An Analytical Solution of the Radiative Transfer Equation for Inhomogeneous Finite Medium with Fresnel Boundary Conditions

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1 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 n nalytical Solution of the Radiative Tranfer Equation for Inhomogeneou Finite Medium with Frenel Boundary Condition. Elghazaly Reactor & Neutron Phyic Department, Nuclear Reearch Center,tomic Energy uthority,cairo, Egypt. Received: 3/5/ ccepted: // BSTRCT The problem of radiation tranfer through an inhomogeneou finite medium with internal energy ource diffuely pecular reflecting boundary condition i conidered (problem ). The partial heat fluxe for thi problem are given in term of the albedo of the ource-free problem with pecular-reflecting boundarie (problem ). We aume that the reflectivity of the urface dependent upon the refractive indice i conidered a Frenel reflection probability function. The Pomaning-Eddington method i ued to calculate firt the albedo for problem then calculate the partial heat fluxe for problem. Reult are obtained for iotropic cattering in a homogeneou an inhomogeneou media for uniform non-uniform internal ource are compared with the publihed calculation. Key Word: Radiative Tranfer / Inhomogeneou Finite Medium / Heat Flux/ Pomraning- Eddington /Specular Reflecting. INTRODUCTION The conideration of generalized boundary condition accommodated variou type of problem uch a radiative heat tranfer, neutron tranport, etc (,). Radiative heat tranfer problem with generalized boundary condition have been tudied uing different technique (3-) which are ued to find accurate value for the partial heat fluxe for thi problem in a homogeneou an inhomogeneou media. few author have been able to include the effect of change in the index of reflection in their analye of radiative tranfer problem involving Frenel boundarie interface. ( 3-7) In thi paper, we find the partial heat fluxe in turbid media with generalized boundary condition (problem ) uch that reflectivity of the urface dependent upon the refractive indice. The connection between problem the ource free problem with pecular boundary condition (problem ) i formulated. The Pomaning-Eddington method i ued to find the olution of the ource free problem the correponding heat fluxe. The calculation are carried out for iotropic cattering in a homogeneou an inhomogeneou media for uniform non-uniform internal ource. BSIC EQUTIONS Radiation tranfer equation for an aborbing, emitting, inhomogeneou, finite lab with iotropic cattering that contain an internal energy ource, i decribed a 4

2 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 (x) ( ) I(x, ) x I(x, ) d Q(x), -, The generalized boundary condition are given by d - I(, ) f (n) I(,- ),, - x a () (a) d I(a,- ) f (n) I(a, ),, (b) Where x i the optical ditance i the coine of the direction of propagation of the radiation intenity I(x, ) of the particle. Q(x) i the internal energy ource, f f are the externally incident d fluxe on the left rigt h urface, repectively. lo, (n) are the pecular diffue reflectivitie of the boundarie n i the relative refractive index. dependent, ingle-cattering albedo of the medium. The partial heat fluxe are defined by - I(a, ) d, i i (x) The pecular reflectivity (, n) i conidered a a function of refractive indice calculated according to the Frenel equation a (3) i the pace (3) where S - S - n (, n ) S n (4a) S S n ( ) (4b) Thee function (, n) are equal to zero for refractive index n equal to unity equal to unity for all angle larger than the critical angle co, which i given by ( n )/ n. c c Thi i the problem, which i connected to the correponding ource free problem with pecular boundarie unit external irradiation on the left urface given by (x) ( ) (x, ) x (x, ) d, - I(,-) d, - x a, With boundary condition (, ) (n) (,- ) (6a) nd (a,- ) (n) (a, ) (6b) c (5) 4

3 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 The connection between problem may be demontrated a follow. Changing to - in eq.(5), multiplying the reultant equation by I(x, ) eq. () by, ubtracting the obtained equation, then integrating the reultant equation over (-,) x (,a) yield Where = H (7) d T, (8a) d R, (8b) 3 Q f T f Q R (8c) a Q( x) ( x, ) d dx, (9a) T (a, ) d (9c) ( x,-) R (,- ) d, (9b) Put =- x = a x in equation (5) multiplying eq. () (5) by ( a x,- ) I(x, ), repectively, ubtracting the reultant equation, integrating over (-,) x (,a) uing the boundary condition, we find for ( n) (n) Where =H () d R, (a) d T, (b) 3 Q f R f T (c) Q a Q( a x) ( x, ) d dx, (d) The partial fluxe - () + (a) are given by () ( a)

4 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 SOLUTION OF THE SOURCE FREE PROBLEM The ource free problem can be repreented by the particle tranfer equation (x) ( ) (x, ) d (x, ) x () - Let u now ue the Pomraning-Eddington approximation (4) to olve equation ( ). In thi approximation the angular irradiance i expreed a ( x, ) (x, ) E(x) (x, ) F(x)O(x, ) (3) where ( x, ) O(x, ) are even odd function of unity -,repectively, which are normalized to d (x, ) d O(x, ) (4), E(x) i the particle denity which i defined by E(x) F(x) i the net flux defined by F(x) (x, ) d (5) - (x, ) d (6) - Subtituting (3) in () uing (4-6) lead to df(x) (x)e(x) (7) dx where d(d(x)e(x) dx ) F(x) (8) ( x) - (x) (9) D(x) i defined a D(x) d ( x, ) () 44

5 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 The function ( x, ) O(x, ) are obtained by ubtituting (7) in () equating the even the odd part of the reulting equation give (x) ( x, ) ( (x) ) () (x) O ( x, ) (x) ( (x) ) () Where (x) (x) (3) D(x) Subtituting () in () for (x) (x) (x) ( x) lead to (x) ln (4) -(x) i the olution of the trancendental equation (4). If D(x) i a lowly varying function of x, equation (7) (8) lead to d E( x) ( x) E( x) dx (5) Equation (5) for the cae i independent of the patial parameter x ha the olution E( x) exp( x) B exp( x) x B Where B are contant to be determined by applying the boundary condition. Uing (6) in (7) lead to F( x) ( exp( x) B exp( x)) (7) - Here we hall ue a weight function method to force the boundary condition to be atified to find the contant B. Weight function method In thi method the boundary condition given by (6) are forced to be atified uing a weight function W () a follow (4-8) d W( )[ (, ) -- ( n ) (,- ) ] (6) (8a) 45

6 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 a (4) d W( )[ (b,- ) ( n ) (a, ) ] (8b) The weight function W H,, for iotropic cattering, i taken in term of Chraekher H-function H H() will be taken in approximate form by (6) C exp (Z ) [ / ( ) ] / (3a) / () ln( ) (3b) [ln( ) Y / ] Z i the extrapolated ditance of the Milne problem. The econd weight function W i repreented by the adjoint of the angular intenity W W ( ) H( ) / (-). (9) (- ) H ( ) C [ - ] ( ) (3a) H( ) ( 3 / ) (3b) Where I (,- ) I (a, ) at left boundary at right boundary By ubtituting the two a weight function on Eq.(8), we get on the contant B then we can calculate the flounce rate E(x) the net fluxe F(x); Thee are ued to evaluate the reflectivity R, the tranmiivity T the partial heat fluxe + -. RESULTS ND DISCUSSION The partial heat fluxe + - at the boundarie for homogeneou an inhomogeneou plane-parallel medium, with diffuely pecularly reflecting boundarie with thickne a=,,5 have been calculated. The pace dependent, ingle cattering albedo ( x) i taken in the exponential form e - x.two weight function method W H W are ued to calculate the partial heat fluxe. The calculation are carried out for inhomogeneou an iotropic cattering with ource-free medium, internal ource Q(x)= in Table (). In Table () the calculation are carried out for internal 46

7 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 - Table (): The partial heat flux for an inhomogeneou, iotropic cattering medium d d with f =,f =,thickne a=..5,. 5 different internal ource Q(x) = + - W W H Ref.[9] W W H Ref.[9] Q(x) = ource Q(x)=-x, Q(x)=-x. The calculated value for + - how good agreement with the exact numerical reult (9). In Table (3) (4) we lit the value of the particle heat fluxe at the boundarie for homogeneou inhomogeneou medium with internal ource Q(x) =,, Q(x)=-x, Q(x)=-x repectively. The calculation done for different ingle cattering albedo, with boundarie which have refractive indice n=.33.5 for thickne a=.. The particle heat fluxe at the boundarie for homogeneou medium, for different reflective index for internal ource Q(x) =, are tabulated in Table (5). 47

8 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 - Table (): The partial heat flux for an inhomogeneou, iotropic cattering medium d d with f =,f =,thickne a=..5,. 5 different internal ource Q(x) = -x + - W W H Ref.[9] W W H Ref.[9] Q(x) = -x

9 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 - Table (3): The partial heat flux for an inhomogeneou, iotropic cattering medium d d with f =,f =,thickne a=..5, with the boundarie have different refractive indice n. different internal ource N=.33 Q(x) = N= W W H W W H W W H W W H Q(x) =

10 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 - Table (4): The partial heat flux for an inhomogeneou, iotropic cattering medium with d d f=,f=,thickne a=..5, with the boundarie have different refractive indice n. different internal ource. Q(x) =-x N=.33 N= W W H W W H W W H W W H Q(x) = -x

11 rab ournal of Nuclear Science pplication, 46(3), (4-5) 3 - Table (5): The partial heat flux for an homogeneou, iotropically cattering medium d d with f =, f =, thickne a=,,5, internal ource Q(x)=,,. 5 with the boundarie have different refractive indice n N=.33 Q(x) = n= W W H W W H W W H W W H Q(x) = CONCLUSIONS The partial heat fluxe at the boundarie of an iotropic, cattering, inhomogeneou, planeparallel medium containing an internal ource with general boundary condition are olved in term of the olution of the correponding ource free problem. The Pomraning-Eddington approximation weight function method are ued to olve the integral equation for the ource free problem. The calculation for an iotropic cattering, inhomogeneou medium without an internal energy ource with an internal energy ource how good agreement with thoe of. Elghazaly M.T. ttia (9). The partial heat fluxe of homogeneou inhomogeneou medium for iotropic, cattering containing an internal energy ource reflectivity of the urface dependent upon the refractive indice have alo been calculated. 5

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