2D Full-core Calculations of PWR by MC Method

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Whole Core Calculations of Power Reactors by Useof Monte Carlo Method

Masayuki NAKAGAWA & Takamasa MORI

To cite this article: Masayuki NAKAGAWA & Takamasa MORI (1993) Whole Core Calculations

of Power Reactors by Use of Monte Carlo Method, Journal of Nuclear Science and Technology,30:7, 692-701, DOI: 10.1080/18811248.1993.9734535

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2/11

journal

of NucLEAR SciENCE and

TEcHNOLOGY,

30[7], pp.

692-701

(July 1993).

TECHNICAL REPORT

Whole

ore alculations

of

Power Reactors

by Use

of onte

arlo Method

Masayuki NAKAGAWA and Takamasa

MORI

japan Atomic Energy Research Institute

Received

December

10 1992

Whole

core

calculations have been performed for a commercial size PWR and a prototype

LMFBR by using vectorized Monte Carlo codes. Geometries of cores

were

precisely

represented

in a pin by pin model.

The

calculated

parameters

were kef , control rod worth power distri

bution and so on. Both multigroup and continuous energy models

were

used and the accuracy

of multigroup approximation was evaluated

through

the comparison of both results. One million

neutron

histories

were tracked to considerably

reduce

variances. It was demonstrated that the

high speed vectorized codes could calculate kef , assembly power and some reactivity

worths

within

practical computation time. For pin

power

and small reactivity

worth

calculations, the

order of 10 million histories would be necessary.

t

would be difficult for the conventional

scalar

code to solve

such large

scale problems while the

present

codes consumed computation

time less than 30 min for a PWR and 1 hour for an LMFBR. Required number of histories to

achieve target design

accuracy

were

estimated for

those neutronic parameters.

KEYWORDS: Monte Carlo method multigroup model,

continuous

energy model,

whole core, neutronics

calculation

PWR

type reactors LMFBR type reactors

con-

trol

rod worth reactivity worth power distribution variance, accuracy

I

INTRODU TION

In the calculations of neutronic character-

istics of reactor core, various approximations

are usually introduced in the conventional

deterministic methods. Those are multigroup

approximation, diffusion theory, finite mesh

size, unit cell homogenization, resonance self-

shielding factor and so on. Therefore some

corrections are necessary to predict neutronic

characteristics with sufficient accuracy. Many

calculation steps are needed to obtain accurate

prediction.

On the other hand, the Monte Carlo meth-

od based on the continuous

energy

model is

essentially free from such approximations and

gives a solution of transport equation by a

single calculation step starting from a point-

wise nuclear data library. It can solve any

problem in principle with complex geometry

such as a power reactor in neutronic sense.

The application of the method to whole reac-

tor core calculations in full detailed geometry

has been limited by the reasons

that

reactor

cores generally have very complex geometry

and their calculations consume much computa-

tion time to achieve a required accuracy.

Recently whole core calculations were per-

formed by Redmond II & Ryskamp for a small

scale reactor such as an experimental or a

research reactor

ANSC1

1

• In their calculations,

an MCNP run takes 157 min on the CRAY-2

for 60,000 neutron histories for the core with

D.O moderator. To realize Monte Carlo cal-

culations for a power reactor, remarkable

progress is necessary in both computation

speed and reduction of work in preparing

input data, especially geometry description

data in order to track a great number of

Tokai-mura

/baraki-ken 319-11.

8

D o w n l o a d e d b y [ F e d e r a l U r d u U n i v e r s i t y o f A r t s S c i e n c e

a n d T e c h n o l o g y ] a t 2 2 : 4 8 0 6 D

e c e m b e r 2 0 1 5


3/11

Vol 30, No. 7 July 1993) TEC I I > IC \L R PORT M. Nakagawa,

T.

Mori)

693

neutron histories in precisely represented ge

ometry. When such disadvantages are over

come, the continuous energy method can not

only give reference solutions for benchmark

type problems but also be used even in core

design calculations.

Most reactor cores consist of nearly iden

tical fuel assemblies and fuel pin cells. It

makes possible to describe a whole core by

defining only once the cells of any structure

that

appear repeatedly in geometry. The

capability to

treat

multiple lattice geometry

lattice-in-lattice geometry)

can significantly re

duce an amount of work required in geometry

data preparation.

As well known, a variance in a Monte

Carlo result is proportional to 1/ where

is the number of particle histories. Fur

thermore, Monte Carlo eigenvalue calculations

are

biased in evaluated eigenvalues, eigen

functions and their variances as investigated

in some works<

2

J< J.

This

problem seems to

be more serious for flux or reaction rate cal

culations in large light

water

reactors. To

overcome it, much more neutron histories

would be actually necessary to achieve a

desired confidence level.

We have successfully developed new Monte

Carlo codes


4/11

694

TEcii '\ICAL REPORT (M. Nakagawa,

T

Mori)

./.

Nuc/ Sci. Techno/.

nance region, cross sections

are

treated by

the probability table method. Computations

were carried out on the FACOM VP-2600

computer.

2. Preparation of Multigroup

Cross Sections

The used multigroup cross section libra

ries

are

the SRAC code libraryc•J in the case

of PWR and the

70

group JFS-3- 3 libraryc'o;

in the case of LMFBR.

In

the PWR calculations, the SRAC system

was

used to generate four group cross sections

for each material by collapsing from the

107

group structure. The energy boundaries

are

10.0 MeV, 67.38 keY, 130.7 eV, 0.6825 eV and

l

5

eV. Ultrafine group spectrum calculations

were made to generate effective resonance

cross sections in the low energy resonance

region

(below 900 eV).

Material-wise cross

· sections involved in the fuel pin cell were

calculated using a

unit

cell model. Those

for control rods were calculated using a super

cell model. The transport cross sections were

defined by the

0

diagonal transport approxi

mation.

In

the LMFBR calculations, the material

wise 70 group cross sections were produced

by the SLAROM code(l'J. Heterogeneity ef

fect on resonance self -shielding factors was

taken into account for fuel materials. All

the cell models are the same as those adopted

in the PWR calculations.

The

0

diagonal

transport approximation was also assumed.

lli WR CORE CALCULATIONS

1. Modeling of Core

We modeled a core used in a commercial

type four loop 1,160 MWe reactor.

The

initial

core consists of 2.1, 2.6 and 3.1% enriched

235

U fuel assemblies. The calculations were

performed for the beginning of cycle (BOC)

fuel configuration. The specifications of mod

el core is presented in Table 1.

The

reactor

core including the core barrel is modeled as

precisely as possible. The upper and lower

core supports, and

water

are

homogenized in

the model. A fuel pin consists of fuel pellet,

cladding

(Zircaloy 4),

plenum and end caps

(stainless steel). Grid spacers

are

smeared into

the neighboring moderator region.

Table 1 Specifications of PWR

core

geometry

Pin diameter

Pellet diameter

Plenum

length

Total

fuel pin

height

Pin pitch

Assembly lattice

Assembly pitch

No. of C. R. guide tube

Height of core

Inner radius of core barrel

Thickness

of core support

9 5mm

8 36

mm

170mm

390cm

12.6 mm

Square 17x17

215 mm

24

Fuel

average temperature

Moderator

average temperature

4.1 m

2 2

m

20cm

900 K

600K

The total numbers of fuel assemblies and

fuel pins are 193 and 55,777, respectively.

Each assembly has

24

control rod channels

and a channel for a detector.

In

geometry description, the core region

including the baffle plate is presented as a

square lattice consisting of unit fuel assembly

(level 1 lattice).

Each square assembly also

consists of water gap and square fuel pin

cells

(level 2 lattice).

The calculation model is

shown in

Fig.

l(a) and

b).

These figures are

drawn by the CGVIEW codec

12

J using input

data for the Monte Carlo calculations. The

latter

is a closeup view of a

part

of core.

As chemical shim, boronic acid is diluted

into water except for control rod channels.

A nearly critical core

was

realized

with

2,000

ppm natural boron content. A control rod

worth was obtained from the difference be

tween kerr values at the critical and at the 12

control rod clusters inserted cores. The com

position of control rod is Ag(80),

IN 15)

and

Cd(5). The location of inserted cluster assem

blies

are

shown in Fig. 1(a).

2. Calculation Results

The eigenvalues of the critical and the

control rod inserted cores were calculated.

The results

are

shown in Table 2. The

number of neutron histories is one million

for all the cases. The variances

(1a)

of

l ert

values

are

0.04'"'-'0.05% by the continuous

energy model and '"'-'0.03% for the four group

model. The

latter method gives smaller va

riances though the same number of histories

8



e c e m b e r 2 0 1 5


5/11

Vol. 30 No. 7

July

1993) TEc l i ; ; I c \L

R PORT

M. Nakagawa, T. Mori)

b) Closeup view of a part of core

Fig

1 Calculation model of PWR

--85-

695

D o w n l o a d e d b y [ F e d e r a l U

r d u U n i v e r s i t y o f A r t s S c i e n c e


e c e m b e r 2 0 1 5


6/11

696

TEcHNIC L REPoRT

(M. Nakagawa,

T Mori) }

Nucl. Sci. Techno .,

were

tracked.

In

any

case,

the

variance is

small enough.

The variances

of

control

rod

worths

are

3 4% of

which

absolute value is

1.58 L1k/

kl?'.

able 2 kerr and

reactivity worth

of

PWR

Reference

core

4

group

model

keff 1.2165

f

rlk/kk

1a C o)

±0. 00029

Critical core

2,000

ppm

boron

0.9930

± 0.

00027

0.185

0.20

Continuous energy

model

keff

1.2233

1a ±0. 00040

r1k/ kk

1a (

1.

0008

±0. 00051

0.

182

0.33

12 C. R.

clusters

inserted

0.9780

±0. 00034

0.0154

2.9

0.9852

±0.

00047

0.0158

4.4

The

four

group

model

gives smaller eigen

values for all the

cores

compared

with

those

by

the continuous

energy

model.

At the

critical core, the difference is 0.6 L1k. Con

siderable

parts of this difference arise in the

fine group (107

groups)

cell calculation

stage.

The control rod worth is also underestimated

by this method, but the difference is almost

within the

statistical error.

The power

distributions of

assembly and

fuel pin are compared in able 3. These

values

are

shown for radial

direction

from

the center to the right

boundary.

Pin power

was

calculated for a single pin located next

to a

central channel

for a

detector within each

assembly.

The

variance of

assembly

power

is within 2%

while that

of pin power ranges

7 12%.

The

differences

between

two models

are given

in

relative

values

to the continuous

energy

model.

The

assembly powers

calcu

lated by

two

models

agree within

about

2a

uncertainty

with each other. The

largest

difference of

--5.5%

is beyond

2a uncertainty

but the

cause

is not due to the model but the

statistical

uncertainty. The pin power

shows

still large differences up to '10

q :;

due to

large variances but

these differences

are

al

most within

la

uncertainty. If we took one

million or more

as the number

of histories,

the

biases

on the

eigenfunction and their vari

ance seem to be not so seriously large in the

present core

as

discussed by Brissenden &

Garlick. We could obtain

reliable

power distri

butions by increasing the

number

of histories.

able

3 Radial

power distribution

of fuel

assembly

and pin in

PWR'

Assembly•t

F21

F26 F21

F26 F21

F26

F21

F31

~ ·

Assembly

power

4 group

model (lO-•)

260

290 250

272 224 237

181

135

Continuous energy

model

(10- )

263 302 260 283 237

241 179

131

Differences (?,;)

-1.1

4 4 4 -5.5

-1.7

1.1

3.

1

Pin power

4

group

model

(10-'')

1.

06

1.

18

1.

04 1.10 0.883 0.992

0.703 0.553

Continuous energy

model (10-'') 0.962

1.

24

1. 08 1. 09

0.951

0.908

0.

790

0.502

Differences (%)

10

5 4 7

9

-11

10

t The

variances

are 1 2 and 7-12 ?d

for

assemblies

and pins,

respectively.

t t

F21,

F26 and F : ~ 1 show the

assemblies

with

2.1, 2.6

and : ~ 1 enriched fuel,

respectively.

These

are

located

in the order from the

center

to the right

boundary

of

the

core. Total

fission

source

is

normalized

to unity.

In order

to

examine the adequacy of effec

tive cross sections used in the four group

method,

the effective

fission

cross sections

of

the thermal group are

compared

in able 4

which

are the averaged values

weighted

with

the volume flux

over each

assembly. It is

found

that the

conventional method

gives

accurate cross sections for the assemblies

with

three different enrichments. It is noted that

the

space dependence is very small as seen

for the

2.1% and

2.6°/?

enriched

fuel assem

blies. This fact

suggests

the adequacy of

group cross section production method using

the fundamental

mode

spectrum

for all assem

blies

with

regular cells.

8 6



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Vol. 30,

No.

7 (July 1993) TECill ICAL

REPORT

(M. Nakagawa,

T.

Mori)

697

Table 4 Comparison of thermal group fission cross sections

averaged in assemblies of PWR (cm-')t

Assembly

4 group model

Continuous energy model

F21

0.127

0.

127

F26

0. 155

0. 155

F21

0.127

0.

127

F26

0. 155

0.154

F21

0.127

0.126

F26

0.155

0. 155

F21

0. 127

0. 126

F31

0.181

0. 180

t

The

locations

of assemblies

are

the

same

as

those

in

Table :1

The computation time consumed by the

continuous energy model is

14

min (without

cross section tally) for 1 million histories and

larger by a factor of 2 compared with

that

by the few group model.

If

effective cross

sections are tallied, the computation time in

creases by several tens percent since the con

siderable parts of tally calculation

are

not

vectorized on present supercomputers. The

speedup by vectorization (CPU time ratio of

scalar calculation to vector one) is a factor of

12 '14 in the present runs.

This

far tor is

similar between the continuous energy and

the few group models. We summarize the

number of histories and

u

variance in Ta

ble

5.

In

addition the estimated number of

histories to achieve target accuracies for core

design

are

also shown. If we assume prac

tical computation time as 1 h. eigenvalues and

assembly power distribution can be calculated

within such CPU.

In

the case of pin power

calculation, higher speed computation, about

10

times, is desired to achieve the target

accuracy. If a core configuration becomes

complex, for example, a core consisting of

mixed MOX and U0

2

fuel assemblies, accurate

prediction of pin power distribution would

be

more difficult. Since the Monte Carlo method

is a powerful tool in such a case,

further

speedup of computation is expected.

Table 5 Required number of histories to achieve target accuracy for PWR

Power distribution

k ff

C.

R.

worth

Assembly

Pin

~ .

Present histories

10

111

(%)

0.05

Target accuracy

(%)

0.

1

Required histories

2. 5 10

IV

PROTOTYPE

LMF R

CORE CALCULATION

1. Modeling

o

Core

The model core of LMFBR consists of

inner and outer cores, control rod assemblies,

radial and axial blanket, and shield surround-

10

10';

10';

1.5 10

4

2

2

2.

2 10';

2.5x10'

4x10

ing the blanket.

The

total number of fuel

assemblies and fuel pins are given in Table 6

The

diameters of fuel pellet and pin in the

core are 0.54 and 0.65 em, respectively. Pin

pitch is 0.787 em in the core and 1.50 em in

the blanket.

The

pitch of fuel assembly is

7.56 em. Fuel and blanket assemblies have 169

Table 6 Number of fuel assemblies and pins in LMFBR

Region

Assembly Pin

Inner core

Outer

core

Control rod channel

Radial blanket

Shielding

Total

108

90

19

174

7

463

8

18,252

15,210

19

(number of channels)

10,614

44, 076+B,C pins



e c e m b e r 2 0 1 5


8/11

698

TECHNICAL REPORT (M. Nakagawa, T Mori)

]. Nucl. Sci. Techno .,

and 61 fuel pins, respectively. These pins

and assemblies

are

precisely modeled in the

calculations except for spiral wire spacers

which are smeared into the neighboring sodi

um region.

The core has a

MOX

fuel part of

93

em

height and, 30 and 35 em thick upper and

lower axial blankets of depleted U

2

,

respec

tively.

The

whole core is presented as a

hexagonal lattice consisting of fuel assemblies

(level 1 lattice).

Each hexagonal assembly also

consists of outer sodium region, wrapper tube

and hexagonal fuel pin cells (level 2 lattice).

The calculation geometry is shown in Fig 2

a) (level

1)

and

b)

(level 2). The latter is a

closeup view of a part of the core. These

figures

are drawn

by the CGYIEW code. The

control rod cluster consists of

19

B.C pins

which are seen in Fig. 2 b). Since this cluster

structure can not divided into unit cells, it is

described in level 1 lattice. The positions of

inserted control rods are shown in Fig. 2(a).

Guide tubes

with

cylindrical shape

are

also

seen at the positions of Na follower

(control

rod

withdrawn).

2 Calculation Results

The number of neutron histories is one

million for the reference core (all control rods

withdrawn) and half million for the other cases.

Sodium void reactivity worth was calculated

by removing whole sodium from the inner

core. The reactivity worths of the central

and the three control rods were calculated

from the differences of

kerr

values between

the reference and the rod inserted cores.

The

calculated results

are

shown in Table

7. The variances

(1a)

of kerr values of the

reference core are about 0.05

%L1k

for both

models, so the prediction accuracies

are

very

good. The variances of reactivity worths are

given by relative values in percent unit. In

the case of sodium void worth of which

absolute value is about 0.8 %L1k

kk ,

the va

riance is about 11 . It is found

that

the

variance decreases with increasing the reac

tivity worth.

The multigroup method gives a larger kerr

value by about 0.3%L1k for the reference core.

All the reactivity worths are slightly over-

estimated due to this approximation. The

comparison of neutron spectra is presented in

Fig

3.

The multigroup method gives smaller

values of spectrum below 1 keY. This is

because the multigroup elastic removal cross

sections

are

overestimated in the energy re

gion where neutron flux gradient is steepc

13

>

A difference is also observed at the 28 keY

iron resonance. The ll rr values are propor

tional to the summation of

1

multiplied

spectra in those cases. Although the values

of spectrum by the multigroup model

are

larger than the continuous energy model below

1 keY, the

kerr

value is smaller because this

is mainly contributed from the peak region

of spectrum 10 keV-1

MeV)

where the former

gives smaller values of spectrum in the some

groups.

In

the core with three control rods insert

ed, power distribution was calculated

at

three

assemblies in the inner and outer cores, and

the radial blanket of which locations are

shown in Fig. 2(a).

The

assembly and single

fuel pin

(located

at the center

of each assembly)

powers obtained are shown in Table 8. The

assembly power can be evaluated

within

the

variance of 1.5 in the core region while

the pin power has about

4

variance.

In

the radial blanket, the variance are still large

(

-lO?o'

for

the

assembly

and -20

for

the

pin).

The

multigroup method gives slightly lower

power compared with the continuous energy

one, but such differences may not attributed

to the approximation

error

if

2a

uncertainty

is assumed.

We

summarize the present number of his

tories and

la

variance in Table 9 Assumed

target accuracies for core design are also

shown. The values of 1a

are

shown in the

table but

2a

uncertainty may be often used

in core design. We can estimate the required

number of histories to achieve these accura

cies as shown in the last low in the table.

In the reference core, the computation time

by the continuous energy model is

51

min for

one million histories and the ratio to that by

the multigroup one is about 1.5. It is found

that criticality and assembly power distri

bution could be calculated within practical




e c e m b e r 2 0 1 5


9/11

Vol

30, No. 7 (July 1993)

TEcll l ' ICAL RI 'ORT (M. Nakagawa, T. Mori)

( -1 0.00,

l = · ~ . O l t i 99 00J t 3 ~ ~ ~ ~ ~ ~ i

ZO £

:MATERIAL

g - r - - - - - - - ~ ~ - ~ ~ · ~ f ~ F ~ L ~ I G H ~ T - · _ ~ ~ ~ - - ~ ~ ~ ~ ~ ~ ~ - - - - - L - E _ v E _ L

__

_-_,

'

•

( -130.00, -150.00, 99.00)

@

Posilions

ol

inserted

C A

0

Na lollower (C.R. Wllhdrawn)

( 130.00,-150.00, 99.00)

®

Assemblies

ol

p calculation

(a) Whole core lattice model

( -90.00. 10.00.

B9 OOJ ( -

30.00. 10.JO. B i uOJ

(b) Closeup view of a

part

of core

Fig 2 Calculation model of prototype LMFBR

9

699




e c e m b e r 2 0 1 5


10/11

700

TECI INtCAL

REPORT (M.

Nakagawa, T .

Mori)

] Nucl. Sci. Techno/.

Table 7

keff

and reactivity worth of LMFBR

Reference core

Reactivity worth

Na follower

I. C. void

Central

C. R.

3 C.R.

No. of

histories

10

5x10'

5x10'

5x10'

Multigroup

model

keff

1. 0411

1.

0503

1. 0232

0.9989

1 ,

±0.

00050

:1:0.00077

±0.

00067

±0.

00066

rlkjkk

0.0084

-0.0168

-0.0405

1u

( ~ ' )

10.5

4.8

2.0

Continuous

energy

model

il ff

1. 0379

1. 0461

1.

0207

0.9982

rJ

±0.

00055

±0.

00066

±0.

00066

±0.

00068

rlkjkk

0.0076

-0.0163

-0.0383

1·;

( 9;,)

11

5. 1

2.3

Jo·•

:::: .::::··:·:::::::,,,,:',,,1 ••..•••...

:·········

.....

-

........ - .....:

..............

: ....... --- .

L

::J

0

'

..

'

.r.

.,

'

::J

IL

10 ' j : j

10

1

10'

10'

1o'

Energy (eVl

Fig

3

Comparison

of

neutron

spectra

calculated

by

multigroup and continuous energy models

Table

8

Fuel assembly

and pin

powers

in LMFBRt

Inner

core

Outer

core

Ass.

Pin

Ass.

Pin

--

Multigroup

model

(10-6)

126 7.84

138

8.13

rT

(%) 1.2

3.6 1.3 4.3

Continuous

energy

model

(10- )

129

8.04

146 8.91

fT

(%) 1.5

4.3 1.4

4. 7

t Total fission source is normalized to unity.

10'

Radial blanket

Ass. Pin

13.9

0. 182

9. 1 19

11.5

0. 271

9.3

25

computation time


11/11

Vol 30, No. 7 (July

1993)

TECIINIC\L REPoRT (M. Nakagawa,

T.

Mori)

701

Table 9 Required number of histories to achieve

target

accuracy for LMFBR

Power distribution

Reactivity worth

keff

·

-

--

Assembly Pin Na void

1 C.R.

-

Present

histories

10

1a

( ~ ; )

0.048

Target

accuracy

o;;

0.

1

Required

histories

3x10'•

Speedup by vectorization (ratio of CPU time

consumed by scalar calculation to vector one)

is a

factor

of

15

for

both multigroup and continu

ous energy models. The factor increases

up

to 18.5 if

the

batch size increases to 20,000

particles.

It is

noted

that the computation

time

required for

an

LMFBR calculation is

much

larger

than that for a

PWR

because the

number of scattering collision is much larger

in

the

former

case.

V SUMM RY

The Monte Carlo

calculations for the whole

cores of power reactors have

been

carried

out

by the use

of

the multigroup and the

continuous

energy

models. Complex

geom

etry of

such cores

can be easily modeled

by using the capability of multiple

lattice

geometry. When

the

total

neutron

histories

are 1 million, the variances of keff and assem

bly powers are satisfactory

compared

with

the required accuracies. On the

other

hand,

about

10

times histories are necessary to

reduce the variance of

a

pin

power

below

a

few percent and of a small reactivity worth

below one percent. Through the present cal

culations,

the accuracy of multigroup approxi

mation could be evaluated by comparing

with

the results by the

continuous

energy model.

Further speedup

and

improvement

of ge-

10';

10"

5x10

5x10'•

1.5

5

10

5

2

3

2xl0 ;

6x 10';

6x

10

1.3xl0

ometry

description capability

enable

us

to

extensively use

the

continuous

energy

Monte

Carlo method

in

whole core

calculations.

In

a

detailed design

stage, this

method

would

provide

us a powerful tool.

K ~ f ~ R E N E S

(1) RELJ.\IU :-J J Il.

E.L .. RYsl,_\\11', ].M.: Nucl. Tech.

no/.

95.

272 (1991) also idem: Trans. Am.

Nucl.

Soc., 61. 377 (1990).

(2) BRissE;\IDE'>. R.I.,

GARLICK,

A. R.: Ann . . Vue/.

Energy

13, 63 (1986).

(:l)

GELBAIW,

E.:

Prog.

Nuc/.

Energy 24,1 (1990).

(I) NAi.;A

Documents

2D Full-core Calculations of PWR by MC Method