| A mathematical analysis was developed for non-equilibrium separation process problems. The analysis led to one-to-one correspondence between the equilibrium and the non-equilibrium models. The analysis was extended from single-stage, binary systems to multistage, multi-component systems; demonstrating solution uniqueness, and to the analysis of the problem of solution multiplicity. The solution uniqueness in the equilibrium model was rigorously demonstrated by Lucia in 1986, and as an extension to non-equilibrium case it was demonstrated that one-to-one correspondence exists between the equilibrium and the non-equilibrium problem, proving the solution uniqueness for the non-equilibrium problem. While the conceptual framework for formulating phase mass and energy balances and equilibrium relationships is the same in equilibrium and non-equilibrium models, it was established that there are clear differences in the way the equations are used. In the equilibrium model formulation, the balance equations are written for the stage as a whole and the phases that exit the stage are assumed to be in equilibrium with each other. However for the non-equilibrium model, the material and energy balance equations were written for each phase allowing for mass and energy transfer across the interface. The temperature, pressure, and chemical potential are equal across the interface. The Maxwell-Stefan equations were applied to the film model, demonstrating that there is a one-to-one correspondence between the bulk and the interface compositions, in non-equilibrium, single-stage problem, and that the imposition of the mass transfer equations does not add to the solution multiplicity. The multistage, non-equilibrium problem was analyzed, too, and it was demonstrated that all multiplicity in results is due to multiplicity that occurs in the phase calculations in the interface. |
