Analytical solution of the entransy dissipation of heat conduction process in isolated system
Authors: Wang Huan-Guang  , Wu Di  , Rao Zhong-Hao  ,   School of Electronic Power Engineering, China University of Mining and Technology, Xuzhou 221116, China
Published: December 2020

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.