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Chemical Engineering 412

Introductory Nuclear Engineering

Lecture 14Nuclear Reactor Theory I

Six Factor Formula

Spiritual Thought

“’Things will work out’ may well be President Hinckley's most repeated assurance to family, friends, and associates. ‘Keep trying’ he will say. ‘Be believing. Be happy. Don’t get discouraged. Things will work out’”

Elder Jeffrey R. Holland

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The Death of Kirk3

OEP #3 (due 2/21/18)

Curse you Kahn! In the epic battle against the USS Vengeance, The Enterprise is badly damaged, and begins careening towards San Francisco on earth (I know, I know… but on this planet earth, San Francisco is the headquarters for Starfleet). Kirk heroically enters the fusion core chamber to realign the broken fission supports. In so doing, he is exposed to a lethal neutron dose, which permeates the chamber he is in. In this scene, Spock is coming to grips with the fact that his friend is dying, and a “touching” moment ensues in which they touch the glass opposite of each other’s hand with their own hand. In order for Spock to not also die from neutron radiation poisoning, the flux (uncollided) must be reduced by 1000 times. Is this possible?

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The BIG Picture5

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)