LWR FUEL PERFORMANCE ANALYSIS

i
TABLE OF CONTENTS
Topic
List of Figures
1. INTRODUCTION
1.1 Objective
1.2 Cracking Scenario
1.3 Crack Patterns
1.4 Major Thermal Effects of Cracking
1.5 Effects of Cracking on Fission Gas Release
1.6 Major Mechanical Effects of Cracking
1.7 Crack Healing
2. THERMAL CALCULATIONS NOT DEPENDENT ON CRACKING
2.1 Outline of Computer Calculations
2.2 Cladding Temperature Profile
2.3 Cladding Thermal Strain
2.4 Cladding Stored Heat
2.5 Fuel Thermal Strain
2.6 Fuel Stored Heat
2.7 Fuel Rod Equivalent Temperature
3. THERMAL CALCULATIONS FOR UNCRACKED FUEL PELLETS
3.1 Pellet Surface Temperature and Gap Conductance
3.2 Pellet Temperature Profile
4. THERMAL CALCULATIONS FOR CRACKED FUEL PELLETS
4.1 Pellet Surface Temperature and Gap Conductance
4.2 Pellet Temperature Profile
5. ILLUSTRATIVE EXAMPLE
5.1 Fuel Pellet Surface Temperature
5.2 Fuel Pellet Centerline Temperature
5.3 Fuel Rod Equivalent Temperature
5.4 Fractional Radius for Grain Growth
5.5 Gap Closure
5.6 Pellet - Clad Contact
6. EXPERIMENTAL EVALUATION
7. CONCLUSIONS
7.1 Relocation Into Gap
7.2 Reduced Fuel Thermal Conductivity
7.3 Fuel Rod Equivalent Temperature
7.4 Crack Healing
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ii
Topic
Additional Figures from Fuel Pellet Thermal
Calculations
APPENDIX B Material Properties
B.1 Fuel Thermal Conductivity
B.2 Fuel Thermal Strain
B.3 Fuel Specific Heat Capacity
B.4 Enthalpy Change for Fuel
B.5 Cladding Thermal Conductivity
B.6 Cladding Thermal Strain
B.7 Cladding Specific Heat and Enthalpy
B.8 Thermal Conductivity of Fill Gas
APPENDIX C Input Parameters
C.1 General Design Parameters
C.2 Outside Cladding Temperature
APPENDIX D Halden Fuel Rod Parameters
APPENDIX E Computer Program Listing
APPENDIX F Table of Symbols
REFERENCES
Page
89
107
111
73
73
73
87
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53
53
59
59
62
64
64
66
71

iii
LIST OF FIGURES
Fig. 2-1 Flow chart for comparative study computer program.
Fig. 4-1 Comparison of circumferential pellet-clad contact models.
Fig. 5-1 Comparison of cracked/uncracked fuel pellet surface temperature (cold radial gap width =
95 m).
Fig. 5-2 Comparison of cracked/uncracked fuel pellet centerline temperature (cold radial gap width = 95 pm).
Fig. 5-3 Comparison of cracked/uncracked fuel rod equivalent temperature (cold radial gap width = 95 um).
Fig. 5-4 Comparison of cracked/uncracked fuel pellet fractional radius for the onset of grain growth (cold radial gap width = 95 m).
Fig. 5-5 Comparison of cracked/uncracked fuel gap closure (cold radial gap width = 95 m).
Fig. 5-6 Circumferential pellet-cladding contact behavior for cracked fuel pellets.
Fig. 6-1 Comparison of computational models to
Halden experimental fuel rod data (experimental data from Ref. 11).
Fig. A-1 Comparison of cracked/uncracked fuel pellet surface temperature (cold radial gap width = 75 m).
Fig. A-2 Comparison of cracked/uncracked fuel pellet surface temperature (cold radial gap width = 115 m).
Fig. A-3 Comparison of cracked/uncracked fuel pellet centerline temperature (cold radial gap width = 75 m).
Fig. A-4 Comparison of cracked/uncracked fuel pellet centerline temperature (cold radial gap width = 115 m).
Fig. A-5 Comparison of cracked/uncracked fuel rod equivalent temperature (cold radial gap width = 75 m).

iv
Fig. A-6 Comparison of cracked/uncracked fuel rod equivalent temperature (cold radial gap width = 115 m).
Fig. A-7 Comparison of cracked/uncracked fuel pellet fractional radius for the onset of grain growth (cold radial gap width = 75 pm).
Fig. A-8 Comparison of cracked/uncracked fuel pellet fractional radius for the onset of grain growth (cold radial gap width = 115 pm).
Fig. A-9 Comparison of cracked/uncracked fuel gap closure (cold radial gap width = 75 pm.
Fig. A-10 Comparison of cracked/uncracked fuel gap closure (cold radial gap width = 115 m).
Fig. B-1 Comparison of EPRI and MATPRO UO
2 conducitivity.
thermal
Fig. B-2 Comparison of EPRI and MATPRO porosity factors.
Fig. B-3 Comparison of EPRI and MATPRO conductivity integrals.
Fig. B-4 MATPRO's version of thermal strain for uranium dioxide.
Fig. B-5 MATPRO's version of specific heat capacity for uranium dioxide.
Fig. B-6 Enthalpy change correlation for uranium dioxide.
Fig. B-7 Comparison of MATPRO and CENPD thermal conductivity for Zircaloy-4.
Fig. B-8 MATPRO's version of thermal strain for
Zircaloy-4.
Fig. B-9 MATPRO's version of specific heat capacity for Zircaloy-4.
Fig. B-10 Enthalpy change correlation for Zircaloy-4.
Fig. B-ll MATPRO's version of thermal conductivity for helium.
Fig. C-1 Correlation for the enthalpy change of water 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% core power.
Fig. C-4 Behavior of hot fuel rod at 100% core power
Fig. C-5 Comparison of nominal fuel rod outside cladding temperature to modeled outside cladding temperature correlation.
Fig. C-6 Comparison of hot fuel rod outside cladding temperature to modeled outside cladding temperature 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 operating 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 generation 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 mechanical and thermal consequences. This report primarily gives a quantitative 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 esimates 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 increasing power, further relocation results from thermo-elastic deformation 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

-4irregular 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 magnitude 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 significant 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 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 expansion of the fuel pellet is larger than the expansion of the clad;
- if the fuel/clad gap is nearly closed at the start of the maneuver, then contact forces between pellet and clad can develop as the power is increased (pellet-clad mechanical interaction, PCMI);
- fission products from the pellet (such as iodine and/or cesium), possibly freshly released, are available for chemical attack on the clad (pellet-clad chemical interaction, PCCI);
- the combined effect of clad stresses from PCMI and local corrosive action from PCCI can lead to clad failure from stress corrosion cracking (SCC); and
- SSC failures require the action of both a corrosive environment and a sufficiently high stress for a sufficiently 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 mandate 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 overconservatisms 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 provide 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

-6relocated fuel could play a role in fuel pellet conditioning. Time dependent 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 1700
0
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 material where previously a cracked surface existed. The local creep deformation of Sec. 1.6, on the other hand, could occur with a change in shape which does not affect the existence of the cracked 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 calculations 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 calculation. 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 temperatures 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"' 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.
(2-1)

-8heat
Calculate fuel rod equivalent temperature
Print out res ults
Figure 2-1.
Flow chart for comparative study computer program.

-9-
If internal heat deposition in the clad is neglected, Eq. (2-1) may be manipulated to give
T k dT c
=
27r
,-q' r n() r
T
(2-2) where and
To = outside cladding temperature; r = outside radius of cladding;
T = temperature at radius r; 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 equation gives
T(r) = -1.015x105 { 1.3959x10 [1.9485x10
+ 1.9704x10-5(1.3959x10-2T + 4.9261x10-6T 2 o o
1+ n(
27r r
} (2-3) 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
2 r
2 r r1
Jo (ST) r dr (2-4)

-10where c = average cladding thermal strain;
(ET) = cladding thermal strain at radius r (given in c Appendix B); and r l
= hot inside cladding radius; r0 = hot outside cladding radius.
To perform the above integration, the trapezoidal rule is employed, using twenty five equally spaced mesh intervals of size M where r
- r
M = 25 (2-5)
The calculation of the average thermal strain requires hot dimensions which are determined by rhot r + 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 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. However, the temperature profiles utilized in the computation are based upon hot dimensions. Therefore hot dimensions are combined, for simplicity, with cold density in Eqs. (2-7) and (2-9). It is felt that only slight error arises from this practice.

-11where SH = cladding stored heat; c
= density of Zircaloy-4 = 6550 kg/m and AH = enthalpy change of cladding at radius r (given in Appendix B).
Utilizing the trapezoidal method with 25 trapezoids, an explicit solution 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
-f r
2
2 f 2 (T) r2 o f r dr (2-8) where and of = average fuel thermal strain;
(T) f
= fuel thermal strain at radius r (given in Appendix B); r
2
= 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 determined for each iteration of the fuel temperature profile calculation 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
SHf = r
2
AHf DPf 27r dr (2-9)
*
See footnote accompanying Eq. (2-7).

-12where and
SHf = fuel stored heat;
D = fractional theoretical density p = fuel density = 10 980 kg/m
3 f r
2
= hot fuel pellet radius
AHf = enthalpy change of fuel at radius r (given in 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 instantaneously stopped. In mathematical terms
(2-10)
SHf + SH = (SHf) + (SHc)eq
At constant temperature, the stored heat equations become
(SHf) = DPf AHf(Teq) rr (2-11)
(SH ) c)eq
P AH (T ) (ro r) c c eq 1
(2-12)
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 (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

-13f(T )
T = T s s f'(Ts)
(2-14) where T
I
T f(Ts)
= slope intercept temperature;
= temperature where the slope is evaluated;
= 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 rr
Ip f
+ p c
(C (T)) c p c
(r
2 o r
1
(2-15) where (C ) = constant pressure specific heat capacity of UO
2 in Appendix B)
(given and (C
) c
= constant pressure specific heat capacity of the cladding
(given in Appendix B).
Beginning with an initial guess for T (fuel pellet surface temperature),
T
I is calculated by Eq. (2-14) and replaces T for another calculation.
Through this successive iteration, the fuel rod equivalent temperature is determined when T
I 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 uncracked UO
2 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
2g where T
2
= pellet surface temperature
T
1
= inside cladding temperature q' = LHGR r
2
= hot fuel pellet radius and h = gap conductance.
The gap conductance is determined by k h = --- g 6+6' where k = thermal conductivity of fill gas (100% helium given in g Appendix B);
6 = hot radial gap width; and 6' = root mean square cladding-fuel surface roughness.
(3-1)
(3-2)
3.2 Pellet Temperature Profile
The fuel temperature profile (based on constant volumetric heat deposition rate) is calculated from the following equation where
T
J kdT o
A
T
2
12 kdT + 1 X
1 r )2
I
T f kdT = conductivity integral for 95% theoretical o density fuel (given in Appendix B);
PF = porosity factor (given in Appendix B);
T = temperature at radius r.
and
(3-3)

-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 temperature profile is then computed from equations of the form:
T(r) = A + Bx + Cx
2
(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 accuracy of the above fit to the conductivity integral for 250 < T < 2800
0
C
0 is ±+0.4
C. 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
T < 585
0
C
(x < 3.441)
585 < T < 1000°C
(3.441 < x < 4.814)
1000 < T < 1405°C
(4.814 < x < 5.866)
1405 < T < 1800
0
C
(5.866 < x < 6.783)
1800 < T < 2205
0
C
(6.783 < x < 7.723)
T > 2205
0
C
(x > 7.723)
A
26.68
B
72.42
'C
26.08
126.7
12.82
35.01
40.26
46.48
31.75
-630.9
274.13
12.43
-1750.15
604.12
-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 determined by
T
2
= T
1
+ 2rrh
2g
However, the gap conductance for a cracked fuel pellet takes the form (Ref. 11)
(4-1) h = (l-F)h + F h g
0 c
(4-2)
.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/r
2
) (4-3) where 6 = hot radial gap width in mm; and r
2
= 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
F = 0.3 + [100() r 2
-1
+ 1.429] and for burnups greater than 0.6 MWd/kgU
(44)

-20-
F = 0.3 + [A(00) r
2
B
+ 1.429]
1
(4-5) 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/r
2 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/r
2
.
Due to the accumulation 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/r
2 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 dr
-q'r
27r
2
2 k
(4-7)
(47)

-21ox
0 O
DVYJAOD do NOIJDY& c -' cn a
UZ i o
01
14 rl
@
H
: 3
0
I e f
0
I 0 M,--

-22-
Using a finite difference method dT T(rn) - T(rn-Ar) dr Ar
Eq. (4-7) is rearranged to
Tn+ = T + q'r Ar
(kf)n
_
fl (4-8)
(4-9) 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 = U0
2 thermal conductivity at radius r .
Beginning at the surface of the pellet, the temperature at the first radial increment is computed using a temperature T
2 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 calculated 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 U0
2 thermal conductivity is employed in Eq. (4-9).
The effective UO2 thermal conductivity is (Ref. 12)
-3
3.175x0
-2 r
2
(0.077
-
+ 0.015) kf
+ 1] (4-10)

-23where and kf U02 thermal conductivity (given in Appendix B); k = fill gas conductivity (given in Appendix B);
6 = hot radial gap width in m; r
2
= 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). Calculations 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 extrapolations 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 consistently shows a higher temperature than the cracked pellet with a maximum 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 improved 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

-26-
)
I
Hi
Ol
.
3 o
0
4- 0
H c o b
V
U)
(Do) mQiVnaaWL n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~c-

-27surface 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 remainsunchanged 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 temperature. With increasing LHGR, the difference in temperature between the cracked and uncracked pellets increases. At 30 kW/m this difference 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 centerline 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-
0
.9
(Do) a tun~za so
0
B a) pQ
0
4)
0
'- g
= o k
Q)
H 0 o
~ r-t
H o S
H 0
0
OH orlj
U 0 o c,
.1

-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 equivalent 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 temperature 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 temperature for columnar grain growth, 1700
0
C, were arbitrarily chosen. However, 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 healing 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 fractional 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
4)'
0
4)
6
.
4 o
0 -q % o
Z
Q' o 0 r-I
0X o
NQ S 0 x
l g
.4
W.4 ..-
0
4
0
\0
(D ) I nsaJmIL
0
CN
O .4

-31-
\0
'* o
SnICIr INOIJDY
Olt2 o
To
0 o
O a
H 0 o
X
0
0
+4 al
-H
(
V:4

-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 temperature 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 undisturbed 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 proportionally 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 pelletcladding 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

-33o
C\
0 mflSO' d JCmImd
0 a
H
.l
0
0o o0
.
: C
H
Ii
I'
0'
0o t
4

-34cO '
CC~~~~~~~~~~:
O
C?
lDVyiXOD Q'aID JsTI?a lNDUrIa o o
\0
-2 iD fH
H
-
0
0
(D
0
A
0
0a
0
8e
H H o t o
E3
C.)
O
CJ H
O m;0

-35is 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 Boiling 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, calculated pellet centerline temperatures are compared to the Halden experimental 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 temperatures 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 inhibited by pellet-clad contact interference. Even though the model utilized in this study for cracked pellet-clad contact does not explicitly 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 determined centerline temperature at 40 kW/m. Assuming the same proportional difference in the Maine Yankee calculations, a 55
0
C 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-
2000
O o
1600 r4
A: pr;
I fr
ER
1200
800
400 o 10 20 30
LINEAR HEAT GENERATION RATE (kW/m)
40
Figure 6-1.
Comparison of computational models to Halden experimental fuel rod data (experimental data from ref. 11).

-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 U0
2 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 uncracked pellet by about 40
0
C. Regardless of initial gap size, the effects of internal cracks are diminished by the relocation improved gap conductance. Thus, cracked pellets always display a cooler outer temperature 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 temperature 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.

-42-
0 rv As
(o o ) miUnaman89a
,f%
0
' I o o
'; s 0-
Ii
0H
0

-43-
4)
4 i
4)
S
_s
'.i ' ml o oN a' o a
I
.I
1
4
S
FN r
Ma

-44-
4
N=(
-Wn I wa
(o,) =,Lvadm
4
4
0 a,
0
C)
-t
.:0
Ga
I
H
'd -Z
ER o e
.3
xr% e C o
0 *

-45o0 o
OQ
-I
(o, amaaam~t a
S (g
-I
Is a,
.4
0
0 e ad
0
Ore o i4
C f
OH. o

-46o
0 oo
I a,
Id g
O ' q.4 w .,
.
vo
(D ) aunlyaaaal o as Cs1.
0
OH
Oe
'IC)

-47-
A
0 o oto oo
4
X
Cp
P-
H
0
(D a
) avllvtyaal o0
0 o
CV
0
=d C o a

-48-
0
Ok a
E o
;s e
\0 siaO
4
SIGYE ~tNOIDYEtIL
C
CQ

-49-
0
It
;
SfnIa rIYINOIDYHA
N o
F-'
'O
0
-P k
E4
E- S
9
-4
V
H O
0
!I
4 t a
0
0
+'
0
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0
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LI
0
!~4
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9
0
I
-4
A n rl
%-
MS
0 i &
H s
H C) ri
.r
0
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-
-50o 0
K&ISO 'ID d1 O J c ed
0o
, i t~~~~~~~~~~~~~~~~~~~~~~~~~.
.1
P4,
0
_ 19 ii c,-
0
H m
I
0
0
0
0
.
0
. It
0

-51-
0
M8~S010 cTVD M
0
MIlc
-4 a ri
H rl
I a
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0
2 y
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0
I o
0
.a
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0
0

-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 concerning 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
PF l+P+10P
= where kf = U02 thermal conductivity in W/mK
PF = porosity factor, normalized to 95% theoretically dense fuel (dimensionless);
P = fractional porosity of fuel (dimensionless);
T = temperature in C.
and
MATPRO's version of the UO
2
0°C < T < 1650
0
C is (Ref. 6) thermal conductivity correlation for
(B-2) kf = PF f 464+T~~~~~~
(B-3) for 1650 < T < 2840
0
C

-54-
- 2 kf = PF [1.91 + 1.216x10
exp(1.867x1 T)] (B-4)
PF = (2.58-5.8x10
PF -4
(1-D)
1 (2.58-5.8x10-4
(B-5)
(B-5where kf = UO2 thermal conductivity in W/m'K;
PF = porosity factor, normalized to 95% theoretically dense 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, 300
0
C to 2000
0
C. Above this range, the MATPRO version increases more sharply than the EPRI correlation, thus resulting in a 4.5% difference at 2000
0
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
-5
- 8.5x10
+ 1.53x10
1 4 4
(T+273)
(B-6)

-55-
0'a 0 0l%
0IN 0
(IXU/R) LLIAIJ
1
DflUIOD 'IYMH L ZOfl
S 0IN r4
,o om
A
E4
I'
"l i
0 o' f.

-56-
A
, V-4
V4 a a1
V-4 aOWVY XIISOZIOd
0 0
IO
CO
'-4
(.4
H
C0
4
O% p4
H
E-4
0 a,
0I~
-H

-57where
0o
T f kf dT = conductivity integral in kW/m;
T = temperature in C.
and
From MATPRO for 0°C < T < 1650°C
T o f 464] 6.513x10
exp(l.867x10-3T) - 6.513x10
- 3 for 1650
0
C < T < 2840°C
T
(B-7) exp(l.867x10-3T) + 2.974 (B-8) where and
T o k dT = conductivity integral in kW/m;
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 difference will result in a discrepancy of about 50
0
C for a fuel centerline temperature calculation.
Due to itsgreater simplicity and ease of mathematical manipulation, it was decided to use the EPRI version of the UO
2 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 o
0
Ct
N
C, id
H
0 o
C o: .u
H
;
0
0 Rl
I
"O
_ rq
0 oV
F;J q
0
0 o
0
07
I I I
N ao
,I e- '3
(//M)
u'v lP
.
I m
I
N
*4
I I so o c m r%4

-59temperature 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 exceedingly burdensome equation. Also, a different reference temperature for the conductivity integral (i.e., 200
0
C) would reduce the calculated pellet centerline temperature differences. Thus, for consistency in the comparative study of cracked and uncracked pellets, the EPRI U0
2 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 c
T
= -4.972x10
- 2
2
+ 1.14x10l T
3
(B-9) where ST = UO
2 thermal strain in %
T = temperature in C.
and
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 calculations, a correlation for uranium dioxide specific heat capacity was taken from MATPRO. The following equation is also plotted in Fig. B-5.

-60-
~4- ~ n
K
MIY&LbS
SH r4l
'tywuawi a r e
,4
H
0 o 4
Ed l
Ed
IQ
H
0
0
0
H
.1

-61coO o
1'70 X)
(xs/r) LLIDYvJYD g
4H
DIAIDSdS
0
N
H
.i
qa§
I .
C'
.
4, a o
.4
IQ
AI
,a

-62p
535 285
85.005 exp( T )
T
2 535 285
T
2
1.6591x106
+
T exp
-18971
T )
-8
(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 U0
2 may be derived from the following equation
T
AHf(T) =
(B-ll)
T ref
C dT
P
Using the MATPRO specific heat capacity, U0
2 below for a reference temperature of 25
0
C.
enthalpy is calculated
AHf(T)
2
.158803 + 1.216x10-8T
535 285)exp( T- )-l
- 2
18971) - 3.2705x10
T where and
AH(T) = U0
2 enthalpy in MJ/kg
T = temperature in K.
Figure B-6 presents the UO2 enthalpy change in graphical form.
(B-12)

-63-
0
(9/rw)
0
IDNYHD xdrEvHma
0
0
U o
Id
U
0
.4
0
0
I
4)
.1

-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
(B-13) c where and k = Zircaloy-4 thermal conductivity in W/m.k
T = temperature in K.
The relation from CENPD-218 is k = 13.959 + 9.8522x10l T c
(B-14) where and k = Zircaloy-4 thermal conductivity in W/m.k
T = temperature in C.
The above two conductivities are presented in Fig. B-7. At 300
0
C the
MATPRO conductivity differs by 4.4% from that of CENPD. With increasing temperature, this difference decreases, where at 500C, the difference 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-
QN
CNl
CO w
(X. /A) LIAIM1nOD IYWdL
4
-I
0 co
4
0
.
0
0
,q
0 a o
0
0 c ' o
0
*--I
0 go
B 4
3a
. F

-66where and
¢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 35
0
C.
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 where and
(C )
= A + BT
AH (T)
C
= C + DT + ET
2
C
P
= specific heat capacity of Zircaloy-4 in J/kgK;
AH(T) = enthalpy change of Zircaloy-4 in MJ/kg;
T = temperature in C;
A,B,C,D,E = constant.
(B-16)
(B-17)
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.

-67-
_
I
__
I I I I I I I I I o f- .....
I .
.
MIYUIS
;
0o
I .
O
I _ I .
No -1
U tq
0r.
I
-I

-68-
0
-4 o
( o
(x.2/r) ;mIayo3 A DIdlodS
I
)
0 o o
0
0
0
N tq qI
0
< d
M
0 0
I
H
C'
0 of

-69-
O
0C
(t3/vr) SDMYHD X&IYHJM¶
N c0
0 o a
V
Is
EH a
<6
ED4

-70-
Range
25<T<127
0
C
126<T<367°C
367<T<817
0
C
817<T<820
0
°C
820<T<840°C
840<T<860
0
C
860<T<880°C
880<T<900
0
C
900<T<920°C
920<T<975
0
°C
T>975
0
C
Table B-1
Constants Used in Specific Heat and Enthalpy
Equations for Zircaloy-4 (Eqs. B-16 and B-17)
A B C D
275
287
295
-34200
- 3110
460
- 3860
- 3550
2890
7690
356
E
.21
.121
-7.67x10
.0978
-9.23x10
-3
3
2.87x10
2.95x10
-4
6.04x10
-8
4.89x10
-8
42.3
4.4
1.25
5.2
4.85
-2.3
-7.53
0.0
14.1
1.33
2.12x10
-3.11xlO
-3
2.2x10
- 6
.222
6.25x10
-7
2.6x10
6
1.68
-3.86x10
-3
1.55
-3.55x10
-3
-1.35
2.42x10 6
2.89x10
3
-1.15x10
6
-3.56
7.69x10
-3
6
1.72x10
-2
3.56x10
- 4
0.0

-71-
B.8 Thermal Conductivity of Fill Gas
A relation for the thermal conductivity of helium was also taken from MATPRO. The correlation is where and
-3 0.668
k = 3.366 x 10 (T) g k = heliunthermal conductivity in W/m.K
T = temperature in K.
(B-18)
The above equation is plotted in Fig. B-ll.

-72-
N O
/0A
(M N/M)
OD
IIAIMflUOD
4
AceZHJ
C
0 o o
,i i
0)
§
I !
0 o 0 x
M m
4) q-I

-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
Fuel Density
Fill Gas
Cladding Material
Fuel-Cladding Surface
Roughness (arithmetic mean)
Fuel Pellet Radius
Cladding Thickness
Cladding Outside Radius
Cladding Inside Radius
Cold Radial Gap Width
Value
95% T.D.
100% Helium
Zircaloy-4
1.5 pm
4.782 mm
711 m
5.568 mm
4.857 mm
75 pm
(Nominal)
5.588 mm
4.877 mm
95 pm
5.608 mm
4.897 mm
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
System Pressure
Fuel Rod Pitch
Coolant Passage Equivalent
Diameter
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 determined (Ref. 17). (All thermodynamic data presented in this Appendix is for water at a pressure of 15 MPa. The error associated with this departure from the system pressure of 15.5 MPa is expected to be insignificant.) Instead of evaluating the specific heat directly at the core average temperature of 304°C, it was decided to make the calculation 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) = w pw AT (C-l) where AHw = H
2
0 enthalpy rise
AT = H20 temperature rise
= (326 282)°C = 44
0
C;

-75-
T-
£
(t2/rw) asNiHr XAdIc&
.
0
..
It
D H i
C3,
0
It tt
1-% , a
+1 oC) a
I q
0 o
0 i i
0
0 b

-76and (Cp) H20 specific heat = 5.67x10
3
MJ/kg-K.
The coolant mass flow rate is also derived for nominal conditions.
Here, where and m core
A w q' core
= core average LHGR = 19.75 kW/m
Q2 = active fuel rod length = 3.472 m;
AH w m
= H20 enthalpy rise = 0.2495 MJ/kg;
= coolant mass flow rate = 0.275 kg/sec.
(C-2)
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= 08 (Cp)w 0 h = 0.023K (Vp
0.4 e for H20 at 300
0
C, 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; and pw
=
H
2
3
0 density = 726.2 kg/m ;
(v)w = H20 viscosity = 9.17 x 10
- 5 kg/m-s results in
(C-3)
The bulk temperature rise across a coolant channel may be expressed as
AT b
= (C) w
(C-4)

-77where and
Thus, q' = average LHGR of a fuel rod in the coolant channel the other parameters as defined above.
AT b
= bulk coolant temperature rise = (2.23 kW/) q' (C-5)
The following convention is used for the various linear heat generation rates q,
%C P
= 35.80 kw/m ~l00
(C-6) qominal nominal,max
= 24.69 kW/m ( P
100
(C-7) q' =q' max
/1.25
(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 344.8
0
C.
Correlations for bulk temperature and outside cladding temperature as a function of axial position are given below
T = Tb b-inlet
[1+ sin( ex) sin( ex)
(C-9)

-78-
'-4
0 0 N a 0 t
-4
'9
0
'-4 o
0) i
U)
0 aa
0 0 o I 0 o u
0 t
0 c
1z I o
To
O
0
~N
.1.
0 a rx.
rz.

-79q' max ( f)
T =T +
To Tb + 2wr h o cos ex
(C-10)
(C-10) where and z = axial position (z = 0 at core midplane);
£ = active fuel length; ex
= extrapolated zero flux fuel length; h = bulk-cladding heat transfer coefficient;
T b
= bulk coolant temperature;
Tbinle t
= inlet bulk temperature;
AT b
= bulk temperature use across channel;
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 q'
1/2
-Z/2 qax cos ( ex
) dz
* - 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

-80sin (Q)
Tb Tbinlet + 1 ATb
1
+ ex
1
(C-12) and T = T + 0.628 q' Cos( o b max ex
(C-13) 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
(C-14) ex where and q = maximum LHGR in kW/m max
(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 correlation used in the study for the cladding outside temperature. Even though based upon an arbitrary choice, this correlation closely reflects the actual operating conditions. In mathematical terms, the outside cladding temperature correlation is

-81-
_
I I
__
I .
I I
·
I
__
I
__
I
·
I
0 to c a
0
N T
C4 .. L
0
I
(Do) a y o
0
CN
1
0
0
H
; ¢
H
PS
H
0
.C
P4
C rz
4

-82r
0
-
I I I I I I I I I cn d vP
D g
H.
E:
X
_
.14) a el
-
4 o
H P
00
.4o
o
; 8
0
-I
0
.6t
0 td)
N wi t
(D,)
I
0
CN
'amfIlY,.Wa
- I t I II
0 co

-83-
0 0
(.o,,)mmlvnn
0
0
'%0
0
SZ
4° to
S 0
=,R

-84-
0
O m
:Et~ -
'
0
_I
C
(3o) Mmlyax;' o
0
C-4
B
,
Ze o
-%
§
H 0 a
.e
o
1
U
Z ' o
0
0
0 ai

-85where and
T = 282 + 2.6 q'
T for 0 q' < 25 kW/m
T = 347 so for q' > 25 kW/m
T = outside cladding temperature in °C o q' = LHGR in kW/m.
(C-15)

-87-
APPENDIX D
HALDEN FUEL ROD PARAMETERS
This appendix presents the Halden Boiling Water Reactor experimental 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% UO
2
, 9.5% PuO
2
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:

-88where and
T = 240 + 0.6 q' for 0 < q' < 25 kW/m
T = 255 o for q' > 25 kW/m
T = outside cladding temperature in °C o q' = LHGR in kW/m.
(D-1)

-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 q
= outside cladding temperature in °C
= 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 program used in this study. For a flow chart of this program see
Chapter 2.

-90a
!4
es ii m mco
4 , 4 s4 4 ni
.6 # * * +
lq I t ...
+
- 4 m
*
.: 4
W C i
*
_. 4i o c c
*
_i 4'
* v IT q I I I I I I q t
0 r4
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4 -4 4 4. 4 o O m 3 3 3 C3
* v C
* * .
.
I I en NNNNNNN
0 C' O .
0.'
F * * 4 ATS * IV q IT v Iq %
N N N N N N N N N N N N
1 0
0 ON 101- C- 10% Dc 0c 0- 0% 0I 0% a -M 0% r
NNNNNNNNNNNNNNNNNNN
4I 4 4T 4T 4T 4 4T 4 4 4T 4 4T 4 4 4 4 4 IT
0
4
O 4
4, i0
I. 'l
G
0000000 00 o.40n
0 '0 '0
*, 44 in in in
40 0 0 in in 101
0 4 0 4 4
'0
00* 0 0 * 0 0 * 0 0 0 0 0 0 0 O0 0 0
41
I
0000000000000000000000 o
..
0 0 0
-
* +
V4
* *
. t
* * * * 4
NY f
M
4 4 4 4 4 * 0n io O iO O n f O nl O * a
* O n OO * a tm
0 0 0 "a
· 000000000000
4* * * f * * *
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0
M
45 NiM mC
* t cP.n~~sww mw rN
4
a a o o o o a o a o
*
0000000000
4 * 4 *
0000
N q M t3 r') r; t ii) in i In 10
3 in in n i n in in Ln Ln i in in i7 in in
00
0 O0 C 0 O0 O0 O0 0
4 r-i
.-t
-
00 0000 00000000000000000
4,k 4 4 4 4 4 4 4 4 _ N 4 4 r
4
~C~
( N C
4 4 4 4 4 4 4
WI CM1
W
CM W CM WI CMt WI Cii WI
Cii
WI
CM
WI ii · Wf· Ci W Cii
W
1
F41
Eq o

-91-
'
2.0
outside clad tem 347,00 c
= 40.00 kw
m cold aP width = 95.00 um
=
radius
mr,
c
=
=
hot
um
fractional radius for columnar rain
= fuel rod euivalent temperature = 891+86 c
0.o 456752
0. 164658
Table E-2.
Sample output from comparative study computer program.

-92i
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APPENDIX F
TABLE OF SYMBOLS
Symbol Quantity Units x
F b
F h h c
T r
6
6' k k e
M
ET
ET
%T.D.
D
P
PF temperature radius
°c,K m m radial gap width root mean square cladding-fuel surface roughness m percent theoretically dense U0
2 none fractional theoretical dense U0
2 none fractional porosity of fuel pellet none
UO
2 porosity factor
(normalized to 95% T.D.) none thermal conductivity W/m.K
effective UO
2
2 thermal conductivity W/m-K mesh interval m thermal strain average thermal strain none none fuel burnup burnup factor fraction of circumferential cracked pellet to cladding contact zero contact pressure gap conductance contact gap conductance
MWd/kgU none none
2
W/m *K
W/m .K
Defining
Equation
(B-2)
(B-5)
(4-10)
(2-5)
(2-4)
(2-8)
(4-5)
(4-5)
(4-2)
(4-2)

-108h g p
C
AH
SH gap conductance density constant pressure specific heat capacity enthalpy change stored heat
W/m *K kg/m
3
J/kg.K
J/kg
J/m
(SH) eq f(T) f'(T)
T eq
%C.P. q"' constant temperature stored heat equivalent fuel rod temperature function derivative of the function f(T) with respect to temperature equivalent fuel rod temperature percent core power rate of heat deposition per unit volume
J/m
J/m
J/m.K
°C none
W/m
3
W/m z
Z e q'(LHGR) linear heat generation rate
D equivalent coolant passage diameter v
(P)w w m h m fuel rod axial position active fuel rod length core average H20 viscosity core average coolant mass flow rate bulk coolant to cladding heat transfer coefficient m m extrapolated zero flux fuel rod length m coolant velocity along fuel rods m/s kg/m s kg/s
2
W/m *K
(4-2)
(3-2)
(2-7)
(2-9)
(2-11)
(2-12)
(2-13)
(2-15)
(2-10)
(C-2)
(C-3)

w g c f o
1
2
52 n
ATb
T b
(T )
-109bulk coolant temperature rise across fuel channel bulk coolant temperature along fuel rod °C outside cladding temperature for the onset of nucleate boiling °C
(C-9)
(C-14)
Subscripts
H20 fill gas cladding fuel outside cladding inside cladding fuel pellet surface fuel pellet centerline radial increment index

-111-
REFERENCES
1. M. Ishida and L. Hosokawa, "Light Water Reactor Fuel Rod Modeling
Study at Hitachi, Ltd.", oral presentation at an EPRI sponsored meeting on Fuel Rod Modeling, April 1977.
2. K.P. Galbraith, "Pellet-to-Cladding Gap Closure from Pellet
Cracking", XN-73-17, August 1973.
3. N. Fuhrman, V. Pasupathi, D.B. Scott, S.M. Temple, S.R. Pati and
T.E. Hollowell, "Evaluation of Fuel Performance in Maine Yankee
Core I, Task C", EPRI NP-218, November 1976.
4. T.B. Bailey and M.D. Freshley, "Internal Gas Pressure Behavior in
Mixed-Oxide Fuel Rods During Irradiation", Nucl. Appl. Tech. 9,
233-241(1970).
5. R.M. Carroll, "Fission-Product Release from UO
2
", Nucl. Safety 4,
35-42(1962).
6. P.E. MacDonald and L.B. Thompson, "MATPRO: Version 09, A Handbook of Materials Properties for Use in the Analysis of Light Water
Reactor Fuel Rod Behavior", TREE-NUREG-1005, December 1976.
7. A.S. Bain,
Fuel Elements", AECL-1827, September 1963.
8. J.B. Ainscough and F. Rigby, "Measurements of Crack Sintering
Rates in UO
2
Pellets", J. Nucl. Matl. 47, 246-250(1973).
9. J.T.A. Roberts and B.J. Wrona, "Crack Healing in U0
2
", J. Am.
Ceramic Soc. 56, 297-299(1973).
10. W.S. Lovejoy and S.K. Evans, "A Crack Healing Correlation Predicting 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 Conductance
Model: Comparison of FRAP-S Calculations with Measured Fuel
Centerline Temperatures", CONF-750360-2, March 1975.
12. P.E. MacDonald and J. Weisman, "Effect of Pellet Cracking on
Light Water Reactor Fuel Temperature", Nucl. Tech. 31, 357-366(1976).
13. Shippingport Atomic Power Station Preliminary Safety Analysis
Report for Light Water Breeder Reactor, 1976.
14. O.R. Olander, "Fundamental Aspects of Nuclear Reactor Fuel
Elements", p. 133, ERDA, 1976.

-112-
15. H.B. Meieran and E.L. Westermann, "Summary of Phase II of the EPRI
LWR 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 Reactor
Fuel 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, and
Solid Phases", John Wiley and Sons, New York, 1969.
18. M.M. El-Wakil, "Nuclear Heat Transport", p. 243, International
Textbook 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 Halden
Project Fuel Behavior Test Program: Experiment Data Report for
Test Assemblies IFA-226 and IFA-239", ANCR-1270, December, 1975.