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Lumped rate model with pores (LRMP)¶

The lumped rate model with pores [3, 4] deviates from the general rate model (see Section General rate model (GRM)) by neglecting pore diffusion. The particle phase $$c^p_j$$ is still there, but no mass transfer happens except for binding and film diffusion. Hence, the model equations are given by

\begin{aligned} \frac{\partial c^\ell_i}{\partial t} &= -u \frac{\partial c^\ell_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^\ell_i}{\partial z^2} - \frac{1}{\beta_c} \sum_{j} d_j \frac{3}{r_{p,j}} k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] + f_{\text{react},i}^\ell\left(c^\ell\right), \end{aligned}
(15)\begin{split}\begin{aligned} \frac{\partial c^p_{j,i}}{\partial t} + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} \frac{\partial}{\partial t} \sum_{m_{j,i}} c^s_{j,i,m_{j,i}} &= \frac{3}{F_{\text{acc},j,i} \varepsilon_{p,j} r_{p,j}}k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] \\ &+ f_{\text{react},j,i}^p\left( c_j^p, c_j^s \right) + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} f_{\text{react},j,i}^s\left( c_j^p, c_j^s \right) \end{aligned}\end{split}

with the same meanings of variables and parameters as in the general rate model. The equations are complemented by Danckwerts boundary conditions [8]

\begin{split}\begin{aligned} u c_{\text{in},i}(t) &= u c^\ell_i(t,0) - D_{\text{ax},i} \frac{\partial c^\ell_i}{\partial z}(t, 0) & \forall t > 0,\\ \frac{\partial c^\ell_i}{\partial z}(t, L) &= 0 & \forall t > 0. \end{aligned}\end{split}

As for the general rate model, both quasi-stationary and dynamic binding models are supported:

\begin{split}\begin{aligned} \text{quasi-stationary: }& & 0 &= f_{\text{ads},j}\left( c^p_j, c^s_j\right), \\ \text{dynamic: }& & \frac{\partial c^s_j}{\partial t} &= f_{\text{ads},j}\left( c^p_j, c^s_j\right) + f_{\text{react},j}^s\left( c_j^p, c_j^s \right). \end{aligned}\end{split}

By default, the following initial conditions are applied for all $$z \in [0,L]$$:

\begin{aligned} c^\ell_i(0, z) &= 0, & c^p_{j,i}(0, z) &= 0, & c^s_{j,i,m_{j,i}}(0,z) &= 0. \end{aligned}

Multiple particle types types are supported. This model can also be used to simulate Size exclusion chromatography. For the specification of flow rate and direction, the same holds as for the general rate model (see Section Specification of flow rate / velocity and direction).

For information on model parameters see Lumped rate model with pores.

The radial flow LRMP describes transport of solute molecules through the interstitial column volume by radial convective flow, band broadening caused by radial dispersion, mass transfer resistance through a stagnant film around the beads, and adsorption to the inner bead surfaces.

The main assumptions are:

• The shells of the column are homogenous in terms of interstitial volume, fluid flow, and distribution of components. Thus, only one spatial coordinate in radial direction $$\rho$$ is needed and axial transport is neglected in the column bulk volume.

• The bead radii $$r_{p}$$ are much smaller than the column radius $$\mathrm{P}-\mathrm{P}_c$$, with $$\mathrm{P}$$ and $$\mathrm{P}_c$$ being the inner and outer column radius respectively, and the column length $$L$$. Therefore, the beads can be seen as continuously distributed inside the column (i.e., at each point there is interstitial and bead volume).

• The fluids are incompressible, i.e. the velocity field $$\mathrm{V} \colon \mathbb{R}^3 \to \mathbb{R}^3$$ submits to $$\operatorname{div}\left( \mathrm{V} \right) \equiv 0$$. That is, the volumetric flow rate at the inner and outer column radius are the same.

Consider a hollow (double walled) column with inner column diameter $$\mathrm{P}_c>0$$ and outer diameter $$\mathrm{P}>\mathrm{P}_c$$, filled with spherical beads of (possibly) multiple types with radius $$r_{p,j} \ll L$$ (see Fig. 2), where $$j$$ is the particle type index. The mass balance in the interstitial column volume is described by

\begin{aligned} \frac{\partial c^\ell_i}{\partial t} &= -\frac{u}{\rho} \frac{\partial c^\ell_i}{\partial \rho} + D_{\text{rad},i} \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial c^\ell_i}{\partial \rho} \right) - \frac{1}{\beta_c} \sum_{j} d_j \frac{3}{r_{p,j}} k_{f,j,i}\left[ c^\ell_i - c^p_{j,i} \right] + f_{\text{react},i}^\ell\left(c^\ell\right), \end{aligned}

The equations are complemented by Eq. ModelParticleLRMP and the Danckwerts boundary conditions [8]

\begin{split}\begin{aligned} u c_{\text{in},i}(t) &= u c^\ell_i(t,0) - D_{\text{rad},i} \frac{\partial c^\ell_i}{\partial \rho}(t, 0) & \forall t > 0,\\ \frac{\partial c^\ell_i}{\partial \rho}(t, \mathrm{P}) &= 0 & \forall t > 0. \end{aligned}\end{split}

The complementing binding equations are described by the same equations as for the axial LRMP.

For information on model parameters see Radial Flow Models in addition to Lumped rate model with pores.