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.