20 Kinetics With Delayed Neutrons

8/9/2019 20 Kinetics With Delayed Neutrons

1/38

2009 Januar 1

Kinetics With Delayed Neutrons

B. Rouben

McMaster University

EP 6D03Nuclear Reactor nalysis

!Reactor Physics"

#00$ %an.&'r.


2/38

2009 Januar 2

Pro('t and Delayed )ractions

We no* have to distin+uish bet*een the 'ro('t anddelayed neutrons. ,et-s *rite

( )

"6!

"/!

"1!

"3!

"#!

"!

d

dp

dp

d

p

fractionneutrondelayedthedefinedhavewewhere

and

Therefore

fissionperreleasedneutronsdelayedofnumberAverage

fissionperreleasedneutronspromptofnumberAverage

fissionperreleasedneutronsofnumbertotalAverage

=

==

+=

=

==


3/38

2009 Januar 3

Pro('t and Delayed 2ources

Without delayed neutrons *e had +otten to the 4ollo*in+evolution e5uation 4or the neutron density

No* *e *ill need to se'arate the neutron source into a

'ro('t 'art

and a delayed 'art.

"!"!"!"!"! # trntrnDBtrn

t

trnaf

=

( ) "7! nffp =


4/38

2009 Januar 4


8he delayed source co(es 4ro( the decay o4 delayed&neutron 'recursors. We sa* last ti(e that there are (any

delayed&neutron 'recursors. 9o*ever *e *ill 4or no*

assu(e only delayed&neutron 'recursor +rou'. We *ill

*rite its concentration as C. By the radioactive&decay la* the decay rate o4 the

'recursor is C *here is the decay constant o4 the

'recursor. 8here is delayed neutron born 4ro( the decay o4

'recursor nuclide. 8here4ore the 'roduction rate o4 the

'recursor (ust be e5ual to the 4ission rate ti(es d

"$!nffd ==


5/38

2009 Januar 5


8here4ore the e5uation 4or the evolution o4 the 'recursorconcentration is !'roduction rate : decay rate"

and the ne* e5uation 4or the evolution o4 the neutrondensity is

8he above cou'led e5uations are the 'oint&;ineticse5uations in ener+y +rou' *ith delayed&neutron&

'recursor +rou'.

"0!Cdt

dCf =

"!"!"!"!"!"! # CtrntrnDBtrn

t

trnaf +=


6/38

2009 Januar 6

Point&Kinetics E5uations *ith Precursor

t the end o4 the 'revious 'resentation *e had


7/382009 Januar 7

Point&Kinetics E5uations *ith Precursor !cont."

2i('li4y E5. !#" by 4actori>in+ 4ro( 4irst 3 ter(s

( )

( )

( )( )

( )( ) ( ) ?"?#!

..

"1!

"3!

"?#!"!"!

#

#

trCtrnt

trn

eiCtrnkt

trn

getto

kDBkand

timegenerationneutron

useweNow

trCtrnDB

t

trn

eff

effa

f

eff

f

f

af

+

=

+

=

=+

=

=

+

+

=

f


8/38

2009 Januar 8

Point&Kinetics E5uations *ith Precursor !cont."

2o let-s re*rite the 4inal 'oint&;inetics e5uations 4or theneutron density and 'recursor concentration in ter(s o4

ti(e only !*rite e5uation 4or n4irst as 'er convention"

We can also re*rite the e5uations in ter(s o4 usin+

but this *ill involve

"!

"6!

Cndt

dC

Cndt

dn

=

+

=

"$!

"7!

Cdt

dC

Cdt

d

=

+

=

n=


9/38

2009 Januar 9

2olution o4 Point&Kinetics E5uation

,et-s try to solve the 'oint&;inetics e5uations !6" @ !". Note that there are # ti(e constants ! and A" *hich enter into

the e5uations so *e (ay e='ect that the ti(e evolution *ill

so(eho* involve these # ti(e constants. We have a syste( o4 t*o di44erential e5uations in st&order in #

variables nand C. 8he usual 4or( o4 solution tried 4or lineardi44erential e5uations is the e='onential 4or( so let-s try

2ubstitutin+ this 4or( into E5s. !6" @ !" yields a4ter division o4

both sides by the co((on 4actor et 4ound in every ter(

"0! tt CeCnen

( )"#!

"!

+

=

=

+

=

nCCnC

Cnn


10/38

2009 Januar 10

2olution o4 Point&Kinetics E5uation

8hese are in 4act # ho(o+eneous e5uations in # un;no*ns

We ;no* that there *ill be a non&trivial solution only i4 the deter(inant

is >ero i.e.

8here are # *ays to attac; this e5uation

ne is to reali>e that this is a 5uadratic e5uation *hose roots can be 4ound. We-llco(e bac; to this (ethod later.

8he other (ethod is +ra'hical. Cn this a''roach the 4or( o4 the e5uation usually

seen is obtained by *ritin+ in ter(s o4 l*. Re(e(ber 4ro( E5. !$" in the'revious 'resentation

( )

"1!0"!

"3!0

=+

=+

Cn

Cn

( ) ( )

( ) ( ) "/!0

..0

=++=++

ei

"!

== effk


11/38

2009 Januar 11

Cnhour E5uation

E5. !/" then beco(es

8his 4or( o4 the e5uation is called the Cnhour e5uation !4or delayed&neutron&'recursor +rou'".

8he Cnhour e5uation is not easy to solve as the le4t&hand side is adiscontinuous 4unction *hich +oes to at # values o4 !at Fvalues o4 i4 *e had done the analysis *ith (any delayed&neutron&'recursor +rou's".

8he *ay to visualise the solutions is to 'lot the le4t&hand side inter(s o4 and see *here it crosses a hori>ontal line at hei+ht.

( )( ) ( )

( )

"6!

0

=++

++

+=+

+

=++

or

followsasisolatedusuallyareintermsThe


12/38

2009 Januar 12

Cnhour E5uation 8o 'lot E5. !6" *e need to note the 4ollo*in+ 4eatures

.

"00!00

"6!

DD

heightatlinehorizontalaissidehandrightThe

asabovefromallyasymptotic

asbelowfromallyasymptotic

asandas

andasandasbecause

fornegativeforpositiveisatthroughgoes

sidehandleftThe

+

+

+

>>

=

=++

++

+

+


13/38

2009 Januar 13

Cnhour E5uation !cont."

8he 'lots o4 the l.h.s. and r.h.s. o4 E5. !6" are then as 4ollo*s

G)ro( ,a(arshH

8here are al*ays #

solutions 4or

WhenI 0 bothvalues are ne+ative

WhenJ 0 one

value is 'ositive and

the other is ne+ative. When 0 one

value is 0 the other is

is ne+ative.

D


14/38

2009 Januar 14

eneral 2olution 4or Precursor rou'

8hus there are al*ays # values o4 as solutions and

conse5uently the +eneral solution 4or the neutron density and the

'recursor concentration is a su( o4 # e='onentials

By convention let-s a+ree that 1is the al+ebraically lar+ersolution i.e. the ri+ht(ost one.

Note that l is very s(all !L (s" there4ore the vertical line at

Al is very (uch to the le4t o4 the line at !i.e. the 4i+ure is 4ar

4ro( bein+ to scale". 8hus althou+h is al+ebraically s(aller !i.e. it is the one on

the le4t" in absolute (a+nitude it is (uch lar+er than 1

( )

( ) "7!

"!

#

#

#

#

tt

tt

eCeCtC

enentn

+=

+=

"ar+!# positiveelisunless >>


15/38

2009 Januar 15

eneral 2olution 4or Precursor rou'

Physically l is the ti(e constant at *hich theneutron density !or 'o*er" *ould evolve *ithout

delayed neutrons !as *e sho*ed be4ore"

*hereas Ais the ti(e constant corres'ondin+to the delayed neutrons.

Because o4 the cou'led nature o4 the ;inetics

e5uations the ti(e constants 4or the evolution o4

the neutron density !and 'o*er" are not e5ual to

l or A but at least one o4 the ti(e constants isinter(ediate bet*een these values.


16/38

2009 Januar 16

eneral 2olution 4or Precursor rou' !cont."

2u((ari>in+ there4ore

Cn the +eneral solution then the e='onential ter( in

*ill 4irst die a*ay !5uite 5uic;ly" and the ter( in 1*ill

re(ain as the asy('totic ti(e evolution o4 the neutrondensity increasin+ e='onentially i4 J 0 decreasin+ e='onentially i4

! 0 and tendin+ to a constant value i4 " 0.

casesallinnegativeelfairlyandand

if

if

if

casesallin

"ar+!

00

00

00

0

#

#

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20 Kinetics With Delayed Neutrons