Study of radiation in spherical media using moment method

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1 Aerican Journal of Physics and Applications 03; (): 5-3 Published online July 0, 03 ( doi: 0.648/j.ajpa Study of radiation in spherical edia using oent ethod T. Sghaier Daa University,Kingdo of Saudi Arabia Eail address: To cite this article: T. Sghaier. Study of Radiation in Spherical edia Using oent ethod. Aerican Journal of Physics and Applications. Vol., No., 03, pp doi: 0.648/j.ajpa Abstract: A oent technique is presented to iprove the perforance of the discrete ordinates ethod when solving the radiation probles in spherical edia. In this approach the angular derivative ter of the discretized -D radiative transfer equation is derived fro an expansion of the radiative intensity on the basis of angular oents. The set of resulting differential equations, obtained by the application of the S N ethod associated to oent ethod, is nuerically solved using the boundary value proble with the finite difference algorith. Results are presented for the different independent paraeters. Nuerical results obtained using the oent approxiation copare well with the benchar approxiate solutions. oreover, the new technique can easily be applied to higher-order S N calculations. Keywords: RTE, Spherical ediu, Angular Derivative Ter, DO, oent ethod. Introduction In practical engineering probles, radiative transfer in participating edia appears in any applications such as cobusting systes, furnaces and reactor nuclear theory. any of these systes can be considered as spherical enclosure. Therefore, it is desirable to have an accurate and reliable odel for solving the radiative transfer equation for this geoetry which contain, absorbing, eitting and scattering ediu. A nuber of studies interested in resolving the radiative transfer equation in such edia have been conducted. These wors included various nuerical techniques: integral transforation techniques [,], spherical haronics ethod [3], Galerin ethod [4-5] and others [6-7]. The discrete ordinates ethod has also been used in solving the radiative transfer equation in spherical edia[ 8 ].This ethod enjoys great popularity owing to its accuracy and to its copatibility with other techniques; Sghaier et al [9] and Trabelsi et al [0] developed a discrete ordinates ethod associated with the finite Legendre transfor. Recently, Aouled-Dlala et al [] used the finite chebyshev transfor to treat the angular derivative ter of the discretized one-diensional radiative transfer equation. Ki et al[]used the cobined finite volue and discrete ordinates ethod to investigate radiative heat transfer between two concentric spheres and Li et al [3] developed a Chebyshev collocation spectral ethod for coupled radiation and conduction in a concentric spherical participating ediu. Very recently, ishra et al [4] used a lattice Boltzann and odified discrete ordinates ethod to study radiative transport in a spherical ediu with and without conduction. In order to apply the discrete ordinates ethod, it is necessary to approxiate the angular derivative ter appearing in the radiative transfer equation in spherical coordinates. This ter is generally approxiated by a classical finite differencing schee [5]. In this wor, we introduce a new approach to evaluate the angular derivative ter using an angular oent technique. This yields a quasi-analytical expression of the discrete angular derivative ter. The obtained higher oents of the radiative intensity are expressed in ter of the incident radiation, the net radiative heat flux and the radiation pressure using the generalized eddington approxiation [6 ]. We therefore adopt the discrete ordinates ethod to study the radiation transfer in such edia. In this paper, the considered ediu is a hollow sphere. The boundaries are aintained at different but unifor teperatures and are considered to be opaque, gray diffusely eitting and diffusely reflecting. The obtained results are copared with those given by the standard discrete ordinates ethod and with those available in the literature. The atheatical forulation is given for

2 6 T. Sghaier: Study of Radiation in Spherical edia Using oent ethod gray edia but its extension to any absorption coefficient based non gray odel is straightforward.. Analysis The radiative transfer equation through an absorbing, eitting and isotropically scattering spherical shell ediu is µ + + r r r µ (, µ ) ( µ ) (, µ ) β (, µ ) r I r I r I r σ + = χ Ib T ( r) + I( r, µ ') dµ ', R < r < R and - µ +. () The above equation is subject to the following boundary conditions ( ) b, ( ) ( ), (a) 0 I R, µ = ε I + ε I R, µ ' µ ' dµ ', µ > 0 ( ) b, ( ) ( ), (b) 0 I R, µ = ε I + ε I R, µ ' µ ' dµ ', µ < 0 In Eqs.(), (a) and (b), r is the space radial variable, µ is the cosine of the angle between the direction s of the radiation intensity I ( r, µ ) and the positive r-axis. χ, σ and β are the absorption, scattering and extinction coefficients, respectively, which are related by β = χ + σ. The blacbody radiation is related to the teperature T ( r ) in the ediu through I b ( ) 4 n σt r =, π where n denotes the refractive index and σ the Stefan Boltzann constant. In the boundary conditions given by Eqs (a) and (b), ε is the isotropic eissivity of the opaque boundaries. The subscripts and refer to the boundaries at r = R and r = R respectively. The geoetry and coordinates for the hollow sphere are shown in Fig.. Fig. Hollow sphere geoetry and notations... Differencing Schee The discrete for of the radiative transfer equation is obtained by evaluating Eq. () at each of the discrete directions and replacing the integral by nuerical quadrature to give

3 Aerican Journal of Physics and Applications 03; (): µ σ ( µ ) β χ ( ( )) (3) b ' ' r r r I + r µ I + I = I T r + w I µ = µ ' = The discrete ordinate representation of the boundary conditions, Eqs. (a) and (b) is given by I ( R ) = ε I + ( ε ) w I µ µ > 0 (4a) b, ' ' ' ' =, µ ', < 0 b, ' ' ' ' =, µ ', > 0 I ( R ) = ε I + ( ε ) w I µ µ < 0 (4b) where subscripts and ' refer to discrete directions, is the total nuber of these directions. If a standard difference schee is used, the discrete for of the ter involving the angular derivative ter is written µ ( µ ) I + / + / / / µ = µ α I α w I, (5) where I + / and I / are the angular intensities in the directions +/ and -/. The constants α + / and α / only depend on the differencing schee and therefore they ay be deterined by exaining the case of an isotropic intensity field as described in Ref. [5]. As far as the curved geoetries are concerned, the differencing schee introduces two angular variables; I + / and I /. They ust be deterined at each space position r. For this purpose, the standard diaond difference approxiation is used to relate the I + / and I /, naely I = [ I+ / + I / ]. In the calculation of the average angular intensity I, we need the starter intensity I /. For spherical geoetry, it is obtained fro the solution of the transfer equation in slab geoetry with starter direction cosine µ =. We develop in what follows an alternative technique based on angular oent equations..the results fro both approaches will be copared... oent ethod We develop a new approach to approxiate the angular derivative ter: D( r, µ ) = ( µ ) I( r, µ ) µ (6) We start by defining the -th order oent of D(r,µ) + µ µ µ, (7) D ( r) = D( r, ) d 0 The application of the angular oent technique to the angular derivative ter, denoted D( r, µ ) yields + + (, ) (, ) + + D r µ µ d µ = I r µ µ dµ I ( r, µ ) µ dµ. (8) The oents of the radiative intensity represent a generalized incident radiation, which for the case n=0, n= and n= reduces the usual definition of the incident radiation, radiative flux and the radiation pressure. The obtained higher oents of the radiative intensity are then expressed in ter of the incident radiation, the net radiative flux and the radiation pressure using the generalized eddington approxiation (6). In order to obtain the angular derivative ter, the integrals over direction are replaced by the discrete for + f( r, µ ) d µ w f( r, µ ), (9) = where the w are the quadrature weights associated with the directions µ. Thus Eq. (8) is approxiated by + wd µ = wi µ wi µ = = =. (0)

4 8 T. Sghaier: Study of Radiation in Spherical edia Using oent ethod Now, the angular derivative ters D are obtained fro Eq. () written for =, -. Thus, this syste is closed by using the obvious relation wd 0. (3) Thus the unnowns D are the solution of a linear algebraic syste: A d = b, with the atrix A given by = A=.. µ µ.. µ µ µ.. µ (4) and the vector d and b given by w D wd d=., b=. w D 0... ( ) wi µ ( ) wi µ = = (5) The angular derivative ters D are then given by where and ( A ) is the set of coponents ( ) inverse of the atrix which is given by A, D = A B, (6) ( ) w j= + = ' ' µ ' ' ' µ ' ' = ' =, B w I w I A the ( ) 0 µ σ r r r w, (7) A = µ. It is called the transpose Vanderonde atrix. The new discrete ordinates representation of Eq. (), for a finite nuber of discrete ordinates ay be written ( ri) + ( A ) B + βi = χib ( T ( r) ) + w' I', =, (8) j= ' = and the discrete ordinates representation of the boundary conditions is given by Eqs.(4a, 4b) Once the directional intensities I are nown, the radiative heat flux qr ( r ) and the incident radiation energy G ( r ) in the ediu are deterined fro their definitions as = + ( ) = q ( r) π I r, µ µ d µ π µ w I, (9a) r = G ( r) = + π I ( r, µ ) d µ = π wi. (9b) = Equations (8) and (4a, 4b) provide the coplete atheatical forulation for the radiation proble in a diensional spherical ediu. A nuerical technique, naely the boundary value proble with finite difference (BVPFD) [7] is used to solve this proble. The new technique called O-DO with eight directions (N=8) is adopted. The weights and quadrature points are those of corresponding Gaussian quadratures.

5 Aerican Journal of Physics and Applications 03; (): Results and Discussion In table, we give the nuerical values of r q (r) for a radiation proble between two concentric spheres with diffusely eitting and reflecting boundaries subject to pure radiative transfer. Results obtained fro the new technique denoted by O-DO have been copared with that of i et al [], Jia et al [7], Sghaier et al [9], standard discrete ordinates ethod, denoted by DO [] and those of ishara et al [4].For the standard discrete ordinates ethod, the coefficient α + / and α / are calculated following the procedure as described by odest [ 6 ]. We present results for r = 0.5, τ = and for various cobinations of the boundary eissivities ε and ε and for different values of outer sphere teperature θ. O results show an excellent coparison. In table, we present the nuerical values of oent of order ( + I( r, µ ) ud µ ) for a purely radiation proble with blac boundaries, inner sphere teperature θ = and outer sphere teperature θ =0, and for different optical thicness of inner and outer sphere τ = βr and τ = βr respectively. Results are copared with those of Visanta et al, Jia et al and Abulwafa et al. The O results copare well with those available in literature [4, 5, 6]. An increase in optical thicnessτ = βr, the ediu becoes ore denser. The net radiative heat flux decreases in the ediu as shown in Fig.. For the optical thicness τ =, θ = 0.5 and for blac boundaries, the effects of the ratio R /R on the radiative heat flux is shown in Fig.3. With an increase in the ratio R /R, the ediu tends to planar one and the radiative heat flux becoes constant. Fig.4 shows the effect of the outer sphere teperature θ on the radiative heat flux distribution in the ediu. When the outer sphere becoes hot, the net radiatif heat flux changes sign and becoes negative. In Fig.5, we study the effect of eissivity of the inner and outer sphere on the radiative heat flux distribution in the ediu. For this study, τ =, θ = 0.5and R /R =0.5. It 'is observed when the hot inner sphere is ore reflecting, the net radiative heat flux in the ediu becoes less. 4. Conclusion An analysis of a radiation proble in one diensional absorbing, eitting and isotropically scattering hollow spherical ediu is investigated. The angular derivative ter appearing in this geoetry is approxiated by aing use of a new approach called O-DO approxiation. This leads to an accurate expression for the angular derivative ter. The set of differential equations is solved using the boundary value proble with finite difference algorith. The accuracy of the new technique has been verified by coparison with benchar approxiate solutions. ε ε Θ Table. Values of r q(r) for various cobinations of ε and ε ε with r q r DO[] r q r FLT[9] r q r O-DO r = 0.5, Θ =0.5 and τ =. r q r Galerin[4] ethod r q r FV[] r q r DO[4] Table. The net radiative heat flux with transparent boundaries, isotropic incidence at the inner surface and ω=.0. τ τ Ref.[4] Ref.[5] Ref.[6] Present wor a a a a a a

6 30 T. Sghaier: Study of Radiation in Spherical edia Using oent ethod Fig. Effect of the optical thicness τ on the diensionless radiative flux with ε =ε =, θ =0.5 and R /R =0.5. Fig 3. Effect of the ratio R/R on diensionless radiative flux with ε=ε=, θ=0.5 and τ=.0.

7 Aerican Journal of Physics and Applications 03; (): Fig 4. Effect of the boundary teperature θ on the diensionless radiative flux with ε=ε=, τ= and R/R=0.5. Fig 5. Effect of the surfaces eissivity on the diensionless radiative flux with θ=0.5, τ= and R/R=0.5. References [] G. C. Poraning, C. E. Siewert, On the integral for of the equation of transfer for a hoogeneous spheres. JQSRT98; 8:503-6 [] C. E. Siewert, J. R Thoas, On coupled conductive radiative heat-transfer probles in a sphere. JQSRT 99; 46:63-7 [3] Hsin-sen Chu, Weng Ling-Chia, Tsen. Cobined conduction and radiation in absorbing, eitting and isotropically-scattering concentric spherical edia. JQSRT 99;46:5-7. [4] G. Jia, Y. Yener, JW. Cipolla, Radiation between two concentric spheres separated by participating ediu. JQSRT 99;46:-9. [5] E.. Abulwafa, Radiative-transfer in a linearly-anisotropic spherical ediu. JQSRT 993;49: [6] R. visanta, A. Grosbie, Radiative transfer between two concentric spheres separated by absorbing an eitting gray ediu. JQSRT 976;6:87-9.

8 3 T. Sghaier: Study of Radiation in Spherical edia Using oent ethod [7] S. T. Thynell,. N. Ozisi Radiation transfer in an isotropically scattering hoogeneous solid sphere. JQSRT; 989; 33: [8] J. R. Tsai,. N. Ozisi, F. J. Santarelli, Radiation in spherical syetry with anisotropic scattering and variable properties. JQSRT 989;4: [9] T. Sghaier,. S. Sifaoui, A. Soufiani, Study of radiation in spherical edia using discrete ordinates ethod associated with the finite Legendre transfor JQRST 000; 64: [0] H. Trabelsi, T. Sghaier,. S. Sifaoui, A theoretical study of radiation between spheres using a odified discrete ordinates ethod associated with Legendre transfor. JQSRT 005; [] N. Aouled-Dlala, T. Sghaier, E. Seddii, Nuerical solution of conductive and radiative heat transfer in concentric spherical and cylindrical ediu. JQSRT 007; []. Y. Ki, J. H. Cho, S. W. Bae, Radiative heat transfer between two concentric spheres separated by a two phase ixture of non gray particules using the odified discrete ordinates ethod. JQSRT; 008; [3] B. W. Li, Y. S. Sun, S. W. Zhang DW SW. Chebyshev collocation spectral ethods for coupled radiation and conduction in concentric spherical partipating ediu. Journal of Heat Transfer 009; vol 3. [4] S C. ishra, C. H. Krishna,. Y. Ki, Lattice Boltzann ethod and odifiede discrete ordinates ethod applied to radiative transport in a spherical ediu with and without conduction. Nuerical Heat Transfer Part A,58,85-8,00. [5]. F. odest, Radiative heat transfer, nd, Acadeic, SanDiego, CA3; 003. [6] K. Sen, S. J. Wilson, Generalised eddington approxiation for radiative transfer probles in spherically syetric oving edia. Astrophysics and space science 003; [7] ISL ath Library. st ed. Houston, TX. (997).