ML20011A428

Review of Densification Effects in Lacbwr

ML20011A428
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Site:	La Crosse File:Dairyland Power Cooperative icon.png
Issue date:	05/15/1973
From:	GULF UNITED NUCLEAR FUELS CORP.
To:
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ML20011A420	List: ML20011A419 ML20011A424 ML20011A426 ML20011A427 ML20011A428 ML20011A431 ML20011A435 ML20011A437 ML20011A438 ML20011A440 ML20011A445
References
SS-1085, NUDOCS 8110130339
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Text

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

1 SS-1085

,t' REVIEW OF DENSIFICATION EFFECTS IIi LA CROSSE BOILING WATER REACTOR May 15,1973

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Work Performed on Gulf United Project 2348 Contract AT(11-1)-1(56 Chicago Operations Office of the United States Atomic Energy Commission GULF UNITED NUCLEAR FUELS CORPORATION Elmsford, New York 6\\

8110130339 810929

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PDR ADOCK 05000409 P

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4 cA r-i TABLE OF CONTENTS 1

1.

INTRODUCTION.....................

3

2., OBSERVED PERFORMANCE OF BWR FUEL.........

3 2.1 LACBWR 3

2.2 Other BWR Fuel.....

3.

ANALYSIS OF FUEL DESIGN. CRITERIA...........

5 5

3.1 Nuclear Design....

3.1.1 Fuel Densificauon..............

5 3.1.2 Fabrication Effects..............

6

-(

3.1.3 Nuclear Peaking Factors...........

6 6

3.2 Fuel Rod Clad Creep Collapse 7

}

3.3 Thermal-Hydraulic Design...............

3.3.1 Critical Heat Flux..............

7 3.3.2 Centerline. Temperature............

7 4.

LOSS-OF-COOLANT ACCIDENT.............

13 5.

R E F EREN C ES.....................

17 h

APPENDIX A - POWER SPIKE. ANALYSIS........... A-1 Ii

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4 I:

i TABLES 1.

Creep Collapse Analy. sis Conditions for LACBWR Fuel Rod....

8 2.

Initial Temperatures for Fuel Assembly Heatup Analysis 16 A.1 Convolution of Spike Factor.with Power Distribution..

A-10 FIGURES 1

LACBWR Fuel Rod. Creep. Buckling Analysis..........

9 2.

Gap Conductance Zircaloy Clad Fuel

. 11 3.

Clad Tempera.tures Intermediate Break, Distilled Fuel

. 15 A.1 Effect of Axial Gap on Power Spike A-5 4

4 l

iii

c c

't s'

1.

INTRODUCTION The USAEC regulatory staff has reviewed data pertaining to fuel densification under reactor operating conditions. Their conclusions, presented in Reference 1, are based on observed shrinkage in fuel column heights, formation of gaps in fuel columns, and immersion density measurements on irradiated PWR fuel.

The principal concerns of fuel densification are:

1.

Local power peaking due to axial gaps in the fuel colamns.

2.

Increased linear power due to reduced stack height.

Clad collapse at the location of axial gaps in the fuel column.

3.

4.

Decreased gap conductance due to diametral pellet shrinkage.

These effects, if present and of sufficient magnitude, could reduce the power capability of the reactor.

f The performance of stainless steel clad fuelin LACBWR has been examined in light of the AEC revie,v of fuel densification. Re-evaluation i

l indicated:

l 1

There is no clad collapse.

l 2.

Fuel centerline temperatures are less than melting at 1250'o design power.

3.

The minimum critical heat flux ratio (CHFR) is 1.5 at 125%

i desip power.

4.

The loss of coolant accident was reanalyzed and the peak clad temperature would be less than 2300* F for the worst break conditions.

t 1

l

-..,_.,c.._

o Based on the analyses to follow, it is concluded that the postulated effects of fuel densification do not significantly alter the performance of

(

LACBWR fuel. Therefore, no modifications to the reactor operation are required to meet present license limitations.

1 l

t l

l e

2

o f

2.

OBSERVED PERFORMANCE OF BWR FUEL 2.1 LACBWR Two fuelinspections were conducted on the irradiated fuelin the La Crosse reactor to assess its performance during the first operating cycle. These inspections were made at average cycle exposures of 3274 and 11 110 Mwd /MTU.

Extensive fuel pin diameter measurements were performed during the first inspection. The measured values ranged from 0.393 to 0.400 inches compared to a nominal Ciameter of 0.396 inches. Tivere was no

(

correlation between diameter variations and the relative linear power at the location of measurement or exposure.

Some of the observed variation is due to fabrication tolerances

(*0.0015 inches) and crud buildup.

These results do not indicate any clad collapse and none was observed in both inspections when the fuel assemblies were scanned.

LACBWR has in-core instrumentation to measure flux profiles at different locations across the core. Measurements are obtained by in-serting the flux monitor into the instrumentation tube and making an axial traverse. The data obtained during the first operating cycle do not show any evidence of pr,wer spikes that would be associated with large axial gaps in the fuel columns.

12 OTHER BWR FUEL 2

GE has reported the behavior of BWR iuel based on extensive examinations of their fuelin several reactors. This experience is applicable to the stainless steel clad LACBWR fuel because the initial e

3 s

cold gaps are the same, allowing for the difference in thermal expansion between stainless steel and Zircaloy. In addition, there is less creepdown g

in the stainless steci clad because of its superior mechanical properties at c

temperatures of interest, so that LACBWR fuel has a smaller propensity for forming axial gaps.

The most sensitive measurements on axial gaps were obtained with neutron radiography. Examination of over 200 fuel rods has yielded a maximum axial gap of 0.18 inches except for some observations at t'm top of rod which were affected by rod reversal and handling. In no instance were large axial gaps and subsequent clad collapse observed.

C J

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4

1 3.

ANALYSIS OF FUEL DESIGN CRITZRIA 3.1 NUCLEAR DESIGN The maximum allowable heat flux for LACBWR at design poher is 2

410,000 Btu /hr-ft which corresponds to 12.4 kw/ft. The resulting hot spot factor for the fuelis 3.76 which accounts for variations in fuel fabrication, nuclear power distributions, and for densification effects.

3.1.1 Fuel Densification One of the effects of the postulated densification phenomenon is a e

change in the linear power density. Fuel densification, if it were to occur, would decrease the active length of the fuel column. This effect is compensated partially by axial thermal expansion. Accorcting to the AEC constralat,1 the percentage decrease of the fuel column due to densification is:

l AL _(0.965 - p + 2a)

L~

2 where p 6 the nominal fuel density (957o T.D.)

a is the dens'.tj fabrication tolerance (assumed to be 0.005)

The fractio ~tl change in length is 1.257o or an effective multiplier on linear power of 1.0125. The axial thermal expansion of a peflet with a linear power of 12.4 kw/f t is 1.17c. Therefore, the net effect on peak linear power is 1.0125 x 0.989 1.0014, which is negligible.

!t 5

A second postulated effect of fuel densification is the creation of an axial gap in the fuel column which produces a local " power spike." The

(

maximum m;ial gap in the power spike analysis was conservatively assumed to be 0.18 hches, consistent with Reference 2.

A conservative model (Appendix A) was constructed consistent with the assumptions of Reference 1. The analysis demonstrates that the power 0

spike penalty is 4.8% More precisely, if P is the peak linear power max without densification, the peak linear power with densification is 033x. At a 95% confidence level this peak power will be exceeded 1.048 P by less than 1 rod in the core.

3.1.2 Fabrication Effects Variations in dimensions allowed by fabrication tolerances can result in higher heat fluxes. These tolerances are combined statistically to obtain a hot spot fabrication factor for the fuel. Fabrication experience has shown that an engineering factor of 1.03 is greater than the 3cr value calcu-lated from fabrication measurements.

3.1.3 Nuclear Peaking Factors f

The allowable nuclear peaking factor (3.56) is the total peaking factor (3.76) divided by the statistical combination of fabrication and densificador. factors, as 1 + (0.048 + 0.03 = 1.057. Detailed nuclear analyses are performed during the cycle to insure that the control cd patterns do not result in nuclear peaks greater than 3.56. These calcula-tions allow for residual fuel burnup distribution; nonuniform void distribution and actual control rod patterns. Previous experience has shown good agreemen+ between measured and predicted power distributions in LACBWR.

3.2 FUEL ROD CLAD CREEP COLLAPSE A mechanical design analysis has been performed to determine if collapse of the LACBWR fuel rod cladding would be predicted during its lifetime. The analysis was performed by means of Gulf United computer program CREBUCK.

CREBUCK solves for the time-dependent creep deformation (uualization) of a long externally pressurized hollow circular cylinder t

d 6

having an initial ovality. CREBUCK is based upon the theory given in Reference 3, but is modified to include stainless steel creep formulations, g

including thermal creep and irradiation enhancement of creep based on m

References 4 throagn 7.

The loading conditions, geometry, and material properties used for the creep collapse analyses are given in Table 1. The analysis conditions given in Table 1 were selected to be very conservative. A pressure differential of 1300 psi is maintained across the clad throughout the analysis, neglecting any internal pressure due to fission gas release.

Temperatures and fast flux levels are far above those expected at the fuel rod hotspot. The clad dimensions used are nominal clad mean diameter, nominal wall thickness, and maximum ovality. The anaNsis includes effects of fast fluence on creep rate. The results of the creep-collapse analyses are shown in Fig.1 and indicate no significant change of ovality during the lifetime of the fuel rod. Therefore, collapse will not occur.

3.3 THERMAL-HYDRAULIC DESIGN 3.3.1 Critical Heat Flux The license limit on the critical heat ficx ratio is 1.5 at 125?o of design

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power. The critical heat flux is based on the Janssen-Levy design correla-tion using the average quality over the cross section of the assembly, assembly flow rates, nnd power distributions. The effect of postulated power spikes is accounted for in the nuclear peaking factors assigned to predicted and measured power distributions. Critical heat flux ratios are computed during each cycle as port of the nuclear design calculations.

The effect of a local power spike of 5?o acting over a short distance

(< 0.25 inch) is not significant on the critical heat flux. Therefore, accounting for postulated power spikes by increasing the local power peaking is conserva-tive in determining critical heat flux ratios. The minimum CHFR will be greater than 1.5 at 125?o design power.

3.3.2 Centerline Temperatures Fuel rod temperatures are dependent on power level and fuelclad i

gap conductance. GE has analyzed irradiation data for grain growth in low 7

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and hich burnup fuel to determine gap conductances. The conclusions i

based on these studies: are:

S' s

TABLE 1 - CREEP COLLAPSE ANALYSIS CONDITIONS FOR LACBWR FUEL ROD LOADING CONDITIONS Pressure Differential Achoss Clad., psi 1300.0 Mean Clad Temperature, *F 655 Flux Rate, Average Over Life, 2

13 N/cm -sec (E > 1 Mev) 2.7 x 10

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TUBE DIMENSIONS Clad OD, in.

0.396 Clad Wall Thickness, in.

0.020 Initial Ovality, 1/4 (OD

- ODmin), in.

0.0005 max 8

ces-7 mc

/N 3

M 1

0 i

0 s

1 p

0 X 0 3

0 55 7 3 6 2 1

=

s i

T oP s

y lanA gn ilkcuB 0

p 00 e

i 0

er 2

C srh d

,e o

m i

R T

leuF RWB CAL 0

0, 1

0 0

1

.g iF 0

7 6

5 S X5.,c50O4 5E o9 t a ":. O 7

__ f e

4

?

Ii!

i; 1!

j =

il!

di1

I

,l' 1.

Gap conductance increases with increased linear power.

2.

Gap conductances increase with decreased initial cold gap, f

3.

Gap conductance is independent of exposure burnups up to 10,000 Mwd /MTU.

l l

These conclusions were based on BWR fuel Zircaloy clad irradia-tions having gap.4hameter (g/D) ratios of 0.013 and 0.033 and linear powers of 5 to 20 kw/ft.

The nominal diametral clearance in LACBWR fuelis 0.006 inches and the pellet diameter is 0.350 inches. Stainless steel has a higher coefficient of thermal expansion (10 x 10 s in./in. - F) than Zircaloy 8

(3.6 x 10 in./!n.

• F). Therefore, the effective cold gap in LACBWR fuel relative to Zircaloy clad fuel should be increased by 0.0013 inches.

l The resulting g/D ratio for LACBWR fuelis 0.0073/0.350 = 0.021 which is similar to that for GE fuel.2 Because the g/D ratio (0.021) and the fuel density (95% T.D. in LACBWR fuel) are similar to those found in the gap conductance study,2 the results are applicable to the LACBWR fuel.

2 The results of the GE study are presented in Fig. 2 which shows gap conductance as a functia 2 of linear power and g/D ratio for Zircaloy clad fuel. The solid lines at g/D ratios of 0.013 and 0.033 represent GE's interpretation of the data. The dashed lines are logarithmic inter-polations for the g/D values of interest to LACBWR fuel.

Peak centerline temperatures at 125% of full power were calculated using the AL'C model for instantaneous densification to 96.5% T.D. The reduction in pellet ctiameter based on a nominal density of 95% T.D. and a standard deviation of 0.5% is:

+

~

AD =

- (0.35) = 0.0029 inches 3

l l

The resulting g/D ratio is 0.0102/0.36 = 0.029 and the linear power at l

125% of full power is 15.5 kw/ft. Using these parameters and Fig.2, it 2

i can be seen that the gap conductance is greater than 1000 Btu /hr-ft _op l

uhich was used in the original design calculationa.

1' i

10 i

i l

r f

4000 j

2000 l

[

g/D = 0.033 u.

g/D = 0.013

/0 f

-- l f f

/

=I 5 1000 f

f l

f T#

~

800 j

/

f o-

^

e

/

/l..

~

~

Sex

/

g/D = 0.02g, /

/

d g/D = 0.029 400

/l

/

300

% Data (Ref. 3)

= = = = Interpolation t

t I

0 4

8 12 16 20 Linear Power, kw/f t.

A Fig. 2 -Gap Conductance Zircaloy Clad Fuel 11 i

A The fuel centerline temperature calculated at 15.5 kw/ft is 3820* F.

which is below fuel melting. This is based on a gap conductance of 1000 Btu /hr-ft - F and a fuel thermal conductivity having 2

melt fg kdT = 93 w/cm.

A t

t X

1

' g l

I 4.

LOSS-OF-COOLANT ACCIDENT In order to assess the effect tf fuel densification on the maximum clad temperature following a loss-of-coolant (LOCAL. the fuel assembly hed.up calculations described in Sections 5 and 6 of Reference 8 were repeated for an intermediate size break (Tabb 6.2 of Reference 8) M this size break led to the highest clad temperatures. All parameters were kept the same as in Reference 8, except for the following:

1.

New, slightly more conservative radiation interchange factors 8

calculated by Gulf United were substituted in place of the original values.

C 2.

Heat generation in the cladding was added to account for the

!ncrease in estimated heat release during a reaction between cladding and steam.s 3.

The gap conductance between fuel and cladding was lowered from 2

1000 to 440 Btu /hr-ft

• F for allfuel rods. Thia gap conductance for densified fuelis obtained from Fig. 2 for g/D = 0.029 (see Section 3.3.2) and a linear power of 6.96 kw/ft which is the power of the center rod of the peak power fuel assembly with a LOCA peaking factor of 0.88 (see pages 5-50 of Reference 8).

2 With a gap conductance of 440 Btu /hr-ft

• F the average clad and fuel temperatures at the start of the fuel assembly heatup period (t = 5 seconds) for the densified fuel, as calculate.i by means of the ARGUS code, are shown l

in Table 2 with the values for normal fuel (gap conductance of 1000 Btu /hr-ft _op) 2 included for comparison.

s As in the or4h.al analysis, the high pressure coolant spray was assumed to reach the core at t = 20 seconds. The shroud wetting time calculated by i

k 13 5

I means of the Yamanouchi relation fee the densified fuelis 2.2 minutes,

'f which after addition of the 1 minute prescribed by the AEC interim criteria, lead to t = 3.5 minutes as the time of shroud wetting. The l

maximum clad temperature with fuel densification is cciculated to be 2296* F at a time of 5.25 minutes. The temperature history of the shroud, corner and center rod for the densified fuel are shown in Fig. 3.

The peak clad teaperature is still below the maximum allownble clad temperature of 2300* F proposed for stainless steel fuel cladding.

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f TABLE 2 - INITIAL TEMPERATUREE FOR FUEL ASSEMBLY HEATUP ANALY5iS (t = 5 seconds)

Normal Fuct Densified Fuel Rod

• Avg Avg Avg Avg Group Clad, *F Fuel, F Clad, *F Fuel, *F 1

940 1155 980 1370 2

940 1160 987 1387 3

940 1165 992 1415 4

941 1185 1011 1454 5

9 41:

1255 1052 1576 b

6 940 1160 992 1398 7

341 1170 1001 1422 8

943 1195 1018 1475 9

970 1260 1057 1590 10 942 1180 1009 1446 11 950 1210 1027 1500 12 980 1280 1072 1635 13 964 1250 1047 1565 14 1005 1340 1106 1721 15 1041 1430 1194 1882

• Rod group numbering scheme is shown in Fig. 5.24 of Reference 8 16

1 f.

(

5.

REFERENCES 1.

"USAEC Technical Report on Densification of Light Water Reactor Fuels," November 1972.

2.

Ditmore, D. C. and Elkins, R. B.: "Densification Considerations in BWR Fuel Rod Design and Performance," NEDM-10735, December 1972.

3.

Wilson, W. K.: "A Method of Analysis for the Creep-Buckling of Tubes under External Pressure," WAPD-TM-956 (October 1970).

4.

Howl, D. A.: Creep Equation for 10-12% Cold Worked AISI Type 304

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Stainless Steel Under Pressurized Water Reactor Conditions, TRG Report 1265(s).

5.

Walters, L. C., et al.: WADCO Quarterly Progress Report, WHAN -FR-40-1.

6.

Walters, L. C., et al.: Battelle Quarterly Progress Report, BNWL-1318-3.

7.

Lewthwaite, G. W. and Mosedale, D.: Irradiation Creep of Helical Springs in DFR, FRDC/FEWP/, P(70) 563.

8.

Technical Evaluation, Adequacy of Lacrosse Boiling Water Reactor Emergency Core Cooling System, Gulf United Report SS-942 (May 31,1972).

9.

Response to Questions by AEC/DL with Regard to Galf tinited Report SS-942, Technical Evaluation, Adequacy of Lacrosse Boiling Water Emergercy Core Cooling System, Gulf Urmed Report SS-1075 (April 30,1973).

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17

=

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APPENDIX A - POWER SPIKE ANALYSIS INTRODUCTION This appendix contains a description of the method used to obtain the power spike in a rod as a function of gap size. For the power spike

'n a ror' due to a gap in another rod, a perturbation method was used.

For the power spike in a rod due to a gap in the same rod, MONTE CARLO calculations accelerated by the use of a reciprocity principle were used. This appendix also contains the statistical analysis required to support the conclusion that at a 95% confidence levelless than one rod in the core will have a power den-ity in excess of the peak power density

{

calculated by applying a power s,.de penalty factor of 1.048 to the peak power density calculated without consideration of fuel densification effects.

The perturbation theory analysis used to determine the power spike in a rod due to a gap in another rod will be discussed first.

GAP IN ADJACENT ROD In a fuel rod with a gap in it, there is an increase in the thermal neutron flux in the vicinity of the gap, due to +he decreased thermal ab-sorption in the rod.

We neglect the effect cf the gap on the fast and epithermal neutron distribution. It will be quite small since the characteristic length for the 4

l diffusion of fast neutrons is the square root of the age, and the effect of l

the gap on the fast neutron distributhn will, therefore, be spread over a large region.

A-1

Consider, therefore a one-group thermal model with a slowing-r-

l down source S from higher energies. We have 4

- V Dvp + E $ = S, (1) a if diffusion theory is assumed. The effect of the gap will be caused pre-dominantly by the change in the thermal absorption cross section in the lattice cell containing the gap. (We use cross sections homogenized over the lattice cell.) If the absorption cross section in the perturbed problem is Za + 6S, then the perturbed fit.x, $', satisfies a

- V DV$' + (I + 6E ) $' = S, (2) a a

or

- V DV6' + 3 c' = S - 6E $'

(3) a a

l Thus, the flux, $', is a linear superposition of the unperturbed flux due to the source S, and the flux 6$ s $'-p due to the source -6E @ -

(.

(Note that -5S p' is positive because 6Ea is less than zero, since the a

a pre.sence cf the gap decreases the thermal absorptic.1 cross section).

Now the source

-6S $' depends on the unknown $'; however, a

6E $

6E ($

60 = E $ + 6E 6 &

(4) a a

a a

Thus, neglecting the second order term 6E 6$, we can approximate the a

source SI p' by 6E @-

a a

We are v.arking in diffusion theory; the point kernel for an infinite 1

I homogeneous medium is X(E,x') = '

(5) 4n D l x-x'l, where x =

Aa is the reciprocal diffusion length. This kernelis D

i A-2 i

appropriate for the case of an infinite lattice of rods, with cross sections I

homogenized over the lattice ce1L We, therefore, have

- xi x-il l

I 5$(x ) = f - 6E (E)$ (f ) 4nD,,a,xl d x' (6) 3 g

i If we approximate the integralby takin;T' at the center of the gap, and x at the center of the rod where we are interested in the power spike, we l

obtain from Eq. 6,

~Y ~ B A

e 6&(I ) * ~0E ( B) @

B)0Y (7)

A a

4nD[x xB (Recall again that 6I < 0, so that 6$ R ) > 0.) HereEA s the center of i

a A

the lattice cellin which we wish to know the power spike, and E is the B

(-

center of the lattice cellin which the gap is. Now $ G ) is just the un-B perturbed flux in the infinite homogeneous medium of homogenized lattice cells; dierefore, $ M ) " @ N ). The quantity 6V is the area, Acell ' UI B

A the lattice cell times the height of the gap, Lgap. We, therefore, have 5$(x }

~#!*A *

~

A e

= -6 E (_x

$(I )

a B cell gap 4;9 y _g k A

B

{

Eq. 8 yields the increase in thermal flux in rod A due to a gap in rod B.

l The increase in the power is somewhat less, since some of the power comes from fissions caused by neutrons with nonthermal energies.

l In the reactor being considered, no more than 88% of the power comes from fissions by thermal neutrons. This is the case for 0% void condition i

l and is conservative for other void conditions. One must multiply 6p/$

by f th, the fraction of power coming from the thermal group, in order to obtain the relative po'vcr spike, 6P/P, due to the gap.

A A-3 i

^

6P(I )

, -x x -IB

= {th (-6I (2 )) A L

(9) f a B cell gap 47tD E -2 P(T )

A A

B The factor

-x i -i g

g x -*B A

plays the role of a die-away factor. Once the power spike has been calculated for a rod adjacent to the rod with a' gap, the power spike in other rods can be computed by scaling downwards by the ratio of the die-away factors.

The numerical results obtained with this model are given in Fig. A.1.

(

A nearest neighbor rod to the rod with a gap suffers a power spike of 1.1% per centimeter of ghp. For a rod 8 pitches away, a 0.58?o power spike per centimeter of gap is produced. For a rod tw pitches away, a 0.27% power spike per centimeter of gap is obtained. For a rod (5 pitches away, a 0.107o spike per centimeter of gap is obtained.

GAP IN SAME ROD The computation of the power spike in the same rod as the gap is considerably more difficult. The reason for this is that transport effects are important. This can be seen by the use of the same model as used above, where the effect of the gap is treated by using a source

-6 E p. (This is valid in transport theory as well.) Now very close to a

a point source diffusion Weory grossly underestimates the flux (see page 200 of Reference 1). Consequently, just below the gap the use of diffusion theory will underestimate the flux.

One can use MONTE CARLO calculations to calculate the power 2

spike in the same ad as the gap. The reciprocity principle is used A-4 L

k

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g l

4 6

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&2 c Ti 8 9 Uzw 03 o

e 3 e d

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3e c

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c M

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3.5 ~ d o

M oa 1

T_ E.

a

. o d

g

=

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U 6.

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.5 ~ S o ~ W

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~

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O c) to n.

c c

M N

e O

O O.

O O

O.

O O

O O

d l

A -5 d/dp *c !as Jamod a

and the source ic put in the region where the flux estimate is desired. The application of the reciprocity principle to the acceleratica of MONTE 3

F CARLO calculations is due to Maynard and is also discussed in a book by Spanier and Gelbard.4 4

The use of the reciprocity urinciple permits the use of MONTE CARLO to obtain average fluxes ia small regions without running into the difficulty of lart,e statistical errors, as would happen if one tried to do the problem directly by MONTE CARLO. The use of the reciprocity principle is valid for one-energy-group fixed-source problems.

Cross sections averaged over the thermal group were obtained.

The geometry was simplified by treating the water region of the lattice cell as cylindrical and by homogenizing the materialin the lattice cells external to the lattice cell containing the gap. The sowce to ths.rmal was assumed proportional to the atom density of hydrogen in each regicn. The thermal flux averaged over a disc just below the gap was obtained and compared to the unperturbed flux obtained from a similar MONTE CARLO problem without the gap.

The height of the disc in which the average flux was computed was 0.05 inches and the radius was equal to the pellet radius. The MONTE CARLO program used was UNC-SAM-G.5 The ratio of perturbed to

(

unperturbed thermal fluxes was converted into a ratio of perturbed to unperturbed power densities by multiplying by the fraction of the power due to fissions caused by thermal nutrone. This fraction was conserva-tively estimated as 0.9. The results are sven in Fig.1. No credit for radiation loss of heat was tasen -these are true power spikes, not effective power spikes.

COMBINED EFFECTS The statistical analysis is a conservative simplification of the model presented in Appendix E of Reference 6. The analysis assumes t'aat the probability of a rod having a gap is unity. The probability of a gap in s i

particular axial segment is assumed to increase linearly with fuel roa height. The fuel rod, which is 83 inches high, is subdivided into 21 axial segments of 3.96 inches each.

A-6 I

=

The probability of a rod having a gap in the mth axial sci' ment is

~

therefore g

r m ", m - 1/2) 2(

3 (ID P

n For the core midplane (m = 11) one obtains P

= 0.0476. Although m

the limiting axiallocation occurs below the core midplane, the probability of a gap in a rod at the limiting axial location was taken as P

= 0.0476.

m Similarly, the maximum gap size at the core midplane is 0.18 inches.

Although the maximum gap size is linear with height, the maximum gap size at the axially limiting location was assumed to be 0.18 inches. The probability values for the various relative gap sizes are consistent with the.AEC probability values given in Reference 6.

Let S be the power spike (measured as a deviation from unity) in the jth rod due to the superposition of the effe s of gaps in the jth rod and in neighboring rods. The power spike S is a random variable since it is a function of the presence or absence of gaps of va:ious sizes in the jth rod and in the rods neighboring the jth rods, and since these

{

gaps are themselves distributed statistically. Furthermore, let PU) be the (linear) power density in the jth rod, at some axial node, whca fuel be the minimum der:sification is not taken into consideration. Let Pmax value of all P' satisfying the inequality.

Ipr {P(l)(1 + S )) > P'} 4 0.223 (11)

U J

Here pr {pO) (1 + SU)) > P'}

l is the probability that the spike factor SU) in the jth rod is greater than j

P'/PO) -1. The sum over j in Eq.11 is a sum over all rods j in the I

reactor. The rabimum value of I satisfying Eq.11 we have called max ; i is the peak power density inebding the effects of densificgon.

P The power spike penalty factor may be defined as the ratio Pmax/Pmax,

i A-7 i

f F

i..-- - -

)

where P is the peak power density in the absence of densification ax effects. The calculation corresponding to Eq.11 must be carried out for

,y i

the axial node which yields the most limiting value of P j

max -

1 l

"he left-hand side of Eq.11 represents the expected number of rods in the core wRh a power density less than P'. If this expected number of rods is less than 0.223, then it is shown in Appendix E of Reference 6 l

that, at a 95% confidence level, less than one rod in the core will have a power density greater than P'. The argument is based on the normal approximation to a binomial distribution, and involves the fact that if one is dealing with a binomial distribution where the probability of success on a single trialis p, then the expected number of successes in n trials is np.

The standard deviation in tise number, r, of successes in n trials is inp(1 - p) which is approximately equal to @ for small values of p. The probability of obtaining a number r:of successes in excess of 1.65 standard deviations from +he mean is 0.05. IIence, in order that the number, r, of successes be less than 1 at a 95"o confidence level, it is necessary that r = np + 1.65V np i 1, which implies that the expected number of successes j

np is less than or equal to 0.223.

The statistical model used assumes that the spike factor in a red j is the linear superposition of the spike factors due to each of gaps in the i

rod j and all surrounding rods. A gap is assumed to affect the power in a rod in a given axial segment only if the gap lies in that axial segment.

The probability distribution of the spike factors was computed as follows. The only rods which contribute significantly to the sum over j in Eq.11 are the corner rods in the assemblies. The reason for this is that the power in a corner pin exceeds by at least 'l% the power in any 4

l other pin in the same assembly.

We must, therefore, compute the power spike distribution only for l

corner rods. It is conservatively assumed that all rods other than the corner rod or its two nearest neighbors have gaps of maximum size. All seven gap sizes and their corresponding probabilities, as given in the AEC Table 4.2.A of Reference 6, are used for the corner rod itself. For the two nearest neighbors of the corner rod, a conservative simplification of the AEC gap size distribution is used. This consisted in assuming that a gap in a nearest neighbor rod is of one of two sizes, with probabilities consietent with the ASC probability valaes. According to the AEC probability tabic, if a gap occurs, the probability is 0.560 that the size of I

i(

A-8 l

. - - _. _ -. _. _ _ _. -. -, _. _.. _ _ _ _. _ _ _. _ _. _. _. _ _ _ _ _ _. ~ _ _ _... _ _ -. _ _ _ _ _ _

e

[

the gap is less than or equal to 0.572 times the maximum size gap. Thus,

\\

in the simplified model, if a gap occurs, the probability of the gap being F

just equal to 0.572 times the mtximum value was taken as 0.560. The l

probability that, if a gap occurs, it is of maximum size, was 1 - 0.560 = 0.4 P Let the probability of a gap of size k in tla corner rod be pk, let the probability of a lower size gap in one of the nearest neighbor roda be p,

p and the probability of an upper size gap be Pu. Then, for example, the probability of a gap cf size k in the sener rod, coupled with a lower size gap in one of the nearest neighbor rods and an upper size gap in the other nearest neighbor rods is:

pk X 2pI u 9

The spike factor cor:esponding to this distribution of gaps is:

Sk+SI+Su+A where.g is the power spike (again measured as a deviation from unity) are the spike factors corresponding to a in the corner rod, St and Su lower and an upper size gap in a nearest neighbor rod, and A is a con-(

stant equal to the spike factor for all rods other than the corner rod or its nearest neighbors, assuming Ulat these other rods all have gaps of maximum size.

It is now necessary to make use of the probability distribution of various spike factors and Eq.11 to determine Pmax, the minimum value of P' satisfying Eq,11. This must be done at the limiting axial location.

In the case considered, this was at the third axial node out of 10 axial nodes (0 ' of the way up the core). ThM part of the calculation correspond-to the " convolution" of the power distribution with the spike factor proba-bility distribution given in Table 3.6 of Appendix E of Reference 6. If one examines the power distribution at this axial node, one finds that eight rods have powers clustered very closely together about a power value cf about 3.34. The next highest power corner rods are a group of eight with a maximum power of 2.99. This is some 12?o different and means that these rods will not contribute to tl:e sum over j in Eq.11.

is 3.509, correspanding to Table A.1 shows that the vage ci P max a power spike penalty of Pmax /P

= 3.509/3.348 = 1.048.

m A

A'

f I

TABLE A.1 - CONVOLUTION OF SPIKE FACTOR WITH POWER DISTRIBUTION Probability of Spike Factor S )

Exceeding U

R guired tc Make This Spike P)

P lJ (1 + SU) ) = 3.509 Factor O

Rod Index j 1

3.348 0.0480 0.0357 2

3.346 0.0486 0.0226 3

3.342 0.0499 0.0226

(

4 3.341 0.0502 0.0223 5

3.338 0.0511 0.0216 6

3.332 0.0530 0.0216 7

3.331 0.0533 0.0216 8

3.329 0.0540 0.0112 sum = 0.179 A-10

\\

(

4

1

~

Table A.1 shows that the expected number of rods with a linear power density exceeding 3.509 is 0.1'i9. Since this expected number of g

rods is less than 0,223, we know that at a 95?o confidence leve11ess than one rod will have a peak power of 3.509. Thus the power spike penalty is 3.509/3.348 = 1.048.

r 5

e A-11 l

t

~. - '...

REFERENCES f

1.

Weinberg, A. M. and Wigner, E. P. : "The Physical Theory of Neutron Chain Reactors," p.217, University of Chicago Press, Chicago,1967.

2.

Bell, G.I. and Glasstone, S.: " Nuclear Reactor Theory," p.108ff, Van Nostrand Reinhold Company, New York,1970.

3.

Maynard, C.W.. Application of the Reciprocity Theorem to the Acceleration of Monte Carlo Calculations, Nuclepr Sci. and Eng.,10:97 (1961).

4.

Spanier, J. and Gelbard, E. M.: " Monte Carlo Principles and Neutron Transport Problems," Chap. 4, Addison-Wesley Publishing Company, Reading, Mass.,1969.

5.

Troube'tzky, E.S.: UNC-SAM-2: A Fortran Monte Carlo Program Treating Time-Dependent Neutron and Photon Transport Transport Through Matter," UNC-5157 (Sept.19S6) and Modification of UNC-SAM-2 to UNC-SAM--3, UNC-5157,(Supplement 1) (Jan.;1970).

~

6.

USAEC Technical Report on Densification of Light Water Reactor Fuels, November 1972.

h t

A-12 A

.