Local adsorption model-local equilibrium theory
Local adsorption theory is divided into two sets of models depending on the rate of adsorption- local equilibrium theory and local kinetic theory [45], [46].
Local equilibrium theory is based on the existence of equilibrium between adsorbate and fluid concentration, and how fast the equilibrium is reached in each stage in the packed bed column. Thus\(\ \frac{\partial q_{i}}{\partial t}\) is defined by using the Langmuir isotherm as follows:
\begin{equation} \frac{\partial q_{i}}{\partial t}=\frac{\partial q_{i}}{\partial C_{i}}\times\frac{{\partial C}_{i}}{\partial t}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ (3.8)\nonumber \\ \end{equation}
and
\begin{equation} \frac{\partial q_{i}}{\partial C_{i}}=\frac{q_{\max}b}{{(1+bC_{i})}^{2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ (3.9)\nonumber \\ \end{equation}
where\(\text{\ q}_{\max}\ \)and \(b\) are from Langmuir isotherm model, see Eq. 3.2.
The local kinetic theory is based on a non-equilibrium adsorption reaction. It is expressed based on the rate of adsorption and desorption of the adsorbate on the surface of adsorbent particles.
In this work an equilibrium surface reaction is governing, so the only model that has been applied is the local equilibrium theory. For the local kinetic theory, see the related literature [45], [46].
Therefore, Eq. 3.4 becomes
\begin{equation} D_{\text{ax}}\frac{\partial^{2}C_{i}}{\partial z^{2}}-v\frac{\partial C_{i}}{\partial z}=\left[1+\rho_{p}\frac{\left(1-\varepsilon_{b}\right)}{\varepsilon_{b}}\times\frac{q_{\max}b}{\left(1+bC_{i}\right)^{2}}\right]\frac{\partial C_{i}}{\partial t}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ (3.10)\ \nonumber \\ \end{equation}
For the model calculations the partial differential equation (3.10) has to be solved. Eq 3.10, is discretized using finite differences and solved using MATLAB® with implicit PDEPE solvers.
The initial condition is a packed bed filled with solvent. Then, at the inlet of the column a step change is applied. The applied boundary conditions are of Danckwert’s type conditions [47] and used at the inlet (z=0) and at the outlet (z=L) of the column (Eq. 3.11).
\begin{equation} \frac{\partial C_{i}}{\partial z}|_{z=0}=\frac{v}{D_{\text{ax}}}\left(C_{i}-C_{i,feed}\right);\ \ \ \frac{\partial C_{i}}{\partial z}|_{z=l}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.11)\ \nonumber \\ \end{equation}
The initial conditions are given by Eq. 3.12
\begin{equation} C_{i}\left(z,t=0\right)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ (3.12)\nonumber \\ \end{equation}
3.3 Modeling the combination of the photo-microreactor and the adsorption column
In this section it is aimed to predict the cis -cyclooctene conversion when there is a continuous closed system of the photo-microreactor with an integrated adsorption column and recycling the unreacted cis -isomer to the reactor feed solution (Figure 1). In order to mathematically model the system, two control volumes are selected: the photo-microreactor and the packed bed (1), and the stirred vessel with solution to be fed to the photoreactor (2).