Numerical Solutions for Nonlinear, Nonideal SMB

By contrast, when the mass transfer resistances and/or axial dispersion are considered, there is no analytical solution for an SMB operated under nonlinear isotherm conditions. A numerical solution of the applicable mathematical model must be used instead to calculate the performance of the SMB, to simulate the influence of the various design and operating parameters, and to search for the optimum flow rates and switching time that give the desired results. In this quest, the selection as a starting point of the optimum set of flow rates and switching time derived from the equilibrium theory permits a considerable reduction of the number of calculations. As discussed earlier by Ruthven and Ching [27], four 

In this approach, an equivalent countercurrent movement of solid is assumed instead of the SMB process. This equivalent true moving bed (TMB) neglects the d3mamics associated with periodic switching and produces mean concentration profiles over a switching period. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

All four mathematical models predict well the steady-state concentration profiles and the performance parameters of the experimental SMB system. The two transient models predict the time evolution of the internal concentration profiles of each component and the number of cycles that is required to reach steady-state. 

However when only the steady-state concentration profiles and the process performance of an SMB are required, the use of the model SSM-1 is recommended because it requires a much shorter computing time and allows the separate, easy adjustment of the adsorption isotherm when needed. 

---