International Journal of Photoenergy

Volume 2018, Article ID 1678385, 16 pages

https://doi.org/10.1155/2018/1678385

## Modelling and Simulation of the Radiant Field in an Annular Heterogeneous Photoreactor Using a Four-Flux Model

Correspondence should be addressed to R. Natividad; moc.liamg@rnanyer and A. Ramírez-Serrano; xm.xemeau@szerimara

Received 7 May 2017; Revised 22 July 2017; Accepted 26 July 2017; Published 28 January 2018

Academic Editor: Detlef W. Bahnemann

Copyright © 2018 O. Alvarado-Rolon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work focuses on modeling and simulating the absorption and scattering of radiation in a photocatalytic annular reactor. To achieve so, a model based on four fluxes (FFM) of radiation in cylindrical coordinates to describe the radiant field is assessed. This model allows calculating the local volumetric rate energy absorption (LVREA) profiles when the reaction space of the reactors is not a thin film. The obtained results were compared to radiation experimental data from other authors and with the results obtained by discrete ordinate method (DOM) carried out with the Heat Transfer Module of Comsol Multiphysics® 4.4. The FFM showed a good agreement with the results of Monte Carlo method (MC) and the six-flux model (SFM). Through this model, the LVREA is obtained, which is an important parameter to establish the reaction rate equation. In this study, the photocatalytic oxidation of benzyl alcohol to benzaldehyde was carried out, and the kinetic equation for this process was obtained. To perform the simulation, the commercial software COMSOL Multiphysics v. 4.4 was employed.

#### 1. Introduction

In the last decades, photocatalytic processes have been the subject of different studies such as wastewater treatment [1–6], air purification in polluted environments with volatile organic compounds [7–9], and synthesis of fine organic compounds such as benzaldehyde [10, 11]. According to literature [1, 3–6, 11–16], the following different variables are crucial in a photocatalytical process efficiency: (a) catalyst type and concentration, (b) reagent type and concentration, (c) geometry and type of reactor, and (d) characteristics of the radiation inside the photoreactor. Because of the number of variables and the interaction among them, the modeling of this type of processes is expected to be rather useful not only for reactor design but also to achieve a better insight and understanding of the process.

The mathematical modeling and simulation of a photocatalytic reactor imply a great challenge due to the numerous involved variables; however, the computational analysis of these variables aids to accomplish such a task. Furthermore, the computational analysis allows evaluating hydrodynamic effects and kinetics without employing physical prototypes. The full modeling of photocatalytic reactors requires to include several submodels to simulate the physical phenomena occurring inside the reactor. Some of these necessary submodels are (a) radiation emission and incidence, (b) radiation absorption and scattering, (c) photoconversion kinetics, and (d) hydrodynamics [5, 13–15, 17, 18]. These are the result of mass, energy, and momentum balances, as well as radiation distribution and optical characterization of reaction space [6, 16, 19, 20]. These submodels are strongly interlinked. For example, the kinetics is a function of radiation absorption, which is in turn a function of catalyst characteristics and hydrodynamics. The conversion and performance of a photocatalytic reaction are a function of the local volumetric rate energy absorption (LVREA), which is defined as the energy due to photons absorbed per time and volume inside the photoreactor [21]. To evaluate the LVREA is necessary to solve the radiation transfer equation (RTE) [22–25].
where is the spectral radiation intensity, represents the wavelength, is the extinction coefficient, which is the sum of the absorption coefficient, and is the scattering coefficient. The ratio is the scattering albedo coefficient which is inherent to each photocatalyst since it represents its photon absorption capacity. *Ω* is the solid angle, and is the phase function representing the redistribution of radiation after the scattering event. According to the first term in the right side of (1), the intensity is diminished by the effect of mainly two phenomena, scattering and absorption. This decrease is characterized by the extinction coefficient. There is also an increase in the intensity due to the scattering from other directions, and it is represented by the second term in the right-hand side of (1) [24, 26, 27].

The analytical solution of the RTE is a rather complex task, unless it is limited to simple reactor geometries with specific assumptions. Even when using specialized software, the radiation field simulation is a task that requires a high computational effort. Comsol Multiphysics v. 4.4 contains the physics of radiation in participating media (rpm), in the Heat Transfer Module, which is designed to solve 3D radiation transfer problems, taking into account the phenomena of emission, dispersion, and absorption of radiation. The Comsol Multiphysics v. 4.4 Heat Transfer Module employs the discrete ordinate method (DOM). This method consists the transformation of the integral-differential RTE into a system of algebraic equations to describe the transport of photons in such way that can be solved following the direction of propagation, starting from the values provided by the boundary conditions. However, RTE is solved by discretizing the solid angle at every discrete position in the 3D domain, which is computationally very demanding and may result in unrealistic results when the discretization of the solid angle is not refined enough.

A viable alternative is to employ numerical computational methods as the statistical method Monte Carlo (MC), which is known as highly accurate but requires a great computational effort [21, 28, 29]. Also, it is possible to employ analytical simplified methods like the two-flux model (TFM) and the six-flux model (SFM). These models consist of several algebraic equations developed for flat slab geometries [15–17, 30], which were obtained by solving a system of differential equations with specific boundary conditions, for example, the outer wall of the reactor is opaque. SFM is very accurate for cylindrical geometries [14] in which the space where the reaction occurs, *δ*, is much smaller than the radius of the reactor,

However, in this investigation, a reactor in which the lamp is immersed in the reaction medium was used, so (2) is not satisfied. The geometry used in this work is shown in Figure 1. This paper aims to evaluate the effectiveness of a modified model based on four flux of radiation (FFM), whose equations are based on a cylindrical geometry, to mathematically represent the radiation field in a stirred annular photoreactor. This model is coupled to a reaction rate model representing the benzyl alcohol oxidation. The FFM evaluates the incident radiation in each point of the reaction space. This model considers that the incident radiation is the sum of radiation fluxes traveling from the light source towards this point and the fluxes due from both axial and radial scattering. As this model is developed from cylindrical geometries, its solution is expected to better represent the radiant field inside an annular photocatalytic reactor than the models developed from slab plane geometries where the reaction space is only a thin film.