use of the fission matrix method for solution of the eigenvalue

Virginia Tech Nuclear Science and Engineering Lab (NSEL) at Arlington

USE OF THE FISSION MATRIX METHOD FOR SOLUTION OF THE EIGENVALUE

PROBLEM IN A SPENT FUEL POOL

William Walters, Nathan Roskoff, Katherine Royston, and AlirezaHaghighat

Virginia Tech

Nuclear Science and Engineering Lab (NSEL) at Arlington

Nuclear Engineering Program, Mechanical Eng. Dept.

Arlington, VA


Outline

• Introduction

• Background• Fission Matrix Method• Reference Spent Fuel Pool

•Methodology

• Results• Full Monte Carlo• Fission Matrix

• Conclusions and Future Work


Introduction

• Spent fuel pool neutronics calculations are important• Criticality safety• Safeguards and verification

• Full Monte Carlo calculations face difficulties in this area• Convergence is difficult to low coupling between regions (due to

absobers)• Convergence can also be difficult to detect

• Computation times are very long, especially to get detailed information• Changing pool configuration requires complete recalculation

• Fission Matrix method can address some of these issues• Fission matrix coefficients are pre-calculated using Monte Carlo• Computation times are much shorter, with no convergence issues• Detailed fission distributions are obtained at pin level• Changing pool assembly configuration does not require new pre-

calculations• No additional Monte Carlo


Fission Matrix (FM) Method• Eigenvalue formulation

𝐹𝑖 =1

𝑘 𝑗=1𝑁 𝑎𝑖,𝑗𝐹𝑗

• k is eigenvalue• 𝐹𝑗 is fission source, 𝑆𝑗 is fixed source in cell j• 𝑎𝑖,𝑗 is the number of fission neutrons produced in cell 𝑖 due to a fission neutron

born in cell 𝑗.

• Subcritical multiplication formulation

𝐹𝑖 =

𝑗=1

𝑁

(𝑎𝑖,𝑗𝐹𝑗 + 𝑏𝑖,𝑗𝑆𝑗) ,

• 𝑏𝑖,𝑗 is the number of fission neutrons produced in cell 𝑖 due to a source neutron born in cell 𝑗.

4


Fission Matrix Formulation

• The main task

Determination of coefficients 𝑎𝑖,𝑗 and 𝑏𝑖,𝑗, especially at the pin-level.

We reduced the number of calculations by considering:• Geometric similarity• Geometric symmetry• Degree of coupling

• Sensitivity of the coefficients to different parameters

5


Reference Spent Fuel Pool

• Being developed for Integral Inherently Safe LWR (I2S-LWR)* reactor design

• 19x19 𝑈3𝑆𝑖2 fuel assemblies

• 4.45 w/o and 4.95 w/o U-235

• Metamic® absorbers (Al-B4C) used between assemblies

• Considering fresh fuel for now

One 19x19 Assembly9x9 segment of spent

fuel pool*Led by Georgia Tech, funded under a DOE-IRP program


Methodology

• As the computational cell size, use a single pin• 𝑁 = 9 × 9 × 336 = 27,216 total fuel pins / fission matrix cells

• Allows for good accuracy and pin-resolved fission rates

• Standard FM would require 27,216 separate fixed-source calculations to determine FM coefficients

• The standard approach is clearly NOT feasible


Fission Matrix Coefficients

• For this work, we performed 49 fixed-source MCNP5 calculation separately for each pin in one octant of an assembly in the middle of the pool• Uniform source radially

• Cosine shape axially

• U-235 fission spectrum

• We use the similar coefficients considering symmetries for the rest of pins in an assembly as follows


Fission Matrix Coefficients for one assembly

Source in single pin Neutron production tallied in all cells (results are x 100)

Repeat for all 49 source pins in octant:


9x9 array of assemblies – Using geometric similarity of coefficients

• Previous calculation only gives 49 rows of the fission matrix

• Obtain the rest using octal symmetry and geometric similarity

• Coefficients with red arrows are identical, as are those with blue arrows


Solution of Fission Matrix Equations• Once the 49 base sets of fission matrix coefficients have been

calculated, the full set of coefficients is obtained using the geometric considerations for any pool configuration.

•𝑁 = 336 ∗ 𝑁𝑎𝑠𝑠𝑒𝑚𝑏𝑙𝑖𝑒𝑠 total equations

• Solve for k and fission density (fundamental eigenfuction) using a power iteration approach is used as follows

𝐹𝑖(0)=1

𝑁

𝐹𝑖(𝑚+1)=1

𝑘(𝑚)

𝑗=1

𝑁

𝑎𝑖,𝑗𝐹𝑗(𝑚)

Where 𝑘(𝑚) = 𝑖=1𝑁 𝐹𝑖

(𝑚)


Test Problems


Fission Matrix vs Full MCNP

*total pre-calculation time for both materials and used for all cases: 4707 min


Case 3 Eigenfunction

Total k difference: -143 pcm

0,00E+00

2,00E-02

4,00E-02

6,00E-02

8,00E-02

1,00E-01

1,20E-01

1,40E-01

1,60E-01

1,80E-01

0 2 4 6 8 10

Fiss

ion

So

urc

e

Assembly number

MCNP FM

Comparison of FM with MC

Reference Solution


Case 11 Eigenfunction


0,00E+00

2,00E-02

4,00E-02

6,00E-02

8,00E-02

1,00E-01

1,20E-01

1,40E-01

1,60E-01

1,80E-01

0 2 4 6 8 10

Fiss

ion

So

urc

e

Assembly number

MCNP FM

Comparison with FM with MC

Reference Solution


Case 4 Eigenfunction distribution

*MCNP Uncertainties <1%


Comparison with FM with MC

Reference Solution

0,00E+00

2,00E-02

4,00E-02

6,00E-02

8,00E-02

1,00E-01

1,20E-01

1,40E-01

1,60E-01

1 2 3 4 5 6 7 8 9

Fiss

ion

So

urc

e

Pool Assembly Column Number

Eigenfunction (sum of rows)

MCNP FM


Conclusions

• The fission matrix method can quickly and accurately calculate the eigenvalue and eigenfunction for a spent fuel pool

• Relatively few pre-calculations, using geometric considerations (reflection, etc)

• Changes in assembly positions or adding assemblies does not require any new Monte Carlo calculations


Ongoing & Future Work

• Filing for a patent for an automated tool for safety and security of spent fuel pool

•Use of spent fuel materials instead of fresh fuel

•Better treatment of axial variable

•Develop a database of coefficients for all possible burnup and cooling times

•Calculate subcritical multiplication factor and detector response

use of the fission matrix method for solution of the eigenvalue

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