Prediction of coupled radiative and conductive heat transfer in concentric cylinders with nonlinear anisotropic scattering medium by spectral collocation method

: Accurate prediction of the angular and spatial distributions of radiative intensity is a very important and challenging issue for the coupled radiation and conduction problem with nonlinear anisotropic scattering medium. Different with the traditional hybrid spectral methods, spectral collocation method associated with discrete ordinate method (SCM-DOM), the spectral collocation method is extended to discretized both angular and spatial domains of governing equations in concentric cylinders. The angular and spatial derivative terms of governing equations in the cylindrical coordinate system are approximated by high order Chebyshev polynomials instead of the low order finite difference schemes. The performance of SCM is evaluated by comparing with available data in literature. Numerical results show that convergence rates of angular and spatial nodes approximately follow the exponential decaying law. In addition, for nonlinear anisotropic scattering medium, the SCM provides smoother results and mitigates the ray effect. The SCM is a successful and efficient method to deal with coupled radiative and conductive heat transfer in concentric cylinders. Furthermore, the effects of various geometric and thermal physical parameters on dimensionless temperature and heat flux are comprehensively investigated.

Abstract: Accurate prediction of the angular and spatial distributions of radiative intensity is a very important and challenging issue for the coupled radiation and conduction problem with nonlinear anisotropic scattering medium. Different with the traditional hybrid spectral methods, spectral collocation method associated with discrete ordinate method (SCM-DOM), the spectral collocation method is extended to discretized both angular and spatial domains of governing equations in concentric cylinders. The angular and spatial derivative terms of governing equations in the cylindrical coordinate system are approximated by high order Chebyshev polynomials instead of the low order finite difference schemes. The performance of SCM is evaluated by comparing with available data in literature. Numerical results show that convergence rates of angular and spatial nodes approximately follow the exponential decaying law. In addition, for nonlinear anisotropic scattering medium, the SCM provides smoother results and mitigates the ray effect. The SCM is a successful and efficient method to deal with coupled radiative and conductive heat transfer in concentric cylinders. Furthermore, the effects of various geometric and thermal physical parameters on dimensionless temperature and heat flux are comprehensively investigated. Greek Symbols: *

Introduction
The combined radiative-conductive heat transfer in participating medium [1] plays a dominant role in high temperature equipment, such as aeroengine combustor, nuclear reactor and industry furnaces, etc. Accurate prediction of temperature and heat flux requires solving radiative transfer equation (RTE) and energy equation simultaneously [2][3][4][5][6]. Different with RTE in Cartesian coordinate, RTE in cylindrical coordinate exists the angular derivative term which would increase the mathematical complexity.
In recent years, the coupled radiative-conductive heat transfer in participating medium of cylindrical geometry has evoked wide interests of many researchers. As early as 1982, Fernandes and Francis [7] gave the rigorous formulations of combined conduction and radiation in concentric cylinders and numerically solved by Galerkin finite element method. Pandey [8] employed undetermined parameters method to solve this coupled problem for gray and nongray gases contained between infinitely long concentric cylinders with black surfaces. Krishnaprakas [9] used the hybrid strategy to analyze combined conduction and radiation in cylindrical geometries. In this paper, energy equation was solved by finite different method, and RTE was solved by discrete ordinates method in conjunction with Crank-Nicolson scheme. The effects of thermal-physical parameters, namely emittance, scattering albedo, scattering phase function, conduction-radiation parameters on heat fluxes were investigated. Dlala et al. [10] investigated coupled radiative-conductive heat transfer in gray hollow spheres and cylinders. They used finite Chebyshev transform (FCT) to improve the performance of discrete ordinates method, and adopted Chebyshev polynomials to approximate the angular derivative term instead of finite difference scheme. The FCT was more accurate than traditional discrete ordinate method. Mishra et al. [11,12] developed the modified discrete ordinate method and lattice Boltzmann method to analyze coupled radiative-conductive heat transfer in infinite and finite concentric cylinders with absorbing, emitting, and scattering medium. Authors claimed this modified discrete ordinate method was not require complicated and intensive calculation to determine the discrete directions and directional weights, allowed freedom of direction selection. Zhou et al. [13] extended the MDOM, which based on superposition technique and considering the contributions of the walls and medium, to the cylindrical medium. And the results showed that the cost computational time was comparable to DOM and the ray effect can be mitigated effectively.
Different with above numerical algorithms, spectral methods [14][15][16] are useful tools to solve ordinary differential equations or partial differential equations with high order accuracy, and usually the best choice for solving problem with smooth solution [17]. Benefiting from high accuracy, simple implementation, and exponential convergence characteristic, they have been widely used to solve problems in many fields, such as computational fluid dynamics [18,19], magnetohydrodynamics [20], and optics [21]. Using spectral methods to solve thermal radiative heat transfer already has a history of two decades [22][23][24][25][26]. Among these researches, several hybrid spectral methods such as spectral collocation method associated with discrete ordinate method (SCM-DOM) [22][23][24] and spectral element method combined with discrete ordinate method (SEM-DOM) [25] were developed, in which the spatial domain was discretized by spectral methods, and the angular domain was discretized by DOM. Recently, Wang et al. [26] further extended the SCM-DOM for solving polarized radiative transfer problems in multi-layered participating media. Different with the above hybrid spectral methods, Zhou et al. [27,28] taken advantage of SCM to discretize the entire spatial and angular domain rather than only the spatial domain. Their numerical test showed that SCM can achieve the high accuracy both in spatial and angular directions. They further developed SCM to solve radiative integro-differential transfer equation [29] in one-dimensional medium which only contains unknown radiative flux and eliminate the effect of angular derivative term. However, in common situations, radiative heat transfer is coupled to other models of heat transfer. To the best of authors' knowledge, no research to date has aimed to using SCM to solving coupled radiative and conductive heat transfer so far.
The objective of this work is to extend the SCM to solve coupled radiative and conductive heat transfer in concentric cylindrical medium. This paper is organized as follow: In section 2, the physical and mathematical models of coupled radiative and conductive heat transfer in concentric cylinders are presented. In section 3, the SCM formulations of RTE and energy equation are deduced. In section 4, the performance of SCM is verified from the available data in the literature. In section 5, the effects of various geometric and thermo-physical parameters are comprehensively investigated. Finally, the conclusions are summarized in section 6.

Physical and mathematical models of concentric cylinders
As shown in Figure 1, the present study considers the coupled radiative and conductive heat transfer processing in cylindrical coordinate system. The absorbing, emitting and anisotropic scattering medium is filled in concentric cylinders. In concentric infinite cylinders, the non-conservation form of RTE for a gray medium is [30][31][32] where ( , ) I r Ω is the radiative intensity at spatial position r along angular direction Ω ; The direction Ω can be expressed by the direction cosines sin cos where  is polar angle and  is azimuthal angle; a  and s  are absorption coefficient and scattering coefficient, respectively. The anisotropic scattering phase function ( , )   Ω Ω represents the probability that a radiative beam along angular direction  Ω is scattered to angular direction Ω , and is approximated by a finite series of Legendre polynomials as where n A is the coefficient of Legendre expansion of order n which are listed in Table 1 [33]; n P is the Legendre polynomials;  is the included angle between the incident direction  Ω and the scattering direction Ω .
where w  is the wall emissivity, , b w I is the blackbody radiative intensity at the wall, w n is unit wall normal. As shown in Figure 2, the following symmetric condition is satisfied Illustration of the symmetry.
For the infinite concentric cylinders, the steady-state energy equation can be written as with the boundary conditions where G is the incident radiative energy, For convenience of analysis, the following dimensionless parameters [10] are introduced , ,

Spectral collocation discretization
As shown in Figure 3, the spatial-angular domain       where i  is the Gauss-Lobatto points. Volume 9, Issue 3, 581-602. Then, dimensionless radiative intensity and dimensionless temperature can be approximated by Lagrange interpolation polynomials and collocation points [16]  where i h are Lagrange interpolation polynomials.
Substituting Eq (14) into Eq (9), weighting by weight function and integrating over the computational domain, Eq (9) can be discretized as The corresponding boundary conditions are discretized as * * , Similarly, the energy equation is discretized as  [14] are the first and second order derivative matrix, respectively.

The accuracy and efficiency of SCM
Based on the above described SCM model for coupled radiative and conductive heat transfer in concentric cylinders with participating medium. In the following, several test cases are adopted to verify the performance of SCM model. Compared with available data in references, the accuracy and efficiency of SCM for coupled radiative-conductive heat transfer in concentric cylinders are validated.
In order to quantitatively evaluate the accuracy of SCM, the integral averaged relative error is defined as  The distribution of dimensionless temperature within concentric cylinders by SCM is plotted in Figure 5, and compared to LBM-FVM results. The SCM results is very close to those of the LBM-FVM results, and the integral average relative error is 0.875%.   Then, we would verify the accuracy in the whole domains by testing whether the condition * * constant is satisfied. This is because as long as the equilibrium is satisfied, the energy equation always writes div * 0 where * is the dimensionless total heat flux and is spatial position. For a one-dimensional system, spatial position is simplified as radial distance and the divergence equation in cylindrical coordinates simply writes * * 0 which means * * constant. In Table 3, the results obtained with different collocation point numbers * * are given, where the standard deviation ∑ * * * * , the * * is the mean value of the * * , i.e, * * ∑ * * .
It can be seen that in all case of different number of collocation points, * * remains almost constant everywhere in the medium, with a very small deviation insuring four or five correct digits. Therefore, the accuracy of SCM is verified everywhere in the domain.
In order to the efficiency of this SCM model, Figure 6 depicts the effect of collocation point numbers on the integral averaged relative error. As shown in Figure 6, the horizontal axes are the number of radial points, azimuthal angle points and polar angle points, respectively. The vertical axis is the integral averaged relative error for the case of

Results and discussions
In order to comprehensively analyze this coupled heat transfer, the effects of geometric and thermo-physical parameters on dimensionless temperature and heat flux are investigated. In the subsection, the effects of different kinds of scattering phase functions are firstly studied, and then investigate the effect of various geometric and thermal physical parameters based on nonlinear anisotropic F 1 scattering phase function. Figure 7 presents the effect of scattering phase function on dimensionless temperature distribution. In these scattering phase functions, F 3 scattering phase function 3

The effect of scattering phase function
is a typical forward scattering phase function, B 3 scattering phase is a typical backward scattering phase function. Figure 8 shows the dimensionless temperature distribution for three kinds of typical cases, . When * r is small, namely the region near the inner wall with high temperature. The dimensionless temperature 3 3 B isotropic scattering F      . While in the region near the outer wall with low temperature, the dimensionless temperature of the three kinds of scattering phase function is almost the same.    Firstly, the region near the high temperature inner wall is investigated. As shown in Figure 9, the dimensionless temperature distribution 3 3 B isotropic scattering F      . The reason is that the forward scattering phase function means more forward transmit power while the backward scattering phase function means more backward scattering energy. At the same time, Figure 10 shows that the ratio of radiative heat flux to total heat flux in the region near the inner wall is more than 0.9. Therefore, for the case of forward scattering phase function F 3 , the temperature near the inner wall is higher than that of the case of isotropic scattering. Meanwhile, compared with the case of isotropic scattering, the case of backward scattering phase function B 3 leads to a lower temperature.
Secondly, the region near the outer wall with the low temperature is analyzed. As shown in Figure 9, the higher temperature distribution near the inner wall leads to the lower temperature distribution near the outer wall, namely 3 3

F isotropic scattering B
     . However, as shown in Figure 10, the ratio of radiative heat flux to total heat flux near the outer wall is smaller than that of other regions. This means the influence of radiation is reduced near the outer wall. Consequently, the increase in temperature caused by the forward scattering phase function and the decrease in temperature caused by the backward scattering phase function are both weakened. Therefore, the dimensionless temperature distributions of the three kinds of scattering phase functions are almost the same in the regions near the outer wall.

The effect of conduction-radiation parameter
As shown in Figure 11, the dimensionless temperature distribution tends to be linear with the increasing of conduction-radiation parameter cr N from 0.01 to 10. The conduction-radiation parameter is defined as the ratio of conductive heat transfer and radiation heat transfer. It is obvious that, for the large value of cr N , conduction plays a dominant role in this coupled heat transfer problem. On the contrary, the small value of cr N means radiation becomes much more pronounced.
For the conduction-dominated problem, the variation of dimensionless temperature tends to linear. Meanwhile, the variation of dimensionless temperature tends to nonlinear for the radiation-dominated problem. Thus, with the increasing of conduction-radiation parameter, the distribution of dimensionless temperature tends to linear.

The effect of radius ratio
For the cases of  Figure 12c and d. In Figure 12c, the dimensionless temperature increases as the radius ratio changes from 0.1 to 0.5. But, in Figure 12d, the dimensionless temperature shows a downward trend instead of continuing to rise, and the dimensionless temperature profiles tends to be linear. These phenomena can be explained by studying the radiative heat flux and the ratio of radiative heat flux to total heat flux, which are depicted in Figures 13 and 14, respectively.
In the three cases of 1, 0.1, 0.01 cr N  , the radiative heat flux would augment steady with the increasing of the radius ratio. But the ratio of radiative heat flux to total heat flux firstly increases with radius ratio changing from 0.1 to 0.5, and then decreases with radius ratio increasing from 0.5 to 0.9.
It's worth to note that, for the case of ) means the radiative heat transfer plays a dominant part. When the radius ratio changes from 0.1 to 0.5, the radiative heat flux is increasing. Consequently, the dimensionless temperature would rise and the profile presents the nonlinear characteristics. When the radius ratio keeps increasing from 0.5 to 0.9, the radiative heat flux continues to increase, but the ratio of radiative heat flux to total heat flux decrease.
Considering the dominant role of radiative heat transfer, the decrease of ratio would definitely change the profile of temperature distribution, namely the profile tends to be linear which shows the increasing influence of conductive heat transfer. These are reasons why the dimensionless temperature rises firstly, then decrease and tends to be linear in the case of 0.01 cr N  .

The effect of scattering albedo
As shown in Figure 15, the dimensionless temperature decreases with the increasing of scattering albedo, and the distribution of dimensionless temperature also tends to be linear. The scattering albedo means the relative magnitude of absorption coefficient and scattering coefficient.
= 0  means the no-scattering medium, and = 1  indicates the pure scattering medium. When scattering albedo approaches to 1, the scattering becomes stronger. This means the less radiant energy is absorbed. Consequently, the distribution of dimensionless temperature shows a downward trend. Meanwhile, for the case of a larger scattering albedo, the less radiant energy is absorbed means that conduction dominates a larger part in this coupled problem. Therefore, the distribution of dimensionless temperature tends to be linear. Figure 15. Effects of scattering albedo  on dimensionless temperature distribution.

Conclusions
SCM is developed to solve the coupled radiation-conduction problem in concentric cylinders with absorbing, emitting and nonlinear anisotropic scattering medium. Both the two involved RTE and steady-state energy equation are solved by SCM. In the solving process, the spatial and angular domains of RTE, and the spatial domain of energy equation are discretized by high order Chebyshev polynomials and Chebyshev collocation points. Compared with available data in references, accuracy and efficiency of the SCM for the coupled radiative-conductive heat transfer are validated. The high order accuracy can be obtained in a few nodes, and the exponential convergence characteristic of SCM exists in both spatial and angular domains. Considering that the SCM can obtain the high order accuracy and exponential convergence rate, the SCM model is an efficient model to solve the coupled radiative-conductive heat transfer in concentric cylinders with nonlinear anisotropic scattering medium. Besides, the effects of scattering phase function, conduction-radiation parameter, radius ratio and scattering albedo on dimensionless temperature and heat flux are comprehensively investigated.