THERMAL TRANSPORT INVESTIGATION AND PARAMETRIC STUDY IN CYLINDRICAL OBLIQUE FIN MINICHANNEL HEAT SINK FAN YAN A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 ____________________________________________________________________ DECLARATION II ____________________________________________________________________ ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my supervisor, Assistant Professor Dr. Lee Poh Seng, for his guidance and support during my years at National University of Singapore (NUS). He has been playing a very important role in my professional development. His professionalism and high standard in research always encourage and inspire me throughout the whole course of my study. I would also like to thank my co-supervisor Dr. Chua Beng Wah from SIMTech for his encouragement and support in this research collaboration. I would like to acknowledge the financial support received form NUS, and the MOE Academic Research Fund (AcRF) – Tier research project for the support in the development of work in various ways. I would like to thank my research group members particularly Dr. Jin Li Wen, Dr. Tamanna Alam, Dr. Karthik Balasubramanian, Dr. Lee Yong Jiun, Dr. Pawan Kumar Singh, Mu Nasi, Mrinal Jagirdar, Kong Xin Xian and Liang Tian Shen for their discussions and inputs to this work. I also would like to thank the fellow graduate students Bernard Saw Lip Huat, Tong Wei, Ye Yong Huang for their friendship. I would like to acknowledge our lab technologist Ms. Roslina Abdullah for her help in purchasing equipments and creating a good environment in Thermal Process Lab 2. I would also like to thank High Performance Computer specialist Wang Junhong for his readily available assistance. III ____________________________________________________________________ I am especially grateful to my husband Dr. Low Soon Chiang and all my family members for their supreme support and encouragement. Without them, my dream would not have come true. IV ____________________________________________________________________ TABLE OF CONTENTS DECLARATION .II ACKNOWLEDGEMENTS III TABLE OF CONTENTS V ABSTRACT IX LIST OF TABLES . XI LIST OF FIGURES XIII NOMENCLATURE . XVII PUBLICATION ARISING FROM THIS THESIS XXIII CHAPTER INTRODUCTION .1 1.1. Background .1 1.2. Objectives .4 1.3. Significance and Scope of the Study 1.4. Organization for Dissertation CHAPTER LITERATURE REVIEW .9 2.1. Thermal Application for Cylindrical Heat Sink 2.1.1. Lithium-ion Batteries 2.1.2. Motors .13 2.2. Single-Phase Heat Transport in Micro/Mini channels 18 2.3. Passive Techniques in Micro/Mini channels .20 2.4. Active Techniques in Micro/Mini channels 30 2.5. Optimization Techniques for Heat Sinks 31 V ____________________________________________________________________ CHAPTER NUMERICAL ANALYSIS OF NOVEL CYLINDRICAL OBLIQUE FIN MINICHANNEL HEAT SINK 37 3.1. CFD Simulation Approach 37 3.1.1. Cylindrical Oblique fin Minichannel Geometry Consideration .38 3.1.2. Simulation Model Setup .41 3.1.3. Governing Equation 46 3.1.4. Boundary Condition .47 3.1.5. Grid Independence Study .47 3.2. Results and Discussion 49 3.2.1. Velocity and Temperature Profile 49 3.2.2. Secondary Flow Distribution 52 3.2.3. Entrance Region Effect .54 3.2.4. Heat Transfer Characteristic .59 3.2.5. Pressure Drop Characteristic 63 3.3. Conclusions 64 CHAPTER EXPERIMENTAL INVESTIGATION OF NOVEL CYLINDRICAL OBLIQUE FIN MINICHANNEL HEAT SINK .67 4.1. Experimental Setup and Procedures .67 4.1.1. Experimental Setup .67 4.1.2. Test Section 69 4.1.3. Experimental Procedure .71 4.1.4. Data Reduction .72 4.1.5. Uncertainties Analysis 76 4.2. Results and Discussion 77 4.2.1. Validation of Numerical Predictions 77 4.2.2. Heat Transfer Characteristic .80 4.2.3. Pressure Drop Characteristic 85 VI ____________________________________________________________________ 4.2.4. Overall Heat Transfer Characteristic 87 4.3. Conclusions 88 CHAPTER PARAMETRIC INVESTIGATION OF HEAT TRANSFER AND FRICTION CHARACTERISTICS IN CYLINDRICAL OBLIQUE FIN MINICHANNEL HEAT SINK 91 5.1. Theoretical Analysis 92 5.1.1 Governing Equation .92 5.1.2 Similarity Analysis of Oblique Fin 96 5.2. Physical Model Assumptions .100 5.2.1 Numerical Solution Method 101 5.2.2 Validation of Numerical Model .104 5.3. Results and Discussion 105 5.3.1 The Effect of Aspect Ratio in Straight Fin Channels 106 5.3.2 Flow Distribution with changing Secondary Channel Gap .108 5.3.3 Flow Distribution with changing Oblique Angle 113 5.3.4 Flow Distribution with changing Reynolds Number .118 5.3.5 The Effect of Oblique Angle .120 5.3.6 The Effect of Secondary Channel Gap .127 5.4. Multiple Correlations .134 5.4.1 Proposed Form of Correlations 134 5.4.2 Correlations of Nuave and fappRe .136 5.5. Conclusions 138 CHAPTER INVESTIGATION ON THE INFLUENCE OF EDGE EFFECT ON FLOW AND TEMPERATURE UNIFORMITIES IN CYLINDRICAL OBLIQUE FIN MINICHANNEL HEAT SINKS 143 6.1. Introduction .144 6.2. Methods 146 VII ____________________________________________________________________ 6.2.1 Experimental Setup and Procedures 146 6.2.2 Numerical Simulation Approach .149 6.3. Results and Discussion 153 6.3.1 Validation of Numerical Simulations 153 6.3.2 Flow Distribution Study 154 6.3.3 Fluid Temperature Profile .161 6.3.4 Edge Effect and Temperature Uniformity Characteristics 163 6.4. Conclusions 169 CHAPTER CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK .173 7.1. Numerical Study 173 7.2. Experimental Investigation 174 7.3. Similarity Analysis and Parametric Study .175 7.4. Edge Effect Investigation .176 7.5. Recommendations for Future Work .177 REFERENCES 179 APPENDIX 191 APPENDIX A: UNCERTAINTY ANALYSIS FOR EXPERIMENTAL DATA 193 Systematic uncertainty 193 Random uncertainty 197 Combining systematic and random uncertainties 198 VIII ____________________________________________________________________ ABSTRACT A novel cylindrical oblique fin minichannel heat sink, in the form of an enveloping jacket, was proposed to be fitted over cylindrical heat sources. The periodic oblique fins cause the hydrodynamic boundary layer development to be reinitialized at the leading edge of each fin. This decreases the thermal boundary layer thickness, enhances the heat transfer performance and incurs negligible pressure drop penalty. Its cooling effectiveness was compared with conventional straight fin minichannel heat sink through experimental and numerical approaches for the Reynolds number ranging from 50 to 500, with excellent agreement. The results show that the average Nusselt number for the cylindrical oblique fin minichannel heat sink increases up to 75.6% and the total thermal resistance decreases up to 59.1% compared with the conventional straight fin minichannel heat sink. Initial findings show that a flow recirculation zone forms at larger Reynolds number in the secondary channel. However, this recirculation is insignificant in the present low Reynolds number study. Furthermore, it was found that the entrance length of oblique fin minichannel is shorter than that in straight fin minichannel. Overall heat transfer characteristics (ENu, Ef) show that the cylindrical oblique fin minichannel enhances heat transfer significantly and reduces pumping power. To optimize and analyze the heat transfer performance of the cylindrical oblique fin heat sink, a similarity analysis and parametric study on the geometric dimensions of the heat sink were performed. Three dimensional conjugated heat transfer simulation using Computational Fluid Dynamics IX ____________________________________________________________________ (CFD) approach was conducted to analyse the laminar convective heat transfer and apparent friction characteristics for 43 different cylindrical heat sinks with varied geometric dimensions. The studies were performed by varying the oblique angle from 20° to 45°, the secondary channel gap from 1mm to 5mm and the Reynolds number from 200 to 900. In this work, the flow distributions of varying oblique angles, secondary channel gaps and Reynolds number were also investigated and reported. Based on the 259 numerical data points, multiple correlations for the average Nusselt number and the apparent friction constant were formulated, verified and presented. These correlations successfully pave the way for optimization of the oblique fin heat sink without the need for numerical simulation analysis or fabrication of the heat sink. The influences of edge effect on flow and temperature uniformities were also investigated for oblique-finned structures on both planar and cylindrical heat source surfaces through numerical and experimental studies. The flow field analysis shows that poor flow mixing exists in the draining and filling regions, while the flow regime between the middle regions is not influenced by the edge effects in the blockaded cylindrical oblique fin heat sink. For regular cylindrical oblique fin heat sink, the flow fields in both the main and secondary channels are distributed uniformly in the spanwise direction. A uniform and lower surface temperature distribution for regular cylindrical oblique fin heat sink is observed as a result of the improved flow mixing due to the absence of the edge effects. This further proves that the cylindrical oblique fin heat sink is an effective cooling solution for cylindrical heat sources. X References [84] López R., Lecuona A., Ventas R., Vereda C., 2012, "A numerical procedure for flow distribution and pressure drops for U and Z type configurations plate heat exchangers with variable coefficients," Journal of Physics: Conference Series, Vol. 395, art. no. 012060. [85] Datta A. B. and Majumdar A. K. , 1980, "Flow distribution in parallel and reverse flow manifolds," International Journal of Heat and Fluid Flow, Vol. 2, pp. 253-262. [86] Bassiouny M. K. and Martin H., 1984, "Flow distribution and pressure drop in plate heat exchangers-I U-type arrangement," Chemical Engineering Science, Vol. 39, pp. 693-700. [87] Harpole G. M. and Eninger J. E., 1991, "Micro-channel heat exchanger optimization," Proceedings - IEEE Semiconductor Thermal and Temperature Measurement Symposium, pp. 59-63. [88] Copeland D., Behnia M., Nakayama W., 1997, "Manifold microchannel heat sinks: Isothermal analysis," IEEE Transactions on Components Packaging and Manufacturing Technology Part A, Vol. 20, pp. 96-102. [89] Ryu J. H., Choi D. H., Kim S. J., 2003, "Three-dimensional numerical optimization of a manifold microchannel heat sink," International Journal of Heat and Mass Transfer, Vol. 46, pp. 1553-1562. [90] Wang Y. and Ding G. F., 2008, "Numerical analysis of heat transfer in a manifold microchannel heat sink with high efficient copper heat spreader," Microsystem Technologies, Vol. 14, pp. 389-395. [91] Sadeghi A., Asgarshamsi A., Saidi M. H., 2009, "Laminar forced convection in annular microchannels with slip flow regime," Proceedings of the 7th International Conference on Nanochannels, Microchannels, and Minichannels, ICNMM2009 PART A, pp. 353-361. [92] Nonino C., Del Giudice S., Savino S., 2010, "Temperature-dependent viscosity and viscous dissipation effects in microchannel flows with uniform wall heat flux," Heat Transfer Engineering, Vol. 31, pp. 682691. [93] Rokni M. and Sundén B., 1996, "Numerical investigation of turbulent forced convection in ducts with rectangular and trapezoidal crosssection area by using different turbulence models," Numerical Heat Transfer; Part A: Applications, Vol. 30, pp. 321-346. [94] Sadasivam R., Manglik R. M., Jog M. A., 1999, "Fully developed forced convection through trapezoidal and hexagonal ducts," International Journal of Heat and Mass Transfer, Vol. 42, pp. 43214331. 186 References [95] Bahrami M., Yovanovich M. M., Culham J. R., 2006, "Approximate solution for pressure drop in microchannels of arbitrary cross-section," Collection of Technical Papers - 9th AIAA/ASME joint Thermophysics and Heat Transfer Conference Proceedings 2, pp. 753-766. [96] McHale J. P. and Garimella S. V., 2010, "Heat transfer in trapezoidal microchannels of various aspect ratios," International Journal of Heat and Mass Transfer, Vol. 53, pp. 365-375. [97] Steinke M. E. and Kandlikar S. G., 2006, "Single-phase liquid friction factors in microchannels," International Journal of Thermal Sciences, Vol. 45, pp. 1073-1083. [98] Xia C., 2002, "Spray/jet cooling for heat flux high to 1kw/cm2," Annual IEEE Semiconductor Thermal Measurement and Management Symposium 18th, pp. 159-163. [99] Cheng L., Luan T., Du W., Xu M., 2009, "Heat transfer enhancement by flow-induced vibration in heat exchangers," International Journal of Heat and Mass Transfer, Vol. 52, pp. 1053-1057. [100] Sung M. K. and Mudawar I., 2008, "Single-phase hybrid microchannel/micro-jet impingement cooling," International Journal of Heat and Mass Transfer, Vol. 51, pp. 4342-4352. [101] Husain A. and Kim K. Y., 2009, "Analysis and optimization of electrokinetic microchannel heat sink," International Journal of Heat and Mass Transfer, Vol. 52, pp. 5271-5275. [102] Heymann D., Pence D., Narayanan V., 2010, "Optimization of fractallike branching microchannel heat sinks for single-phase flows," International Journal of Thermal Sciences, Vol. 49, pp. 1383-1393. [103] Baodong S., Lifeng W., Jianyun L., Heming C., 2011, "Multi-objective optimization design of a micro-channel heat sink using adaptive genetic algorithm," International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 21, pp. 353-364. [104] Hu G. and Xu S., 2009, "Optimization design of microchannel heat sink based on SQP method and numerical simulation," 2009 International Conference on Applied Superconductivity and Electronmaganetic Devices, ASEMD 2009, art. no. 5306686, pp. 89-92. [105] Kou H. S., Lee J. J., Chen C. W., 2008, "Optimum thermal performance of microchannel heat sink by adjusting channel width and height," International Communications in Heat and Mass Transfer, Vol. 35, pp. 577-582. [106] Tonomura O., Takase T., Kano M., Hasebe S., 2007, "Optimal shape design of pressure-driven microchannels using adjoint variable method," Proceedings of the 5th International Conference on 187 References Nanochannels, Microchannels, and Minichannels, ICNMM2007, pp. 427-434. [107] Tsai T. H. and Chein R., 2012, "Simple model for predicting microchannel heat sink performance and optimization," Heat and Mass Transfer/Waerme- und Stoffuebertragung, Vol. 48, pp. 789-798. [108] Türkakar G. and Okutucu-Özyurt T., 2012, "Dimensional optimization of microchannel heat sinks with multiple heat sources," International Journal of Thermal Sciences, Vol. 62, pp. 85-92. [109] Wang Z. H., Wang X. D., Yan W. M., Duan Y. Y., Lee D. J., Xu J. L., 2011, "Multi-parameters optimization for microchannel heat sink using inverse problem method," International Journal of Heat and Mass Transfer, Vol. 54, pp. 2811-2819. [110] Canhoto P. and Reis A. H., 2011, "Optimization of fluid flow and internal geometric structure of volumes cooled by forced convection in an array of parallel tubes," International Journal of Heat and Mass Transfer, Vol. 54, pp. 4288-4299. [111] Kim D. K., 2012, "Thermal optimization of plate-fin heat sinks with fins of variable thickness under natural convection," International Journal of Heat and Mass Transfer, Vol. 55, pp. 752-761. [112] Hsieh C. T. and Jang J. Y., 2012, "Parametric study and optimization of louver finned-tube heat exchangers by Taguchi method," Applied Thermal Engineering, Vol. 42, pp. 101-110. [113] Qi Z. g., Chen J. P., Chen Z. J., 2007, "Parametric study on the performance of a heat exchanger with corrugated louvered fins," Applied Thermal Engineering, Vol. 27, pp. 539-544. [114] Steinke M. E. and Kandlikar S. G., 2004, "Single-phase Heat Transfer Enhancement Techniques in Microchannel and Minichannel Flows," Proceedings of the 2nd International Conference on Microchannels and Minichannels (ICMM2004), pp. 141-148. [115] Suga K. and Aoki H., 1995, "Numerical study on heat transfer and pressure drop in multilouvered fins," Journal of Enhanced Heat Transfer, Vol. 2, pp. 231-238. [116] Wong M., Owen I., Sutcliffe C. J., 2009 "Pressure loss and heat transfer through heat sinks produced by selective laser melting," Heat Transfer Engineering, Vol. 30, pp. 1068-1076. [117] Fluent v.12.1, "ANSYS FLUENT User's Guide," ANSYS, Inc., 200901-29. [118] DeJong N. C. and Jacobi A. M., 2003, "Flow, heat transfer, and pressure drop in the near-wall region of louvered-fin arrays," Experimental Thermal and Fluid Science, Vol. 27, pp. 237-250. 188 References [119] Shah R. K., 1978, Laminar flow forced convection in ducts : a source book for compact heat exchanger analytical data. Academic Press, New York. [120] Alharbi A. Y., Pnce D. V., Cullion R. N., 2003, "Fluid flow through microscale fractal-like branching channel networks," Journal of Fluids Engineering, Transactions of the ASME, Vol. 125, pp. 1051-1057. [121] Liu D. and Garimella S. V., 2005, "Analysis and optimization of the thermal performance of microchannel heat sinks," International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 15, pp. 7-26. [122] Incropera F. P., 2011, Fundamentals of heat and mass transfer, 7th ed. Hoboken, Wiley, N.J. [123] Qu W., 2004, "Transport phenomena in single-phase and two-phase micro-channel heat sinks," Ph.D. 3154718, Purdue University, United States - Indiana. [124] Kays W. M., 1984, Compact heat exchangers, 3rd ed. Vol. 153987. McGraw-Hill, New York. [125] Winoto S. H., Tandionol, Shah D. A., Mitsudharmadi H., 2010, "On the development of concave surface boundary layer flows," AIP Conference Proceedings, Vol. 1225, pp. 10-26. [126] Rosaguti N. R., Fletcher D. F., Haynes B. S., 2006, "Laminar flow and heat transfer in a periodic serpentine channel with semi-circular crosssection," International Journal of Heat and Mass Transfer, Vol. 49, pp. 2912-2923. [127] Fan Y.,Lee P. S., Jin L. W., Chua B. W., 2011, "Numerical simulations of forced convection in novel cylindrical oblique-finned heat sink," 2011 IEEE 13th Electrical Package Technology Conference, EPTC 2011, Dec 7-9, pp. 647-652. [128] Nellis G. and Klein S., 2009, Heat transfer, Cambridge University Press, Cambridge. [129] Rahman M., Comini G., Brebbia C. A., 1998, Advances in fluid mechanics, Computational Mechanics Pub., Southampton. [130] Kays W. M., 2005, Convective heat and mass transfer, 4th ed., McGraw-Hill Higher Education, New York. [131] Holman J. P., 2010, Heat transfer, 10th international ed., McGraw Hill Higher Education, Boston. [132] Fan Y., Lee P. S., Jin L. W., Chua B. W., 2013, "A simulation and experimental study of fluid flow and heat transfer on cylindrical 189 References oblique-finned heat sink," International Journal of Heat and Mass Transfer, Vol. 61, pp. 62-72. [133] Wu H. Y. and Cheng P., 2003, "An experimental study of convective heat transfer in silicon microchannels with different surface conditions," International Journal of Heat and Mass Transfer, Vol. 46, pp. 2547-2556. [134] Alharbi A. Y., Pence D. V., Cullion R. N., 2004, "Thermal characteristics of microscale fractal-like branching channels," Journal of Heat Transfer, Vol. 126, pp. 744-752. [135] Navidi W. C., 2009, Principles of statistics for engineers and scientists. McGraw-Hill, New York. [136] Rhie C. M. and Chow W. L., 1983, "Numerical study of the turbulent flow past an airfoil with trailing edge separation," AIAA journal, Vol. 21, pp. 1525-1532. [137] Sparrow E. M., Abraham J. P., Minkowycz W. J., 2009, "Flow separation in a diverging conical duct: Effect of Reynolds number and divergence angle," International Journal of Heat and Mass Transfer, Vol. 52, pp. 3079-3083. [138] Menter F. R., 1994, "Two-equation eddy-viscosity turbulence models for engineering applications," AIAA journal, Vol. 32, pp. 1598-1605. [139] Lee G. G., Allan W. D. E., Goni Boulama K., 2013, "Flow and performance characteristics of an Allison 250 gas turbine S-shaped diffuser: Effects of geometry variations," International Journal of Heat and Fluid Flow, Vol. 42, pp. 151-163. [140] Bardina J. E., Huang P. G., Coakley T. J., 1997, "Turbulence Modeling Validation Testing and Development," NASA Technical Memorandum 110446, Moffett Field California. [141] Lee Y. J., Lee P. S., Chou S. K., 2013, "Numerical study of fluid flow and heat transfer in the enhanced microchannel with oblique fins," Journal of Heat Transfer, Vol. 135, art. no. 041901. [142] Taylor J. R. 1997, An introduction to error analysis : the study of uncertainties in physical measurements, 2nd ed. Vol. 901910, University Science Books, Sausalito, Calif. 190 Appendix A APPENDIX APPENDIX A: UNCERTAINTY ANALYSIS FOR EXPERIMENTAL DATA 191 Appendix A 192 Appendix A APPENDIX A: UNCERTAINTY ANALYSIS FOR EXPERIMENTAL DATA Measurement uncertainties can be divided into two components: systematic uncertainty and random uncertainty. Systematic uncertainties are biases in measurement which lead to the situation where the mean of many separate measurements differs significantly from the actual value of the measured attribute. Random uncertainties show up as different results for ostensibly the same repeated measurement. They can be estimated by comparing multiple measurements, and reduced by averaging multiple measurements. Therefore, uncertainty analyses are conducted based on the principles proposed by Taylor [142] in the present study. The quantities used in data reduction and their associated uncertainty analyses are listed below: Systematic uncertainty Suppose that x, , z are measured with uncertainties δx, ., δz, and the measured values used to compute the function q(x, …, z). If the uncertainties in x,…, z are independent and random, then the uncertainty in q is q  ( q q x)    ( z ) x z (A-1) In any case, it is never larger than the ordinary sum q  q q x  .  z x z (A-2) The quantities used for the calculation of the single-phase Nusselt number and their associated systematic uncertainties are listed below: 193 Appendix A Cross section area of minichannel: Ac  Ac Ac where ( wb  wt ) H  [ (A-3)  ( wb  wt ) wb  wt ]2  ( H H ) =0.4%  ( wb  wt ) H =0.32% , =0.25%, wb  wt  H  5m wb  wt H Fluid velocity: u u u where 0.5%  Q Ac  ( (A-4) Q Q )2  ( Ac Ac ) = [0.64%, 5.01%] Q  5% and Q =5ml/min Q Hydraulic diameter: Dh  Ac P Dh  ( Dh where (A-5) Ac Ac )2  ( P P ) =0.49% P  ( wb  wt  H )  =0.28% P wb  wt  H Reynolds number: Re   Re Re uDh (A-6)   ( u u )2  ( Dh Dh ) = [0.81%, 5.04%] 194 Appendix A Sensible heat gain by coolant: q  C pQ(T f ,o  T f ,i ) (A-7)  (T f ,o  T f ,i ) q Q  ( )2  [ ] = [14.15%, 15%] q Q T f ,o  T f ,i where  (T f ,o  T f ,i )  T f ,o  T f ,i T f ,o  T f ,i T f ,o  T f ,i =14.14% Convective heat transfer area: Atot  Ab   Afin Atot Atot  ( Ab Ab (A-8) )2  ( A fin A fin ) = 0.44% Ab A fin =0.37%, =0.25% Ab A fin where Ab  36  wb  L , Afin  36  2H  L , Average fluid temperature: T f ,ave  T f ,i  T f ,o (A-9) T f ,ave  ( T f ,i )  ( T f ,o ) =0.35 °C where T f ,ave =25 °C Average wall temperature Tw, ave  T w, i i 1 (A-10) Tw,i  Tcu  q ln(17 / 12.5) 2Lkcu Tw,ave  Tw,i  (Tcu )  [  ( 195 q ln(17 / 12.5) )] =0.5°C 2Lkcu (A-11) Appendix A where Tw,ave =40 °C Difference in average wall and fluid temperatures  (Tw,ave  T f ,ave )  (Tw,ave )  (T f ,ave ) =0.61°C Heat transfer coefficient: have  q (A-12) Atot (Tw,ave  T f ,ave )  (Tw,ave  T f ,ave ) have A q  ( )  ( tot )  [ ] = [14.73%, 15.55%] have q Atot Tw,ave  T f ,ave Nusselt number: Nuave  have Dh kf Nu ave Nu ave (A-13)  ( have have )2  ( Dh Dh ) = [14.74%, 15.56%] Total thermal resistance: Rtot  Tw,ave  T f ,i (A-14) q where T f ,i =21°C  (Tw,ave  T f ,i ) q Rtot  [ ]  ( ) = [14.63%, 15.45%] Rtot Tw,ave  T f ,i q Pressure drop (minor losses): Pc  K  f (uin  us ,in )  c  f uin 2 (A-15) Pe  K  f (us ,o  uo )  e  f uo 2 (A-16) 196 Appendix A Pc  Pe = [6.55E-06, 4.82E-06] Pa Pressure drop: Pch  P  Pc  Pe (A-17) where P = 10Pa Pch Pch (P)  (Pc )  (Pe ) = [2%, 20%] Pch  Friction factor: f  f f  ( Pch Dh u L Pch Pch )2  ( (A-18) Dh Dh )  (2 u u )2  ( L L ) = [2.43%, 22.38%] Random uncertainty The statistical methods described in the followings give a reliable estimate of the random uncertainty, providing a well-defined procedure for reducing them. Generally, suppose we make N measurements of the quantity x (all using the same equipment and procedures), and find the N values x1 , x2 , ., xN . The best estimate for x is usually the average of x1 , x2 , ., xN . That is, xbest  x , where x  (A-19) x1  x2  .  xN  xi  N N The standard deviation of the measurements x1 , x2 , ., xN is an estimate of average uncertainty of the measurements x1 , x2 , ., xN and can be obtained by 197 Appendix A x   N  xi  x N  i 1  (A-20) Therefore, the uncertainty of x can be given by the standard deviation σx divided by N . This quantity is called the standard deviation of the mean, and is denoted  x (other common names for this are standard error and standard error of the mean): r   x  x N (A-21) Equation (A-21) indicates that the uncertainty of quantity x due to data reduction results mainly from the use of finite data in calculating a statistical mean. This type of uncertainty depends on the reduction scheme. To minimize the reduction uncertainty, data with 100-cycle length was collected for the present experiments. Therefore, the uncertainty of the measured quantities can be reduced. For the velocity, pressure drop and temperature value are the standard deviation of the value obtained by cycle-averaging. Combining systematic and random uncertainties Recently, there has been a move by research journals and standards organizations to require use of empirical rules for combining systematic and random uncertainties to give a single total uncertainty. Since we now have estimated both the systematic and random components of  , our only problem is to combine them to give  T itself. While no rigorous confidence level can be associated with the total uncertainty,  T , coverage analogous to the 95 percent confidence level can be given for the recommended total uncertainty models. Thus 198 Appendix A  T  ( S )  ( R ) where (A-22)  S and  R are systematic and random uncertainty respectively. Using the method described above, total uncertainty calculation (a sample) of Nusselt number and friction factor for conventional straight fin heat sink at various Reynolds number are listed in Table A and Table B. 199 Appendix A Table A Sample total uncertainty calculation of Nusselt number Nuave Nuave Nu Re 48.07 Nuave 3.77 Averaged (%) 15.15 δNuave-S δNuave-R δNuave-T 0.57 0.29 0.64 73.99 4.68 15.15 0.71 0.02 0.71 99.91 5.30 15.15 0.80 0.27 0.85 125.53 6.04 15.15 0.92 0.13 0.92 151.70 6.90 15.15 1.05 0.20 1.06 177.07 6.56 15.15 0.99 0.25 1.03 203.29 6.98 15.15 1.06 0.43 1.14 228.71 6.60 15.15 1.00 0.76 1.25 253.95 7.86 15.15 1.19 0.28 1.22 280.11 8.63 15.15 1.31 0.31 1.34 304.79 8.05 15.15 1.22 0.19 1.23 330.37 8.67 15.15 1.31 0.25 1.34 356.42 8.58 15.15 1.30 0.50 1.39 382.10 8.70 15.15 1.32 0.11 1.32 408.42 8.56 15.15 1.30 0.25 1.32 434.18 8.91 15.15 1.35 0.08 1.35 460.47 8.59 15.15 1.30 0.16 1.31 200 Appendix A Table B Sample total uncertainty calculation of friction factor f ave f ave f Re 48.07 fave 8.20 Averaged (%) 22.38 δfave-S δfave-R δfave-T 1.83 0.93 2.06 73.99 4.93 16.80 0.83 0.52 0.98 99.91 3.31 11.22 0.37 0.19 0.42 125.53 2.61 9.37 0.24 0.04 0.25 151.70 1.91 7.51 0.14 0.03 0.15 177.07 1.49 6.59 0.10 0.05 0.11 203.29 1.22 5.67 0.07 0.05 0.08 228.71 1.10 5.12 0.06 0.03 0.06 253.95 0.91 4.57 0.04 0.04 0.06 280.11 0.81 4.21 0.03 0.05 0.06 304.79 0.70 3.84 0.03 0.04 0.04 330.37 0.62 3.59 0.02 0.04 0.04 356.42 0.53 3.33 0.02 0.03 0.04 382.10 0.45 3.14 0.01 0.02 0.02 408.42 0.42 2.95 0.01 0.03 0.03 434.18 0.39 2.80 0.01 0.03 0.03 460.47 0.37 2.66 0.01 0.02 0.03 201 [...]... conventional straight fin and cylindrical oblique fin heat sinks 71 Figure 4-4 Cross section of the mini channel heat sink 73 Figure 4-5 Wall temperature comparison (a) Conventional straight fin minichannel (b) Cylindrical oblique fin minichannel 78 Figure 4-6 Local Nusselt number for conventional straight fin and cylindrical oblique- finned minichannel heat sink 80 Figure... 53 Figure 3-8 Velocity streamline profile for (a) Conventional straight fin minichannel (b) Oblique fin minichannel 55 Figure 3-9 Local Nusselt number comparison between conventional straight fin minichannel and cylindrical oblique fin minichannel 56 Figure 3-10 local heat flux profile between conventional straight fin minichannel and cylindrical oblique fin minichannel 58 Figure 3-11... angle -βπ 93 Figure 5-2 Single oblique fin structure 96 Figure 5-3 Full domain configuration for cylindrical oblique fin minichannel heat sink 100 Figure 5-4 Simplified computation domain for cylindrical oblique fin minichannel heat sink 101 Figure 5-5 Comparison of numerical and experimental results for cylindrical oblique fin minichannel 105 Figure... obtained from experiments and numerical analyses for conventional straight fin and cylindrical oblique- finned minichannel heat sink 81 Figure 4-8 Total thermal resistance obtained from experiments and numerical analyses for conventional straight fin and cylindrical oblique- finned minichannel heat sink 83 Figure 4-9 Local wall temperature distribution for conventional straight fin. .. experimental study of fluid flow and heat transfer on cylindrical oblique- finned heat sink, ” International Journal of Heat and Mass Transfer, Vol 61, pp 62-72 2 Fan Y., Lee P S., Jin L W., Chua B W., Zhang D C., 2014, “A parametric investigation of heat transfer and friction characteristics in cylindrical oblique fin minichannel heat sink, ” International Journal of Heat and Mass Transfer, Vol 68, pp 567-584... W., 2014, Investigation on the influence of edge effect on flow and temperature uniformities in cylindrical oblique- finned minichannel array,” International Journal of Heat and Mass Transfer, Vol 70, pp 651-663 4 Fan Y., Lee P S., Jin L W., Chua B W., 2014, “Experimental investigation on heat transfer and pressure drop of a novel cylindrical oblique fin heat sink, ” International Journal of Thermal Sciences,... literature relevant to the present study These include thermal applications for cylindrical minichannel heat sink, single-phase heat transport in micro/mini channels, passive and active techniques for heat transfer enhancement, and optimization techniques for heat sinks Chapter 3 presents the numerical simulation investigations on both conventional and cylindrical oblique fin micro/mini channels It gives the... optimize the dimensions of cylindrical oblique fin heat sink for good overall heat transfer performance using similarity analysis and parametric numerical investigations 4 _Chapter 1 Introduction  Obtain generalized correlations to predict the heat transfer performance and pressure drop characteristics of the cylindrical oblique fin minichannel heat sink when the parameter values... used in the parametric computations  Examine and investigate the influences of edge effect on flow and temperature uniformity in cylindrical oblique fin minichannel heat sinks through systematic numerical and experimental studies 1.3 Significance and Scope of the Study The results of this present study would have significant impact on both providing an innovative cooling solution for cylindrical heat. .. application and understanding the flow physics behind oblique fin structure The novel cylindrical oblique fin minichannel heat sink could enhance the heat transfer performance significantly and make the heat source temperature more homogeneous and not compromise with high pumping power The proposed technique could lead to a smaller and lighter cooling system, increases the longevity of the cylindrical heat . and cylindrical oblique fin minichannel heat sink 84 Figure 4-10 Pressure drop for conventional straight fin and cylindrical oblique- finned minichannel heat sink 85 Figure 4-11 Average heat. Figure 5-3 Full domain configuration for cylindrical oblique fin minichannel heat sink 100 Figure 5-4 Simplified computation domain for cylindrical oblique fin minichannel heat sink 101 Figure. the cylindrical oblique fin minichannel heat sink increases up to 75.6% and the total thermal resistance decreases up to 59.1% compared with the conventional straight fin minichannel heat sink.