An updated approach to calculation of diffusion
coefficients
E. Fridman, J. Leppänen
Text optional: Institutsname  Prof. Dr. Hans Mustermann  www.fzd.de  Mitglied der Leibniz-Gemeinschaft
Outline
• Diffusion coefficient from P1 equations
• Different forms of transport correction
• Numerical example
• Conclusions
Page 2
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Getting diffusion coefficient from P1 equations
• Multi-group P1 equation in 1D:
•
•
•
•
Page 3
d
1,g  t,g0,g 
dx

1 d
0,g  t,g1,g 
3 dx


 S0,g

 S1,g
s0,g' g 0,g'
g'
s1,g' g 1,g'
g'
ϕ0 and ϕ1 – 0th and 1st flux moments
Σ0 and Σ1 – 0th and 1st moments of scattering XS
Σt – total XS
S - sources
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Getting diffusion coefficient from P1 equations
• Multi-group P1 equation in 1D:
d
1,g  t,g0,g 
dx

1 d
0,g  t,g1,g 
3 dx


 S0,g

 S1,g
s0,g' g 0,g'
g'
s1,g' g 1,g'
g'
• Diffusion coeff. can be derived from the 2nd equation:
– Assuming isotropy of the sources  S1,g= 0
– Using Fick’s law:
Page 4
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Getting diffusion coefficient from P1 equations
• Multi-group P1 equation in 1D:
d
1,g  t,g0,g 
dx

1 d
0,g  t,g1,g 
3 dx


 S0,g

 S1,g
s0,g' g 0,g'
g'
s1,g' g 1,g'
g'
• Diffusion coeff. can be derived from the 2nd equation:
– Assuming isotropy of the sources  S1,g= 0
– Using Fick’s law:
Jg
 
1,g  
Page 5
D


3 t,g


d
0,g
dx
1
d
0,g
s1,g' g1,g'  dx


g'


1,g


E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Getting diffusion coefficient from P1 equations
• Multi-group P1 equation in 1D:
d
1,g  t,g0,g 
dx

1 d
0,g  t,g1,g 
3 dx


 S0,g

 S1,g
s0,g' g 0,g'
g'
s1,g' g 1,g'
g'
• Diffusion coeff. can be derived from the 2nd equation:
– Assuming isotropy of the sources  S1,g= 0
– Using Fick’s law:
Jg
 
1,g  
In-scatter Σtr,g
Page 6
D


3 t,g


d
0,g
dx
1
d
0,g
s1,g' g1,g'  dx


g'


1,g


E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Complexities in calculations of diffusion coefficient
t , g 


s1, g ' g 1, g '
g'
1, g
• Current spectra is needed - not easy to calculate
• What can be done? Examples:
–
–
–
–
Page 7
Out-scatter approximation
Replacing ϕ1 by ϕ0
Hydrogen transport correction
P1 spectral calculations
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Some relevant references:
• Yamamoto et al. “Simplified Treatments of Anisotropic
Scattering in LWR Core Calculations”, Journal of Nuclear
Science and Technology 45, 2008.
• Herman et al. “Improved Diffusion Coefficients Generated
From Monte Carlo Codes”, Proc. MC2013, Sun Valley, 2013.
• Choi et al., “Impact of inflow transport approximation on light
water reactor analysis,” Journal of Computational Physics
299, 2015.
Page 8
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Out-scatter approximation
Page 9
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Out-scatter approximation
• Additional assumption:
– P1 in-scatter and out-scatter sources are equal

g'
Page 10

s1,g' g 1,g'



s1,g  g' 1,g
g'
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Out-scatter approximation
• Additional assumption:
– P1 in-scatter and out-scatter sources are equal


s1,g' g 1,g'

g'


s1,g  g' 1,g
g'
• Then:


s1,g'g 1,g'
g'
1,g



s1,gg' 1,g
g'
1,g



s1,gg' 1,g
g'
1,g
   s1,gg'   s1,g
g'
• Typical out-scatter form of Σtr,g
– Knowledge of current spectra is not required
– Current Serpent approach
 tr ,g   t,g   s1,g   t,g  ˆ g s0,g
Page 11
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Out-scatter approximation
2.5
P1 Out-scatter/In-scatter
2
1.5
1
0.5 -3
10
•
•
-2
10
-1
10
0
10
1
2
10
Energy, eV
10
3
10
4
10
5
10
6
10
7
Out-scatter and in-scatter sources are not everywhere close (ratio is shown)
Difference in fast region can result in “too strong” transport correction
–
Page 12
10
Serpent results: UO2 PWR assembly, current spectra from 0-D P1 equation (modified B1)
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Hydrogen transport correction
Page 13
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Hydrogen transport correction
• Idea:
– Obtain H-transport correction curve: Σtr Σ
T
– From H-only slab with a fixed fission source
– Use the H-correction curve to modify Σtr from lattice calculations
• Assumption:
– H is a major source for scattering anisotropy (μ ≈ 2/3A)
• Procedure:
1. Calculate
2. Calculate
3. Calculate
 ,  ,
    Correction curve
     
tr ,all
tr ,H
T,H
corr
tr .H
T,H
corr
tr .all
corr
tr ,all
tr ,H
tr .H
• Described in details in MC2013 paper by Bryan Herman
– “Improved Diffusion Coefficients Generated From Monte Carlo Codes”
Page 14
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Hydrogen transport correction curve
1
Tr - out-scatter
Tr - H-corr.
0.9
0.8
Tr
 /
Tot
0.7
0.6
0.5
0.4
0.3
0.2 -10
10
10
-8
10
-6
-4
10
Energy, eV
10
-2
• Available in version 2.1.27
• H2O curve is shown
Page 15
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
10
0
10
2
Diffusion coefficient: Serpent vs. Casmo
Casmo
Page 16
Serpent
Out-scatter
H-correction
B1
g1
1.47534
1.55880
1.48000
1.44966
g2
0.42139
0.41320
0.43040
0.44533
g1
-
5.7%
0.3%
-1.7%
g2
-
-1.9%
2.1%
5.7%
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Some test problems
Page 17
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
2D PWR core
• Serpent - few-group XS + reference solution
• DYN3D - nodal diffusion calculations
• Verify DYN3D results vs. full core Serpent solution
Page 18
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
2D PWR core: Radial power distribution
Page 19
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
3D PWR core – OECD MOX benchmark
• Serpent - few-group XS + reference solution
• DYN3D - nodal diffusion calculations
• Verify DYN3D results vs. full core Serpent solution
Page 20
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
3D PWR core: Radial power distribution
Page 21
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Summary and future work
• H-transport correction was implemented in Serpent 2.1.27
• LWR diffusion coefficients are consistent with Casmo
• Somewhat improved nodal diffusion results…
• But some other factors should be accounted for
– Discontinuity factors
– Reflector models
– Leakage correction
• H-like correction can be used for other scatters
– deuterium, graphite, …
– importance (anisotropy) decreases with A (μ≈2/3A)
– further investigation is required
Page 22
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Thank you!
Page 23
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Generation of few-group diffusion coefficients
• Σtr,g can be used for the generation of DG in two ways:
Option 1:
Collapsing of Σtr,g
 tr ,G
 


gG
tr ,g g
gG
Page 24
g
 DG 
1
3 tr ,G
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
Generation of few-group diffusion coefficients
• Σtr,g can be used for the generation of DG in two ways:
Option 1:
Collapsing of Σtr,g
 tr ,G
 


gG
tr ,g g
gG
Page 25
g
1
 DG 
3 tr ,G
Option 2:
Collapsing of Dg
1
Dg 
 DG
3 tr ,g
E. Fridman | Serpent I UGM | Milan, 26-29.09.2016
D 


gG
gG
g g
g