Constructal design of multi-scale diamond-shaped pin fins cooled by mixed convection

Abstract

In the present paper, the heat transfer density rate from a multi-scale diamond-shaped pin fins cooled by mixed convection is maximized under fixed pressure drop based on constructal design. Instead of Richardson number, the ratio of Rayleigh number to Bejan number (Ra/Be) arises due to the imposed pressure drop. Two sizes of diamond-shaped pin fins are placed in a fixed volume in vertical cross flow. The smaller fins are placed in the entrance region of the unused fluid between the larger fins with the same leading-edge of the larger fins. All the larger and the smaller fins are kept at constant temperature. The equations of the buoyancy-pressure driven flow are solved by finite volume method (FVM) for steady, two-dimensional, laminar, and incompressible flow. The leading-edge angles of the larger and the smaller fins are varied from (30°) to (60°). Rayleigh number is kept constant at (Ra = 105), the range of Bejan number is (104 ≤ Be ≤ 106), the range of the ratio of Rayleigh number to Bejan number is (0.1 ≤ Ra/Be ≤ 10), and the air is used as a working fluid with (Pr = 0.71). The heat transfer density is maximized twice, the first maximization is done for the larger fins in the absence of the smaller fins, and the second maximization is done with the presence of the smaller fins at constant leading-edge angle of the larger fins (λ = 30°). The optimal spacing between the larger fins is found in the first maximization, and the optimal height of the smaller fins is found in the second maximization. In the first maximization, the design indicated that the optimal spacing between the larger fins is constant for all fin angles (λ) at constant (Ra/Be). And based on this indication, the number of fins installed in fixed area must be reduced as (λ) increases. As well as, the results show that the heat transfer density can be maximized twice for all the studied angles of the smaller fins (30°, 45°, and 60°), and for all ratios (Ra/Be = 0.1, 1, and 10). Also, the highest value of the second maximized heat transfer density is found at the angle of the smaller fins (λ1 = 30°) for all ratios (Ra/Be).