LIFE-II — A computer analysis of fast-reactor fuel-element behavior as a function of reactor operating history

Paper C4/1"

NUCLEAR ENGINEERING AND DESIGN 18 (1972) 83-96 NORTH-HOLLAND PUBLISHING COMPANY

F~ Int~

LIFE-II - A COMPUTER FAST-REACTOR AS A FUNCTION

ANALYSIS

FUEL-ELEMENT

OF REACTOR

OF

II!1

BEHAVIOR

OPERATING

HISTORY *

V.Z. JANKUS and R.W. WEEKS Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Received 2 August 1971

The LIFE-II computer code provides a detailed thermal, radiation, and mechanical analysis of cylindrical fast-reactor fuel-element behavior as a function of the actual reactor operating history. Materials behavior models of the effects of fuel restructuring, fuel cracking and hot pressing, migration of fuel constituents, fuel swelling due to solid and gaseous fission products, fission-gas release, fuel-cladding interactions, and cladding swelling due to void nucleation and growth, are included in a generalized plane-strain iterative procedure. As a result, LIFE-II predicts the thermo-elastic, creep, and swelling deformations of the fuel element as a function of the reactor power and coolant temperature history. Comparison of LIFE-II calculations with results from actual fuel-element irradiations in the EBR-I1 reactor demonstrate the sensitivity of fuel-element performance to reactor operating cycles. Sensitivity analyses are also being performed to indicate the most critical areas in which improved fuel-element material and performance data are needed. The LIFE-II code is a realistic and versatile design tool for the prediction of fast-reactor fuel-element behavior.

1. Introduction 1.1. Objectives Computer simulation of fast-reactor fuel-element behavior has received much attention at Argonne National Laboratory because of the lack o f adequate fast-flux testing facilities, the length of time required for fuel-element tests, and the great difficulties in instrumenting in-pile tests. The fuel-element codes are being developed to accomplish important practical objectives. Through parametric design studies, the ANL LIFE-II code for mixed-oxide, stainless steel clad fast-reactor elements aids in the definition of the most critical experiments to be run in-pile, and, because the simulation is based on materials behavior models, LIFE-II helps to focus materials research efforts on those properties most critically related to fuel-element performance. LIFE-II has been designed to follow the actual reactor operating history in predicting in-pile behavior, and this ability makes the * This work was performed under the auspices of the U.S. Atomic Energy Commission.

code especially useful in postirradiation analyses of fuel elements. This history-following ability, coupled with fuel-element failure criteria, also makes possible the monitoring of cumulative cladding damage in operating fuel elements to predict their residual lifetime. This last function will become increasingly important as fuel-element burnup is extended for improved reactor economics. Finally, LIFE-11 provides the initial conditions for accident analyses in which the fuel-element response to extremely rapid transients is desired at various burnup conditions. 1.2. Problem definition L1FE-II is designed to predict the in-pile thermal, mechanical, and nuclear performance of cylindrical, fast-reactor fuel elements, as a function of the reactor operating history. Although other systems can be treated within the same general framework, the specific models to be discussed in this paper are for mixed-oxide fueled, stainless-steel clad fuel-element designs. As a function of time and axial position, LIFE-II has been designed to predict the following: tempera-

84

V.Z. Jankus, R. W. Weeks,Analysis o.f fast-reacmr fuel-element behavior

ture distribution in the element; dimensions of the fuel and cladding, including length changes; extent of fuel cracking and crack healing; extent of fuel restructuring; amount of gas released to the plenum; plenum pressure; distribution of fuel constituents; hot-prersing of the fuel ; stress and strain distribution including a breakdown into thermoelastic, thermal, and flux-enhanced creep and swelling deformations for each r,~.gion of fuel and cladding; interface pressure between the fuel and cladding; and amount of cumulative damage that has occurred in the cladding to reduce the expected fuel-element lifetime. 1.3. General method o f solution LIFE-I1 is an extension of the LIFE-I analysis [1 3], to include fuel cracking during power transients and subsequent crack healing, and cumulative damage failure criteria for the cladding. Several minor changes have also been accomplished, for example, to make the fuel-cladding contact thermal conductance more realistic, allow prediction of capsule as well as cladding swelling for encapsulated elements, and gradually reduce the power of the specific fuel element in relation to the total reactor operating power as burnup increases. The fuel-element geometry is assumed to be that shown in figs. 1 and 2. The cladding is loaded exter-

AXIAL

i

r• • • •

T • •~_~Pc

T

NODES

pp

1~w T ' ;

I I '

t

~

t. ; ~'

1

r

PLENUM i i i

~ ~

/

F wf

d

]

'

~ FUELCLADf GAPor INTERFACE '/FRICTION

f

-]wfp

I I

I-

I

CENTRAL

I ~ _1 I ~ ~/[ • .A ~, A AAA A& A AA ; PB

CB

Fig. l. Fuel-element axial cross-section.

VOID

Fig. 2. Oxide fuel-element radial cross-section. nally by the coolant pressure Pc, and an additional axial load P may also be specified. An initial gap between the fuel and cladding may be specified for pellet fuel, and this gap can open or close any number of times, depending on the fuel-element operating history. When the gap is open, the plenum pressure Pp is exerted between the fuel and cladding. A finite coefficient of friction is assumed to act at the fuelcladding interface, and thus the axial strain of the fuel column may differ from that of the cladding. The central void, which develops in the fuel during operation, is assumed to communicate with the plenum so that the plenum pressure loads the fuel column internally. The radial temperature distribution is obtained by dividing the fuel and cladding into any specified number of radial nodes, so that the temperature and temperature gradient are known as finally as desired. This enables the code to predict diffusion effects and local transient thermal stresses in detail This procedure is followed in assessing the extent of fuel cracking, as will be discussed 1. . . . For the gross mechanical analysis, the fuel is always divided into three radial zones, corresponding physically to the colunmar grain region, the equiaxed grain region, and the unrestructured or "undisturbed" region. The cladding

v.z. Jankus, R. W. Weeks, Analysis of fast-reactor fuel-element behavior

then forms a fourth radial zone. For each of these zones, the integral averages for the stresses, strains (including swelling), and displacements are determined using a generalized plane-strain analysis with materials properties evaluated at the average temperature of each region. This is combined with an iteration procedure that allows the code to follow changing boundary conditions resulting from reactor-power fluctuations. The gross mechanical analysis is limited to four regions because the resultant equations may be readily solved algebraically, which requires less computer time. To account for axial variations in the flux, power,

IINITIAL INPUT I

f

~-'~OPERATING CONDITIONS~ TEMPERATURE DIST. MASS TRANSFER

t

GAS RELEASE PLENUMPRESSURE

No

t

[CHOOSE AXIAL REGION}<

Y"

f IASSUMEtSr~,NSl ~__ I COMPUTE STRESSES~ - ~

f

COMPUTE CREEP AND SWELLING INCREMENTS

t

DIFF EQS. J u i = f (pf=,/zf¢,,=f,,z= )

t GAP OPEN,'~ J STICK, ) pf=,.f~,~l,..~ OR SL P J

....

t COMPUTE Uh ~ ,~ EACH REGION

No Yes

J

/COMPARE STRESSES AND \STRAINS TO ASSUMPTIONS/~' t Yes j~LAST AXIAL SECTION1') No

t No~D,D POWERC~NGEr~r¥.. suoER MPOSE IT.ERMAL STRESSES

~ADFjUEL CRACKIN~

UST MATERIALSL.J PROPERTIES I

t

COMPUTE CUMULATIVE CLAD DAMAGE

t t GO TO NEXTI

I WRITE OUTPUT

I

TIME STEP I

Fig. 3. Simplified LIFE-II flow chart.

85

coolant temperature, coolant pressure, and fuel-element geometry, LIFE-II divides the element into as many as ten axial sections, and repeats the deformation analysis under boundary conditions appropriate to each specific section. A simplified flow-chart of LIFE-II is shown in fig. 3. After operating conditions for a time step are determined, the radial temperature distribution is established for each axial section, assuming the geometry is that existing at the end of the previous step. The gas release from each radial and axial fuel segment is determined and added to that already released to establish the plenum pressure, which then forms a boundary condition for the deformation analysis. At the start of the deformation analysis, an initial approximation is made for the change in total strains expected to occur in each region during the time step, based on the length of the time step and the changes that occurred in the previous step. From this assumption, first approximations of the average deviatoric and hydrostatic stresses in each concentric cylinder are determined using empirical creep laws and the swelling and hot-pressing relationships (to be discussed later). These first approximations are incorporated into the complete constitutive equations, which are then combined with the equations of equilibrium, compatibility, and the kinematical relations, and solved subject to the boundary conditions appropriate for the time step. This solution yields a second approximation of the change in total strains in each region, which is then compared with the initial approximation. If convergence criteria on the stress and strain are not met, iteration continues. If the maximum changes in stress, strain, or restructuring exceed specified values, the code cuts the time step internally, and the power and coolanttemperature changes for the step are adjusted accordingly. The time step may also be cut if the convergence is too slow. If the power has changed from the previous step, the local thermal stresses in each fuel region are computed and superimposed on the global stresses to determine whether fuel fracture occurs. If the power has not changed, the amount of cracking or crack healing is determined from the average condition in each region prevailing for the time step. The mechanical properties of the fuel are adjusted accordingly, and finally the cumulative damage to

V.Z. Jankus, R. W. Weeks, Analysis of fast-reactor fuel-element behavior

86

the cladding is computed.

shown schematically in fig. 4 with various linear fits, a LIFE-I parametric study was made [8] with a pressure and temperature-dependent gap conductance hfc of the form

2. Thermal analysis

2.1. Temperature distribution

1 --

The radial, quasi-static temperature distribution is established by the same methods as in LIFE-I [1], with provision for treatment of encapsulated elements at the user's option. Two thermal conductivity options were allowed for the fuel with LIFE-I I4, 5], and another has been added for LIFE-II. This function, attributed to Biancheria [6], is given by k f = ( 3 . 1 1 + 0 . 0 2 7 2 T ) - I + 5.39X 10 13 T 3 ,

i ....

--

1 ~

hgap

(2)

h0(l +Pfc/PO) '

where hgap

= kgas/ZS.x, = the conductivity of the gas in the fuelcladding gap when a gap exists (taken as helium), zXx = the width of the gap, ho,Po = constants used to fit the data (see fig. 4), = the interface pressure between the fuel Pfc and cladding. The results, computed for steady-state operation of a mixed-oxide element at 11 kW/ft with an initial 1-mil radial gap, are shown in figs. 5 - 7 . Both shortterm fuel restructuring and long-term cladding deformations were affected, and, although fuel cracking was not included, it is evident that the mechanical and the thermal performance can be strongly inkgas

(1)

where kf is the conductivity in W cm 1 K - I , and T is the temperature in K. At present, this conductivity expression is preferred. The conductance for the fuel-cladding gap remains debatable, since most "data" for this quantity are obtained by postirradiation analysis, assuming fixed restructuring temperatures and no fuel-cladding pressure dependence. Using data of Robertson et al. [7], 8000.

1 -

hfc

I

l

I

I

ROBERTSONet ol. (1962) CONDUCTANCE FOR UO2 ON ZIRCALOY, SMOOTH SURFACES (0.4-0.8p.), WITH HELIUM OR XENON GAS BOND / /-'1

6000. - -

/ J /// / /

/. / /

i H'e

{

/ / ~

I ~ , .i, ~

ho=1728 , po=4 0 0 0

m

ui

4000. i//

//-~

L

O

/

z

o

" / ~ / / / "

j.....--ho--,,28, po:8OOO

:;21;2 o

I'" ~

~ . . . . . ~ . ~ ho = 1440, po:8000

el

'~ (.9

200o.

L.I ~

hfc:ho(I +--~-j

I

I

1°

2000

4000

6000

l

8000 INTERFACE PRESSURE Pfc' psi

Fig. 4. Linear fitting to the data of Robertson et al. [ 1 ].

I0000

12000

V.Z. Jankus, R.W. Weeks, Analysis of fast-reactor fuel-element behavior

87

60 ho'6.0 8tu- kr'I-*F'L(In.t)"l 864 Btu-hr'l-*F-I-(fli) "1

<~

2.'=

4o

hfc=h-(l+ ) vPfc Po

2,0

4

~

ho=8

at

ho'12.0 117281

8

6

Po=8000~ / /

<3

I.,5 20

ho= 24.0 (3456)

- -

/ / /

Pf¢ hfc=h o (1+ - ~ o )

"~Po

o

ooo

=4000

I.C

Po " 8 0 0 0 psi

IOhr

IOOhr

2JOhr

i IOd

300hr ]

mOOd

TIME

Fig. 5. Columnar grain formation as a function of fuel-clad gap conductance.

Fig. 7. C l a d d i n g - d i a m e t e r

200d TIME

300d

i n c r e a s e as a f u n c t i o n

of fuel-clad

gap conductance parameters.

2.0

hf "h O+ p/c- )

c o Po Po,800() p~

2oo'o.,I lOOd

'

4000hr

I 200d TrME

I

6000hr

8000hr 300d

Fig. 6. Cladding-diameter increase as a function of fuel-clad gap conductance. fluenced by the assumed gap conductance. In L1FE-I1, 1 0 - 9 ( T - 6 8 ) , where T is the temperature in degrees the pressure and temperature-dependent gap conducFahrenheit. tance is included as an option, but to reduce the required computer time, the conductance after gap 2.2. Fuel restructuring closure is usually set constant at hgap ~ 0.97 W c m - 1 The rate of fuel restructuring is based on the K - l . Minor errors in the thermal coefficieats of extheory of pore migration [9] with the migration pansion used in LIFE-I have been corrected as follows: velocity given by 1) For the fuel, af = 8.192 X 10 -6 + 2.430 X 10 -9 ( T - 2 0 ) , where T is the temperature in degrees centigrade; V = C 1 T -3/2 [exp ( - C 2 / T ) ] d T / d r , 2) For the cladding, a c = 8.878 X 10 - 6 + 1.481 X (3)

88

V.Z. Jankus, R. W. Weeks, Analysis o f fast-reactor fuel-element behavior

LIFE-I CALCULATIONS (XGO'I-FIE)

12

LIFE-I CALCULATIONS (XGOI-FIE)

~2o

/EBR-II

.w:

/ . . . . . . . . . . . . . EQUIAXED . . . . . . . ZONE . . . . . . . . . . . .

"o

OPERATING HISTORY

/ I

) COLUMNAR ZONE

II

I/

0 (3. b,J

i , Jf

n; W

>~ EQUIAXED ZONE

CENTRAL VOIO

,

Ao I

~-" io

I

L

i

J

I

,

"

1

~

I

___ ..... £__

o

)/

o_ x

--

i'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m

8

COLUMNAR ZONE .....................................................................

/ ......................

= ~t-', ,i "~ 4 F I 2/I

/

CENTRAL VOID

, .........

:

--

~'~

er

0

40

80

TIME

120

IN EBR-Tf

160

200

40

240

80

TIME

(hours)

120

IN EBR-TI

160

200

240

(hours)

Fig. 8. Reactor operating history and fuel restructuring for element XG01-F 1E.

Fig. 9. Fuel zones and boundary temperatures for element XG0 I-F 1E.

where C 1 and C 2 involve properties of the fuel, and d T / d r is the temperature gradient. This expression, with different constants for the columnar and equiaxed zones, is evaluated at the zone boundaries and nmltiplied by the time interval to determine the growth of the regions. Calibration of the restructuring constants for LIFE-I was performed using a low burnup General Electric element [10] F1E, with the results shown in figs. 8 and 9, as reported earlier [3]. A mass balance then determines the size of the central void. In LIFE-I, constant densities were assigned to each restructured region, however, and the discrepancy between predicted and actual restructuring values increased as the initial fuel density varied from the 87% initial smeared density of F 1E. Work is in progress to make the final densities of the restructured zones a function of the initial fuel density in LIFE-If to resolve this problem and bring the physical model more into agreement with reality.

the LIFE-II deformation analysis reduce to the following: a) E q u i l i b r i u m : ~o r

or

O~- +

--

r

o 0

- 0,

(4)

where o r and o 0 are the stress components, and axial symmetry is assumed. b) K i n e m a t i c s : C rT

= Ou/Or

,

eT = u/r

,

ezT = constant

,

(5)

where eT r,o,z are the total strain components, u is the radial displacement, and a state o f generalized plane strain is assumed. c) C o n s t i t u t i v e relations:

V = E - 1 [Or- aOo +Oz)] + ~ r + ~y + a~P + d + a 4 ,

6r

Co T = E - 1 [oo - V(Or+Oz)] + ~ r + ~op + aCop + ~ + a 4 ,

3. Mechanical analysis

~z~ = e - l [Oz- aOr+Oo)l + ~ r + ~zp + A~p + 4 + a ~ , (6)

3.1. F i e l d e q u a t i o n s

The field equations for each concentric cylinder in

where

V.Z. Jankus, R.W. Weeks,Analysis of fast-reactor fuel-element behavior e F,O,z T °r o,z eVo,z_,

= components of total strain at time t + At,

= components of stress at time t + At, = components of the creep strain accumulated up to time t, AeP, o,z = components of creep strain that occur during the time interval At, er,o,zS = components of swelling strain accumulated up to time t, AeS,o,z = components of swelling strain that occur during time interval At, E, v = elastic modulus and Poisson's ratio, aT = f a d T the linear thermal expansion from room temperature. The creep-strain increments Aep, determined as shown later, are combined with the flow laws in the following form: d) P r a n d t l - R e u s s 1~7ow Laws:

Aep (2Or _ °o _ az)

Ae p

2a e

_

Aep

aeP

( 2 0 0 - Or -- O z ) ,

_ Aep

AePz - ~

(20 z -- o r - o0) ,

(7)

Oe = 2

ary of each region when power is increasing, and at the inner boundaries when the power is decreasing. The resultant stresses, evaluated as [11] t h _- azO th = ~o~E (T-- TO) at the outer boundaries, aoO

(10) o0ith = Oz ith = laEv (~- Ti) at the inner boundaries, where coefficient of thermal expansion, = modulus of elasticity, p = Poisson's ratio, the average temperature of the region, To, Ti = the deviation in temperature at the outer and inner boundaries of the region from the steady-state values, are then superimposed on the average stresses for each fuel region to determine whether the fuel has cracked. The conditions for fuel fracture are discussed in subsection 3.2.5. Both thermal and flux-enhanced creep are considered for the fuel, and the following formulation for the creep-strain increments is based on in-pile data for UO 2 and out-of-pile data for (U, Pu)O 2 [12].

E

' e4.5 exp(-Q/RT) A((p)a

Aep=

where the equivalent stress is given by 1/2 [(Or_o 0 ) 2 + ( o r _ O z ) 2 + ( o 0 _ a z ) 2 1 1 / 2 , (8)

f

AI(~)Oeexp(-QI/RT)+BOe~)At d2

and the equivalent creep-strain increment Aep,

ACp = ~1 N/r2[(ACrp

AeP) 2

+ (Aep _ AeP)2 + (Aep _ AeP)2] 1/2,

(9)

is found from the creep law for the fuel or cladding, as required. 3.2. Fuel treatment

where .~ep ~b A(~) AI(~) Q Q1 B d Oe

3.2.1. Thermoelastic and creep deformations Thermoelastic deformations of the three fuel regions are accounted for in the usual manner, except during power transients when the local thermal stresses are required for the fuel cracking analysis. These local stresses are evaluated at the outer bound-

89

R T

= the equivalent creep-strain increment, = the fission rate = 1.83X 10 - 3 + 6 . 1 0 X 10-164, = 2.89 X 107 + 9.40 X 10-64, = 140 kcal/mole, -- 110 kcal/mole, = 6 . 0 × 10 -23 , = grain size (~- 10/2m), = the equivalent uniaxial stress [psi], = the universal gas constant, = temperature [K].

3.°,2.2. Fuel swelling due to fission products Fuel swelling due to solid fission products is simply treated by specifying the net solid volume

(11)

90

V.Z. Jankus, R. W. Weeks, Analysis of fast-reactor fuel-element behavior

created per fission and multiplying by the number of fissions occurring in each fuel region. Gaseous fissionproduct swelling is treated as in LIFE-I ll1, with a simple gas-law approach for the restructured regions, ( A v ] g as

\ v 7i

niRTi

vi(Pi g )

(12)

C 1 equals 4.71, Q/R equals 8000°R, and T is the average fuel temperature in region K [o R]. This is presently used in LIFE-II. The plenum pressure is determined by volume averaging the temperature of the gas in both the central void and the plenum, so that the average temperature becomes

where AV i

NF = change in volume of the ith region,

r,

= original volume of the ith region, number of gas moles remaining in the region, :q = average temperature of the gas, assumed equal to the average temperature of the region, P( = - (o'r + Ooi + 4 ) 1 3 , O/0,Z = the average stresses in the ith region, r Pi = a constant similar to the usual surface-tension correction term, but referring to the entire ith region, R = the universal gas constant. In the unrestructured region, the gas bubbles are so small that they are assumed incompressible, and they contribute to swelling in a manner similar to solid fission products. Considerable progress has been made by Poeppel et al. [13 15] on a detailed fuel-swelling model, which includes the production, migration, coalescence, and eventual release of gas-bubble distributions. Poeppel's model also includes interaction of the bubbles with structural defects and fission-gas resolution. The subroutine for this advanced fuel-swelling model, called "GRASS", is now operational and will be incorporated into the LIFE code in the near future.

Tav =

Vpl +/~1"= Vv°id(J) NF ' -V-pl+ G Vv°id (]')

(14)

Tpl i=1 Tvoid(/) where Tp! is the plenum temperature, Vpl is the plenum volume, NF is the number of axial fuel sections, and Tvoid(J) and rvoid(] ) are the central void temperature and volume in axial section i. The plenum pressure then becomes nR Tav Ppl =

(15)

NF Vpl +i~l.: Vvoid(]')

where n is the number of gas moles initially in the plenum plus the moles of fission gas released. 3.2.4. Hot pressing Hot pressing of the fuel decreases both the porosity initially present in the fuel and that generated due to fuel cracking. The diffusional portion of this process is taken as in LIFE-I, with [17] AehP = _ 14f2biD 0 exp ( - QIR T)

3.2.3. Fission-gas release and plenum pressure LIFE-1 required constant fission-gas-release rates to be specified for each fuel region. This led to an overprediction of gas release when following the reactor operating history because the release rate of each region was temperature independent. A temperature-dependent release rate solved this problem [161 by FRPT(K) = exp ( - C 1 - Q / R T ) ,

(13)

where FRPT(K) is the gas-release rate in fuel region K,

k Td2 X Pi

(I pl

- _

)

etga

At ,

Pth where p

pm

= Po/(l + 3 ~ ) ,

e ,.s

: L AVl \

vii

'

(16)

V.Z. Jankus, R. W. Weeks, Analysis o f fast-reactor fuel-element behavior

f2 bi D Oexp(-Q/RT)

k T d Pi

Pi P0 a At

= volume of a vacancy (~ 4 X 10 -23 cm3), = a stress concentration factor for the ith region (~ 4), = diffusion coefficient, with D O ~ 6.8 X 10 - 5 cm 2 sec-l, and Q ~ 98.3 kcal/mole, = Boltzmann's constant, = absolute temperature [K], = average grain diameter (~ 10/am), = hydrostatic pressure in the ith region, = density in the ith region, = theoretical density of the fuel, = coefficient of thermal expansion, = time interval.

3.2.5. Fuel cracking and healing At the end of each time step, LIFE-II computes whether fuel cracking would occur in any of the three principal directions. If the power has changed, the local thermal stresses are computed as previously described in subsection 3.2.1 and superimposed on the average stresses for each fuel region. The cracking criterion is based on the fracture strength data of Canon et al. [ 18] for UO2, which is represented by

AeP0 = - O z / 6 G ,

(18)

AeP = Oz/3G ,

where B is the bulk modulus, and G is the shear modulus. The mechanical properties of the cracked fuel are then modified in an attempt to represent the cracked material as an isotropic material. Again using a crack perpendicular to the axial direction for illustration, after cracking, the elastic behavior of the cracked fuel is described by 0 z

=0,

e r = (o r - VOo)/E , e o = (o 0 - VOr)/E , e z = - v ( o r + Oo)/E.

(19)

It is desired to represent the behavior of this material as an isotropic material, or by er =(o~-- v n a i o - p n o i z ) / E n , --p

of = 1 5 0 0 0 + 3 . 7 T k ,

91

O r --

(17) e z = (0~ - - v n Or_ i vnoio)/E n ,

is the ultimate tensile stress, and Tk is the temperature in K. When any of the principal stresses in a fuel region reach of, the fuel cracks in that region. The stress component perpendicular to a crack vanishes in the region, and the elastic strain existing prior to cracking is then redistributed. For instance, when a crack occurs perpendicular to the axial direction, of vanishes and the other stresses are assumed to remain momentarily constant. The axial elastic strain then decreases by of~E, and the strains in the other principal directions increase by v o f / E each. It is assumed that these changes in elastic strain are compensated by equal changes in plastic strain and volumetric expansion

(20)

where of

Ae S = Oz/9 B , AePr = -- O z / 6 G ,

where E n and v n are the new elastic moduli to be determined. Eqs. (19) and (20) are now solved for the stresses and compared for the same strains by considering the average square deviation D =(o~-or) 2+(0i0-o0) 2 +(o~-Oz) 2.

(21)

Averaging over all values of the strains considering positive and negative values equally probable, and minimizing D yields /)n

-

P

2+v' 22-v E n - 3 2+v

E l-v"

(22)

Since v is small, these relationships have been simpli-

v.z. Jankus, R.W. Weeks,Analysis of fast-reactor fuel-element behavior

92 fled to the following: En -~E _2

and

vn = 1

(23)

LIFE-II allows for repeated cracking in all fuel sections (and in all principal directions) so that for a region with N cracks, the elastic constants become E C = E ( ] ) N,

and

vc=v(1) N .

(24)

LIFE-II also allows the cracked fuel to creep faster than the uncracked fuel by modifying the creep rate

cal treatment based on the enhanced vacancy formation rate [20]. Nonlinear time dependence is not included, although the code framework would allow such treatment. 3.3.2. Clad swelling Both empirical and theoretical calculations of clad swelling are available as options in LIFE-If. The empirical expression currently used for AISI solutiontreated type 304 stainless steel, is [21]

AV/V[%] = ((pt) (2"05 27/0+78/02) [ ( T - 4 0 ) X 10 10]

as

X exp ( - 0 . 0 1 5 T - 5100/T+ 32.6),

ep.C = Cpt 2:-

where m is the stress exponent for the thermal creep. This assumes that the creep rate is related to the shear modulus [19]. When the fuel element operates without a power change during a time step, crack healing is allowed. Presently, cracks are assumed to heal whenever a) the stresses in the region become compressive, b) the temperature exceeds 1400°C, and c) the time at steady state exceeds 1 hr. When the cracks heal, the mechanical properties of the fuel are again restored to their usual (temperature dependent) values. 3.3. Cladding treatment 3.3.1. Thermoelastic and creep The thermoelastic and creep deformations of the cladding in LIFE-II are dealt with as in L1FE-I [1 ]. Basically, Aep = [AOem exp ( - Q / k T ) + Ban(p]At,

(27)

(25)

(26)

whe re Aep = equivalent creep-strain increment, Q = average activation energy for thermal creep k = Boltzmann's constant, T = absolute temperature, q~ = neutron flux, At = time interval, A, B, m, n = experimentally determined constants. Several options are allowed to handle the fluxenhanced creep of the cladding, including a theoreti-

where (pt = fast fluence ((p>0.1 MeV) [1022 n/cm2], T = temperature [K], 0 = T 623. This is known as the "0-equation". For 20°A coldworked type 316 stainless steel [22],

AV/V[%] = 9 X 10-35((pt) 1.5 [ 4 . 0 2 8 - 3.712 X 1 0 - 2 T + 1.0145 X 1 0 - 4 T 2 - 7.879 X 1 0 - 8 T 3] ,

(28)

where (pt is the fast fluence, and T is in °C. The theoretical treatment of clad swelling is that of Harkness and Li [23] based on void nucleation, growth, and preferential attraction to sinks. 3.3.3. Failure criterion The failure criterion currently used in LIFE-II is a linear cumulative damage law of the form n

ti ~Ft i=1

.=1'

(29)

where t i is the time that the cladding is at the ith hoop-stress level, rFi is the time required for cladding rupture when at the ith level of hoop stress, and n is the total number of stress levels the cladding is subjected to in its linearized stress history. The denominator TFi is, of course, the most critical part of this failure criterion. It has been shown [24] that the stress-rupture data for unirradiated, 20% cold-worked, type 316 stainless-steel

93

v.z. Jankus, R. I¢. Weeks, Analysis o f fast-reactor fuel-element behavior

tubing correlates very well with the empirical expression TF=[exp-(

62'49

ll'6-X104]ll.987T ]J

(30)

× [sinh(1.79 X 10-5a)] -8.806 ,

where TF --- time to rupture [hr], T = temperature [K], o = hoop stress [psi]. Eqs. (29) and (30) are then used to compute the cumulative damage to each axial section o f the cladding, leading to eventual failure. Radiation damage, fatigue damage, and corrosion due to fission products and sodium interactions will be factored into this criterion as more data become available. Combining the data of Yaggee and Li [25] with L1FE-II calculations, it is also possible to infer the mode of cladding failure, i.e., whether the failure will be a relatively passive pinhole formation, or a more explosive type of rupture. This information is of obvious importance to LMFBR safety studies.

4. Results 4.1. XGOS-F2H calibrations When LIFE-I became operational, a sensitivity study was made to determine the influence of following the reactor operating history [ 1 - 2 ] . The Pi'

parameters in the fuel-swelling equations were adjusted so that the predicted deformations of element XG05-F2H agreed with experiment [2]. Then the code was run at steady state using first the average operating power and then the peak operating power up to the same final total fluence. The results of this study are shown in table 1. It was determined that the increase in the predicted deformations when following the operating history were primarily the result of fuel ratchetting during power fluctuations [3]. 4.2. XO72-SOPC5 calculations LIFE-I did not allow fuel cracking, and it was speculated that cracking might reduce the amount of fuel ratchetting. It was also believed, however, that fuel cracking might be required as a mechanism to move the fuel out to the clad early in life. This was because in an Argonne mixed-oxide element (X072-SOPC5), which had an exaggerated initial diametral gap of 14 mils and was run at 11 kW/ft to 3 atm % burnup, the fuel closed the gap completely. Element X072-SOPC5 has now been calculated four ways with LIFE-II; the full operating history was run with and without fuel cracking, and the "equivalent" steady-state average operating power history was also run with and without fuel cracking. The results are summarized in table 2 and figs. 10 and 1 I, where the full history with fuel cracking was calibrated to agree with the experimental values. The steady-state runs for SOPC5 underpredicted fuel restructuring, and since they ran cooler, they also

Table 1 LIFE-I midplane results for fuel element XG05-F2H.

1. Total fluence 2. Clad swelling strain [%] 3. Clad creep strain [%] 4. £xD/D final [%] 5. Outer radius of columnar zone [IN] 6. Fission gas released [%]

Experimental values

Actual history a Steady peak power (911 days) (284.5 days) (542 days at 0 kW/ft) (45 mW or 14.5 kW/ft)

--0.47 X 1023 1.12

0.47 X 1023 0.72 0.46 1.12

0.47 × 1023 0.72 0.21 0.92

varied

0.072

0.069

0.039

75

75

75

74

Fuel: mixed-oxide pellets (20 wt % Pu); Clad: 316 SS; Burnup: 6.75 at.%. a Actual history predictions calibrated to agree with experimental values shown.

Steady operating power (369 days) (35.96 mW or 11.6 kW/ft) 0.48 X 1023 0.78 < 0.001 0.78

V.Z. Jankus, R. IV. Weeks, Analysis of fast-reactor ~tel-element behavior

94

Table 2 LIFE-II results for fuel element X072-SOPC5 a Full history (274 days) with without fuel cracking fuel cracking

Experimental values 1. Total fluence [NVT] 2. Clad swelling strain [%] 3. Clad creep strain [(/~] 4. ~D/D final [%] 5. Time of gap b closure [ HR] 6. Outer radius of columnar zone [INI 7. Fission gas released [%]

- 2.2 X 1022 - 0.13

2.21 X 1022 0.14 < 0.001 0.17 850

~ 0.17 gap closed ~ 0.100

2.21 x 1022 0.14 0.21 0.37 125

0.106

56.4

Steady operating power (275 days) (8.1 kW/ft) with without l~ael cracking fuel cracking 2.24 x 1022

0.16 < 0.001 0.18 - 1620

0.095

56.2

2.24 x 1022 0.16 < 0.001 0.19 - 3200

0.043

55.3

0.043

50.8

51.4

Fuel: Mixed-oxide pellets (20 wt % Pu); Clad: 304t1 SS; Burnup: 3.0 at.%. a Results shown are for section just below midplane. b SOPC5 had an initial diametral fuel-clad gap of 14 mils.

LIFE-rl

/. . . . . . . . . . . . . . . EQUA I XEDZONE~

12-IC--

x

RESTRUCTURING

(X072-SOPC5)

/

ANALYSIS

COLUMNARZONE

~____~ --FULL HISTORY

8--

.....

AVERAGE P O W E R

6--

(3 nr

, . _ _ # E NTR.A_L_VO

CENTRAL VOID ..............................................

4-2

IIiI It

0

.~

1

L

]

L

I

L

t

1

I

I

l

I

i

I

=

I

i

J

, _ - _ _ _C_E_NTR__AL _VOLD_

CENTRAL VOID

=

/,,"

COLUMNARAND

1/ III

II

COLUMNAR ZONE

I

EQUA I XEDZONE

o/'

,io'

,ko' TIME IN E B R - ~ I

i 200 (hours)

A

I 240

,

I 280

i

320

I 360

Fig. 10. Zone boundaries and temperatures for element X072-SOPC5, run both steady-state and with full operating history.

F.Z. Jankus, R. W. Weeks, Analysis of fast-reactor fuel-element behavior

95

LIFE-II CALCULATIONS (X072- SOPC5)

I-AVERAGEPOWER

16 -.at

\

___.~_CTUAL POWER IN EBR-II

12 8 I-Z laJ

~E bJ i ILl

4 0

~,~ !

I

I

I

HISTORY FOLLOWlNG~

I

~

I ........

AVERAGEPOWER~

I0

I

//

I

WITH FUEL CRACKING WITHOUT FUEL CRACKING (RESULTS SHOWN FOR SECTION JUST BELOW MIDPLANE]

8 mo

o

6 4

-r__ITS.... ~ . . . . . . .

....

_

:::222..22

e- . . . . . . . . . . . . .

llt~'r . . . . .

............

__,j

.... I I

i i

=:=--~::=---:--:::-"-----=== "--i:-:=-i

. . . . . . . . . . .

il

....dl

z

-+

.....

fl

JL__II

i L . . . . . . . . . . . . . . . . .

. JL.J

I i

I i i i t.J

2

d 0 -e

i

2:30

i

235

i

240 TIME IN EBR-TT (days)

i

245

I

250

I

255

Fig. 11. Element power and cladding hoop stress for element X07 2-SOPC5, run both steady-state and with full operating history.

underpredicted gas release; this, by chance, increased the final diametral changes to near the experimental value. In table 2 it is shown that without ratchetting (i.e., the steady-state runs), fuel cracking does allow faster fuel-cladding gap closure. It is especially interesting to note that when following the full power history, fuel-cladding contact is made earlier when fuel cracking is not allowed. This is because fuel ratchetting is more severe without cracking, when higher stresses can be supported in each of the fuel regions. Thus, LIFE-I overpredicted ratchetting effects in fullhistory runs due to the neglect of fuel cracking, whereas in steady-state runs the fuel had too much self-restraint resulting in delayed gap closure. From fig. 10 it is evident that the steady-state analysis for element SOPC5 has markedly different fuel restructuring characteristics, developing no columnar region at all early in life; the temperatures at the boundaries of the restructured regions are also grossly different from those computed with the fulloperating history. In fig. 11, the cladding hoop stress is plotted for all four runs of SOPC5 during a segment of the reactor operating history. The cladding stress

levels are lower during the steady-state runs, and it is evident that cumulative damage calculations may succeed only when the complete cyclic stress-strain history is known.

5. Conclusions Most of the initial objectives of this work have been accomplished. LIFE-II is a versatile design tool capable of yielding a detailed description of mixedoxide, stainless-steel clad, fuel-element behavior in fast-breeder reactors. Much work remains, however, in improving the input reliability, the models for fuelswelling, fuel constituent migration, primary creep and strain hardening of the clad, and in the development of fully realistic cladding-failure criteria. As more calculations are made and confidence increases in the code predictions, it is expected that LIFE-II will play an important role in defining fast-reactor operations as well as in design studies. The potential of the code is only beginning to be realized.

96

V.Z. Jankus, R. W. Weeks,Analysis of fast-reactor fuel-element behavior

Acknowledgements The authors are very grateful t o their many colleagues in the Materials Science Division at Argonne National Laboratory who have contributed substantially to this work.

References [ I] V.Z. Jankus and R.W. Weeks, LIFE-I, a FORTRAN-IV Computer Code for the Prediction of Fast-reactor Fuelelement Behavior, USAEC Report ANL-7736 (November 1970). 12] V.Z. Jankus and R.W. Weeks, LIFE-I, A History-Dependent Analysis of Fast Reactor Fuel Element Behavior, ANS Trans. t3, 2 (November 1970) 830. [3] R.W. Weeks, V.Z. Jankus, M. Katsuragawa and J.D.B. Lambert, Analysis of Mixed-Oxide Fuel Element Irradiations Using the LIFE-I Computer Code, Proc. ANS Topical Meeting on Fast Reactor Fuel Element Technology, New Orleans, April, 1971, to be published. [4] C.W. Sayles, A Three-Region Analytical Model for the Description of the Thermal Behavior of Low-Density Oxide Fuel Rods in a Fast-Reactor Environment, Trans. Am. Nucl. Soc. 10, 2 (1967) 458. [5] W.E. Baily, F.A. Aitken, R.R. Asamoto and C.N. Craig, Thermal Conductivity of U-Pu Oxide Fuels, Intern. Symp. on Plutonium Fuels Tech., Scottsdale, Arizona, 1967, Nucl. Met. 13 (1968) 293-308. [6] A. Biancheria et al., Westinghouse (USA) Report WARD-4135-1 (September 1969) p. 6. [7] J.A.L. Robertson, A.M. Ross, M.J.F. Notley and J.R. MacEwan, Temperature Distribution in UO2 Fuel Elements, J. Nucl. Mater. 7, 3 (1962) 225. [8] V.Z. Jankus and R.W. Weeks, USAEC Report ANL7758, Reactor Development Program Progress Report (November 1970) p. 67. 19t F.A. Nichols, Behavior of Gaseous Fission Products in Oxide Fuel Elements, WAPD-TM-570 (October 1966). [ 10] J.R. Honekamp, EBR-I1 Project, Argonne National Laboratory, private communication.

[ 11 ] S. Timoshenko and J.N. Goodier, Theory of Elasticity (McGraw-Hill Book Co., N.Y., 1951) 2nd ed., p. 412. [ 12] J.L. Routbort and A.A. Solomon, private communication. [13] Che-Yu Li, S.R. Pati, R.B. Poeppel, R.O. Scattergood and R.W. Weeks, Some Considerations of the Behavior of Fission Gas Bubbles in Mixed-Oxide Fuels, Nucl. Appl. Tech. 9 (1970) 188. [ 14] R.B. Poeppel, An Advanced Gas Release and Swelling Subroutine, Proc. ANS Topical Meeting on Fast Reactor Fuel Element Technology, New Orleans, April, 1971, to be published. [ 15] L.C. Michels, B.J. Makenas and R.B. Poeppel, Behavior of Fission-gas Bubbles in Mixed-Oxide Fuel, submitted to ANS Winter Meeting (October 1971). [ 16] V.Z. Jankus and R.W. Weeks, USAEC Report ANL-7776, Reactor Development Program Progress Report (January 1971) p. 63. [17] R.C. Rossi and R.M. Fulrath, Final Stage Densification in Vacuum ttot-Pressing of Alumina, J. Am. Ceram. Soc. 48 (1965) 558. [ 18] R.F. Canon, J.T.A. Roberts and R.J. Beals, Deformation of UO2 at High Temperatures, J. Am. Ceram. Soc. 54 (1971) 105. [ 19] J.A. Tesk, private communication. [20] S.D. Harkness, J.A. Tesk and Che-Yu Li, An Analysis of Fast Neutron Effects on Void Formation and Creep in Metals, Trans. Am. Nucl. Soc. 12 (1969) 523. [21] R.D. Leggett, E.O. Ballaxd, F.E. Bard, J.W. Weber, L.A. Pember and R.J. Jackson, Correlation of Predictions and Observations in Mixed Oxide Fuels, ANS Trans. 13, 2 (November 1970) 574. [22] W.H. Sutherland, WADCO Corporation, private communication (April 1971). [23] S.D. Harkness and Che-Yu Li, A Study of Void Formation in Fast Neutron Irradiated Metals, Trans. AIME 2 (1971) 1457. [ 24] C.Y. Cheng and R.W. Weeks, USAEC Report, Reactor Development Program Progress Report (July 1971), to be punished. [25] F.L. Yaggee and Che-Yu Li, Failure Mechanisms for Internally Pressurized Thin-wall Tubes and their Relationship to Fuel-element Failure Criteria, USAEC Report ANL-7805, to be pubLished.