熱管具備優越傳熱能力，廣被作為傳熱裝置使用，而近來電子產品因在空間上受限，因此能有效增加熱交換面積的平板式熱管(又稱為蒸汽腔室)開始受到重視。對一般的電子散熱產業而言，目前的分析模式都過於複雜，因此本研究的目的是建立一個實用的分析模式。本研究假設平板式熱管中的汽體為毛細結構之間的單一熱傳介面，建立了三維的non-lumped模式。該模式具備分析軸向熱傳導與顯示局部溫度分佈的能力，且能併於整體散熱模組分析中。而且Non-lumped模式也可簡化為更簡便的lumped模式，計算結果也與實驗相契合。
此外，本研究利用風洞進行平板式熱管及其同尺寸之金屬板的散熱表現量測。結果顯示，接觸小面積熱源時，平板式熱管的散熱表現比金屬板優秀。之後將non-lumped模式應用到整體散熱模組的模擬上，並把板式鰭片與加熱塊列入考慮，使其暫態變化更接近量測結果。計算結果顯示，平板式熱管的熱阻主要是由受熱側管壁所貢獻，計算出的最高溫升與實驗之間的最大差異為6.3 %。
另一方面，本研究藉由變數分離法求得三維穩態熱傳導方程式的解析解，針對矩形板之擴散熱阻進行探討。經比對後，解析解與數值模擬結果幾乎完全相同。當與熱源之間的邊長比很小時，擴散熱阻會比一維傳導熱阻要大許多。並藉由非等向性擴散熱阻分析之概念，計算平板式熱管等效熱傳導係數。分析後發現，本研究使用的平板式熱管之徑向熱傳導係數僅48.74 W/m-K，但軸向熱傳導係數會是徑向的數十倍。將其併入整體模組模擬，其表面溫度分佈也與non-lumped模式類似。
本研究發展了能分析散熱表現的non-lumped模式，而且也另行完成矩形板擴散熱阻的理論解析，並應用到平板式熱管上，提出非等向性的等效熱傳導係數之計算方式。熱管以及與之整合的散熱模組逐漸受到散熱工業的重視，但目前仍無評估整體散熱模組表現的有效模式提出。本研究發展的non-lumped模式，以及平板式熱管等效熱傳導係數之計算方式，相信會對相關設計開發工作有所助益。

Heat pipes are widely used as heat-transport devices because of their superiority of transferring heat within low temperature differences. Recently, the vapor chambers (flat plate heat pipes) are applied on the electronic cooling. However, most of the available analysis models are too complicated to real applications. In the proposed non-lumped model, vapor is assumed as a single interface between wick structures, so the local temperature distribution and the heat spreading effect can be therefore investigated. Furthermore, the non-lumped model can be simplified as a lumped model, which contains four control volumes only. The calculated results are in line with the results in previous literatures and the experiments conducted in this study.
Experimental comparisons between copper/aluminum plates and a vapor chamber having the same thickness have been also conducted. The spreader plates integrated with a plate-fin heat sink are tested in a wind tunnel. For small-area heat sources, the vapor chamber shows a lower thermal resistance and much uniform temperature distribution than the metal plates. Then, the heat sink integrated with the vapor chamber is simulated by using non-lumped model. In order to obtain more accurate transient responses, the heating block and insulation are also considered in simulation domain. From the simulations, the major thermal resistance of the vapor chamber is contributed by the heated wall. The maximum difference of the hotspot temperature rises between the simulation and experiments is 6.3 %.
Additionally, a steady-state three-dimensional heat conduction equation is analytically solved by using separation of variables. The temperature distribution within a partially heated rectangular plate is solved and the spreading resistance is therefore obtained. The analytical solutions show very good agreement with numerical simulations, and they are also in line with the previous correlation and approximation. The spreading resistance of the thin plates with a small aspect ratio is much higher than the one-dimensional conduction resistance.
By adapting the ideas of isotropic or anisotropic heat spreading, the effective thermal conductivities of the vapor chamber have been calculated. The radial conductivity of the vapor chamber is merely 48.74 W/m-K, but the axial conductivity is more than ten times of the radial conductivity. The high axial conductivity can sufficiently enhance the heat spreading along the axial direction, and thereby resulting in a lower total thermal resistance. Combining into the whole module simulation, the surface temperature distribution is similar as that obtained by non-lumped model.
In this study, the non-lumped model is developed and presented. Moreover, the thermal spreading resistances have also been analytically investigated. An anisotropic method to calculate the effective thermal conductivities of vapor chamber is also proposed. Heat pipes, vapor chambers and integrated thermal modules draw lots of attention from cooling industries. However, there was no sufficient model to estimate the performances of the integrated thermal modules. The proposed non-lumped model and calculation of effective thermal conductivities will be helpful to evaluate the performances of vapor chambers, and they will be useful in further designing work of vapor chambers and integrated modules.

1. Faghri, Amir, Heat pipe science and technology, Taylor & Francis, 1995.
2. Zuo, Z. J. and Faghri, A., A network thermodynamic analysis of the heat pipe, Int. J. Heat Mass Transfer, vol. 41, pp. 1473-1484, 1998.
3. Chang, Won Soon and Colwell, Gene T., Mathematical modeling of the transient operating characteristic of a low-temperature heat pipe, Numerical Heat Transfer, vol. 8, pp. 169-186, 1985.
4. Tournier, J. M. and El-Genk, M. S., A heat pipe transient analysis model, Int. J. Heat Mass Transfer, vol. 37, pp. 753-762, 1994.
5. Tournier, J. M. and El-Genk, M. S., Segregated solution technique for simulating the transient operation of heat pipes, Numerical Heat Transfer, Part B, vol. 25, pp.331-355, 1994.
6. Tournier, J. M. and El-Genk, M. S., Transient analysis of the start-up of a water heat pipe from a frozen state, Numerical Heat Transfer, Part A, vol. 28, pp.461-486, 1995.
7. Vadakkan, Unnikrishnan, Garimella, Suresh V. and Murthy, Jayathi Y., Transport in Flat Heat Pipes at High Heat Fluxes from Multiple Discrete Sources, J. Heat Transfer, vol. 126, pp.347-354, June 2004.
8. Koito, Y., Imura, H., Mochizuki, M., Saito, Y., and Torii, S, Numerical analysis and experimental verification on thermal fluid phenomena in a vapor chamber, Applied Thermal Engineering, vol. 26, pp. 1669-1676, 2006.
9. Tan, B. K., Huang, X. Y., Wong, T. N. and Ooi, K. T., A study of multiple heat sources on a flat plate heat pipe using a point source approach, Int. J. Heat Mass Transfer, vol. 43, pp. 3755-3764, 2000.
10. Vafai, K. and Wang, W., Analysis of flow and heat transfer characteristics of an asymmetrical flat plate heat pipe, Int. J. Heat Mass Transfer, vol. 35, pp. 2087-2099, 1992.
11. Zhu, N. and Vafai, K., Vapor and liquid flow in an asymmetrical flat plate heat pipe: a three-dimensional analytical and numerical investigation. Int. J. Heat Mass Transfer, vol. 41, pp. 159-174, 1998.
12. Wang, Y. and Vafai, K., Transient characterization of flat plate heat pipes during starup and shutdown operations. Int. J. Heat Mass Transfer, vol. 43, pp. 2641-2655, 2000.
13. Wang, Y. and Vafai, K., An experimental investigation of the thermal performance of an asymmetrical flat plate heat pipe, Int. J. Heat Mass Transfer, vol. 43, pp. 2657-2668, 2000.
14. Kennedy, D.P., Spreading Resistance in Cylindrical Semiconductor Devices, J. Applied Physics, 31, pp. 1490-1497, 1960.
15. Negus, K.J., and Yovanovich, M.M., Constriction Resistance of Circular Flux Tubes with Mixed Boundary Conditions by Linear Superposition of Neumann Solutions, ASME Paper, 84-HT-84, pp. 1-6, 1984.
16. Negus, K.J., and Yovanovich, M.M., Application of the Method of Optimised Images to Steady Three-Dimensional Conduction Problems, ASME Paper 84-WA/HT-110, pp. 1-11, 1984.
17. Krane, Matthew John M., 1991, Constriction Resistance in Rectangular Bodies, ASME, J. Electronic Packing, 113, pp. 392-396, 1991.
18. Song, S., Lee, S. and Au, V., Closed-Form Equation for Thermal Constriction/Spreading Resistances with Variable Resistance Boundary Condition, Proceedings of the 1994 IEPS Conference, pp. 111-121, 1994.
19. Lee, S., Song, S. and Au, V., Constriction/Spreading Resistance Model for Electronic Packaging, Proceedings of the 4th ASME/JSME thermal Engineering Joint Conference, 4, pp. 199-206, 1995.
20. Yovanovich, M.M., Culham, J.R. and Teertstra, P., Analytical Modeling of Spreading Resistance in Flux Tubes, Half Spaces, and Compound Disks, IEEE Transactions on Components, Packaging, and Manufacturing Technology-Part A, 21, pp. 168-176, 1998.
21. Yovanovich, M.M., Muzychka, Y.S. and Culham, J.R., Spreading Resistance of Isoflux Rectangles and Strips on Compound Flux Channel, J. Thermophysics and Heat Transfer, 13, pp. 495-500, 1999
22. Yovanovich, M.M., Thermal Resistances of Circular Source on Finite Circular Cylinder with Side and End Cooling, ASME, J. Electronic Packing, 125, pp. 169-177, 2003.
23. Muzychka, Y.S., Yovanovich, M.M. and Culham, J.R., Influence of Geometry and Edge Cooling on Thermal Spreading Resistance, J. Thermophysics and Heat Transfer, 20, pp. 247-255, 2006.
24. Muzychka, Y.S., Culham, J.R. and Yovanovich, M.M., Thermal Spreading Resistance of Eccentric Heat Sources on Rectangular Flux Channels, ASME, J. Electronic Packing, 125, pp. 178-185, 2003.
25. Lam, T.T., Fischer, W.D. and Cabral P.S., Analysis of Thermal Resistance of Orthotropic Materials Used for Heat Spreaders, J. Thermophysics and Heat Transfer, vol. 18, No.2, pp. 203-208, 2004.
26. Ma, H.B. and Peterson, G.P., The Influence of the Thermal Conductivity on the Heat Transfer Performance in a Heat Sink,” ASME, J. Electronic Packing, 124, pp. 1-6, 2002.
27. Borgmeyer, B.V. and Ma, H.B., Heat Spreading Analysis of a heat Sink Base Embedded with a Heat Pipe, Proceedings of IPACK2005, ASME InterPACK ’05, pp. 1-4, 2005.
28. Kline, S.J. and McClintock, F.A., Describing uncertainties in single sample experiments, Mechanical Engineering, 75, pp. 3-8, 1953.
29. Air Movement and Control Association International Inc. and American Society of Heating, Refrigerating and Air Conditioning Engineers, Inc., Laboratory Method of Testing Fans for Aerodynamic Performance Rating, 1999.
30. Yagi, S. and Kunii, D., Studies on effective thermal conductivities in packed beds, AIChE Journal, Vol. 3, No. 3, pp. 373-381, 1957.
31. Japan Association for Heat Pipes (JAHP), Jitsuyou Heat Pipe, second ed. (in Japanese), Nikkan Kogyo Shimbun, Ltd., Tokyo, 2001.
32. Issa, R.I., Solution of Implicitly Discretized Fluid Flow Equation by Operator Splitting, J. Comput. Phys., vol. 62, pp.40-65, 1986.
33. Maxwell, J.C., A Treatise on Electricity and Magnetism, vol. I, 3rd edition, Dover Publications, 1954.
34. Chi, S. W., Heat Pipe Theory and Practice, Hemisphere, Washington, D.C., 1976.
35. Yimim Xuan, Yuping Hong and Qiang Li, Investigation on transient behavior of flat plate heat pipes, Experimental Thermal and Fluid Science, vol.28, pp. 249-255, 2004.
36. Cooper, M.G., Saturation nucleate pool boiling - a simple correlation, Int. Chem. Engng. Symp. Ser. 86, pp. 785-792, 1984.
37. Gorenflo, D., Sokol, P. and Caplanis, S., Pool boiling heat transfer from single plain tubes to various hydrocarbons, Int. J. Refrig., 13, pp. 286-292, 1990.