ht 2005t8: chain reaction1 chain reaction multiplication criticality conditions
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HT 2005 T8: Chain Reaction 1
Chain Reaction
Multiplication
Criticality Conditions
HT 2005 T8: Chain Reaction 2
Overview• Chain Reaction
• Conversion and Breeding
• Doubling Time
• Neutron Cycle in a Thermal Reactor
• Multiplication Factor k
• Simplified Fuel Cycle
• Fuel at Discharge
HT 2005 T8: Chain Reaction 3
Chain Reactionν
β
γ
γ
β
ν
n
23592 U
2 MeVn
0.1 eVn
23592 U
23592 U
HT 2005 T8: Chain Reaction 4
Uranium Fuel
238U
4.5 109 yr
235U0.7 109 yr
234U247000 yr
Application
Unat 99.27% 0.72% 0.006% CANDU (D2O)
UE 98-95% 2-5% 0.006% LWR (H2O)
DU (UD) 99.8% 0.2% 0.006%Fast Reactor (FR, FBR)
Armour for tanksArmour-piercing
HT 2005 T8: Chain Reaction 5
Conversion or Breeding
238 23992 92U Un
23993 Np
23.5min
23994 Pu
2.35 d
24110 yr
Fertile:
Fissile:
232 23390 90Th Thn
23391Pa
22.3min
23392 U
27.0 d
159200 yr
HT 2005 T8: Chain Reaction 6
Conversion
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
238 U
capture
total
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
239 Pu
capture
fission
HT 2005 T8: Chain Reaction 7
Conversion
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
232 Th
capture
fission
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
233 U
capture
fission
HT 2005 T8: Chain Reaction 8
Neutron Cycle in Reactor
1
2
N
Nk
n/
fissi
on
N1
N2Leakage
Fast fission
Resonance abs.
Non-fuel abs.
Leakage
Non-fissile abs.
Fission
Slo
win
g d
ow
n
Ene
rgy
E
2 MeV
1 eV
200 MeV/fission
ν ≈ 2.5
1N 1FNLP N
1FNLp P N
1FNLf pP N
1TNL FNLP fpP N 1f TNL FNLP P fpP N
2 1f TNL FNLN P P fpP N
HT 2005 T8: Chain Reaction 9
Fast Fission
HT 2005 T8: Chain Reaction 10
Simple Criticality CalculationsNumber of fission neutrons due to both fast and thermal fission
Number of fission neutrons due to only thermal fission
Resonance escape probability
thermal utilizationFa
F Othera a
F Fff
f Fa
p
f
P
F
a
2 1
2
1
1
FNL TNL f
Ff
f
FNL TNL
Fa
FNL TNL
N N P pfP P
P
Nk
Nfp P
P k pP
P
f
1.65 0.71 0.87 0.97 0.99
1.04 1.00FNL TNLf p P P
k k
HT 2005 T8: Chain Reaction 11
Fuel Cycle (Thermal)
FuelFactory
ThermalReactor
nat
UE
or U
DU
FissileMaterials
Actinides(MA)
Fiss. Pr.(FP)
Spent Fuel
natU
HT 2005 T8: Chain Reaction 12
Fuel Cycle (Fast)
FuelFactory
FastReactor238 239
Initial Loading
U PuFP+MA
Spent Fuel
Spent Fuel
239 Pu
238 UDU
HT 2005 T8: Chain Reaction 13
Neutron Economy
HT 2005 T8: Chain Reaction 14
Breeding Factor
1 neutronabsorbed
in fuel
η neutronsproduced
B neutronscapturedin fertile
Non-fuel captureand lossesC+L
1 neutron tomaintain
chain reaction
2 ( ) 2
1 ( )
1 ( ) 1
2.2
B C L
B C L C L
HT 2005 T8: Chain Reaction 15
η Factor
η
Isotope(100%)
Thermal (0.025eV)
Fast (>0.5 MeV)
235U 2.07 2.35233U 2.29 2.40
239Pu 2.15 2.90
HT 2005 T8: Chain Reaction 16
Fission Neutrons239Pu
235U
233U
235
239
233
2.432 0.066
2.349 0.1
2.482 0.
50
075
2.41
2.8
2 0.136
44 0.138 E
E
E
E
E
E = 1
HT 2005 T8: Chain Reaction 17
Conversion Factor
If B < 1 it is called conversion factor, C.
Number of 239Pu created per 1 neutron absorbed
238235
235 1aFNL TNL
a
C p P P
ReactorSystem
InitialFuel
ConversionCycle
ConversionRation
BWR 235U (2-4%) 238U → 239Pu 0.6
PWR 235U (2-4%) 238U → 239Pu 0.6
PHWR (CANDU) Unat238U → 239Pu 0.8
HTGR 235U (5%) 232Th → 233U 0.8
LMFBR 239Pu (10-20%) 238U → 239Pu 1.0 – 1.6
HT 2005 T8: Chain Reaction 18
Neutron Cycle in Reactor
1
2
N
Nk
n/
fissi
on
N1
N2Leakage
Fast fission
Resonance abs.
Non-fuel abs.
Leakage
Non-fissile abs.
Fission
Slo
win
g d
ow
n
Ene
rgy
E
2 MeV
1 eV
200 MeV/fission
ν ≈ 2.5
1N 1FNLP N
1FNLp P N
1FNLf pP N
1TNL FNLP fpP N 1f TNL FNLP P fpP N
2 1f TNL FNLN P P fpP N
HT 2005 T8: Chain Reaction 19
Chain Reaction in Unat?
235 235 235
235 235 238 235 235 235 235 238 238
235 235
235 235 238 238 235 235 235 238 1
5792.42 0.547 1.32
579 101 2.72 139
ff
f c c f c c
ff
f c c f c c
N
N N N
N N e e
2
2
Moderator = H O not possible (LWR)
Moderator = D O possible (PHWR)
Moderator = Graphite possible (RBMK)
HT 2005 T8: Chain Reaction 20
Doubling TimeTd for a breeder reactor is the amount of time required for the original fissile loading to double
3
atoms[ ]
gW
Volume Rating [ ]cm
WMass Rating [ ]
g is Mass of Fissile Fuel
1
F
ff ff
f F f
F
C F F a
P F F a
net P C F F a
N
R N
R N
M
atomsR M N
satoms
R BM Ns
R R R B M N
1f F
d net F
fd
F
a
NT
B
T M
R
R N
HT 2005 T8: Chain Reaction 21
Doubling Time for FR
11
21
239
3.36 10
2.52 10
2.15 ; 1.8
500Pu
1
22
f F fd
a
f
d
F
a f
Jfission
atomsN
gbarn barn
MWR
ton
NT
B R
nT
eyr
HT 2005 T8: Chain Reaction 22
Multiplication Factor
2
1
1: subcritical
1: critical
1: supercritical
Nk
N
0
( ) is Production rate
( ) is Loss rate
( ) ( ); Lifetime of a neutron generation
( ) ( )
( ) ( ) 1( ) ( ) 1 ( ) ( )
( )
1( ) exp
P t
L t
P t N tk l
L t L t
dN t P t kP t L t L t N t
dt
kN t N t
l
L t l
t
( )N t
HT 2005 T8: Chain Reaction 23
Chain Reaction by Fast N.
235U
( 1)0 0
3
9
23 241
99
9
( )
Reactor Period1
19 ; 1.2 ; 6
( 2 ) 2 10
19 6 10 6 100.29
2353.4
1 3.4 1.7 102 10
15 8.5 10
5
t k l t T
f t
t t A t
tt t
f
t
N t N e N e
lT
kg cm b b
v E MeV cm s
N N cmA
cm sv
l s
9
8
8
8.5 101.85 10
0.85( )
exp(0) 10
k T s
N t tN
HT 2005 T8: Chain Reaction 24
23
2424
23
235 24
24 78
235
1 6 10 235g
3 103 10 can (theoretically) fission 235 1175g
6 10
10% leackage 1kg U is fissioned by 3 10 neutrons
N(t)3 10 exp 5.6 10 s
1 10
Fission of 1 g of U 1MWd 8.
gatom atoms
tt
4
235 7
6 10 MJ 20 t TNT
1 kg of U 20kt TNT within 5.6 10 s
Time to Fission 1 kg 235U
HT 2005 T8: Chain Reaction 25
Energy Comparison
10
235 4
235
235 -7 14
TNT = Tri Nitro Toluol
1 kg of TNT = 4.2 MJ
1W = 3.1 10 fission s
Fission of 1 g of U 8.6 10 MJ 20 t TNT
Fission of 1 kg of U 20kt TNT
Bomb : 1 kg of U is fissioned within 6 10 s P=1.5 10 MW
NPP (
235 3F-3): 1 kg of U is fissioned within 8 hours P=3 10 MW
F-3 consumes 3 bombs a day (as that droped on Nagasaki)
HT 2005 T8: Chain Reaction 26
Distribution of FP Mass
HT 2005 T8: Chain Reaction 27
Nuclide Half-life (year)
Discharge from 3000 MWt LWR
(kg/a)
Actinides 237Np 2 140 000 14.5 238Pu 80 4.5 239Pu 24 000 166.0 240Pu 6 600 76.7 241Pu 14 25.4 242Pu 374 000 15.5
241Am 433 16.6 243Am 7 400 3.0 244Cm 18 0.6
FISSION PRODUCTS 79Se 65 000 0.2 85Kr 11 0.1 90Sr 28 13.4 93Zr 1 500 000 13.4 99Tc 210 000 23.2
107Pd 6 500 000 7.3 126Sn 100 000 1.0
129I 15 700 000 5.8 135Cs 3 000 000 9.4 137Cs 30 31.8 151Sm 90 0.4
HT 2005 T8: Chain Reaction 28
Long-Lived Radio Isotopes
237N
p
238P
u
239P
u
240P
u
241P
u
242P
u
241A
m
243A
m
244C
m
79S
e
85K
r
90S
r
93Z
r
99T
c
107P
d
126S
n
129I
135C
s
137C
s
151S
m
0,1
110
100
0,1
110
100
Long
-live
d ra
dioa
ctiv
e is
otop
es
in L
WR
spe
nt fu
el
Isotope
Dis
char
ge fr
om 3
000M
Wth L
WR
(kg
/yea
r)
HT 2005 T8: Chain Reaction 29
Half-Lives of L.L. Isotopes
237N
p23
8Pu
239P
u24
0Pu
241P
u24
2Pu
241A
m24
3Am
244C
m79
Se
85K
r90
Sr
93Z
r99
Tc
107P
d12
6Sn
129I
135C
s13
7Cs
151S
m 100
101
102
103
104
105
106
107
Hal
f-lif
e of
long
-liv
ed is
otop
esin
LW
R s
pent
fuel
Isotope
Hal
f-lif
e (y
ears
)
HT 2005 T8: Chain Reaction 30
The END