Numerical Analysis of a Tree-Shaped Cooling Structure for a 2-D Slab: A Validation of a “Constructally Optimal” Configuration

A 2-d rectangular slab, internally cooled by circular tubes of different diameters, is used as a benchmark to perform a numerical thermal optimization procedure on a heat transfer configuration obtained by “constructal” considerations. Three different geometric arrangements are investigated: in the first one, all cooling holes have the same diameter; in the second, there are two hierarchies of holes with two different diameters; in the third, yet another series of smaller holes is added. To assist the user in systematically assembling multiple configurations, a custom user interface utility for generating constructal geometries has been developed. Thermal boundary conditions are assigned as if the slab were the axial cross-section of an elongated prismatic body, and as if the ducts were internally cooled by forced convection. For each case (single diameter of the cooling holes, 2 and 3 different diameters) a series of numerical simulations has been run to assess and compare the performance of different values of the geometrical parameters (location of the holes, slab aspect ratio, etc.). The best performing configurations for each case have then been compared with those identified for the other cases, the objective function being the average temperature of the slab “volume”. All results consistently indicate that, under the specified design conditions, the configuration obtained by a simple constructal optimization is the best performing one at each configuration level (1, 2 or 3 series of cooling holes).