RERTR Publications:
Analysis Methods for Thermal Research and Test Reactors
ANL/RERTR/TM-29
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 mesh-dependent effective diffusion parameters can be used
to calculate reactivity effects of the absorber. An alternate approach is to
isolate the absorber from the diffusion-theory calculation by applying group-dependent
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 = ltr / 3 d
where d º extrapolation distance into the absorber,
ltr º the neutron transport mean free path in the diffusing medium, and
A º the internal boundary condition defined as the current-to-flux ratio at the
absorber surface.
Transport calculations are used to determine the group-dependent current-to-flux ratios. It is assumed that the material-dependent internal boundary condition is the same at every point on the surface of the absorber. However, current-to-flux ratios used in diffusion-theory calculations should never exceed the black absorber limit.
3.1 Current-to-Flux Ratios for Black Absorbers
For a plane black boundary (i.e. no return neutrons) the extrapolation distance
has the value d = 0.7104ltr which corresponds
to a current-to-flux ratio A = ltr / 3d
= 0.4692. The extrapolation distance for a black cylindrical rod is a function
of the radius, R, and the transport mean free path, ltr
, of neutrons in the surrounding medium. Figure 1 is a plot of the current-to-flux
ratio evaluated at the surface of a black cylindrical absorber of infinite length
as a function of R/ltr .These black-rod
internal boundary conditions were calculated from extrapolation distances given
in Ref. 6. Note that in the limit as R/ltr
approaches infinity, the current-to-flux 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 ai (R/l tr)i
where the coefficients are:
i |
ai |
|
0 |
2.49993E-01 |
|
1 |
2.50031E-01 |
|
2 |
-1.87744E-01 |
|
3 |
7.70505E-02 |
|
4 |
-1.56659E-02 |
|
5 |
1.00392E-03 |
|
6 |
1.17961E-04 |
|
7 |
-1.52381E-05 |
3.2 Current-to-Flux Ratios for Non-Black Absorbers
For non-black absorber materials the surface current-to-flux ratios are energy-dependent.
For this case the internal boundary conditions can be calculated from a discrete-ordinate
transport calculation using finely spaced mesh intervals in order to determine
group-dependent 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 Sa / Ss
<< 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,
fl = fr
and Jl = Jr. The surface current-to-flux ratio then equals
the a blackness coefficient discussed earlier.
3.3 Example
Until recently, the shim-safety 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 shim-safety
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 shim-safety 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. Full-density 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 WIMS-D4M code7. One-dimensional P1
S8 transport calculations were performed with the TWODANT code8
in order to determine group-dependent current-to-flux ratios on the surface
of the equivalent cylindrical rod. Table 5 summarizes the current-to-flux ratios
obtained in this manner. This table also shows the internal boundary conditions
calculated for the TiB2(95% 10B)-Al6351 shim rods now
used in the Ford Nuclear Reactor. Based on Sa
/ Ss, t / L,
and Sat 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 group-4 neutrons. Therefore,
the polynomial fit to Fig. 1 was used to determine the group-4 current-to-flux
ratios.
TABLE 4 COMPOSITION AND GEOMETRY OF THE FNR SHIM-SAFETY RODS |
|||
Composition (atoms/b-cm) |
|||
Nuclide |
Borated Stainless Steel |
TiB2(95%10B)-Al6351 |
|
10B |
1.108E-3 |
1.5749E-3 |
|
11B |
5.184-3 |
7.5390E-5 |
|
Mg |
4.0978E-4 |
||
Al |
5.7161E-2 |
||
Si |
5.9104E-4 |
||
Ti |
1.0158E-3 |
||
V |
1.6293E-5 |
||
Cr |
1.640E-2 |
1.5962E-5 |
|
Mn |
1.8129E-4 |
||
Fe |
5.644E-2 |
1.4862E-4 |
|
Ni |
1.130E-2 |
1.4131E-5 |
|
Cu |
2.6122E-5 |
||
Zn |
5.0734E-5 |
||
Geometry |
|||
Dimensions (cm) |
Borated SS |
TiB2-Al6351 |
|
Effective Length |
60.960 |
60.960 |
|
Rectangular-Like 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 SHIM-SAFETY RODS |
||||
Borated Stainless Steel Rods |
||||
Quantity |
Group 1 |
Group 2 |
Group 3 |
Group 4 |
Eu(eV) |
1.0000E+07 |
8.2100E+05 |
5.5300E+03 |
6.2500E-01 |
Sa/Ss |
6.8727E-03 |
6.4574E-03 |
1.5990E-01 |
2.6788E+00 |
D(cm) |
1.6583E+00 |
8.8032E-01 |
3.1743E-01 |
9.5010E-02 |
Sa(cm-1) |
1.8958E-03 |
3.3825E-03 |
1.4756E-01 |
2.5676E+00 |
L(cm) |
2.9576E+01 |
1.6132E+01 |
1.4667E+00 |
1.9236E-01 |
t/L |
7.4318E-02 |
1.3625E-01 |
1.4986E+00 |
1.1426E+01 |
Sat |
4.1669E-03 |
7.4347E-03 |
3.2434E-01 |
5.6436E+00 |
IBC = J/f |
2.5331E-02 |
-1.1123E-02 |
8.1122E-02 |
4.0956E-01 |
TiB2(95%10B)-Al6351 Rods |
||||
Sa/Ss |
3.6927E-03 |
1.4154E-02 |
1.7049E+00 |
3.7923E+01 |
D(cm) |
3.1275E+00 |
1.6415E+00 |
1.1959E+00 |
9.3876E-02 |
Sa(cm-1) |
6.6305E-04 |
3.6092E-03 |
1.7942E-01 |
3.4624E+00 |
L(cm) |
6.8680E+01 |
2.1326E+01 |
2.5818E+00 |
1.6466E-01 |
t/L |
3.2360E-02 |
1.0422E-01 |
8.6082E-01 |
1.3498E+01 |
Sat |
1.4736E-03 |
8.0214E-03 |
3.9876E-01 |
7.6952E+00 |
IBC = J/f |
1.3262E-02 |
-6.9260E-03 |
1.1656E-01 |
4.1097E-01 |
NOTE: These tables show that:
1. IBC's are not needed for groups 1 and 2 since Sa/Ss
<<1.
2. IBC's are needed for groups 3 and 4.
3. IBC for group 4 is that of a black rod since Sat >>1.