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

18
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

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

Page 1: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

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

Page 2: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

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 << size of individual regions of cell • Heterogeneous reactor

  If λt >> cell regions • E.g. MTR fuel element

– Plate-type fuel ⇒ Quasi-homogeneous reactor

Page 3: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 3

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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…

Page 4: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 4

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Page 5: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

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. ⇒

Page 6: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 6

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Page 7: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 7

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Page 8: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 8

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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)

Page 9: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 9

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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)

Page 10: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 10

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Page 11: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 11

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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.

Page 12: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 12

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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…

Page 13: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 13

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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” ↓

Page 14: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 14

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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 (%)

Page 15: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 15

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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…)

Page 16: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

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)

Page 17: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 17

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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

Page 18: Lesson 12: Heterogeneous Reactors, Reactor Kinetics-1...Heterogeneous Reactors, Reactor Kinetics-1.. 6 Laboratory for Reactor Physics and Systems Behaviour Neutronics ε, Fast Fission

Heterogeneous Reactors, Reactor Kinetics-1.. 18

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

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