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Six Factor Formula

(Reactor Physics)

Presented by:Seraj ul Haq, PE

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Contents

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Revision

Fast Neutrons, Thermal Neutrons?

Fission?

Fast, Thermal Cross-section?

Resonance? Resonance Peaks?

Probability? Sun will set today: Probability =? [0, +0.x, >1]

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Multiplication Factor k

k =

k < 1, Dieing reaction k = 1, Sustained reaction k > 1, Increasing (uncontrollable?) reaction

k = 0 (??), k < 0 (???)

Neutron production from fission in one generation

Neutron absorption in the preceding generation

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Fast Fission Factor,

=

Cross section of fast fission is small Also effected by fuel and Mod arrangement Normally 1 in homo, b/c fuel atoms surrounded by mod 1.03 in hetro

T, P, Enrichment, poisons effect ???

Number of fast neutrons produced by all fissions

Number of fast neutrons produced by thermal fissions

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Resonance Escape Probability, p

p =Number of fast neutrons that reach thermal energy

Number of fast neutrons that start slow down

Resonance range for U-238 is 6 eV 200 eV

In homo, p is small compared to hetro (why ?)

p mainly effected by fuel temp due to Doppler effect (?), mod tempdue to change in density (?)

Will poison effect?

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Thermal Utilization Factor, f

f =

Effected by change in T, not directly but due to moderator inventory

(How?)Poison (How?), Enrichment (?)

Value always < 1 (?)

Number of thermal neutrons absorbed in fuel

Number of thermal neutrons absorbed in all Rx materials

p

a

m

a

U

a

U

a

ppp

a

mmm

a

UUU

a

UUU

a

VVV

V

p

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Reproduction Factor,

=Number of fast neutrons produced by all fissions

Number of fast neutrons produced by thermal fissions

Will Temp effect (How?)

Poison (How?), Enrichment (?)

Value always < 1 (?)

238238235235

235235235

aa

f

UU

a

UU

f

NN

N

Single nucleus

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Multiplication Factor k

k =

Supposed all neutrons remain within the assembly

Neutron production from fission in one generation

Neutron absorption in the preceding generation

fpk

Four factor formula

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Effective Multiplication Factor

kef =Neutron production from fission in one generation

Neutron absorption in thepreceding generation

Will Four factor formula work?

Neutron Leaked in thepreceding generation+

Neutron production from fission in one generation

Neutron production from all fissions in preceding generation (?)kef =

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Fast Non-Leakage Probability, Lf

L =Number of fast neutrons that do not leak off from the Rx

Number of fast neutrons produced by all fissions

Always less than 1 in practical Rx?

Will Temp. effect?

Will Geometry effect

f

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Thermal Non-Leakage Probability, Lt

L =Number of thermal neutrons that do not leak off from the Rx

Number of neutrons that reach thermal energies

Always less than 1 in practical Rx?

Will Temp. effect?

Will Geometry effect

t

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Effective Multiplication Factor

k = Neutron production from fission in one generation

Neutron absorption in the preceding generation

tfLLfpk

Six factor formula

eff

eff

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Neutrons in the nth Generation

Knowing the value of Keff, we can find

Where

Nn neutrons in nth generation

N0 neutrons in firstgeneration

n

eff0n )(kNN

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Critical, Sub/Super Critical

If Keff = 1, Rx is Critical

Can be critical at any power value

Remember, power doesn't depend upon k

For Keff < 1, Rx is Sub-Critical For Keff > 1, Rx is Super-Critical

How can we increase or decrease Keff of apractical reactor? Think!

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SixFactorFo

rmula

(Explanation)

140 FAST NEUTRONS LEAK OUT

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NOpLf

100 THERMALNEUTRONS LEAK

OUT FROM THE

CORE

180

NEUTRONS

ABSORBED

BY

RESONAN

CE PEAKSFAST

FISSION

= 1.04

FAST

NON-

LEAKAGE

Lf= 0.865

RESONAN

CE

ESCAPE

p = 0.80

THERAM

NON-

LEAKAGE

Lt = 0.881

THERMA

UTILIZATIO

N

f = 0.799

REPRODUCATION

= 2.02

1000 NEUTRONS AT START OF

GENERATION

NET INCREASE OF

40 NEUTRONS

FROM FAST

FISSION

NO

NO

1040

NEUTRONS

140 FAST NEUTRONS LEAK OUT

FROM THE CORE

900NEUTRONS

NOLf

720

NEUTRONS

620

NEUTRONS

NOpLfLt

120 NEUTRONS ABSORBED IN

NON FUEL

495

NEUTRONS

NOpLfLtf

NET INCREASE OF 505

NEUTRONS FROM

THERMAL FISSION

1000 NEUTRONS

k NO

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Reactivity, The gain or loss in neutron population (N0keff N0),

expressed as a fraction of the present generation (N0keff)

Fractional variation of keff from 1

eff0

0eff0

kN

NkN

eff

eff

k

1k

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Reactivity, Reactivity () could be 0,

>0 or

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Units of Reactivity,

A friction; dimensionless

Can be expressed as simple

fraction, percent Some times expressed as

pcm (Percent milli) = 10-5

Dollar ($) [reactivity when Rx is

prompt critical; = 0.0068

k

1k

k

k

k

k

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For Tomorrow

1. Reactivity Coefficients

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Thank you for your attention