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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 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)
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
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
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)