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Chemical Engineering 412

Introductory Nuclear Engineering

Lecture 14Nuclear Reactor Theory I

Nuclear Criticality

Core Nomenclature

• Fuel – fissile material in core• Heavy atoms – generally fissile, fertile, and fissionable

material• Moderator – collision material that slows neutrons• Cladding – fuel containment• Reflector – core wrapping that minimizes neutron

leakage• Blanket – material used to produce useful isotopes from

core• Shielding – neutron (and other radiation) absorbing

material to protect personnel and instrumentation

Fast Neutron Life Cycle

• What happens to fast neutrons?

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Fast Neutron Factors

• Fast Fission, ε• Resonance Escape Probability, p• thermal utilization, f• Fission factor, η• Fast non-leakage probability, 𝑃𝑃𝑁𝑁𝑁𝑁

𝑓𝑓

• Thermal non-leakage probability, 𝑃𝑃𝑁𝑁𝑁𝑁𝑡𝑡𝑡

• Investigate for two reactor types:– Homogenous– Real (heterogeneous)

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Six Factor Formula

𝑛𝑛′ ≡ 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑃𝑃𝑁𝑁𝑁𝑁𝑓𝑓 𝑃𝑃𝑁𝑁𝑁𝑁𝑡𝑡𝑡

𝑛𝑛′ next generation neutrons𝑛𝑛 neutrons produced per thermal fission𝑛𝑛 ratio of total neutrons to thermal neutrons (1.0-1.08)𝑛𝑛 resonance escape probability (0.8-0.9)𝑛𝑛 thermal utilization = Σ𝑎𝑎𝐹𝐹𝜙𝜙𝐹𝐹𝑉𝑉𝐹𝐹

Σ𝑎𝑎𝐹𝐹𝜙𝜙𝐹𝐹𝑉𝑉𝐹𝐹+Σ𝑎𝑎𝑁𝑁𝐹𝐹𝜙𝜙𝑁𝑁𝐹𝐹𝑉𝑉𝑁𝑁𝐹𝐹(0-1)

𝑛𝑛 fission factor = 𝜈𝜈Σ𝑓𝑓𝐹𝐹

Σ𝑎𝑎𝐹𝐹(2.0-2.2)

𝑃𝑃𝑁𝑁𝑁𝑁𝑓𝑓 non-leakage of fast neutrons = exp−𝐵𝐵𝑐𝑐2𝜏𝜏 (near 1)

𝑃𝑃𝑁𝑁𝑁𝑁𝑡𝑡𝑡 non-leakage of thermal neutrons = 11+𝑁𝑁2𝐵𝐵𝑐𝑐2

(near 1)

Quantitative Neutron Cycle

𝑘𝑘 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑃𝑃𝑁𝑁𝑁𝑁𝑓𝑓 𝑃𝑃𝑁𝑁𝑡𝑡𝑡

𝑘𝑘∞ = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛

Thermal Fission Factor, η

𝑛𝑛 ≡ 𝜈𝜈Σ𝑓𝑓Σ𝑎𝑎

= 𝜈𝜈𝜎𝜎𝑓𝑓235𝑁𝑁235

𝜎𝜎𝑎𝑎235𝑁𝑁235 + 𝜎𝜎𝑎𝑎238𝑁𝑁238 =𝑣𝑣235𝜎𝜎𝑓𝑓235

𝜎𝜎𝑎𝑎235 + 𝜎𝜎𝑎𝑎238𝑁𝑁238

𝑁𝑁235

𝑒𝑒 =𝑁𝑁235

𝑁𝑁235 + 𝑁𝑁238

⇒𝑁𝑁238

𝑁𝑁235 =1 − 𝑒𝑒𝑒𝑒

𝑛𝑛 =𝑣𝑣235𝜎𝜎𝑓𝑓235

𝜎𝜎𝑎𝑎235 + 𝜎𝜎𝑎𝑎2381 − 𝑒𝑒𝑒𝑒

For a homogenous reactor: ϕF=ϕNF

𝜼𝜼 variation with fuel enrichment

Conceptual Definitions

• 𝐿𝐿 = ½ distance thermal neutron travels from point of thermalization to absorption

• 𝜏𝜏 = 1/6 mean squared distance from initial (birth) point to thermalization point

• Neutrons• Travel further in fast spectrum

• 30 vs. 6 cm in light water• Spend far more time as thermal neutrons

• Probability of non-leakage• Approaches 100% as reactor dimensions

become infinite• Both fast and thermal neutrons.

Non-leakage probabilities

𝑃𝑃𝑁𝑁𝑁𝑁𝑡𝑡𝑡 =1

1 + 𝐿𝐿2𝐵𝐵𝑐𝑐2

𝐿𝐿2 ≡𝐷𝐷Σ𝑎𝑎

=𝐷𝐷𝑀𝑀

Σ𝑎𝑎𝑀𝑀Σ𝑎𝑎𝑀𝑀

Σ𝑎𝑎𝑀𝑀 + Σ𝑎𝑎𝐹𝐹= 𝐿𝐿𝑀𝑀2 1 −

Σ𝑎𝑎𝐹𝐹

Σ𝑎𝑎𝑀𝑀 + Σ𝑎𝑎𝐹𝐹= 𝐿𝐿𝑀𝑀2 1 − 𝑛𝑛

𝐷𝐷 thermal diff. coefficient𝐿𝐿 thermal diff. length

• 𝐵𝐵𝑐𝑐 critical buckling – comes from reactor geometry• Table 10.6• Derived from Neutron Diffusion Equation• Balance of size and geometry of reactor

Σ𝑎𝑎 absorption cross-section𝑛𝑛 fuel utilization factor

Non-leakage probabilities

𝑃𝑃𝑁𝑁𝑁𝑁𝑓𝑓 = exp −𝐵𝐵𝑐𝑐2𝜏𝜏

τ ≈ 𝜏𝜏𝑀𝑀

𝜏𝜏 Fermi age

𝐵𝐵𝑐𝑐 critical buckling – comes from reactor geometry (Table 10.6)

Critical Bare Reactor Summary

29.34/cos

64.3/63.3cos405.2J405.23

32.2/738.0405.2J405.21

88.3/85.3coscoscos3

57.1/57.1cos1

22

0

22

20

2

222

2

fR

fR

fR

fR

fR

ERPRr

rA

Rsphere

VEPHz

RA

Hz

RDcylinder

ERPR

AR

Dcylinder

EVPcz

by

axA

cbaDplate

aEPaxA

aDplate

Σ

Σ

+

−

Σ

−

Σ

+

+

−

Σ

−

ππ

π

ππππππ

ππ

geometry Buckling (𝐵𝐵𝑔𝑔2) Flux Aavφ

φmax=Ω

Multiplication Factor

𝑘𝑘𝑒𝑒𝑓𝑓𝑓𝑓 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑃𝑃𝑁𝑁𝑁𝑁𝑓𝑓 𝑃𝑃𝑁𝑁𝑁𝑁𝑡𝑡𝑡

𝑘𝑘∞ = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛

𝑘𝑘𝑒𝑒𝑓𝑓𝑓𝑓 ≡𝑛𝑛𝑒𝑒𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑛𝑛 𝑛𝑛𝑛𝑛𝑝𝑝𝑛𝑛𝑛𝑛 𝑝𝑝𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑒𝑒

𝑛𝑛𝑒𝑒𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑛𝑛 𝑛𝑛𝑎𝑎𝑠𝑠𝑒𝑒 𝑛𝑛𝑛𝑛𝑝𝑝𝑛𝑛𝑛𝑛 𝑝𝑝𝑛𝑛 𝑛𝑛𝑛𝑛𝑒𝑒𝑣𝑣𝑝𝑝𝑛𝑛𝑛𝑛𝑛𝑛 𝑔𝑔𝑒𝑒𝑛𝑛𝑒𝑒𝑛𝑛𝑎𝑎𝑛𝑛𝑝𝑝𝑛𝑛𝑛𝑛

𝑘𝑘𝑒𝑒𝑓𝑓𝑓𝑓 =𝑛𝑛′

𝑛𝑛

𝒌𝒌∞variation with fuel:modifier ratio (HM/H)

𝒌𝒌∞variation with (HM/H)

Neutron Cycle Parameters

Natural uranium and moderator in homogeneous reactor

A Few Parameters

neutronsoriginalgenerationoneafterneutronsk

fissionneutrons

f

f

a

f

f

=

+=

+==

=

=

αν

σσσν

σσνη

σσ

α

ν

γ

γ

1

/ Total (prompt and delayed) neutrons produced per fission

Capture to fission ratio

Neutrons released per absorption(> 1 converter, > 2 breeder)

Multiplication cycle

consumedatomsfissileproducedatomsfissileC =

conversion/breeding ratio(>1 breeder reactor)

Typical Parameter Variation

Homogeneous reactor, 2% enrichment