本文是以數值方法探討以矽作為基板且水為冷卻流體之梯形微渠道之三維不可壓縮層流的流場與熱傳特性。在數值計算方法上在以控制體積法為基礎，求解那維爾-史托克方程式(Navier-Stokes equations)與共軛能量方程式。以SIMPLE法來求解動量方程式中壓力項與速度項之耦合問題。網格設計則採用正交非均勻的交錯式網格。本研究中，數值計算的參數範圍為50≦Re≦600 , 0.05W≦P≦0.8W , 20W/cm2≦q"≦40W/cm2。數值結果首先會與文獻中已獲得的實驗結果作驗證且發現一致的結果。
本研究使用表面曲面法配合基因演算來展現梯形微渠道之數值最佳化。三個設計參數分別為渠道上底寬度與總寬度之比α(Wc/Ww+Wc)、渠道深度與總深度之比β(Hc/Hb+Hc)與梯形上下底之比γ(Wb/Wc)。在等熱通量與等泵浦功率的條件下，將微渠道散熱器的熱阻最小化，經由最佳化方法後的結果獲得一最佳幾何形狀，其α=0.8、β=0.586和γ=0.79。由結果可看出平均紐賽數隨雷諾數或泵浦功率增加而增加，而其熱阻則隨泵浦功率增加而變小。

This study presents the numerical simulation of three-dimensional incompressible steady and laminar fluid flow and conjugate heat transfer of a trapezoidal micro-channel heat sink using water as a cooling fluid in a silicon substrate. Navier-Stokes equations with conjugate energy equation are discretized by finite-volume method. The coupling of the velocity and the pressure terms of momentum equations are solved by SIMPLE algorithm. An orthogonal non-uniform staggered grid are used for the establishment of mesh grids. In this study, numerical computations are performed for a range of 50≦Re≦600, 0.05W≦P≦0.8W, 20W/cm2≦q"≦40W/cm2. The numerical results are first validated with the available experimental results in the literature, and a good agreement has been found.
The present study demonstrates the numerical optimization of a trapezoidal micro-channel heat sink design using response surface methodology (RSM) and genetic algorithm method (GA).Three design variables are selected from the geometric variables, ratio of the upper width of the micro-channel to the whole widthα(Wc/Ww+Wc)、the depth of the micro-channel to the whole depth and the ratio of upper width β(Hc/Hb+Hc) and lower width of the micro-channelγ(Wb/Wc). The thermal resistance of a trapezoidal micro-channel is minimized for a constant heat flux and constant pumping power. Based on the optimal results, the optimum condition is α=0.8、β=0.586 and γ=0.79. It can be observed that the averaged Nusselt number increases with increase in Reynolds number or pumping power, and thermal resistance decreases as the pumping power increases.

[1]Tuckermann D.B., Pease R.F.W., “High-performance heat sinking for VLSI, ” IEEE Electronic Device Letters, pp. 126-129, 1981.
[2]Wu P.Y., and Little W.A., “Measurement of friction factors for the flow of gases in very fine channels used for microminiature Joule-Thomson refrigerators, Cryogenics , Vol.24, No. 8, pp. 273-277, 1983.
[3]Rahman M.M., and Gui F., “Experimental measurements of fluid flow and heat transfer in microchannel cooling passages in a chip substrate,” Advances in Electronic Packaging:Thermal Management, Solder Technology, and Optoelectronic Packaging, ASME EEP-Vol.4-2, pp.685-692,1993.
[4]Rahman M.M., and Gui F., “Design fabrication and testing of microchannel heat sink for aircraft avionics cooling,” Proceedings of the 28th Intersociety Energy Conversion Engineering Conference, Vol. 1, pp.1-6,1993.
[5]Peng X.F., and Peterson G.P., “Forced convection heat transfer of single-phase binary mixtures through microchannels,” Experimental Thermal and Fluid Science, Vol. 12, pp. 98-104, 1996.
[6]Fedorov A.G., Viskanta R., “Three-dimensional conjugate heat transfer in the microchannel heat sink for electronic packaging ”, International Journal of Heat and Mass Transfer , Vol. 43, pp. 399-415, 2000.
[7]Kawano K., Minakami K., Iwasaki H., and Ishizuka M., “Development of microchannels heat exchanging, ” in: Proceedings of the ASME Heat Tranfer Division: Application of Heat Transfer in Equipment, Systems, and Education, Vol. 3, pp. 173-180, 1998.
[8]Qu W., Mala G.M., and Li D., ”Pressure-driven water flows in trapezoidal silicon microchannels,” International Communications in Heat and Mass Transfer, Vol. 43, pp. 353-364, 2000.
[9]Qu W., Mala G. M., and Li D., “Heat transfer for water flow in trapezoidal silicon microchannels,” International Communications in Heat and Mass Transfer, Vol. 43, pp. 3925-3936, 2000.
[10]Ambatipudi K. K., and Rahman M. M., “Analysis of conjugate heat transfer in microchannel heat sinks,” Numerical Heat Transfer, Part A, Vol. 37, pp. 711-731, 2000.
[11]Toh K.C., Chen X.Y., and Chai J.C., “Numerical computation of fluid flow and heat transfer in microchannels, ” International Journal of Heat and Mass Transfer, Vol.45, pp. 5133-5141, 2002.
[12]Zhao C.Y., and Lu T.J., “Analysis of microchannel heat sinks for electronics cooling,” International Journal of Heat and Mass Transfer, Vol. 45, pp. 4857-4869, 2002.
[13]Wu H.Y., and Cheng P., “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, 2003.
[14]Wu H.Y., and Cheng P., “Friction factors in smooth trapezoidal silicon microchannels with different aspect ratios,” International Journal of Heat and Mass Transfer, Vol. 46, pp. 2519-2525, 2003.
[15]Hsieh S.S., Lin C.Y., Huang C.F. and Tsai H.H., “Liquid flow in a micro-channel, ” Journal of Micromechanics and Microengineering, Vol. 14, pp. 436-445, 2004.
[16]Hsieh S.S., Lin C.Y., “Convective heat transfer in liquid microchannels with hydrophobic and hydrophilic surfaces”, International Journal of Heat and Mass Transfer, Vol. 52, pp. 260-270, 2009.
[17]Chein R., and Chen J., “Numerical study of the inlet/outlet arrangement effect on microchannel heat sink Performance”, International Journal of Thermal Sciences, Vol. 48, pp. 1627-1638, 2009.
[18]Liu, D., and Garimella, S.V., “Analysis and optimization of the thermal performance of microchannel heat sinks,” International Journal of Numerical Methods Heat Fluid Flow, Vol.15, No. 1, pp. 7-26, 2005.
[19]Li J. and Peterson G. P. “Bud”, “Geometric optimization of a micro heat sink with liquid flow,” IEEE Transaction on Components and Packaging Technologies, Vol. 29, No. 1, pp. 145-154, 2006.
[20]Husain A., and Kim K.Y., “Shape optimization of micro-channel heat sink for micro-electronic cooling,” IEEE Transaction on Components and Packaging Technologies, Vol. 31, No.2, pp. 322-330, 2008.
[21]Husain A., and Kim K.Y., “Thermal optimization of a microchannel heat sink with trapezoidal cross section,” Journal of Electronic Packaging, Vol. 131, 021005 (1-6), 2009.
[22]Husain A., and Kim K.Y., “Enhanced multi-objective optimization of a microchannel heat sink through evolutionary algorithm coupled with multiple surrogate models,” Applied Thermal Engineering, Vol. 30, pp. 1683-1691, 2010.
[23]Patankar S.V., Numerical Heat Transfer and Fluid Flow, New York: McGraw-Hill, pp. 124-134 , 1980.
[24]Patankar S.V., Spalding D.B., “A calculation processure for heat, mass and momentum transfer in three-dimension parabolic flows,” International Journal Heat Mss Transfer, Vol. 15, pp. 1787-11806, 1972.
[25]葉怡成, 高等實驗計畫法(advanced design of experiments), 五南圖書公司,2009.
[26]Bagley J.D., “The Behavior of Adaptive System which Employ Genetic and Correlation Algorithm,” Dissertation Abstracts International, Vol. 28, 1967.
[27]De Jong K.A., “An analysis of the behavior of a class of genetic adaptive systems,” PhD Dissertation, University of Michigan, No. 76~9381, 1975.
[28]Goldberg D.E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989.
[29]Davis L.D., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991.
[30]Koza J.R., Genetic Programming, on the Programming of Computers by Means of Natural Selection, MIT Press, 1992.
[31]周明, 孫樹棟, 遺傳算法原理及應用(Genetic Algorithms: theory and applications), 國防工業出版社, 1999.
[32]Langhaar H.L., “Steady flow in the transition length of a straight tube”, Journal of Applied Mechanics, Vol.9, pp. 55-58, 1942.