### RERTR Publications:

Analysis Methods for Thermal Research and Test Reactors

#### ANL/RERTR/TM-29

COMPUTING CONTROL ROD WORTHS

IN THERMAL RESEARCH REACTORS

##### 2. EFFECTIVE DIFFUSION PARAMETERS

Methods used to determine effective diffusion parameters depend on the
geometry of the absorber. If the control rod can be described by one or
more thin slab absorbers (i.e. thickness << transverse dimensions),
effective diffusion parameters can be written in terms of a pair of "blackness
coefficients" which are defined below. For non-slab-like absorbers
effective diffusion parameters can be determined from a reaction rate ratio
matching requirement. In either case, the effective diffusion parameters
depend on the neutronic properties of the absorber and on the mesh structure
within it, but not on the outside media. For those high energy groups for
which S_{s} >> S_{a}
for the control material, normal diffusion theory is valid so the determination
of effective diffusion parameters is unnecessary for these groups.

**2.1 Thin Slab Geometry
**

For control materials in the shape of thin slabs, mesh-dependent effective
diffusion parameters can be expressed in terms of a pair of energy-dependent
blackness coefficients, a and b.
For an absorber slab of thickness t the blackness
coefficients are defined by the equations

a º (J_{l}
+ J_{r}) / (f_{l }+f_{r}),
and b º (J_{l} - J_{r})
/ (f_{l} - f_{r})

where f and J are the asymptotic neutron fluxes
and currents __into__ the slab on the left-hand and right-hand surfaces
of the slab. Because of these definitions, the blackness coefficients depend
only on the properties of the absorber slab.

**2.1.1 Blackness Coefficients
**

Blackness theory^{1-3} provides a mechanism for evaluating the
blackness coefficients. The theory assumes:

1. The coolant slab is uniform and of infinite lateral extent.

2. There are no neutron sources within the control slab due to fission,
n2n, or

scattering from other energies.

3. Neutron scattering within the slab is isotropic.

4. Diffusion theory is applicable to all regions within the reactor except
for the

control slab.

5. Blackness coefficients evaluated for infinite slabs are applicable
to finite slabs

whose transverse dimensions are very large relative to the thickness.

Because of the first two assumptions, the one-dimensional monoenergetic
Boltzmann transport equation can be solved by expanding the angular flux
within the slab into spherical harmonics in order to determine surface fluxes
and currents. The fourth assumption is normally violated at locations just
outside a strongly absorbing slab. Therefore, the flux shape determined
by using blackness-modified diffusion parameters is likely to be erroneous
at such locations. The last assumption is necessary because quantities analogous
to a and b for finite
slabs do not exist. However, it is reasonable to expect this assumption
to provide a good approximation.

To illustrate how the blackness coefficients are calculated, a
and b are evaluated in the P_{1} approximation,
with and without scattering, in the Appendix. Although the algebra is lengthy,
the Appendix outlines how the same methods are extended to calculate the
blackness coefficients in the P_{3} and P_{5} approximations.
More details on these higher order approximations are given in Ref. 3. For
very strong absorbers (S_{a} / S_{s}
>> 1) a modified form of the zero-scatter P_{1} approximation,
namely

a_{0m}(P_{1}) = 0.4692 [1
- 2E_{3}(S_{a}t)]
/ [1 + 3E_{4}(S_{a}t)]

b_{0m}(P_{1}) = 0.4692 [1
+ 2E_{3}(S_{a}t)]
/ [1 - 3E_{4}(S_{a}t)]
,

gives good results. In these equations E_{3} and E_{4}
are the exponential integrals of the absorber thickness expressed in absorption
mean free paths. The blackness coefficients, especially b,
are very sensitive to neutron scattering and so these approximations fail
when scattering becomes significant. In such cases the coefficients should
be evaluated in the P_{5} approximation.

Broad-group blackness coefficients are most accurate if they are obtained
by flux-weighting the fine-group values. Thus,

<a> = S_{i} a_{i} (f_{l}
+ f_{r})_{i} / S_{i}
(f_{l} + f_{r})_{i
}

<b> = S_{i}
b_{i} (f_{l}
- f_{r})_{i} / S_{i}
(f_{l} - f_{r})_{i
}

where the summations are over the number of fine groups which make up
the broad group. The fine group surface fluxes (f_{l}
and f_{r}) may be obtained from a one-dimensional
P_{1}, S_{8} transport calculation.

It is usually sufficient to calculate <a>
and <b> only for the thermal and epithermal
groups. Normally, blackness coefficients for the fast groups are not needed
because for these energies S_{s} >>
S_{a} for the absorber and so normal
diffusion theory applies without any need for effective diffusion parameters.

**2.1.2 Effective Diffusion Parameters
**

The a and b blackness
coefficients form a pair of internal boundary conditions applicable on the
surfaces of the absorber slab. However, most diffusion codes are not programmed
to handle internal boundary conditions of this type. Therefore, it is convenient
to determine a set of effective diffusion parameters (D_{eff} and
S_{a-eff}) in terms of the blackness
coefficients which preserve the current-to-flux ratios on each side of the
absorber slab. These effective diffusion parameters depend on the mesh interval
size h and therefore allow the use of a very course mesh in the absorber
for the diffusion-theory calculations.

Expressions for the effective diffusion parameters are derived in the
Appendix, so only the results are given here. For the case where the diffusion
code, such as DIF3D^{4}, determines fluxes at the center of the
mesh intervals,

k = (1/t) [ b^{1/2}
+ a^{1/2} ] / [ b^{1/2}
- a^{1/2} ] ,

D_{eff} = (h/2) (a + b)
[(1 + cosh kh)/2] (tanh kt) / (sinh kh) , and

S_{a-eff} = D_{eff} [cosh
kh - 1] / h^{2 }.

The equations for k and S_{a-eff}
are also valid for use in diffusion codes which calculate fluxes on the
mesh boundaries. For this case, however, the expression for D_{eff}
becomes (see Ref. 3)

D_{eff} = (h/2) (a + b)
(tanh kt) / (sinh kh)

For an effectively black absorber a ® b ®
0.4692 and k tends to infinity. Effective diffusion parameters can be obtained
for this limiting case by setting kt equal to
an arbitrarily large, but finite, value such as kt
= 10. In this limit the effective diffusion parameters corresponding to
mesh-centered fluxes reduce to

D_{eff} = at/(2n) and S_{a-eff}
= [an/(4t)] e^{kt/n
}

where n=1,2,... determines the mesh interval size h = t/n.

**2.1.3 Examples
**

Blackness coefficients and effective diffusion parameters have been evaluated
for several control materials commonly used in research reactors in slab
geometry. Tables 1-3 summarize the results for slabs of natural cadmium,
a Ag-In-Cd alloy, and natural hafnium.

The thickness of the cadmium sheet (t = 0.1016
cm) corresponds to that used in the 30-MW Oak Ridge Research Reactor (now
shut down) and the Swedish R2 Reactor. Flat blades of the Ag-In-Cd alloy
were assumed to be 0.310 cm thick with a density of 9.32 g/cm^{3}
and a composition of 4.9 wt % Cd, 80.5 wt % Ag, and 14.6 wt % In. The natural
hafnium slab is 0.50 cm thick, which equals the thickness of the square
hafnium annulus used in the control elements of Japan's JRR-3 reactor.

For each of these materials broad-group blackness coefficients were evaluated
in the P_{1}, P_{3} and P_{5} approximations. Fine-group
flux weighting was used to determine the P_{5} average values <a(P_{5})>
and <b(P_{5})> and the modified
zero-scattering P_{1} average values <a_{0m}(P_{1})>
and <b_{0m}(P_{1})>. For
comparison purposes, Table 1 also includes the unmodified zero-scattering
P_{5} average values <a_{0}(P_{5})>
and <b_{0}(P_{5})>. Note
that the effective diffusion parameters (D_{eff} and S_{a-eff})
given in these tables depend on the mesh interval size h and are based on
the <a(P_{5})> and <b(P_{5})>
values. Recall that the effective diffusion parameters, and the blackness
coefficients upon which they depend, are functions only of the properties
of the slab and are independent of the surrounding media.

Unmodified diffusion parameters (D and S_{a})
may be used for those groups for which S_{a}
/ S_{s} << 1 and/or t
/ L <<1, where L is the diffusion length of neutrons in the absorber.
On this basis effective diffusion parameters are needed for groups 3-5 for
slabs of natural hafnium and the Ag-In-Cd alloy. For cadmium, however, effective
diffusion parameters are really needed only for group 5. For this group
the thickness of the cadmium sheet in absorption mean free paths (S_{a}t)
is about 5.1, which means that the cadmium sheet absorbs over 99% of all
the incident group 5 neutrons (i.e. group 5 is approximately black). Table
1 shows that in the P_{5} approximation a(P_{5})
= 0.4698, which nearly equals the black limit of 0.4692. This unique property
of cadmium results from the very large absorption resonance at about 0.18
eV.

These tables also show that for S_{a}t
> 1 the modified-zero-scattering approximation gives remarkably good
values for the blackness coefficients. Even for S_{a}t
as low as about 0.2, <a_{0m}(P_{1})>
is quite satisfactory. However, in this range of S_{a}t
values <b_{0m}(P_{1})>
is badly over-calculated because of its sensitivity to neutron scattering
effects. Where applicable, however, the modified-zero-scattering approximation
is very useful because these blackness coefficients can be easily calculated.

**2.2 Other Geometries
**For control rod geometries which cannot be approximated by a one-dimensional
slab treatment, quantities analogous to the a
and b blackness coefficients do not exist so
other methods are needed to determine effective diffusion parameters. Since
analytical expressions for the effective diffusion parameters cannot be
obtained for multi-dimensional control rods, an iterative technique is used
to determine D

_{eff}and S

_{a-eff}. It is assumed that a set of effective diffusion parameters can be found which depend on the nuclear cross sections of the absorber, its dimensions, and the mesh spacing used in diffusion-theory calculations to describe the control rod, but which are independent of the environment outside the lumped absorber.

To determine the effective diffusion parameters a control cell characteristic of the rod and its surroundings is defined. This cell, with reflective boundary conditions, explicitly models the lumped absorber, its immediate environment, and a surrounding fuel region. For this cell Monte Carlo calculations are performed to determine for each energy group the capture rate in the absorber lump relative to the fission rate in the fuel region. This same cell is used for diffusion-theory calculations choosing the same mesh structure in the absorber which will be used later for global diffusion calculations. Beginning with the highest energy group, D and S

_{a}values are adjusted in a series of diffusion-theory calculations until the absorption rate in the absorber lump relative to the fission rate in the surrounding fuel region equals that obtained from the Monte Carlo calculation. This process is repeated on a group-by-group basis. Effective diffusion parameters are those adjusted values of D and S

_{a}which result in a match to the Monte Carlo reaction rate ratios. An arbitrary relationship between D

_{eff}and S

_{a-eff}may be defined such as

D

_{eff}= [3 S

_{a-eff}]

^{ -1}.

As for the slab case, effective diffusion parameters are not needed for those high-energy groups for which S

_{s}>> S

_{a}.

This method is commonly used by the University of Michigan in two-group calculations of the worths of the shim-safety rods in the Ford Nuclear Reactor (FNR)

^{5}. However, it is a rather laborious procedure when more energy groups are used in the diffusion-theory calculations. Therefore, other procedures, described below, are normally used in the ANL-RERTR program to calculate control rod worths within the framework of diffusion theory.