Annals of Nuclear Energy 36 (2009) 575–582

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Annals of Nuclear Energy

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TN approximation on the critical size of time-dependent, one-speed andone-dimensional neutron transport problem with anisotropic scattering

Hakan Öztürk a,*, Süleyman Güngör b

a Osmaniye Korkut Ata University, Technical and Vocational School of Higher Education, Osmaniye 80000, Turkeyb Çukurova University, Faculty of Science and Letters, Department of Physics, Adana 01330, Turkey

a r t i c l e i n f o

Article history:Received 29 July 2008Received in revised form 2 January 2009Accepted 9 January 2009Available online 23 February 2009

0306-4549/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.anucene.2009.01.012

* Corresponding author. Tel.: +90 328 825 1818; faE-mail addresses: hozturk@cu.edu.tr, hakanozturk3

a b s t r a c t

The criticality problem is studied based on one-speed time-dependent neutron transport theory, for auniform and finite slab, using the Marshak boundary condition. The time-dependent neutron transportequation is reduced to a stationary equation. The variation of the critical thickness of the time-dependentsystem is investigated by using the linear anisotropic scattering kernel together with the combination offorward and backward scattering. Numerical calculations for various combinations of the scatteringparameters and selected values of the time decay constant and the reflection coefficient are performedby using the Chebyshev polynomials approximation method. The results are compared with those previ-ously obtained by other methods which are available in the literature.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Criticality type eigenvalues are needed for various applicationsin reactor physics such as pulsed neutron experiments. In thisphenomena, the number of neutrons decreases with time after ashort neutron pulse. Then, by considering the behavior of theangular flux of neutron population at later times, fundamentaland higher order time eigenvalues (decay constants) are com-puted to determine the criticality conditions of the system. Fun-damental time eigenvalue is defined as the smallest number ofk’s for the time-dependent system which have a neutron distribu-tion with a term exp(kt) (Sahni and Sjöstrand, 1990). In homoge-neous systems, determining the criticality conditions andcomputing the time decay constant can be considered to be iden-tical. If a relation between the decay constant and criticalityparameters is established then the results obtained for a criticalsystem could be used for a time-dependent system. Therefore,when the neutron flux density in a pulsed neutron experimentdecays exponentially it is possible to remove the time depen-dence of the neutron flux from the transport equation. Then, thetime-dependent transport problem can be reduced to a stationaryone. The time eigenvalues of many systems can be determined bycalculating the critical size of those systems in stationary condi-tion (Carlvik, 1968; Dahl et al., 1983; Sahni and Sjöstrand, 1990;Sahni et al., 1992; Yildiz, 1999).

ll rights reserved.

x: +90 328 825 0097.@hotmail.com (H. Öztürk).

There are many cases in which an extended knowledge aboutthe scattering of neutrons through the media should be taken intoconsideration in order to investigate the solution of the transportequation for a more accurate analysis. Among the polynomialexpansion based techniques, the spherical harmonics or PN methodis the most widely used in transport theory applications. However,in some circumstances, the PN expansion of angular flux near mate-rial boundaries may be a poor representation. Hence, Aspelund(1958), Conkie (1959) and Yabushita (1961) presented some disad-vantages of the PN method, especially in the case of anisotropicscattering and computation of the extrapolated end points, andthus they used the Chebyshev polynomials approximation (TN

method) in their studies. Recently, Anli et al. (2006a,b) applied amodified version of the TN method to the critical and reflected crit-ical slab problems (Öztürk et al., 2007).

In this work, the reflected critical slab problem is studied byusing Chebyshev polynomials of first kind, i.e., the TN method.Therefore, the work previously carried out by Öztürk et al. (2007)is extended to a time-dependent system with linearly anisotropicscattering together with both forward and backward scattering.In the method, first the time-dependent transport equation is re-duced to a stationary one, and then the neutron angular flux is ex-panded in terms of the Chebyshev polynomials of first kind. Finally,the critical slab thicknesses are calculated for various values of theanisotropy parameters a, b, and b1, selected values of time decayconstant K, and the reflection coefficient R.

The numerical results obtained for the reflected critical slabthickness of the time-dependent system will be tabulated togetherwith the corresponding results obtained by various other methodsand the ones previously reported in the literature.

576 H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582

2. Transport equation for a time-dependent system

The linear transport equation for a time-dependent neutronpopulation without source can be written as (Bell and Glasstone,1970),

1t@wðr;X; tÞ

@t¼ �X � rwðr;X; tÞ � r�Twðr;X; tÞ

þ r�SZ

f ðX0 �XÞwðr;X0; tÞdX0; ð1Þ

where X0 is the direction of neutron velocity before (and X after) acollision and w(r, X, t) is the angular flux of neutrons at position r indirection X. r�T and r�S denote the macroscopic total and scatteringdifferential cross-sections of the time-dependent system, respec-tively; t is the neutron velocity. f ðX0 �XÞ is the scattering kernelwhich is assumed to be of the form of a combination of linearlyanisotropic scattering and backward–forward isotropic scattering,

f ðX0 �XÞ ¼ 1� a� b4p

ð1þ 3b1X0 �XÞ þ a

2pdðX0 �X� 1Þ

þ b2p

dðX0 �Xþ 1Þ: ð2Þ

Here a and b are parameters which vary over the range of 0 6 a,b 6 1, a + b 6 1 and denote the forward and backward scatteringprobabilities in a collision, respectively. The parameter b1 is theaverage cosine of the scattering angle and is restricted to the rangejb1j � 1=3 to ensure the positivity of the scattering function for allangles (Sahni et al., 1992).

The angular flux of the neutrons is assumed to decay exponen-tially with time and given by (Sahni et al., 1992),

wðr;X; tÞ ¼ wðr;XÞ exp ð�ktÞ: ð3Þ

Substituting Eq. (2) and Eq. (3) into Eq. (1), it is obtained for one-dimensional case,

l @wðx;lÞ@x

þ rTð1� acÞwðx;lÞ

¼ crT

2ð1� a� bÞ

Z 1

�1wðx;l0Þð1þ 3b1ll0Þdl0 þ bcrTwðx;�lÞ;

ð4Þ

subject to free space boundary and symmetry conditions:

wða;lÞ ¼ 0; ð5aÞwðx;lÞ ¼ wð�x;lÞ; l > 0; ð5bÞ

where a is the critical half thickness of the finite homogeneous slabin units of mean free path. By following the same general notationas those of Sahni et al. (1992), one can easily see that Eqs. (4 and 1)are formally identical if

rT ¼ r�T �kt; ð6aÞ

crT ¼ r�S; ð6bÞ

where rT is the total macroscopic cross-section and c is the meannumber of secondary neutrons per collision in the critical system.By introducing a generalized decay constant K,

K ¼ ktr�S� r�a

r�S; ð7Þ

one can obtain from Eq. (6)

K ¼ 1� 1c: ð8Þ

If d denote the characteristic dimension of the system, then the sizeof the system can be expressed as dr�S for the time-dependent case

and drT for the stationary case (Sahni and Sjöstrand, 1990; Carlvik,1968). Thus, one can obtain from Eq. (6),

dr�S ¼ cðdrTÞ ¼ cð2aÞ: ð9Þ

Therefore, the time-dependent transport equation is reduced to astationary one which can be solved by the usual methods. In the fol-lowing section of this study, the critical thickness of the homoge-neous slab surrounded by the same reflecting material from bothsides is calculated by Chebyshev polynomials approximation.

3. Chebyshev polynomials approximation (TN method)

By following the preceding studies (Anli et al., 2006a,b; Öztürket al., 2007), the angular flux of the neutrons is expanded in a seriesof Chebyshev polynomials of first kind as,

wðx;lÞ ¼ /0ðxÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p T0ðlÞ þ2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p XN

n¼1

/nðxÞTnðlÞ ð10Þ

where l is the cosine of the angle between the neutron velocity vec-tor and the positive x-axis. When Eq. (10) is substituted into Eq. (4)and the resulting equation is multiplied by Tm(l) and integratedover l 2 ð1;1Þ by using the orthogonality and recurrence relationsof Chebyshev polynomials of first kind (Arfken, 1985),

Z 1

�1TmðlÞTnðlÞð1� l2Þ�1=2dl ¼

0; m – n;p2 ; m ¼ n – 0;p; m ¼ n ¼ 0;

8><>: ð11aÞ

Tnþ1ðlÞ � 2lTnðlÞ þ Tn�1ðlÞ ¼ 0; ð11bÞ

and after some algebra one can obtain the TN moments of the angu-lar flux:

d/1ðxÞdx

þrTð1� cÞ/0ðxÞ ¼ 0; ð12aÞ

d/2ðxÞdx

þd/0ðxÞdx

þ2rT ½1� cða�bÞ�/1ðxÞ

�2rT b1cð1�a�bÞ/1ðxÞ ¼ 0; ð12bÞd/3ðxÞ

dxþd/1ðxÞ

dxþ2rT ½1� cðaþbÞ�/2ðxÞ

þ23

crTð1�a� bÞ/0ðxÞ ¼ 0; ð12cÞ

d/nþ1ðxÞdx

þd/n�1ðxÞdx

þ2rTf1� c½aþð�1Þnb�g/nðxÞ

þ crTð1�a� bÞ ½1þð�1Þn�n2�1

/0ðxÞþ3b1½1þð�1Þnþ1�

n2�4/1ðxÞ

( )¼ 0;

n� 3: ð12dÞ

In order to obtain the eigenvalue spectrum, a well known solutionfor the homogeneous Eq. (12) is employed, of the form (Davison,1958),

/nðxÞ ¼ GnðmÞ expðrT x=vÞ: ð13Þ

The analytic expressions for all Gn(m) are obtained when Eq. (13) issubstituted into Eq. (12),

G1ðvÞ ¼ �vð1� cÞ; ð14aÞG2ðvÞ ¼ �2m½1� cða� bÞ � b1cð1� a� bÞ�G1ðmÞ � 1; ð14bÞ

G3ðvÞ ¼ �2m½1� cðaþ bÞ�G2ðmÞ � G1ðmÞ �23mcð1� a� bÞ; ð14cÞ

Gnþ1ðvÞ þ 2m½1� cðaþ ð�1ÞnbÞ�GnðmÞ þ Gn�1ðvÞ

þ mcð1� a� bÞ ½1þ ð�1Þn�n2 � 1

þ 3b1½1þ ð�1Þnþ1�

n2 � 4G1ðmÞ

( )¼ 0;

n � 3: ð14dÞ

Table 1The slab thickness dr�S for selected values of K and different degrees of anisotropy parameters, a, b and b1 as calculated by T11 approximation (R = 0.0).

a b1 K

0.00990 0.15525 0.16666 0.34999 0.50000 0.65999 0.98990

0.00 �0.3 14.89573 2.95958 2.81974 1.61946 1.17969 0.88650 0.534550.0 16.83124 3.25437 3.09625 1.74379 1.25209 0.92736 0.546340.3 19.85110 3.67839 3.49229 1.91082 1.34445 0.97653 0.55913

0.01 �0.3 14.96274 2.96916 2.82862 1.62224 1.18016 0.88550 0.532440.0 16.90562 3.26427 3.10536 1.74617 1.25200 0.92580 0.543910.3 19.92794 3.68850 3.50150 1.91247 1.34346 0.97416 0.55631

0.34 �0.3 18.02400 3.34439 3.17232 1.68809 1.14565 0.79858 –0.0 20.32209 3.64371 3.44984 1.78663 1.18890 0.81577 –0.3 23.85773 4.06200 3.83534 1.91095 – 0.83437 –

0.65 �0.3 23.90694 3.76873 3.53003 1.50593 – – –0.0 26.84271 4.02148 3.75510 1.53834 – – –0.3 31.31638 4.35035 4.04464 1.57341 – – –

0.70 �0.3 25.55447 3.80373 3.54526 – – – –0.0 28.65818 4.03057 3.74344 – – – –0.3 33.37673 4.31810 3.99120 – – – –

0.75 �0.3 27.61643 3.78920 3.50654 – – – –0.0 30.92278 3.97740 3.66608 – – – –0.3 35.92758 4.20711 3.85758 – – – –

0.99 �0.3 53.44032 – – – – – –0.0 54.61850 – – – – – –0.3 55.89642 – – – – – –

b b1

0.01 �0.3 14.85576 2.95093 2.81142 1.61265 1.17287 0.87943 0.528120.0 16.73402 3.23960 3.08218 1.73405 1.24323 0.91881 0.539250.3 19.71030 3.65185 3.46724 1.89616 1.33248 0.96594 0.55126

0.34 �0.3 13.72959 2.65425 2.52044 1.33706 0.88084 0.59932 –0.0 14.63883 2.78922 2.64618 1.38288 0.89905 0.60551 –0.3 15.77535 2.95048 2.79604 1.43509 0.91880 0.61193 –

0.65 �0.3 12.82370 2.20409 2.06172 0.79829 – – –0.0 13.20062 2.24752 2.10057 0.80193 – – –0.3 13.61377 2.29411 2.14215 0.80563 – – –

0.70 �0.3 12.67834 2.06030 1.91134 – – – –0.0 12.98923 2.09150 1.93850 – – – –0.3 13.32518 2.12448 1.96713 – – – –

0.75 �0.3 12.52684 1.85399 1.69605 – – – –0.0 12.77626 1.87332 1.71193 – – – –0.3 13.04164 1.89344 1.72839 – – – –

0.99 �0.3 5.14621 – – – – – –0.0 5.14686 – – – – – –0.3 5.14752 – – – – – –

H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582 577

where the normalizations G�1(m) = 0 and G0(m) = 1 are used. Since/N+1(x) = 0 in TN approximation as in PN approximation, the discreteand continuum eigenvalues (m) may be calculated by settingGN+1(m) = 0 for various values of c, a, b and b1.

The algebraic equations given in Eq. (14) may also be solved bywriting them in a homogeneous matrix form,

½MðmÞ�G ¼ 0; ð15Þ

where M(m) is the (N + 1) � (N + 1) coefficient matrix and G = [G0,G1,. . .,Gn]T. Non-trivial solution for the eigenvalues can be obtainedby setting det[M(m)] = 0.

With discrete eigenvalues computed, the nth Chebyshevmoment of the angular flux for odd numbers of N can be expressedas,

/nðxÞ ¼XðNþ1Þ=2

k¼1

kkGnðmkÞ½exp ðrT x=mkÞ þ ð�1Þn exp ð�rT x=mkÞ�; ð16Þ

where kk are arbitrary coefficients and can be determined from thephysical boundary conditions of the system, and the parity relationof Gn(�m) = (�1)nGn(m) is used.

The Marshak boundary condition is based on the condition ofzero incoming current at the vacuum boundary and gives some-what more accurate results than those of Mark, at least for smallN (Bell and Glasstone, 1970). Therefore in this work, for the slabcriticality in the case of reflected boundary,

wða;�lÞ ¼ Rwða;lÞ; l > 0; ð17Þ

Marshak boundary condition is applied by using the Chebyshevpolynomials of first kind,

Z 1

0½wða;�lÞ � Rwða;lÞ�T2k�1ð�lÞdl ¼ 0;

k ¼ 1;2; . . . ; ðN þ 1Þ=2: ð18Þ

The critical equation is obtained when Eq. (16) is inserted into Eq.(10) and then replacing the resulting equation in the boundary

Table 2The slab thickness dr�S for selected values of K and different degrees of anisotropy parameters, a, b and b1 as calculated by T11 approximation (R = 0.50).

a b1 K

0.00990 0.15525 0.16666 0.34999 0.50000 0.65999 0.98990

0.00 �0.3 12.89440 1.69420 1.58816 0.76272 0.50733 0.35232 0.182360.0 14.23746 1.75312 1.64057 0.77575 0.51299 0.35486 0.182820.3 16.06448 1.82005 1.69973 0.78964 0.51891 0.35746 0.18328

0.01 �0.3 12.95604 1.69580 1.58948 0.76212 0.50618 0.35082 0.180870.0 14.30791 1.75435 1.64154 0.77500 0.51176 0.35330 0.181310.3 16.19502 1.82079 1.70024 0.78873 0.51757 0.35584 0.18175

0.34 �0.3 15.06564 1.73648 1.61908 0.71463 0.43486 0.27127 –0.0 16.52813 1.77987 1.65708 0.72199 0.43721 0.27192 –0.3 18.61215 1.82745 1.69852 0.72963 – 0.27259 –

0.65 �0.3 18.61750 1.67201 1.53766 0.51154 – – –0.0 20.15395 1.69382 1.55599 0.51278 – – –0.3 22.21934 1.71673 1.57517 0.51404 – – –

0.70 �0.3 19.49406 1.62088 1.48185 – – – –0.0 21.02288 1.63815 1.49609 – – – –0.3 23.04571 1.65613 1.51086 – – – –

0.75 �0.3 20.52060 1.53412 1.38920 – – – –0.0 22.02664 1.54644 1.39902 – – – –0.3 23.98596 1.55915 1.40911 – – – –

0.99 �0.3 18.22992 – – – – – –0.0 18.27573 – – – – – –0.3 18.32196 – – – – – –

b b1

0.01 �0.3 12.87389 1.69144 1.58560 0.76084 0.50546 0.35038 0.180700.0 14.27416 1.74947 1.63721 0.77365 0.51102 0.35285 0.181140.3 15.83673 1.81525 1.69537 0.78729 0.51681 0.35539 0.18158

0.34 �0.3 12.02271 1.58302 1.48075 0.67074 0.41164 0.25902 –0.0 12.69992 1.61470 1.50884 0.67674 0.41360 0.25958 –0.3 13.50365 1.64876 1.53893 0.68294 0.41560 0.26014 –

0.65 �0.3 11.31604 1.36927 1.26587 0.44209 – – –0.0 11.59987 1.38069 1.27567 0.44283 – – –0.3 11.90765 1.39247 1.28575 0.44357 – – –

0.70 �0.3 11.19885 1.29501 1.18989 – – – –0.0 11.43364 1.30341 1.19695 – – – –0.3 11.68498 1.31202 1.20416 – – – –

0.75 �0.3 11.07500 1.18806 1.08073 – – – –0.0 11.26380 1.19350 1.08511 – – – –0.3 11.46326 1.19903 1.08956 – – – –

0.99 �0.3 4.72837 – – – – – –0.0 4.72892 – – – – – –0.3 4.72946 – – – – – –

578 H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582

condition given in Eq. (18) with the parity relation of Chebyshevpolynomials of first kind, Tn(l) = (�1)nTn(�l),

XðNþ1Þ=2

k¼1

ð�1Þmkk G0ðvkÞ½1� R� coshrT avk

� �Im

�

þXN

n¼1

GnðvkÞ½ð�1Þn � R� ð1þ ð�1ÞnÞ coshrT avk

� ��

þð1� ð�1ÞnÞ sinhrT avk

� ��In;m

�¼ 0; ð19Þ

where Im and In,m are defined by

Im ¼Z 1

0

TmðlÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p dl

¼p=2; m ¼ 0;sinðmp=2Þ

k ; m � 1

(and

Z 1

0

Tmð�lÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p dl ¼ ð�1ÞmIm;

ð20Þ

In;m ¼Z 1

0

TnðlÞTmðlÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p dl

¼p=2; n ¼ m ¼ 0;p=4; n ¼ m – 0;sinððnþmÞp=2Þ

2ðnþmÞ þ sinððn�mÞp=2Þ2ðn�mÞ ; n – m;

8><>: ð21aÞ

Z 1

0

Tnð�lÞTmð�lÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

p dl ¼ ð�1ÞnþmIn;m: ð21bÞ

The critical Eq. (19) may also be written in the matrix form as,

½MkmðaÞ�L ¼ ½0�; ð22Þ

where L is the vector of elements kk, k = 1, 2, . . . , (N + 1)/2 andMk

mðaÞ is a ðN þ 1Þ=2� ðN þ 1Þ=2 square matrix and 0 is a null vec-tor. The reflected critical slab thicknesses can be calculated by set-ting det½Mk

mðaÞ� ¼ 0.As an example, an analytic solution for the critical half thickness

a, can be obtained from Eq. (19) for T1 approximation and when theresult is inserted into Eq. (9), an expression for the critical size ofthe time-dependent system can be obtained as,

Table 3The slab thickness dr�S for selected values of K and different degrees of anisotropy parameters, a, b and b1 as calculated by T11 approximation (R = 0.99).

a b1 K

0.00990 0.15525 0.16666 0.34999 0.50000 0.65999 0.98990

0.00 �0.3 0.50541 0.03060 0.02838 0.01246 0.00804 0.00547 0.002760.0 0.50575 0.03060 0.02838 0.01246 0.00804 0.00547 0.002760.3 0.50723 0.03060 0.02838 0.01246 0.00804 0.00547 0.00276

0.01 �0.3 0.50518 0.03058 0.02836 0.01244 0.00801 0.00544 0.002730.0 0.50549 0.03058 0.02836 0.01244 0.00801 0.00544 0.002730.3 0.50642 0.03058 0.02836 0.01244 0.00801 0.00544 0.00273

0.34 �0.3 0.50461 0.02961 0.02738 0.01127 0.00669 0.00410 –0.0 0.50474 0.02961 0.02738 0.01127 0.00669 0.00410 –0.3 0.50470 0.02962 0.02738 0.01127 – 0.00410 –

0.65 �0.3 0.50250 0.02674 0.02445 0.00773 – - –0.0 0.50253 0.02674 0.02445 0.00773 – – –0.3 0.50255 0.02674 0.02445 0.00773 – – –

0.70 �0.3 0.50171 0.02561 0.02329 – – – –0.0 0.50174 0.02561 0.02329 – – – –0.3 0.50177 0.02561 0.02329 – – – –

0.75 �0.3 0.50060 0.02391 0.02155 – – – –0.0 0.50062 0.02391 0.02155 – – – –0.3 0.50064 0.02391 0.02155 – – – –

0.99 �0.3 0.27565 – – – – – –0.0 0.27565 – – – – – –0.3 0.27565 – – – – – –

b b1

0.01 �0.3 0.50524 0.03058 0.02836 0.01244 0.00801 0.00544 0.002730.0 0.50439 0.03058 0.02836 0.01244 0.00801 0.00544 0.002730.3 0.50863 0.03058 0.02836 0.01244 0.00801 0.00544 0.00273

0.34 �0.3 0.50439 0.02961 0.02738 0.01127 0.00669 0.00410 –0.0 0.50424 0.02961 0.02738 0.01127 0.00669 0.00410 –0.3 0.50448 0.02961 0.02738 0.01127 0.00669 0.00410 –

0.65 �0.3 0.50198 0.02674 0.02445 0.00773 – – –0.0 0.50201 0.02674 0.02445 0.00773 – – –0.3 0.50204 0.02674 0.02445 0.00773 – – –

0.70 �0.3 0.50115 0.02561 0.02329 – – – –0.0 0.50118 0.02561 0.02329 – – – –0.3 0.50121 0.02561 0.02329 – – – –

0.75 �0.3 0.50000 0.02391 0.02155 – – – –0.0 0.50002 0.02391 0.02155 – – – –0.3 0.50004 0.02391 0.02155 – – – –

0.99 �0.3 0.27505 – – – – – –0.0 0.27505 – – – – – –0.3 0.27505 – – – – – –

H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582 579

dr�S ¼2c

rT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� cÞf1� c½a� bþ b1ð1� a� bÞ�g

p� tanh�1 �2

ffiffiffi2p

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c a� bþ b1ð1� a� bÞ½ �

1� c

r1� R1þ R

!:

ð23Þ

4. Numerical results and discussion

As mentioned in the Introduction, the criticality or the eigen-value problem is identical with the fundamental decay constantproblem. Thus, the transport equation for a time-dependent sys-tem is first reduced to a stationary one with the assumption thatthe angular flux is decaying exponentially with time. Then, the crit-ical thickness for the neutrons in a homogeneous finite slab sur-rounded by identical reflectors on both sides is investigated withthe reduced linear transport equation. The collision parameters care calculated from Eq. (8) for selected values of the decay constantK. The numerical results for the critical thickness of the stationarysystem (drT) are computed from Eq. (19), or (22) and then the

thickness dr�S of the time-dependent system is determined fromEq. (9) by the order of T11 approximation. They are obtained byusing Marshak boundary condition for different values of collision,anisotropy and reflection parameters and given in Tables 1–4. Allnumerical results are performed using Maple software. The totalmacroscopic cross-section is assumed to be normalized,rT = 1 cm�1.

After the reduction procedure described in Section 2, the neu-tron angular flux is expanded in terms of the Chebyshev polynomi-als of first kind similar to the one in Anli et al. (2006a,b) and themoment equations are obtained and given in Eq. (12). Then, the de-sired eigenvalues for TN approximation are calculated from Eq. (14)by setting GN+1(m) = 0 similar to the PN approximation for variousvalues of c, a, b and b1. Because of its consistency for the transporttheory calculations, Marshak boundary condition is applied to theproblem to present the criticality condition for the reflectiveboundary, which may easily be reduced to vacuum boundary byletting R = 0 in the criticality equations. The tables present not onlythe results obtained by the TN method but also the ones obtainedby other methods for comparison. As a result, they reveal the rela-tively simple derivation and rapid convergence of the TN method

Fig. 2. The thickness dr�S as a function of forward scattering parameter a, for twovalues of time decay constant K and for different degrees of linear anisotropicscattering b1, as calculated by T11 approximation.

Table 4Comparison of the slab thickness dr�S obtained by different orders of TN approximation with the literature values (R = 0.0).

a, b K This work Dahl et al. Carlvik Sahni et al.

T3 T7 T11 P11

a = 0.00, b = 0.00 0.296738 2.07374 2.00724 2.00249 2.00237 2.0 – –0.025350 9.99132 10.00263 10.00128 10.00116 10.0 – –0.007180 19.99121 20.00258 19.99614 20.00114 20.0 – –0.453510 1.50333 1.39350 1.37805 1.37749 – 1.37239 –0.097850 4.44019 4.43711 4.43544 4.43538 – 4.43387 –0.071990 5.38731 5.39073 5.38916 5.38911 – 5.38787 –

a = 0.70, b = 0.00 0.294780 2.65181 2.06713 1.84064 1.82119 – - 1.00.155250 4.51627 4.10359 4.03057 4.02732 – – 4.00.123940 5.38692 5.05395 5.01431 5.01308 – – 5.0

a = 0.00, b = 0.70 0.202590 1.95570 1.61021 1.53060 1.52639 – – 1.50.104280 3.13519 3.00655 3.00166 3.00156 – – 3.00.050990 5.02440 5.00235 5.00118 5.00112 – – 5.0

a = 0.35, b = 0.35 0.266780 1.95269 1.47168 1.28674 1.27124 – – 1.00.191840 2.50540 2.11503 2.03143 2.02723 – – 2.00.133320 3.27537 3.02572 3.00505 3.00455 – – 3.00.095500 4.15243 4.00893 4.00284 4.00269 – – 4.00.071040 5.08458 5.00526 5.00223 5.00212 – – 5.0

580 H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582

once more, as previously mentioned in Anli et al. (2006a,b) andÖztürk et al. (2007).

The numerical results obtained for the thickness dr�S of thetime-dependent system by the T11 approximation are listed in Ta-bles 1–3 for reflection coefficients of R = 0.0, 0.50 and 0.99. Further-more, some results in Table 4 are obtained for different orders of TN

approximation. The results which are obtained by other methods,which are accepted as the benchmark values, are also shown inthe table to display the consistency and effectiveness of the presentmethod. In addition, the variation of the thickness dr�S as functionsof the anisotropy parameters and the time decay constant is illus-trated in Figs. 1–3 for the case of vacuum boundary.

Tables 1–3 present the calculated critical thicknesses of thetime-dependent system for selected values of time decay constantK and different degrees of anisotropy parameters a, b and b1, withreflection coefficients R = 0.0, 0.50 and 0.99, respectively. The topsections of these tables represent the critical thicknesses for for-ward scattering (b = 0) and their bottom sections represent thecritical thicknesses for backward scattering (a = 0). Unlike other ta-bles, in Table 4 the critical thicknesses obtained by different ordersof the TN approximation are presented for forward and backwardscattering together, i.e., a = 0.35 and b = 0.35 in the case of vacuumboundary, i.e., R = 0. Besides the numerical results obtained in thiswork, the results obtained by the PN approximation together with

Fig. 1. The thickness dr�S as a function of time decay constant K for differentdegrees of linear anisotropic scattering b1 and forward (F) a = 0.70 and backward (B)b = 0.70 scattering, as calculated by T11 approximation.

Fig. 3. The thickness dr�S as a function of backward scattering parameter b, for twovalues of time decay constant K and for different degrees of linear anisotropicscattering b1, as calculated by T11 approximation.

the literature values quoted in Carlvik (1968), Sahni et al. (1992)and Dahl et al. (1983), are also given for comparison. From thenumerical results tabulated in the tables for the critical thicknessof time-dependent system, it is seen that the behavior of the crit-

H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582 581

ical thickness depends on the anisotropy parameters and the decayconstant K (or the collision parameter c).

In Fig. 1, the time decay constant decreases with increasingthe critical thickness in all cases. Since the neutrons tend to goaway from the center to the boundary of the slab for forwardscattering, the critical thickness is greater than the one for back-ward scattering, as expected. For thin slabs, b1 practically has noeffect on the critical size of the system, while for thick slabs b1

can change the critical size of the system significantly. In addi-tion, the contribution of the linear anisotropy parameter b1 onthe critical thickness is seen to be significant in the case of for-ward scattering and it is found to be insignificant in the case ofbackward scattering. This phenomenon is observed in Fig. 1 asoverlapping the curves of backward scattering for all values ofb1 ranging from �0.3 to 0.3.

The change in the thickness with respect to forward scatteringparameter a is given in Fig. 2 for two values of K and three differ-ent degrees of linear anisotropic scattering b1. According to this fig-ure, the behavior of the critical thickness with increasing a is non-monotonic. The critical thickness of the time-dependent systemdr�S for a fixed value of K first increases with forward scatteringparameter a (b = 0) and then decreases. Therefore for a given valueof K, the slab thickness may strongly depends on the value of b1,and the variation of the thickness depends on the choice of the va-lue of K. For example for K = 0.15525 in Table 1, the maximum va-lue of the thickness with b1 = 0.3 is observed at aboutdr�S ¼ 4:35035 for a = 0.65. However, for b1 = �0.3 and 0.0 itreaches the maximum values at about a = 0.70. Neutrons are ex-pected to tend to go forward direction towards the boundary inthe case of positive b1, and to scatter back into the center of theslab in the case of negative b1. This manifest as an increase or de-crease in the thickness of the slab. Hence, the addition of linearanisotropic scattering increases or decreases the thickness with re-spect to the positive or negative selected values of b1, respectively.In other words, neutron leakage is less favored by isotropic scatter-ing than by linear anisotropic scattering. These general trends areobserved from the tables and thus the figures.

The behavior of the thickness of the decaying system starts tobe monotonic with increasing a for all values of b1 when K is largerthan about 0.50000 in the case of vacuum boundary, R = 0. Thethickness is decreasing continuously for larger values of K. Thetransition between the two trends of the thickness is observed atabout dr�S ¼ 1:17 and 1.34 for isotropic scattering (a = 0). However,this continuous behavior of the thickness is seen for smaller valuesof K in the presence of reflection, i.e., R – 0. For example in the caseof a percentage of reflection R = 0.50 as given in Table 2, the thick-ness decreases with increasing a for all values of b1 when K is lar-ger than about 0.16666. The transition is observed at aboutdr�S ¼ 1:61 and 1.69 for a = 0.34.

In the case of backward scattering b – 0 (a = 0), the variation ofthe thickness is illustrated in Fig. 3 for two values of K and linearanisotropy parameters b1 ranging from �0.3 to 0.3, and detailednumerical results are given in the tables. As seen from this figureand all the tables, the thickness decreases uniformly with increas-ing the backward anisotropy parameter b regardless of the value ofthe linear anisotropy parameter b1.

Since there is no mechanism for loss of neutrons, the criticalthickness of the slab decreases and seems to go to zero when thereflection coefficient approaches unity, as expected. This may beinterpreted as the neutron distribution of the system is completelydense in the normal plane. This condition is easily seen in Tables 1–3.

The variation of the thickness may be understood by taking thechange in the angular distribution into account. Since the maxi-mum angular flux always occurs at l = 0 at the origin, a maximumflux can only be obtained when c (or K) is greater than a certain

value and it changes from higher to lower values of l when c in-creases at the boundary.

In this work, the critical thickness for neutrons in a time decay-ing system with forward, backward and linear anisotropic scatter-ing is investigated by using the Chebyshev polynomials of first kindin the angular distribution of neutrons. As seen from the numericalresults tabulated in the tables, for a fixed time decay constant,anisotropic scattering affects the critical size of the system consid-erably. The thickness first increases with increasing forward scat-tering and then decreases. Furthermore, the thickness decreasescontinuously and goes to zero in the limit of R ? 1. However, inthe case of backward scattering, the critical thickness always de-creases with increasing b regardless of the linear anisotropyparameter b1 and the reflection coefficient R. This result is generalfor many systems involving the particle scattering. Hence, the ef-fect of backward scattering on the variation of the thickness maybe said to be negligible in practice and this conclusion can be seenfrom Fig. 1.

As can be realized from Table 4, the numerical results obtainedfor thicker slab converge faster to the benchmark values than forthinner slab. The results which are almost the same with thebenchmark values are obtained even with the T3 approximationwhen the slab is thick. A perfect agreement is observed betweenthe numerical results obtained for the critical thickness of decayingsystem in this work and those previously obtained by the PN meth-od, and the ones presented by Carlvik (1968), Sahni et al. (1992)and Dahl et al. (1983).

An appropriate conclusion can be reached about the TN approx-imation used in this study: the TN method converges rapidly andcan easily be applied to the solution of transport problems, withthe results as good as other methods. The rapid convergence andapplicability of the TN method can be realized from Section 3. Itis relatively simple with respect to the derivation of equations,and requires less computational effort for this type of transportproblems. In addition, the numerical results tabulated in the tablesindicate that the TN method works as good as other polynomialexpansion based techniques such as PN method in transport theorycalculations.

5. Conclusions

In this work, the criticality problem was studied based on one-speed time-dependent neutron transport theory, for a uniform andfinite slab, using the Marshak boundary condition. The time-dependent neutron transport equation was reduced to a stationaryequation. The variation of the critical thickness of the time-depen-dent system was then investigated by using the linear anisotropicscattering kernel together with the combination of forward andbackward scattering. Numerical calculations for various combina-tions of the scattering parameters and selected values of the timedecay constant and the reflection coefficient were performed byusing the Chebyshev polynomials approximation technique. Theresults were then compared with those previously obtained byother methods and available in the literature. It was shown thatthe TN approximation used in this study can easily be applied tothe solution of transport problems, with the results that are asgood as other methods.

References

Anli, F., Yasa, F., Güngör, S., Öztürk, H., 2006a. TN approximation to neutrontransport equation and application to critical slab problem. J. Quant. Spectrosc.Radiat. Transfer 101, 129–134.

Anli, F., Yasa, F., Güngör, S., Öztürk, H., 2006b. TN approximation to reflected slaband computation of the critical half thickness. J. Quant. Spectrosc. Radiat.Transfer 101, 135–140.

582 H. Öztürk, S. Güngör / Annals of Nuclear Energy 36 (2009) 575–582

Arfken, G., 1985. Mathematical Methods for Physicists. Academic Press, Inc.,Orlando.

Aspelund, O., 1958. On a new method for solving the (Boltzmann) equation inneutron transport theory. PICG 16, 530–534.

Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory. VNR Company, New York.Carlvik, I., 1968. Monoenergetic critical parameters and decay constants for small

homogeneous spheres and thin homogeneous slabs. Nucl. Sci. Eng. 31, 295–303.Conkie, W.R., 1959. Polynomial approximations in neutron transport theory. Nucl.

Sci. Eng. 6, 260–266.Dahl, B., Protopopescu, V., Sjöstrand, N.G., 1983. On the relation between decay

constants and critical parameters in monoenergetic neutron transport. Nucl.Sci. Eng. 83, 374–379.

Davison, B., 1958. Neutron Transport Theory. Oxford University Press, New York.

Öztürk, H., Anli, F., Güngör, S., 2007. TN method for the critical thickness of one-speed neutrons in a slab with forward and backward scattering. J. Quant.Spectrosc. Radiat. Transfer 105, 211–216.

Sahni, D.C., Sjöstrand, N.G., 1990. Criticality and time eigenvalues for one-speedneutron transport. Prog. Nucl. Energy 23, 241–289.

Sahni, D.C., Sjöstrand, N.G., Garis, N.S., 1992. Criticality and time eigenvalues forone-speed neutrons in a slab with forward and backward scattering. J. Phys. D:Appl. Phys. 25, 1381–1389.

Yabushita, S., 1961. Tschebyscheff polynomials approximation method of theneutron transport equation. J. Math. Phys. 2, 543–549.

Yildiz, C., 1999. Influence of anisotropic scattering on the size of time-dependentsystems in monoenergetic neutron transport. J. Phys. D: Appl. Phys. 32, 317–325.