Core axial power distribution

8/10/2019 Core axial power distribution

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journal of NUCLEAR SciENCE and

TECHNOLOGY,

19[6), pp. 449-459 (June 1982).

Simple

Functional Method for

Calculating

Axial Power Distribution of PWR Core

Hiroshi TOCHIHARA

Mitsubishi Atomic Power Industries, Inc.

Received August 5,

1981

Revised November

2

1981

A

new,

far

simpler, method is presented to replace

the

laborious one-dimensional dif

fusion code calculations currently practised for deriving the axial power distribution through

reactor core-an important factor in determining

the

hot channel factor of PWR cores.

With the new method, the axial power distribution is approximated by a combination of

simple functions, using as intermediate parameter the axial offset , representing the extent

of imbalance in aggregate power output between the upper and lower core halves.

The new simplified method reduces almost to

l/50

the computer time required for deriv

ing the axial power distribution for any given condition of reactor operation, compared

with that

demanded by

1 D

diffusion code calculations,

with

deviation of resulting values

at peak power position hardly exceeding 2%. The proposed method promises to serve

usefully for plant-side calculations and analytical

treatment

of the axial power distribution.

KEYWORDS: PWR type

reactors

reactor core axial

power

distribution axial

offset functional representation xenon oscillation on-site calculat ion computer

calculation

errors

I. INTRODUCTION

449

The

axial power distribution through reactor core is an important factor in determining

the

hot channel factor and

DNBR

of PWR cores, and is currently calculated individually

to

cover various core conditions by means of one-dimensional

1-D)

diffusion code. Detailed

-calculation by this method is known to provide realistic axial power distributions under

various core conditions including xenon oscillation

oJ

czJ On the other hand, the recent

trend in PWR core, design is to limit the variation of axial power distribution during

pow r

change by

what

is known as constant axial offset control operation

csJ,

aimed

at

maintaining the hot channel factor within given limits, and this makes it important in

PWR operation to acquire precise knowledge of the axial power distribution and the axial

.offset. With this mode of operation, a requirement calling for operation in an unforeseen

pattern

necessitates knowing the axial power distribution in advance of the actual change

brought upon the power level and control rod movement. Against this necessity, the

-computers currently equipping the reactor plants are only capable of calculating the axial

pow r

distribution created under normal conditions of operationc

4

l but for other situations,

these computers are not provided to perform within the short time available the many

.sets of requisite calculations in so far as they are to be based on the 1-D diffusion code,

which demands much computer time as well as additional time for preparation.

The power distribution of a PWR is usually measured by in-core movable neutron

detectors only once or twice a month, whereas the axial offset can easily be measured

continuously by means of ex-core neutron detectors. It should therefore be most expedient

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110.

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450 ]. Nucl. Sci. Techno .,

if a simple functional method could be devised to rapidly derive the axial power distribu

tion from the information thus acquired on the axial offset. An

attempt

has been reported

of estimating the axial power distribution, from the signals of multi-positioned fixed in

core detectors through Fourier fittingc

5

J.

But this method cannot be applied to the case

where the reference information is supplied

in

the form of axial offset signals by ex-core

detectors.

The present report proposes a method by which the axial power distribution is

represented by cosine, exponential or similar simple function, with the axial offset adopted

as intermediate parameter.

The present method enables simple and timely calculation of the axial power distri

bution under various conditions of reactor operation solely based on axial offset data,

which are continually obtainable from ex-core neutron detectors.

The

application could

eventually be further developed into systems for monitoring power distribution and DNBR

in real time.

The following chapters present the theory of this simple functional method and the

results obtained from calculations therewith. Comparisons made with corresponding results

obtained with calculations by

1-D

diffusion code reveal the present simple method to be

capable of calculating the axial power distribution of a PWR core with ample precision.

ll

FUNCTIONAL METHOD FOR

REPRESENTING

AXIAL POWER

CONFIGURATION

OF

CORE

The current practice for approximating the axial power distribution of a PWR core

is to represent it (a)

at

the beginning of core life BOL) by a cosine curve with its peak

coinciding with the mid-height of core, and (b) beyond the middle of core life by a

double-peaked shape with one peak each in the upper and lower core halves. The

positions of the two peaks should not vary

appreciably with eventual distortion of the

axial power distribution by xenon oscillation

at any given burnup if a constant control

rod position is assumed. Under this assump

tion, the heights of the two peaks

vary

linearly with axial offset, which is defined

by

A 0 = Qr- Qs X 100 ( ),

Qr s

where

qr:

Integral power of

upper core half

q

8

:

Integral power of

lower core half.

(1)

A typical example of the relations

holding between the peak heights and value

of axial offset is shown in

Fig.

1, which

has been derived with calculations based

on 1-D diffusion code. Using this relation,

the axial power distribution could be

4

; ;

0:::

:I. 5

'

.c:

'

First Cycle,

9, 000

t1WD/T, at

Full

Power,

All

ROds

Out

0

+50

Axial offset ( l

Fig.

1 Relation between peak power along

core axis and values of axial offset

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452

]. Nucl. Sci. Technol.

The values of

Z,

and Z

2

are

calculated for different values of burnup by 1-D diffusion

code, assuming the conditions of hot full power, equilibrium xenon, all control rods out.

It is found that

Z,

and Z2 are not sensitive to changes in power level, their values being

much the same in different plants of similar design at the same burnup.

For

BOL, Z

1

and

Z

2

are

calculated based on xenon oscillation, using 1-D diffusion code.

The resulting data

are

normalized to

upon which we obtain

7

where

K,

and K

2

are

coefficients.

From the definition ( 1) of axial offset, this value can be described in the form

and

from the forms defined in Eqs. ( 2 )rv( 5) for expressing the axial power distribution,

~ O ~

=KsFz,+K4Fz

2

(

9)

where K

3

and K

4

are

coefficients. Combining Eqs. ( 7) and (

9),

we have

Fz = -K2

__

_ ~ 0

+__ _+ K

2

K

3

1

K,K

4

2

Ks 100 K,

K, K,K4-K2Ks),

10)

11)

On the other hand, from Fig. 1,

12)

where a l> a

0

, {3

1

and

0

are constants, while KI>

K2,

K

3

and

K

are determined from b

ao ft and

3o

using Eqs. (10)rv(12).

To determine b and b

2

we obtain from Eqs. ( 7) and ( 9 ),

2e-brcz2-Z

1

+2

{ b

1

/2rra

2

)

COs[2rra

2

(Z

1

-0.5)] +sin[Zrra2(Z -0.5)]}

e-b

co.s-z,J (

13

)

K,+

Ks=

[ b , / 2 r r a 2 1]

2e-b2CZ2-Z1 + 2 {(b2/2rras) C0S[27ras(0.5-Z 2)] +sin[2rras(0.5-Z

2)]}

eb

2

0.S-Z

2

)

14)

K

2

- K

4

= ~ 2rra

3

[(b2/2rras)

2

+ 1]

Then, C and C

4

are determined from the relative extrapolation distances at core bottom

and core top :

15)

from which,

16)

C _ Zrra4A2 sin[-2rr(1-Z2)a4]+cos[

-2rr(1-Z2)a4]

2

- ..:l2+(1-Z2)

17)

26

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Vol. 19, No. 6 (June 1982) 453

In practice, while a

1

and

a

4

can

be

determined from the above equations, these

coefficients are chosen empirically to reproduce realistic axial power distributions.

For

example, in the case of a 3 loop PWR, the values adopted would

be

where A.

0.

is the axial offset

( ),

while

where again

{

-0.01

hr=

-0.005

{

0.01

h.=

0.005

l ={

1.0

BOL

to 5,000

MWD/T

and 5,000

MWD/T

after

5,000 MWD/T,

BOL to 5,000

MWD/T

and 5,000

MWD/T

after

5,000 MWD/T,

BOL to 5,000

MWD/T

and 5,000

MWD/T

after 5,000 MWD/T,

HU: Burnup in MWD/T.

(18)

(19)

(20)

21)

Thus, the axial power distributions with arbitrary axial offset can

be

calculated using

the predetermined constants ar, ao,

f3r,

(3.,

Z

r,

Z

2

Ar and

A2.

The next step is to calculate the axial power distribution change during xenon

transient, and for this purpose the changes in the number densities of xenon and iodine

are expressed by

d f t ~ ) = - A I t

Z)+Y

U Z)

dt I I f t 1

(22)

23)

where

l t, Z),

X t,

Z):

Number densities of iodine and xenon, respectively,

at

timet, axial position Z

~ I , J.x:

Decay constants of iodine and xenon, respectively

Y I Yx: Fission yields of iodine and xenon, respectively

l JrfJ

Z)

:

Fission number

at

axial position Z, calculated by means of formula

I:JrfJ(Z)

=

(l JrrfJr+IJ

2

t/J2)P(Z)PREL,

;r> Z):

Xenon micro-absorption cross section multipled by neutron flux

at

axial

position Z calculated by means of formula

a;,rp(Z)

=

a:_

1

t/Jr

+a:_

2

t/J2)P(Z)PREL,

in above equations

I h I

2 :

Core-averaged fast and thermal fission cross sections

t/Jr 2:

Core-averaged fast and thermal neutron fluxes

a:_

1

a:_

2

:

Fast and thermal xenon micro absorption cross sections

P Z)

:

Relative power

at

axial position

Z

PREL:

Relative power level of reactor PREL=l.O

at

full power).

The

above cross sections and neutron fluxes are defined for a given burnup

under hot full power condition.

The procedure is to first calculate

P Z)

by the proposed functional method, and then

the xenon distribution, using Eqs. 22) and 23). The xenon distribution thus obtained is

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454

].

Nucl. Sci. Techno .,

further

utilized for determining the effective multiplication factors Kffr for the upper

core half, and

K fiB

for the lower core

half:

K e f f r = - - - ( - > - - ~ ~ +Dl

B " + - - v r - ( l ~ f t +

~ + ~ R ~ : ~ ~ r )

'a

1

' 'R 1 'a

1

.Xe a

2

2 a

2

.Xe ,

24)

25)

where

~ a , . ~ R .

l ~ h l'a

2

and

I ~ J z

are the core-averaged macro cross sections in the 2-

group diffusion equation, expressed in

the

usual notations ;

D

and

D

2

the

core-averaged

fast and thermal group diffusion coefficients; B

2

the

transverse

buckling; . x e and ~ ~ , . x .

the fast group xenon macro absorption cross sections for the top and bottom half cores.

The

values of

~ ~

. x e

and

~ ~ , . x .

are calculated by the formulas

X t , z)a{,P Z)dZ

- - - - 26)

[ / Z ) d Z

X t , z)a{,P Z)dZ

P Z)dZ

27)

All macro cross sections are determined for a given burnup and for hot full power

Condition. The cross sections ~ ~

. x e

and ~ ~

. x e

relevant to the thermal group correspond

ing to ~ ~

. x e

and

~ ~ , . x .

for the fast group.

Now the difference in reactivity due to the xenon distribution between

the

upper and

lower core halves is calculated:

Llpx=ln( KKffr

)x 100.

fiB

28)

It is in proportion to this reactivity

difference

that

the change in axial offset

30 ------------.------------

End

of

Life

is assumed to be caused by the xenon

20

transient during a given time step. This

assumption is justified by calculations based

10

on the effective one-group diffusion equa-

tion for the case of limited reactivity dif

ference and uniform reactivity distribution:

where

LlA. 0.: Axial offset change

during

time

step

considered

Llpx 1: Reactivity difference at end

of time step

Llpx o: Reactivity difference

at

beginning of time step.

Using Eq.

29),

the new value of axial offset

at the end of

the

time step under consid

eration is obtained, from which the axial

power

distribution at

the

end of

the

same

'

/)

0

0

- -10

0

x

-20

-30

4 o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

-1.0

0

1,0

React vi ty dl ffer ence, ~ f X

(%)

Fig. 3 Variation in axial offset induced by reac

tivity difference caused between

upper

and

lower core halves due

to

xenon

transient

-28-

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Vol.

19,

No. 6 (June 1982)

455

time step is calculated by

the

functional calculation method. Reiteration of this procedure

yields the total change brought to

the

axial power distribution during whole xenon

transient.

The

coefficient, g

1

in Eq. (29) is determined for each burnup using the 1-D

diffusion calculations, with the reactivity difference given individually for each step. The

relation between

A

0. and lpx is shown in

Fig

3 for the representative cases of

beginning and end of life.

The axial power distributions under various core conditions including xenon transients

are thus calculated with this method based solely on axial offset data.

ill RESULTS OF

XI L

POWER DISTRIBUTION CALCULATION

We calculate

the

axial power distributions of a typical 3-loop PWR core under various

core conditions using the present functional method, and compare the results with those

obtained from

1-D diffusion code calcula-

tions. The principal parameters of the

PWR plant taken up for th is study are as

presented in

Table

1 The axial power

distributions of this core under various

conditions

are

calculated

with

the

1-D

diffusion TWINKLE

codeC J,

which is also

used for deriving the input

data

1

, Z

2

a1o

ao f31o f3o

for the functional calcula

tion.

Then

the axial power distributions

are calculated

with the

functional method

using the input data shown in

Table

2

Table 1

Main core parameters

of PWR plant studied

Core thermal power

Core equivalent diameter

Core height

Number of assemblies

Assembly lattice

arrangement

Power density

Coolant pressure

Inlet temperature

Average temperature

2,432 MWt

304cm

366

em

157

15

x

15

fuel lattice

91.6W/cc

158

kg/cm

2

a

288.6c

306.4

Table 2

Parameters used in present method for deriving axial power distribution

Parameters

Case 1

Case 2 Case 3

Case 4

(Fig. 4 a))

(Fig. 4(b))

(Fig. 4(c)) (Fig. 4(d))

Burnup (MWD/T)

O(CYl)

9,000(CY1) O(CYl)

O(Eq. CY)

Power level( )

100

100

100 50

Bank D inser tion( ) 0 0 33.3 0

Axial offset( )

-13.6

-0.6 -21.9

16.4

Z1

0.30 0.15 0.30 0.20

Zz

0.65 0.80 0.65 0.80

a1

-1.333

-1.717 -1.333

-1.417

ao

1.230

1.135 1.230

1.085

PI

1.250

1.433 1.

250

1.

583

Bo

1.295

1.150 1.295 1.055

.ll

0.025

0.060 0.025 0.050

Az

0.025

0.060 0.025 0.050

al 0.686

1.003 0.669

0.918

a

0.414

0.997 0.231 1.082

bl

1.596

1.739

1.536 0.469

bz

1.010

1.129 0.354 1.394

cl 0.525

1.328

0.630

0.566

c

1.499

-0.171

2.283

-0.491

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456 ]. Nucl. Sci. Techno .,

The results of calculation

are

shown in Fig. 4(a) -'(d), together

with

the corresponding

data obtained

with

the

1-D

diffusion calculations.

The

curves representing the axial

power distribution at the first cycle are, in Fig. 4(a) for BOL, hot full power, equilibrium

condition, in

Fig.

4(b) for

9,000

MWD/T, hot full power, equilibrium condition, and in Fig.

4(c) for BOL, hot full power, bank D

33.3

inserted condition, while Fig. 4(d) shows the

curves for the equilibrium cycle, BOL, 50 power, equilibrium condition.

N

c

0

First Cycle, BOL

Hot Full

Power

All

Rods Out

A.O. -13.6

%

:;::;

1.0

'

D

0

X

0

'- '

0

Q

0

D

0. 5

This method

Q

>

;::;

c..

0

1.0

(Top)

1-D diffusion

code

0.5

Relative core height

Fig. 4(a) First cycle

at

BOL, in hot full power, all rods

out

0

(Bottom)

1 .5 . - - - - - - - - - - - - - - - -- - - - - - - - - - - - -- - - - - - - - - - - - -- - ,

First Cycle, 9, 000

MHD T

Hot

Full Power, All

Rods Out

A.O.

-0,6

1.0

This method

1-D diffusion code

0

1.0 0.5

(Top)

Relative

core height

0

(Bottom)

Fig. 4 (b) First cycle

at 9,000 MWD/T,

in hot full power, all rods out

Fig.

4(a)-(d)

Comparison

of power

distributions obtained

by present

method and

by 1 D

diffusion code

3

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458

]. Nucl. Sci. Techno .,

the results of

1 D

diffusion calculation. Thus establishing the capability of this simple

method to predict axial power distributions during the xenon transients such as xenon

oscillation, with ample precision for all practical purposes.

3

2

B

10

'

I

0

0

-

x


12/12

Vol. 19, No. 6 June

1982) 459

that

these

shortcomings-which

could, furthermore, be overcome

by

input data refinement

should not

detract

in any way from

the

practical applicability of the method to axial

power distribution analyses.

V. CONCLUSIONS

A simple functional method has been presented, which permits derivation of the PWR

axial power distribution solely from information on axial offset, acquirable continually

with ex-core neutron monitors.

The

method possesses the merits of:

(1)

Providing amply precise axial power distributions under

arbitrary

core conditions.

2)

Reducing

the

computer time to almost 1/50 of that required for 1-D diffusion

calculations, which should permit practical application to plant-side power distribution

determination, with possible development eventually to provide real-time power distri

bution monitoring and DNBR monitoring systems.

3) Convenient utilization for analytical

treatment

of the axial power distributions

required for accident and reactor transient analyses.

Other applications should be found for the method upon extending its use.

Text edited grammatically

by

Mr.

M. Yoshida.)

---REFERENCES---

( ) Verification

of

Mitsubishi

PWR

nuclear design procedure,

MAP

1-1004,

Rev. 1,

1976).

2) AoKI

N.,

et

al :

] At Energy Soc. japan

in

Japanese),

22 10],

718

1980).

3)

MoRITA,

T. et

at :

WCAP-8403 1974).

4) SHIMAZU, Y., et al.: ]. Nucl. Sci. Techno/. 19 1],

39

1977).

5)

TERNEY, W. B. et al : Trans.

Am

Nucl. Soc.

22,

682 1975).

6)

BARRY,

R.F.

RISHER,

D. H.: WCAP-8028-A

1975).

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