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Analysis of conjugate heat transfer in tube-in-block heat exchangers for some engineering applications

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Title:
Analysis of conjugate heat transfer in tube-in-block heat exchangers for some engineering applications
Physical Description:
Book
Language:
English
Creator:
Gari, Abdullatif Abdulhadi
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Microchannels
Magnetic refrigerators
Electronic cooling
Nanofluids
Snow melting
Dissertations, Academic -- Mechanical Engineering -- Doctoral -- USF
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: This project studied the effect of different parameters on the conjugate heat transfer in tube-in-block heat exchangers for various engineering applications. These included magnetic coolers (or heaters) associated with a magnetic refrigeration system, high heat flux coolers for electronic equipment, and hydronic snow melting system embedded in concrete slabs. The results of this research will help in designing the cooling/heating systems and select their appropriate geometrical dimensions and material for specific applications. Types of problems studied in this project are: steady state circular microchannels with heat source in the gadolinium substrate, transient heat transfer in circular microchannels with time varying heat source in a gadolinium substrate, transient heat transfer in composite trapezoidal microchannels of silicon and gadolinium with constant and time varying heat source, steady state heat transfer in microchannels using fluids suspended with nanoparticl es, and analysis of steady state and transient heat transfer in a hydronic snow melting system. For each of these problems a numerical simulation model was developed. The mass, momentum, and energy conservation equations were solved in the fluid region and energy conservation in the solid region of the heat exchanger to arrive at the velocity and temperature distributions. Detailed parametric study was carried out for each problem. Parameters were Reynolds number, heat source value, channel diameter or channel height, solid materials and working fluids. Results are presented in terms of solid-fluid interface temperature, heat flow rate, heat transfer coefficient, and Nusselt number along the length of the channel and with the progression of time. The results showed that an increase in Reynolds number decreases the interface temperature but increases the heat flow rate and Nusselt number. When the heat source varied with time, by applying and removing the magnetic field, the interface^ temperature, heat flow rate, and Nusselt number attained a periodic variation with time. The decrease in the diameter at constant Reynolds number decreases the interface temperature and increases the heat flow rate at the fluid-solid interface.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Abdullatif Abdulhadi Gari.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 194 pages.
General Note:
Includes vita.

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University of South Florida Library
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University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001910846
oclc - 173609294
usfldc doi - E14-SFE0001716
usfldc handle - e14.1716
System ID:
SFS0026034:00001


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Analysis of conjugate heat transfer in tube-in-block heat exchangers for some engineering applications
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ABSTRACT: This project studied the effect of different parameters on the conjugate heat transfer in tube-in-block heat exchangers for various engineering applications. These included magnetic coolers (or heaters) associated with a magnetic refrigeration system, high heat flux coolers for electronic equipment, and hydronic snow melting system embedded in concrete slabs. The results of this research will help in designing the cooling/heating systems and select their appropriate geometrical dimensions and material for specific applications. Types of problems studied in this project are: steady state circular microchannels with heat source in the gadolinium substrate, transient heat transfer in circular microchannels with time varying heat source in a gadolinium substrate, transient heat transfer in composite trapezoidal microchannels of silicon and gadolinium with constant and time varying heat source, steady state heat transfer in microchannels using fluids suspended with nanoparticl es, and analysis of steady state and transient heat transfer in a hydronic snow melting system. For each of these problems a numerical simulation model was developed. The mass, momentum, and energy conservation equations were solved in the fluid region and energy conservation in the solid region of the heat exchanger to arrive at the velocity and temperature distributions. Detailed parametric study was carried out for each problem. Parameters were Reynolds number, heat source value, channel diameter or channel height, solid materials and working fluids. Results are presented in terms of solid-fluid interface temperature, heat flow rate, heat transfer coefficient, and Nusselt number along the length of the channel and with the progression of time. The results showed that an increase in Reynolds number decreases the interface temperature but increases the heat flow rate and Nusselt number. When the heat source varied with time, by applying and removing the magnetic field, the interface^ temperature, heat flow rate, and Nusselt number attained a periodic variation with time. The decrease in the diameter at constant Reynolds number decreases the interface temperature and increases the heat flow rate at the fluid-solid interface.
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Analysis of Conjugate Heat Transfer in Tube-in-Block Heat Ex changers for Some Engineering Applications by Abdullatif Abdulhadi Gari A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Roger Crane, Ph.D. Yehia Hammad, Ph.D. Autar Kaw, Ph.D. Stanley Kranc, Ph.D. Date of Approval: June 22, 2006 Keywords: microchannels, magnetic refrigerat ors, electronic cooling, nanofluids, snow melting Copyright 2006, Abdullatif Abdulhadi Gari

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Dedication To Dad for his guidance (1944-2003) and To Mom for her patience.

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Acknowledgements I would like to thank Dr. Rahman for his gui dance, support, and patience. I also would like to thank the committee members and faculty members of the Mechanical Engineering Department at the University of South Florida for their teaching and support. I also would like to thank my colleagues who worked with me under Dr. Rahman’s supervision for their cooperation and help.

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i Table of Contents List of Figures iv List of Nomenclature xvii Abstract xix Chapter 1 – Introduction and Literature Review 1 1.1 Introduction 1 1.2 Literature Review 4 1.2.1 Steady State Heat Transfer in Rectangular and Trapezoidal Microchannels 4 1.2.2 Steady State Heat Transf er in Circular Microchannels 7 1.2.3 Transient Heat Transfer in Microchannels 10 1.2.4 Heat Transfer Using Nanofluids 11 1.2.5 Hydronic Snow Melting System 14 1.3 Objectives 16 Chapter 2 – Steady State Heat Transfer in Circular Microchannels During Magnetic Heating or Cooling 19 2.1 Mathematical Model 19 2.2 Numerical Simulation and Parametric Study 22 2.3 Results and Discussion 23 2.4 Conclusions 40 Chapter 3 – Transient Heat Transfer in Circular Microcha nnels Under Time Varying Heat Source 41 3.1 Mathematical Model 41 3.2 Numerical Simulation and Parametric Study 44 3.3 Results and Discussion 44 3.4 Conclusions 52 Chapter 4 – Transient Heat Transfer in Trapezoidal Microchannels During Activation of Magnetic Heating 55 4.1 Mathematical Model 55 4.2 Numerical Simulation and Parametric Study 58 4.3 Results and Discussion 59 4.4 Conclusions 86

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ii Chapter 5 – Transient Heat Transfer in Trapezoidal Micr ochannels Under Time Varying Heat Source 91 5.1 Mathematical Model 91 5.2 Numerical Simulation and Parametric Study 94 5.3 Results and Discussion 95 5.4 Conclusions 112 Chapter 6 – Steady State Heat Transf er Using Nanofluids in Circular Microchannels 114 6.1 Mathematical Model 114 6.2 Numerical Simulation and Parametric Study 117 6.3 Results and Discussion 117 6.4 Conclusions 134 Chapter 7 – Steady State Analysis of Hydr onic Snow Melting System 136 7.1 Mathematical Model 136 7.2 Numerical Simulation and Parametric Study 139 7.3 Results and Discussion 140 7.4 Conclusions 149 Chapter 8 – Transient Analysis of Hydroni c Snow Melting System 151 8.1 Mathematical Model 151 8.2 Numerical Simulation and Parametric Study 155 8.3 Results and Discussion 155 8.4 Conclusions 164 Chapter 9 – Conclusions 165 9.1 Conclusion 165 9.2 Recommendations for Future Research 167 References 168 Appendices 172 Appendix A: FIDAP Program for Steady State Heat Transfer in Circular Microchannels During Magnetic Heating or Cooling Simulation 173 Appendix B: FIDAP Program for Transient Heat Transfer in Circular Microchannels Under Time Varying Heat Source Simulation 175 Appendix C: FIDAP Program for Transient Heat Transfer in Trapezoidal Microchannels During Activation of Magnetic Heating Simulation 177 Appendix D: FIDAP Program for Transient Heat Transfer in Trapezoidal Microchannels Under Time Varying Heat Source Simulation 179

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iii Appendix E: FIDAP Program for St eady State Heat Transfer Using Nanofluids in Circular Microc hannels Simulation 181 Appendix F: FIDAP Program for St eady State Analysis in Hydronic Snow Melting System Simulation 184 Appendix G: FIDAP Program for Tr ansient Analysis in Hydronic Snow Melting System Simulation 188 Appendix H: Design of Experiment on Parameters and Quantitative Error Analysis of the Grid Test 192 About the Author End Page

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iv List of Figures Figure 1.1: Schematic drawing of the bed heat exchanger assembly for magnetic refrigerating 2 Figure 1.2: Schematic drawing for a mi crochannel test fixture used for electronic cooling 2 Figure 1.3: Schematic drawing for hydronic snow melting system 4 Figure 2.1: Schematic drawing for the circular microchannel model 20 Figure 2.2: Schematic drawing for th e circular microchannel simulated model 20 Figure 2.3: Local dimensionless periphera l average interface temperature for different grid sizes (Re = 1600, G = 5 T, d = 0.036 cm, Water) 24 Figure 2.4: Local Nusselt number variation for different axial locations along the angular direction (Re = 1600, G = 5 T, d = 0.036 cm, Water) 24 Figure 2.5: Local dimensionless peri pheral average interface temperature along axial coordinate at various Reynolds number (d = 0.036 cm, Water) 26 Figure 2.6: Local peripheral average Nusselt number along axial coordinate at various Reynolds numbers (d = 0.036 cm, Water) 26 Figure 2.7: Local dimensionless peri pheral average interface temperature along axial coordinate at vari ous diameters and constant Reynolds numbers (Re = 1600, G = 5 T, Water) 28 Figure 2.8: Local dimensionless periphera l average heat transfer coefficient along axial coordinate at vari ous diameters and constant Reynolds numbers (Re = 1600, G = 5 T, Water) 28

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v Figure 2.9: Local peripheral average Nusselt number along axial coordinate at various diameters and consta nt Reynolds numbers (Re = 1600, G = 5 T, Water) 29 Figure 2.10: Local dimensionless peri pheral average interface temperature along axial coordinate at various diameters and constant inlet velocity (Vin = 4.367 m/s, G = 5 T, Water) 30 Figure 2.11: Local peripheral average Nusselt number along axial coordinate at various diameters and c onstant inlet velocity (Vin = 4.367 m/s, G = 5 T, Water) 30 Figure 2.12: Local Nusselt number varia tion for different working fluids along the angular direction (Re = 1600, G = 5 T, d = 0.036 cm) 32 Figure 2.13: Local dimensionless peri pheral average interface temperature along axial coordinate for differe nt fluids (Re = 1600, G = 5 T, d = 0.036 cm) 32 Figure 2.14: Local dimensionless periphera l average heat transfer coefficient along axial coordinate for differe nt fluid (Re = 1600, G = 5 T, d = 0.036 cm, Water) 33 Figure 2.15: Local peripheral average Nusselt number along axial coordinate for different fluids (Re = 1600, G = 5 T, d = 0.036 cm, Water) 34 Figure 2.16: Dimensionless maximum temp erature in the system at various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 35 Figure 2.17: Pressure drop in the model at various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 35 Figure 2.18: Local average Nusselt number at z = 0.4 cm for various diameter (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 37 Figure 2.19: Local average Nusselt nu mber at z = 0.8 cm for various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 37 Figure 2.20: Comparison for friction factor results to existing experimental results in the literature 38 Figure 2.21: Comparison for Nusselt number results to existing experimental results in the literature 38

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vi Figure 3.1: Schematic for microchannel heat exchanger model 42 Figure 3.2: Maximum temperature over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm) 46 Figure 3.3: Peripheral average interf ace temperature at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm) 46 Figure 3.4: Peripheral average heat flow rate at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm) 47 Figure 3.5: Peripheral average Nusselt number at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm) 47 Figure 3.6: Maximum temperature over 9 seconds with G = 4 T (Re = 1600, d = 0.036 cm) 49 Figure 3.7: Peripheral average heat flow rate at different axial locations over 9 seconds with G = 4 T (Re = 1600, d = 0.036 cm) 49 Figure 3.8: Peripheral average Nusselt number at different axial locations with G = 4 T (Re = 1600, d = 0.036 cm) 50 Figure 3.9: Peripheral average heat flow ra te at different axial locations with d = 0.012 cm (Re = 1600, G = 2 T) 51 Figure 3.10: Peripheral average Nusselt number at different axial locations with d = 0.012 cm (Re = 1600, G = 2 T) 51 Figure 3.11: Peripheral average heat flow ra te at different axial locations with Re = 1000 (G = 2 T, d = 0.036 cm) 53 Figure 3.12: Peripheral average Nusselt number at different axial locations with Re = 1000 (G = 2 T, d = 0.036 cm) 53 Figure 4.1: Schematic draw for the m odel of trapezoidal microchannel 56 Figure 4.2: Peripheral average interface temperature along the axial direction at the fluid-silic on interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 60 Figure 4.3: Peripheral average heat flow rate along the axial direction at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 60

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vii Figure 4.4: Peripheral average interface temperature along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 61 Figure 4.5: Peripheral average heat flow rate along the ax ial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 61 Figure 4.6: Peripheral average heat tr ansfer coefficient along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 63 Figure 4.7: Peripheral average Nusselt number along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 63 Figure 4.8: Peripheral average interface temperature along the axial direction at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 64 Figure 4.9: Peripheral average heat flow rate along the ax ial direction at different time steps at the fl uid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 65 Figure 4.10: Peripheral average heat tr ansfer coefficient along the axial direction at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 66 Figure 4.11: Peripheral average Nusselt number along the axial direction at different time steps at the fl uid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 66 Figure 4.12: The effect of Reynolds number on the peripheral average interface temperature along the axia l direction at different time periods at the fluid-gadolinium interface (G = 5 T, Hfl = 0.03 cm) 68 Figure 4.13: The effect of Reynolds nu mber on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (G = 5 T, Hfl = 0.03 cm) 68 Figure 4.14: The effect of Reynolds number on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (G = 5 T, Hfl = 0.03 cm) 69

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viii Figure 4.15: The effect of Reynolds number on the peripheral average interface temperature along the axia l direction at different time periods at the fluid-silicon interface (G = 5 T, Hfl = 0.03 cm) 69 Figure 4.16: The effect of Reynolds numb er on the Peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (G = 5 T, Hfl = 0.03 cm) 70 Figure 4.17: The effect of Reynolds number on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (G = 5 T, Hfl = 0.03 cm) 70 Figure 4.18: The effect of magnetic fiel d on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-gadolinium interface (Re = 2000, Hfl = 0.03 cm) 71 Figure 4.19: The effect of magnetic fiel d on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, Hfl = 0.03 cm) 71 Figure 4.20: The effect of magnetic fiel d on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, Hfl = 0.03 cm) 72 Figure 4.21: The effect of magnetic fiel d on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-silicon inte rface (Re = 2000, Hfl = 0.03 cm) 72 Figure 4.22: The effect of magnetic fiel d on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, Hfl = 0.03 cm) 73 Figure 4.23: The effect of magnetic fiel d on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, Hfl = 0.03 cm) 73 Figure 4.24: The effect of channel dept h on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T) 75 Figure 4.25: The effect of channel dept h on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T) 75

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ix Figure 4.26: The effect of channel de pth on the periphera l average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T) 76 Figure 4.27: The effect of channel dept h on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-silicon interfac e (Re = 2000, G = 5 T) 76 Figure 4.28: The effect of channel dept h on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T) 77 Figure 4.29: The effect of channel de pth on the periphera l average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T) 77 Figure 4.30: The effect of height of ga dolinium substrate on the peripheral average interface temperature along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 78 Figure 4.31: The effect of height of ga dolinium substrate on the peripheral average heat flow rate along the ax ial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 78 Figure 4.32: The effect of height of ga dolinium substrate on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gado linium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 79 Figure 4.33: The effect of height of ga dolinium substrate on the peripheral average interface temperature along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 80 Figure 4.34: The effect of height of ga dolinium substrate on the peripheral average heat flow rate along the ax ial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 80 Figure 4.35: The effect of height of ga dolinium substrate on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silic on interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 81

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x Figure 4.36: The effect of channel spaci ng on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 82 Figure 4.37: The effect of channel sp acing on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 82 Figure 4.38: The effect of channel sp acing on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 83 Figure 4.39: The effect of channel spaci ng on the peripheral average interface temperature along the ax ial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 84 Figure 4.40: The effect of channel sp acing on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 84 Figure 4.41: The effect of channel sp acing on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 85 Figure 4.42: The effect of changing the fluid on the peripheral average interface temperature along the axia l direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 87 Figure 4.43: The effect of changing the fluid on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 87 Figure 4.44: The effect of changing the fluid on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 88 Figure 4.45: The effect of changing the fluid on the peripheral average interface temperature along the axia l direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 88

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xi Figure 4.46: The effect of changing the fluid on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 89 Figure 4.47: The effect of changing the fluid on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 89 Figure 5.1: Schematic for microchannel heat exchanger model 92 Figure 5.2: Local variation of magnetic field with time 95 Figure 5.3: Local Nusselt along the flui d-gadolinium interface at different axial locations after 1 s econd (Re = 2000, G = 5 T, Hfl = 0.03 cm) 97 Figure 5.4: Local Nusselt along the fluidsilicon interface at different axial locations after 1 second (Re = 2000, G = 5 T, Hfl = 0.03 cm) 97 Figure 5.5: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 99 Figure 5.6: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluidgadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 99 Figure 5.7: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 101 Figure 5.8: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-sili con interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 101 Figure 5.9: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-so lid interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 102 Figure 5.10: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-gadolinium interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 103

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xii Figure 5.11: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-ga dolinium interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 103 Figure 5.12: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-silicon interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 105 Figure 5.13: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-sili con interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 105 Figure 5.14: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-soli d interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 106 Figure 5.15: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-gadolinium interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 106 Figure 5.16: Peripheral average heat flow rate over 9 seconds at different axial locations at the flui d-gadolinium interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 107 Figure 5.17: Peripheral average interf ace temperature over 9 seconds at different axial locations at th e fluid-silicon interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 107 Figure 5.18: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 108 Figure 5.19: Peripheral average Nusselt number over 9 seconds at different axial locations at the fl uid-solid interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 108 Figure 5.20: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fl uid-gadolinium interface at Re = 1000 (G = 5 T, Hfl = 0.03 cm) 110 Figure 5.21: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-gadol inium interface at Re = 1000 (G = 5 T, Hfl = 0.03 cm) 110

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xiii Figure 5.22: Peripheral average interf ace temperature over 9 seconds at different axial locations at the fluid-silicon interface at Re = 1000 (G = 5 T, Hfl = 0.03 cm) 111 Figure 5.23: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-sili con interface with at Re = 1000 (G = 5 T, Hfl = 0.03 cm) 111 Figure 5.24: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-solid interface with at Re = 1000 (G = 5 T, Hfl = 0.03 cm) 113 Figure 6.1: Schematic of nanofluid circular microchannel model 115 Figure 6.2: Peripheral average interface temperature along the axial direction (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 118 Figure 6.3: Peripheral average Nusse lt number along the axial direction (Silicon, Water + 4% volume frac tion of Alumina, Re = 500, d = 0.06 cm) 118 Figure 6.4: Local interface temperatur e along the angular direction at different axial locations (Silic on, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 120 Figure 6.5: Local heat flow rate along th e angular direction at different axial locations (Silicon, Water + 4% vol ume fraction of Alumina, Re = 500, d = 0.06 cm) 120 Figure 6.6: Local heat transfer coeffi cient along the angular direction at different axial locations (Silic on, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 121 Figure 6.7: Local Nusselt number along th e angular direction at different axial locations (Silicon, Wate r + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 121 Figure 6.8: Peripheral average interface temperature and heat flow rate along the axial direction (Silic on, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 123 Figure 6.9: Peripheral average heat tran sfer coefficient and Nusselt number along the axial direction (Silic on, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 123

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xiv Figure 6.10: Interface temperature along the axial direction at different angular location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 124 Figure 6.11: Heat flow rate along the ax ial direction at different angular location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 124 Figure 6.12: Nusselt number along the ax ial direction at different angular location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 125 Figure 6.13: Peripheral average interface temperature along the axial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm) 126 Figure 6.14: Peripheral average heat flow rate along the ax ial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm) 127 Figure 6.15: Peripheral average Nusselt number along the axial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm) 127 Figure 6.16: Peripheral average interface temperature along the axial direction at different solid s ubstrates (Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 129 Figure 6.17: Peripheral average heat flow rate along the ax ial direction at different solid substrates (W ater + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 129 Figure 6.18: Peripheral average Nusselt number along the axial direction at different solid substrates (W ater + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 130 Figure 6.19: Peripheral average interface temperature along the axial direction at different Reynolds numbers (Silicon, Water + 4% volume fraction of Alumina, d = 0.06 cm) 130 Figure 6.20: Peripheral average heat flow rate along the ax ial direction at different Reynolds numbers (Silicon, Water + 4% volume fraction of Alumina, d = 0.06 cm) 131 Figure 6.21: Peripheral average Nusselt number along the axial direction at different Reynolds numbers (Silicon, Water + 4% volume fraction of Alumina, d = 0.06 cm) 131

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xv Figure 6.22: Peripheral average interface temperature along the axial direction at different tube di ameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500) 132 Figure 6.23: Peripheral average heat flow rate along the ax ial direction at different tube diameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500) 133 Figure 6.24: Peripheral average heat tr ansfer coefficient along the axial direction at different tube di ameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500) 133 Figure 6.25: Peripheral average Nusselt number along the axial direction at different tube diameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500) 134 Figure 7.1: The three dimensional s now melting system model for steady state analysis 137 Figure 7.2: Heat flow rate at the outer pipe surface along the angular direction 141 Figure 7.3: Heat flow rate at the snow boundary surface along the width of the slab 141 Figure 7.4: Average temperature at the concrete-ground interface along the width of the slab 142 Figure 7.5: Heat flow rate at the co ncrete-ground interface along the width of the slab 142 Figure 7.6: Average interface temperature at the inner pipe surface along the axial direction 144 Figure 7.7: Average bulk temperature of the fluid along the axial direction 144 Figure 7.8: Average interface temperature at the outer pipe surface along the axial direction 146 Figure 7.9: Average heat flow rate at the outer pipe surface along the axial direction 146 Figure 7.10: Average heat flow rate at the snow surface along the axial direction 148

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xvi Figure 7.11: Average heat flow rate lost to the ground along the axial direction 149 Figure 8.1: The three dimensional snow melting system model for transient analysis 152 Figure 8.2: Average periphera l interface temperature al ong axial direction at different times 157 Figure 8.3: Average peripheral interface heat flow rate along axial direction at different times 157 Figure 8.4: Average surface temperature along axial direction at different times 158 Figure 8.5: Average ground interface te mperature along axial direction at different times 159 Figure 8.6: Average ground interface heat flow rate along axial direction at different times 159 Figure 8.7: Average periphera l interface temperature al ong axial direction at different times 161 Figure 8.8: Average peripheral interface heat flow rate along axial direction at different times 161 Figure 8.9: Average surface temperature along axial direction at different times 162 Figure 8.10: Average ground interface te mperature along axial direction at different times 163 Figure 8.11: Average ground interface heat flow rate along axial direction at different times 163

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xvii List of Nomenclature d channel diameter, m D dimensionless channel diameter, d/H go heat generation rate, W/m3 G magnetic field strength, T h heat transfer coefficient, W/m2-K h* dimensionless heat tran sfer coefficient, h*L/ks H height of the substrate, m k thermal conductivity, W/m-K L channel length, m nr number of intervals in r-direction within the tube nx number of intervals in x-direction ny number of intervals in y-direction nz number of intervals in z-direction p pressure, Pa r distance in radial direction, m Re Reynolds number, Vd/ S volume of the solid substrate, m3 T temperature, oC V velocity of fluid, m/s W half of the tube spacing, m x distance along x-direction, m y distance along y-direction, m z distance along z-direction, m Z dimensionless distance along axial direction, x/L

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xviii Greek Symbols thermal diffusivity, m2/s density, kg/m3 kinematic viscosity, m2/s angular coordinate, radian dimensionless temperature, (T-Tin)/[(go.S)/(ks.d)] Subscripts f fluid in inlet max maximum r radial s solid z axial angular

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xix Analysis of Conjugate Heat Transfer in Tube-in-Block Heat Ex changers for Some Engineering Applications Abdullatif Abdulhadi Gari ABSTRACT This project studied the effect of di fferent parameters on the conjugate heat transfer in tube-in-block h eat exchangers for various e ngineering applications. These included magnetic coolers (or heaters) associat ed with a magnetic refrigeration system, high heat flux coolers for electronic e quipment, and hydronic snow melting system embedded in concrete slabs. The results of this research will help in designing the cooling/heating systems and select their appr opriate geometrical dimensions and material for specific applications. Types of problems studied in this proj ect are: steady state circular microchannels with heat source in the gadolinium substrate, transient heat transfer in circular microchannels with time varying heat source in a gadolinium substrate, transient heat transfer in compos ite trapezoidal microchannels of silicon and gadolinium with constant and time varying he at source, steady state heat transfer in microchannels using fluids suspended with na noparticles, and anal ysis of steady state and transient heat transfer in a hydronic s now melting system. For each of these problems a numerical simulation model was deve loped. The mass, momentum, and energy conservation equations were solved in the fluid region and energy conservation in the

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xx solid region of the heat exchanger to arrive at the velocity and temperature distributions. Detailed parametric study was carried out for each problem. Parameters were Reynolds number, heat source value, channel diameter or channel height, solid materials and working fluids. Results are presented in term s of solid-fluid interface temperature, heat flow rate, heat transfer coefficient, and Nu sselt number along the length of the channel and with the progression of time. The resu lts showed that an increase in Reynolds number decreases the interface temperature but increases the heat flow rate and Nusselt number. When the heat sour ce varied with time, by applying and removing the magnetic field, the interface temperature, heat flow rate, and Nusselt number attained a periodic variation with time. The decrease in the di ameter at constant Re ynolds number decreases the interface temperature and increases the he at flow rate at the fluid-solid interface.

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1 Chapter 1 – Introduction a nd Literature Review 1.1 Introduction Tube-in-block is the most popular heat exchangers in the i ndustry. Some of the applications that use this type of heat exch angers are: Magnetic refrigeration, electronic cooling, and hydronic snow melting system. Magnetic refrigeration and electronic cooling use tube-in-block heat exch angers at a microscopic scale. Magnetic refrigeration profits from the fact that the temperature of certain materials increases when placed in a magnetic field, and likewise decreases when the magnetic field is removed. This phenomenon is known as the “magnetocaloric effect”. Figure 1.1 shows a schematic drawing of the bed and heat exchanger assembly. This study is concerned with the magnetocalori c beds only. The magnocaloric beds are microchannels fabricated in a magnocaloric mate rial substrate. The material used in this project is gadolinium. When a microchannel consists of Gadolinium substrate is exposed to a magnetic field, it generates heat in the Ga dolinium substrate. This heat generation in the system is dissipated with flui d flowing through the microchannels. Electronic cooling is a technique that removes heat from a silicon wafer. It consists of microchannels embedded in a chip substrate on one side and electronic circuits located on the other si de of the wafer. A cover plate made of pyrex glass is bonded on the microchannels to form the closed channel construction. The wafer with the

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2 Figure 1.1: Schematic drawing of the be d heat exchanger assembly for magnetic refrigerating [1] Figure 1.2: Schematic drawing for a microc hannel test fixture used for electronic cooling [2]

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3pyrex cover plate was mounted on a Plexig las plate containing inlet and outlet connections for fluid flow (figure 1.2). The increase in power dissi pation of electronic circuits led researchers to st udy varieties of different desi gns to improve and come up with better microchannel performances. Some researchers introduced a way to achieve high heat dissipation by increasing the th ermal conductivity of the fluid with the suspension of nanoparticles of a solid that has much higher therma l conductivity than the fluid. Different microchannel geometries as we ll as substrate materi als and working fluid coolants have great effect on th e heat dissipation performance. Hydronic snow melting system is widely used in the industry for a range of applications such as side walks, driveways, bridges and airplane runways. The purpose of this system is to melt snow and clear the ro ad pedestrians and vehi cles. It consists of piping system embedded in concrete slab wher e hot fluid pumped in to warm up the slab. The pipes are placed in spiral pattern to distri bute heat equally within the slab. Hot fluid runs through the pipes to heat up the slab and melt the accumulated snow on the surface. Figure 1.3 shows a schematic drawing for the hydronic snow melting system. This work presents a parametric study fo r fluid flow and heat transfer in the applications of magnetic re frigeration, electronic cooli ng, and hydronic snow melting application. Some of these parameters incl ude Reynolds number, magnetic field and tube diameter. The results are presented as the peri pheral average interfac e temperature, heat flow rate, heat transfer coe fficient, and Nusselt number. Stea dy state as well as transient cases were considered in this study to arri ve to an understanding about the performance of tube-in-block heat exchangers.

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4 Figure 1.3: Schematic drawing for hydronic snow melting system [3] 1.2 Literature Review 1.2.1 Steady State Heat Transf er in Rectangular and Tr apezoidal Microchannels Peng and Peterson [4] experimentally inve stigated single-phase forced convective heat transfer of water in small rectangular microchannels for different diameters. The results stated that the geometry had a signi ficant effect on the si ngle-phase convective heat transfer. It was also found that the lami nar heat transfer is dependent upon the aspect ratio. The turbulent was found to be a function of a non-dimensi onal variable Z, such that Z=0.5 is the optimum configur ation regardless the channe l’s aspect ratio. Empirical correlations were suggested fo r calculating both the heat transfer and the pressure drop. Papautsky et al. [5] described the effect of a rect angular microchannel aspect ratio on

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5laminar friction constant. The experimental data obtained for the water showed an approximate 20% increase in the friction cons tant for a specified driving potential when compared to micro-scale predictions from the classical Navier-Stokes theory. Lower aspect ratio also showed a substantial increase of 20 % in friction constant. The experiment data also showed a similar increase when lo w Reynolds numbers are used (less than 100). Qu et. al. [6] investigated heat transfer ch aracteristics of water flowing through trapezoidal silicon microchannels. A num erical analysis was also carried out by solving a conjugate heat transfer problem to determine the temperature field in both solid and fluid regions. When comparing the e xperimental results to those numerical predictions it was found that Nusselt number found experimentally is much lower. This may be due to the effects of surface roughne ss of the microchannel walls. A modified relation that was established based on r oughness-viscosity model was suggested to account for the roughness-viscosity effect s in future experimental work. Rahman [2] presented new experimental measur ements for pressure drop and heat transfer coefficient in microchannel heat sinks Two different channel patterns were used: the parallel pattern and the series pattern. Ch annels of different de pths or aspect ratio were studied while water was used as the working fluid. Flow rate, pressure and temperature were measured at several locati ons in the wafer to calculate the local and average Nusselt number and coefficient of fric tion in the device for different flow rate, channel size, and channel configuration. Yang et al. [7] studied the entry flow induced by an applied electrical potent ial through microchannels betw een two parallel plates. A nonlinear, two dimensional Poisson equation, zeta potential and Nernst-Planck equation were numerically solved using a finite difference method. A body force was included in

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6the full Navier-Stokes equations. The effect of the entrance region on the fluid velocity distribution, charge density boundary layer, entrance le ngth, and shear stress are discussed. It was found that the thickness of electrical doub le layer (EDL) in the entry region is thinner than that in the fully de veloped region. The change in the velocity profile was apparent in the en try region and the ax ial velocity profile is no longer flat across the channel height when the Reynolds number is large. Quadir et. al. [8] applied a finite element method to evaluate the perfor mance of microchannel heat exchangers used in electronic packaging. The finite element method was prove d satisfactory to predict the surface temperature and fluid temperature when compared with other results obtained from different concepts. This method allows us to calculate the total thermal resistance of the microchannel heat sink from the surf ace and fluid temperature fields. This methodology added an advantage of modeling a non-uniform surface heat flux distribution and could be used as an a lternative to massive CFD calculations. Rahman and Shevade [9] studied square and recta ngular microchannel with heat generation in the substrate. Water was the wo rking fluid in a gado linium substrate. The governing equations in both so lid and fluid regions were solved numerically for the velocity and temperature profiles. Varying the aspect ratio, Reynolds number, and heat generation, it was found that Nu sselt number is larger near the entrance and decreases downstream due to the development of the th ermal boundary layer. It was also noticed that for larger Reynolds number the outlet temperature decreased and the average heat transfer coefficient increased. Rahman et. al. [ 10] has studied the analys is of heat transfer processes during the h eat up and cool down phases of a ma gnetic material substrate when subjected to a magnetic field. A computer simulation of fluid flow and heat transfer are

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7carried out. Rectangular and square microcha nnels were considered where water was the working fluid and Gadolinium was the substrat e material. The parametric study included heat generation, aspect ratio and Reynolds num ber. They found a sinus oidal behavior for the heat transfer coefficient along time. Tunc and Bayazitoglu [11] studied the convection heat transfer in rectangular microchannel and assumed a fully de veloped both thermally and hydrodynamically fluid flow. Because of the rectangular cross-section shape momentum equations were solved first and then substituted in the energy equation. The integral transform technique is applied twice, once for velocity and once for temperature. This gives a uniform temperature and unifo rm heat flux boundary conditions. The results showed similar behavior to previous studi es on circular microt ubes. The values of Nusselt number were presented by varying the as pect ratio. Pfund et al. [12] determined friction factor for high aspect ratio microc hannels. Reynolds number was between 60 and 3450. Pressure drop were measured within th e channel. Transition to turbulence was observed with flow visualizati on. Kohl et al. [13] experime ntally examined the pressure drop in microchannels. Straight channel test section with rectangul ar microchannels was used. Koo and Kleinstreuer [14] investigated the effects of visc ous dissipation on the temperature field and ultimately on the friction factor using dimensional analysis and computer simulation. It turned out that fo r microconduits, viscous dissipation is a strong function of the channel aspect ratio, Reynolds number, Eckert number, Prandtl number, and conduit hydraulic diameter.

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81.2.2 Steady State Heat Transfer in Circular Microchannels Lelea et al. [15] conducted an experimental and numerical research on heat transfer and fluid flow in microtubes. Th e diameters selected were 0.1, 0.3 and 0.5 mm and Reynolds number range used was up to 800. Distilled water was used as the working fluid and stainless steel as the substrate mate rial. The experimental results confirmed the conventional or classical theo ries are applicable for wate r flow through microchannel of above size. Yu et. al. [16] investigated fluid flow and h eat transfer characteristics of dry nitrogen gas and water in microtubes with diameters of 19, 52, and 102 micrometers were investigated with Reynolds number rangi ng from 250 20,000 and Prandtl number ranging from 0.7 to 5. Lower values for the product f*Re were observed for smaller diameters. The reduction in f*Re was more for laminar flow. The heat transfer and Nusselt number were much higher in the turbulent regime for smaller diameters. Adams et. al. [17] investigated turbulent, single phase forced convention of wate r in circular microchannels with diameters of 0.76 and 1.09 mm. It was found that Nusselt number resulted are higher than those predicted by traditional large channel correlations. Based on data presented and earlier data for smaller diameter channe ls, a generalized correlation was developed for the Nusselt number for turbulent, single-phase, forced convection in circular microchannels. Owhaib and Palm [18] experimentally investigated the heat transfer characteristics of single-phase forced convention of R-134a through microtubes. The diameters were 1.7, 1.2, and 0.8 mm and both laminar and turbul ent flows were employed with Reynolds number from less than 1,000 to 17,000. The results were compared to both correlations suggested for macroscale and microscale ch annels. The experiment al results in the

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9turbulent regime showed agreement with correlations of the classical macroscale correlations but not with any of the sugge sted correlations for microchannels. The experimental results for the heat transfer co efficient in the laminar region were almost identical for all three diameters. Celata et. al. [19] studied the expe rimental researches carried out in single phase heat transfer a nd flow in capillary tubes. The laminar and transition regimes were analyzed in detail to clarify the discrepa ncies among the results obtained by different researchers. The exam ined experiments showed that the friction factor is in good agreement with the Hage n-Poiseuille theory for Reynolds numbers below 600-800. The transition from laminar to turbulent regime occurred at Reynolds number between 1900 and 2500. Diabatic expe riments showed that heat transfer correlations in laminar and turbulent regimes, developed for conventional tubes, were not properly adequate for heat transfer coeffi cients predictions in microtubes. Rao and Rahman [20] investigated a steady state lamina r flow for a circular microtube in a rectangular substrate. Silicon, Silicon Carbide, and stainless steel were used for substrate materials while water and FC-72 were used as the coolants. In add ition, Reynolds number and geometrical dimensions were varied fo r a thorough investigati on. The results showed that Nusselt number was highest for the Sili con substrate and FC72 coolant case and lowest for the stainless substr ate and water coolant case. It also showed that increasing the hydraulic diameter and Reynolds number results in higher average Nusselt number. Nield and Kuznetsov [21] analytically examined steady laminar flow of an incompressible fluid through a tube of almost circular cross-sec tion, first for the case of a tube whose wall is wavy in the azimuthal direction, and then for one wavy in the axial direction. Giulio and D’Agaro [22] investigated the roughness effects on heat transfer and

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10pressure losses in microscale tubes and ch annels using a finite element CFD code. Surface roughness was explicitly modeled th rough a set of randomly generated peaks along the ideal smooth surface. Grohmann [23] presented an experimental technique and experimental results of heat transfer measurements in microtubes of 250 and 500 m diameter. The data obtained with single pha se argon showed no physical difference of heat transfer mechanism between microand macrotubes. Broderick et al. [24] analyzed the thermally developing electro-osmotically generated flow within circular microtubes with finite Debye-layer thickness. The effect of variations in the rela tive microtube radius and strength of the Joule and viscous heating on the thermal transport were explored over the possible ranges of the governing parameters Chakraborty [25] developed closed form expressions for Nusselt number variation in a thermally fully developed microtube flow, under a combined influence of electroosmoti c forces and imposed pressure gradient. Significant insights were also developed regarding the infl uence of adverse pressure gradients on the thermal transport in the presence of aiding electroosmotic effects. Hwang and Kim [26] investigated the pressu re drop characteristics in microtubes using R-134a as the test fluid. A new correlation to predict the twophase flow pressure drop in microtubes was developed. 1.2.3 Transient Heat Transfer in Microchannels Yang et. al. [27] studied transient aspe cts of electroosmotic flow in a slit microchannel. Exact solution for the electr ical potential profile and the transient electroosmotic flow field are obtained. This was done by solving the complete PoissonBoltzmann equation and the Navier-Stokes eq uation under an analytical approximation

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11for the hyperbolic sine function. The characteri stics of the transien t electroosmotic flow were discussed under influences of the el ectric double layer and ge ometric size of the microchannel. 1.2.4 Heat Transfer Using Nanofluids There are researchers who worked on nanof luids such as Eastman et. al. [28] who studied nanofluids for ethylen e glycol suspended with copp er and oxide nanoparticles. Based on his calculations, there was a si gnificant increase in the fluid thermal conductivity by 40% for 0.3% volum e fraction of copper in ethylene glycol for diameters less than 10 nm. The effects are anomalous base d on previous theoretical calculations that had predicted a strong effect of partic le shape on effective nanofluid thermal conductivity, but no effect on either partic le size or particle thermal conductivity. Keblinski et.al. [29] explai ned the increase in the composite thermal conductivity due to grain size reduction by explor ing four possible explanatio ns: Brownian motion of the particles, molecular-level layering of the liquid at the liquid/particle interface, the nature of the heat transport in the nanoparticles, a nd the effects of nanopart icles clustering. He concluded that the key factor to understand the thermal properties of nanofluids are the ballistic, rather than diffusi ve, nature of heat transport in the nanoparticles, combined with direct or fluid-mediated clustering effect s that provide paths for rapid heat transport. Cheng and Law [30] have introduced an e xponential model to calcu late the effective viscosity for a fluid suspended with nanoparticles. Theoretical consideration is restricted to the dilute condition without effects of dynamic particle interactions and fluid turbulence. This led to power series expressed in terms of particle concentration. The

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12derivation then extended using an exponential model for the condition of high particle concentration. His model was f ound to be comparab le to various em pirical formulas available in the literature. Putra et. al. [31] has investigated natural convection for nanofluids inside horizontal cylinder heated from one end and cooled from the other. His experiment resulted in paradoxical behavior of heat transfer deterior ation. He investigated this deterioration in the heat transfer and its dependence on parameters such as the particle concentration, material of the particles a nd the geometry of the containing cavity. He suggested more investigation done before prac tical applications of cooling system using nanofluids to understand the physical phenomenon. Bang and Chang [32] have studied the boiling heat transfer charac teristics of water suspended with nanoparticles of alumina with different volume concentration. Pool boiling heat transfer coefficients and phenomena were compared with those of pure wa ter. The experimental results show that these nanofluids have poor heat transfer perf ormance compared to pure water in natural convection and nucleate boiling. On the other hand, CHF has been enhanced in not only horizontal but also vertical pool boiling. Comparisons to the Rhosenow correlation showed that the correlation can potentially pr edict the performance with an appropriate modified liquid-surface combination factor and changed physical properties of the base liquid. Vadasz et. al. [33] has investigated theoretically the impressive increase in thermal conductivity of nanofluids achie ved experimentally in the li terature. His work was done on a macroscale level aiming to explain the possible mechanism behind the impressive increase in the composite thermal conductiv ity. He explained that the thermal wave effects via hyperbolic heat conduction could have been the reason behind the excessive

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13improved effective thermal conductivity. He suggested that alte rnative po ssibilities should be investigated before reaching an ultimate conclusion. Wen and Ding [34] examined the effect of particle migration on heat transfer under a fully developed laminar flow regime The flow model takes into account the effects of shear-induced and visc osity-gradient-induced particle migration as well as self diffusion due to Brownian motion, which is co upled with an energy equation. The results suggest a significant non-uniformity in partic le concentration. This leads to a higher Nusselt number compared to constant thermal conductivity assumption. Further improvement of the model is needed to take into account other factor s such as entrance effects, dynamics of particles and particle-wa ll interactions. Maiga et. al.[35] has studied the forced convection flow of water-Al2O3 and ethylene glycol-Al2O3 nanofluids inside a uniformly heated tube that is submitted to a co nstant and uniform heat flux at the wall. It was found that nanoparticles enha nced the heat transfer at the tube wall for both laminar and turbulent flows. The improvement of heat transfer increases even more with the increase in nanoparticles c oncentration. The presence of nanoparticles also produces adverse effect on the wall frict ion that increases with the particle volume concentration. Results also showed that the ethylene glycolAl2O3 mixture gives a far better heat transfer enhancement than water-Al2O3 mixture. Wang and Xu [36] measured the effective thermal conductivities for fluids su spended with nanopartic les by a steady state parallel method. They tested water, vacuum pump fluid, engine oil and ethylene glycol suspended with Al2O3 and CuO. Their experimental results showed that nanofluids have higher thermal conductivities than those of base fluids. Comp ared to theoretical models, the predicted thermal co nductivities of mixtures are much lower than the measured data.

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14They suggested that more comprehensive theory is needed to fully explain the nanofluid behavior. 1.2.5 Hydronic Snow Melting System Before 1952, researchers only considered th e energy required to melt the snow. In 1952, Chapman published two articles consideri ng the requirements of heat and mass transfer. The first [37] one described five energy requirements fo r the snow melting process. These five energy requirements are th e heat of fusion, sensible heat gain from snowfall, heat of vaporization, heat transfer by radiation and convection, and back loss to the ground. The sum of the first four terms e quals the required heat output at the upper surface. The second [38] article suggested considering a frequency distribution of the loads. The article confirms th at the actual load should be de termined on an hourly basis. Then make a frequency distribution analysis to set the design capacity. The author introduced the concept of free area ratio in th is article. Chapman [ 39] also showed that four factors contribute to the to tal load. None of the four f actors can be singled out. Only the frequency analysis of combined four loads is sufficient to determine the required thermal output of the system. He also addresse d the issue of relating the inches of water equivalent to inches of snowfall topic. Updated design guidelines and recommendations were presented by Ramsey et al [40]. He re viewed and identified recommended revisions to the current ASHRAE snow melting load calculation procedures. It also provided sample results based on the revised procedure. The load at the melting surface included sensible and snow melting loads along with th e heat losses due to convection, radiation, and evaporation. The correlations used are accep ted for turbulent convection heat transfer

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15coefficient. The convection an d evaporation losses are calcu lated as functions of wind speed and the characteristic di mensions of the slab. Calcul ations are performed for a baseline case using the wind speed as re ported from the meteorological data. Kilkis [41] studied the complex hy dronic space conditioning and snow melting circuits. These systems operate at moderate supply temperature at the expense of large temperature drop and wide dive rsity of generally oversized equipment. He presented a simple analytical algorithm that can accurate ly calculate equipment temperature drop and heating capacity, energy loss and fluid flow ra te in a complex circuit. Kilkis has also published two papers in 1994. In the first article [42], he discussed the need to develop a universal and simple technique that doe s not require extens ive manipulation of meteorological data in the first publication. He recommended that the design algorithm should require only the air temperature, wind speed and maximum recorded daily snowfall in order to calculate the heat requi rement. In the second publication [43] he developed a simple analytical technique to pred ict the transfer of h eat in the snow melting slab while retaining sufficient accurac y. Bounded by ASHRAE guidelines, his method was applied for a limited design range using meta l pipes. Results of sample designs were compared to solutions used finite elemen t method. The comparison indicated sufficient accuracy for engineering calculations. Rees et al [44] addressed the issue that the transient response of snow melting systems fo r pavement are operated intermittently and that it has a significant effect on the overall system performance. He developed a twodimensional numerical transient analysis method that includes a boundary condition model able to deal with di fferent snow melting conditions. Parametric study was carried out examining the transient effects using real storm data for a real storm event and the

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16results were presented. He found that effects of storm structur e were of most significance. He also presented calculations of the back and edge losses under transient conditions. 1.3 Objectives From the microchannel literature review, it appears that none of the previous studies have addressed the heat generation with in the substrate material for circular micro tubes. The main emphasis of this study is to develop a simulation model for fluid flow and heat transfer in circular microchannels by taking into account the heat generation in the solid due to applied magnetic field, conducti on in the solid and convection of heat to the fluid. Gadolinium was chosen to be the solid substrate material and water as the working fluid. Detailed parametric study is carried out to study th e effect of Reynolds number, heat generation rate and diameter of the tube. Steady state and transient cases were studied to better understand the effect of different parameters on the performance of the system. Nanofluids were investigated in the literature review. Researchers presented experimental and theoretical work on nanoflu ids. One researcher presented a molecular dynamic simulation based on mathematical mode l to explore four possible explanations for the significant increase in the nanoflui ds thermal conductivity. The simulation did not show a parametric study to understand the eff ect of different parameters on the system. This project suggests that nanofluid is applied into an existing microtube model developed by Rao and Rahman [20] and st udy the nanofluids under different conditions. This could result in new understanding to th e behavior of nanoflu ids under conditions existing in the literature. Different solid substrate material and working nanofluids were

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17chosen. Parametric study was carried out to study the effect of Reynolds number, and diameter of the tube. Literature has provided a number of hydronic snow melting researches to calculate the heat load required for snow melting at different steady and transient conditions. Results were tabulated for differe nt conditions and para meters. These results of previous work are based on a two dimensi onal calculations ignori ng the fact that the fluid looses heat and drops in temperature wh ile transferring through the slab in the third direction. As the fluid temperature drops along th e slab in the axial di rection, it decreases the heat flow rate to the system and thus the snow melting process. Design parameters were studied in previous work for calculati on of the heat load required for snow melting at different conditions. Their direct effect on the heat flow rate and snow melting performance was not emphasized. In this project, a three dimensional model was presented to study the heat fl ow rate process within the slab and at the snow melting surface. A numerical simulation was develo ped to solve the nonlinear system of discretisized equations of conservation laws in the model. Parametric study was carried out studying the effect of several parameters on the performance of steady state system. The transient effect was analyzed under two different storm conditions. The main objectives of the current work are: To develop a numerical model for fluid fl ow and heat transfer in tube-in-block heat exchangers exposed to different kinds of boundary conditions. To conduct a parametric study to understand how different parameters affect the performance of tube-in-block heat exchangers.

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18 To explore both steady state and transient models to understand how the time factor affects the performance of tube-in-block heat exchangers. To study different designs of tube-in-bloc k heat exchangers expands our choices to use the appropriate desi gn for a certain application.

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19 Chapter 2 – Steady State Heat Transfer in Circular Microchannels During Magnetic Heating or Cooling 2.1 Mathematical Model The problem in hand is a microchannel a ssembly with circular channels in a rectangular solid (Gadolinium) substrate. Th e channel/substrate exte nds to a length L. The thickness of the substrate (o r the distance between tubes in the vertical direction) is H and distance between tubes in the horiz ontal direction is 2W. Figure 2.1 shows a schematic drawing of the model. Fluid (wat er) flows through circul ar channels with diameter D and length L as si ngle pass from inlet to outlet manifold. Due to symmetry, the analysis can be done by considering half of a tube and associated solid material as shown in figure 2.2. Magnetic field (G) is applied and the heat (go) is generated throughout the substrate. Heat is conducted through the substrate solid material then convected to the working fluid in the microchannels. The governing equations for the conserva tion of mass, momentum, and energy in the liquid region are [45]: 0 1 1 z V V r V r r Vz r r (1) V r r V z V V r r V r r V r p V r z V V V r V r V Vr r r r r f r z r r r 2 2 2 2 2 2 2 2 2 22 1 1 1 1 (2)

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20 Figure 2.1: Schematic drawing for the circular microchannel model Figure 2.2: Schematic drawing for the circular microchannel simulated model

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21 r f r z rV r r V z V V r r V r r V p r r V V z V V V r V r V V2 2 2 2 2 2 2 2 22 1 1 1 (3) 2 2 2 2 2 2 21 1 1 z V V r r V r r V z p z V V V r V r V Vr z z z f z z z z r (4) 2 2 2 2 2 2 21 1 z T T r r T r r T z T V T r V r T Vf f f f f f z f f r (5) Considering constant thermal conductivity, th e energy conservation e quation in the solid region with heat generation is [46]: 02 2 2 2 2 2 k g z T y T x To s s s (6) Note that a cylindrical coordinate system was used to model convection within the circular tube while a cartesian coordinate system was used to model conduction within the solid substrate material. Equations (1) to (6) are subject to the following boundary conditions: At z = 0, 0 r < d/2: in f f in z z rT T V V V V, ,, 0 0 (7) At z = 0, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (8) At z = L, r < d/2: 0 p (9) At z = L, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (10) At x = 0, -d/2 y +d/2, 0 z L: 0 0 0 0 x T x V x V Vf z r (11)

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22At x = 0, -H/2 y -d/2, 0 z L: 0 x Ts (12) At x = 0, +d/2 y +H/2, 0 z L: 0 x Ts (13) At x = W, -H/2 y +H/2, 0 z L: 0 x Ts (14) At y = -H/2, 0 x W, 0 z L: 0 y Ts (15) At y = +H/2, 0 x W, 0 z L: 0 y Ts (16) At 0 z L, r = d/2: dr T k dr T k T Ts s f f s f (17) 2.2 Numerical Simulation and Parametric Study The governing equations along with the boundary conditions were solved using the Galerkin finite element method. Four-node quadrilateral elements were used. In each element, the velocity, pressure, and temperat ure 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 te mperature fields. The solution was considered converged when the field values did not chan ge from one iteration to the next, and the residuals for each variable became negligible. For the numerical computations, the hei ght (H) and the half width (W) of the model were set to 0.2 cm and 0.236 cm respect ively. The original (repeating) case used a diameter of 0.036 cm. Diameter was change d from 0.012 cm to 0.048 cm having a (d/H)

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23ratio of 0.06 to 0.24. Reynolds number was varied between 1000 and 2200. Magnetic field (G) value of 5 and 10 T were used to study the effects of changing the magnetic field. Magnetic field value was translated into heat generation per unit volume in the substrate material (go) by using experimental data of Pechasky and Gschneider [47]. Gadolinium was used as the solid substrate material. Gadolinium is one of the materials that generate significant heating and cooli ng when exposed to a magnetic field under room temperature. Water was used as the primary working fluid. Two more fluids, namely Ammonia and R134a, were used to comp are their results to those of water. The local solid-fluid interface temp erature, heat transfer coefficient and Nusselt number were calculated from the computed velocity and temperature distributions. 2.3 Results and Discussion The distribution of cells in the comput ational domain was determined from a series of tests with different number of elemen ts in the x, y, r (within the tube), and z directions. The results obtained by using (nx = 6, ny = 12, nr = 6, nz = 20) that had an average error of 1.29% and a maximum error less than 2% compared to (nx = 10, ny = 20, nr = 10, nz = 40). Figure 2.3 shows th e dimensionless periphe ral average interface temperature for different grid systems. Quanti tative error analysis is one way to validate the accuracy of the results from a grid independent study. This was performed and presented in Appendix H. Figure 2.4 shows the local Nusselt number variation around the tube periphery for different axial locations. The effect of the substrate’s recta ngular shape is obvious along the angular direction of the tube. It ma y be noticed that local Nusselt number is

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24 0 0.2 0.4 0.6 0.8 1 1.2 1.4 00.20.40.60.81 Dimensionless Axial DirectionDimensionless Average Peripheral Interface Temperature, nx = 4, ny = 8, nr = 4, nz = 20 nx = 6, ny = 12, nr = 6, nz = 30 nx = 8, ny = 16, nr = 8, nz = 40 nx = 10, ny = 20, nr = 10, nz = 40 Figure 2.3: Local dimensionless peripheral average interface temperature for different grid sizes (Re = 1600, G = 5 T, d = 0.036 cm, Water) 0 2 4 6 8 10 12 14 16 18 -90-60-300306090 Angular direction, Local Nusselt numbe r Z = 0.1 Z = 0.2 Z = 0.4 Z = 0.6 Z = 0.8 Z = 0.9 Figure 2.4: Local Nusselt number variation for different axial locations along the angular direction (Re = 1600, G = 5 T, d = 0.036 cm, Water)

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25minimum at angular locations corresponding to the corners of the substrate where thermal resistance within the solid is maximum. It ch anges slightly in the middle part of the tube, whereas becomes larger at bot h top and bottom ends. The vari ation indicates that shape of the substrate influences the local heat tran sfer rate as expected in a conjugate problem. Figure 2.5 shows the effects of both the change in Reynolds number and magnetic field strength on the dimensionless peripheral average interface temperature. Heat is generated in the substrate causi ng the temperature to rise. The low inlet temperature cools down the substrate. As th e fluid travels through the channel, its temperature increases providing less cooling to the substrate. This causes the interface temperature to increase along the axial dire ction. Higher Reynolds number decreases the interface temperature by providing more coo ling to the interface. Higher magnetic field generates more heat in the substrate causing an increase in the interface temperature. On the other hand, the dimensionless periphe ral average interface temperature does not change with the magnetic field. Figure 2.6 shows the effects of both the change in Reynolds number and magnetic field value on the Nusselt number. Interface temperature increases along the axial direc tion causing the temperature gradient in the solid to decrease. As a result, the heat flow rate de creases along the axial direction. The Nusselt number directly depends on the heat flow rate and thus it decreases along the axial direction. At lower Reynolds number, a lowe r fluid velocity cause s both the fluid and interface temperatures to increase. The increase in the interface temperature is larger than the increase in the fluid bulk temperature causing an increase in the bulk-interface temperature difference. This increase in the bulk-interface temperature difference decreases the Nusselt number w ith the lowering of Reynolds number. The change in the

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26 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 00.20.40.60.81 Dimensionless Axial Coordinate, ZDimensionless Peripheral Average Interface Temperature, Re=1000, G=5T Re=1600, G=5T Re=2200, G=5T Re=1000, G=10T Re=1600, G=10T Figure 2.5: Local dimensionless peripheral average interface te mperature along axial coordinate at various Reynolds numbers (d = 0.036 cm, Water) 0 2 4 6 8 10 12 14 00.20.40.60.81 Dimensionless Axial Coordinate, ZPeripheral Average Nusselt Number Re=1000, G=5T Re=1600, G=5T Re=2200, G=5T Re=1000, G=10T Re=1600, G=10T Figure 2.6: Local peripheral average Nusselt number along axial c oordinate at various Reynolds numbers (d = 0.036 cm, Water)

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27magnetic field value had no significant effect on the heat transfer coefficient. This is because with the increase of magnetic field st rength, both heat flux at the interface and difference between interface and bulk fluid temperature increases by the same amount. Therefore, there was no significant change in Nusselt number with the change in the magnetic field. The change in diameter of the channe l is examined next. When changing the diameter, the velocity will change for a constant Reynolds number or the Reynolds number will change for a constant veloci ty. Figure 2.7 shows the local dimensionless peripheral average interface temperature along the axial direction for different diameters at constant Reynolds number. Smaller diam eter at the same Reynolds number gives higher fluid velocity. This shou ld decrease the interface te mperature. On the contrary, smaller diameter also results in larger soli d volume causing more h eat generated from the solid body. This increases the heat flow to the fluid. This study s hows that the interface temperature is decreasing with diameter, wh ich means that the interface temperature is affected more by the change of the volume of the solid substrate. Figure 2.8 shows the variation of the dimensionless peripheral average heat transfer coefficient for different diameters of the channel. As the tube di ameter decreases, the interface temperature increases causing the interface-bulk temperat ure difference to increase. Temperature in the solid substrate increases due to more heat generation caused by the increase in the solid volume. This causes an increase in heat flow rate. The heat transfer coefficient is the ratio of the heat flow rate to the interface-bulk temp erature difference. This study showed that the increase in the heat flow ra te is much more than the increase in the interface-bulk temperature difference. The net re sult is an increase in the heat transfer

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28 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.20.40.60.81 Dimensionless Axial Coordinate, ZDimensionless Peripheral Average Interfac e Temperature, D=0.12, V=6.55 m/s D=0.18, V=4.36 m/s D=0.24, V=3.27 m/s Figure 2.7: Local dimensionless peripheral average interface te mperature along axial coordinate at various diameters and cons tant Reynolds numbers (Re = 1600, G = 5 T, Water) 8 10 12 14 16 18 20 00.20.40.60.81 Dimensionless Axial Direction, ZDimensionless Peripheral Average Heat Transfer Coefficient, h* D=0.12, V=6.55 m/s D=0.18, V=4.36 m/s D=0.24, V=3.27 m/s Figure 2.8: Local dimensionles s peripheral average heat transfer coefficient along axial coordinate at various diameters and cons tant Reynolds numbers (Re = 1600, G = 5 T, Water)

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29 0 2 4 6 8 10 12 14 00.20.40.60.81 Dimensionless Axial Coordinate, ZPeripheral Average Nusselt Number D=0.12, V=6.55 m/s D=0.18, V=4.36 m/s D=0.24, V=3.27 m/s Figure 2.9: Local peripheral average Nusselt number along axial c oordinate at various diameters and constant Reynolds nu mbers (Re = 1600, G = 5 T, Water) coefficient as the diameter becomes smalle r. Figure 2.9 shows the variation of the Peripheral average Nusselt number for diffe rent channel diameters. Nusselt number directly depends on the heat transfer coeffi cient and the channel diameter. The decrease in the channel diameter overcomes the increase in the heat transfer coefficient. The net result is a decrease in Nusselt number due to the decrease in channel diameter. Figure 2.10 shows the effect of changi ng the diameter on the dimensionless peripheral average interface temp erature while keeping the flui d velocity at the entrance constant. Smaller diameter results in la rger solid substrate providing more heat generation. This increases the heat flow rate to the fluid, and hence, increases the interface temperature. Figure 2.11 shows the effect of changing the diameter on the peripheral average Nusselt number while keep ing the fluid velocity at the entrance

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30 0.0 0.5 1.0 1.5 2.0 00.20.40.60.81 Diensionless Axial Coordinate, ZDimensionless Peripheral Average Interface Temperature, D=0.12, Re=1066 D=0.18, Re=1600 D=0.24, Re=2133 Figure 2.10: Local dimensionless periphera l average interface temperature along axial coordinate at various diameters and constant inlet velocity (Vin = 4.367 m/s, G = 5 T, Water) 0 2 4 6 8 10 12 14 16 00.20.40.60.81 Dimensionless Axial Coordinate, ZPeripheral Average Nusselt Number D=0.12, Re=1066 D=0.18, Re=1600 D=0.24, Re=2133 Figure 2.11: Local Peripheral average Nusselt number along axial coordinate at various diameters and constant inlet velocity (Vin = 4.367 m/s, G = 5 T, Water)

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31constant. As the tube diameter decreases, the interface temperature increases causing the interface-bulk temperature difference to incr ease. Temperature in the solid substrate increases due to more heat generation caused by the increase in the solid volume. This causes an increase in heat flow rate. The heat transfer coefficient is the ratio of the heat flow rate to the interface-bulk temperature difference. The increase in the interface-bulk temperature difference overcomes the increase in the heat flow rate resulting in a decrease in the heat transfer coefficient. Nu sselt number has similar trend as those of the heat transfer coefficient because of its direct dependence on the heat transfer coefficient. Note that both the diameter and the heat tran sfer coefficient are decreasing together as the tube diameter decreases. Figure 2.12 shows the local peripheral average Nusselt number variation for different working fluids at two different axial locations. As explained above, Nusselt number is smaller around larger solid area b ecause of the smaller temperature gradient in the larger solid region. Nusselt number for water is the highest. Figure 2.13 shows the dimensionless peripheral average interf ace temperature changes along the axial coordinate for different worki ng fluids at the same Reynolds number. To run the fluids as liquid at the appropriate te mperature range the inlet temperature was chosen as –30 oC for both ammonia and R-134a refrigerants. Inlet velocities for these fluids were selected in such a way that Reynolds number remains cons tant. The lowest velocity was obtained for R-134a and the highest for water. This contribu ted to different performance of these three working fluids. A higher fluid velocity results in lower rate of increase of interface temperature along the axial direction. A mu ch higher rate of increase of interface temperature is given by R-134a wh ich has the lowest inlet veloci ty for the same Reynolds

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32 0 2 4 6 8 10 12 14 -90-60-300306090 Angular direction, Local Nusselt numbe r water at Z = 0.2 water at Z = 0.8 Ammonia at Z = 0.2 Ammonia at Z = 0.8 R-134a at Z = 0.2 R-134a at Z = 0.8 Figure 2.12: Local Nusselt number variati on for different work ing fluids along the angular direction (Re = 1600, G = 5 T, d = 0.036 cm) -6 -5 -4 -3 -2 -1 0 1 2 3 00.20.40.60.81 Dimensionless Axial Coordinate, ZDimensionless Peripheral Average Interface Temperature, Water Amonia R-134a Figure 2.13: Local dimensionless periphera l average interface temperature along axial coordinate for different fluids (Re = 1600, G = 5 T, d = 0.036 cm)

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33 0 2 4 6 8 10 12 14 16 18 20 00.20.40.60.81 Dimensionless Axial Direction, ZDimensionless Peripheral Average Heat Transfer Coefficient, h* Water Amonia R-134a Figure 2.14: Local dimensionless peripheral av erage heat transfer coefficient along axial coordinate for different fluids (Re = 1600, G = 5 T, d = 0.036 cm, Water) number. Figure 2.14 shows the dimensionless pe ripheral average heat transfer coefficient changes along the axial coordinate for diffe rent working fluids at the same Reynolds number. The interface temperature increases alo ng the axial direction. This decreases the temperature gradient at the interface, and hence, the heat flow rate to the fluid. It also increases the interface-bulk temperature differen ce. Both of these cause the heat transfer coefficient to decrease along the axial directi on. An increase in inlet velocity decreases the interface temperature. This increases th e temperature gradient at the interface, and hence, the heat flow rate to the fluid. It also decreases the interface-bulk temperature difference. Both of these cause the heat transf er coefficient to increase. Therefore, water has the highest heat transfer coefficient while R-134a has the lowest. It may be noted that the decrease in the interface-bulk temperatur e difference was much larger than the

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34 0 2 4 6 8 10 12 00.20.40.60.81 Dimensionless Axial Coordinate, ZPeripheral Average Nusselt Number Water Amonia R-134a Figure 2.15: Local peripheral average Nusselt number along axial coordinate for different fluids (Re = 1600, G = 5 T, d = 0.036 cm, Water) increase in the heat flow rate. Figure 2. 15 shows the peripheral average Nusselt number changes along the axial coordinate for diffe rent working fluids at the same Reynolds number. Nusselt number depends directly on th e heat transfer coefficient and the fluid thermal conductivity. R-134a has lower thermal conductivity compared to water and ammonia. This causes its Nusselt number to be higher than it is for ammonia. The maximum dimensionless temperature in the substrate occurs in the farthest corner from the channel entrance. A higher Re ynolds number enhances the heat flow rate to the fluid and, thus, decreases the maxi mum temperature in the system. A smaller diameter at constant Reynolds number result s in higher fluid velocity but lower mass flow rate. A smaller mass flow rate means higher temperature in the system and hence higher maximum temperature. This is show n in figure 2.16. Figure 2.17 shows the

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35 0 0.5 1 1.5 2 2.5 3 3.5 700100013001600190022002500 Reynolds NumberDimensionless Maximum Temperature, max D=0.12 D=0.18 D=0.24 Figure 2.16: Dimensionless maximum temperat ure in the system at various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 0 10 20 30 40 50 60 70 80 700100013001600190022002500 Reynolds NumberPressure Difference, kPa D=0.012 D=0.018 D=0.024 Figure 2.17: Pressure drop in the model at various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water)

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36pressure drop from inlet to exit of the ch annel that resulted fo r various diameters and Reynolds numbers. The pressure drop is highe r for smaller diameter This is expected because the flow encounters larger frictional resistance from the walls. The pressure drop also increases with Reynolds number. This is because higher velocities cause higher frictional resistance. Figures 2.18 and 2.19 show the dimensionless periphera l average heat transfer coefficient and peripheral average Nusselt number at Z = 0.4 and Z = 0.8 along the axial direction from the inlet for various Reynolds number and diameters. For the same diameter, the increase in Reynolds number, or the inlet velocity, de creases the interface temperature. This increases the temperature gr adient at the interface, and hence, the heat flow rate to the fluid. It also decreases the interface-bulk temperature difference. The increase in the heat flow rate and the decrease in the interface-bulk temperature difference both causes the heat transfer coe fficient to increase, and hence, the Nusselt number. For the same Reynolds number, smalle r diameter gives higher velocity which results in lower interface temperature. Lowe r interface temperature, as previously discussed, results in an increase the heat flow rate and a decreas e in the interface-bulk temperature difference. This increases the heat transfer coefficient for smaller diameter. Nusselt number is the product of the heat tr ansfer coefficient and the diameter. The decrease in the diameter overcomes the increas e in the heat transfer coefficient. As a result, Nusselt number decreases as the diameter becomes smaller. It can be observed that an overall decrease of the Nusselt number occurs along the tube axial direction. Figures 2.20 and 2.21 show a comparison with previous experimental results for flow in a microtube with constant wall heat flux to validate the numerical results for this

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37 0 2 4 6 8 10 12 14 700100013001600190022002500 Reynolds NumberPeripheral Average Nusselt Number D=0.12 D=0.18 D=0.24 Figure 2.18: Local average Nusselt number at z = 0.4 cm for various diameter (with constant Reynolds number) and Reynolds number (G = 5 T, Water) 0 2 4 6 8 10 12 700100013001600190022002500 Reynolds NumberPeripheral Average Nusselt Number D=0.12 D=0.18 D=0.24 Figure 2.19: Local average Nusselt number at z = 0.8 cm for various diameters (with constant Reynolds number) and Reynolds number (G = 5 T, Water)

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38 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0100200300400500600 Reynolds NumberFriction Facto r Numerical friction factor Experimental friction factor [13] Figure 2.20: Comparison for fricti on factor results to existing experimental results in the literature 10 100 1000 1,00010,000100,000 Reynolds NumberNusselt Number Nu, numerical Nu, experimental [17] Figure 2.21: Comparison for Nusselt number resu lts to existing experimental results in the literature

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39project. In the absence of any test data fo r magnetic heaters (or coolers), these are most appropriate comparisons that could be done. Figure 2.20 shows a good agreement between the numerical and experimental result s for the friction coefficient at different Reynolds numbers in Kohl et al. [13]. Small difference appeared between the two results and could be referred to the approximations used to conduct the numerical simulation as well as errors involved in experimental meas urements. Circular microtube with diameter equals to the hydraulic diameter of the channel used in th e experiment was simulated. Only one channel was considered in this comparison out of the 5 channels in the experimental work. Adams et al. [17] examined turbulent, single-phase forced convection of water in circular microchannels. A genera lized correlation for the Nusselt number for turbulent, single phase forced convection in circular microchannels was developed. Figure 2.21 shows a comparison of their Nusselt number data at different Reynolds number with predictions by numerical simulation. Interpolation was made to obtain the error range for the experimental data presented in Kohl et al. [13] and Adams et al. [17] compar ed to our work. It was found that the error range for Kohl et al. [13] was 5.58 % and 16.27 % at Reynolds number of 170 and 450 respectively. Also the error range for Adams et al. [17] was 7.71 % and 16.34 % at Reynolds number of 6,500 and 22,000 respectively. Adams et al. [17] presented the difference between his experime ntal and predicted Nusselt number values were less than 18.6%. Design of experiment is one way to study the effect of different parameters and parameter combinations. A detailed was done in Appendix H. The percentage difference for the entrance effect was found to be ranging from 42.0 % to 43.4 % along the microchannel axial direction. When compared to the large tube correlation it

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40was found that the percentage difference ranges from 25.7 % to 22.4 % along the microchannel axial direction. 2.4 Conclusions A study of parameters that affect c onvective heat transfer in circular microchannels during heat generation in the substrate due to an applied magnetic field was done. Generally the interface temperature increases along the axial direction of the fluid flow due to heat generation in the s ubstrate. Lower Reynolds number results in higher interface temperature but lower Nusselt number. A higher magnetic field strength also results in higher interface temperature but has no effect on Nusselt number. Smaller diameter for a constant Reynolds number gives higher interface temperature and heat transfer coefficient but it decreases Nusselt num ber. Decreasing the diameter at constant inlet velocity increases the interface temper ature but decreases the Nusselt number. The maximum temperature occurs at the farthest corner of the substrate from the channel entrance. A higher Reynolds number decrease s the maximum temperature in the system. A smaller diameter at constant Reynolds num ber results in higher maximum temperature. Among the working fluid examined, water has the highest Nusselt number while ammonia has the lowest. Pressure drop was fou nd to be higher for smaller diameter at constant Reynolds number. The peripheral aver age heat transfer coefficient was found to decrease with diameter at constant Re ynolds number and increase with increasing Reynolds number. This study provided a founda tion for the prediction of heat transfer coefficient during magnetic heatin g or cooling of the substrate material. It will be used for the design of experiments for practical demonstration of a magnetic microcooler.

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41 Chapter 3 – Transient Heat Transfer in Circular Microcha nnels Under Time Varying Heat Source 3.1 Mathematical Model The problem in hand consists of circul ar microchannels inside a rectangular magnetic material substrate. Fluid is entering the microchannels at constant inlet temperature. The magnetic material substrate is subjected to a magnetic field resulting in heat source and heat sink periodically. The boundary c onditions at all sides are considered adiabatic. Because of the symmetry along the width of the wafer, only half of the pipe is simulated. The pipe at the left e dge was considered to ta ke into consideration the side end effect. The material of the s ubstrate was Gadolinium while the working fluid was water. Figure 3.1 shows a 3-D model studied in this project. Boundary conditions at the bottom, left and top are adiabatic while it is symmetrical at the right side. The governing equations for the conserva tion of mass, momentum, and energy for the fluid in Cartesian coordinate are [45]: 0 1 1 z V V r V r r Vz r r (1) V r r V z V V r r V r r V r p V r z V V V r V r V V t Vr r r r r f r z r r r r 2 2 2 2 2 2 2 2 2 22 1 1 1 1 (2)

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42 Figure 3.1: Schematic for microchannel heat exchanger model r f r z rV r r V z V V r r V r r V p r r V V z V V V r V r V V t V2 2 2 2 2 2 2 2 22 1 1 1 (3) 2 2 2 2 2 2 21 1 1 z V V r r V r r V z p z V V V r V r V V t Vz z z z f z z z z r z (4) 2 2 2 2 2 2 21 1 z T T r r T r r T z T V T r V r T V t Tf f f f f f z f f r f (5) For transient heat transfer with uniform heat generation and constant thermal conductivity, the energy conservation equa tion in the solid region is [46]:

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43 t T k g z T y T x Ts o s s s 12 2 2 2 2 2 (6) Equations (1) to (6) are subject to th e following Initial and boundary conditions: At t = 0: in f sT T T (7) At z = 0 and r d/2: in f f in z z rT T V V V V, ,, 0 0 (8) At z = 0 and r > d/2: 0 z Ts (9) At z = L and r d/2: 0 p (10) At z = L and r > d/2: 0 z Ts (11) At x = 0, 0 z L, 0 y H: 0 x Ts (12) At x = W, 0 z L, (H-d)/2 y (H+d)/2: 0 0 0 0 x T x V x V Vf z r (13) At x = W, 0 z L, 0 y (H-d)/2: 0 x Ts (14) At x = W, 0 z L, 0 y (H+d)/2: 0 x Ts (15) At y = 0, 0 x W, 0 z L: 0 y Ts (16) At y = H, 0 x W, 0 z L: 0 y Ts (17) At r = d/2 and 0 z L: r T k r T k T T V V Vs s f f s f z r , 0 0 0 (18)

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443.2 Numerical Simulation and Parametric Study The governing equations along with the boundary conditions were solved using the Galerkin finite element method. The Newton-Raphson algorithm was used to solve the nonlinear system of discretized equations. An iterative pr ocedure was used to arrive at the solution for the velocity and temperature fields. The solution was considered converged when the field values did not chan ge from one iteration to the next, and the residuals for each variable became negligible. For the numerical computations, the hei ght (H) and the half width (W) of the model were set to a constant value, 0.2 cm and 0.236 cm respectively. The length of the channel (L) was also a constant value of 2. 5 cm. To represent the heat generation and cooling process numerically, the magnetic fi eld value (G) was fluctuated between the values of +2 and -2 T. Time period of 2 seconds was used. The first period was chosen to be one half of the selected time length since it started from the initial condition. Diameter of 0.036 cm and Reynolds number of 1600 were used for this case. One parameter was changed while all other parameters kept cons tant to study the effect of each parameter separately. Parameters included the magnetic fi eld that was changed to the values of +4 and -4 T. Time length was one parameter and changed to 10 seconds. To study the effect of diameter, it was reduced to 0.012 cm. Re ynolds number effect was studied and was changed to 1000. 3.3 Results and Discussion The local interface temperature, heat fl ow rate, heat transfer coefficient and Nusselt number were calculated from the result ed velocity and temperature distribution.

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45Figure 3.2 shows the maximum temperature in the model. The heat generation occurred in the 1st, 3rdand 5th periods while the cooling process occurred in the 2nd and 4th periods. The maximum temperature in the model occurred in the solid region at the outlet edge corner of the substrate. It can be observed that the maximum temperature in the system increases when heat is generated in the syst em while it decreases when the system is cooled. Because the inlet temperature to the system is 20 oC, the maximum temperature does not drop below this value. Figure 3.3 sh ows the interface temperature at different axial locations of the microc hannel. Sinusoidal behavior is observed for the interface temperature as the heat generation and coolin g process alternates. As heat is generated the interface temperature incr eases while it decreases when cooling process takes over. The interface temperature incr ease along the axial direction is slow at the beginning of the transient process because of the initial condition effect. The temperature range gets larger along the axial direction due to larger fluid bulk temperature which slows down the heating or cooling process. Figure 3.4 shows th at the heat flow rate is supplied to and rejected from the fluid as the heat generation and cooling process alternates in the system. Similarly to the interface temperature, a sinus oidal behavior is observed in the heat flow rate as heat generation and cooling process alternates. The heat flow rate increase in value for each of the heat generation and c ooling process and the in itial condition effect diminishes over time. As the fluid travel s through the channel its bulk temperature increases providing heat transfer resistan ce. This decreases the interface-solid temperature difference and thus the heat flow rate. Figure 3.5 shows the Nusselt number behavior over a period of 9 s econds. The Nusselt number behavi or is affected by the heat flow rate and the interface-bulk temperature diffe rence. The increase in the heat flow rate

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46 19 20 21 22 23 24 25 26 27 012345678910 Time, sMaximum Temperature, oC Tmax Figure 3.2: Maximum temperature over 9 seco nds (Re = 1600, G = 2 T, d = 0.036 cm) 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm z = 2.5 cm Figure 3.3: Peripheral average interface temper ature at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm)

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47 -60 -40 -20 0 20 40 60 012345678910 Time, sPeripharel Average Heat Flow Rate, kW/m2 z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm z = 2.5 cm Figure 3.4: Peripheral average he at flow rate at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm) 0 4 8 12 16 20 24 0123456789 Time, sPeripheral Average Nusselt Number z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm z = 2.5 cm Figure 3.5: Peripheral average Nusselt number at different axial locations over 9 seconds (Re = 1600, G = 2 T, d = 0.036 cm)

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48increases the Nusselt number. As heat gene ration and cooling proc ess alternates the interface-bulk temperature difference alternat es positive and negative values. This temperature difference becomes zero at cert ain time in each time period. This causes Nusselt number to be infinite at that time. After the heating and cooling processes alternates the system Nusselt number increases from negative infinite value to the actual value. The increase in the heat flow rate in creases Nusselt number thereafter. Since the Nusselt number behavior is understood, only fewer points along time will be plotted to the results of the other cases. Figure 3.6 shows the maximum temperatur e in the model when magnetic field of 4 T is used. Comparing to figure 3.2 the maxi mum temperature is higher because of the higher value of heat generation used. Figure 3.7 shows the heat flow ra te at the interface using magnetic field of 4 T. Higher heat generation results in higher interface and solid temperatures in the model. Interface-solid te mperature difference also increases with the increase in heat generation. The value of the heat generation diminished the effect of the initial condition and increased the heat fl ow rate along time. The interface-solid temperature difference is higher closer to the tube inlet because the inlet fluid temperature increases the interface-solid temper ature difference. Similarly to the previous case, the heat flow rate decreases along the axial direction. Figure 3.8 shows the Nusselt number over 9 seconds time period using magnetic field of 4 T. The Nusselt number shows similar behavior to the previous case. Compared to figure 3.5, Nusselt number is not affected by the change in the magnetic field value.

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49 18 20 22 24 26 28 30 32 0246810Time, sMaximum Temperature, oC Tmax Figure 3.6: Maximum temperature over 9 s econds with G = 4 T (Re = 1600, d = 0.036 cm) -80 -60 -40 -20 0 20 40 60 80 012345678910 Time, sPeripheral Average Heat Flow Rate, kW/m2 z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm z = 2.5 cm Figure 3.7: Peripheral average he at flow rate at different axial locations over 9 seconds with G = 4 T (Re = 1600, d = 0.036 cm)

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50 0 2 4 6 8 10 12 14 16 012345678910 Time, sPeripheral Average Nusselt Number z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 3.8: Peripheral average Nusselt number at different axial locations with G = 4 T (Re = 1600, d = 0.036 cm) Figure 3.9 shows the heat flow rate in th e model when the diameter of 0.012 cm is used. Compared to figure 3.4, the values of heat flow rate is higher. As the diameter decreases, the solid volume increases causing more heat generation. This will increase the value of the heat flow rate at the interface. Th is is similar to the effect of increasing the magnetic field in the previous case. Compar ing to figure 3.7, doubling the magnetic field has higher effect on the heat flow rate increase. Figure 3.10 shows Nusselt number behavior when diameter of 0.012 cm is used. The increase in Nusselt number is a direct result for the heat flow rate increase. The Nusselt number has similar behavior as the case of higher magnetic field value. The higher Nu sselt number value at the beginning of each time period has similar explanation of t hose discussed in the previous case.

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51 -80 -60 -40 -20 0 20 40 60 80 012345678910 Time, sPeripheral Average Heat Flow Rate, kW/m2 z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 3.9: Peripheral average he at flow rate at different ax ial locations with d = 0.012 cm (Re = 1600, G = 2 T) 0 4 8 12 16 20 24 012345678910 Time, sPeripheral Average Nusselt Number z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 3.10: Peripheral average Nusselt number at different axial locations with d = 0.012 cm (Re = 1600, G = 2 T)

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52 Figure 3.11 shows the heat flow rate of the model at the fluid-solid interface surface for Reynolds number of 1000. The absolute values of the heat flow rate are lower compared to figure 3.4. Lower velocity means higher bulk temperature and thus less interface-bulk heat transfer. This results in less heat flow rate to the fluid. Figure 3.12 shows the Nusselt number behavior in the model for Reynolds number of 1000. Nusselt number directly decreases with the heat flow rate. Therefore, Nusse lt number is also less for lower Reynolds numbers. This is clearly sh own when the figure is compared to figure 3.5. 3.4 Conclusions Study of the parameters of the circular microchannel model w ith transient heat generation was done. Mathematical model was developed and computations were done to solve the system. The effect of different para meters was studied. The results showed the behavior of the maximum temperature, inte rface temperature. The maximum temperature in the model does not drop below 20 oC because of the inlet temperature condition. The interface temperature, heat fl ow rate, and Nusselt number showed sinusoidal behavior as heat generation and cooling process alternates. Th e heat flow rate increase in value within each time period. The absolute value of heat flow rate is used when calculating the Nusselt number. This causes Nusselt number to increase during each time period. Notice that Nusselt number reaches high values at the beginning of each time period. This is because the interface-bulk temperatures differe nce becomes very small in value at which both temperatures alternate in value. Higher magnetic field increases the heat flow rate to the fluid as well as Nusselt number. Larger diameter decreases fluid velocity causing

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53 -50 -40 -30 -20 -10 0 10 20 30 40 50 012345678910 Time, sPeripheral Average Heat Flow Rate, kW/m2 z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 3.11: Peripheral average h eat flow rate at different ax ial locations w ith Re = 1000 (G = 2 T, d = 0.036 cm) 0 2 4 6 8 10 12 14 012345678910 Time, sPeripheral Average Nusselt Number z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 3.12: Peripheral average Nusselt number at different axial locations with Re = 1000 (G = 2 T, d = 0.036 cm)

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54higher fluid bulk temperature. This increase s the heat flow rate and, thus, Nusselt number. Lower Reynolds number results in lo wer heat flow rate and Nusselt number.

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55 Chapter 4 – Transient Heat Transfer in Trapezoidal Microchannels During Activation of Magnetic Heating 4.1 Mathematical Model The model in hand is a trapezoidal cro ss-sectional microchanne l consisting of two different substrates gadolinium on top a nd silicon on bottom while the channel was manufactured into the bottom one. The gadolin ium substrate on the top is joined to the silicon substrate and the primary working flui d selected was water. Figure 4.1 shows the schematic drawing of the model. Heat is generated in the gadolinium substrate and convected to the water while part of it is c onducted to the silicon and then convected to water. The applicable differential equations for the conservation of mass, momentum, and energy in the Cartesian coordinate system for the fluid can be written as [45], 0 z w y v x u (1) 2 2 2 2 2 21 z u y u x u x p z u w y u v x u u t u (2) 2 2 2 2 2 21 z v y v x v x p z v w y v v x v u t v (3) 2 2 2 2 2 21 z w y w x w x p z w w y w v x w u t w (4)

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56 Figure 4.1: Schematic draw for the model of trapezoidal microchannel 2 2 2 2 2 2z T y T x T z T w y T v x T u t Tf f f f f f f (5) The energy conservation equation in the solid gadolinium substrate is [46]: t T k g z T y T x Tgd gd gd gd gd 10 2 2 2 2 2 2 (6) and for the solid silicon substrate is: t T z T y T x Tsi si si si 12 2 2 2 2 2 (7) Equations (1) – (7) are subject to following initial and boundary conditions: At t = 0: Tf = Tgd = Tsi = Tin (8) At z = 0, at fluid inlet: u=0, v=0, w=win, T=Tin (9)

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57At z = 0, on solid surface: 0 z Tsi, 0 z Tgd (10) At z = L, at fluid outlet: p=0 (11) At z = L, on solid surface: 0 z Tsi, 0 z Tgd (12) At x = 0, 0 < y < (Hsi-Hfl), 0 < z < L: 0 z Tsi (13) At x = 0, (Hsi-Hfl) < y < Hsi, 0 < z < L: u=0, 0 x v 0 x w 0 x Tf (14) At x = 0, Hsi < y < (Hsi+Hgd), 0 < z < L: 0 x Tgd (15) At x = B, 0 < y < Hsi, 0 < z < L: 0 x Tsi (16) At x = B, Hii < y < (Hsi+Hgd), 0 < z < L: 0 x Tgd (16) At y = 0, 0 < x < B, 0 < z < L: 0 y Tsi (17) At y = (Hsi-Hfl), B-Bs < x < B, 0 < z < L: u=0, v=0, w=0, Tf = Tsi, and y T k y T ksi si f f (18) At y = Hsi, 0 < x < B-Bc, 0 < z < L: u=0, v=0, w=0, Tf = Tgd, and y T k y T kgd gd f f (19) At y = Hsi, B-Bc < x < B, 0 < z < L: y T k y T kgd gd si si (20) At y = (Hsi+Hgd), 0 < x < B, 0 < z < L: 0 y Tgd (21)

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58The inclined fluid-silicon surface, 0
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594.3 Result and Discussion Figures 4.2 and 4.3 show a grid inde pendence study carried out in the crosssectional area of the channel to determine th e optimum grid size and to insure accurate results. The number of intervals in the axial direction kept constant because it does not affect the results significantly The results obtained by usin g (nx = 24, ny = 16, nz = 10) that had an average error of 0. 08% in the silicon side and a maximum error of 0.1% in the interface temperature compared to (nx = 36, ny = 24, nz = 10). The heat flow rate also shows a small error comparing th e above two grid systems. Th e average error for the heat flow rate was 0.78% while the maximum was 1.27%. Figures 4.2 and 4.3 show that the grid size selected was sufficient enough to reach optimum results. Figure 4.4 shows that gadolinium interface temperature increases by time. This is because of the heat generated in the system. After 4 seconds the system almost reaches steady state. The first 2 seconds has mu ch larger increase in average interface temperature. Heat is generated in the gado linium substrate and then transferred to the fluid through convection and to the silicon substrate through conduction. The figure shows that there is an increase in the aver age interface temperature at the gadolinium interface along the axial direction which is caused by the heat generation. The rate of increase in the average interface temperatur e decreases when reaching the end of the channel. This is because the fluid temperature rises resulting in less heat rejection. Figure 4.5 shows that the heat flow rate at the gadolinium interface increases with time. This is

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60 19.5 20 20.5 21 21.5 22 22.5 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature, oC nx=12, ny=8, nz=10 nx=24, ny=16, nz=10 nx=36, ny=24, nz=10 Figure 4.2: Peripheral average interface temperature along the axial direction at the fluidsilicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 60 70 80 00.511.52 Axial Direction, cmPerpheral Average Heat Flow Rate, W/m2 nx=12, ny=8, nz=10 nx=24, ny=16, nz=10 nx=36, ny=24, nz=10 Figure 4.3: Peripheral av erage heat flow rate along the axia l direction at th e fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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61 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC 0 s 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.4: Peripheral average interface temper ature along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 60 70 80 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.5: Peripheral average heat flow rate along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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62because of the heat generation in the ga dolinium substrate. The average interface temperature increases along the axial direc tion. The fluid bulk temperature increases along the axial direction at a higher rate than th e interface temperature. This causes the heat flow rate to decrease al ong the axial direction. From th e figure one can see that heat flow rate across the interface reaches steady st ate closer to the inlet before locations further away along the axial dire ction. Figures 4.6 and 4.7 show the average heat transfer coefficient and Nusselt number respectively. Both, heat flow rate and interface-bulk temperature difference, increase by time. The h eat transfer coefficient is a ratio of the heat flow rate to the interf ace-bulk temperature difference. Th e figure shows that the heat transfer coefficient increases with time clos er to the channel entrance while it decreases with time towards the end of the channel. This means that the rate of change in the heat flow rate to the interface-bulk temperatur e difference ratio decreases along the axial direction. This could be e xplained by the increase in th e fluid bulk temp erature which causes more increase the interface-bulk temp erature difference further away from the channel entrance. The net result for the combin ed effects cause the average heat transfer coefficient and thus the Nusselt number to increase by time at closer to the inlet but decrease by time closer to the outlet. The heat flow rate decreases along the axial direction due to the increase in the interface -bulk temperature difference. This insures a decrease in the heat transfer coeffici ent and thus Nusselt number along the axial direction. Figure 4.8 shows that the silicon interface temperature increases by time. This is because the heat is conduc ted through the silicon substrate from the heat generated in the gadolinium substrate. Similarly to the gadolinium side, the system almost reaches steady state after 4 seconds. The heat that is conducted from the gadolinium to the

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63 8 10 12 14 16 18 20 00.511.52 Axial Direction, cmPeripheral Average Heat Transfer Coefficient kW/m2.oC 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.6: Peripheral average heat transfer coefficient along the axial direction at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 8 10 12 14 16 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.7: Peripheral average Nusselt number along the axial directi on at different time steps at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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64 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC 0 s 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.8: Peripheral average interface temper ature along the axial direction at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) silicon causes the interface temperature increas e at the fluid-silicon interface. The figure also shows that there is an increase in the average interface temperature at the silicon interface along the axial direction which is caused by the heat conducted to the silicon from the heat generated in the gadolinium substrate. The rate of increase in the average interface temperature decreases wh en reaching the end of the channel. This is because the fluid temperature rises resulting in less heat rejection. Figure 4.9 shows that the heat flow rate at the silicon inte rface increases with time. This is because of the heat conducted to the silicon from the heat gene ration in the gadolinium substrate. The average interface temperature increases along the axial direction. The heat flow rate is less in the silicon interface compared to the gadol inium interface. This is b ecause the heat flow travels through the silicon substrate by conduction before facing the fluid-silic on interface. This

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65 0 10 20 30 40 50 60 70 80 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.9: Peripheral average heat flow rate along the axial direction at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) slows down the heat transfer process in the silicon substrate resulting less temperature gradient than in the gadolinium substrate where the heat is generated. As the fluid travels along the axial direction it gains more heat from the gadolinium substrate and less heat from the silicon side, especially with the rising bulk fluid temperature. This result in lower rate of decrease in the heat flow ra te along the axial dire ction at the silicon interface compared to the gadolinium interface. Figures 4.10 and 4.11 show that after 2 seconds the heat transfer coe fficient and Nusselt number ch anges could be ignored. The heat flow rate and the interface-bulk temper ature difference increase with time. The heat flow rate increase is higher which results in an increase in the heat transfer coefficient and Nusselt number with time. The heat flow rate decreases along th e axial direction due to the increase in the fluid bulk temperature. Nusselt number follow similar trend to those

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66 4 8 12 16 20 24 00.511.52 Axial Direction, cmPeripheral Average Heat Transfer Coefficient kW/m2.oC 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.10: Peripheral average heat transfer coefficient along the axial direction at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 6 8 10 12 14 16 18 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number 0.2 s 0.4 s 0.6 s 0.8 s 1 s 2 s 4 s 10 s Figure 4.11: Peripheral average Nusselt number along the axial directi on at different time steps at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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67of heat transfer coefficient. It decreases al ong the axial direction due to the increase in the fluid bulk temperature. Figures 4.12, 4.13 and 4.14 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different Reynolds number along the axial direction at the gadolinium-fluid interface. Higher fluid velocity increases the heat tr ansfer coefficient at the interface causing lower inte rface temperature and lower flui d bulk temperature. It will also increase the temperature gradient in the solid substrate increasi ng the heat flow rate at the interface. Nusselt number increases du e to the increase in the heat transfer coefficient. Figures 4.15, 4.16 and 4.17 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different Reynolds number along the axial direction at the silicon-fluid interface. Similar to the gadolin ium side, higher fluid velocity increases the heat transfer coefficient at the interface causing lower in terface temperature and lower fluid bulk temperature but higher heat flow rate and Nusselt number. Figures 4.18, 4.19 and 4.20 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different magnetic fields along the axial direction at the gadolinium-fluid interface. Higher heat gene ration increases the temperature profile in the solid substrate causing an increase in th e average interface temp erature and heat flow rate. The increase in both, interface temperatur e and heat flow rate, cancel their effects resulting in negligible change in Nusselt number. Figures 4.21, 4.22 and 4.23 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different magnetic fields along the axial direction at the silicon-fluid interface. Similar to the gadolinium side, higher heat generation increases

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68 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.12: The effect of Reynolds num ber on the peripheral average interface temperature along the axial dire ction at different time periods at the fluid-gadolinium interface (G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 60 70 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.13: The effect of Reynolds number on the peripheral average heat flow rate along the axial direction at different time pe riods at the fluid-gado linium interface (G = 5 T, Hfl = 0.03 cm)

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69 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.14: The effect of Reynolds numbe r on the peripheral av erage Nusselt number along the axial direction at different time pe riods at the fluid-gado linium interface (G = 5 T, Hfl = 0.03 cm) 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.15: The effect of Reynolds num ber on the peripheral average interface temperature along the axial direc tion at different time periods at the fluid-silicon interface (G = 5 T, Hfl = 0.03 cm)

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70 0 10 20 30 40 50 60 70 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.16: The effect of Reynolds number on the Peripheral averag e heat flow rate along the axial direction at different time peri ods at the fluid-silic on interface (G = 5 T, Hfl = 0.03 cm) 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number Re=2000, time=0.4 s Re=2000, time=0.8 s Re=1500, time=0.4 s Re=1500, time=0.8 s Re=1000, time=0.4 s Re=1000, time=0.8 s Figure 4.17: The effect of Reynolds numbe r on the peripheral av erage Nusselt number along the axial direction at different time peri ods at the fluid-silic on interface (G = 5 T, Hfl = 0.03 cm)

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71 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.18: The effect of magnetic field on the peripheral average interface temperature along the axial direction at different time pe riods at the fluid-ga dolinium interface (Re = 2000, Hfl = 0.03 cm) 0 10 20 30 40 50 60 70 80 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.19: The effect of magne tic field on the peripheral av erage heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, Hfl = 0.03 cm)

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72 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.20: The effect of magnetic field on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, Hfl = 0.03 cm) 19 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.21: The effect of magnetic field on the peripheral average interface temperature along the axial direction at different time pe riods at the fluid-sili con interface (Re = 2000, Hfl = 0.03 cm)

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73 0 10 20 30 40 50 60 70 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.22: The effect of magne tic field on the peripheral av erage heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, Hfl = 0.03 cm) 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number G=2 T, time=0.4 s G=2 T, time=0.8 s G=5 T, time=0.4 s G=5 T, time=0.8 s G=10 T, time=0.4 s G=10 T, time=0.8 s Figure 4.23: The effect of magnetic field on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, Hfl = 0.03 cm)

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74the temperature profile in the solid substrat e causing an increase in the average interface temperature and heat flow rate. The increase in both, interface temp erature and heat flow rate, cancel their effects resulting in no change in Nusselt number. Figures 4.24, 4.25 and 4.26 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different channe l height along the ax ial direction at the gadolinium-fluid interface. The reduction in the channel height with maintaining the same Reynolds number results in higher fluid velocity. This enhances the cooling rate and decreases the interface temperature as well as the fluid bulk temperature. On the other hand, the heat flow rate increases due to the higher temperature gradient in the solid substrate. The increase in the heat flow rate increases Nusselt number. Figures 4.27, 4.28 and 4.29 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different channe l height along the ax ial direction at the silicon-fluid interface. Similarly to the ga dolinium side, the channel height reduction increases the fluid velocity. This decrease s the interface temperat ure and the fluid bulk temperature. The heat flow rate increases due to the higher temperature gradient in the solid substrate. The increase in the heat flow rate increases Nusselt number. Figures 4.30, 4.31 and 4.32 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different heig ht of gadolinium substrate along the axial direction at the gadolinium-fl uid interface. Higher gadolinium height results in higher heat generation but the heat generation per unit volume remain s the same. Since the heat generation is calculated per unit volume, it is expected that the increase in gadolinium height does not affect the temp erature gradient in the solid. Results showed a small

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75 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.24: The effect of channel depth on the peripheral average interface temperature along the axial direction at different time pe riods at the fluid-ga dolinium interface (Re = 2000, G = 5 T) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.25: The effect of channel depth on th e peripheral average heat flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T)

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76 0 4 8 12 16 20 24 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.26: The effect of channel depth on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T) 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.27: The effect of channel depth on the peripheral average interface temperature along the axial direction at different time pe riods at the fluid-sili con interface (Re = 2000, G = 5 T)

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77 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.28: The effect of channel depth on th e peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T) 0 5 10 15 20 25 30 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number Hfl=0.03 cm, time=0.4 s Hfl=0.03 cm, time=0.8 s Hfl=0.02 cm, time=0.4 s Hfl=0.02 cm, time=0.8 s Figure 4.29: The effect of channel depth on the peripheral average Nusselt number along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T)

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78 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.30: The effect of height of gadolinium substr ate on the peripheral average interface temperature along the axial direction at different time periods at the fluidgadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.31: The effect of hei ght of gadolinium substrate on the peripheral average heat flow rate along the axial direction at diffe rent time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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79 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.32: The effect of height of gadolinium substr ate on the peripheral average Nusselt number along the axial direction at diffe rent time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) increase in the average interface temperature which indicates a small decrease in the solid temperature gradient. This means that the ga dolinium height has a ve ry low effect on the system. The heat flow rate, on the other ha nd, decreased with higher gadolinium height. This is a direct result for the decrease in the solid temperature gradient. The decrease in the heat flow directly decreases the heat tr ansfer coefficient and thus the Nusselt number. Figures 4.33, 4.34 and 4.35 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different heig ht of gadolinium substrate along the axial direction at the silicon-fluid interface. Hi gher gadolinium height results in higher heat generation. Similarly to the gadolinium si de, this increases the average interface temperature was barely noticeable. The temper ature gradient decrease d in the gadolinium

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80 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.33: The effect of height of gadolinium substr ate on the peripheral average interface temperature along the axia l direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.34: The effect of hei ght of gadolinium substrate on the peripheral average heat flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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81 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number Hgd=0.3 cm, time=0.4 s Hgd=0.3 cm, time=0.8 s Hgd=0.4 cm, time=0.4 s Hgd=0.4 cm, time=0.8 s Hgd=0.5 cm, time=0.4 s Hgd=0.5 cm, time=0.8 s Figure 4.35: The effect of height of gadolinium substr ate on the peripheral average Nusselt number along the axial direction at di fferent time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) substrate due to larger solid volume. Unlike the gadolinium side, the temperature gradient in the silicon substrate decrease d at earlier times (t = 0.4 s) but increased at later times (t = 0.8 s). This is because of the constant heat conduction from the gadolinium to the silicon substrate which increased the temperatur e gradient at later times. Both the heat transfer coefficient and Nusselt number are decreasing with larger gadolinium height. This is due to increase in the interfacebulk temperature difference as the interface temperature increases. Figures 4.36, 4.37 and 4.38 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different channe l spacing along the axial direction at the gadolinium-fluid interface. Larger spacing for the same model width results in smaller channel. This increases the fluid velocity for the same Reynolds number. Higher fluid

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82 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.36: The effect of channel spac ing on the peripheral average interface temperature along the axial dire ction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate kW/m2 sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.37: The effect of channel spacing on th e peripheral average he at flow rate along the axial direction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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83 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.38: The effect of channel spaci ng on the peripheral average Nusselt number along the axial direction at different time pe riods at the fluid-ga dolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) velocity increases the cooling process and us ually results in lower interface temperature and higher heat flow rate. In this case, the solid volume also increases as the channel spacing increase. This provides more heat gene ration in the solid substrate and increases the interface temperature. Both effects cancel each other and result in negligible change in the interface temperature and heat flow ra te. The heat transfer coefficient and Nusselt number depend directly on the heat flow rate. The change in the heat transfer coefficient and the Nusselt number are also negligible. Figures 4.39, 4.40 and 4.41 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different channe l spacing along the axial direction at the silicon-fluid interface. Larger spacing for the same model width results in smaller channel. This increases the fluid velocity for the same Reynolds number. Higher fluid

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84 19 20 21 22 23 24 25 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.39: The effect of channel spac ing on the peripheral average interface temperature along the axial direc tion at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.40: The effect of channel spacing on th e peripheral average he at flow rate along the axial direction at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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85 0 2 4 6 8 10 12 14 16 18 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number sp=0.28 cm, time=0.4 s sp=0.28 cm, time=0.8 s sp=0.29 cm, time=0.4 s sp=0.29 cm, time=0.8 s sp=0.3 cm, time=0.4 s sp=0.3 cm, time=0.8 s Figure 4.41: The effect of channel spaci ng on the peripheral average Nusselt number along the axial direction at different time pe riods at the fluid-sili con interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) velocity increases the cooling process and us ually results in lower interface temperature and higher heat flow rate. The solid volum e also increases as the channel spacing increase. This provides more heat generati on in the solid substrate and increases the interface temperature. The heat generation in the gadolinium substrate increases and is conducted to the silicon substrate. Silicon ha s high thermal conductivity and this transfers some of the added heat generation in the gadolinium. This additional heat generation conducted to the silicon substr ate overcomes the effect of the increase in the fluid velocity and the net result can be observed in higher interface temperature. The additional heat generation conducted to the silicon substrat e also enhances the heat flow rate to the fluid at the interface. This in crease in the heat flow rate increases the heat transfer coefficient and Nusselt number.

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86 Figures 4.42, 4.43 and 4.44 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different worki ng fluids along the ax ial direction at the gadolinium-fluid interface. Water has the lo west rate of increase in the interface temperature along the axial direction but the highest heat flow rate and Nusselt number. This is because water has the highest conduc tivity among the fluids tested in this research. The high fluid thermal conductivity redu ces the fluid resistance in heat transfer and enhances the heat transfer to the fluid re sulting in lower interface temperature. On the other hand, R134a has the lowest fluid ther mal conductivity. This cau se higher resistance to the heat flow to the fluid resulting in a decr ease in the heat flow rate but an increase in the interface temperature. Nusselt number dire ctly depends on the heat flow rate and, therefore, has similar behavior. Figures 4.45, 4.46 and 4.47 show the resu lts of interface temper ature, heat flow rate and Nusselt number for different worki ng fluids along the ax ial direction at the silicon-fluid interface. Similarly to the ga dolinium side, water has lowest interface temperature but the highest heat flow ra te and Nusselt number. R134a have highest interface temperature but the lowest heat flow rate and Nusselt number. 4.4 Conclusions Study of the parameters in transien t case of the trapezoidal composite microchannel model with heat generation in gadolinium was done. Mathematical model was developed and computations were done to solve the system. Th e study showed that the model reaches steady state after approx imately 4 seconds. It also showed that gadolinium and silicon sides have similar results and values for interface temperature and

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87 20 21 22 23 24 25 26 27 28 29 00.511.5 Axial direction, cmPeripheral Average Interface Temperature,oC Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.42: The effect of changing the fluid on the peripheral average interface temperature along the axial dire ction at different time periods at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.43: The effect of changing the fluid on the peripheral average heat flow rate along the axial direction at different time pe riods at the fluid-ga dolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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88 0 2 4 6 8 10 12 14 16 18 20 00.511.52 Axial Direcation, cmPeripheral Average Nusselt Number Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.44: The effect of changing the flui d on the peripheral average Nusselt number along the axial direction at different time pe riods at the fluid-ga dolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 20 21 22 23 24 25 26 27 00.511.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.45: The effect of changing the fluid on the peripheral average interface temperature along the axial direc tion at different time periods at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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89 0 10 20 30 40 50 00.511.52 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.46: The effect of changing the fluid on the peripheral average heat flow rate along the axial direction at different time pe riods at the fluid-sili con interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) 0 4 8 12 16 20 24 00.511.52 Axial Direction, cmPeripheral Average Nusselt Number Water, time=0.4 s Water, time=0.8 s Ammonia, time=0.4 s Ammonia, time=0.8 s R-134a, time=0.4 s R-134a, time=0.8 s Figure 4.47: The effect of changing the flui d on the peripheral average Nusselt number along the axial direction at different time pe riods at the fluid-sili con interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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90heat flow rate. This is b ecause the silicon has higher c onductivity that makes up for the heat generation in the gadolinium side. Th e increase in Reynolds number increases the heat flow rate and Nusselt number. The incr ease in the magnetic field increases the heat flow rate and interface temperature. The increa se in both cancels each other and results in no change in the Nusselt number. Lowering th e channel height for the same Reynolds number increases the heat flow rate and Nusselt number. The change in gadolinium height and channel spacing had the least eff ect on both the heat fl ow rate and Nusselt number. Using water results in lower interface temperature but higher heat flow rate while R-134a results in higher interface temperature but lower heat flow rate. The magnetic field and the working fluid had the highest effect on the interface temperature and heat flow rate. Nusselt number was a ffected the most by the channel height (Hfl) and the change in the working fluid.

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91 Chapter 5 – Transient Heat Transfer in Trapezoidal Microc hannels Under Time Varying Heat Source 5.1 Mathematical Model The model in hand is the same as the one de scribed in the previous chapter. It is a trapezoidal cross-sectional microchannel manufactured into a silicon substrate with gadolinium substrate attached on the top. Heat is generated in the Gadolinium substrate and convected to the water while part of it is conducted to the Sili con and then convected to water. In this chapter a time varying heat source is studied rather than a constant one. The magnetic material substrate is subjected to a magnetic field resul ting in a heat source and sink periodically. All boundary conditions at all sides are considered adiabatic. Symmetrical condition along only on e axis is considered. Figure 5.1 shows the schematic draw of the model. Boundary c onditions at the bottom, left and top are adiabatic while it is symmetrical at the right side. The differential equations for the conser vation of mass, momentum and heat in the Cartesian coordinate system for transien t case for the fluid are given as [45], 0 z w y v x u (1) 2 2 2 2 2 21z u y u x u x p z u w y u v x u u t u (2)

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92 Figure 5.1: Schematic for microchannel heat exchanger model 2 2 2 2 2 21z v y v x v x p z v w y v v x v u t v (3) 2 2 2 2 2 21z w y w x w x p z w w y w v x w u t w (4) 2 2 2 2 2 2z T y T x T z T w y T v x T u t Tf f f f f f f (5) The energy conservation equation in the solid Gadolinium substrate is [46]: t T k g z T y T x Tgd gd gd gd gd 10 2 2 2 2 2 2 (6) And for the solid Silicon substrate is [46]:

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93 t T z T y T x Tsi si si si 12 2 2 2 2 2 (7) Equations (1) – (7) are subject to following initial and boundary conditions: At t = 0: Tf = Tgd = Tsi = Tin (8) At z = 0, at fluid inlet: u=0, v=0, w=win, T=Tin (9) At z = 0, on solid surface: 0 z Tsi, 0 z Tgd (10) At z = L, at fluid outlet: p=0 (11) At z = L, on solid surface: 0 z Tsi, 0 z Tgd (12) At x = 0, 0 < y < (Hsi-Hfl), 0 < z < L: 0 z Tsi (13) At x = 0, (Hsi-Hfl) < y < Hsi, 0 < z < L: u=0, 0 x v 0 x w 0 x Tf (14) At x = 0, Hsi < y < (Hsi+Hgd), 0 < z < L: 0 x Tgd (15) At x = B, 0 < y < Hsi, 0 < z < L: 0 x Tsi (16) At x = B, Hii < y < (Hsi+Hgd), 0 < z < L: 0 x Tgd (16) At y = 0, 0 < x < B, 0 < z < L: 0 y Tsi (17) At y = (Hsi-Hfl), B-Bs < x < B, 0 < z < L: u=0, v=0, w=0, Tf = Tsi, and y T k y T ksi si f f (18) At y = Hsi, 0 < x < B-Bc, 0 < z < L: u=0, v=0, w=0, Tf = Tgd, and

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94 y T k y T kgd gd f f (19) At y = Hsi, B-Bc < x < B, 0 < z < L: y T k y T kgd gd si si (20) At y = (Hsi+Hgd), 0 < x < B, 0 < z < L: 0 y Tgd (21) At the inclined channel surface be tween fluid and silicon, 0
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95 Figure 5.2: Local variation of magnetic field with time was chosen to be one half of the selected time length sin ce it started from the initial condition. The parameters to be changed were selected to be: magnetic field, height of fluid microchannel, and Reynolds number. Initially these para meters were selected as follows: Re = 2000, G = 5 Tesla, Hfl = 0.03 cm, and water as the working fluid. One parameter was changed at a time while all ot her parameters kept c onstant to study the effect of each parameter separately. Magnetic fi eld that was changed to the values of +10 and -10 T, Hfl was changed to 0.02 cm, and Reynolds number was changed to 1000. 5.3 Results and Discussion The local interface temperature, heat fl ow rate, heat transfer coefficient and Nusselt number were calculated from the result ed velocity and temperature distribution.

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96The heat generation occurred in the 1st, 3rd, and 5th periods while th e cooling process occurred in the 2nd and 4th periods. Figure 5.3 and 5.4 show local Nusselt number along different channel edges. Curves for different axial locations were pl otted. Figure 5.3 shows the local Nusselt along the fluid-gadolinium interface at different ax ial locations at the end of the first time period. It shows that Nusselt number is highe r closer to the center of the trapezoidal microchannel. This is because the fluid volume is lesser at the edges of the trapezoidal microchannel which results in lesser resistance in heat transfer. This lesser resistance gives smaller temperature gradient in the solid region resulting in smaller heat flow rate. The heat flow rate increases along the edge to wards the center of the channel. This causes the heat transfer coefficient, and thus Nusselt number, to increase. Along the axial direction, the temperature gradient decreases causing the heat flow rate to decrease. The interface-bulk temperature differe nce increases along the axial direction. Both cause the Nusselt number to decrease along the axial direction. Figure 5.4 shows the local Nusselt along th e fluid-silicon interface at different axial locations at the end of the first time period. The silicon edge is tapered at closer to the side ends of the channel. At the tapered edge, there is small fluid volume where the heat flow rate almost transfers without a ny resistance. As a result, the temperature gradient is smaller and causes smaller heat flow rate values, and thus, smaller Nusselt number. The heat flows from the gadolinium to the silicon by conduction and then to the fluid by convection. At the very far tip of th e tapered edge, the heat flow finds away to travel through the fluid and to the silicon ov ercoming the small fluid resistance. Further away from the tip of the tapered edge, the heat flow from the silic on takes control giving

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97 0 2 4 6 8 10 12 14 16 18 0.10.150.20.250.30.35 Side Direction, cmLocal Nusselt Numbe r z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 5.3: Local Nusselt along the fluid-gadolin ium interface at different axial locations after 1 second (Re = 2000, G = 5 T, Hfl = 0.03 cm) -10 -5 0 5 10 15 20 25 30 35 0.10.150.20.250.30.35 Side Direction, cmLocal Nusselt Numbe r z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 5.4: Local Nusselt along the fluid-silicon interface at different axial locations after 1 second (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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98positive results. The heat flow rate increases quickly along the tapered edge reaching reasonable values resulting in adequate Nusselt number values. Along the horizontal edge, the temperature gradient and the heat flow rate decrease. This is because part of the heat flow rate that tr avel through the silicon transfers by convection to the fluid before it reaches the center of the cha nnel. This eventually causes Nusselt number to decrease along the horizontal side of the silicon edge. Similarly to the gadolinium edge, the temperature gradient decreases al ong the axial direction. This causes the heat flow rate to decrease along the axial direction. The in terface-bulk temperature difference increases along the axial direction. Both cause the Nusselt number to de crease along the axial direction. Figure 5.5 shows the Peripheral average interface temperature over 9 seconds at different axial locations at the fluid-gadoliniu m interface. Sinusoidal behavior is observed for the interface temperature as the heat ge neration and cooling process alternates. As heat is generated the interface temperature increases while it decreases when cooling process takes over. The interface temperature ra te of change along the axial direction is slow at the beginning of the transient proce ss because of the initial condition effect. The interface temperature rate of change also increases along the axial direction because of the inlet condition effect. Figure 5.6 shows the average heat fl ow rate over 9 seconds at different axial locations at the fluid-gadolinium interface. It shows that the heat flow rate is supplied to and rejected from the fluid as the heat generation and cooling process alternates in the system. The ra te of change in the heat flow rate increases with each new period of heating or cooling process. This is because of the ini tial condition where the temperature distribution provide s the highest interface-solid temperature difference and

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99 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.5: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-gadoliniu m interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.6: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-gadolinium interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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100thus higher heat flow rate. Th e initial condition effect diminishes over time. The heat flow rate is higher closer to the inlet because of the constant temperature inlet condition. Figure 5.7 shows the Peripheral average interface temperature over 9 seconds at different axial locations at the fluid-silicon interface. Sinusoidal behavior is also observed for the interface temperature as the gadolin ium interface. Heat is generated in the gadolinium substrate then convected to the fluid directly and c onducted to the silicon substrate. Even though this process slows down the heat transfer in the silicon substrate the silicon side has a bit smaller interface temperature than the gadolinium because its high thermal conductivity. Figure 5.8 shows the average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface. Similar to the gadolinium side, the heat flow rate is supplied to and rejected fr om the fluid over time. The silicon side shows a less heat flow rate at its interface than the gadolinium. This is because the heat generation occurs in th e gadolinium substrate. Figure 5.9 shows the average Nusselt number over 9 seconds at different axial locations at the total interface including both si des of the channel, gadolinium and silicon. The Nusselt number behavior is shown to be increasing for each time period regardless of the process. At the beginning of each peri od, the interface and bulk temperature switch because the cooling and heating processes a lternate. At certain point of the switching process Nusselt number will be infinite, that is when the interface and bulk temperature are equal. That is why at each time period (3.5 s, 5.5 s and 7.5 s) Nusselt number increase to a high value then decreases to reasonable values where it starts to increase again. Nusselt number is always positive and increasing for both heating and cooling processes because only the absolute value of the heat flow rate is considered. The switch between

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101 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.7: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.8: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface (Re = 2000, G = 5 T, Hfl = 0.03 cm)

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102 6 8 10 12 14 16 18 012345678910 Time, sAverage Nusselt Number z = 1.50 cm z = 2.00 cm Figure 5.9: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-solid interface (Re = 2000, G = 5 T, Hfl = 0.03 cm) interface and bulk temperatures seems to occu r early in the second period causing lower Nusselt number at time 1.5 s. Nusselt number s eems to be lower further away from the inlet because of the lower heat flow rate that results from lower interface-bulk temperature. Figure 5.10 shows the peripheral average interface temperature over 9 seconds at different axial locations at the fluid-gadolin ium interface with magne tic field alternates between +10 and -10. The interface temperatur e reaches higher values when heating and lower values when cooling than those of the pr evious case. This is because of the larger magnetic field applied during heating and c ooling. Figure 5.11 shows the average heat flow rate over 9 seconds at different axial lo cations at the fluid-ga dolinium interface with

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103 10 12 14 16 18 20 22 24 26 28 30 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.10: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-gadolinium interface with G = 10 T (Re = 2000, Hfl = 0.03 cm) -100 -80 -60 -40 -20 0 20 40 60 80 100 012345678910 Time, sAverage Heat Flow Rate, kw/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.11: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-gadolinium interface with G = 10 T (Re = 2000, Hfl = 0.03 cm)

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104magnetic field alternates between +10 and -10. The heat flow rate is higher when heating and lower when cooling compared to the previous case. This is because of the higher heat source and sink caused by higher magnetic field at both processes. Fi gures 5.12 and 5.13 show results at the silicon edge similar to those at the gadolinium when compared to the original case. Both sides, gadolinium and silic on, has an increase in the absolute value of the interface temperature and heat flow rate s during the heating a nd cooling processes. Figure 5.14 shows the average Nusselt number ove r 9 seconds at different axial locations at the fluid-solid interface with magnetic field alternates between +10 and -10. Our experience from previous work tells us that the change in the magnetic field does not affect the Nusselt number. The figure shows very close results to those of the original case. Figures 5.15 to 5.19 show the effect of using smaller ch annel height on the heating and cooling performance of the tr apezoidal microchannel system. Figure 5.15 shows the peripheral average interface temper ature over 9 seconds at different axial locations at the fluid-ga dolinium interface with Hfl = 0.02 cm. Compared to figure 5.5, the interface temperature is less for smalle r channel height. This is because smaller channel with same Reynolds number have hi gher axial velocity which enhances the convection heat transfer to the fluid. This results in lower interface temperatures during the heating process but higher interface temp erature during the cooling process. The absolute value of heat flow rate is higher for smaller channels as shown in figure 5.16. This is because the temperature gradient in the solid is increasing due to higher fluid velocity. Figures 5.17 and 5.18 show similar results of the silicon side to those of the gadolinium side. Figure 5.19 shows the av erage Nusselt number over 9 seconds at

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105 12 14 16 18 20 22 24 26 28 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.12: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-silicon in terface with G = 10 T (Re = 2000, Hfl = 0.03 cm) -100 -80 -60 -40 -20 0 20 40 60 80 100 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.13: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface with G = 10 T (Re = 2000, Hfl = 0.03 cm)

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106 6 8 10 12 14 16 18 012345678910 Time, sAverage Nusselt Number z = 1.50 cm z = 2.00 cm Figure 5.14: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-solid inte rface with G = 10 T (Re = 2000, Hfl = 0.03 cm) 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.15: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-gadolinium interface with Hfl = 0.02 cm (Re = 2000, G = 5 T)

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107 -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.16: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-gadolinium interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.17: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluidsilicon inte rface with Hfl = 0.02 cm (Re = 2000, G = 5 T)

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108 -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.18: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface with Hfl = 0.02 cm (Re = 2000, G = 5 T) 0 4 8 12 16 20 24 012345678910 Time, sAverage Nusselt Number z = 1.50 cm z = 2.00 cm Figure 5.19: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluidsolid interface with Hfl = 0.02 cm (Re = 2000, G = 5 T)

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109different axial locations at th e fluid-solid interface with Hfl = 0.02 cm. It shows more irregular Nusselt number than the original case The temperature gradient is the change in temperature in the solid region at the interface. When the h eating process takes over, the solid has higher temperature than the interface. When the cooling process takes place the interface temperature becomes higher than th e solid. This switch between values takes place at the beginning of every time period. Th is affects the sign of the heat flow rate indicating a heating or cooling process. The heat flow rate is the nominator for Nusselt number. In addition another switch between the interface and bulk temperatures is taking place. The interface-bulk temperature differe nce is the denominator of the Nusselt number. The time when the temperature gr adient switch has took place while the interface-bulk temperatures are still in the switch process Nusselt number may be negative. After the interface-bu lk temperature switch Nusselt number starts its increase to reasonable values. Higher fluid velocity that enhances the heat flow rate and results in earlier temperature gradient switch in the so lid might be the reason behind the reason for having low Nusselt numbers. Compared to the original case, Nusselt number is lower with smaller channel height. Figure 5.20 shows the peripheral average interface temperature over 9 seconds at different axial locations at the fluid-gado linium interface at Re = 1000. Lower Reynolds number, or lower fluid velocity, slows down the heating or cooling process and thus results in lower interface temperature. It also results in lower absolute value of heat flow rate. This is shown in figur e 5.21 in comparison with figur e 5.6. Figures 5.22 and 5.23 show the peripheral average interface temperat ure and heat flow rate over 9 seconds at

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110 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.20: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-gadolinium interface at Re = 1000 (G = 5 T, Hfl = 0.03 cm) -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 c m z = 1.00 c m z = 1.50 c m z = 2.00 c m Figure 5.21: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-gadolinium interface at Re = 1000 (G = 5 T, Hfl = 0.03 cm)

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111 14 16 18 20 22 24 26 012345678910 Time, sPeripheral Average Interface Temperature, oC z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.22: Peripheral average interface temp erature over 9 seconds at different axial locations at the fluid-silicon in terface at Re = 1000 (G = 5 T, Hfl = 0.03 cm) -60 -40 -20 0 20 40 60 012345678910 Time, sAverage Heat Flow Rate, kW/m2 z = 0.50 cm z = 1.00 cm z = 1.50 cm z = 2.00 cm Figure 5.23: Peripheral average heat flow rate over 9 seconds at different axial locations at the fluid-silicon interface w ith at Re = 1000 (G = 5 T, Hfl = 0.03 cm)

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112different axial locations at the fluid-sili con interface at Re = 1000 respectively. They show similar results of those of gadolinium si de when compared to figures 5.7 and 5.8. Nusselt number behavior in figure 5.24 is e xplained in the same fashion as in the previous case. The value of Nusselt number is less than the original case because of the lower absolute values of the heat flow rate resulted from lower fluid velocity. 5.4 Conclusions This study of a time varying heat generation in a trapezoidal composite microchannel model with showed the effect of different parameters. The behavior of the interface temperature, heat fl ow rate, and Nusselt number were mostly sinusoidal. The results showed that the local Nusselt number was higher at the center of the channel. It also showed that Nusselt number decrease d along the axial direction. The interface temperature showed sinusoidal behavior as th e heating and cooling processes alternates. The heat flow rate increases in absolute value in each time period. Nusselt number found to increase in each time period regardless the process. Higher magnetic field increases the interface temperature and the absolute value of the heat flow rate. It does not have any effect on the behavior of Nusselt number. Sma ller channel height decreased the interface temperature but increased the absolute heat flow rate. Nusselt number decreased because of its dependence on the hydraulic diameter of the channel. Lower Reynolds number decreases the interface temperature, heat flow rate, and Nusselt number.

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113 0 2 4 6 8 10 12 14 012345678910 Time, sAverage Nusselt Number z = 1.50 cm z = 2.00 cm Figure 5.24: Peripheral average Nusselt number over 9 seconds at different axial locations at the fluid-solid interf ace with at Re = 1000 (G = 5 T, Hfl = 0.03 cm)

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114 Chapter 6 – Steady State Heat Transf er Using Nanofluids in Circular Microchannel 6.1 Mathematical Model The problem in hand is a microchannel assembly with circular channel in a rectangular solid substrate. Th e channel substrate extends to a length (L). The thickness of the substrate is (H) and distance between tubes in the horizontal direction is 2W. Nanofluid flows through circular channels wi th diameter (D) and length (L) as single pass from inlet to outlet manifold. Nanofluids are fluids suspended with solid particles. These nanometer-sized particles have high thermal conductivity which enhances the overall fluid’s thermal conductivity. The heat is supplied at the bottom of the substrate. Heat is conducted through the solid substrate material and co nvected to the working fluid. Figure 6.1 shows a schematic drawing for the model simulated. The governing equations for the conserva tion of mass, momentum, and energy in the liquid region are [45]: 0 1 1 z V V r V r r Vz r r (1) V r r V z V V r r V r r V r p V r z V V V r V r V Vr r r r r f r z r r r2 2 2 2 2 2 2 2 2 22 1 1 1 1 (2)

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115 Figure 6.1: Schematic of nanoflu id circular microchannel model r f r z rV r r V z V V r r V r r V p r r V V z V V V r V r V V2 2 2 2 2 2 2 2 22 1 1 1 (3) 2 2 2 2 2 2 21 1 1z V V r r V r r V z p z V V V r V r V Vr z z z f z z z z r (4) 2 2 2 2 2 2 21 1 z T T r r T r r T z T V T r V r T Vf f f f f f z f f r (5) Considering constant thermal conductivity, th e energy conservation e quation in the solid region with heat generation is [46]: 02 2 2 2 2 2 z T y T x Ts s s (6)

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116It may be noted that a cylindrical coordinate system was used to model convection within circular tube while a Cartesian coordinate system was used to model conduction within the solid substrate material. Equations (1) to (6) are subject to the following boundary conditions: At z = 0, 0 r < d/2: in f f in z z rT T V V V V, ,, 0 0 (7) At z = 0, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (8) At z = L, r < d/2: 0 p (9) At z = L, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (10) At x = 0, -d/2 y +d/2, 0 z L: 0 0 0 0 x T x V x V Vf z r (11) At x = 0, -H/2 y -d/2, 0 z L: 0 x Ts (12) At x = 0, +d/2 y +H/2, 0 z L: 0 x Ts (13) At x = W, -H/2 y +H/2, 0 z L: 0 x Ts (14) At y = -H/2, 0 x W, 0 z L: 0 y Ts (15) At y = +H/2, 0 x W, 0 z L: 0 y Ts (16) At 0 z L, r = d/2: dr T k dr T k T Ts s f f s f (17) At y = -H/2, 0 x W, 0 z L: s sk q dy T" (18)

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1176.2 Numerical Simulation and Parametric Study The governing equations along with the boundary conditions were solved using the Galerkin finite element method. Four-node quadrilateral elements were used. In each element, the velocity, pressure, and temperat ure fields were approximated which led to a set of equations that defined the continuum. 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 did not chan ge from one iteration to the next, and the residuals for each variable became negligible. For the numerical computations, the hei ght (H) and the half width (W) of the model were set to 0.2 cm and 0.1 cm respec tively. The channel lengt h (L) was set to 2.3 cm. The diameter was changed from 0.06 cm to 0.14 cm. Reynolds number was varied between 500 and 1500. Silicon was used as th e solid substrate material. Another two solid substrate were used: Silicon carbide and stainless steel. Water suspended with 4% volume fraction of Alumina (AL2O3) was used as the working fluid. 1% and 2% volume fraction of Alumina suspended into water we re used as well. The local solid-fluid interface temperature, heat flow rate, heat transfer coefficient and Nusselt number were calculated from the resulted velocity and temperature distributions. 6.3 Results and Discussions To establish an independent mesh, several mesh sizing were studied and compared. Interface temperatures and Nusselt numbers were plotted. Figure 6.2 and 6.3 show the interface temperature and Nusse lt number for different mesh sizing,

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118 20 25 30 35 40 45 50 55 60 65 024681012 Axial Direction, cmPeripheral Average Interface Temperature,oC nx=4, ny=8, nr=4, nz=40 nx=6, ny=12, nr=6, nz=40 nx=8, ny=16, nr=8, nz=40 nx=10, ny=20, nr=10, nz=40 Figure 6.2: Peripheral average interface temp erature along the axial direction (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 4.0 4.5 5.0 5.5 6.0 024681012 Axial Direction, cmPeripheral Average Nusselt Number nx=4, ny=8, nr=4, nz=40 nx=6, ny=12, nr=6, nz=40 nx=8, ny=16, nr=8, nz=40 nx=10, ny=20, nr=10, nz=40 Figure 6.3: Peripheral average Nusselt number along the axial direction (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm)

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119respectively. From both figures the mesh si ze (nx = 8, ny = 16, nr = 8, nz = 40) was found to be sufficient for mesh independence. The local interface temperature, heat fl ow rate, heat transfer coefficient and Nusselt number were calculated from the result ed velocity and temperature distribution. Figure 6.4 shows the interface temperature alon g the angular direction from bottom to top at different axial locations. This is done because of the one sided heat flux boundary condition from the bottom. The interface temperature is slightly higher at the bottom of the tube, which is closer to the heat source, at all axial locations. Figure 6.5 shows the heat flow rate along the angular direction from bottom to top at different axial locations. The heat flow rate is higher closer to the bo ttom of the tube, which is closer to the heat source. There is a slight increase in the heat flow rate between the side and bottom of the tube. This is the area where the heat flow st ream lines are parallel to the channel tube. The high conductance of the solid material tran sfers parts the heat towards the sides and top of the tube. This where it slightly re -increase but then continue to decrease afterwards. Notice that the heat flow rate is higher closer to the tube entry. This is expected because of the low inlet fluid temperature that results in higher temperature difference. Figure 6.6 shows the heat transfer coefficient al ong the angular direction from bottom to top at different axial locations. The heat transfer coefficient is generally decreasing from the bottom of the tube to the top. Similarly to the heat flow rate, the heat transfer coefficient is higher closer to the tube entry. This is because of the fluid low inlet temperature. Figure 6.7 shows the Nusselt nu mber along the angular direction from bottom to top at different axial locations. The Nusselt number follows similar trend of those of heat transfer coefficient because of its direct dependence.

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120 18 20 22 24 26 28 30 32 -90-60-300306090 Angular Direction, Local Interface Temperature, oC z = 0 cm z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 6.4: Local interface temperature along the angular direction at different axial locations (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 30 40 50 60 70 80 90 -90-60-300306090 Angular Direction, Local Heat Flow Rate, kW/m2 z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 6.5: Local heat flow ra te along the angular direction at different axial locations (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm)

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121 6 12 18 24 30 36 -90-60-300306090 Angular Direction, Local Heat Transfer Coefficient, kW/m2.K z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 6.6: Local heat transfer coefficient along the angular direction at different axial locations (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 0 5 10 15 20 25 30 -90-60-300306090 Angular Direction, Local Nusselt Numbe r z = 0.5 cm z = 1.0 cm z = 1.5 cm z = 2.0 cm Figure 6.7: Local Nusselt number along the angula r direction at differe nt axial locations (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm)

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122 Figure 6.8 shows the average peripheral in terface temperature and heat flow rate along the axial direction for Silicon substrate and water suspended with 4% volume fraction of Alumina as the working flui d. Reynolds number was 500 and the tube diameter was 0.06 cm. The interface temperature increases along the axial direction because of the heat flux. The heat flow rate is decreasing along the axial direction because the fluid temperature rises along th e axial direction. Figure 6.9 shows the peripheral average heat transfer coefficien t and Nusselt number along the axial direction. The Nusselt number follows similar trend of t hose of heat transfer coefficient because of its direct dependence. The average periphera l heat transfer coefficient and Nusselt number decrease along the axial direction because of the inte rface-bulk temperature difference that is increasing along the axial direction. Detailed analysis is represented to study the angular effect on the interface temperature, heat flow rate, and Nusselt number. Because of the one sided heat flux boundary condition the top, side and bottom lines of the tube were analyzed separately. It is expected that the bottom side will have th e highest interface temperature and heat flow rate. Figure 6.10 shows the interface temperature along the axial direction at different angular locations. The bottom of the tube ha s higher interface temp erature while the top of the tube has the lowest. This is because of the heat flux boundary condition at the bottom of the substrate. Figure 6.11 shows the heat flow rate along the axial direction at different angular locations. The heat flow rate is decreasing along the axial because of the fluid bulk temperature rises. At the bottom of the tube the h eat flow rate decreasing rate along the axial directi on decreases because of the heat source. Figure 6.12 shows Nusselt number along the axial direction at different angular locations. Nusse lt number decreases

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123 0 5 10 15 20 25 30 35 00.511.522.5 Axial Direction, cmPeripheral Average Interface Temperature, oC20 30 40 50 60 70 80 90 100Peripheral Average Heat Flow Rate kW/m2 Temperature Heat flow rate Figure 6.8: Peripheral average interface temper ature and heat flow rate along the axial direction (Silicon, Water + 4% volume frac tion of Alumina, Re = 500, d = 0.06 cm) 0 5 10 15 20 25 30 35 40 00.511.522.5 Axial Direction, cmPeripheral Average Heat Transfer Coefficient, kW/m2.K0 4 8 12 16 20 24Peripheral Average Nusselt Number Heat transfer coeffcient Nusselt number Figure 6.9: Peripheral average heat transfer coefficient and Nusselt number along the axial direction (Silicon, Water + 4% volume fr action of Alumina, Re = 500, d = 0.06 cm)

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124 20 22 24 26 28 30 00.511.522.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Top of the tube Side of the tube Bottom of the tube Figure 6.10: Interface temperature along the axia l direction at different angular location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 0 20 40 60 80 100 0.811.21.41.61.822.2 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Top of the tube Side of the tube Bottom of the tube Figure 6.11: Heat flow rate along the axial dire ction at different angul ar location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm)

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125 0 4 8 12 16 20 24 28 32 00.511.522.5 Axial Direction, cmLocal Nusselt Numbe r Top of the tube Side of the tube Bottom of the tube Figure 6.12: Nusselt number along the axia l direction at differe nt angular location (Silicon, Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) along the axial direction because the interf ace-bulk temperature difference decreases along the axial direction. Figure 6.13 shows the interface temper ature along the axial direction using different working fluids. Four fluids used are: pure water and water suspended with Alumina (Al2O3) at different volume fractions of 4% 2% and 1%. To maintain the same Reynolds number the fluid velocity must be adjusted for each fluid. The water suspended with higher volume fraction of Alumina has higher thermal conductivity. This thermal conductivity enhancement is caused by the high thermal conductivity of the solid particles. This causes more heat rejection to the working fluid and thus lower interface temperatures. On the other hand, the fluid ve locity decreases with higher volume fraction, because of the change in properties and ma intaining the same Reynolds number, which

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126 20 21 22 23 24 25 26 27 28 29 30 00.511.522.5 Axial Direction, cm Peripheral Average Interface Temperature, oC water + Al2O3 (4%) water + Al2O3 (2%) water + Al2O3 (1%) pure water Figure 6.13: Peripheral average interface temper ature along the axial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm) cause an increase to the interf ace temperature. The net result is a decrease in the interface temperature. Figure 6.14 shows the heat flow rate along the axial direction using different working fluids. The heat flow rate decreas es along the axial direction because of the increase in the fluid bulk temperature. It is expected that higher volume fraction of the nanoparticles in the fluid will enhance the heat flow rate at the inte rface of the tube. On the other hand, the heat flow rate decreases b ecause of the lower fluid velocity due to the change in properties and main taining the same Reynolds nu mber. The net result is an increase in the heat flow rate with lo wer volume fraction. Nusselt number directly depends on the heat flow rate and the flui d conductivity. Figure 6. 15 shows the Nusselt number along the axial directi on at different angular locati ons. Nusselt number depends on the heat transfer coefficient and the flui d thermal conductivity. Both are changing but

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127 50 55 60 65 70 0.811.21.41.61.822.2 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 water + Al2O3 (4%) water + Al2O3 (2%) water + Al2O3 (1%) pure water Figure 6.14: Peripheral average heat flow rate al ong the axial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm) 8 10 12 14 16 18 20 22 24 26 00.511.522.5 Axial Direction, cmPeripheral Average Nusselt Number water + Al2O3 (4%) water + Al2O3 (2%) water + Al2O3 (1%) pure water Figure 6.15: Peripheral average Nusselt numbe r along the axial direction at different working fluids (Silicon, Re = 500, d = 0.06 cm)

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128the heat flow rate has highe r effect on Nusselt number. This results in a decreasing Nusselt number with higher nanoparticles volume fraction. Figure 6.16 shows the interface temper ature along the axial direction using different solid materials. The three solids used are: Silicon, silicon carbide, and stainless steel. Silicon and silicon carbide has close pr operty values and hence the effect of this change was negligible. Stainless steel, on the other hand, has much lower thermal conductivity. The lower thermal conductivity wo rks as a thermal resistance and reduces the heat flow rate increase s the interface temperature at the interface. Figure 6.17 shows the heat flow rate along the axial direction using different solid materials. The stainless steel has lower thermal conductivity which leads to lower heat flow ra te at the interface. This results in lower Nusselt number for stainless steel as shown in figure 6.18. Figure 6.19 shows the interface temper ature along the axial direction using different Reynolds numbers. Highe r Reynolds number enhances the cooling rate and thus decreases the interface temperature. This is b ecause it increases the heat rejection in the system causing the heat flow rate at the in terface to increase as shown in figure 6.20. Nusselt number is directly dependent on the heat flow rate and thus it also increases with Reynolds number. The relatively higher Nusse lt numbers shown in figure 6.21 resulted from higher Reynolds number is because the thermal development of the fluid depends on the value of Reynolds number. Higher Reynolds numbers require longer tube which causes the Nusselt number to be higher at the same axial location. Figure 6.22 shows the interface temper ature along the axial direction using different diameters. Larger diameter reduces the fluid velocity to maintain its Reynolds number and hence slows down the cooling proces s. It is expected that this causes the

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129 20 21 22 23 24 25 26 27 28 29 30 00.511.522.5 Axial Direction, cmPeripheral Average Interface Temperature, oC Silicon Silicon Carbide Stainless Steel Figure 6.16: Peripheral average interface temper ature along the axial direction at different solid substrates (Water + 4% volume frac tion of Alumina, Re = 500, d = 0.06 cm) 50 55 60 65 70 0.811.21.41.61.822.2 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Silicon Silicon Carbide Stainless Steel Figure 6.17: Peripheral average heat flow rate along the axial direction at different solid substrates (Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm)

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130 6 8 10 12 14 16 18 20 22 00.511.522.5 Axial Direction, cmPeripheral Average Nusselt Number Silicon Silicon Carbide Stainless Stee l Figure 6.18: Peripheral average Nusselt number along the axial directio n at different solid substrates (Water + 4% volume fraction of Alumina, Re = 500, d = 0.06 cm) 20 21 22 23 24 25 26 27 28 29 30 00.511.522.5 Axial Direction, cmPeripheral Average Interface Temperature,oC Re = 500 Re = 1000 Re = 1500 Figure 6.19: Peripheral average interface temper ature along the axial direction at different Reynolds numbers (Silicon, Water + 4% vol ume fraction of Alumina, d = 0.06 cm)

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131 50 55 60 65 70 75 80 0.811.21.41.61.822.2 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 Re = 500 Re = 1000 Re = 1500 Figure 6.20: Peripheral average heat flow rate al ong the axial direction at different Reynolds numbers (Silicon, Water + 4% vol ume fraction of Alumina, d = 0.06 cm) 0 5 10 15 20 25 30 35 40 45 00.511.522.5 Axial Direction, cmPeripheral Average Nusselt Number Re = 500 Re = 1000 Re = 1500 Figure 6.21: Peripheral average Nusselt numbe r along the axial direction at different Reynolds numbers (Silicon, Water + 4% vol ume fraction of Alumina, d = 0.06 cm)

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132 20 21 22 23 24 25 26 27 28 29 30 00.511.522.5 Axial Direction, cmPeripheral Average Interface Temperature,oC d = 0.06 cm d = 0.10 cm d = 0.14 cm Figure 6.22: Peripheral average interface temper ature along the axial direction at different tube diameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500) interface temperature to increase It also reduces the volume of the solid substrate. Less solid volume results in less temperature rise in the model. In this case, the interface temperature decreased with higher diameter. Figure 6.23 shows the heat flow rate along the axial direction using diffe rent diameters. Higher diamet ers results in lower fluid velocity which slows down the cooling proce ss. Thus, the heat flow rate reduces with larger diameter. The heat transfer coefficient has similar trend to those of the heat flow rate as shown in figure 6.24 because of its direct depe ndence. Nusselt number, on the other hand, depends on the heat transfer coefficient and diameter. As the diameter increases, the heat transfer coefficient decreases. This decrease is not enough to overcome the diameter increase. Thus Nusselt number increases with the diameter increase as shown in figure 6.25. This is also the reason why the change of Nusselt number is smaller in ratio than t hose of heat transfer coefficient.

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133 0 10 20 30 40 50 60 70 0.811.21.41.61.822.2 Axial Direction, cmPeripheral Average Heat Flow Rate, kW/m2 d = 0.06 cm d = 0.10 cm d = 0.14 cm Figure 6.23: Peripheral average heat flow rate along the axial direction at different tube diameters (Silicon, Water + 4% volum e fraction of Alumina, Re = 500) 0 5 10 15 20 25 30 00.511.522.5 Axial Direction, cmPeripheral Average Heat Transfer Coefficient, kW/m2.K d = 0.06 cm d = 0.10 cm d = 0.14 cm Figure 6.24: Peripheral average heat transfer coefficient along the axial direction at different tube diameters (Silicon, Water + 4% volume fraction of Alumina, Re = 500)

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134 6 10 14 18 22 26 30 00.511.522.5 Axial Direction, cmPeripheral Average Nusselt Number d = 0.06 cm d = 0.10 cm d = 0.14 cm Figure 6.25: Peripheral average Nusselt number along the axial direction at different tube diameters (Silicon, Water + 4% volum e fraction of Alumina, Re = 500) 6.4 Conclusions Grid test was performed for the nanoflu id system to select an appropriate meshing. The model was simulated and so lved for the velocity and temperature distribution. The results were presented in te rms of the interface temp erature, heat flow rate, and Nusselt number along the axial dire ction and at different parameters. The interface temperature increased along the axial direction while the heat flow rate decreased. Nusselt number, on the other ha nd, decreased along th e axial direction. Angular analysis was carried out to understa nd the effect of one-s ided heating boundary condition. It showed that the heat flow rate and the Nusselt number are higher on the side of the tube where it is closer to the heated edge. Increasin g the volumetric fraction of the nanoparticles decreased the interface temperat ure and the heat flow rate as well as

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135Nusselt number. Stainless steel has a higher interface temperature but lower heat flow rate and Nusselt number than the Silicon. Th e increase in the Reynol ds number decreases the interface temperature but increases the h eat flow rate and Nu sselt number. Larger diameters decreased the inte rface temperature and the heat flow rate. Nusselt number depends on the heat flow rate and diameter. The increase in diameter overcomes the decrease in the heat transfer coe fficient and increases Nusselt number.

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136 Chapter 7 – Steady State Analysis of Hydronic Snow Melting System 7.1 Mathematical Model The problem in hand is a hydronic snow melting system. It consists of stainless steel pipe embedded in a concrete slab exposed to snow weather condition. Hot fluid is passing through the pipe to heat the slab a nd melt the snow accumulated. The slab is mounted over a soil ground. Because of the symm etry in the system, only half of one pipe is modeled. To consider the heat loss to the ground it was assumed to be of height ten times the height of the concrete slab. The assumption of isothermal boundary condition at the bottom of the ground is now valid. The symmetry lines also have isothermal boundary condition. Figure 7.1 shows the mode l to be studied. The model has a slab width of 15 cm and height of 12 cm. The dept h of the ground was considered to be ten times of the slab height (120 cm) to insure ta king into account the heat loss to the ground. The governing equations for the conservati on of mass, momentum, and energy in the liquid region are [45]: 0 1 1 z V V r V r r Vz r r (1) V r r V z V z V r r V r r r r p V r z V V V r V r V Vr r r r t f r z r r r2 2 2 22 1 1 1 1 (2)

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137 Figure 7.1: The three dimensional snow meltin g system model for steady state analysis r t f r z rV r r V z V z V r r V r r r p r r V V z V V V r V r V V2 2 22 1 1 1 (3) z V z V r r V r r r z p z V V V r V r V Vz z z t f z z z z r 21 1 1 (4) The kmodel was used for simulation of turbul ence. In this model, equations governing the conservation of turbulence kinetic ener gy and its rate of dissipation were solved. These equations can be expressed as:

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138 2 2 2 2 2 2 21 1 1 1z z r r t k t k t k t z rV r r V z V r V r V z V V r z k z k r r k r r r z k V k r V r k V (5) 2 2 2 2 2 2 1 2 2 21 1 1 1 z z r r t t t t z rV r r V z V r V r V z V V r k C z z r r r r r k C z V r V r V (6) where 2k Ct (7) The empirical constants appearing in equati ons (5) to (7) are given by the following values, C= 0.09, C1 = 1.44, C2 = 1.92, k = 1.0, = 1.3. The energy equation in the fluid region is: 2 2 2 2 2 2 21 1 Pr z T T r r T r r T z T V T r V r T Vf f f f t t f f z f f r (8) Considering constant thermal conductivity, th e energy conservation e quation in the solid region is [46]: 02 2 2 2 2 2 z T y T x Ts s s (9) Note that a cylindrical coordinate system was used to model convection within the circular tube while a cartesian coordinate system was used to model conduction within the solid substrate material.

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139Equations (1) to (6) are subject to the following boundary conditions: At z = 0, 0 r < d/2: in f f in z z rT T V V V V, ,, 0 0 (10) At z = 0, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (11) At z = L, r < d/2: 0 p (12) At z = L, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (13) At x = W, (H-d)/2 y (H+d)/2, 0 z L: 0 0 0 0 x T x V x V Vf z r (14) At x = W, -E y (H-d)/2, 0 z L: 0 x Ts (15) At x = W, (H+d)/2 y H, 0 z L: 0 x Ts (16) At x = 0, -E y H, 0 z L: 0 x Ts (17) At y = -E, 0 x W, 0 z L: 0 y Ts (18) At y = H, 0 x W, 0 z L: 0 y Ts (19) At 0 z L, r = d/2: dr T k dr T k T T V V Vs s f f s f z r , 0 0 0 (20) 7.2 Numerical Simulation and Parametric Study The governing equations along with the boundary conditions were solved using the Galerkin finite element method. An iter ative procedure was used to arrive at the solution for the velocity and temperature fi elds. The solution was considered converged

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140when the field values did not change from one iteration to the next, and the residuals for each variable became negligible. This study is a parametric study that ha s an original case an d several other cases to study the effect of each parameter. In e ach case only one parameter is changed. The parameters to be changed are the working fluid, Re ynolds number, inlet fluid temperature, pipe diameter and pipe spacing. The working fluid for the original case is water and the fluid was consid ered turbulent at Reynolds nu mber = 3000. Fluid entered at 80 oC to a in. stainless steel pipe. The spaci ng was taken as 15 cm. The working fluid was changed to Ethylene Glycol in one case. Variable properties were used for the Ethylene Glycol while water properties were considered constant. Reynolds number was increased to 5000 and the inlet fluid temperature was changed to 60 oC. The pipe diameter was changed to 1.0 in. keeping th e Reynolds number cons tant. Finally, the spacing was increased to 20 cm. 7.3 Results and Discussion Studying the results for the temperatur e distribution in the model one can calculate the heat flow rate at various locations in the model. The following figures show the results of all cases with th e effect of each parameter. Figures 7.2 to 7.5 represent the original case. Figure 7.2 shows the heat flow rate along the angular direction. It is clear that the heat flow rate is higher at the top of the pipe and lower at the bottom. This is because the top of the pipe is facing th e snow boundary condition which causes higher temperature difference resulting in higher heat flow rate. Notice that the heat flow rate drops along the axial direction. Th is is a direct result for the drop of the fluid temperature

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141 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 -90-60-300306090 Angular Direction, Peripheral Average Heat Flow Rate, W/m2 z = 0 cm z = 10 cm z = 20 cm z = 30 cm z = 40 cm z = 50 cm z = 60 cm Figure 7.2: Heat flow rate at the outer pipe surfa ce along the angular direction 0 200 400 600 800 1,000 1,200 1,400 03691215 Slab Width Direction, cmAverage Heat Flow Rate, W/m2 z = 0 cm z = 10 cm z = 20 cm z = 30 cm z = 40 cm z = 50 cm z = 60 cm Figure 7.3: Heat flow rate at the snow boundary surface along the width of the slab

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142 20 25 30 35 40 45 50 03691215 Slab Width Direction, cmAverage Concrete-Ground Interface Temperure,oC z = 0 cm z = 20 cm z = 40 cm z = 60 cm Figure 7.4: Average temperatur e at the concrete-ground interface along the width of the slab -50 0 50 100 150 200 250 300 03691215 Slab Width Direction, cmAverage Heat Flow Rate, W/m2 z = 0 cm z = 10 cm z = 20 cm z = 30 cm z = 40 cm z = 50 cm z = 60 cm Figure 7.5: Heat flow rate at the concrete -ground interface along the width of the slab

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143along the axial direction of the pipe due to heat rejection. Figure 7.3 shows the heat flow rate at the snow boundary condition along the widt h of the slab. It is obvious that the heat flow rate is higher at the snow surface where it is closer to the pipe (x = 15 cm). This is because that location is closer to the heat source, hot fluid. It is also shown that the heat flow rate drops in the axial direction. This is a direct result for the drop of the fluid temperature axially. Figure 7.4 shows the interface temperature at the concrete-ground interface along the width of slab. The temperatur e is higher at locations closer to the pipe and it is decreasing along the axial direction. Figur e 7.5 is the heat flow rate at the concrete-ground interfac e along the width of slab. The h eat flow rate is higher at locations closer to the pipe because it is clos er to the heat source. Notice that it becomes a negative value at the midway between two pi pes. This is because the heat flow is escaping through the space between two pipe s to support the snow melting process. Using less spacing or inlet temperature may not allow it. The above figures were for the original case explained above. To study the effect of each parameter alone, several cases we re studied where only one parameter was changed for each case while the rest are kept constant. Figures present the results for all cases compared to the original one. Figures 7.6 to 7.7 represent the interface temperature at the inner pipe surface a nd the fluid bulk temperature along the axial direction at different parameters. Figure 7.6 shows the aver age interface temperature al ong the axial direction. Ethylene Glycol has a slightly higher density and very high vi scosity compared to water. The ratio of density to viscosity for water is 7 times that of Ethylene Glycol. This means for the same Reynolds number Ethylene Glycol gives much higher fluid velocity. This

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144 35 40 45 50 55 60 65 70 75 80 0102030405060 Axial Direction, cmPeripheral Average Interface Temperature, oC Original Ethyl. Glycol Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.6: Average interface temperature at the inner pipe surface along the axial direction 35 40 45 50 55 60 65 70 75 80 85 0102030405060 Axial Direction, cmAverage Fluid Bulk Temperature, oC Original Ethyl. Glycol Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.7: Average bulk temperature of the fluid along the axial direction

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145enhances the convection heat transfer and cause the wall temperature at the inner pipe surface to drop drastically. Hi gher Reynolds number means higher fluid velocity. The higher velocity enhances the heat flow rate from the fluid to the solid. This results in higher temperature at the inner pipe surface. Lower inlet temp erature have same trend as the original case with higher temperature. On th e contrary, its rate of decrease is less than the original case along the axial direction. This is because the temperature difference between the lower inlet temperature and the boundary condition is less which results in lower heat flow rate through the system. Enla rging the pipe diameter reduces the solid material which acts as a resistance to the heat flow rate. Larger pipe also increases the pipe surface area enhancing the heat flow rate. This increase s the interface temperature at the fluid-pipe interface. Larger spacing, on the other hand, enlarges the solid material and adds more resistance to the heat flow rate. There was a slight decrease in the interface temperature with larger spacing. Figure 7.7 shows the average bulk temp erature along the axial direction. The results of figure 7.7 are similar to those of figure 7.6. Both the bulk and inner interface temperature has similar trend along the axia l direction. This is because the interface temperature is a direct result for the change in the fluid bul k temperature. Using Ethylene Glycol is the only case when the bulk temper ature has a different slope than the interface temperature because it has different thermal conductivity. Figure 7.8 shows the averag e interface temperature along the axial direction at the pipe-concrete surface. It has ve ry similar trend as the averag e interface temperature at the inner pipe surface in figure 7.6. This is be cause the pipe is very thin and has high conductivity. Figure 7.9 shows the heat flow rate at the outer pipe surface, pipe-concrete

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146 35 40 45 50 55 60 65 70 75 80 85 0102030405060 Axial Direction, cmPeripheral Average Interface Temperature, oC Original Ethyl. Glyco l Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.8: Average interface temperature at the outer pipe surface along the axial direction 0 500 1000 1500 2000 2500 0102030405060 Axial Direction, cmPeripheral Average Heat Flow Rate, W/m2 Original Ethyl. Glyco l Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.9: Average heat flow rate at th e outer pipe surface along the axial direction

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147interface, along the axial direction. Ethylene Glycol causes smaller temperature gradient in the solid substrate as explained before. Ther efore, it decreases the heat flow rate at the interface. For higher Reynolds number the fluid ve locity is higher. This increases the heat flow rate from the fluid to the solid. Hence it in creases the heat flow rate at the outer pipe surface. Lower inlet temperature gives less te mperature gradient in the solid causing the heat flow rate to decrease. It is obvious that the heat flow rate decreases in a lower rate in the axial direction. This is because lower inlet temperature causes less interface-bulk temperature difference and thus reduces the h eat flow rate from the working fluid to the system. Larger diameter results in a reduc tion in the solid volume and an increase the pipe surface area. Both increase in the heat flow rate to the model. Larger spacing between the pipes means more heating load pe r pipe. This cool down the pipe walls and decreases the pipe wall temp erature. This results in a high interface-bulk temperature difference and increases the heat flow rate to the system. Figure 7.10 shows the average heat flow rate at the snow surface along the axial direction. As explained above Ethylene Glycol decreases the temperature gradient in the substrate. This results in a decrease in th e heat flow rate on th e snow surface. Higher Reynolds number with higher fluid velocity increa ses the heat flow rate from the fluid to the solid. The increase in the heat flow rate at the snow surface is a direct result. Lower inlet temperature results in a decrease in the h eat flow in the system. This also reflects on the snow surface heat flow. Using larger diam eter results in an increase the pipe surface area. This increases the heat flow rate to the snow surface. With larger spacing between the pipes the amount of heat flow is distri buted over a larger area reducing the average heat rate per unit area.

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148 0 50 100 150 200 250 300 350 400 450 500 0102030405060 Axial Direction, cmAverage Heat Flow Rate, W/m2 Original Ethyl. Glyco l Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.10: Average heat flow rate at the snow surface along the axial direction Figure 7.11 shows the average heat flow rate at the ground-concrete interface along the axial direction. This represents the heat loss to the ground. Ethylene Glycol causes smaller temperature gradient in the soli d. This results in a decrease in the heat flow rate to the ground. Higher Reynolds number with higher fluid ve locity increases the heat flow rate from the fluid to the solid a nd hence the heat flow rate to the system. The increase in the heat flow rate to the ground is a direct result. Lower inlet temperature results in a decrease in the heat flow in the system. This also reflects on the heat loss to the ground. As explained above the increase in diameter increases the heat flow rate. This increases the heat loss to the ground as we ll. Larger spacing increases the heat flow rate from as explained above. It also give s space to the heat flow to pass through the pipes to the top. The increase in the heat fl ow rate overcomes the heat amount passes to the top between the pipes and incr eases the heat loss to the ground.

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149 0 20 40 60 80 100 120 140 0102030405060 Axial Direction, cmAverage Heat Flow Rate, W/m2 Original Ethyl. Glyco l Re = 5000 T = 60 C d = 1 in sp = 8 in Figure 7.11: Average heat flow rate lo st to the ground along the axial direction 7.4 Conclusions Three dimensional steady state snow me lting model was developed. A numerical scheme was applied to come up with the heat flow rate calculations within the system. Parametric study was carried out to understandi ng the effect of different parameters. The heat flow rate was higher at the top of the pipe where it faces the snow boundary condition. Using Ethylene Glycol, as the working fluid, results in lower heat flow rate at the fluid interface and the snow boundary c ondition. The increase in Reynolds number increases the heat flow rate at the fluid interface and the snow boundary condition. Lower inlet temperature results in less heat flow rate at the fluid interface and the snow boundary condition. Larger diameter for the same Reynolds number results in larger heat flow rate at the fluid interface and snow boundary condition. Larger spacing between

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150pipes increases the heat flow rate at the fl uid interface but reduces it at the snow surface. The heat loss to the ground was analyzed a nd plotted. Ethylene Gl ycol and lower inlet temperature decrease the heat loss to the ground. On th e other hand, higher Reynolds number, larger diameter and spacing in crease the heat loss to the ground.

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151 Chapter 8 – Transient Analysis of Hydronic Snow Melting System 8.1 Mathematical Model The problem in hand is a hydronic snow melting system. It consists of stainless steel pipe embedded in a concrete slab exposed to snow weather condition. Hot fluid is passing through the pipe to heat the slab a nd melt the snow accumulated. The slab is mounted over a soil ground. Because of the symm etry in the system, only half of one pipe is modeled. To consider the heat loss to the ground it was assumed to be of height ten times the height of the concrete slab. The assumption of isothermal boundary condition at the bottom of the ground is now valid. The symmetry lines also have isothermal boundary condition. Figure 8.1 shows the model to be studied. The model in figure 8.1 has a width, or pipe spacing, of 15 cm and height of 12 cm. The pipe depth was at the center of th e model at 6 cm. The depth of the ground was considered to be ten times of the slab hei ght (120 cm) to insure taking into account the heat loss to the ground. The governing equations for the conserva tion of mass, momentum, and energy in the liquid region are [45]: 0 1 1 z V V r V r r Vz r r (1)

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152 Figure 8.1: The three dimensional snow me lting system model for transient analysis V r r V z V z V r r V r r r r p V r z V V V r V r V V t Vr r r r t f r z r r r r 2 2 2 22 1 1 1 1 (2) r t f r z rV r r V z V z V r r V r r r p r r V V z V V V r V r V V t V2 2 22 1 1 1 (3) z V z V r r V r r r z p z V V V r V r V V t Vz z z t f z z z z r z 21 1 1 (4)

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153The kmodel was used for simulation of turbul ence. In this model, equations governing the conservation of turbulence kinetic ener gy and its rate of dissipation were solved. These equations can be expressed as: 2 2 2 2 2 2 21 1 1 1z z r r t k t k t k t z rV r r V z V r V r V z V V r z k z k r r k r r r z k V k r V r k V t k (5) 2 2 2 2 2 2 1 2 2 21 1 1 1 z z r r t t t t z rV r r V z V r V r V z V V r k C z z r r r r r k C z V r V r V t (6) where 2k Ct (7) The empirical constants appearing in equati ons (5) to (7) are given by the following values, C= 0.09, C1 = 1.44, C2 = 1.92, k = 1.0, = 1.3. The energy equation in the fluid region, considering Prt = 1, is: 2 2 2 2 2 2 21 1 Pr z T T r r T r r T z T V T r V r T V t Tf f f f t t f f z f f r f (8) Considering constant thermal conductivity, th e energy conservation e quation in the solid region is [46]:

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154 t T z T y T x Ts s s s 12 2 2 2 2 2 (9) Note that a cylindrical coordinate system was used to model convection within the circular tube while a cartesian coordinate system was used to model conduction within the solid substrate material. Equations (1) to (9) are subject to the following boundary conditions: At t = 0: Ts = Tf = Tin (10) At z = 0, 0 r < d/2: in f f in z z rT T V V V V, ,, 0 0 (11) At z = 0, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (12) At z = L, r < d/2: 0 p (13) At z = L, r d/2, 0 < x < W, 0 < y < H: 0 z Ts (14) At x = W, (H-d)/2 y (H+d)/2, 0 z L: 0 0 0 0 x T x V x V Vf z r (15) At x = W, -E y (H-d)/2, 0 z L: 0 x Ts (16) At x = W, (H+d)/2 y H, 0 z L: 0 x Ts (17) At x = 0, -E y H, 0 z L: 0 x Ts (18) At y = -E, 0 x W, 0 z L: 0 y Ts (19) At y = H, 0 x W, 0 z L: 0 y Ts (20)

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155At 0 z L, r = d/2: dr T k dr T k T T V V Vs s f f s f z r , 0 0 0 (21) 8.2 Numerical Simulation and Parametric Simulation The governing equations along with the boundary conditions were solved using the Galerkin finite element method. An iter ative procedure was used to arrive at the solution for the velocity and temperature fi elds. The solution was considered converged when the field values did not change from one iteration to the next, and the residuals for each variable became negligible. This work is a parametric study that has an original case and several other cases to study the effect of each parameter. In each case only one parameter is changed. The parameters to be changed are the Reynolds num ber, pipe diameter, pipe depth and pipe spacing. The original case used water as th e working fluid and the fluid was considered turbulent at Reynolds number = 3000. Fluid en tered at 80 oC to a in stainless steel pipe. The spacing was taken as 15 cm. Reynolds number was increased to 45000 and 6000 while the pipe diameter was changed to and 1 in keeping the Reynolds number constant. The pipe depth was changed to 3 a nd 9 cm while pipe spacing was changed to 12 and 9 cm. 8.3 Results and Discussion To study the transient eff ect a storm scenario was se lected and applied to the model. This provides us with understanding about how the model will perform during a real storm. Another storm was selected with 6 times of the snow fall of the first one to

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156make sure the model will handle severe storm. Figures 8.2 to 8.6 re present results for the first storm scenario while figures 8.7 to 8.11 represent results for the second storm scenario. The second storm had 6 times the s now fall than the first and lasted for 9 hours longer. Figure 8.2 shows the average periphera l interface temperature along the axial direction over the storm tim e length of 5 hours. Two more hours added for both storm scenarios to understand how the model performs after the storm is over. The fluid enters at 80 oC and cools down along the axia l direction because of the storm. The fluid heats up the model over time and thus the interface temp erature. Notice that the temperature curve along the axial direction comes into a steady state over the 5 hours storm length. The last tow hours represent the time when the stor m is over and the weather condition is removed. The temperature curve increases as the model warms up with the end of the storm. Figure 8.3 shows the heat flow rate at the interface along th e axial direction over time. The heat flow rate seems to be consta nt along the axial direction. This is because both the interface and bulk temperature drop together along the axial direction maintaining constant temperature difference. Th e transient behavior is a little different where the bulk temperature seems to be increasing more rapidly than the interface temperature causing the heat flow rate at the interface to sl ow down over time. This could be understood because the interface temperat ure increasing rate drops with more accumulated snow where the fluid could maintain decent bulk temperature. Figure 8.4 shows average temperature at the surface along the axial direction over time. The surface temperature is decreasing along the axial direction because the fluid is at higher temperature at the inlet. On the ot her hand, the surface temp erature is increasing

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157 60 62 64 66 68 70 72 74 76 78 80 0102030405060 Axial Direction, cmAverage Peripheral Fluid-Pipe Interfac e Temperature, oC 1 hour 2 hour 3 hour 4 hour 5 hour 6 hour 7 hour Figure 8.2: Average peripheral interface temp erature along axial direction at different times 0 1000 2000 3000 4000 5000 6000 0102030405060 Axial Direction, cmAverage Peripheral Fluid-Pipe Interface Hea t Flow Rate, W/m2 1 hour 2 hour 3 hour 4 hour 5 hour 6 hour 7 hou r Figure 8.3: Average peripheral interface heat fl ow rate along axial direction at different times

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158 -2 -1 0 1 2 3 4 5 6 0102030405060 Axial Direction, cmAverage Surface Temperature, oC 1 hour 2 hour 3 hour 4 hour 5 hour 6 hour 7 hour Figure 8.4: Average surface temperature al ong axial direction at different times with respect to time. This is because the fl uid enters at a temper ature high enough to overcome the storm condition and heat up the system and the surface temperature. This could be different if a lower inlet temperature is used or more snow fall occurred. This shows the importance of trying a different storm scenario with more snow fall. Similarly to previous cases, the last tow hours of the time length studied has higher surface temperature as it increases w ith the end of storm condition. Figure 8.5 shows the average ground interface temperature at the bottom of the concrete slab along the axial direction over time. The ground temperature decreases along the axial direction as the fluid temperature drops. This is because as the flui d temperature drop the heat transfer to the ground decrease reducing temperature at the ground interface. Notice that the rate of ground temperature drop is less than the rate of surface temperature drop along the axial direction. This is because the surface temper ature is exposed to the cold weather while

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159 -2 0 2 4 6 8 10 0102030405060 Axial Direction, cmAverage Ground-Sandstone Interface Temperature, oC 1 hour 2 hour 3 hour 4 hour 5 hour 6 hour 7 hour Figure 8.5: Average ground interface temperatur e along axial direction at different times 0 5 10 15 20 0102030405060 Axial Direction, cmAverage Ground-Sandstone Interface Heat Flow Rate, W/m2 1 hour 2 hour 3 hour 4 hour 5 hour 6 hour 7 hou r Figure 8.6: Average ground interface heat flow rate along axial dire ction at different times

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160the ground is kept warmer under the heated slab. The ground temperature increases with time because of the high inlet temperature th at heat up the model. Figure 8.6 shows the average heat flow rate at th e ground interface along th e axial direction ov er time. This is the heat loss to the ground. Because the flui d temperature drops along the axial direction the heat flow rate also drops. As the model h eat up with time the heat flow rate to the ground increases. The second storm studied has 6 times the snow fall of the previous storm. It lasted 14 hours and the time length considered 2 more hours to observe th e model performance after the storm is over. Figure 8.7 shows th e average peripheral interface temperature along the axial direction over time. The fluid inlet temperature used was higher at 95 oC. This is to study the highest possible inlet te mperature use and check if it will give good results. The temperature curve along the axial direction comes into a steady state within the first 2 hours. This could be caused by the severity of st orm that cools the model fast enough to reach the steady state in a short time. Steady state occurred fast in both axial direction and over time. It is also noticed that the temperature drop along the channel is lesser than the previous storm which means using 95 oC overcome six times the snow fall of the previous storm. The temperature curv e increases as the model warms up with the end of the storm. Figure 8.8 shows the heat flow rate at the in terface along the axial direction over time. The heat flow rate is constant along the axial direction because both the interface and bulk temperature drop toge ther along the axial direction maintaining constant temperature difference. Because of the storm condition the heat flow rate from the fluid to the model slows down. Figure 8. 9 shows average temperature at the surface along the axial direction over time. The surface temperature is decrea sing along the axial

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161 80 82 84 86 88 90 92 94 96 98 100 0102030405060 Axial Direction, cmAverage Peripheral Fluid-Pipe Interfac e Temperature, oC 2 hour 4 hour 6 hour 8 hour 10 hour 12 hour 14 hour 16 hour Figure 8.7: Average peripheral interface temp erature along axial direction at different times 0 1000 2000 3000 4000 5000 6000 7000 8000 0102030405060 Axial Direction, cmAverage Peripheral Fluid-Pipe Interfac e Heat Flow Rate, W/m2 2 hour 4 hour 6 hour 8 hour 10 hour 12 hour 14 hour 16 hour Figure 8.8: Average peripheral interface heat fl ow rate along axial direction at different times

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162 -10 -8 -6 -4 -2 0 2 4 6 8 10 0102030405060 Axial Direction, cmAverage Surface Temperature, oC hour 2 hour 4 hour 6 hour 8 hour 10 hour 12 hour 14 hour 16 Figure 8.9: Average surface temperature al ong axial direction at different times direction because the fluid is at higher temperature at the inlet. Comparing the rate of decrease in the surface temperature along th e axial direction for both storm scenarios shows that the rate of decrea se is less for the second storm. This also shows that using inlet temperature of 95 oC was sufficient to warm up th e concrete slab. The surface temperature maintains almost constant value wi th respect to time. This also supports our conclusion of sufficient inlet temperature. One can see how fast the surface temperature rises at the last tow hours when the stor m ends. Figure 8.10 shows the average ground interface temperature at the bottom of the co ncrete slab along the axial direction over time. The ground temperature is almost consta nt along the axial direction with a slight decrease as the fluid temperat ure drops. This could be explai ned by the small drop in the channel temperature along the axial direction ca used by the fact that the inlet temperature at 95 oC overcomes the accumulated snow for the second storm, which is six times the first storm scenario. The ground temperature incr eases with time because of the high inlet

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163 -10 -8 -6 -4 -2 0 2 4 6 8 10 0102030405060 Axial Direction, cmAverage Ground-Sandstone Interface Temperature, oC 2 hour 4 hour 6 hour 8 hour 10 hour 12 hou r 14 hour 16 hou r Figure 8.10: Average ground interface temperatur e along axial direction at different times 0 2 4 6 8 10 12 14 16 18 20 0102030405060 Axial Direction, cmAverage Ground-Sandstone Interface Heat Flow Rate, W/m2 2 hour 4 hour 6 hour 8 hour 10 hour 12 hour 14 hour 16 hour Figure 8.11: Average ground interface heat flow rate along axial direction at different times

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164temperature that heat up the model. Figure 8. 11 shows the average heat flow rate at the ground interface along the axial direction over time. Because the ground temperature gradient was negligible at th e ground interface al ong the axial directio n, the heat flow rate also remains negligible. The heat flow rate to the ground remains constant with time because of its negligible value. 8.4 Conclusions Three dimensional transient snow me lting system model was developed. A numerical scheme was applied to come up with the heat flow rate calculations within the system. Calculated heat flow rates were plotted and presented. Two different storm scenarios were studied. The heat flow rate at the fluid interface showed a constant behavior along the axial direction for heavie r storms. The temperature at the surface was decreasing along the axial direct ion. The heat loss to the ground decrease in the axial direction too. The rate of ch ange in the surface temperat ure was less for the heavier storm. Results presented showed that c onsidering a higher in let temperature can overcome as much as six times the snow fall. The third dime nsion provided much understanding of the model behavi or in the axial direction.

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165 Chapter 9 – Conclusions 9.1 Conclusion Different types and designs of microcha nnels were studied in this project. Transient as well as steady state cases with different boundary conditions were implemented in this study to understand the microchanne l cooling performance under different conditions and designs Numerical simulation models were developed for this project with the desired desi gns and boundary conditions. Computer software was used to solve the conservation equations for the velocity and temperature distribution. A parametric study was conducted for each of th e designs and results were presented in terms of interface temperature, heat flow ra te and Nusselt number at the fluid-solid interface. The parameters that were varied are Reynolds number, Magnetic field strength, channel diameter or hei ght, and working fluid. The results showed that an increase in Reynolds number decreases the interface temperature but increases the heat flow rate and Nusselt num ber. The change in magnetic field value had the greatest impact on the inte rface temperature and the heat flow rate. On the other hand, it did not have any significan t effect on the Nusselt number at the fluidsolid interface. When the heat source vari ed with time, by applying and removing the magnetic field, the interface temperature, heat flow rate, and Nusselt number attained a periodic variation with time. The absolute heat flow rate and the Nusselt number were

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166increasing with time at each period. The decr ease in the diameter at constant Reynolds number decreases the interface temperature and increases the heat flow rate at the fluidsolid interface. Nusselt number depends on the diameter and the heat flow rate. Therefore, it will eith er increase or decrease with the ch ange in diameter according to the change in the heat flow rate. Using different fluids revealed that water results in the lower interface temperature and highest heat flow ra te. In nanofluids, the increase in the solid volumetric ratio decreases the heat flow ra te and Nusselt number. Using stainless steel substrate results in lower heat flow rate and Nusselt number compared to Silicon. The increase in Reynolds number increases both the heat flow rate and the Nusselt number. Larger diameter decreases the heat flow rate and the heat flow rate coefficient. The Nusselt number could decrease due to lower heat transfer coefficient or increase due to larger diameter. This is determined separa tely for each case. Nusselt number increased with diameter in this study. Hydronic snow melting system was also si mulated to carry out a parametric study in order to arrive to an unde rstanding how different paramete rs affect the melting process. Ethylene Glycol showed less heat flow rate in the model compared to water. The increase in Reynolds number increased the heat flow ra te in the model. Higher heat flow rate could be achieved by increasing the inlet temper ature or using smaller pipe diameter. The heat loss to the ground increased with higher Reynolds number or larger diameter. The most important addition to the literature is the three-dimensional model that showed a great drop in the bulk fluid temperature along the axial direction. This temperature drop reduces the quality the perfor mance of the energy transfer to melt the snow along the axial direction. This temperature reduction must be taken into consideration when

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167designing the system. Transient cases were studied by analyzing two different snow storm scenarios. They were simulated to understand their effect on the melting system with time. The heat flow rate at the fluid interface showed constant behavior along the axial direction for the heavier storm. The ra te of change in the surface temperature was less for the heavier storm. It was also found that the system could also work in heavy storms with a slight increase in the inlet temperature. 9.2 Recommendations for Future Research Researchers could continue studying di fferent designs for microchannels. Other channel geometries or parameters that are not included in this work could be good material for future studies. Only one compos ite material was studied in this work. There are many other combinations of other composite material and design that could be used in an effort to find the optimum design. Steady state parametric study was done on nanofluids in this work. Transient problem was not considered for this research as well as other nanoparticles which might have better thermal conductivity. Concrete was the only solid used for the hydronic snow melting system and only circular pipes were considered. Other material such as asphalt and other ch annel geometries can be included in future research.

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168 References [1] Rahman, M. M. and Rosario, L., “Thermodyn amic Analysis Magnetic Refrigeration,” ASME Int. Mechanical Engineering Congress and Exposition Anaheim, CA, November 13-19, 2004. [2] Rahman, M. M., “Measurements of Heat Transfer in Microchannel Heat Sinks,” Int. Comm. Heat Mass Transfer Vol. 27, 2000, pp. 495-506. [3] ASHRAE Inc., ASHRAE Handbook HVA C Application. ASHRAE Inc., 1995. [4] Peng, X. F., and Peterson, G. P., “Conv ective Heat Transfer and Flow Friction for Water Flow in Microchannel Structures,” Int. J. Heat Mass Transfer Vol. 39, 1996, pp. 2599-2608.. [5] Papautsky I., Gale B., Mohanty S., Ameel T., and Frazier B., “E ffects of Rectangular Microchannel Aspect Ratio on Laminar Friction Constant,” Proceedings of SPIE, The Int. Society of Optical Engineering Vol. 3877, 1999, pp. 147-158. [6] Qu W., Mala G. M., and Lee D., “Heat Transfer for Water Flow in Trapezoidal Silicon Microchannels,” Int. J. Heat and Mass Transfer Vol. 43, No. 21, 2000, pp. 39253936. [7] Yang, R. J., Fu. L. M., and Hwang, C. C., “Electroosmotic Entry Flow in a Microchannel,” J. of Colloid and Interface Science Vol. 244, No. 1, 2001, pp. 173-179. [8] Quadir G. A., Mydin A., and Seetharamu K. N., “Analysis of Microchannel Heat Exchangers Using FEM,” Int. J. of Numerical Met hods for Heat and Fluid Flow Vol. 11, 2001, pp. 59-75. [9] Rahman, M. M. and Shevade, S. S., “Microchannel Thermal Management during Volumetric Heating or Cooling,” Proc. First International Energy Conversion Engineering Conference, 2003. [10] Rahman M. M. and Shevade S. S., “Flow in Microchannel with Time Varying Heat Source,” Proceedings of IMECE, ASME Internat ional Mechanical Engineering Congress and Exposition Washington, D. C., November 16-21, 2003. [11] Tunc, G. and Bayazitoglu, Y., “Heat Transfer in Rectangular Microchannels,” Int. J. of Heat and Mass Transfer Vol. 45, 2002, pp. 765-773. [12] Pfund, D., Rector, D., Shekarriz, A., Popescu, A ., Welty, J. “Pressure Drop Measurements in a Microchannel,” American Institute of Chemical Engineers Journal Vol. 46, No. 8, 2000, pp. 1496-1507.

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169[13] Kohl, M. J., Abdel-Khalik, S. I., Jeter, S. M., and Sadowski, D. L. “An Experimental Investigation of Microcha nnel Flow with Internal Pressure Measurements,” Int. J. of Heat and Mass Transfer Vol. 48, 2005, pp. 1518-1533. [14] Koo, J., Kleinstreuer, C., “Visco us Dissipation Effects in Microtubes and Microchannels,” International Journal of Heat and Mass Transfer Vol. 47, No. 14/16, 2004. pp. 3159-3169. [15] Lelea, D., Nishio, S., and Takano, K ., “The Experimental Research on Microtube Heat Transfer and Fluid Flow of Distilled water,” Int. J. of Heat and Mass Transfer Vol. 47, 2004, pp. 2817-2830. [16] Yu, D., Warrington, R., Barron, R ., and Ameel, T., “An Experimental and Theoretical Investigation of Fluid Flow and Heat Transfer in Microtubes,” ASME/JSME Thermal Engineering Conference Vol. 1, 1995, pp. 523-530. [17] Adams, T. M., Abdel-Khalik, S. I., Jeter, S. M., and Qureshi, Z. H., “An Experimental Investigation of Single-Pha se Forced Convection in Microchannels,” Int.Chemical Engineering Vol. 41, 1998, pp. 851-857. [18] Owhaib, W., and Palm, B., “Experimental of Single-Phase Forced Convection Heat Transfer in Circular Microchannels,” Experimental Thermal and fluid Science Vol. 28, 2004, pp. 105-110. [19] Celata, G. P., Cumo, M., and Zummo, G., “Thermalhydraulic Characteristics of Single-Phase Flow in Capillary Pipes,” Experimental Thermal and fluid Science Vol. 28, 2004, pp. 87-95. [20] Rao, P. S. C. and Rahman, M. M., “A nalysis of Steady State Conjugate Heat Transfer in a Circular Micr otube Inside a Substrate,” Proc. 2004 ASME International Mechanical Engineering Congress and Exposition Anaheim, CA, November 13-19, 2004. [21] Nield, D. A., and Kuznetsov, A. V., “Investigation of Forced Convection in an Almost Circular Microt ube with Rough Walls,” International Journal of Fluid Mechanics Research Vol. 30, No. 1, 2003, pp. 1-10. [22] Giulio, C., and D’Agaro, P., “Num erical Simulation of Roughness Effect on Microchannel Heat Transfer and Pressure Drop in Laminar Flow,” Journal of Physics. D: Applied Physics Vol. 38, 2005, pp. 1518-1530. [23] Grohmann, S., “Measurements and M odeling of Single-Phase and Flow-Boiling Heat Transfer in Microtubes,” International Journal of Heat and Mass Transfer Vol. 48, No. 19/20, 2005, pp. 4073-4089. [24] Broderick, S. L., Webb, B. W., and Maynes, D., “Thermally Developing ElectroOsmotic Convection in Microchannels with Finite Debye-Layer Thickness,” Numerical Heat Transfer; Part A: Applications Vol. 48, No. 10, 2005, pp. 941-964.

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170[25] Chakraborty, S., “Analytical Solutions of Nusselt Number for Thermally Fully Developed Flow in Microtubes under a Comb ined Action of Electroosmotic Forces and Imposed Pressure Gradients,” International Journal of Heat and Mass Transfer Vol. 49, No. 3/4, 2006, pp. 810-813. [26] Hwang, Y. W., and Kim, M. S., “The Pressure Drop in Micr otubes and Correlation Development,” International Journal of Heat and Mass Transfer Vol. 49, No. 11/12, 2006, pp. 1804-1812. [27] Yang C., Beng C., and Chan V., “Transient Analysis of Electroosmotic Flow in a Slit Microchannel,” J. of Colloid and Interface Science Vol. 248, No. 2, 2002, pp. 524527. [28] Eastman J. A., Choi S. U. S., Li S ., Yu W., and Thompson L. J., “Anomalously Increased Effective Therma l Conductivities of Ethylene Glycol-Based Nanofluids Containing Copper Nanoparticles,” App. Phys. Lett. Vol. 78, No. 6, 2001, pp. 718-720. [29] keblinski P., Phillpot S. R., Choi S. U. S., and Eastman J. A., “Mechanism of Heat Flow in Suspensions of Nano-si zed Particles (Nanofluids),” Int. J. Heat and Mass Transfer Vol. 45, No. 21, 2002, pp. 855-863. [30] Cheng N. S. C., and Law A. W. K., “Exponential Formula for Computing Effective Viscosity,” Powder Tech., Vol. 129, 2003, pp. 156-160. [31] Putra N., Roetzel W., and Das S. K. “Natural Convection of Nano-fluids,” Int. J. Heat and Mass Transfer Vol. 39, 2003, pp. 775-784. [32] Bang I., C. and Chang S. H., “Boiling Heat Ttransfer Performance and Phenomena of Al2O3-water Nano-fluids from a Plain Surface in a Pool,” Int. J. Heat and Mass Transfer Vol. 48, 2005, pp. 2407-2419. [33] Vadasz J. J., Saneshan G., and Vadasz P., “Heat Transfer Enhancement in NanoFluids Suspensions: Possible M echanism and Explanations,” Int. J. Heat and Mass Transfer Vol. 48, 2005, pp. 2673-2683. [34] Wen D. and Ding Y., “Effect of Particle Migration on Heat Transfer in Suspensions of Nanoparticles Flowing through Minichannels,” Microfluids Nanofluids Vol. 1, 2005, pp. 183-189. [35] Maiga S. E. B., Nguyen C. T., Galanis N., and Roy G., “Heat Tr ansfer Behaviors of Nanofluids in a Uniformly Heated Tube,” Superlattices and Microstructures Vol. 35, 2004, pp. 543-557. [36] Wang X., Xu X., and Choi S. U. S., “Thermal Conduc tivity of Nanoparticles-Fluid Mixture,” Journal of Thermophysics and Heat Transfer Vol. 13, No. 4, 1999, pp. 474480. [37] Chapman, W. P., “Design of Snow Melting Systems,” Heating and Ventilating Vol. 49, No. 11, 1952, pp. 88-91. [38] Chapman, W. P., “Design of Snow Melting Systems,” Heating and Ventilating Vol. 49, No. 4, 1952 pp. 96-102.

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171[39] Chapman, W. P., “A Review of Snow Melting Systems Design,” ASHRAE Transactions 1999. [40] Ramsey, J. W., Hewett, M. J., Kuehn, T. H., and Peterson, S. D., “Updated Design Guidelines for Snow Melting Systems.” ASHRAE Transactions 1999. [41] Kilkis, B. I., “An Analytical Algor ithm for Hydronic Circuit Analysis and Assessment of Equipment Performance.” ASHRAE Transaction Vol. 105, Part 1, 1999, pp 368-374. [42] Kilkis, I. B., “Design of Em bedded Snow Melting Systems part 1.” ASHRAE Transaction Vol. 100, Part 1, 1994, pp 423-433. [43] Kilkis, I. B., “Design of Em bedded Snow Melting Systems part 2.” ASHRAE Transaction Vol. 100, Part 2, 1994, pp 434-441. [44] Rees, S. J., Spitler, J. D., and Xiao, X., “Transient Anal ysis of Snow Melting System Performance.” ASHRAE Transaction Vol. 108, Part 2, 2002, pp 406-423. [45] White F. M., 1991, Viscous Fluid Flow. 2nd Ed., McGraw-Hill, New York. [46] Ozisik, M. N., 1993, Heat Conduction. 2nd Ed., John Wiley and Sons, New York. [47] Pechasky, V. K., and Gschneider, K. A., “Magnetocaloric Effect and Magnetic Refrigeration.” Journal of Magnetism and Magnetic Materials Vol. 200, 1999, pp. 4456.

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172 Appendices

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173Appendix A: FIDAP Program for Steady State Heat Transfer in Circular Microchannels During Magnetic Heating or Cooling Simulation FIPREP( ) CONDUCTIVITY( ADD, SET = "Fluid", CONS = 0.0014435, ISOT ) CONDUCTIVITY( ADD, SET = "Solid", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "Fluid", CONS = 0.9974 ) DENSITY( ADD, SET = "Solid", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "F luid", CONS = 0.9988 ) SPECIFICHEAT( ADD, SET = "Solid", CONS = 0.054971 ) 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 = "Gadolinium", 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 = "Gd-in", PLOT ) ENTITY( ADD, NAME = "Gd-top", PLOT ) ENTITY( ADD, NAME = "Gd-lwall", PLOT ) ENTITY( ADD, NAME = "Gd-bottom", PLOT ) ENTITY( ADD, NAME = "Gd-bsym", PLOT ) ENTITY( ADD, NAME = "Gd-tsym", PLOT ) ENTITY( ADD, NAME = "Gd-out", PLOT ) ENTITY( ADD, NAME = "f-bwall1", PLOT, ATTA = "Gadolinium", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-bwall2", PLOT, ATTA = "Gadolinium", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-twall1", PLOT, ATTA = "Gadolinium", NATT = "Fluid-t" ) ENTITY( ADD, NAME = "f-twall2", PLOT, ATTA = "Gadolinium", 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 = 436.690952 ) BCNODE( ADD, UZ, ENTI = "f-tin", CONS = 436.690952 ) 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 ) SOURCE( ADD, HEAT, CONS = 5.13, ENTI = "Gadolinium" ) BCFLUX( ADD, HEAT, ENTI = "Gd-top", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gd-lwall", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gd-bsym", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "Gd-tsym", CONS = 0 ) DATAPRINT( ADD, CONT ) EXECUTION( ADD, NEWJ )

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174Appendix A: (Continued) 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, ACCF = 0.05 ) WIND = 0.75, NOFI = 3 ) CLIPPING( ADD, MINI ) 0, 0, 0, 0, 20, 0 END( )

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175Appendix B: FIDAP Program fo r Transient Heat Transfer in Circular Microchannels Under Time Varying Heat Source Simulation FIPREP( ) /========================================= / MATERIAL PROPERTIES /========================================= CONDUCTIVITY( ADD, SET = "Fluid", CONS = 0.0014435, ISOT ) CONDUCTIVITY( ADD, SET = "Solid", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "Fluid", CONS = 0.9974 ) DENSITY( ADD, SET = "Solid", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "F luid", CONS = 0.9988 ) SPECIFICHEAT( ADD, SET = "Solid", CONS = 0.054971 ) VISCOSITY( ADD, SET = "Fluid", CONS = 0.0098 ) /========================================= / ENTITY NAMES /========================================= ENTITY( ADD, NAME = "Fluid-b", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Fluid-t", FLUI, PROP = "Fluid" ) ENTITY( ADD, NAME = "Gadolinium", SOLI, PROP = "Solid" ) ENTITY( ADD, NAME = "f-bin", PLOT ) ENTITY( ADD, NAME = "f-bsym", PLOT ) ENTITY( ADD, NAME = "f-bout", PLOT ) ENTITY( ADD, NAME = "f-tin", PLOT ) ENTITY( ADD, NAME = "f-tsym", PLOT ) ENTITY( ADD, NAME = "f-tout", PLOT ) ENTITY( ADD, NAME = "Gd-in", PLOT ) ENTITY( ADD, NAME = "Gd-top", PLOT ) ENTITY( ADD, NAME = "Gd-lwall", PLOT ) ENTITY( ADD, NAME = "Gd-bottom", PLOT ) ENTITY( ADD, NAME = "Gd-bsym", PLOT ) ENTITY( ADD, NAME = "Gd-tsym", PLOT ) ENTITY( ADD, NAME = "Gd-out", PLOT ) ENTITY( ADD, NAME = "f-bwall1", PLOT, ATTA = "Gadolinium", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-bwall2", PLOT, ATTA = "Gadolinium", NATT = "Fluid-b" ) ENTITY( ADD, NAME = "f-twall1", PLOT, ATTA = "Gadolinium", NATT = "Fluid-t" ) ENTITY( ADD, NAME = "f-twall2", PLOT, ATTA = "Gadolinium", NATT = "Fluid-t" ) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= BCNODE( ADD, UX, ENTI = "f-bin", ZERO ) BCNODE( ADD, UY, ENTI = "f-bin", ZERO ) BCNODE( ADD, UZ, ENTI = "f-bin", CONS = 436.69 ) BCNODE( ADD, TEMP, ENTI = "f-bin", CONS = 20 )

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176Appendix B: (Continued) BCNODE( ADD, UX, ENTI = "f-tin", ZERO ) BCNODE( ADD, UY, ENTI = "f-tin", ZERO ) BCNODE( ADD, UZ, ENTI = "f-tin", CONS = 436.69 ) 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 ) SOURCE( ADD, HEAT, CONS = 2.56, ENTI = "Gadolinium" ) SOURCE( ADD, HEAT, CONS = -2.56, ENTI = "Gadolinium" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "Fluid-b" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "Fluid-t" ) ICNODE( ADD, TEMP, CONS = 20 ENTI = "Gadolinium" ) ICNODE( ADD, VELO, READ) ICNODE( ADD, TEMP, READ ) /========================================= / EXECUTION COMMANDS /========================================= 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 ) PROBLEM( ADD, 3-D, INCO, TRAN, LAMI, NONL, NEWT, NOMO, ENER, FIXE, SING ) SOLUTION( ADD, N.R. = 100, VELC = 0.001) TIMEINTEGRATION( ADD, BACK, NSTE = 200, TSTA = 0.0, DT = 0.005, FIXE) TIMEINTEGRATION( ADD, BACK, NSTE = 400, TSTA = 1.0, DT = 0.005, FIXE) POSTPROCESS( ADD, NBLO = 2, NOPT, NOPA ) 2, 10, 2 20, 200, 20 POSTPROCESS( ADD, NBLO = 1, NOPT, NOPA ) 20, 400, 20 END( )

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177Appendix C: FIDAP Program for Transi ent Heat Transfer in Trapezoidal Microchannels During Activation of Magnetic Heating Simulation FIPREP( ) /========================================= / MATERIAL PROPERTIES /========================================= CONDUCTIVITY( ADD, SET = "si", CONS = 0.29637, ISOT ) DENSITY( ADD, SET = "si", CONS = 2.329 ) SPECIFICHEAT( ADD, SET = "si", CONS = 0.16778 ) CONDUCTIVITY( ADD, SET = "fl", CONS = 0.0014435, ISOT ) DENSITY( ADD, SET = "fl", CONS = 0.9974 ) SPECIFICHEAT( ADD, SET = "fl", CONS = 0.9988 ) VISCOSITY( ADD, SET = "fl", CONS = 0.0098 ) CONDUCTIVITY( ADD, SET = "gd", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "gd", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "gd", CONS = 0.054971 ) /========================================= / ENTITY NAMES /========================================= ENTITY( ADD, NAME = "si1", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "si2", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "si3", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "fl", FLUI, PROP = "fl" ) ENTITY( ADD, NAME = "gd", SOLI, PROP = "gd" ) ENTITY( ADD, NAME = "si1-in", PLOT ) ENTITY( ADD, NAME = "si1-out", PLOT ) ENTITY( ADD, NAME = "si1-bot", PLOT ) ENTITY( ADD, NAME = "si1-right", PLOT ) ENTITY( ADD, NAME = "si1-top", PLOT ) ENTITY( ADD, NAME = "si1-left", PLOT ) ENTITY( ADD, NAME = "si2-in", PLOT ) ENTITY( ADD, NAME = "si2-out", PLOT ) ENTITY( ADD, NAME = "si2-bot", PLOT ) ENTITY( ADD, NAME = "si2-right", PLOT ) ENTITY( ADD, NAME = "si2-int", PLOT, ATTA = "fl", NATTA = "si2" ) ENTITY( ADD, NAME = "si3-in", PLOT ) ENTITY( ADD, NAME = "si3-out", PLOT ) ENTITY( ADD, NAME = "si3-bot", PLOT ) ENTITY( ADD, NAME = "si3-axi", PLOT ) ENTITY( ADD, NAME = "si3-int", PLOT, ATTA = "fl", NATTA = "si3" ) ENTITY( ADD, NAME = "fl-in", PLOT ) ENTITY( ADD, NAME = "fl-out", PLOT ) ENTITY( ADD, NAME = "fl-axi", PLOT )

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178Appendix C: (Continued) ENTITY( ADD, NAME = "fl-int", PLOT, ATTA = "fl", NATTA = "gd" ) ENTITY( ADD, NAME = "gd-in", PLOT ) ENTITY( ADD, NAME = "gd-out", PLOT ) ENTITY( ADD, NAME = "gd-axi", PLOT ) ENTITY( ADD, NAME = "gd-top", PLOT ) ENTITY( ADD, NAME = "gd-left", PLOT ) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= BCNODE( ADD, UX, ENTI = "fl-in", ZERO ) BCNODE( ADD, UY, ENTI = "fl-in", ZERO ) BCNODE( ADD, UZ, ENTI = "fl-in", CONS = 403.91) BCNODE( ADD, TEMP, ENTI = "fl-in", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "si2-int", ZERO ) BCNODE( ADD, VELO, ENTI = "si3-int", ZERO ) BCNODE( ADD, VELO, ENTI = "fl-int", ZERO ) BCNODE( ADD, UX, ENTI = "fl-axi", ZERO ) SOURCE( ADD, HEAT, CONS = 5.12, ENTI = "gd" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si1" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si2" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si3" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "fl" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "gd" ) /========================================= / EXECUTION COMMANDS /========================================= 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. = 100, VELC = 0.001) /TIME LENGTH = 1s TIMEINTEGRATION( ADD, BACK, NSTE = 200, TSTA = 0.0, DT = 0.005, FIXE) POSTPROCESS( ADD, NBLO = 2, NOPT, NOPA ) 2, 10, 2 20, 200, 20 END( )

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179Appendix D: FIDAP Program for Transi ent Heat Transfer in Trapezoidal Microchannels Under Time Vary ing Heat Source Simulation FIPREP( ) /========================================= / MATERIAL PROPERTIES /========================================= CONDUCTIVITY( ADD, SET = "si", CONS = 0.29637, ISOT ) DENSITY( ADD, SET = "si", CONS = 2.329 ) SPECIFICHEAT( ADD, SET = "si", CONS = 0.16778 ) CONDUCTIVITY( ADD, SET = "fl", CONS = 0.0014435, ISOT ) DENSITY( ADD, SET = "fl", CONS = 0.9974 ) SPECIFICHEAT( ADD, SET = "fl", CONS = 0.9988 ) VISCOSITY( ADD, SET = "fl", CONS = 0.0098 ) CONDUCTIVITY( ADD, SET = "gd", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "gd", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "gd", CONS = 0.054971 ) /========================================= / ENTITY NAMES /========================================= ENTITY( ADD, NAME = "si1", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "si2", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "si3", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "fl", FLUI, PROP = "fl" ) ENTITY( ADD, NAME = "gd", SOLI, PROP = "gd" ) ENTITY( ADD, NAME = "si1-in", PLOT ) ENTITY( ADD, NAME = "si1-out", PLOT ) ENTITY( ADD, NAME = "si1-bot", PLOT ) ENTITY( ADD, NAME = "si1-right", PLOT ) ENTITY( ADD, NAME = "si1-top", PLOT ) ENTITY( ADD, NAME = "si1-left", PLOT ) ENTITY( ADD, NAME = "si2-in", PLOT ) ENTITY( ADD, NAME = "si2-out", PLOT ) ENTITY( ADD, NAME = "si2-bot", PLOT ) ENTITY( ADD, NAME = "si2-right", PLOT ) ENTITY( ADD, NAME = "si2-int", PLOT, ATTA = "fl", NATTA = "si2" ) ENTITY( ADD, NAME = "si3-in", PLOT ) ENTITY( ADD, NAME = "si3-out", PLOT ) ENTITY( ADD, NAME = "si3-bot", PLOT ) ENTITY( ADD, NAME = "si3-axi", PLOT ) ENTITY( ADD, NAME = "si3-int", PLOT, ATTA = "fl", NATTA = "si3" ) ENTITY( ADD, NAME = "fl-in", PLOT ) ENTITY( ADD, NAME = "fl-out", PLOT ) ENTITY( ADD, NAME = "fl-axi", PLOT )

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180Appendix D: (Continued) ENTITY( ADD, NAME = "fl-int", PLOT, ATTA = "fl", NATTA = "gd" ) ENTITY( ADD, NAME = "gd-in", PLOT ) ENTITY( ADD, NAME = "gd-out", PLOT ) ENTITY( ADD, NAME = "gd-axi", PLOT ) ENTITY( ADD, NAME = "gd-top", PLOT ) ENTITY( ADD, NAME = "gd-left", PLOT ) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= BCNODE( ADD, UX, ENTI = "fl-in", ZERO ) BCNODE( ADD, UY, ENTI = "fl-in", ZERO ) BCNODE( ADD, UZ, ENTI = "fl-in", CONS = 403.91) BCNODE( ADD, TEMP, ENTI = "fl-in", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "si2-int", ZERO ) BCNODE( ADD, VELO, ENTI = "si3-int", ZERO ) BCNODE( ADD, VELO, ENTI = "fl-int", ZERO ) BCNODE( ADD, UX, ENTI = "fl-axi", ZERO ) SOURCE( ADD, HEAT, CONS = 5.12, ENTI = "gd" ) SOURCE( ADD, HEAT, CONS = -5.12, ENTI = "gd" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si1" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si2" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "si3" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "fl" ) ICNODE( ADD, TEMP, CONS = 20, ENTI = "gd" ) ICNODE( ADD, VELO, READ ) ICNODE( ADD, TEMP, READ ) /========================================= / EXECUTION COMMANDS /========================================= 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 ) PROBLEM( ADD, 3-D, INCO, TRAN, LAMI, NONL, NEWT, NOMO, ENER, FIXE, SING )SOLUTION( ADD, N.R. = 100, VELC = 0.001) TIMEINTEGRATION( ADD, BACK, NSTE = 200, TSTA = 0.0, DT = 0.005, FIXE) TIMEINTEGRATION( ADD, BACK, NSTE = 400, TSTA = 1.0, DT = 0.005, FIXE) POSTPROCESS( ADD, NBLO = 2, NOPT, NOPA ) 2, 10, 2 20, 200, 20 POSTPROCESS( ADD, NBLO = 1, NOPT, NOPA ) 20, 400, 20 END( )

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181Appendix E: FIDAP Program for Steady St ate Heat Transfer Using Nanofluids in Circular Microchannels Simulation FIPREP( ) /========================================= / MATERIAL PROPERTIES /========================================= CONDUCTIVITY( ADD, SET = "wtr", CONS = 0.001434, ISOT ) DENSITY( ADD, SET = "wtr", CONS = 0.9982 ) SPECIFICHEAT( ADD, SET = "wtr", CONS = 0.9995 ) VISCOSITY( ADD, SET = "wtr", CONS = 0.01003 ) /composite of Alumina suspended in water with 1% CONDUCTIVITY( ADD, SET = "wa1", CONS = 0.0016491, ISOT ) DENSITY( ADD, SET = "wa1", CONS = 1.0281 ) SPECIFICHEAT( ADD, SET = "wa1", CONS = 0.96954 ) VISCOSITY( ADD, SET = "wa1", CONS = 0.010331 ) /composite of Alumina suspended in water with 4% CONDUCTIVITY( ADD, SET = "wa4", CONS = 0.0017925, ISOT ) DENSITY( ADD, SET = "wa4", CONS = 1.1180 ) SPECIFICHEAT( ADD, SET = "wa4", CONS = 0.87958 ) VISCOSITY( ADD, SET = "wa4", CONS = 0.011033 ) CONDUCTIVITY( ADD, SET = "si", CONS = 0.29637, ISOT ) DENSITY( ADD, SET = "si", CONS = 2.329 ) SPECIFICHEAT( ADD, SET = "si", CONS = 0.16778 ) CONDUCTIVITY( ADD, SET = "gd", CONS = 0.0250956, ISOT ) DENSITY( ADD, SET = "gd", CONS = 7.895 ) SPECIFICHEAT( ADD, SET = "gd", CONS = 0.054971 ) /========================================= / ENTITY NAMES /========================================= ENTITY( ADD, NAME = "fb", FLUI, PROP = "wa4" ) ENTITY( ADD, NAME = "ft", FLUI, PROP = "wa4" ) ENTITY( ADD, NAME = "s", SOLI, PROP = "si" ) ENTITY( ADD, NAME = "sin", PLOT ) ENTITY( ADD, NAME = "sout", PLOT ) ENTITY( ADD, NAME = "sbot", PLOT ) ENTITY( ADD, NAME = "sbaxi", PLOT ) ENTITY( ADD, NAME = "int1", PLOT, ATTA = "s", NATT = "fb" ) ENTITY( ADD, NAME = "int2", PLOT, ATTA = "s", NATT = "fb" ) ENTITY( ADD, NAME = "int3", PLOT ATTA = "s", NATT = "ft" ) ENTITY( ADD, NAME = "int4", PLOT ATTA = "s", NATT = "ft" ) ENTITY( ADD, NAME = "staxi", PLOT ) ENTITY( ADD, NAME = "stop", PLOT ) ENTITY( ADD, NAME = "sleft", PLOT )

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182Appendix E: (Continued) ENTITY( ADD, NAME = "fbin", PLOT ) ENTITY( ADD, NAME = "fbout", PLOT ) ENTITY( ADD, NAME = "fbaxi", PLOT ) ENTITY( ADD, NAME = "fint", PLOT ) ENTITY( ADD, NAME = "ftin", PLOT ) ENTITY( ADD, NAME = "ftout", PLOT ) ENTITY( ADD, NAME = "ftaxi", PLOT ) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= /for 4% composite Vin=164.48 ===> Re=1000 / vin=32.896 ---> Re=200 /for 1% composite Vin=167.48 ===> Re=1000 / vin=33.496 ---> Re=200 /for pure water Vin=167.47 ===> Re=1000 / vin=33.494 ---> Re=200 BCNODE( ADD, UX, ENTI = "fbin", ZERO ) BCNODE( ADD, UY, ENTI = "fbin", ZERO ) BCNODE( ADD, UZ, ENTI = "fbin", CONS = 32.896 ) BCNODE( ADD, TEMP, ENTI = "fbin", CONS = 20 ) BCNODE( ADD, UX, ENTI = "ftin", ZERO ) BCNODE( ADD, UY, ENTI = "ftin", ZERO ) BCNODE( ADD, UZ, ENTI = "ftin", CONS = 32.896 ) BCNODE( ADD, TEMP, ENTI = "ftin", CONS = 20 ) BCNODE( ADD, VELO, ENTI = "int1", ZERO ) BCNODE( ADD, VELO, ENTI = "int2", ZERO ) BCNODE( ADD, VELO, ENTI = "int3", ZERO ) BCNODE( ADD, VELO, ENTI = "int4", ZERO ) BCNODE( ADD, UX, ENTI = "fbaxi", ZERO ) BCNODE( ADD, UY, ENTI = "fbaxi", ZERO ) BCNODE( ADD, UX, ENTI = "ftaxi", ZERO ) BCNODE( ADD, UY, ENTI = "ftaxi", ZERO ) /These 7 lines are default but include them any way just in case BCFLUX( ADD, HEAT, ENTI = "sbot", CONS = 7.1702 ) BCFLUX( ADD, HEAT, ENTI = "sbaxi", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "fbaxi", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "ftaxi", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "staxi", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "stop", CONS = 0 ) BCFLUX( ADD, HEAT, ENTI = "sleft", CONS = 0 ) /========================================= / EXECUTION COMMANDS /========================================= DATAPRINT( ADD, CONT )

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183Appendix E: (Continued) EXECUTION( ADD, NEWJ ) PRINTOUT( ADD, NONE ) OPTIONS( ADD, UPWI ) PROBLEM( ADD, 3-D, INCO, STEA, LAMI, NONL, NEWT, MOME, ENER, FIXE, SING ) SOLUTION( ADD, S.S. = 100, VELC = 0.001) /SOLUTION( ADD, N.R. = 100, VELC = 0.001) END( )

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184Appendix F: FIDAP Program for Steady St ate Analysis in H ydronic Snow Melting System Simulation FIPREP /========================================= / EXECUTION COMMANDS /========================================= EXECUTION (NEWJOB) PROBLEM (ADD, 3-D, TU RB, NONLINE, ENERGY) SOLUTION (ADD, SEGR=10) DATAPRINT (ADD,CONT) PRINTOUT (ALL) /========================================= / ENTITY NAMES /========================================= /continium ENTITY (ADD, NAME = "fluid-t", FLUID, PROP = "fl") ENTITY (ADD, NAME = "fluid-b", FLUID, PROP = "fl") ENTITY (ADD, NAME = "pipe-t", SOLID, PROP = "pi") ENTITY (ADD, NAME = "pipe-b", SOLID, PROP = "pi") ENTITY (ADD, NAME = "solid-t", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -b", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -1", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -2", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -3", SOLID, PROP = "so") ENTITY (ADD, NAME = "ground", SOLID, PROP = "gr") /inlet ENTITY( ADD, NAME = "ft-in", PLOT) ENTITY( ADD, NAME = "fb-in", PLOT) ENTITY( ADD, NAME = "pt-in", PLOT) ENTITY( ADD, NAME = "pb-in", PLOT) ENTITY( ADD, NAME = "st-in", PLOT) ENTITY( ADD, NAME = "sb-in", PLOT) ENTITY( ADD, NAME = "s1-in", PLOT) ENTITY( ADD, NAME = "s2-in", PLOT) ENTITY( ADD, NAME = "s3-in", PLOT) ENTITY( ADD, NAME = "g-in", PLOT) /MS1 ENTITY( ADD, NAME = "ft-out", PLOT) ENTITY( ADD, NAME = "ft-axi", PLOT) ENTITY( ADD, NAME = "ft-intt", WALL, AT TA = "fluid-t", NATT = "pipe-t") ENTITY( ADD, NAME = "ft-in tb", WALL, ATTA = "fluid -t", NATT = "pipe-t") ENTITY( ADD, NAME = "ft-bot", PLOT) /MS2 ENTITY( ADD, NAME = "fb-out", PLOT) ENTITY( ADD, NAME = "fb-axi", PLOT) ENTITY( ADD, NAME = "fb-intt", WALL, ATTA = "fluid-b", NATT = "pipe-b") ENTITY( ADD, NAME = "fb-intb", WALL, ATTA = "fluid-b", NATT = "pipe-b") /MS3 ENTITY( ADD, NAME = "pt-out", PLOT) ENTITY( ADD, NAME = "pt-axi", PLOT) ENTITY( ADD, NAME = "pt-intt", PLOT)

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185Appendix F: (Continued) ENTITY( ADD, NAME = "pt-intb", PLOT) ENTITY( ADD, NAME = "pt-bot", PLOT) /MS4 ENTITY( ADD, NAME = "pb-out", PLOT) ENTITY( ADD, NAME = "pb-axi", PLOT) ENTITY( ADD, NAME = "pb-intt", PLOT) ENTITY( ADD, NAME = "pb-intb", PLOT) /MS5 ENTITY( ADD, NAME = "st-out", PLOT) ENTITY( ADD, NAME = "st-axi", PLOT) ENTITY( ADD, NAME = "st-intt", PLOT) ENTITY( ADD, NAME = "st-intb", PLOT) ENTITY( ADD, NAME = "st-bot", PLOT) /MS6 ENTITY( ADD, NAME = "sb-out", PLOT) ENTITY( ADD, NAME = "sb-axi", PLOT) ENTITY( ADD, NAME = "sb-intt", PLOT) ENTITY( ADD, NAME = "sb-intb", PLOT) /MS7 ENTITY( ADD, NAME = "s1-out", PLOT) ENTITY( ADD, NAME = "s1-axi", PLOT) ENTITY( ADD, NAME = "s1-top", PLOT) ENTITY( ADD, NAME = "s1-bot", PLOT) /MS8 ENTITY( ADD, NAME = "s2-out", PLOT) ENTITY( ADD, NAME = "s2-left", PLOT) ENTITY( ADD, NAME = "s2-bot", PLOT) /MS9 ENTITY( ADD, NAME = "s3-out", PLOT) ENTITY( ADD, NAME = "s3-axi", PLOT) ENTITY( ADD, NAME = "s3-bot", PLOT) /MS10 ENTITY( ADD, NAME = "g-out", PLOT) ENTITY( ADD, NAME = "g-axi", PLOT) ENTITY( ADD, NAME = "g-left", PLOT) ENTITY( ADD, NAME = "g-bot", PLOT) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= /inlet BCNODE (ADD, UX, CONS=0, ENTITY = "ft-in") BCNODE (ADD, UY, CONS=0, ENTITY = "ft-in") BCNODE (ADD, UZ, CONS=6.1 869, ENTITY = "ft-in") BCNODE (ADD, UX, CONS=0, ENTITY = "fb-in") BCNODE (ADD, UY, CONS=0, ENTITY = "fb-in") BCNODE (ADD, UZ, CONS=6.1 869, ENTITY = "fb-in") /interface BCNODE (VELOCITY, ZERO, ENTITY = "ft-intt") BCNODE (VELOCITY, ZERO, ENTITY = "ft-intb") BCNODE (VELOCITY, ZERO, ENTITY = "fb-intt") BCNODE (VELOCITY, ZERO, ENTITY = "fb-intb") /axi

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186Appendix F: (Continued) BCNODE( ADD, UX, ENTI = "ft-axi", ZERO ) BCNODE( ADD, UX, ENTI = "fb-axi", ZERO ) /temperature BCNODE (ADD, TEMPERATURE, CONS = 80, ENTITY = "ft-in") BCNODE (ADD, TEMPERATURE, CONS = 80, ENTITY = "fb-in") BCNODE (ADD, TEMPERATURE, CONS = 0, ENTITY = "s1-top") /insulation/symmetry BCFLUX (HEAT, CONS = 0. 0, ENTITY = "ft-axi") BCFLUX (HEAT, CONS = 0. 0, ENTITY = "fb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "pt-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "pb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "st-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-left") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-left") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-bot") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-axi") /in face and out face BCFLUX (HEAT, CONS = 0. 0, ENTITY = "st-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "st-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-out") /special BCNODE( KINE, CONS = 0. 00, ENTI = "ft-in" ) BCNODE( KINE, CONS = 0. 00, ENTI = "fb-in" ) BCNODE( DISS, CONS = 0.00, ENTI = "ft-in" ) BCNODE( DISS, CONS = 0.00, ENTI = "fb-in" ) /BCNODE( KINE, CONS = 0.001, ENTI = "ft-in" ) /BCNODE( KINE, CONS = 0.001, ENTI = "fb-in" ) /BCNODE( DISS, CONS = 0.00045, ENTI = "ft-in" ) /BCNODE( DISS, CONS = 0.00045, ENTI = "fb-in" ) /IC /ICNODE( KINE, CONS = 0.003, ALL ) /ICNODE( DISS, CONS = 0.00045, ALL ) /========================================= / MATERIAL PROPERTIES /========================================= /fluid=fl (water) DENSITY (ADD, SET = "fl", CONS = 0.98) CONDUCTIVITY (ADD, SET = "fl", CONS = 0.0015750478)

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187Appendix F: (Continued) SPECIFICHEAT (ADD, SET = "fl", CONS = 1.000717) VISCOSITY( ADD, SET = "fl", TWO-, CONS = 0.00423 ) /EDDY(BOUS) /pipe=pi (Carbon Steel 1%) DENSITY (ADD, SET = "pi", CONS = 7.801) CONDUCTIVITY (ADD, SET = "pi", CONS = 0.102772467) SPECIFICHEAT (ADD, SET = "pi", CONS = 0.113049713) /solid=so (concrete) DENSITY (ADD, SET = "so", CONS = 2.3) CONDUCTIVITY (ADD, SET = "so", CONS = 0.00239005736) SPECIFICHEAT (ADD, SET = "so", CONS = 0.210325048) /ground=gr (sandstone) DENSITY (ADD, SET = "gr", CONS = 1.5) CONDUCTIVITY (ADD, SET = "gr", CONS = 0.00239005736) SPECIFICHEAT (ADD, SET = "gr", CONS = 0.454110899) END

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188Appendix G: FIDAP Program fo r Transient Analysis in Hydr onic Snow Melting System Simulation FIPREP /========================================= / EXECUTION COMMANDS /========================================= EXECUTION (NEWJOB) PROBLEM (ADD, 3-D, TRAN TURB, NONLINE, ENERGY) SOLUTION (ADD, SEGR=1000, CR, CGS, VELC=0.01, RESC=0.01, SCHANGE=0) DATAPRINT (ADD,CONT) PRINTOUT (NONE) TIMEINTEGRATION( ADD, BACK, NSTE = 5040, TSTA = 0.0, DT = 5, FIXE) POSTPROCESS( NBLO = 2 ) 120, 600, 120 720, 5040, 720 /========================================= / ENTITY NAMES /========================================= /continium ENTITY (ADD, NAME = "fluid-t", FLUID, PROP = "fl") ENTITY (ADD, NAME = "fluid-b", FLUID, PROP = "fl") ENTITY (ADD, NAME = "pipe-t", SOLID, PROP = "pi") ENTITY (ADD, NAME = "pipe-b", SOLID, PROP = "pi") ENTITY (ADD, NAME = "solid-t", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -b", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -1", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -2", SOLID, PROP = "so") ENTITY (ADD, NAME = "solid -3", SOLID, PROP = "so") ENTITY (ADD, NAME = "ground", SOLID, PROP = "gr") /inlet ENTITY( ADD, NAME = "ft-in", PLOT) ENTITY( ADD, NAME = "fb-in", PLOT) ENTITY( ADD, NAME = "pt-in", PLOT) ENTITY( ADD, NAME = "pb-in", PLOT) ENTITY( ADD, NAME = "st-in", PLOT) ENTITY( ADD, NAME = "sb-in", PLOT) ENTITY( ADD, NAME = "s1-in", PLOT) ENTITY( ADD, NAME = "s2-in", PLOT) ENTITY( ADD, NAME = "s3-in", PLOT) ENTITY( ADD, NAME = "g-in", PLOT) /MS1 ENTITY( ADD, NAME = "ft-out", PLOT) ENTITY( ADD, NAME = "ft-axi", PLOT) ENTITY( ADD, NAME = "ft-in tt", WALL, NATTA = "fluid -t", ATTA = "pipe-t") ENTITY( ADD, NAME = "ft-in tb", WALL, NATTA = "fluid-t", ATTA = "pipe-t") ENTITY( ADD, NAME = "ft-bot", PLOT) /MS2 ENTITY( ADD, NAME = "fb-out", PLOT) ENTITY( ADD, NAME = "fb-axi", PLOT) ENTITY( ADD, NAME = "fb-intt", WALL, NATTA = "fluid-b", ATTA = "pipe-b")

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189Appendix G: (Continued) ENTITY( ADD, NAME = "fb-intb", WALL, NATTA = "fluid-b", ATTA = "pipe-b") /MS3 ENTITY( ADD, NAME = "pt-out", PLOT) ENTITY( ADD, NAME = "pt-axi", PLOT) ENTITY( ADD, NAME = "pt-intt", PLOT) ENTITY( ADD, NAME = "pt-intb", PLOT) ENTITY( ADD, NAME = "pt-bot", PLOT) /MS4 ENTITY( ADD, NAME = "pb-out", PLOT) ENTITY( ADD, NAME = "pb-axi", PLOT) ENTITY( ADD, NAME = "pb-intt", PLOT) ENTITY( ADD, NAME = "pb-intb", PLOT) /MS5 ENTITY( ADD, NAME = "st-out", PLOT) ENTITY( ADD, NAME = "st-axi", PLOT) ENTITY( ADD, NAME = "st-intt", PLOT) ENTITY( ADD, NAME = "st-intb", PLOT) ENTITY( ADD, NAME = "st-bot", PLOT) /MS6 ENTITY( ADD, NAME = "sb-out", PLOT) ENTITY( ADD, NAME = "sb-axi", PLOT) ENTITY( ADD, NAME = "sb-intt", PLOT) ENTITY( ADD, NAME = "sb-intb", PLOT) /MS7 ENTITY( ADD, NAME = "s1-out", PLOT) ENTITY( ADD, NAME = "s1-axi", PLOT) ENTITY( ADD, NAME = "s1-top", PLOT) ENTITY( ADD, NAME = "s1-bot", PLOT) /MS8 ENTITY( ADD, NAME = "s2-out", PLOT) ENTITY( ADD, NAME = "s2-left", PLOT) ENTITY( ADD, NAME = "s2-bot", PLOT) /MS9 ENTITY( ADD, NAME = "s3-out", PLOT) ENTITY( ADD, NAME = "s3-axi", PLOT) ENTITY( ADD, NAME = "s3-bot", PLOT) /MS10 ENTITY( ADD, NAME = "g-out", PLOT) ENTITY( ADD, NAME = "g-axi", PLOT) ENTITY( ADD, NAME = "g-left", PLOT) ENTITY( ADD, NAME = "g-bot", PLOT) /========================================= / BOUNDARY CONDITION COMMANDS /========================================= /inlet BCNODE (ADD, UX, CONS=0, ENTITY = "ft-in") BCNODE (ADD, UY, CONS=0, ENTITY = "ft-in") BCNODE (ADD, UZ, CONS=6.1 869, ENTITY = "ft-in") BCNODE (ADD, UX, CONS=0, ENTITY = "fb-in") BCNODE (ADD, UY, CONS=0, ENTITY = "fb-in") BCNODE (ADD, UZ, CONS=6.1 869, ENTITY = "fb-in")

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190Appendix G: (Continued) /interface BCNODE (VELOCITY, ZERO, ENTITY = "ft-intt") BCNODE (VELOCITY, ZERO, ENTITY = "ft-intb") BCNODE (VELOCITY, ZERO, ENTITY = "fb-intt") BCNODE (VELOCITY, ZERO, ENTITY = "fb-intb") /axi BCNODE( ADD, UX, ENTI = "ft-axi", ZERO ) BCNODE( ADD, UX, ENTI = "fb-axi", ZERO ) /temperature BCNODE (ADD, TEMPERATURE, CONS = 80, ENTITY = "ft-in") BCNODE (ADD, TEMPERATURE, CONS = 80, ENTITY = "fb-in") /variable heat transfer/storm data BCFLUX (HEAT, CONS = 1, CURV = 1, FACT = 1, ENTITY = "s1-top") TMFUNCTION(SET=1, NPOI = 8) /CURVE=16) 0000 -0.008537614 3600 -0.008686007 7200 -0.008811232 10800 -0.009111684 14400 -0.010630406 18000 -0.012986843 21600 -0.005518975 25200 -0.00667844 /insulation/symmetry BCFLUX (HEAT, CONS = 0. 0, ENTITY = "ft-axi") BCFLUX (HEAT, CONS = 0. 0, ENTITY = "fb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "pt-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "pb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "st-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-axi") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-left") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-left") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-bot") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-axi") /in face and out face BCFLUX (HEAT, CONS = 0. 0, ENTITY = "pt-in") BCFLUX (HEAT, CONS = 0. 0, ENTITY = "pb-in") BCFLUX (HEAT, CONS = 0. 0, ENTITY = "st-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-in") BCFLUX (HEAT, CONS = 0.0, ENTITY = "pt-out") BCFLUX (HEAT, CONS = 0. 0, ENTITY = "pb-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "st-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "sb-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s1-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s2-out") BCFLUX (HEAT, CONS = 0.0, ENTITY = "s3-out")

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191Appendix G: (Continued) BCFLUX (HEAT, CONS = 0.0, ENTITY = "g-out") /special BCNODE( KINE, CONS = 0. 00, ENTI = "ft-in" ) BCNODE( KINE, CONS = 0. 00, ENTI = "fb-in" ) BCNODE( DISS, CONS = 0.00, ENTI = "ft-in" ) BCNODE( DISS, CONS = 0.00, ENTI = "fb-in" ) /IC ICNODE( ADD, TEMP, CONS = 0, ENTI = "fluid-t" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "fluid-b" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "pipe-t" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "pipe-b" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "solid-t" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "solid-b" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "solid-1" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "solid-2" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "solid-3" ) ICNODE( ADD, TEMP, CONS = 0, ENTI = "ground" ) /========================================= / MATERIAL PROPERTIES /========================================= /fluid=fl (water) DENSITY (ADD, SET = "fl", CONS = 0.98) CONDUCTIVITY (ADD, SET = "fl", CONS = 0.0015750478) SPECIFICHEAT (ADD, SET = "fl", CONS = 1.000717) VISCOSITY( ADD, SET = "fl", TWO-, CONS = 0.00423 ) /pipe=pi (Carbon Steel 1%) DENSITY (ADD, SET = "pi", CONS = 7.801) CONDUCTIVITY (ADD, SET = "pi", CONS = 0.102772467) SPECIFICHEAT (ADD, SET = "pi", CONS = 0.113049713) /solid=so (concrete) DENSITY (ADD, SET = "so", CONS = 2.3) CONDUCTIVITY (ADD, SET = "so", CONS = 0.00239005736) SPECIFICHEAT (ADD, SET = "so", CONS = 0.210325048) /ground=gr (sandstone) DENSITY (ADD, SET = "gr", CONS = 1.5) CONDUCTIVITY (ADD, SET = "gr", CONS = 0.00239005736) SPECIFICHEAT (ADD, SET = "gr", CONS = 0.454110899) END

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192Appendix H: Design of Experiment on Paramete rs and Quantitative Error Analysis of the Grid Test Design of experiment Design of experiment is a technique to study the effect of different parameters on a specific data output by applying a comb ination of parameter changes. Using combinations of three parameters will end up with 8 runs and 8 data outputs. This study was performed at two different locations: Z = 0.4 and Z = 0.8. The parameters used in this study are: Reynolds number, magnetic field intensity, and tube diameter. The parameter combinations used are listed toge ther with the resulting data as follows: Z = 0.4 Z = 0.8 Re = 1000, G = 5, d = 0.036 Nu = 7.55 6.64 Re = 1600, G = 5, d = 0.036 Nu = 8.92 7.71 Re = 1000, G = 10, d = 0.036 Nu = 7.55 6.64 Re = 1600, G = 10, d = 0.036 Nu = 8.92 7.71 Re = 1000, G = 5, d = 0.048 Nu = 9.23 8.01 Re = 1600, G = 5, d = 0.048 Nu = 10.94 9.34 Re = 1000, G = 10, d = 0.048 Nu = 9.23 8.01 Re = 1600, G = 10, d = 0.048 Nu = 10.94 9.34 The contrast constants for each parameter and parameter combinations are obtained by performing a series of ma thematical operations, adding and subtracting the data according to the run they were generated from. The contrast for the parameters and parameter combinations are shown as follows: Z = 0.4 Z = 0.8 Contrast Re = 6.16 4.80 Contrast G = 0 0 Contrast Re, G = 0 0 Contrast d = 7.40 6.00 Contrast Re, d = 0.68 0.52 Contrast G, d = 0 0 Contrast Re, G, d = 0 0 Sum of the squares for the effects are calculated using the following formula: n Contrast SSeff 3 22 ) ( where, n = number of runs

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193Appendix H: (Continued) Then, Total sum of squares is calculate by adding all the sum of squares as follows: d G G G TSS SS SS SS SS, Re, Re, Re... .......... This is done for each effect ending up with the following results: Z = 0.4 Z = 0.8 SSRe = 0.5929 0.3600 SSG = 0 0 SSRe, G = 0 0 SSd = 0.8556 0.5625 SSRe, d = 0.007225 0.004225 SSG, d = 0 0 SSRe, G, d = 0 0 The percentage contribution of each of the ef fects is obtained by taking the ratio of the sum of squares of the effect to the total sum of squares and then multiplying the results with 100. For example, the percentage c ontribution of effect Re is given as: 100 ReRe TSS SS of on contributi Percentage The factor having the highest percentage contri bution is said to have the most effect on the experiment. The results were as follows: Z = 0.4 Z = 0.8 % contribution of Re = 40.73 % 38.85 % % contribution of G = 0 % 0 % % contribution of Re, G = 0 % 0 % % contribution of d = 58.78 % 60.70 % % contribution of Re, d = 0.496 % 0.456 % % contribution of G, d = 0 % 0 % % contribution of Re, G, d = 0 % 0 % The results clearly show that the change in Reynolds number and diameter have the most effect on Nusselt number but the change in the magnetic field is the least.

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194Appendix H: (Continued) Quantitative error analysis To insure accurate results, the number of elem ents that were used to mesh the geometry had to be deemed adequate. This was done by performing computations for several combinations of elements in the axial and cr oss-sectional directions. These combinations were: 4 x 8 x 20, 6 x 12 x 30, and 8 x 16 x 40. The quantitative difference in grid independence can be calculat ed using the equation: e N D C T N is the number of elements along an axis, and C, D, and e are constants to be evaluated. Three different grid tests provides three equatio ns with known T and N. This results in a set of non-linear equations with three variab les. An initial value of e was assumed, and after performing a number of it erations, a correct value for e is determined. Afterwards, the percentage error could be obtained by: 100 C C T This test is performed at two different axial locations: Z = 0.4 and Z = 0.8. The results are shown below. Z = 0.4 Z = 0.8 e = 0.561867 1.204276 C = 27.49269 30.61695 D = 8.22457 41.05001 % error for 4 x 8 x 20 = 5.56 % 3.63 % % error for 6 x 12 x 30 = 4.42 % 2.23 % % error for 8 x 16 x 40 = 3.76 % 1.58 %

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About the Author Abdullatif Gari was born in Saudi Arabia and lived there until he achieved his Bachelor’s degree in April of 1994 in M echanical Engineering Depart ment at King Abdul Aziz University in Jeddah. He received a scholarsh ip from his country to continue his higher education career abroad. He finished his Ma ster’s degree in Mechanical Engineering at Oklahoma State University in December of 199 9. He also finished his Ph.D. degree in Mechanical Engineering at the University of South Florida. He is going to work as an Assistant Professor in the thermal Engineering Department at King Abdul Aziz University in Jeddah, Saudi Arabia.