Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations

3.19

Oxide Fuel Performance Modeling and Simulations

P. Van Uffelen
European Commission, Joint Research Centre, Institute for Transuranium Elements, Eggenstein-Leopoldshafen, Germany

M. Suzuki
Japan Atomic Energy Agency, Tokai-mura, Ibaraki, Japan

ß 2012 Elsevier Ltd. All rights reserved.

3.19.1
3.19.1.1
3.19.1.2
3.19.1.3
3.19.2
3.19.2.1
3.19.2.1.1
3.19.2.1.2
3.19.2.1.3
3.19.2.1.4
3.19.2.1.5
3.19.2.2
3.19.2.2.1
3.19.2.2.2
3.19.2.2.3
3.19.2.2.4
3.19.2.3
3.19.2.3.1
3.19.2.3.2

3.19.3
3.19.3.1
3.19.3.1.1
3.19.3.1.2
3.19.3.2
3.19.3.2.1
3.19.3.2.2
3.19.4
3.19.4.1
3.19.4.2
3.19.4.3
3.19.4.4
3.19.5
References

Introduction
Importance of Fuel Performance Modeling
Geometrical Idealization and Size of the Problem
Uncertainties and Limitations
Basic Equations and State of the Art
Heat Transfer
Axial heat transfer in the coolant
Heat transport through the cladding
Heat transport from cladding to the fuel pellet
Heat transport in fuel pellets
The structure of the thermal analysis
Mechanical Analysis
Main assumptions and equations
Calculation of strains
Boundary conditions

Pellet–cladding interaction
Fission Gas Behavior
Basic mechanisms
Modeling the fission gas behavior
Design Basis Accident Modeling
Loss-of-Coolant Accident
Specific LOCA features
Specific LOCA modeling requirements
Reactivity-Initiated Accidents
Specific RIA features
Specific RIA modeling requirements
Advanced Issues and Future Needs
Deterministic Versus Probabilistic Analyses
The High Burnup Structure
Mixed Oxide Fuels
Multiscale Modeling
Summary and Conclusions

Abbreviations
bcc
BOL
BWR
CANDU
(C)SED
CSR

Body-centered cubic
Beginning of life
Boiling water reactor
CANada Deuterium Uranium

(Critical) strain energy density
Volatile fission product release

CZP
DFT
DNB
ECCS
ECR
EOL
EPMA
fcc

536
536
536
537
538
538
538
539
539
540
541
541
541
542
546
547
551
551

554
557
557
557
558
561
561
562
564
564
566
568
570
572
574

Cold zero power
Density functional theory
Departure from nucleate boiling
Emergency core cooling systems
Equivalent cladding reacted
End of life
Electron microprobe analysis
Face-centered cubic

535


536

Oxide Fuel Performance Modeling and Simulations

FDM
FEM
FGR
HBS
HCP
HM
HZP
IAEA
IFA
LEFM
LHR
LOCA
LWR
MC
MD
MIMAS
MOX
NEA
O/M
OECD
PAS
PCI
PCMI
PWR
RIA
S/V
SANS
SCC

SM
TAD
TEM

Finite difference method
Finite element method
Fission gas release
High-burnup structure
Hexagonal closed packed
Heavy Metals
Hot zero power
International Atomic Energy Agency
Instrumented fuel assembly
Linear elastic fracture mechanics
Linear heating rate
Loss-of-coolant accident
Light water reactor
Monte Carlo
Molecular dynamics
Micronized master blend
Mixed oxide
Nuclear Energy Agency
Oxygen-to-metal ratio
Organisation for Economic Cooperation
and Development
Positron annihilation spectroscopy
Pellet–cladding interaction
Pellet–cladding mechanical interaction
Pressurized water reactor
Reactivity-initiated accident

Surface-to-volume ratio
Small-angle neutron scattering
Stress corrosion cracking
Shell model
Temperature-accelerated dynamics
Transmission electron microscopy

3.19.1 Introduction
3.19.1.1 Importance of Fuel Performance
Modeling
In order to ensure the safe and economical operation
of fuel rods, it is necessary to be able to predict their
behavior and lifetime. The accurate description of
the fuel rod’s behavior, however, involves various
disciplines ranging from the chemistry, nuclear and
solid-state physics, metallurgy, ceramics, and applied
mechanics. The strong interrelationship between
these disciplines, as well as the nonlinearity of many
processes involved, calls for the development of computer codes describing the general fuel behavior. Fuel
designers and safety authorities rely heavily on these
types of codes since they involve minimal costs in
comparison with the costs of an experiment or an
unexpected fuel rod failure. The codes are being

used for R&D purposes, for the design of fuel rods,
new products, or modified fuel cycles, and for supporting loading of fuel into a power reactor, that is,
to verify compliance with safety criteria in safety case
submissions. A list of commonly used fuel performance codes is provided in Table 1.
3.19.1.2 Geometrical Idealization and
Size of the Problem

In principle, our spatial problem is three-dimensional
(3D). However, the geometry of a cylindrical fuel rod (a
very long, very thin rod) suggests that any section of a
fuel rod may be considered as part of an infinite body:
that is, neglecting axial variations. By further assuming
axially symmetric conditions because of the cylindrical
geometry, the original 3D problem is reduced to a 1D
one. Analyzing the fuel rod at several axial sections with
a (radially) 1D description is sometimes referred to as
quasi-2D or-11=2D. Most fuel rod performance codes
fall into this category. Real 2D codes such as, for
instance, the FALCON code,1 offer the possibility to
analyze r–z problems (no azimuthal variation) and r–’
problems (no variation in axial direction). An example of
a 3D code is TOUTATIS2 and DRACCAR3 and is dealt
with in Chapter 3.22, Modeling of Pellet–Cladding
Interaction. DRACCAR is addressed later in Section
3.19.3.1.2. Generally, 2D or 3D codes are used for the
analysis of local effects, whereas the other codes have
the capability to analyze the whole fuel rod during a
complicated, long power history.
In order to estimate the ‘size’ of the problem at
hand, the number of time steps must also be specified.
For a normal irradiation under base load operation, that
is, under no-load follow operation, $100–500 time steps
are sufficient. However, for an irradiation in a research
reactor, such as the heavy-water boiling water reactor
(BWR) of the Organisation for Economic Cooperation
and Development (OECD), Halden, many more variations of the linear rating with time are recorded. In such
a situation, one must either simplify the complicated

power history or increase the number of time steps to
the order of several thousands. The simplest geometrical
idealization needs $20 radial and 20 axial nodes; a 2D
representation of a single pellet would approximately
need several hundred nodes. Therefore, local models,
which are in almost all cases nonlinear, must be very
carefully constructed, since even for the simplest geometrical idealization the number of calls may easily
reach the order of millions:
15 radial  15 axial nodes  5000 time steps  3 iterations
¼ 3:4 Â 106 calls


Oxide Fuel Performance Modeling and Simulations

Table 1

537

List of fuel performance codes

COMETHE
COPERNICa
ENIGMA

FALCON, FREY

FEMAXI
FRAPCON

METEORa

PIN-micro
START

TRANSURANUS

Hoppe, N.; Billaux, M.; van Vliet, J.; Shihab, S. COMETHE version 4D release 021 (4.4-021), Vol. 1,
general description; Belgonucleaire Report, BN-9409844/220 A; Apr 1995
Bonnaud, E.; Bernard, C.; Van Schel, E. Trans. Am. Nucl. Soc. 1997, 77
Kilgour, W. J.; Turnbull, J. A.; White, R. J.; Bull, A. J.; Jackson, P. A.; Palmer, I. D. Capabilities and
validation of the ENIGMA fuel performance code. In Proceedings of the ENS Meeting on LWR Fuel
Performance, Avignon, France, 1992
Rashid, J.; Montgomery, R.; Yagnik, S.; Yang, R. Behavioral modeling of LWR fuel as represented in the
FALCON code. In Proceedings of the Workshop on Materials Modelling and Simulations for Nuclear
Fuel, New Orleans, LA, Nov 2003
Suzuki, M.; Saitou, H. Light Water Reactor Fuel Analysis Code FEMAXI-6 (Ver. 1); JAEA-Data/Code
2005-003, Feb 2006
Berna, G. A; Beyer, C. E.; Davis, K. L.; Lanning, D. D. FRAPCON-3: A computer code for the calculation of
steady-state, thermal-mechanical behaviour of oxide fuel rods for high burnup; NUREG/CR-6534,
PNNL-11513; Dec 1997
Struzik, C.; Moyen, M.; Piron, J. High burnup modelling of UO2 and MOX fuel with METEOR/
TRANSURANUS Version 1.5. In Proceedings of the International Topical Meeting on LWR Fuel
Performance, Portland, OR, Mar 1997
Pazdera, F.; Strijov, P.; Valach, M.; et al. User’s guides for the computer code PIN-micro; UJV 9512-T,
Rez; Nov 1991
Bibilashvili, Y. K.; Medvedev, A. V.; Khostov, G. A.; Bogatyr, S. M.; Korystine, L. V. Development of
the fission gas behaviour model in the START-3 code and its experimental support. In Proceedings of the
International Seminar on Fission Gas Behaviour in Water Reactor Fuels, Cadarache, France, Sept 2000
Lassmann, K. The TRANSURANUS code – past, present and future; Review article, ITU Activity Report
2001 – EUR 20252, ISBN 92-894-3639-5; 2001

a

Based on TRANSURANUS.

3.19.1.3

Uncertainties and Limitations

In general, the uncertainties to be considered may be
grouped into four categories. The first category deals
with the prescribed or input quantities for the fuel
rod performance code: fuel fabrication parameters
(rod geometry, composition, etc.), which are often
available with an acceptable precision and are subject
to specification limits. The second category covers
irradiation parameters (reactor type, coolant conditions, irradiation history, etc.). Although they contain
a certain level of uncertainty, they can be properly
managed in actual analyses. The third category of

Deformed geometry

Nondeformed
geometry
(fresh fuel)
Fuel pellet

One-dimensional
description

Cladding

Even with the computer power of today, a full
3D analysis of, for instance, a simulation of a complex
irradiation history in an experimental reactor is
practically impossible with deterministic models. In
some cases, it is possible, but in a limited part of a
rod, such as a certain fraction of the axial length or of
the azimuth angle of a rod. In addition, such an
analysis is limited by the fact that the shape and
positions of the fuel fragments are determined by
a stochastic process. Nevertheless, attempts toward
3D analysis tools exist, such as the simplified 3D
model DRACCAR which is useful in predicting
the assembly-wise behavior during a loss-of-coolant
accident (LOCA).

Two-dimensional
description

Figure 1 Schematic view of a deformed fuel pellet;
comparison between a one-dimensional and a
two-dimensional description.

uncertainties is related to the material properties,
such as the fuel thermal conductivity or the fission
gas diffusion coefficients. The fourth and last category of uncertainties is the so-called model uncertainties. A good example of such an uncertainty is
the plain strain assumption in the axial direction as
illustrated in Figure 1, representing the interaction
of the deformed and cracked fuel with the cladding.

Intuitively, it is clear that for a detailed analysis


538

Oxide Fuel Performance Modeling and Simulations

of such problems, 2D or even 3D models are
indispensable.
One of the most important consequences of all
uncertainties is that one must implement models of
‘adequate’ complexity.

3.19.2 Basic Equations and
State of the Art
3.19.2.1

Heat Transfer

The objective of this section is to describe how the
temperature distribution in a nuclear fuel rod is calculated in a fuel rod performance code. The scope is
limited to a description of the important physical
phenomena, along with the basic equations and the
main assumptions. Detailed numerical aspects as
well as mathematical derivations are provided in
some reference works.4–6
The temperature distribution in a fuel rod is
of primary importance for several reasons. First
of all, the commercial oxide fuels have poor
thermal conductivities, resulting in high temperatures even at modest power ratings. Second, the

codes are used for safety cases where one has
to show that no fuel melting will occur, or that
the internal pressure in the rod will remain below
a certain limit. Finally, many other properties
and mechanisms are exponentially dependent on
temperature.
The most important quantity is of course the local
power density q 000, which is the produced energy per
unit volume and time. It is usually assumed that q 000
depends only on the radius and on time. The linear
rating is then simply given by
ð rcl;o
q 000 ðr Þ2pr dr
q0 ¼
rf ;i

ð rcl;o
ð rf ;o
000
q 000
¼ qf f ðr Þ2pr dr þ
cl 2pr dr
rf ;i

rcl;i

½1Š

where rf ;i /rcl;i is the inner fuel/cladding radius,
rf ;o /rcl;o is the outer fuel/cladding radius, q000

f and
q000
are
the
average
power
density
in
the
fuel
and
cl
cladding, respectively, and f ðr Þ is a radial distribution (form) function (see below). Generally, the
linear rating is a prescribed quantity and is a
function of the axial coordinate z and the time t.
For some phenomena (e.g., cladding creep), the fast
neutron flux is also needed. It can be prescribed as
well but may also be calculated from the local
power density q 000.

3.19.2.1.1 Axial heat transfer in the coolant

In general, three regimes must be covered in a light
water reactor (LWR):
1. The subcooled regime, where only surface boiling
occurs. This regime is typical for pressurized
water reactors (PWRs) under normal operating
conditions.
2. The saturated, two-phase regime. This regime
is typical for BWRs under normal operating

conditions.
3. The saturated or overheated regime. This regime
may be reached in all off-normal situations. A typical
example is a LOCA.
The fuel rod performance codes use 1D (axial) fluid
dynamic equations that can only cope with the first
two regimes. For simulating the third type of regime,
the whole reactor coolant system needs to be analyzed by means of thermohydraulic system codes
such as RELAP, TRACE, or ATHLET in order to
provide adequate boundary conditions to the fuel rod
performance code.
The temperature calculation in the coolant serves
two purposes. First of all, the axial coolant temperature in the basic (fictional) channel provides the
(Dirichlet) boundary condition for the radial temperature distribution in the fuel rod. It results from
the combined solution of the mass, momentum, and
energy balance equations. The simplified equation
used in fuel performance codes reads
cr

@T
@T
00 2prcl;o
þ qc000
þ crw
¼ qcl;c
A
@t
@z

½2Š

where c represents the heat capacity, r the density,
00
the heat
w the velocity, T the temperature, qcl;c
flux from the cladding to the coolant, A the channel
cross-sectional area, rcl,o the cladding outer radius,
and qc000 the power density in the coolant. In general,
00
the heat flux from cladding to coolant qcl;c
should be
computed by means of a thermohydraulic code.
Mathematically, the boundary condition is of the
convective type:

@T ðr ; t Þ
00
¼ Àl
¼ afT ðr ¼ rcl;o ÞÀTc g
qcl;c
@r rcl;o
where a is the heat transfer coefficient between the
cladding and the coolant and Tc ¼ Tc ðz; t Þ is the
(bulk) coolant temperature. Only for a steady-state
condition
crw

dT q 0
¼ þ qc000
A

dz

½3Š


Oxide Fuel Performance Modeling and Simulations

the heat flux from the cladding to the coolant is
known and is given by
00
¼
qcl;c

q0
2prcl;o

Under normal operational conditions, the mass flow
rate m_ ¼ Arw, and the coolant inlet temperature and
pressure are prescribed. In an off-normal or accidental situation, the normal operational condition is the
initial condition, but the boundary conditions must
be provided by the thermohydraulic system codes.
The second objective of the heat flow calculation
in the coolant is the derivation of the radial temperature drop between the coolant and the cladding
Tcl À Tc, resulting from convection:
q00 ¼ afilm ðTcl À Tc Þ ¼

qc000
2prcl;o

The heat transfer coefficient in the film depends

on the type of convection (forced or natural) and
the type of coolant (gas, liquid, liquid metal). In the
subcooled regime of a PWR, the Dittus–Boelter correlation is largely applied, whereas in the saturated
regime of a BWR, the Jens–Lottes correlation is
applied (see separate lecture on thermohydraulics).
3.19.2.1.2 Heat transport through the
cladding

The heat transport through the cladding occurs
through conduction:


1@
@T
r lc
þ qcl000 ¼ 0
r @r
@r
where lc is the cladding conductivity ($20 W mKÀ1 for
Zircaloy), and the heat generation in the cladding is
generally neglected (the g-heating as well as the exothermic clad oxidation process are generally disregarded). In order to account for the presence of an
outside oxide layer with a thermal conductivity on the
order of 2 W mKÀ1 for ZrO2 (thickness <100 mm), the
total equivalent cladding conductivity can be obtained
by applying the formula for serial thermal resistances.
In a similar manner, the appearance of crud on the
outer cladding surface is sometimes accounted for
through an additional heat transfer coefficient.
3.19.2.1.3 Heat transport from cladding to
the fuel pellet

The temperature difference in the pellet–cladding
gap, DTgap , is calculated as follows7,8:
DTgap ¼

q 00
hgap

539

where q 00 is the heat flux in watts per unit area and
hgap is the heat transfer coefficient between the fuel
and the cladding (gap conductance). Heat transfer by
convection can be neglected. In general, the heat
transfer coefficient hgap depends on
1. gap width or contact pressure between fuel and
cladding
2. gas pressure and composition
3. Surface characteristics of cladding and fuel
In fact, there are three parallel conduction routes:
hgap ¼ hrad þ hcon þ hgas
The contribution of the radiative component is
given by

 4
Cs
Tf À Tcl4
hrad ¼
1=ef þ 1=ecl À 1 Tf À Tcl
where Cs is the Stefan–Boltzmann constant, e is the

emissivity, and T is the temperature. The radiative
component is <1% during normal operating conditions because of the limited difference between the
two surface temperatures.
The component hcon reproduces the improvement
in heat transfer due to contact pressure:


P b
hcon ¼ ald 2
dH
where l and d are the mean values of the thermal
conductivity and the arithmetic mean roughness,
respectively, P is the contact pressure, H is the
Meyer hardness of the softer material, and a and b
are model parameters.
The heat transfer through conduction in the gas is
often based on the model of Ross and Stoute9:
hgas ¼

lgas
d þ s þ gf þ gcl

where the thermal conductivity of a multicomponent
gas is only composition-dependent and calculated by
means of
2
3

lgas

6
7
6
7
6
7
6
7
n 6
X
7
l
j
6
7
0
1
¼
6
7
7
j ¼1 6
6B
7
n
6 B 1 þ P w ck C
7
C
jk cj A5
4@

k¼1
j 6¼k

with c and w being the molar concentrations and
weighting factors, respectively. The gas extrapolation


Oxide Fuel Performance Modeling and Simulations

lengths gf and gcl (or temperature jump distance)
account for the imperfect heat transport across the
solid–gas interface, which is dependent on the material and gas pressure. Detailed formulations are discussed in Lassmann and Hohlefeld.7,8
It is important to note that, despite very detailed
formulations for the gap conductance, there is an
unavoidable uncertainty in the gap size s due to
input uncertainties, but also due to uncertainties in
the mechanical computation (e.g., cracking, radial
relocation of cracked fragments, and fuel swelling,
see Section 3.19.2.2).
3.19.2.1.4 Heat transport in fuel pellets

The heat produced by the slowing down of the fission
fragments in the fuel pellets is removed through
conduction in the pellets:


@T 1 @
@T
¼
lr

þ q 000
½4Š
rc
@t
r @r
@r
where c is the specific heat at constant pressure for
fuel. The boundary conditions are
Inner boundary:

@T ðr ¼ ri ; t Þ
¼ 0 ðradial symmetryÞ
@r

q00
ðpellet surface
Outer boundary: DTgap ¼
hgap
temperature is knownÞ
The temperature distribution in the pellets is therefore affected by two terms: the heat source and the
fuel thermal conductivity. At beginning of life (BOL),
the heat production in LWRs is subject to a slight
(typically in the range of 10%) depression, that is,
000
ffi I0 ðr Þ, where I0 ðr Þ is the modified Bessel funcqBOL
tion. During the irradiation of the fuel, epithermal
neutrons are captured preferentially near the surface
of the fuel by 238U. This leads to an enrichment of
239
Pu at the outer periphery of the fuel. At end of life

(EOL), qf000;o % ð2 À 3Þqf000;i , that is, the power density
distribution is a steep function of the radius (see
Figure 2). This effect therefore needs to be considered, and a specific model for the radial power density such as TUBRNP is a prerequisite for any
temperature analysis at high burnup.
Conduction of heat in the pellets occurs by
phonons or by the kinetic energy of electrons:
l ¼ lph þ lel . At temperatures below 1500 K, the
phonon contribution predominates. Above this temperature, the electronic contribution becomes important. When applying the kinetic gas theory to the
propagation of atomic vibrations (phonons) or

Radial form factor of power density qЈЈЈ ( / )

540

3
EOL
average burnup
60 MWd kgU–1
2

BOL

1

0

0

1

2
3
Radius (mm)

4

5

Figure 2 Radial form factor of the power density q000 at the
beginning and end of life for ‘typical’ light water reactor
conditions according to the TUBRNP model. The radial
distribution of the power density depends on enrichment,
rod diameter, neutron spectrum, and other parameters.
Reproduced from Lassmann, K.; Walker, C. T.;
van de Laar, J. J. Nucl. Mater. 1998, 255, 222–233,
with permission from European Commission.

quasiparticles, it appears that the phonon conductivity
in the temperature range of interest can be expressed as
lph ¼

1
A þ BT

where A corresponds to the scattering of phonons
by imperfections such as point defects, line and planar
defects, fission gas bubbles, etc. The parameter B corresponds to the scattering by phonon–phonon (Umklapp) interactions. When the burnup in the pellets
increases, the accumulation of point defects and fission
products will increase the phonon scattering (the A
term). The same happens if the fuel (e.g., UO2) is

doped with a neutron absorber such as Gd2O3, or if a
deviation from stoichiometry occurs (x 6¼ 0, where
x ¼|2 – O/M|and O/M is the oxygen-to-metal ratio
in UO2), that is, in general A ¼ A(BU, Gd, Pu, x), where
BU denotes the local burnup.
The temperature-dependent creation of electronic carriers, that is, excitation of free electrons,
leading to lel is typically expressed as


C
W
lel ¼ 2 exp À
T
kT
Besides the temperature and composition, the phonon contribution is also strongly dependent on the
fuel density. Several different formulations exist to
account for the reduction of the thermal conductivity
of a theoretically dense fuel (lTD) with the porosity
level (P) in the pellets10:


Oxide Fuel Performance Modeling and Simulations
1ÀP
Maxwell–Eucken: l ¼ lTD 1þaP
, where a is a
function of pore shape.
Loeb: l ¼ lTD ð1 À aPÞ, where a ¼ 1.7–3.

3.19.2.1.5 The structure of the thermal
analysis

The structure of the thermal analysis in a fuel performance code can best be summarized as follows: The
material properties l, r, and c are organized in an
independent database, whereas the power density q 000,
the gap conductance hgap , and the heat transfer coefficient between the cladding and coolant a are formulated in a model. The ‘rest’ is in a numerical algorithm,
solving the heat conduction equation in the pellets and
the convection problem in the coolant. A typical resulting temperature distribution calculated by means of
the TRANSURANUS code is shown in Figure 3.
3.19.2.2

Mechanical Analysis

The first barrier against release of radioactive fission
products to the environment is the cladding of the
nuclear fuel rod. The stress and associated deformation assessment of the cladding are therefore essential
in fuel performance calculations. Furthermore, the
deformation of both the pellets and the cladding
affects the gap width, which in turn affects the conductance of the gap and hence the temperature

541

distribution in the pellets. The thermal and mechanical
analyses are therefore equally important and closely
coupled. In principle, both problems should therefore
be solved simultaneously. In practice, however, all fuel
performance codes solve them separately but provide
coupling through an iterative scheme. This important
numerical aspect will not be dealt with in this chapter.
The interested reader is referred to a general discussion on this issue in various references.11–13
The next sections summarize how stress and

strains are calculated in both the ceramic pellets
and the metallic cladding, while underlining the
main assumptions and limitations.
3.19.2.2.1 Main assumptions and equations

The main assumptions generally made in fuel performance codes are the following:
1. The system is axisymmetric, that is, variables do
not vary tangentially.
2. Although the fuel and cladding move axially (not
necessarily at the same rate), planes perpendicular
to the z-axis remain plane during deformation
(generalized plain strain condition), that is, the
rod remains cylindrical.
3. Dynamic forces (inertia) are in general not treated,
and the time dependence inherent in the analysis
(creep) is handled incrementally (quasisteady).

IFA-504; linear rating qЈ = 20 kW m–1
2000
Fuel

Temperature (ЊC)

1500 Data

Cladding

Xe-filled rod

1000

Data
He-filled rod

500

0

0

2.5
5.0
Radial position (mm)

7.5

Figure 3 Radial temperature distribution in a boiling water reactor rod at the beginning of life. Comparison between the
range of experimental results and predictions of the TRANSURANUS code for two different fill gases (He, Xe). The data refers
to a thermocouple measurement in the central hole of the fuel pellet, indicated by the dashed line. From van Uffelen, P.;
Konings, R. J. M.; Vitanza, C.; Tulenko, J. In: Handbook of Nuclear Engineering; Cacuci, D.G., Ed., Springer: Germany, 2010;
pp 1520–1627, Chapter 13, with permission from Springer ScienceþBusiness Media B.V.


542

Oxide Fuel Performance Modeling and Simulations

4. Elastic constants are often isotropic and constant
within a cylindrical ring, but can be anisotropic in
some codes (e.g., Suzuki and Saitou14).
5. The total strain can be written as the sum of elastic

and nonelastic components.

3.19.2.2.2 Calculation of strains
3.19.2.2.2.1

1
¼ ½sr À nðst þ sa ފ
eelastic
r
E
1
¼ ½st À nðsr þ sa ފ
eelastic
t
E
1
eelastic
¼ ½sa À nðsr þ st ފ
a
E

The first two assumptions reduce the problem to one
dimension. The third assumption indicates that the
stresses are related through a local equilibrium condition for the radial force in the following form:
dsr st À sr
¼
dR
R
where sr and st represent the normal radial and
tangential stresses, respectively, and R corresponds

to the radius of the deformed geometry.
Since the fuel stack and cladding are treated as a
continuous, uncracked medium, no discontinuities
are allowed in their displacements. This is translated
by the compatibility relations for the strains:
du
dR
u
et ¼
R
ea ¼ constant ¼ C3
er ¼

½5Š

where u represents the radial deformation and ei are
the normal strains.
Finally, the last equation relates the stresses to the
strains. Based on the fifth assumption, the constitutive
relations read
X
etotal ¼ eelastic þ
enonelastic
½6Š
8 9
< er =
e ¼ et
: ;
ea

Elongation (–)

where

0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
300

½7Š

UO2

Elastic strain

The elastic strains for an isotropic material are
reversible and given by

3.19.2.2.2.2

Nonelastic strain

3.19.2.2.2.2.1 Thermal strain The nonelastic
strains consist of various contributions. First of all,
there is the thermal strain resulting from temperature
differences, which is assumed to be isotropic and
reversible:
eti ¼ aðT À T0 Þ i 2 fr; t; ag
The thermal expansion coefficients depend on the
material and the temperature itself, as shown in
Figure 4.
The larger thermal expansion of UO2 with respect
to that of Zircaloy explains why thermal expansion is
one of the largest contributions to the gap closure in a
nuclear fuel rod at BOL.
3.19.2.2.2.2.2 Swelling The second contribution to
the nonelastic strain in the fuel pellets comes from
swelling, and is also assumed to be isotropic. The fuel
swelling in turn has four contributions:
" 
 
1
DV
DV
s
þ
efuel ¼
3
V solid FP
V gaseous FP
#
 

 
DV
DV
À
À
V densification
V hot pressing

Zircaloy

500

700
Temperature (ЊC)

900

1100

Figure 4 Elongation of UO2 fuel and Zircaloy cladding due to thermal expansion (À) as a function of the temperature ( C).


where the first term is attributed to the inexorable
swelling of solid fission products:
!
 
X
DV
vi
¼ BU

Yi À 1
vU
V solid FP
solid FP
which is linearly dependent on burnup, on the fission
product yield (Yi), and on the partial volume of
the species (vi). In general, the solid fission product
swelling is on the order of 1% per 10 GWd tÀ1.
The second term comes from gaseous fission product
swelling:
 
ð
DV
4p Rmax 3
¼
R N ðRÞdR
V gaseous FP
3 0
and requires a model to predict the gaseous fission
product behavior, more precisely the gas bubble
formation due to the low solubility of rare gases in
UO2 (see Section 3.19.2.3.2). During the initial stages
of the irradiation (BU < 10 MWd kgÀ1 heavy metals
(HM)), the density increases as some fabrication
porosity disappears as a result of the impact of fission
fragments on the (small) pores. In general, the shrinkage process depends on the temperature, burnup, and
fission rate, as well as a combination of the initial
density, the pore size distribution, and the grain size.
The ideal situation is thus to have a fundamental
model for densification, like those proposed by

Assmann and Stehle15 and Suk et al.16 However,
values for the parameters involved are not always
well known. Therefore, many code developers have
implemented empirical correlations for the fraction
of the original porosity which has annealed out as a
function of the local burnup, the temperature, and the
grain size: for example17,18
DP
¼ a½1 À b expðÀa1 BU Þ À ð1 À bÞexpðÀa2 BU ފ
P0
where a ¼ [T (in  C) – 83]/(288Dgr), ab ¼ 5.12 exp
[À5100/T (in K)], a2 ¼ 0.0016 tUO2 per MWd, and
a1 ¼ 100a2. The densification, together with the solid
fission product swelling, is illustrated in Figure 5.
Under the influence of large temperatures, stress
levels, and defect production rates during irradiation, a
fraction of the fabrication porosity will disappear. This
fourth contribution to fuel swelling is referred to as
‘hot pressing’ and is similar to creep (see Section
3.19.2.2.2.2.3). Therefore, either the vacancy diffusion


dP
ODvol P
¼ ÀK
s
kT R2gr
dt

Pellets stack length change (mm)

Oxide Fuel Performance Modeling and Simulations

543

−2.5

−3.0

0

5

10
15
20
Burnup (MWd kg–1 UO2)

25

Figure 5 Change of the fuel pellet stack length (mm) at
beginning of life as a function of the burnup (MWd kgÀ1
UO2), showing the combined effect of densification and
solid fission product swelling. Reproduced from White,
R. J. Measurements of pellet and clad dimensional changes
in the Halden reactor; HWR-678; OECD Halden Reactor
Project; Halden, 2001, with permission from Halden Reactor
Project.

or the plastic flow (i.e., dislocation climb or other

model of creep)
dP
9
¼ À asP
dt
4
is considered.4
The isotropic swelling strain in the cladding is due
solely to void formation; hence it requires a model for
the evolution of voids in the metal.
3.19.2.2.2.2.3 Plasticity and creep The third contribution to the nonelastic strain in the fuel is viscoplastic in nature. It consists of the instantaneous
plastic deformation when the yield stress is exceeded
and of the time-dependent creep. For the fuel and
cladding, a simple isotropic plastic flow model can be
applied. Nevertheless, as creep is the main contributor to stress relaxation after cracking (see Section
3.19.2.2.2.2.4) in the oxide pellets, it is often only
considered in the cladding.
In a multiaxial state of stress, a method of relating
the onset of plastic deformation to the results of a
uniaxial test is required. Furthermore, when plastic
deformation takes place, one needs to determine (1)
how much plastic deformation has occurred and (2)
how that deformation is distributed among the individual components of strain in the principal directions. For the first requirement, a so-called yield
function is needed. This may be 1D like the Von
Mises criterion, assuming that the shear stress is
neglected and the material is isotropic14,19:


Oxide Fuel Performance Modeling and Simulations

1
seff ¼ pffiffiffi½ðsr À st Þ2 þ ðsr À sa Þ2 þ ðst À sa Þ2 Š1=2
2
so that yielding only occurs when the effective or
equivalent stress (seff) exceeds the yield stress determined from a uniaxial tensile test. Others have introduced the anisotropic factors according to Hill’s
methodology.6 Finally, a multidimensional yield surface20,21 has also been proposed. In order to account
for work-hardening, one generally assumes that the
yield stress changes with the total permanent deformation. The plastic strain is therefore computed
incrementally.
In order to answer the second question, each
increment of effective plastic strain is related
to the individual plastic strain components by a
flow rule:
Dei ¼ Deeff

@seff
@si

i 2 fr; t; ag

When using the above-mentioned definition of the
equivalent stress, one obtains the Prandtl–Reuss
flow rule22:
p

p

Dei ¼

3Deeff

Si
2seff

i 2 fr; t; ag

indicating that the plastic strain increment is
proportional to the deviatoric stress Si ¼ si À sh,
where sh ¼ (sr þ st þ sa)/3 is the hydrostatic stress.
For the time-dependent creep, one needs strain
rate equations although the total creep strain is also
computed incrementally by multiplying the strain
rate with the time step length. For primary creep,
typically an empirical expression is applied:
e_ eff ¼ K sneff t m
where K, n, m are constants.
For the secondary or steady-state creep, there are
three parallel processes. The vacancy diffusion or
Nabarro–Herring creep and the dislocation climb
dominate at high temperature and high stresses,
respectively:


Bq 000 seff
Ed
c
vacancy diffusion
Þ¼
exp
À
ð_eeff

R2gr
kT
 0
E
c
4:5
ð_eeff
Þ ¼ B 0 q 000 seff
exp À d dislocation climb
kT
The third process is irradiation-induced creep, dominating at low temperatures and assumed to be proportional to the effective stress and the local fission
rate density or q 000.

3.19.2.2.2.2.4 Pellet cracking The fourth and last
nonelastic strain component stems from the pellet
cracking. Pellet cracking already occurs at startup
with fresh fuels due to the differential thermal expansion since the hot pellet center expands more than
the cold periphery. In order to assess the linear heat
generating rate at which cracking in cylindrical pellet
occurs, the maximum thermal stress (¼st,max ¼ sa,max
at pellet periphery) in an uncracked pellet submitted
to a parabolic temperature gradient
st;max ¼ À

aEq 0
8pð1 À uÞl

must be compared with the (uniaxial) fracture stress,
which is $130 MPa. When using E ¼ 200 GPa,
u ¼ 0.31, the thermal diffusivity a ¼ 10À5 KÀ1, and

an average thermal conductivity of l ¼ 3 W mKÀ1,
radial cracks are predicted to be initiated in the pellet
periphery at a linear heat rate q0 of the order of
5 kW mÀ1. The number of cracks (Ncr) is dependent
on the linear heat rate. Oguma23 proposed a linear
model for the number of radial cracks, which is illustrated in Figure 6. In addition to radial cracks, also
axial and (especially under ramping conditions) circumferential cracks are formed.
The consequences of cracking are very important in fuel performance modeling. Owing to the
larger thermal expansion of the fuel fragments in
comparison with that of a monolithic cylinder, and
to vibration-induced motion, they move outward.
This is called pellet ‘relocation’ and has a strong
impact on the thermal behavior, as shown in Figure 7.
It reduces the pellet–cladding gap size, thereby
reducing the temperature levels in the fuel at BOL.
This constitutes the largest contribution to the gap
closure ($30–50%, depending on q0 ) but is also the
one which is subject to the largest uncertainty
because of the stochastic nature of the cracking

Crack pattern
Number of fragments

544

16
8
0
0

10

20

30

40

Rod power (kW m–1)

Figure 6 Calculated crack pattern from thermoelastic
stress. Reproduced from Oguma, M. Nucl. Eng. Des. 1983,
76, 35–45.


Oxide Fuel Performance Modeling and Simulations

process. This also raises questions about the usefulness of applying 3D stress calculations.
The contribution from relocation is generally
accounted for in the tangential strain component as a
(linear) function of the linear heat rate: et ¼ u/r, where
u ¼ sdg, s being the initial radial gap size and dg the
fraction of the gap size closing as a result of relocation.
An example based on the relocation model in the
FRAPCON3 code24 is illustrated in Figure 8.25

545

When the pellets swell large enough so that they
come into contact with the cladding, which creeps

down under influence of the pressure difference
between the coolant pressure and the fill gas pressure,
then relocation may be (partly) reversed. If both
pellet swelling and cladding creep-down continue,
the gap is closed and the pellet fragments are pushed
inward, so that the relocation approaches total
elimination.

IFA-504; linear rating qЈ = 20 kW m–1
2000
Cladding

Fuel

Temperature (ЊC)

1500 Data

Without relocation

Xe-filled rod

With relocation
1000
Data
He-filled rod
500

0

0

2.5

5.0

7.5

Radial position (mm)

Figure 7 Radial temperature distribution in a boiling water reactor rod (instrumented fuel assembly-505) at beginning of
life calculated by the TRANSURANUS code. Comparison between the analysis of a Xe-filled rod with (full line) and without
(dashed line) taking relocation into account. van Uffelen, P.; Konings, R. J. M.; Vitanza, C.; Tulenko, J. In: Handbook of Nuclear
Engineering; Cacuci, D.G., Ed., Springer: Germany, 2010; pp 1520–1627, Chapter 13, with permission from Springer
ScienceþBusiness Media B.V.

FRAPCON-3 modified
≥40 kW m–1
50
35 kW m–1
Relocation (%)

30 kW m–1
25 kW m–1
40

≤20 kW m–1

30

0

10

20
30
Slice averaged burnup (MWd kg–1 HM)

40

50

Figure 8 Fraction of gap closure due to pellet fragment relocation (dg), derived from the relocation model in the FRAPCON3
code. van Uffelen, P.; Konings, R. J. M.; Vitanza, C.; Tulenko, J. In: Handbook of Nuclear Engineering; Cacuci, D.G., Ed.,
Springer: Germany, 2010; pp 1520–1627, Chapter 13, with permission from Springer ScienceþBusiness Media B.V.


546

Oxide Fuel Performance Modeling and Simulations

The effect of relocation on the mechanical behavior is also of primary importance since it reduces the
overall thermal stress in the pellets and may even
change the sign of the stress in pellet centers from
compression (in a cylinder) to traction (in fragments).26 The exact location and size of every crack
would be required to accounting for the effects of the
cracks exactly and to solve a 3D stress–strain problem
in each block. Instead, one simply modifies either the
material constants14,25 or modifies the constitutive
equations. An example of the former approach is

that of Jankus and Weeks (Figure 9),27 where a
reduction of the elastic constants is proposed:
 Ncr
2
E
E0 ¼
3
 Ncr
1
0
n ¼
n
2
which means that an equivalent continuous and
homogeneous solid body with directionally dependent
(anisotropic) properties is considered. As the pellet–
clad gap closes during irradiation, the contact pressure
can press the fragments inward, thereby reducing the
relocated radius to a minimum value. Some codes
also account for the restoration of the elastic constants as the relocation is reversed (partially).14
In order to modify the constitutive equations, a
plane stress condition has been proposed20: that is,
the tangential stress is set equal to the fill gas pressure
once the radial crack appears. Both types of approaches, however, do not account for crack healing.
3.19.2.2.3 Boundary conditions

In order to solve the main equations summarized in
Section 3.19.2.2.1, boundary conditions are required.

3.19.2.2.3.1

Radial boundary conditions

In general, continuity of the radial stress and displacement at each radial zone is imposed and the
radial stress at the outer cladding surface is determined by the coolant pressure: sr(rcl,o) ¼ Àpcool.
The boundary condition in the rod depends on
the configuration. When pellet–cladding mechanical
interaction (PCMI) is not established, the radial stress
at the pellet periphery is determined by the fill gas
pressure in the fuel rod (pgas): sr(rf,o) ¼ Àpgas. For the
boundary condition in the pellet center, two possibilities exist. In hollow pellets, the radial stress at the
pellet center is equal to fill gas pressure: sr(rf,i) ¼ Àpgas,
whereas in the case of full cylindrical pellets, the radial
and tangential stresses are equal at the pellet center.
When the fuel and cladding are in contact, a fuel
pellet interfacial pressure exists (pfc), which determines the boundary condition at the pellet surface:
sr(rf,o) ¼ sr(rcl,i) ¼ Àpfc. The other radial boundary
conditions remain unchanged.
3.19.2.2.3.2

Axial boundary conditions

The plane strain assumption entails that the axial strain
is constant in the plane perpendicular to the axis. The
axial strain is therefore determined by an axial force
balance equation including the fill gas pressure, the
plenum spring pressure, the fuel column weight, the
friction forces, and the coolant pressure imposed on
both end plugs of the cladding. The friction forces
depend on the fuel–cladding interaction and can only

be taken into account iteratively. Indeed, when one of
the axial sections i is analyzed, it is not known whether
the frictional forces between fuel and cladding originating from a section above/below i need to be considered in the axial balance of forces. This is schematically

1.2
EЈ/E

Elastic constant ratio

1

VЈ/V

0.8
0.6
0.4
0.2
0
0

2

4

6
Number of cracks

8

10

12

Figure 9 Effect of the number of cracks on the elastic modulus and the Poission’s ratio according to Jankus and Weeks.27


Oxide Fuel Performance Modeling and Simulations

No radial contact
‘slip’
Cladding

Case
1

Fuel

Static friction
‘no slip’
Cladding

2

Fuel

Trapped

Static friction
‘no slip’

‘Slip’

Cladding

Fuel

Trapped

3

Sliding friction
‘slip’

‘Slip’

Cladding

Fuel

4

Figure 10 Four possible modes of an interaction between
fuel and cladding. van Uffelen, P.; Konings, R. J. M.; Vitanza,
C.; Tulenko, J. In: Handbook of Nuclear Engineering;
Cacuci, D.G., Ed., Springer: Germany, 2010; pp 1520–1627,
Chapter 13, with permission from Springer Scienceþ
Business Media B.V.

shown in Figure 10. In the case of a radial contact
between the fuel and cladding, both bodies may stick to

each other, but some sliding may be possible in specific
conditions (sticking or static vs. sliding friction). Part of
the fuel rod may be ‘trapped,’ which means that rather
high axial forces may act on cladding and fuel.
One advantage of 2D and 3D finite element models is that such effects are automatically included in
the analysis through the use of specific gap elements,
as explained in more detail in Chapter 3.22, Modeling of Pellet–Cladding Interaction.
3.19.2.2.4 Pellet–cladding interaction
3.19.2.2.4.1 Pellet–cladding mechanical
interaction

As dealt with in detail in Chapter 3.22, Modeling of
Pellet–Cladding Interaction, PCMI is one of

547

the important analytical targets in predicting fuel
behavior, since it can be a cause of fuel failure.
The mechanical interaction of the fuel with the
cladding is mainly caused by the different thermal
expansion rates of the two components (see Section
3.19.2.2.2). However, the poor thermal conductivity
of the fuel results in a strong temperature gradient
across the pellet. This leads to an increase of the gap
closure rate at both ends of a pellet because even
in the early stages of irradiation, the differential
thermal expansion in the pellet causes its so-called
hour-glass shape (Figure 11). The resulting internal
thermal stresses exceed the fracture stress of UO2
(around 100 MPa) causing pellet fracture. Once the

pellet has fractured, it is able to deform much more
readily under the effects of the temperature field,
causing the pellet ends to bow outward leading to
the hour-glass shape.
PCMI in the early stage of irradiation is therefore
caused by the differential thermal expansion of the
pellets, leading to a bamboo or ridging deformation
of the cladding. The ridges in the cladding, the height
of which can reach 20 mm at high powers, coincide
with the pellet ends and can cause hoop stresses of
around 400 MPa, which is close to the clad yield
stress. When aggressive fission products such as
iodine are in their vicinity, these local concentrations
of stress and strain lead to the so-called stress corrosion cracking (SCC) in LWR fuel systems.
The onset of PCMI is affected by a number of
design and fabrication parameters. First of all, the
pellet geometry is adapted to extend the onset of
and to mitigate PCMI. In particular, the pellet height
has been reduced with respect to its diameter, while
chamfers were applied for delaying PCMI and reducing the probability of fuel chipping. Second, a rod
geometry is important. More precisely, the width of
the original clearance between the pellets and the
cladding should be large enough. Nevertheless, the
gap size is subject to many uncertainties, the largest
being the pellet fragment relocation. In addition, it is
generally assumed in fuel performance modeling that
the pellets are located concentrically within the cladding, although this is seldom the case. Any eccentricity in the stacking arrangement, resulting from
fabrication or fuel rod handling, is likely to lead to
premature onset of PCMI even though its effect will
diminish as gap closure occurs.

Once PCMI has started, both the pellet geometry
and the material properties of the interacting components influence the maximum stresses and strains
as well as their evolution. Flat-ended pellet stacks


548

Oxide Fuel Performance Modeling and Simulations

will generate larger axial expansion in comparison
with dished stacks, especially in fresh fuel. This is
due to the fact that in flat-ended pellets, the hot
central part determines the maximum length change,
whereas for dished pellets there is no contact between
pellets along the central axis (when the power is not too
high) and hence the axial expansion will be controlled
by the outer (cooler) regions of the pellet. In practice,
the ratio of axial to tangential strain can vary between
0.2 for large dishes and 2 for undished pellets.28 Nevertheless, the axial expansion from flat-ended pellets
can diminish with burnup because of in-pile dishing,
for instance caused by densification and creep in the
hot central parts of the pellets during PCMI.
In addition to the difficulties to reproduce 2D
(local) pellet deformations during PCMI by means
of 1D fuel performance codes, there are other challenges to be dealt with. Apart from pellet and cladding
eccentricity and slanting, which mostly affect the onset
of PCMI, there are uncertainties related to the assessment of the stress in cracked pellets, to pinching by
assembly grids, as well as to the (static and dynamic)
friction coefficient between pellets and cladding. In
particular, the slipping is severely restricted by the

interaction layer that establishes between both components after gap closure at higher burnups.
Therefore, for these PCMI evaluations, the (local
2D) finite element method (FEM) is in principle a
more advantageous method than the finite difference
method (FDM). FEM puts the reaction forces exerting among all the ring elements of the pellet and
cladding into one entire matrix to obtain a numerical
solution,14 while in FDM displacements of each ring,
Pellet hourglassing

Ridges

As fabricated

Prior to PCMI

After PCMI

Figure 11 Schematic presentation of pellet–cladding
mechanical interaction. P. van Uffelen, R.J.M. Konings,
C. Vitanza and J. Tulenko, Analysis of Reactor Fuel Rod
Behavior. In: Handbook of Nuclear Engineering
(D.G. Cacuci, Ed.), Chapter 13, p. 1520-1627, Springer
Germany, 2010, with permission from Springer Scienceþ
Business Media B.V.

elements of the pellet and cladding are summed up
independently to determine the diameter change,
granting that FDM is actually modified to some extent
to cope with mechanical interactions among the ring
elements. There have been attempts to develop 3D

fuel performance codes,29 as explained in Chapter
3.22, Modeling of Pellet–Cladding Interaction.
3.19.2.2.4.2

Irradiation-induced SCC

Failures during power variations are not only attributed to stress. Stresses increase at the intersection of
radial crack planes, the interpellet planes, and the
inner surface of the cladding. The brittle nature of
the failure site, however, has led to the general consensus that, although stress is the primary initiator, propagation of the crack is chemically assisted, and thus the
process is termed SCC. The chemical assistance for
brittle cracking in all probability arises from the environment established by the release of fission products
into the fuel–clad interspace, with isotopes of iodine as
the main corroding species. There is some discussion as
to whether the corrodent must be freshly released from
the fuel or whether it is sufficient for it to have accumulated in the fuel–clad gap throughout the irradiation period prior to the overpower transient.
Since its discovery in the Canada deuterium
uranium (CANDU) reactor and the BWR in the
1970s, the incidence of pellet–cladding interaction
(PCI)-SCC failure is thought to be affected by
four factors, as reviewed by Cox30: stress, time, material, and chemical environment. Understanding the
important variables for PCI-SCC will help fuel
designers to propose solutions. These include adapting the appropriate fuel geometry (reducing the
height to diameter ratio, introducing dishes and
chamfers), imposing restrictions on the power variations, modifying the fuel assembly design, and applying coatings on the inner cladding surface in order to
reduce the friction coefficient and to accommodate in
a ductile manner the biting of pellet ends. More
precisely, in BWRs, Cu and Zr barriers were applied
in the form of thin metallic layers as an integral part
of the tube fabrication process. Pure zirconium (of

either crystal bar or sponge origin)31,32 has been
adopted as the standard barrier because irradiation
experience showed that copper offered less protection after high irradiation doses. However, pure zirconium oxidizes more rapidly in comparison with
Zircaloy, while the terminal solubility of hydrogen
is lower in the liner.33 Accordingly, when a primary
defect occurs in the cladding, for instance, due to
debris fretting, water ingress will oxidize both the


Oxide Fuel Performance Modeling and Simulations

fuel and the liner. This occurs typically in the lower
(cooler) part of the fuel rod. The hydride formation
near the cladding inner diameter can lead to a sunburst hydride and would, due to the volume increase,
set up a tangential stress in the Zircaloy part of the
cladding, which can promote crack formation. The
local hydrogen absorption can cause more severe
hydriding and therefore secondary defects in the
form of long axial splits and circumferential cracks.
Secondary cladding defects in LWR fuel can cause
large releases of uranium and fission products to the
primary coolant,34 and seem to be correlated with PCI
caused, for instance, by control blade movements in
BWRs. Mitigation of these secondary failures has been
achieved by increasing the number of rods per assembly35 (e.g., from 6 Â 6 to 10 Â 10 in BWR assemblies)
in order to reduce the linear heat generation rate per
rod and by alloying the liner with either Fe36 or Sn.37
A more recent component of PCI-resistant fuel
designs is the use of softer fuel pellets obtained by
means of large-grained doped UO2 for both BWR

and PWR applications. Since the early 1990s, AREVA
has developed an optimized chromia-doped UO2 fuel
that exhibits significantly higher performance compared to standard UO2.38 The grains are on the order
of 60 mm compared to 10 mm in standard UO2. As a
result, the fuel releases less fission gas and is less prone
to PCI failures during ramp tests thanks to a larger
creep rate.39 Postirradiation examinations after ramp
tests revealed a larger number of radial cracks on the
pellet periphery. Recent 2D finite element simulations
of PCI have shown that this can be attributed to the
larger friction due to fuel–clad bonding in highburnup fuel and the reduced fracture stress of the
doped fuel.40 The simulations also indicate that, unlike
the hoop stress, the shear stress distribution in the
cladding is smoother thanks to the reduced fracture
stress of the doped fuel: that is, thanks to the larger
number of radial cracks in the periphery of the pellets.
Modeling PCI has evolved since the first attempts
aiming at the fitting of the time to failure (or failure
probability), maximum power, and power increase to
experimental data, although the uncertainty could
reach a factor of 5.41
The initial step is assumed to occur either at
preexisting flaws42 according to a frequency distribution, or spontaneously when a threshold such as the
iodine concentration is exceeded.43 More recently,
Park et al.44 postulated that pits would generate preferentially around grain boundaries and coalesce to
form a microcrack, referred to as grain boundary
pitting coalescence. The microcrack is assumed to

549

develop into an incipient crack, starting and propagating along the grain boundary.
For the simulation of crack propagation, most
authors apply linear elastic fracture mechanics
(LEFM), in line with Kreyns et al.45 By fitting the
experimental data of Wood,46 they advocated that the
crack velocities could be related to the fourth power
of the stress intensity factor KI:
da
¼ CKI4
dt
where a is the crack length and C is a constant. The
crack intensity factor was shown to be controlled by
the elastic stress field at a flaw tip as described by KI,
rather than the nominal applied stress:
pffiffiffi
KI ¼ s a Y
where s represents the nominal hoop stress and the
factor Y incorporates a correction factor to account
for the finite width of a defect-bearing component.
Nevertheless, Kreyns et al.45 pointed out that smallscale plasticity will occur near the crack tip, in a cone
with radius rp:
 
1 KI 2
rp ¼ pffiffiffiffiffi
6p sy
where sy corresponds to the yield stress and
pffiffiffiffiffiffiffi
KI ¼ s aeff Y , which results in an effective crack
length:
aeff ¼ a þ rp

and the effective correction factor Yeff ¼ Y
becomes

paffiffiffiffi
ffi
eff
a

Y
Yeff ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 À Áffi
2
1 À ssy Y6p
Anderson47 provided a general correction factor so
that KI can be expressed more generally:

rffiffiffiffiffi 
pR pa
a a R
F
; ;
KI ¼
t Q
2c t t
where p is the internal pressure on the tube (MPa), R
is the mean tube radius, t is the tube wall thickness,
and Q is the shape factor for an elliptic crack
a 1:65
Q ¼ 1 þ 1:464
c

where c is the half length of the crack and F is a
boundary correction factor which depends on the
shape of the initial crack formed at the inner cladding


550

Oxide Fuel Performance Modeling and Simulations

surface. Park et al.44 applied an expression for F within
the range of 5 R/t 20, 2c/a 12, and a/t 0.8:
F ¼ 1:12 þ 0:053x þ 0:0055x2 þ
À
Á
R 2
2 20 À t
ð1 þ 0:02x þ 0:0191x Þ
1400
aa .
where x ¼
t 2c
The KISCC value provided by Park et al. was
3.3 and 4.8 MPa m1=2 for stress-relieved and recrystallized Zircaloy-4, respectively.
Zhou et al.42 replaced the constant C in the equation
for the crack velocity by a function of the iodine
concentration and an Arrhenius-type temperature
dependency. More importantly, however, they also
involved the pellet–cladding contact pressure in the
estimation of the stress intensity factor. The local effect
of the frictional shear forces is accounted for by adopting the Coulomb friction model, according to which

the friction force is proportional to the contact pressure. The extension by Zhou et al. is in line with recent
findings of Michel et al.48 Based on 2D and 3D finite
element computations, they showed that the tangential
stress concentration in the cladding is proportional
to the shear loading transmitted at the pellet–clad
interface. As a result, the peak hoop stress at the inner
surface of the cladding depends on the interfacial shear
stress and the uniform loading in the hoop direction.
Nevertheless, the simulations did not account either
for cladding anisotropy or for stress relaxation.
Stress relaxation was accounted for in an empirical
manner by Mattas et al.49 They assumed the chemical
(intergranular) crack growth rate to have an initial
value and to decrease exponentially as the crack
depth increases. Chemical crack growth was postulated to continue until a critical stress intensity for
cleavage and fluting was achieved, at which point
intragranular cleavage initiated until failure.
As pointed out by Rousselier et al.,50 stress relaxation and the ensuing crack arrest are necessary to
explain a so-called discontinuity observed during
ramp experiments: depending on the maximum
power of the fuel rod, either a through crack is obtained
within maximum 10 min or the SCC damage is limited
to a few micrometers, even after hours of operation at
the maximum power. This discontinuous behavior was
observed above the SCC initiation threshold of about
300 MPa in Zircaloy-4.51 Rousselier et al. attributed this
to the stress relaxation and to the fact that the intergranular crack could leave the stress concentration
zone at the crack tip. Crack arrest was postulated to
occur when at the same time the stress intensity factor

would be below a certain threshold (KI KIA) and the
stress intensity factor would decrease. However, quantitative information cannot be directly inferred from
their analysis, since the critical stress corresponding to
KIA should depend on the material (stress relieved vs.
recrystallized), the irradiation, as well as the loading
history. Furthermore, the LEFM should not be applied
without corrections for the viscoplastic behavior and,
finally, because of the local inhomogeneities of the
material at the scale of the crack, one should apply
local rather than global criteria such as KI KIA.
With the advent of improved hardware and software on one hand and more detailed experimental
data on the other, more detailed models are being
developed at various scales. At the electronic and
atomic scale, first-principle computations should
enable analyzing the individual effect of impurities
such as iodine on the binding energies, in much the
same way as Xin et al.52 have studied the properties of
point defects and their interactions with Nb in Zr, or
Kaji and Tsuru53 analyzed the clustering of Ni in Fe.
By means of finite element computations at the level
of crystallographic grains, Rousselier et al. suggest
avoiding the limitations associated with LEFM for
crack initiation and propagation. Because of the excessive computation time and the lack of precise data of
some model parameters, their model is not applicable
in fuel performance codes but should enable analyzing
the effect of corrosive fission products on the intergranular damage by coupling the mechanical problem
with a diffusion problem. A similar tool has already
been developed by Musienko and Cailletaud,54 albeit
the corrosive environmental parameters are accounted
for by a phenomenological approach: that is, via an

effective diffusion coefficient. At the mesoscopic scale,
Kaji et al. developed a 2D model for SCC growth, in
order to analyze qualitatively the effect of load (normal
vs. shear stress) and grain boundary corrosion on the
branching aspect of crack growth. The macroscopic
models are mostly based on finite element simulations.40,48 These tools provide deeper insight and qualitative information about the various parameters
affecting PCI as explained above. Nevertheless, these
models are not yet capable of replacing the simplified
1D models implemented in fuel performance codes,
despite the improvements of hardware and software.
Marchal et al.40 are trying to develop analytical
‘enrichments’ for 1D models in fuel performance
codes based on the 2D models. The stochastic nature
of cracking and the complex evolution of material
properties and boundary conditions during irradiation
remain the most important difficulties to be tackled.


Oxide Fuel Performance Modeling and Simulations

3.19.2.2.4.3 Outside-in cracking caused by
power ramps

High-burnup fuel rods in LWRs are characterized by
the absence of a clearance between the pellets and
their metallic containment. As a direct result, PCMI
in the power ramp of high-burnup fuel is characterized by a direct stretching of the cladding by the
pellet, in which cladding is subjected to both hoop
tensile stress and axial stress: that is, to a biaxial stress
state.55 During power ramp tests with high-burnup

BWR rods, a failure mechanism therefore occurred
from the outside of the cladding toward the inner
surface of the Zircaloy56 as opposed to the standard
PCI-SCC as discussed above. Postirradiation examination revealed that the process started an axial split
with cracking of radial hydrides that had formed
during the power ramp test, followed by a step-bystep cracking of hydrides at the crack tip. The process
bears similarities with secondary hydride failures discussed above. The radial temperature gradient in the
cladding wall and the hoop tensile stress due to the
ramp test facilitate the hydrogen diffusion and precipitation of radial hydrides on the outer surface of
the tube. These hydrides can crack under the influence of stress caused by PCMI and progressing
toward the inner tube surface. The main difference
with secondary hydride failures that start on the
inner surface is that hydrogen is already absorbed
and accumulated at the outer surface due to clad
oxidation during normal operation.
3.19.2.3

Fission Gas Behavior

On average, each fission event produces 0.3 Xe and
Kr atoms. These inert fission gas atoms have a very
low solubility limit ($0.3 wt% for Xe) causing two
important life-limiting phenomena in the fuel rod:
either they remain in the pellets and contribute to the
swelling, or they are released from the pellets and
increase internal gas pressure of the rod while reducing the thermal heat transfer in the gap. Fuel swelling
may lead to PCMI and even cladding failure under
certain conditions. Likewise, the fission gas release
(FGR) may lead to higher fuel rod temperatures,
which in turn could lead to higher FGR (positive

feedback loop) until the rod fails due to clad ballooning and clad burst.
Because of its implications for fuel performance, the
basic mechanisms involved in the FGR and swelling in
LWR fuel will be summarized first, before outlining
how these phenomena are implemented in a code.
The interested reader will find more details in

551

Chapter 3.20, Modeling of Fission-Gas-Induced
Swelling of Nuclear Fuels and in Van Uffelen.57
3.19.2.3.1 Basic mechanisms
3.19.2.3.1.1

Recoil, knockout, and sputtering

In general, a fission event entails – among others –
two fission fragments that convey their kinetic energy
to the fuel lattice. A fission fragment close enough to a
free surface (<6–7 mm) can escape from the fuel due to
its high kinetic energy (60–100 MeV). This is called
recoil release. When fission fragments make elastic
collisions with the nuclei of the lattice atoms, a collision
cascade begins. The interaction of a fission fragment, a
collision cascade, or a fission spike with a stationary gas
atom near the surface can also cause the latter to be
ejected if it happens within a distance close enough to
the surface. This process is called ‘release by knockout.’
Finally, a fission fragment traveling through oxide loses
energy, causing a high local heat pulse. When this

happens close to the fuel surface, a heated zone will
evaporate or sputter, thereby releasing any fission
product contained in the evaporated zone.
Recoil, knockout, and sputtering can only be
observed at temperatures below 1000  C, when thermally activated processes (see Sections 3.19.2.3.1.2,
3.19.2.3.1.3, 3.19.2.3.1.5, 3.19.2.3.1.6, 3.19.2.3.1.7,
3.19.2.3.1.8) do not dominate. They are almost temperature independent and therefore called ‘athermal
mechanisms.’ It is generally of little importance in a
reactor at intermediate burnup levels. The fraction of
athermal release is roughly under 1% for rod burnups below 45 MWd kgUÀ1, and increases to roughly
3% when the burnup reaches about 60 MWd kgUÀ1.
3.19.2.3.1.2

Lattice diffusion of single gas atoms

The first and basic step in FGR is single gas atom
diffusion in the lattice. Possible mechanisms by which
the inert gas atoms migrate through the fuel have been
studied by Grimes58 by considering low-energy migration pathways between solution sites as well as the
stability of gas atoms at a variety of solution sites within
a defective UO2Æx lattice (x ¼ |O/M – 2|, the deviation from the stoichiometry). He postulates a cationvacancy-controlled migration pathway for Xe atoms.
Indeed, according to his calculations, Xe is trapped
at a uranium vacancy in UO2 þ x, at a trivacancy
cluster in UO2Àx and at a di- or trivacancy in UO2.
Since the local environment of the migrating Xe atoms
is supposed to become the charged tetravacancy
for all stoichiometries, the mechanism for diffusion
only considers the association of a cation vacancy to
the trap sites (Figure 12). (Uranium vacancies as the

552

Oxide Fuel Performance Modeling and Simulations

Uranium vacancy

Oxygen vacancy

Divacancy

Trivacancy

Oxygen vacancy

Uranium ion

Interstitial site

Tetravacancy

Uranium vacancy

Interstitial site
58

Figure 12 Possible solution sites for fission products in UO2Æx according to Grimes.

slower moving species are rate-controlling for most
diffusion-related processes in UO2.)

The lattice diffusion coefficient is influenced by
the temperature, deviations from stoichiometry and
additives (e.g., Cr, Nb), phase changes, and, therefore,
also indirectly by the burnup. Also, the fission fragments are assumed to contribute to the diffusion
process, which is referred to as ‘irradiation-enhanced
diffusion.’ This is due to the interaction of the fission
fragments and the associated irradiation damage
cascades with the fission gas atoms in the lattice,
resulting in a displacement of the gas atoms. This
effect dominates the diffusion process at temperatures
below 1000  C and is temperature independent. For
temperatures between 1000 and 1400  C, vacancies
necessary for the gas atom diffusion are assumed to
be created both thermally and by the damage cascades
related to fission fragments. Above 1400  C, a purely
thermally activated diffusion coefficient is applied:
that is, thermally created vacancies for diffusion are
predominant. These three temperature regimes are
reflected in the three components of the single gas
atom diffusion coefficient (m2 sÀ1) often applied in
the fuel performance codes59:
D ¼ D1 þ D2 þ D3
where

with F_ the fission rate density and T the absolute
temperature.
Unperturbed (intrinsic) diffusion of single inert
gas atoms (Xe, Kr) can be observed at low damage
and gas concentration (10À5 at.%). At higher gas and
damage concentrations, other effects should be taken

into account.
3.19.2.3.1.3

D3 ¼ 8 Â 10À40 F_

Trapping

In nuclear fuels, either natural (e.g., impurities, dislocation lines, closed pores, etc.) or irradiation-produced
imperfections in the solid (e.g., vacancy clusters in
fission tracks, fission gas bubbles, solid fission product
precipitates, etc.) depress the amount of fission products available for diffusion by temporarily or permanently trapping the migrating atoms. The experiments
show that, for the burnup characteristic of power reactors, gas atom trapping due to (intragranular) fission
gas bubbles in the grains is predominant. The trapping
rate depends on the number density and size of the
intragranular bubbles, and hence on temperature, fission rate, and burnup. A second important effect of
trapping occurs at the grain boundaries. It deals with
the delay for the onset of thermal FGR, via the bubble
interconnection mechanism (see Section 3.19.2.3.1.8).
3.19.2.3.1.4



35000
À6
D1 ¼ 7:6 Â 10 exp À
T


pffiffiffi
13800

À25
_
D2 ¼ 5:6 Â 10
F exp À
T

Courtesy of Professor Robin Grimes.

Irradiation-induced resolution

A fraction of the gas atoms trapped in bubbles can be
redissolved in the surrounding matrix through the
interaction of a fission fragment with the bubble.
Two different types of mechanisms are proposed to
explain the experimental observations. On one hand,
microscopic models consider the resolution of one
gas atom at a time when interacting with a fission


Oxide Fuel Performance Modeling and Simulations

fragment or an energetic atom from the collision
cascade. Macroscopic models on the other hand consider the complete bubble destruction, but there is
still discussion about the detailed mechanisms. For
(larger) grain boundary bubbles, resolution is supposed to be less effective.
3.19.2.3.1.5 Grain boundary diffusion

Grain boundary diffusion is the most commonly
observed route for solute migration in polycrystalline
materials. It is generally accepted that diffusion in

crystalline solids proceeds more rapidly along grain
boundaries than through the lattice. This is due to the
atomic jump frequency in these planar defects, which is
about a million times greater than the jump frequency
of regular lattice atoms in stoichiometric materials at
0.6 times the absolute melting temperature. Nevertheless, there is a switch from release assisted by grain
boundary diffusion in trace-irradiated UO2 to trapping
and eventual interlinkage of the intergranular bubbles
(see Section 3.19.2.3.1.8). This switch occurs early in
life, so that grain boundary diffusion is only considered
to contribute to the precipitation of fission gas atoms in
grain boundary bubbles, rather than to the long-range
transport along grain boundaries to the free surface of
the pellets.60
3.19.2.3.1.6 Grain boundary sweeping or
grain growth

In LWR fuel under normal operating conditions,
only normal grain growth is observed, that is, large
grains grow at the expense of smaller ones. It affects
the FGR in two ways. First of all, grain boundary
sweeping provides another mechanism for the collection of gas at these internal surfaces from which
release can occur. The collection results from the
low solubility of the fission gas, and hence the sweeping grain boundary does not redeposit any gas in the
newly formed crystal behind it. The moving grain
boundary acts as a fission gas collecting filter. At the
same time, grain boundary bubbles hinder grain
growth to some extent.
Second, the average diffusion distance for the
fission products created in the grain increases. Unlike

the first consequence, this tends to reduce the release
rate. Grain boundary sweeping occurs at temperatures above roughly 1600  C.
3.19.2.3.1.7 Intragranular bubble migration

The migration of intragranular fission gas bubbles provides an alternative to the sequence ‘bubble formation–
resolution–gas atom diffusion’ in order to describe

553

fission product release from nuclear fuels. Migration
of bubbles in the oxide fuels has two other important
consequences, namely the columnar grain growth
with the concomitant central void formation (observed in fast breeder reactor fuel) and the coalescence of the bubbles which gives rise to fuel swelling.
Under normal operating conditions, however, intragranular fission gas bubbles remain small (typically
below 20 nm) due to resolution, and show a small
mobility at least up to 1800  C.61 This is partly
explained by the pinning by dislocations and other
crystal defects.
3.19.2.3.1.8 Grain boundary (intergranular)
bubble interconnection

Fission gas bubbles appear along grain boundaries
after a certain burnup, depending on the temperature
history. When bubbles interconnect, they form a socalled tunnel network through which the gas can be
released. The bubble interconnection is a reversible
process, for the tunnel network can close again under
the influence of the surface tension when the outgoing flux of gas atoms outweighs their supply.
The bubble interconnection has two important
consequences. First of all, it determines the onset of
release as the release remains small (due to athermal

release) before grain boundary bubbles interconnect
with open grain edge tunnels. This incubation period
is reflected in the Vitanza threshold for FGR, which is
shown in Figure 13. The ensuing release corresponds
to a seepage process. Second, when grain face bubbles interconnect and form snake-like tunnels, there
will be a sudden release of the gas accumulated in
these bubbles, referred to as ‘burst release.’ This can
also be interpreted as a sudden interconnection or
opening of grain face bubbles due to microcracking
along grain boundaries during abrupt power variations. Cracking results in a sudden opening of a fraction of the grain boundaries with the instantaneous
venting of the corresponding fraction of the accumulated gas atoms.
Interconnection of gas-filled bubbles takes place
in general where diffusion-controlled precipitation
occurs at the grain boundaries: that is, when both the
temperature and the burnup are high enough. The
conditions correspond roughly to the Vitanza62,63
threshold:
Tc ð CÞ ¼

9800
À BU Á
ln 0:005

where Tc represents the central temperature in ( C)
and BU the burnup in (MWd kgÀ1) UO2.


554

Oxide Fuel Performance Modeling and Simulations

1800
Original 1% data
Siemens 2% data

Central temperature (°C)

1700
1600

New 1% data
Empirical Halden threshold

1500
1400
1300
1200
1100
1000
900
800
0

10

20

30
40
50

Burnup (MWd kg–1 UO2)

60

70

Figure 13 Original Halden (or Vitanza) criterion for the onset of fission gas release and supporting data. Data from
Vitanza, C.; Graziani, U.; Fordestrommen, N. T.; Vilpponen, K. O. Fission Gas Release from In-Pile Measurements;
HPR-221.10; OECD Halden Reactor Project; 1978; Vitanza, C.; Kolstad, E.; Graziani, C. Fission gas release from UO2 pellet at
high burnup. In Topical Meeting on Light Water Reactor Fuel Performance, Portland, OR, May 1979; American Nuclear
Society: Portland, OR, 1979.

3.19.2.3.2 Modeling the fission gas behavior

There are various approaches in FGR and swelling
modeling. They can be classified into two categories.
On one hand, there are purely empirical models,
including those based on soft computing techniques
such as neural networks. These models are inexpensive
to use and provide an efficient tool for the design of fuel
rods within a limited range of application. However,
they are not suitable for gaining knowledge about the
underlying mechanisms, nor do they enable us to extend
their range of application to higher discharge burnup
values as required by the industry. On the other hand,
there are mechanistic models which aim at the physical
description of the underlying phenomena as explained
in Chapter 3.20, Modeling of Fission-Gas-Induced
Swelling of Nuclear Fuels. Despite their great data
needs, such models provide an excellent basis both for

the analysis of the mechanisms and for the extension
of the models beyond their range of calibration.
Fuel performance codes nowadays tend to implement more and more mechanistic models, based on
very detailed but stand-alone models (i.e., models that
are not implemented in a fuel performance code, such
as Van Uffelen57 and Noirot64). They all consider
FGR to be a two-step process. The first step deals
with the gas behavior in the grains (intragranular part),
whereas the second step deals with the gas behavior
along the grain boundaries (intergranular part).
3.19.2.3.2.1 Intragranular behavior

For the behavior in the fuel grains, the following
scenario is generally adopted. The gas atoms are

created by fission in the fuel matrix. They then diffuse in the grains toward grain boundaries by thermal
and irradiation-enhanced diffusion. Small intragranular bubbles with a diameter of 1–2 nm are observed
in irradiated fuel. They are created in the wake of
fission spikes and then grow by diffusion (trapping).
They are continuously destroyed by fission spikes
(resolution). There is no bubble migration except at
temperatures above roughly 1800  C. The bubbles act
as sinks for gas atoms, thereby reducing the amount
of gas available for release.
This scenario leads to solving a diffusion equation
in a sphere with a source term proportional to
the local fission rate density (S ¼ Yfp F_ ), which
is based on the pioneering model of Booth.65 He
proposed the equivalent sphere model. This theory
considers a polycrystalline sinter as a collection of

uniform spheres with an equivalent radius in order
to simplify the mathematical problem. The hypothetical sphere radius (RB) is defined so that the
effective surface-to-volume ratio of the fuel (S/V) is
preserved:
 
V
RB ¼ 3
S t
where (S/V)t accounts for the sum of the geometric
surface of the pellets as well as the surface due to
open porosity. As irradiation proceeds, fission gases
are generated within the Booth sphere and migrate to
the surface, where the concentration is taken to be
zero. He proposed that the fractions of stable gas
release can be approximated by


Oxide Fuel Performance Modeling and Simulations

sffiffiffiffiffiffiffiffiffi
Dt
Dt
fann ðt Þ ffi 6
À3 2
2
pRgr
Rgr
for so-called annealing conditions (i.e., without
source term, but with an initial nonzero concentration), and
sffiffiffiffiffiffiffiffiffi

Dt
3 Dt
À
firr ðt Þ ffi 4
pR2gr 2R2gr
for irradiation conditions (nonzero source term, but no
initial concentration). In a second model, he proposed
the approximation for the release-to-birth ratio for
unstable gas release under steady-state conditions66:
rffiffiffiffi!
rffiffiffiffi"
rffiffiffiffi#
R
3
D
l
1
D
À
¼
coth Rgrain
B Rgrain l
D
Rgrain l
where l represents the decay constant of the species
under consideration. It should be underlined that the
diffusion coefficient to be used is subject to an order of
magnitude uncertainty. The expression in Section
3.19.2.3.1 is often being used with a multiplication or
reduction factor of about 5.

Regardless of the uncertainty on the diffusion
coefficient, the Booth models themselves suffer
from several limitations:
1. They consider a constant temperature and fission
rate density.
2. They do not account for resolution and trapping at
intragranular bubbles.
3. They do not account for grain boundary sweeping.
4. They cannot reproduce an incubation period
(see Vitanza curve).
5. They do not account for resolution at grain
boundary bubbles.
All these limitations have been alleviated over time.
First, several numerical techniques have been proposed to cope with time-varying conditions, which
have been compared in Lassmann and Benk.67 In
order to deal with trapping and resolution, Speight68
found that, instead of solving one diffusion equation
coupled with an equation for the gas balance in the
traps, one could solve a single diffusion equation for
the sum of the concentration in the matrix and in the
traps with an effective diffusion coefficient (Deff):
b
Deff ¼ D
bþg
where g ¼ 4pRbubble D corresponds to the trapping
rate coefficient and b corresponds to the resolution rate
coefficient. Whatever model is being considered for

555

resolution, fission gas behavior models generally introduce a simple resolution rate coefficient that is proportional to the local fission rate density and depends
on the bubble size:
b ¼ 2pðRbl þ dÞ2 mff F_
where it is assumed that a bubble can be destroyed if
its center lies within a distance d from the fission
fragment track of length mff ¼ 7–10 mm. The condition
for applying Deff is that the traps are saturated. Experiments show that small intragranular bubbles stabilize
rapidly both in size and in diameter. Intragranular
bubbles can therefore be considered saturated for irradiation times of practical interest (beyond 0.5 MWd
kgUÀ1). Nevertheless, the difference between D and
Deff is only important for temperatures above
$1100  C. It should be underlined, however, that during a power ramp the application of Deff provides an
overestimation of the trapping effect.69
Over time, several models have been proposed
wherein the Booth sphere radius was taken to be
equal to the average grain radius of the fuel, in order
to be able to account for grain growth. However, it must
first be pointed out that there is no consensus about
which grain growth model should be applied, even
though the Ainscough et al.70 model is often applied:


dRgr
1
1
¼k
À
dt
Rgr Rmax
where k is a temperature-dependent rate constant and

Rmax ¼ Rmax(BU) is the grain size at which growth stops.
This burnup-dependent quantity is introduced in
order to account for the retarding effect of fission
products on grain growth as burnup proceeds.
Most FGR models only account for the increase of
the average diffusion distance when normal grain
growth occurs. Some other models only take into
consideration the sweeping effect, assuming either
that the fractional release is proportional to the
grain boundary velocity, or that the gas in the total
fraction of grain volume swept by the grain boundaries is released. So they all fail to properly incorporate boundary motion into the intragranular diffusion
equation and artificially separate the two aspects of
grain growth on FGR. Only some stand-alone models
have been proposed so far that account for both
simultaneously by solving the diffusion equation in
a sphere with a moving boundary (e.g., Suzuki and
Saitou14 and Forsberg and Massih71).
For alleviating the fourth and fifth limitations of
the Booth models, an intergranular module has to be
introduced.


556

Oxide Fuel Performance Modeling and Simulations

3.19.2.3.2.2 Intergranular behavior

Three main different concepts are being applied.
First of all, an intergranular model that does not

model the kinetics at the grain boundaries directly.
In a way, the Booth model is a special case of this
type, in that it considers gas atoms to be released as
soon as they arrive at the grain boundary. The other
models in this category consider gas arriving at the
grain boundaries to precipitate straight away in grain
boundary bubbles. An open tunnel network is assumed
to be established along the grain boundaries once a socalled saturation value for the intergranular gas atom
concentration (Nmax) has been collected. In order to
derive this saturation value, one assumes the following:
1. Intergranular bubbles are lenticular, with y being
the dihedral angle between the grain boundary
and the bubble surface (see Figure 14).
2. A mechanical equilibrium exists between the bubble gas pressure (pgas), the surface tension (g), and
the hydrostatic pressure (sh) in the surrounding
matrix: pgas ¼ r2g þ sh , where rbl is the radius of
bl
curvature of the grain boundary bubble.
3. A perfect gas law can be applied as the equation
of state.
Under the above assumptions, the following expression is obtained for Nmax:


4rbl f ðyÞ Ã 2g
f
þ sH
Nsat ¼
rbl
3kT ðsin yÞ2
where f ðyÞ ¼ 1 À 32 cos y þ 12 ðcos yÞ3 and f* stands

for the fraction of the grain face surface occupied by

P
r sin q

r

q
Q h

q

S

O

r cos q

R
Figure 14 Schematic representation of a lenticular grain
face bubble with radius of curvature r. From Handbook of
Nuclear Engineering, Dan Gabriel CACUCI, doi: 10.1007/
978-0-387-98149-9_13. Copyright 2010, with permission
from Springer ScienceþBusiness Media B.V.

the bubbles at the interconnection. As soon as Nmax is
achieved, any excess gas atoms arriving at the
grain boundaries are deemed released. It must be
pointed out that no consensus has been reached as to
what value should be applied for the hydrostatic stress.

Often it is neglected, or a constant and uniform value is
used.26 Obviously this is a rough approximation, and in
the FEMAXI code the calculated hydrostatic stress
resulting from thermal stress can be used.14
In a second category of models, the intergranular
kinetics is considered directly. As such, account is
made of the reversible character of the tunnel establishment. The first model was proposed by White and
Tucker.72 They considered two parallel processes for
release from the Booth sphere: intragranular diffusion
and gaseous diffusion through tunnels along grain
boundaries. To this end, they solve two different
diffusion equations in the equivalent Booth sphere.
More recently, Koo et al.73 proposed two different
contributions for the S/V used to compute the
equivalent Booth sphere radius. One contribution
was attributed to macroscopic radial cracks in the
pellet periphery, while the second contribution was
ascribed to a fraction of the tunnel network along
grain boundaries that was in contact with the open
grain corner porosity (based on percolation theory in
two dimensions). Recently, several other models have
been proposed wherein the gaseous diffusion through
the open tunnel network is modeled according
to Darcy’s or Poisseuille’s law in a tube (e.g., Van
Uffelen57 and Kogai74). These models enable the
effect of the hydrostatic pressure on the release
kinetics to be accounted for. More recently, White75
went even further with the details, and modeled the
evolution of the bubble morphological relaxation
through differential absorption/emission of vacancies, which is surface-curvature driven. Nevertheless,

these models are not yet implemented as standard
models in fuel performance codes.
In a third category of models, a comprehensive
list of mechanisms (see Section 3.19.2.3.1) has been
implemented in the form of a set of ordinary differential equations; that is, the intra- and intergranular
parts of the model are solved simultaneously (e.g.,
MARGARET64). Some of these models were essentially produced to deal with fission product behavior
under severe accident conditions. Up to now, only the
FASTGRASS model of Rest and Zawadski76 is
applied in the VICTORIA code.77 More recently, a
detailed fission gas behavior model has been coupled
to FALCON78 and is not only being applied to
the analysis mostly of reactivity-initiated accident


Oxide Fuel Performance Modeling and Simulations

557

(RIA)79 and LOCA,80 but also provides some novel
results in the description of power ramps.81

3.19.3 Design Basis Accident
Modeling

3.19.2.3.2.3 Coupling intra- and intergranular
behavior

Fuel designers and safety authorities rely heavily on
fuel performance codes since they require minimal

costs in comparison with the costs of an experiment
or an unexpected fuel rod failure. Nevertheless, two
types of fuel performance codes are generally being
applied, corresponding to the normal operating and
the design basis accident conditions (see Chapter
2.22, Transient Response of LWR Fuels (RIA)
and Chapter 2.23, Behavior of LWR Fuel During
Loss-of-Coolant Accidents, respectively). In order
to simplify the code management by limiting the
number of programs and to take advantage of the
hardware improvements, there is a tendency to generate a single fuel performance code that can cope
with the different conditions, which requires different types of developments.
On one hand, extending the application range of
a fuel performance code originally developed for
steady-state conditions to accident conditions requires
modifications to the basic equations in the thermomechanical description of the fuel rod behavior,6 stable
numerical algorithms, and a proper time-step control,
in addition to the implementation of specific models
dealing with the high-temperature behavior of cladding such as observed under LOCA conditions.85 For
dealing with RIA events, one should check carefully
the thermal expansion model because of the edgepeaked power distribution, as well as the other models
affecting the effective cold gap width,86 and the model
for thermal heat transfer in the plenum. On the other
hand, for fuel performance codes developed to simulate various aspects of the nuclear fuel behavior under
accident conditions, such as TESPA,87 MFPR,88
FRAPTRAN,89 SCANAIR,90 FALCON,91 or
RANNS92 code, the thermomechanical behavior of
the fuel must be incorporated and/or the extension
of models to normal operating conditions is necessary
to consider burnup-dependent phenomena such as

thermal conductivity degradation, FGR, gap closure
and swelling, as well as cladding corrosion. Such a
posteriori modifications of the fuel performance code
may entail difficulties in terms of convergence and
calculation time.

In general, the intra- and intergranular modules of
a fission gas behavior model are coupled in two
directions. On one hand, the intragranular module
provides the source term for the intergranular module.
On the other hand, the intergranular module provides
the boundary condition for the diffusion equation
in the spherical grains and/or the supplementary
source term near the grain boundary. In fact, most
models make use of the Booth approximations; that is,
they assume zero boundary conditions, meaning that
the grain boundary is considered to be a perfect sink.
Some models consider a finite segregation factor. In
order to account for the resolution effect on grain
boundary bubbles, three different approaches are
being utilized. The first group considers a correction
factor for the Booth flux, accounting for the fact
that the resolution opposes the gaseous diffusion out
of the grains. The second group of models applies a
time-varying boundary condition that makes use of
a time-dependent flux balance. In the third and last
group, a supplementary source term is defined in a fine
layer adjacent to the grain face, either as a uniform
source in a fine layer or as a Dirac distribution at a
given distance from the grain boundary. Mainly the

first two approaches have been implemented in fuel
performance codes.
3.19.2.3.2.4 Swelling

Theoretically speaking, the FGR and the fission gas
swelling models should be closely related. In most
codes, however, semiempirical relations are being
used for the gaseous swelling as a function of the
temperature and burnup (e.g., MATPRO correlation82,83), or as an empirical function of the released
fraction, for instance Billaux84:
 
DV
¼ Að1 À aFG FGR À aCs CSRÞBU
V gaseous FP
where A is a constant for solid swelling, FGR is
the local fraction FGR from the grain, CSR is the
fractional release of the volatile fission products
(Cs, I) from the grain, BU is the local fuel burnup,
aFG ¼ 0.37, and aCs ¼ 0.45. The empirical nature
reflects the uncertainty pertaining to both the release
and swelling models (in particular during power
ramps at high burnup), as can also be inferred from
the large variety of models presented above.

3.19.3.1

Loss-of-Coolant Accident

3.19.3.1.1 Specific LOCA features

As explained in Chapter 2.23, Behavior of LWR
Fuel During Loss-of-Coolant Accidents, a LOCA


558

Oxide Fuel Performance Modeling and Simulations

in a water-cooled reactor consists of a large break of
the coolant primary system and the consequent loss
of core cooling capacity. It is a design basis accident:
that is, one that the plant design must account and
accommodate for in terms of ensuring that a core
coolable configuration is maintained. Post-LOCA
coolability is achieved by an immediate reactor
scram, that is, a rapid insertion of control rods in
the core which ends the fission power production in
the nuclear fuel, and by the activation of emergency
core cooling systems (ECCS) which enters into function upon depressurization of the primary system,
within a relatively short time after the start of the
transient.
As schematically shown in Figure 15, a typical
LOCA sequence in a PWR consists of three main
phases. During the first phase, depressurization of the
primary system occurs, which is accompanied by a
reactor scram. In the course of this phase, the fuel
temperature may oscillate depending on position,
timing of the power reduction, and timing of the
depressurization. The cladding temperature may
even decrease since the water still has enough cooling

capacity and because the water temperature is
decreasing as pressure decreases.
In the second phase, the core heats up because of
the remaining heat generation in the fuel (stored and
decay heat) and the loss of water. Normally, the fuel
cladding temperature increases at moderate rate, approximately some few degrees per second in the earlier phase of the temperature escalation. When the
temperature exceeds 600  C, the cladding becomes
more and more prone to plastic deformation and,
under the effect of the rod inner pressure – which is

ECR

Temperature (ЊC)

Fuel
1200

PCT

Clad
Oxidation

800

Cooling
Quenching

Burst
400

Rupture

Ballooning

50

100
Time (s)

150

Figure 15 Schematic evolution of the cladding
temperature in a fuel rod during a loss-of-coolant
accident.

appreciably larger than in the depressurized reactor
vessel – it starts deforming and ballooning in the
space between the fuel assembly grid spacers. If
the ballooning exceeds a certain size, the fuel rod
cladding will burst open, breaking the first barrier
to radiation containment. In addition, the Zircaloy
cladding undergoes a crystallographic transition from
the a-phase to a b-phase at temperatures exceeding
900  C (see Section 3.19.3.1.2.1), although the transition temperature is affected by the amount of hydrogen contained in the metal cladding as well as the
heating rate.
In the third phase, the cladding temperature
continues to increase but gradually due to the
increasing radiant and convective heat transfer,
exhibiting a relatively flat temperature trend for
some period normally lasting for few minutes.

Under the ECCS effect, the cladding temperature trend is then reversed and, after a period
of gradual temperature decrease, the cladding is
finally quenched by the ECCS water rising in
the core.
3.19.3.1.2 Specific LOCA modeling
requirements

The major concern from the fuel safety criteria viewpoint is the rapid cladding oxidation taking place
when the cladding is heated up to high temperature
and exposed to steam environment, as it occurs in a
LOCA. In fact, the safety criteria for LOCA are stated
as five requirements (although small variations exist
in each country) dealing with the calculated performance of the cooling system under the most severe
LOCA conditions:
1. In order to preserve cladding ductility, its oxidation shall be limited to 17% of the metallic wall
thickness (i.e., equivalent cladding reacted, ECR),
where the oxidation level is computed by means of
the Baker–Just correlation (see Section 3.19.3.1.2.2)
assuming conservatively that the inner and outer
surfaces of cladding are oxidized concurrently, that
is, two-side oxidation.
2. In order to prevent rapid cladding embrittlement
as well as runaway oxidation, the cladding peak
temperature shall remain below 1204  C.
3. The total amount of hydrogen generation shall
not exceed 1% of the hypothetical amount generated by the reaction of all the metal in the
surrounding fuel.
4. The calculated changes in core geometry shall
leave the core amenable to cooling.

Oxide Fuel Performance Modeling and Simulations

5. After any operation of the ECCS, the core temperature shall be maintained at an acceptable low
value and decay the heat removed for the extended
period required by long-lived radioactivity.
The first two criteria can be assessed by means of fuel
performance codes containing some specific models
such as high-temperature oxidation and deformation,
as well as the crystallographic phase transition and
the rupture criterion. These specific models are treated in the next four sections.
3.19.3.1.2.1 Phase transition model

At high temperatures, the cladding may undergo a
phase transition from the hexagonal closed packed
(HCP) a-phase to the face-centered cubic (fcc)
b-phase. The a ! b crystallographic phase transition
of zirconium takes place principally in the temperature range of 800–1000  C, which is typical in a
LOCA. The phase transformation is not only temperature dependent but is also influenced by the composition and the impurities of the alloy, the corrosion
level (oxygen and hydrogen contents) of the cladding,
and the temperature increase rate.93 Quasiequilibrium conditions are generally assumed in the static
model, where the fraction of the b-phase (’) is a
simple function of the temperature: ’ ¼ f (T ). Specific
functions are available for each alloy. Figure 16
illustrates these functions for Zircaloy-4 and for
Zr1%Nb,94 indicating that due to the niobium content the phase transition of zirconium is completed
at temperatures about 100  C lower. In the case
of a dynamic model, as proposed, for example,

Zr1%Nb

Fraction of β-phase

by Forgeron et al.,95 the effect of the temperature
variation rates on the phase transitions are directly
taken into account by means of a differential
equation:
d’
¼ K ðT Þ’ð1 À ’Þ
dt
where K(T) is an appropriate function of the temperature in the form

 À

Á
K ¼ ÆT À f À1 ð’Þexp C1 þ C2 T À f À1 ð’Þ
with t as the time, and C1 and C2 as empirical constants. During temperature variations, the fraction of
the b-phase can therefore be calculated through a
numerical algorithm:
’i ¼ ’iÀ1 þ K ðTi ; ’iÀ1 Þ’iÀ1 ð1 À ’iÀ1 ÞDt
where the indexes i and i À 1 denote the actual
and the previous time steps, respectively. As an
example, Figure 16 also represents a hysteresis loop
of the phase transition calculated for Zircaloy-4 by
means of the dynamic model at Æ10  C sÀ1 heating/
cooling rates.
3.19.3.1.2.2

High-temperature oxidation

The cladding–steam reaction model is often based on

parabolic kinetic correlations for both the oxygen
mass gain and/or the ZrO2 layer thickness growth.
The actual reaction rate constant is defined as a
function of the temperature through an Arrhenius
relation:
Km ¼ Am eÀQm =T
where Km is the oxygen mass gain rate in mg cmÀ2 s1/2,
Am is the pre-exponential factor, Qm is the activation
energy divided by the perfect gas constant, and T
is the cladding temperature in K. The kinetics
of high-temperature oxidation of Zircaloy were
described by Baker and Just in 196296 by the following equation:

1.0
Zircaloy

0.8

559

0.6

w ¼ 2029t 1=2 eÀ11450=T
0.4
+10 ЊC s–1
0.2
0
700

−10 ЊC s–1

800

900
1000
Temperature (°C)

1100

Figure 16 Fraction of the b-phase as a function of
temperature calculated by the equilibrium model for
Zircaloy-4 and Zr1%Nb and by the dynamic model for
Zircaloy-4 at Æ10  C sÀ1 temperature rates.

where w is the weight of metal zirconium reacted per
unit surface in mg cmÀ2, T is the temperature in K,
and t is the time (at the constant temperature T)
in seconds. Although a more precise correlation was
derived subsequently by Cathcart and Powel (C–P),97
the Baker–Just (B–J) correlation is quoted here because
it is the reference correlation for expressing the cladding zero ductility criterion for LOCA accidents. An
improved correlation was developed by Cathcart and
Powel97: