The Effect of Thermal Radiation on the Surface Catalytic Chemical Reaction

The effect of thermal radiation on the two dimensional, steady-state, coupled heat and mass transfer from a fluid flow to a sphere in the presence of an exothermal catalytic chemical reaction on the surface of the sphere is investigated in the present work. The P1 approximation models the radiative transfer. The finite difference method was used to discretize the mathematical model equations. The discrete equations were solved by the defect correction multigrid method. The influence of thermal radiation on the sphere surface temperature, concentration and reaction rate was analysed. It was found that, for high values of the radiation conduction parameter, thermal radiation has a significant effect on the surface reaction.

The coupled heat and mass transfer to a sphere with a first-order, non-isothermal chemical reaction on the surface of the sphere was analysed numerically by Juncu and Mihail [18] without the boundary layer assumptions. Paterson [19] investigated the case of isothermal surface reaction in spherical geometry using the film model for mass transfer.
The aim of the present work is to make a first step in the investigation of the effect of thermal radiation on the forced convection heat and mass transfer to a sphere in the presence of an exothermal catalytic surface chemical reaction. The flow past the sphere is considered steady, laminar and incompressible. To the best of our knowledge, this problem is reported for the first time here. The P1 approximation [20,21] was used to model the radiative transfer. In spite of its simplicity, the P1 approximation is the best compromise between accuracy and computational efficiency. This paper is organized as follows. The next section describes the mathematical model of the problem. Section Method of solution presents the numerical algorithm. The numerical experiments made and the results obtained are presented in section Results. The concluding remarks of this work are briefly mentioned in the final section.

Mathematical model
Let us consider a sphere placed in a laminar, incompressible, steady flow of a Newtonian fluid. The flow configuration and coordinate system are shown in Figure 1. The diameter of the sphere d is assumed considerably larger than the mean free path of the surrounding fluid. The free stream velocity, temperature and concentration are U∞, T∞ and C∞. On the surface of the sphere an exothermic, first-order catalytic chemical reaction occurs. The physical properties are constant and isotropic. The effects of buoyancy, viscous dissipation and work done by pressure forces are considered negligible. The fluid is assumed to be a gray, emitting, absorbing and isotropic scattering medium.
For the assumptions discussed previously, the dimensionless energy and chemical species balance equations for a participating medium (the radius of the sphere a is considered the length scale and the free stream velocity U the velocity scale), expressed in dimensionless spherical coordinate system (r, θ), are: -Chemical species balance equation -Energy balance equation where qr,r and qr,θ are the dimensionless normal and tangential components of the radiative heat flux vector qr, qr = qr (qr,r, qr,θ). In the energy balance equation (2), the effect of thermal radiation is quantified by the divergence of the radiative heat flux vector qr. The relation for the computation of the radiative heat flux depends on the mathematical model used for the radiative transfer.

2.Method of solution
The mathematical model equations were solved numerically with the nested multigrid defectcorrection method [22]. The mass and energy balance equations (1,5) were discretized with the upwind and centered finite difference schemes (a double discretization required by the defect -correction iteration) on a vertex-centered grid, [23]. The spatial derivatives of the radiative transfer equation (6) were approximated by the centered finite difference scheme. Numerical experiments were made on meshes with the discretization steps, Δθ = π / 64, Δr = 1 / 64, Δθ = π / 128, Δr = 1 / 128 and Δθ = π / 256, Δr = 1 / 256. The external boundary conditions (3d) and (7c) are assumed to be valid at a large but finite distance from the center of the sphere.
The defect -correction iteration was applied only to the discrete approximation of the mass and heat balance equations. Two multigrid cycles were used inside the defect -correction iteration step. The structure of the MG cycle is: 1) cycle of type V; 2) smoothing by alternating line Gauss Seidel method; 3) two smoothing steps are performed before the coarse grid correction and one after; 4) prolongation by bilinear interpolation for corrections; 5) restriction of residuals by full weighting. The radiative transfer equation was solved by the multigrid method previously presented. The velocity field (VR, Vθ) were calculated solving numerically the Navier-Stokes equations. More information about the hydrodynamic computations can be viewed in [22].
The error criteria employed are: the discrete L2 norm of the residuals and the discrete L∞ norm of the difference between the numerical solutions of two consecutive defect -correction iterations are smaller than 10 -8 . Results that can be used to validate the accuracy of the present computations are not available in literature. The mesh independence of the Nu / Sh number and dimensionless sphere average surface temperature and concentration was the accuracy test used in the present computations.

3.Results and discussions
The dimensionless groups of the present problem can be divided into the following three classes: (1) convectiondiffusion dimensionless groups, Pr, Re and Sc; (2) radiative dimensionless groups, , ℰ, K and Rd; (3) surface reaction dimensionless groups B, φ 2 and γ.
The fluid flow past a rigid sphere is steady, laminar and axisymmetric for Re ≤ 210, Johnson and Patel [24]. The numerical values considered for the Pr and Sc numbers are, Pr = 1.0 and Sc = 2.0. The values of the radiative dimensionless groups , ℰ and K are given by the values of a, ka, β and ε. The values of ka and β were taken from [21]. The emissivity ε takes values in the range, 0.5 ≤ ε ≤ 0.9. The values considered for the radiation -conduction parameters Rd are, 0 ≤ Rd ≤ 10 4 . The numerical values assumed for the radius of the sphere are, a ~ 0.01 m, Wijngaarden et al. [25]. The values for the Prater number, B, the dimensionless activation energy, γ and the dimensionless reaction parameter, φ 2 , were selected following the recommendations made by Aris [26] and Froment et al. [27].
The main aspect analysed in this section is the influence of the radiation dimensionless groups on the surface catalytic chemical reaction. The values of the reaction parameters were chosen such that, when thermal radiation is neglected, the consumption of the reactant on the surface of the sphere is almost complete.
The influence of the radiation -conduction dimensionless group Rd on the sphere surface average dimensionless concentration, ̅ , and temperature, ̅ , is presented in Figures 2 and 3, respectively. The three curves plotted in each figure were computed for three different values of the sphere radius. Figures  2 and 3 show that the increase in Rd increases ̅ and decreases ̅ . The variation of ̅ and ̅ versus Rd is monotonous but not smooth. It is similar to a step function. The decrease in the sphere radius increases the Rd value for which the jump occurs. One can state that the ignition / extinction of the surface catalytic chemical reaction also depends on the intensity of the radiative transfer. In the next paragraphs, the Rd values for which ̅ and ̅ vary abruptly will be named critical domain.
The effect of the radiation -conduction parameter Rd on the dimensionless average heat and mass transfer rates and on the external effectiveness factor is presented in Figure 4.    The dimensionless directed -integrated intensity of radiation G is the solution of the diffusionreaction equation (6). Figure 9 shows that the surface profiles of G do not have the usual characteristics of the solution of a diffusion -reaction equation. The coupling between equations (6) and (7) and implicitly the interaction between Z and G determine the shape of the G profiles.
Concerning the effect of the other radiation parameters on the surface catalytic chemical reaction, the numerical experiments made have shown the followings: the increase in the absorption coefficient ka increases the effect of thermal radiation while the increase in the total attenuation factor β and the emissivity coefficient ε decreases the effect of thermal radiation. In the previous sentences, the increase in the effect of thermal radiation means the decrease in the Rd value for which the jump occurs.

Conclusions
The effect of thermal radiation on the steady-state, heat and transfer from a fluid flow to a sphere with an exothermic catalytic chemical reaction on the surface of the sphere is the topic of the present work. The radiative transfer is modeled by the P1 approximation. The case when the conversion of the reactant on the surface of the sphere is almost complete in the absence of thermal radiation is investigated in detail.
The effect of the thermal radiation on the surface catalytic chemical reaction consists of the decrease in the sphere surface temperature. The increase in the radiation -conduction group Rd increases the effect of the thermal radiation on the surface catalytic chemical reaction. The variation of the sphere surface average dimensionless temperature and concentration versus Rd is similar to a step function. The values of Rd for which the jump occurs depend on the radius of the sphere.
The extension of the present analysis to other values of the reaction parameters and convection rates (the products Re Pr and Re Sc) will be a challenge for future works. G * -dimensional directedintegrated intensity of the radiation ka -absorption coefficient k0pre-exponential factor Kdimensionless group, K = ka a nindex of refraction Nuaverage Nusselt number defined by relation (12) Nuθlocal Nusselt number defined by relation (11) Pr -Prandtl number, Pe =  / ρ α