# 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

\[\begin{aligned}
\frac{\partial c^l_i}{\partial t} &= -u \frac{\partial c^l_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^l_i}{\partial z^2} - \frac{1}{\beta_c} \sum_{j} d_j \frac{3}{r_{p,j}} k_{f,j,i}\left[ c^l_i - c^p_{j,i} \right] + f_{\text{react},i}^l\left(c^l\right),
\end{aligned}\]

\[\begin{split}\begin{aligned}
\frac{\partial c^p_{j,i}}{\partial t} + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} \frac{\partial}{\partial t} \sum_{m_{j,i}} c^s_{j,i,m_{j,i}} &= \frac{3}{F_{\text{acc},j,i} \varepsilon_{p,j} r_{p,j}}k_{f,j,i}\left[ c^l_i - c^p_{j,i} \right] \\
&+ f_{\text{react},j,i}^p\left( c_j^p, c_j^s \right) + \frac{1 - \varepsilon_{p,j}}{F_{\text{acc},j,i} \varepsilon_{p,j}} f_{\text{react},j,i}^s\left( c_j^p, c_j^s \right)
\end{aligned}\end{split}\]

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

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

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 \(z \in [0,L]\):

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

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.