DIFIS 2.0 – 3D FINITE ELEMENT NEUTRON KINETIC CODE
A.I.Zhukov and A.M.Abdullayev
NSC Kharkov Institute of Physics and Technology
1 Akademicheskaya, Kharkov 61108 Ukraine
ABSTRACT
The paper presents new 3-dimensional neutron kinetics code for VVER type core. DiFis 2.0
is an extension of previous steady-state version of the code DiFis for transition processes.
Nuclear model includes time-dependent diffusion equations in two-group approximation. The
code takes into account six delayed group neutron precursors. Cross-sections with all
feedbacks (including Doppler, moderator density, etc) are pre-calculated by a well-known
Westinghouse code PHOENIX-H. Finite-element technology for formulation of problem and
its solution is used. Normally the code uses 24 three-angles finite elements per fuel assembly
cross-section and 26 axial meshes. The code can be run with a coarse mesh – with 6 finite
elements and fine mesh – with 54 finite elements per assembly.
Thermal-hydraulic part is based on an “average” fuel rod model. Each fuel assembly contains
such a rod, which provides heat transfer from a fuel pellet through gap and cladding into
coolant. 14 radial meshes and 26 axial ones are normally used for each “average” fuel rod.
Besides “average” fuel rods, a probe “hot” fuel rod – one per fuel assembly – is used.
The code provides outputs of nuclear power versus time as well as spatial temperature
distribution in fuel, cladding and coolant. DiFis 2.0 can be used for control rod ejection
accident, and other reactivity insertion transients.
INTRODUCTION
Existing 1D neutron kinetic codes are too conservative. 3D codes are necessary for realistic
analysis of the reactivity insertion accidents.
DiFis 2.0 use PHOENIX-H code [1] for generating nuclear cross-sections with feedback
corrections.
DiFis 2.0 based on a previous version DiFis 1.0 and applies Finite Element Method (FEM) for
solution of neutron kinetic equations. FEM is widely used in different branches of Physics
and Engineering. A lot of well-known codes for Stress and Strain Analysis, Fluid Dynamics,
Hear t Transfer, Electromagnetic Analysis. Modern Nodal Method used in nuclear
calculations can be treated as kind of FEM.
NUCLEAR MODEL
The following well-known equations describe time-dependent diffusion of neutrons in twogroup approximation:
1
 D11    a1   r 1  Q1
v1t
 2
 D2 2    a 2  2  Q2
v2 t
Here indexes g = 1, 2 denote the fast and thermal group correspondingly. The source of the
fast neutrons


6
Q1  1   1 f 11   2  f 2  2  I   i Ci
i 1
consist of two terms: the first one is a source of prompt neutrons and the second one describes
the generating of six groups of delayed neutrons.
Equations for precursors concentration are


dCi
 i 1 f 11   2  f 2  2   i Ci , i = 1,…,6
dt
The source of the thermal neutrons
Q2   r 1
Albedo boundary conditions are
Dg  g  n   g  g  J g  0
THERMAL-HYDRAULICS MODEL
Equations for heat transfer (enthalpy rise) are [2]

h
 div j  q
t
- for fuel, gap, and cladding, and

h
h p
G

 q  divj
t
z t
- for coolant.
FEEDBACK MODELS
PHOENIX provides with feedback corrections:
1. Moderator density correction includes:
a. Diffusion coefficient correction
b. Moderator absorption correction
c. Boron concentration correction
d. Spectrum correction
2. Doppler correction – because of changing fuel temperature
3. Xe, Sm, Pm correction
4. Control Rod corrections
FINITE ELEMENT TECHNIQUE
Let us expand neutron fluxes into series
x, y, z     j t   F j x, y, z 
j
Where
F j x, y, z    j x, y    j z  ,
 j x, y  is a set of simplex-functions,  j z  - is a set of linear functions. After applying
weighted residual method
 

 1

  v 1t  D11   a1  r 1  F j dV   Q1  F j dV
one can obtain matrix equation for flux values in mesh points:
 1k  Lˆ jk 1k  Sˆ jk Q1k
Tˆ jk 
Typical shape of a Finite Element (FE) is depicted in Fig. 1.
Meshes available for Nuclear Model in DiFis 2.0 are shown in Fig. 2. Mesh with 24 triangles
per FA was used for presented examples of calculation.
Radial mesh for heat transfer is depicted in Fig. 2.
DiFis 2.0 uses 24 axial FEs.
6
5
4
3
2
1
Fig. 1 Shape of a Finite Element.
12
34
11
36
2
14
1
13
10
9
38
39
15
3
2
1
37
12
14
33
35
32
31
10
9
11
13
30
29
8
27
24
25
28
8
40
15
42
17
44
45
2
1
4
5
20
21
48
49
6
16
3
17
6
4
7
5
41
24
16
43
4
5
18
19
22
20
21
23
18
46
3
6
19
22
47
7
26
23
52
50
54
53
51
Fig. 2 Types of meshes are used in the code.
Fig. 3 Radial mesh for heat transfer. Normally DiFis 2.0 uses 7 radial zones for fuel pellet, 2
zone for gap (if exists), and 5 zones for cladding.
EXAMPLE OF STEADY-STATE CALCULATIONS
Before starting with transient the code finds steady-state fluxes distribution. Fig.4 shows fuel
assembly (FA) average power for typical loading pattern VVER-100 calculated by ANC-H
code and DiFis 2.0
Fig. 4 FA average power calculated by ANC-H and difference between ANC-H and DiFis
2.0. Maximum difference = 4%, rms = 1.6%.
EXAMPLE OF TRANSIENT CALCULATIONS
DiFis 2.0 can calculate detailed power and enthalpy/temperature spatial and time distribution
at any stage of reactivity insertion accidents. Example of analysis for Control Rod ejected
accident is given below.
There was calculated a steady-state fluxes for a case when Control Bank was completely
inserted. Then Rod Control Cluster Assembly (RCCA) in FA # 85 was ejected. Calculated
Rod worth is 0.176%. Two cases were analyzed. Typical sequence of events were
1. Case 1 (without scram): 0 s – rod ejection starts, 0.1 s RCCA gets the top of the core
2. Case 2 (scram): 0 s – rod ejection starts, 0.1 s RCCA gets the top of the core, 0.4 s –
all banks start movement to the bottom of the core, 3.0 s – all banks get the bottom of
the core.
Core power vs. time for this accident is shown on Fig. 5.
1,4
1,2
Core Power
1,0
without scram
scram
0,8
0,6
0,4
0,2
0,0
0
0,5
1
1,5
2
2,5
3
3,5
time, s
Fig. 5 Control Rod ejection accident. Core Power vs. time for Cases 1and 2.
Control Rod worth = 0.176%.
The FA # 85, which is located under ejected Control Rod, was chosen for demonstration.
Figs. 6 and 7 show power and fuel temperature distribution at different time moment for
Case 1. Figs. 8 and 9 show the similar distribution for Case2.
CONCLUSION
Presented code is compatible with the well-known codes such as PHOENIX-H and ANC-H.
DiFis 2.0 provides accuracy ~4% (in compare with ANC) for steady-state calculations.
The code is capable to calculate spatial and time power and enthalpy/temperature distribution
in the Core.
DiFis 2.0 justification and benchmarking will be continued.
2.5
1.00 s
0.40 s
2.0
0.12 s
0.05 s
node power
0.00 s
1.5
1.0
0.5
0.0
1
3
5
7
9
11
13
15
17
19
21
23
# node
Fig. 6 Case 1. Relative linear power (node power) vs. axial location for FA under ejected
Control Rod at different stage of the accident.
1200
2.00 s
1.00 s
0.40 s
0.12 s
0.05 s
0.00 s
Average Fuel Temperature, C
1100
1000
900
800
700
600
500
400
1
3
5
7
9
11
13
15
17
19
21
23
# node
Fig. 7 Case 1. Fuel temperature averaged over pellet radius vs. axial location for FA under
ejected Control Rod at different stage of the accident.
4.0
2.00 s
1.00 s
0.40 s
0.12 s
0.05 s
0.00 s
3.5
node power
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1
3
5
7
9
11
13
15
17
19
21
23
# node
Fig. 8 Case 2. Relative linear power (node power) vs. axial location for FA under ejected
Control Rod at different stage of the accident.
1200
2.00 s
1.00 s
0.40 s
0.12 s
0.05 s
0.00 s
Average Fuel Temperature, C
1100
1000
900
800
700
600
500
400
1
3
5
7
9
11
13
15
17
19
21
23
# node
Fig. 9 Case 2. Fuel temperature averaged over pellet radius vs. axial location for FA under
ejected Control Rod at different stage of the accident.
LIST OF NOMENCLATURE
Dg
- concentration of precursors of group i
- diffusion coefficients
G
h
I
Jg
- mass velocity
- enthalpy density (enthalpy per unit mass)
- efficiency of delayed neutrons
- neutron currents for boundary conditions: J1  0 , J 2  12 2
j
q
p
Qg
- heat flux
- heat source
- pressure
- neutron sources
vg
- neutron velocities

1 i
- albedo coefficient
- life time for precursors of group i
Ci
6
  I  i
i
g
- effective fraction of delayed neutrons
i 1
- fraction of delayed neutrons for precursors of group i
- scalar fluxes
 g  fg - macroscopic -fission cross-sections
 g  fg - macroscopic -fission cross-sections

 ag
- water density
- macroscopic absorption cross-sections
r
- macroscopic removal cross-section
REFERENCES
1. Rudi J.J. Stamm'ler, Maximo J. Abbate. "Methods of Steady-State Reactor Physics in
Nuclear Design". 1983, Academic Press
2. L.S.Tong, Joel Weisman. "Thermal Analysis of Pressurized Water Reactors." Third
edition. 1996, American Nuclear Society, La Grande Park, Illinois, USA