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WTRLGD—A computer program for the transient analysis of waterlogged fuel rods under the RIA condition
Nuclear Engineering and Design 66 ( 198 I) 223-232 North-Holland Publishing Company
223
WTRLGD--A COMPUTER PROGRAM FOR THE TRANSIENT ANALYSIS OF WATERLOGGED FUEL RODS UNDER THE RIA CONDITION Masa-Aki OCHIAI
Tokai Branch, Ship Research Institute, Ministry of Transport, Tokai-Mura, Naka-Gun, Ibaraki-Ken, Japan Received 30 April 1981
A computer code WTRLGD has been developed to describe the transient internal pressure of a waterlogged fuel rod during power burst and also to predict the possibility of the rod failure in the mode of cladding rupture. The code predicts transient thermal behavior of the fuel rod on the basis of an assumption of axisymmetry, and thermal-hydraulic transients of the internal water on the basis of a homogeneous volume-junction model modified so as to involve the cladding deformation. Calculated transients of the rod pressure are in fairly good agreement with those measured in the NSRR experiments, simulating the fuel rod behavior under an RIA condition. The comparison between calculation and experiment verifies that the code is an effective tool for the prediction of the failure of a waterlogged fuel rod.
1. Introduction Light water reactors (LWRs) contain a very small number of defective fuel rods. Some of them become waterlogged fuel rods due to water sinking through their penetrating defects after reactor shut down. Instantaneous increase in reactor power, such as power burst in a reactivity initiated accident (RIA), results in quick rise in the internal pressure of waterlogged fuel rods causing their cladding tubes to rupture. The behavior of fuel rods under the RIA condition has been studied experimentally in the Nuclear Safety Research Reactor (NSRR) at JAERI [l], Japan, the SPERT-CDC [2] and the TREAT [3], USA. The NSRR experiments on the failure behavior of waterlogged fuel rods were studied more minutely and extensively, revealing that [4]: (a) The failure of waterlogged fuel rods is by cladding rupture, while that of intact fuel rods is by melting of the cladding tubes. (b) Waterlogged fuel rods require much less energy deposition for failure than intact rods do. (c) Mechanical energy is released by the failure of waterlogged fuel rods. From the above finding the failure behavior of waterlogged fuel rods is one of important subjects to be studied for LWR safety under the R I A condition, since they rupture easily with a considerable mechanical energy release. It is, therefore, very profitable to devise a computer 0029-5493/81/0000-0000/$02.75
code for prediction of the transient behavior of the internal pressure and the possibility of the failure of waterlogged fuel rods. For this purpose the author developed a computer code (WTRLGD) for such prediction and for analizing the test results from the NSRR or the SPERT-CDC. The present paper describes both the outline and the mathematical models of the code W T R L G D as well as a comparison of the calculation with the NSRR test results.
2. Outfine of WTRLGD
The computer code W T R L G D is developed for the analysis of a waterlogged fuel rod under the RIA condition. The features of the code are as follows. (a) o n e is to analyze the transient internal pressure of a waterlogged fuel rod and predict the possibility of its failure. (b) The other is its application to any kind of a waterlogged fuel rod irrespective of the m o u n t of internal water and of the pin holes on its cladding tube. The internal pressure of a waterlogged fuel rod increases with enthalpy of the internal water when reactor power increases rapidly. For this kind of transient, the volume-junction model in a RELAP code [5] has been well known to be valid. The model, however, must be modified for this problem in two respects. One is that the internal pressure should be calculated involving the deformation of a cladding tube, since a little expansion
© 1981 N o r t h - H o l l a n d
M. Ochiai / WTRLGD-- Waterloggedfuel rods
224
can easily suppress the rod pressure because of less compressibility of water. The other is that a backward difference approximation should be a d o p t e d for the calculation of internal ~ater flow regardless of introduction of an iteration procedure for evaluation of nonlinear terms, since a forward approximation is apt to yield the divergency of flow brought by the rapid pressure transient of less compressible fluid. Consequently additional features of the code are, (c) The internal pressure is determined thermodynamically, involving the effects of the cladding deforma. tion. (d) Hydro-dynamic equations are calculated by a backward difference approximation. The code performs evaluation of the fuel rod divided into 5 parts, i.e., the fuel pellets, the cladding tube, the gap region, and the upper and the lower plenum regions as shown in fig. I. The heat generated in the fuel pellets is transferred through gap region and a cladding tube into cooling water. Internal water can fill the three void regions, the gap region and the both plenum regions depicted in the figure. The transient pressure model in the code is based on the following assumptions: (a) one dimensional heat transfer in the radial direction, (b) one dimensional internal water flow in the axial direction and (c) quasisteady transients. The flow chart of the code is shown in fig. 2. A subroutine (INPUT) reads input data on the dimension of a fuel rod, the amount of internal water, the time and space distribution of heat generation, etc. A subroutine (HEAT) treats heat fluxes at the surface of fuel pellets and at the inner and outer surfaces of a cladding tube. Then, a subroutine (FLTMP) determines the temperature distribution in a fuel rod by the one dimensional heat conduction equation. Next, a subroutine (WTRSTT) determines the state of internal water involving the deformation of a cladding tube. This process is one of characteristic of the code. The slight difference in evaluating the specific volume of internal water resuits in a significant error in the rod pressure due to less compressible property of water. A severe convergence criterion is, therefore, required for calculation of the
state of water and the cladding deformation. When the pressure distribution within a rod is determined, the leak flow rates from pin holes are calculated in a
Fig. 2. Flow chart of WTRLGD.
subroutine (FLWHOL). In addition the transient interhal water flow is also calculated in a subroutine (WTRFLW) based on the pressure distribution. Finally, the mass and enthalpy of water axe redistributed according to the flow field. Then the calculation goes to the next time step.
3.
Mathematical
models
3.1. Fuel rod temperature Neglecting the heat conduction along a rod axis, we obtained the temperature distribution from the following one-dimensional equation:
aT pC at
1 a [krar~+q,,,.] r ar ~ ~r
(l)
The initial condition is
T = To(r ),
g0s
1,~
,
.
at t = 0.
(2)
~ , p ~
watlerI gap rregionI 1 u00er ~lenum regi0n / fuel pellets Fig. 1. Schematic of a fuel rod.
j
pressuresensor
Eq. (1) is rewritten in terms of the thermal conductivity integral F as
lowerplenumregiOn
/
cJo~
T ~
~be
1 aF_ irma r-~; a at
(3)
225
M. Ochiai / WTRLGD--Waterlogged fuel rods where
(4)
F= frk dT 'IT0
and a = k/pC.
(5)
Expression of eq. (3) at radial nodes results in the form of a forward difference equation F,. - F/° _ F/° i + F/°l - 2 F/°
a°i(At)
(Ar) 2 O I 7 0 _ _
+
F~2;,(Ar" i)-, + q,,,0,.
(6)
where subscript i denotes the node number and superscript 0 the prior time step. The thermal conductivity integral F~ can be obtained from eq. (6) at all the nodes except for the boundary nodes. The temperature T~ is determined by the numerical integration of the following equation: tF, d F
Ti = T°(r') + Jo k-'~) "
(7)
The temperatures at the boundary nodes are determined from the boundary conditions of
aT/ar = 0
at r : O,
-kOT q't' ~r =
at r = r t ,
kar= Or qci ,,
_ k aT
water is less than that in the NSRR experiments. The heat fluxes (q~' and q~) in the gap region are calculated based on such correlations for the water flow in a circular tube as those by Dittus and Boelter for the forced convective heat transfer, by Jens and Lottes for nucleate boiling and by Bromley for film boiling.
"~r=q~"
at r = rci,
(8)
at r = r=o,
where q" is the surface heat flux, and the subscripts f, ci and co denote the fuel pellet surface, the inner and outer surfaces of' a cladding tube, respectively. The heat flux (qco) at the outer surface of a cladding tube is calculated based on such correlations as those by O~trach for the free convective heat transfer with a laminar boundary layer, by lakob for the free convective heat transfer with a turbulent boundary layer and by Jens and Lottes for the nucleate boiling heat transfer. The transition and film boiling correlations are not required for the analysis of the NSRR experiments owing to occurrence of the waterlogged rod failure in the NSRR experiments before its cladding temperature reaches 150°C [4]. Some correlations, however, are to be added to the code for the analysis of the failure behavior of a waterlogged rod if subcooling temperature of cooling
3. 2. Pressure distribution of internal water The interior void space of a rod is divided into several volumes in the subroutine WTRSTT as shown in fig. 3. Both the upper and lower plenum regions are treated as one volume respectively, and the gap region is divided into several volumes. In each volume the state of the water is assumed to be homogeneous. Knowing the specific enthalpy i and the mass W of the water in each volume, fixed for a given time step, we can determine both the temperature T and pressure P in the volume, taking into account the deformation of a cladding tube, which was performed in the subroutine. Once the increment of the effective strain de e of a cladding tube is determined for a given load step, the stresses and strains in a cladding tube are evaluated in the subroutine D E F O R M described in the next section, which also determines the capacity V and the pressure in each volume. The specific volume v~ of water in each volume is determined from the fixed specific enthalpy and the pressure determined in the subroutine D E F O R M using a steam table. Consequently the temporary mass of the water in each volume Wx defined by W~ ----V/v x
(9)
is compared with the fixed mass of the water W. When Wx is greater than W, de= must be decreased, while when W~ is less than W, de e must be increased. This
m .
J=Jm=-1 -
--upper plenumregion
fuel pellets ~gop region
R t.
clodding tube 3 2 O = I
-:--lower
plenum region
Fig. 3. Model of the interior void space of. a fuel rod.
226
M. Ochiai / WTRLGD--Waterlogged fuel rods
procedure is iterated until the convergence criterion is fulfilled:
W-
Wx
- - - - i f - - < 10
(10)
then the pressure distribution of the internal water is determined.
where doi are the principal stress increments, E the modulus of elasticity, v the Poison's ratio and a the thermal expansion coefficient. Considering a hollow thin wall tube with the closed ends, we can write the equilibrium equation in the forms for respective components as ol =0~ = 0 ,
o2=oo=Prm/h=2o,
3.3. Deformation of a cladding tube
03 ----o: = Prm/2h = o,
The deformation of a cladding tube is calculated in the subroutine D E F O R M based on the following assumptions: (a) a thin wall tube, (b) axisymmetric loading and deformation of a cladding tube, (c) incremental theory of plasticity, (d) Prandfl-Reuss flow rule, (e) no creep deformation, (f) insignificant bending strains and stresses in a cladding tube. The constitutive equation is obtained from the material properties package subcode MATPRO [6], which describes the equation in the form of true strains and stresses. Then, strains and stresses are treated in the same form in WTRLGD. The relationship between the principal plastic strain increments d ep and the effective plastic strain increment deep is provided by the Prandtl-Reuss flow rule:
where rm is the radius of a cladding tube and h the wall thickness. Substituting eq. (16) into eqs. (12) and (13), we obtain S i and % as
3 d~ deP~ = ~ -~-e Si,
i=1,2,3.
(ll /
Here S~ are the deviatoric stress components in principal coordinates defined by eq. (12) and oe is the effective stress defined by eq. (13):
Si=oi-½~o
oo
1
,,
i=1,2,3
(12)
((0, - 0 2 ) 2 + (02 - o3)2 + (03 - 0 , ) 2} ' :
= ~2-2( ]~ Si2}m/2,
s, = - o ,
=o,
(16)
=0,
oo
(17)
Once de e is given, the effective strain ee is determined as
ee(t) = ee(t -- At) + dee,
(18)
and the effective stress % is determined from the stressstrain curve calculated from the MATPRO based on the equation of oe = r n i n
{
eee,K(ee)
At ]
j'
(19)
where K is the strength coefficient, n the strain hardening exponent and m the strain rate sensitivity. The effective plastic strain eep and its increment deep are determined from
eep = eo - ( o , / e ) ,
(20)
deep = eep(t) - eep(t - At).
(21)
Then the principal plastic strain increments de r are calculated from eq. (11), while the principal total strain increments de~ also from eq. (15). Consequently strains and stresses are determined as follows:
e,(t)=e,(t-At)+de,,
i = 1,2,3,
(22)
o I = 0,
(23)
(13) 02 = 20 = 2 0 e / ¢ 3 ,
(24)
where o~ are the principal stresses. Substituting eq. (13) into eq. (11), we obtain the expression for deep as
o3 = o = o,/v~-.
(25)
deep = {~ 2 (de/P) 2 ) 1/2
Finally, the dimensions and the pressure are determined from the expressions given by
(14)
The total strain increments de~ are described by a generalized form of Hooke's law given by
de, --der + ½ (doi-,(doj +dot,)) +a dT, i= 1,2,3,
(15)
h = h o exp(eI),
(26)
rm = rmo exp(e2),
(27)
l= 1o exp(e3),
(28)
h/2,
(29)
rei =
rna - -
M. Ochiai / WTRLGD-- Waterloggedfuel rods V = ~r(d -- r2)l,
(30)
2o__hh_ 2%___hh
p_
rm
tj-t
_ ti
__
227
tpt
(31)
vf3rm '
where l is the length of each volume and the subscript 0 denotes the initial value.
M : MOss G :Flow rote A : Flow oreo
3.4. Transient internal water flow
i : specific entholpy
Transient flow of the water between the two volumes can be calculated based on the following assumptions: (a) one dimensional flow in the axial direction, (b) negligible body force (except gravity), (c) negligible works done by pressure and viscous force, (d) negligible kinetic and potential energy. For this calculation we use the mass, energy and momentum conservation equations. These equations are
Fig. 4. Internal water flow model.
into a cladding, and
ja =
~ 0),
jb= (~i+l (Oj ~0).
(a) Mass conservation
30 4- 3pu
n
(32)
-G-_-~T=-,
where p is the fluid density and u the fluid velocity in the axial direction.
Integration of the momentum equation over the shaded portion depicted in fig. 5 results in
(
i lj
lj÷, ~dGj ( ~ - ~ + , ) + a
(b) Energy conservation
api at
aq",
~-~
2
api_
-1- u-~-z - O.
GjIGjI
-g(pflj + .j+,lj+,)/2.
(33)
(37)
where (c) Momentum conservation
~(1#2)
apu t - a p u 2 + a P + 1 a r r + at az -~z r Or p g = 0,
(34)
where 1" is the viscous stress and g the gravitational acceleration constant. Integrating the mass equation over a fixed volume, we obtain dWj = @ , - a j , dt
fJ- pjajAy ' Rj is the hydraulic radius (rci) - r f j ) and ~ the friction factor. The second term of the right hand side of eq. (37) is the momentum flux term, which is approximately given
(351
where Wj is the mass in the j th volume and Gj the mass flow rate from thejth volume to the ( j + l)th as shown in fig. 4. Integrating the energy equation over the j th volume, we obtain dlj
0-7=
O.~.j -
(Lo. + , j o % ,
-ij~a~,
(36)
where I is the total enthalpy. Q.i. the rate of heat transfered from fuel pellets and Qo.t the rate of heat
P
Pressure
A
Flow oreo
G
Flow rote
Fig. 5. Controle volume for momentum conservation.
M. Ochiai/ WTRLGD--Waterloggedfuel rods
228
leak flow rates from pin holes, which are also limited by two-phase critical flow rate obtained by Chisholm et al. [8]. Finally the mass of the water in each volume is redistributed by eq. (35) and the enthaipy by eq. (36).
by
_ g-,/=lg-,/=l
pjAj
pa a ]j.j+,
•
2
x g+,/2lg+,/21 ,
(38)
oj+,A~+,
where
g_+,/= = ( g + g+,)/2. Friction factor fj is calculated based on such correlations as Hagen-PoiseuiUe law for laminar flow, Blasius correlation for turbulent flow and Martinelli-Nelson's parameters for two-phase flow. Thus transient flow rate Gj can be calculated from the backward difference form of eq. (37), which is given by
1
OA2 /j,j+l
-t)glgl-g(oj6 +pj+llj+,)/2}(At),
(39)
where
L/A=5
A--;+ Ai- ,
and
b
+
The backward difference aproximation is chosen to eliminate divergency in the flow rate, which inevitably requires an iteration process. Critical flow rate based on Moodie's modcl [7] is adopted as the upper limit for the flow rate. Consequently Gj evaluated from eq. (39) is replaced by the critical flow rate Gcr, if Gj is greater than Gcr. Assuming pin holes to be orifices, we calculate the
Top End Fitting Magnetic Iron Core Sr~ Spring Adaptor / D!sk Fuel Pellets
Extensive experiments have been performed on the failure behavior of a waterlogged fuel rod in the NSRR. A waterlogged test fuel rod was prepared by evacuating a test fuel rod (fig. 6) and allowing water to enter the evacuated interior. The fuel rod was weighed before and after waterlogging to determine the amount of water entering its interior. A fast response pressure transducer was attached to the bottom end of the fuel rod contained in the capsule inserted into the experiment space in the center of the NSRR test facility as shown in fig. 7. In order to examine the validity of WTRLGD, the calculated results were compared with the experimental results of two NSRR waterlogged fuel rod tests, Test 401-4C [9] and Test 432-1B [4]. In Test 401-4C a fully waterlogged fuel rod (99 vol.%) was subjected to a 3.82 ms period power burst which would have resulted in a total energy deposition of 150 c a l / g of UO 2 unless it had failed. However, it failed in the form of a 70 mm split in its cladding tube before deposition of the total energy. The failure occurred about 8 msec after a peak power at an energy deposition of 116.5 cal/g of UO> The time of failure was distinguished by a sharp pressure pulse in the capsule and a sudden decrease in the rod internal pressure. The transient rod internal pressure for Test 401-4C calculated with WTRLGD is compared with the measured pressure in fig. 8, where a peak power appears about 30 ms after the initiation of the power burst adopted as the origin of time. From the figure it is found that the measured rod pressure lags about 15 ms behind the power burst, reaches a peak pressure of 93 MPa about 6 ms after the peak power
Bottom Claddin~ Disk Spacer End Fitting
r--I 265 mm (overoll length) Fig. 6. Standard test fuel rod in the NSRR.
4. Comparison with experiments
o (octive length)
II
229
M. Ochiai / WTRLGD--Waterlogged fuel rods Capsule hold-down device
Contt
Wot
Verlicol Offset I
pit Neutro
utron rodiography ~rn
device
pressure from 91 MPa to the ambient pressure about 8 ms after the peak power. Fairly good agreement is seen in the figure between the measured and the calculated pressure in the lower plenum region where the pressure transducer was attached. It should be noted that the calculation predicts well the measured pressure, in particular both the pressure behavior before reaching the peak pressure and the time of rod failure defined by the occurrence of instability destruction of a cladding tube. However, a little discrepancy is observed between the measured and the calculated pressure in the slow attenuation phase (34 ~ 38 ms). One of the causes is ascribable to the thin wall tube assumption adopted in the calculation since the strains in the cladding tube become large in this phase. Nevertheless the good prediction for the time of rod failure indicates that such an inadequate assumption employed in the calculation yields no significant error in the analysis of the failure behavior of a waterlogged fuel rod. The NSRR experiments reveal that the internal pressure of a fully waterlogged fuel rod is almost uniformly distributed along the axis of a cladding tube [10]. The calculation also shows the almost uniform pressure distribution as seen in fig. 9, though the pressure in the upper plenum region is slightly lower than those in the others because of a small amount of remaining gas (1
vol.%). The curves drawn in fig. 10 designating the transient states of internal water are considerably different from
Fig. 7. General arrangement of NSRR facilities.
and then decreases slowly from 93 MPa to 91 MPa within 2 ms. Also it is found that the rod failure is illustrated as an abrupt decrease in the measured rod
i j=5[]~ upper plenum
MPalooL-"~-i~]< lower plenum .... 100
reactor power rod pressure (experiment) rodr°d'Dressure.~ilu(calculation) re
0
80hO msec ~-80
~
I
'
401
.zl msee
b
|
I
I
',, O(
0
~
i
20 40 Time (msec)
I
"" . . . .
I
60
Fig. 8. Comparison with experimental results in the case of a fully waterlogged fuel rod.
I
1
I
I
I
2 3 4 5 volume number (j)
Fig. 9. Axial distribution of the internal pressure of a fully waterlogged fuel rod.
230
M. Ochiai / WTRLGD-- Waterloggedfuel rods
i00
MPa
__ 100i n°
32 msec
/ 60
......
m
~
J
,i....
-- 80
=3 ~gap j=2 ~ = 1 ~ 1 ..... lenum
~ 6O
j=5~upper
#
®
o
reoctor power rod pressure (experiment) rod pressure (calculationJ
40
,_ 4 _ O, 0
-
i
-
i 20
i
I 40
J
i 60
i 80
Time (msec) 2" ....-->~ j=3
saturation
20
62~i ~4~ 0
ri
200
I 400
I
I 600
800 (KJ/Kg)
specific enthalpy Fig. 10. Tr~sient states of internal water of a fully waterlo~ed fuel rod.
the saturation curve. The curve denoting the state of water in the lower plenum region is almost parallel to the vertical axis as shown in the figure, which indicates that the pressure increase in the plenum region results from almost adiabatic compression of water. The figure also designates that the pressure in the gap region increases in proportion with enthalpy of water. For Test 432-1B, a partially waterlogged fuel rod (70 vol.%) was subjected to the same power burst as in Test 401-4C. The measured rod pressure lags aboiat 25 ms behind the power burst, reaches a peak pressure of 72 MPa about 9 ms after the peak power and then slowly decreases as shown in fig. 11. The rod failure was not observed during Test 432-1B. Good agreement is also obtained between the measured and the calculated fuel rod pressures as shown in the figure. Again it should be emphasized that the calculation can predict fairly well the measured pressure, in particular the pressure behavior before reaching its peak and both the time and intensity of its peak. Detailed examination, however, indicates that the calculation shows a rapid decrease compared with the measurement after passing the peak pressure. It is found from fig. 12 that the calculated pressure in the upper plenum region scarcely increases during Test 432-1B due to compressibility of remaining gas (30
Fig. 1I. Comparison with experimental results in the case of a partially waterlogged fuel rod.
vol.~) in a partially waterlogged fuel rod, as revealed by the NSRR partially waterlogged fuel rod experiments [10]. The axial distribution of the calculated rod pressure decreases monotonously going from the bottom to the top end after the time of the peak pressure (38 ms) as shown in the figure. From this result it is obviously inferred that the flow rate of water from the gap region into the upper plenum region should strongly control the decreasing rate of the rod pressure. The flow rate must have become smaller because of the two-phase choking, if two-phase state had been considered in the calculation. The homogeneous model used in the calculation, however, is apt to suppress the local two-phase state arising locally at the top of the gap region. Conse-
MPa 60
O
~lncreasin~
peak
40
20
.
u er
~ J
lower plenum~
I I i 2 ~ 4 volume number (j) Fig. 12. Axial distribution of the internal pressure of a partially waterlogged fuel rod.
M. Ochmi / WTRLGD--Waterloggedfuel rods quently the overestimation of the flow rate into the upper plenum region can be considered as one of the causes for the difference between the experiment and the calculation in the decreasing rate of the rod pressure. Nevertheless the model is considered to be effective enough for analyzing the transient rod pressure during a power burst, since the prediction of the rod failure does not require any precise description of the rod pressure after the.. time of its peak. The curves plotted in fig. 13 show the transient states of water in a partially waterlogged rod. The curves are closer to the saturation curve than those plotted in fig. l0 for a fully waterlogged rod, and in particular the closest curve is that of the state of the water in the top volume ( j = 4 in fig. 13) in the gap region. Therefore, had we chosen a very small volume for the top volume so as to evaluate adequately the local two-phase state, the behavior of the rod pressure in the decreasing phase would have been well predicted by the calculation. Adoptation of such a model in the calculation, however, requires too much computor running time in order to avoid divergency in flow rate.
231
5. Conclusions The W T R L G D has been developed to analyze the internal pressure of a waterlogged fuel rod under the condition of an RIA and also to predict the possibility of its failure in the mode of cladding rupture. The calculated results are compared with the results of two tests made in the NSRR for a fully and a partially waterlogged fuel rods. As a result it is found that the calculation predicts adequately the internal pressure behavior of both the fully and the partially waterlogged rods, in particular the pressure behavior in the increasing phase and both time and intensity of its peak pressure. Therefore, the code can predict well the occurrence of the rod failure although there arises a slight discrepancy between the measured and the calculated rod pressure in the decreasing phase.
Acknowledgement We would like to express our deepest appreciation to Mr. T. Yoshimura, Mr. N. Ohnishi and Dr. K. Takeuchi for their valuable advices and consultations, and to staff of the NSRR for valuable experimental data.
j=5~pper
M:: I
plen~
Nomenclature
~=~J(lower~len~
38 msec
60
j=l
j 34 j=4
40
32
saturation
20 3O 28 0 0
200
400
600
800 (KJ/rg)
specific enthalpy
Fig. 13. Transient states of internal water of a partially waterlogged fuel rod.
A a C E F f G Ccr g h ! i K k ! m n P {~i, Qout
q'" q"
flow area thermal diffusivity specific heat modulus of elasticity thermal conductivity integral defined by eq. (4) friction factor mass flow rate critical mass flow rate gravitational acceleration wall thickness of a cladding tube total enthalpy specific enthalpy strength coefficient thermal conductivity length strain rate sensitivity strain hardening exponent pressure rate of heat transfered into gap region rate of heat transfered out of gap region density of heat generation heat flux
M. Ochiai / WTRLGD-- Waterloggedfuel rods
232
r,O,z S T t U
V W
cylindrical coordinates deviatoric stress temperature time velocity capacity specific volume mass
Greek symbols thermal expansion coefficient strain Poison's ratio density stress viscous stress
Subscripts ci CO
f i ]
0
inner surface of a cladding tube outer surface of a cladding tube surface of a fuel pellet radial node axial node initial value
Superscripts p 0
plastic prior time step
References
[1] M. Ishikawa and S. Shiozawa, J. Nucl. Mater. 95 (1980) 1-30. [2] R.W. Miller et al., IDO-16883 (1964). [3] L. Harrison et al., ANL-7325 (1966) 158-163. • [4] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 7304 (1977). [5] W.H. Rettig et al., IN-1321 (1970). [6] P.E. MacDonald and L.B. Thomson, (ed.), TREENUREG- 1005 (1976). [7] F.J. Moody, Trans. ASME, Ser. C 87 (1966). [8] K. Akagawa, Gas-Liquid Two-Phase Flow (Corona, Tokyo, 1974). [9] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 7051 (1977). [10] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 9319 (1981).
223
WTRLGD--A COMPUTER PROGRAM FOR THE TRANSIENT ANALYSIS OF WATERLOGGED FUEL RODS UNDER THE RIA CONDITION Masa-Aki OCHIAI
Tokai Branch, Ship Research Institute, Ministry of Transport, Tokai-Mura, Naka-Gun, Ibaraki-Ken, Japan Received 30 April 1981
A computer code WTRLGD has been developed to describe the transient internal pressure of a waterlogged fuel rod during power burst and also to predict the possibility of the rod failure in the mode of cladding rupture. The code predicts transient thermal behavior of the fuel rod on the basis of an assumption of axisymmetry, and thermal-hydraulic transients of the internal water on the basis of a homogeneous volume-junction model modified so as to involve the cladding deformation. Calculated transients of the rod pressure are in fairly good agreement with those measured in the NSRR experiments, simulating the fuel rod behavior under an RIA condition. The comparison between calculation and experiment verifies that the code is an effective tool for the prediction of the failure of a waterlogged fuel rod.
1. Introduction Light water reactors (LWRs) contain a very small number of defective fuel rods. Some of them become waterlogged fuel rods due to water sinking through their penetrating defects after reactor shut down. Instantaneous increase in reactor power, such as power burst in a reactivity initiated accident (RIA), results in quick rise in the internal pressure of waterlogged fuel rods causing their cladding tubes to rupture. The behavior of fuel rods under the RIA condition has been studied experimentally in the Nuclear Safety Research Reactor (NSRR) at JAERI [l], Japan, the SPERT-CDC [2] and the TREAT [3], USA. The NSRR experiments on the failure behavior of waterlogged fuel rods were studied more minutely and extensively, revealing that [4]: (a) The failure of waterlogged fuel rods is by cladding rupture, while that of intact fuel rods is by melting of the cladding tubes. (b) Waterlogged fuel rods require much less energy deposition for failure than intact rods do. (c) Mechanical energy is released by the failure of waterlogged fuel rods. From the above finding the failure behavior of waterlogged fuel rods is one of important subjects to be studied for LWR safety under the R I A condition, since they rupture easily with a considerable mechanical energy release. It is, therefore, very profitable to devise a computer 0029-5493/81/0000-0000/$02.75
code for prediction of the transient behavior of the internal pressure and the possibility of the failure of waterlogged fuel rods. For this purpose the author developed a computer code (WTRLGD) for such prediction and for analizing the test results from the NSRR or the SPERT-CDC. The present paper describes both the outline and the mathematical models of the code W T R L G D as well as a comparison of the calculation with the NSRR test results.
2. Outfine of WTRLGD
The computer code W T R L G D is developed for the analysis of a waterlogged fuel rod under the RIA condition. The features of the code are as follows. (a) o n e is to analyze the transient internal pressure of a waterlogged fuel rod and predict the possibility of its failure. (b) The other is its application to any kind of a waterlogged fuel rod irrespective of the m o u n t of internal water and of the pin holes on its cladding tube. The internal pressure of a waterlogged fuel rod increases with enthalpy of the internal water when reactor power increases rapidly. For this kind of transient, the volume-junction model in a RELAP code [5] has been well known to be valid. The model, however, must be modified for this problem in two respects. One is that the internal pressure should be calculated involving the deformation of a cladding tube, since a little expansion
© 1981 N o r t h - H o l l a n d
M. Ochiai / WTRLGD-- Waterloggedfuel rods
224
can easily suppress the rod pressure because of less compressibility of water. The other is that a backward difference approximation should be a d o p t e d for the calculation of internal ~ater flow regardless of introduction of an iteration procedure for evaluation of nonlinear terms, since a forward approximation is apt to yield the divergency of flow brought by the rapid pressure transient of less compressible fluid. Consequently additional features of the code are, (c) The internal pressure is determined thermodynamically, involving the effects of the cladding deforma. tion. (d) Hydro-dynamic equations are calculated by a backward difference approximation. The code performs evaluation of the fuel rod divided into 5 parts, i.e., the fuel pellets, the cladding tube, the gap region, and the upper and the lower plenum regions as shown in fig. I. The heat generated in the fuel pellets is transferred through gap region and a cladding tube into cooling water. Internal water can fill the three void regions, the gap region and the both plenum regions depicted in the figure. The transient pressure model in the code is based on the following assumptions: (a) one dimensional heat transfer in the radial direction, (b) one dimensional internal water flow in the axial direction and (c) quasisteady transients. The flow chart of the code is shown in fig. 2. A subroutine (INPUT) reads input data on the dimension of a fuel rod, the amount of internal water, the time and space distribution of heat generation, etc. A subroutine (HEAT) treats heat fluxes at the surface of fuel pellets and at the inner and outer surfaces of a cladding tube. Then, a subroutine (FLTMP) determines the temperature distribution in a fuel rod by the one dimensional heat conduction equation. Next, a subroutine (WTRSTT) determines the state of internal water involving the deformation of a cladding tube. This process is one of characteristic of the code. The slight difference in evaluating the specific volume of internal water resuits in a significant error in the rod pressure due to less compressible property of water. A severe convergence criterion is, therefore, required for calculation of the
state of water and the cladding deformation. When the pressure distribution within a rod is determined, the leak flow rates from pin holes are calculated in a
Fig. 2. Flow chart of WTRLGD.
subroutine (FLWHOL). In addition the transient interhal water flow is also calculated in a subroutine (WTRFLW) based on the pressure distribution. Finally, the mass and enthalpy of water axe redistributed according to the flow field. Then the calculation goes to the next time step.
3.
Mathematical
models
3.1. Fuel rod temperature Neglecting the heat conduction along a rod axis, we obtained the temperature distribution from the following one-dimensional equation:
aT pC at
1 a [krar~+q,,,.] r ar ~ ~r
(l)
The initial condition is
T = To(r ),
g0s
1,~
,
.
at t = 0.
(2)
~ , p ~
watlerI gap rregionI 1 u00er ~lenum regi0n / fuel pellets Fig. 1. Schematic of a fuel rod.
j
pressuresensor
Eq. (1) is rewritten in terms of the thermal conductivity integral F as
lowerplenumregiOn
/
cJo~
T ~
~be
1 aF_ irma r-~; a at
(3)
225
M. Ochiai / WTRLGD--Waterlogged fuel rods where
(4)
F= frk dT 'IT0
and a = k/pC.
(5)
Expression of eq. (3) at radial nodes results in the form of a forward difference equation F,. - F/° _ F/° i + F/°l - 2 F/°
a°i(At)
(Ar) 2 O I 7 0 _ _
+
F~2;,(Ar" i)-, + q,,,0,.
(6)
where subscript i denotes the node number and superscript 0 the prior time step. The thermal conductivity integral F~ can be obtained from eq. (6) at all the nodes except for the boundary nodes. The temperature T~ is determined by the numerical integration of the following equation: tF, d F
Ti = T°(r') + Jo k-'~) "
(7)
The temperatures at the boundary nodes are determined from the boundary conditions of
aT/ar = 0
at r : O,
-kOT q't' ~r =
at r = r t ,
kar= Or qci ,,
_ k aT
water is less than that in the NSRR experiments. The heat fluxes (q~' and q~) in the gap region are calculated based on such correlations for the water flow in a circular tube as those by Dittus and Boelter for the forced convective heat transfer, by Jens and Lottes for nucleate boiling and by Bromley for film boiling.
"~r=q~"
at r = rci,
(8)
at r = r=o,
where q" is the surface heat flux, and the subscripts f, ci and co denote the fuel pellet surface, the inner and outer surfaces of' a cladding tube, respectively. The heat flux (qco) at the outer surface of a cladding tube is calculated based on such correlations as those by O~trach for the free convective heat transfer with a laminar boundary layer, by lakob for the free convective heat transfer with a turbulent boundary layer and by Jens and Lottes for the nucleate boiling heat transfer. The transition and film boiling correlations are not required for the analysis of the NSRR experiments owing to occurrence of the waterlogged rod failure in the NSRR experiments before its cladding temperature reaches 150°C [4]. Some correlations, however, are to be added to the code for the analysis of the failure behavior of a waterlogged rod if subcooling temperature of cooling
3. 2. Pressure distribution of internal water The interior void space of a rod is divided into several volumes in the subroutine WTRSTT as shown in fig. 3. Both the upper and lower plenum regions are treated as one volume respectively, and the gap region is divided into several volumes. In each volume the state of the water is assumed to be homogeneous. Knowing the specific enthalpy i and the mass W of the water in each volume, fixed for a given time step, we can determine both the temperature T and pressure P in the volume, taking into account the deformation of a cladding tube, which was performed in the subroutine. Once the increment of the effective strain de e of a cladding tube is determined for a given load step, the stresses and strains in a cladding tube are evaluated in the subroutine D E F O R M described in the next section, which also determines the capacity V and the pressure in each volume. The specific volume v~ of water in each volume is determined from the fixed specific enthalpy and the pressure determined in the subroutine D E F O R M using a steam table. Consequently the temporary mass of the water in each volume Wx defined by W~ ----V/v x
(9)
is compared with the fixed mass of the water W. When Wx is greater than W, de= must be decreased, while when W~ is less than W, de e must be increased. This
m .
J=Jm=-1 -
--upper plenumregion
fuel pellets ~gop region
R t.
clodding tube 3 2 O = I
-:--lower
plenum region
Fig. 3. Model of the interior void space of. a fuel rod.
226
M. Ochiai / WTRLGD--Waterlogged fuel rods
procedure is iterated until the convergence criterion is fulfilled:
W-
Wx
- - - - i f - - < 10
(10)
then the pressure distribution of the internal water is determined.
where doi are the principal stress increments, E the modulus of elasticity, v the Poison's ratio and a the thermal expansion coefficient. Considering a hollow thin wall tube with the closed ends, we can write the equilibrium equation in the forms for respective components as ol =0~ = 0 ,
o2=oo=Prm/h=2o,
3.3. Deformation of a cladding tube
03 ----o: = Prm/2h = o,
The deformation of a cladding tube is calculated in the subroutine D E F O R M based on the following assumptions: (a) a thin wall tube, (b) axisymmetric loading and deformation of a cladding tube, (c) incremental theory of plasticity, (d) Prandfl-Reuss flow rule, (e) no creep deformation, (f) insignificant bending strains and stresses in a cladding tube. The constitutive equation is obtained from the material properties package subcode MATPRO [6], which describes the equation in the form of true strains and stresses. Then, strains and stresses are treated in the same form in WTRLGD. The relationship between the principal plastic strain increments d ep and the effective plastic strain increment deep is provided by the Prandtl-Reuss flow rule:
where rm is the radius of a cladding tube and h the wall thickness. Substituting eq. (16) into eqs. (12) and (13), we obtain S i and % as
3 d~ deP~ = ~ -~-e Si,
i=1,2,3.
(ll /
Here S~ are the deviatoric stress components in principal coordinates defined by eq. (12) and oe is the effective stress defined by eq. (13):
Si=oi-½~o
oo
1
,,
i=1,2,3
(12)
((0, - 0 2 ) 2 + (02 - o3)2 + (03 - 0 , ) 2} ' :
= ~2-2( ]~ Si2}m/2,
s, = - o ,
=o,
(16)
=0,
oo
(17)
Once de e is given, the effective strain ee is determined as
ee(t) = ee(t -- At) + dee,
(18)
and the effective stress % is determined from the stressstrain curve calculated from the MATPRO based on the equation of oe = r n i n
{
eee,K(ee)
At ]
j'
(19)
where K is the strength coefficient, n the strain hardening exponent and m the strain rate sensitivity. The effective plastic strain eep and its increment deep are determined from
eep = eo - ( o , / e ) ,
(20)
deep = eep(t) - eep(t - At).
(21)
Then the principal plastic strain increments de r are calculated from eq. (11), while the principal total strain increments de~ also from eq. (15). Consequently strains and stresses are determined as follows:
e,(t)=e,(t-At)+de,,
i = 1,2,3,
(22)
o I = 0,
(23)
(13) 02 = 20 = 2 0 e / ¢ 3 ,
(24)
where o~ are the principal stresses. Substituting eq. (13) into eq. (11), we obtain the expression for deep as
o3 = o = o,/v~-.
(25)
deep = {~ 2 (de/P) 2 ) 1/2
Finally, the dimensions and the pressure are determined from the expressions given by
(14)
The total strain increments de~ are described by a generalized form of Hooke's law given by
de, --der + ½ (doi-,(doj +dot,)) +a dT, i= 1,2,3,
(15)
h = h o exp(eI),
(26)
rm = rmo exp(e2),
(27)
l= 1o exp(e3),
(28)
h/2,
(29)
rei =
rna - -
M. Ochiai / WTRLGD-- Waterloggedfuel rods V = ~r(d -- r2)l,
(30)
2o__hh_ 2%___hh
p_
rm
tj-t
_ ti
__
227
tpt
(31)
vf3rm '
where l is the length of each volume and the subscript 0 denotes the initial value.
M : MOss G :Flow rote A : Flow oreo
3.4. Transient internal water flow
i : specific entholpy
Transient flow of the water between the two volumes can be calculated based on the following assumptions: (a) one dimensional flow in the axial direction, (b) negligible body force (except gravity), (c) negligible works done by pressure and viscous force, (d) negligible kinetic and potential energy. For this calculation we use the mass, energy and momentum conservation equations. These equations are
Fig. 4. Internal water flow model.
into a cladding, and
ja =
~ 0),
jb= (~i+l (Oj ~0).
(a) Mass conservation
30 4- 3pu
n
(32)
-G-_-~T=-,
where p is the fluid density and u the fluid velocity in the axial direction.
Integration of the momentum equation over the shaded portion depicted in fig. 5 results in
(
i lj
lj÷, ~dGj ( ~ - ~ + , ) + a
(b) Energy conservation
api at
aq",
~-~
2
api_
-1- u-~-z - O.
GjIGjI
-g(pflj + .j+,lj+,)/2.
(33)
(37)
where (c) Momentum conservation
~(1#2)
apu t - a p u 2 + a P + 1 a r r + at az -~z r Or p g = 0,
(34)
where 1" is the viscous stress and g the gravitational acceleration constant. Integrating the mass equation over a fixed volume, we obtain dWj = @ , - a j , dt
fJ- pjajAy ' Rj is the hydraulic radius (rci) - r f j ) and ~ the friction factor. The second term of the right hand side of eq. (37) is the momentum flux term, which is approximately given
(351
where Wj is the mass in the j th volume and Gj the mass flow rate from thejth volume to the ( j + l)th as shown in fig. 4. Integrating the energy equation over the j th volume, we obtain dlj
0-7=
O.~.j -
(Lo. + , j o % ,
-ij~a~,
(36)
where I is the total enthalpy. Q.i. the rate of heat transfered from fuel pellets and Qo.t the rate of heat
P
Pressure
A
Flow oreo
G
Flow rote
Fig. 5. Controle volume for momentum conservation.
M. Ochiai/ WTRLGD--Waterloggedfuel rods
228
leak flow rates from pin holes, which are also limited by two-phase critical flow rate obtained by Chisholm et al. [8]. Finally the mass of the water in each volume is redistributed by eq. (35) and the enthaipy by eq. (36).
by
_ g-,/=lg-,/=l
pjAj
pa a ]j.j+,
•
2
x g+,/2lg+,/21 ,
(38)
oj+,A~+,
where
g_+,/= = ( g + g+,)/2. Friction factor fj is calculated based on such correlations as Hagen-PoiseuiUe law for laminar flow, Blasius correlation for turbulent flow and Martinelli-Nelson's parameters for two-phase flow. Thus transient flow rate Gj can be calculated from the backward difference form of eq. (37), which is given by
1
OA2 /j,j+l
-t)glgl-g(oj6 +pj+llj+,)/2}(At),
(39)
where
L/A=5
A--;+ Ai- ,
and
b
+
The backward difference aproximation is chosen to eliminate divergency in the flow rate, which inevitably requires an iteration process. Critical flow rate based on Moodie's modcl [7] is adopted as the upper limit for the flow rate. Consequently Gj evaluated from eq. (39) is replaced by the critical flow rate Gcr, if Gj is greater than Gcr. Assuming pin holes to be orifices, we calculate the
Top End Fitting Magnetic Iron Core Sr~ Spring Adaptor / D!sk Fuel Pellets
Extensive experiments have been performed on the failure behavior of a waterlogged fuel rod in the NSRR. A waterlogged test fuel rod was prepared by evacuating a test fuel rod (fig. 6) and allowing water to enter the evacuated interior. The fuel rod was weighed before and after waterlogging to determine the amount of water entering its interior. A fast response pressure transducer was attached to the bottom end of the fuel rod contained in the capsule inserted into the experiment space in the center of the NSRR test facility as shown in fig. 7. In order to examine the validity of WTRLGD, the calculated results were compared with the experimental results of two NSRR waterlogged fuel rod tests, Test 401-4C [9] and Test 432-1B [4]. In Test 401-4C a fully waterlogged fuel rod (99 vol.%) was subjected to a 3.82 ms period power burst which would have resulted in a total energy deposition of 150 c a l / g of UO 2 unless it had failed. However, it failed in the form of a 70 mm split in its cladding tube before deposition of the total energy. The failure occurred about 8 msec after a peak power at an energy deposition of 116.5 cal/g of UO> The time of failure was distinguished by a sharp pressure pulse in the capsule and a sudden decrease in the rod internal pressure. The transient rod internal pressure for Test 401-4C calculated with WTRLGD is compared with the measured pressure in fig. 8, where a peak power appears about 30 ms after the initiation of the power burst adopted as the origin of time. From the figure it is found that the measured rod pressure lags about 15 ms behind the power burst, reaches a peak pressure of 93 MPa about 6 ms after the peak power
Bottom Claddin~ Disk Spacer End Fitting
r--I 265 mm (overoll length) Fig. 6. Standard test fuel rod in the NSRR.
4. Comparison with experiments
o (octive length)
II
229
M. Ochiai / WTRLGD--Waterlogged fuel rods Capsule hold-down device
Contt
Wot
Verlicol Offset I
pit Neutro
utron rodiography ~rn
device
pressure from 91 MPa to the ambient pressure about 8 ms after the peak power. Fairly good agreement is seen in the figure between the measured and the calculated pressure in the lower plenum region where the pressure transducer was attached. It should be noted that the calculation predicts well the measured pressure, in particular both the pressure behavior before reaching the peak pressure and the time of rod failure defined by the occurrence of instability destruction of a cladding tube. However, a little discrepancy is observed between the measured and the calculated pressure in the slow attenuation phase (34 ~ 38 ms). One of the causes is ascribable to the thin wall tube assumption adopted in the calculation since the strains in the cladding tube become large in this phase. Nevertheless the good prediction for the time of rod failure indicates that such an inadequate assumption employed in the calculation yields no significant error in the analysis of the failure behavior of a waterlogged fuel rod. The NSRR experiments reveal that the internal pressure of a fully waterlogged fuel rod is almost uniformly distributed along the axis of a cladding tube [10]. The calculation also shows the almost uniform pressure distribution as seen in fig. 9, though the pressure in the upper plenum region is slightly lower than those in the others because of a small amount of remaining gas (1
vol.%). The curves drawn in fig. 10 designating the transient states of internal water are considerably different from
Fig. 7. General arrangement of NSRR facilities.
and then decreases slowly from 93 MPa to 91 MPa within 2 ms. Also it is found that the rod failure is illustrated as an abrupt decrease in the measured rod
i j=5[]~ upper plenum
MPalooL-"~-i~]< lower plenum .... 100
reactor power rod pressure (experiment) rodr°d'Dressure.~ilu(calculation) re
0
80hO msec ~-80
~
I
'
401
.zl msee
b
|
I
I
',, O(
0
~
i
20 40 Time (msec)
I
"" . . . .
I
60
Fig. 8. Comparison with experimental results in the case of a fully waterlogged fuel rod.
I
1
I
I
I
2 3 4 5 volume number (j)
Fig. 9. Axial distribution of the internal pressure of a fully waterlogged fuel rod.
230
M. Ochiai / WTRLGD-- Waterloggedfuel rods
i00
MPa
__ 100i n°
32 msec
/ 60
......
m
~
J
,i....
-- 80
=3 ~gap j=2 ~ = 1 ~ 1 ..... lenum
~ 6O
j=5~upper
#
®
o
reoctor power rod pressure (experiment) rod pressure (calculationJ
40
,_ 4 _ O, 0
-
i
-
i 20
i
I 40
J
i 60
i 80
Time (msec) 2" ....-->~ j=3
saturation
20
62~i ~4~ 0
ri
200
I 400
I
I 600
800 (KJ/Kg)
specific enthalpy Fig. 10. Tr~sient states of internal water of a fully waterlo~ed fuel rod.
the saturation curve. The curve denoting the state of water in the lower plenum region is almost parallel to the vertical axis as shown in the figure, which indicates that the pressure increase in the plenum region results from almost adiabatic compression of water. The figure also designates that the pressure in the gap region increases in proportion with enthalpy of water. For Test 432-1B, a partially waterlogged fuel rod (70 vol.%) was subjected to the same power burst as in Test 401-4C. The measured rod pressure lags aboiat 25 ms behind the power burst, reaches a peak pressure of 72 MPa about 9 ms after the peak power and then slowly decreases as shown in fig. 11. The rod failure was not observed during Test 432-1B. Good agreement is also obtained between the measured and the calculated fuel rod pressures as shown in the figure. Again it should be emphasized that the calculation can predict fairly well the measured pressure, in particular the pressure behavior before reaching its peak and both the time and intensity of its peak. Detailed examination, however, indicates that the calculation shows a rapid decrease compared with the measurement after passing the peak pressure. It is found from fig. 12 that the calculated pressure in the upper plenum region scarcely increases during Test 432-1B due to compressibility of remaining gas (30
Fig. 1I. Comparison with experimental results in the case of a partially waterlogged fuel rod.
vol.~) in a partially waterlogged fuel rod, as revealed by the NSRR partially waterlogged fuel rod experiments [10]. The axial distribution of the calculated rod pressure decreases monotonously going from the bottom to the top end after the time of the peak pressure (38 ms) as shown in the figure. From this result it is obviously inferred that the flow rate of water from the gap region into the upper plenum region should strongly control the decreasing rate of the rod pressure. The flow rate must have become smaller because of the two-phase choking, if two-phase state had been considered in the calculation. The homogeneous model used in the calculation, however, is apt to suppress the local two-phase state arising locally at the top of the gap region. Conse-
MPa 60
O
~lncreasin~
peak
40
20
.
u er
~ J
lower plenum~
I I i 2 ~ 4 volume number (j) Fig. 12. Axial distribution of the internal pressure of a partially waterlogged fuel rod.
M. Ochmi / WTRLGD--Waterloggedfuel rods quently the overestimation of the flow rate into the upper plenum region can be considered as one of the causes for the difference between the experiment and the calculation in the decreasing rate of the rod pressure. Nevertheless the model is considered to be effective enough for analyzing the transient rod pressure during a power burst, since the prediction of the rod failure does not require any precise description of the rod pressure after the.. time of its peak. The curves plotted in fig. 13 show the transient states of water in a partially waterlogged rod. The curves are closer to the saturation curve than those plotted in fig. l0 for a fully waterlogged rod, and in particular the closest curve is that of the state of the water in the top volume ( j = 4 in fig. 13) in the gap region. Therefore, had we chosen a very small volume for the top volume so as to evaluate adequately the local two-phase state, the behavior of the rod pressure in the decreasing phase would have been well predicted by the calculation. Adoptation of such a model in the calculation, however, requires too much computor running time in order to avoid divergency in flow rate.
231
5. Conclusions The W T R L G D has been developed to analyze the internal pressure of a waterlogged fuel rod under the condition of an RIA and also to predict the possibility of its failure in the mode of cladding rupture. The calculated results are compared with the results of two tests made in the NSRR for a fully and a partially waterlogged fuel rods. As a result it is found that the calculation predicts adequately the internal pressure behavior of both the fully and the partially waterlogged rods, in particular the pressure behavior in the increasing phase and both time and intensity of its peak pressure. Therefore, the code can predict well the occurrence of the rod failure although there arises a slight discrepancy between the measured and the calculated rod pressure in the decreasing phase.
Acknowledgement We would like to express our deepest appreciation to Mr. T. Yoshimura, Mr. N. Ohnishi and Dr. K. Takeuchi for their valuable advices and consultations, and to staff of the NSRR for valuable experimental data.
j=5~pper
M:: I
plen~
Nomenclature
~=~J(lower~len~
38 msec
60
j=l
j 34 j=4
40
32
saturation
20 3O 28 0 0
200
400
600
800 (KJ/rg)
specific enthalpy
Fig. 13. Transient states of internal water of a partially waterlogged fuel rod.
A a C E F f G Ccr g h ! i K k ! m n P {~i, Qout
q'" q"
flow area thermal diffusivity specific heat modulus of elasticity thermal conductivity integral defined by eq. (4) friction factor mass flow rate critical mass flow rate gravitational acceleration wall thickness of a cladding tube total enthalpy specific enthalpy strength coefficient thermal conductivity length strain rate sensitivity strain hardening exponent pressure rate of heat transfered into gap region rate of heat transfered out of gap region density of heat generation heat flux
M. Ochiai / WTRLGD-- Waterloggedfuel rods
232
r,O,z S T t U
V W
cylindrical coordinates deviatoric stress temperature time velocity capacity specific volume mass
Greek symbols thermal expansion coefficient strain Poison's ratio density stress viscous stress
Subscripts ci CO
f i ]
0
inner surface of a cladding tube outer surface of a cladding tube surface of a fuel pellet radial node axial node initial value
Superscripts p 0
plastic prior time step
References
[1] M. Ishikawa and S. Shiozawa, J. Nucl. Mater. 95 (1980) 1-30. [2] R.W. Miller et al., IDO-16883 (1964). [3] L. Harrison et al., ANL-7325 (1966) 158-163. • [4] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 7304 (1977). [5] W.H. Rettig et al., IN-1321 (1970). [6] P.E. MacDonald and L.B. Thomson, (ed.), TREENUREG- 1005 (1976). [7] F.J. Moody, Trans. ASME, Ser. C 87 (1966). [8] K. Akagawa, Gas-Liquid Two-Phase Flow (Corona, Tokyo, 1974). [9] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 7051 (1977). [10] Reactivity Accident Laboratory and NSRR Operation Section, JAERI-M 9319 (1981).