ALBEDO PROBLEM FOR PURE-TRIPLET

SCATTERING IN FINITE MEDIUM WITH OPTICALLY SMOOTH AND ROUGH SURFACES

A. EL-Depsy and A. M. Shawky

Physics Department, Faculty of Science, Damietta University,

34517 New Damietta City, Egypt. Rec. 14/10/2017 In final form 19/03/2018 Accept. 02/06/2018

An exact integral formulation of radiative transfer equation in a medium with optically smooth and rough surfaces, absorbing and pure–triplet scattering is developed. The resulting integral equations are solved using Galerkin approximation. The effect of scattering coefficients, optical thickness and the roughness of the boundary on the reflection coefficients are investigated. The results are calculated and compared to the available experimental and theoretical data for both smooth and rough boundaries.

INTRODUCTION

Plane-parallel media with constant refractive index (like the atmosphere) have

been extensively studied. Degheidy etal [1,2] have solved albedo problem for pure-triplet scattering in both semi-infinite and finite medium with specular reflectivity at the boundaries. Francois A. etal [3] studied specular reflection of rough surfaces by describing the rough surface as constituted of multiple unresolved facets. Sallah etal [4,5] have calculated the heat fluxes for pure–triplet scattering of radiative transfer in binary discrete random finite media, continuous stochastic neutron transport in finite media and the time dependent radiative transfer in a finite slab.

El-Wakil etal [6] have investigated the reflection and transmission coefficients in biological tissues for anisotropic scattering with specular and diffuse reflectivity at the boundary.

Few studies consider plane-parallel media with reflective index changes. Most of these studies deal with Fresnel interface in plane-parallel media with specific simple configurations [7-10]: single layer with isotropic scattering [11, 12], and partial anisotropic scattering of a single slab under diffuse incident flux [13, 14], etc. Caron

Journal of Nuclear and Radiation Physics, Vol. 13 no. 2 (2018) 103-115

etal [15] used discrete ordinate method to treat a multi-layer system with smooth or rough interface.

The deviation characteristics of the specular reflectivity of micro-rough surfaces from that predicted by the Fresnel׳s equation under the assumption of smooth surface are examined by Zang etal [16].

In this paper, we will consider the pure triplet scattering in a medium with smooth and rough surfaces. In section 2, the system of radiative transfer equations are formulated. In section 3, we used the Galerkin method to solve the system of equations and obtain the coefficients of roughness. Numerical results with comparisons (for available smooth surfaces data) are presented with conclusions in section 4.

ANALYSIS

The equation of radiative transfer for an absorbing, gray, and pure triplet scattering, for plane-parallel medium of optical thickness b, is

1

1

( , ) ( , ) ( , ) ( , ) , 0 (1)2

x x P x d x bx

where

3 3( , ) 1 (5 3 )(5 3 ) (2)4fP

with boundary conditions

1(0, ) 1 (0, ) , 0 (3)

2( , ) ( , ) , 0 (4)b b Here (f = 7a, -1 ≤ a ≤ 1) where a is the anisotropic factor, where x is the optical space variable, µ is the direction cosine of the radiation intensity ψ(x, µ);ω is the single scattering albedo , ρ1 and ρ2 are the reflectivities at x=0 and x=b, respectively.

Solving Eq. (1) using the integrating factor yields

104 A. EL-Depsy et al.

/ ( )/12 /

1 2 0

2 ( )/1

0

2 /( )/1 2

2 /1 2 0

2 ( )/

0

( )/

1( , ) ( )1 2

(5 3) [5 ( ) 3 ( )]8

( ) , 0 51 2

(5 3) [5 ( ) 3 ( )]8

2

bx x y

b

bx y

bbx y

b

bx y

x y

x e e y dye

f e q y P y dy

e e y dye

f e q y P y dy

e

2 ( )/

0 0

( ) (5 3) [5 ( ) 3 ( )]8

x xx yfy dy e q y P y dy

and ( ) /

/ ( )/2 12 /

1 2 0

2 ( ) /1

0

( ) / 2 ( ) /

0 0

( )/

( , ) ( )1 2

(5 3) [5 ( ) 3 ( )] , 0 (6)8

( ) (5 3) [5 ( ) 3 ( )]2 8

( ) (52 8

bx bb y b

b

by b

b bb y b y

by x

x

ex e e y dye

f e q y P y dy

fe y dy e q y P y dy

fe y dy

2 ( ) /3) [5 ( ) 3 ( )]b

y x

x

e q y P y dy

where φ(x), P(x) and q(x) are integral functions which are defined as:

1

1

( ) ( , ) (7)x x d

1

1

( ) ( , ) (8)P x x d

and

(9)1 3( ) ( , )1

q x x d

Introducing Eqs. (5) and (6) into Eqs. (7-9), we obtain the corresponding integral forms of the radiative transfer equation which is written as follows:

ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 105

1 4 20 0

1 1(2 )/ /

2 2 / 2 /1 2 1 20 0

(2 )/ (2 )/ (2 )/ ( )/1 2 2 1

2 /1 2

( ) ( ) ( ) ( )[5 ( ) 3 ( )][5 ( ) 3 ( )]2 8

1 1

( )2 (1

b b

b x x

b b

b y x b y x b y x x y

b

fx E x y y dy sign x y E x y E x y q y P y dy

e ed de e

e e e ee

1

0 0

1 (2 )/ (2 )/ (2 )/ ( )/2 1 2 2 1

2 /1 20 0

( ) (10))

( )(5 3) [5 ( ) 3 ( )]8 1

b

b b y x b y x b y x x y

b

d y dy

e e e ef d q y P y dye

2 5 30 01 1(2 )/ /

2 2 / 2 /1 2 1 20 0

(2 )/ (2 )/ (2 )/ ( )/1 2 2 1

2 /1 2

( ) ( ) ( ) ( ) [5 ( ) 3 ( )][5 ( ) 3 ( )]2 8

1 1

( )2 1

b b

b x x

b b

b x y b y x b y x x y

b

fP x sign x y E x y y dy E x y E x y q y P y dy

e ed de e

e e e ee

1

0 0

1 (2 )/ (2 )/ (2 )/ ( )/3 1 2 2 1

2 /1 20 0

( ) (11)

( )(5 3 ) [5 ( ) 3 ( )]8 1

b

b b x y b y x b y x x y

b

d y dy

e e e ef d q y P y dye

and

4 7 50 0

1 13 (2 )/ 3 /

2 2 / 2 /1 2 1 20 0

(2 )/ (2 )/ (2 )/ ( )/2 1 2 2 1

1 2

( ) ( ) ( ) ( ) [5 ( ) 3 ( )][5 ( ) 3 ( )]2 8

1 1

( )(

2 1

b b

b x x

b b

b x y b y x b y x x y

fq x sign x y E x y y dy E x y E x y q y P y dy

e ed de e

e e e e

1

2 /0 0

1 (2 )/ (2 )/ (2 )/ ( )/5 3 1 2 2 1

2 /1 20 0

) ( ) (12)

( )(5 3 ) [5 ( ) 3 ( )]8 1

b

b

b b x y b y x b y x x y

b

d y dye

e e e ef d q y P y dye

where 1

/2

0

(x) xn n

dE e

106 A. EL-Depsy et al.

METHOD OF SOLUTION

To find solutions for φ(x), P(x) and q(x), we will use a set of trial functions in the form

0(x) , (13)

Nn

nn

A x

0

(x) , (14)N

nn

n

P B x

and

0(x) , (15)

Nn

nn

q D x

where An, Bn and Dn are the expansion coefficients. Substituting Eqs. (13)-(15) into Eqs. (10)-(12), multiplying each one of the resulting equations by xm, m=0,1,2,…N, and then integrating over x from 0 to b, we obtain the unknown coefficients, which are used to calculate some quantities such as, the reflectance and transmittance. These quantities are defined as

1

01

0

(0, ), (16)

(0, )

dR

d

and 1

01

0

( , ), (17)

(0, )

b dT

d

where 1

21 22 3 1

0 00

2 5

3 5 31

2

0

( )(0, ) ( (2 1)) { (1, (2 1))! 2

( 1, (2 1)) (5 3 )[{5 ( 1, (2 1))8

3 ( 1, (2 1)) {5 (1, (2 1)) 3 (1, (2 1)) (18)

(1, 0)2

i N

n ni n

n n n n

n n n

n nn

d E b i A b ii

fb i D B b i

b i b i b i

A

5 3

0

(5 3 )[5 (1, 0) 3 (1, 0)8

N N

n n n nn

f D B

ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 107

1

2 21 2 13 2

0 00

5 312

5 3

( )(0, ) (2 ) (1,2 ) ( 1, (2 1))! 2

(5 3 ) 5 ( 1, (2 1)) 3 ( 1, (2 1)) (19)8

5 (1,2 ) 3 (1,2 )

i N

n n ni n

n n n n

n n

d E ib A ib b ii

fD B b i b i

ib ib

and

12 21 2

3 10 00

5 3

5 31

( )( , ) ((2 1) ) ( 1, (2 1) ) (1, (2 1))

! 2

(5 3 ) 5 ( 1, (2 1)) 3 ( 1, (2 1)) (20)8

5 (1, (2 1) ) 3 (1, (2 1) )

i N

n n ni n

n n n n

n n

b d E i b A i b b ii

f D B b i b i

i b i b

We have used the definition of the reflectivity ρ1 and ρ2 which are defined in Murphy

[18].

The reflection functions of rough surfaces are considered for collimated and diffused incident radiation respectively. Francois A. et al. are considered a semi-analytical model to simulate bidirectional reflectance distribution function (BRDF) spectra of a rough slab layer containing impurities.[3] Also, He etal [19] gave expressions for the Bidirectional Reflectance Distribution Function (BRDF) as a sum of specular component and diffuse component respectively given as:

( )( )

, (21)g

c Fji e

i i

r e zrcos d

( )( ) , (22)d F

ji ei r

r G Z Drcos cos

1 / 2 , (23)r icos k k

where j = 1 and 2 refer to the left boundary and the right boundary, respectively to the internal surface, e to the external surface, and is the Fresnel reflection coefficient at the bisecting angle. and are respectively the unit vectors in the direction of the incident and reflected light, and are respectively the polar angles of incidence and reflectance, and the delta function =1 in the cone of specular reflection and ∆=0 elsewhere. Here i stands for incident and r for reflected light.

108 A. EL-Depsy et al.

The geometric factor G is defined as:

2

2

4 1 cos cos sin sin cos, (24)

cos cosi r i r r

i r

G

The surface roughness function g is given by

22 cos cos , (25)i rg

The distribution function is given by

2 22 2

2

1 exp , (26)44xyD

g g

where

1 22 22 sin 2sin sin cos sin , (27)xy i i r r r

And τ is the autocorrelation length, which is a measure of the spacing between surface peaks. The effective roughness σ was introduced by He etal [19] to allow averaging over only the illuminated (non-shadowed) parts of the surface. Particularly for grazing angles of incidence or reflection, it can be considerably smaller than the root mean square roughness σ0. They are related by

1/ 22 20 0 0= 1 / , (28)z

where z is the root of the transcendental equation

2

02

0

( )exp , (29)2 4 2i r

zz K K

and

0tan cot / 2 , (30)i i iK erfc 0tan cot / 2 , (31)r r rK erfc

The shadowing function z is given by

, (32)i i r rz z z where

0

11 cot / 22 , (33)

cot 1

i

i ii

erfcz

ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 109

0

11 cot / 22 , (34)

cot 1

r

r rr

erfcz

and

0

0

cot21cot , (35)2 2cot

erfc

The reflection coefficient ρ is defined as

/ 2

0

, (36)i r r r rr cos cos sin d d

The diffused component of ρ is given by / 2

0

, (37)d di r r r rr cos cos sin d d

where is the azimuthal angle of reflection. It is assumed that the azimuthal angle of incidence is . The specular reflectance coefficient is

( ) , (38)c gF ir e z

For normal incidence, the diffused component of is given by / 2

0

1 ( / 2) , (39)dF r r rr G Z D sin d

where 2

0

2(x) , (40)x

terfc e dt

RESULTS AND CONCLUSIONS

The reflectance and transmittance are calculated for specified values of

refractive index (n), coefficients of pure-triplet scattering (a), single scattering albedo (ω) and the thickness (b) for isotropic incidence of unit intensity f*(μ), on the transparent left boundary.

For smooth surfaces with constant refractive indices we calculated the reflection at the boundary given in Table 1.

110 A. EL-Depsy et al.

Table 1. Values of the reflection coefficients for smooth surface

n 1

1,20

=2 ,Fr n d 1

00

1=2 ,Fr dn

1.1 0.194343 0.0251574 1.2 0.336305 0.0442803 1.3 0.444457 0.0611318 1.4 0.528980 0.0768115 1.5 0.596342 0.0917780 1.6 0.650879 0.1062460 1.7 0.695617 0.1203200 1.8 0.732726 0.1340540 1.9 0.763841 0.1474760 2.0 0.790151 0.1605970

For smooth and rough surfaces of conducting material, we calculated the

reflection at the boundary given in Table 2.

Table 2. Values of the reflection coefficients for smooth , and for rough surfaces

n 1 2 r 1.1 0.0703964 0.00035600 1.2 0.0957300 0.00129809 1.3 0.1146950 0.00267357 1.4 0.1309080 0.00436750 1.5 0.1456270 0.00629279 1.6 0.1594130 0.00838310 1.7 0.1725430 0.01058780 1.8 0.1851710 0.01286810 1.9 0.1973810 0.01519500 2.0 0.2092260 0.01754710

Table 3 presents the effect of (n), (a) and (b = 1) on reflectance for smooth

surfaces are calculated and compared with available data from Ref. [2].

ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 111

Table 3. Converged values of reflections for Smooth surface of thickness (b = 1), ,

In Table 4, the transmittance for smooth surfaces are calculated and compared

with available data from Ref. [2].

Table 4. Converged values of transmission for smooth surface of thickness (b = 1), ,

In tables 5 to 7 the reflectance and transmittance are calculated for rough

surfaces with different values of the coefficients of pure-triplet scattering (a).

a

Ref.[2] Present Ref.[2] Present Ref.[2] Present

0.020598 0.043654 0.069685 0.099375 0.133647 0.173775 0.221577 0.279742 0.352432

0.020875 0.044287 0.070784 0.101097 0.136216 0.177528 0.227033 0.287739 0.364388

0.020696 0.043841 0.069953 0.099715 0.134048 0.174224 0.222062 0.280256 0.352978

0.020973 0.044475 0.071053 0.101435 0.136612 0.177966 0.227498 0.288217 0.364878

0.020749 0.043944 0.070101 0.099902 0.134266 0.174467 0.222323 0.280529 0.353266

0.021027 0.044578 0.071201 0.101620 0.136827 0.178202 0.227747 0.288471 0.365135

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

b

Ref.[2] Present Ref.[2] Present Ref.[2] Present

0.231769 0.246111 0.262881 0.282708 0.306448 0.335304 0.371006 0.416145 0.474763

0.231883 0.246518 0.263817 0.284483 0.309485 0.340185 0.378556 0.427574 0.491924

0.231788 0.246137 0.262901 0.282703 0.306399 0.335186 0.370794 0.41581 0.474284

0.231912 0.246563 0.263859 0.284503 0.309461 0.340090 0.378365 0.427258 0.491464

0.231797 0.24615 0.26291 0.2827

0.306372 0.335123 0.370681 0.415633 0.474032

0.231928 0.246587 0.263882 0.284515 0.309450 0.340042 0.378266 0.427094 0.491225

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

112 A. EL-Depsy et al.

Table 5. Converged values of reflections and transmissions for rough surface of thickness (b = 1) , a = 0.

n

Reflectance Transmittance Reflectance Transmittance Reflectance Transmittance

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.020701 0.043849 0.069961 0.099716 0.134031 0.174173 0.221953 0.280055 0.352642

0.231849 0.246261 0.263085 0.282947 0.306699 0.335538 0.371191 0.416244 0.474748

0.016309 0.034796 0.056043 0.080718 0.109723 0.144309 0.186255 0.238194 0.304177

0.231854 0.246271 0.263104 0.282977 0.306743 0.335602 0.371283 0.416378 0.474944

0.021028 0.044238 0.070411 0.100229 0.134612 0.174834 0.222716 0.280959 0.353755

0.231865 0.246297 0.263149 0.283048 0.306849 0.335757 0.371506 0.416700 0.475417

n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.021291 0.044529 0.070737 0.100599 0.135038 0.175336 0.223322 0.281710 0.354719

0.231880 0.246331 0.263208 0.283140 0.306987 0.335957 0.371795 0.417119 0.476032

0.021586 0.044855 0.071102 0.101013 0.135517 0.175899 0.224000 0.282552 0.355802

0.231897 0.246369 0.263275 0.283244 0.307142 0.336183 0.372120 0.417590 0.476725

0.021895 0.045196 0.071483 0.101446 0.136017 0.176488 0.224710 0.283433 0.356935

0.231915 0.246410 0.263344 0.283353 0.307305 0.336419 0.372461 0.418083 0.477450

Table 6. Converged values of reflection and transmission for rough surface of thickness (b = 1), a = 0.05

n

Reflection Transmission Reflection Transmission Reflection Transmission

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.020687 0.043818 0.069910 0.099643 0.133934 0.174052 0.221808 0.279885 0.352440

0.231837 0.246243 0.263069 0.282942 0.306716 0.335588 0.371289 0.416403 0.474979

0.020772 0.043912 0.070015 0.099762 0.134071 0.174214 0.222003 0.280126 0.352749

0.231842 0.246254 0.263089 0.282972 0.306760 0.335653 0.371382 0.416538 0.475177

0.020974 0.044135 0.070265 0.100046 0.134398 0.174598 0.222466 0.280701 0.353487

0.231854 0.246280 0.263134 0.283043 0.306866 0.335808 0.371605 0.416860 0.475649

ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 113

n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.021239 0.044428 0.070592 0.100417 0.134827 0.175103 0.223074 0.281454 0.354455

0.231869 0.246315 0.263194 0.283137 0.307006 0.336010 0.371896 0.417283 0.476269

0.021534 0.044754 0.070957 0.100831 0.135305 0.175665 0.223752 0.282295 0.355537

0.231886 0.246353 0.263260 0.283241 0.307161 0.336236 0.372223 0.417755 0.476963

0.021842 0.045095 0.071338 0.101264 0.135804 0.176253 0.224461 0.283176 0.356670

0.231903 0.246393 0.263330 0.283351 0.307324 0.336473 0.372564 0.418250 0.477690

Table 7. Converged values of reflections and transmissions for rough surface of thickness (b = 1), a = 0.14

n

Reflection Transmission Reflection Transmission Reflection Transmission

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.020590 0.043633 0.069645 0.099309 0.133543 0.173617 0.221343 0.279398 0.351929

0.231816 0.246212 0.263042 0.282935 0.306749 0.335686 0.371477 0.416710 0.475422

0.020675 0.043726 0.069750 0.099428 0.133680 0.173779 0.221537 0.279639 0.352238

0.231821 0.246223 0.263061 0.282965 0.306794 0.335751 0.371571 0.416845 0.475621

0.020877 0.043949 0.069999 0.099711 0.134006 0.174163 0.222000 0.280213 0.352975

0.231833 0.246250 0.263107 0.283037 0.306901 0.335907 0.371795 0.417169 0.476096

n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.021142 0.044242 0.070326 0.100081 0.134434 0.174666 0.222606 0.280965 0.353942

0.231848 0.246284 0.263168 0.283131 0.307042 0.336110 0.372088 0.417593 0.476719

0.021437 0.044567 0.070690 0.100495 0.134911 0.175228 0.223283 0.281805 0.355022

0.231865 0.246323 0.263235 0.283237 0.307198 0.336338 0.372416 0.418068 0.477415

0.021745 0.044908 0.071071 0.100927 0.135410 0.175815 0.223991 0.282684 0.356154

0.231883 0.246364 0.263306 0.283347 0.307362 0.336577 0.372760 0.418565 0.478146

CONCLUSION

From the previous results we conclude the following points:

1. Smooth surfaces are ideal, while the real surfaces are almost rough. 2. Human tissues, metal surfaces, the boundaries of nuclear reactors, etc. are

examples of rough surfaces

114 A. EL-Depsy et al.

3. The roughness of the human tissues can be used as indicator to differentiate between healthy and ill organs, like liver, kidneys, lunges etc.

4. This model has also many applications to examine materials, like paint, pigmented plastics, decorative and protective coating, crystalline materials, etc.

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ALBEDO PROBLEM FOR PURE-TRIPLET SCATTERING … 115