Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium

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Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 210–219 www.elsevier.com/locate/jqsrt

Application of Synthetic Kernel (SKN) method to participating linearly anisotropically scattering solid spherical medium Mesut Tekkalmaza, Zekeriya Altac- b, Metallurgical and Materials Engineering Department, Eskis- ehir Osmangazi University, 26480 Eskis- ehir, Turkey b Mechanical Engineering Department, Eskis- ehir Osmangazi University 26480 Eskis- ehir, Turkey

a

Received 15 February 2007; accepted 29 August 2007

Abstract The Synthetic Kernel (SKN) method is applied to a solid spherical absorbing, emitting and linearly anisotropically scattering homogeneous and inhomogeneous medium. The SKN method relies on approximating the integral transfer kernels by Synthetic Kernels. The radiative integral transfer equation is then reducible to a set of coupled second-order differential equations. The SKN method, which uses Gauss quadratures, is tested against integral equation and the discreteordinates S8 solutions for various optical radius and scattering albedo variations. r 2007 Elsevier Ltd. All rights reserved.

1. Introduction A number of engineering applications involve radiative transfer in spherical medium such as heatgenerating gray gas radiation, heat transfer analysis through fibrous and foam-insulating materials, molten glass and glassy material. The radiative transfer problems of an arbitrary distribution of interior and exterior sources are also of interest in astrophysical applications and neutron transport theory. Radiative transfer in a participating homogeneous or inhomogeneous solid spherically symmetric medium with isotropic or linearly anisotropic scattering has been the subject of numerous studies [1–9]. The discrete ordinate method (DOM) became the dominant means for obtaining numerical solutions of radiative transfer problems for almost two decades. Although the method yields results of sufficient accuracy for most engineering problems, the DOM is plagued with the so-called ray effect. The main cause of the ray effect is angular discretization. As a remedy, one may simply increase the number of directions; however, the ray effect is still persistent [10]. Additionally, in curvilinear geometries, a great deal of difficulties have to be dealt with due to RTE streaming term, which involves the directional derivative of the intensity. The ray effect in curvilinear geometries yields over- or under-estimated incident energy and/or heat fluxes at either the core or outer boundary. On the other hand, the incident energy and the radiative heat flux in the radiative integral transfer equations (RITEs) contain only spatial variables due to integrations over all solid angles. A major disadvantage of treating the RITEs is that the numerical solution of RITE leads to dense matrices, which is a Corresponding author.

E-mail addresses: [email protected] (M. Tekkalmaz), [email protected] (Z. Altac- ). 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.08.021

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Nomenclature En(x) nth-order exponential integral functionR 1 G(t) dimensionless incident radiation ð¼ 2p 1 Iðt; mÞ dmÞ Gn(t) expression defined by Eq. (12) I(t, m) dimensionless radiative intensity G KG 1 ; K 2 transfer kernels of incident energy given by Eqs. (5) and (6) q q K 1 ; K 2 transfer kernels of heat flux given by Eqs. (7) and (8) S0(t) dimensionless isotropic source function given by Eq. (3) S1(t) dimensionless anisotropic source function given by Eq. (4) T(t) temperature a1 coefficient of linear anisotropy f1(t), f2(t) boundary terms defined by Eqs. (9) and R 1 (10) q(t) dimensionless radiative heat flux ð¼ 2p 1 Iðt; mÞm dmÞ qn(t) expression defined by Eq. (14) wn Gauss quadrature weights Greek symbols O0(t) b k(t) mn y(t) s(t) t

scattering albedo ( ¼ s(t)/b) extinction coefficient absorption coefficient Gauss quadrature abscissas dimensionless temperature ( ¼ T(t)/Tref) scattering coefficient dimensionless optical variable ( ¼ rb)

Subscripts n ref w

nth component of the SKN equations reference wall

matter of concern in multidimensional geometries. This consequence may restrict either the computational memory or the execution time or both. A second disadvantage of dealing with RITEs is that the integral transfer kernels for a particular geometry must be derived analytically, and for complicated geometries the derivation of kernels is a difficult task, not to mention the numerical difficulties of treating the singularities of these kernels. The Synthetic Kernel (SKN) method stems from the RITEs, and it consists of approximations to RITE kernels in the form of exponentials, similar to the exponential kernel approximation. Then, the RITEs, in multidimensional geometries, can be cast as a set of coupled elliptic second-order partial differential equations. The method was employed in neutron transport and thermal radiative problems of homogeneous and inhomogeneous media [11–17], the SKN solutions—of benchmark problems established agreed well with those of the spherical harmonics, DOM and Monte-Carlo. Most recently, the SKN method was also employed to participating homogeneous and inhomogeneous one- and two-dimensional cylindrical media [18–20]. These studies show that the method can be used to obtain very accurate solutions with less computational efforts. In this study, the SKN method is extended to radiative transfer of linearly anisotropically scattering, homogeneous and inhomogeneous, participating solid spherical media, and the performance of the SKN solutions is investigated in comparison to the DOM S8 and the exact RITE solutions.

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2. Derivation of the RITE for solid spherical medium For general coordinate systems, the dimensionless form of the RITE for a homogeneous absorbing, emitting and anisotropically scattering medium is given in Ref. [21]. The derivations of RITEs for solid spherical media is carried out using the general forms—Eqs. (12) and (13) of Ref. [21]. To obtain RITEs for a one-dimensional solid spherical medium, we perform the integrations of Eqs. (12) and (13) over the azimuthal and polar angles. Then the dimensionless RITEs can be cast as; for the incident energy Z t0 Z t0 0 0 0 0 0 0 GðtÞ ¼ f 1 ðtÞ þ KG ðt; t ÞS ðt Þ dt þ a KG (1) 0 1 1 2 ðt; t ÞS 1 ðt Þ dt t0 ¼0

t0 ¼0

and for the net radiative heat flux Z t0 Z qðtÞ ¼ f 2 ðtÞ þ K q1 ðt; t0 ÞS 0 ðt0 Þ dt0 þ a1 t0 ¼0

t0

t0 ¼0

K q2 ðt; t0 ÞS1 ðt0 Þ dt0 ,

(2)

where S0(t) and S1(t) are the isotropic and anisotropic source functions, which are defined as S0 ðtÞ ¼ 4p½1  O0 ðtÞy4 ðtÞ þ O0 ðtÞGðtÞ

(3)

S1 ðtÞ ¼ O0 ðtÞqðtÞ,

(4)

and

where y(t) ¼ T(t)/Tref is the dimensionless temperature; O0(t) is the space-dependent scattering albedo given by s(t)/b and b ¼ s(t)+k(t), which is the extinction coefficient; s(t) and k(t) are the space-dependent scattering and absorption coefficients, respectively; and [1O0(t)]y4(t) is the dimensionless blackbody radiation intensity. The integral transfer kernels of incident energy are expressed with superscript G, and for the heat flux, expressed with superscript q, can be written as follows: 0 KG 1 ðt; t Þ ¼

0 KG 2 ðt; t Þ

t0 fE 1 ðjt  t0 jÞ  E 1 ðt þ t0 Þg, 2t

(5)

  1 ðt  t0 Þ 0 0 0 0 0 0 t E 2 ðjt  t jÞ þ t E 2 ðt þ t Þ  E 3 ðjt  t jÞ þ E 3 ðt þ t Þ , ¼ 2t jt  t0 j

(6)

  t0 ðt  t0 Þ 0 0 0 0 tE ðjt  t jÞ  tE ðt þ t Þ þ E ðjt  t jÞ  E ðt þ t Þ , 2 2 3 3 2t2 jt  t0 j

(7)

K q1 ðt; t0 Þ ¼ K q2 ðt; t0 Þ ¼

1  0 tt E 3 ðjt  t0 jÞ þ tt0 E 3 ðt þ t0 Þ  jt  t0 j 2t2  E 4 ðjt  t0 jÞ þ ðt þ t0 ÞE 4 ðt þ t0 Þ  E 5 ðjt  t0 jÞ þ E 5 ðt þ t0 Þ .

ð8Þ

The boundary source terms are obtained as f 1 ðtÞ ¼ f 2 ðtÞ ¼

Iw ft0 E 2 ðt0  tÞ  t0 E 2 ðt0 þ tÞ þ E 3 ðt0  tÞ  E 3 ðt0 þ tÞg, 2t Iw ftt0 E 3 ðt0  tÞ  tt0 E 3 ðt0 þ tÞ þ ðt0  tÞE 4 ðt0  tÞ 2t2  ðt0 þ tÞE 4 ðt0 þ tÞ þ E 5 ðt0  tÞ  E 5 ðt0 þ tÞg,

(9)

ð10Þ

where t0 is the optical radius in mean-free-path (mfp) and Iw is the incident diffuse radiation intensity on the outer surface.

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3. Derivation of the SKN equations The rationale behind the Synthetic Kernel exponential approximation and quadrature selection is given in detail in Ref. [15]. For the mth-order exponential integral functions, we use the following finite sum obtained by the N-point Gaussian integration over mA(0,1) interval: Z 1 N X E m ðxÞ ¼ mm2 ex=m dm ffi wn mm2 ex=mn for m ¼ 1; 2; . . . ; 5, (11) n 0

n¼1

where (mn, wn) sets are quadrature abscissas and weights over the prescribed interval. To begin derivations, we substitute Eq. (11) in Eqs. (5) and (6) altogether in Eq. (1) and define Z t0 t0 S0 ðt0 Þ dt0 0 0 G n ðtÞ ¼ fejtt j=mn  eðtþt Þ=mn g 2tmn t0 ¼0  Z t0  0 0 ðt  t0 Þ 0 jtt0 j=mn 0 ðtþt0 Þ=mn jtt0 j=mn ðtþt0 Þ=mn S 1 ðt Þ dt þ a1 e þ t e  m e þ m e t . ð12Þ n n 0 2t t0 ¼0 jt  t j Eq. (1) can now be written as GðtÞ ¼ f 1 ðtÞ þ

N X

wn G n ðtÞ.

(13)

n¼1

Similarly, substituting Eq. (11) into Eqs. (7)–(8) and Eq. (2), we define  0 Z t0  0 0 ðt  t0 Þ jtt0 j=mn ðtþt0 Þ=mn jtt0 j=mn ðtþt0 Þ=mn t S 0 ðt Þ dt te qn ðtÞ ¼  te þ mn e  mn e 0 2 2t t0 ¼0 jt  t j 8 9 0 0 0 Z t0 < tt0 ejtt j=mn þ tt0 eðtþt Þ=mn  m jt  t0 jejtt j=mn = n mn S 1 ðt0 Þ dt0 þ a1 . 0 0 0 0 ðtþt Þ=mn 2t2  m2n ejtt j=mn þ m2n eðtþt Þ=mn ; t0 ¼0 : þmn ðt þ t Þe

ð14Þ

Then the net radiative heat flux can be simply written as qðtÞ ¼ f 2 ðtÞ þ

N X

wn qn ðtÞ.

(15)

n¼1

It could be shown that Eqs. (12) and (14) satisfy the following coupled first-order differential equations: qn ðtÞ ¼ m2n

dG n ðtÞ þ a1 m2n S1 ðtÞ dt

(16)

and d 2 ½t qn ðtÞ ¼ t2 G n ðtÞ þ t2 S0 ðtÞ. dt

(17)

When taking the derivative of the RHS of Eq. (16), we have to deal with the following term: d dO0 2 d ½O0 ðtÞt2 qðtÞ ¼ t qðtÞ þ O0 ðtÞ ½t2 qðtÞ. dt dt dt

(18)

We can use the energy balance given by Eq. (19) on the second term of Eq. (18). div q ¼

1 d 2 ½t qðtÞ ¼ ½1  O0 ðtÞ½4py4 ðtÞ  GðtÞ. t2 dt

(19)

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We use Eqs. (3), (4) and (19), and rearrange the remaining expressions to eliminate qn(t) from Eqs. (16) and (17). We then finally obtain the so-called SKN equations as follows:   m2n d 2 dGn ðtÞ t  2 þ G n ðtÞ ¼ 4p½1  O0 ðtÞ½1  a1 m2n O0 ðtÞy4 ðtÞ dt t dt dO0 ðtÞ , þ O0 ðtÞ½1 þ a1 m2n ½1  O0 ðtÞGðtÞ  a1 m2n qðtÞ dt for n ¼ 1; 2; :::; N,

ð20Þ

which is subject to the following boundary conditions: radial symmetry at the center and at the surface of the sphere  dG n ðt0 Þ 1 1 þ þ (21) G n ðt0 Þ ¼ a1 O0 ðt0 Þqðt0 Þ, dt t0 mn which is obtained by taking the limits of Gn(t) and dGn(t)/dt at t ¼ t0. These boundary conditions are used regardless of the physical wall conditions since the physical boundary conditions are already included in boundary source terms of f1(t) and f2(t). To find an expression for the net radiative heat flux, we substitute qn(t) from Eq. (16) in Eq. (17) along with the use of Eqs. (3) and (4), and then solving it for q(t), we obtain ! !, N N X X 2 dG n 2 qðtÞ ¼ f 2 ðtÞ  wn mn wn mn . (22) 1  a1 O0 ðtÞ dt n¼1 n¼1 Eq. (22) allows us to compute the net radiative heat flux with only Gn(t) information. 4. Results and discussion The following benchmark problems have been set for the comparison of the SKN solutions with the exact integral equation and S8 solutions. Benchmark problem 1 (BMP-1): The problem of radiative transfer in a solid spherical medium with a constant scattering albedo (homogeneous medium) and transparent outer boundary is considered for both forward (a1 ¼ 1) and backward (a1 ¼ 1) scattering. The medium is cold and the only source in the medium is due to the externally isotropic unit incidence of radiation (Iw ¼ 1) at the surface of the sphere. The incident energy and the net heat flux are computed for solid spherical media of t0 ¼ 1 (Case A) and t0 ¼ 2.5 (Case B), for constant scattering albedo values of O0 ¼ 0.5 and 0.995 and for coefficient of linear anisotropy values of a1 ¼ –1 and 1. Benchmark problem 2 (BMP-2): The medium is inhomogeneous via the space-dependent scattering albedo. As in BMP-1, the medium is also cold and is subject to the externally isotropic unit incidence of radiation at the surface (Iw ¼ 1) with the same optical geometrical dimensions. For the two solid spherical geometries (Cases A and B), the following space-dependent scattering albedos are considered: (linear variations) O0 ¼ 0.2+0.4F1, O0 ¼ 0.80.4F1, (quadratic variations) O0 ¼ 0.44/15F1+0.5F2 and O0 ¼ 1.016/ 15F1+0.5F2, where Fn ¼ (t/t0)n. The average value of O0(t) over the medium is equal to 0.5 in all the cases. Similarly, the incident energy and the net heat flux solutions for the coefficient of linear anisotropy values of a1 ¼ –1 and 1 are examined. In this study, 0ptpt0 interval is equally divided into N grid elements in numerical solutions of the RITE, S8 [22] and SKN equations. The RITEs are solved using the subtraction of singularity technique [15–18]. The convergence criterion based on relative errors for all methods was zo106. The incident radiation and the net radiative heat flux profiles obtained with the exact RITE, SK2, SK3 and S8 for Case A (BMP-1 and O0 ¼ 0.5) of forward and backward scattering media are depicted in Fig. 1. The incident radiation profiles of the SK3 solutions exhibit very good agreement with those of RITE in all scattering cases, while the incident energy profile with the SK2 slightly deviates from the exact solutions

ARTICLE IN PRESS M. Tekkalmaz, Z. Altac- / Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 210–219

0.04

0.9

0.7

a1 = -1, Ω0 = 0.5 Exact SK2 SK3 S8

0.00

a1 = -1, Ω0 = 0.5 Exact SK2 SK3 S8

q(τ)

G(τ)

0.8

0.6

-0.04

-0.08

0.5

-0.12 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

τ

0.6

0.8

1.0

τ 0.04

0.9

a1 = 1, Ω0 = 0.5 Exact SK2 SK3 S8

0.7

a1 = 1, Ω0 = 0.5 Exact SK2 SK3 S8

0.00

q(τ)

0.8

G(τ)

215

-0.04

-0.08

0.6

-0.12

0.5 0.0

0.2

0.4

0.6 τ

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

τ

Fig. 1. The incident radiation and the net radiative heat flux profiles for a1 ¼ 1 and O0 ¼ 0.5 of Case A (a and b) and for a1 ¼ 1 (c and d).

towards the origin. The incident radiation profile with the S8 solutions also show deviations of larger magnitude (ray effect) near the center of the sphere. The net radiative heat flux profiles of the SKN and S8 solutions exhibit very good agreement with those of RITE in all scattering cases. For the backward scattering of Case A, using 800 grid intervals, the cpu times for the exact RITE, SK2, SK3, DOM S8 and S16 are 31.7, 0.20, 0.28, 0.02 and 0.04 s, respectively. For other benchmark problems, the cpu time is in the same order of magnitude. It is possible to reduce the cpu time of DOM or SKN solutions by using better initial guesses; however, in this study, the initial guesses for all quantities of the all methods used were assigned the same value to justify comparisons. In Fig. 2(a) and (b), the incident radiation profiles obtained with the exact RITE, SKN and S8 for Case A and O0 ¼ 0.995, and for forward and backward anisotropic scattering media, are depicted. In both cases, the SK3 solutions are in good agreement with those of the exact RITE, with only exception that, at the outer shell surface, the SK2 approximation exhibits slight deviations in forward and backward scattering media of Case A. The net heat flux profile exhibits very good agreement with the exact solution; in general, the deviations for the incident energy are larger in magnitude than those of heat flux. For that reason, comparisons solely based on the incident energy are adapted from this point on. In Fig. 2(c) and (d), the exact RITE, the SKN and S8 solutions of the incident radiation of Case B, for O0 ¼ 0.5 and for forward and backward anisotropic scattering media are depicted. The SK3 solutions are in excellent agreement with those of exact RITE. The S8 solutions exhibit slight deviations for a1 ¼ 1 and 1 at the origin due to ray effect. In Fig. 2(e) and (f), the

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Fig. 2. The incident radiation of Case A for (a) a1 ¼ 1, (b) a1 ¼ 1 (O0 ¼ 0.995), of Case B for (c) a1 ¼ 1, (d) a1 ¼ 1 (O0 ¼ 0.5), and of Case B for (e) a1 ¼ 1 and (f) a1 ¼ 1 (O0 ¼ 0.995).

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0.9

0.9

a1 = 1, Ω0 = 0.2+0.4F1 Exact SK2 SK3 S8

G(τ)

0.7

a1 = -1, Ω0 = 0.8-0.4F1 Exact SK2 SK3 S8

0.8 0.8 G(τ)

0.8

0.7

0.6 0.7 0.5

0.6 0.6

0.4 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

τ 0.8

0.8

1.0

2.0

2.5

0.7

a1 = -1, Ω0 = 0.2+0.4F1 Exact SK2 SK3 S8

a1 = 1, Ω0 = 0.8-0.4F1 Exact SK2 SK3 S8

0.6

0.5 G(τ)

0.6

G(τ)

0.6 τ

0.4

0.4 0.2 0.3

0.0

0.2 0.0

0.5

1.0

1.5 τ

2.0

2.5

0.0

0.5

1.0

1.5 τ

Fig. 3. The incident radiation of Case A for (a) a1 ¼ 1 and O0 ¼ 0.2+0.4F1, (b) a1 ¼ 1 and O0 ¼ 0.80.4F1, of Case B for (c) a1 ¼ 1 and O0 ¼ 0.2+0.4F1 and (d) a1 ¼ 1 and O0 ¼ 0.80.4F1.

exact RITE, the SKN and S8 solutions of the incident radiation profiles of Case B, for O0 ¼ 0.995 and for forward and backward anisotropic scattering media are depicted. In both cases, the SK3 and S8 solutions are in good agreement with those of exact RITEs, only the SK2 approximation exhibits slight deviations in forward and backward scattering media. The exact RITE, the SKN and S8 solutions of the incident radiation for Cases A and B, and for O0 ¼ 0.2+0.4F1 and 0.80.4F1 are depicted in Fig. 3. The SK3 solutions are in good agreement with those of exact RITE, while the SK2 approximation exhibits slight deviations in forward and backward scattering media of Case A. DOM S8 solutions show ray-effect deviations in larger magnitude near the t ¼ 0. In Fig. 4, the exact RITE, the SKN and S8 solutions of the incident radiation for Cases A and B for quadratically varying (O0 ¼ 0.44/15F1+0.5F2 and 1.016/15F1+0.5F2) scattering albedos are presented. In Figs. 4(a) and (b), the SK3 solutions are in good agreement with those of exact RITE, and the SK2 approximation exhibits slight deviations in forward and backward scattering optically very thin media (Case A). In this case, S8 solutions exhibit ray effect near the core. In Figs. 4(c) and (d), the SK2 and the SK3 incident energy solutions are almost identical. For a1 ¼ 1 of O0 ¼ 0.44/15F1+0.5F2 case, with respect to the exact solution, the incident energy is underestimated by 2–3% while for a1 ¼ 1 of O0 ¼ 1.016/ 15F1+0.5F2, it is overestimated. On the other hand, S8 solutions at the core also deviate from the exact

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0.9

0.9 a1 = 1, Ω0 = 0.4-4/15F1+0.5F2 Exact SK2 SK3 S8

Exact SK2 SK3 S8

0.8 G(τ)

G(τ)

0.8

a1 = -1, Ω0 = 1.0-16/15F1+0.5F2

0.8

0.7

0.7 0.7

0.6 0.6 0.5

0.6 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

τ 0.8

0.8

1.0

0.8 a1 = -1, Ω0 = 0.4-4/15F1+0.5F2 Exact SK2 SK3 S8

a1 = 1, Ω0 = 1.0-16/15F1+0.5F2 Exact SK2 SK3 S8

0.7 0.6 G(τ)

0.6

G(τ)

0.6 τ

0.4

0.5 0.4

0.2 0.3 0.0

0.2 0.0

0.5

1.0

1.5 τ

2.0

2.5

0.0

0.5

1.0

1.5

2.0

2.5

τ

Fig. 4. The incident radiation of Case A for (a) a1 ¼ 1 and O0 ¼ 0.44/15F1+0.5F2, (b) a1 ¼ 1 and O0 ¼ 1.016/15F1+0.5F2, of Case B for (c) a1 ¼ 1 and O0 ¼ 0.44/15F1+0.5F2 and (d) a1 ¼ 1 and O0 ¼ 1.016/15F1+0.5F2.

solution (with larger errors); however, towards the outer surface, S8 profile, along with SK2 and SK3 solutions, converges to the exact solution. 5. Conclusion The SKN approximation is applied to solid spherical participating homogeneous and inhomogeneous media. The SKN solutions for the test cases are compared with the exact and S8 solutions. Based on the results presented, the following conclusions have been drawn: (a) the method is computationally very easy to implement and memory efficient. Literature on the numerical solution and acceleration techniques of the second-order differential equations is plentiful, (b) the method could be extended and successfully employed to a linearly anisotropically scattering medium and (c) the ray affect is not observed in the SKN solutions, while S8 deviations at the core of the sphere are clearly due to the ray effect. References [1] Traugott SC. An improved differential approximation for radiative transfer with spherical symmetry. AIAA J 1969;7:1825–32.

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