# Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1 ... Heterogeneous Reactors, Reactor...

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Heterogeneous Reactors, Reactor Kinetics-1.. 1

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Heterogeneous Reactors, Reactor Kinetics-1.. 2

Laboratory for Reactor Physics and Systems Behaviour

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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 > cell regions • E.g. MTR fuel element

– Plate-type fuel ⇒ Quasi-homogeneous reactor

Heterogeneous Reactors, Reactor Kinetics-1.. 3

Laboratory for Reactor Physics and Systems Behaviour

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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…

Heterogeneous Reactors, Reactor Kinetics-1.. 4

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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

Heterogeneous Reactors, Reactor Kinetics-1.. 5

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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. ⇒

Heterogeneous Reactors, Reactor Kinetics-1.. 6

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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

Heterogeneous Reactors, Reactor Kinetics-1.. 7

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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

Heterogeneous Reactors, Reactor Kinetics-1.. 8

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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)

Heterogeneous Reactors, Reactor Kinetics-1.. 9

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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)

Heterogeneous Reactors, Reactor Kinetics-1.. 10

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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

Heterogeneous Reactors, Reactor Kinetics-1.. 11

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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.

Heterogeneous Reactors, Reactor Kinetics-1.. 12

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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 (%)

Heterogeneous Reactors, Reactor Kinetics-1.. 15

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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…)

Heterogeneous Reactors, Reactor Kinetics-1.. 16

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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)