A mathematical analysis for constructal design of tree flow networks under unsteady flow

Tree flow networks play an important role in both natural and man-made systems. In an effort to develop a deeper understanding of the optimal design of these networks, we have developed a simple analytical approach to deal with steady and unsteady flows. As a result, optimal relations for the homothetic ratio of tube sizes and optimal angles between daughter tubes are obtained. The obtained optimum homothetic ratios satisfy the criterion of the minimization procedure of flow impedance based on geometry and the svelteness ratio. The robustness, accuracy and convergence of model are also proved mathematically in order to validate the results.