Absorber blackness and extrapolation distance

Progress in Nuclear Energy.

Printedin GreatBritain.

1981,Vol. 8, pp. 135-144

0079-6530/81/030135-10505.00/0

PergamonPressLtd

ABSORBER BLACKNESS A N D EXTRAPOLATION DISTANCE R. ARONSON" Physics Department, Harvard University, Cambridge, Massachusetts 02138, U.S.A. *Permanent Address: Nuclear Engineering Department, Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201, U.S.A.

ABSTRACT There is a simple and elegant relation, first given by Spinks and proved by Pellaud, between the blackness B of a cylindrical gray control rod surrounded by a pure scattering medium, the average angle of scattering fl in the medium, and the increase Ad in the extrapolation distance for the medium due to the fact that the rod is not black. The relation is exact under certain conditions given by Pellaud. We have extended the result, showing that it is valid for more general scattering laws, and for plane and spherical geometry as well. We get another simple result in a case not considered by Pellaud. We also derive a formula for Ad which holds generally in all three geometries in a one-speed model with no other assumptions.

KEYWORDS Transport theory; neutron transport; control rods; Milne problem; extrapolation distance.

INTRODUCTION In the course of investigating the effect of control rods, Spinks (1965) came up with an interesting numerical observation, which he took to be approximate, about the extrapolation distance for a gray rod in a one-speed model. This result was later shown by Pellaud (1968) to be exact under certain assumptions. The result is: Ad ~ d-d o

4 3(l-f I)

i-~ B '

(i)

where d = extrapolation distance for cylindrical gray rod, do = extrapolation distance for black rod of same radius, ~ = blackness of rod, and fl = average value of cosine of scattering angle in moderator. This remarkable result shows that Ad depends only on the blackness and on fl and is independent of rod radius. Pellaud showed that this is rigorously true for a gray rod embedded in a infinite nonabsorbing moderator medium in a one-group model if it is assumed that the angular distribution of the neutrons returned from rod to moderator is isotropic. The scattering in the moderator was assumed linearly anisotropic in the laboratory

135

136

R. ARONSON

system. The present work had a two-fold motivation: to understand and simplify Pellaud's argument, and to see how far it could be generalized. In particular, we looked at plane and spherical geometry as well as at cylindrical geometry. In addition, we considered general anisotropic scattering (still in the one-group model) and a general directional distribution for the return from the rod. The results were the following: i.

We obtained a completely general formal expression for Ad (as well as for do,

which followed immediately from previous work (Aronson,

1970)), in the PN-approxi-

mation for arbitrary N. The realization of this expression requires matrix operations using the eigenvalues and eigenvectors for PN-approximation. 2. We verified Pellaud's results for the cylinder since the result is independent of radius). 3.

Equation

(and by extension to the plane,

(1) holds also in spherical geometry for isotropic return.

4. In all three geometries, if the return is isotropic.

Eq.(1)

is valid for arbitrary anisotropic scattering,

5. In plane geometry, there is an equally simple result for cosine return, valid for arbitrary anisotropic scattering, to wit: 2d Ad

o 3(l_fl ) •

(2)

UNIVERSAL FORMULATION We consider four cases: slab geometry in an exact formulation, and slab, spherical and cylindrical geometry in PN-approximation. In each case we wish to solve the Milne problem

(Davison,

1957) for the infinite region r > a.

We want to use the notation of the Transfer Matrix Method (Aronson and Yarmush, 1966; Aronson, 1970, 1972) because the operator equations in this notation are the same for all the problems. The starting point is the observation that in each case the flux can be written in the operator form

@+(r) = B+(r)A+(r) + B*(r)A + --

~ _

~--

_

(r)

'

(3)

with

A+(r) = e-A(r-r')A+(r'),

(4)

A_(r) = e--A(r'-r)A+(r').

(5)

Explicit forms for ~±, A±, B± and B E are given for each of the four cases in Appendices A - D.

Absorber Blackness

Here S±(r ) and A±(r)

137

are vectors and BE(r), B~ are integral operators in the exact

case and matrices in the spherical harmonic representations. ~ is a diagonal operator whose elements are the attenuation coefficients, i.e., the reciprocals of the attenuation lengths ~ (Re ~ > 0). Equations (4) and (5) are true for all r,r" but we use Eq. (4) only when r > r" and Eq. (5) only when r < r ~. That is, everything is described in terms of decreasing exponentials. Basically,

S± and S_ represent

the flux travelling in the direction of increasing

and decreasing r respectively and A+ and A_ are mode amplitudes defined in the same respective senses. Whether we are doing an exact calculation or an approximate one, there is always a discrete dominant mode, which we designate by the subscript 0. That is~ ~ is diso crete and greater than any other value of v. For the nonabsorbing medium considered here, ~ is infinite. The corresponding components of the mode amplitudes are o A±0(r ). For the Milne problem there is only one inward-traveling mode, so Eq. (3) becomes S_+(r) = B±(r)A+(r) + B*0(r)A_o

Here we have noted that A±0 are independent

(6)

.

of r (because i/~ = 0). o

The asymptotic

flux can be written as $± (r) = Bt0(r)A+0 + B~0(r)A_o.

(7)

The forms of B±(r) and B~(r) for the four cases considered are given in Appendices A - D. The only formulas from the appendices that we need explicitly here are the asymptotic formulas for the total flux and current and the form of B_(r) at the boundary.

In each case, we can write

SaS(r) = A+0 + f(r)A_o

jaS(r )

where

(Day±son,

i 3(l-f I)

,

. Sas (r)

(8)

f ~(r)A 0 3(l-f I)

,

(9)

1957) f(r) = r-a, r = -%n ~,

=

a - 1, r

- -

plane, cylinder, sphere.

(io)

138

R. ARONSON

EXTRAPOLATION LENGTH The solution of the Milne problem for the nonabsorbing region r > a can be obtained from Eq. (6) along with the shape of the return flux ~+(a) and the blackness

B

J(a) J_(a)"

(ii)

Here J+(a) and J_(a) are the partial currents into and out of the non-absorbing medium at the absorber-medium interface and J(a) is the net current. Since the current in a nonabsorbing medium is purely asymptotic and J(a) = J+(a) - J_(a), we have

J+(a) = -

I-B J(a) = - -I-B - jaS(a). B B

(12)

The extrapolation length is defined by

d = ~aS(a)

(13)

G as ('a) ' which from Eq.(8) can be written

d = f(a) __A+0 i f~(a) + A_0 f'(a) Note that f(a) = 0 in all three geometries.

(14)

The ratio A+0/A_0 is obtained by solv-

ing Eq. (6) for A+(a) and evaluating the zeroth component.

A+0

[B+l(a)

A0

B_,(a)]00 + i 0[B+I(a)~+(a)]0" ~

Putting everything together and noting that d of ~ + ( a ) ,

we o b t a i n

the general

Thus

o

(15)

is the part of d that is independent

result

[B+l(a)B*(a)]00 d

= o

Ad

(16) f'(a)

i I-B 3(l-f I) B

[B+l~+(a)]0 J+(a)

(17)

Absorber Blackness

SIMPLE CONSEQUENCES Plane Geometry~ From Eqs.

139

OF THE UNIVERSAL EXPRESSIONS

Exact Formulation

(A.7), (i0) and (16),

do

i fl B~l(vo,~)~d~ ' 2(l-fl) o

(18)

while from Eq. (17),

Ad

i 3(l-f I)

i-8 B

f

l

_

B+l(vo'~)~+(a'~)d~ o fl ~+(a,~)d~

(19)

o Now Eq.

(A.6) implies that

(~o,~)B+(P,~o)dP

Thus, if ~+(a,~) is isotropic, formula follows immediately.

From Eqs.

(20)

i.e., if ~+(a,~) = const, 0 < ~ ! i, then Spinks' Also, the assumption ~+(a,~) = const x ~, 0 < U ! i,

leads immediately to Eq. (2). No other return distribution for Ad, although of course, Eq. (19) is always valid.

All Three Geometries,

= 2.

leads to a simple form

PN-Approximation

(B.2) and (B.5), we have for a return proportional

to P£(~),

[~+(a)] i = const x ~i£ = (~i£/ei£) J+(a).

(21)

Inserting this into Eq.(17) gives

[B+l(a)~]O~ ~d

Now from the definition,

1 l-~ 3(l-f I) 8

(22) ~i£

Eq.(B.4), ~i0 = 1/4 and all = 1/2 for Marshak conditions.

Further, since in all three geometries

(B+)io = ~io' we have for isotropic return

(B~I ~)O0/elO = 4, so that Eq. (22) is again Spinks' formula, our Eq. (i). For cosine return, £=i, and Eq. (22) becomes in all three geometries

140

R. ARONSON

Ad

2

3(l_fl )

Again, Eq.(2) follows immediately

I-B [BTI ~

~

(23)

(a)~]01.

for plane geometry.

SUMMARY The expressions

for d

and Ad, Eqs. (16) and (17) respectively, are universal. o Pellaud's result follows when the return from the absorbing rod is isotropic. In plane geometry, the expression for do, given by Eq. (18) or (24), is simple and leads to the simple relation, geometries,

Eq.

(2), between d

and Ad for cosine return. In the curved o is not simple - it can be computed as a series in

the expression for d

o i/a - and there is no simple relation between d

o

and Ad.

It is pointed out in Appendices C and D that it would have been more natural in the curved geometries to interchange the definitions of [Bi(r)]iowith those of [B~(r)]io,_ since then [B+(r)]i j ~ _ limiting case ~ • o looked different.

would have the same form for j = 0 as for j > 0, even in the The expressions

for d

and Ad in terms of the B's would have ~

o

We would have found

d

= -

o

2

f.(a)[B~l(a)~]O0 2

Ad

(24)

3(1_fi )

1-B [~71(a)~+(a)]o ~

(25) J+(a)[B+l(a)~]00

The difference in the appearance of these equations from that of Eq. (16) and (17) or (22) is due entirely to the difference in the definition of B+(a). The numerical values of d

APPENDIX A.

o

and Ad would of course be unchanged.

PLANE GEOMETRY,

EXACT

Here the flux can be written in terms of the Case eigenfunctions 1967) ¢9(~) as

~(x,~) = Z { A ( 9 ) e - X / ~ ( ~ ) 9>0

(Case and Zweifel,

+ A(-9)eX/9@_9(~) }

(A.I)

-I where cos ~ is the angle between the neutron direction and the x-axis. We identify x with r and take the interface at a. The sum is a generalized sum. This is of the form of Eq. (3) with ~±(x,w) = ¢(x,±~),

~ > 0

(A.2)

Absorber Blackness B±(~,~)

= B+*(U,~)

=

141

~v(±~),

A± (x,~)) = A(±~)e ± (x-a)/~

0 < ~ i i, 0 < ~

(A.3)

~>0.

(A. 4)

For the asymptotic part of the flux, which is a discrete term in the sum, we have i i $~s(~) = B±(x,~,Vo)A+ ° + B_(x,~,9o)A_ ° = 2 A(x-a$ i~_-~) + ~ B, + where A and B are two constants.

(A.5)

Whether we identify A with A+o and B with A

or -o the reverse is purely a matter of convenience. Here we will define things so that when Uo ÷ ~, B+- assume their common limiting value of 1/2. Thus we take (here r = x)

1

B±o(~) ~ B±(~,9 o) = ~

B*o(r,~)

(A.6)

1 + ~__) -- B*o(r,~,~o) = ~(r - a _ l_fl •

(A. 7)

We see that the asymptotic total flux and current are ~aS(r) = A+o + (r-a)A jas (r) = -

APPENDIX B.

PLANE

i 3(l-f I)

(A.8)

-o

A

-o

.

(A. 9)

PN-APPROXIMATION.

GEOMETRY,

The spherical harmonic expansion of the flux can be written as N

~(r,~) = [ 2£+1 P~ (~)~£(r). £=0

(B.I)

In general, the forward and backward parts of the flux will be represented by linear combinations of the 4£, differing only in the sign of the odd-parity terms: N

[$+(r)] i = ~ e -

£=0

(+l)£~z(r) i£

i=1,7,. '

(N+I)/2.

(B.2)

""

The values of the ~i£ depend on the type of boundary conditions used

(Davison,1957).

With Marshak conditions, which we will use for simplicity, [~+(r)] i =

oP2i_l(~),(r,~)d~,

(B.3)

from which a comparison of Eqs. (B. i) and (B.2) gives

~i£ =

2£+1 2

I1

o P2i-I (~)P£ (~) d~"

(B.4)

142 Note that

Ro ARONSON with

Marshak

conditions,

the

forward

and backward

contributions

to

the

current J at r are respectively J±(r) = [@±(r)] I.

(B.51

This discussion makes no reference to geometry other than that the flux depends only on the two variables r and ~. Thus, these results continue to hold for spherical (but not cylindrical) geometry. In plane geometry

-xl~ ~(x)

= [ {A.e j>0 J

xl~

J g£(~j) + A .e -J

J(-l)Ig£(~j)}

(B.6) "

The g~(~j) are the standard polynomials discussed, for instance, by Case and Zweifel (1967).

Comparison with Eqs. (3)-(5) gives N

(B+)ij~_ = (B~)ij_~ = E~o~iE(±l)Eg~(~J)'

i = i ..... (N+I)/2, j>0

(B. 7)

In the same way as in Appendix A, we have (B+)io~- = ~'lO

(B.8)

1 [B*(r)]io = (r-a)mio + 3(l-fl--------~ mil"

(B.9)

The asymptotic total flux and current are given by Eqs. (A.8) and (A.9) respectively. APPENDIX C.

SPHERICAL GEOMETRY, PN-APPROXIMATION.

Here the known form of the ~n(r) (Davison, 1957) leads to N r/~. [B±(r)]ij = ~o~i~(±l) ~ g~(~j)k~(r/~j)e J

[~(r)]ij

(C.I)

-r/~. = ~0~i£(±i) £ g~(~j)i£ (r/~j)e 3, i = i, ...,(N+I)/2, j>O,(C.2)

where kz(z) and i£(z) are the modified spherical Bessel functions.

For j=0 it is

convenient for most purposes to use the limiting forms when v. ÷ ~. However, for 3 the purpose of obtaining our result, it is simpler to interchange the natural way of choosing the "inward" and "outward" modes. That is, [B+(r)]io~_ and [B~(r)]i ° are chosen to be the limiting forms of Eqs. (C.2) and (C.I) respectively. Now

iOur g~ are the G~ of Case and Zweifel (P.221).

Absorber Blackness

143

(C.3)

lim g£(Vo)i~(r/Vo )= $io O

and lim gE (Vo)kE(r/Vo)=

~! (2C+I) r ~+I

(c. 4)

H (l-fk)-I. k=l

Thus we take

[B*(r)]io = ~-

(c. 5)

c~.

10

N

y.

[B*(r)]io = a ~0~i~(+i)%

(c. 6)

.. (2~+l)r~+l k~l(l-fk )-I - ( ~11.0

The asymptotic total flux and current are given by ~aS(r) : A+o + A_o(a - I)

jas (r)

APPENDIX D.

a 3(1-fl)r2

A

(C.7)

(c. 8)

. --O

PN-APPROXIMATION,

CYLINDRICAL GEOMETRY

Without going into detail here, we will adopt the coordinate convention and the results of Kofink (1959). In exactly the same way as for spherical geometry, we have

N [B+(r)]iJ- =E=OZ m~O ~i'Em(-+l)m nZcv(n)%m gE(n)(vj)Km(r/~j)' J > 0

(D.I)

N ~0~i,~m(±l) [B~(r)]ij = ~ ~~=0 m

(D.2)

TM

(n)~(n)t~ C~m ~ ~ jrI Im(r/~ j) , j > 0.

Here there are two angular indices, ~ and m. g(n)(v) are polynomials of degree (~-2n).

(n) The C~m are a set of constants and the

Both are defined by Kofink.

The K

m

and

I are the modified Bessel functions. m Under the assumption of symmetry with respect to both any plane perpendicular to the axis and any plane through the axis (basically this says that things are the same looking North as looking South and are also the same looking West as looking East), m and ~ have the same parity. The index n is determined by j, one or more eigenvalues ~. corresponding to each value of n. J (0) = 1 and The largest eigenvalue Vo corresponds to n=O. Since (Kofink, 1959) c ~m g~O)(~j)t_ = g~(~J)' one can determine the limiting forms from those of the Bessel functions.

Just as in the spherical case, we choose as [B+(r)]i ° ~ -

of Eq. (D.2) and for [B~(r)]i ° - ±

the limiting value of Eq. (D.I).

the limiting value Thus

R. ARONSON

144

[B+(r)]i ° = ~.io ~_

(D.3)

[B~(r)]io = - e. Zn ~ + [ a i (-i)£2~-i£!(g-i)! ~_ lo a ~>0 '~£ ~)'2£+i'!!r£

~ (l-fk)-l. k=l

(D. 4)

The asymptotic total flux and current are given by ~aS(r) = A+o - £n a A -o jas (r) =

I 3(l-fl)r

A

-o

(D.5)

.

(D. 6)

The index i in this Appendix indexes a function of two angles, not one as for the other geometries. It is not difficult to generalize Eqs. (B.I)-(B.5) to take this into account. With Marshak conditions, Eq. (B.5) can be thought of as a definition of the i = i component of ~+._ Then one can show easily that ~.x,oo = ~' ~i,ll = ½" Since m ~ £ and has the same parity as ~, there are no other ei,£m for £=0 or i. We can just write these quantities as e'lO and eil respectively. identical with the aio and ~il obtained from Eq.

(B.4).

Their values are

This is the only result

about the ~'s that we need explicitly.

ACKNOWLEDGEMENT The author wishes to express his appreciation for the hospitality given him by Harvard University, where this work was done.

REFERENCES Aronson, R. (1970). Transfer Matrix Solutions of One- and Two-Medium Transport Problems in Slab Geometry, J. Math. Phys., Ii, 931-940. Aronson, R. (1972). General Solution for Polarized Radiation in a Homogeneous-Slab Atmosphere, Ap. J., 177, 411-421. Aronson, R. and D.L. Yarmush (1966). Transfer Matrix Method for Gamma-Ray and Neutron Penetration, J. Math. Phys., 7, 221-237. Case, K. M. and P.F. Zweifel (1967). Linear Transport Theory. Addison-Wesley, Reading, Mass. Davison, B. (1957). Neutron Transport Theory. Oxford University Press, London. Kofink, W. (1959). Complete Spherical Harmonics Solution of the Boltzmann Equation for Neutron Transport in Homogeneous Media with Cylindrical Geometry, Nucl. Sci. Eng., 6, 475-486. Pellaud, B. (1968). The Extrapolation Distance for a Black or Grey Cylindrical Neutron Absorber by the Spherical Harmonics Method, Nucl. Sci. Eng.,33, 169-186. Spinks, N. (1965). The Extrapolation Distance at the Surface of a ~rey Cylindrical Control Rod, Nucl. Sci. Eng., 22, 87-93.