Chain Reaction Multiplication Criticality Conditions

HT 2005 T8: Chain Reaction 1

Chain Reaction

Multiplication

Criticality Conditions


Overview• Chain Reaction

• Conversion and Breeding

• Doubling Time

• Neutron Cycle in a Thermal Reactor

• Multiplication Factor k

• Simplified Fuel Cycle

• Fuel at Discharge


Chain Reactionν

β

γ

γ

β

ν

n

23592 U

2 MeVn

0.1 eVn

23592 U

23592 U


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


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


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


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


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


Fast Fission


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


Fuel Cycle (Thermal)

FuelFactory

ThermalReactor

nat

UE

or U

DU

FissileMaterials

Actinides(MA)

Fiss. Pr.(FP)

Spent Fuel

natU


Fuel Cycle (Fast)

FuelFactory

FastReactor238 239

Initial Loading

U PuFP+MA

Spent Fuel

Spent Fuel

239 Pu

238 UDU


Neutron Economy


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


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


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


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


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


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)


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


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


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


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


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


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)


Distribution of FP Mass


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


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)


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

)


The END

Chain Reaction Multiplication Criticality Conditions

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