RERTR Publications:
Analysis Methods for Thermal Research and Test Reactors
ANL/RERTR/TM-29
COMPUTING CONTROL ROD WORTHS
IN THERMAL RESEARCH REACTORS
4. CONTROL ROD WORTH EVALUATIONS
Because of their strong neutron-absorbing 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 mesh-dependent effective diffusion parameters for the absorber rod. An alternate method is to isolate the absorber material from the diffusion calculation by applying a group-dependent set of internal boundary conditions (current-to-flux 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 water-filled
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 width-to-thickness 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
(En < 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 current-to-flux
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 spectrum-weighted P5 blackness coefficients (<a(P5)>
and <b(P5)> and calculated using
the group-5 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 diffusion-theory worth calculations agree within 1s
of the corresponding VIM9-Monte Carlo results. These values are taken
from Ref. 3, which includes a description of the R2 reactor core configuration.
The ORR 179-AX5 core10 was water-reflected with all-fresh U3Si2
(4.8 gU/cm3) LEU fuel. It contained 14 standard (19-plate) fuel elements
and 4 shim rods each with an upper cadmium poison section and a lower 15-plate
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 179-AX5
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 (En <0.625
eV) in the diffusion-theory calculations. Table 7 also shows that the DIF3D-diffusion
and the VIM-Monte 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 rod-in and rod-out 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 |
|||||
Fuela |
Rod Config. |
Keff -DIF3Db |
Drc-%dk/k |
keff -VIM |
Drc-%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 |
bThe
DIF3D calculations were done for group 5 of cadmium made black. cDr = (kout - kin)/koutkin. |
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 |
R-out, In. |
R-in, In. |
R-bank, In. |
k-out |
k-in |
% 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 |
aIntegration of the differential rod worth from the lower to the upper limit gives the total rod worth. |
ORR WATER-REFLECTED LEU CRITICAL
179AX5
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
FE |
FE |
FE |
H2O |
H2O |
H2O |
H2O |
H2O |
FE |
FFD4 |
FE |
FFD6 |
FE |
H2O |
H2O |
H2O |
H2O |
FE |
FE |
FE |
FE |
FE |
H2O |
H2O |
H2O |
H2O |
FE |
FFF4 |
FE |
FFF6 |
FE |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
H2O |
FE = 19-plate standard fuel element
FF = 15-plate fuel follower element
worths are not expected to be exactly the same because of differences in the
rod bank elevations.
4.2 Ag-In-Cd Control Elements
Some research reactors use control elements consisting of flat forked blades
composed of the Ag-In-Cd alloy described earlier. For this alloy and for a thickness
t = 0.310 cm, Table 2 gives the effective diffusion
parameters based on the flux-weighted P5 blackness coefficients,
<a(P5)> and <b(P5)>,
for mesh intervals of h = t and h = t/2.
These blackness-modified diffusion parameters were used in 3D diffusion-theory
calculations to determine control rod worths in the 10-MW IAEA Generic Reactor11.
For this reactor the 23-plate fuel elements use fresh LEU U3Si2
- Al dispersion fuel with a 235U loading of
390 g per fuel element.
DIF3D-diffusion and VIM-Monte 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 Ag-In-Cd 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 20-MW JRR-3 reactor12 consist
of square water-filled 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/cm3.
The effective diffusion parameters were evaluated using the spectrum-averaged
P5 blackness coefficients.
Three-dimensional DIF3D diffusion and VIM Monte Carlo calculations were used
to calculate control rod worths in the JRR-3 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 JRR-3 reactor.
TABLE 8 XYZ CALCULATIONS FOR THE 10-MW IAEA GENERIC REACTOR FOR FRESH LEU U3Si2 FUEL WITH Ag-In-Cd CONTROL BLADES |
||||
Rod Configuration |
Code |
h-cm |
keff |
Dra-% 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 |
DIF3Db |
h = t/2 |
1.0309 |
12.99 |
Rod 3 Out |
VIM |
1.0838±0.0033 |
8.39±0.36 |
|
Rod 3 Out |
DIF3Db |
h = t |
1.0790 |
8.66 |
Rod 3 Out |
DIF3Db |
h = t/2 |
1.0813 |
8.47 |
Rod 3 Out |
DIF3Db |
h = t/3 |
1.0818 |
8.43 |
Rod 3 Out |
DIF3Db |
h = t/4 |
1.0816 |
8.44 |
bDr = (kout - kin)/koutkin. |
LOCATIONS of the Ag-In-Cd CONTROL
BLADES in the 10-MW IAEA GENERIC REACTOR
C |
C |
C |
C |
C |
C |
SFE |
SFE |
CFE-1 |
SFE |
SFE |
SFE |
SFE |
SFE |
SFE |
SFE |
CFE-2 |
SFE |
SFE |
CFE-3 |
SFE |
Irr. Pos. |
SFE |
SFE |
SFE |
SFE |
SFE |
SFE |
CFE-4 |
SFE |
Irr Pos. |
SFE |
CFE-5 |
SFE |
SFE |
SFE |
C |
C |
C |
C |
C |
C |
SFE = 23-plate standard fuel element
CFE = 17-plate control fuel element
TABLE 9 EIGENVALUES AND HAFNIUM CONTROL ROD WORTHS FOR THE JRR-3 REACTOR |
||||
Rod Config. |
Keff -DIF3D |
Dra-% dk/k |
keff -VIM |
Dra-% 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 |
aDr = (kout - k)kout k. |
Ford Nuclear Reactor Core Congiguration
With Fresh LEU Fuel
D2O TANK |
|||||||
H2O |
H2O |
SFE |
SFE |
SFE |
SFE |
SFE |
H2O |
H2O |
H2O |
SFE |
CFE-A |
SFE |
CFE-C |
SFE |
H2O |
H2O |
H2O |
SFE |
SFE |
SFE |
SFE |
SFE |
H2O |
H2O |
H2O |
SFE |
CFE-B |
SFE |
CFE-RR |
SFE |
H2O |
H2O |
H2O |
SFE |
SFE |
SFE |
SFE |
SFE |
H2O |
H2O |
H2O |
H2O |
SFE |
SFE |
H2O |
H2O |
H2O |
SFE = 18-plate standard fuel element
CFE = 9-plate control fuel element
Figure 2. FNR 27-Element Water-Reflected
LEU Core
4.4 Borated Stainless Steel Control Elements
Shortly after the conversion of the Ford Nuclear Reactor (FNR) from HEU to
LEU fuel, full-length rod worth measurements were made in the 27-element 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 shim-safety rods used
in this UAlx-Al core are described in Table 4. This table shows that
the FNR shim-safety 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 27-element core configuration using the MCNP Monte Carlo
code14, 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 ratios15.
Table 10 compares measured and calculated rod worths and includes the University
of Michigan results based on their two-group reaction rate matching process
described in Ref. 5. For all these calculations the rod worth was determined
by computing the reactivity difference between rod-in and rod-out cases with
the regulating rod (RR) fully withdrawn. The Monte Carlo and diffusion calculations
with IBC's are three-dimensional whereas the diffusion calculations with EDP's
are two-dimensional XY calculations. None of the calculations account for the
beam tubes associated with the D2O 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 shim-safety rods were replaced
with borated aluminum rods composed of an alloy of titanium diboride in 6351
aluminum. The boron in the TiB2 has a 10B enrichment >
95%. Table 4 compares the geometry and compositions of the TiB2(95%10B)-Al6351
shim-safety rods with the former borated stainless steel ones while Table 5
compares the internal boundary conditions. These IBC's were used in 3-dimensional
DIF3D calculations to determine the worths of the borated aluminum rods in the
27-element FNR core (Fig. 2). These rod-out and rod-in 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 SHIM-SAFETY ROD WORTHS FOR THE 27-ELEMENT FRESH UAlX 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 |
||||
UM-2DB: 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 |
||||
UM-2DB: 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 TiB2-Al SHIM-SAFETY ROD WORTHS IN THE FNR 27-ELEMENT CORE |
|||||||
Rod |
Withdrawala |
Code |
Eigenvalue |
Rod Worth % dk/k |
Ratio |
||
% |
(TiB2-Al) |
TiB2-Al |
B-SS |
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 |
bThe TiB2-Al/B-SS rod worth ratio was calculated for the 27-element core (Fig. 2). However, the measured ratio is based on rod calibrations made in the October 1995 core when the borated stainless steel shim-safety rods were replaced with TiB2(95%10B)-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 10B concentration (42%) of the TiB2-Al6351 shim-safety rods relative to the borated stainless steel ones (see Table 4). Although both shim rod materials are black to thermal neutrons (En<0.625 eV), the higher 10B concentration results in greater epithermal absorption in the TiB2-Al6351 (compare the group-3 IBC's in Table 5). Shim-safety rod worths for both materials were measured in the FNR October 1995 core16. Table 11 compares the measured TiB2-Al/B-SS worth ratios in this core with the values calculated for the 27-element 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.