Figure 1 Schematic overview of the complete photo-microreactor and the adsorption system control volumes; (1): photo-microreactor and packed bed, (2): inlet vessel with reactor feed solution
The rate of photoisomerization of the cis -isomer into thetrans -isomer has been given in the previous section (3.1). It is assumed that the amount of the cis - isomer adsorbed in the packed bed is negligible. Therefore, for the first control volume and a single pass of the flow, if \(C_{t}=\frac{n_{\text{cis}}}{V}\), where\(\ n_{\text{cis}}\) denotes the molar amount of thecis -isomer in the inlet vessel and V the volume of the photo-microreactor, the rate equation becomes (only if the outlet concentration of the trans -isomer from the packed bed is zero):
\begin{equation} -\frac{dC_{\text{cis}}}{\text{dt}}={(k}_{1}+k_{-1})\times C_{\text{cis}}-k_{-1}\times\frac{n_{\text{cis}}}{V}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.13)\nonumber \\ \end{equation}
In order to model the second control volume, a mass balance is set up over the inlet vessel, see Eq. 3.14. The inlet vessel is assumed to be ideally mixed.
\begin{equation} \left[n_{\text{cis}}\right]_{t+t}-\left[n_{\text{cis}}\right]_{t}=Q\times C_{cis,\ iv}\times t-Q\times\frac{n_{\text{cis}}}{V}\times t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ (3.14)\nonumber \\ \end{equation}
where \(\text{Q\ }\text{and}\ C_{cis,\ iv}\) stand respectively for the total volumetric flow rate and the cis concentration at the inlet of the vessel. Division the right and left hand side of Eq. 3.18 by\(t\ \)and taking the limit \(t\ \rightarrow 0,\) leads to:
\begin{equation} \frac{dn_{\text{cis}}}{\text{dt}}+\frac{Q}{V}n_{\text{cis}}=Q\times C_{cis,\ iv}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.15)\nonumber \\ \end{equation}
Eq. 3.15 expresses the change in the amount of cis -isomer with respect to time. The concentration of cis -isomer at the outlet of the first control volume, (i.e. the outlet of the packed bed fortrans -isomer adsorption) is equal to the concentration of thecis -isomer at the inlet of control volume 2, (i.e. the vessel with the reactor feed solution)\(\ C_{cis,\ iv}=C_{\text{cis}}\).
Eq. 3.13 and 3.15 are first order ordinary differential equations (ODEs) which are solved simultaneously by employing numerical methods (in MATLAB®) whereby the final concertation of cis -isomer or the final conversion of the cis -isomer with respect to time is obtained. The initial values for numerically solving Eq. 3.13 and 3.15 are \(C_{\text{cis}}=C_{0}\) and \(n_{\text{cis}}=\frac{C_{0}}{V}\), where \(C_{0}\) is the initial concentration of the cis -isomer in the reactor feed vessel.