Multi-group Model

Nuclear Reactor Theory, JU, Second Semester, 2008-2009 (Saed Dababneh).

1

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

\\ rDrrrrv gsgaggsgfg

g

Multi-group Model

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

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

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsggag

extg

G

gggsg

G

ggfgggg

g

Calculate group-averaged:

Or for,

we need group-averaged )(),( rr sggrg

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

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

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsgggrg

extg

G

gggsg

G

ggfgggg

g


2

Multi-group Model• Group-averaged parameters?• ENDF.

• Integrate term by term over groups and equate to equation of multi-group model.

),,(),(

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

),,(),(

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

)(

1

0

\\\

\

0

\\\

tErErD

tErErtErEr

SdEtErEEr

dEtErErEEt

tEr

Ev

sa

exts

f

�

Units!


3

Multi-group Model

1

),,()(

1),(),(

1 g

g

E

Eg

gg

g

dEtErEvtv

tr

ttr

tv

1

),,(),(g

g

E

E

g dEtErtr • Define group flux

1

1

),,(

),,()(

1

1g

g

g

g

E

E

E

E

gdEtEr

dEtErEv

v


4

Multi-group Model

1

),,(),(),()(g

g

E

E

gg dEtErErDtrrD

1

1

),,(

),,(),(

)(g

g

g

g

E

E

E

E

g

dEtEr

dEtErErD

rD


5

Multi-group Model

dEtErErtrrg

g

E

E

agag

1

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

dEtEr

dEtErEr

rg

g

g

g

E

E

E

E

a

ag

1

1

),,(

),,(),(

)(


6

Multi-group Model

dEtErErtrrg

g

E

E

sgsg

1

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

dEtEr

dEtErEr

rg

g

g

g

E

E

E

E

s

sg

1

1

),,(

),,(),(

)(


7

Multi-group Model

G

g

E

E

E

E

s

E

E

G

g

E

E

s

E

E

s

G

gggsg

g

g

g

g

g

g

g

g

g

g

dEdEtErEEr

dEdEtErEEr

dEdEtErEErtrr

1

\\\

1

\\\

0

\\\

1

\

1 1\

\

1

\

1\

\

1

\

\\

),,(),(

),,(),(

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

1 1\

\\

\\\\ ),,()(

),(

1)(

g

g

g

g

E

E

E

E

s

ggsg

dEdEtErEEtr

r


8

Multi-group Model

G

g

E

E

fg

fg

f

E

E

E

E

f

G

ggfggg

g

g

g

g

g

g

dEtErEE

dEtErEE

dEtErEEdEE

dEdEtErEEEtrr

1

\\\\

\

0

\\\

\

0

\\\

\

0

\\\

1

\

1\

\

1

1

\

\\\

),,()()(

),,()()(

),,()()()(

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

1\

\\

\\\\\\ ),,()()(

),(

1)(

g

g

E

E

f

gfgg

dEtErEEtr

r

1

)(g

g

E

E

g dEE


9

Multi-group Model

ENDF

High G, few mesh

points.

Small G, more mesh

points.

Poison, burnup (or better consumption), temperature, control rod position, etc…

Flux

Flux


10

Multi-group ModelWhat could make life a little easier?!• No upscattering

set group G to include neutrons up to ~1 eV.

• No group skipping when scattering down (directly coupled).

.for 0)( \\ ggrgsg

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

11 \

\\

\

\\ trrtrrtrr gsgg

g

gggsg

G

gggsg

Your choice of how to tackle in-scattering.

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

\\ trrtrrtrr gsgggggs

G

gggsg

HW 27HW 27 How can we pledge this? What about H?


11

Multi-group ModelCriticalityCriticality

G

ggfggg

g

ggg

ggsggrggg

trrK

trrtrrtrrD

1

1

1

\

\\\

\

\

\\

),()(1

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

Fk

M1

No upscatter

Not only sinks

Not all sources, only fission.

Iterations.Iterations.

),()(),()(

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

\

\

\\

\

\\\

11

trrDtrr

Strrtrrtrtv

gggrg

extg

G

ggg

ggsg

G

ggfgggg

g

Redundant when no upscatter.


12

Multi-group Model

3

2

1

332313

2212

11

0

00

rss

rs

r

D

D

D

M

333223113

332222112

331221111

fff

fff

fff

F

No upscatter



13

Multi-group Model

3

2

1

3323

2212

11

0

0

00

rs

rs

r

D

D

D

M

333223113

332222112

331221111

fff

fff

fff

F

No upscatter

Directly coupled



14

Multi-group ModelMulti-group Multi-group one-group one-group

0

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

dEtErtrdEtErtrg

g

E

E

g

0

0

),,(

),,()(

1

1

),,(

),,()(

1

11

1

dEtEr

dEtErEv

vdEtEr

dEtErEv

v g

g

g

g

E

E

E

E

g


15

Multi-group Model

0

0

),,(

),,(),(

)(

),,(

),,(),(

)(1

1

dEtEr

dEtErErD

rD

dEtEr

dEtErErD

rDg

g

g

g

E

E

E

E

g

dEtEr

dEtErEr

r

dEtEr

dEtErEr

ra

aE

E

E

E

a

ag g

g

g

g

0

0

),,(

),,(),(

)(

),,(

),,(),(

)(1

1


16

Multi-group Model

1 when 0),()(),()(1\

\\

Gtrrtrr gsg

G

gggsg

1)()(0

1

dEEdEEg

g

E

E

g

1 when ),()(),()(1\

\\\

Gtrrtrr f

G

ggfgg

Substituting all of the above into

yields

which is the one-group diffusion equation.


17

Multi-group Model

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

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

11 \

\\

\

\\\

trrDtrrtrr

Strrtrrtrtv

gggsggag

extg

G

gggsg

G

ggfgggg

g

),()(),()(

),()(),(1

trrDtrr

Strrtrtv

a

extf


18

Multi-group Model

Project 3Project 3

Work out the multi-group to two-group multi-group to two-group collapsing and investigate criticality.

Write down the appropriate matrices.


19

Poisoning

Saturates

135Xe106 b


20

Poisoning

Continuously accumulates

149Sm105 b


21

Poisoning• Not anticipated! Reactor shut down! Time scale:Time scale:

Hours and days.Hours and days.135Xe106 b

149Sm105 b

XeI

Ia

Xea

mXe

a


22

HW 28HW 28

Poisoning

eratora

clada

fuela

poisona

poisona

eratora

clada

fuela

fuela

eratora

clada

fuela

fuela

f

f

kk

k

mod12

mod2

mod1

that Show

(critical)

reactor). (Infinite use uslet ,1

Reactivity

Negative reactivity due to poison buildup. It is proportional to the amount of poison.


23

Poisoning

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

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

trtrXetrXetrItrt

trXe

trtrItrItrt

trI

XeaXeIfXe

IaIfI

Initial conditions?Initial conditions?• Clean Core Startup.Clean Core Startup.• Shutdown (later).

constant.)0()( assume uslet and

Fuel.Fresh 0)0()0(

t

XeI

small

Assume no spacial dependence.


24

Poisoning

)(

)1()(

)(

)(

0

0

)(

0

0

0

0

ttXeaIXe

fI

tXeaXe

fXeI

IXeaXe

XeaXe

ee

etXe

)1()( 0 t

I

fI IetI

HW 29HW 29 Show that:

and

)(I)(Xe


25

Poisoning

)(I)(Xe

eratora

clada

fuela

Xea

eratora

clada

fuela

poisona tXet

modmod

)()(

• Now, we know Xe(t)


26

Poisoning• Shutdown. Shutdown. After the reactor has been operating for a “long” time.

.0)0()(

)()0(

)()0(

t

XeXe

II

),(),(),(

),(),(

trXetrIt

trXe

trIt

trI

XeI

I

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

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

trtrXetrXetrItrt

trXe

trtrItrItrt

trI

XeaXeIfXe

IaIfI


27

PoisoningHW 30HW 30 Show that

)()(

)()(

)()(

tt

XeI

It

t

IXeXe

I

eeI

eXetXe

eItI

Height of the peak depends on I() and Xe(), i.e. depends on .

> 0 ?


28

Poisoning

If, the available excess reactivity can compensate for less than 30 minutes of poison buildup, can’t startup again after ~30 minutes of shutdown, because you can’t achieve criticality. Need to wait some 40 hours (in this case) for Xe to decay down.

Shutdown Xe negative try to add positive reactivity move control rods out need to have enough reserve costly to do that.


29

Poisoning


30

PoisoningStrategiesStrategies• If you plan to shut down for “short maintenance”, think about stepback.• Examine different scenarios using this code from

http://www.nuceng.ca/ • Prepare your own report, code, calculations, graphs, comments, conclusions etc…..• Be creative and use whatever experience you gained during your study in this program.

http://www.nuceng.ca/


31

PoisoningXe OscillationsXe Oscillations• (r,t) (spacial dependence) flux locally Xe burnup (reactivity) flux further control rods globally in flux elsewhere Xe burnup ….. Xe oscillation but limited by opposite effect due to increase (decrease) of I in the high (low) flux region.• In large reactors (compared to neutron diffusion length) local flux, power and temperature could reach unacceptable values for certain materials safety issues. • Think of one sensor and one control rod feel average flux apparently OK more sensors and control rods to locate and deal with local changes.


32

Poisoning

Permanent PoisonsPermanent Poisons• 149Sm has sizeable but lower cross section than 135Xe.• It does not decay.

• Accumulates with time.• Consequences?????????

....................).........,(),(),(

trtrt

trSmfSm

????

Multi-group Model

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