## 4 Heterogeneous Systems: Ozonation
### Optical Path Length  
As discussed in Section 2, light intensity along the channel can be considered constant or can be modeled by the Lambert-Beer law. In this case, when the light source emission is perpendicular to a rectangular channel, the optical path length can be simply assumed to be equal to the distance between the two irradiated walls of the microdevice, i.e., the height of the channel. In the case of channels of different cross-sectional shapes, the proposed solution [23; 67] is to estimate a mean optical path length defined as the integral mean of the cross-sectional chords in the direction perpendicular to the light irradiance. According to Figure 2, the mean optical length \(\bar{l}\) is defined by:  
\[\bar{l}=\frac{\int_{a}^{b}\overline{AB}(x)\ dx}{b-a} \tag{24}\]  
which, for a circular channel, is  
\[H_{2}O_{2}\xrightarrow{h_{V}}2 HO^{\bullet}\] (r.5) \[HO^{\bullet}+H_{2}O_{2}\xrightarrow{k_{H}}H_{2}O+HO_{2}^{\bullet}\] (r.6) \[A+HO^{\bullet}\xrightarrow{k_{H}O}byproducts+HO^{\bullet} \xrightarrow{k_{H}O}byproducts\dots\] (r.7) \[2 HO_{2}^{\bullet}\to H_{2}O_{2}+O_{2}\] (r.8)  
in which direct photolysis of the substrate is also taken into account. According to this simplified scheme, by-products can be further oxidized with the same intrinsic reaction constant \(k_{HO}\) (r.7). A detailed mechanistic study with the identification, isolation, and investigation of by-products can be used for further detailed mechanistic models and better parameter estimation. Under these simplified assumptions, the kinetic model for the plug flow reactor assumption can be written as [52]:  
\[\frac{d[A]}{d\tau}=\ -\frac{I_{254}}{V}\ \Phi_{UVC}^{A}\ \ \Big{(}1-\exp\Big{(} -2.3\,\overline{I}\,Abs_{tot}\Big{)}\Big{)}\frac{\epsilon_{UVC}^{A}\cdot[A]}{ Abs_{tot}}-\frac{2k_{HO}[A]F_{H_{2}O_{2}}}{k_{HO}[A]_{0}+k_{H}[H_{2}O_{2}]} \tag{26}\] \[\frac{d[H_{2}O_{2}]}{d\tau}=\ -F_{H_{2}O_{2}}-\frac{F_{H_{2}O_{2}} \cdot k_{H}\cdot[H_{2}O_{2}]}{k_{HO}[A]_{0}+k_{H}[H_{2}O_{2}]} \tag{27}\]  
where  
\[F_{H_{2}O_{2}}=\frac{L_{UVC}}{V}\Phi_{UVC}^{H_{2}O_{2}}\Big{(}1-\exp\Big{(}-2.3\,\overline{I}\,Abs_{tot}\Big{)}\Big{)}.\frac{\epsilon_{UVC}^{H_{2}O_{2}}}{ Abs_{tot}} \tag{28}\]  
where \(\frac{I_{254}}{V}\) is the incident light intensity per unit volume, \(\Phi_{UVC}^{A}\) is the quantum yield of photolysis the species \(A\), \(\epsilon_{UVC}^{A}\) is the molar absorption coefficient of the species \(A\), \(\left[A\right]_{0}\) is the initial concentration of the species \(A\), \(\Phi_{UVC}^{H_{2}O_{2}}\) is the quantum yield of photolysis of hydrogen peroxide, \(\epsilon_{UVC}^{H_{2}O_{2}}\) is the molar absorption coefficient of hydrogen peroxide, \(\overline{I}\) is the mean optical path, \(\tau\) is the residence time, and \(Abs_{tot}\) is the absorbance of the reacting mixture. In pure water and under the assumption that by-products have negligible molar absorption coefficients, it becomes:  
\[Abs_{tot}=\epsilon_{UVC}^{H_{2}O_{2}}[H_{2}O_{2}]+\epsilon_{UVC}^{A}[A] \tag{29}\]  
In the presence of more mechanistic information, the above-mentioned model can be further detailed by relaxing some hypotheses. The reaction rates can be also used in full descriptive models by taking into account the influence of axial dispersion using Equation (30):  
\[u\frac{\partial[A]}{\partial x}-D\frac{\partial^{2}[A]}{\partial x^{2}}+r_{A }=0 \tag{30}\]  
where \(x\) is the axial spatial coordinate, \(r_{A}\) is given by the above-mentioned equations, and \(D\) is evaluated through Equations (21)-(22), reported in Section 4. At present, there are no models including axial dispersion in microreactors for the estimation of intrinsic kinetic constants for UVC/H\({}_{2}\)O\({}_{2}\) processes.