Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations

Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations Comprehensive nuclear materials 3 19 oxide fuel performance modeling and simulations Comprehensive nuclear materials 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.

BOL Beginning of life

BWR Boiling water reactor

CANDU CANada Deuterium Uranium

(C)SED (Critical) strain energy density

CSR Volatile fission product release

CZP Cold zero power DFT Density functional theory DNB Departure from nucleate boiling ECCS Emergency core cooling systems ECR Equivalent cladding reacted EOL End of life

EPMA Electron microprobe analysis fcc Face-centered cubic

535

FDM Finite difference method

FEM Finite element method

FGR Fission gas release

HBS High-burnup structure

HCP Hexagonal closed packed

HM Heavy Metals

HZP Hot zero power

IAEA International Atomic Energy Agency

IFA Instrumented fuel assembly

LEFM Linear elastic fracture mechanics

LHR Linear heating rate

LOCA Loss-of-coolant accident

LWR Light water reactor

MC Monte Carlo

MD Molecular dynamics

MIMAS Micronized master blend

MOX Mixed oxide

NEA Nuclear Energy Agency

O/M Oxygen-to-metal ratio

OECD Organisation for Economic Cooperation

and Development

PAS Positron annihilation spectroscopy

PCI Pellet–cladding interaction

PCMI Pellet–cladding mechanical interaction

PWR Pressurized water reactor

RIA Reactivity-initiated accident

S/V Surface-to-volume ratio

SANS Small-angle neutron scattering

SCC Stress corrosion cracking

SM Shell model

TAD Temperature-accelerated dynamics

TEM Transmission electron microscopy

3.19.1 Introduction

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

com-puter 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 sup-porting loading of fuel into a power reactor, that is,

to verify compliance with safety criteria in safety casesubmissions A list of commonly used fuel perfor-mance codes is provided inTable 1

Size of the Problem

In principle, our spatial problem is three-dimensional(3D) However, the geometry of a cylindrical fuel rod (avery long, very thin rod) suggests that any section of afuel rod may be considered as part of an infinite body:that is, neglecting axial variations By further assumingaxially symmetric conditions because of the cylindricalgeometry, the original 3D problem is reduced to a 1Done Analyzing the fuel rod at several axial sections with

a (radially) 1D description is sometimes referred to asquasi-2D or-11=2D Most fuel rod performance codesfall into this category Real 2D codes such as, forinstance, the FALCON code,1 offer the possibility toanalyzer–z problems (no azimuthal variation) and r–’problems (no variation in axial direction) An example of

a 3D code is TOUTATIS2and DRACCAR3and is dealtwith inChapter3.22, Modeling of Pellet–CladdingInteraction DRACCAR is addressed later inSection3.19.3.1.2 Generally, 2D or 3D codes are used for theanalysis of local effects, whereas the other codes havethe capability to analyze the whole fuel rod during acomplicated, long power history

In order to estimate the ‘size’ of the problem athand, 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 stepsare sufficient However, for an irradiation in a researchreactor, such as the heavy-water boiling water reactor(BWR) of the Organisation for Economic Cooperationand Development (OECD), Halden, many more varia-tions of the linear rating with time are recorded In such

a situation, one must either simplify the complicatedpower history or increase the number of time steps tothe order of several thousands The simplest geometricalidealization needs
20 radial and 20 axial nodes; a 2Drepresentation of a single pellet would approximatelyneed several hundred nodes Therefore, local models,which are in almost all cases nonlinear, must be verycarefully constructed, since even for the simplest geo-metrical idealization the number of calls may easilyreach the order of millions:

15 radial 15 axial nodes  5000 time steps  3 iterations

¼ 3:4  106calls

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)

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

condi-tions, irradiation history, etc.) Although they contain

a certain level of uncertainty, they can be properly

managed in actual analyses The third category of

uncertainties is related to the material properties,such as the fuel thermal conductivity or the fissiongas diffusion coefficients The fourth and last cate-gory of uncertainties is the so-called model uncer-tainties A good example of such an uncertainty isthe plain strain assumption in the axial direction asillustrated in Figure 1, representing the interaction

of the deformed and cracked fuel with the cladding.Intuitively, it is clear that for a detailed analysis

Table 1 List of fuel performance codes

COMETHE 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 COPERNIC a Bonnaud, E.; Bernard, C.; Van Schel, E Trans Am Nucl Soc 1997, 77

ENIGMA 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

FALCON, FREY 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

FEMAXI Suzuki, M.; Saitou, H Light Water Reactor Fuel Analysis Code FEMAXI-6 (Ver 1); JAEA-Data/Code

2005-003, Feb 2006 FRAPCON 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

METEORa Struzik, C.; Moyen, M.; Piron, J High burnup modelling of UO 2 and MOX fuel with METEOR/

TRANSURANUS Version 1.5 In Proceedings of the International Topical Meeting on LWR Fuel Performance, Portland, OR, Mar 1997

PIN-micro Pazdera, F.; Strijov, P.; Valach, M.; et al User’s guides for the computer code PIN-micro; UJV 9512-T,

Rez; Nov 1991 START 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 TRANSURANUS Lassmann, K The TRANSURANUS code – past, present and future; Review article, ITU Activity Report

2001 – EUR 20252, ISBN 92-894-3639-5; 2001

aBased on TRANSURANUS.

Deformed geometry

Two-dimensional description

One-dimensional description

Nondeformed geometry (fresh fuel) Fuel pellet Cladding

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

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

The objective of this section is to describe how the

temperature distribution in a nuclear fuel rod is

calcu-lated 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

tempera-tures 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 densityq000, which is the produced energy per

unit volume and time It is usually assumed thatq000

depends only on the radius and on time The linear

rating is then simply given by

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

cl are the average power density in the fuel and

cladding, respectively, and f ðrÞ is a radial

distribu-tion (form) funcdistribu-tion (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 q000

3.19.2.1.1 Axial heat transfer in the coolant

In general, three regimes must be covered in a lightwater reactor (LWR):

1 The subcooled regime, where only surface boilingoccurs This regime is typical for pressurizedwater reactors (PWRs) under normal operatingconditions

2 The saturated, two-phase regime This regime

is typical for BWRs under normal operatingconditions

3 The saturated or overheated regime This regimemay be reached in all off-normal situations A typicalexample is a LOCA

The fuel rod performance codes use 1D (axial) fluiddynamic equations that can only cope with the firsttwo regimes For simulating the third type of regime,the whole reactor coolant system needs to be ana-lyzed by means of thermohydraulic system codessuch as RELAP, TRACE, or ATHLET in order toprovide adequate boundary conditions to the fuel rodperformance code

The temperature calculation in the coolant servestwo purposes First of all, the axial coolant tempera-ture in the basic (fictional) channel provides the(Dirichlet) boundary condition for the radial tem-perature distribution in the fuel rod It results fromthe combined solution of the mass, momentum, andenergy balance equations The simplified equationused in fuel performance codes reads

w the velocity, T the temperature, q00

cl ;c the heatflux from the cladding to the coolant,A the channelcross-sectional area, rcl,o the cladding outer radius,andq000

c the power density in the coolant In general,the heat flux from cladding to coolantq00

cl;cshould becomputed by means of a thermohydraulic code.Mathematically, the boundary condition is of theconvective type:

q00 cl;c¼ l @Tðr; tÞ

@r



rcl;o

¼ afTðr ¼ rcl;oÞTcgwhere a is the heat transfer coefficient between thecladding and the coolant and Tc¼ Tcðz; tÞ is the(bulk) coolant temperature Only for a steady-statecondition

crwdT

q0

the heat flux from the cladding to the coolant is

known and is given by

q00 cl;c¼ q02prcl ;oUnder normal operational conditions, the mass flow

rate _m ¼ Arw, and the coolant inlet temperature and

pressure are prescribed In an off-normal or

acciden-tal 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

tem-perature drop between the coolant and the cladding

Tcl Tc, resulting from convection:

q00¼ afilmðTcl TcÞ ¼ qc000

2prcl ;oThe 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

cor-relation is largely applied, whereas in the saturated

regime of a BWR, the Jens–Lottes correlation is

applied (see separate lecture on thermohydraulics)

forZircaloy), and the heat generation in the cladding is

generally neglected (the g-heating as well as the

exo-thermic clad oxidation process are generally

disre-garded) In order to account for the presence of an

outside oxide layer with a thermal conductivity on the

order of 2 W mK1for 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

the fuel pellet

The temperature difference in the pellet–cladding

gap,DTgap, is calculated as follows7,8:

DTgap¼ q00

hgap

where q00is the heat flux in watts per unit area and

hgapis the heat transfer coefficient between the fueland the cladding (gap conductance) Heat transfer byconvection can be neglected In general, the heattransfer coefficienthgapdepends on

1 gap width or contact pressure between fuel andcladding

2 gas pressure and composition

3 Surface characteristics of cladding and fuel

In fact, there are three parallel conduction routes:

hgap¼ hradþ hconþ hgasThe contribution of the radiative component isgiven by

Tf  Tclwhere Csis the Stefan–Boltzmann constant, e is theemissivity, and T is the temperature The radiativecomponent is<1% during normal operating condi-tions because of the limited difference between thetwo surface temperatures

The componenthconreproduces the improvement

in heat transfer due to contact pressure:

The heat transfer through conduction in the gas isoften based on the model of Ross and Stoute9:

dþ s þ gfþ gclwhere the thermal conductivity of a multicomponentgas is only composition-dependent and calculated bymeans of

lgas¼Xn

j ¼1

lj

1þ Pn k¼1

j 6¼k

wjkc k

c j

0B

@

1CA

2666664

3777775with c and w being the molar concentrations andweighting factors, respectively The gas extrapolation

lengths gf and gcl (or temperature jump distance)

account for the imperfect heat transport across the

solid–gas interface, which is dependent on the

mate-rial and gas pressure Detailed formulations are

dis-cussed 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,

seeSection 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:

rc @T

1r

where c is the specific heat at constant pressure for

fuel The boundary conditions are

Inner boundary:@Tðr ¼ ri;tÞ

Outer boundary: DTgap¼ q00

hgapðpellet surfacetemperature is knownÞThe temperature distribution in the pellets is there-

fore 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,

q000

BOLffi I0ðrÞ, where I0ðrÞ is the modified Bessel

func-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

Figure 2) This effect therefore needs to be

consid-ered, and a specific model for the radial power

den-sity 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:

l¼ lphþ lel At temperatures below 1500 K, the

phonon contribution predominates Above this

tem-perature, the electronic contribution becomes

impor-tant When applying the kinetic gas theory to the

propagation of atomic vibrations (phonons) or

quasiparticles, it appears that the phonon conductivity

in the temperature range of interest can be expressed as

A þ BTwhere A corresponds to the scattering of phonons

by imperfections such as point defects, line and planardefects, fission gas bubbles, etc The parameterB cor-responds to the scattering by phonon–phonon (Umk-lapp) interactions When the burnup in the pelletsincreases, the accumulation of point defects and fissionproducts will increase the phonon scattering (the Aterm) The same happens if the fuel (e.g., UO2) isdoped with a neutron absorber such as Gd2O3, or if adeviation 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 generalA ¼ A(BU, Gd, Pu, x), where

BU denotes the local burnup

The temperature-dependent creation of tronic carriers, that is, excitation of free electrons,leading to lelis typically expressed as

pho-of a theoretically dense fuel (lTD) with the porositylevel (P) in the pellets10

:

0 1 2 3

EOL average burnup

van de Laar, J J Nucl Mater 1998, 255, 222–233, with permission from European Commission.

Maxwell–Eucken: l¼ lTD 1P

1þaP, where a is afunction of pore shape

Loeb: l¼ lTDð1  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

perfor-mance code can best be summarized as follows: The

material properties l, r, and c are organized in an

independent database, whereas the power densityq000,

the gap conductancehgap, and the heat transfer

coeffi-cient between the cladding and coolant a are

formu-lated 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

result-ing temperature distribution calculated by means of

The first barrier against release of radioactive fission

products to the environment is the cladding of the

nuclear fuel rod The stress and associated

deforma-tion 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

con-ductance of the gap and hence the temperature

distribution in the pellets The thermal and mechanicalanalyses are therefore equally important and closelycoupled In principle, both problems should therefore

be solved simultaneously In practice, however, all fuelperformance codes solve them separately but providecoupling through an iterative scheme This importantnumerical aspect will not be dealt with in this chapter.The interested reader is referred to a general discus-sion on this issue in various references.11–13

The next sections summarize how stress andstrains are calculated in both the ceramic pelletsand the metallic cladding, while underlining themain assumptions and limitations

The main assumptions generally made in fuel mance codes are the following:

perfor-1 The system is axisymmetric, that is, variables donot vary tangentially

2 Although the fuel and cladding move axially (notnecessarily at the same rate), planes perpendicular

to the z-axis remain plane during deformation(generalized plain strain condition), that is, therod remains cylindrical

3 Dynamic forces (inertia) are in general not treated,and the time dependence inherent in the analysis(creep) is handled incrementally (quasisteady)

0 500 1000 1500 2000

Data

Data

Cladding Fuel

He-filled rod Xe-filled rod

Radial position (mm)

IFA-504; linear rating q⬘ = 20 kW m –1

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.

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

The first two assumptions reduce the problem to one

dimension The third assumption indicates that the

stresses are related through a local equilibrium

con-dition for the radial force in the following form:

dsr

st srRwhere 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:

er¼dudR

et¼uR

whereu represents the radial deformation and eiare

the normal strains

Finally, the last equation relates the stresses to the

strains Based on the fifth assumption, the constitutive

strains consist of various contributions First of all,there is the thermal strain resulting from temperaturedifferences, which is assumed to be isotropic andreversible:

et

i ¼ aðT  T0Þ i 2 fr; t; agThe thermal expansion coefficients depend on thematerial and the temperature itself, as shown inFigure 4

The larger thermal expansion of UO2with respect

to that of Zircaloy explains why thermal expansion isone of the largest contributions to the gap closure in anuclear fuel rod at BOL

3.19.2.2.2.2.2 Swelling The second contribution tothe nonelastic strain in the fuel pellets comes fromswelling, and is also assumed to be isotropic The fuelswelling in turn has four contributions:

esfuel¼13

DVV

where the first term is attributed to the inexorable

swelling of solid fission products:

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 t1

The second term comes from gaseous fission product

R3NðRÞdR

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(seeSection 3.19.2.3.2) During the initial stages

of the irradiation (BU < 10 MWd kg1

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

shrink-age 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

P0

¼ a½1  b expða1BUÞ  ð1  bÞexpða2BUÞ

where a¼ [T (in C) – 83]/(288Dgr), ab¼ 5.12 exp

[5100/T (in K)], a2¼ 0.0016 tUO2 per MWd, and

a1¼ 100a2 The densification, together with the solid

fission product swelling, is illustrated inFigure 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

ODvolkT

P

R2 grs

or the plastic flow (i.e., dislocation climb or othermodel of creep)

3.19.2.2.2.2.3 Plasticity and creep The third tribution to the nonelastic strain in the fuel is visco-plastic in nature It consists of the instantaneousplastic deformation when the yield stress is exceededand of the time-dependent creep For the fuel andcladding, a simple isotropic plastic flow model can beapplied Nevertheless, as creep is the main contribu-tor to stress relaxation after cracking (see Section3.19.2.2.2.2.4) in the oxide pellets, it is often onlyconsidered in the cladding

con-In a multiaxial state of stress, a method of relatingthe onset of plastic deformation to the results of auniaxial test is required Furthermore, when plasticdeformation takes place, one needs to determine (1)how much plastic deformation has occurred and (2)how that deformation is distributed among the indi-vidual components of strain in the principal direc-tions For the first requirement, a so-called yieldfunction is needed This may be 1D like the VonMises criterion, assuming that the shear stress isneglected and the material is isotropic14,19:

UO 2 ), 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.

seff¼ 1ffiffiffi

2

p ½ðsr stÞ2þ ðsr saÞ2þ ðst saÞ21=2

so that yielding only occurs when the effective or

equivalent stress (seff) exceeds the yield stress

deter-mined from a uniaxial tensile test Others have

intro-duced the anisotropic factors according to Hill’s

methodology.6Finally, a multidimensional yield

sur-face20,21has also been proposed In order to account

for work-hardening, one generally assumes that the

yield stress changes with the total permanent

defor-mation 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:

Dei¼ Deeff@seff

@si i 2 fr; t; agWhen using the above-mentioned definition of the

equivalent stress, one obtains the Prandtl–Reuss

flow rule22:

Dep

i ¼3Dep

eff2seff Si i 2 fr; t; agindicating that the plastic strain increment is

proportional to the deviatoric stress Si¼ si sh,

where sh¼ (srþ stþ 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:

_eeff¼ K sn

efftmwhereK, 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,

inating at low temperatures and assumed to be

pro-portional to the effective stress and the local fission

rate density orq000

3.19.2.2.2.2.4 Pellet cracking The fourth and lastnonelastic strain component stems from the pelletcracking Pellet cracking already occurs at startupwith fresh fuels due to the differential thermal expan-sion since the hot pellet center expands more thanthe cold periphery In order to assess the linear heatgenerating rate at which cracking in cylindrical pelletoccurs, the maximum thermal stress (¼st,max¼ sa,max

at pellet periphery) in an uncracked pellet submitted

to a parabolic temperature gradient

st ;max¼  aEq0

8pð1  uÞlmust be compared with the (uniaxial) fracture stress,

u ¼ 0.31, the thermal diffusivity a ¼ 105

K1, and

an average thermal conductivity of l¼ 3 W mK1,radial cracks are predicted to be initiated in the pelletperiphery at a linear heat rate q0 of the order of

5 kW m1 The number of cracks (Ncr) is dependent

on the linear heat rate Oguma23 proposed a linearmodel for the number of radial cracks, which is illu-strated inFigure 6 In addition to radial cracks, alsoaxial and (especially under ramping conditions) cir-cumferential cracks are formed

The consequences of cracking are very tant in fuel performance modeling Owing to thelarger thermal expansion of the fuel fragments incomparison with that of a monolithic cylinder, and

impor-to vibration-induced motion, they move outward.This is called pellet ‘relocation’ and has a strongimpact on the thermal behavior, as shown inFigure 7

It reduces the pellet–cladding gap size, therebyreducing the temperature levels in the fuel at BOL.This constitutes the largest contribution to the gapclosure (
30–50%, depending on q0) but is also theone which is subject to the largest uncertaintybecause of the stochastic nature of the cracking

Rod power (kW m –1 ) 0

0 8 16

process This also raises questions about the

useful-ness 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: et¼ u/r, where

u ¼ sdg, s being the initial radial gap size and dgthe

fraction of the gap size closing as a result of relocation

An example based on the relocation model in the

FRAPCON3 code24is illustrated inFigure 8.25

When the pellets swell large enough so that theycome into contact with the cladding, which creepsdown under influence of the pressure differencebetween the coolant pressure and the fill gas pressure,then relocation may be (partly) reversed If bothpellet swelling and cladding creep-down continue,the gap is closed and the pellet fragments are pushedinward, so that the relocation approaches totalelimination

40

30 50

500

0

1000 1500 2000

He-filled rod Xe-filled rod

Radial position (mm)

IFA-504; linear rating q⬘ = 20 kW m –1

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.

The effect of relocation on the mechanical

behav-ior 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

frag-ments).26The 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:

3

 NcrE

n0¼ 12

 Ncrnwhich 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

con-stants 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

ap-proaches, however, do not account for crack healing

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 placement at each radial zone is imposed and theradial stress at the outer cladding surface is deter-mined by the coolant pressure: sr(rcl,o)¼ pcool.The boundary condition in the rod depends onthe configuration When pellet–cladding mechanicalinteraction (PCMI) is not established, the radial stress

dis-at the pellet periphery is determined by the fill gaspressure in the fuel rod (pgas): sr(rf,o)¼ pgas For theboundary condition in the pellet center, two possibi-lities exist In hollow pellets, the radial stress at thepellet center is equal to fill gas pressure: sr(rf,i)¼ pgas,whereas in the case of full cylindrical pellets, the radialand tangential stresses are equal at the pellet center.When the fuel and cladding are in contact, a fuelpellet interfacial pressure exists (pfc), which deter-mines the boundary condition at the pellet surface:

sr(rf,o)¼ sr(rcl,i)¼ pfc The other radial boundaryconditions 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 Theaxial strain is therefore determined by an axial forcebalance equation including the fill gas pressure, theplenum spring pressure, the fuel column weight, thefriction forces, and the coolant pressure imposed onboth end plugs of the cladding The friction forcesdepend on the fuel–cladding interaction and can only

be taken into account iteratively Indeed, when one ofthe axial sectionsi is analyzed, it is not known whetherthe frictional forces between fuel and cladding origi-nating from a section above/belowi need to be consid-ered in the axial balance of forces This is schematically

0 0.2 0.4 0.6 0.8 1 1.2

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

mod-els is that such effects are automatically included in

the analysis through the use of specific gap elements,

as explained in more detail inChapter3.22,

Model-ing of Pellet–CladdModel-ing Interaction

3.19.2.2.4 Pellet–cladding interaction

3.19.2.2.4.1 Pellet–cladding mechanical

interaction

As dealt with in detail inChapter3.22, Modeling of

Pellet–Cladding Interaction, PCMI is one of

the important analytical targets in predicting fuelbehavior, since it can be a cause of fuel failure.The mechanical interaction of the fuel with thecladding is mainly caused by the different thermalexpansion rates of the two components (seeSection3.19.2.2.2) However, the poor thermal conductivity

of the fuel results in a strong temperature gradientacross the pellet This leads to an increase of the gapclosure rate at both ends of a pellet because even

in the early stages of irradiation, the differentialthermal expansion in the pellet causes its so-calledhour-glass shape (Figure 11) The resulting internalthermal stresses exceed the fracture stress of UO2

(around 100 MPa) causing pellet fracture Once thepellet has fractured, it is able to deform much morereadily under the effects of the temperature field,causing the pellet ends to bow outward leading tothe hour-glass shape

PCMI in the early stage of irradiation is thereforecaused by the differential thermal expansion of thepellets, leading to a bamboo or ridging deformation

of the cladding The ridges in the cladding, the height

of which can reach 20mm at high powers, coincidewith the pellet ends and can cause hoop stresses ofaround 400 MPa, which is close to the clad yieldstress When aggressive fission products such asiodine are in their vicinity, these local concentrations

of stress and strain lead to the so-called stress sion cracking (SCC) in LWR fuel systems

corro-The onset of PCMI is affected by a number ofdesign and fabrication parameters First of all, thepellet geometry is adapted to extend the onset ofand to mitigate PCMI In particular, the pellet heighthas been reduced with respect to its diameter, whilechamfers were applied for delaying PCMI and reduc-ing the probability of fuel chipping Second, a rodgeometry is important More precisely, the width ofthe original clearance between the pellets and thecladding should be large enough Nevertheless, thegap size is subject to many uncertainties, the largestbeing the pellet fragment relocation In addition, it isgenerally assumed in fuel performance modeling thatthe pellets are located concentrically within the clad-ding, although this is seldom the case Any eccentric-ity in the stacking arrangement, resulting fromfabrication or fuel rod handling, is likely to lead topremature onset of PCMI even though its effect willdiminish as gap closure occurs

Once PCMI has started, both the pellet geometryand the material properties of the interacting compo-nents influence the maximum stresses and strains

as well as their evolution Flat-ended pellet stacks

No radial contact

‘slip’

Cladding

Case 1 Fuel

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.

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

Nev-ertheless, 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

chal-lenges 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

assess-ment 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

com-ponents 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

exert-ing among all the rexert-ing elements of the pellet and

cladding into one entire matrix to obtain a numerical

solution,14while in FDM displacements of each ring,

elements of the pellet and cladding are summed upindependently to determine the diameter change,granting that FDM is actually modified to some extent

to cope with mechanical interactions among the ringelements There have been attempts to develop 3Dfuel performance codes,29 as explained in Chapter3.22, Modeling of Pellet–Cladding Interaction

3.19.2.2.4.2 Irradiation-induced SCC

Failures during power variations are not only uted to stress Stresses increase at the intersection ofradial crack planes, the interpellet planes, and theinner surface of the cladding The brittle nature ofthe failure site, however, has led to the general consen-sus that, although stress is the primary initiator, propa-gation of the crack is chemically assisted, and thus theprocess is termed SCC The chemical assistance forbrittle cracking in all probability arises from the envi-ronment established by the release of fission productsinto the fuel–clad interspace, with isotopes of iodine asthe main corroding species There is some discussion as

attrib-to whether the corrodent must be freshly released fromthe fuel or whether it is sufficient for it to have accu-mulated in the fuel–clad gap throughout the irradia-tion period prior to the overpower transient

Since its discovery in the Canada deuteriumuranium (CANDU) reactor and the BWR in the1970s, the incidence of pellet–cladding interaction(PCI)-SCC failure is thought to be affected byfour factors, as reviewed by Cox30: stress, time, mate-rial, and chemical environment Understanding theimportant variables for PCI-SCC will help fueldesigners to propose solutions These include adapt-ing the appropriate fuel geometry (reducing theheight to diameter ratio, introducing dishes andchamfers), imposing restrictions on the power varia-tions, modifying the fuel assembly design, and apply-ing coatings on the inner cladding surface in order toreduce the friction coefficient and to accommodate in

a ductile manner the biting of pellet ends Moreprecisely, 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 (ofeither crystal bar or sponge origin)31,32 has beenadopted as the standard barrier because irradiationexperience showed that copper offered less protec-tion after high irradiation doses However, pure zir-conium oxidizes more rapidly in comparison withZircaloy, while the terminal solubility of hydrogen

is lower in the liner.33Accordingly, when a primarydefect occurs in the cladding, for instance, due todebris fretting, water ingress will oxidize both the

Pellet hourglassing

Ridges

After PCMI Prior to PCMI

As fabricated

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.

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

sun-burst 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,34and 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

assem-bly35(e.g., from 6 6 to 10  10 in BWR assemblies)

in order to reduce the linear heat generation rate per

rod and by alloying the liner with either Fe36or Sn.37

A more recent component of PCI-resistant fuel

designs is the use of softer fuel pellets obtained by

and PWR applications Since the early 1990s, AREVA

has developed an optimized chromia-doped UO2fuel

that exhibits significantly higher performance

com-pared to standard UO2.38The grains are on the order

of 60mm 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.39Postirradiation 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

high-burnup fuel and the reduced fracture stress of the

doped fuel.40The 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 flaws42according to a frequency

distribu-tion, or spontaneously when a threshold such as the

iodine concentration is exceeded.43 More recently,

Parket al.44

postulated that pits would generate

pref-erentially around grain boundaries and coalesce to

form a microcrack, referred to as grain boundary

pitting coalescence The microcrack is assumed to

develop into an incipient crack, starting and gating along the grain boundary

propa-For the simulation of crack propagation, mostauthors apply linear elastic fracture mechanics(LEFM), in line with Kreyns et al.45

By fitting theexperimental data of Wood,46they advocated that thecrack velocities could be related to the fourth power

of the stress intensity factorKI:

da

dt ¼ CKI4wherea is the crack length and C is a constant Thecrack intensity factor was shown to be controlled bythe elastic stress field at a flaw tip as described byKI,rather than the nominal applied stress:

KI¼ s ffiffiffi

a

pYwhere s represents the nominal hoop stress and thefactor Y incorporates a correction factor to accountfor the finite width of a defect-bearing component.Nevertheless, Kreyns et al.45

pointed out that scale plasticity will occur near the crack tip, in a conewith radiusrp:

small-rp ¼ 1ffiffiffiffiffi6p

apbecomes

Yeff ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY

s y

 2 Y2 6p

 r

Anderson47 provided a general correction factor sothatKIcan be expressed more generally:

KI¼pRt

ffiffiffiffiffipaQ

wherep is the internal pressure on the tube (MPa), R

is the mean tube radius, t is the tube wall thickness,andQ is the shape factor for an elliptic crack

Q ¼1 þ 1:464 ac 1:65where c is the half length of the crack and F is aboundary correction factor which depends on theshape of the initial crack formed at the inner cladding

surface Parket al.44

applied an expression forF withinthe range of 5 R/t 20, 2c/a 12, and a/t 0.8:

F ¼ 1:12 þ 0:053x þ 0:0055x2þ

ð1 þ 0:02x þ 0:0191x2Þ 20Rt

1400where x¼a

t

a

2c.

The KISCC value provided by Park et al was

3.3 and 4.8 MPa m1=2 for stress-relieved and

recrys-tallized Zircaloy-4, respectively

Zhouet al.42

replaced the constantC 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

adopt-ing the Coulomb friction model, accordadopt-ing to which

the friction force is proportional to the contact

pres-sure The extension by Zhouet al is in line with recent

findings of Michelet al.48

Based on 2D and 3D finiteelement 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 Mattaset 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

postu-lated 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 Rousselieret al.,50

stress tion and the ensuing crack arrest are necessary to

relaxa-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-4.51Rousselieret al attributed this

to the stress relaxation and to the fact that the

inter-granular 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 thestress intensity factor would decrease However, quan-titative information cannot be directly inferred fromtheir analysis, since the critical stress corresponding to

KIAshould depend on the material (stress relieved vs.recrystallized), the irradiation, as well as the loadinghistory Furthermore, the LEFM should not be appliedwithout corrections for the viscoplastic behavior and,finally, because of the local inhomogeneities of thematerial at the scale of the crack, one should applylocal rather than global criteria such asKI KIA.With the advent of improved hardware and soft-ware on one hand and more detailed experimentaldata on the other, more detailed models are beingdeveloped at various scales At the electronic andatomic scale, first-principle computations shouldenable analyzing the individual effect of impuritiessuch as iodine on the binding energies, in much thesame way as Xinet al.52

have studied the properties ofpoint defects and their interactions with Nb in Zr, orKaji and Tsuru53analyzed the clustering of Ni in Fe

By means of finite element computations at the level

of crystallographic grains, Rousselier et al suggestavoiding the limitations associated with LEFM forcrack initiation and propagation Because of the exces-sive computation time and the lack of precise data ofsome model parameters, their model is not applicable

in fuel performance codes but should enable analyzingthe effect of corrosive fission products on the inter-granular damage by coupling the mechanical problemwith a diffusion problem A similar tool has alreadybeen developed by Musienko and Cailletaud,54albeitthe corrosive environmental parameters are accountedfor by a phenomenological approach: that is, via aneffective diffusion coefficient At the mesoscopic scale,Kajiet al developed a 2D model for SCC growth, inorder to analyze qualitatively the effect of load (normal

vs shear stress) and grain boundary corrosion on thebranching aspect of crack growth The macroscopicmodels are mostly based on finite element simula-tions.40,48These tools provide deeper insight and qual-itative information about the various parametersaffecting PCI as explained above Nevertheless, thesemodels are not yet capable of replacing the simplified1D 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 performancecodes based on the 2D models The stochastic nature

of cracking and the complex evolution of materialproperties and boundary conditions during irradiationremain the most important difficulties to be tackled

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

character-ized 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 Zircaloy56as opposed to the standard

PCI-SCC as discussed above Postirradiation

exami-nation 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-by-step cracking of hydrides at the crack tip The process

bears similarities with secondary hydride failures

dis-cussed above The radial temperature gradient in the

cladding wall and the hoop tensile stress due to the

ramp test facilitate the hydrogen diffusion and

pre-cipitation of radial hydrides on the outer surface of

the tube These hydrides can crack under the

influ-ence 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

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

reduc-ing the thermal heat transfer in the gap Fuel swellreduc-ing

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

balloon-ing 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

Swelling of Nuclear Fuelsand in Van Uffelen.57

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 afree surface (<6–7 mm) can escape from the fuel due toits high kinetic energy (60–100 MeV) This is calledrecoil release When fission fragments make elasticcollisions with the nuclei of the lattice atoms, a collisioncascade begins The interaction of a fission fragment, acollision cascade, or a fission spike with a stationary gasatom near the surface can also cause the latter to beejected if it happens within a distance close enough tothe surface This process is called ‘release by knockout.’Finally, a fission fragment traveling through oxide losesenergy, causing a high local heat pulse When thishappens close to the fuel surface, a heated zone willevaporate or sputter, thereby releasing any fissionproduct contained in the evaporated zone

Recoil, knockout, and sputtering can only beobserved at temperatures below 1000C, when ther-mally activated processes (seeSections 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 tem-perature independent and therefore called ‘athermalmechanisms.’ It is generally of little importance in areactor at intermediate burnup levels The fraction ofathermal release is roughly under 1% for rod burn-ups below 45 MWd kgU1, and increases to roughly3% when the burnup reaches about 60 MWd kgU1

3.19.2.3.1.2 Lattice diffusion of single gas atoms

The first and basic step in FGR is single gas atomdiffusion in the lattice Possible mechanisms by whichthe inert gas atoms migrate through the fuel have beenstudied by Grimes58by considering low-energy migra-tion pathways between solution sites as well as thestability of gas atoms at a variety of solution sites within

a defective UO2 lattice (x ¼ |O/M – 2|, the tion from the stoichiometry) He postulates a cation-vacancy-controlled migration pathway for Xe atoms.Indeed, according to his calculations, Xe is trapped

devia-at a uranium vacancy in UO2 þ x, at a trivacancycluster in UO2xand at a di- or trivacancy in UO2.Since the local environment of the migrating Xe atoms

is supposed to become the charged tetravacancyfor all stoichiometries, the mechanism for diffusiononly considers the association of a cation vacancy tothe trap sites (Figure 12) (Uranium vacancies as the

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

frag-ments 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 1000C and is temperature independent For

temperatures between 1000 and 1400C, vacancies

necessary for the gas atom diffusion are assumed to

be created both thermally and by the damage cascades

related to fission fragments Above 1400C, 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 (m2s1) often applied in

the fuel performance codes59:

D ¼ D1þ D2þ D3where

3.19.2.3.1.3 Trapping

In nuclear fuels, either natural (e.g., impurities, cation lines, closed pores, etc.) or irradiation-producedimperfections in the solid (e.g., vacancy clusters infission tracks, fission gas bubbles, solid fission productprecipitates, etc.) depress the amount of fission pro-ducts available for diffusion by temporarily or perma-nently trapping the migrating atoms The experimentsshow that, for the burnup characteristic of power reac-tors, gas atom trapping due to (intragranular) fissiongas bubbles in the grains is predominant The trappingrate depends on the number density and size of theintragranular bubbles, and hence on temperature, fis-sion rate, and burnup A second important effect oftrapping occurs at the grain boundaries It deals withthe delay for the onset of thermal FGR, via the bubbleinterconnection mechanism (seeSection 3.19.2.3.1.8)

dislo-3.19.2.3.1.4 Irradiation-induced resolution

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

fragment or an energetic atom from the collision

cascade Macroscopic models on the other hand

con-sider the complete bubble destruction, but there is

still discussion about the detailed mechanisms For

(larger) grain boundary bubbles, resolution is

sup-posed 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

Neverthe-less, there is a switch from release assisted by grain

boundary diffusion in trace-irradiated UO2to trapping

and eventual interlinkage of the intergranular bubbles

(seeSection 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

collec-tion 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

sweep-ing 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

tempera-tures above roughly 1600C

3.19.2.3.1.7 Intragranular bubble migration

The migration of intragranular fission gas bubbles

pro-vides an alternative to the sequence ‘bubble formation–

resolution–gas atom diffusion’ in order to describe

fission product release from nuclear fuels Migration

of bubbles in the oxide fuels has two other importantconsequences, namely the columnar grain growthwith the concomitant central void formation (ob-served in fast breeder reactor fuel) and the coales-cence of the bubbles which gives rise to fuel swelling.Under normal operating conditions, however, intra-granular fission gas bubbles remain small (typicallybelow 20 nm) due to resolution, and show a smallmobility at least up to 1800C.61 This is partlyexplained by the pinning by dislocations and othercrystal defects

3.19.2.3.1.8 Grain boundary (intergranular)bubble interconnection

Fission gas bubbles appear along grain boundariesafter a certain burnup, depending on the temperaturehistory When bubbles interconnect, they form a so-called tunnel network through which the gas can bereleased The bubble interconnection is a reversibleprocess, for the tunnel network can close again underthe influence of the surface tension when the outgo-ing flux of gas atoms outweighs their supply.The bubble interconnection has two importantconsequences First of all, it determines the onset ofrelease as the release remains small (due to athermalrelease) before grain boundary bubbles interconnectwith open grain edge tunnels This incubation period

is reflected in the Vitanza threshold for FGR, which isshown inFigure 13 The ensuing release corresponds

to a seepage process Second, when grain face bles interconnect and form snake-like tunnels, therewill be a sudden release of the gas accumulated inthese bubbles, referred to as ‘burst release.’ This canalso be interpreted as a sudden interconnection oropening of grain face bubbles due to microcrackingalong grain boundaries during abrupt power varia-tions Cracking results in a sudden opening of a frac-tion of the grain boundaries with the instantaneousventing of the corresponding fraction of the accumu-lated gas atoms

bub-Interconnection of gas-filled bubbles takes place

in general where diffusion-controlled precipitationoccurs at the grain boundaries: that is, when both thetemperature and the burnup are high enough Theconditions correspond roughly to the Vitanza62,63threshold:

TcðCÞ ¼ 9800

ln0:005BU whereTcrepresents the central temperature in (C)andBU the burnup in (MWd kg1

) UO2

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

inChapter3.20, 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

imple-ment 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 fuse in the grains toward grain boundaries by thermaland irradiation-enhanced diffusion Small intragra-nular bubbles with a diameter of 1–2 nm are observed

dif-in irradiated fuel They are created dif-in the wake offission spikes and then grow by diffusion (trapping).They are continuously destroyed by fission spikes(resolution) There is no bubble migration except attemperatures above roughly 1800C 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 tothe local fission rate density (S ¼ Yfp_F), which

is based on the pioneering model of Booth.65 Heproposed the equivalent sphere model This theoryconsiders a polycrystalline sinter as a collection ofuniform spheres with an equivalent radius in order

to simplify the mathematical problem The thetical sphere radius (RB) is defined so that theeffective surface-to-volume ratio of the fuel (S/V) ispreserved:

S

 twhere (S/V)taccounts for the sum of the geometricsurface of the pellets as well as the surface due toopen porosity As irradiation proceeds, fission gasesare generated within the Booth sphere and migrate tothe surface, where the concentration is taken to bezero He proposed that the fractions of stable gasrelease can be approximated by

800

900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Burnup (MWd kg –1 UO2)

Original 1% data Siemens 2% data New 1% data Empirical Halden threshold

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 UO 2 pellet at high burnup In Topical Meeting on Light Water Reactor Fuel Performance, Portland, OR, May 1979; American Nuclear Society: Portland, OR, 1979.

fannðtÞ ffi 6

ffiffiffiffiffiffiffiffiffiDt

pR2 gr

s

 3Dt

R2 grfor so-called annealing conditions (i.e., without

source term, but with an initial nonzero

concentra-tion), and

firrðtÞ ffi 4

ffiffiffiffiffiffiffiffiffiDt

pR2 gr

s

32

Dt

R2 grfor 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:

Rgrain

ffiffiffiffiDl

r

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

3.19.2.3.1is 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

pro-posed to cope with time-varying conditions, which

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):

Deff ¼ D b

b þ gwhere g ¼ 4pRbubbleD corresponds to the trapping

rate coefficient andb corresponds to the resolution rate

coefficient Whatever model is being considered for

resolution, fission gas behavior models generally duce a simple resolution rate coefficient that is pro-portional to the local fission rate density and depends

intro-on the bubble size:

b ¼ 2pðRblþ dÞ2

mff_Fwhere it is assumed that a bubble can be destroyed ifits center lies within a distance d from the fissionfragment track of length mff¼ 7–10 mm The conditionfor applyingDeffis that the traps are saturated Experi-ments show that small intragranular bubbles stabilizerapidly both in size and in diameter Intragranularbubbles can therefore be considered saturated for irra-diation times of practical interest (beyond 0.5 MWdkgU1) Nevertheless, the difference between D and

1100C It should be underlined, however, that ing a power ramp the application ofDeffprovides anoverestimation of the trapping effect.69

dur-Over time, several models have been proposedwherein the Booth sphere radius was taken to beequal to the average grain radius of the fuel, in order

to be able to account for grain growth However, it mustfirst be pointed out that there is no consensus aboutwhich grain growth model should be applied, eventhough the Ainscoughet al.70

model is often applied:

wherek is a temperature-dependent rate constant and

Rmax¼ Rmax(BU) is the grain size at which growth stops.This burnup-dependent quantity is introduced inorder to account for the retarding effect of fissionproducts on grain growth as burnup proceeds.Most FGR models only account for the increase ofthe average diffusion distance when normal graingrowth occurs Some other models only take intoconsideration the sweeping effect, assuming eitherthat the fractional release is proportional to thegrain boundary velocity, or that the gas in the totalfraction of grain volume swept by the grain bound-aries is released So they all fail to properly incorpo-rate boundary motion into the intragranular diffusionequation and artificially separate the two aspects ofgrain growth on FGR Only some stand-alone modelshave been proposed so far that account for bothsimultaneously by solving the diffusion equation in

a sphere with a moving boundary (e.g., Suzuki andSaitou14and Forsberg and Massih71)

For alleviating the fourth and fifth limitations ofthe Booth models, an intergranular module has to beintroduced