- Email: [email protected]
transient conditions
Nuclear Engineering and Design 80 (1984) 39-48 North-Holland, Amsterdam
39
FARST - A COMPUTER CODE FOR THE EVALUATION UNDER STEADY-STATE/TRANSIENT CONDITIONS
M. N A K A M U R A
OF FBR FUEL ROD BEHAVIOR
a n d M. S A K A G A M I
Hitachi Energy Research Laboratory, Hitachi Lid, 1168 Moriyama-cho, Hitachi-shi, Ibaraki-ken, Japan Received 21 November 1983
FARST, a computer code for the evaluation of fuel rod thermal and mechanical behavior under steady-state/transient conditions has been developed. The code characteristics are summarized as follows: (i) FARST evaluates the fuel rod behavior under the transient conditions. The code analyzes thermal and mechanical phenomena within a fuel rod, taking into account the temperature change in coolant surrounding the fuel rod. (ii) Permanent strains such as plastic, creep and swelling strains as well as thermoelastic deformations can be analyzed by using the strain increment method. (iii) Axial force and contact pressure which act on the fuel stack and cladding are analyzed based on the stick/slip conditions. (iv) FARST used a pellet swelling model which depends on the contact pressure between pellet and cladding, and an empirical pellet relocation model, designated as "jump relocation model". The code was successfully applied to analyses of the fuel rod irradiation data from pulse reactor for nuclear safety research in Cadarache (CABRI) and pulse reactor for nuclear safety research in Japan Atomic Energy Research Institute (NSRR). The code was further applied to stress analysis of a 1000 MW class large FBR plant fuel rod during transient conditions. The steady-state model which was used so far gave the conservative results for cladding stress during overpower transient, but underestimated the results for cladding stress during a rapid temperature decrease of coolant.
I. Introduction As cladding is the first barrier against the release of radioactive fission products, evaluation of cladding integrity is critical to good fuel rod design. Many computer codes [1,2] have been developed for such evaluation and careful consideration has been given to modelling thermo-mechanical behavior, during steady-state burnup irradiation. Recently, fuel rod design codes have been required for behavior evaluation during transient conditions. This requires knowledge about the fuel rod integrity under transient conditions, as well as under steady state conditions, and has led to the development of FARST. F A R S T code can be applied to thermal and mechanical design, in-pile performance prediction and postirradition analysis of fuel rods, which are mixed-oxide fuelled, stainless-steel cladding tube for sodium-cooled fast breeder reactors. However, other similar systems (e.g.: light water reactor fuel rods) can be analyzed with F A R S T , subject to minor modification. Analytical models describing thermal behavior of fuel rod under the steady-state/transient conditions are
presented in section 2, while those for mechanical behavior are presented in section 3. Applications of this code to analyses of experimental data from pulse reactor for nuclear safety research in Cadarache (CABRI) and pulse reactor for nuclear safety research in Japan Atomic Energy Research Institute (NSRR) [9] are presented in section 4. Finally applications to evaluation of cladding stress of in a large F B R plant fuel rods under the various kinds of transient conditions are also presented in section 5.
2. Analytical models for fuel rod and coolant thermal behavior
2.1. Basic equations Thermal behavior analysis is performed using one-dimensional (radial) thermal conductance model for a fuel rod and one-dimensional (axial) incompressible flow model for the coolant surrounding the fuel rod. These models are widely accepted and have been utilized in other codes [1,2].
0 0 2 9 - 5 4 9 3 / 8 4 / $ 0 3 . 0 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
40
Basic equations are written as follows.
1 O/ OT\ ,,,~ -r or[rKf~r )+q t
0T Cfpf Ot
(at)
OT
lr OrO rK~r
CcPc Ot
ture, eq. (2) is rewritten as follows: CNa T = ( C N a T ) z 0
(fuel pellet)
PNa/tNa S = Jo = c o n s t a n t
0
( p N~CN~TS)
)
(1)
1 [ (2, rooqs- (CN.o arS) dz
(cladding)
(coolant
now)
(2)
0
+ ~ (PN~CN~TSUN~)= 2,rr~oq~
(5)
It should be noted that the value of q, in eq. (5) is a function of cladding temperature [eq. (3)1. The coolant pressure change along the axial direction is neglected in the numerical calculation, because the coolant pressure in a FBR core is about 10°-10 ~ atm and any effect of the coolant pressure change on fuel rod thermal and mechanical behavior is negligible.
and qs
= - K ¢ ( OT ~-r ) too
(3)
The thermal behavior is related to the thermal conductance at a pellet-cladding gap. The gap width and contact pressure are determined by analyzing the mechanical behavior of the fuel rod.
2.2. A nalysis method for fuel rod thermal behavior For convenience in calculation, eq. (1) can be rewritten as follows by using the boundary condition that the radial fuel temperature gradient at the fuel pellet centerline vanishes:
r(r,
t) = T ( r n , t) +f~r, 2 ~ r K f d r
rR - dr T(r~i, t) + f~o2,rrK¢
T(r, t)
//
(fuel pellet)
(4)
f
The mechanical analysis is based upon the quasistatic, axisymmetric, generalized plane strain approximations both for fuel stack and cladding. These approximations are widely accepted as producing good numerical results. A more rigorous treatment is outside the scope of an integral fuel rod behavior analysis code. The validity of the quasi-static approximation is seen in the following consideration. The equilibrium equations of stress under transient conditions are written as follows:
02u, = ~.~ Ooij POt ~ J ~xj ( i , j = x , y , z ) .
(6)
Assuming the time scale of transient events to be of the order of 0.1-10 s and using the material properties of the fuel pellet, we have
OZui , O°ij
[ [ C R Or-q")2~rrdr f
3.1. Assumptions and approximations
(cladding)
where Q =L,i[
3. Analytical models for fuel rod mechanical behavior
P ~ t 2 / OXj = 1 0 - 9 - 1 0 - 1 3
0t
and
(7)
In this evaluation, the following relation is used.
R =J. C¢oc--~2~rrdr+ 2~rrKc rci
r¢i.
For the numerical calculation of eq. (4), a new calculation method was developed for the purpose of the improvement of calculation time and numerical stability. A detailed explanation is given in the Appendix.
2.3. Analytical method for coolant thermal behavior For the numerical calculation of coolant tempera-
oij - EuJrro. The quasi-static approximation can be used for actual cases of the transient states in FBR. Basic equations for the stress tensors are written as follows: OOij
E-g j = o (;,j=x,y,z).
(8)
In the mechanical analysis, eight strain components are taken into account; elastic/plastic strain, creep strain, thermal expansion, swelling strain, fuel cracking,
M. Nakamura, M. Sakagami / FARST
-
fuel relocation, and pore migration in the fuel pellet. The FARST code treats the axial friction force as well as contact pressure between pellets and cladding when pellet/cladding mechanical interaction occurs. Detailed explanations are presented in the following section. 3.1. Analysis method for fuel rod mechanical behavior
code for the evaluation of FBR fuel rod behavior
ILl p-
1400 .........................~
AG=(]-/)60
(0__
(ii) The difference between the fuel-cladding gap volume before and after the jump relocation is transfered to the crack void volume distributed in the bulk of cracked fuel and a central hole volume introduced due to the jump relocation. The effect of pellet relocation on the fuel temperature is shown in fig. 2 [6]. Calculated data with the jump relocation model are found to be in good agreement with the measurement data within the experimental uncertainty.
• :EXPERIMENTAL DATA
CALC .Wl THOUT RELOCATION MODEL
oe ILl I1. p-
1200
jii 'j .............
W
_z 3.1.1. Pellet relocation Measurements of fuel centerline temperature have established that a large fraction of as-fabricated diametrical gap volume is transferred into the bulk of fuel pellet as a crack void during the first power application. This is shown by fig. 1 which was obtained from the Halden Reactor [5]. A sudden decrease in gap width occurs at the linear heat rate of about 100 W / c m . This change or "Jump Relocation", is considered to occur due to fuel pellet cracking. In FARST code, this phenomena is modelled as follows: (i) When pellet is broken into many pieces due to cracking at first power application, pellet diameter increases suddenly and the change of gap width (AG) can be written as follows [4]:
41
.................. .....
1000 d: W
lZ U.I O -I UJ
CA LC. Wl TH RELOCAT I ON
800 i
O
U,.
i
i
k
J
J
i
i
:::)
"
~
I
100
I
200
I
m - CALCULATION -.e.- " EXPERIMENT
400
LINEAR HEAT RATE(W/cm} Fig. 1. Change in gap width at first power application in the Halden Reactor [5].
n.T"
3.1.2. Pellet swelling Pellet swelling has two components: gas bubble swelling and solid FP (Fission Product) swelling. The latter swelling rate does not depend on the restrained force acting on the fuel pellet, while the former does. If restrained force acts on the fuel pellet, the growth of the gas bubble in the pellet is supressed. The FARST code includes the following models for pellet swelling. (i) When pellet/cladding mechanical interaction does not occur (the gap is open), pellet swelling has two component: gas bubble swelling and solid FP swelling. These swelling rate are determined from experimental data. (ii) When pellet/cladding mechanical interaction occurs (the gap is closed), only solid FP swelling is taken into account, and a smaller value for the pellet swelling rate is adopted compared with the swelling rate when the gap is open. The effect of this pellet swelling model on the diametrical change of cladding is shown in fig. 3. Irradiation data were obtained in EBR-II [7].
1.0
300
i
Fig. 2. Effect of relocation on fuel centerline temperature.
iz <
0
MODEL i
4 o 8 10 12 ROD AVERAGE BURNUP (GWD/TU)
-
UMP O N
i
BOTTOM
'
I
0.2
0.4
1.
06
I.
08
1.0
TOP AXIAL POSITION(Z/L ) Fig. 3. Diametrical changes in cladding.
42
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
If the decrease of pellet swelling rate due to pellet/cladding mechanical interaction is neglected, the predicted values of A D / D become about 2-3%. This result shows the validity of our pellet swelling model. 3.1.3. Axial force For axial forces acting on pellets and cladding, their contact states are classified into three cases, as shown in fig. 4. Case 1 shows the state in which the system has an open gap and pellet/cladding mechanical interaction does not occur in any pellets. In this case, axial force is determined from spring force and plenum gas pressure. The weight of the fuel pellets is neglected. Case 2 shows the state in which pellet/cladding mechanical interaction occurs. In this case, friction force is determined from the following equations for the slip/stick conditions between pellet and cladding: Fz ~
AE~ = A ¢ ~ ( s t i c k ) ,
Fz =/~P,,
A¢f = Ac~(slip).
(9)
Case 3 shows the state in which pellet/cladding mechanical interaction occurs at an axial position and an axial gap exists between the pellets. In this case, the axial force acting on the fuel pellets below the position where pellet/cladding mechanical interaction occurs is determined only by the plenum gas pressure. The calculation algorithm is summarized in the following: (1) If pellet/cladding mechanical interaction does not occur, the axial friction force equals zero.
(2) When pellet/cladding mechanical interaction occurs, axial force is determined using the stick condition, such that the axial increment of pellet and cladding become equal. (3) If the axial force determined above is greater than the maximum friction force, the stick condition cannot be satisfied. In this case, the slip condition is adopted: i.e. axial force is reset to the maximum friction force, which is given by the product of a friction coefficient and contact pressure. (4) If the obtained axial force is less than the product of plenum gas pressure and cross section of the fuel pellet, then the axial compression force between the pellets does not occur and the axial force acting on fuel pellets is determined by only gas pressure. 3.1.4. Stress-strain analysis For the stress-strain analysis, strain components cj, ( j = r, 0, z) are divided into two parts: mean strain i and deviatoric strain ej ( j = r, O, z):
ej=,j-i
J
(10)
Stress components oj ( j = r, 0, z) are similarly divided into two parts: ~ = ½( Or'~- oe ~l- Oz), 5=0)-5
(j=r,O,z).
(11)
Mean strain is expressed as follows: = ~
soring
(j=r,O,z).
+ %h + % , ,
(12)
where Kv
E 1 - 2u "
cladding /
Deviatoric strain is calculated from the elastic strain, plastic strain and creep strain: ej = e E + e ~ + ec
f U~let J
(j=r,O,z).
(13)
m
Elastic strain is calculated using Hooke's law. Plastic and creep strains are calculated using the yon Mises' condition and the Prandtl-Reuss rule:
m
1
C4me. 1 PCMI does not oocur,
Cede. 2
Case.3
PCMI occurs.
Axial gad exists.
Fig. 4. Contact states between pellet and cladding.
d4=TgSj, , dec
- 1sj. -
(14)
M. Nakamura, M. Sakagami / FARST - codefor the evaluation of FBR fuel rod behavior The yielding condition is written as follows
(3 j~.S2)1/2= Y= constant.
(15)
The equilibrium equation of the stress component can be written as follows: a o, o o a r o, + - - r = O. -
(16)
Stress components in a plane perpendicular to an axial position are given as functions of radial coordinate r and time t, because of the axisymmetric generalized plane strain approximation. The relations between strain and displacement can be written as follows: aU
"=ar'
Ur
'a=--r '
%=C(t),
(17)
where, C(t) is a function of time. The stress-strain distribution is obtained by numerically by ~olving eq. (10)-(17) using the finite difference method. The change in compressibility of a fuel pellet due to cracking is treated by introducing a modified Young's modulus and Poisson's ratio. In the FARST code, the following modified elastic moduli are used [4]: P(")= 2
2p
J'
E(')=(2) " EI+~,"
(18)
3.1.5. Cladding failure probability One of the objectives of the FARST code is to evaluate the failure probability of a fuel rod under the steady-state/transient conditions. Cladding failure probability is calculated using the cumulative damage fraction method [8]:
CDF=f~,
tions. CABRI is a light-water-cooled pulse reactor. A capsule including one fuel rod and coolant flow (sodium) surrounding it was used for irradiation of the fuel rods. In this section, the CABRI B2 loss of coolant experiment was selected for the verification of the FARST code. The experimental conditions and fuel pin parameters are shown in table 1. In this experiment, coolant temperature and fuel temperature were measured continuously with a thermocouple. Linear heat rate at a thermocouple position was held at 235 W / c m during the transient experiment. The calculated coolant temperature and the measured values are compared in fig. 5; values agree within 4-5%. Fig. 6 shows the comparison of calculated fuel temperature at the thermocouple position and the measured one; values agree within 4-5%. In these calculations, heat flux loss from the capsule was taken into consideration. This effect leads to a decrease in coolant temperature by about 50 o C at the thermocouple position in the fuel rod. 4.2. NSRR experiment analysis The FARST code was applied to the analysis of the experimental data obtained in NSRR [9]. Reactivity-initiated accident (RIA) experiments using the light water reactor fuel rods were selected for the code verification. The purpose of this experiment is to evaluate the rela-
Table 1 Fuel rod parameters and coolant flow conditions in the CABRI reactor CLADDING ITEM
UNIT
FUEL P E L L E T
(19)
where, time to rupture t t is obtained from the out-of pile experiments on the cladding tube and it depends on stress and temperature. The value of CDF indicates the cladding failure probability and if the value of the CDF exceeds 1.0, the cladding is thought to have failed.
4. Application to transient experiments analysis 4.1. CABRI experiment analysis CABRI experiments were performed for the purpose of evaluating fuel rod behavior under transient condi-
43
INNER RADIUS
VALUE
mm
1.94, 0.0
OUTER RADIUS mm
6.40, 635
DENSITY
96.2, 92.4
%TD
FUEL STACK LENGTH mm CLADDING INNER RADIUS mm OUTER RADIUS PEAK POWER COOLANT FLOW RATE
lOO
(A), (B)
W/er~ g/s
TT0.0
F
129
~"
650
6,61 7.60
~,00
462.0
L 102.7 l + t/? t : TI ME( s
420
UNIT : mm • : THERMOCOUPLE
44
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior =
900
CORE
" CALCULATION ~.
1500 -o-
~E hi
" EXPERIMENT
o~ 800 v ILl n,"
1000 700
J
o
LLI
Q.
~E ~E
F- 600
X
~E
<, 8
500
I 0.5
I 1.0
I 1.5
2.0
VELOCITY (m/s)
Fig. 7. Maximum cladding temperature versus coolant velocity. 400
_
_
I 20
0
I 40
I 60
I 80
tion between cladding maximum temperature and coolant inlet velocity. The cladding surface temperature was measured continuously with a thermocouple. Fuel rod parameters are shown in table 2. The experimental data were analyzed with a modification of materal properties from FBR type to LWR type. The calculated and measured values for maximum cladding temperature versus coolant velocity are shown in fig. 7. It can be seen in fig. 7 that the calculation is in good agreement with the experiment. The results mentioned above indicate that the FARST code can predict the transient behavior of at least low burnup fuel rods.
100
DISTANCE FROM BOTTOM (cm) Fig. 5. Comparison of coolant temperature.
1600
i
v bJ
1400
Z)
uJ 12oo a. 1000 .,J W =) h
00 "
COOLANT
O
0::
500
--
: CALCULATION
--
: EXPERIMENT
800
SCRAM i :
I
1
5
1O
TIME
I 15
(s)
5. A p p l i c a t i o n to a large F B R plant fuel rod
Fig. 6. Comparison of fuel centerline temperature at the thermocouple position. Table 2 Fuel rod parameters and coolant flow conditions in NSRR ITEM FUEL P E L L E T INNER RADIUS OUTER RADIUS DENSITY FUEL STACK LENGTH
UNIT
VALUE
mm mm %TD
95.
mm
135~}
CLADDING INNER RADIUS mm OUTER RADIUS ram INPUT ENERGY kcaljg COOLANT VELOCITY rrv's INLET C TEMPERATURE
0.0
4.65
4.74 530
IgO.O 0,3-1~ 2O
The FARST code was applied to the analysis of the cladding integrity of a 1000 MWe class large FBR plant fuel rod. The conceptual design of this fuel is shown in table 3. This fuel rod has a gas plenum in its lower section and the fuel pellet density is 94.5% TD. High pellet density is selected for the reduction of fuel doubling time. In this analysis, attention is paid especially to the cladding hoop stress under steady-state/transient conditions. Cladding stress strongly affects the fuel rod integrity. 5.1. Steady-state analysis
The calculated results for the cladding hoop stress in an assumed steady-state irradiation history are shown in fig. 8. We can see that the fuel temperature has its
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
Table 3 Calculation conditions CLADDING AXIAL BLANKET
150
UNIT VALUE
ITEM FUEL
ILIIIJ 3,0
PELLET
INNER RADIUS mm
OUTER RADIUS
..,
FUEL STACK LENGTH CLADDING INNER RADIUS OUTER RADIUS
COOLANT FLOW RATE
Ill .Ill
6.4;
DENSITY
PEAK POWER
/ ~/.
III I 000
at cycle 2 BOC exceeds the power level at cycle 1 EOC, Since the linear heat rate increases at the rate of 100%/10 h at cycle 2 BOC, stress relaxation due to creep deformation during power increase can hardly be expected. The cladding hoop stress can be seen to increase gradually during operation with increasing fission product gas pressure. These results indicate that the cladding hoop stress has the most severe value if transient events occur at cycle 2 BOC. The calculations for various transient conditions are performed in the following section.
.oo I Ill III1 6.60
mm
III-E-F
~
,.,o I
w~c~,~o.o [ g/s
45
ill_ill j_ ,,o
I
II lit
1151S~ 0 7 0 if) hi {E t--
PLENUM
maximum value at the beginning of cycle 1. After that, the fuel temperature decreases mainly due to the central void formation and fuel pellet swelling. Fig. 8 indicates that the cladding hoop stress has its maximum value at cycle 2 BOC, because the power level
W
25 w
0 I
500 n~ bJ T
z
450 n~
<
bJ
-J
~
20
4I
8[
400 5
~ 12
T I M E (s)
1.0 . , - ~
-,-..__.___
L
ot.,-
W
3000
t,,,, I n , "
W
z
\ -,t
W~
m LtJ (3 Z
.jn" w~
,, },- 18(~
2500
0 o
g Q_ Q~
2000 292
584
876
I 4
I
8
I
12
16
TIME (S)
TIME (day)
Fig. 8. Changes in fuel centerline temperature and cladding hoop stress under steady-state irradiation.
Fig. 9. (a, top) Change in cladding hoop stress under overpower conditions. (b) Change in gap conductance under overpower conditions.
46
M. Nakamura, M. Sakagami / F A R S T - code for the evaluation of FBR fuel rod behavior to~-to E
5.2. Transient analysis
The following transient phenomena are analyzed, which are considered to affect the fuel rod integrity significantly: (i) change in linear heat rate (overpower condition), (ii) change in coolant flow rate or coolant temperature. Change in the cladding hoop stress under overpower transient is shown in fig. 9(a). Overpower continuation time is set 6 s and its height is assumed to be 14% of nominal power. The figure shows that the time at which cladding hoop stress has its maximum value is delayed by a few seconds from the time for a maximum linear heat rate. In fig. 9(a), gap conductance increases by about 40% at the overpower transient due to an increase of contact pressure between pellet and cladding as shown in fig. 9(b). If the change in gap conductance during the power transient is not considered, fuel temperature and cladding hoop stress are overestimated by about 100 o C and 5 k g / m m 2, respectively. The relation between the maximum cladding hoop stress and overpower continuation time is shown in fig. 10. The maximum cladding hoop stress increases with increasing overpower continuation time and approaches an asymptotic value for times longer than 40 s. This asymptotic value is obtained by a steady-state model analysis. The steady-state model gives conservative resuits and overstimates the cladding hoop stress. When the coolant inlet temperature drops suddenly, cladding hoop stress increases in a few seconds due to the thermal contraction of the cladding. The change in cladding hoop stress is shown in fig. 11. In this case, the
to to LU nI-to n O
M I N A L POWER /" :
\
\ ~
20
~STEADY-STATE ~CALCULATION
a,
o z
0. Po
°
/.
<
.J 0
i 0
dT .
I
20 OVERPOWER
Po
/I ~
I I
DURATION
I
dT
__
- 30~C/s
__-
- 15°C/s
CL
o o I
/
2s
t
121 1:3 < .J 20 = 0
f
I 10
I
I 20
I
l/0 30
4o0
ZcY .
TIME ( s )
Fig. 11. Changes in cladding hoop stress under decreased coolant inlet temperature.
increment in cladding hoop stress attains about 12 k g / m m 2. The steady-state model, however, cannot predict this overshoot and underestimates the cladding hoop stress.
6. Conclusion
A computer code FARST was developed for the evaluation of FBR fuel rod behavior under steadystate/transient conditions. The FARST code has the functions of analyzing the thermal and mechanical behavior of a fuel rod over its whole length and predicting the cladding failure probability during normal operation and transient conditions. For code verification, FARST was applied to the analysis of transient experimental data from CABRI and NSRR. Calculation results showed that the FARST code could predict the fuel rod behavior under the transient conditions well and confirmed its application to FBR fuel rod design. The FARST code was applied to the evaluation of a large FBR plant fuel rod. The calculations showed that the steady-state model could possibly underestimate cladding hoop stress for coolant inlet temperature decrease. Additional code development and calibration using the experimental data will be carried out in the future.
I Acknowledgements
I
40 TIME ( s )
60
Fig. 10. M a x i m u m cladding hoop stress versus overpower duration time.
The authors would like to express thanks to Drs K. Taniguchi, S. Yamada, S. Kobayashi, R. Takeda and K. Miki of the Energy Research Laboratory, Hitachi Ltd for encouragement throughout this study.
M. Nakamura, M. Sakagami / FARST - codefor the evaluation of FBR fuel rod behavior Appendix In this section, a numerical calculation method for fuel and cladding temperature under the transient conditions is shown. Basic equations for the fuel pellet are written as follows: r
0
r ( r , t) = T(rfi, t) +fr,12~-~rKcdr,
(iv) The values of material properties at time t + t/2 are used for a time interval between t and t + At (Crank-Nicholson model). Using these assumptions eqs. (A1) and (A2) can be written as follows:
T ;t++Ia , /[ ~1 _ ~ N"~
(AI)
where
47
)= t~N+ TN(-- TI+I _ r ; + a t + r , ~ N / A]_-t --½(TI+,-- T~v+at--T~v), (A4)
where
Q=f~i(CfPr~t - q " ) 2 ~ ' r d r .
(A2) r~
Radial mesh division is shown in fig. A1. Eq. (A1) is rewritten as
T, rN+,,t)=
T(rN,
t)+ frr;+'2QK dr.
8 N I n rN+l +
"fN =
rN
CNP N
At
qN
,A3)
~,, )]' In order to carry out the numerical integration in eq. (A3), the following assumptions are used. (i) Constant values of the material properties are used for the region between two mesh points. (ii) Between two mesh points, fuel temperature is expressed as follows:
T=TN+(TN+I_T~v)
E j=l
cioj,
"
At
-
T/++Ia t - T j t ~ - Tj' + a ' - Tj' +
At
r--rN rN+ 1 -- rN
× -~(~.+, - ~) x (2~.+, +
~)1 /
(rN~r
(iii) The time derivative of fuel temperature is approximated as follows:
[ j*l
-qj
The calculation is performed from the inside to the outside of the pellet. When the time interval t goes to the infinity, eq. (A4) is reduced to the finite difference
a T / ' =__1 (Tt+zt at ]N A t ~ ' N T~).
CN-~ PN-I, aN-~ 0.8
C~ . P~. q('
O3
r
~
°
~
i
r2 ................ rN-1,rN , rN+l ................ rM T1 , 1"2 ................ TN-1,TN , TN+I ................TM rl
,
Fig. A1. Radial mesh division (fuel pellet).
/11
o., -_:,,r.,°,ytio°,
Solution jI f I • :Calculated / / I 0.2-[ Results L/ / ].= 0
09
0.4
0.6
0J
1.0
r/Fro Fig. A2. Comparison between numerical and analytical solutions.
48
M. Nakamura, M. Sakagami / F A R S T - code for the evaluation of FBR fuel rod behavior
form of the steady-state model. Eq. (A4) has no singularity at r = 0 (if r goes to zero, the value of r2+a × In § + l / r j goes to zero). The validity of this calculation method is shown by comparing the numerical results with the analytical solution. In one such problem, an initially isothermal (0 o C) fuel pellet (inner radius = 0, outer radius = rfo ) is stepped to TO ( ° C ) at the outer radius. Fuel heat conductance coefficient K, heat capacity C and pellet density pf are assumed to be constant. An analytical solution can easily be obtained as an expansion of the Bessel function [10]. A comparison of the calculated results and the analytical solution is shown in fig A2. A satisfactory agreement is obtained, indicating the validity of the numerical method described above. Nomenclature C CDF D Dp
qs S s T t
heat capacity ( J / g o C) cumulative damage fraction cladding diameter (cm) a factor of either 0 or 1 depending on whether plastic flow is prevented or takes place Young's modulus ( k g / m m 2) deviatoric strain (/-component) axial force (kg) G = E / 2 ( 1 + v) ( k g / m m 2) initial gap width (mm) mass flux of coolant flow (kg/s) thermal conductivity ( W / c m 2 g o C) K , = E / ( 1 - 2v) k g / m m 2 number of cracks power density ( W / c m 3) heat flux at cladding surface ( W / c m z) coolant flow are per one rod (cm 2) deviatoric stress (/-component) ( k g / m m 2 s) temperature ( o C) time (s)
t*
t* = Kt/Cprt2o
tf ui t~ Y At AD c~j c ~, /~
time to rupture (s) displacement (i-component) (mm) coolant velocity ( c m / s ) yielding criterion ( k g / m m 2) time increment (s) diameter change (cm) strain ( / j - c o m p o n e n t ) mean strain Poisson's ratio friction coefficient plastic flow parameter (mm2/kg s) coefficient of viscosity (mm2/kg s) stress ( / j - c o m p o n e n t ) ( k g / m m 2) mean stress ( k g / m m 2)
E ei Fz G GO J0 K K.
n q"
7/ Oij
o
p
x/
density ( g / c m 3)
y z
coordinate system)
0 z
coordinate component system)
component
(Cartesian
coordinate
(cylindrical coordinate
Subscript
c ci co f fi fo Na
cladding cladding inner radius cladding outer radius fuel pellet fuel pellet inner radius fuel pellet outer radius sodium
Superscript
C E P
creep strain elastic strain plastic strain
References [1] J.B. Newman et al., The CYGRO-4 fuel rod analysis computer program Nucl. Engrg. Des. 46 (1978) 1. [2] J. Wordsworth, IAMBUS-1 - A digital computer code for the design, in-pile performance prediction and post-irradiation analysis of arbitrary fuel rods, Nucl. Engrg. Des. 31 (1974) 309. [3] P. Royl et al., Transient fuel-cladding deformation analysis of the in-pile H3 'TREAT' test with the 'SAS2A/DEFORM-II' code Nucl. Engrg. Des. 27 (1974) 299. [4] M. Ishida et al., An Analysis of Mechanical Interaction between Pellet and cladding, Proc. Int. Conf. on Fast Breeder Reactor Fuel Performance, Monterey, California (March, 1979). [5] M. Oguma and T. Hosokawa, private communication. [6] EPRI, Light Water Reactor Fuel ROd Modelling Code Evaluation, NP-369 (1977). [7] P.J. Levine et al., Irradiation Performance of WSA-3, -4, and -8 Mixed Oxide Fuel Pins in Grid-Spaced Assemblies, Proc. Int. Conf. on Fast Breader Reactor Fuel Performance, Monterey, California (March, 1979). [8] A.J. Lovel et al., Observations of In-Reactor Endurance and Rupture Life for Fueled and Unfueled FTR Cladding, Trans. ANS 32 (1979) 217. [9] T. Fujishiro et al., Effects of coolant flow on light water reactor fuel behaviors during reactivity initiated accident, J. Nucl. Sci. Tech. 18 (1981) 196. [10] H.S. Carlow et al., Conduction of Heat in Solids, 2nd Ed. (Clarendon Press, Oxford).
39
FARST - A COMPUTER CODE FOR THE EVALUATION UNDER STEADY-STATE/TRANSIENT CONDITIONS
M. N A K A M U R A
OF FBR FUEL ROD BEHAVIOR
a n d M. S A K A G A M I
Hitachi Energy Research Laboratory, Hitachi Lid, 1168 Moriyama-cho, Hitachi-shi, Ibaraki-ken, Japan Received 21 November 1983
FARST, a computer code for the evaluation of fuel rod thermal and mechanical behavior under steady-state/transient conditions has been developed. The code characteristics are summarized as follows: (i) FARST evaluates the fuel rod behavior under the transient conditions. The code analyzes thermal and mechanical phenomena within a fuel rod, taking into account the temperature change in coolant surrounding the fuel rod. (ii) Permanent strains such as plastic, creep and swelling strains as well as thermoelastic deformations can be analyzed by using the strain increment method. (iii) Axial force and contact pressure which act on the fuel stack and cladding are analyzed based on the stick/slip conditions. (iv) FARST used a pellet swelling model which depends on the contact pressure between pellet and cladding, and an empirical pellet relocation model, designated as "jump relocation model". The code was successfully applied to analyses of the fuel rod irradiation data from pulse reactor for nuclear safety research in Cadarache (CABRI) and pulse reactor for nuclear safety research in Japan Atomic Energy Research Institute (NSRR). The code was further applied to stress analysis of a 1000 MW class large FBR plant fuel rod during transient conditions. The steady-state model which was used so far gave the conservative results for cladding stress during overpower transient, but underestimated the results for cladding stress during a rapid temperature decrease of coolant.
I. Introduction As cladding is the first barrier against the release of radioactive fission products, evaluation of cladding integrity is critical to good fuel rod design. Many computer codes [1,2] have been developed for such evaluation and careful consideration has been given to modelling thermo-mechanical behavior, during steady-state burnup irradiation. Recently, fuel rod design codes have been required for behavior evaluation during transient conditions. This requires knowledge about the fuel rod integrity under transient conditions, as well as under steady state conditions, and has led to the development of FARST. F A R S T code can be applied to thermal and mechanical design, in-pile performance prediction and postirradition analysis of fuel rods, which are mixed-oxide fuelled, stainless-steel cladding tube for sodium-cooled fast breeder reactors. However, other similar systems (e.g.: light water reactor fuel rods) can be analyzed with F A R S T , subject to minor modification. Analytical models describing thermal behavior of fuel rod under the steady-state/transient conditions are
presented in section 2, while those for mechanical behavior are presented in section 3. Applications of this code to analyses of experimental data from pulse reactor for nuclear safety research in Cadarache (CABRI) and pulse reactor for nuclear safety research in Japan Atomic Energy Research Institute (NSRR) [9] are presented in section 4. Finally applications to evaluation of cladding stress of in a large F B R plant fuel rods under the various kinds of transient conditions are also presented in section 5.
2. Analytical models for fuel rod and coolant thermal behavior
2.1. Basic equations Thermal behavior analysis is performed using one-dimensional (radial) thermal conductance model for a fuel rod and one-dimensional (axial) incompressible flow model for the coolant surrounding the fuel rod. These models are widely accepted and have been utilized in other codes [1,2].
0 0 2 9 - 5 4 9 3 / 8 4 / $ 0 3 . 0 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. (North-Holland Physics Publishing Division)
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
40
Basic equations are written as follows.
1 O/ OT\ ,,,~ -r or[rKf~r )+q t
0T Cfpf Ot
(at)
OT
lr OrO rK~r
CcPc Ot
ture, eq. (2) is rewritten as follows: CNa T = ( C N a T ) z 0
(fuel pellet)
PNa/tNa S = Jo = c o n s t a n t
0
( p N~CN~TS)
)
(1)
1 [ (2, rooqs- (CN.o arS) dz
(cladding)
(coolant
now)
(2)
0
+ ~ (PN~CN~TSUN~)= 2,rr~oq~
(5)
It should be noted that the value of q, in eq. (5) is a function of cladding temperature [eq. (3)1. The coolant pressure change along the axial direction is neglected in the numerical calculation, because the coolant pressure in a FBR core is about 10°-10 ~ atm and any effect of the coolant pressure change on fuel rod thermal and mechanical behavior is negligible.
and qs
= - K ¢ ( OT ~-r ) too
(3)
The thermal behavior is related to the thermal conductance at a pellet-cladding gap. The gap width and contact pressure are determined by analyzing the mechanical behavior of the fuel rod.
2.2. A nalysis method for fuel rod thermal behavior For convenience in calculation, eq. (1) can be rewritten as follows by using the boundary condition that the radial fuel temperature gradient at the fuel pellet centerline vanishes:
r(r,
t) = T ( r n , t) +f~r, 2 ~ r K f d r
rR - dr T(r~i, t) + f~o2,rrK¢
T(r, t)
//
(fuel pellet)
(4)
f
The mechanical analysis is based upon the quasistatic, axisymmetric, generalized plane strain approximations both for fuel stack and cladding. These approximations are widely accepted as producing good numerical results. A more rigorous treatment is outside the scope of an integral fuel rod behavior analysis code. The validity of the quasi-static approximation is seen in the following consideration. The equilibrium equations of stress under transient conditions are written as follows:
02u, = ~.~ Ooij POt ~ J ~xj ( i , j = x , y , z ) .
(6)
Assuming the time scale of transient events to be of the order of 0.1-10 s and using the material properties of the fuel pellet, we have
OZui , O°ij
[ [ C R Or-q")2~rrdr f
3.1. Assumptions and approximations
(cladding)
where Q =L,i[
3. Analytical models for fuel rod mechanical behavior
P ~ t 2 / OXj = 1 0 - 9 - 1 0 - 1 3
0t
and
(7)
In this evaluation, the following relation is used.
R =J. C¢oc--~2~rrdr+ 2~rrKc rci
r¢i.
For the numerical calculation of eq. (4), a new calculation method was developed for the purpose of the improvement of calculation time and numerical stability. A detailed explanation is given in the Appendix.
2.3. Analytical method for coolant thermal behavior For the numerical calculation of coolant tempera-
oij - EuJrro. The quasi-static approximation can be used for actual cases of the transient states in FBR. Basic equations for the stress tensors are written as follows: OOij
E-g j = o (;,j=x,y,z).
(8)
In the mechanical analysis, eight strain components are taken into account; elastic/plastic strain, creep strain, thermal expansion, swelling strain, fuel cracking,
M. Nakamura, M. Sakagami / FARST
-
fuel relocation, and pore migration in the fuel pellet. The FARST code treats the axial friction force as well as contact pressure between pellets and cladding when pellet/cladding mechanical interaction occurs. Detailed explanations are presented in the following section. 3.1. Analysis method for fuel rod mechanical behavior
code for the evaluation of FBR fuel rod behavior
ILl p-
1400 .........................~
AG=(]-/)60
(0__
(ii) The difference between the fuel-cladding gap volume before and after the jump relocation is transfered to the crack void volume distributed in the bulk of cracked fuel and a central hole volume introduced due to the jump relocation. The effect of pellet relocation on the fuel temperature is shown in fig. 2 [6]. Calculated data with the jump relocation model are found to be in good agreement with the measurement data within the experimental uncertainty.
• :EXPERIMENTAL DATA
CALC .Wl THOUT RELOCATION MODEL
oe ILl I1. p-
1200
jii 'j .............
W
_z 3.1.1. Pellet relocation Measurements of fuel centerline temperature have established that a large fraction of as-fabricated diametrical gap volume is transferred into the bulk of fuel pellet as a crack void during the first power application. This is shown by fig. 1 which was obtained from the Halden Reactor [5]. A sudden decrease in gap width occurs at the linear heat rate of about 100 W / c m . This change or "Jump Relocation", is considered to occur due to fuel pellet cracking. In FARST code, this phenomena is modelled as follows: (i) When pellet is broken into many pieces due to cracking at first power application, pellet diameter increases suddenly and the change of gap width (AG) can be written as follows [4]:
41
.................. .....
1000 d: W
lZ U.I O -I UJ
CA LC. Wl TH RELOCAT I ON
800 i
O
U,.
i
i
k
J
J
i
i
:::)
"
~
I
100
I
200
I
m - CALCULATION -.e.- " EXPERIMENT
400
LINEAR HEAT RATE(W/cm} Fig. 1. Change in gap width at first power application in the Halden Reactor [5].
n.T"
3.1.2. Pellet swelling Pellet swelling has two components: gas bubble swelling and solid FP (Fission Product) swelling. The latter swelling rate does not depend on the restrained force acting on the fuel pellet, while the former does. If restrained force acts on the fuel pellet, the growth of the gas bubble in the pellet is supressed. The FARST code includes the following models for pellet swelling. (i) When pellet/cladding mechanical interaction does not occur (the gap is open), pellet swelling has two component: gas bubble swelling and solid FP swelling. These swelling rate are determined from experimental data. (ii) When pellet/cladding mechanical interaction occurs (the gap is closed), only solid FP swelling is taken into account, and a smaller value for the pellet swelling rate is adopted compared with the swelling rate when the gap is open. The effect of this pellet swelling model on the diametrical change of cladding is shown in fig. 3. Irradiation data were obtained in EBR-II [7].
1.0
300
i
Fig. 2. Effect of relocation on fuel centerline temperature.
iz <
0
MODEL i
4 o 8 10 12 ROD AVERAGE BURNUP (GWD/TU)
-
UMP O N
i
BOTTOM
'
I
0.2
0.4
1.
06
I.
08
1.0
TOP AXIAL POSITION(Z/L ) Fig. 3. Diametrical changes in cladding.
42
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
If the decrease of pellet swelling rate due to pellet/cladding mechanical interaction is neglected, the predicted values of A D / D become about 2-3%. This result shows the validity of our pellet swelling model. 3.1.3. Axial force For axial forces acting on pellets and cladding, their contact states are classified into three cases, as shown in fig. 4. Case 1 shows the state in which the system has an open gap and pellet/cladding mechanical interaction does not occur in any pellets. In this case, axial force is determined from spring force and plenum gas pressure. The weight of the fuel pellets is neglected. Case 2 shows the state in which pellet/cladding mechanical interaction occurs. In this case, friction force is determined from the following equations for the slip/stick conditions between pellet and cladding: Fz ~
AE~ = A ¢ ~ ( s t i c k ) ,
Fz =/~P,,
A¢f = Ac~(slip).
(9)
Case 3 shows the state in which pellet/cladding mechanical interaction occurs at an axial position and an axial gap exists between the pellets. In this case, the axial force acting on the fuel pellets below the position where pellet/cladding mechanical interaction occurs is determined only by the plenum gas pressure. The calculation algorithm is summarized in the following: (1) If pellet/cladding mechanical interaction does not occur, the axial friction force equals zero.
(2) When pellet/cladding mechanical interaction occurs, axial force is determined using the stick condition, such that the axial increment of pellet and cladding become equal. (3) If the axial force determined above is greater than the maximum friction force, the stick condition cannot be satisfied. In this case, the slip condition is adopted: i.e. axial force is reset to the maximum friction force, which is given by the product of a friction coefficient and contact pressure. (4) If the obtained axial force is less than the product of plenum gas pressure and cross section of the fuel pellet, then the axial compression force between the pellets does not occur and the axial force acting on fuel pellets is determined by only gas pressure. 3.1.4. Stress-strain analysis For the stress-strain analysis, strain components cj, ( j = r, 0, z) are divided into two parts: mean strain i and deviatoric strain ej ( j = r, O, z):
ej=,j-i
J
(10)
Stress components oj ( j = r, 0, z) are similarly divided into two parts: ~ = ½( Or'~- oe ~l- Oz), 5=0)-5
(j=r,O,z).
(11)
Mean strain is expressed as follows: = ~
soring
(j=r,O,z).
+ %h + % , ,
(12)
where Kv
E 1 - 2u "
cladding /
Deviatoric strain is calculated from the elastic strain, plastic strain and creep strain: ej = e E + e ~ + ec
f U~let J
(j=r,O,z).
(13)
m
Elastic strain is calculated using Hooke's law. Plastic and creep strains are calculated using the yon Mises' condition and the Prandtl-Reuss rule:
m
1
C4me. 1 PCMI does not oocur,
Cede. 2
Case.3
PCMI occurs.
Axial gad exists.
Fig. 4. Contact states between pellet and cladding.
d4=TgSj, , dec
- 1sj. -
(14)
M. Nakamura, M. Sakagami / FARST - codefor the evaluation of FBR fuel rod behavior The yielding condition is written as follows
(3 j~.S2)1/2= Y= constant.
(15)
The equilibrium equation of the stress component can be written as follows: a o, o o a r o, + - - r = O. -
(16)
Stress components in a plane perpendicular to an axial position are given as functions of radial coordinate r and time t, because of the axisymmetric generalized plane strain approximation. The relations between strain and displacement can be written as follows: aU
"=ar'
Ur
'a=--r '
%=C(t),
(17)
where, C(t) is a function of time. The stress-strain distribution is obtained by numerically by ~olving eq. (10)-(17) using the finite difference method. The change in compressibility of a fuel pellet due to cracking is treated by introducing a modified Young's modulus and Poisson's ratio. In the FARST code, the following modified elastic moduli are used [4]: P(")= 2
2p
J'
E(')=(2) " EI+~,"
(18)
3.1.5. Cladding failure probability One of the objectives of the FARST code is to evaluate the failure probability of a fuel rod under the steady-state/transient conditions. Cladding failure probability is calculated using the cumulative damage fraction method [8]:
CDF=f~,
tions. CABRI is a light-water-cooled pulse reactor. A capsule including one fuel rod and coolant flow (sodium) surrounding it was used for irradiation of the fuel rods. In this section, the CABRI B2 loss of coolant experiment was selected for the verification of the FARST code. The experimental conditions and fuel pin parameters are shown in table 1. In this experiment, coolant temperature and fuel temperature were measured continuously with a thermocouple. Linear heat rate at a thermocouple position was held at 235 W / c m during the transient experiment. The calculated coolant temperature and the measured values are compared in fig. 5; values agree within 4-5%. Fig. 6 shows the comparison of calculated fuel temperature at the thermocouple position and the measured one; values agree within 4-5%. In these calculations, heat flux loss from the capsule was taken into consideration. This effect leads to a decrease in coolant temperature by about 50 o C at the thermocouple position in the fuel rod. 4.2. NSRR experiment analysis The FARST code was applied to the analysis of the experimental data obtained in NSRR [9]. Reactivity-initiated accident (RIA) experiments using the light water reactor fuel rods were selected for the code verification. The purpose of this experiment is to evaluate the rela-
Table 1 Fuel rod parameters and coolant flow conditions in the CABRI reactor CLADDING ITEM
UNIT
FUEL P E L L E T
(19)
where, time to rupture t t is obtained from the out-of pile experiments on the cladding tube and it depends on stress and temperature. The value of CDF indicates the cladding failure probability and if the value of the CDF exceeds 1.0, the cladding is thought to have failed.
4. Application to transient experiments analysis 4.1. CABRI experiment analysis CABRI experiments were performed for the purpose of evaluating fuel rod behavior under transient condi-
43
INNER RADIUS
VALUE
mm
1.94, 0.0
OUTER RADIUS mm
6.40, 635
DENSITY
96.2, 92.4
%TD
FUEL STACK LENGTH mm CLADDING INNER RADIUS mm OUTER RADIUS PEAK POWER COOLANT FLOW RATE
lOO
(A), (B)
W/er~ g/s
TT0.0
F
129
~"
650
6,61 7.60
~,00
462.0
L 102.7 l + t/? t : TI ME( s
420
UNIT : mm • : THERMOCOUPLE
44
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior =
900
CORE
" CALCULATION ~.
1500 -o-
~E hi
" EXPERIMENT
o~ 800 v ILl n,"
1000 700
J
o
LLI
Q.
~E ~E
F- 600
X
~E
<, 8
500
I 0.5
I 1.0
I 1.5
2.0
VELOCITY (m/s)
Fig. 7. Maximum cladding temperature versus coolant velocity. 400
_
_
I 20
0
I 40
I 60
I 80
tion between cladding maximum temperature and coolant inlet velocity. The cladding surface temperature was measured continuously with a thermocouple. Fuel rod parameters are shown in table 2. The experimental data were analyzed with a modification of materal properties from FBR type to LWR type. The calculated and measured values for maximum cladding temperature versus coolant velocity are shown in fig. 7. It can be seen in fig. 7 that the calculation is in good agreement with the experiment. The results mentioned above indicate that the FARST code can predict the transient behavior of at least low burnup fuel rods.
100
DISTANCE FROM BOTTOM (cm) Fig. 5. Comparison of coolant temperature.
1600
i
v bJ
1400
Z)
uJ 12oo a. 1000 .,J W =) h
00 "
COOLANT
O
0::
500
--
: CALCULATION
--
: EXPERIMENT
800
SCRAM i :
I
1
5
1O
TIME
I 15
(s)
5. A p p l i c a t i o n to a large F B R plant fuel rod
Fig. 6. Comparison of fuel centerline temperature at the thermocouple position. Table 2 Fuel rod parameters and coolant flow conditions in NSRR ITEM FUEL P E L L E T INNER RADIUS OUTER RADIUS DENSITY FUEL STACK LENGTH
UNIT
VALUE
mm mm %TD
95.
mm
135~}
CLADDING INNER RADIUS mm OUTER RADIUS ram INPUT ENERGY kcaljg COOLANT VELOCITY rrv's INLET C TEMPERATURE
0.0
4.65
4.74 530
IgO.O 0,3-1~ 2O
The FARST code was applied to the analysis of the cladding integrity of a 1000 MWe class large FBR plant fuel rod. The conceptual design of this fuel is shown in table 3. This fuel rod has a gas plenum in its lower section and the fuel pellet density is 94.5% TD. High pellet density is selected for the reduction of fuel doubling time. In this analysis, attention is paid especially to the cladding hoop stress under steady-state/transient conditions. Cladding stress strongly affects the fuel rod integrity. 5.1. Steady-state analysis
The calculated results for the cladding hoop stress in an assumed steady-state irradiation history are shown in fig. 8. We can see that the fuel temperature has its
M. Nakamura, M. Sakagami / FARST - code for the evaluation of FBR fuel rod behavior
Table 3 Calculation conditions CLADDING AXIAL BLANKET
150
UNIT VALUE
ITEM FUEL
ILIIIJ 3,0
PELLET
INNER RADIUS mm
OUTER RADIUS
..,
FUEL STACK LENGTH CLADDING INNER RADIUS OUTER RADIUS
COOLANT FLOW RATE
Ill .Ill
6.4;
DENSITY
PEAK POWER
/ ~/.
III I 000
at cycle 2 BOC exceeds the power level at cycle 1 EOC, Since the linear heat rate increases at the rate of 100%/10 h at cycle 2 BOC, stress relaxation due to creep deformation during power increase can hardly be expected. The cladding hoop stress can be seen to increase gradually during operation with increasing fission product gas pressure. These results indicate that the cladding hoop stress has the most severe value if transient events occur at cycle 2 BOC. The calculations for various transient conditions are performed in the following section.
.oo I Ill III1 6.60
mm
III-E-F
~
,.,o I
w~c~,~o.o [ g/s
45
ill_ill j_ ,,o
I
II lit
1151S~ 0 7 0 if) hi {E t--
PLENUM
maximum value at the beginning of cycle 1. After that, the fuel temperature decreases mainly due to the central void formation and fuel pellet swelling. Fig. 8 indicates that the cladding hoop stress has its maximum value at cycle 2 BOC, because the power level
W
25 w
0 I
500 n~ bJ T
z
450 n~
<
bJ
-J
~
20
4I
8[
400 5
~ 12
T I M E (s)
1.0 . , - ~
-,-..__.___
L
ot.,-
W
3000
t,,,, I n , "
W
z
\ -,t
W~
m LtJ (3 Z
.jn" w~
,, },- 18(~
2500
0 o
g Q_ Q~
2000 292
584
876
I 4
I
8
I
12
16
TIME (S)
TIME (day)
Fig. 8. Changes in fuel centerline temperature and cladding hoop stress under steady-state irradiation.
Fig. 9. (a, top) Change in cladding hoop stress under overpower conditions. (b) Change in gap conductance under overpower conditions.
46
M. Nakamura, M. Sakagami / F A R S T - code for the evaluation of FBR fuel rod behavior to~-to E
5.2. Transient analysis
The following transient phenomena are analyzed, which are considered to affect the fuel rod integrity significantly: (i) change in linear heat rate (overpower condition), (ii) change in coolant flow rate or coolant temperature. Change in the cladding hoop stress under overpower transient is shown in fig. 9(a). Overpower continuation time is set 6 s and its height is assumed to be 14% of nominal power. The figure shows that the time at which cladding hoop stress has its maximum value is delayed by a few seconds from the time for a maximum linear heat rate. In fig. 9(a), gap conductance increases by about 40% at the overpower transient due to an increase of contact pressure between pellet and cladding as shown in fig. 9(b). If the change in gap conductance during the power transient is not considered, fuel temperature and cladding hoop stress are overestimated by about 100 o C and 5 k g / m m 2, respectively. The relation between the maximum cladding hoop stress and overpower continuation time is shown in fig. 10. The maximum cladding hoop stress increases with increasing overpower continuation time and approaches an asymptotic value for times longer than 40 s. This asymptotic value is obtained by a steady-state model analysis. The steady-state model gives conservative resuits and overstimates the cladding hoop stress. When the coolant inlet temperature drops suddenly, cladding hoop stress increases in a few seconds due to the thermal contraction of the cladding. The change in cladding hoop stress is shown in fig. 11. In this case, the
to to LU nI-to n O
M I N A L POWER /" :
\
\ ~
20
~STEADY-STATE ~CALCULATION
a,
o z
0. Po
°
/.
<
.J 0
i 0
dT .
I
20 OVERPOWER
Po
/I ~
I I
DURATION
I
dT
__
- 30~C/s
__-
- 15°C/s
CL
o o I
/
2s
t
121 1:3 < .J 20 = 0
f
I 10
I
I 20
I
l/0 30
4o0
ZcY .
TIME ( s )
Fig. 11. Changes in cladding hoop stress under decreased coolant inlet temperature.
increment in cladding hoop stress attains about 12 k g / m m 2. The steady-state model, however, cannot predict this overshoot and underestimates the cladding hoop stress.
6. Conclusion
A computer code FARST was developed for the evaluation of FBR fuel rod behavior under steadystate/transient conditions. The FARST code has the functions of analyzing the thermal and mechanical behavior of a fuel rod over its whole length and predicting the cladding failure probability during normal operation and transient conditions. For code verification, FARST was applied to the analysis of transient experimental data from CABRI and NSRR. Calculation results showed that the FARST code could predict the fuel rod behavior under the transient conditions well and confirmed its application to FBR fuel rod design. The FARST code was applied to the evaluation of a large FBR plant fuel rod. The calculations showed that the steady-state model could possibly underestimate cladding hoop stress for coolant inlet temperature decrease. Additional code development and calibration using the experimental data will be carried out in the future.
I Acknowledgements
I
40 TIME ( s )
60
Fig. 10. M a x i m u m cladding hoop stress versus overpower duration time.
The authors would like to express thanks to Drs K. Taniguchi, S. Yamada, S. Kobayashi, R. Takeda and K. Miki of the Energy Research Laboratory, Hitachi Ltd for encouragement throughout this study.
M. Nakamura, M. Sakagami / FARST - codefor the evaluation of FBR fuel rod behavior Appendix In this section, a numerical calculation method for fuel and cladding temperature under the transient conditions is shown. Basic equations for the fuel pellet are written as follows: r
0
r ( r , t) = T(rfi, t) +fr,12~-~rKcdr,
(iv) The values of material properties at time t + t/2 are used for a time interval between t and t + At (Crank-Nicholson model). Using these assumptions eqs. (A1) and (A2) can be written as follows:
T ;t++Ia , /[ ~1 _ ~ N"~
(AI)
where
47
)= t~N+ TN(-- TI+I _ r ; + a t + r , ~ N / A]_-t --½(TI+,-- T~v+at--T~v), (A4)
where
Q=f~i(CfPr~t - q " ) 2 ~ ' r d r .
(A2) r~
Radial mesh division is shown in fig. A1. Eq. (A1) is rewritten as
T, rN+,,t)=
T(rN,
t)+ frr;+'2QK dr.
8 N I n rN+l +
"fN =
rN
CNP N
At
qN
,A3)
~,, )]' In order to carry out the numerical integration in eq. (A3), the following assumptions are used. (i) Constant values of the material properties are used for the region between two mesh points. (ii) Between two mesh points, fuel temperature is expressed as follows:
T=TN+(TN+I_T~v)
E j=l
cioj,
"
At
-
T/++Ia t - T j t ~ - Tj' + a ' - Tj' +
At
r--rN rN+ 1 -- rN
× -~(~.+, - ~) x (2~.+, +
~)1 /
(rN~r
(iii) The time derivative of fuel temperature is approximated as follows:
[ j*l
-qj
The calculation is performed from the inside to the outside of the pellet. When the time interval t goes to the infinity, eq. (A4) is reduced to the finite difference
a T / ' =__1 (Tt+zt at ]N A t ~ ' N T~).
CN-~ PN-I, aN-~ 0.8
C~ . P~. q('
O3
r
~
°
~
i
r2 ................ rN-1,rN , rN+l ................ rM T1 , 1"2 ................ TN-1,TN , TN+I ................TM rl
,
Fig. A1. Radial mesh division (fuel pellet).
/11
o., -_:,,r.,°,ytio°,
Solution jI f I • :Calculated / / I 0.2-[ Results L/ / ].= 0
09
0.4
0.6
0J
1.0
r/Fro Fig. A2. Comparison between numerical and analytical solutions.
48
M. Nakamura, M. Sakagami / F A R S T - code for the evaluation of FBR fuel rod behavior
form of the steady-state model. Eq. (A4) has no singularity at r = 0 (if r goes to zero, the value of r2+a × In § + l / r j goes to zero). The validity of this calculation method is shown by comparing the numerical results with the analytical solution. In one such problem, an initially isothermal (0 o C) fuel pellet (inner radius = 0, outer radius = rfo ) is stepped to TO ( ° C ) at the outer radius. Fuel heat conductance coefficient K, heat capacity C and pellet density pf are assumed to be constant. An analytical solution can easily be obtained as an expansion of the Bessel function [10]. A comparison of the calculated results and the analytical solution is shown in fig A2. A satisfactory agreement is obtained, indicating the validity of the numerical method described above. Nomenclature C CDF D Dp
qs S s T t
heat capacity ( J / g o C) cumulative damage fraction cladding diameter (cm) a factor of either 0 or 1 depending on whether plastic flow is prevented or takes place Young's modulus ( k g / m m 2) deviatoric strain (/-component) axial force (kg) G = E / 2 ( 1 + v) ( k g / m m 2) initial gap width (mm) mass flux of coolant flow (kg/s) thermal conductivity ( W / c m 2 g o C) K , = E / ( 1 - 2v) k g / m m 2 number of cracks power density ( W / c m 3) heat flux at cladding surface ( W / c m z) coolant flow are per one rod (cm 2) deviatoric stress (/-component) ( k g / m m 2 s) temperature ( o C) time (s)
t*
t* = Kt/Cprt2o
tf ui t~ Y At AD c~j c ~, /~
time to rupture (s) displacement (i-component) (mm) coolant velocity ( c m / s ) yielding criterion ( k g / m m 2) time increment (s) diameter change (cm) strain ( / j - c o m p o n e n t ) mean strain Poisson's ratio friction coefficient plastic flow parameter (mm2/kg s) coefficient of viscosity (mm2/kg s) stress ( / j - c o m p o n e n t ) ( k g / m m 2) mean stress ( k g / m m 2)
E ei Fz G GO J0 K K.
n q"
7/ Oij
o
p
x/
density ( g / c m 3)
y z
coordinate system)
0 z
coordinate component system)
component
(Cartesian
coordinate
(cylindrical coordinate
Subscript
c ci co f fi fo Na
cladding cladding inner radius cladding outer radius fuel pellet fuel pellet inner radius fuel pellet outer radius sodium
Superscript
C E P
creep strain elastic strain plastic strain
References [1] J.B. Newman et al., The CYGRO-4 fuel rod analysis computer program Nucl. Engrg. Des. 46 (1978) 1. [2] J. Wordsworth, IAMBUS-1 - A digital computer code for the design, in-pile performance prediction and post-irradiation analysis of arbitrary fuel rods, Nucl. Engrg. Des. 31 (1974) 309. [3] P. Royl et al., Transient fuel-cladding deformation analysis of the in-pile H3 'TREAT' test with the 'SAS2A/DEFORM-II' code Nucl. Engrg. Des. 27 (1974) 299. [4] M. Ishida et al., An Analysis of Mechanical Interaction between Pellet and cladding, Proc. Int. Conf. on Fast Breeder Reactor Fuel Performance, Monterey, California (March, 1979). [5] M. Oguma and T. Hosokawa, private communication. [6] EPRI, Light Water Reactor Fuel ROd Modelling Code Evaluation, NP-369 (1977). [7] P.J. Levine et al., Irradiation Performance of WSA-3, -4, and -8 Mixed Oxide Fuel Pins in Grid-Spaced Assemblies, Proc. Int. Conf. on Fast Breader Reactor Fuel Performance, Monterey, California (March, 1979). [8] A.J. Lovel et al., Observations of In-Reactor Endurance and Rupture Life for Fueled and Unfueled FTR Cladding, Trans. ANS 32 (1979) 217. [9] T. Fujishiro et al., Effects of coolant flow on light water reactor fuel behaviors during reactivity initiated accident, J. Nucl. Sci. Tech. 18 (1981) 196. [10] H.S. Carlow et al., Conduction of Heat in Solids, 2nd Ed. (Clarendon Press, Oxford).