RERTR Publications:
Analysis Methods for Thermal Research and Test Reactors
ANL/RERTR/TM29
COMPUTING CONTROL ROD WORTHS
IN THERMAL RESEARCH REACTORS
3. INTERNAL BOUNDARY CONDITIONS
Since diffusion theory fails near the surface of a strong neutron absorber,
a pair of group and meshdependent effective diffusion parameters can be used
to calculate reactivity effects of the absorber. An alternate approach is to
isolate the absorber from the diffusiontheory calculation by applying groupdependent
internal boundary conditions at the absorber surface.
The internal boundary condition, A, is of the form
D f¢ + A f = 0
where f¢ is the normal derivative of the flux at
the absorber surface. Thus,
A = D f¢ / f = J / f
= D / d = l_{tr} / 3 d
where d º extrapolation distance into the absorber,
l_{tr} º the neutron transport mean free path in the diffusing medium, and
A º the internal boundary condition defined as the currenttoflux ratio at the
absorber surface.
Transport calculations are used to determine the groupdependent currenttoflux ratios. It is assumed that the materialdependent internal boundary condition is the same at every point on the surface of the absorber. However, currenttoflux ratios used in diffusiontheory calculations should never exceed the black absorber limit.
3.1 CurrenttoFlux Ratios for Black Absorbers
For a plane black boundary (i.e. no return neutrons) the extrapolation distance
has the value d = 0.7104l_{tr} which corresponds
to a currenttoflux ratio A = l_{tr} / 3d
= 0.4692. The extrapolation distance for a black cylindrical rod is a function
of the radius, R, and the transport mean free path, l_{tr}
, of neutrons in the surrounding medium. Figure 1 is a plot of the currenttoflux
ratio evaluated at the surface of a black cylindrical absorber of infinite length
as a function of R/l_{tr} .These blackrod
internal boundary conditions were calculated from extrapolation distances given
in Ref. 6. Note that in the limit as R/l_{tr}
approaches infinity, the currenttoflux ratio becomes that of a planar black
boundary. Using the least squares process, data from Fig. 1 were fit to a polynomial
of the form
(J/f)_{black} = S_{ i }a_{i} (R/l_{ tr})^{i }
where the coefficients are:
i 
a_{i} 

0 
2.49993E01 

1 
2.50031E01 

2 
1.87744E01 

3 
7.70505E02 

4 
1.56659E02 

5 
1.00392E03 

6 
1.17961E04 

7 
1.52381E05 
3.2 CurrenttoFlux Ratios for NonBlack Absorbers
For nonblack absorber materials the surface currenttoflux ratios are energydependent.
For this case the internal boundary conditions can be calculated from a discreteordinate
transport calculation using finely spaced mesh intervals in order to determine
groupdependent fluxes and currents at the surface of the absorber. If any of
the values exceed the values given in Fig. 1, they should be replaced with the
black absorber limits. Internal boundary conditions are not needed for those
upper energy groups for which S_{a} / S_{s}
<< 1 and/or t / L << 1 where L is the
diffusion length and t is the minimum transverse
dimension of the absorber material.
For the very special case of an absorber slab placed within a symmetric neutron
field,
f_{l} = f_{r}
and J_{l} = J_{r}. The surface currenttoflux ratio then equals
the a blackness coefficient discussed earlier.
3.3 Example
Until recently, the shimsafety rods used in the University of Michigan Ford
Nuclear Reactor consisted of a borated stainless steel material. Table 4 (from
Ref. 5) gives the transverse dimensions and composition of these shimsafety
rods. To obtain a set of internal boundary conditions, the rod, the surrounding
moderator, and fuel were represented in cylindrical geometry. Since most thermal
neutrons are absorbed near the surface of the shimsafety rod, the radius of
the "equivalent" cylindrical rod was chosen so as to preserve the
surface area of the actual rod. This cylindrical rod was divided into three
concentric zones with the outer and middle zones having thicknesses of 1 mm
and
3 mm, respectively. Fulldensity borated stainless steel was used in the outer
and middle zones, but the density in the inner zone was reduced so as to preserve
the total amount of material in the equivalent rod. Multigroup cross sections
were generated for each control zone and for all the other regions outside the
shim rod using the WIMSD4M code^{7}. Onedimensional P_{1}
S_{8} transport calculations were performed with the TWODANT code^{8}
in order to determine groupdependent currenttoflux ratios on the surface
of the equivalent cylindrical rod. Table 5 summarizes the currenttoflux ratios
obtained in this manner. This table also shows the internal boundary conditions
calculated for the TiB_{2}(95% ^{10}B)Al6351 shim rods now
used in the Ford Nuclear Reactor. Based on S_{a}
/ S_{s}, t / L,
and S_{a}t values,
where t is the minimum thickness of the actual shim
rod, Table 5 shows that internal boundary conditions are not needed for the
fast groups 1 and 2 and that the rod is black to group4 neutrons. Therefore,
the polynomial fit to Fig. 1 was used to determine the group4 currenttoflux
ratios.
TABLE 4 COMPOSITION AND GEOMETRY OF THE FNR SHIMSAFETY RODS 

Composition (atoms/bcm) 

Nuclide 
Borated Stainless Steel 
TiB_{2}(95%^{10}B)Al6351 

^{10}B 
1.108E3 
1.5749E3 

^{11}B 
5.1843 
7.5390E5 

Mg 
4.0978E4 

Al 
5.7161E2 

Si 
5.9104E4 

Ti 
1.0158E3 

V 
1.6293E5 

Cr 
1.640E2 
1.5962E5 

Mn 
1.8129E4 

Fe 
5.644E2 
1.4862E4 

Ni 
1.130E2 
1.4131E5 

Cu 
2.6122E5 

Zn 
5.0734E5 

Geometry 

Dimensions (cm) 
Borated SS 
TiB_{2}Al6351 

Effective Length 
60.960 
60.960 

RectangularLike Cross Section 

Major Axis 
5.668 
5.715 

Minor Axis 
2.198 
2.222 

Radius of Rounded Ends 
1.099 

Radius of Rounded Corners 
0.635 
TABLE 5 INTERNAL BOUNDARY CONDITIONS FOR THE FNR SHIMSAFETY RODS 

Borated Stainless Steel Rods 

Quantity 
Group 1 
Group 2 
Group 3 
Group 4 
E_{u}(eV) 
1.0000E+07 
8.2100E+05 
5.5300E+03 
6.2500E01 
S_{a}/S_{s} 
6.8727E03 
6.4574E03 
1.5990E01 
2.6788E+00 
D(cm) 
1.6583E+00 
8.8032E01 
3.1743E01 
9.5010E02 
S_{a}(cm^{1}) 
1.8958E03 
3.3825E03 
1.4756E01 
2.5676E+00 
L(cm) 
2.9576E+01 
1.6132E+01 
1.4667E+00 
1.9236E01 
t/L 
7.4318E02 
1.3625E01 
1.4986E+00 
1.1426E+01 
S_{a}t 
4.1669E03 
7.4347E03 
3.2434E01 
5.6436E+00 
IBC = J/f 
2.5331E02 
1.1123E02 
8.1122E02 
4.0956E01 
TiB_{2}(95%^{10}B)Al6351 Rods 

S_{a}/S_{s} 
3.6927E03 
1.4154E02 
1.7049E+00 
3.7923E+01 
D(cm) 
3.1275E+00 
1.6415E+00 
1.1959E+00 
9.3876E02 
S_{a}(cm^{1}) 
6.6305E04 
3.6092E03 
1.7942E01 
3.4624E+00 
L(cm) 
6.8680E+01 
2.1326E+01 
2.5818E+00 
1.6466E01 
t/L 
3.2360E02 
1.0422E01 
8.6082E01 
1.3498E+01 
S_{a}t 
1.4736E03 
8.0214E03 
3.9876E01 
7.6952E+00 
IBC = J/f 
1.3262E02 
6.9260E03 
1.1656E01 
4.1097E01 
NOTE: These tables show that:
1. IBC's are not needed for groups 1 and 2 since S_{a}/S_{s}
<<1.
2. IBC's are needed for groups 3 and 4.
3. IBC for group 4 is that of a black rod since S_{a}t >>1.