Maximal Heat Transfer Density: Optimal Distribution of Discrete Heat Sources on Vertical Walls in Channels and Enclosures with Natural Convection

This paper is a review of current work on the optimal distribution of discrete heat sources cooled by laminar natural convection. The global objective is to maximize the global conductance between the wall and the fluid, or to minimize the hot-spot temperatures when the total heat generation rate and global system dimensions are specified. Three scenarios are investigated: (i) a large number of small heat sources mounted on a vertical wall facing a fluid, (i) a small number of finite-size heat sources mounted on the side wall in a two-dimensional enclosure, and (iii) a heated area on the wall of a vertical diverging or converging channel with chimney flow. For (i) and (ii), it is shown that the optimal distribution is not uniform (the sources are not equidistant), and that as the Rayleigh number increases the heat sources placed near the tip of a boundary layer should have zero spacings. In configuration (iii), the geometry is free to change in three directions: by varying the space between the walls, the distribution of heating along the walls, and the angle between the two walls. Numerical simulations cover the Rayleigh number range 105 ≤ RaH ≤ 107, where H is the channel height. Nonuniform wall heating is modeled as an isothermal patch of varying height, which is placed near the entrance of the channel, or near the exit. It is shown that for maximal heat transfer rate density it is better to place the heated sections at the channel entrance. The optimal angle between the two walls is approximately zero when RaH is large. The optimized spacing is of the same order of magnitude as the optimal spacing reported earlier for parallel isothermal walls. The robustness of flow architectures with optimized distribution of heat sources is discussed.