## 2 Heterogeneous Systems: TiO\({}_{2}\) Photocatalysis
### Reactor Modeling  
Most papers dealing with reactions occurring over immobilized TiO\({}_{2}\)[47; 49; 57; 58; 59] adopt a Langmuir-Hinshelwood reaction mechanism for the study of intrinsic reaction kinetics, expressing the reaction rate of the generic compound \(A\) as  
\[r=\ \frac{kK_{ad}[A]_{surf}}{1+K_{ad}[A]_{surf}} \tag{1}\]  
where \(k\) is the intrinsic kinetic constant for the surface reaction of adsorbed \(A\); \(K_{ad}\) is the adsorption equilibrium constant of \(A\) on the catalytic surface, under the Langmuir adsorption assumptions, and \([A]_{surf}\) is the concentration of \(A\) at the liquid/solid interface. The \([A]_{surf}\) value is usually lower than the bulk liquid concentration because of the presence of mass transfer limitations. At a steady state, this gives:  
\[\frac{kK_{ad}[A]_{surf}}{1+K_{ad}[A]_{surf}}=k_{g}a([A]_{bulk}-[A]_{surf}) \tag{2}\]  
where \(k_{g}\) is the global mass transfer coefficient, which is a function of the Schmidt and Reynolds numbers and, therefore, of the physical properties of the fluid, the velocity profile inside the microreactor channel, the geometry of the channel, and the diffusion coefficient of \(A\) in the liquid phase. The general theoretical approach to solve the problem would be to find a semi-empirical correlation between the non-dimensional Sherwood, Schmidt, and Reynolds numbers to express \(k_{g}=f(Sc,\ Re)\) and then couple the semi-empirical equation with Equations (1) and (2) for the estimation of the intrinsic kinetic  
Figure 1: Main mechanisms controlling pollutant degradation in immobilized catalyst microreactors. parameter \(k\). The procedure is, in general, a lengthy process that relies on accurate estimation of diffusion coefficients, often not available in the literature. A common correlation for laminar flow in microchannels is [24]:  
\[Sh=\frac{k_{g}d_{H}}{L}=7.54\bigg{(}1+0.095\frac{d_{H}}{L}Re\cdot Sc\bigg{)}^{0.45} \tag{3}\]  
In the context of water treatment studies, a constant value for water viscosity is commonly used, without considering any possible effect of small diameters and water matrices. Recent direct use of Equations (1)-(3) for the determination of intrinsic parameters for the degradation of pollutants in water is proposed by [PERSON] et al. [24]. For this aim, Rhodamine B degradation was studied in a microchannel over immobilized TiO\({}_{2}\). The kinetic parameters were determined in a macroscopic batch reactor with immobilized TiO\({}_{2}\) and then the results were validated using the microreactor modeled according to Equations (1)-(3), with a reported maximum error of 20% in the observed conversion. Another study [57] also followed the same general approach, using a different correlation for rectangular microchannels. However, most papers make direct use of Equation (1) in the following form:  
\[r=\ \frac{k_{a}K_{ad}[A]_{bulk}}{1+K_{ad}[A]_{bulk}} \tag{4}\]  
where \(k_{a}\) is generally known as an apparent kinetic constant and it does not take into account the effect of mass transfer limitations on the observed overall reaction rate. Of course, Equation (4) reduces to Equation (1) under kinetic regime assumption. In fact, if mass transfer is much faster than intrinsic reaction rate, \(k_{g}\gg k\), as a result, \([A]_{bulk}=[A]_{surf}\) and \(k_{a}=k\). Apparent constants estimation by Equation (4) is a common practice in the literature to screen different materials and compare the results for different target compounds. However, these constants are dependent on the specific adopted device, lamp source, and operating conditions, and they cannot be used to translate models to full scale and to design and optimize real water treatment systems. The few papers addressing this point tackle the estimation of \(k\) values using two main approaches: (_i_) design devices and adopt conditions so that radial mass transfer limitations can be considered negligible or (_ii_) describe the system with a full set of equations taking into account mass transfer limitations and solve them numerically.