Unstable 3D pin-cell burnup problem

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In certain burnup problems such as long, axially symmetric 3D assemblies or fuel rods or large symmetric core geometries traditional Monte Carlo burnup schemes may run into instabilities[1][2]. This page describes the simulation of one such case first using the traditional burnup schemes (to showcase the instability) and then using a stable burnup scheme.

Problem description

Base input for unstable 3D pin-cell problem

 ** Run options **

% --- 50k neutrons per cycle, 100 inactive cycles

set pop 20000 1000 200 

% --- 200 W/cm linear power

set power 60000

 ** Geometry definition **

% --- Fuel Pin definitions:

pin p1
fuel  0.47
void  0.48
clad  0.54

% --- Lattice 

lat l1  1  0.0 0.0 1 1 1.5

% --- Surrounding surfaces:

% Boundary of geometry:

surf 3 cuboid -0.75 0.75 -0.75 0.75 -160 160

% Lower boundary of fuel

surf 4 pz -150

% Upper boundary of fuel

surf 5 pz  150

% --- Cell definitions:

% Active fuel pin 

cell  3  0  fill l1  -3  4 -5

% Coolant below active fuel (bottom reflector)

cell  4  0  cool     -3 -4

% Coolant above active fuel (top reflector)

cell  5  0  cool     -3  5

% outside world 

cell 99  0  outside   3     % Outside world

% --- Reflective boundary conditions in XY, black in Z:

set bc 3 3 1

 ** Material definitions **

% --- Fuel material (4.85 % enrichment):

mat fuel    -10.283 vol 208.19 rgb 200 200 125
92235.09c      0.016166667
92238.09c      0.317166667
8016.09c       0.666666667

% --- Cladding (Zr-4)

mat clad -6.56000E+00 rgb 180 180 180
 8016.06c     -1.19276E-03
 8017.06c     -4.82878E-07
24050.06c     -4.16117E-05
24052.06c     -8.34483E-04
24053.06c     -9.64457E-05
24054.06c     -2.44600E-05
26054.06c     -1.12572E-04
26056.06c     -1.83252E-03
26057.06c     -4.30778E-05
26058.06c     -5.83334E-06
40090.06c     -4.97862E-01
40091.06c     -1.09780E-01
40092.06c     -1.69646E-01
40094.06c     -1.75665E-01
40096.06c     -2.89038E-02
50112.06c     -1.27604E-04
50114.06c     -8.83732E-05
50115.06c     -4.59255E-05
50116.06c     -1.98105E-03
50117.06c     -1.05543E-03
50118.06c     -3.35688E-03
50119.06c     -1.20069E-03
50120.06c     -4.59220E-03
50122.06c     -6.63497E-04
50124.06c     -8.43355E-04

% --- Coolant:

mat cool     -0.75 moder lwtr 1001 rgb 50 50 255
 1001.06c    0.666666667
 8016.06c    0.333333333

% --- Thermal scattering data for light water:

therm lwtr lwj3.11t

The base input for the problem is given above. The input describes a 300 cm long fuel rod in infinite lattice geometry. Axially the fuel rod is reflected from top and bottom with 10 cm water layers after which a black boundary condition is applied. The radial geometry is shown here:

xy-plot of the pin-cell geometry.

Let's say that we want to calculate the axial power distribution and flux distribution using 100 axial bins over the active fuel length at burnups between 0 MWd/kgU and 20 Mwd/kgU. In order to capture the axial burnup distribution we will divide the fuel material axially into a rather small number of division zones (10) using the div card.

Unstable solution using explicit Euler's scheme

Yz-meshplot of the fission heat deposition and thermal flux as a function of burnup (explicit Euler's scheme). The instability of the burnup algorithm is clearly visible.

We'll first run the calculation using the explicit Euler's scheme for the discretization of the Bateman equations. This scheme is equal to using the constant values for the beginning-of-step (BOS) flux and cross sections to burn the materials through each of the burnup steps. Explicit Euler's scheme can be chosen with

set pcc ce

Where "ce" refers to "constant extrapolation".

Unstable solution using predictor-corrector scheme

Stable solution using SIE burnup scheme

Stable solution using a symmetry boundary


  1. ^ Dufek, J. and Hoogenboom, E. "Numerical Stability of Existing Monte Carlo Burnup Codes in Cycle Calculations of Critical Reactors", Nucl. Sci. Eng., 162 (2009) 307-311
  2. ^ Dufek, J. et al. "Numerical stability of the predictor–corrector method in Monte Carlo burnup calculations of critical reactors", Ann. Nucl. Energy, 56 (2013) 34-38