Optimal concentrations in transport systems

Abstract

Many biological and man-made systems rely on transport systems for the distribution of material, for example matter and energy. Material transfer in these systems is determined by the flow rate and the concentration of material. While the most concentrated solutions offer the greatest potential in terms of material transfer, impedance typically increases with concentration, thus making them the most difficult to transport. We develop a general framework for describing systems for which impedance increases with concentration, and consider material flow in four different natural systems: blood flow in vertebrates, sugar transport in vascular plants and two modes of nectar drinking in birds and insects. The model provides a simple method for determining the optimum concentration c[subscript opt] in these systems. The model further suggests that the impedance at the optimum concentration μ[subscript opt] may be expressed in terms of the impedance of the pure (c = 0) carrier medium μ[subscript 0] as μ[subscript opt]∼2[superscript α]μ[subscript 0], where the power α is prescribed by the specific flow constraints, for example constant pressure for blood flow (α = 1) or constant work rate for certain nectar-drinking insects (α = 6). Comparing the model predictions with experimental data from more than 100 animal and plant species, we find that the simple model rationalizes the observed concentrations and impedances. The model provides a universal framework for studying flows impeded by concentration, and yields insight into optimization in engineered systems, such as traffic flow.