lwr fuel performance analysis fuel cracking and …
TRANSCRIPT
LWR FUEL PERFORMANCE ANALYSISFuel Cracking and Relocation
byJ.T. Maki and J.E. Meyer
Energy Laboratory Report No: MIT-EL 78-038
October, 1978
LWR FUEL PERFORMANCE ANALYSIS
Fuel Cracking and Relocation
by
J.T. Maki and J.E. Meyer
Energy Laboratoryand
Department of Nuclear Engineering
Massachusetts Institute of TechnologyCambridge, Massachusetts 02139
Final Report for Research Project
sponsored by
Northeast Utilities Service Co.Yankee Atomic Electric Co.
under theMIT Energy Laboratory Electric Utility Program
Energy Laboratory Report No. MIT-EL 78-038
October, 1978
i
TABLE OF CONTENTS
Topic Page
List of Figures iii
SUMMARY I
1. INTRODUCTION 3
1.1 Objective 31.2 Cracking Scenario 31.3 Crack Patterns 31.4 Major Thermal Effects of Cracking 41.5 Effects of Cracking on Fission Gas Release 4
1.6 Major Mechanical Effects of Cracking 51.7 Crack Healing 6
2. THERMAL CALCULATIONS NOT DEPENDENT ON CRACKING 72.1 Outline of Computer Calculations 72.2 Cladding Temperature Profile 72.3 Cladding Thermal Strain 92.4 Cladding Stored Heat 102.5 Fuel Thermal Strain 112.6 Fuel Stored Heat 112.7 Fuel Rod Equivalent Temperature 12
3. THERMAL CALCULATIONS FOR UNCRACKED FUEL PELLETS 153.1 Pellet Surface Temperature and Gap Conductance 153.2 Pellet Temperature Profile 15
4. THERMAL CALCULATIONS FOR CRACKED FUEL PELLETS 194.1 Pellet Surface Temperature and Gap Conductance 194.2 Pellet Temperature Profile 20
5. ILLUSTRATIVE EXAMPLE 25
5.1 Fuel Pellet Surface Temperature 255.2 Fuel Pellet Centerline Temperature 275.3 Fuel Rod Equivalent Temperature 295.4 Fractional Radius for Grain Growth 295.5 Gap Closure 325.6 Pellet - Clad Contact 32
6. EXPERIMENTAL EVALUATION 37
7. CONCLUSIONS 39
7.1 Relocation Into Gap 39
7.2 Reduced Fuel Thermal Conductivity 397.3 Fuel Rod Equivalent Temperature 397.4 Crack Healing 40
ii
Topic
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
APPENDIX E
APPENDIX F
Additional Figures from Fuel Pellet ThermalCalculations
Material PropertiesB.1 Fuel Thermal ConductivityB.2 Fuel Thermal StrainB.3 Fuel Specific Heat CapacityB.4 Enthalpy Change for FuelB.5 Cladding Thermal ConductivityB.6 Cladding Thermal StrainB.7 Cladding Specific Heat and EnthalpyB.8 Thermal Conductivity of Fill Gas
Input ParametersC.1 General Design ParametersC.2 Outside Cladding Temperature
Halden Fuel Rod Parameters
Computer Program Listing
Table of Symbols
REFERENCES
Page
41
53
53
59
59
62
64
64
66
71
73
73
73
87
89
107
111
iii
LIST OF FIGURES
Fig. 2-1 Flow chart for comparative study computerprogram.
Fig. 4-1 Comparison of circumferential pellet-cladcontact models.
Fig. 5-1 Comparison of cracked/uncracked fuel pelletsurface temperature (cold radial gap width =95 m).
Fig. 5-2 Comparison of cracked/uncracked fuel pelletcenterline temperature (cold radial gapwidth = 95 pm).
Fig. 5-3 Comparison of cracked/uncracked fuel rodequivalent temperature (cold radial gapwidth = 95 um).
Fig. 5-4 Comparison of cracked/uncracked fuel pelletfractional radius for the onset of graingrowth (cold radial gap width = 95 m).
Fig. 5-5 Comparison of cracked/uncracked fuel gapclosure (cold radial gap width = 95 m).
Fig. 5-6 Circumferential pellet-cladding contactbehavior for cracked fuel pellets.
Fig. 6-1 Comparison of computational models toHalden experimental fuel rod data (experi-mental data from Ref. 11).
Fig. A-1 Comparison of cracked/uncracked fuel pelletsurface temperature (cold radial gapwidth = 75 m).
Fig. A-2 Comparison of cracked/uncracked fuel pelletsurface temperature (cold radial gapwidth = 115 m).
Fig. A-3 Comparison of cracked/uncracked fuel pelletcenterline temperature (cold radial gapwidth = 75 m).
Fig. A-4 Comparison of cracked/uncracked fuel pelletcenterline temperature (cold radial gapwidth = 115 m).
Fig. A-5 Comparison of cracked/uncracked fuel rodequivalent temperature (cold radial gapwidth = 75 m).
iv
Fig. A-6 Comparison of cracked/uncracked fuel rodequivalent temperature (cold radial gapwidth = 115 m).
Fig. A-7 Comparison of cracked/uncracked fuel pelletfractional radius for the onset of graingrowth (cold radial gap width = 75 pm).
Fig. A-8 Comparison of cracked/uncracked fuel pelletfractional radius for the onset of graingrowth (cold radial gap width = 115 pm).
Fig. A-9 Comparison of cracked/uncracked fuel gapclosure (cold radial gap width = 75 pm.
Fig. A-10 Comparison of cracked/uncracked fuel gapclosure (cold radial gap width = 115 m).
Fig. B-1 Comparison of EPRI and MATPRO UO2 thermalconducitivity.
Fig. B-2 Comparison of EPRI and MATPRO porosity factors.
Fig. B-3 Comparison of EPRI and MATPRO conductivityintegrals.
Fig. B-4 MATPRO's version of thermal strain foruranium dioxide.
Fig. B-5 MATPRO's version of specific heat capacityfor uranium dioxide.
Fig. B-6 Enthalpy change correlation for uranium dioxide.
Fig. B-7 Comparison of MATPRO and CENPD thermalconductivity for Zircaloy-4.
Fig. B-8 MATPRO's version of thermal strain forZircaloy-4.
Fig. B-9 MATPRO's version of specific heat capacityfor Zircaloy-4.
Fig. B-10 Enthalpy change correlation for Zircaloy-4.
Fig. B-ll MATPRO's version of thermal conductivityfor helium.
Fig. C-1 Correlation for the enthalpy change ofwater at 15 MPa pressure.
v
Fig. C-2 Behavior of H 0 coolant at a system
pressure of 1 MPa.
Fig. C-3 Behavior of nominal fuel rod at 100% corepower.
Fig. C-4 Behavior of hot fuel rod at 100% core power
Fig. C-5 Comparison of nominal fuel rod outsidecladding temperature to modeled outsidecladding temperature correlation.
Fig. C-6 Comparison of hot fuel rod outside claddingtemperature to modeled outside claddingtemperature correlation.
-1-
SUMMARY
It is expected that virtually all fuel pellets in a pressurized
water reactor (PWR) are cracked during power operation. This report
provides quantitative estimates of the thermal effects of cracking for
a case of interest (relatively new fuel in the Maine Yankee Reactor).
The method to account for cracking has been previously developed by
others.
Results show that it is important to account for cracking. The
equivalent temperature that represents heat stored in a fuel rod oper-
ating at 40 kW/m is 892°C for the cracked pellet versus 1016°C for a
no-crack calculation, a decrease of 124 K.
Methods used in this study were designed to isolate cracking as a
single effect for a representative case. The method used for obtaining
fuel rod outside surface temperature as a function of linear heat gen-
eration rate for this representative case is felt to be especially
suited for use in this and other isolated effect studies.
-3-
1. INTRODUCTION
1.1 Objective
During normal power changes in light water reactors, thermal
stresses within the fuel rod cause the uranium dioxide fuel pellets
to crack. Associated with cracking is relocation, defined as the
geometry changes of the fuel pellet by displacement of the cracked
pieces. Pellet cracking and relocation has several important mech-
anical and thermal consequences. This report primarily gives a quan-
titative comparison of those thermal effects for cracked and uncracked
fuel pellets at the beginning of life. Even though all ceramic fuel
pellets crack, calculations for uncracked pellets permit making esi-
mates of the cracking contribution to fuel behavior. The estimates
can also be used to indicate the nature of errors in fuel rod modeling
codes in which the cracking/relocation effects are neglected.
1.2 Cracking Scenario
During the first rise to power, a temperature gradient within the
fuel rod causes the center of the pellet to expand more than the cooler
periphery. Thermo-elastic deformation proceeds until the fracture
stress of the pellet is exceeded. For typical light water reactors,
the fracture stress of the fuel is reached at a linear heat generation
rate of about 5 kW/m. Once the fracture stress is exceeded, sudden
jumping of the pellet pieces occurs upon cracking (Ref. 1). With in-
creasing power, further relocation results from thermo-elastic deform-
ation of each cracked pellet piece. If cladding contact is made, the
crack void in the bulk of the fuel is under compressive forces and
accommodates some fraction of the pellet thermal expansion. After the
crack void is consumed, radial expansion is approximately equal to the
free thermal expansion of the pellet.
1.3 Crack Patterns
Photomacrographs display distinctive crack patterns that seem
different for fast than for thermal reactor fuel (Refs. 2 and 3). After
cooldown, fast reactor fuel exhibits radial cracking from the central
void to the periphery. Light water power reactor fuel displays an
-4-
irregular crack pattern. It is apparent, therefore, that findings
concerning cracking from fast reactor research may not be applicable
for light water power reactor cases (see also Sec. 1.7).
1.4 Major Thermal Effects of Cracking
Due to nonperfect radial cracking, the fuel to cladding gap width
will be circumferentially nonuniform. The gap width may vary between
that calculated for a non-cracked operating fuel rod and essentially
zero. Relocation of the pellet pieces into the gap causes improved
heat transfer between fuel and cladding. The effective fuel to cladding
gap conductance corresponds to a value reflecting both contact and open
gap conductance. However, cracking also causes fuel to fuel gaps to
open within the fuel pellet. Some of these gaps are oriented so as to
degrade heat conduction, counteracting at least some of the effect of
improved heat transfer at the pellet surface. Estimates of the magni-
tude of change resulting from both these effects are given later in
this report.
1.5 Effects of Cracking on FissiOn Gas Release
One possible mode of fission gas release concerns gases formed
in the columnar grain growth region of the pellet. Such gases diffuse
to the center of the pellet and are contained under pressure by the
plastic central region. When the fuel is cooled, thermal stresses
crack the pellet and the gas is released (Ref. 4). Another postulated
mechanism is that gas is released from all free surfaces of the fuel
pellet. When pellet cracking occurs, the free surface area increases
by an order of magnitude. Thus, pellet cracking apparently playsa sig-
nificant role in fission gas release (Ref. 5). At least at the time of
preparation of the MATPRO Version 09 report, the state-of-the-art was
represented by quoting an Argonne National Laboratory quarterly report
and indicating that the effects of fuel cracking on gas release are not
well documented (Ref. 6, p. 146). Such effects will be omitted from
the remainder of this report as well.
-5-
1.6 Major Mechanical Effects of Cracking
There have been a number of fuel rod failures by clad cracking
during plant up-power maneuvers. It is thought that physical features,
which characterize such failures, include:
- as the power level is increased, the thermal expansionof the fuel pellet is larger than the expansion of theclad;
- if the fuel/clad gap is nearly closed at the start ofthe maneuver, then contact forces between pellet andclad can develop as the power is increased (pellet-cladmechanical interaction, PCMI);
- fission products from the pellet (such as iodine and/orcesium), possibly freshly released, are available forchemical attack on the clad (pellet-clad chemicalinteraction, PCCI);
- the combined effect of clad stresses from PCMI and localcorrosive action from PCCI can lead to clad failure fromstress corrosion cracking (SCC); and
- SSC failures require the action of both a corrosive en-vironment and a sufficiently high stress for a suf-ficiently long period of time.
If the reactor is brought to power very slowly, it is thought
that PCMI forces are reduced. This action is called "conditioning"
the fuel. Existing fuel vendor conditioning recommendations can man-
date power ascensions as long as one or two days (rather than few hour
startups that could otherwise be employed). High costs of power to
replace that lost during the long startups and load dispatching plant
readiness considerations emphasize the importance of eliminating over-
conservatisms in the conditioning recommendations.
Fuel cracking and relocation can influence the SCC situation in
several ways. Edges of broken pieces which contact the clad can pro-
vide locations for highly concentrated PCMI forces to act (and perhaps
also to break protective clad oxide layers). The cracks themselves
may act as routes by which fission products can reach the clad. Finally,
the broken pieces of the fuel may not fit together easily during the
pellet compaction caused by the PCMI forces. But the compaction could
be accomplished by smaller PCMI forces if very local creep deformations
of the fuel occur. Thus the time dependent compaction of cracked and
-6-
relocated fuel could play a role in fuel pellet conditioning. Time de-
pendent compaction information is not now available. It has also been
omitted from the present study.
1.7 Crack Healing
During steady state reactor operation, healing of cracks may
occur simultaneously with grain growth if the pellet exceeds 1400°C
(approximately the onset of equiaxed grain growth) (Refs. 7,8,9 and 10).
Typical light water reactor fuel rarely exceeds this temperature and
thus, healing essentially does not occur. Fast reactor fuel often
exceeds 17000 C (the onset of columnar grain growth) whereby extensive
healing occurs. Healing and columnar grain formations contribute to
the crack pattern differences noted in Sec. 1.3.
Healing is used to describe a re-establishment of solid materialwhere previously a cracked surface existed. The local creep de-formation of Sec. 1.6, on the other hand, could occur with achange in shape which does not affect the existence of thecracked surface.
-7-
2. THERMAL CALCULATIONS NOT DEPENDENT ON CRACKING
The computational procedures used in the comparative study of
cracked/uncracked fuel pellets are presented in this and the next two
chapters. This chapter deals with thermal calculations needed for both
cracked and uncracked fuel pellets.
2.1 Outline of Computer Calculations
Before describing the cracked and uncracked fuel model calcula-
tions in detail, an overview of the computational method is presented.
Serving as a general outline for this and the following two chapters,
Figure 2-1 presents a simplified flow chart of the computer program
developed for this study.
The computer program is initialized by reading in the input data
whereby the cladding temperature profile, thermal strain, and stored
heat are computed. Not evident by the diagram is that a branch exists
after the cladding stored heat calculation is made. Each model
(cracked/uncracked) has its own computational procedure for the fuel
pellet temperature profile calculations and fuel stored heat calcula-
tion. Regardless of model, however, an iterative process for computing
the fuel pellet temperature profile is utilized. An initial guess of
the gap width (the cold gap width) is made whereby the pellet tempera-
tures are calculated, followed by the fuel thermal strain calculation
which gives the new hot gap width. The above procedure is repeated
until the hot gap converges within 0.05 pm where the iterative process
is stopped. Following this, each model undergoes the same fuel rod
equivalent temperature calculation.
2.2 Cladding Temperature Profile
Beginning with the steady-state heat conduction equation
-V(k VT) = q"' (2-1)c
where q' = rate of heat deposition per unit volume;
k = cladding thermal conductivity; and
T = temperature,
a relation for the cladding temperature profile may be derived.
-8-
heat
<ii
Flow chart for comparative study computer program.
Calculate fuelrod equivalenttemperature
Print out
res ults
Figure 2-1.
-9-
If internal heat deposition in the clad is neglected, Eq. (2-1)
may be manipulated to give
T,-q' r
k dT = n() (2-2)c 27r rT
where To = outside cladding temperature;
r = outside radius of cladding;
T = temperature at radius r;
and q' = linear heat generation rate (LHGR) of the fuel rod.
With a relation for the thermal conductivity of Zircaloy-4 given
in Appendix B, the left side of Eq. (2-2) is evaluated to yield a
second order expression in temperature. Solving the quadratic equa-
tion gives
T(r) = -1.015x105 { 1.3959x10 - [1.9485x10
+ 1.9704x10-5(1.3959x10-2T + 4.9261x10-6T 2o o
1+ n( } (2-3)27r r
where
T(r) = temperature of cladding at radius r in C;
To = outside cladding temperature in C;
and
q' = LHGR in kW/m.
2.3 Cladding Thermal Strain
The average cladding thermal strain is calculated from the
equation
c 2 r2 Jo (ST) r dr (2-4)r - r1
-10-
where c = average cladding thermal strain;
(ET) = cladding thermal strain at radius r (given inc Appendix B);
rl = hot inside cladding radius;
and r0 = hot outside cladding radius.
To perform the above integration, the trapezoidal rule is em-
ployed, using twenty five equally spaced mesh intervals of size M
where
r° - r1M = 25 (2-5)
The calculation of the average thermal strain requires hot dimensions
which are determined by
rhot r cold( + Ec (2-6)
Since the hot dimensions and the average thermal strain are dependent
upon one another, an iterative approach is taken in which the hot inside
cladding radius converges within 0.5 pm.
2.4 Cladding Stored Heat
The cladding stored heat is defined by the equation
SH = A H p 2wr dr (2-7)
Note that either cold dimensions and cold density or hot dimensions
and hot local density should be employed in this calculation. How-ever, the temperature profiles utilized in the computation are basedupon hot dimensions. Therefore hot dimensions are combined, forsimplicity, with cold density in Eqs. (2-7) and (2-9). It is feltthat only slight error arises from this practice.
-11-
where SH = cladding stored heat;c
= density of Zircaloy-4 = 6550 kg/m
and AH = enthalpy change of cladding at radius r (givenin Appendix B).
Utilizing the trapezoidal method with 25 trapezoids, an explicit solu-
tion to Eq. (2-7) is obtained. The value for the stored heat is used
in the equivalent fuel rod temperature calculation.
2.5 Fuel Thermal Strain
The average fuel thermal strain calculation is similar to that for
the cladding thermal strain. For the fuel pellet
r2-f 2 2f 2 (T) f r dr (2-8)
r2 o
where of = average fuel thermal strain;
(T) f = fuel thermal strain at radius r (givenin Appendix B);
and r2 = hot fuel pellet radius.
Fifty equally spaced mesh intervals are employed in the trapezoidal
method to solve the above integral. The fuel thermal strain is de-
termined for each iteration of the fuel temperature profile calcula-
tion until hot gap width convergence is reached.
2.6 Fuel Stored Heat
The fuel stored heat calculation is also similar to that for the
cladding. For the fuel
r2SHf = AHf DPf 27r dr (2-9)
*See footnote accompanying Eq. (2-7).
-12-
where SHf = fuel stored heat;
D = fractional theoretical density
p = fuel density = 10 980 kg/m3f
r2 = hot fuel pellet radius
and AHf = enthalpy change of fuel at radius r (givenin Appendix B).
Again, fifty trapezoids are used to numerically solve this integral.
2.7 Fuel Rod Equivalent Temperature
The actual stored heat of a fuel rod may be equated to the stored
heat if the rod were at some constant temperature. That constant
temperature may then be defined as the fuel rod equivalent temperature
Teq. In physical terms, Teq is the temperature that would be reached
by the fuel rod if all internal heating and external cooling were in-
stantaneously stopped. In mathematical terms
SHf + SH = (SHf) + (SHc)eq (2-10)
At constant temperature, the stored heat equations become
(SHf) = DPf AHf(Teq) rr2 (2-11)
(SH ) P AH (T ) (ro - r) (2-12)c)eq c c eq 1
It should be noted that the stored heat from the fill gas has been
deleted from consideration because it is insignificant compared to
that of the fuel and cladding. Rearranging Eq. (2-10),
f(Tq) = (SHf)eq + (SHc) -q SHf (2-13)
where T = fuel rod equivalent temperature, as yet unknown.eq
The root of Eq. (2-13), T , may be found by using Newton's method of
slope intercept. The slope intercept is determined by
-13-
f(T )T = T - s (2-14)
s f'(Ts)
where TI = slope intercept temperature;
T = temperature where the slope is evaluated;
f(Ts) = value of the function f at T ;
f'(Ts) = the derivative of the function f, evaluated
at T;
and where the derivative of the function f is
f'(T) = D pf (C (T)) rr2 + p (C (T))c (r2 r (2-15)Ip f c p c o 1
where (C ) = constant pressure specific heat capacity of UO2 (givenin Appendix B)
and (C ) = constant pressure specific heat capacity of the claddingpn c (given in Appendix B).
Beginning with an initial guess for T (fuel pellet surface temperature),
TI is calculated by Eq. (2-14) and replaces T for another calculation.
Through this successive iteration, the fuel rod equivalent temperature
is determined when TI converges upon itself within 0.05°C where
eq I
-15-
3. THERMAL CALCULATIONS FOR UNCRACKED FUEL PELLETS
This chapter contains the computational methods used for un-
cracked UO2 pellets employed in the comparative thermal effects study.
3.1 Pellet Surface Temperature and Gap Conductance
Solving the steady-state heat conduction equation for the gap
yields an equation for the pellet surface temperature
T2 T1+ 27rr h (3-1)2g
where T2 = pellet surface temperature
T1 = inside cladding temperature
q' = LHGR
r2 = hot fuel pellet radius
and h = gap conductance.
The gap conductance is determined by
kh = --- (3-2)g 6+6'
where k = thermal conductivity of fill gas (100% helium given ing Appendix B);
6 = hot radial gap width;
and 6' = root mean square cladding-fuel surface roughness.
3.2 Pellet Temperature Profile
The fuel temperature profile (based on constant volumetric heat
deposition rate) is calculated from the following equation
T T2 1 I
J kdT 12 A kdT + 1 X r )2 (3-3)
o
T
where f kdT = conductivity integral for 95% theoreticalo density fuel (given in Appendix B);
PF = porosity factor (given in Appendix B);
and T = temperature at radius r.
-16-
The temperature of the fuel pellet at radius r may be determined
by equating the value of the conductivity integral (a function of r)
to its corresponding temperature. This is accomplished by fitting
second order equations to the conductivity integral. The pellet tem-
perature profile is then computed from equations of the form:
T(r) = A + Bx + Cx2 (3-4)
where x = fT kdT in kW/m
and A, B, C = constants.
The constants A, B, and C are presented in Table 3-1. The ac-
curacy of the above fit to the conductivity integral for 250 < T < 28000C
is ±+0.40C. This error is deemed insignificant compared to the overall
assumptions made in the study.
-17-
Table 3-1
Constants for Temperature of Conductivity Integral Equation (Eq. 3-4)
Range A
T < 5850C(x < 3.441)
26.68
B
72.42
'C
26.08
585 < T < 1000°C
(3.441 < x < 4.814)
1000 < T < 1405°C
(4.814 < x < 5.866)
1405 < T < 18000 C
(5.866 < x < 6.783)
1800 < T < 22050 C
(6.783 < x < 7.723)
T > 22050C(x > 7.723)
126.7
40.26
-630.9
-1750.15
12.82
46.48
274.13
604.12
35.01
31.75
12.43
-11.91
-2589.9 822.27 -26.08
-19-
4. THERMAL CALCULATIONS FOR CRACKED FUEL PELLETS
This chapter deals with the computational methods used for
cracked pellet thermal effects.
4.1 Pellet Surface Temperature and Gap Conductance
As in Section 3, the pellet surface temperature may be deter-
mined by
T2 = T1 + 2rrh (4-1)2g
However, the gap conductance for a cracked fuel pellet takes the
form (Ref. 11)
h = (l-F)h + F h (4-2)g 0 c
.k
where h = zero contact pressure conductance = g
h = contact conductance = -g .c 6'
k = thermal conductivity of fill gas (given in Appendix B);
and F = fraction of pellet circumference lying against the cladding.
Two models for the fraction of circumferential cracked pellet
contact were investigated. The Kjaerheim & Rolstad model (Ref. 12)
may be expressed as
F = 0.3 + 0.7[0.2](5 0 6/r2) (4-3)
where 6 = hot radial gap width in mm;
and r2 = hot fuel pellet radius in mm.
The FRAP-S model is given below for fuels with burnups less
than 0.6 MWd/kgU (Ref. 11).
1006 4 -1F = 0.3 + [100() + 1.429] (44)
r 2
and for burnups greater than 0.6 MWd/kgU
-20-
F = 0.3 + [A(00) B + 1.429] 1 (4-5)r2
where A = 100-98 Fb;
B = 4-0.5 Fb;
Fb = 14 (x-0.6)4 + 1]-1
and x = burnup in MWd/kgU.
It should be noted that the burnup factor Fb, and thus, the
fraction of pellet-clad contact, F, reaches an asymptotic limit near
10 MWd/kgU burnup.
Figure 4-1 presents the Kjaerheim & Rolstad model and the FRAP-S
model for burnups under 0.6 MWd/kgU and for 10 MWD/kgU. Only at
larger ratios of 6/r2 does the Kjaerheim & Rolstad model predict a
larger fraction of pellet contact than the FRAP-S model (regardless
of burnup). With increasing burnup, the FRAP-S model predicts a larger
fraction of pellet contact for all ratios of 6/r2. Due to the accumu-
lation of gaseous fission products at the grain boundaries in the fuel,
the fuel pellet tends to crack more and thus causes greater contact
with the cladding.
Noting that the FRAP-S model is more conservative in the usual
operating range of 6/r2 than the Kjaerheim & Rolstad model, and that
the FRAP-S model is capable of including burnup considerations, the
FRAP-S model was chosen for incorporation into the study.
4.2 Pellet Temperature Profile
The cracked fuel pellet temperature profile is calculated from
the steady state heat conduction equation
- V(kVT) = q'" (4-6)
which can be put into the form
-q'r=T - (4-7)
dr 2 (47)27r2 k
-21-
c -'0 O
DVYJAOD aYID - JITda do NOIJDY&
a
UZ i
o
14
'r
0 a001 rlrlo @
H 0: 3
I
o fe 0
0I M,--
ox
cn
-22-
Using a finite difference method
dT T(rn) - T(rn-Ar)dT _ n fl (4-8)
dr Ar
Eq. (4-7) is rearranged to
q'r Ar
Tn+ = T + (4-9)
27rr2 (kf)n
where Tn = fuel pellet temperature at radius r ;
n = index with range 2 < n < 52;
Ar = r2/50 = radial increment;
q' = LHGR;
and (kf)n = U02 thermal conductivity at radius r .
Beginning at the surface of the pellet, the temperature at the
first radial increment is computed using a temperature T2 obtained from
gap calculations. Equation (4-9) is then used successively to step
toward the center of the pellet and obtain the temperature profile of
the fuel.
The cracked pellet temperature profile is calculated in 50 radial
increments. Comparison of uncracked pellet centerline temperatures cal-
culated from the conductivity integral and from the finite difference
equation (also using 50 radial increments) displayed a maximum deviation
of 0.1°C. From these calculations it can be assumed that 50 increments
would produce sufficient accuracy for the cracked pellet temperature
calculations.
To account for the reduction in heat transfer due to cracks in the
fuel, an effective U02 thermal conductivity is employed in Eq. (4-9).
The effective UO2 thermal conductivity is (Ref. 12)
-3 -2k = kf[ - 3.175x0 + 1] (4-10)e kk+
r2(0.077 - + 0.015)kf
-23-
where kf - U02 thermal conductivity (given in Appendix B);
k = fill gas conductivity (given in Appendix B);
6 = hot radial gap width in m;
and r2 = hot fuel pellet radius in mm.
-25-
5. ILLUSTRATIVE EXAMPLE
This chapter presents the results of the computer calculations
for cracked/uncracked fuel pellets with a nominal cold radial gap width
of 95 m. This and other dimensions are based on the nominal fuel rod
dimensions for the Maine Yankee reactor (see also Appendix C). Calcu-
lations were also made for initial radial gap widths of 75 and 115 pm
and are presented in Appendix A. The 75 and 115 m radial gap widths
are estimates which correspond to the extreme tolerances expected in
the determination of the nominal gap width (Ref. 13).
Note that the calculations of this chapter have been performed
for linear heat generation rates (LHGR) between zero and 60 kW/m. The
local peak LHGR for Maine Yankee is 35.8 kW/m. In addition, occurrences
such as columnar grain growth and central hole formation occur when
temperatures rise above '1700°C. Therefore, portions of the plots
above approximately 40 kW/m should be regarded as mathematical extrap-
olations from the lower LHGR results. These portions are included for
the purposes of illustrating how the calculations behave, but may be
significantly in error because of the neglect of some significant high
temperature phenomena.
5.1 Fuel Pellet Surface Temperature
Figure 5-1 shows the fuel surface temperature as a function of
LHGR for a cracked and uncracked pellet. The uncracked pellet consis-
tently shows a higher temperature than the cracked pellet with a maxi-
mum difference of about 210°C at 40 kW/m. This effect is due to the
fact that the cracked pellet relocates into the gap, resulting in im-
proved gap conductance and a lower fuel surface temperature.
The uncracked fuel surface temperature decreases with increasing
LHGR above 45 kW/m. Beyond this LHGR, the pellet core becomes hotter
where thermal strains expand the pellet into the gap and improved gap
conductance results in a lower surface temperature. Within 60 kW/m,
the cracked fuel pellet does not exhibit this effect.
The surface temperature for uncracked fuel displays a concave
downward function of LHGR, while the cracked pellet displays a nearly
linear function. For both cases, a kink exists at 25 kW/m in the
to0
)
04- 0
I . 3
Hi o HOl
c V Cmo 0 0b U) n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c-(Do) mQiVnaaWL
-26-
-27-
surface temperature plot. This kink corresponds to the LHGR at which
the outside cladding temperature changes from an increasing function of
LHGR to a constant temperature at the onset of nucleate boiling (see
Appendix C for further details on the determination of the outside
cladding temperatures).
The general shape of the surface temperature curves remainsun-
changed with different gap sizes. However, with increasing initial
gap size, the uncracked pellet surface temperature increases at all
linear heat generation rates. The differences between the cracked/
uncracked pellet surface temperaturesalso increases with increasing
cold gap size. The maximum difference occurs at higher temperatures
for larger cold gap widths. Remaining within 1°C of themselves, the
surface temperature of the cracked pellets is unaffected by cold gap
sizes up to linear heat generation rates of 40 kW/m. The largest
calculated difference is 6.5°C (at 60 kW/m for the two extreme cold
gap widths).
5.2 Fuel Pellet Centerline Temperature
Figure 5.2 shows the fuel centerline temperature as a function
of LHGR for a cracked and an uncracked pellet. For all linear heat
generation rates, the cracked pellet exhibits a lower centerline tem-
perature. With increasing LHGR, the difference in temperature between
the cracked and uncracked pellets increases. At 30 kW/m this dif-
ference is only 20°C, while at 60 kW/m the difference increases to
about 130°C.
With increasing cold gap size, the centerline temperature for
uncracked and cracked pellets increases for all linear heat generation
rates. Only for the larger cold radial gap width of 115 pm does the
cracked pellet centerline temperature exceed that of the uncracked
pellet. That is, from 10 kW/m up to 45 kW/m the cracked pellet center-
line temperature is higher than that for the uncracked pellet with a
maximum difference of about 40°C at 24 kW/m. Beyond 45 kW/m, the
uncracked pellet temperature is higher with a centerline difference
of about 60°C at 60 kW/m.
-28-
o 80 00 0
(Do) a tun~za
0so
B
a)pQ
04)
r4H0
Q)
r-to~
ao0
o=gk
'- 0H
H
o SH (IF1
OH
orlj
U 0o c,
.1
0
00.9
-29-
5.3 Fuel Rod Equivalent Temperature
The fuel rod equivalent temperature as a function of LHGR for
cracked/uncracked pellets is presented in Fig. 5-3. For all linear
heat generation rates, the uncracked pellet displays a higher equiva-
lent temperature. At 30 kW/m, the difference is 104°C, at 60 kW/m the
equivalent temperature difference increases to 170°C.
As the cold gap size increases, the equivalent fuel rod tempera-
ture increases for both the uncracked/cracked cases. The relative
difference between the equivalent temperatures for the two cases is
unaffected by the cold gap size at each LHGR.
5.4 Fractional Radius for Grain Growth
A temperature for equiaxed grain growth, 1400°C, and a tempera-
ture for columnar grain growth, 17000C, were arbitrarily chosen. How-
ever, these temperatures are within the range of those determined by
other investigators (Ref. 14). In addition, a fixed temperature for
grain growth in U02 is a gross assumption in itself because grain
growth is a function of burnup, fuel stoichiometry, and duration of
irradiation. The 1400°C and 1700°C grain growth temperatures do serve
to indicate the extent of microstructural change (including crack
healing) regions, however.
Crack healing occurs simultaneously with grain growth. At the
lower pellet temperatures (within a grain growth regime), the time at
a steady state operating temperature may not be long enough to allow
for the diffusionally controlled grain growth and healing processes
to occur. However, higher fuel temperatures (as found in fast reactor
fuel) do allow equiaxed grain growth, columnar grain growth, and heal-
ing to occur quite extensively during normal reactor operating cycles.
Figure 5-4 presents the fractional radii for grain growth in
cracked and uncracked fuel pellets. For this consideration, a frac-
tional radius of 1.0 is defined as the surface of the fuel pellet.
Consistent with the higher temperature profile for the uncracked pellet,
grain growth occurs throughout more of the uncracked pellet than for
the cracked pellet at all linear heat generation rates. Crack healing
may be expected to begin at the pellet center at about 30 kW/m. At
40 kW/m, about 21% of the cracked pellet is within the grain growth/
heaing regime, while at 60 kW/m, 41% of the pellet is in this range.
-30-
0 0 00 0 0.4 0 \0W.4 ..- 4
(D ) I nsaJmIL
0
4)'
0
4)
.e4o 6
Z
r-I
o0 -q %
Q'
o 0
C.-4H0X oE: NQ S 0 x
OHlg
a0o
O .4
0O.CN
To
0
+4
al
-H
O0 o
40 aH 0
o
X 0
( V:4
'* Olt2
o oSnICIr INOIJDY
-31-
\0
-32-
With increasing cold gap size, a lower LHGR is required to
initiate grain growth, and more of the pellet is within the grain
growth area at each LHGR. This is consistent with the higher tempera-
ture profiles for larger initial gap pellets.
Only at the larger cold radial gap width of 115 m does the
cracked pellet initiate grain growth at a lower LHGR than the uncracked
pellet. This effect reflects the fact that the cracked pellet has a
higher central temperature profile over the lower range of linear heat
generation rates than the uncracked pellet.
5.5 Gap Closure
Figure 5-5 displays the percentage of gap closure for uncracked
and cracked fuel pellets. The gap width considered here is an undis-
turbed gap, where the fuel pellet is treated as uniformly circular and
free from pellet piece relocation. Regardless of the cold gap size,
the uncracked pellet achieved greater gap closure than the cracked
pellet for all linear heat generation rates. Even though the cracked
pellet relocates into the gap, the improved heat transfer results in a
lower average pellet temperature and lower thermal strain. Thus, from
reduced thermal expansion, the cracked pellet has a larger undisturbed
hot gap width than the uncracked pellet at all linear heat generation
rates.
For both cracked and uncracked cases, the percentage of gap closure
decreases with increasing cold gap size at each LHGR. Even though the
larger cold gap pellets have higher temperature profiles, and larger
absolute changes in pellet size, thermal expansion alone cannot pro-
portionally close up the original gap as that of a smaller cold gap pellet.
5.6 Pellet - Clad Contact
The uncracked pellet model assumes a perfectly circular pellet.
Thus, only with 100% gap closure, where the gap width equals the pellet-
cladding surface roughness, does pellet-clad contact exist. Up to
60 kW/m, pellet to cladding contact was not reached under the conditions
employed in this report for uncracked pellets.
Figure 5-6 presents the calculated circumferential pellet-cladding
contact for cracked pellets. With increasing LHGR, the fraction of
contact increases. This is to be expected since the amount of contact
-33-
0 0mflSO' d JCmImd
a
H
.l
00o
o0
.
: CH
IiI'
0'
0o
t 4
oC\
-34-
-2o iDo fH
\0 H
AH
0
0a
0
0
0- fr(D
0A
08eH H
o t o
CJ H"d
C.)'4X E3 O
.9IPa;
0O liI02
m;0
cO ' OCC~~~~~~~~~~: C?lDVyiXOD Q'aID - JsTI?a lNDUrIa
-35-
is a function of the hot gap width. With higher linear heat generation
rates, the hot gap becomes smaller which results in more contact. Also,
with larger cold gap pellets, the hot gap width is larger which results
in less pellet-clad contact at each LHGR.
-37-
6. EXPERIMENTAL EVALUATION
Beginning of life experimental fuel performance data was obtained
from an instrumented fuel assembly irradiated at the Halden Heavy Boil-
ing Water Reactor (Ref. 11). The fuel rods in this assembly are typical
of pressurized light water reactor design in dimensions and materials
(see Appendix D). Utilizing the models described in this report, cal-
culated pellet centerline temperatures are compared to the Halden ex-
perimental data in Fig. 6-1.
The comparisons indicate that, for these experiments, only slight
differences exist between the uncracked (non-relocation affected gap
conductance, solid pellet thermal conductivity) and cracked (relocation
affected gap conductance, cracked pellet thermal conductivity) model
predictions. Both models produce reasonable pellet centerline tempera-
tures for the third and fourth power cycles. However, when pellet piece
relocation is considered in the gap conductance and a solid pellet
thermal conductivity is assumed, the model drastically underestimates
the centerline temperature for all power cycles. This indicates that a
model incorporating pellet piece relocation in the gap conductance must
also consider the crack induced heat transfer barriers within the fuel
pellet to obtain reasonable results.
As displayed in Fig. 6-1, experimental evidence indicates that
thermal stabilization of a fuel rod is obtained approximately after the
third power cycle (Refs. 2 and 11). This asymtopic effect may result
because the internal thermal stresses which cause the major amount of
pellet cracking are essentially relieved during the heatup-cooldown
phases of cycling, and because further pellet piece relocation is in-
hibited by pellet-clad contact interference. Even though the model
utilized in this study for cracked pellet-clad contact does not ex-
plicitly incorporate cycling effects, the model correlates quite well
to fuel rods thermally conditioned by power cycling.
Extrapolating a best fit curve through the third and fourth power
ramp experimental data points, an estimated difference of 50°C exists
between the cracked pellet model prediction and the experimentally de-
termined centerline temperature at 40 kW/m. Assuming the same propor-
tional difference in the Maine Yankee calculations, a 550C discrepancy
would result between the cracked pellet model and actual beginning of life
pellet centerline temperature at 40 kW/m, with the calculated temperatures
being higher.
-38-
10 20 30
LINEAR HEAT GENERATION RATE (kW/m)
Figure 6-1. Comparison of computational models to Halden experimental
fuel rod data (experimental data from ref. 11).
2000
1600
Oo
frIER
A:pr;
r4
1200
800
400
o 40
-39-
7. CONCLUSIONS
This chapter presents the conclusions from this study based upon
Maine Yankee light water reactor fuel under steady-state, beginning of
life operation.
7.1 Relocation Into Gap
Once the fuel pellet cracks, a fraction of the fuel pieces lie
against the cladding. This contact results in better heat transfer
conditions than for an assumed uncracked pellet which has not relocated
into the gap. When compared to an uncracked pellet under the same
operating conditions, the cracked fuel pellet exhibits a lower surface
temperature. This difference may be as much as 210°C at 40 kW/m.
7.2 Reduced Fuel Thermal Conductivity
Some of the cracks within the fuel pellets form heat barriers
and degrade overall heat transfer conditions. To account for this heat
transfer degradation, an effective fuel thermal conductivity is used
which essentially serves to reduce the normal U02 thermal conductivity.
Generally, the adverse thermal effects of internal cracks are countered
by the crack induced relocations into the gap. Only for large cold gap
widths does the cracked pellet centerline temperature exceed that of
the uncracked pellet. At 35 kW/m with an initial radial gap size of
115 m, the cracked fuel centerline temperature exceeds that of an un-
cracked pellet by about 400C. Regardless of initial gap size, the ef-
fects of internal cracks are diminished by the relocation improved gap
conductance. Thus, cracked pellets always display a cooler outer tem-
perature distribution resulting in a lower average thermal strain, less
undisturbed gap closure, and less stored heat than for an uncracked
fuel pellet.
7.3 Fuel Rod Equivalent Temperature
Due to the cooler outer temperature profile which results in less
stored heat, the cracked fuel pellet always exhibits a lower fuel rod
equivalent temperature than the uncracked pellet. At any particular
LHGR, the difference between the equivalent temperatures for cracked/
uncracked pellets is approximately the same for any initial gap width.
-40-
During a LOCA, the cladding and the fuel will equilibrate to
roughly the same temperature since the heat transfer to coolant is
poor and the heat deposition rate is low. The lower equivalent tem-
perature indicates less severe LOCA conditions. Thus, the stored heat
of an uncracked pellet at 30 kW/m will produce the same equivalent
temperature as a cracked pellet at 37.5 kW/m.
7.4 Crack Healing
Significant crack healing is expected when extensive grain growth
has occurred. Only at 35 kW/m does 10% of a fuel pellet with a nominal
initial gap width reach into the grain growth regime (fuel temperature
greater than 1400°C). Under normal reactor operating conditions, LWR
fuel pellets seldom exceed this LHGR. Thus, extensive crack healing
is not expected in typical light water reactor fuel.
-41-
APPENDIX A
ADDITIONAL FIGURES FROM FUEL PELLET THERMAL CALCULATIONS
This appendixpresents the figures for cracked/uncracked fuel
pellet thermal effects at initial radial gap widths of 75 and 115 m.
These two gap widths were chosen as the extreme tolerances for the
determination of the nominal cold gap width. The figure captions
should be self-explanatory, but if further explanation is required,
refer to Chapter 5.
0
' - I
';
o o
Ii0-s
0
0H
0 0 00 0 0rv As ,f%
-42-
(o o ) miUnaman89a
4)
i 4
4)WSID
la0
'.i 'ml_s ,H
o oN
a'
o a
I .I
S14FN rMa
o o oo 0 0
(Do ) alp
-43-
-44-
o o 0o 0 04 4 4N=( -Wn I wa
(o,) =,Lvadm
.:0Ga
H
I
ER
4
0a,
0
q-H
C)
0-t0HFP0p4H
0 C)
'd -Z
o e.3 xr%
e C
o 0 *
1.4
-45-
o oo 0-I
(o, amaaam~t
a
S (g0
-I
.4Is a,
-i0Ht0 e
ad
(DfOre0
E 0o oH i4fCOH.
o0
oOQ
oo
0Ia,
oId
g
0Hi
OH
Oe
'IC)
E 10
o o o .0O ' oo oq.4 . as Cs1.
w ., vo
-46-
(D ) aunlyaaaal
0
4X
Cp
P-
oo
0H O H
0
I C H=d -
o a
0oCV
o O 0o o ooto o0
(D a ) avllvtyaal
-47-
A
0
Ok
.Coe-H
aH D
E e
o ;s
0 rlC)
CQ
\0 4 C
siaO yxOIDyaSIGYE ~tNOIDYEtIL
-48-
-49-
It N; o
SfnIa rIYINOIDYHA
F-'
0.0-P
k
$40t¢CN490LI0
!~4
A 0HC)rlnMS
%- 0HE Hi &Hs s
H C)
E4E- S
9 -4
0cc
VaH O
4!It
a
0 I0
0
I 0eq Xr
-4
I
.r
ri
V
HC3
0I0+'0
0
'O
-50-
i- , t~~~~~~~~~~~~~~~~~~~~~~~~~.
o 0 0o
K&ISO 'ID d1 O J w U A c ed
rlIsi.1-H0P4,0
_ 19
0
iic,-
00_ H0
mH I.S fm
0
* aj B-
. 0
. It
i<(H0
-51-
-4
ari
0 0
M8~S010 cTVD M MIlc
H
rl
Ia.0
2 y
H
0
.aH
d0
Io0
0
0
O(11
-53-
APPENDIX B
MATERIAL PROPERTIES
This appendS presents the material properties for uranium dioxide,
Zircaloy-4, and helium used in this study. In some cases, a comparison
between two relations for a particular property, and a discussion con-
cerning the choice of a relation incorporated into the work will be
given.
B.1 Fuel Thermal Conductivity
Two relations for the thermal conductivity of uranium dioxide were
investigated. The thermal conductivity used by EPRI is (Ref. 15)
kf = PF [402.4+T + 6.12x10 (T+273) (B-1)
PF 1.1316(1-P)PF = (B-2)l+P+10P
where kf = U02 thermal conductivity in W/mK
PF = porosity factor, normalized to 95% theoreticallydense fuel (dimensionless);
P = fractional porosity of fuel (dimensionless);
and T = temperature in C.
MATPRO's version of the UO2 thermal conductivity correlation for
0°C < T < 16500 C is (Ref. 6)
kf = PF 464+T + 1.216x102 exp(l.867xlO-3T) (B-3)f 464+T~~~~~~for 1650 < T < 28400 C
-54-
kf = PF [1.91 + 1.216x10- 2 exp(1.867x1 T)] (B-4)
PF = - (2.58-5.8x10 -4T) (1-D) (B-5)PF -4 (B-5-
1 - (2.58-5.8x10-4 T) (0.05)
where kf = UO2 thermal conductivity in W/m'K;
PF = porosity factor, normalized to 95% theoreticallydense fuel (dimensionless);
D = %T.D./100 (dimensionless);
and T = temperature in C.
The above correlations for 95% T.D. fuel are plotted in Fig. B-1.
Both relationships give essentially the same thermal conductivity over
the temperature range of primary concern, 3000 C to 20000C. Above this
range, the MATPRO version increases more sharply than the EPRI corre-
lation, thus resulting in a 4.5% difference at 20000 C and increasing
to a 29% difference at the melting point (2800°C).
Figure B-2 presents the porosity factors from EPRI and MATPRO at
500°C and 2000°C. At low temperatures, small differences exist between
the two porosity factors. With increasing temperatures, the difference
increases. For fuel near 95% T.D., both porosity factors equate quite
well for all operating temperatures.
Using the two thermal conductivities, the following conductivity
integrals were derived for 95% T.D. fuel with an arbitrarily choosen
reference temperature of 0°C. From EPRI:
T
kf dT = 3.824 n 1 + 40 + 1.53x10 1 4 (T+273)4
-5- 8.5x10 (B-6)
-55-
0IN 0 ull, 0 0'a 0 0l% S 0IN
(IXU/R) LLIAIJ1DflUIOD 'IYMH L ZOfl
f.
o'
om,oA
E4
I'
i"l
0
r4
-56-
O UA O IOA o o ai , 0 a1 0' COV-4 V4 V-4 0 0 0
aOWVY XIISOZIOd
HC0
H
E-4
(.4
4O%
p4
0
a,0I~
-H
'-4
-57-
T
where f kf dT = conductivity integral in kW/m;
0o
and T = temperature in C.
From MATPRO for 0°C < T < 1650°C
T
f 464] 6.513x10o
exp(l.867x10-3T) - 6.513x10- 3 (B-7)
for 16500 C < T < 2840°C
T
fkf dT = 1.91x10-T + 6.513x10-3
exp(l.867x10-3T) + 2.974 (B-8)
T
where k dT = conductivity integral in kW/m;
o
and T = temperature in C.
The values of the above integrals are plotted in Fig. B-3. Over
most of the fuel operating temperature range, the difference between
the two conductivity integrals is approximately constant. This dif-
ference will result in a discrepancy of about 500C for a fuel center-
line temperature calculation.
Due to itsgreater simplicity and ease of mathematical manipulation,
it was decided to use the EPRI version of the UO2 thermal conductivity
in this study. Furthermore, since most fuel operating temperatures are
below 2000°C, minute differences would result from cracked pellet
-58-
I I -I I I I I I I
I I I ,I - I I I .
N ao e- '3 u'v (- m N *4
(//M) lP
00
uoN.
o
0 idCt H
C,N 0
o: o .-uHo ;
C 0
I
0
_ 0rq E-
F9oV o
0
0 rHS e0 Rl
"O
F;J q0
07
so o
m cr%4
__
I
I
-59-
temperature profile calculations if the MATPRO thermal conductivity
were used. If only 95% T.D. fuel were to be considered for uncracked
pellet calculations (where the porosity factor is normalized to one),
then the MATPRO correlation would have been used. However, to ensure
greater flexibility, incorporation of the porosity factor was deemed
necessary. Integration of the complete MATPRO relation (with its
temperature dependent porosity factor) would have resulted in an ex-
ceedingly burdensome equation. Also, a different reference temperature
for the conductivity integral (i.e., 2000C) would reduce the calculated
pellet centerline temperature differences. Thus, for consistency in
the comparative study of cracked and uncracked pellets, the EPRI U02
thermal conductivity correlation was employed.
B.2 Fuel Thermal Strain
A relation used for uranium dioxide thermal strain was taken from
MATPRO and is presented below
cT = -4.972x10- 2 + 7.107x10 4T
+ 2.581x10 -7T2 + 1.14x10l T3 (B-9)
where ST = UO2 thermal strain in %
and T = temperature in C.
Figure B-4 displays this correlation in graphical form. The
root of the thermal strain equation lies at about 65°C.
B.3 Fuel Specific Heat Capacity
For use in stored heat and equivalent fuel rod temperature calcu-
lations, a correlation for uranium dioxide specific heat capacity was
taken from MATPRO. The following equation is also plotted in Fig. B-5.
-60-
~4- ~ ~ K n N( KUI r4l SH MIY&LbS 'tywuawi a r e
,4H
CH
0
o 4
IQHEd l 0Ed 0
0H
.1
-61-
0 0 0 0O0 o coO gX) 1'70 4H N
(xs/r) LLIDYvJYD IS DIAIDSdS
.iH
qa§
I .C'
. 4,a
o
o0
.4
IQAIa,a
-62-
535 28585.005 exp( T )
T
2 535 285 2
T
-8+ 2.432 x 10 T
1.6591x106 -18971+ 2 exp T )T
(B-10)
where C = U 2 specific heat capacity in MJ/kg-K
and T = temperature in K.
B.4 Enthalpy Change for Fuel
The enthalpy change for U02 may be derived from the following
equation
T
AHf(T) =
Tref
C dTP
(B-ll)
Using the MATPRO specific heat capacity, U02 enthalpy is calculated
below for a reference temperature of 250 C.
AHf(T) .158803 + 1.216x10-8T2
535 285)-exp( T- )-l
+ 87.455 exp(- 18971) - 3.2705x10- 2
T
AH(T) = U02 enthalpy in MJ/kg
T = temperature in K.
(B-12)
Figure B-6 presents the UO2 enthalpy change in graphical form.
p
where
and
-63-
0 0
(9/rw) IDNYHD xdrEvHma
U
U
0
Ido
0.-4CH
0
0
I
4)
.1
0
-64-
B.5 Cladding Thermal Conductivity
Taken from MATPRO and CENPD-218 (Ref. 16), two correlations for
the thermal conductivity of Zircaloy-4 were investigated. MATPRO's
version takes the form
k = 7.51 + 2.09x10-2 T - 1.45x10-5 T + 7.67x10 T (B-13)c
where k = Zircaloy-4 thermal conductivity in W/m.k
and T = temperature in K.
The relation from CENPD-218 is
k = 13.959 + 9.8522x10l T (B-14)c
where k = Zircaloy-4 thermal conductivity in W/m.k
and T = temperature in C.
The above two conductivities are presented in Fig. B-7. At 3000C the
MATPRO conductivity differs by 4.4% from that of CENPD. With increas-
ing temperature, this difference decreases, where at 500C, the dif-
ference is only 1.8%.
To allow for a simple and explicit cladding temperature profile
relation, the CENPD thermal conductivity was chosen. The use of the
MATPRO thermal conductivity is expected to produce similar results for
inside cladding temperature calculations. However, the complexity of
the MATPRO relation does not lend itself towards a direct computation
of the cladding temperature profile; this made its application unwarranted.
B.6 Cladding Thermal Strain
The correlation used for the cladding thermal strain was taken
from MATPRO. For 27 < T < 800°C, the thermal strain relation is
ET = -2.373x10-2 + 6.721xlO-4T (B-15)
-65-
00co
40
0,q
0 a0.
0o a0
00*--I0c 'o go
B 4
3a
. F
QN CO 4CNl w -I
(X. /A) LIAIM1nOD IYWdL
-66-
where ¢T = Zircaloy-4 thermal strain in %
T = temperature in C.
The above correlation is presented in Fig. B-8. The root of the
thermal strain equation lies at about 350C.
B.7 Cladding Specific Heat and Enthalpy
From data points supplied by MATPRO, the following equations were
derived from linear interpolation for specific heat capacity and
enthalpy of Zircaloy-4. These equations may be expressed as
(C )p c
AH (T)C
where CP
= A + BT
= C + DT +
(B-16)
ET2 (B-17)
= specific heat capacity of Zircaloy-4 in J/kgK;
AH(T) = enthalpy change of Zircaloy-4 in MJ/kg;
T = temperature in C;
and A,B,C,D,E = constant.
The values of A, B, C, D, and E are given in Table B-1. The specific
heat of the cladding is plotted in Fig. B-9, and the enthalpy change
is presented in Fig. B-10.
and
-67-
I I I I I I I I I I
f- ..... I .. I . I _ I .0o; O
MIYUIS A'IDVO14H: IU( d
00No -1
"-I ;d
U tq
0r.oH r4I -I
o
_ __
-68-
o 0o o( ) I
(x.2/r) ;mIayo3 A DIdlodS
0-4
0
0o N8 tq0
qI0
d
< M
I0 0
H
0C'
of
00
-69-
.4. N0r c0(t3/vr) SDMYHD X&IYHJM¶
0
o
IsV 1
a C4.0EH
0-:A
Pim
V.A
a
<6
ED4
O0C
-70-
Table B-1
Constants Used in Specific Heat and EnthalpyEquations for Zircaloy-4 (Eqs. B-16 and B-17)
A B C D
25<T<1270C
126<T<367°C
367<T<8170C
817<T<820 0°C
820<T<840°C
840<T<8600C
860<T<880°C
880<T<9000C
900<T<920°C
920<T<975 0°C
T>9750C
275
287
295
-34200
- 3110
- 460
- 3860
- 3550
2890
7690
356
.21
.121
.0978
42.3
4.4
1.25
5.2
4.85
-2.3
-7.53
0.0
-7.67x103
-9.23x10-3
14.1
1.33
.222
1.68
1.55
-1.35
-3.56
1.72x10-2
2.87x10
-42.95x10
-3.42x102
-3-3.11xlO- 3
-4.6x104
-3.86x10-3
-3.55x10-3
2.89x103
7.69x10-3
3.56x10- 4
6.04x10-8
4.89x10-8
2.12x10
2.2x10- 6
6.25x10-7
2.6x106
2.42x10 6
-1.15x106
-3.76x106
0.0
Range E
-71-
B.8 Thermal Conductivity of Fill Gas
A relation for the thermal conductivity of helium was also taken
from MATPRO. The correlation is
-3 0.668k = 3.366 x 10 (T) (B-18)g
where k = heliunthermal conductivity in W/m.K
and T = temperature in K.
The above equation is plotted in Fig. B-ll.
-72-
O 4 C
N)3 /0A OD 0(M N/M) IIAIMflUOD AceZHJ
i
0)
§
o ,io
I !o0 0
x
1fM ma
4)
q-I
N
-73-
APPENDIX C
INPUT PARAMETERS
The input parameters were supplied by Yankee Atomic Electric Company
from base case design and operating data. The data is characteristic
of Maine Yankee PWR fuel. However, the conclusions from this study
are applicable to other light water reactor fuel.
C.1 General Design Parameters
The following is a list of design features of the MaineYankee
fuel incorporated into this study. In addition, the tolerances on the
initial radial gap have been estimated for this study to be ±20 pm
(similar to LWBR, Ref. 13, Table 4.2.1-1). Therefore, the nominal and
the extreme tolerances are included in the calculations. The diameter
of the fuel pellet remained unchanged, whereby the cladding dimensions
were altered (at constant thickness) to allow for the differing cold
gap widths.
Parameter Value
Fuel Density 95% T.D.
Fill Gas 100% Helium
Cladding Material Zircaloy-4
Fuel-Cladding SurfaceRoughness (arithmetic mean) 1.5 pm
Fuel Pellet Radius 4.782 mm
Cladding Thickness 711 m (Nominal)
Cladding Outside Radius 5.568 mm 5.588 mm 5.608 mm
Cladding Inside Radius 4.857 mm 4.877 mm 4.897 mm
Cold Radial Gap Width 75 pm 95 pm 115 pm
C.2 Outside Cladding Temperature
The following data from Yankee Atomic Electric Company is used in
developing an input model for the cladding surface temperature as a function
of linear heat generation rate.
-74-
Parameter
Active Fuel Length
Core Average LHGR
Average LHGR for Peak Rod(average x 1.45)
Axial Peaking Factor(peak/average)
Peak LHGR for Peak Rod(28.64x1. 25)
System Pressure
Fuel Rod Pitch
Coolant Passage EquivalentDiameter
Coolant Velocity
Coolant Inlet Temperature
Coolant Average Temperature
Value
3.472 m
19.75 kW/m
28.64 kW/m
1.25
35.8 kW/m
15.5 MPa
14.73 mm
13.55 mm
4.63 m/sec.
282°C
304°C
Other parameters consistent with the core average temperature at
100% core power are derived below. From enthalpy data which is presented
in Fig. C-1, the specific heat capacity of the coolant may be deter-
mined (Ref. 17). (All thermodynamic data presented in this Appendix is
for water at a pressure of 15 MPa. The error associated with this de-
parture from the system pressure of 15.5 MPa is expected to be insig-
nificant.) Instead of evaluating the specific heat directly at the
core average temperature of 304°C, it was decided to make the calcula-
tion at the inlet and outlet bulk temperatures. At 100% core power,
the temperature rise from the inlet to the core average is equal to
the rise from the core average to the bulk outlet temperature for a
nominal channel. Thus,
AH
(C) = wpw AT (C-l)
where AHw = H20 enthalpy rise
= (1.4905 - 1.241) MJ/kg = 0.249.5 NJ/kg;
AT = H20 temperature rise
= (326 - 282)°C = 440C;
i00o
00 i
Ittt1-% , +1
oC) aa
I q
-0
04
00
3-b
CH9
It
D H
i
C3,0
qa&OU
C4U 0£ . . ..
(t2/rw) asNiHr XAdIc&
-75-
T-
-76-
and (Cp) H20 specific heat = 5.67x10 3 MJ/kg-K.
The coolant mass flow rate is also derived for nominal conditions.
Here,
q' Icore
m A (C-2)w
where q' = core average LHGR = 19.75 kW/mcore
Q2 = active fuel rod length = 3.472 m;
AH = H20 enthalpy rise = 0.2495 MJ/kg;w
and m = coolant mass flow rate = 0.275 kg/sec.
Using the Dittus-Boelter equation with Maine Yankee and published
thermodynamic data, the bulk-cladding heat transfer coefficient may be
calculated (Refs. 17,18). Rearranging the Dittus-Boelter equation
h= 0.6 08 (Cp)w0h = 0.023K (Vp ) 0.4 (C-3)0.2 ( w ()
e
for H20 at 3000C, 15 MPa pressure
where k = H20 thermal conductivity = 0.559 W/m..K;w 2
De = equivalent coolant passage diameter = 13.55 mm;
V = coolant velocity = 4.63 m/sec;
pw = H20 density = 726.2 kg/m 3;
and (v)w = H20 viscosity = 9.17 x 10- 5 kg/m-s results in
h = bulk-cladding heat transfer coefficient = 33.2 kW/m --K.
The bulk temperature rise across a coolant channel may be ex-
pressed as
ATb = (C) (C-4)p w
-77-
q' = average LHGR of a fuel rod in the coolant channel
the other parameters as defined above.
ATb = bulk coolant temperature rise = (2.23 kW/) q' (C-5)
The following convention is used for the various linear heat generation
rates
%C Pq, = 35.80 kw/m ~l00
qominal = 24.69 kW/m ( Pnominal,max 100
q' =q' /1.25max
(C-6)
(C-7)
(C-8)
where %C.P. = percent core power.
Using Eqs. (C-4) through (C-8), Fig. C-2 displays the coolant behavior
as a function of core power with a constant inlet temperature of 282°C
and a saturation temperature of 344.80 C.
Correlations for bulk temperature and outside cladding temperature
as a function of axial position are given below
Tb = Tbb b-inlet
+ ATb [1+ sin( ex) sin( ex)
where
and
Thus,
(C-9)
-78-
0 0 0'0o( ) gaa N a
o0'-4
-4t
'90
iU)'-4 0o 0)
aa0 0o I 0
0o t4 0u c
1z I
oTo
IVO +-
0m0Of *r-~N
00)
.1.Cg0I0
arx.
rz.
-79-
q' max ( f)T =T + cos (C-10)To Tb + 2wr h (C-10)
o ex
where z = axial position (z = 0 at core midplane);
£ = active fuel length;
= extrapolated zero flux fuel length;ex
h = bulk-cladding heat transfer coefficient;
Tb = bulk coolant temperature;
Tbinle t = inlet bulk temperature;
ATb = bulk temperature use across channel;
and T = cladding outside temperature.
Assuming an axial cosine power distribution and an axial power
peaking factor of 1.25, the extrapolated zero flux fuel rod length
may be determined by trial and error from the equation
1/2
qax cos ( ) dz
q' -Z/2 ex
* - Q/2
f dz
which results in
2 = 4.82 m.ex
With this value and others previously given, the bulk and outside
cladding temperature profiles reduce to
-80-
sin (Q) 1Tb Tbinlet + 1 ATb 1 + ex (C-12)
and T = T + 0.628 q' Cos( (C-13)o b max
ex
where q' = maximum LHGR of the fuel rod in kW/m.max
For conditions when the coolant is undergoing nucleate boiling,
the outside cladding temperature is determined by the Jens and Lottes
correlation (Ref. 19). For the Maine Yankee Parameters, this relation
reduces to
(To)& L 344.8 + 0.845 [qma cos( Z)] 0.25 (C-14)ex
where q = maximum LHGR in kW/mmax
and (T )J&L = outside cladding temperature in °C.
Figures C-3 and C-4 show the bulk and outside cladding axial
temperature distribution at 100% core power for a nominal fuel rod and
a hot fuel rod. At 100% core power, the nominal rod does not undergo
nucleate boiling while the hot fuel rod does.
For various core powers, Figs. C-5 and C-6 display the cladding
outside temperature as a function of linear heat generation rate for a
nominal and hot fuel rod. Superimposed on each figure is the corre-
lation used in the study for the cladding outside temperature. Even
though based upon an arbitrary choice, this correlation closely re-
flects the actual operating conditions. In mathematical terms, the
outside cladding temperature correlation is
-81-
I I I . I I I I I I
to
c a
0
N T C4
.. L .. I .. . I 0
(Do) a y
o0CN
0
Z0H
; ¢H
PS
00
0
0.C
P4
H
C
rz4
0'0
_ __ · __ __ ·
1
-82-
I I I I I I I I I
cn
d vP
D E:
g a H. X
_
.14)
0 0 0CN co
(D,) 'amfIlY,.Wa
-
4
0 oH P
00
.4oo
; 80
-I
0
.6t
0
el
r
- t I - I I t II
td)Nwi
-83-
0 0 0(.o,,)mmlvnn
0'%0
0
4°EDo SZ to
S 0
=,R
-84-
O 0 0:Et~ - _I om ' C C-4
(3o) Mmlyax;'
0
0o
§ o.-% p 4w IO
,'% E* q B Ze o
H
H 0a .e
1 0o U. #
o
Z '
0o ai' :
0
0
-85-
T = 282 + 2.6 q'T
T = 347so
for 0 q' < 25 kW/m
for q' > 25 kW/m
T = outside cladding temperature in °Co
q' = LHGR in kW/m.
where
and
(C-15)
-87-
APPENDIX D
HALDEN FUEL ROD PARAMETERS
This appendix presents the Halden Boiling Water Reactor experi-
mental fuel rod and operating parameters discussed in Chapter 6 (Ref. 20).
These parameters also represent the fuel rod conditions inputted to
the calculational models.
Parameter
Fuel assembly
Fuel rod
Core active length
Axial power peaking factor
Coolant
System pressure
Coolant inlet temperature
Coolant saturation temperature
Fill gas
Fill gas pressure
Cladding
Fuel
% T.D.
Cladding outside diameter
Cladding inside diameter
Fuel pellet diameter
Cold radial gap width
Burnup
Value
IFA-226
AA
1.7 m
1.2 to 1.7
D20
3.45 MPa
237°C
240°C
Helium
10.1 kPa
Zircaloy-4
90.5% UO2, 9.5% PuO2
92.1
10.69 mm
9.51 mm
9.30 mm
105 pm
0 - 0.1 MWd/kg U
An outside cladding temperature has again been represented as a
two line straight function of linear heat generation rate. Using the
Jens-Lottes equation for nucleate boiling (as in Appendix C), the
following representation seems reasonable:
-88-
T = 240 + 0.6 q'
T = 255o
for 0 < q' < 25 kW/m
(D-1)
for q' > 25 kW/m
T = outside cladding temperature in °Co
q' = LHGR in kW/m.
where
and
-89-
APPENDIX E
COMPUTER PROGRAM LISTING
A sample listing of input parameters is presented in Table E-1.
Not a complete tabulation, this table merely presents an example of
the form for inputting data to initialize the computer program. The
variable names are explained below.
xmodel = the model designation, 1 being the uncracked model,2 being the cracked model
den = fractional theoretical density of the fuel
tco = outside cladding temperature in °C
q = linear heat generation rate in kW/m
gapc = cold radial gap width in pm
radco = outside cold clad radius in mm
radci = inside cold clad radius in mm
radfc = cold fuel pellet radius in mm
Table E-2 displays the computer output for one set of input data.
The input parameters are listed first which identifies the calculation.
These are followed by the parameters computed by the program.
Following Table E-2 is the listing of the Fortran computer pro-
gram used in this study. For a flow chart of this program see
Chapter 2.
-90-
ii . , s4 ni - 4 .: i _. 4 4 4 4 r4 _i 4 4i 4' 4 -4 4 4. 4
m mco m W C o c c 0 o O m 3 3 3 C3 -
.6 # * * + + - * * * * * 0 * * * * . .lq I t ... v IT q I I I I I I q t t T C v I I
en NNNNNNN 0 C' O O- . 0.'
F * * 4 * ATS IV q IT v Iq %
O 44,
I. 'l
'0
i0 G
in m n 3
N N N N N N N N N N N N r N 1 0 0 ON 101- C- 10% Dc 0c 0- 0% 0I 0% a -M 0% *I 0
NNNNNNNNNNNNNNNNNNN a a ( 0 0 * a * *4I 4 4T 4T 4T 4 4T 4 4 4T 4 4T 4 4 4 4 4 IT 4
00000004 40 0 o.40n 4 40
00 0 '0 '0*, 44
in in in
03 0 0 a 0 3 0 030 3 0000000000040 0 0 0 4 0 4 0 4 0 4.0 4
in in 101 m 3 1) in in in in
00000000000000000000000000oooooo o ooooo00* 0 * 0 c * 0 0 0 * 0 0 0 0 0 0 0 O0 0 0
I41
o
4
r-i
0000000000000000000000o 0 0 0 * * * * * * 4 4 4 4 4 4 * * *.. * + * 0n io O iO O n f O nl O O n OO0 - V4 %S _t4 V4 . t NY N m m M M f a a tm i 0 0 0 "a
· 0000000000004* * * O4 * * f * * 4
0 45 NiM mC f l t 11 ) 1 rN8 w m C- , -o 0 * rN rm mw -T -M - cP.n~~sww
ii) in i In b
s 0o 0000
0000000000a a o o o o a o a o* 4 * 4 *NNNNNNNNNN
q O v v iv 3 :; %r M q t3
0000N s Nr') r; t
10 3 in in n i n in in Ln Ln n i in in i7 in in in in in
00 0 O0 0 C O O0 C O- 0 -- O- OO O0 O0 O0 0 0 0
1
F41
Eq00 0000 00000000000000000- 4,k 4 4 4 4 4 4 4 4 _ N 4 4 4 4 4 4 4 4 4 4
WI CM1 W CM W CM WI CMt WI Cii WI Cii WI CM WI ~C~ W r ( N C ii Wf· · I~ Ci W Cii W
a
!4es
.-t
-91-
model 1 traditional model 2 cracked this is
fractional densit 0.9500
outside clad tem - 347,00 c
linear heat eneration rate = 40.00 kw er
cold aP width = 95.00 um
outside cold clad radius = 5.588 nmm
inside cold clad radius = 4.877 mr,
cold Pellet radius = 4781 mm
inside clad temrP 396,15 c
fuel surface tem = 419.54 c
fuel centerline temp = 1749.62 c
hot fuel radius 4+829 mm
hot aP width = 58.890 um
Percent Pellet contact 30.449
fractional radius for eeuiaxed rain growth
fractional radius for columnar rain Srowth =
fuel rod euivalent temperature = 891+86 c
2.0
0.o 456752
0. 164658
Sample output from comparative study computer program.
'
m
Table E-2.
-92-
i
t
03
r- oC
I40C.
-0 r-ii cL
c13 0O
* f-i 1-3
s 3 >t 3a-i m
-r -i .4
C(3LII'
3_
a + (3 (30 -0
C M
m u
rf 0
s C^ r- >0
13 C W-.c tai
' C- .q' 0 C3=..
U) 12s >
U S.g -3-o (~-JUo~r 3E 3 ia Ci
(3
-0'0ra
(3o
O'0;-Lft
* rC- a
0 0(.to..-- L@' 4i.-:
Ch oU
*1-4 -03 a
O0-*,i .. n.H r
F_ F_-
- O3 O
m il-f-i Ilin:.fJ O 13
viC0 ato ul!U) 4
Ci~ 3
0C
0
.100
.f E
t
a;
U)
t.
C-,-o C-, r 'i -r
C aC -. 4CC
r- 0t3 0 Ct C2 t al
E U -.-i 0 r. ' 0 i tO critlC-x Ust-a
r' 0 ! c -.. ' '
Ck O it O O'C.nu'- - V" C C-
f--if.I (3£
o Us _S i . 0
(3( (jC} .- .Cei 0.f XX Us t ototov;
(u O O O O O O u U CJ O ( 3 3 ( 3 ( 3 3 33tJO u £ u(
0-tO
0O_-S.
-J
C:
f3C:-t,
. 33
>-<-L
W-JC-04:2tI-
-'4
t
OC.-I
.+~Ct
Ir
-3
r%oi
(3
o
0 u u u u u u 0 u u u u
-II
4
0J1.c-
I
1
I C'j
CC
i + t'.
)J "to
il+ t +C
-0 l I- ,N UU t0e-*3 * iCi t r- C1tC Oi4
Nr r (Ni ( 3 : i
Cin P
UI 5-0 11 N 0t Ut I)5$ in 4 t
.r li .i .-4 -4 .- i.!
0
O (JL ,L
-93-
r0 o
r0
C u
0 0 *.i3CV S.
*0 . ttt a -'. f
.- -,r 03 0 ai
e m
O- a U U21 '2 * 2
Cra4
--I
rS:
Cs
-roU
i-
tM
C.CiCi
-074
to
aCt
U
0to'.
r
0OItof.
r
.O
t ;:
-O
* .
Cisr
O
Us m
0
n O
-4 -4
0' 0*
0I 0I 3
.el14,ILI
.
10'd
*'* 0(31r'4,.t
'.43
.
0IN O* "
S.
0 'X ra *,s "h
Nv a's i~1"5 + a .i
4. 4."a.'3 r~u ',~'( aii :'
r-4 gi -4"La '' -ij a
* ·4 · m
a· to0 It
4.-t?U
j w j 4,.
3 ' CiMi a
t 3 c -00w-i
t Ot CI .CI 0..
(a 07 U1 t ma a
0
0
U t J i
.I- --- A0 C-+ * i ON
.-
.9-
'.
0 -.4 . 0
0 -0 .
5- 5..5C.C . II~
-'0
* 11
-C Ctt4 00 'C't to;- -
* 0
-
tl .0 > .C .Cs.4
i X, .Oo -rq %.
C 0-r-
0-)'-4
EJ
-94-
U= i-I I
t fOO+
i * *
'L i r; UI .
N I f*
t-, C D
O- 11 E 10
0 ., * * 0
ctt >2t'* 4 4, 4
C-It, N C4 INo a -4L Li U U('4 C11 C-. (" 4
.
C 4,('-i I t i \V CTS C Li
o Oc3 O c+ + + 4
C 04 4)4)4t _0 Os .- s .~.-.**** .+ * 4 u
.+ . . + 'l g I i o.1 t: II If U `4 4ri 4ri IC.
N N n N Li Lm mr m 9* w
* m 5a t4 4, 44) ('i-i' si Lir-+ 1-
o 03 ·.4U r U., -.- 4-
f- i- i--,I .21 -ri
; Ii- !I
·rr- C-A Z V !
0.
0C?
.4to
f.
0
-4ri
r.UIC-It O
U !1F '
4) f V
00
r;C4.)C
X4
3-'
rIV4
.3-f
tCi
04-
E
C.
f,;X:
C"-3
U 'i O E.. LJ
.C
'riii1!
r3
o0~
41
ri!(0.C
-c
,-
'13ito
.C
.-4*0
larof0
vC.3-44,)co
3-.,
(-3
in
-C0
rU0(0C-.'-0
Oi
EJ ur t t U r
-95-
,0
,0
--
oi3 ""
.41:3 C j
C . -'i.:s v M;"a .3La~
--
3s
; 4-*
N a NI%. f-
* C-_ ~%
1, 1.
.,- i 40 - IC'-i 0 0s. v
C C:. C+.
9-
OC-
-oC-
..%t';
.- r
· '3 !,O- -
O +'.'-CC-L
0" vj
,
£ri:- .. t* ' .
r-'!
'0
.t-J
3
-o
.:
r'0
.t
3la
to
.i-
,
I n
--
r.t
',0i *
Il. 4, -.
· . '. .-1
·t . 11 iI ", v.-
_-"~ ' I -'Z
= O q- .s.. -tsi1'ff £ C, .
3-ri .
4. *
* *t C
*i r r
Iv .lin (* *-
. .'.... --. I , !'- - -
N (-4 sh --
' -u *.
t: 4 r
Os-- ' *I C-
O 0(--
£J J ( 0
r-i;S
0E
,t
4-
r.0
s_,t
r:.,O0eI
r3
I'+
V..
(4
Os
4-
a;f-1-i-
Cr
4,
-'
to
..
-i
'-to
i
f-
mCa
10
.ri
C -(-4 C
0 -. -P
I, II .
.t9
uCl O
-96-
.,
· -. ,,t:. 0
0-4 -1
'~0 ., "-4 0 I
rl O 4 -· 0 -tii -C + * > 0 P ; 7 s 4 . I 0 C ' *0 ', i 0 ! ,C:0 . o . - vt * a (N CJ
>m .4) C 0 4- 0 -Ir' C r4 ' -i
0 (0 - V v "JC O I" O
JC: r~ A m - -0 " 0nUa ' r N -t N-
7r1 C - - 3 - I 3 0 '- 0 * , 0L * C0 ( N a i i (im r* V.u U, 4 1 Z P H fn0 1 v '- O (i 00 ' ,i
IC i i; E J 3 · 1-P j
o. O .0 1_ '3 *j - -V'e- - j 4'
: m.- m v ' . 4 0 01) * 00 N -P _ t -!O ff sOt.- r 4 I
Ai£-O C Ni4IN *- NO H C ,*
,. i -. - 4 C 0 tl !1 (-'v -t- i vr-' . B r II '
U; 'f U , * ' 04 *L GI- Xtr - O2. ' 0 O.i Ci C , O .! _. 11 *r4 IIC ( _j J1 i ri d m O
UUf UO( U Ci Cs . @ C U _<£J Ot L, OU O Oi O
-97-
_L :
C CC.t .
s _
L .
x + .· v I g _t Gc- .- ' ..- t i *
o - t '=',' " O ,,, . 'io 0 * 0 t ' c
L - v - +n o R c
tn + "c C t.- .I ! .L3 !" * O ' ,' 3 m an Ct e i
0- + t C· 44 4C b-
T , - -. .: - o O . V O.,4-f3t - -- *34-!I C + I flu ~ I -i a '" -~.e.I. .Uo r ̂ t tv i , O .+ I ^.. - r* 0J
O I./3 ! C-i '.3 ' ,I CO It .~j C -C' ti .! * - .a .'1 'O E -If 'O f. -v-O i 3 r- If M o ID , - + r-. C E E . .,n *. .E O * -. · i , U * .,!4 , a u i v 'O4 O O X- tCi 01 * (Ii - C_t s Ct ~ IO '2* IC - .0r U D ° H ci-C 3 ^-3 u* ** C .0 CS ? : 'f XO v3 .H_ ;-N c(n · 4 I3 tH C' to w 4 c4-~
to et &t iv0 +*i '' I 4 + 3 0 ( ' 0 1 ' * 11i
rn o
-lC41i~11 * 110. _ " 02L0 0.e _C tu-i r e ,
-98-
+4 d_%
C
+ +4Q *c
O- fi .-* 4i
·-* * O"O C- CV '. O -': x, +
· 3 ' -h t - -* *4 * O
-E + .'2 ! {- ·i r * * ') * i1
U-
a* -. ri
*- +* - *,i
C 4
ul x. ('-I .* C0 '* '* 0 :;:
-0 , 2 4-I C' l ru !s _-'*j
s- ! ~ +
, i i ..r. r
0'-4
VOJO oi
-i
C-
a-s
-r0t
I-
.O-9C-
.Jj -
1.
4,'-3-P
'-4
Ca
o
r'
C0f.
O0
*1
3
4'
-C
C-o
.41
00C
-00
I 0
I 0
i -
4C- la *4
O !uLs
-99-
_
Nr r
at a. +O C
O O ..O 0
v
' r'O O-'C
.! , i
iC iC
ri C *H C.. ~Cv -,,*eo *o
.i . tiO 0 C -1- wn C7--t n+
'v -m a- C a 03 'I 0
UILO LICP1 n n
4.
1-.v-i
r4-
-,'t0.0.,
C
O
,-ptn;to)r-1-1W'i
,
.C)
..
"CI
f-
.C
4C
.. '31;E
r- . *r,
0 -t- C-
A ) -3 - 0f: tin i
E O) E f
0V3r
t:.
.I
Ep-i
oC00
-l0
.-
'rlV.-c
f-
.C
,
C
'CI
'tO
rI
v-IIti
C-i'O
T'-
-P
C-..I
-oC
s-i
V-
0 0 .ii Ii
.,-fl 1
* 4-
_ a· .-,G,..gI
u U u :U U UU
-100-
f-
Ci
_.. ' , Q-,* * *!~ ,O ** -* 0 r *
* *,-4h'J-' :* .O I S o iall NO I C, i t I I '' "iLC I.'nb'J· & I + * * , t i* . i L*
3 Ci ef i P 3 in in in Ln:3 C O * *;-l * n * in
ipX i; } \ f4-ur *IN l II
C1 I *- 4 · b r.11 V II- 10 I- '
W 3-Co3 . . v oo3 .v-4 v i r iC 4 Cf + 0 .r-f U -4 Ci r-4 i, *+N ++C O. ti L £ t -
inUi*CJ. *. u U C VC 4 , C - --OC - X -O ,N N N - C
L v co i (B .N - - . .> ' i HX- + _'-4 - - Cor I - I * N, - '4 C +3 - ' , I -\ 1, at* a C4 W D * t + >*. i *. ^ ~i'- . m % m I m C. ,
c! + @ t 's? * n+ +
C *is-&i: 1n Ti * .C COn W LO O W 3 o al lal(V WN*4,Ciin3Ci4,Ci i-3 tCiWCT1 C.- A. C. C.-
W-a)un - C , 4 _ C c lt- * wiI r- m . + + g -all a 1 "nhi t- I C'a4 'C* 1) r * CN -I C m C C C4C C. C
CO itoD *t VE o ru +- tOr toCit - r4 * * o *+i * +
C c f W A 4 b * ' , i + I i L - * ( 43 * *i * W . o -4 o N V _rt .£ ;: c a3 ££ * v 6 al CD3M CD Ql i- co ci *Ci *.* Cr4 * w4 A
N N > C o 4 * t 0k * C + IT '- 4e+ j _t 4 Y O at_ r 4 ¢i C 4 + r- i Y .
o -C i'Jt -C *t * + 0 .f O .1 } + St e U O -W I 11 * ti * * N l C O £1
4, 4 ¢ H W C -. ' (i C1 W t al + Ci 010J1 CI ( i Ci Vi 4 If r- *Q u ) 9, -' - t * l I- -:- i 4 C- y
W- Y1r 4 r4 L) .V * -P s- wi
C UCr
C't' rJ
-101-
* !%C0 I- Cii In
in In
r- =.O , 4
*..-,. C Ii-4-*
L* 4-
r- e4 Ct -O ( -O -4 - *,In +
i- e~ C- 0) }
* "0
i
in
'0 Ii
r~ a
,-
o .* 4-
0 0"'0 '0
1* * 4
' 0 *
.i
',0
4. i .,
· + ,-V q, V3
4..-
!*
ICiInWn
'0
a-I A
4
+' OCi .' iC * 0*
';
* 41 L:N3 -.
-1: - · ' *v .* , ..- .'- -- .v _ .
1 t *in C In
I In I
W sN0 - la
a CI a
0 0
0 0 :00W 0 0
pi Ci; C
V + V.C -
C C I
0 N )1a . LO O. o% D
pi LI LIL) a .
C- * o*04)n0 in Cio0*C3J CiC'JC02V 02 *4i 4 4e 4 .1123r4)('lJ4) 4
-4 - -4 I- -i -4 -4 -4 -4 -4
-o(0
- ,t
0
.C .C
! I
- ,C
": r..t
it INJ U
;ir o
4 4
r3f r.s
4 *
c- tt- *
4- _ D
ii ii i:1
** "03
%O If
4 "0
.4 - r.004e
C-CCl c
(0 ellO
s
"N "--4-1r
-*
t3
_ W
in
10 O
_ I'
r ,
3 4 4
v °t .
000
r _3
l O.-00
04 0 14-
Ci --i
.--i 4-'-ci
-102-
0 0oo ,+ .
0: 0 ', O-
OO-l
3 ,O~~~~~~~~~~~1 N, ,C C >.at a1 E ri rt
on On a.il:0 a ' a
- to 3t C-,
r ¢ ~ D ' X -3 M@ ~ ~ ~ ~~~~~~~~v r-' 1 a * O CO C' " C C -
c b, $10 i. m"1 O O' ', '~4a -0 C 1" a · , C . S
. 0 k' 3 3 r4,. C O- .- ( 3
tD .- Zr3 ,, · tD m h 3 U 0 0 .*.., U *r ;._ , -PB1 .C3* m ~ -. . .- W - 0 = = W
·J -CD _! '. E .J C0 J ' 'J 0 J
t3 W 0 c C3 o *^ - ir o M r-i V ' i C r4 pi m - 4
C C C C r4 &H O C 3 t3 r4 _ - M -P a£ X~~?3 U rt 3 3 fL 1)4 'P nL v XE'-. 4. il E, a * 4v*t D
(J O s H X C a w a X50 at Ca ri 'O? Ct 3 N i · 4l t 0 033 t3 a i t 19 t a , il at ;t C-t aC C -0 E : -J I I C.- - Mnal @ r . aj . 3 Fl t ;X~ ~~~~ C>:@ v s,,Ct Ct s' -D F r· arz ft " D
3 C) - X , e; -f QC J * @ 3 U, U: ua k at *rt C- * <E -C L *tt o~ a 4-· p 7X u
e t1 . 8- eF) 3 v +1 t) -> rf r3 o ;;
tt~~~~~ ~ ~ ~ fUs t3 wt .S U L3 tw3 t!H~
-103-
1.0
O--*C
4- - ' 00,
0t Z - i b
.
E -. )0 E - I I
C 4 0 C-"i f ; i C
q- .- _i C Cs
5s
C-
0
.=_ClsI
'4-
Ci
-0CfCotrc,
4-C3 0
-,t n'-(4
et
C
C-.
V - V-*Ct *<_C _3C - 0
- - .0 53 C rt Cs
5OI 4- 4.3 %C' -.0 Csl.p
00..
V4't-
'. Ci
-4
,00000: -h C '
IM 4i 4O 4
* ~Ini- .- .O4 . 4. 4 4.
* Cs Cs 4 '4'- r. - i r- r-i "i
0 * * 4 4 4
0£ N N N N N
II 4- C- - - -tN *e- .- I .- I -ri *u-
(i0 i 0 oui O0 ' O OU
,'-
,4
s-.t-
IciCs
'aO
r4
ib
.,ri
'-
i-
4
Ff-
0.
0
3
0C.-
T,
C
f.
0O ' C0O C.rS.P - -aai R C
.dC-
0C
3r'
if
0ci
.-
Irt-.
.r4
..J
0N
.J-i-N
Co
n0 ,n
FO ii
0 i 0
i i- "U
OOI
00 u
-104-
C11-4-, -t-, -* * N NN N * 0'
'-4 Ul 4S 'N !
-t M IN N 0
CM ' .N N
OC' , 0 *.,C 'o*l - *C- ) 0 O N i 0 JI t ix " 3
, AO Q 3 3 3 -' -e - 4 ,3 'i m ' 4-
*t.
N
013
0r~
0.-
M
3 4 4
3 II -3 it aS14) L
0
'O*C I
C+ i-
L-g
. -:,
I: n
r-.r-
-t ; In -. I.,. '"I '- -- ,' 0
.
C.r
.,
4 ·
-o
COC;
i
4-
.-mr -4
Z- f
CM
i -41
O, is. , *' i+ %-.
V0o *C Cs +ri
C C- _-4
_{ '- *i
i4
CC-
; 0{E
* S
-i
t-
0
0rC-
t-4
4-.43
.i
;Ci-r4
.,-C
. ;
4 -
4, - 4,-_. *-tf l r
·. -Or 0Ja '
s 4, 0. +. '1
ro *4 :3
3 cnS _ ,n
N-4
o 0 0 0 00'- C ' I.03 3IO Ij1) VI Ln
0 0u
C-
3C .4,-'C--I
00%0
u
*0 *- *
* ) .* 0- 4 * * rI t
'i 00 CM 5 n + H4)4: ) + - 0 '0"0 A4 0mm * * *0 C
C C. 0 'ti N
o f 3 C' C 40v + + .' I o; CM 4V 4
4 i! ' O i 0 C C 0 0 N O 0t- tE .; CM -0 .- -a ·- i M -
H. i m3 @1 0 m m 03 m 0 i
0 0 0 0 0 00C tr CM IT I1 . 1
0o3 . O03 0 0303
-105-
-4
-
-
to4
:2
4-
s.4
,4
m.3
C-:2r.
C0 EOJ
4-'.0al
0.I
as--r-
40
a-c0.-t3
o(3
(3
-
a-
i-
Ne,
104--*
,.
C ciW. C
n -0
·3 -a.
f- II i * 0:24- 0,j
IO,0-l
' - -"
., - X_ *-
4 :o _'. '
4;- - j
U 4-"' L iy ,_.
^ ii iCi m s,
000
O4 4
. 0i-Q
:;; - C"0 -r.i-4 4- W C" rt- ib
000 C
· i o v
-.
L.
3
04)
Z
'_
co
;0%04
ai0-0
il
C.
4
00'M -0
0No
N
CD
t- C
C-J ao 0J ~or-.o
-107-
APPENDIX F
TABLE OF SYMBOLS
DefiningSymbol Quantity Units Equation
T temperature °c,K
r radius m
6 radial gap width m
6' root mean square cladding-fuelsurface roughness m
%T.D. percent theoretically dense U02 none
D fractional theoretical dense U02 none
P fractional porosity of fuel pellet none
PF UO2 porosity factor (B-2)(normalized to 95% T.D.) none (B-5)
k thermal conductivity W/m.K
k effective UO2 thermal conductivity W/m-K (4-10)e 2
M mesh interval m (2-5)
ET thermal strain none
ET average thermal strain none (2-4)(2-8)
x fuel burnup MWd/kgU
Fb burnup factor none (4-5)
F fraction of circumferentialcracked pellet to claddingcontact none (4-5)
h zero contact pressure gap 2conductance W/m *K (4-2)
h contact gap conductance W/m .K (4-2)c
-108-
h gap conductance W/m *K (4-2)
g (3-2)
p density kg/m3
C constant pressure specificheat capacity J/kg.K
AH enthalpy change J/kg
SH stored heat J/m (2-7)(2-9)
(SH) constant temperature J/m (2-11)eq stored heat (2-12)
f(T) equivalent fuel rod J/m (2-13)temperature function
f'(T) derivative of the function f(T)with respect to temperature J/m.K (2-15)
T equivalent fuel rodeq temperature °C (2-10)
%C.P. percent core power none
q"' rate of heat depositionper unit volume W/m3
q'(LHGR) linear heat generation rate W/m
D equivalent coolant passagediameter m
z fuel rod axial position m
Z active fuel rod length m
e extrapolated zero fluxfuel rod length m
v coolant velocity along fuel rods m/s
(P)w core average H20 viscosity kg/m sw
m core average coolant massflow rate kg/s (C-2)
h bulk coolant to cladding 2heat transfer coefficient W/m *K (C-3)
-109-
ATb bulk coolant temperature riseacross fuel channel (C (C-4)
Tb bulk coolant temperaturealong fuel rod °C (C-9)
(T ) outside cladding temperaturefor the onset of nucleate boiling °C (C-14)
Subscripts
w H20
g fill gas
c cladding
f fuel
o outside cladding
1 inside cladding
2 fuel pellet surface
52 fuel pellet centerline
n radial increment index
-111-
REFERENCES
1. M. Ishida and L. Hosokawa, "Light Water Reactor Fuel Rod ModelingStudy at Hitachi, Ltd.", oral presentation at an EPRI sponsoredmeeting on Fuel Rod Modeling, April 1977.
2. K.P. Galbraith, "Pellet-to-Cladding Gap Closure from PelletCracking", XN-73-17, August 1973.
3. N. Fuhrman, V. Pasupathi, D.B. Scott, S.M. Temple, S.R. Pati andT.E. Hollowell, "Evaluation of Fuel Performance in Maine YankeeCore I, Task C", EPRI NP-218, November 1976.
4. T.B. Bailey and M.D. Freshley, "Internal Gas Pressure Behavior inMixed-Oxide Fuel Rods During Irradiation", Nucl. Appl. Tech. 9,233-241(1970).
5. R.M. Carroll, "Fission-Product Release from UO2", Nucl. Safety 4,35-42(1962).
6. P.E. MacDonald and L.B. Thompson, "MATPRO: Version 09, A Handbookof Materials Properties for Use in the Analysis of Light WaterReactor Fuel Rod Behavior", TREE-NUREG-1005, December 1976.
7. A.S. Bain, "Cracking and Bulk Movement in Irradiated Uranium OxideFuel Elements", AECL-1827, September 1963.
8. J.B. Ainscough and F. Rigby, "Measurements of Crack SinteringRates in UO2 Pellets", J. Nucl. Matl. 47, 246-250(1973).
9. J.T.A. Roberts and B.J. Wrona, "Crack Healing in U02", J. Am.Ceramic Soc. 56, 297-299(1973).
10. W.S. Lovejoy and S.K. Evans, "A Crack Healing Correlation Pre-dicting Recovery of Fracture Strength in LMFBR Fuel", Trans. Am.Nucl. Soc. 23, 174-176(1976).
11. P.E. MacDonald and J.M. Broughton, "Cracked Pellet Gap ConductanceModel: Comparison of FRAP-S Calculations with Measured FuelCenterline Temperatures", CONF-750360-2, March 1975.
12. P.E. MacDonald and J. Weisman, "Effect of Pellet Cracking onLight Water Reactor Fuel Temperature", Nucl. Tech. 31, 357-366(1976).
13. Shippingport Atomic Power Station Preliminary Safety AnalysisReport for Light Water Breeder Reactor, 1976.
14. O.R. Olander, "Fundamental Aspects of Nuclear Reactor FuelElements", p. 133, ERDA, 1976.
-112-
15. H.B. Meieran and E.L. Westermann, "Summary of Phase II of the EPRILWR Fuel Rod Modeling Code Evaluations Project Using the CYGRO-3,LIFE-THERMAL-1 and FIGRO Computer Programs", ODAI-RP0397-2,RP-397-2, March 1976.
16. M.G. Andrews, H.R. Freeburn, and S.R. Pati, "Light Water ReactorFuel Rod Modeling Code Evaluation, Phase II, Topical Report",CENPD-218, April 1976.
17. J.H. Keenan, F.G. Keyes, P.G. Hill and J.G. Moore, "Steam Tables,Thermodynamic Properties of Water Including Vapor, Liquid, andSolid Phases", John Wiley and Sons, New York, 1969.
18. M.M. El-Wakil, "Nuclear Heat Transport", p. 243, InternationalTextbook Co., New York, 1971.
19. J.R. Lamarsh, "Introduction to Nuclear Engineering", p. 346,Addison-Wesley Publishing Co., 1975.
20. E.T. Laats, P.E. MacDonald and W.J. Quapp, "USNRC-OECD HaldenProject Fuel Behavior Test Program: Experiment Data Report forTest Assemblies IFA-226 and IFA-239", ANCR-1270, December, 1975.