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## Analytical solution of the entransy dissipation of heat conduction process in isolated system

Wang Huan-Guang, Wu Di, Rao Zhong-Hao
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• #### 摘要

(火积)耗散与熵增均可以作为传热不可逆性的度量, 当前(火积)理论的反对者认为(火积)是不必要的. 为说明(火积)的必要性, 从有效性的角度进行了论证, 即在描述传热过程不可逆性的变化上, (火积)的严格解析解存在, 而熵的严格解析解难以得到. 本文构建了孤立系内的一维及多维热传导模型, 求解了温度及其梯度的级数型解析解, 将其代入(火积)耗散的求解式, 得到其最初的形式为一多重级数的多重积分, 交换积分与级数计算顺序, 并利用特征函数的正交性, 将(火积)耗散求解式中的积分运算求出, 并使级数的维数降低, 最终将其表示为一稳态项与一瞬态项加和的形式, 其极限与文献中的结果一致. 通过对孤立系内(火积)耗散解析解的求解可以得出: 由于热传导过程熵与(火积)的解析解求解难度不同, 在描述传热过程不可逆性变化上, (火积)更加有效; 对于孤立系内不同维数的热传导问题, 只要温度场解析解存在, (火积)耗散解析解均可以应用特征函数正交性求解得到.

#### Abstract

Entransy dissipation and entropy generation both can be used as measures of the irreversibilities of heat transfer problems. Nowadays those who oppose the entransy theory insist that the entransy is needless. In order to illustrate the necessity of the entransy theory, demonstration is made from the viewpoint of effectiveness which is based on the fact that when describing the variation of the irreversibility in the process of heat transfer, the exact analytical solution of the entransy dissipation exists, but that of the entropy generation is difficult to obtain. In this paper, one-dimensional (1D) and multi-dimensional heat conduction models within isolated systems are constructed, among which, the former is to illustrate the deriving process concisely, and the latter is to verify the universal existence of the analytical solution of entransy dissipations. In the 1D model, two bodies with the same geometrical and thermophysical properties but different initial temperatures transfer heat through the contacting surfaces; while in the three-dimensional (3D) model, the initial condition is arbitrary. According to the literature, the primary expression of the total entransy dissipation is derived when substituting the series-typed expression of temperature gradient into the universal calculating equation, which is in the form of a multi integral of a multi series. To reduce such an expression to the simplest form without performing any integral calculation, the orders of the integral and the series are exchanged, and considering the independence between the concerning variables and functions, the multi integral calculation is simplified into the product of several 1D integrals, one relates to time and is easily solved, and the others are dependent on space, of which the dimension is reduced by using the inherent orthogonality of the characteristic functions. The ultimate solutions of the entransy dissipation for all the models are expressed as the summation of a stationary item and a transient item, and the former is consistent with the result obtained from the viewpoint of thermodynamics given by the literature, and the latter has the limitation of zero when time tends to infinity. In order to verify the correctness of the universal solution of the entransy dissipation, a concrete 2D heat transfer problem is constructed, in which four bodies transfer heat through connecting faces, of which the initial temperature is centrosymmetric in the isolated system, and uniform within each body. The analytical solution of the entransy dissipation to the 2D problem has the same tendency and limitation as those of the 1D model, but varies faster on condition that the thermopysical properties of the bodies in both models are identical. In order to make comparison, the calculating equation of the entropy generation for the 1D model is also derived, which has the form of the integral of the logarithm of the series-typed temperature, and such an integral is hard to solve mathematically, which suggests the limitation of entropy when describing the variation of irreversibility from the viewpoint of heat transfer instead of thermodynamics. Through the derivation and comparison shown in this paper, the following conclusions are reached: owing to the differences in complicity between obtaining analytical solutions of the entransy dissipation and those of the entropy generation, the former is more effective when describing variation of the irreversibility in heat transfer process; for heat transfer problems of different dimensions in isolated systems, analytical solutions of the entransy dissipation are expected to be obtained when the precondition that the analytical solutions of the temperature exist, is satisfied.

#### 作者及机构信息

###### 通信作者: 饶中浩, raozhonghao@cumt.edu.cn
• 基金项目: 中央高校基本科研业务费(批准号: 2014QNB07)资助的课题.

#### Authors and contacts

###### Corresponding author: Rao Zhong-Hao, raozhonghao@cumt.edu.cn
• Funds: Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2014QNB07).

#### 参考文献

 [1] Guo Z Y, Zhu H Y, Liang X G 2007 Int. J. Heat Mass. Trans. 50 2545 [2] Chen L G, Wei S H, Sun F R 2008 J. Phys. D: Appl. Phys. 41 1 [3] Feng H J, Chen L G, Sun F R 2012 Sci China: Tech. Sci. 55 779 [4] Guo J F, Cheng L, Xu M T 2009 Chin. Sci. Bull. 54 2708 [5] Chen Q, Ren J X 2008 Chin. Sci. Bull. 53 3753 [6] Chen Q, Fu R H, Xu Y C 2015 Appl. Energ. 139 81 [7] Cheng X T, Zhang Q Z, Xu X H, Liang X G 2013 Chin. Phys. B 22 020503 [8] Cheng X T, Liang X G 2014 Acta Phys. Sin. 63 190501 (in Chinese) [程雪涛, 梁新刚 2014 物理学报 63 190501] [9] Tao Y B, He Y L, Liu Y K 2014 J. Engineer. Thermophys. 35 973 (in Chinese) [陶于兵, 何雅玲, 刘永坤2014工程热物理学报 35 973] [10] Zhang L, Liu X H, Jiang Y 2013 Energy 53 332 [11] Feng H J, Chen L G, Xie Z H, Sun F R 2014 Int. Commun. Heat Mass 52 26 [12] Zhang Y, Chen Q, Zhang Y P, Wang X 2013 Int. J. Heat Mass. Trans. 65 265 [13] Kim K H, Kim K J 2015 Int. J. Heat Mass. Trans. 84 80 [14] Zhang T, Liu X H, Jiang Y 2013 Energ. Convers. Manag. 75 51 [15] Cheng X, Liang X, Guo Z 2011 Chin. Sci. Bull. 56 847 [16] Cheng X T, Liang X G 2012 Chin. Sci. Bull. 57 3244 [17] Chen Q, Liang X G, Guo Z Y 2013 Int. J. Heat Mass. Trans. 63 65 [18] Grazzini G, Borchiellini R, Lucia U 2013 J. Non-equil. Thermody 38 259 [19] Herwig H 2014 J. Heat Trans. T. ASME. 136 045501 [20] Bejan A 2014 Ind. Eng. Chem. Res. 53 18352 [21] Oliveira S D R, Milanez L F 2014 Int. J. Heat Mass. Trans. 79 518 [22] Sekulic D P, Sciubba E, Moran M J 2015 Energy 80 251 [23] Awad M M 2014 J. Heat Trans. T. ASME. 136 095502 [24] Özisik M N 1993 Heat Conduction (2nd Ed.) (Hoboken: John & Sons. Inc.)

#### 施引文献

•  [1] Guo Z Y, Zhu H Y, Liang X G 2007 Int. J. Heat Mass. Trans. 50 2545 [2] Chen L G, Wei S H, Sun F R 2008 J. Phys. D: Appl. Phys. 41 1 [3] Feng H J, Chen L G, Sun F R 2012 Sci China: Tech. Sci. 55 779 [4] Guo J F, Cheng L, Xu M T 2009 Chin. Sci. Bull. 54 2708 [5] Chen Q, Ren J X 2008 Chin. Sci. Bull. 53 3753 [6] Chen Q, Fu R H, Xu Y C 2015 Appl. Energ. 139 81 [7] Cheng X T, Zhang Q Z, Xu X H, Liang X G 2013 Chin. Phys. B 22 020503 [8] Cheng X T, Liang X G 2014 Acta Phys. Sin. 63 190501 (in Chinese) [程雪涛, 梁新刚 2014 物理学报 63 190501] [9] Tao Y B, He Y L, Liu Y K 2014 J. Engineer. Thermophys. 35 973 (in Chinese) [陶于兵, 何雅玲, 刘永坤2014工程热物理学报 35 973] [10] Zhang L, Liu X H, Jiang Y 2013 Energy 53 332 [11] Feng H J, Chen L G, Xie Z H, Sun F R 2014 Int. Commun. Heat Mass 52 26 [12] Zhang Y, Chen Q, Zhang Y P, Wang X 2013 Int. J. Heat Mass. Trans. 65 265 [13] Kim K H, Kim K J 2015 Int. J. Heat Mass. Trans. 84 80 [14] Zhang T, Liu X H, Jiang Y 2013 Energ. Convers. Manag. 75 51 [15] Cheng X, Liang X, Guo Z 2011 Chin. Sci. Bull. 56 847 [16] Cheng X T, Liang X G 2012 Chin. Sci. Bull. 57 3244 [17] Chen Q, Liang X G, Guo Z Y 2013 Int. J. Heat Mass. Trans. 63 65 [18] Grazzini G, Borchiellini R, Lucia U 2013 J. Non-equil. Thermody 38 259 [19] Herwig H 2014 J. Heat Trans. T. ASME. 136 045501 [20] Bejan A 2014 Ind. Eng. Chem. Res. 53 18352 [21] Oliveira S D R, Milanez L F 2014 Int. J. Heat Mass. Trans. 79 518 [22] Sekulic D P, Sciubba E, Moran M J 2015 Energy 80 251 [23] Awad M M 2014 J. Heat Trans. T. ASME. 136 095502 [24] Özisik M N 1993 Heat Conduction (2nd Ed.) (Hoboken: John & Sons. Inc.)
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##### 出版历程
• 收稿日期:  2015-05-29
• 修回日期:  2015-08-14
• 刊出日期:  2015-12-05

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