RERTR Publications:
Analysis Methods for Thermal Research and Test Reactors
ANL/RERTR/TM29
COMPUTING CONTROL ROD WORTHS
IN THERMAL RESEARCH REACTORS
4. CONTROL ROD WORTH EVALUATIONS
Because of their strong neutronabsorbing character, special methods are needed to determine control rod worths in diffusion theory calculations. As discussed in the above paragraphs, one such method is to determine a pair of group and meshdependent effective diffusion parameters for the absorber rod. An alternate method is to isolate the absorber material from the diffusion calculation by applying a groupdependent set of internal boundary conditions (currenttoflux ratios) at the absorber surface. Using these methods, control rod worths have been calculated for several control rod materials.
4.1 Cadmium Control Elements
Control elements used in the Oak Ridge Research Reactor (ORR) and the Swedish
R2 Reactor have the same design. The poison section consists of a waterfilled
square cadmium annulus 2.345 in. (5.956 cm) on a side, 30.5 in. (77.47 cm) long,
and 0.040 in. (0.1016 cm) thick. Since the widthtothickness ratio is very
large, effective diffusion parameters obtained earlier for a cadmium slab of
this thickness (see Table 1) are applicable. For the reasons given earlier,
effective diffusion parameters are needed only for the group 5 thermal neutrons
(E_{n} < 0.625 eV). Table 1 shows that for this group the cadmium
thickness in absorption mean free paths is greater than 5. Therefore, the cadmium
sheet is effectively black to group 5 neutrons with a corresponding currenttoflux
ratio equal to 0.4692. It was shown in Ref. 3 that the worth of a cadmium slab
of this thickness calculated with effective diffusion parameters obtained from
the spectrumweighted P_{5} blackness coefficients (<a(P_{5})>
and <b(P_{5})> and calculated using
the group5 black internal boundary condition gave nearly identical results
both of which agreed with the result of a Monte Carlo calculation within 1s
statistics. Therefore, control rod worths for the ORR and R2 reactors were calculated
using the black internal boundary condition for group 5 neutrons incident on
the cadmium absorber and normal diffusion theory for all the other groups.
Table 6 summarizes 3D calculations for control rod worths in the R2 reactor.
Note that all the diffusiontheory worth calculations agree within 1s
of the corresponding VIM^{9}Monte Carlo results. These values are taken
from Ref. 3, which includes a description of the R2 reactor core configuration.
The ORR 179AX5 core^{10} was waterreflected with allfresh U_{3}Si_{2}
(4.8 gU/cm^{3}) LEU fuel. It contained 14 standard (19plate) fuel elements
and 4 shim rods each with an upper cadmium poison section and a lower 15plate
fuel follower section. Differential rod worths were measured by the positive
period method. The integral rod worth was obtained by integrating the differential
measurements from the lower to the upper limit of the shim rod displacement.
Measured and calculated integral worths for the D6 shim rod in ORR Core 179AX5
are compared in Table 7. As with the Swedish R2 reactor, the black internal
boundary condition (J/f = 0.4692) was applied at
the surface of the cadmium absorber for the thermal group (E_{n} <0.625
eV) in the diffusiontheory calculations. Table 7 also shows that the DIF3Ddiffusion
and the VIMMonte Carlo total worth calculations are in good agreement, but
are about 5.6% larger than the measured value. This difference is typical of
ORR worth measurements discussed in Ref. 10 and is partly the result of approximate
corrections to the experimental data for delayed photoneutrons and temperature
feedback effects. Note that the integrated worth and the total worth, based
on rodin and rodout eigenvalue calculations, agree to within less that 1%.
However, these integral and total
TABLE 6 EIGENVALUES AND CADMIUM CONTROL ROD WORTHS FOR THE SWEDISH R2 REACTOR 

Fuel^{a} 
Rod Config. 
K_{eff }DIF3D^{b} 
Dr^{c}%dk/k 
k_{eff }VIM 
Dr^{c}%dk/k 
HEU 25019 
All Out 
1.1602 
1.1662±0.0025 

" 
All In 
0.9654 
17.39 
0.9700±0.0022 
17.34±0.30 
" 
At 50% 
1.0826 
6.18 
1.0862±0.0024 
6.32±0.27 
" 
Only G3 Out 
1.0233 
11.53 
1.0266±0.0024 
11.66±0.29 
LEU 32616 
All Out 
1.1562 
1.1537±0.0020 

" 
All In 
0.9655 
17.09 
0.9656±0.0025 
16.89±0.31 
" 
At 50% 
1.0816 
5.97 
1.0790±0.0026 
6.00±0.27 
" 
Only G3 Out 
1.0184 
11.70 
1.0191±0.0025 
11.45±0.28 
^{b}The
DIF3D calculations were done for group 5 of cadmium made black. ^{c}Dr = (k_{out}  k_{in})/k_{out}k_{in}. 
TABLE 7 D6 INTEGRAL ROD WORTH FOR ORR CORE 179AX5 

Integration Limits, In.^{a} 
Integral worth, % dk/k 

LL = 0.0 
UL = 26.56 
Calc. 
Exp. 
C/E 

7.239 
6.855 
1.056 

Total Worth 

Code 
Rout, In. 
Rin, In. 
Rbank, In. 
kout 
kin 
% dk/k 
VIM 
26.56 
0.0 
17.72 
1.0400±0.0018 
0.9666±0.0020 
7.299±0.273 
DIF3D 
26.56 
0.0 
17.72 
1.0371 
0.9641 
7.309 
^{a}Integration of the differential rod worth from the lower to the upper limit gives the total rod worth. 
ORR WATERREFLECTED LEU CRITICAL
179AX5
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
FE 
FE 
FE 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
FE 
FFD4 
FE 
FFD6 
FE 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
FE 
FE 
FE 
FE 
FE 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
FE 
FFF4 
FE 
FFF6 
FE 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
FE = 19plate standard fuel element
FF = 15plate fuel follower element
worths are not expected to be exactly the same because of differences in the
rod bank elevations.
4.2 AgInCd Control Elements
Some research reactors use control elements consisting of flat forked blades
composed of the AgInCd alloy described earlier. For this alloy and for a thickness
t = 0.310 cm, Table 2 gives the effective diffusion
parameters based on the fluxweighted P_{5} blackness coefficients,
<a(P_{5})> and <b(P_{5})>,
for mesh intervals of h = t and h = t/2.
These blacknessmodified diffusion parameters were used in 3D diffusiontheory
calculations to determine control rod worths in the 10MW IAEA Generic Reactor^{11}.
For this reactor the 23plate fuel elements use fresh LEU U_{3}Si_{2}
 Al dispersion fuel with a ^{235}U loading of
390 g per fuel element.
DIF3Ddiffusion and VIMMonte Carlo results for eigenvalues and reactivities
are compared in Table 8. They all agree within the 1s
Monte Carlo statistics. Based on effective diffusion parameters calculated from
the AgInCd blackness coefficients, the worth of Rod 3 was determined for mesh
intervals of h = t, t/2,
t/3, and t/4. Table 8
shows that the calculated rod worth is nearly independent of the mesh interval
size. However, the results suggest a maximum value of h=t/2
be used for determining the effective diffusion parameters from the blackness
coefficients.
4.3 Hafnium Control Elements
Control elements for the Japanese 20MW JRR3 reactor^{12} consist
of square waterfilled natural hafnium boxes 6.36 cm on a side and 0.50 cm thick.
Using the methods described earlier, Table 3 shows the evaluated blackness coefficients
for a hafnium slab of this thickness and a density of 13.3 g/cm^{3}.
The effective diffusion parameters were evaluated using the spectrumaveraged
P_{5} blackness coefficients.
Threedimensional DIF3D diffusion and VIM Monte Carlo calculations were used
to calculate control rod worths in the JRR3 reactor. The standard fuel element
has 20 plates whereas the control rod follower element has 16 plates of fresh
LEU fuel. Using effective diffusion parameters corresponding to a mesh interval
h = t/2, Table 9 compares DIF3D and VIM eigenvalues
and control rod worths for the JRR3 reactor.
TABLE 8 XYZ CALCULATIONS FOR THE 10MW IAEA GENERIC REACTOR FOR FRESH LEU U_{3}Si_{2} FUEL WITH AgInCd CONTROL BLADES 

Rod Configuration 
Code 
hcm 
k_{eff} 
Dr^{a}% dk/k 
All Out 
VIM 
1.1922±0.0031 

All Out 
DIF3D 
1.1903 

All In 
VIM 
1.0296±0.0031 
13.25±0.36 

All In 
DIF3D^{b} 
h = t/2 
1.0309 
12.99 
Rod 3 Out 
VIM 
1.0838±0.0033 
8.39±0.36 

Rod 3 Out 
DIF3D^{b} 
h = t 
1.0790 
8.66 
Rod 3 Out 
DIF3D^{b} 
h = t/2 
1.0813 
8.47 
Rod 3 Out 
DIF3D^{b} 
h = t/3 
1.0818 
8.43 
Rod 3 Out 
DIF3D^{b} 
h = t/4 
1.0816 
8.44 
^{b}Dr = (k_{out}  k_{in})/k_{out}k_{in}. 
LOCATIONS of the AgInCd CONTROL
BLADES in the 10MW IAEA GENERIC REACTOR
C 
C 
C 
C 
C 
C 
SFE 
SFE 
CFE1 
SFE 
SFE 
SFE 
SFE 
SFE 
SFE 
SFE 
CFE2 
SFE 
SFE 
CFE3 
SFE 
Irr. Pos. 
SFE 
SFE 
SFE 
SFE 
SFE 
SFE 
CFE4 
SFE 
Irr Pos. 
SFE 
CFE5 
SFE 
SFE 
SFE 
C 
C 
C 
C 
C 
C 
SFE = 23plate standard fuel element
CFE = 17plate control fuel element
TABLE 9 EIGENVALUES AND HAFNIUM CONTROL ROD WORTHS FOR THE JRR3 REACTOR 

Rod Config. 
K_{eff }DIF3D 
Dr^{a}% dk/k 
k_{eff }VIM 
Dr^{a}% dk/k 
All Out 
1.2291 
1.2227±0.0023 

At 50% 
1.1224 
7.74 
1.1143±0.0024 
7.96±0.25 
All In 
0.8689 
33.74 
0.8763±0.0028 
32.33±0.39 
^{a}Dr = (k_{out}  k)k_{out} k. 
Ford Nuclear Reactor Core Congiguration
With Fresh LEU Fuel
D_{2}O TANK 

H_{2}O 
H_{2}O 
SFE 
SFE 
SFE 
SFE 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
SFE 
CFEA 
SFE 
CFEC 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
SFE 
SFE 
SFE 
SFE 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
SFE 
CFEB 
SFE 
CFERR 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
SFE 
SFE 
SFE 
SFE 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
H_{2}O 
SFE 
SFE 
H_{2}O 
H_{2}O 
H_{2}O 
SFE = 18plate standard fuel element
CFE = 9plate control fuel element
Figure 2. FNR 27Element WaterReflected
LEU Core
4.4 Borated Stainless Steel Control Elements
Shortly after the conversion of the Ford Nuclear Reactor (FNR) from HEU to
LEU fuel, fulllength rod worth measurements were made in the 27element fresh
LEU core (Fig. 2) in December 1981. Total rod worths were obtained by integrating
incremental worths, measured by the positive period technique, from the lower
to the upper limit of rod movement. The results of these measurements are reported
in Ref. 13.
The shape and composition of the borated stainless steel shimsafety rods used
in this UAl_{x}Al core are described in Table 4. This table shows that
the FNR shimsafety rods cannot be approximated by a thin slab treatment and
so blackness theory cannot be used to evaluate their worth. However, rod worths
were evaluated for this 27element core configuration using the MCNP Monte Carlo
code^{14}, the DIF3D code with internal boundary conditions (IBC's)
for groups 3 and 4 (see Table 5), and the DIF3D code with effective diffusion
parameters (EDP's) obtained by matching reaction rate ratios^{15}.
Table 10 compares measured and calculated rod worths and includes the University
of Michigan results based on their twogroup reaction rate matching process
described in Ref. 5. For all these calculations the rod worth was determined
by computing the reactivity difference between rodin and rodout cases with
the regulating rod (RR) fully withdrawn. The Monte Carlo and diffusion calculations
with IBC's are threedimensional whereas the diffusion calculations with EDP's
are twodimensional XY calculations. None of the calculations account for the
beam tubes associated with the D_{2}O tank on the north side of the
core, nor do the calculations account for the rod bank and regulating rod elevations
corresponding to the differential worth measurements, since this data is unavailable.
Nevertheless, Table 10 shows very acceptable agreement among the measured and
calculated rod worth values. Note that the rod worths are increased by about
7% when they are evaluated for the case where the regulating rod is withdrawn
50% and the rod bank is withdrawn 67%.
4.5 Titanium Diboride Aluminum Control Elements
In October 1995 the FNR borated stainless steel shimsafety rods were replaced
with borated aluminum rods composed of an alloy of titanium diboride in 6351
aluminum. The boron in the TiB_{2} has a ^{10}B enrichment >
95%. Table 4 compares the geometry and compositions of the TiB_{2}(95%^{10}B)Al6351
shimsafety rods with the former borated stainless steel ones while Table 5
compares the internal boundary conditions. These IBC's were used in 3dimensional
DIF3D calculations to determine the worths of the borated aluminum rods in the
27element FNR core (Fig. 2). These rodout and rodin calculations were done
with the rod bank and the regulating rod fully withdrawn.
Table 11 summarizes the results and compares the rod worths with those obtained
earlier
TABLE 10 FNR EIGENVALUES AND SHIMSAFETY ROD WORTHS FOR THE 27ELEMENT FRESH UAl_{X} LEU CORE 

% Rod Withdrawal 
Code 
Geom. 
Eigenvalue 
Rod Worth, % dk/k 

Reg. Rod 
Bank 
Rod 
Meas 
Calculated 

100.0 
100.0 
A: 100.0 
MCNP 
XYZ 
1.03234±0.00070 

DIF3D: IBC's 
XYZ 
1.03632 

DIF3D: EDP's 
XY 
1.02208 

100.0 
100.0 
A: 0.0 
MCNP 
XYZ 
1.00999±0.00074 
2.220 
2.144±0.098 
DIF3D: IBC's 
XYZ 
1.0145 
2.109 

DIF3D: EDP's 
XY 
0.99910 
2.250 

UM2DB: EDP's 
XY 
2.279 

50.0 
66.7 
A: 100.0 
DIF3D: IBC's 
XYZ 
1.02035 

A: 0.0 
DIF3D: IBC's 
XYZ 
0.99751 
2.244 

100.0 
100.0 
B: 0.0 
MCNP 
XYZ 
1.00938±0.00070 
2.320 
2.203±0.095 
DIF3D: IBC's 
XYZ 
1.01176 
2.342 

DIF3D: EDP's 
XY 
0.99782 
2.379 

UM2DB: EDP's 
XY 
2.648 

50.0 
66.7 
B: 100.0 
DIF3D: IBC's 
XYZ 
1.02106 

B: 0.0 
DIF3D: IBC's 
XYZ 
0.99529 
2.535 

100.0 
100.0 
C: 0.0 
MCNP 
XYZ 
1.01084±0.00073 
2.283 
2.060±0.097 
DIF3D: IBC's 
XYZ 
1.01450 
2.075 

DIF3D: EDP's 
XY 
0.99944 
2.216 

UM2DB: EDP's 
XY 
2.247 

50.0 
66.7 
C: 100.0 
DIF3D: IBC's 
XYZ 
1.02040 

C: 0.0 
DIF3D: IBC's 
XYZ 
0.99778 
. 
2.222 
TABLE 11 COMPARISON OF THE BORATED STAINLESS STEEL AND THE TiB_{2}Al SHIMSAFETY ROD WORTHS IN THE FNR 27ELEMENT CORE 

Rod 
Withdrawal^{a} 
Code 
Eigenvalue 
Rod Worth % dk/k 
Ratio 

% 
(TiB_{2}Al) 
TiB_{2}Al 
BSS 
Calc'd 
Meas.^{b} 

A A B C 
100.0 0.0 0.0 0.0 
DIF3D: IBC's DIF3D: IBC's DIF3D: IBC's DIF3D: IBC's 
1.03568 1.01074 1.00801 1.01113 
2.383 2.650 2.345 
2.109 2.342 2.075 
1.130 1.131 1.130 
1.148 1.095 1.096 
^{b}The TiB_{2}Al/BSS rod worth ratio was calculated for the 27element core (Fig. 2). However, the measured ratio is based on rod calibrations made in the October 1995 core when the borated stainless steel shimsafety rods were replaced with TiB_{2}(95%^{10}B)Al6351. 
(Table 10) for the borated stainless steel material. The worth of the newer rods is about 13% larger than that of the original rods. This increase in rod worth is the result of the larger circumference (6.8%) and the larger ^{10}B concentration (42%) of the TiB_{2}Al6351 shimsafety rods relative to the borated stainless steel ones (see Table 4). Although both shim rod materials are black to thermal neutrons (E_{n}<0.625 eV), the higher ^{10}B concentration results in greater epithermal absorption in the TiB_{2}Al6351 (compare the group3 IBC's in Table 5). Shimsafety rod worths for both materials were measured in the FNR October 1995 core^{16}. Table 11 compares the measured TiB_{2}Al/BSS worth ratios in this core with the values calculated for the 27element core configuration. The average calculated and measured ratios are 1.13 and 1.11, respectively. Different core configurations and core burnup may be responsible for the somewhat different worth ratios.