3.2.3 Packed bed Modeling
In order to predict the dynamic behavior of the adsorption column, a mass balance over the liquid phase of an infinitesimal sized slice of the packed bed leads with perfect radial mixing and negligible pressure drop to Eq. 3.4, see [40]–[42].
\begin{equation} D_{\text{ax}}\frac{\partial^{2}C_{i}}{\partial z^{2}}-v\frac{\partial C_{i}}{\partial z}-\frac{1-\varepsilon_{b}}{\varepsilon_{b}}N_{i}=\frac{\partial C_{i}}{\partial t}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ (3.4)\nonumber \\ \end{equation}
In Eq. 3.4 \(C_{i},\)\(D_{\text{ax}},\ v,\ \varepsilon_{b},\ and\ N_{i}\) are the concentration of TCO in the liquid phase, the axial dispersion coefficient, the interstitial liquid velocity, the bed porosity, and the mass transfer rate into the particle per unit volume of the particle phase, respectively. The first and second term in Eq. 3.4 stand for the dispersive transport and the convective transport in the column, respectively. The third term represents the mass transfer between liquid phase and adsorbent. The last term is related to accumulation of the adsorbate. This model is based on the following assumptions: isothermal adsorption and spherical adsorbent particles packed uniformly in the bed.
As it was stated, \(N_{i}\ \)is the mass transfer rate between solid and fluid phase and can be represented as [40]–[42]:
\(N_{i}=\rho_{p}\frac{\partial q_{i}}{\partial t}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ (3.5)\)
\(q_{\text{i\ }}\left[\frac{\text{mol}}{\text{kg}_{s}}\right]\)is the concentration of TCO on the adsorbent surface and\(\rho_{p}\left[\frac{\text{kg}_{s}}{\text{dm}_{s}^{3}}\right]\)is the adsorbent particle density. In order to define\(\frac{\partial q_{i}}{\partial t}\) mathematically, it is important to know whether the mass transfer resistance or the intrinsic adsorption kinetics dominates the adsorption rate.