Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.....
Transcript of Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.....
Heterogeneous Reactors, Reactor Kinetics-1.. 1
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Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1
Quasi-homogeneous, Heterogeneous Reactors “Homogenised” Macroscopic Cross-sections f , p , ε for a Unit Cell k∞ for a Thermal Reactor Lattice
Reactor Kinetics (Point Reactor Model) Kinetics without Delayed Neutrons
• Prompt Neutron Lifetime
Kinetics with Delayed Neutrons • Kinetics Parameters, Point Kinetics Equations
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Quasi-homogeneous and Heterogeneous Reactors
In general, reactor core consists of lattice of uniformly spaced, fuel rods … • Repetitive “unit cell” pattern (fuel / clad / coolant / moderator…) • Distance between the central axes of 2 neighbouring cells : lattice pitch
Ususally λt << size of individual regions of cell • Heterogeneous reactor
If λt >> cell regions • E.g. MTR fuel element
– Plate-type fuel ⇒ Quasi-homogeneous reactor
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Homogenised Σi ’s
In the quasi-homogeneous case, one simply needs to weight by volume…
Situation more complex for the heterogeneous case • Need to consider flux depression in the fuel • E.g. for a homogeneous thermal reactor…
• For a heterogeneous lattice…
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f for a thermal lattice
For a cell “ fuel / moderator ”
⇒
Considering equivalent cylindrical cell (Wigner-Seitz approximation),
⇒ F , E : “lattice functions”
e.g. on a diffusion-theory basis (not strictly valid):
with x = a/Lc , y = a/Lm , z = b/Lm
and
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p for a lattice
We have seen:
- Ieff independent of moderator (depends on fertile material and lattice geometry)
Semi-empirical results available for Ieff , e.g.
For cylindrical rods:
(Nc pertains to fertile…)
e.g. from Lamarsh: Introduction to Nucl. Engg. ⇒
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ε , Fast Fission Factor ε takes account of fissions in fertile material (E > Es ~ 1.35 MeV for U238)
For a heterog. lattice, one may show:
Typical values for ε : • Large lattices (e.g. Unat / graphite) ~ 1.03 - 1.05 • Tight lattices (e.g. enr. UO2 / H2O) ~ 1.05 - 1.10
Pcc : probability for n emitted in fuel to have 1st
collision there … depends on ( )fuel
- σ ’s all refer to U238 (1-group, fast)
- large dependence on Vc/Vm
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k∞ for a Lattice (Thermal Reactor)
For example, for Unat + graphite ,
→ For a heterogeneous sytem, one can have k∞ ~ 1,1 • First critical reactor configuration… “Fermi Pile” at Chicago, 2nd Dec., 1942
N.B.: For keff , one needs to estimate for the lattice…
For an enrichment less than ~ 5%,
(as before) and
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Reactor Kinetics
In general, one seeks to determine • Time-dependent diffusion equation needs to be solved numerically
For the global behaviour, a simplification can be made • “Point kinetics” equations for the total neutron population • Does not describe spatial effects in large complex systems, but very useful…
Two cases may be considered for the time-dependent behaviour • Without delayed neutrons (hypothetical) • Real situation (with delayed neutrons)
One particular case, can be considered analytically • Step change in keff → Leads to Reactivity Equation (Inhour Equation)
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Point Kinetics without Delayed Neutrons
For the neutron population:
Using:
, i.e.
Prompt Kinetics Equation
For a constant keff :
⇒ If keff > 1… P ↑ (supercritical system) If keff < 1… P ↓ (subcritical system)
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Physical Significance of l For a hypothetical passive medium with same cross-sections but keff = 0 (e.g. = 0…)
l is same and
Result is analogous to the law of radioactive decay : 1/l is like λ , i.e. l is like T…
Thus, l is neutron lifetime • Measure of time taken for “disappearance” of the n’s (P ↓) , in face of absorption, leakage…
Like keff , l may be calculated on the basis of different theories (diffusion, 1-group, multigroup, multizone,…, transport,…) • Consider particular case: bare homogeneous reactor, analysed via 1-group diffusion theory
€
ν
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l (contd.) One has: with
Leakage ~ supplementary absorptions corresponding to:
Thus,
For an infinite system: (thermal diff. time; slowing-down time negligible…)
With (independent of v)
One may write:
(independent of P) i.e.
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Prompt Period of a Reactor
For the reactor without delayed neutrons,
td for different moderators:
If ,
For a fast reactor, ⇒ Factor of 148 in < 1ms !
Thus, typically,
⇒ Reactors would be almost impossible to control…
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Delayed Neutrons
Small fraction of the neutrons, not prompt (~ 0.6% for U235) • Produced by disintegration of FP’s, e.g.
Many different “precursors”
• ~ 6 groups (of precursors, i.e. of delayed neutrons)
• yi, Ti ⇒ βi, λi (i = 1,6)
created “with delay” ↓
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Delayed Neutron Parameters
(U235)
- Eavg of delayed n’s ~ 0.4MeV
- λi’s relatively constant
- βi’s depend on nuclide, e.g.
β = Sum (βi) = 0.21% for Pu239
= 0.26% for U233 … other “fissiles”
- β small, but very important for control of the chain reaction ⇒ kinetic behaviour
- Response of a reactor which becomes slightly supercritical, much slower
Gp. Precursors T1/2 (s) λi (s-1) βi (%)
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Point Kinetics with Delayed Neutrons
Fraction β of n’s in reactor are delayed, so that the neutron production rate
It is, in fact :
Thus,
As before, substituting
keff , l : reactor characteristics indep. of P, may be calculated (e.g. 1-gp. diff. theory…)
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Point Kinetics with Delayed Neutrons (contd.)
One has:
Supplementary eqns. needed for Ci ’s (precursor equations)
⇒
With the definitions:
prompt neutron lifetime
… (1)
… (2)
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Point Kinetics with Delayed Neutrons (contd.2) ⇒ (1) and (2) : complete system of 7 linear differential eqns. (Point Kinetics Equations)
(Very important basis for studies of kinetics, reactor stability, nuclear safety, etc.)
N.B.: ρ… deviation of keff from 1 (normally very small, but very wide range: - ∞ to 1)
Stationary Case:
Substituting for Ci (0) into ,
Because , so that the only non-trivial solution is:
⇒ Stationary states correspond to ρ = 0 (keff = 1), and the delayed neutrons have no effect…
and
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Summary, Lesson 12
“Homogenisation” of cross-sections for a reactor lattice
Lattice parameters (f , p , ε , k∞ for unit cell)
Reactor kinetics (Point Reactor)
Kinetics without delayed neutrons • Prompt neutron lifetime
Kinetics with delayed neutrons • Role of precursors
Point Kinetics Equations