A new study area of constructal theory is presented by applying constructal design on multiple heat sources which connected to and cooled in a square-shaped fin. Analyzing the optimal location of the heat sources in the fin for several configurations is the primary goal of this study. According to the constructal theory, favorable distribution of the heat sources is achieved by minimizing the global thermal resistance of the fin body. Since the heat sources might be contemplated as heat-generating electronic bodies or hot-water pipes, two boundary conditions of uniform heat flux and uniform temperature are considered for the heat sources. The minimization of the thermal resistance in the uniform heat flux and uniform temperature cases leads to the minimization of the hot spot temperature and the maximization of the total heat removal, respectively. Temperature fields in the square-shaped fin are calculated using a finite element approach of MATLAB PDE Toolbox. The optimized configurations are numerically compared with each other. It is shown that the most commonly used configurations of the heat sources in a cooling industry are unfavorable and must be avoided. Favorable configurations such as triangular arrangement of heat sources are proposed to replace the unfavorable configurations.

A new algorithm is proposed based on semi-analytical methods to solve the conjugate heat transfer problems. In this respect, a problem of conjugate forced-convective flow over a heat-conducting plate is modeled and the integro-differential equation occurring in the problem is solved by two lately-proposed approaches, Adomian decomposition method and differential transform method. The solution of the governing integro-differential equation for temperature distribution of the plate is handled more easily and accurately by implementing Adomian decomposition method/differential transform method rather than other traditional methods such as perturbation method. A numerical approach is also performed via finite volume method to examine the validity of the results for temperature distribution of the plate obtained by Adomian decomposition method/differential transform method. It is shown that the expressions for the temperature distribution in the plate obtained from the two methods, Adomian decomposition method and differential transform method, are the same and show closer agreement to the results calculated from numerical work in comparison with the expression obtained by perturbation method existed in the literature.

Optimization of a highly conductive insert embedded into a heated rectangular chip has been lately investigated. The role of the insert was to gather the heat current within the chip body and remove it to a minimum temperature heat sink. The central goal of this paper is to invoke several reconsiderations, which result in excessive reduction of the peak temperature in the heated chip in comparison with the lowest peak temperature existed in the archival literature. It is proved that for the configuration under study with its bottom surface receiving a heat flux, the branching patterns of the insert must be avoided, in case the appropriate revisions in the architecture (width, location and cross section area) of the simple branchless patterns are considered. An analytical solution for predicting the peak temperatures in the chip is also addressed. It is demonstrated that under the same volume fraction and thermal conductivity of the cooling insert, the peak temperature is reduced to 2.9℃, which is 94% below the lowest temperature existed in the archival literature, which was around 50℃.