APPLICATION OF THE T
NMETHOD FOR THE
CRITICAL SLAB PROBLEM WITH FORWARD,
BACKWARD AND LINEAR ANISOTROPIC
SCATTERING
Hakan Öztürk
aand Süleyman Güngör
baOsmaniye Korkut Ata University,
Technical & Vocational School of Higher Education, Osmaniye/Turkey bÇukurova University, Faculty of Science & Letters,
INTRODUCTION
•
In transport theory, criticality problems in different geometries with different scattering kernels have been studied by polynomial expansion based techniques. Among these methods, although the Legendre polynomials expansion is the most commonly used method, it is not the unique one valid for all cases. As an alternative to PN method, Aspelund, Conkie and Yabushita have used the Chebyshev polynomials approximation method (TN method) and presented the advantages of using Chebyshev polynomials in the angular distribution.•
In this work, the approximation with Chebyshev polynomials of the first kind, i.e. TN method, is developed for the solution of the critical slab problem. One-speed neutrons are assumed to be scattered anisotropically in forward and backward directions along the slab. In the method, the angular distribution is defined in terms of Chebyshev polynomials of the first kind similar to previous works, and then critical slab thicknesses are computed for various values of c, the mean number of secondary neutrons per collision, and the anisotropy parameters.•
It is observed from the tabulated results that the TN method with its rapid convergence and comparatively simple algebraic manipulation gives accurate results as well as the other polynomial expansion based techniques in transport theory calculations.THE CRITICAL SLAB PROBLEM
The time-independent stationary transport equation with forward, backward and linear anisotropic scattering is given by,
subject to free space boundary and symmetry conditions: 0 ) , ( a 0 ), , ( ) , ( x x
Here (x,) is the angular flux of the neutrons at position x traveling in direction , cosine of the angle between the neutron velocity vector and the positive x axis. T is the total macroscopic cross section and c is the number of secondary neutrons per collision. and are the parameters varied over the range of 0 , 1, + 1 and denote the forward and backward scattering probabilities in a collision, respectively. b1 is the average cosine of the scattering angle, [10]. It is assumed that the slab material is to be homogeneous, 2a is the slab thickness, and the origin of the x-axis is chosen such that the slab extends from x = a to x = a.
(1) (2a) (2b)
1 1 1 )d ( , ) 3 1 ( ) , ( ) 1 ( 2 ) , ( ) 1 ( ) , (
x c b x c x c x x T T TBy following the previous works [7-9], the angular flux is expanded in terms of the Chebyshev polynomials of first kind:
Inserting Eq. (3) into Eq. (1) by using recurrence and orthogonality relations of the Chebyshev polynomials of first kind given below [11],
(3) (4) (5)
1 2 0 2 0(
)
(
)
1
2
)
(
1
)
(
)
,
(
l l lx
T
T
x
x
m
n
n
m
n
m
d
T
T
n m0
0
2
/
0
1
)
(
)
(
1 1 2
)
(
)
(
)
(
2
T
n
T
n1
T
n1
one can obtain the equations of moments;
n 3. The general solutions to Eqs. (6) are in the form of [1]
)
/
exp(
)
(
)
(
x
G
n
Tx
n
(6a) (6b)0
)
1
(
d
)
(
d
0 1
c
x
x
T
1 ( )
2 (1 ) 0 2 d ) ( d d ) ( d 1 1 1 0 2
T c b c T x x x x
(1 ) 0 3 2 ) ( 1 2 d ) ( d d ) ( d 0 2 1 3
T c c T x x x x
, 0 ) ( 4 ) 1 ( 1 3 ) ( 1 ) 1 ( 1 ) 1 ( ) ( ) 1 ( 1 2 d ) ( d d ) ( d 1 2 1 1 0 2 1 1 x n b x n c x c x x x x n n T n n T n n
(6c) (6d) (7)(9)
and by substituting this into Eqs. (6) it is possible to obtain analytic expressions for all Gn()
)
(
)
1
(
)
(
0 1
c
G
G
2 (1 ) 1 ( ) (1 ) 1
( ) ) ( 2 1 0 2
c c
b c
G
G
, 0 ) ( 4 ) 1 ( 1 3 ) ( 1 ) 1 ( 1 ) 1 ( ) ( ) 1 ( 1 2 ) ( ) ( 1 2 1 1 0 2 1 1 G n b G n c G c G G n n n n n n (8a) (8b) (8c)where G1() = 0 and G0() = 1, n 3. As seen from Eqs. (8), the functions Gn() depend on and one can obtain the discrete and continuum eigenvalues by setting GN+1() = 0 in this method as in PN approximation for various c, b1, and values. With so computed discrete eigenvalues, the general solution of Eq. (7) in the present approximation can be expressed as
With so computed discrete eigenvalues, the general solution of Eq. (7) in the present approximation can be expressed as
2 1 1)
/
exp(
)
1
(
)
/
exp(
)
(
)
(
N k k T n k T k n k nx
G
v
x
v
x
v
BOUNDARY AND CRITICALITY CONDITIONS
In this work, the Marshak boundary conditions are used for the critical slab problem. It is therefore for TN approximation,
1
00
)
,
(
)
(
a
d
T
k k = 1,3,...,N. (10)The criticality condition can be accomplished from Eq. (10) for a finite slab of half-thickness a. Thus, using Eq. (9) in Eq. (3) and then replacing the resulting equation in the boundary condition Eq. (10), one can obtain the critical equation as
2 1 1 1 , / / 0( ) cosh( / ) ( ) ( 1) 0 N j N n k n v a n v a j n j j T j j kG v a v G v e e I I
T j T j (11)(12)
(13)
(14) where Ik and In,k are given by
1 0 2 , 1 ) 2 / sin( 0 , 2 / 1 ) ( k k k k d T I k k
1 0 2 , . , ) ( 2 ) 2 / ) sin(( ) ( 2 ) 2 / ) sin(( 0 , 4 / 0 , 2 / 1 ) ( ) ( k n k n k n k n k n k n k n d T T I n k k n Eq. (11) can also be written in the matrix form as
]
[
)]
(
M
[
kja
B
k
0
where Bk is a vector with elements k, is the square matrix with elements
[(N+1)/2]2 and 0 is a null vector. For the matrix system in Eq. (13) to have a unique solution, the determinant of must vanish, which is the criticality condition. For example for T1 approximation, one can obtain an analytic expression for the critical half thickness of the slab by setting N = 1 in Eq. (11) as
)] ( M [ k a j )] ( M [ k a j
c b c b c c a T 1 ) 1 ( 1 2 2 tanh ) 1 ( 1 ) 1 ( 2 1 1 1 1 (15)NUMERICAL RESULTS AND CONCLUSION
•
We carry out the calculations using vacuum boundary condition from the criticality condition given in Eq. (11) in order to determine the critical thickness as a function of the anisotropy parameters. All the numerical results are computed using Maple software. For all calculations the total macroscopic cross section is assumed to be its normalized value, T = 1 cm1. The discrete eigenvalues j, are calculated from Eqs. (8) and the numerical results obtained for the critical thickness are presented in Tables 1-3 for various values of c, b1, and . Meanwhile, one can also fix the critical size of the system and seek the quantity of c.•
In this work, first the TN moments given in Eqs. (6) are derived by expanding the angular flux in terms of the Chebyshev polynomials of first kind as in Eq. (3). Then, the discrete eigenvalues are computed by setting GN+1() = 0 in Eqs. (8) similar to PN approximation. The critical slab thicknesses are computed for various values of c, b1, and and with different orders of TN approximation and they are listed in Tables 1-3 together with those already obtained by the traditional PN method and the ones available in literature side by side for comparison. The literature values tabulated in the tables are quoted from Sahni et al. [10] and Dahl and Sjöstrand [12]. The calculations are performed up to N = 9 which is reported as to be sufficient in the case of using Marshak boundary conditions [2,13].Table 1. Critical thicknesses for isotropic scattering ( = = b1 = 0.0) c P3 T3 P5 T5 P7 T7 P9 T9 Sahni Dahl 1.01 16.6700 16.6500 16.6636 16.6662 16.6616 16.6616 16.6607 16.6612 - -1.10 4.2426 4.2306 4.2318 4.2354 4.2294 4.2294 4.2284 4.2288 - -1.40 1.5154 1.5230 1.4844 1.4910 1.4776 1.7480 1.4756 1.4762 - -1.60 1.0768 1.0878 1.0410 1.0484 1.0308 1.0318 1.0272 1.0280 - -2.00 0.6842 0.6962 0.6478 0.6556 0.6342 0.6360 0.6282 0.6269 - -1.61538 1.0534 1.0646 1.0174 1.0250 1.0070 1.0080 1.0034 1.0042 1.00 -1.07757 5.0145 4.9997 5.0051 5.0085 5.0029 5.0028 5.0019 5.0022 5.00 -1.1631293 3.0217 3.0163 3.0059 3.0100 3.0030 3.0030 3.0019 3.0023 - 3.00 1.1084678 4.0168 4.0056 4.0053 4.0089 4.0029 4.0028 4.0018 4.0022 - 4.00
Table 2. Critical thicknesses for forward scattering ( = 0.3 and = 0.0) c b1 P3 T3 P5 T5 P7 T7 P9 T9 Sahni 1.01 0.3 0.0 0.3 17.3855 19.6081 23.0386 17.3546 19.5792 23.0126 17.3782 19.5988 23.0254 17.3817 19.6027 23.0301 17.3760 19.5959 23.0213 17.3759 19.5958 23.0214 17.3750 19.5947 23.0184 17.3753 19.5949 23.0195 -1.10 0.3 0.0 0.3 4.3561 4.8056 5.4597 4.3425 4.7955 5.4551 4.3357 4.7837 5.4349 4.3418 4.7896 5.4409 4.3322 4.7796 5.4296 4.3320 4.7795 5.4298 4.3309 4.7780 5.4274 4.3314 4.7785 5.4280 -1.40 0.3 0.0 0.3 1.5335 1.6444 1.7910 1.5472 1.6645 1.8208 1.4773 1.5813 1.7181 1.4903 1.5946 1.7319 1.4590 1.5617 1.6969 1.4609 1.5639 1.6996 1.4522 1.5546 1.6896 1.4539 1.5562 1.6911 -2.00 0.3 0.0 0.3 0.6870 0.7231 0.7674 0.7039 0.7448 0.7958 0.6306 0.6595 0.6940 0.6438 0.6736 0.7093 0.6029 0.6288 0.6596 0.6076 0.6342 0.6659 0.5869 0.6114 0.6404 0.5908 0.6155 0.6446 -1.58573 1.61833 1.65572 0.3 0.0 0.3 1.1088 1.1234 1.1431 1.1255 1.1462 1.1746 1.0487 1.0552 1.0640 1.0627 1.0700 1.0798 1.0246 1.0280 1.0326 1.0279 1.0321 1.0377 1.0134 1.0153 1.0180 1.0160 1.0182 1.0211 1.00 1.00 1.00 1.08085 1.09365 1.11166 0.3 0.0 0.3 5.0233 5.0291 5.0391 5.0055 5.0176 5.0375 5.0065 5.0084 5.0120 5.0121 5.0142 5.0182 5.0035 5.0045 5.0065 5.0033 5.0045 5.0067 5.0023 5.0029 5.0042 5.0027 5.0034 5.0049 5.00 5.00 5.00
Table 3. Critical thicknesses for backward scattering ( = 0.0 and = 0.3) c b1 P3 T3 P5 T5 P7 T7 P9 T9 Sahni 1.01 0.3 0.0 0.3 13.7235 14.7197 15.9775 13.7046 14.7019 15.9610 13.7188 14.7143 15.9712 13.7208 14.7164 15.9734 13.7175 14.7128 15.9693 13.7174 14.7128 15.9694 13.7169 14.7121 15.9686 13.7170 14.7123 15.9688 -1.10 0.3 0.0 0.3 3.5301 3.7473 4.0151 3.5215 3.7401 4.0099 3.5167 3.7334 4.0007 3.5204 3.7370 4.0041 3.5146 3.7311 3.9980 3.5145 3.7311 3.9981 3.5138 3.7302 3.9970 3.5140 3.7305 3.9973 -1.40 0.3 0.0 0.3 1.2600 1.3161 1.3823 1.2761 1.3356 1.4060 1.2107 1.2637 1.3262 1.2209 1.2740 1.3365 1.1967 1.2494 1.3116 1.1981 1.2510 1.3132 1.1923 1.2450 1.3072 1.1933 1.2460 1.3081 -2.00 0.3 0.0 0.3 0.5697 0.5883 0.6091 0.5883 0.6097 0.6340 0.5143 0.5286 0.5444 0.5266 0.5414 0.5579 0.4866 0.4991 0.5130 0.4916 0.5046 0.5189 0.4707 0.4825 0.4956 0.4743 0.4862 0.4994 -1.48010 1.49981 1.52152 0.3 0.0 0.3 1.0825 1.0897 1.0983 1.1004 1.1113 1.1246 1.0291 1.0317 1.0348 1.0403 1.0433 1.0470 1.0116 1.0126 1.0140 1.0137 1.0151 1.0169 1.0052 1.0057 1.0064 1.0066 1.0072 1.0079 1.00 1.00 1.00 1.05797 1.06407 1.07167 0.3 0.0 0.3 5.0115 5.0136 5.0160 4.9963 5.0006 5.0058 5.0034 5.0043 5.0050 5.0063 5.0073 5.0081 5.0018 5.0024 5.0027 5.0017 5.0024 5.0028 5.0011 5.0016 5.0017 5.0013 5.0018 5.0020 5.00 5.00 5.00
• In this work, the critical slab problem for neutrons of one-energy group is considered in a homogeneous finite slab in the case of forward, backward and linearly anisotropic scattering. TN method is established in detail using the Chebyshev polynomials of first kind in the angular part of the neutron angular flux.
•It is observed from the numerical results obtained for all values of the collision parameter c listed in Tables 2 and 3, the critical thickness increases with increasing values of the linear anisotropy parameter b1 ranging from 0.3 to +0.3. This means that leakage is less favoured by isotropic scattering than by linear anisotropy. In addition, this point can be seen obviously that the critical size of the system decreases when all neutrons tend to scatter in backward direction along the slab. In contrast, the slab thickness increases in the case of forward scattering, as expected.
• The convergence of the numerical results tabulated in the tables obtained by TN method is seen to be good. Therefore, a complete agreement can said to be exist between the results obtained in this work and those already obtained by PN approximation and the ones existed in literature. Furthermore, as can be followed in Section 2, this method can easily be applied to the solution of the neutron transport equation as well as the other polynomial expansion based techniques.
