Using the Lumped Finite Adsorption Rate Model in CADET

Chromatography Modeling
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This has come up a few times recently, so I thought it’d be useful to write it up here. It’s also exactly the kind of model relationship we’re trying to clarify in the unified and modular modeling framework we’re currently putting together for an upcoming publication.

We show that, under a simple parameter transformation, the Lumped Finite Adsorption Rate Model is numerically equivalent to the CADET Lumped Rate Model without Pores.

The Lumped Finite Adsorption Rate Model is introduced by Schmidt-Traub (2020) as follows:

Another subgroup of the lumped rate approach consists of considering, besides axial dispersion, the reaction dispersion model where, in addition, the adsorption kinetics are rate limiting. It is an extension of the reaction model (Section 6.2.4.2). Like the mass transfer coefficient in the TDM, the adsorption and desorption rate constants are considered as effective lumped parameters, k_{\mathrm{ads,eff}} and k_{\mathrm{des,eff}}.
Since no film transfer resistance is considered (c_{p,i}=c_i), the solid-phase material balance can be described by Eq. (6.88):

\frac{\partial q_i^{*}}{\partial t} = k_{\mathrm{ads,eff},i}\, q_{\mathrm{sat},i} \left( 1-\sum_{j=1}^{n}\frac{q_j^{*}}{q_{\mathrm{sat},j}} \right) c_i - k_{\mathrm{des,eff},i}\, q_i^{*} \tag{6.88}

and the material balance of the mobile phase by:

\frac{\partial c_i}{\partial t} + u_{\mathrm{int},e} \cdot \frac{\partial c_i}{\partial x} + \frac{1-\varepsilon_e}{\varepsilon_e} \left( \varepsilon_p \frac{\partial c_i}{\partial t} + (1-\varepsilon_p) \frac{\partial q_i}{\partial t} \right) = D_{\mathrm{ax}} \cdot \frac{\partial^2 c_i}{\partial x^2} \tag{6.89}



To show the equivalence, we introduce an apparent velocity \tilde{u} and an apparent axial dispersion coefficient \tilde{D}^{\mathrm{ax}}_{i},

\tilde{u} \coloneqq \frac{\varepsilon^{\mathrm{b}}}{\varepsilon^{\mathrm{t}}} u_{\mathrm{int},e}, \qquad \tilde{D}^{\mathrm{ax}}_i \coloneqq \frac{\varepsilon^{\mathrm{b}}}{\varepsilon^{\mathrm{t}}} D_{\mathrm{ax}},

and associate

c^\ell_i = c_i, \qquad c^\mathrm{s}_i = q_i, \qquad z=x, \qquad \varepsilon^\mathrm{t} = \varepsilon_e + (1-\varepsilon_e)\varepsilon_p.

With these substitutions, the Lumped Rate Model without Pores becomes

\begin{aligned} \frac{\partial c_i^\ell}{\partial t} + \frac{1-\varepsilon^{\mathrm t}}{\varepsilon^{\mathrm t}} \frac{\partial c_i^\mathrm s}{\partial t} &= -\tilde u \frac{\partial c_i^\ell}{\partial z} + \tilde D_i^{\mathrm{ax}} \frac{\partial^2 c_i^\ell}{\partial z^2} \\ &= - \frac{\varepsilon_e}{\varepsilon^{\mathrm t}} u_{\mathrm{int},e} \frac{\partial c_i}{\partial x} + \frac{\varepsilon_e}{\varepsilon^{\mathrm t}} D_{\mathrm{ax}} \frac{\partial^2 c_i}{\partial x^2}. \end{aligned}

where

\varepsilon^{\mathrm t} = \varepsilon_e+(1-\varepsilon_e)\varepsilon_p \quad \Rightarrow \quad \frac{1-\varepsilon^{\mathrm t}}{\varepsilon^{\mathrm t}} = \frac{(1-\varepsilon_e)(1-\varepsilon_p)} {\varepsilon_e+(1-\varepsilon_e)\varepsilon_p}.

Multiplying the entire equation by

\frac{\varepsilon^\mathrm{t}}{\varepsilon_e} = 1+\frac{1-\varepsilon_e}{\varepsilon_e}\varepsilon_p

gives

\frac{\partial c_i}{\partial t} + u_{\mathrm{int},e} \frac{\partial c_i}{\partial x} + \frac{1-\varepsilon_e}{\varepsilon_e} \left( \varepsilon_p \frac{\partial c_i}{\partial t} + (1-\varepsilon_p) \frac{\partial q_i}{\partial t} \right) = D_{\mathrm{ax}} \frac{\partial^2 c_i}{\partial x^2}.

Likewise, substituting the binding law with the kinetic Langmuir formulation yields

\frac{\partial q_i}{\partial t} = k_{\mathrm{ads,eff},i}\, q_{\mathrm{sat},i} \left( 1-\sum_{j=1}^{n} \frac{q_j}{q_{\mathrm{sat},j}} \right) c_i - k_{\mathrm{des,eff},i}\, q_i.
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