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 P₁ 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 P₃ and P₅ 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 P₁ approximation, namely

a_0m(P₁) = 0.4692 [1 - 2E₃(S_at)] / [1 + 3E₄(S_at)]

b_0m(P₁) = 0.4692 [1 + 2E₃(S_at)] / [1 - 3E₄(S_at)] ,

gives good results. In these equations E₃ and E₄ 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 P₅ 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 P₁, S₈ 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 DIF3D⁴, 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] / h² .

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.

Table 1 - Effective Diffusion Parameters for a Cadmium Slab

Table 2 - Effective Diffusion Parameters for a Ag-In-Cd Slab

Table 3 - Effective Diffusion Parameters for a Hafnium Slab

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/cm³ 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 P₁, P₃ and P₅ approximations. Fine-group flux weighting was used to determine the P₅ average values <a(P₅)> and <b(P₅)> and the modified zero-scattering P₁ average values <a_0m(P₁)> and <b_0m(P₁)>. For comparison purposes, Table 1 also includes the unmodified zero-scattering P₅ average values <a₀(P₅)> and <b₀(P₅)>. 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(P₅)> and <b(P₅)> 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_at) 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 P₅ approximation a(P₅) = 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_at > 1 the modified-zero-scattering approximation gives remarkably good values for the blackness coefficients. Even for S_at as low as about 0.2, <a_0m(P₁)> is quite satisfactory. However, in this range of S_at values <b_0m(P₁)> 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.