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 sixflux 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.
In the last decades, photocatalytic processes have been the subject of different studies such as wastewater treatment [
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 [
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 integraldifferential 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 [
However, in this investigation, a reactor in which the lamp is immersed in the reaction medium was used, so (
Schematic representation of the geometry of the assessed annular photocatalytic reactor.
The main objective of this work was to validate the proposed fourflux model, which is specifically designed for annular photocatalytic reactors with a relationship, that is, the reactor is not thinwalled. FFM is tested against the results with experimental data of the photocatalytic and selective oxidation of benzyl alcohol towards benzaldehyde. Moreover, the radiation profiles were compared to those calculated by MC, DOM, and SFM. The FFM and DOM were carried out with commercial software Comsol Multiphysics 4.4, which is a powerful differential equation solver.
The main objective of this work was to test a proposed FFM to efficiently represent the radiant field inside an annular reactor when the reaction space is not a thin film. In order to validate the proposed model, the profiles obtained with FFM were compared to those previously reported in the literature. Also, the FFM was applied to describe the radiant field in a batch annular photoreactor employed to experimentally obtain benzyl alcohol oxidation data. Then, the kinetics of this reaction was established as function of LVREA.
The profiles obtained in a thinfilm slurry reactor of inner wall (TFSIW) reported by Li Puma et al. [
Characteristics of systems.
Catalyst  Lamp characteristics  Reactor characteristics  

System 1 
TiO_{2} DP 25 
Power: 4 W 
Length: 0.225 m 


System 2 
TiO_{2} anatase 
Power: 8 W 
Length: 0.445 m 


System 3  LiVMoO_{6} 
Power: 8 W 
Length: 0.25 m 
This photoreactor was previously reported [
Once the radiation model was validated with data reported in systems 1 and 2, this model was applied to simulate the radiation field in system 3 during benzyl alcohol selective oxidation towards benzaldehyde. Experimental data of benzyl alcohol oxidation were obtained in an annular cylindrical photocatalytic reactor. It is worth pointing out that in this reaction system, the lamp was placed at the center of the reactor without any additional physical protection (e.g., quartz sleeve). For this reason, the relationship
The employed catalyst was LiVMoO_{6}, and a detailed characterization has been previously reported [
The emission of radiation from the cylindrical lamp is modeled using the linear source spherical emission (LSSE). This model considers that the lamp is a linear source, and each point on the line emits radiation isotropically and in every direction. It is assumed that the radiation emitted by each point of the lamp is constant along the axial length of the lamp [
The experimental emitted radiation was measured by a UVX radiometer equipped with a sensor of 254 nm placed at the lamp wall and 0.01 m from the lamp.
To establish the mathematical FFM, the following assumptions were made: (a) reactor with slurry catalyst, (b) heterogeneous model, (c) isothermal process, (d) perfect mixing and therefore the catalyst concentration is homogeneous at all reaction space, (e) photons are absorbed only by catalyst particles, (f) the flux of photons occurs only in four directions, two radial, and two axial directions, (g) the emission of photons by the lamp is isocratic, (h) oxygen bubbles do not affect the radiation fluxes, and (i) the scattering of photons by the catalyst is isotropic.
FFM was employed to evaluate the incident radiation on a given point inside the reaction space. In this model, the total radiation flux is taken as the sum of the flux of photons traveling from the light source towards that point and flux of photons from scattering in both two axial directions and two both radial directions. In concordance, a photon balance was performed in a differential volume element shell shaped in cylindrical coordinates (Figure
Directions of the fluxes of photons in the fourflux model.
The flux of incident radiation
By reordering and applying
The term
Equations (
(1) BC 1: at wall lamp or inner wall:
(2) BC 2: at external reactor wall (opaque wall):
(3) BC 3: at upper and bottom wall:
These boundary conditions are shown in Figure
Boundary conditions used in the fourflux model.
The LVREA using the fourflux model can be calculated by the following expression:
The software COMSOL Multiphysics version 4.4 and subroutines performed in Matlab® were employed to solve the FFM and kinetic models, respectively. To carry out the simulation, the geometric domain of both, reaction space and lamp, was established. The model is twodimensional and symmetric with respect to the axial axis. A nonuniform mesh was used, with a size of element calibrated to plasma, giving major emphasis on the inner wall of the annulus, using a fine mesh at this boundary and coarser in the outer wall of the reactor to accurately assess each border (Figure
Graphical representation of photocatalytic reactor and geometry employed to solve (a) fourflux model and (b) discrete ordinate method, in COMSOL Multiphysics.
The results obtained by FFM were compared with the following.
Discrete ordinate method (DOM) carried out with the physics of radiation in participating media of the Heat Transfer Module of Comsol Multiphysics 4.4. To do so, the geometric domain of reaction space was established as 3D model. Several preliminary simulations were run using this method. In these trials, the mesh in all domains was refined incrementally until the physical ram limit of the workstation (8 Gb) was reached. Geometry and mesh employed are shown in Figure
Sixflux model (SFM) was implemented in programming language Matlab according to the methodology reported by Li Puma [
Monte Carlo Method (MC) was also implemented in programming language Matlab based on Moreira et al. [
The codes to solve the applied models, SFM and MC, are rather lengthy. However, they can be provided upon request.
To determine the radiation effect on reaction rate, a kinetic expression as function of LVREA can be obtained.
Figure
Emission model results. Calculated and experimental incident radiation profiles.
The results of the proposed model ((
The first analyzed photocatalytic reactor was a TFSIW reported by Li Puma et al. [
Radial profiles of local volumetric rate energy absorption (LVREA) obtained with fourflux model at different concentrations of catalyst for system 1 and its comparison with the other models.
The results obtained by FFM method are in agreement with the data previously reported by Moreira et al., which were obtained from MC for PhotoCREC water II [
Radial profiles of local volumetric rate energy absorption (LVREA) obtained with fourflux model at different catalyst concentration for system 2 and its comparison with the other models.
This system was theoretically and experimentally studied. Figure
Radial profiles of local volumetric rate energy absorption (LVREA) obtained by fourflux model at different concentration of catalyst (LiVMoO_{6}) for system 2 and its comparison with the other models.
In Figure
Table
Comparison of correlation coefficients of different radiation absorption models for studied systems.
System  Catalyst 

DOM  SFM  FFM  


%A_{MC} 

%A_{MC} 

%A_{MC}  
1 
TiO_{2}  0.20  0.9530  101.69  0.9611  103.21  0.9458  80.78 
0.40  0.9602  120.54  0.9583  110.80  0.9167  84.22  
0.60  0.9595  143.91  0.9242  116.33  0.8906  94.20  


2 
TiO_{2} anatase  0.04  0.9943  91.83  0.9945  80.09  0.9868  114.16 
0.09  0.9833  88.33  0.9744  73.70  0.9642  110.29  
0.14  0.9629  87.47  0.9421  73.52  0.9285  113.02  


3 
LiVMoO_{6}  0.20  0.8089  237.11  0.8468  81.79  0.9605  100.88 
0.40  0.8004  257.06  0.8518  76.16  0.9553  95.18  
0.60  0.7984  258.14  0.8571  70.79  0.9564  95.73 
Through Figures
Effect of catalyst concentration on the reactor radial section where photon absorption occurs (LVREA map) in system 3.
Simulated results of incident radiation as function catalyst loading.
To obtain a kinetic expression for photocatalytic oxidation of benzyl alcohol, the integral method was employed. An adjust by least squares was performed for different models, including the LHHW model, and it was found that the better adjustment is at pseudo first order in respect of concentration of benzyl alcohol. This result is in agreement with the results reported by [
Test for a pseudo firstorder kinetics as a function of concentration.
Taking into account the data of concentration—time obtained at each catalyst loading, an adjustment by least squares was performed to obtain the dependence of rate constants with the LVREA (Table
Firstorder kinetic constants and values of LVREA at different catalyst loadings.


LVREA (W/m^{3}) 

0.00  0.0101  0.00 
0.01  0.0587  630.00 
0.10  0.0750  5300.00 
0.40  0.0856  13600.00 
0.70  0.0882  16700.00 
1.00  0.0900  18000.00 
Figure
Adjustment for dependence of
Therefore, the kinetic equation that describes the photocatalytic oxidation of benzyl alcohol to benzaldehyde is
Equation (
Comparison of experimental concentration profiles (dots) with those obtained by the proposed model.
In Figure
The proposed mathematical model (FFM) describes the radiant field in a photocatalytic annular reactor. Its numerical solution corresponds appropriately with experimental and numerical data, and it requires a minor computational effort than the other models, such as DOM, which is very robust and accurate but requires a high RAM capacity. The FFM was specifically designed for cylindrical geometries with the lamp located at the axial axis of the reactor submerged in reaction medium. The FFM predicts the LVREA profiles better than the other models when
The obtained kinetic equation describes the reaction rate in the photocatalytic reactor for selective oxidation of benzyl alcohol as function of the LVREA. The FFM allows the evaluation of LVREA at different catalyst loadings, power lamp, or reactor dimensions. Therefore, it allows the calculation of reaction rates at different experimental setups.
Within the range of studied variables, the reaction rate of the selective oxidation of benzyl alcohol adequately fits a firstorder kinetics, where the kinetic coefficient is a function of LVREA, and this depends on catalyst loading, power lamp, and annulus width.
Model 2D: from the File menu, choose New. In the New window, click Model Wizard and select a 2D axisymmetric model.
Interface for ordinary differential equations, ODE: in the Select Physics tree, select Mathematics > ODEs Interface > ODEs in general form (g). Click Add. The Study is Stationary.
Parameters: go to Global definitions section and insert Parameters. In the Settings window for Parameters, locate the Parameters section and add the necessary parameters, such as reactor dimensions (reactor length, internal radius, and external radius), dispersion probabilities (
Global variable: go to Global definitions section and insert Variables. In the Settings windows for Variable, write the expression for radiation intensity (Irz), according to (
Geometry: set reactor geometry as rectangular section that represents the 2D axial section of the reactor. It can be drawn as a simple rectangle.
Model definitions: in model definitions, insert section for variables. In the Settings windows for Variable, insert the LVREA expression. It can be introduced with (
Set of differential equations: in the ODE interfaces, select all domains. Go to the ODE general form window settings and introduce the differential equations system defined for (
Incident intensity boundary: in ODE interfaces, insert a Dirichlet boundary condition. In this setting section, select the internal boundary (inner radius of reaction section). Locate boundary condition field and introduce (
External wall boundary condition: in ODE interfaces, insert a Dirichlet boundary condition. In this setting section, select the external boundary (external radius of reaction section). Locate boundary condition field and introduce (
Upper and bottom wall boundary condition: in ODE interfaces, insert a Dirichlet boundary condition. In this setting section, select the upper and bottom boundary. Locate boundary condition field and introduce (
Weak form of ODE: in ODE interfaces, insert a Weak Form for ODE condition. In this setting section, select all domains. In Weak expression field, introduce the following expressions:
WEAK = 0; 0; −test(gaz)^{∗}gaz + test(ga); −test(gcz)^{∗}gcz + test(gc).
Mesh: in the mesh section, introduce a mesh using the option of free quadratic mesh. You can try different mesh sizes. In free quadratic mesh, add distribution, select inner wall and locate the input section, and introduce 500 in number of elements.
Go to the study section and Run model.
Model 3D: from the File menu, choose New. In the New window, click Model Wizard and select a 3D model.
Radiation in participating media: in the Select Physics tree, select Heat Transfer > Radiation > Radiation in Participating Media (rpm). Click Add. The Study is Stationary.
Parameters: go to Global definitions section and insert Parameters. In the Settings window for Parameters, locate the Parameters section and add the necessary parameters, such as reactor dimensions (reactor length, internal radius, and external radius), characteristics of the lamp (power, dimensions, and wavelength), catalyst charge, and optical properties of catalyst (absorption, kappa_s; dispersion, sigma_s; and extinction coefficients, beta_s)
Global variable: go to Global definitions section and insert variables. In the Settings windows for Variable, write the expression for radiation intensity (Irz), according to (
Geometry: set reactor geometry as annular section. It can be drawn as a Boolean difference from two cylinders.
Model definitions: in model definitions, insert section for variables. In the Setting windows for Variable, insert the LVREA expression. It can be introduced as LVREA = rpm.G
Radiation in participating media (rpm): in the physics for radiation in participating media go to Radiation with participating media window settings and introduce dispersion and absorption coefficients in the model input sections.
In radiation in participating media, insert Incident intensity section. In this setting section, select the internal boundaries. Locate the incident intensity field and introduce Irz variable.
Opaque surface: in radiation in participating media, insert opaque surface. In the setting section, select external boundaries of contours of the domain. In wall adjust, select Black Wall.
Mesh: in the mesh section, introduce a mesh using the option of free tetrahedral mesh. You can try different mesh sizes. A too fine mesh can cause the available RAM to be exceeded.
Go to the study section and Run model.
Area of lamp (m^{2})
Catalytic particle area (m^{2})
Benzylic acid concentration (mol·dm^{−3})
Catalyst concentration (kg·m^{−3})
Flux incident radiation (Watts·m^{−2})
Upwards flux scattering radiation (Watts·m^{−2})
Flux backscattering radiation (Watts·m^{−2})
Downwards flux scattering radiation (Watts·m^{−2})
Spectral radiation intensity (Watts·m^{−2}·sr^{−1})
Apparent reaction constant (s^{−1})
Intrinsic reaction constant without catalyst (s^{−1})
Reaction constant with catalyst (s^{−1})
Intrinsic reaction constant with catalyst (Watts^{(−m)}·m^{(3m)})
Length (m)
Length of lamp (m)
Length of the reactor (m)
Local volumetric rate of absorption of energy (Watts·m^{−3})
Reaction order with respect to LVREA
Number of catalyst particles (m^{−3})
Probabilities of scattering toward up
Probabilities of backscattering
Probabilities of forward scattering
Phase function
Radial coordinate (m)
Inner radius of the annulus (m)
Radius of lamp (m)
Radius of reactor (m)
Time (h)
Volume (m^{3})
Conversion
Coordinate axial (m).
Extinction coefficient (m^{−1})
Thickness of the annulus (m)
Absorption coefficient (m^{−1})
Wavelength
Scattering coefficient (m^{−1})
Albedo coefficient
Solid angle (sr).
Discrete ordinate method
Fourflux model
Local volumetric rate of energy absorption
Langmuir–Hinselwwod–Hougen–Watson kinetic model
Monte Carlo model
Radiation transfer equation
Sixflux model
Twoflux model
Thinfilm slurry reactor of inner wall.
The authors declare that they have no conflicts of interest.
The authors are grateful to PRODEP for the financial support through Project 103.5/13/5257 and CONACYT through Project 269093. Mr. O. AlvaradoRolon is grateful to CONACYT for the financial support (Scholarship 401273) to conduct postgraduate studies. Citlalit Martínez Soto is acknowledged for the technical support.