本文應用微分轉換法模擬垂直通道內混合對流的流場和溫度場，透過改變通道的曲率半徑，可以模擬不同幾何形狀的垂直通道，且同時考慮熱輻射的效應可以更近一步貼近現實的狀況。此外，以熱力學第二定律的觀點，討論熱力系統中經常存在的不可逆性，本文將利用熵增量來進行不可逆性程度的衡量，並且利用比贊數來評估不可逆性是由摩擦或熱傳所主導。
用於求解本問題的微分轉換法有別於積分轉換法，微分轉換法可用於求解非線性問題，這是拉普拉斯轉換法和傅立葉轉換法無法達到的。為了驗證微分轉換法可以用於解混合對流之問題，先將垂直通道之幾何近似於平板並與先前的研究進行比對，結果無論是在速度場、溫度場或者是其他無因次參數上都相符。
透過改變無因次混合對流參數(Ξ)可以模擬不同浮力效應，當無因次混合對流參數變大時接近高溫壁面的速度會明顯增快，而靠近低溫壁面的流速則會下降。布林克曼數(Br)是用來表示黏性耗散之強度，在黏性耗散效應越強時，流體溫度上升且速度加快，熵增數在高溫及低溫的壁面上也都會增大。無因次幾何參數(b)利用改變內半徑和通道的比值來描述垂直通道之曲率，改變通道之幾何對壁面上的溫度沒有明顯影響，因此在壁面上無因次局部紐賽數沒有明顯差異，但通道彎曲程度較大，則速度會比較快，而且熵增數會在低溫壁面上大於彎曲程度較小者。熱輻射參數(N_r)用於描述熱輻射效應的強度，當熱輻射參數變大時可以使得溫度分佈較平緩，同時讓速度整體下降。
綜合以上，主要會使不可逆性增大的關鍵為黏性耗散效應，同時較大的黏性耗散也會使溫度有較大的差異。無因次混合對流參數則是可以讓流體的流場有明顯的變化。熱輻射對整體有一定影響力，但是相較於其他參數，改變熱輻射參數能影響的幅度相對有限。

In this study, differential transformation method is used to investigate the velocity and temperature profile of mixed convection in a vertical channel. The geometry of the channel is not limited to parallel plates, different curvatures are discussed in this paper. With the effects of viscous dissipation and thermal radiation taken into consideration. Based on second law of thermodynamic, it is well known that irreversibility existed in thermodynamic systems. In order to estimate such irreversibility, entropy generation number is used and Bejan number is applied to evaluate either heat transfer or friction dominated the irreversibility.
The result shows that with bigger mixed convection dimensionless parameter, the velocity of the fluid can increase near the hot wall and decrease at the cool wall. Brinkman number is correlated to velocity and temperature, which caused more entropy generated. Also, smaller dimensionless geometry parameter has higher velocity in the middle and result in higher irreversibility. Last but not least, thermal radiation parameter can reduce the velocity and smoothen the temperature profile.
In conclusion, thermal radiation is an important way of heat transfer even it only makes a small difference compare to other factors. The one factor that can significantly shift the velocity profile is mixed convection dimensionless parameter and Brinkman number can affect temperature profile most dramatically. As for irreversibility, increasing Brinkman number can considerably increases entropy generation number.

[1] L. N. Tao, "On Combined Free and Forced Convection in Channels," Journal of Heat Transfer, vol. 82, no. 3, pp. 233-238, 1960, doi: 10.1115/1.3679915.
[2] W. Aung and G. Worku, "Developing Flow and Flow Reversal in a Vertical Channel With Asymmetric Wall Temperatures," Journal of Heat Transfer, vol. 108, no. 2, pp. 299-304, 1986, doi: 10.1115/1.3246919.
[3] W. Aung and G. Worku, "Theory of Fully Developed, Combined Convection Including Flow Reversal," Journal of Heat Transfer, vol. 108, no. 2, pp. 485-488, 1986, doi: 10.1115/1.3246958.
[4] T. T. Hamadah and R. A. Wirtz, "Analysis of Laminar Fully Developed Mixed Convection in a Vertical Channel With Opposing Buoyancy," Journal of Heat Transfer, vol. 113, no. 2, pp. 507-510, 1991, doi: 10.1115/1.2910593.
[5] C. o.-K. Chen, H.-Y. Lai, and C.-C. Liu, "Numerical analysis of entropy generation in mixed convection flow with viscous dissipation effects in vertical channel," International Communications in Heat and Mass Transfer, vol. 38, no. 3, pp. 285-290, 2011/03/01/ 2011, doi: https://doi.org/10.1016/j.icheatmasstransfer.2010.12.016.
[6] C.-K. Chen and S.-H. Ho, "Application of differential transformation to eigenvalue problems," Applied Mathematics and Computation, vol. 79, no. 2, pp. 173-188, 1996/10/01/ 1996, doi: https://doi.org/10.1016/0096-3003(95)00253-7.
[7] J. S. Chiou and J. R. Tzeng, "Application of the Taylor Transform to Nonlinear Vibration Problems," Journal of Vibration and Acoustics, vol. 118, no. 1, pp. 83-87, 1996, doi: 10.1115/1.2889639.
[8] C. L. Chen and Y. C. Liu, "Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method," Journal of Optimization Theory and Applications, vol. 99, no. 1, pp. 23-35, 1998/10/01 1998, doi: 10.1023/A:1021791909142.
[9] S. H. Ho and C. o. K. Chen, "Analysis of general elastically end restrained non-uniform beams using differential transform," Applied Mathematical Modelling, vol. 22, no. 4, pp. 219-234, 1998/04/01/ 1998, doi: https://doi.org/10.1016/S0307-904X(98)10002-1.
[10] 朱心平, "應用微分轉換法於強非線性物理系統之研究," 碩士論文 學術論文, 機械工程學系, 國立成功大學, 2004. [Online]. Available: https://hdl.handle.net/11296/6j83cc
[11] C. o.-K. Chen, H. Y. Lai, and C.-C. Liu, "Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam," Microsystem Technologies, vol. 15, no. 6, pp. 813-820, 2009/06/01 2009, doi: 10.1007/s00542-009-0834-1.
[12] Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, "A multi-step differential transform method and application to non-chaotic or chaotic systems," Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1462-1472, 2010/02/01/ 2010, doi: https://doi.org/10.1016/j.camwa.2009.11.005.
[13] A. Barletta, "Laminar mixed convection with viscous dissipation in a vertical channel," International Journal of Heat and Mass Transfer, vol. 41, no. 22, pp. 3501-3513, 1998/11/01/ 1998, doi: https://doi.org/10.1016/S0017-9310(98)00074-X.
[14] C.-C. Liu and C.-Y. Lo, "Numerical analysis of entropy generation in mixed-convection MHD flow in vertical channel," International Communications in Heat and Mass Transfer, vol. 39, no. 9, pp. 1354-1359, 2012, doi: 10.1016/j.icheatmasstransfer.2012.08.001.
[15] C. o.-K. Chen, B.-S. Chen, and C.-C. Liu, "Heat transfer and entropy generation in fully-developed mixed convection nanofluid flow in vertical channel," International Journal of Heat and Mass Transfer, vol. 79, pp. 750-758, 2014/12/01/ 2014, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2014.08.078.
[16] C. o.-K. Chen, B.-S. Chen, and C.-C. Liu, "Entropy generation in mixed convection magnetohydrodynamic nanofluid flow in vertical channel," International Journal of Heat and Mass Transfer, vol. 91, pp. 1026-1033, 2015/12/01/ 2015, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2015.08.042.
[17] A. Bejan, Entropy generation through heat and fluid flow. Wiley New York, 1982.
[18] 趙家奎, 微分變換及其在電路中的應用. 華中理工大學出版社, 1988.
[19] M.-J. Jang, C.-L. Chen, and Y.-C. Liu, "Two-dimensional differential transform for partial differential equations," Applied Mathematics and Computation, vol. 121, no. 2, pp. 261-270, 2001/06/15/ 2001, doi: https://doi.org/10.1016/S0096-3003(99)00293-3.
[20] L.-T. Yu and C. o.-K. Chen, "Application of Taylor transformation to optimize rectangular fins with variable thermal parameters," Applied Mathematical Modelling, vol. 22, no. 1, pp. 11-21, 1998/01/01/ 1998, doi: https://doi.org/10.1016/S0307-904X(98)00004-3.
[21] 陳柏佑, "混合微分轉換/有限差分法在非線性暫態熱傳問題之研究," 碩士論文 學術論文, 機械工程學系, 國立成功大學, 2005. [Online]. Available: https://thesis.lib.ncku.edu.tw/thesis/detail/ae73278b94744619d3320694f723a795/
[22] A. Bejan, "Second-Law Analysis in Heat Transfer and Thermal Design," in Advances in Heat Transfer, vol. 15, J. P. Hartnett and T. F. Irvine Eds.: Elsevier, 1982, pp. 1-58.
[23] A. Bejan, G. Tsatsaronis, and M. J. Moran, Thermal design and optimization. John Wiley & Sons, 1995.
[24] A. Bejan, "Entropy generation minimization: The new thermodynamics of finite‐size devices and finite‐time processes," Journal of Applied Physics, vol. 79, no. 3, pp. 1191-1218, 1996, doi: 10.1063/1.362674.
[25] A. Bejan, "The method of entropy generation minimization," in Energy and the Environment: Springer, 1999, pp. 11-22.
[26] A. Datta, "Entropy Generation in a Confined Laminar Diffusion Flame," Combustion Science and Technology, vol. 159, no. 1, pp. 39-56, 2000/10/01 2000, doi: 10.1080/00102200008935776.
[27] M. Famouri and K. Hooman, "Entropy generation for natural convection by heated partitions in a cavity," International Communications in Heat and Mass Transfer, vol. 35, no. 4, pp. 492-502, 2008/04/01/ 2008, doi: https://doi.org/10.1016/j.icheatmasstransfer.2007.09.009.
[28] M. H. Matin, R. Hosseini, M. Simiari, and P. Jahangiri, "Entropy generation minimization of nanofluid flow in a MHD channel considering thermal radiation effect," Mechanics, vol. 19, no. 4, pp. 445-450, 2013.
[29] S. Bhattacharjee and W. L. Grosshandler, "The formation of a wall jet near a high temperature wall under microgravity environment," 1988.
[30] S. Paoletti, F. Rispoli, and E. Sciubba, "Calculation of exergetic losses in compact heat exchanger passages," in Asme Aes, 1989, vol. 10, no. 2, pp. 21-29.
[31] A. Bejan and E. Sciubba, "The optimal spacing of parallel plates cooled by forced convection," International Journal of Heat and Mass Transfer, vol. 35, no. 12, pp. 3259-3264, 1992/12/01/ 1992, doi: https://doi.org/10.1016/0017-9310(92)90213-C.
[32] A. Bejan, Advanced Engineering Thermodynamics, Forth Edition. 2016, pp. 1-746.
[33] M. Bouabid, M. Magherbi, A. McHirgui, and A. Brahim, "Entropy Generation at Natural Convection in an Inclined Rectangular Cavity," Entropy [electronic only], vol. 5, 12/01 2011, doi: 10.3390/e13051020.
[34] M. Rashid, M. I. Khan, T. Hayat, M. I. Khan, and A. Alsaedi, "Entropy generation in flow of ferromagnetic liquid with nonlinear radiation and slip condition," Journal of Molecular Liquids, vol. 276, pp. 441-452, 2019/02/15/ 2019, doi: https://doi.org/10.1016/j.molliq.2018.11.148.
[35] A. Barletta, "Combined forced and free convection with viscous dissipation in a vertical circular duct," International Journal of Heat and Mass Transfer, vol. 42, no. 12, pp. 2243-2253, 1999/06/01/ 1999, doi: https://doi.org/10.1016/S0017-9310(98)00343-3.
[36] 張桂豪, "波形板上對流熱傳與熵增之研究," 博士 博士論文, 機械工程學系, 國立成功大學, 2011. [Online]. Available: https://hdl.handle.net/11296/hqqv48
[37] S. Rosseland, Theoretical Astrophysics: Atomic Theory and the Analysis of Stellar Atmospheres and Envelopes (no. 第 1 卷). Clarendon Press, 1936.
[38] E. Magyari and A. Pantokratoras, "Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flows," International Communications in Heat and Mass Transfer, vol. 38, no. 5, pp. 554-556, 2011/05/01/ 2011, doi: https://doi.org/10.1016/j.icheatmasstransfer.2011.03.006.
[39] K. Das, "Radiation and melting effects on MHD boundary layer flow over a moving surface," Ain Shams Engineering Journal, vol. 5, 05/01 2014, doi: 10.1016/j.asej.2014.04.008.
[40] M. Gnaneswara Reddy, "Influence of thermal radiation, viscous dissipation and Hall current on MHD convection flow over a stretched vertical flat plate," Ain Shams Engineering Journal, vol. 5, no. 1, pp. 169-175, 2014/03/01/ 2014, doi: https://doi.org/10.1016/j.asej.2013.08.003.
[41] A. Sahoo and R. Nandkeolyar, "Entropy generation and dissipative heat transfer analysis of mixed convective hydromagnetic flow of a Casson nanofluid with thermal radiation and Hall current," Scientific Reports, vol. 11, no. 1, p. 3926, 2021/02/16 2021, doi: 10.1038/s41598-021-83124-0.
[42] M. Hatami, D. D. Ganji, and M. Sheikholeslami, Differential transformation method for mechanical engineering problems. Academic Press, 2016.
[43] A. Khatib, "Differential Transform Method for Differential Equations," Master Master Thesis, Palestine Polytechnic University, 2016.
[44] A. Bejan, Convection heat transfer. John wiley & sons, 2013.
[45] M. F. Modest and S. Mazumder, Radiative heat transfer. Academic press, 2021.
[46] L. Chen, C. Wu, and F. Sun, "Finite time thermodynamic optimization or entropy generation minimization of energy systems," 1999.