## 2 Heterogeneous Systems: TiO\({}_{2}\) Photocatalysis
### Reactor Modeling
In this case, the definition of \(Pe\) becomes:  
\[Pe^{\prime}=\frac{ud_{H}^{2}}{\mathfrak{D}L}=Pe\bigg{(}\frac{d_{H}}{L}\bigg{)} ^{2} \tag{8}\]  
In the rare situation in which \(Pe\) and \(Pe^{\prime}\) are both \(\ll 1\), molecular diffusion is very efficient in equalizing concentrations in bulk liquid, in both the radial and longitudinal dimensions of the reactor, within the residence time. The occurrence can only be verified when very short reactors or extremely low flow rates are adopted; if \(Da_{II}\) is \(<0.1\), there are ideally no spatial concentration gradients; if \(Da_{II}\gg 1\), the reaction rate is even faster than molecular diffusion with respect to the space-time in the reactor, and therefore, the reactor can be considered oversized at the adopted flow rate. If \(Pe\gg 1\) and \(Pe^{\prime}\ll 1\), diffusion can efficiently equalize concentrations in the radial direction, but advection is the only significant mechanism along the length of the capillary; depending on the relative order of magnitude of the non-dimensional numbers and the velocity profile, a plug flow reactor or an axial dispersion model might be adequate to describe bulk liquid. In this condition, when \(Da_{II}\) is \(<0.1\), no radial concentration gradients will be present, but the time-space in the reactor can be low compared to the diffusion and reaction characteristic times, and low conversion can be observed; when \(Da_{II}\) is larger than unity, mass transfer limitations have to be considered around the solid layer. If \(Pe\) and \(Pe^{\prime}\) are both \(\gg 1\), an axial dispersion model or a pure convection laminar model might be adequate to describe bulk liquid, depending on the situation. Characterizing non-ideal flow patterns using a tracer can help to discriminate between different situations; in this case, if \(Da_{II}\) is \(>1\), then mass transfer limitations in the ra dial direction will cause a concentration gradient in the proximity of the catalytic surface; if \(Da_{II}\) is \(<\) 0.1, very low conversion will be observed since most of the pollutants will reach the outlet of the reactor without coming into contact with the catalytic surface where, in addition, the reaction is extremely slow. This is schematized in Scheme 1. When a kinetic regime cannot be assumed or estimating non-dimensional groups _a priori_ is not easy, different approaches have been used. [PERSON] et al. [49] examined the degradation of salicylic acid using a 70-mm-long channel, with a cross-sectional rectangular section of about 1 \(\times\) 0.5 mm, with immobilized TiO\({}_{2}\) and low flow rates of the order of 2 mL-h\({}^{-1}\). Navier-Stokes equations were coupled with Equation (2) for the mass transfer limitations and the Langmuir-Hinshelwood rate law and solved using the numerical software COMSOL. Multiphysics The simulations showed a significant concentration gradient in both the radial and longitudinal directions, which is in line with the fact that the calculated apparent kinetic constants were a function of the adopted flow rate. In a more recent study, [PERSON] et al. [9] investigated the degradation of Reactive Red 106 dye in a photo-electrocatalytic microreactor with a TiO\({}_{2}\) (anatase) anode. In this case, the photo-microreactor consisted of a cylindrical UV transparent poly(methyl methacrylate) glass channel. The microreactor was chosen for the determination of the intrinsic kinetics of decomposition because of the simplicity of a detailed model development compared to a scaled-up batch system.