ML20008D765

App 3A to Midland 1 & 2 PSAR, Modal Analysis of Xenon- Induced Oscillations. Includes Revisions 1-36

ML20008D765
Person / Time
Site:	Midland
Issue date:	01/13/1969
From:	CONSUMERS ENERGY CO. (FORMERLY CONSUMERS POWER CO.)
To:
References
NUDOCS 8007300648
Download: ML20008D765 (17)

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v 6

A APPENDIX 3A MODAL ANALYSIS OF XENON-INDUCED OSCILLATIONS 4

l II;TE DUCTION A nedal analysis of xencn-induced oscillations in large pcuer corer has been cenple:ed.

Axial and scinuthal oscillations were investigated for beginning of life, flattened and slightly dished power distributions. The data were generated for a range of core paraneters typical of current central newer, cta icn (FpR) light-rater moderated reacter designs. This work was reported in 3AW-305.

The conclusions in that report have been verified by the recent analysis and are listed below:

1 l.

Ter a fixed dinension., the tendency toward spatial xenon escillaticns is increased as the flux increases.

I 2.

Fer a fixed flux, the tendency toward spatial oscillation is increased i

as the dimension of the core increases.

3 The lar6e size of current FKR designs permits an adecuate xenon descrip-j icn using one-grcup theory.

j h.

Flattened and dished power distributions have a greater tendency toward

'])

instability than ner al beginning-of-life distributions.

5 In sodal analysis of typical FWR designs, codal coupling can be ignored.

j In addition, current cores are not large enough to permit second-harnonic instability.

t Che large negative power coefficient present in large central station FWR e.

designs tends to danpen oscillations.

If this coefficient is sufficiently large, oscillations cannot occur regardless of core size or flux levul.

7 The critical dianeter for azimuthal oscillations is larger than the cri31 cal height for axial oscillaticus.

Current cores are not large encush to excite radial oscillations.

5.

Ex1 ination of the dianeter, height, and power ccefficient for a typical F,.? design indicates that oscillations shculd not cccur at the reginning cf life with unflattened pcuer distributions.

E:vever, the prchabili ;.

f :scillations at scne later tine is hicher since ccre depletien tends

flatten the pcwer distribution.

i 9

The period of oscillation (25 to 30 hours3.472222e-4 days <br />0.00833 hours <br />4.960317e-5 weeks <br />1.1415e-5 months <br />) is lonc enouch tc pernit easy l

cen:rcl of the oscillations if such control is provided in the design.

The n:ial analysis repcrted in Reference 1 has been reviced to include it.e aciers:cr temperature effect en the pcVer ccefficient and expanded c include g

the ccncept of stability index. These two quantities are defined a:.d are fcli; red t.. a discussics of the

• r--+ance of the scierator tenparc mr-e:-

b-

= friz _2n: in the axial and actru:hal ncces.

The acsunr.i:nz uce:: _. e:

THIS DOCUMENT CGfdAINS g ggOh[ 00%O 3A-1 POOR QUAUTY PAGES

'k recent modal analysis are discussed, and the uncertainties in the basic parsreters are defined in regard to their effect en the stability index.

Finally, the results of the axial end azicuthal todal analyre of the design

nder consideration are precented. The rodal cethods d

scribed in thir report uce cere-averaged quantitiet such as flux, power coefficient, and reactivity held by caturation xenon.

.s.-

.. n Cr tr t 12,..,1

-,7,.-.

vs In subsequent sections, the power coefficient is shown to have a strong influence on stability; if it is sufficiently negative, variation in these parareters is of nincr importance.

The power coefficient is defined at fcilows:

l co ) f BTf) f i-

)

BT \\

-cc ec

_=

+

=

g-cP

(

1

( c., /

i ')

(cr- /

cx c:

T 2

where T,. fuel terperature i

T = noderator temperature T

p = reactivity PT*Ew#

Equation 1 vill be used in conjunction with fuel and roderator terperature distributions to examine the reactivity changes asscciated with unifer:

Erinuthal and axial power shifts.

STABILITY iNDEX Stability index is as defined as 1

• y'( Xj x

-Y)

,1. ( A. + Ax)

=P x j cc

^+

2 1

2 (2) p,

- a,c j

g J

a vhere A

= decay constant of iodine-135 g

A

= decay constant of xenon-135 x

o

- ricrosecpic cross section cf xenon-1v x

-)

00341

A-2

l

<])

0

- product of unperturbed flux distribution and the square of the first mode buchling weighted over volume a

= reactivity per unit flux held by caturation xenon divided b3 x

core migration area X

= product of unperturbed xenon distribution and square cf first U

=cde buckling integrated over volume Yx

= yield of xenon divided by sum of iodine and xenon yields a

= pov r coefficients in units of reactivity per unit flux divided T

by core migration area t

2 p

= constant proportional to difference in fundamental and first 0

mode buckling It is possible to make varicus simplifying assunptions concerning the vari-ables in Equation 2 and to perform a parametric study.

This simplification j

is shown in one of the following sections.

In Reference 1 a ceasure of stability, b, is used, which is equal to -22.

)

An equivalent definition of the stability index which is useful in interpret-ing the results of digital calculations is

}

o -

z(t-t )

2r p_p

_g sin - (t - to) e U

(,)

3 i.

where P

= power at oscillation node g

t

= time of oscillatiens node ecuivalent to p o

o P

= power at any tine t u

= oscillation period Given the results of a digital calculation, the stability index can be fcund by fitting to Equation 3 These definiticns are frcn References 2 and 3 and can be ~ easily derived frcs wcr?. fcund in Reference 1.

MODERATOR TEMPEFARJRE Reactivity variations in the axial and atinuthal directions differ because of the differing moderator terperature distribution. The assurptionc rade are as fcllcus:

1.

The reacter is naintained at constant total pcwer.

4 2.

C:re inlet temperature is a ccnctant.

't-:-

g.,9 3 ry t' f '

\\j r

}

3 Core outlet temperature on the average is a constant.

4.

Coolant ficw is unifor.

j 5

During a xenon oscillation, symmetrical points have power changes that are equal but opposite in sign.

f.:cderator temperature increase is proportional to power fracticn.

c.

3 7

Puel temperature increase is proportional to power ratio.

I The ninor importance of the raderator coefficient in axial stabilit;. analysis is demonstrated by this simple derivation:

i Axial Gecretry Regic: 1 Region 2 Region 3 Region 4 i

P P

P P

1 2

3 4

i T'

5 Y

T" T

y 0

1 2

3 4

D f{='moderatortemperatureatplaceshownabove; 1 - 1, 2, 3, 4, where T is re inlet and T O

g is core outlet.

P

= power in Region 1.

4P1 = power change in Regien 1.

T{ = averaEe te=perature in Region 1.

P

= average power.

The initial fuel and =cderator terperature distribution is described as follows:

e

't "n"

A L

I P = 1.0 Tg-TO" T i=1 7" = T" + P C T' = 1/2 (7" + T"0) 1 O

1T 1

1

= 1/2 (Th" + ()

T=

+PC2T T = T* + P.,C

'T = 1/2 (T +T)

U 3

2 3 T 3

3 2

m T" = ] + P CgT T{ = 1/2 (T{ + 7 )

g F,

= Average Fuel Temperature Ir. crease at P e

T{=AverageFuelTemperatureinRegion l1 T' = T"1 +

1 p

FP

• 2 m

.3 T~ = T' +

2 2

y FP s

T3

= T" +

T^3 3

p Ifh "T=P+

b h

P Tne average core fuel and coderator temperature is fcund as follows:

F = Average Core Moderator Temperature C

_e T = Average Core Fuel Temperature c

h h

P = 1/4 E P

[ = 1/h I Y c

i=1 i

c i=1 1

3^-5 0071 d.

I h

? = 1/8 (2T=1 2 d + 2T + T* + T )

t c

e 3

0 4

= 1/8 (2T~ - 2P C + 2T"1 + 2P C + 2T~ + EP_ C c

3T T~4 + 7~-)

(L) 0 1T 2T v

+

= 1/L ( T ~ + T + T*) + 1/8 (T[ + T'0) + fT (P+P

+ P. )

0 1

2 1

2 f

T

=T+F e

c T

(5)

Equitien L shevs that for a syssetrical power distribution the core average tscerator temperature is the average of the inlet and the cutlet.

Tne avera.;e rederator tecperature decreases as pcVer shifts toward the inlet and increases as it shifts toward the o :tlet.

Let the pcVer distributien te perturbed by AP ; i = 1, 2, 3 k; then 1

N h

)

r-i-

? + L? = 1.0 1

i

=1 T = T * (P1 + AP )C = T*1 + AP C 1

O 1 T 1T T'

= T' + (P2 + AP )C = T~ + ( a?

AP,)C

+

2 1

2 T 2

1 c T t

T ' ~ = T ' + ( P + LP_ ) C = T + (SP + SP2 + LP )C 3

e 3 T 3

1

T m.g _- nt e

2g Tne fellering ecuations show that the avc age te perature at syttetrical points in the axial core direction changes as the power shifts, but that there is nc-net te perature change between the sytretrical points.

One power change is assured to te the negative cf the other.

T*1 T'

AP C

=

1T (6) 1

.)

00N5

A-C

d

')

T

-T (AP + aP )C

=

2 y

2 T (Il Tj'-T (L P + AP + AP )C (3)

=

2 3 7 9

n T

- T?. = T - T~ + C ( P_ - P )

3 c

0 T 3 1

e T1 T'~ = T:

C (LP 3

T 2 + AP_)

+

3 1

3 If AP2

- A P,

=

3 T: - T~1 T'~

T'"

=

3 3

1 (o) t Equaticn 9 shows that any reactivity change resulting fre a tederater te:pera-ture change during a xenen oscillatien at constant pcVer is conpensated t;. a similar change at a syntetrical point.

This can te shevn in a e4-4'n"

~="ner for any two sytretrical axial points or regicn averages.

Also the reactivity associated with fuel te perature changes behaves in a direct canner.

The preceding assumptions and conclusiens were checked by 1-D calculations

. utilizing nuclear-the("tal iteration techni ues. A chance in =cderatcr cc.effi-cient frem -0 5 x 10~

60 0F to +0. 8 x 10- op "F changed the power coefficient by +5%.

The icportance of the coderator coefficient in considering atituthal stability is shown by the following derivation:

5 average temperature in left half of ccre (unperturted).

=

5 average terperature in right half of ccre (unperturbed).

=

L v

=

T constant ccre inlet.

t

=

i T

ccre outlet at average full power ccndi icns.

=

P average power.

=

P power in left core half.

=

P pcVer in right core half.

=

N s'

00M6 3A-7

T

-T 4

2 P /P.

(10) o A

TL=T.+

1 L

5 - T'4 (assuming that the axial power shape is L

sy metrical).

Iet the power distribution be perturbed AP at ccnotant total power:

T

-T g

1 TL=T+

(PL + AP) 4 gur T

-T g

1 Yg=T1+

(P

-0

}

R 2P

+

T

-T o

1

-T R=-

AP 11

-T R

2P T

-T

+

o 1

AP 12

--TL=+

TL 2P T

-T T

o 1

(13)

-TR"+

P Equation 13 shows that the net temperature difference between core halves is nenzero in the animuthal direction.

Equations 11 and 12 show the cor-relation between a power change and the resulting terperature chan6e fcr average conditions or the radial slice at the axial ridpoint.

Also, with a positive moderator coefficient the reactivity changes ass ciated with a power shift vill be largest in the top cf the ccre and Ersilest in the bottcm of the core. With a negative moderator coefficient this is reversed.

In su-nary, the moderator temperature coefficient is very i pcriant in 1

f considering azimuthal stability, but of secondary importance in ecnsider-ing axial stability.

The partial differential of the moderator tempera-ture to the partial differential of the power used in correlating the moderator terperature coefficient for the consideratien cf ari.uthal sta-bility is approximately O.3 FK'/cc. Tne reference reactivity variaticn associated with a power change is -3,15 x 10-0 Ap/g t in the 177-fmel ele.r.:

i core.

tv 0 0s**),..A k,.f i

3A-S

')

Modal Analysis Parameter Study Details of the simplifications used in the stability paraceter stucy are given here.

Basic assu ptions are listed below.

Ass =ption 1.

=c For a flat flux ? /4 = 1.0 For a sine-shaped flux i /c 1.0667

=

3 0 Ascusttien 2 - c__

p

= Ie.v o

P = power density in watts /cc.

t If = cne-group core sacroscopic fission cross secticn.

-10 K = 0 320kO6 x 10 watt-s/fissicn.

?

H th f

+ I.,

f Assumption 3 -IfnIf t

o

~

I,h n thermal caercscopic fission cross section.

t r

I^f = fast tacrosecpic fission cross secticn.

~

= fast average flux.

T?

E7 = cne-group core sacroscopic fission cross section at reference enrich =ent.

Ti 7 Assunption k -If = I'f

~N E

E = core average enrichtent.

E

= reference care average enrichment.

5 0098 3A-9

O YT T'I Assumption 5 - a =--

x E,th -'a K

e contribution of thermal group to criticality.

Y

= sum of iodine and xenon yield.

T I

= average thermal absorption cross section.

The equation for a was fit in the following form for use in the parametric study:

- Y (8.469E + 70.672)(10~4).

a T

x Assumption 0 - -0=X 1

X 0

A /4 o +1 x 0x Assumption 7 - u h = 9.8696 A(axial) d k

critical dimension includin5 extrapciation distance.

A = shape factor, A = 3, sine; A = 1.0, flat.

Assumption 8 - p k = ll.626h A + 2 516 (azituthal)

VEK(10-6)

• I#

30 0

Assumption 9 -aT" ao f

ao 3",T power coefficient in units of reactivity per regawatt.

=

V

= active core volume.

Assumptions 5 and 8 are extensions of the procedures fcund in Reference 1.

The basic parase:ers used in the modal analysis are as fcilows:

A, hr' O.10368 y

A

-1 x, hr 0.07590 Y

0.062 1

Y C.C01 3A-lo 0034.9

If.,c=~

0.067586 at 2.65 vt %

4

, n/c= -s k.035 x 10 at 67 377./cc Kh 0 76Dh o

2 0.181 x 10 T o, c=

x I

0.025504 F.

th

-1 I

c=

0.090 a,

h D

c 0 395 2

t.

c=

53.;p 2

2 L, c=

4.12 i

1.22375 x 10" $

g Active fuel height of cores, c=

365 76 Equivalent dia=eter, 177-fuel element core, c=

327.k2SS Reflector savings, cm 15 I

Volume, 177-fuel element core, c=3 3 079781 x 10 s

1 3A-ll 00:~'.0

1 i

i-Ih STABILITY U7CERTAIIEY Possible uncertainty exists in stability because of uncerta.inties in the basic parameters.

I Uncertainty in the Doppler coefficient has been evaluated by cenparing cal-culations with Hellstrand's experitental results. An uncertainty cf 12.2%

i between the two is possible.

The following uncertainty correlatien allows i:

a possible variation of 13%.

I Several effects are involved in the fuel temperature uncertainty used in this j

analysis: (1) the relatienship between power density and fuel temperature; l

(2) the relaticnship between the effective fuel temperature in a fuel rod i:

cc pared to the distributed values; (3) the relationship between the effective l

fuel temperature and the change that would result from averaging spatial effects in the uncalculated directien. The power coefficient as used in this analysis has been calculated t-an effective value for the axial directien.

An un-certainty of 10% in the fuel te=perature is used to allow for the uncertainty j;

.in the power density correlation and distributive effects in the uncalculated direction. The. effective fuel rod te=perature is not considered because it increases the fuel temperature. The 10% uncertainty reduces the reference fuel temperatures f.o minical values.

j The yields ~of iodine-135 frc: the important fissioning isotopes are within I

' ! 3%. This' value is used as a possible variation. The direct yield cf

}.

xenon has been fcund to have a small effect on stability (a possible variatien 7

oro! 3% is allowed). The cucted value cf the xen0n cross section is within

!;h%, but a value of 1. 5% is used in the following analysis.

Available calculations of. the' moderator coefficient are within 5% of the ex-

~

l perimental values. : A possible variation of I 20% is used to allow for dis-tributive effects at power.

The para eters varied are listed in the --following table along with their effects on the stability index.

Change in Variation Stability Index Parameter

(

asl Loppler,LE/K/F

!3 (1) r Fuel Te perature

! ~ 10_

(1)

Iodine Yieldl 1 -3 1 0.005 Xenon Yield'-

1 30-

+ 0.001 Xenon CrcssiSection

!5

! 0.011 Moderater '.Ceefficient -

+ 20 L (1).

h*

[.10

! 0.01:

Peve- - istribution '

Tetal

0.027 0. 0*J '.

.:._ w r.

1 The parameters noted by (1) have uncertainties incorporated through the local

)

power coefficient as shown in Ecuation 14.

I l 9T I l

I f *$

I

+ ( 20%) If l a[m fp = ( 16%)

f C

( g^f) (

'T )

( #^s ) ( ##Tj T

The variation is applied to decrease stability.

Note that in the consideration of axial stability the second ter= is set to zero.

Axial Stability Figures 3A-1 and 3A-2 depict the predicted axial stability index for the lhh-inch active fuel height.

The first graph shows the results of using the ncrinal cr reference value for the tasic paraceters, and the second graph reflects the effects of compounding all the uncertainties listed in an earlier secticn.

For the 460-day cycle the following threshold conditions are predicted.

% Flatness

'EOL EOL Reference Paraceters 67 92 Compounded Parameters 30 47 Referring to Figure 3-37 in Section 3 of the PSAR, the axial power shape teccces flat and slightly dished over the cycle length considered where flatness is defined as the ratic of the length of the flat portion of the pcuer shape to the total dimension under study.

Axial instability in the 144-inch active fuel height core is therefore probable.

Part length centrol rods (XCRA) have been included in the design to provide the techanis: to thwart unacceptable axial peaking ccaditions.

The effectiveness of the XCRA in controlling axial xenen oscillations has been depicted graphically in Figure 3-8 in the PSAR.

A21 uthal Stability As stated in the introduction, a large negative power ccefficient tends to danpen oscillaticns.

If this coefficient is sufficiently large, cscilinticns cannot occur regardless of core size er flux level.

A negative coderator coefficient tends to increase the negative pcVer coeffi-cient and therefore increase the stability of the core.

In this design, the tederatcr c^ efficient at power vill always be negative starting with a value cf

-0.h2 x 10-Ak/k/ F ap rated power for the teginning of the first cycle and increasing to -3 x 10

• Ak/k/OF at the end of the equilibrium cycle.

The flat-ness of the core at the beginning of the first cycle vill be less than Shg, r%g vnich is the predicted threshold when the uncerteinties are cc pour.ded tc result

%j in the tinitur statility index.

The threshold 2:nditi:nc citained frc Pi;;re

(

_ A-3 are CL;vn t el v.

() [),,',..,,

3 O.*.

1 0 s

J s

'N

% Flatness EOL EOL (1)

Reference Parameters 83 Cc pounded Parameters Sh 72 (1)Pelow Threshold at 100% Flatness It is concluded that the core design under exm:ination vill not be sus-ceptible to diverging atinuthal oscillations even if the uncertainties of the basic parameters are compounded in the cost adverse manner.

--,---...,5 nstanta c 1.

Neuhold, R. J. and Clark, R.

E.,

Xenon Oscillations, The Babcock &

Wilcox Company, PAW ROS, September 1966.

2.

Control of Xenon Instabilities in Large FJR's, WCAP-3680-h, July 1967 3

Poncelet, C. G. and Christie, A.

M.,

The Effect of a Finite Time Step Length en Calculated Xenon Stability Characteristics in large FJR's, paper presented at Winter AUS Meeting, november 1967 e

\\..

00353 3A-lh

.. s.

+ 0.10

-0.05 100', ECL N

100': B0L 63.1. N

~

~~__---__'~~~~

t EOL f-

,./ ' s x '

0.0 l

-g E

39 4r E0L OL p

- G 19 6': Sh

~ ~ ~ _ __ _ _,

~ > -

19.6'. EOL

~~~~~--_._---_7__

~

' ~ - -4

~ ~ ~ ~ ~ ~ ~ ~ - - - - - ' - ~ - - - - - - - - -

0',

EDL 0.10

-0 15 300

-350

-400 450 500 550-f Core Full Poner Days (Design)

REFERENCE AxlAL.STADILITY INDEI VERSUS CORE ENRICHWENT FOR VAR 10'JS

' POWER SHAPES AT BOL AND EOL Figur e 31 1 1.

j-I 003.54

.]

+0.20 l

-0115 1001 EOL

-0.10

'd w

s s

~~s x[

j g ' " '

N

- a 'oL

+ 0. 0a s

s E

N

___}'

~

~ ~63. l' SOL E

%[39 45 EOL

].

s

~__

~-

~

00 i

N

39. 4',

20L

~~

19.6~ EOL

%' ~~~~~

19.64 SOL f

_. f' ' 05 E0L

- 0 05 0

EDL 2

~

~ ~ _ _, _ _

~~

-0.10 300 350 400 450 500 550 Core Full Poner Days (Design)

HIMIMUM AXIAL STABILITY lh0EX YERSUS CORE ENRICHMENT FOR VARIOUS P0nER SH APES ' AT. BOL AND EOL

).

.J '

Figure 3A-2 00355

t' S 0.15;

+0.10 0 05 E

/

E=

H.in BOL

/

}

- 0. 05 7

Min BOL g /

Re' EOL Ref BOL

- 0.- 10 0

25 50 75 100 Flatness,$

STABILITY INDEX VERSUS. FLATNESS BEGINNING AND END OF LIFE-2452 FAT (AZIMUTHAL)

F.i gu r e. 3 A-3 x,3.

..r.

~

00356