Dynamic difference equation model for chromatography

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as Madonna. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of V and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the liquid phase volume of any element, n, as indicated in Fig. 4.24 below. Here j represents the diffusive flux, L the liquid flow rate and and Cla the concentration of any species A in both the solid and liquid phases, respectively. 

For each element, the material balance in the liquid phase, here for component A, is 

As before, the concentrations are taken as the average in each segment, and the diffusion fluxes are related to the concentration gradients at the segment boundaries. 

The convective mass flows in and out of the segment are calculated by multiplying the respective concentrations by the constant volumetric flow rate, L,n. The diffusive mass flows are calculated from Pick s Law, using the inlet 

The transfer rate of A, Tr (g/s), from liquid to solid is given by 

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