Analysis of Reactor Static Characteristics
2008년 2학기
Project 2: One-D, P3 Code
Due Nov 25, 2008
The objectives of Part A of this project is to write a multigroup 1-D FDM P3 code and then examine the improvement attainable by the higher order angular flux treatment over the diffusion theory calculation.
Use your own 1-D multigroup FDM diffusion code as the base. The base code needs to be extended to solve the P3 equations which are given by the following simplified form:
i kjjjj j
Sr-Dx2SD0 -2Sr - 2Sr
3 -SD2Dx2+ 4Sr
3 + 5St
3
y {zzzz zi
kjjf`
0HxL f2HxL y {zz=i
kjjjj qHxL - 2qHxL
3
y {zzzz
(1)
where φ ξˆ ( )0 =φ ξ0( )+2 ( )φ ξ2 is the pseudo zeroth flux moment and the diffusion cross section is defined in terms of the diffusion coefficients and node width as the following:
SD0= D0
h2 ; SD2= D2
h2 , D0 = 1
3 Str ;D2 = 3
7 St , (2)
2 2
Dξ 2
ξ
= ∂
∂ is the second order derivative operator.
The modal currents are related by the following Fick’s law
JJ0HxL
J2HxLN=-DxJb0 0 0 b2Ni
kjjf` 0HxL f2HxL y {zz
(3)
where relative diffusivity is defined as:
b0 = D0
h ; b2 = D2
h (4)
while the modal currents at the vacuum boundary are related with the flux moments with the albedo matrix as:
Analysis of Reactor Static Characteristics
2008년 2학기
JJ0HxL J2HxLN=i
kjjjj j
1 2 - 3
8
- 1
8 7 8
y {zzzz zi
kjjf`
0HxL f2HxL y {zz
(5)
Equations (3) and (5) should be used to derive the finite difference approximation for the currents at the interior surface and the boundary surface, respectively. Follow the steps below to write a 1-D FDM code.
1. Modify the input processing routine to define the additional second order diffusion coefficient which requires the total cross section. Modify the linear system setup routine such that the 2x2 matrix appearing in Eq. (1) is used in the finite differenced nodal balance equation. Note that the off-diagonal terms which are required to discretize Eq. (3) become also a 2x2 diagonal matrix.
2. Write a LU factorization routine for a block tridiagonal matrix. The block size is 2x2.
Write also the forward and backward substitution routine to the block LU factors in solving the block tridiagonal linear system. Establish the outer iteration scheme with the solution obtained from the LU solver. Note that the solution for Eq. (1) is for the pseudo zero-th order flux so that the true zero-th order flux needs to be obtained by subtracting two times the second order flux from the pseudo flux. The true zeroth order flux should be used in the fission and scattering source determination.
3. Solve the 1-D C5G2 problem which consists of UOX(Composition 2), MOX(Composition 3), and Reflector(Composition 1) strip as shown in the following input.
3 3 ! nregions, ncomps
10.71 21.42 21.42 ! region width 2 3 1 !composition assignment 2 4 4 !number of nodes per region
0 2 !boundary condition, 0 for reflective, 2 for vacuum 1.e-6 1.e-5 !k and fission source convergence criterion 500 !max number of outers
DC Removal nu-fission Chi 1
1 1.200000E+00 5.100000E-02 0.000000E+00 1.000000E+00 2 2.000000E-01 4.000000E-02 0.000000E+00 0.000000E+00 1 2.267778E-01
2 5.000000E-02 1.626667E+00 2
1 1.200000E+00 2.965599E-02 4.569983E-03 1.000000E+00 2 4.000000E-01 9.265912E-02 1.135299E-01 0.000000E+00
Analysis of Reactor Static Characteristics
2008년 2학기
1 2.481218E-01 2 2.043002E-02 7.406742E-01 3
1 1.200000E+00 2.965462E-02 6.852401E-03 1.000000E+00 2 4.000000E-01 2.316396E-01 3.445001E-01 0.000000E+00 1 2.481232E-01
2 1.586347E-02 6.016937E-01
4. Examine the node size dependence of the solution by trying several refined mesh calculations. Estimate the true value of k-eff and regionwise fission source distribution.
Use the Mathematica program attached, which employs the P3 NEM, for verification of your results.
5. Compare the P3 results with the diffusion code result and discuss the reasons for the differences. Compare then the result with the Monte Carlo result that you can obtain by using your previously written Monte Carlo code. Use sufficient number of histories in the Monte Carlo calculation. Plot the angular flux shape as a polar diagram at the two region interfaces.
6. Prepare a self explanatory report about the method, coding and the results.
