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The effect of forward, backward and linear anisotropic scattering on the critical slab thickness
J. Quant. Spectrosc. Radiut. Trons/er Vol. 53, NO. 6, pp. 687-691.
1995
Copyright c 1995 Elsevier Science Ltd
Pergamon
0022-4073(95)00019-l
Printed
in Great
Britain. All rights reserved 0022-4073195 $9.50 + 0.00
THE EFFECT OF FORWARD, BACKWARD AND LINEAR ANISOTROPIC SCATTERING ON THE CRITICAL SLAB THICKNESS CEVDET TEZCANt,f IDepartment
of Physics,
and CEMAL YILDIZg
Ankara University, 01600, Tandogan, Ankara and §Department Istanbul Technical University, 80626, Istanbul, Turkey
of Physics.
(Received 6 July 1994; received for publication 20 February 1995)
Abstract-The critical slab problem is studied for, one-speed time independent, plane symmetric linear transport equation, using a scattering kernel which consists of linear combinations of backward, forward and linear anisotropic scattering. The moment equations of P, approximation written for anisotropic scattering kernel are transformed into moment equations for linear anisotropic scattering. This approach is based on the use of scale transformations. Moment solutions are connected with a subgroup of 2 x 2 reel symmetric matrices. Using the solution procedure given for linear anistropic scattering, the contribution of the combination of the linear anisotropy and 8-distributions to the critical slab thickness is obtained up to the P,,-approximation.
INTRODUCTION
In single speed time-independent, neutron transport theory, extremely anisotropic scattering kernels which combine 6 -functions and isotropic scattering’,’ are applied to many problems with various methods3m3’in order to understand its contribution on the physical parameters such as Milne’s problem, the criticality problems, . . . , etc. In the usual development of neutron transport theory the variation of the critical sizes with linear, quadratic,. . . , etc. anisotropy or with the degree of anisotropy were always monotonic. The studies with the kernel expressed in terms of 6-functions led to the non-monotonic variations of the critical sizes in plane and cylindrical geometries. The kernel which combine 6-functions and linear anisotropic scattering is also used to see the contribution of linear anisotropy.‘.” Here we want to investigate the contribution of linear anisotropy with 6 functions on the critical slab thickness using the well-known P,, approximation. We consider the kernel
f=M
+Y~~‘)+bs(l*‘--)+d~(~‘+~)
(1)
which can be written as
(2) where
fn =a&
+b +(-l)“d+fa$,,,
(3)
b, d give the fraction of neutrons which emerge from a collision in the forward backward directions respectively. In addition a = 1-b -d and y is the coefficient of linear anisotropic scattering. tTo whom OSRT53.b”
all correspondence
should
be addressed 687
Cevdet Table
I. The values
Texan
and Cemal
of a’ for backward scattering c=1.1,yf=3(1-y’)=2.95.
Ytldtz (a = I - a, b = 0, d = a) for
1.2,y * = 3( 1 - y ‘) = 2.95.
Table 2. The values of a’ for backward
scattering
for c =
Table 3. The values of a for backward
scattering
for c = 1.1, y * = 3( 1 - y ‘) = 2.95.
Table 4. The values of a for backward
scattering
for c =
1.2,y * = 3( 1 - y ‘) = 2.95.
Table 5. The values of a’ for backward
scattering
for c =
I. 1, Y* = 3( I - Y‘) = 2.97.
Table 6. The values of a’ for backward
scattering
for c =
I .2, y * = 3( I - y’) = 2.97.
The effect of forward, backward and linear anisotropic scattering Table 7. The values of a for backward scattering for c = I .I,
IO.4546 1
THE
1.76571
1.75631
TRANSFORMATION
i.7%7[ 1.75501 ,31
1.79641 1.75481 1.63391
7*
1.79631 1.75461 1.63371
OF ANISOTROPIC
= 3(1 -
Y') = 2.97.
1.5 1.9 1.1
PN-EQUATIONS
In order to study the effect of linear anisotropy with 6-functions on the critical slab of half thickness t we consider the one-speed neutron transport equation32,34 +I d P--_(x,P)++(x?P)=c +(x3 P’Y(P’-+P) dp, dx s -I with the boundary conditions ‘+(t,-~)P2a-I(~)d~=0 a=1,2...,(N+1)/2. s0 The expansion of the angular flux in terms of Legendre polynomials
(5)
(6) and the kernel in Eq. (1) in plane geometry leads to3’ (k+l)~~,,,+k~~,_,+(2k+1)11-c’6,-7.6x,)OI=0,
(7)
where c’=ac/(l
-bc
-cd),
(84
y = acy/3( 1 - bc + cd), x’ = [(l - bc)* - c*d*]“*x, l(/k= &/[l - bc - (- l)%d].
W) (9) (10)
From Eq. (10) it follows that 1(/i’)= ~$c’/(l - bc - cd)“*,
for even k,
(11)
$(kl)= (bp’/(l - bc + cd)“*,
for odd k,
(12)
or
=(l
-bcl_cd)“‘(; ;)($)’
(13)
where q = (1 - bc - cd)“*/(l - bc + cd)“*.
(14)
Cevdet Tezcan and Cemal Yhz
690
The differential equations (7) are similar to the equations for linear anistropic scattering and fission. Only r and c are replaced by r’ and c’. Therefore the solutions for extremely anistropic kernel with linear anistropy in Eq. (1) in plane geometry can be written as35 4;’ = Ak cash wx,
for even k,
r#~f)= A,, sinh wx,
for odd k,
(15)
where x’ is given in Eq. (9). If the coefficients Ak of the moment solutions satisfy (k + l)wA,+, + kcoAk_, + (2k + l)A,(l - CC&,)= 0, (I - 6,,)(k
+ I)coAp+, + kd-,
(16)
k even, k odd.
+ (2k + l)Ak(l - y&,) = 0,
(17)
Eqs. (16) and (17) can be written in matrix form as [D(w)]A
(18)
= 0
where (1 -c’)
0 3(1-y’)
Co
P(w)1 =
0
0
0
.
.
0
2W
0
.
.
0
5
30
’
.
0
.
.
0
0
20
0
0
(19)
NW
.
.
No
(2N+
1)
For a nontrivial solution of Eq. (18),
(20)
det[D(o)] = 0, gives the roots 0,.
The general solution of C&(X), for the m th root, w, has the form
(21) m=I
where $T = cosh(o,x’), = sinh(o,x’),
for even k
(22)
for odd k
(23)
Ap is the solution to [D(o,)];ir
with At = for all m.
The general solution for the angular distribution W/[l
= ; n$o ‘r$r
in Eq. (6) is given as
- bc - (-
W4”2)RW
(24)
(2n + l)h, A:c#$‘Pn (/i)/[l - bc - ( - 1W]
.
(25)
Using Marshak boundary conditions in Eq. (5) for a slab of half thickness t, the criticality conditions is obtained for the scattering kernel in Eq. (1). In P, approximation one can obtain the following analytical expressions for the critical half thickness t t=-
1 tanh-li[=]li2[! {[(l - bc)2 - c*d*][3(1 - c)(l -v)]}“*
1::
Tzr,
(26)
691
The effect of forward, backward and linear anisotropic scattering
or in terms of unprimed quantities, we get I=-
{[(I - c)[3(1 - ble +cd) - acy]j”2tanh-’
1 3(1 - bc + cd) - aq 4
(27)
If the scattering kernel represent for example isotropic scattering with backward scattering, that is, if a = 1 - CIb = 0, d = a, then the expression for the critical half thickness t turns to be
f = - ([(l - C)[3(1 + 2) - (1 - 441:2tanh-’
13(1 +ac)-(1 I-c 4
(28)
-a)cy
or since c > 1, it becomes
f = - {[(c - 1)[3(1 +a:,
-(l
13(1 +ac)-(1 - a)cy]}r’Ztan-’ [ 4 c-l
-a)q
I7 ‘I2
(29)
One can similarly obtain the analytical expressions of the critical half thickness for isotropic scattering with forward scattering (a = 1 - a, b = a, d = 0) or for forward scattering with backward scattering for specified isotropic scattering (a = 1 - 2k, b = k + a, d = k - a). The solution of the linear transport equation with the combination d-functions with isotropic scattering and the numerical values of the critical half thickness were given up to the P,, approximation 3’. Here the contribution on the variation of the critical half thickness with the combination of 6-functions with linear anistropic scattering is obtained. The linear anistropy parameter in Eq.(l) is expressed by y. The numerical values of the critical half thickness for backward anisotropy and for different values of y shown in tables are obtained for c = 1.1 and 1.2, respectively,
up to P,, approximation. The convengence of the numerical values is seen to be good, except for strong backward anisotropy.
REFERENCES 1. E. in6nii, Transp. Theory Statist. Phys. 3, 107 (1973). 2. 3. 4. 5. 6. 7. 8. 9. IO. 11. 12. 13. 14.
15. 16. 17. 18.
19. 20. 21. 22. 23. 24.
25. 26. 27. 28.
29. 30. 31. 32. 33. 34. 35.
E. in&G, Phys. Fluids 19, 1332 (1976). C. E. Siewert and M. M. R. Williams, J. Phys. D: Appl. Phys. 10, 2031 (1977). C. Tezcan, Transp. Theory Statist. Phys. 6, 91 (1977). C. E. Siewert and C. Devaux, JQSRT 20, 29 (1978). E. fndnii and C. Tezcan, Transp. Theory Statist. Phys. 7, 51 (1978). G. C. Pomraning, Trunsp. Theory Statist. Phys. 7, 161 (1978). M. M. R. Williams, J. Phys. D: Appl. Phys. 11, 2455 (1978). C. Tezcan, Transp. Theory Statist. Phys. 8, 181 (1979). C. Tezcan, Transp. Theory Statist. Phys. 8, 187 (1979). C. Devaux, C. E. Siewert, and Y. Yener, JQSRT. 21, 505 (1979). S. A. Elwakil, M. H. Haggag, and E. A. Saad, JQSRT 23, 553 (1980). S. A. Elwakil, JQSRT 24, 179 (1980). G. C. Pomraning, Trunsp. Theory Statist. Phys. 9, 1 (1980). G. Spiga and P. Vestrucci, Ann. Nucl. Energy 8, 165 (1980). C. Tezcan, Trunsp. Theory Statist. Phys. 10, 105 (1981). R. D. M. Garcia and C. E. Siewert, J. Phys D: Appl. Phys. 14, L65 (1981). R. T. Ackyord and M. M. R. Williams, Ann. Nucl. Energy 11, 269 (1984). C. Tezcan and R. Sever, Ann. Nucl. Energy 12, 573 (1985). A. Shafig, H. E. De Meyer, and C. C. Grosjean, Ann. Nucl. Energy 12, 9 (1985). A. Shafig, H. E. De Meyer, and C. C. Grosjean, Ann. Nucl. Energy 12, 23 (1985). C. Tezcan and R. Sever, Ann. Nucl. Energy 13, 223 (1986). C. Tezcan and C. Ytldtz, Ann. Nucl. Energy 13, 345 (1986). J. S. Cassell, Ann. Nucl. Energy 13, 219 (1986). C. Yildiz, I1 Nuouo Cimento 103B, (1989). C. Yildiz, If Nuouo Cimento 105B, (1990). D. C. Sahni and N. G. Sjostrand, Trunsp. Theory Statist. Whys. 20, 339 (1991). C. Tezcan, J. Math. Phys. 32, January (1991). K. Razi Naqvi, A. El-Shahat, and S. A. El-Wakil, Whys. Reo. A 49, 4967 (1992). C. Tezcan and C. Yildiz, II Nuovo Cimento 106A (1993). C. Tezcan and C. Ytldiz, JQSRT 49, 41 I (1993). G. I. Bell and S. Glasstone, Nuclear Renctor Theory, Van Nostrand-Reinhold, New York, (1972). K. M. Case and P. F. Zweifel, Lineur Trnnsport Theory, Addison-Wesley, New York, (1967). B. Davison, Neutron Transport Theory, London (1958). C. E. Lee and M. P. Dias, Ann. Nucl. Energy 11, 515 (1984).
1995
Copyright c 1995 Elsevier Science Ltd
Pergamon
0022-4073(95)00019-l
Printed
in Great
Britain. All rights reserved 0022-4073195 $9.50 + 0.00
THE EFFECT OF FORWARD, BACKWARD AND LINEAR ANISOTROPIC SCATTERING ON THE CRITICAL SLAB THICKNESS CEVDET TEZCANt,f IDepartment
of Physics,
and CEMAL YILDIZg
Ankara University, 01600, Tandogan, Ankara and §Department Istanbul Technical University, 80626, Istanbul, Turkey
of Physics.
(Received 6 July 1994; received for publication 20 February 1995)
Abstract-The critical slab problem is studied for, one-speed time independent, plane symmetric linear transport equation, using a scattering kernel which consists of linear combinations of backward, forward and linear anisotropic scattering. The moment equations of P, approximation written for anisotropic scattering kernel are transformed into moment equations for linear anisotropic scattering. This approach is based on the use of scale transformations. Moment solutions are connected with a subgroup of 2 x 2 reel symmetric matrices. Using the solution procedure given for linear anistropic scattering, the contribution of the combination of the linear anisotropy and 8-distributions to the critical slab thickness is obtained up to the P,,-approximation.
INTRODUCTION
In single speed time-independent, neutron transport theory, extremely anisotropic scattering kernels which combine 6 -functions and isotropic scattering’,’ are applied to many problems with various methods3m3’in order to understand its contribution on the physical parameters such as Milne’s problem, the criticality problems, . . . , etc. In the usual development of neutron transport theory the variation of the critical sizes with linear, quadratic,. . . , etc. anisotropy or with the degree of anisotropy were always monotonic. The studies with the kernel expressed in terms of 6-functions led to the non-monotonic variations of the critical sizes in plane and cylindrical geometries. The kernel which combine 6-functions and linear anisotropic scattering is also used to see the contribution of linear anisotropy.‘.” Here we want to investigate the contribution of linear anisotropy with 6 functions on the critical slab thickness using the well-known P,, approximation. We consider the kernel
f=M
+Y~~‘)+bs(l*‘--)+d~(~‘+~)
(1)
which can be written as
(2) where
fn =a&
+b +(-l)“d+fa$,,,
(3)
b, d give the fraction of neutrons which emerge from a collision in the forward backward directions respectively. In addition a = 1-b -d and y is the coefficient of linear anisotropic scattering. tTo whom OSRT53.b”
all correspondence
should
be addressed 687
Cevdet Table
I. The values
Texan
and Cemal
of a’ for backward scattering c=1.1,yf=3(1-y’)=2.95.
Ytldtz (a = I - a, b = 0, d = a) for
1.2,y * = 3( 1 - y ‘) = 2.95.
Table 2. The values of a’ for backward
scattering
for c =
Table 3. The values of a for backward
scattering
for c = 1.1, y * = 3( 1 - y ‘) = 2.95.
Table 4. The values of a for backward
scattering
for c =
1.2,y * = 3( 1 - y ‘) = 2.95.
Table 5. The values of a’ for backward
scattering
for c =
I. 1, Y* = 3( I - Y‘) = 2.97.
Table 6. The values of a’ for backward
scattering
for c =
I .2, y * = 3( I - y’) = 2.97.
The effect of forward, backward and linear anisotropic scattering Table 7. The values of a for backward scattering for c = I .I,
IO.4546 1
THE
1.76571
1.75631
TRANSFORMATION
i.7%7[ 1.75501 ,31
1.79641 1.75481 1.63391
7*
1.79631 1.75461 1.63371
OF ANISOTROPIC
= 3(1 -
Y') = 2.97.
1.5 1.9 1.1
PN-EQUATIONS
In order to study the effect of linear anisotropy with 6-functions on the critical slab of half thickness t we consider the one-speed neutron transport equation32,34 +I d P--_(x,P)++(x?P)=c +(x3 P’Y(P’-+P) dp, dx s -I with the boundary conditions ‘+(t,-~)P2a-I(~)d~=0 a=1,2...,(N+1)/2. s0 The expansion of the angular flux in terms of Legendre polynomials
(5)
(6) and the kernel in Eq. (1) in plane geometry leads to3’ (k+l)~~,,,+k~~,_,+(2k+1)11-c’6,-7.6x,)OI=0,
(7)
where c’=ac/(l
-bc
-cd),
(84
y = acy/3( 1 - bc + cd), x’ = [(l - bc)* - c*d*]“*x, l(/k= &/[l - bc - (- l)%d].
W) (9) (10)
From Eq. (10) it follows that 1(/i’)= ~$c’/(l - bc - cd)“*,
for even k,
(11)
$(kl)= (bp’/(l - bc + cd)“*,
for odd k,
(12)
or
=(l
-bcl_cd)“‘(; ;)($)’
(13)
where q = (1 - bc - cd)“*/(l - bc + cd)“*.
(14)
Cevdet Tezcan and Cemal Yhz
690
The differential equations (7) are similar to the equations for linear anistropic scattering and fission. Only r and c are replaced by r’ and c’. Therefore the solutions for extremely anistropic kernel with linear anistropy in Eq. (1) in plane geometry can be written as35 4;’ = Ak cash wx,
for even k,
r#~f)= A,, sinh wx,
for odd k,
(15)
where x’ is given in Eq. (9). If the coefficients Ak of the moment solutions satisfy (k + l)wA,+, + kcoAk_, + (2k + l)A,(l - CC&,)= 0, (I - 6,,)(k
+ I)coAp+, + kd-,
(16)
k even, k odd.
+ (2k + l)Ak(l - y&,) = 0,
(17)
Eqs. (16) and (17) can be written in matrix form as [D(w)]A
(18)
= 0
where (1 -c’)
0 3(1-y’)
Co
P(w)1 =
0
0
0
.
.
0
2W
0
.
.
0
5
30
’
.
0
.
.
0
0
20
0
0
(19)
NW
.
.
No
(2N+
1)
For a nontrivial solution of Eq. (18),
(20)
det[D(o)] = 0, gives the roots 0,.
The general solution of C&(X), for the m th root, w, has the form
(21) m=I
where $T = cosh(o,x’), = sinh(o,x’),
for even k
(22)
for odd k
(23)
Ap is the solution to [D(o,)];ir
with At = for all m.
The general solution for the angular distribution W/[l
= ; n$o ‘r$r
in Eq. (6) is given as
- bc - (-
W4”2)RW
(24)
(2n + l)h, A:c#$‘Pn (/i)/[l - bc - ( - 1W]
.
(25)
Using Marshak boundary conditions in Eq. (5) for a slab of half thickness t, the criticality conditions is obtained for the scattering kernel in Eq. (1). In P, approximation one can obtain the following analytical expressions for the critical half thickness t t=-
1 tanh-li[=]li2[! {[(l - bc)2 - c*d*][3(1 - c)(l -v)]}“*
1::
Tzr,
(26)
691
The effect of forward, backward and linear anisotropic scattering
or in terms of unprimed quantities, we get I=-
{[(I - c)[3(1 - ble +cd) - acy]j”2tanh-’
1 3(1 - bc + cd) - aq 4
(27)
If the scattering kernel represent for example isotropic scattering with backward scattering, that is, if a = 1 - CIb = 0, d = a, then the expression for the critical half thickness t turns to be
f = - ([(l - C)[3(1 + 2) - (1 - 441:2tanh-’
13(1 +ac)-(1 I-c 4
(28)
-a)cy
or since c > 1, it becomes
f = - {[(c - 1)[3(1 +a:,
-(l
13(1 +ac)-(1 - a)cy]}r’Ztan-’ [ 4 c-l
-a)q
I7 ‘I2
(29)
One can similarly obtain the analytical expressions of the critical half thickness for isotropic scattering with forward scattering (a = 1 - a, b = a, d = 0) or for forward scattering with backward scattering for specified isotropic scattering (a = 1 - 2k, b = k + a, d = k - a). The solution of the linear transport equation with the combination d-functions with isotropic scattering and the numerical values of the critical half thickness were given up to the P,, approximation 3’. Here the contribution on the variation of the critical half thickness with the combination of 6-functions with linear anistropic scattering is obtained. The linear anistropy parameter in Eq.(l) is expressed by y. The numerical values of the critical half thickness for backward anisotropy and for different values of y shown in tables are obtained for c = 1.1 and 1.2, respectively,
up to P,, approximation. The convengence of the numerical values is seen to be good, except for strong backward anisotropy.
REFERENCES 1. E. in6nii, Transp. Theory Statist. Phys. 3, 107 (1973). 2. 3. 4. 5. 6. 7. 8. 9. IO. 11. 12. 13. 14.
15. 16. 17. 18.
19. 20. 21. 22. 23. 24.
25. 26. 27. 28.
29. 30. 31. 32. 33. 34. 35.
E. in&G, Phys. Fluids 19, 1332 (1976). C. E. Siewert and M. M. R. Williams, J. Phys. D: Appl. Phys. 10, 2031 (1977). C. Tezcan, Transp. Theory Statist. Phys. 6, 91 (1977). C. E. Siewert and C. Devaux, JQSRT 20, 29 (1978). E. fndnii and C. Tezcan, Transp. Theory Statist. Phys. 7, 51 (1978). G. C. Pomraning, Trunsp. Theory Statist. Phys. 7, 161 (1978). M. M. R. Williams, J. Phys. D: Appl. Phys. 11, 2455 (1978). C. Tezcan, Transp. Theory Statist. Phys. 8, 181 (1979). C. Tezcan, Transp. Theory Statist. Phys. 8, 187 (1979). C. Devaux, C. E. Siewert, and Y. Yener, JQSRT. 21, 505 (1979). S. A. Elwakil, M. H. Haggag, and E. A. Saad, JQSRT 23, 553 (1980). S. A. Elwakil, JQSRT 24, 179 (1980). G. C. Pomraning, Trunsp. Theory Statist. Phys. 9, 1 (1980). G. Spiga and P. Vestrucci, Ann. Nucl. Energy 8, 165 (1980). C. Tezcan, Trunsp. Theory Statist. Phys. 10, 105 (1981). R. D. M. Garcia and C. E. Siewert, J. Phys D: Appl. Phys. 14, L65 (1981). R. T. Ackyord and M. M. R. Williams, Ann. Nucl. Energy 11, 269 (1984). C. Tezcan and R. Sever, Ann. Nucl. Energy 12, 573 (1985). A. Shafig, H. E. De Meyer, and C. C. Grosjean, Ann. Nucl. Energy 12, 9 (1985). A. Shafig, H. E. De Meyer, and C. C. Grosjean, Ann. Nucl. Energy 12, 23 (1985). C. Tezcan and R. Sever, Ann. Nucl. Energy 13, 223 (1986). C. Tezcan and C. Ytldtz, Ann. Nucl. Energy 13, 345 (1986). J. S. Cassell, Ann. Nucl. Energy 13, 219 (1986). C. Yildiz, I1 Nuouo Cimento 103B, (1989). C. Yildiz, If Nuouo Cimento 105B, (1990). D. C. Sahni and N. G. Sjostrand, Trunsp. Theory Statist. Whys. 20, 339 (1991). C. Tezcan, J. Math. Phys. 32, January (1991). K. Razi Naqvi, A. El-Shahat, and S. A. El-Wakil, Whys. Reo. A 49, 4967 (1992). C. Tezcan and C. Yildiz, II Nuovo Cimento 106A (1993). C. Tezcan and C. Ytldiz, JQSRT 49, 41 I (1993). G. I. Bell and S. Glasstone, Nuclear Renctor Theory, Van Nostrand-Reinhold, New York, (1972). K. M. Case and P. F. Zweifel, Lineur Trnnsport Theory, Addison-Wesley, New York, (1967). B. Davison, Neutron Transport Theory, London (1958). C. E. Lee and M. P. Dias, Ann. Nucl. Energy 11, 515 (1984).