The problem of the boundary conditions in diffusion theory calculation is rather difficult because diffusion theory is not valid in the vicinity of a boundary.

As has been seen previously these boundaries represent either an external empty space or a “non-diffusion region” defining a control rod. In the first case for a convex body the vacuum is equivalent to a “black” absorber from which no neutron can return to the reactor. In the case of a region representing a control rod it is possible to define its “blackness” as the probability that a neutron entering this region will be absorbed in it (after any number of collisions). The blackness is in general an energy-dependent quantity, most control rods being “black” at thermal energy and “grey” for neutrons of higher energy. In the diffusion calculations the boundary condition is usually given in the form of an extrapolation length

d = —- calculated at the boundary, (5.9)

grad ф

d is in general an energy dependent quantity.

In order to obtain exact boundary conditions one has to perform transport calcula­tions on the region surrounding the boundary where diffusion theory is not valid. The extrapolation length of a control rod is then obtained performing a transport theory calculation over a control-rod cell. This cell is usually defined as the part of the reactor core corresponding to this control rod.

What complicates considerably the problem is that eqn. (5.9) cannot be calculated using the exact fluxes obtained at the boundary with a transport calculation. From the diffusion calculation performed over the complete reactor it is usually required to obtain an accurate overall power distribution and reactivity. The boundary conditions used should satisfy this condition and should not try to reproduce the exact flux distribution in the immediate vicinity of a boundary where diffusion theory is not valid.

If we consider the exact solution of the transport equation we see that it is possible to split it into two parts: an asymptotic neutron distribution valid far from the sources (or from the localized absorbers) and a transient part which is only important near the source (or absorber)

Ф(Г) = 4>As(r) + <(>trans(r);

this transient part vanishes within a few mean free paths from the boundary.

The asymptotic solution of the transport equation is practically coincident with the diffusion theory result in a weakly absorbing medium. As an example we can consider the monoenergetic case of a point source in an infinite homogeneous medium. The result can be extended to absorbers which can be represented by an appropriate distribution of negative point sources.

The diffusion equation takes the form

Ф(г) = S

with S = source strength

The transport theory solution is (see ref. 36, p. 236) ф(г) = <Mr) + фtrans(r) = -^-r є "

with

к = V32,2«(l-£) (1-22./52,),

2′ = 12, r = 1 2«=2,-2,

The asymptotic solution is smaller by a factor у than the diffusion theory solution. This is due to the fact that the diffusion theory flux is too low near the source, and must therefore be higher far from the source, in order to give the same number of absorbed neutrons

2 0(/>(r)dr = S.

J r = 0

In reality for usual cases у ~ 1 because 2„ <^2(.

Applying these considerations to the opposite case of an absorber rod we have the situation shown in Fig. 5.2.

The extrapolation length to be used in diffusion calculations should not be obtained from the exact transport theory solution, but from the asymptotic solution

d =—

grad ф As

In practice if the transport calculation has been performed with numerical methods it is impossible to separate the asymptotic from the transient solution. This is only possible when analytical methods are used. For example, in the analytical solution of the Pi equations in multi-region cylindrical cells the total flux and current are obtained as a sum of Bessel functions of different arguments out of which it is possible to separate the asymptotic terms (see ref. 37).

In practice for the external boundary of the reactor it is possible to use the extrapolated length

0.

71 0.71

2tr 2Д1-Д)

which is obtained from the asymptotic solution of the so-called Milne problem’38’ (vacuum boundary in slab geometry with sources at — °°). In the case of control rods it is possible to use expressions derived by Kusheriuk and McKay relating the blackness with the extrapolated length.’39’40’ These expressions are based on the following considerations. Under the assumption of validity of the diffusion theory on the boundary it is possible to relate the blackness to the extrapolation length. The Pt expansion of the angular flux is [see eqn. (4.19)]

ф(г, ІІ) = ^ф(г) + ^Ш(г)

which in one-dimensional cylindrical geometry takes the form

The neutron current is [see eqn. (4.13)]

J(r)=f ф(г, fl) cos 0 dfl = 27Г f ф(г, fl) cos в sin в d6. (5.12)

J4ir Jo

At a boundary the current can be resolved into two components (remembering that the positive J is in the direction of increasing r, i. e. the outward direction),

	j J out J in	(5.13)
where from (5.12) we have
	rn
J out	2ir J ф(г, fl) cos в sin в dd,	(5.14)
Jin =	— 2ir j ф(г, XI) cos в sin в d6. Jn/2	(5.15)

Substituting (5.11) in (5.14) and (5.15) we have

J = — + —

•/out ^ ‘ 2>

Ф

4

Defining the blackness /3 as the probability that a neutron entering the rod is absorbed in it

(5.18)

Substituting (5.16) and (5.17) in (5.18) under consideration of Fick’s law (4.31) and of (5.9) we obtain for the extrapolation length the expression

d=ikИ] <5,9)

this expression assumes the validity of diffusion theory on the boundary. Considering the asymptotic solution of the transport equation Kushneriuk and McKay found that the value I is more accurately replaced by a function of the radius R measured in moderator mean free paths. The expression takes then the form

£[£-*<*-4 <5’20)

or in presence of an air gap between rod and moderator

sb ‘*’■)] <5’21)

where Ra is the external radius of the gap, R the rod radius and 2,r refers to the surrounding medium. This formula is very often used to calculate control rods of HTRs. The blackness /3(R) is usually calculated with collision probability codes based on the numerical solution of the integral form of the Boltzmann equation (e. g. the MINOTAUR code, ref. 41).