Hyperbolic Averaged Models for Describing Dispersion Effects in Chromatographs

Multi-mode Hyperbolic Averaged Models for Describing Dispersion Effects in Chromatographs 

t) is the scaled solute concentration in the fluid phase, Cw the solute concentration at the wall, 6 the normalized adsorbed concentration (O 0 1), K the adsorption equilibrium constant, p the transverse Peclet number, T represents the adsorption capacity (ratio of adsorption sites per unit tube volume to the reference solute concentration), and Da is the local Damkohler number (ratio of transverse diffusion time to the characteristic adsorption time). We shall assume that p 4Cl while T and Da are order-one parameters. (In physical terms, this implies that transverse molecular diffusion and adsorption processes are much faster compared to the convection.) 

Transverse averaging of the above model using the procedure outlined in Section II gives the following averaged model to order p  

The averaged model is a four-mode model, the four modes (or variables) being the fluid-phase cup-mixing concentration (Cm), the fluid-phase average concentration ((C)), the solute concentration at the wall (Cw), and the adsorbed 

For the case of p — 0 (which corresponds to adsorption, desorption, and transverse diffusion time scales going to zero), we have 

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