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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.
8/9/2019 20 Kinetics With Delayed Neutrons
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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
=
==
+=
=
==
8/9/2019 20 Kinetics With Delayed Neutrons
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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 =
8/9/2019 20 Kinetics With Delayed Neutrons
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2009 Januar 4
Pro('t and Delayed 2ources
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
"$!nffd ==
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2009 Januar 5
Pro('t and Delayed 2ources
8here4ore the e5uation 4or the evolution o4 the 'recursorconcentration is !'roduction rate : decay rate"
and the ne* e5uation 4or the evolution o4 the neutrondensity is
8he above cou'led e5uations are the 'oint&;ineticse5uations in ener+y +rou' *ith delayed&neutron&
'recursor +rou'.
"0!Cdt
dCf =
"!"!"!"!"!"! # CtrntrnDBtrn
t
trnaf +=
8/9/2019 20 Kinetics With Delayed Neutrons
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2009 Januar 6
Point&Kinetics E5uations *ith Precursor
t the end o4 the 'revious 'resentation *e had
8/9/2019 20 Kinetics With Delayed Neutrons
7/382009 Januar 7
Point&Kinetics E5uations *ith Precursor !cont."
2i('li4y E5. !#" by 4actori>in+ 4ro( 4irst 3 ter(s
( )
( )
( )( )
( )( ) ( ) ?"?#!
..
"1!
"3!
"?#!"!"!
#
#
trCtrnt
trn
eiCtrnkt
trn
getto
kDBkand
timegenerationneutron
useweNow
trCtrnDB
t
trn
eff
effa
f
eff
f
f
af
+
=
+
=
=+
=
=
+
+
=
f
8/9/2019 20 Kinetics With Delayed Neutrons
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2009 Januar 8
Point&Kinetics E5uations *ith Precursor !cont."
2o let-s re*rite the 4inal 'oint&;inetics e5uations 4or theneutron density and 'recursor concentration in ter(s o4
ti(e only !*rite e5uation 4or n4irst as 'er convention"
We can also re*rite the e5uations in ter(s o4 usin+
but this *ill involve
"!
"6!
Cndt
dC
Cndt
dn
=
+
=
"$!
"7!
Cdt
dC
Cdt
d
=
+
=
n=
8/9/2019 20 Kinetics With Delayed Neutrons
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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
8/9/2019 20 Kinetics With Delayed Neutrons
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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
8/9/2019 20 Kinetics With Delayed Neutrons
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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
8/9/2019 20 Kinetics With Delayed Neutrons
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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
+
+
+
>>
=
=++
++
+
+
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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
8/9/2019 20 Kinetics With Delayed Neutrons
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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 >>
8/9/2019 20 Kinetics With Delayed Neutrons
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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.
8/9/2019 20 Kinetics With Delayed Neutrons
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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
#
#