Optimization of ramified absorber networks doing desalination

An iterated function system is used to generate fractal-like ramified graph networks of absorbers, which are optimized for desalination performance. The diffusion equation is solved for the boundary case of constant pressure difference at the absorbers and a constant ambient salt concentration far from the absorbers, while constraining both the total length of the network and the total area of the absorbers to be constant as functions of generation G. A linearized form of the solution was put in dimensionless form which depends only on a dimensionless membrane resistance, a dimensionless inverse svelteness ratio, and G. For each of the first nine generations G=2,...,10, the optimal graph shapes were obtained. Total water production rate increases parabolically as a function of generation, with a maximum at G=7. Total water production rate is shown to be approximately linearly related to the power consumed, for a fixed generation. Branching ratios which are optimal for desalination asymptote decreasingly to r=0.510 for large G, while branching angles which are optimal for desalination asymptote decreasingly to 1.17 radians. Asymmetric graphs were found to be less efficient for desalination than symmetric graphs. The geometry which is optimal for desalination does not depend strongly on the dimensionless parameters, but the optimal water production does. The optimal generation was found to increase with the inverse svelteness ratio.