Solution enrichment using concentration gradients and relaxation times control: A mathematical model that takes into account cross-diffusion effects
Keywords:
mathematical models, mass transport processes, solution enrichment, perturbation theory, cross diffusion effects, systems of partial differential equations, modal analysisAbstract
Mathematical models are used to investigate mass transport processes in a solution of many components. The solution is in a chamber limited by two semipermeable barriers, proximal and distal. Through these barriers, a constant flow of solvent is imposed, which drags the solutes towards the distal barrier and produces diffusive counter flows. An extraction region adjacent to the distal barrier is defined and an inequality is established, related to the temporal behavior of the number of moles of the solutes in the extraction region, such that if it is met, it ensures that the relative enrichment function presents a transient phase with an overshoot relative to its final stationary value. Taking into due account cross-diffusion effects, two mathematical models of convection-diffusion processes are constructed: a three-dimensional model and a one-dimensional global model. The transport equations are linearized and solved in the framework of both models. The solutions of the mass transport equations are simplified by a procedure of regular perturbations when the cross effects are weak. Based on the solutions of the equations of both mathematical models, the conditions which ensures that an extraction at an appropriately selected moment of the transient improves the relative enrichment with respect to its steady state value are investigated.
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