also, the rate at which the interfacial heat flux decreases along the length is slower when compared to the gain in fluid temperature. Hence at the exit the fluid attains the highest temperature and the lowest Nusselt number. The Nusselt number is higher for = 0.5 compared to that for = 0.25, and = 0.15. Since the Reynolds number is kept constant, the diameter of the larger tube results in higher value of Nusselt number. 40

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05101520253000.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Figure 3.3 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.15, Re=1500) Figures 3.6, 3.7, and 3.8 show the local Nusselt number variation along the periphery of the tube diameter at different sections along the tube length for three different combinations of substrates and working fluids. All the wafers were tested for = 0.25 and Re = 1500. The pattern/trend in variation of Nusselt number along the -direction is similar in both cases: same coolant flowing in different substrates, and different coolants flowing in a substrate. In all the cases, the fluid has a high Nusselt number at the entrance and at the exit the values stabilize and become fairly constant. Silicon has a higher thermal conductivity compared to Silicon Carbide and the thermal conductivity of water is ten times that of FC-72. Therefore, SiFC-72 (= 2658) 41

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combination attained higher Nusselt values compared to SiCFC-72 (= 2020), and SiCWater (= 189) combinations. 0510152025303500.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Figure 3.4 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 42

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010203040506000.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Figure 3.5 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=Water, =248, =0.5, Re=1500) Figure 3.9 shows the variation of dimensionless local peripheral average interface temperature along the tube length when Silicon is the substrate and water is the coolant. The flow has been tested for Re = 1000, 1500 and 1900. As the fluid enters the tube it tends to take away the heat from the tube walls. In the process it gets heated and leaves the tube at a higher temperature. As expected, the rise in temperature decreases with Reynolds number because a larger mass of fluid is available to carry the same amount of heat. As the flow rate decreases the fluid remains in contact with the solid for a longer duration thus attaining higher temperature. Hence the maximum outlet temperature is attained when = 0.15 and Re = 1000. The least temperature is obtained in the case of = 0.5 and Re = 1900. For a constant Re, tube with the bigger aspect ratio attains a 43

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lower interface temperature compared to the smaller ones. The higher mass flow rate in the larger tube allows greater mass of fluid to take the heat from the walls and hence at the exit the fluid passing though the larger diameter tube attains lower interface temperature compared to the smaller tube. 0510152025303500.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Figure 3.6 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 44

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0102030405060708000.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Figure 3.7 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon, Coolant=FC-72, =2658, =0.25, Re=1500) Figure 3.10 shows the variation of dimensionless local peripheral average interface temperature along the tube length for five different combinations of substrates and working fluids. The configurations have been tested for = 0.25 and Re = 1500. For a given substrate, FC-72 attains lower dimensionless interface temperature compared to water. It can be observed from the figure that SS-Water (= 27) and SiFC-72 (= 2658) obtained the highest and lowest dimensionless interface temperatures. A much larger heat transfer is realized when water is used as the working fluid, since its thermal conductivity is more than 10 times that of FC-72. As the dimensionless interface temperature is directly proportional to the product of temperature difference and thermal conductivity of the fluid, substrate with water as the coolant attains higher dimensionless 45

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interface temperature. The effect of the solid properties is found to be smaller compared to that of the fluid. As the value of increases, the range of variation of dimensionless interface temperature decreases. 01020304050607000.20.40.60.81Dimensionless angular coordinate, Loca l N usse l t num b er, N u Figure 3.8 Variation of local Nusselt number around the periphery of the tube at different sections (along the tube length) (Substrate=Silicon Carbide, Coolant= FC-72, =2020, =0.25, Re=1500) 46

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00.050.10.150.20.250.30.350.40.4500.20.40.60.8Dimensionless axial coordinate, 1 Dimensionless local peripheral average interface temperature, intf 0.15 (Re = 1000)0.5 (Re = 1900)0.5 (Re = 1500)0.5 (Re = 1000)0.15 (Re = 1900)0.15 (Re = 1500)0.25 (Re = 1000)0.25 (Re = 1900)0.25 (Re = 1500) Figure 3.9 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water, =248) Figure 3.11 shows the variation of dimensionless local peripheral average interface heat flux at different locations along the length of the tube for different inlet sizes for Silicon and water combination. Figure 3.12 shows the variation of dimensionless local peripheral average interface heat flux along the length of the tube for five combinations of substrates and coolants. At the entrance, the values of interface heat flux are higher because of the larger temperature difference between the solid and fluid. As the fluid nears the exit the temperature difference decreases and consequently the interface heat flux decreases. As the aspect ratio increases, the interfacial heat flux decreases along the tube length. This can be directly related to the inner surface area (or perimeter) of the tube that is available for convective heat transfer. It can be noted that 47

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interface heat flux does not change significantly with Reynolds number or properties of the fluid and solid. 00.050.10.150.200.20.40.60.81Dimensionless axial coordinate, Dimensionless local peripheral average interface temperature, intf 2658 (Pr = 12.68)27 (Pr = 6.78)189 (Pr = 6.78)2020 (Pr = 12.68)248 (Pr = 6.78) Figure 3.10 Variation of dimensionless local peripheral average interface temperature along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) Figure 3.13 shows the peripheral average Nusselt number distribution along the tube length for different tube diameters with Silicon and water combination. Figure 3.14 shows variation of peripheral average Nusselt number along the length of the tube for five combinations of substrates and coolants. The Nusselt number was calculated using peripheral average interface temperature and heat flux and fluid bulk temperature at that location. It can be observed that the Nusselt value is higher near the entrance and decreases downstream because of the development of a thermal boundary layer. As expected, Nusselt number value is very high close to the entrance and it approaches a 48

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constant asymptotic value as the flow attains the fully developed condition. As the tube diameter is increased, the thermal entrance length becomes larger. It is interesting to note that a fully developed condition is attained for smaller diameters, whereas for larger diameters, the Nusselt values keep decreasing all the way to the exit. Therefore the smaller diameter tube (= 0.15) attained Nu = 4.33 and larger tube (= 0.5) attained 0.511.522.500.20.40.60.81Dimensionless axial coordinate, Dimensionless peripheral average interface heat flux, Q 0.15 ( Re = 1900 ) 0.25 (Re = 1900)0.25 (Re = 1000)0.15 (Re = 1500)0.5 ( Re = 1000 ) 0.5 ( Re = 1500 ) 0.5 ( Re = 1900 ) 0.25 ( Re = 1500 ) 0.15 ( Re = 1000 ) Figure 3.11 Variation of dimensionless local peripheral average interface heat flux along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water=248) Nu = 9.84 at the exit. A significant variation in Nusselt number is observed along the length of the tube when aspect ratio is higher. This can be attributed to lesser substrate available between the heater and the coolant to smooth out the temperature distribution. When the ratio is small, conduction within the substrate results in more uniform distribution of solid-fluid interface temperature. Thus it can be seen that maximum heat 49

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transfer occurs for the tube with a larger diameter as it can carry larger mass of fluid. FC-72s lower thermal conductivity causes it to attain higher Nusselt numbers compared to water. The difference in the Nusselt numbers for a coolant flowing in two different substrates was not very significant. The lowest value of SSWater combination is one of the reasons for it to attain the lowest Nusselt number compared to other substratecoolant combinations. 00.20.40.60.811.21.41.600.20.40.60.81Dimensionless axial coordinate, Dimensionless peripheral average interface heat flux, Q 27 ( Pr = 6.78 ) 189 ( Pr = 6.78 ) 2020 ( Pr = 12.68 ) 248 ( Pr = 6.78 ) 2658 ( Pr = 12.68 ) Figure 3.12 Variation of dimensionless local peripheral average interface heat flux along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) 50

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051015202530354000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu 0.15 ( Re = 1000 ) 0.15 (Re = 1900)0.25 (Re = 1000)0.15 ( Re = 1500 ) 0.5 ( Re = 1900 ) 0.25 ( Re = 1900 ) 0.5 ( Re = 1500 ) 0.25 (Re = 1500)0.5 ( Re = 1000 ) Figure 3.13 Variation of Nusselt number along the length of the tube for different tube diameters (Substrate=Silicon, Coolant=Water, =248) 0510152025303540455055606500.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu 27 ( Pr = 6.78 ) 189 ( Pr = 6.78 ) 2658 ( Pr = 12.68 ) 248 ( Pr = 6.78 ) 2020 ( Pr = 12.68 ) Figure 3.14 Variation of Nusselt number along the length of the tube for different combinations of substrates and coolants (=0.25, Re=1500) 51

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Figure 3.15 shows the peripheral average Nusselt number distribution along the tube length for different Graetz numbers with Silicon and water combination. Figure 3.16 shows the variation of peripheral average Nusselt number along the tube length for different Graetz numbers for five combinations of substrates and coolants. Graetz number is the ratio of heat transferred by convection to the thermal capacity of the fluid. We can observe that fluid flowing through different opening diameters and at different velocities results in overlapping of the patterns. Similar observation can be made when two different fluids are made to flow in different substrates. Thus it can be concluded that different fluids flowing in different diameter tubes and at different velocities will always result in similar Nu vs (Gz) -1 profile. 051015202530354000.0020.0040.0060.0080.010.0120.014Inverse Graetz number, (Gz)-1Nusselt number, Nu 0.15 (Re = 1000)0.15 (Re = 1900)0.25 (Re = 1000)0.15 (Re = 1500)0.5 (Re = 1900)0.25 (Re = 1900)0.5 (Re = 1500)0.25 (Re = 1500)0.5 (Re = 1000) Figure 3.15 Variation of Nusselt number along the length of the tube for different Graetz numbers (Substrate=Silicon, Coolant=Water, 0.15< < 0.5, =248) 52

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010203040506000.0010.0020.0030.0040.005Inverse Graetz number, (Gz)-1Nusselt number, Nu 27 (Pr = 6.78)189 (Pr = 6.78)2658 (Pr = 12.68)248 (Pr = 6.78)2020 (Pr = 12.68) Figure 3.16 Variation of Nusselt number along the length of the tube for different Graetz numbers (=0.25, Re=1500, 6.78 Pr 12.68, 27 2658) The trends of the heat transfer enhancement with diameter, thermal conductivity ratio, Prandtl number and Reynolds number to accommodate most of the flow characteristics in the microtube was sought in the following form: Nu = (Re) 0.225 (Pr) 0.465 () 0.015 () -0.675 () 0.585 (12) Figure 3.17 gives a comparison of numerical Nusselt numbers to values of Nusselt number predicted by equation (12). An analysis of the errors between numerical and predicted values showed that, the differences between the two values are in the range: -22% to +6.9%. The mean value of the error is 6.5%. The range of validity of equation (12) is 1000 Re 1900, 6.78 Pr 12.68, 27 2658, 0 L 0.025 m, and 300 m D 1000 m. It can be noted from Figure 3.17 that a large number of data points are very well correlated with equation (12). The deviation is primarily in the entrance region 53

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where Nusselt number values are larger. So the correlation will have a higher level of confidence for the prediction of local peripheral average heat transfer coefficient for distances somewhat away from the inlet section where the flow and heat transfer is somewhat developed. This may not be a severe drawback since a microchannel heat exchanger is expected to have a much smaller developing length compared to conventional large size heat exchangers. 01020304050051015202530354045Predicted Nusselt numberNumerical Nusselt number 0.15 ( Re = 1500, Pr = 6.78, = 248 ) +6.9%-22%0.25 ( Re = 1500, Pr = 6.78, = 27 ) 0.25 ( Re = 1000, Pr = 6.78, = 248 ) 0.25 ( Re = 1500, Pr = 6.78, = 248 ) 0.25 ( Re = 1900, Pr = 6.78, = 248 ) 0.25 ( Re = 1500, Pr = 12.68, = 2658 ) 0.5 ( Re = 1500, Pr = 6.78, = 248 ) Figure 3.17 Comparison of numerical to predicted Nusselt number based on equation (12) (1000 Re 1900, 6.78 Pr 12.68, 27 2658, 0 L 0.025 m, and 300 m D 1000 m) Figure 3.18 shows the comparison of average Nusselt number with experimentally obtained Nusselt numbers by Bucci et al. [41] and other classical correlations developed for macro-scale channels (Hagen-Poiseuille [38], Sieder and Tate 54

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[42]). It can be noted that the numerically obtained Nusselt numbers are in reasonably good agreement with the experimentally obtained ones. The difference was within 3.4% at Re = 1500 and 1900 whereas a larger deviation of 15.4% is seen at Re =1000. The correlation of Hagen-Poiseuille, valid for thermally developing flow, under predicts the experimental data for the microtube as well as our numerical prediction by a very significant amount. Therefore, classical correlations for convection heat transfer may not be adequate for the prediction of conjugate heat transfer in micromechanical devices. 7891011121390011001300150017001900Reynolds number, ReAverage Nusselt number, Nuavg D = 0.029cm Bucci et al. [41] Hagen-Poiseuille [38] Sieder and Tate [42] Figure 3.18 Comparison of average Nusselt number with experimental and macro-scale correlations (Substrate = Stainless Steel, Coolant = Water, D=290 m, L=0.026 m, q=150 kW/m 2 ) Table 3.1 shows the maximum temperature in the substrate, average heat transfer coefficient, and average Nusselt number values for different inlet diameters and Reynolds numbers. In all the cases Silicon was the substrate and water was the working fluid. The 55

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maximum temperature occurs on the plane adjacent to the heater. When the tube diameter is large it holds a larger mass of fluid which in turn takes away the dissipated heat with a smaller rise in temperature. Thus a lower temperature at the solid-fluid interface results in a smaller maximum temperature in the substrate. As expected, a higher Reynolds number results in smaller maximum temperature in the substrate as well as higher values of average heat transfer coefficient for the heat exchanger. Table 3.2 shows the maximum temperature in the substrate, average heat transfer coefficient, and average Nusselt number values for five different combinations of substrates and coolants. It can be noted that lowest maximum temperature is achieved when Silicon is used as the substrate material and water is used as the coolant. The maximum temperature is a very useful parameter in the design of microelectronic devices which can be related to the reliability of the device. It can be also noticed that even though the Nusselt number is higher for FC-72, the average heat transfer coefficient for the heat exchanger is higher for water. Therefore, water can be a better coolant for microelectronics thermal management. 56

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57 Table 3.1 Maximum temperature in the substrat e, average heat transfer coefficient, and average Nusselt number for different tube diameters (Substrate = Silicon, Coolant = Water, q=300 kW/m 2 ) D (m) Re T s-max ( o C) Average h (W/m 2 o K) Average Nu 300 1000 83.81 19498.50 9.72 300 1500 75.45 20896.09 10.38 500 1900 71.66 22196.76 11.03 500 1000 71.55 14426.45 11.94 500 1500 66.10 15423.97 13 500 1900 62.45 16533.36 14 1000 1000 61.3 9429.60 15.61 1000 1500 55.86 10490.28 17.37 1000 1900 53.03 11406.65 18.89 Table 3.2 Maximum temperature in the substrat e, average heat transfer coefficient, and average Nusselt number for different combina tions of substrates and coolants (D=500m, Re=1500, q=40 kW/m 2 ) Substrate Coolant T s-max ( o C) Average h (W/m 2 o K) Average Nu Silicon Water 26.14 15382.26 12.73 Silicon Carbide Water 26.41 15133.07 12.53 Stainless Steel Water 29.40 13851.57 11.47 Silicon FC-72 46.92 2470.54 21.89 Silicon Carbide FC-72 50.81 2285.01 20.25

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CHAPTER FOUR TRANSIENT CONJUGATE HEAT TRANSFER IN A CIRCULAR MICROTUBE INSIDE A RECTANGULAR SUBSTRATE 4.1 Mathematical Model The physical configuration of the system used in the present investigation is schematically shown in Figure 3.1. Because of the symmetry of the adjacent channels and uniform heat flux at the bottom, the analysis is performed by considering a cross-section of the heat sink containing half of distance between tubes in the horizontal direction. It is assumed that the fluid enters the tube at a uniform velocity and temperature and hence the effects of inlet and outlet plenums are neglected. The differential equations were solved using dual coordinate systems. In the solid substrate a Cartesian coordinate system is used. In the fluid region, differential equations in cylindrical coordinate system were solved. The applicable differential equations in cylindrical coordinate system for the conservation of mass, momentum, and energy in the fluid region for incompressible flow are [38], 0ZzVVR1rVRrVR (1) 2Zr22R22r22R1RRVR1RVReRPZrzR2rVRRrVrVrV1VVVVVVVV22rr2r2 (2) 2ZV2rV2R22V22R212RVRVR12RV2PR1ZzVRVrVVRVRVrVReVV1 (3) 58

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2ZzV22zV22R21RzVR12RzV2ZPZzVzVzVRVRzVrVRezV1 (4) 2Zf22f22R21RfR12Rf2ZfzVfRVRfrVPrRef1 (5) The equation for steady state heat conduction in the solid region is [40], 2Zs22Ys22Xs2PrRe2s (6) The following initial condition and boundary conditions have been employed, 0,At 0 fs (7) ,2R0 0,At Z V r = 0, V = 0, V z = 1, f = 0 (8) ,2R0 ,HLAt Z 0Z 0,ZV ,0 Pfz (9) ,HLZ0 ,21R2-1 ,HBXAt 0Xf,0XzV,0XrV0,V (10) ,HLZ0 ,HBX0 0,YAt 1Ys (11) ,HLZ0 ,2RAt RR ,fsfs (12) The remaining sides comprising the solid substrate were symmetric or insulated where the temperature gradient normal to the surface is zero. It can be observed that the non-dimensionalization of governing transport equations and boundary conditions were carried out using height of the substrate as the 59

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length scale and the inlet velocity as the velocity scale. All dimensionless groups have been defined in the Nomenclature section. The Reynolds number is the most important flow parameter in the governing equations. The transport properties give rise to three important dimensionless groups, namely, Prandtl number Pr, solid to fluid thermal conductivity ratio and solid to fluid thermal diffusivity ratio The important geometrical parameters are: L/H, B/H, channel aspect ratio and dimensionless axial coordinate The dimensionless time has been defined with D/v in as the time scale. It can be related to Fourier number (Fo) as = Fo Re Pr. The dependent variables selected to specify the results are the dimensionless temperature the dimensionless interfacial heat flux Q, and the Nusselt number Nu. 4.2 Results and Discussion A thorough investigation for velocity and temperature distribution was performed by varying the tube diameter. Silicon (Si) and Silicon Carbide (SiC) were the substrates and water (W) and FC-72 (FC) were the working fluids. The length of the microtube is kept constant for all the configurations viz. 0.025 m. When water is used as the working fluid a constant heat flux of 300 kW/m 2 is applied to the bottom of the wafer. A constant heat flux measuring 40 kW/m 2 is applied when FC-72 is used. The fluid enters the tube at a uniform velocity and constant inlet temperature, T in = 20 o C. Interfacial temperature, interfacial heat flux, heat transfer coefficient, and Nusselt number were calculated at different sections along the length of the tube. The configuration was tested for diameters D: 300 m, 500 m, 1000 m and heat flux q: 40 kW/m 2 300 kW/m 2 60

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The dimensionless local interfacial heat flux was calculated for different time intervals. Figures 4.1 and 4.2 show the variation of dimensionless local interfacial heat flux along the periphery of the tube diameter at different time intervals for Silicon and water combination. The aspect ratios are: = 0.15 and 0.5 respectively. The Reynolds number of the flow is 1500. All the plots have been generated for = 0.4 section of the microtube. It can be observed that at all time intervals, the interface heat flux varies over a significant range and follows the same distribution pattern from the beginning of the transient to the final steady state. At smaller aspect ratio ( = 0.15), a higher heat flux is seen in the lower portion of the tube and somewhat lower heat flux in the upper portion of the tube. There is a gradual decrease of heat flux between = 0.4 to = 0.6. At a larger aspect ratio ( = 0.5), the maximum heat flux happens around = 0.4 and there is a steep decrease to the minimum around = 0.6. The large variation of heat flux around the tube periphery for this case is believed to be the result of smaller solid volume available for conduction and thermal energy storage that smooth out the temperature distribution at the tube periphery. From the initial stages until the heat transfer reaches the steady state, the difference in substrate and fluid temperature increases. Thus the highest interface heat flux is obtained when the fluid reaches the steady state. The fluid flowing in smaller diameter tube reaches a higher fluid and substrate temperature. The difference between the fluid and substrate temperature of = 0.15 microtube is 10 times that of = 0.5. Hence, a smaller diameter tube obtains higher interface heat flux. It was found that a smaller diameter tube ( = 0.15, t = 2.35s) takes a longer time to reach steady state compared to a larger tube ( = 0.5, t = 1.94s). 61

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0.511.5200.20.40.60.8Dimensionless angular coordinate, Dimensionless local interface heat flux 1 Q Steady State Fo = 8.5 Fo = 6.099 Fo = 3.698 Fo = 2.258 Fo = 1.391 Fo = 0.547 Figure 4.1 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water=248, =0.15, =0.4, Re=1500) 00.20.40.60.8100.20.40.60.81Dimensionless angular coordinate, Dimensionless local interface heat flux Q Steady State Fo = 0.279 Fo = 0.229 Fo = 0.122 Fo = 0.072 Fo = 0.029 Figure 4.2 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water=248, =0.5, =0.4, Re=1500) 62

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Figure 4.3 shows the variation of dimensionless local interfacial heat flux along the periphery of the tube diameter at different time intervals for SiFC ( = 2658) combination of substrate and coolant. The wafer was tested for: = 0.25 and Re = 1500. The plot has been generated for = 0.4 section of the microtube. The graph follows a similar trend as recorded in the previous cases. From the figure it can be observed that FC-72s lower thermal conductivity results in lower interface heat flux values compared to water (Figure 4.1 and Figure 4.2). It was also found that in the case of same coolant flowing in different substrates, the difference in the interface heat flux values is not very significant. SiCW ( = 189) combination attained higher interface heat flux compared to SiW ( = 248), SiCFC ( = 2020), SiFC ( = 2658) combinations. This can be attributed to the decreasing thermal conductivity ratios of the corresponding combinations. SiCFC combination takes a longer time (t = 10.46s) to reach the steady state and SiW combination takes the least time (t = 2.05s). SiFC combination attained steady state in 8.21s. 63

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00.20.40.60.811.21.41.61.800.20.40.60.81Dimensionless angular coordinate, Dimensionless local interface heat flux, Q Steady State Fo = 2.705 Fo = 2.137 Fo = 0.674 Fo = 0.312 Fo = 0.217 Figure 4.3 Variation of dimensionless local interface heat flux around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=FC-72=2658, =0.25, =0.4, Re=1500) Figures 4.4 and 4.5 show the variation of local Nusselt number along the periphery of the tube diameter for Silicon and water combination at different time intervals for = 0.15 and 0.25. The local Nusselt number was calculated at = 0.4. At all time intervals, as one moves along the periphery of the tube in the -direction, a sinusoidal trend in the Nusselt number values is observed. During the initial stage a significant variation in the Nusselt number is observed and as the fluid reaches the steady state the values vary over a much smaller range. As time progresses, the temperature of the fluid increases in the -direction at all sections of the tube unlike the interfacial heat flux which keeps varying all around the tube periphery and increases gradually. The rate at which the interfacial heat flux increases along the length is slower when compared to 64

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the gain in fluid temperature. Thus, as time increases a decreasing trend in Nusselt number values is recorded at all sections of the microtube. When the fluid reaches the steady state it attains the highest temperature and records the lowest average Nusselt number. The Nusselt number is higher for = 0.25 compared to = 0.15. Since the Reynolds number is kept constant, the diameter of the larger tube results in the higher Nusselt number. In addition, a larger diameter or consequently smaller solid volume between the heater and the fluid results in larger fluctuation of Nusselt number around the tube periphery during the entire transient process. 0510152025303500.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Fo = 0.547 Fo = 0.6762 Fo = 0.8372 Fo = 1.391 Fo = 2.258 Fo = 6.099 Fo = 8.5 Steady State Figure 4.4 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water, =248, =0.15, =0.4, Re=1500) 65

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05101520253000.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Fo = 0.191 Fo = 0.308 Fo = 0.62 Fo = 1.997 Fo = 2.317 Steady State Figure 4.5 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=Water, =248, =0.25, =0.4, Re=1500) Figure 4.6 shows the local Nusselt number variation along the periphery of the tube diameter for SiFC combination of substrate and working fluid at different time intervals (at = 0.4). The wafer was tested for: = 0.25 and Re = 1500. The pattern/trend in variation of Nusselt number along the -direction is similar in both cases: same coolant flowing in different substrates and different coolants flowing in a substrate. In all the cases, it starts with a high Nusselt number and this value keeps decreasing with time and as the flow reaches the steady state the values vary over a much smaller range. In the case of different coolants flowing in the same substrate (Figure 4.5 and Figure 4.6), SiFC combination attains higher Nusselt values compared to SiW. The lower thermal conductivity of FC-72 compared to water results in the higher Nusselt number of the SiFC combination. The difference in the Nusselt numbers at any given time was not found 66

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to be very significant in the case of same coolant flowing in different substrates. Higher values yield higher Nusselt numbers at all sections of the tube and at all times. Thus the highest Nusselt values are obtained by the combinations in the following order: SiFC > SiCFC > SiW > SiCW. 010203040506000.20.40.60.81Dimensionless angular coordinate, Local Nusselt number, Nu Fo = 0.217 Fo = 0.312 Fo = 0.674 Fo = 2.137 Fo = 2.705 Steady State Figure 4.6 Variation of local Nusselt number around the periphery of the tube at different time intervals (Substrate=Silicon, Coolant=FC-72, =2658, =0.25, =0.4, Re=1500) Figures 4.7 and 4.8 show the variation of dimensionless local peripheral average interface temperature along the tube length for SiCW and SiCFC combinations at different time intervals. As the fluid enters the tube it tends to take away heat from the tube wall. In the process it gets heated and leaves the tube at a higher temperature. The fluid tends to take heat from the walls at all times. In addition, thermal energy is stored in both the solid and the fluid until the steady state is reached. It may be noticed that during the earlier part of the transient, the dimensionless interface temperature increases almost 67

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uniformly along the entire length of the tube; during the later part of the transient, the temperature increases are larger at larger locations. Because of the smaller thermal diffusivity of the fluid, the thickness of the thermal boundary layer remains very thin and uniform during the earlier part of the transient. As Fourier number increases, the boundary layer thickens and approaches the steady characteristics of zero thickness at the leading edge and gradually increases with distance from the leading edge over the entire length of the developing flow region. For a given substrate, FC-72 attains lower interface temperature compared to water. As the thermal capacity and conductivity of water are higher than FC-72, it results in higher interface temperature attainment. When water/FC-72 was made to flow in different substrates the variation in interface temperature was not as significant as observed in the earlier case. 00.020.040.060.080.10.120.140.160.180.200.20.40.60.81Dimensionless axial coordinate, Dimensionless local peripheral average interface temperature, intf Steady State Fo = 3.385 Fo = 2.12 Fo = 1.621 Fo = 1.175 Fo =0.464 Fo = 0.191 Figure 4.7 Variation of dimensionless local peripheral average interface temperature along the length of the tube at different time intervals (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 68

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SiCW (= 189) and SiFC-72 (= 2658) obtained the highest and lowest dimensionless interface temperatures. When FC-72 is used as the fluid, an order of magnitude higher temperature difference is seen at the interface. A much larger heat transfer is realized when water is used as the working fluid, since its thermal conductivity is more than 10 times that of FC-72. As the dimensionless interface temperature is directly proportional to the product of temperature difference and thermal conductivity of the fluid, substrate with water as the coolant attains higher dimensionless interface temperature. The effect of the solid properties is found to be smaller compared to that of the fluid. It was also found that: For a constant Re, tube with the bigger attains lower interface temperature compared to the smaller ones. 00.010.020.030.040.050.060.070.08 0 09 00.20.40.60.8Dimensionless axial coordinate, 1 Dimensionless local peripheral average interface temperature, intf Steady State Fo = 4.17 Fo = 2.362 Fo = 1.762 Fo = 1.182 Fo = 0.674 Fo = 0.384 Figure 4.8 Variation of dimensionless local peripheral average interface temperature along the length of the tube at different time intervals (Substrate=Silicon Carbide, Coolant=FC-72, =2020, =0.25, Re=1500) 69

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The fluid mean temperature at the exit has been used to present the transient response of the substrate and coolant combinations. As the temperature at the inlet remains constant, ( m ) o essentially represents the total heat transfer rate in the microtube for any given mass flow rate. Figure 4.9 shows the variation of dimensionless fluid mean temperature at the exit with Fourier number for different inlet diameters of Silicon and water combination. All the configurations were tested for Re = 1500. It may be noticed that enlarging the tube from 300 m (= 0.15) to 1000 m (= 0.5) leads to lowering of the fluid mean temperature at the exit. This can be understood by recognizing that the energy storage capacity of the system becomes higher as the diameter is increased, which reduces the fluid temperature, when part of the Silicon (Cp = 715 J/kg-K) is substituted by water (Cp = 4179 J/kg-K) having higher thermal capacity. 00.050.10.150.2024681012 1 4 Fourier number, FoDimensionless fluid mean temperature at the exit, (m)o 0.250.5 0.15 Figure 4.9 Variation of dimensionless transient fluid mean temperature at the exit for different inlet diameters (Substrate=Silicon, Coolant=Water, =248, Re=1500) 70

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Figure 4.10 shows the variation of dimensionless fluid mean temperature at the exit with Fourier number for four combinations of substrates and coolants. All the configurations were tested for = 0.25 and Re = 1500. In the case of two different coolants flowing in a substrate: FC-72s lower thermal conductivity results in lower dimensionless exit temperature. In the case of same coolant flowing in two different substrates: higher combination yields higher dimensionless fluid temperature at the exit. Though, SiCW/SiCFC-72 attains higher fluid temperature at the exit during later part of the transient compared to SiW/SiFC-72 combinations, they require more time to attain the steady state condition. 00.020.040.060.080.1012345 6 Fourier number, FoDimensionless fluid mean temperature at the exit, (m)o 2658 (Pr = 12.68)2020 (Pr = 12.68)248 (Pr = 6.78)189 (Pr = 6.78) Figure 4.10 Variation of dimensionless transient fluid mean temperature at the exit for different combinations of substrates and coolants (=0.25, Re=1500) Figures 4.11, 4.12, and 4.13 show the peripheral average Nusselt number distribution along tube length for different tube sizes with Silicon and water combination 71

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at different time intervals. The Nusselt number was calculated using peripheral average interface temperature, peripheral average heat flux, and fluid bulk temperature at that location. As the fluid moves from the inlet to the outlet the Nusselt number decreases along the tube length. During the transit the fluid absorbs heat all along its path. But the amount of heat absorbed, decreases all along the tube length. This can be attributed to the development of thermal boundary layer along the tube wall. As the thickness of the boundary layer increases, the resistance to heat transfer from the wall to the fluid increases. As expected, Nusselt number is very high near the entrance, and it approaches a constant asymptotic value as the flow approaches the fully developed condition. It can also be seen that Nusselt number decreases with time because of the increment of thermal boundary layer thickness with time as more heat is transmitted from the solid wall to the fluid. 0102030405060708000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu Fo = 0.547 Fo = 0.676 Fo = 1.391 Fo = 2.258 Fo = 3.698 Fo = 6.099 Fo = 8.5 Figure 4.11 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon, Coolant=Water, =248, =0.15, Re=1500) 72

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As the flow reaches the steady state a very small variation in the Nusselt number values is recorded. It was also found that: as the tube diameter is increased, the thermal entrance length becomes larger which leads to higher Nusselt values. It is interesting to note that a fully developed condition is approached for smaller diameters, whereas for larger diameters, the Nusselt number keeps decreasing all the way to the exit. Therefore the smaller diameter tube (= 0.15) attained Nu = 4.33 and larger tube (= 0.5) attained Nu = 9.84 at the steady state. It can also be observed that a larger diameter tube results in better thermal transport as it can carry larger mass of fluid than a smaller diameter tube for a given Reynolds number. 0102030405060708000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu Fo = 0.191 Fo = 0.308 Fo = 0.62 Fo = 1.186 Fo = 1.997 Fo = 2.317 Figure 4.12 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon, Coolant=Water, =248, =0.25, Re=1500) 73

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01020304050607000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu Fo = 0.029 Fo = 0.046 Fo = 0.072 Fo = 0.122 Fo = 0.229 Fo = 0.279 Figure 4.13 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon, Coolant=Water, =248, =0.5, Re=1500) Figures 4.14 and 4.15 show the variation of peripheral average Nusselt number along the length of the tube for SiCW and SiCFC combinations at different time intervals. It can be observed that the trend is similar to the earlier ones. The lower thermal conductivity of FC-72 is the main reason for it to attain higher Nusselt number. It was observed that lower combination attained fully developed state and yielded a smaller entrance length compared to higher combinations. In the case of same coolant flowing in different substrates: no significant variation in Nusselt numbers were recorded. SiCW ( = 189) and Si-FC ( = 2658) combinations attained the lowest and highest Nusselt numbers during the entire transient period. 74

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0510152025303540455000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu Fo = 0.191 Fo =0.464 Fo = 1.621 Fo = 2.12 Fo = 2.704 Fo = 3.385 Figure 4.14 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon Carbide, Coolant=Water, =189, =0.25, Re=1500) 010203040506070809000.20.40.60.81Dimensionless axial coordinate, Nusselt number, Nu Fo = 0.384 Fo = 0.674 Fo = 1.182 Fo = 2.362 Fo = 3.033 Fo = 4.17 Figure 4.15 Variation of Nusselt number along the length of the tube at different time intervals. (Substrate=Silicon Carbide, Coolant=FC-72, =2020, =0.25, Re=1500) 75

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Figure 4.16 shows the variation of average Nusselt number for different inlet diameters for Silicon and water combination at different time intervals. It can be seen that though smaller diameter tubes take a longer time to attain steady state they do attain fully developed state unlike the larger diameter tubes which take lesser time and have bigger thermal entrance lengths. The average Nusselt number decreases rapidly in the earlier part of the transient and only gradually as the heat transfer approaches the steady state condition. The figure also makes a comparison with experimentally obtained Nusselt number by Bucci et al. [41]. Bucci et al. [41] conducted a steady state analysis of flow inside a D = 290m microtube. Stainless Steel was the substrate and water was the coolant. It can be noted that the numerically obtained Nusselt number is in reasonably good agreement with the experimentally obtained one. The difference was within 3.1%. 5101520253035404550012345678Fourier number, FoAverage Nusselt number, Nuavg 9 Bucci et al. [41]0.5 0.250.15 Figure 4.16 Variation of average Nusselt number for different inlet diameters at different time intervals (Substrate=Silicon, Coolant=Water, =248, Re=1500) 76

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Figure 4.17 shows the variation of average Nusselt number at different time intervals for four combinations of substrates and coolants. It can be observed from the figure that higher Prandtl number fluids attain higher Nusselt numbers compared to lower ones. It can also be seen that for a given Prandtl number, a lower value of results in lesser time to attain steady state. 010203040506000.511.522.533.544 5 Fourier number, FoAverage Nusselt number, Nuavg 2658 (Pr = 12.68)2020 (Pr = 12.68)248 ( Pr = 6.78 ) 189 ( Pr = 6.78 ) Figure 4.17 Variation of average Nusselt number for different combinations of substrates and coolants at different time intervals (=0.25, Re=1500) Figure 4.18 shows the variation of maximum substrate temperature for different inlet diameters of Silicon and water combination at different time intervals. The trend is very similar to the one found in Figure 4.9. The maximum temperature occurs on the plane adjacent to the heater. The fluid flowing in smaller diameter tube attains higher maximum substrate temperature compared to the larger diameter tubes because the bigger diameter tube has larger volume of fluid (against the volume of the substrate) to take the 77

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heat from the walls of the substrate. The magnitude of this temperature is important for the design of cooling systems for microelectronics. 152025303540455055606570758000.511.522. 5 Time, t (s)Maximum substrate temperature, (Ts)max (oC) 0.250.5 0.15 Figure 4.18 Variation of maximum substrate temperature for different inlet diameters at different time intervals (Substrate=Silicon, Coolant=Water, =248, Re=1500) Figure 4.19 shows the variation of maximum substrate temperature at different time intervals for four combinations of substrates and coolants. It can be observed that higher Prandtl number fluids attain higher maximum substrate temperature. It can also be seen that there is a large variation in the maximum substrate temperature for different fluids flowing in the same substrate. Therefore, the selection of coolant is very important for the design of thermal management systems. It can be also noted that for a given coolant, Si provides higher maximum temperature in the earlier part of the transient, but lower maximum temperature in the later part of the transient when compared to SiC. This 78

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is due to the difference in thermal storage capacity of the two materials. The magnitude for Cp is 1654.3 kJ/m 3 -K for Si, whereas 2259.4 kJ/m 3 -K for SiC. 1520253035404550550246810Time, t (s)Maximum substrate temperature, (Ts)max (oC) 12 2658 (Pr = 12.68)2020 (Pr = 12.68)248 (Pr = 6.78)189 (Pr = 6.78) Figure 4.19 Variation of maximum substrate temperature for different combinations of substrates and coolants at different time intervals (=0.25, Re=1500) 79

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CHAPTER FIVE CONCLUSIONS 5.1 Analysis of Cryogenic Storage The conclusions gathered from the results of this investigation can be summarized as follows: The incoming fluid from the cryo-cooler penetrates the fluid in the tank as a submerged jet and diffuses into the fluid medium as it loses its momentum. When the gravity is present, the fluid adjacent to the wall rises upward due to buoyancy and also mixes with the colder fluid due to the forced circulation. In the absence of gravity, the incoming fluid jet expands and impinges on the wall of the tank and then the fluid moves downward along the tank wall and carries heat with it. The mixing of hot and cold fluids takes place at the bottom portion of the tank. The temperature of the fluid is highest at the wall and it decreases rapidly towards the axis of the tank. The discharge of the incoming fluid from the cryo-cooler at several locations and/or at an angle to the axis results in better mixing compared to single inlet at the bottom of the tank. The inlet pipe through which the fluid is discharged radially from a single opening attained the maximum fluid temperature. The C-channel geometry and flow through openings of same diameters proposed in this study provides a better heat transfer from the tank wall to the cold fluid. 80

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5.2 Steady State Analysis of Circular Microtube The numerical simulation for conjugate heat transfer in microtubes was performed by varying the aspect ratios and allowing different flow rates through the tube. The configuration was also tested for different combinations of substrate and coolant. The local distribution of Nusselt number around the tube diameter was obtained at different sections along the tube length. The highest interface temperature is obtained in the case of smaller aspect ratio and lower Reynolds number. For a constant Re, tube with the bigger aspect ratio attains a lower interface temperature. The Nusselt number is large near the entrance because of the development of the thermal boundary layer, and it approaches a constant asymptotic value as the flow approaches a fully developed condition. The range of variation of Nusselt number along the length of the tube is more for larger inlet diameter as lesser substrate is available between the heater and the coolant to smooth out the temperature distribution. The peripheral average interface temperature decreased and Nusselt number increased with increase of Reynolds number, Prandtl number, solid to fluid thermal conductivity ratio, and tube diameter to wafer thickness ratio. A correlation to accommodate the heat transfer characteristics of the fluid flow within the microtube was developed. The differences between the numerical and predicted Nusselt number values using equation (12) are in the range: -22% to +6.9%. The numerically obtained Nusselt numbers are higher than those predicted by the Hagen-Poiseuille, Sieder and Tate correlations. But they are in reasonably good agreement with the experimentally obtained Nusselt numbers for micro tube. The maximum temperature of the substrate and the outlet temperature of the fluid decreases as the Reynolds number increases. 81

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82 5.3 Transient Analysis of Circular Microtube The numerical investigation for transient conjugate heat transfer in microtubes was performed by varying the geometric dimens ions and for different combinations of substrates and coolants. The distribution of local dimensionless interfacial heat flux and local Nusselt number around the tube diameter was obtained at different time intervals. For a constant Re, tube with the larger diam eter attains a lower interface temperature. The Nusselt number is larger near the entrance because of the development of the thermal boundary layer and they approach a constant as ymptotic value as the flow reaches a fully developed condition. For a constant Reynolds number the follo wing specific conclusions can be made: 1) A larger aspect ratio ( ) tube requires lesser amount of time to attain steady state. 2) During the earlier part of the transient, the dimensionless interface temperature increases almost uniformly along the entire le ngth of the tube; during the later part of the transient, the temperature increases are larger at larger locations. 3) The dimensionless interface heat flux in creases with time and attains the maximum at the steady state. 4) At all locations, Nusselt number decrea ses with time and approaches the minimum at the steady state condition. 5) Enlarging the tube from 300 m ( = 0.15) to 1000 m ( = 0.5) results in lowering of the fluid mean temperature at the exit and increasing the Nusselt number. 6) A higher Prandtl number fluid attains hi gher maximum substrate temperature as well as Nusselt number.

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REFERENCES 83 [1] L. J. Hastings, D. W. Plachta, L. Salerno, and P. Kittel, An overview of NASA efforts on zero boiloff storage of cryogenic propellants, Cryogenics, vol. 41, pp. 833-839, 2002. [2] M. S. Haberbusch, R. J. Stochl, and A. J. Culler, Thermally optimized zero boiloff densified cryogen storage system for space, Cryogenics, vol. 44, pp. 485-491, 2004. [3] R. Levenduski and R. Scarlotti, Joule-Thomson cryocooler for space applications, Cryogenics, vol. 36, pp. 859-866, 1996. [4] S. M. Aceves, J. M. Frias, and O. G. Villazana, Analytical and experimental evaluation of insulated pressure vessels for cryogenic hydrogen storage, Int. J. Hydrogen Energy, vol. 25, pp. 1075-1085, 2000. [5] P. Mueller and T. Durrant, Cryogenic propellant liquefaction and storage for a precursor to a human Mars mission, Cryogenics, vol. 39, pp. 1021-1028, 1999. [6] S. C. Russo and R. S. Sugimura, NASA IN-STEP Cryo system experiment flight test, Cryogenics, vol. 36, pp. 721-730, 1996. [7] E. D. Marquardt, Cryocooler reliability issues for space applications, Cryogenics, vol. 41, pp. 845-849, 2002. [8] I. Jun, K. Kenjiro, and O. K. Mamoru, Numerical simulation on boiling twophase flow of liquid nitrogen, Computational Technologies for Fluid/Thermal/Structural/Chemical Systems With Industrial Applications, ASME/PVP, vol. 377, no.2, pp. 219-225, 1998. [9] K. M. Akyuzlu and S. Malipeddi, Numerical simulation of laminar film boiling heat transfer from vertically suspended smooth surfaces in cryogenic fluid subjected to constant wall heat flux, Proceedings of ASME HTD, vol. 334, no. 3, pp. 355-360, 1996. [10] D. Boukeffa, M. Boumaza, M. X. Francois, and S. Pellerin, Experimental and numerical analysis of heat losses in a liquid nitrogen cryostat, Applied Thermal Engineering, vol. 21,pp. 967-975, 2001.

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84 [11] S. Kamiya, K. Onishi, E. Kawagoe, and K. Nishigaki, A large experimental apparatus for measuring thermal conductance of LH2 storage tank insulations, Cryogenics, vol. 40, pp. 35-44, 2000. [12] S. Kamiya, K. Onishi, N. Konshima, and K. Nishigaki, Thermal test of the insulation structure for LH 2 tank by using the large experimental apparatus, Cryogenics, vol. 40, pp. 737-748, 2001. [13] Z. Li, L. Xu, H. Sun, and Y. Xiao, and J. Zhang, Investigation on performances of non-loss storage for cryogenic liquefied gas, Cryogenics, vol. 44, pp. 357-362, 2004. [14] P. Kittel, Propellant preservation using re-liquefiers, Cryogenics, vol. 41, pp. 841-844, 2001. [15] T. M. Harms, M. Kazmierczak, and F. M. Gerner, Developing convective heat transfer in deep rectangular microchannels, International Journal of Heat and Fluid Flow, vol. 20, pp. 149-157, 1999. [16] K. K. Ambatipudi and M. M. Rahman, Analysis of conjugate heat transfer in microchannel heat sinks, Numerical Heat Transfer, Part A, vol. 37, pp. 711-731, 2000. [17] W. Qu, G. M. Mala, and D. Li, Heat Transfer for water flow in trapezoidal silicon microchannels, International Journal of Heat and Mass Transfer, vol. 43, no. 21, pp. 3925-3936, 2000. [18] A. G. Fedorov and R. Viskanta, Three-dimensional conjugate heat transfer in the microchannel heat sink for electronic packaging, International Journal of Heat and Mass Transfer, vol. 43, pp. 399-415, 2000. [19] D. Lelea, S. Nishio, and K. Takano, The experimental research on microtube heat transfer and fluid flow of distilled water, International Journal of Heat and Mass Transfer, vol. 47, pp. 2817-2830, 2004. [20] D. Yu, R. Warrington, R. Barron, and T. Ameel, An experimental and theoretical investigation of fluid flow and heat transfer in microtubes, Thermal Engineering 1995: Proceedings of the 4 th ASME/JSME Thermal Engineering Joint Conference, Maui, Hawaii, vol. 1, pp. 523-530, 1995. [21] T. M. Adams, S. I. Abdel-Khalik, S. M. Jeter, and Z. H. Qureshi, An experimental investigation of single-phase forced convection in microchannels, International Chemical Engineering, vol. 41, nos. 6-7, pp. 851-857, 1998.

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85 [22] V. Gnielinski, New equations for heat and mass transfer in turbulent pipe and channel flow, International Journal of Heat and Mass Transfer, vol. 16, pp. 359368, 1976. [23] W. Owhaib and B. Palm, Experimental investigation of single-phase convective heat transfer in circular microchannels, Experimental Thermal and Fluid Science, vol. 28, pp. 105-110, 2004. [24] P. Y. Wu and W. A. Little, Measuring of the heat transfer characteristics of gas flow in fine channel heat exchangers for micro miniature refrigerators, Cryogenics, vol. 24, pp. 415-420, 1984. [25] G. P. Celata, M. Cumo, and G. Zummo, Thermal-hydraulic characteristics of single-phase flow in capillary pipes, Experimental Thermal and Fluid Science, vol. 28, pp. 87-95, 2004. [26] G. Tunc and Y. Bayazitoglu, Heat Transfer in rectangular microchannels, International Journal of Heat and Mass Transfer, vol. 45, pp. 765-773, 2002. [27] G. A. Quadir, A. Mydin, and K. N. Seetharamu, Analysis of microchannel heat exchangers using FEM, International Journal of Numerical Methods for Heat and Fluid Flow, vol. 11, no. 1, pp. 59-75, 2001. [28] K. C. Toh, X. Y. Chen, and J. C. Chai, Numerical computation of fluid flow and heat transfer in microchannels, International Journal of Heat and Mass Transfer, vol. 45, pp. 5133-5141, 2002. [29] D. B. Tuckerman, Heat transfer microstructures for integrated circuits, Ph.D. thesis, Stanford University, 1984. [30] T. A. Ameel, X. Wang, R. F. Barron, and R. O. Warrington, Laminar forced convection in a circular tube with constant heat flux and slip flow, Microscale Thermophysical Engineering, vol. 1, pp. 303-320, 1997. [31] S. S. Shevade and M. M. Rahman, Transient analysis of microchannel heat transfer with volumetric heat generation in the substrate, Multiphase Phenomena and CFD Modeling and Simulation in Materials Processes as held at the 2004 TMS Annual Meeting, Charlotte, North Carolina, pp. 367-380, 2004. [32] J. R. Rujano and M. M. Rahman, Transient response of microchannel heat sinks in a silicon wafer, Journal of Electronic Packaging, vol. 119, pp. 239-246, 1997. [33] G. A. Quadir, S. L. Beh, K. N. Seetharamu, and A. Y. Hassan, A transient finite element analysis of microchannel heat exchangers, International Journal of Heat Exchangers, vol. III, no. 1, pp. 67-88, 2002.

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86 [34] D. B. Tuckerman and R. F. W. Pease, High performance heat sink for VLSI, IEEE Electron Device Letters, EDL-2(4), pp. 126-129, 1981. [35] L. Jiang, M. Wong, and Y. Zohar, Un steady characterist ics of a thermal microsystem, Sensors and Actuators, vol. 82, pp. 108-113, 2000. [36] G. Karimi and J. R. Culham, Transien t electro-osmotic pumping in rectangular microchannels, ICMENS 2003: Proceedings of the Internati onal Conference on MEMS, NANO, and Smart Systems, Banff, Alberta, Canada, pp. 364-372, 2003. [37] D. Brutin, F. Topin, and L. Tadrist, Tr ansient method for the liquid laminar flow friction factor in microtubes, The American Institute of Chemical Engineers Journal, vol. 49, no. 11, pp. 2759-2767, 2003. [38] A. Bejan, Convection Heat Transfer, 2 nd ed., Wiley, New York, 1994. [39] W. M. Kays and M. E. Crawford, C onvective Heat and Mass Transfer, 3 rd ed., McGraw Hill, New York, 1993. [40] M. N. Ozisik, Heat Conduction, 2 nd ed., pp. 67, Wiley, New York, 1993. [41] A. Bucci, G. P. Celata, M. Cumo, E. Serra, and G. Zummo, Water single-phase fluid flow and heat transfer in capillary tubes, Proceedings of 1 st International Conference on Microchannels and Mini channels, Rochester, N.Y., 2003. [42] E. N. Sieder and G. E. Tate, Indus trial Engineering Chemical, vol. 28, pp. 1429, 1936.

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APPENDICES 87

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Appendix A: Analysis of Liquid Nitrogen Flow in a Spherical Tank TITLE( ) Spherical Tank with inlet located at the bottom of the tank. (Diameter of the inlet = 0.02m, Flow rate=0.0059 kg/s, g=0, q =3.75 W/m 2 ) /*** The problem is designed using the FI-GEN module FI-GEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1 ) WINDOW(CHANGE= 1, MATRIX ) 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 -10.00000 10.00000 -7.50000 7.50000 -7.50000 7.50000 POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 0, Y = 1 ) POINT( ADD, COOR, X = 10, Y = 1 ) POINT( ADD, COOR, X = 10, Y = 0 ) POINT( ADD, COOR, X = 80, Y = 1 ) POINT( ADD, COOR, X = 80, Y = 0 ) POINT( ADD, COOR, X = 80, Y = 70 ) POINT( ADD, COOR, X = 80, Y = 80 ) POINT( ADD, COOR, X = 150, Y = 1 ) POINT( ADD, COOR, X = 150, Y = 0 ) POINT( ADD, COOR, X = 160, Y = 1 ) POINT( ADD, COOR, X = 160, Y = 0 ) POINT( ADD, COOR, X = 0, Y = 85 ) POINT( ADD, COOR, X = 160, Y = 85 ) POINT( ADD, COOR, X = 18.9979, Y = 34.3404 ) POINT( ADD, COOR, X = 10.2369, Y = 39.1615 ) POINT( ADD, COOR, X = 12.8121, Y = 43.431 ) POINT( ADD, COOR, X = 15.6483, Y = 47.5318 ) POINT( SELE, COOR, X = 141.321, Y = 51.3817 ) POINT( ADD, COOR, X = 141.321, Y = 51.3817 ) POINT( ADD, COOR, X = 26.451, Y = 45.0873 ) POINT( ADD, COOR, X = 133.612, Y = 45.0129 ) POINT( ADD, COOR, X = 8.73, Y = 0 ) POINT( ADD, COOR, X = 8.73, Y = 1 ) POINT( ADD, COOR, X = 80, Y = 71.27 ) POINT( ADD, COOR, X = 151.27, Y = 1 ) POINT( ADD, COOR, X = 151.27, Y = 0 ) POINT( ADD, COOR, X = 25.473, Y = 45.8975 ) POINT( ADD, COOR, X = 134.591, Y = 45.8218 ) POINT( ADD, COOR, X = 27.212381, Y = 44.44623 ) POINT( SELE, LOCA, WIND = 1 ) 0.101983, 0.455146 0.103399, 0.545798 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.100567, 0.543909 0.735127, 0.549575 CURVE( ADD, LINE ) 88

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Appendix A: (Continued) POINT( SELE, LOCA, WIND = 1 ) 0.733711, 0.543909 0.732295, 0.449481 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.730878, 0.453258 0.106232, 0.449481 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.733711, 0.545798 0.828612, 0.549575 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.824363, 0.547686 0.827195, 0.455146 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.825779, 0.457035 0.73796, 0.447592 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.441926, 0.666667 0.372521, 0.611898 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.369688, 0.615675 0.433428, 0.685552 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.436261, 0.691218 0.44051, 0.502361 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.441926, 0.691218 0.630312, 0.700661 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.623229, 0.687441 0.628895, 0.504249 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.623229, 0.498584 0.44051, 0.498584 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.623229, 0.68933 0.848442, 0.509915 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.852691, 0.508026 0.854108, 0.426818 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 89

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Appendix A: (Continued) 0.848442, 0.432483 0.347025, 0.436261 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.483003, 0.496695 0.308782, 0.551464 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.304533, 0.562795 0.436261, 0.485364 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.508499, 0.545798 0.137394, 0.564684 0.502833, 0.796978 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.502833, 0.793201 0.866856, 0.562795 0.529745, 0.611898 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.325779, 0.606232 0.365439, 0.610009 0.521246, 0.594901 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.552408, 0.532578 0.55949, 0.78187 0.604816, 0.441926 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.490085, 0.594901 0.423513, 0.525024 0.536827, 0.602455 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.528329, 0.604344 0.51983, 0.68933 0.567989, 0.623229 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.0226629, 0.830973 0.981586, 0.832861 0.477337, 0.455146 0.542493, 0.485364 SURFACE( ADD, POIN, ROWW = 2, NOAD ) CURVE( SELE, LOCA, WIND = 1 ) 0.264873, 0.613787 0.393768, 0.574127 CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.634561, 0.515581 90

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Appendix A: (Continued) 0.558074, 0.394712 CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.270538, 0.723324 0.133144, 0.566572 CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.732295, 0.71577 0.864023, 0.562795 CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.811615, 0.545798 0.828612, 0.519358 POINT( SELE, NEXT = 1 ) CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.854108, 0.455146 0.81728, 0.513692 CURVE( SPLI ) CURVE( SELE, LOCA, WIND = 1 ) 0.0878187, 0.472144 0.24221, 0.693107 0.141643, 0.462701 0.253541, 0.619452 0.15864, 0.474032 0.23796, 0.593012 0.675637, 0.751653 0.905099, 0.498584 0.681303, 0.672332 0.862606, 0.458924 0.640227, 0.679887 0.879603, 0.387158 POINT( SELE, LOCA, WIND = 1 ) 0.449008, 0.604344 0.402266, 0.568461 CURVE( ADD, LINE ) 0.342776, 0.474032 0.502833, 0.67611 0.756374, 0.477809 0.168555, 0.334278 0.767705, 0.345609 MEDGE( ADD, SUCC, INTE = 100, RATI = 0, 2RAT = 0, PCEN = 0 ) MEDGE( SELE, LOCA, WIND = 1 ) 0.379603, 0.173749 0.566572, 0.173749 0.474504, 0.156752 MEDGE( SELE, NEXT = 1 ) 0.627479, 0.154863 0.5, 0.360718 MEDGE( DELE ) CURVE( SELE, LOCA, WIND = 1 ) 0.497167, 0.364495 0.439093, 0.168083 91

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Appendix A: (Continued) 0.603399, 0.168083 0.467422, 0.156752 0.586402, 0.156752 MEDGE( ADD, SUCC, INTE = 100, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.145892, 0.491029 0.749292, 0.494806 0.834278, 0.485364 0.160057, 0.517469 0.249292, 0.519358 0.872521, 0.519358 MEDGE( ADD, SUCC, INTE = 20, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.450425, 0.570349 0.436261, 0.436261 0.589235, 0.644004 0.614731, 0.489141 MEDGE( ADD, SUCC, INTE = 8, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.345609, 0.528801 0.330028, 0.349386 0.481586, 0.785647 0.491501, 0.608121 MEDGE( ADD, SUCC, INTE = 4, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.205382, 0.564684 0.347025, 0.617564 0.696884, 0.576015 0.613314, 0.534466 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) CURVE( SELE, LOCA, WIND = 1 ) 0.695467, 0.560907 0.716714, 0.604344 CURVE( SELE, NEXT = 1 ) 0.770538, 0.570349 0.747875, 0.534466 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) CURVE( SELE, LOCA, WIND = 1 ) 0.771955, 0.553352 0.896601, 0.613787 0.467422, 0.568461 0.569405, 0.475921 0.46034, 0.398489 0.501416, 0.511804 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 2, EDG3 = 1, EDG4 = 2 ) CURVE( SELE, LOCA, WIND = 1 ) 0.174221, 0.523135 0.206799, 0.564684 0.263456, 0.528801 0.232295, 0.479698 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) CURVE( SELE, LOCA, WIND = 1 ) 0.259207, 0.517469 92

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Appendix A: (Continued) 0.412181, 0.562795 0.835694, 0.526912 0.750708, 0.477809 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) CURVE( SELE, LOCA, WIND = 1 ) 0.259207, 0.710104 0.0750708, 0.449481 0.188385, 0.647781 0.635977, 0.770538 0.909348, 0.485364 0.730878, 0.549575 0.902266, 0.341832 0.779037, 0.589235 0.443343, 0.723324 0.126062, 0.428706 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 4, EDG3 = 1, EDG4 = 4 ) CURVE( SELE, LOCA, WIND = 1 ) 0.409348, 0.292729 0.31728, 0.500472 0.26204, 0.627007 0.650142, 0.68933 0.871105, 0.438149 0.71813, 0.551464 0.893768, 0.347498 0.749292, 0.604344 0.430595, 0.706327 0.160057, 0.475921 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 4, EDG3 = 1, EDG4 = 4 ) CURVE( SELE, LOCA, WIND = 1 ) 0.0991501, 0.294618 CURVE( SELE, NEXT = 1 ) 0.252125, 0.602455 CURVE( SELE, NEXT = 1 ) 0.5, 0.532578 0.511331, 0.511804 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) CURVE( SELE, LOCA, WIND = 1 ) 0.5, 0.474032 0.444759, 0.528801 0.623229, 0.232295 0.695467, 0.491029 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) SURFACE( SELE, LOCA, WIND = 1 ) 0.558074, 0.496695 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.342776, 0.502361 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.624646, 0.500472 MFACE( MESH, MAP, NOSM, ENTI = "LN-1" ) SURFACE( SELE, LOCA, WIND = 1 ) 93

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Appendix A: (Continued) 0.685552, 0.500472 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.509915, 0.549575 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.498584, 0.553352 MFACE( MESH, MAP, NOSM, ENTI = "LN-2" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.601983, 0.500472 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.463173, 0.542021 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.563739, 0.545798 MFACE( MESH, MAP, NOSM, ENTI = "LN-3" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.609065, 0.498584 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.634561, 0.506138 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "LN-6" ) MFACE( SELE, LOCA, WIND = 1 ) 0.491501, 0.491029 SURFACE( SELE, LOCA, WIND = 1 ) 0.694051, 0.498584 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.555241, 0.462701 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.756374, 0.464589 MFACE( MESH, MAP, NOSM, ENTI = "LN-7" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.542493, 0.498584 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.283286, 0.727101 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.61898, 0.778093 MFACE( MESH, MAP, NOSM, ENTI = "Insulator" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.613314, 0.498584 UTILITY( HIGH = 9 ) 94

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Appendix A: (Continued) MLOOP( SELE, LOCA, WIND = 1 ) 0.575071, 0.243626 UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.586402, 0.319169 MFACE( MESH, MAP, NOSM, ENTI = "Alum" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.535411, 0.494806 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.417847, 0.17186 MLOOP( SELE, NEXT = 1 ) UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.419263, 0.169972 MFACE( SELE, NEXT = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "LN-4" ) SURFACE( SELE, LOCA, WIND = 1 ) 0.575071, 0.496695 UTILITY( HIGH = 9 ) MLOOP( SELE, LOCA, WIND = 1 ) 0.82153, 0.168083 MLOOP( SELE, NEXT = 1 ) UTILITY( HIGH = 3 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.675637, 0.169972 MFACE( SELE, NEXT = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "LN-5" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.412181, 0.441926 ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( MESH, MAP, ENTI = "Inlet" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.531161, 0.615675 MEDGE( MESH, MAP, ENTI = "I-lintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.410765, 0.385269 0.526912, 0.570349 0.5, 0.740321 0.624646, 0.557129 MEDGE( MESH, MAP, ENTI = "I-wintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.66289, 0.700661 MEDGE( MESH, MAP, ENTI = "I-rintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.410765, 0.421152 MEDGE( MESH, MAP, ENTI = "Al-lintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.498584, 0.576015 0.22238, 0.430595 95

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Appendix A: (Continued) 0.352691, 0.738432 0.613314, 0.341832 MEDGE( MESH, MAP, ENTI = "Al-wintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.419263, 0.477809 MEDGE( MESH, MAP, ENTI = "Al-rintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.116147, 0.634561 0.572238, 0.632672 0.941926, 0.644004 0.355524, 0.551464 0.63881, 0.555241 0.828612, 0.553352 MEDGE( MESH, MAP, ENTI = "Axi-sym" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.0509915, 0.373938 0.213881, 0.670444 0.67847, 0.753541 0.927762, 0.447592 MEDGE( MESH, MAP, ENTI = "O-wall" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.478754, 0.532578 MEDGE( MESH, MAP, ENTI = "Outlet" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.362606, 0.509915 0.518414, 0.753541 ELEMENT( SETD, EDGE, NODE = 2 ) MEDGE( MESH, MAP, ENTI = "Al-wlintf" ) MEDGE( SELE, LOCA, WIND = 1 ) 0.485836, 0.661001 0.600567, 0.487252 MEDGE( MESH, MAP, ENTI = "Al-wrintf" ) END( ) /*** End of FI-GEN. FI-PREP (specifying solid and fluid properties and Boundary conditions) started *** FIPREP( ) CONDUCTIVITY( ADD, SET = "LN", CONS = 0.000267686, ISOT ) CONDUCTIVITY( ADD, SET = "Alum", CONS = 0.42304, ISOT ) CONDUCTIVITY( ADD, SET = "Insulator", CONS = 2.82026e-06, ISOT ) DENSITY( ADD, SET = "LN", CONS = 0.74627 ) DENSITY( ADD, SET = "Alum", CONS = 2.77 ) SPECIFICHEAT( ADD, SET = "LN", CONS = 0.51147 ) SPECIFICHEAT( ADD, SET = "Alum", CONS = 0.20913 ) VISCOSITY( ADD, SET = "LN", CONS = 0.0011, MIXL, CLIP = 10000000 ) PRESSURE( ADD, PENA = 1e-08, DISC ) EDDYVISCOSITY( ADD, SPEZ ) TURBOPTIONS ( ADD, STAN ) GRAVITY( ADD, MAGN = 0, THET = 270, PHI = 0 ) ENTITY( ADD, NAME = "LN-1", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "LN-2", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "LN-3", FLUI, PROP = "LN" ) 96

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Appendix A: (Continued) ENTITY( ADD, NAME = "LN-4", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "LN-5", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "LN-6", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "LN-7", FLUI, PROP = "LN" ) ENTITY( ADD, NAME = "Insulator", SOLI, PROP = "Insulator" ) ENTITY( ADD, NAME = "Alum", SOLI, PROP = "Alum" ) ENTITY( ADD, NAME = "Inlet", PLOT ) ENTITY( ADD, NAME = "O-wall", PLOT ) ENTITY( ADD, NAME = "Axi-sym", PLOT ) ENTITY( ADD, NAME = "Outlet", PLOT ) ENTITY( ADD, NAME = "I-lintf", PLOT, ATTA = "Insulator", NATT = "LN-1") ENTITY( ADD, NAME = "I-wintf", PLOT, ATTA = "Insulator", NATT = "Alum") ENTITY( ADD, NAME = "I-rintf", PLOT, ATTA = "Insulator", NATT = "LN-7") ENTITY( ADD, NAME = "Al-lintf", PLOT, ATTA = "Alum", NATT = "LN-2" ) ENTITY( ADD, NAME = "Al-wlintf", PLOT, ATTA = "Alum", NATT = "LN-4" ) ENTITY( ADD, NAME = "Al-wrintf", PLOT, ATTA = "Alum", NATT = "LN-5" ) ENTITY( ADD, NAME = "Al-rintf", PLOT, ATTA = "Alum", NATT = "LN-6" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-1" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-2" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-3" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-4" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-5" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-6" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "LN-7" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "Alum" ) ICNODE( ADD, TEMP, CONS = 40, ENTI = "Insulator" ) BCNODE( ADD, URC, ENTI = "Inlet", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "Inlet", CONS = 2.5 ) BCNODE( ADD, TEMP, ENTI = "Inlet", CONS = 40 ) BCNODE( ADD, URC, ENTI = "O-wall", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "O-wall", CONS = 0 ) BCNODE( ADD, URC, ENTI = "Axi-sym", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "O-wall", CONS = 8.962715e-05 ) EXECUTION( ADD, NEWJ ) DATAPRINT( ADD, CONT ) PRINTOUT( ADD, NONE ) OPTIONS( ADD, UPWIND ) PROBLEM( ADD, AXI-, INCO, STEA, TURB, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, N.R. = 750, VELC = 0.05, RESC = 0.05, ACCF = 0 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 40, 0 END( ) /*** End of FI-PREP. Program is tested for syntax errors CREATE( FISO ) /*** Program is run in the background RUN( FISOLV, IDEN = "v25ss", BACK, AT = "", TIME = "NOW", COMP ) 97

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Appendix B: Steady State Conjugate Heat Transfer in a Circular Microtube Inside a Rectangular Substrate TITLE() Steady state analysis of the microtube. (Substrate=Silicon, Coolant=Water, Diameter = 300 m, Re = 1500, Pr = 6.78, =248) FI-GEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1 ) WINDOW(CHANGE= 1, MATRIX ) 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 -10.00000 10.00000 -7.50000 7.50000 -7.50000 7.50000 WINDOW( CHAN = 1, MATR ) 1, 0, 0, 0 0, 1, 0, 0 0, 0, 1, 0 0, 0, 0, 1 -10, 10, -7.5, 7.5, -7.5, 7.5 WINDOW( CHAN = 1, MATR ) 1, 0, 0, 0 0, 1, 0, 0 0, 0, 1, 0 0, 0, 0, 1 -10, 10, -7.5, 7.5, -7.5, 7.5 POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 0, Y = 0.1 ) POINT( ADD, COOR, X = 0, Y = 0.2 ) POINT( ADD, COOR, X = 0.1, Y = 0.2 ) POINT( ADD, COOR, X = 0.1, Y = 0.1 ) POINT( ADD, COOR, X = 0.1, Y = 0 ) POINT( ADD, COOR, X = 0.1, Y = 0.085 ) POINT( ADD, COOR, X = 0.085, Y = 0.1 ) POINT( ADD, COOR, X = 0.1, Y = 0.115 ) POINT( ADD, COOR, X = 0.089444, Y = 0.089343 ) POINT( ADD, COOR, X = 0.089343, Y = 0.110556 ) POINT( ADD, COOR, X = 0.1, Y = 0.2, Z = 2.5 ) POINT( ADD, COOR, X = 0.1, Y = 0.115, Z = 2.5 ) POINT( ADD, COOR, X = 0.1, Y = 0.085, Z = 2.5 ) POINT( SELE, LOCA, WIND = 1 ) 0.32287, 0.0219233 0.325859, 0.960638 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.68012, 0.424514 0.64275, 0.444444 0.624813, 0.490284 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.328849, 0.966617 98 0.681614, 0.96861

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Appendix B: (Continued) 0.325859, 0.0219233 0.681614, 0.0179372 SURFACE( ADD, POIN, ROWW = 2, NOAD ) CURVE( SELE, LOCA, WIND = 1 ) 0.665172, 0.432486 0.662182, 0.432486 0.443946, 0.426507 0.38864, 0.542103 0.472347, 0.649726 0.55157, 0.518186 MEDGE( ADD, SUCC, INTE = 6, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.535127, 0.829098 0.428999, 0.687593 0.42003, 0.47434 0.533632, 0.308919 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.424514, 0.502242 0.44843, 0.974589 MFACE( MESH, MAP, NOSM, ENTI = "Si-in" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bin" ) MFACE( MESH, MAP, NOSM, ENTI = "f-tin" ) MFACE( SELE, LOCA, WIND = 1 ) 0.484305, 0.490284 CURVE( SELE, LOCA, WIND = 1 ) 0.430493, 0.721475 MSOLID( PROJ ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, NOSM, ENTI = "Fluid-b", ALG1 ) MSOLID( MESH, MAP, NOSM, ENTI = "Fluid-t", ALG1 ) MSOLID( MESH, MAP, NOSM, ENTI = "Silicon", ALG1 ) ELEMENT( SELE, ALL ) ELEMENT( MODI, INVI, NOSH ) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "Si-lwall" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-bottom" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bwall1" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bwall2" ) MFACE( SELE, LOCA, WIND = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "f-twall1" ) MFACE( MESH, MAP, NOSM, ENTI = "f-twall2" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bsym" ) MFACE( SELE, LOCA, WIND = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "f-tsym" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bout" ) MFACE( MESH, MAP, NOSM, ENTI = "f-tout" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-bsym" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-tsym" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-top" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-out" ) END( ) 99

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Appendix B: (Continued) FIPREP( ) CONDUCTIVITY( ADD, SET = "Fluid", CONS = 0.0014435, ISOT ) CONDUCTIVITY( ADD, SET = "Solid", CONS = 0.3585, ISOT ) DENSITY( ADD, SET = "Fluid", CONS = 0.9974 ) DENSITY( ADD, SET = "Solid", CONS = 2.33 ) SPECIFICHEAT( ADD, SET = "Fluid", CONS = 0.9988 ) SPECIFICHEAT( ADD, SET = "Solid", CONS = 0.16969 ) VISCOSITY( ADD, SET = "Fluid", CONS = 0.0098 ) ENTITY( ADD, NAME = "Fluid-b", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Fluid-t", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Silicon", SOLI, PROP = "Solid" ) ENTITY( ADD, NAME = "f-bin", PLOT ) ENTITY( ADD, NAME = "f-tin", PLOT ) ENTITY( ADD, NAME = "f-bsym", PLOT ) ENTITY( ADD, NAME = "f-tsym", PLOT ) ENTITY( ADD, NAME = "f-bout", PLOT ) ENTITY( ADD, NAME = "f-tout", PLOT ) ENTITY( ADD, NAME = "Si-in", PLOT ) ENTITY( ADD, NAME = "Si-top", PLOT ) ENTITY( ADD, NAME = "Si-lwall", PLOT ) ENTITY( ADD, NAME = "Si-bottom", PLOT ) ENTITY( ADD, NAME = "Si-bsym", PLOT ) ENTITY( ADD, NAME = "Si-tsym", PLOT ) ENTITY( ADD, NAME = "Si-out", PLOT ) ENTITY( ADD, NAME = "f-bwall1", PLOT, ATTA = "Silicon", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-bwall2", PLOT, ATTA = "Silicon", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-twall1", PLOT, ATTA = "Silicon", NATT = "Fluid-t" ) ENTITY( ADD, NAME = "f-twall2", PLOT, ATTA = "Silicon", NATT = "Fluid-t" ) BCNODE( ADD, UX, ENTI = "f-bin", ZERO ) BCNODE( ADD, UX, ENTI = "f-tin", ZERO ) BCNODE( ADD, UY, ENTI = "f-bin", ZERO ) BCNODE( ADD, UY, ENTI = "f-tin", ZERO ) BCNODE( ADD, UZ, ENTI = "f-bin", CONS = 491.277321 ) BCNODE( ADD, UZ, ENTI = "f-tin", CONS = 491.277321 ) BCNODE( ADD, TEMP, ENTI = "f-bin", CONS = 20 ) BCNODE( ADD, TEMP, ENTI = "f-tin", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "f-bwall1", ZERO ) BCNODE( ADD, VELO, ENTI = "f-bwall2", ZERO ) BCNODE( ADD, VELO, ENTI = "f-twall1", ZERO ) BCNODE( ADD, VELO, ENTI = "f-twall2", ZERO ) BCNODE( ADD, UX, ENTI = "f-bsym", ZERO ) BCNODE( ADD, UY, ENTI = "f-bsym", ZERO ) BCNODE( ADD, UX, ENTI = "f-tsym", ZERO ) BCNODE( ADD, UY, ENTI = "f-tsym", ZERO ) BCFLUX( ADD, HEAT, ENTI = "Si-bottom", CONS = 7.170172084 ) BCFLUX( ADD, HEAT, ENTI = "Si-top", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Si-lwall", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Si-bsym", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Si-tsym", CONS = 0 ) 100

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Appendix B: (Continued) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE ) OPTIONS( ADD, UPWI ) PROBLEM( ADD, 3-D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, N.R. = 7500, VELC = 0.02, RESC = 0.02 ) RELAXATION( RESI, MAXI ) 0.1, 0.1, 0.1, 0, 0.05, 0, 0, 0, 0.05 RELAXATION( MINI ) 0.05, 0.05, 0.05, 0, 0.005, 0, 0, 0, 0.005 CLIPPING( ADD, MINI ) 0, 0, 0, 0, 20, 0 END( ) CREATE( FISO ) RUN( FISOLV, IDEN = "mod315", BACK, AT = "", TIME = "NOW", COMP ) 101

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Appendix C: Transient Conjugate Heat Transfer in a Circular Microtube Inside a Rectangular Substrate TITLE() Transient analysis of the microtube. (Substrate=Silicon, Coolant=Water, Diameter = 300 m, Re = 1500, Pr = 6.78, =248) FI-GEN( ELEM = 1, POIN = 1, CURV = 1, SURF = 1, NODE = 0, MEDG = 1, MLOO = 1, MFAC = 1, BEDG = 1, SPAV = 1, MSHE = 1, MSOL = 1, COOR = 1 ) WINDOW(CHANGE= 1, MATRIX ) 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 0.000000 1.000000 -10.00000 10.00000 -7.50000 7.50000 -7.50000 7.50000 WINDOW( CHAN = 1, MATR ) 1, 0, 0, 0 0, 1, 0, 0 0, 0, 1, 0 0, 0, 0, 1 -10, 10, -7.5, 7.5, -7.5, 7.5 WINDOW( CHAN = 1, MATR ) 1, 0, 0, 0 0, 1, 0, 0 0, 0, 1, 0 0, 0, 0, 1 -10, 10, -7.5, 7.5, -7.5, 7.5 POINT( ADD, COOR, X = 0, Y = 0 ) POINT( ADD, COOR, X = 0, Y = 0.1 ) POINT( ADD, COOR, X = 0, Y = 0.2 ) POINT( ADD, COOR, X = 0.1, Y = 0.2 ) POINT( ADD, COOR, X = 0.1, Y = 0.1 ) POINT( ADD, COOR, X = 0.1, Y = 0 ) POINT( ADD, COOR, X = 0.1, Y = 0.085 ) POINT( ADD, COOR, X = 0.085, Y = 0.1 ) POINT( ADD, COOR, X = 0.1, Y = 0.115 ) POINT( ADD, COOR, X = 0.089444, Y = 0.089343 ) POINT( ADD, COOR, X = 0.089343, Y = 0.110556 ) POINT( ADD, COOR, X = 0.1, Y = 0.2, Z = 2.5 ) POINT( ADD, COOR, X = 0.1, Y = 0.115, Z = 2.5 ) POINT( ADD, COOR, X = 0.1, Y = 0.085, Z = 2.5 ) POINT( SELE, LOCA, WIND = 1 ) 0.32287, 0.0219233 0.325859, 0.960638 CURVE( ADD, LINE ) POINT( SELE, LOCA, WIND = 1 ) 0.68012, 0.424514 0.64275, 0.444444 0.624813, 0.490284 CURVE( ADD, ARC ) POINT( SELE, LOCA, WIND = 1 ) 0.328849, 0.966617 102 0.681614, 0.96861

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Appendix C: (Continued) 0.325859, 0.0219233 0.681614, 0.0179372 SURFACE( ADD, POIN, ROWW = 2, NOAD ) CURVE( SELE, LOCA, WIND = 1 ) 0.665172, 0.432486 0.662182, 0.432486 0.443946, 0.426507 0.38864, 0.542103 MEDGE( ADD, SUCC, INTE = 6, RATI = 0, 2RAT = 0, PCEN = 0 ) CURVE( SELE, LOCA, WIND = 1 ) 0.535127, 0.829098 0.428999, 0.687593 0.42003, 0.47434 0.533632, 0.308919 MLOOP( ADD, MAP, VISI, NOSH, EDG1 = 1, EDG2 = 1, EDG3 = 1, EDG4 = 1 ) MFACE( ADD ) MFACE( SELE, LOCA, WIND = 1 ) 0.424514, 0.502242 0.44843, 0.974589 MFACE( MESH, MAP, NOSM, ENTI = "Si-in" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bin" ) MFACE( MESH, MAP, NOSM, ENTI = "f-tin" ) MFACE( SELE, LOCA, WIND = 1 ) 0.484305, 0.490284 CURVE( SELE, LOCA, WIND = 1 ) 0.430493, 0.721475 MSOLID( PROJ ) ELEMENT( SETD, BRIC, NODE = 8 ) MSOLID( MESH, MAP, NOSM, ENTI = "Fluid-b", ALG1 ) MSOLID( MESH, MAP, NOSM, ENTI = "Fluid-t", ALG1 ) MSOLID( MESH, MAP, NOSM, ENTI = "Silicon", ALG1 ) ELEMENT( SELE, ALL ) ELEMENT( MODI, INVI, NOSH ) ELEMENT( SETD, QUAD, NODE = 4 ) MFACE( MESH, MAP, NOSM, ENTI = "Si-lwall" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-bottom" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bwall1" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bwall2" ) MFACE( SELE, LOCA, WIND = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "f-twall1" ) MFACE( MESH, MAP, NOSM, ENTI = "f-twall2" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bsym" ) MFACE( SELE, LOCA, WIND = 1 ) MFACE( MESH, MAP, NOSM, ENTI = "f-tsym" ) MFACE( MESH, MAP, NOSM, ENTI = "f-bout" ) MFACE( MESH, MAP, NOSM, ENTI = "f-tout" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-bsym" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-tsym" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-top" ) MFACE( MESH, MAP, NOSM, ENTI = "Si-out" ) END( ) 103

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Appendix C: (Continued) FIPREP( ) CONDUCTIVITY( ADD, SET = "Fluid", CONS = 0.0014435, ISOT ) CONDUCTIVITY( ADD, SET = "Solid", CONS = 0.3585, ISOT ) DENSITY( ADD, SET = "Fluid", CONS = 0.9974 ) DENSITY( ADD, SET = "Solid", CONS = 2.33 ) SPECIFICHEAT( ADD, SET = "Fluid", CONS = 0.9988 ) SPECIFICHEAT( ADD, SET = "Solid", CONS = 0.16969 ) VISCOSITY( ADD, SET = "Fluid", CONS = 0.0098 ) ENTITY( ADD, NAME = "Fluid-b", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Fluid-t", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Silicon", SOLI, PROP = "Solid" ) ENTITY( ADD, NAME = "f-bin", PLOT ) ENTITY( ADD, NAME = "f-tin", PLOT ) ENTITY( ADD, NAME = "f-bsym", PLOT ) ENTITY( ADD, NAME = "f-tsym", PLOT ) ENTITY( ADD, NAME = "f-bout", PLOT ) ENTITY( ADD, NAME = "f-tout", PLOT ) ENTITY( ADD, NAME = "Si-in", PLOT ) ENTITY( ADD, NAME = "Si-top", PLOT ) ENTITY( ADD, NAME = "Si-lwall", PLOT ) ENTITY( ADD, NAME = "Si-bottom", PLOT ) ENTITY( ADD, NAME = "Si-bsym", PLOT ) ENTITY( ADD, NAME = "Si-tsym", PLOT ) ENTITY( ADD, NAME = "Si-out", PLOT ) ENTITY( ADD, NAME = "f-bwall1", PLOT, ATTA = "Silicon", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-bwall2", PLOT, ATTA = "Silicon", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-twall1", PLOT, ATTA = "Silicon", NATT = "Fluid-t" ) ENTITY( ADD, NAME = "f-twall2", PLOT, ATTA = "Silicon", NATT = "Fluid-t" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "Fluid-b" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "Fluid-t" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "Silicon" ) BCNODE( ADD, UX, ENTI = "f-bin", ZERO ) BCNODE( ADD, UX, ENTI = "f-tin", ZERO ) BCNODE( ADD, UY, ENTI = "f-bin", ZERO ) BCNODE( ADD, UY, ENTI = "f-tin", ZERO ) BCNODE( ADD, UZ, ENTI = "f-bin", CONS = 491.277321 ) BCNODE( ADD, UZ, ENTI = "f-tin", CONS = 491.277321 ) BCNODE( ADD, TEMP, ENTI = "f-bin", CONS = 20 ) BCNODE( ADD, TEMP, ENTI = "f-tin", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "f-bwall1", ZERO ) BCNODE( ADD, VELO, ENTI = "f-bwall2", ZERO ) BCNODE( ADD, VELO, ENTI = "f-twall1", ZERO ) BCNODE( ADD, VELO, ENTI = "f-twall2", ZERO ) BCNODE( ADD, UX, ENTI = "f-bsym", ZERO ) BCNODE( ADD, UY, ENTI = "f-bsym", ZERO ) BCNODE( ADD, UX, ENTI = "f-tsym", ZERO ) BCNODE( ADD, UY, ENTI = "f-tsym", ZERO ) BCFLUX( ADD, HEAT, ENTI = "Si-bottom", CONS = 7.170172084 ) BCFLUX( ADD, HEAT, ENTI = "Si-top", CONS = 0 ) 104

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Appendix C: (Continued) BCFLUX( ADD, HEAT, ENTI = "Si-lwall", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Si-bsym", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Si-tsym", CONS = 0 ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE ) OPTIONS( ADD, UPWI ) PROBLEM( ADD, 3-D, INCO, TRAN, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, N.R. = 7500, VELC = 0.02, RESC = 0.02 ) TIMEINTEGRATION( ADD, TRAP, NSTE = 4000, TSTA = 0, DT = 0.025, VARI, WIND = 0.75, NOFI = 3 ) POSTPROCESS( ADD, NBLO = 4, NOPT, NOPA ) 1, 49, 4 50, 140, 15 141, 1000, 25 1000, 4000, 100 RELAXATION( RESI, MAXI ) 0.1, 0.1, 0.1, 0, 0.05, 0, 0, 0, 0.05 RELAXATION( MINI ) 0.05, 0.05, 0.05, 0, 0.005, 0, 0, 0, 0.005 CLIPPING( ADD, MINI ) 0, 0, 0, 0, 20, 0 END( ) CREATE( FISO ) RUN( FISOLV, IDEN = "tSW3151", BACK, AT = "", TIME = "NOW", COMP ) 105

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106 Appendix D: Thermodynamic Pr operties of Different Solids and Fluids Used in the Analysis Table A1 Thermodynamic properties of different solids Sl. No. Properties Insulator Aluminum Stainless Steel 1 Density (kg/m 3 ) -2770 8027.2 2 Thermal Conductivity (W/m-K) 0.00118 176.99 16.26 3 Specific heat (J/kg-K) -874.99 502.09 Sl. No. Properties Silicon Carbide Silicon 1 Density (kg/m 3 ) 3160 2330 2 Thermal Conductivity (W/m-K) 113.99 149.99 3 Specific heat (J/kg-K) 714.99 709.98 Table A2 Thermodynamic and transpor t properties of different fluids Sl. No. Properties Liquid Nitrogen Water FC-72 1 Density (kg/m 3 ) 746.27 997.4 1691.54 2 Thermal Conductivity (W/m-K) 0.112 0.604 0.0564 3 Specific heat (J/kg-K) 2139.99 4178.98 1041.88 4 Absolute Viscosity (N-s/m 2 ) 0.00011 0.00098 0.000687


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Rao, P. Sharath Chandra.
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Analysis of fluid circulation in a spherical cryogenic storage tank and conjugate heat transfer in a circular microtube
h [electronic resource] /
by P. Sharath Chandra Rao.
260
[Tampa, Fla.] :
University of South Florida,
2004.
502
Thesis (M.S.M.E.)--University of South Florida, 2004.
504
Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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ABSTRACT: The study considered development of a finite-element numerical simulation model for the analysis of fluid flow and conjugate heat transfer in a zero boil-off (ZBO) cryogenic storage system. A spherical tank was considered for the investigation. The tank wall is made of aluminum and a multi-layered blanket of cryogenic insulation (MLI) has been attached on the top of the aluminum. The tank is connected to a cryocooler to dissipate the heat leak through the insulation and tank wall into the fluid within the tank. The cryocooler has not been modeled; only the flow in and out of the tank to the cryocooler system has been included. The primary emphasis of this research has been the fluid circulation within the tank for different fluid distribution scenario and for different level of gravity to simulate all the potential earth and space based applications. The steady-state velocity, temperature, and pressure distributions were calculated for different inlet positions, inlet velocities, and for different gravity values. The simulations were carried out for constant heat flux and constant wall temperature cases. It was observed that a good flow circulation could be obtained when the cold entering fluid was made to flow in radial direction and the inlet opening was placed close to the tank wall. The transient and steady state heat transfer for laminar flow inside a circular microtube within a rectangular substrate during start up of power has also been investigated. Silicon, Silicon Carbide and Stainless Steel were the substrates used and Water and FC-72 were the coolants employed. Equations governing the conservation of mass, momentum, and energy were solved in the fluid region. Within the solid wafer, the heat conduction was solved. The Reynolds number, Prandtl number, thermal conductivity ratio, and diameter ranges were: 1000--1900, 6.78--12.68, 27--2658, and 300 µ m--1000 µ m respectively. It was found that a higher aspect ratio or larger diameter tube and higher thermal conductivity ratio combination of substrate and coolant requires lesser amount of time to attain steady state. It was seen that enlarging the tube from 300 µ m to 1000 µ m results in lowering of the fluid mean temperature at the exit. Nusselt number decreased with time and finally reached the steady state condition. It was also found that a higher Prandtl number fluid attains higher maximum substrate temperature and Nusselt number. A correlation for peripheral average Nusselt number was developed by curve-fitting the computed results with an average error of 6.5%. This correlation will be very useful for the design of circular microtube heat exchangers.
590
Adviser: Muhammad M. Rahman.
653
zero-boiloff.
numerical simulation transient thermal management.
cryocooler.
electronic cooling.
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Dissertations, Academic
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x Mechanical Engineering
Masters.
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t USF Electronic Theses and Dissertations.
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