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

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Transcript of Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1 ... Heterogeneous Reactors, Reactor...

  • 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

    Neutronics

    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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

    Neutronics

    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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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

    Laboratory for Reactor Physics and Systems Behaviour

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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…

  • Heterogeneous Reactors, Reactor Kinetics-1.. 13

    Laboratory for Reactor Physics and Systems Behaviour

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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” ↓

  • Heterogeneous Reactors, Reactor Kinetics-1.. 14

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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

    Laboratory for Reactor Physics and Systems Behaviour

    Neutronics

    Point Kinetics with Delayed Neutrons (contd.)

      One has:

      Supplementary eqns. needed for Ci ’s (precursor equations)

      With the definitions:

    prompt neutron lifetime

    … (1)

    … (2)