ARTICLE IN PRESS. Received 20 June 2007; received in revised form 14 November 2007; accepted 20 November 2007

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1 Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) Radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and particles using the modified discrete-ordinates method Man Young Kim a,, Ju Hyeong Cho b, Seung Wook Baek c a Department of Aerospace Engineering, Chonbuk National University, Jeonju, Chonbuk , Republic of Korea b Environment & Energy Research Division, Korea Institute of Machinery and Materials, Daejeon , Republic of Korea c Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon , Republic of Korea Received 0 June 007; received in revised form 4 November 007; accepted 0 November 007 Abstract The radiative heat transfer between two concentric spheres separated by a two-phase mixture of non-gray gas and a cloud of particles is investigated by using the combined finite-volume and discrete-ordinates method, named modified discrete-ordinates method (MDOM), which integrates the radiative transfer equation (RTE) over a control volume and a control angle simultaneously like in the finite-volume method (FVM) and treats the angular derivative terms due to spherical geometry as the conventional discrete-ordinates method (DOM). The radiative properties involving non-gray gas and particle behavior are modeled by using the extended weighted sum of gray gases model (WSGGM) with particles. Mathematical formulation and final discretization equations for the RTE are introduced by considering the behavior of a two-phase mixture of non-gray gas and particles in a spherically symmetric concentric enclosure. The present approach is validated by comparing with the results of previous works including gray and non-gray radiative heat transfer. Finally, a detailed investigation of the radiative heat transfer with non-gray gases and/or a two-phase mixture is conducted to examine the dependence of the radiative heat transfer upon temperature ratio between inner and outer spherical enclosure, particle concentration, and particle temperature. r 007 Elsevier Ltd. All rights reserved. Keywords: Radiative heat transfer; Concentric spheres; Modified discrete ordinates method; Non-gray WSGGM; Particles. Introduction For many engineering applications such as droplet combustion, spherical reacting systems, and droplet radiator systems for spacecraft thermal control, spherically symmetric assumption is usually made due to its geometric and theoretical simplicity and thereby computational benefits because it physically describes threedimensional phenomena with one-dimensional procedure []. Therefore, substantial efforts have been exerted to analyze the spherically symmetric problems in the field of radiation as well as flow and heat transfer Corresponding author. Tel.: ; fax: address: manykim@chonbuk.ac.kr (M.Y. Kim) /$ - see front matter r 007 Elsevier Ltd. All rights reserved. doi:0.06/j.jqsrt

2 608 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) Nomenclature a m I coefficient of the discretization equation in the direction m at nodal point I C p particle concentration (kg/m 3 ) D m i directional weight in the direction m at surface i, Eq. (7) D m r directional weight in the direction m at the surface normal to *r-direction, Eq. (7) ~e r r-direction base vector G incident radiation, Eq. (4) I radiative intensity, (W/(m sr)) I b blackbody radiative intensity, ¼ st 4 /p (W/(m sr)) K,M total number of gray gases and radiation direction N w 0N r number of control angle and control volume, respectively ~n w unit normal vector at the wall towards the medium q radiative heat flux (W/m ), Eq. () ~r position vector ~s direction vector w weight Greek symbols DA, DV surface area and volume of the control volume, respectively DO m discrete control angle (sr) a m= coefficient of the angular derivative term, Eq. (5) b extinction coefficient, Eq. (6b) w polar angle measured from the r-axis, see Fig. e wall emissivity k,s s absorption and scattering coefficients, respectively (m ) m direction cosine in the r-direction, m ¼ cos w, see Fig. Subscripts, inner and outer spherical walls, respectively, see Fig. N, S north and south neighbors of P P nodal point in which intensities are located g gas k kth gray band n, s north and south control volume faces p particle r r-direction Superscript m radiation direction including combustion [ 9]. During the past few decades, numerous methods have been proposed to solve the radiative transfer equation (RTE) between two concentric spheres separated by radiatively active medium. Among others, while Viskanta and Crosbie [] considered a non-scattering, gray, and heat-generating medium between two concentric spheres with the differential approximations, Tsai et al. [3] investigated the thermal radiation in spherical symmetry with anisotropic scattering and variable properties by using the discreteordinates method (DOM). Jia et al. [4] extended the Galerkin method to investigate the radiative heat transfer

3 between two concentric spheres separated by an absorbing, emitting, and isotropically scattering gray medium. Recently, Sghaier et al. [5] developed a new method for the solution of the RTE in spherical media based on the DOM with finite Legendre transform (FLT) to model the angular derivative term. As a neutron travels through a curved geometry such as cylindrical or spherical coordinate system, the propagating direction relative to the coordinate system is constantly varying, even though the neutron does not physically change its direction. This angular redistribution [6] makes it difficult to handle the angular derivative term appearing in these coordinates. To overcome this phenomenon in spherical symmetric systems two different approaches are suggested. The first one is the conventional artifice of Carlson and Lathrop [7], followed by Lewis and Miller [6], and Tsai et al. [3], where a recursive relation for the coefficients a m= is modeled by examining the divergenceless flow condition. Another procedure that approximates the angular redistribution is FLT DOM by Sghaier et al. [5], where the angular streaming derivative term is derived from a series expansion of the radiative intensity on the basis of Legendre polynomials. Recently, Aouled-Dlala et al. [8] suggested a new finite Chebyshev transform (FCT) to improve the performance of the DOM when solving coupled conduction and radiation problems in a spherical or a cylindrical media. For many engineering applications, there exist non-gray radiating gases such as H OandCO or a twophase mixture with gas and particles in the medium of combustion products or heat transfer promoters [,9]. Especially, since a gas medium with suspended particles has a higher heat capacity than a medium with gas only, a two-phase mixture of gas and particles is expected to be used for enhancing heat transfer in engineering applications []. To the author s knowledge, however, only limited researches have been conducted to examine the effect of the behavior of non-gray gas and/or a two-phase mixture of gas and particles on the calculation of spherically symmetric radiative heat transfer. While Baek et al. [] analyzed the unsteady cooling problem of conduction and radiation with two-phase gray gas and particles exposed to the rarefied cold environment, Baek et al. [9] investigated the single droplet combustion characteristics with non-gray gas radiation effects. They did not, however, provide any benchmark solutions, which can be used in validating the radiative heat transfer of non-gray gas and/or a two-phase mixture of non-gray gas and particles with spherically symmetric applications. Therefore, a benchmark solution for the RTE with non-gray gas and/or particles in spherically symmetric geometry is highly demanded. The objective of this work is to provide benchmark solutions with non-gray gas and particles by calculating a two-phase radiation with non-gray gas and a cloud of particles in a concentric spherical enclosure. In order to solve the RTE in a spherically symmetric coordinate, a modified DOM using the concept of the spatial and angular discretization of the conventional FVM is introduced following the work of Baek and Kim [0], who suggested the hybrid solution method of a combined DOM and FVM for the calculation of radiative heat transfer in an axisymmetric cylindrical geometry. The angular derivative terms, which appear in orthogonal cylindrical and spherical coordinates, are treated as the conventional DOM. The radiative properties involving non-gray gas and particle behavior are modeled by using the extended WSGGM [] with particles []. This paper is organized by the following. Mathematical formulation and the corresponding discretization equation for RTE are derived by considering the behavior of the two-phase mixture of non-gray gas and particles using the modified DOM in a spherically symmetric enclosure. The present approach is then validated by comparing the present results with those of previous works. Finally, detailed investigations of the radiative heat transfer with non-gray gases and/or two-phase media are conducted and examined by changing such various parameters as temperature ratio between inner and outer spherical enclosure, particle concentration, and particle temperature.. Mathematical formulation.. Radiative transfer equation M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) The geometry and coordinates for two concentric spheres are illustrated in Fig., where subscripts and refer to each wall boundary at r ¼ R and R, respectively. w ¼ cos m is the polar angle measured from outward direction of r, ranging from 0 to p. For a non-gray mixture of gas and particles in a spherically symmetric enclosure as shown in Fig., the r-directional radiative heat flux is defined as the summation of all

4 60 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) e r χ = cos - μ s T T ε R ε R Fig.. Schematic of two concentric spheres and its coordinate system. the different gray band heat fluxes such as q R r ¼ X Z I k ð~r;~sþð~s ~e r Þ do, () k O¼4p where I k ð~r;~sþ is the kth band radiative intensity at position ~r in the direction ~s. ~e r is the base vector in the radial direction, and O is the solid angle. To obtain the radiative heat flux for a two-phase mixture of non-gray gas and particles in a spherically symmetric coordinate, the radiative intensity at the kth gray band at any position ~r along a path ~s through a two-phase absorbing, emitting, and scattering gas mixture with particles can be evaluated from the following RTE [8,]: m q r qr ½r I k ð~r;~sþš þ q r qm ½ð m ÞI k ð~r;~sþš ¼ ½k g;k ð~rþþk p ð~rþþs sp ð~rþši k ð~r;~sþ þ w g;k ð~rþk g;k ð~rþi b;g ð~rþþw p;k ð~rþk p ð~rþi b;p ð~rþþ s Z spð~rþ I k ð~r;~s 0 Þ do 0, ðþ 4p O 0 ¼4p where the scattering comes from the evenly distributed and randomly oriented particle cloud and therefore isotropic scattering is assumed for simplicity of the analysis that follows. For a diffusely emitting and reflecting wall the above RTE is subject to the following boundary condition: Z I k;w ðr w ;~sþ ¼ w w g;k I b ðr w Þþ w I k ðr w ;~s 0 Þj~n w ~s 0 j do 0 for ~n w ~s40. (3) p ~n w ~s 0 o0 In Eqs. () and (3), subscripts k, g, p, w, and b denote the kth gray band, gas, particle, wall, and black body, respectively. ~n w is the unit normal vector towards the medium at the wall boundary. k g,k is the absorption coefficient of mixture gases, and k p and s sp are the absorption and the scattering coefficients of particle, respectively. In addition, w g,k and w p,k are the weighting factor related to the kth gray band and are functions of gas temperature, T g, and particle temperature, T p, respectively. As suggested and validated by Yu et al. [], if the two-phase mixture of non-gray gas and particles is not in thermal equilibrium, and if the gas and particles share all the same gray bands, the weighting factor for particles has the same type as that for the gas so that w p;k ðt p Þ¼w g;k ðt p Þ.

5 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) e r N P n S s R R e r χ m- χ m+ Fig.. Schematics of (a) control volume and (b) control angle. To close the above two-phase non-gray RTE, the absorption coefficients and weights of non-gray gas are calculated by using the WSGGM from Smith et al. []. The particle absorption and scattering, however, are assumed gray to utilize the formulations obtained from Yu et al. [] and Chui et al. [3] in the form: k p ¼ p X i N i pd i 4, (4a) s sp ¼ð p Þ X i N i pd i 4, (4b)

6 6 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) where e p is the particle emissivity, N i and pd i =4 are the number density and the projected area, respectively, of the particle pertaining to group i... Modification of the discrete ordinates method (MDOM) Following the conventional artifice of the DOM [3,6,7], which maintains neutron conservation and permits minimal directional coupling, the angular derivative term can be modeled as follows: q qm ½ð m ÞI k Š m¼m m am = I m = k a mþ= I mþ= k DO m, (5) where DO m is a discrete solid angle and a m= are the coefficients for the angular derivative term to be determined. To obtain the discretized form of the RTE, Eq. (5) is substituted into Eq. (), which is then integrated over a control volume, DV, and a control angle, DO m, assuming that the magnitude of intensity is constant within DV and DO m, but allowing its direction to vary by following the conventional practice of the FVM [3 5]. Thereby, the following equation can be obtained: X I m k;i DA id m i þ DA n DA s ða m = I m = a mþ= I mþ= Þþb k I m DVDOm ¼ S m DVDOm, (6a) i¼n;s where b k ¼ k g;k þ k p þ s sp, S m ¼ w g;kk g;k I b;g þ w p;k k p I b;p þ s X sp M I m0 (6c) 4p DOm0 m 0 ¼ are the extinction coefficient and the source term due to emission and scattering, respectively. Also, DA n ¼ 4pr n and DA s ¼ 4pr s are the surface area, DV ¼ 4pðr3 n r3 s Þ=3 is the volume of the incremental spherical shell with an inner radius r s and an outer radius r n, while DO m ¼ pðcos w m cos w mþ Þ is the discrete solid angle as shown in Fig.. Note that the directional weight, D m i, which determines an inflow or outflow of radiant energy across the control volume face according to its sign, is evaluated as follows: Z D m n ¼ Dm s ¼ D m r ¼ ð~s ~e r Þ do ¼ pðsin w mþ sin w m Þ. (7) DO m Now, a recursive relation for the coefficients of the angular derivative term, a m=, can be determined by examining the divergenceless flow condition of Carlson and Lathrop [6]. For this case, I m k;i ¼ I m= and b k I m ¼ Sm, and Eq. (6a) is reduced to X DA i D m i i¼n;s Therefore, we have þ DA n DA s ða m = a mþ= Þ¼0. (8) a mþ= a m = ¼ D m r, (9) with a = ¼ a Mþ= ¼ 0. Then, this expression provides a recursive relation that determines the constants, a m=. Note that, since the directional weights are analogous to the multiplication of direction cosine by quadrature weight in the conventional DOM, Eq. (9) corresponds to another form of the recursive relation [3,6,7]. To relate the facial intensity, I m m= k;i, and the edge intensity of the angular range, I, to the nodal intensity, I m k;i, the following simple step scheme is introduced to ensure positive intensity: I m k;i Dm i ¼ I m maxðdm i ; 0Þ I m k;i maxð Dm i ; 0Þ, (0a) I m = ¼ I m, (0b) (6b)

7 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) with I Mþ= ¼ I M. In Eq. (0a), the subscript i represents n and s, while I denotes the corresponding N and S. By using this scheme, Eq. (6a) can be recast into the following general discretization equation: a m I m ¼ am k;n I m k;n þ am k;s I m k;s þ bm, (a) where a m ¼ maxðda nd m n ; 0ÞþmaxðDA sd m s ; 0Þþb kdvdo m þ DA n DA s a m =, a m k;n ¼ maxð DA nd m n ; 0Þ, a m k;s ¼ maxð DA sd m s ; 0Þ, b m ¼ Sm DVDOm þ DA n DA s a mþ= I mþ. The boundary condition in Eq. (3) for a diffusely reflecting and emitting wall can be arranged to where I m k;w ¼ ww g;k I b;w þ w p D m w ¼ ZDO m ð~s ~n w Þ do X I m0 k;w jdm0 w j for Dm w 40, m 0 ;D m0 w o0 (b) (c) (d) (e) (a) (b) is the directional weight at the wall. It becomes D m r and D m r at the outer and inner spherical walls, respectively. The iterative solution is terminated when the following convergence criterion is attained: maxðji m m;old I j=i m Þp0 6, (3) where I m;old is the previous iteration value of I m. This completes the formulation of the present solution technique for the calculation of spherically symmetric radiative heat transfer..3. Supplemental equations If there exists a non-radiative volumetric heat source S nr in the medium, it has to be equal to the divergence of the radiative heat flux through the following radiation balance equation [6,7]: S nr ¼rq R ¼ XK k¼ ðr q R k Þ¼XK k¼ k g;k ½4pðw g;k I b;g Þ G k Š, (4) where G k ¼ P M m¼ I m k DOm is the incident radiation of the kth gray band gas. It is noted that when the medium is in radiative equilibrium, i.e., S nr ¼ 0, temperature distribution of the medium can be obtained directly from 4pI b;g ¼ P K k¼ k g;kg k = P K k¼ k g;kw g;k [7]. When the medium is gray, its temperature is calculated by simply equating the blackbody function, 4pI b;g, to the incident radiation G ¼ P M m¼ I m DO m [6]. Once the intensity field is calculated, the radiative heat flux at the wall can be estimated by q R r ¼ XK X M k¼ m¼ 3. Results and discussion I m k Dm r. (5) The solution procedures presented above are applied to pure radiative problems in two concentric spheres with various wall temperature ratio, T =T, wall emissivity, optical thickness, non-gray gas composition, and particle temperatures. For all cases presented below, equally spaced control volumes of N r ¼ 50 are used. The total solid angle 4p is divided into N w directions with uniform Dw ¼ w mþ w m.

8 64 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) The radiative behavior in the gray gas medium To validate the present formulations for the analysis of spherically symmetric radiative heat transfer in two concentric spheres, a benchmark problem for gray gas medium without particles is firstly considered. By simply setting k p ¼ s sp ¼ 0 in Eq. (), it is possible to analyze the gas radiation without particles by using the same formulation presented above. Fig. 3 shows the effect of different order of angular control angle on the non-dimensional radiative heat flux, q ¼q R r =st 4, in the medium. The gray medium confined between two black concentric spheres with R =R ¼ 0:5 and T =T ¼ 0:5 is in radiative equilibrium with t ¼ br ¼ :0. As the number of control angle increases from N w ¼ 4 to, the present solutions approach those of Jia et al. [4] by the Galerkin method and Aouled-Dlala et al. [8] by the FCT DOM. Because further refinement of control angle does not significantly influence the results, N w ¼ is adopted hereafter. In Fig. 4, the temperature distributions in the medium are compared with the predictions based on FLT DOM by Sghaier et al. [5] for various wall temperature ratio of T =T ¼ :0,.5, and 0.5 for the case of ¼ ¼ 0:5, R =R ¼ 0:5, and t ¼ :0. Since only radiative heat transfer is involved, we can see that there exists a temperature jump at the bounded walls. It also shows that present predictions are in good agreement with those obtained by using the FLT DOM [5]. Fig. 5 shows the effect of optical thickness on the temperature profile, i.e., ðt=t Þ 4 and T/T. In this case, the conditions of ¼ ¼ 0:5, R =R ¼ 0:5, T =T ¼ 0:5 are used, while t ¼ br has three different values of 0.,.0, and 0.0. When the medium is optically thin, the temperature of the medium is more uniform and the temperature jump near the wall is more pronounced through the far-reaching effect of radiative heat transfer. As the optical thickness increases to 0.0, the medium has a steeper temperature gradient, which is closer to the profile of conduction-only case due to the heat blockage effect of optically thick medium; hence, the temperature jump at the walls is reduced compared with optically thinner cases. Galerkin (Jia et al., 99) S-4 (Aouled-Dlala et al., 007) S- (Aouled-Dlala et al., 007) Nχ= Nχ=0 Nχ=8 Nχ=6 Nχ=4 q* Fig. 3. Effect of angular control angle on the non-dimensional radiative heat flux distribution for the case of ¼ ¼ :0, R =R ¼ 0:5, T =T ¼ 0:5, and t ¼ :0. r / R

9 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) T /T =.0.5 T /T =.5 T / T S- FLT-DOM (Sghaier et al., 000) present MDOM T /T = r / R Fig. 4. Effect of temperature ratio between outer and inner spheres on the non-dimensional temperature distribution for the case of ¼ ¼ 0:5, R =R ¼ 0:5, and t ¼ : S- FLT-DOM (Sghaier et al., 000) τ =0. τ =.0 τ = (T/T ) T / T conduction only r / R 0.5 Fig. 5. Effect of optical thickness on the non-dimensional temperature distribution for the case of ¼ ¼ 0:5, T /T ¼ 0.5, and R =R ¼ 0:5.

10 66 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) Table Radiative heat flux for various combinations of the boundary wall temperature ratio T /T and emissivities e and e with R =R ¼ 0:5 and t ¼ :0 T =T e e ðr=r Þ ðq R r =st 4 Þ Galerkin [4] S-DOM [5] S FLT-DOM [5] Present ðn w ¼ Þ Table summarizes the results of radiative heat flux of ðr=r Þ ðq R r =st 4 Þ for various combinations of the wall temperature ratio T /T and boundary emissivities e and e, and compares the present results with the other solutions by Galerkin method [4], S DOM, and S FLT DOM [5] for the case of R =R ¼ 0:5 and t ¼ :0. It can be seen that the present results are in good agreement with other predictions. 3.. The radiative behavior in a non-gray gas medium In practical combustion engineering and combined heat transfer systems, participating media usually display non-gray characteristics which is wavelength-dependent. Therefore, it is necessary to model and investigate the non-gray gas behavior in two concentric spherical systems. To the author s knowledge, however, since there is no benchmarking data in the literature to validate the present formulations in spherically symmetric system, it is firstly investigated in the one-dimensional planar geometry. By setting the coefficients of angular derivative terms to zero, i.e., a m= ¼ 0, in Eq. () for a spherically symmetric case, the discretization equation can be reduced to the following in one-dimensional planar geometry: I m ¼ I m k;n maxð DA nd m n ; 0ÞþIm k;s maxð DA sd m s ; 0ÞþSm DVDOm jd m r jþb kdvdo m, (6) where the geometric relations of DA n ¼ DA s ¼ are used and DV ¼ r n r s : The directional weights are the same as Eq. (7). Also, only b k ¼ k g;k is used since the non-gray gas radiation without particles is considered here. The test problem for non-gray radiative heat transfer employs a participating medium of 40%H O and 0%CO at 50 K bounded by plain walls 0. m apart with wall emissivities ¼ ¼ 0:8. The wall temperatures are kept at 400 and 500 K at x ¼ 0andx ¼ 0:m; respectively. Fig. 6 shows the predicted radiative source distribution, r q R, which is compared with the benchmark solution of Denison [8]. It shows that the present solution with four gray gases WSGGM agrees well with the benchmark data obtained by the SLW model [8]. The effect of absorption coefficient on gas temperature distribution for the spherically symmetric case is depicted in Fig. 7 using three different gray gas absorption coefficient of k g ¼ 0:,.0, and 0.0 m and three different temperature ratio of T =T ¼ :0,.5, and 0.5 with ¼ ¼ 0:8, R =R ¼ 0:5, and gas composition of 0%H O, 0%CO. As the gray absorption coefficient (namely, optical thickness) increases from 0. to 0.0 m in the cases of T =T ¼ :0 and.5, i.e., when the outer wall temperature is higher than the inner wall temperature, the gas temperature near the inner wall decreases while the temperature near the outer wall increases, and the temperature jump at both walls is reduced. This is because the mean penetration path decreases with increasing absorption coefficient so that more radiation energy emitted from the hot wall is locally absorbed by the medium and, therefore, less energy penetrates towards the cold wall. Consequently,

11 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) dq / dx, kw / m SLW (Denison, 994) present WSGG x, m Fig. 6. Radiative source distribution for 40% H O and 0%CO. The medium temperature is 50 K, while wall temperature is 400 K and 500 K at x ¼ 0 and x ¼ 0. m, respectively. Wall emissivities are ¼ ¼ 0:8. T /T =.0.5 T /T =.5 T / T nongray, WSGG gray, κ g =0. m - gray, κ g =.0 m - gray, κ g =0.0 m - T /T = r / R Fig. 7. Effect of gray absorption coefficient and non-gray behavior on the non-dimensional temperature distribution for the case of ¼ ¼ 0:8 and R =R ¼ 0:5. The non-gray gas composition is 0% H O, 0% CO, and 70% transparent gas.

12 68 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) the temperature near the wall becomes closer to the wall temperature, which leads to a larger gradient of temperature distribution in the medium. Such temperature gradient is positive for T =T 4; while it is negative for T =T o. Note also in Fig. 7 that, with increasing absorption coefficient from 0. to 0.0 m, the gas temperature deviates more at the inner wall than at the outer wall. This may result from the fact that a smaller surface area of the inner wall makes the radiative heating and/or cooling more effective than a larger surface area of the outer wall does. If the medium behaves like non-gray gas, however, the predicted temperature distribution is somewhat different from the gray case. Especially, the temperature gradient of non-gray gas medium near the inner wall is noticeably larger than that of gray gas medium of k g ¼ 0 m. Such quantitative difference between gray and non-gray gases suggests that non-gray gas modeling must be accounted for to conduct better simulations of real gas behavior The radiative behavior in a two-phase mixture of non-gray gas and particles Two-phase mixtures of gas and particles are very often encountered in many engineering practices such as particle gas heat exchangers [], droplet combustion [9], and pulverized coal flames [3]. In this section, the radiative heat transfer in a two-phase mixture of non-gray gas and particles with b k ¼ k g;k þ k p þ s sp is considered. The gray absorption and scattering coefficients of particle depend on the diameter, d i, number density, N i, and emissivity of the particle, e p, as shown in Eq. (4). The particle size distribution is assumed to be in the range of 50, 60, 70, 80, and 00 mm in diameter with 0% each by mass. The particle density is 300 kg/m 3 with the particle emissivity of 0.8 that is typical of coal particles []. Therefore, s sp takes the value of m in case of C p ¼ 0:0 kg=m 3. It is assumed that the particle scattering function is isotropic, which implies that equal amounts of energy are scattered in all directions by the particles. The gas and particle temperatures are kept the same as T g ¼ T p. T /T =.0.5 T /T =.5 T / T no particle Particle concentration=0.00 kg/m 3 Particle concentration=0.0 kg/m 3 Particle concentration=0. kg/m 3 T /T = r / R Fig. 8. Effect of particle concentration on the non-dimensional temperature distribution for the case of ¼ ¼ 0:8 and R =R ¼ 0:5. The non-gray gas composition is 0% H O, 0% CO, and 70% transparent gas. The particle temperature is the same as gas temperature, T p ¼ T g.

13 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) Fig. 8 and Table show the effect of particle concentration on temperature distribution for the case of ¼ ¼ 0:8 and R =R ¼ 0:5 with the gas mixture of 0% H O, 0% CO, and 70% transparent gas. The particle concentration in the medium takes the value of 0 (no particle), 0.00, 0.0, and 0. kg/m 3. As the particle concentration increases to 0. kg/m 3, the temperature distribution in the medium behaves as if the Table Dimensionless temperature for various combinations of the boundary temperature ratio T /T and particle concentration C p with e ¼ e ¼ 0.8 and R /R ¼ 0.5. The non-gray gas composition is 0% H O, 0% CO, and 70% transparent gas. The particle temperature is the same as gas temperature, T p ¼ T g T =T C p T/T r/r ¼ r/r ¼ r/r ¼ r/r ¼ r/r ¼ T /T =.0.5 T /T =.5 T / T T p =T g +00 K T p =T g T p =T g -00K T /T = r / R Fig. 9. Effect of particle temperature on the non-dimensional temperature distribution for the case of ¼ ¼ 0:8 and R =R ¼ 0:5. The non-gray gas composition is 0% H O, 0% CO, and 70% transparent gas. The particle concentration is set to 0.0 kg/m 3.

14 60 M.Y. Kim et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 09 (008) Table 3 Dimensionless temperature for various combinations of the boundary temperature ratio T /T and particle temperature T p with e ¼ e ¼ 0.8 and R /R ¼ 0.5. The non-gray gas composition is 0% H O, 0% CO, and 70% transparent gas. The particle concentration is set to 0.0 kg/m 3 T =T T p T/T r/r ¼ r/r ¼ r/r ¼ r/r ¼ r/r ¼ T g T g T g T g T g T g T g T g T g gray absorption coefficient increases as shown in Fig. 7. This is because the optical thickness of the medium increases to b k ¼ k g;k þ k p þ s sp ; and hence more radiant energy is absorbed by the gas and particle mixture. In some cases, the gas temperature may be different from the particle temperature when they are not in thermal equilibrium with each other. Fig. 9 shows the effect of the particle temperature on the medium temperature when particle concentration is kept at 0.0 kg/m 3, and representative values are listed in Table 3. If the particle temperature increases to T p ¼ T g þ 00 K, the medium temperature also increases because more radiant energy is supplied by hotter particles. When the particle temperature is less than the gas temperature, i.e., T p ¼ T g 00 K, however, gas temperature in the medium decreases because the gas loses energy to the particles of relatively low temperature. 4. Conclusions The radiative heat transfer involving non-gray gas and particle behavior is modeled by using the extended weighted sum of gray gases model with particles to provide benchmark solutions for the RTE with non-gray gas and particles in a spherically symmetric enclosure. The MDOM is proposed utilizing the concepts of the conventional DOM and FVM, i.e., the spatial and angular integration is performed by the FVM, while the angular derivative term is evaluated by formulating the recursive relation for the coefficients of the angular derivative term in the form of directional weights, which are analogous to the multiplication of direction cosine by quadrature weight in the conventional DOM. The mathematical formulation and final discretization for the RTE accounting for the behavior of the two-phase mixture of non-gray and particles are validated by comparing the present results with those of previous works involving gray and non-gray medium. Detailed investigations of the radiative heat transfer with non-gray gases and/or a two-phase medium are conducted to examine the effects of non-gray gas, wall temperature ratio, particle concentration, and particle temperature. The temperature distribution of the non-gray gas is somewhat different from the gray case, especially near the inner wall where the gas temperature drops noticeably and such temperature drop increases with increasing wall temperature ratio. The effect of particle concentration acts like that of gas absorption coefficient. A higher particle temperature causes a higher temperature of the medium. Acknowledgments This work was supported by the Research Center of Industrial Technology at Chonbuk National University, Korea.

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