Argonne National Laboratory
RERTR
Reduced Enrichment for Research and Test Reactors
Nuclear Engineering Division at Argonne
U.S. Department of Energy

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 extrapolation distance into the absorber,

ltr the neutron transport mean free path in the diffusing medium, and

A 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.

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Last modified on July 29, 2008 11:34 +0200