Constructal entropy generation rate minimization for X-shaped vascular networks

Based on constructal theory, an X-shaped vascular network model in a rectangular area is built in this paper. Subjected to the constraints of the total volume of the tubes and space occupied by the rectangular area, the model is optimized by taking the minimizations of the dimensionless entropy generation rate and dimensionless entropy generation ratio, respectively. For the specified heat flow per unit length, the results show that there exist optimal ratios of the elemental diameters (22/3) and lengths of the tubes which lead to the minimum dimensionless entropy generation rate of the X-shaped vascular network. They are obviously different from the optimal results of the H-shaped vascular network. Compared the performances of the X-shaped vascular network with those of the corresponding order of the H-shaped vascular network, the dimensionless entropy generation rate and the dimensionless entropy generation ratio are reduced by 30.78% and 31.55% for the fixed mass flow rate, respectively. However, these reductions are small for the fixed pumping power. Moreover, the performance of the X-shaped vascular network can further be improved by removing the perpendicular constraint of the tubes. The results obtained can provide some new guidelines for the designs of the vasculatures.