nuclear reactor theory, ju, first semester, 2010-2011 (saed dababneh). 1 reactor model: one-group...

Nuclear Reactor Theory, JU, First Semester, 2010-2011 (Saed Dababneh).

1

Reactor Model: One-Group• That was for the bare slab reactor.• What about more general bare reactor models?

• For steady state, homogeneous model:

• BC: (extrapolated boundary) = 0.

),()(),()(),(),(1

trrDtrrtrStrtv a

0),(1

),(),(),(2

22

trL

ktrtr

Dtr af

• R0, H0 are the extrapolated dimensions.

• BC’s:

• Let

• Solve the problem and discuss criticality condition.Solve the problem and discuss criticality condition.


2

Reactor Model: One-Group

01 2

2

2

Bdzdr

rrr

0),(

0),(

2

0

0

Hr

zR

)()(),( zrzr cosBesselHW 26HW 26

Reactor

R

H

z

xy

r


3


Reactor

R

H

• Briefly, we go through HW 26.

01 2

2

2

Bdzdr

rrr

)()(),( zrzr

0

coscos0

2

0

22

2

dr

dr

dr

d

H

zz

dz

d

z

xy

r

)()( 00 rCYrAJ


4


0)(0 CxYx n

00 4048.20)4048.2( RJ


5


Reactor

R

H

z

xy

r

000 cos)

4048.2(,...)(

H

z

R

rJPA

Criticality condition?

Criticality condition?

Do it.Do it.


6


R0

H0/2


7

Reflected Slab: One-Group

x

aa/2

Core

zReflected Slab ReactorReflected Slab Reactor

bb

Re

flect

or

Re

flect

or

),()(),()(

),()(),(1

trrDtrr

trrtrtv

a

f

For steady-state, homogeneous, 1-D

0)()(

0)()()(

2

2

2

2

xdx

xdD

xdx

xdD

RRa

RR

CCa

Cf

CC

C

C Core

R Reflector

)2

()2

(),2

()2

(,0)2

(BCsa

Ja

Jaa

ba CRCRR

0)(1

)(2

2 rL

r

Recall:


8

Ra

RR

RRR

C

Ca

CfC

mCm

CC

DL

L

xba

A

DBxBAx

2

2

)(2sinh

)()cos()(

Verify.

RR

R

RCmCC

mC

RR

CmC

L

bA

L

DaBABD

L

bA

aBA

cosh)2

sin(

sinh)2

cos(

BC

RR

RCmC

mC

L

b

L

DaBBD coth)

2tan(



9

RR

RCmC

mC

L

b

L

DaBBD coth)

2tan(Criticality condition.Criticality condition.

• For bare slab CC was / 2.• Smaller core for reflected reactor (even with a0 > a).• Save fuel.



10

Criticality “Calculation”• Can we solve “real” reactor problems analytically?• The previous discussion provides understanding of the concepts but also indicates the need for computational techniques.computational techniques.

• Assume:

• Adjust parameters so that = 0 (Steady-state).

• What parameters and how to adjust them?

),()(),()(),()(),(1

trrDtrrtrrtrtv af

)(),( retr t

)()()()()()()( rrDrrrrrv af


11

)()()()()()()( rrDrrrrrv af

Criticality “Calculation”

• Fixed design and geometry one free variable is k

• As we did earlier (be guided by HW 20):As we did earlier (be guided by HW 20):

)()()()()()( rrk

rrrrD ffudge

a

221 LBDk af

a

ffudge

operators are ,1

FMFK

Mfudge


12

Criticality “Calculation”

221 LBk af

fudge

FK

Mfudge

1

• Build an algorithm.• “Guess” (reasonably) initial kfudge and (or ) for the zeroth iteration.• Calculate the initial source term.• Iterate:

converges.flux until .....on so and

get 1

. and Guess

0

101

11

10

00

01

00

S

Skk

FS

k

SF

kM

k


13

Criticality “Calculation”• Or:

• If for example k > 1, take action to reduce source or increase absorption.• How?How?

volume

ii

volume

i

volume

i

volume

i

i

dVSk

dVS

dVM

dVF

k

M

Fk

1

sinks

sourcesfission

1

1

1

1


14

Reactor Kinetics Reactor Kinetics Reactor kinetics refers to the manipulation of parameters that affect k and to the subsequent direct response of the reactor system. Examples are:

• Absorber rods or shim movements to compensate for fuel burnup. • Safety scram rods to rapidly shutdown the chain reaction. • Control rods to provide real-time control to keep k = 1 or to maneuver up and down in power.• …..

Reactor Dynamics Reactor Dynamics Reactor dynamics refers to the more indirect feedback mechanisms due to power level effects and other overall system effects such as:

• Temperature feedback. • VoidVoid feedback. • Pump speed control (affects water density and temperature). • …

How to Adjust Criticality

Negative or positive reactivity.reactivity.


15

How to Adjust Criticality

Before all:Before all:

Core Design Core Design The transient response of the reactor to the above direct and indirect changes in basic parameters is highly dependent on the design details of the reactor. Sample issues are:

• Where should the control rods be placed for maximum effectiveness? • Will the power go up or down if a void is introduced into the reactor? • Will the power go up or down if core temperature goes up? • How often should the reactor be refueled? • and so on...


16

Multi-group Model• Wide neutron spectrum.• One-group, two-group? Should be generalized.

),()(),()(),()(

),()(),()(),(1

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsggag

extg

G

gggsg

G

ggfgggg

g

Identify the terms, NOW.

Fraction of an eV

1

),,(),(g

g

E

E

g dEtErtr

Flux-averaged quantities.


17

Multi-group Model

),()(),()(),()(

),()(),()(),(1

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsggag

extg

G

gggsg

G

ggfgggg

g

Total fission

Fra

ctio

n

Scattering in

Oth

er s

ourc

es

LeakageScattering outAbsorption

Fraction of an eV


18

Multi-group Model

5-group example.5-group example.

Maxwellian

1/EFission


19

Multi-group Model

111222333444555

5

1\

\\\ ),()(

fffffg

ggfggg trr

Total fission

Thermal fission (~ 97%)

Fast fission (~ 3%)


20

Multi-group Model

5544332211

5

1\

\\ ),()(

gsgsgsgsgs

gggsg

trr

Scattering in

Upscattering!!???

3g

Skipping!!???


21

Multi-group Model

gsggsggsggsggsg

gsg trr

54321

),()(

Scattering out

3g


22

Multi-group Model

553443333223113

335334333332331

3333

3

5

133

3

),()(),()(

),()(),(1

\

\\\

sssss

sssss

a

ext

ggfgg

trrDtrr

Strrtrtv

Group 3

Removal cross section

5

31

33

353432313

33333

\

\

\

gg

gsa

ssssa

ssar


23

),()(),()(),()(

),()(),()(),(1

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsgggrg

extg

G

gggsg

G

ggfgggg

g

Multi-group Model

Total fissionFra

ctio

nScattering in O

ther

sou

rces

LeakageRemoval In-group Scattering

),()(),()(

),()(),()(),(1

\

\

\\

\

\\\

11

trrDtrr

Strrtrrtrtv

gggrg

extg

G

ggg

ggsg

G

ggfgggg

g

Net Scattering in

nuclear reactor theory, ju, first semester, 2010-2011 (saed dababneh). 1 reactor model: one-group...

Documents

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