## 2 Heterogeneous Systems: TiO\({}_{2}\) Photocatalysis
### Reactor Modeling
The first approach is more common and it is based on of the main advantages of microreactors. Their radial dimension, in fact, is often small enough to consider characteristic time of radial diffusion as much lower than residence time, so that diffusive mass transfer can effectively equalize the concentration along the radius. However, this assumption is not often adequately justified. [PERSON] et al. [60], for example, used a fused silica circular capillary with immobilized TiO\({}_{2}\) for the removal of several dyes from water, and they claim that mass transfer limitations are negligible for an internal diameter of 530 \(\upmu\)m, without giving any indication of the adopted length and flow rates and of the relative order of magnitude of the different transport phenomena in the system. On the other hand, in a recent study, [PERSON] et al. [44] report a fully characterized microdevice, giving a mathematical derivation for the estimation of intrinsic kinetic parameters when the reactor is operated under a kinetic regime, i.e., in the absence of external mass limitations. However, the complete mathematical derivation is only valid for reactions following a first-order rate law. Specifically, they verified assumptions and a model for the degradation of clofibric acid using computational fluid dynamics (CFD) simulations in OpenFOAM and experimental results in a scaled-up system. Different non-dimensional groups have been proposed in the literature for the evaluation of the relative significance of mass transfer mechanisms and intrinsic reaction kinetics. In the field of microreactors for wastewater treatment, these are reported by [PERSON] et al. [42] as the second Damkohler number \(Da_{II}\), defined as the ratio of the rate of a chemical reaction to diffusive mass transfer rate. In particular, [PERSON] et al. [47] report the following expression for Langmuir-Hinshelwood kinetic laws with apparent reaction rate constant \(k_{a}\):  
\[Da_{II}=\frac{reaction\ rate}{diffusive\ mass\ transfer\ rate}=\frac{k_{a}}{\frac{k_{a}a}{k_{ad}}+k_{g}a[A]_{bulk}} \tag{5}\]  
Considering that a laminar flow regime is always present in microreactors and, for low flow rates, the Sherwood number approaches the asymptotic value of 3.66 [48],  
\[Da_{II}=\frac{k_{a}d_{H}}{3.66\ a\ \mathfrak{D}\ \left(K_{ad}^{-1}+[A]_{bulk}\right)} \tag{6}\]  
Equation (6) enables to estimate \(Da_{II}\) once the interfacial area per unit volume \(a\) and the molecular diffusion coefficient \(\mathfrak{D}\) can be evaluated. The term \(d_{H}\) is the hydrodynamic diameter. It is important to stress that this asymptotic value of the Sherwood number is different from the one reported in Equation (3). As a general rule of thumb, a kinetic regime can be assumed if \(Da_{II}<0.1\). In addition to the relative magnitude of molecular diffusion and intrinsic reaction, the influence of advection is also of crucial importance in modeling the concentration profile along the length of microreactors and, therefore, the dependence of the measured concentration at the outlet of the reactor on residence time. In this case, the ratio between the characteristic time of diffusion and advection is given by the non-dimensional Peclet number \(Pe\). \[Pe=\frac{uL}{\mathfrak{D}} \tag{7}\]  
where \(u\) is the characteristic mean bulk velocity, defined as the ratio between the flow rate and the cross-sectional area, and \(L\) is the characteristic length of the problem, usually the length of the reactor. However, it is important to stress that in the specific case of microreactors, characterized by exceptionally high aspect ratios (\(L/d_{H}\)), it can also be useful to define the Peclet number by comparing the efficiency of the diffusive equalization in the radial direction with the one of advection along the length of the reaction [42].