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Asymptotic PN and double PN approximations
Journal
of Nuclear
Energy, Vol.
ASYMPTOTIC
26, pp. 231 to 236.
PN AND
Pergamon
Press 1972.
DOUBLE
Printed
in Northern
Ireland
PN APPROXIMATIONS
SONG TEH HUANG* and E. E. LEWIS Nuclear Engineering Program, The Technological Institute, Northwestern Evanston, Illinois 60201, U.S.A.
University,
(Received 20 January 1971 and in revisedform 3 November 1971)
Abstract-The Prr and double PN approximations
to the monoenergetic transport equation are modified to improve their asymptotic behaviour. Free parameters are introduced into the PrJ and double PN equations through the use of a pseudo-scattering kernel. The parameters are evaluated by stipulating that the resulting scalar flux approximations to the infinite-medium Green’s function yield exact asymptotic solutions while conserving specified spatial moments. Asymptotic P, and double PI solutions are presented for the infinite-medium and half-space Green’s functions and for the sourcefree and constant-source Milne problems. INTRODUCTION
CONSIDERABLE attention has been given to the well-known inability of the PI equations to correctly predict the asymptotic diffusion length (DAVISON, 1956; SELENGUT,1962; POMRANING and CLARK, 1963). In asymptotic diffusion theory the situation is remedied by appropriately modifying the diffusion coefficient. Even for the infinite-medium Green’s function, however, asymptotic diffusion theory correctly predicts neither the root-mean-squared distance travelled by the diffusing neutrons, as the PI equations do, nor the magnitude of the asymptotic flux. Higher order approximations are required if these or other properties of exact transport solutions are to be conserved while determining the diffusion length exactly. POMRANING (1965a, 1965b) introduced a modification to the PN truncation procedure which yields the exact diffusion length. HENRYSONand SELENGUT(1969) have employed synthetic kernels in the integral transport equation to improve the behaviour of asymptotic solutions. In the following sections pseudo-scattering kernels (LEWIS, 1968) are used to introduce free parameters into the PN and double PN(DPN) equations. This procedure permits the asymptotic behaviour of the equations to be modified while at the same time preserving some of the desirable features of the standard equations. The asymptotic forms of the P, and DPl equations that result from the application of the pseudoscattering kernel correctly predict the diffusion length, the magnitude of the asymptotic flux for the infinite-medium Green’s function, and the root-mean-squared distance travelled by diffusing neutrons. The pseudo-anisotropic corrections are given only in terms of the material properties, and moreover, may be expressed analytically for the P, and DPl equations. THEORY For slab geometry and isotropic sources, the transport equation is
w p - (x2pu)+ ax
wx,pu)=
dfi’Y(x,
c
s
,u’)f($ . 0’)
+ z,
(1)
where c is the number of secondary neutrons per collision, x is measured in mean free paths, and p is the directional cosine of fi. The asymptotic PN approximations are * Present Address: Jersey 07101, U.S.A.
Public Service. Electric and Gas Company,
231
80 Park Place, Newark,
New
232
TEH HUANGand E. E. LEWIS
SONG
formulated in terms of the standard PN equations, 1 21+ 1 [
-OfJl =WY 1+(1
ax
aY1+r
la“‘+(l+l)x
1= 1,2, . . . , N,
Y~.+_~ = 0,
where the angular flux and scattering function are approximated by
(2) They, in the PN approximation are just the Legendre coefficients of the scattering function. In the formulation of asymptotic PN(AP,) approximations, however, we shall use thesef, as free parameters in order to improve the asymptotic behaviour of the equations. To formulate the APN equations we need only to stipulate the values of the N + 1 parameters so, fi . . .fN that appear in the pseudo-scattering kernel of equation (2). We require that these parameters by functions only of the properties of the material medium, and therefore not of the boundary condition encountered in specific problems. They are, therefore, functions only of c, the number of secondary neutrons per collision. In addition to requiring that neutrons be conserved, by setting f.= 1, we require that the asymptotic PN(APN) approximation yield the exact asymptotic scalar flux.
for the plane isotropic-source Green’s function in an infinite medium. In the case of isotropic scattering in the laboratory system the diffusion length is the positive solution of the transcendental equation cLtanh_’ (l/L) = 1,
(3)
while the magnitude of the asymptotic solution is given by (DAVISON, 1958) (L” -
K=
1)
cL(cL2 + 1 -
L2) .
(4)
Because only one free parameter, fi, is available in the API equations, only the diffusion length can be exactly specified. Moreover by doing this, the root-meansquared distance travelled by neutrons is no longer preserved as it is in the standard P, equations. In the APB approximation, however, the three free parameters, fi, f2,f3 may be used to specify L, K, and the root-mean-squared distance, 2, exactly. After solving for the Green’s function in the AP, approximation and requiring that L, K and ;$z be exact 3 we find l-
Cfi = -&
1 - cf3 =
(1 -
(1 -
c)y2[3 + ~LK(v-~ - L-2)J)-1,
9 (1 - (1 - c)v2[3 + ~LK(v-~ - L-2)]}-1, 28(1 - c)?
Asymptotic PN and double PN approximations
233
where 1 Y’= ~ [l - 6(1 - c)?_%]/[l 3(1 - c)
- 2(1 - c)LK].
Media with anisotropic scattering may be treated by replacing equations (3) and (4) with the values of L and K obtained from the exact Green’s function solution with anisotropic scattering; the value of 2 is conserved by using the true Legendre expansion of the scattering kernel to obtainf,. For higher order APN approximations the moments z
=/_~dnn’&)/~dxgl(x)
may be conserved for it = 0, 1, . . . N - 1 by using the true Legendre expansion coefficients for fo, fi...fn_2, while modifying only fN_land fn to obtain the exact diffusion length and magnitude of the asymptotic flux. Asymptotic DPN approximations may be derived in a similar manner. In the DPN approximation a discontinuity is permitted at ,u = 0 by expanding the angular flux in two Legendre series for p 3 0. Thus Y(x, P) = iO(V
lu 3 0.
YV(X)P,(2P T l),
To obtain the standard DPN equations, equations (2) and (5) are substituted into equation (l), and the standard orthogonality procedure is then applied (CASE and ZWEIFEL, 1967). A difficulty arises, however, if we attempt to use the fiappearing in the resulting equations for the formulation of an asymptotic DPN(ADPN) approximation. The DPN and P2N+I a pp roximations consist of the same number of first order differential equations (i.e. 2N + 2). Moreover, the DP, approximation is normally considered to be at least as accurate as the corresponding P,N+l approxiavailable for formulating the mation. Considering the number of free parameters, fi, corresponding asymptotic approximations, however, we find that there are only half as many available for the ADP, approximation as for the corresponding APzN+, approximation [i.e. N + 1 values for the ADP, approximation and 2(N + 1) for the AP2N+1 approximation]. This situation exists because the scattering kernel has been expanded in a single set of full-range Legendre polynomials. To eliminate this difficulty we expand the scattering kernel in two, half-range Legendre series. Thus, “21+
1
f(P3 = L:---3*p&b z-0 17r
p S 0.
T 11,
Using the orthogonality properties of the Legendre polynomials it may then be shown that the modified form of the DPN equations is
.a=
1 (2j
+
1) [
J
ax
+ (j
1
ay;+, (2-i+ 1)a\T,*
+ 1) 7
f
+ %~I-) +
$
ax
yj*
=
sj*
h-A;o (c;X+ + d$X-)] , j=O,
1,2...N,
SONG TEH HUANC and E. E. LEWIS
234
with ay;+, ax=
0 ’
where the coefficients are given by
s *1
b& = d&1+
1)
0
W-‘&
*1 c;l = It:(2il+1) dPP,c% s0
s ?? 1
d;, = +W + 0
0
s
‘f 1)
n
n
dk?U’,@ * i.2’- l)P,(2$
+ l),
. '>O p-0
7 1) *,* d&P,(2fi 'X0 s
. P/’ + l)P,(2,~’ -
l),
s
’ fi’+ 1)P,(2p’+ 1). druPX2iu F 1) n,n,<,da’P,t2fi /I’<0
The same criteria are used in formulating the ADPN equations and the AP2N+1 equations. Thus, asymptotic diffusion theory results if the exact value of the diffusion length is specified in the ADP,, equations. To fix the free parameters f,* , fi* in the ADP, approximation, we require that neutrons be conserved, that the asymptotic scalar flux from the plane-source Green’s function be exact, and that the root-meansquared distance travelled by diffusing neutrons be exact. After solving the modified DPl equations for the plane isotropic-source Green’s function and requiring that K, L and 2 be exact, we find
and
where y = ‘$ [l - 6(1 - c)“L~K]-~[~ - 3@(1 - c)]“. RESULTS
A series of infinite-medium and half-space calculations have been carried out in which the PB, AP,, DP1, and ADP, are compared to exact solutions (HUANG, 1969). The errors in the plane-source infinite-medium Green’s functions are plotted in Figs. 1 and 2 for c = O-9 and 0.7 respectively. For infinite-medium problems the AP, and ADP, approximations give identical results: both of these approximations may be reduced to fourth-order self-adjoint equations for the scalar flux. Thus the requirement that they conserve the same scalar flux properties of the exact solution results in the equations being identical.
AsymptoticPrrand double PI approximations
235
c=o9
A< _
06
1---
AD? P,
De - -
-1 2
4
--
I
I
1
I
I
1
6
8
IO
12
14
16
i I8
x,(mfp)
Fig. 1.-Errors for the infinite-mediumGreen’s function.
I--u
I
2
4
6
8
x,(m.f.p
IO
12
I
14
1
16
3
1
Fig. 2.-Errors for the infinite-mediumGreen’s function.
Figures 1 and 2 are typical illustrations of the improved flux behaviour at large distances from the source that results from replacing the P, and DPl equations by their asymptotic counterparts. Because the P, approximation converges rapidly to the correct asymptotic solution as (1 - c) + 0, the improvement resulting from the asymptotic formulation is not as pronounced for the P, equations as for the DP, equations. There appears to be little advantage in using the asymptotic formulations in situations where the neutron distribution near the source is of primary interest, for as the asymptotic improvement from the AP, and ADP, equations becomes more pronounced with decreasing c, the accuracy at short and intermediate distances from the source tends to deteriorate.
236
SONG TEH HUANG and E. E. LEWIS
In order to examine the effects of boundaries on the asymptotic approximations, the emerging currents were calculated for half-space Green’s functions with the plane isotropic source at varying numbers of mean-free-paths, x0, from the vacuum interface. Once again the asymptotic formulations are found to represent an improvement over their standard counterparts only when the emerging neutrons have travelled large distances from the source. For c = 0.7, 0.9, the ADP, approximations yield more accurate results for x0 > 5; the AP, approximation (using Marshak boundary conditions) results in an improvement for x0 > 10 where c = 0.7, but shows no improvement even for x0 = 40 when c = 0.9. Thus an increase in improvement with deceased c is again observed. The Milne problem and the half-space with a uniform source have been used to compare approximate solutions. The uniform-source results may be obtained by superposition from the half-space Green’s function, while the Mime problem is equivalent to the Green’s function as x,, -+ co. No improvement results from the use of the asymptotic formulations in the uniform-source problem, for the average neutron emerging from the vacuum interface does not originate from a large number of mean-free-paths within the half-space. The relative error in the emerging current for the half-space Green’s function tends to 100 per cent at x,, + co for the standard approximations, while it converges to errors of a few per cent for the AP3 and ADP, approximations. However in applying the standard PN or DPN approximations to the Milne problem the asymptotic boundary condition y(x) - e”‘L, x -+ cc is normally modified by replacing the exact diffusion length, L, by a value that is consistent with the approximation. This artifice nullifies the improvements that would otherwise result from the use of the asymptotic approximations. REFERENCES CASE K. M. and ZWEIFELP. F. (1967) Linear Transport Theory. Addison-Wesley, Massachusetts. DAVISONB. (1957) Neutron Transport Theory. Oxford University Press, London. HENRYSON H. and SELENGUTD. S. (1969) Nucl. Sci. Engng 37, 1. HUANG S. T. (1970) Studies of asymptotic PX and double PN approximations to the monoenergetic neutron transport equation, Dissertation, Northwestern University. LEWIS E. E. (1968) Trans. Am. nucl. Sot. 11, 551. POMRANING G. C. and CLARK M. J. (1963) Nucl. Sci. Engng 17,227. POMRANING G. C. (1965a) Nucleonik 6,348. POMRANING G. C. (1965b) Nucl. Sci. Engng 22, 328. SELENGUT D. S. (1962) Trans. Am. nucl. Sot. 5,40.
of Nuclear
Energy, Vol.
ASYMPTOTIC
26, pp. 231 to 236.
PN AND
Pergamon
Press 1972.
DOUBLE
Printed
in Northern
Ireland
PN APPROXIMATIONS
SONG TEH HUANG* and E. E. LEWIS Nuclear Engineering Program, The Technological Institute, Northwestern Evanston, Illinois 60201, U.S.A.
University,
(Received 20 January 1971 and in revisedform 3 November 1971)
Abstract-The Prr and double PN approximations
to the monoenergetic transport equation are modified to improve their asymptotic behaviour. Free parameters are introduced into the PrJ and double PN equations through the use of a pseudo-scattering kernel. The parameters are evaluated by stipulating that the resulting scalar flux approximations to the infinite-medium Green’s function yield exact asymptotic solutions while conserving specified spatial moments. Asymptotic P, and double PI solutions are presented for the infinite-medium and half-space Green’s functions and for the sourcefree and constant-source Milne problems. INTRODUCTION
CONSIDERABLE attention has been given to the well-known inability of the PI equations to correctly predict the asymptotic diffusion length (DAVISON, 1956; SELENGUT,1962; POMRANING and CLARK, 1963). In asymptotic diffusion theory the situation is remedied by appropriately modifying the diffusion coefficient. Even for the infinite-medium Green’s function, however, asymptotic diffusion theory correctly predicts neither the root-mean-squared distance travelled by the diffusing neutrons, as the PI equations do, nor the magnitude of the asymptotic flux. Higher order approximations are required if these or other properties of exact transport solutions are to be conserved while determining the diffusion length exactly. POMRANING (1965a, 1965b) introduced a modification to the PN truncation procedure which yields the exact diffusion length. HENRYSONand SELENGUT(1969) have employed synthetic kernels in the integral transport equation to improve the behaviour of asymptotic solutions. In the following sections pseudo-scattering kernels (LEWIS, 1968) are used to introduce free parameters into the PN and double PN(DPN) equations. This procedure permits the asymptotic behaviour of the equations to be modified while at the same time preserving some of the desirable features of the standard equations. The asymptotic forms of the P, and DPl equations that result from the application of the pseudoscattering kernel correctly predict the diffusion length, the magnitude of the asymptotic flux for the infinite-medium Green’s function, and the root-mean-squared distance travelled by diffusing neutrons. The pseudo-anisotropic corrections are given only in terms of the material properties, and moreover, may be expressed analytically for the P, and DPl equations. THEORY For slab geometry and isotropic sources, the transport equation is
w p - (x2pu)+ ax
wx,pu)=
dfi’Y(x,
c
s
,u’)f($ . 0’)
+ z,
(1)
where c is the number of secondary neutrons per collision, x is measured in mean free paths, and p is the directional cosine of fi. The asymptotic PN approximations are * Present Address: Jersey 07101, U.S.A.
Public Service. Electric and Gas Company,
231
80 Park Place, Newark,
New
232
TEH HUANGand E. E. LEWIS
SONG
formulated in terms of the standard PN equations, 1 21+ 1 [
-OfJl =WY 1+(1
ax
aY1+r
la“‘+(l+l)x
1= 1,2, . . . , N,
Y~.+_~ = 0,
where the angular flux and scattering function are approximated by
(2) They, in the PN approximation are just the Legendre coefficients of the scattering function. In the formulation of asymptotic PN(AP,) approximations, however, we shall use thesef, as free parameters in order to improve the asymptotic behaviour of the equations. To formulate the APN equations we need only to stipulate the values of the N + 1 parameters so, fi . . .fN that appear in the pseudo-scattering kernel of equation (2). We require that these parameters by functions only of the properties of the material medium, and therefore not of the boundary condition encountered in specific problems. They are, therefore, functions only of c, the number of secondary neutrons per collision. In addition to requiring that neutrons be conserved, by setting f.= 1, we require that the asymptotic PN(APN) approximation yield the exact asymptotic scalar flux.
for the plane isotropic-source Green’s function in an infinite medium. In the case of isotropic scattering in the laboratory system the diffusion length is the positive solution of the transcendental equation cLtanh_’ (l/L) = 1,
(3)
while the magnitude of the asymptotic solution is given by (DAVISON, 1958) (L” -
K=
1)
cL(cL2 + 1 -
L2) .
(4)
Because only one free parameter, fi, is available in the API equations, only the diffusion length can be exactly specified. Moreover by doing this, the root-meansquared distance travelled by neutrons is no longer preserved as it is in the standard P, equations. In the APB approximation, however, the three free parameters, fi, f2,f3 may be used to specify L, K, and the root-mean-squared distance, 2, exactly. After solving for the Green’s function in the AP, approximation and requiring that L, K and ;$z be exact 3 we find l-
Cfi = -&
1 - cf3 =
(1 -
(1 -
c)y2[3 + ~LK(v-~ - L-2)J)-1,
9 (1 - (1 - c)v2[3 + ~LK(v-~ - L-2)]}-1, 28(1 - c)?
Asymptotic PN and double PN approximations
233
where 1 Y’= ~ [l - 6(1 - c)?_%]/[l 3(1 - c)
- 2(1 - c)LK].
Media with anisotropic scattering may be treated by replacing equations (3) and (4) with the values of L and K obtained from the exact Green’s function solution with anisotropic scattering; the value of 2 is conserved by using the true Legendre expansion of the scattering kernel to obtainf,. For higher order APN approximations the moments z
=/_~dnn’&)/~dxgl(x)
may be conserved for it = 0, 1, . . . N - 1 by using the true Legendre expansion coefficients for fo, fi...fn_2, while modifying only fN_land fn to obtain the exact diffusion length and magnitude of the asymptotic flux. Asymptotic DPN approximations may be derived in a similar manner. In the DPN approximation a discontinuity is permitted at ,u = 0 by expanding the angular flux in two Legendre series for p 3 0. Thus Y(x, P) = iO(V
lu 3 0.
YV(X)P,(2P T l),
To obtain the standard DPN equations, equations (2) and (5) are substituted into equation (l), and the standard orthogonality procedure is then applied (CASE and ZWEIFEL, 1967). A difficulty arises, however, if we attempt to use the fiappearing in the resulting equations for the formulation of an asymptotic DPN(ADPN) approximation. The DPN and P2N+I a pp roximations consist of the same number of first order differential equations (i.e. 2N + 2). Moreover, the DP, approximation is normally considered to be at least as accurate as the corresponding P,N+l approxiavailable for formulating the mation. Considering the number of free parameters, fi, corresponding asymptotic approximations, however, we find that there are only half as many available for the ADP, approximation as for the corresponding APzN+, approximation [i.e. N + 1 values for the ADP, approximation and 2(N + 1) for the AP2N+1 approximation]. This situation exists because the scattering kernel has been expanded in a single set of full-range Legendre polynomials. To eliminate this difficulty we expand the scattering kernel in two, half-range Legendre series. Thus, “21+
1
f(P3 = L:---3*p&b z-0 17r
p S 0.
T 11,
Using the orthogonality properties of the Legendre polynomials it may then be shown that the modified form of the DPN equations is
.a=
1 (2j
+
1) [
J
ax
+ (j
1
ay;+, (2-i+ 1)a\T,*
+ 1) 7
f
+ %~I-) +
$
ax
yj*
=
sj*
h-A;o (c;X+ + d$X-)] , j=O,
1,2...N,
SONG TEH HUANC and E. E. LEWIS
234
with ay;+, ax=
0 ’
where the coefficients are given by
s *1
b& = d&1+
1)
0
W-‘&
*1 c;l = It:(2il+1) dPP,c% s0
s ?? 1
d;, = +W + 0
0
s
‘f 1)
n
n
dk?U’,@ * i.2’- l)P,(2$
+ l),
. '>O p-0
7 1) *,* d&P,(2fi 'X0 s
. P/’ + l)P,(2,~’ -
l),
s
’ fi’+ 1)P,(2p’+ 1). druPX2iu F 1) n,n,<,da’P,t2fi /I’<0
The same criteria are used in formulating the ADPN equations and the AP2N+1 equations. Thus, asymptotic diffusion theory results if the exact value of the diffusion length is specified in the ADP,, equations. To fix the free parameters f,* , fi* in the ADP, approximation, we require that neutrons be conserved, that the asymptotic scalar flux from the plane-source Green’s function be exact, and that the root-meansquared distance travelled by diffusing neutrons be exact. After solving the modified DPl equations for the plane isotropic-source Green’s function and requiring that K, L and 2 be exact, we find
and
where y = ‘$ [l - 6(1 - c)“L~K]-~[~ - 3@(1 - c)]“. RESULTS
A series of infinite-medium and half-space calculations have been carried out in which the PB, AP,, DP1, and ADP, are compared to exact solutions (HUANG, 1969). The errors in the plane-source infinite-medium Green’s functions are plotted in Figs. 1 and 2 for c = O-9 and 0.7 respectively. For infinite-medium problems the AP, and ADP, approximations give identical results: both of these approximations may be reduced to fourth-order self-adjoint equations for the scalar flux. Thus the requirement that they conserve the same scalar flux properties of the exact solution results in the equations being identical.
AsymptoticPrrand double PI approximations
235
c=o9
A< _
06
1---
AD? P,
De - -
-1 2
4
--
I
I
1
I
I
1
6
8
IO
12
14
16
i I8
x,(mfp)
Fig. 1.-Errors for the infinite-mediumGreen’s function.
I--u
I
2
4
6
8
x,(m.f.p
IO
12
I
14
1
16
3
1
Fig. 2.-Errors for the infinite-mediumGreen’s function.
Figures 1 and 2 are typical illustrations of the improved flux behaviour at large distances from the source that results from replacing the P, and DPl equations by their asymptotic counterparts. Because the P, approximation converges rapidly to the correct asymptotic solution as (1 - c) + 0, the improvement resulting from the asymptotic formulation is not as pronounced for the P, equations as for the DP, equations. There appears to be little advantage in using the asymptotic formulations in situations where the neutron distribution near the source is of primary interest, for as the asymptotic improvement from the AP, and ADP, equations becomes more pronounced with decreasing c, the accuracy at short and intermediate distances from the source tends to deteriorate.
236
SONG TEH HUANG and E. E. LEWIS
In order to examine the effects of boundaries on the asymptotic approximations, the emerging currents were calculated for half-space Green’s functions with the plane isotropic source at varying numbers of mean-free-paths, x0, from the vacuum interface. Once again the asymptotic formulations are found to represent an improvement over their standard counterparts only when the emerging neutrons have travelled large distances from the source. For c = 0.7, 0.9, the ADP, approximations yield more accurate results for x0 > 5; the AP, approximation (using Marshak boundary conditions) results in an improvement for x0 > 10 where c = 0.7, but shows no improvement even for x0 = 40 when c = 0.9. Thus an increase in improvement with deceased c is again observed. The Milne problem and the half-space with a uniform source have been used to compare approximate solutions. The uniform-source results may be obtained by superposition from the half-space Green’s function, while the Mime problem is equivalent to the Green’s function as x,, -+ co. No improvement results from the use of the asymptotic formulations in the uniform-source problem, for the average neutron emerging from the vacuum interface does not originate from a large number of mean-free-paths within the half-space. The relative error in the emerging current for the half-space Green’s function tends to 100 per cent at x,, + co for the standard approximations, while it converges to errors of a few per cent for the AP3 and ADP, approximations. However in applying the standard PN or DPN approximations to the Milne problem the asymptotic boundary condition y(x) - e”‘L, x -+ cc is normally modified by replacing the exact diffusion length, L, by a value that is consistent with the approximation. This artifice nullifies the improvements that would otherwise result from the use of the asymptotic approximations. REFERENCES CASE K. M. and ZWEIFELP. F. (1967) Linear Transport Theory. Addison-Wesley, Massachusetts. DAVISONB. (1957) Neutron Transport Theory. Oxford University Press, London. HENRYSON H. and SELENGUTD. S. (1969) Nucl. Sci. Engng 37, 1. HUANG S. T. (1970) Studies of asymptotic PX and double PN approximations to the monoenergetic neutron transport equation, Dissertation, Northwestern University. LEWIS E. E. (1968) Trans. Am. nucl. Sot. 11, 551. POMRANING G. C. and CLARK M. J. (1963) Nucl. Sci. Engng 17,227. POMRANING G. C. (1965a) Nucleonik 6,348. POMRANING G. C. (1965b) Nucl. Sci. Engng 22, 328. SELENGUT D. S. (1962) Trans. Am. nucl. Sot. 5,40.