Radiation Models for Computational Fluid Dynamics Simulations of Photocatalytic Reactors

2.1 Radiation Models for CFD

The Radiation Models described in this work are those implemented in the commercial software package ANSYS/Fluent. Table 2 summarizes the advantages and limitations of the radiation models.

2.1.1 P-1 Model

The P-1 radiation model is a widely used numerical method to solve the RTE in heat transfer. The P-1 approximation is the simplest case of the more general P-N model, which is based on the expansion of the radiation intensity into an orthogonal series of spherical harmonics ⁷⁹, ⁸⁰, ⁸⁶. The direction dependence in RTE is integrated out, transforming the RTE into a diffusion equation, which is easy to solve and compatible with the finite volume method ⁸⁰, ⁸⁷.

In the P-1 method, for gray radiation the transport equation for the incident radiation (G) (W m⁻²) is ⁸⁰:

(2)

where n is the refractive index of the medium, α (m⁻¹) is the absorption coefficient, σ (5.67 × 10⁻⁸ W m⁻²K⁻⁴) is the Stefan-Boltzmann constant, T (K) is the local temperature, S_G (W m⁻³) is a user-defined radiation source, and Γ (m) is the diffusivity of the incident radiation, which incorporates the effect of absorption, scattering, and phase function, and can be expressed by:

(3)

Where σ_s is the scattering coefficient and C is the linear-anisotropic phase function.

The radiation flux (q_r) is obtained by the following expression:

(4)

Combining Eqs. 3 and 5, the expression for is obtained and it can be directly substituted into the energy equation to account for heat sources (or sinks) due to radiation ⁸⁰.

(5)

Therefore, incident radiation can be obtained by solving the simpler diffusion equation ⁸⁷. The non-gray radiation can be modeled using the P-1 model, with a gray-band model.

The first reported work of P-1 model was in 1996, when Sazhin et al. presented the advantages and limitations of the P-1 model for thermal radiation, demonstrating that the P-1 model is expected to make reliable predictions in optically thick and thin media in a simple geometry ⁸⁶. The P-1 approximation was used to model radiative transfer in a rectangular enclosure in participating media ⁸⁸, in an annular photoreactor ⁷⁷, ⁸⁷, and a closed-conduit reactor ⁸⁹.

2.1.2 Discrete Ordinates (DO) Model

The DO model transforms the integro-differential form of the RTE into a system of algebraic equations ⁷². The RTE is solved for a finite number of discrete solid angles, each associated with a vector direction () ⁸⁰. Thus, the DOM solves the RTE through directional and spatial discretization, giving a set of linear simultaneous equations for the radiation intensity at various points ⁷⁸. The DO model solves for as many transport equations as there are directions () ⁷⁹, ⁸⁰.

The DO model does not involve any simplified assumption other than the discretization of pertinent partial differential equations ⁷⁸. Therefore, this model is widely used because the obtained results have high accuracy. The DO model considers the integration of the radiative transfer equation calculated as ⁸⁰, ⁹⁰:

(6)

where the variables are: the radiation intensity, the position vector, the direction vector, the scattering direction vector, α the absorption coefficient of the medium, σ_s the scattering coefficient, n the refractive index of the medium, σ the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻²K⁻⁴), T the local temperature, ϕ the phase function, and the solid angle.

As for the P-1 model, the non-gray radiation can be performed using the DO model, with a gray-band model, which divides the radiation spectrum into N wavelength bands.

In the DO model, the RTE is integrated over the control volume (existing mesh) using a finite volume method ⁸⁰, ⁸⁴. For the case of anisotropic scattering at complex surfaces, the DO model does not conserve radiant energy. Some CFD codes implemented a conservative variant of the DO to tackle this issue ⁹¹.

Through the years, the DOM was used in the simulation of several types of photochemical or photocatalytic (immobilized or slurry) reactors due to its accuracy. Romero et al. proposed the DOM to solve the RTE in a heterogeneous reaction medium in a photocatalytic system with an annular geometry ⁹². Subsequently, many researchers studied the radiation field using the DOM in annular photoreactors ¹¹, ⁷⁸, ⁹⁰, ⁹³. The radiation field in a flat-plate reactor using DOM was investigated by Brandi et al. ⁹⁴.

Sgalari et al. computed the radiative transfer with DOM within an annular photoreactor for typical situations of photocatalytic processes, investigating the effects of the parameters on the local rate of radiant energy absorption and the angular distribution of the radiation intensity ⁹³. The radiation transport equation was solved using a finite-volume-based DO method for an annular bubble-column photocatalytic reactor with a three-phase flow for photodegradation of Bayer liquor ¹¹. Pareek et al. applied the MC approach to estimate light intensity distribution (LVREA) in an annular photocatalytic reactor and the DO model to assess the effect of wall reflectivity, catalyst loading, and phase function parameter on the light intensity distribution ⁷⁸.

Bagheri and Mohseni developed a CFD model for simulating annular VUV/UV photoreactors considering reflection, absorption, and refraction of photons by the wall and slurry media in radiation modeling using the DO method ⁹⁰. Li et al. used the DOM to study the wall reflection on an annular single-lamp UV reactor, which directly affects the radiation distribution ⁹⁵. Asadollahfardi et al. presented a comparison of the reliability of the DOM with MC predictions for solving RTE ⁷⁶. The DOM is widely applied to model steady-state radiative transfer in media ⁹⁶.

3.1 Geometry Model and Computational Grid

The case study reactor has an annular geometry with 645 mm of length, 15 mm of external radius, and 2.25 mm of internal radius. In this configuration of the photoreactor, a UV lamp was placed in the middle of the cylindrical vessel. In the laboratory case, the lamp is enclosed in a quartz sleeve to isolate it from the reacting medium, but in the simulated system the quartz medium was not considered. This reactor has been used in several experimental and numerical works ⁷⁷, ⁸⁵, ⁸⁷. The photoreactor shown in Fig. 4a was simulated in 2D and 3D. The inlet and outlet correspond to the circular crowns at the ends of the cylindrical reactor. The geometry model was generated using ANSYS DesignModeler®.

After the generation of the geometry, the computational grid was generated using ANSYS Meshing®. The geometric model was discretized, subdividing it into small finite volumes that define the computational grid and where the equations are solved. The generation of the grid is one of the most critical steps of the CFD analysis because the size of the elements influences the accuracy of the results due to numerical approximations. The mesh structure is quite relevant for the simulation of radiation. Thus, there is a compromise between the accuracy of the results and the computational effort required.

Figs. 4b and 4c represent the computational grid of the 3D model and 2D model, respectively. The 2D meshes are made of regular quadrilateral elements. Refinement in this mesh consists of the division of the quadrilaterals. The 3D mesh is made mainly of hexahedral elements, which start as quadrilaterals in the top face and are repeated in layers along the heigh. This repetition creates a regular mesh.

Tab. 3 summarizes the properties of the studied meshes. The 2D meshes were generated using several element sizes (meshes 1, 2, and 3). The lamp wall was identified as the most critical region of the mesh. Thus, meshes 4 and 5 were generated to refine the region near the lamp wall by adding several layers. The 2D domain was generated using an axisymmetric model due to the symmetric condition of the photoreactor. The 3D mesh 7 was refined in Ansys Fluent using different levels of refinement near the lamp wall.

3.2 Selection of Radiation Model

The DO model using the finite volume method (FVM) of ANSYS Fluent R2020® was chosen to simulate the RTE. This is a promising method for modeling radiative transfer due to its extensive applicability and high accuracy.

The P-1 model was not employed because the chosen case had low optical thicknesses. The medium which defines the optical thickness is air and the maximum length in the geometry of the problem is approximately 0.15 m. The MC model was excluded due to its ergodic treatment of the problem. To obtain good approximations of the physical quantities, a high number of histories of the photons was necessary, which results in a high computational cost.

3.3 Lamp Characteristics

The presence of radiation is the distinguishing factor of the photoreactors, so it is necessary to consider the characteristics of the radiation source (lamp). The reactor simulated has a single lamp concentric to the body reactor.

Considering the real characteristics of a UV lamp, the rays pass through the lamp body and lamp sleeve until they reach a point in the medium. This is a complex phenomenon because of absorbance, reflection, and refraction of radiation from/through the lamp's sleeve and body ¹⁰¹. The presented model assumes that there is no sleeve, the UV source behaves as a diffuse radiant emitter, and refraction and reflection through and from the surface of UV lamp are neglected. The sufurce emission model was used to determine the boundary condition for the tubular lamps ¹⁰² and the average value of irradiance computed from Eq. 7 was 127.3 W m⁻² considering a power output of the lamp P_lamp = 6.58 W ⁸⁵.

(7)

Where d_lamp (m) is the lamp diameter and L (m) is the lamp length.

3.4 Boundary Conditions

In the CFD analysis, air was considered as the fluid in the annular space, and its properties were set using the standard models available in Fluent. For the external reactor wall and the wall of the lamp, the properties of fused quartz were considered. The properties of the materials are provided as Supporting Information.

The simulation of the radiation distribution implies the calculation of irradiation, which corresponds to the radiation emitted by the lamp to set as boundary condition, considering the lamp a semi-transparent wall. Then, the simulations were carried out simulating an infinite cylinder, defining as radiation boundary condition the intensity of the radiation emitted by the lamp.

For both 2D and 3D cases, the flow was set as stagnant. For 2D cases, air inlet and outlet were considered as interface boundaries linked by a periodic condition. Because the pressure gradient between the boundaries was set to ΔP/x = 0 Pa m⁻¹, there is no flow field in this case. In 3D cases, air inlet and outlet were considered as walls, with the three velocity components set to zero. The boundary conditions used in the 2D and 3D models are summarized in Tabs. 4 and 5, respectively.

To obtain the radiation field from the RTE without the effect of thermal radiation, the temperature of the domain was set to1 K (minimal temperature supported in ANSYS Fluent R2020®). The flow was stagnant, so although all sets of equations were defined, as well as the boundary conditions (BCs), the actual flow equations were not solved, and so the flow variables were all set to zero. This enables to focus solely on the issues pertaining to the simulation of radiation in CFD, particularly with ANSYS/Fluent.

3.5 Numerical Solution and Convergence Criteria

In the chosen case the flow is stagnant and the boundary conditions for the radiation and the energy are constant. So, the solvers were set to steady state and only the sets of equations for radiation and energy are solved. The discretization scheme is second-order upwind for the energy and second-order upwind for discrete ordinates. The convergence criterion for the numerical solution was residuals should be less than 10⁻²⁰. The numerical mesh validity is checked in Sect. 4.1, for the several meshes described in Sect. 3.1.

4.1 Study of the Computational Grid

Assuming a case with a medium that does not absorb light, the mesh element size and the number of layers in the lamp wall were assessed. The 2D meshes were refined using ANSYS Meshing®, one of the 3D meshes was refined in Ansys Fluent using the refinement tool and another in ANSYS Meshing®. In 3D cases, the simulations were carried out using 20 × 20 directions and 10 × 10 pixilation of angular discretization Figs. 5 and 6 present the light distribution along the radius at a line in the mid-axial position, for 2D and 3D models, respectively.

It is observed that the refinement of the grid in the lamp wall influences the result of the radiation distribution. Only the more refined meshes around the lamp sleeve can describe accurately the radiation intensity. Away from the lamp, the local values in the neighbouring of the sleeve do not influence the overall radiation profile. For the simulation of radiative transfer, it is used a mesh able to accurately describe the full radiation profile. The selected mesh has elements with a size of 3.0 mm where refinement of the 67-layers in the lamp wall was considered for the 2D axisymmetric model. In the 3D model, a grid with an element size of 10 mm and refinement of the 40-layers shows the best results.

These findings indicate that the 2D model describes the same radiation profile behavior as the 3D model. This accordance between geometric models was expected because the 3D case is axisymmetric, with complete homogeneity through the angular direction.

4.2 Study of the Angular Discretization

Angular discretization and pixelation play a major role to obtain a reliable solution for RTE. At any spatial location, each octant of the angular space 4π is divided into solid angles (N_θ × N_ϕ), in which θ and ϕ represent polar and azimuthal angles. In 2D calculations, there are only 4N_θ × N_ϕ directions since only four octants are solved due to symmetry. In 3D cases, 8N_θ × N_ϕ directions are solved. For Cartesian meshes, it is possible to align the global angular discretization with the control volume face. However, when generalized unstructured meshes are used, control volume faces do not align with the angular discretization, leading to control angle overhang. Control angle overhang can be corrected using pixelation. The overhanging control angle is discretized into N_θp × N_ϕp ⁸⁰.

The RTE is simulated for two angular discretizations: (i) 5 × 5 directions (solid angles) and pixilation of 3 × 3 and (ii) 20 × 20 directions and pixilation of 10 × 10 were performed considering the non-absorption case (α = 0 m⁻¹) for 2D and 3D models. Figs. 7 and 8 represent the radiation profile and radiation intensity results for each case of angular discretization for 2D and 3D models, respectively.

The results of angular discretization indicate that for the 2D case the angular discretization does not have a role in the distribution of radiation results, because radial homogeneity is already implicit when simulating only a slice of the reactor. For the 3D case, an increase in angular discretization enables more uniform distribution of light, namely in the lamp wall.

Fig. 8 displays the radiation intensity map for two angular discretization values. The visual effect of the discretization is striking even at the lamp surface with the occurrence of stripes of larger intensity. This heterogeneity is propagated into the medium, as seen in the axial cut of the reactor. The effect of angular discretization is quantified in Fig. 8, where the angular distribution of the radiation intensity is shown at five radial locations. For the less dense angular discretization, the radiation has a wavy pattern originating at the source boundary. Thus, the simulations for 3D cases were performed using an angular discretization of 20 × 20 directions and 10 × 10 pixilation.

4.3 Effect of Absorption Coefficients in Radiation Profiles

For the study of absorption coefficients, only 2D simulation was performed, since for the non-absorption case the 2D and 3D results were very similar. The RTE was solved by discretizing each control volume into 5 × 5 directions (solid angles) and pixelation of 3 × 3 for the 2D model. After the numerical validation of the simulation conditions, the models were used for absorption coefficients in the range of α = 0 to 357.6 m⁻¹. The 2D light distribution profiles for the several absorption coefficients are depicted in Fig. 9.

The incident radiation map inside the annular reactor shows the overall profile from the lamp towards the outer wall. The distribution presented in Fig. 9 can be expected for the annular configuration of the photoreactor due to the radial increase of the section for radiation associated with the medium absorption. The radiation profile is flat throughout the reactor length due to the periodic BCs, so a representative profile can be obtained along any axial position. The dimensionless light distribution profiles along the radius are depicted in Fig. 9, which conveys a more detailed picture of the effect of the absorption coefficient.

The profiles presented in Fig. 9 show that the light intensity decreases rather sharply with the increase in the absorption coefficient. For the case with zero absorption coefficient, the decrease of incident radiation is due to the increment of the photonic flow area. When the absorption coefficient is equal to 357.6 m⁻¹, 90 % of the incident light is absorbed in the initial ≈ 1.5 mm away from the lamp. This indicates that the exposure of the catalyst to radiation, when there are heterogeneities of absorption in the medium, will depend on the flow patterns and degree of mixing within the reactor in the photocatalytic systems. If the flow is very segregated promoting large gradients in the light absorption throughout the reactor, the illumination of the catalyst in the outer wall will be very heterogeneous.

4.4 Comparison with Reported Data

The presented models are compared with experimental data of Quan et al. (2004) ⁸⁵ and simulation data from Yu et al. ⁸⁷ and Huang et al. (2011) ⁷⁷ whose works used the same reactor presented in this work. Quan et al. (2004) ⁸⁵ made experiments in a light-attenuating medium (ozone+air) using potassium ferrioxalate actinometry to observe the absorption effect on the light intensity profile. Simulation data from Yu et al. (2008) ⁸⁷ using the P-1 model to assess the effect of absorption coefficient, scattering coefficient, wall reflectivity, and phase function parameter in an annular photoreactor and data from Huang et al. ⁷⁷ using the FVM to perform the radiative transfer in absorption and scattering media for a wide range of optical thickness were considered in this work. Also, the results were compared with the profile obtained from the Lambert-Beer equation for annular geometries ¹⁰³,

(8)

where I is the radiation intensity (W m⁻²), which depends on initial radiation intensity (W m⁻²), I₀, the reactor radius (m), r₀, the local radius (m), r, and absorption coefficient (m⁻¹), α. The results are shown in Fig. 10.

From Fig. 10 it was concluded that the models to simulate radiation distribution inside an annular reactor described the same behavior as reported in the literature. Also, the proposed method to solve the RTE by the Lambert-Beer law proposed by Abu-Ghararah ¹⁰³ displays the same curve shape for all absorption coefficients. Thus, this review presents a reliable CFD method to simulate the radiation distribution inside an annular reactor.