For the two-dimensional Brusselator model, this paper obtains the steady-state solution of Turing patterns by using a time-evolution algorithm. Based on the research of the Brusselator model, we propose a new time-evolution algorithm for the boundary value problem of partial differential equations (PDEs). By introducing a virtual time parameter, the original boundary value problem is transformed into an initial value problem. Combining the finite difference method and the implicit iterative algorithm, the system evolution is simulated to approximate the steady-state solution, that is, the solution of the boundary value problem of the original equation set. The research shows that compared with the traditional Jacobi iterative method and Gauss-Seidel iterative method, the time-evolution algorithm exhibits stronger stability and convergence in complex non-linear problems. Further, this method is extended to the solution of Poisson's equation and Helmholtz equation. The results show that although the accuracy and efficiency of the time-evolution algorithm in general linear problems are slightly inferior to those of the Gauss-Seidel iterative method, in specific scenarios such as the Helmholtz equation with large wave numbers, its convergence speed and error accuracy are better. The time-evolution algorithm provides a new solution idea for the boundary value problem of non-linear PDEs. In the future, its performance can be further improved by combining high-order difference schemes and parallel computing strategies.

In response to the increasing demand for accurate prediction of gas thermal properties in engineering applications, this paper compares and studies the application effects of three state equations, namely van der Waals equation, Redlich-Kwong equation and Peng-Robinson equation, in the calculation of thermal properties of common gases such as nitrogen. The advantages and disadvantages of different state equations in the calculation of thermal properties of different gases are discussed. In addition, we propose a modified optimization method for the parameter b of the Redlich-Kwong equation. By adjusting the value of b, the calculated results of the modified Redlich-Kwong equation have significantly improved their consistency with the reference data, providing new ideas and methods for theoretical research and industrial applications.

Bénard Convection is a natural convection phenomenon in the plane horizontal layer of fluid heated from below. The study of this phenomenon is of great significance to deeply understand the physical image of dissipative structure and the motion of fluid in chaotic system. In order to study the conditions and physical properties of its generation, this paper establishes a fluid model, derives the characteristic equations satisfied by physical quantities under fixed boundary conditions from the basic set of control equations, uses an important dimensionless number, the Rayleigh number R_a, to characterize whether convection occurs or not, and deduces a critical value of convection from the static mutation to the stable convection R_cl=1708. The theoretical value is obtained and simulated using COMSOL Multiphysics^® to verify that it meets the theoretical prediction and the motion characteristics of the liquid are further analyzed by deriving motion animations, In order to investigate the second instability of convection when the temperature gradient is large, Lorentz equations theory is introduced, and the second critical value R_c2 can be solved by substituting the parameter, and under the assumption of the present paper, R_c2=46177. Accordingly, the conditions for the generation of Benard convection are R_c1＜R_a＜R_c2.

The article focuses on optimizing heat transfer efficiency in heat pipes using entransy theory. It begins by contrasting the application of traditional entropy theory and entransy theory in analyzing heat transfer efficiency. The paper then provides a detailed introduction to entransy theory, including its physical significance and application as an optimization function in engineering. Subsequently, through COMSOL simulation experiments, the practical application of entransy theory in optimizing heat pipe design is demonstrated, validating the theory's correctness. Relationship between the temperature difference and the entransy dissipation rate and a number of parameters, such as the outer diameter of the heat pipe and the thickness of the core layer, is compared. The article provides a new perspective for the study of heat transfer models, such as the heat pipe, and the optimization of the design of the production.

This paper gives an introduction on the basic concepts of quantum heat engine and quantum thermodynamics while exploring relevant properties of the quantum heat engine system based on coupled harmonic oscillators. The authors drew references from classical thermodynamics and introduces Bogoliubov transformation so as to diagonalize the potential term in coupled quantum harmonic oscillators. The authors also based the quantum heat engine system on said oscillators and gives a detailed demonstration of acquiring the efficiency of Otto cycle utilized by this system, arriving at the conclusion that there is a unifying and cohesive factor that interconnects certain corresponding values between classical and quantum heat engines.