Lesson 9: Multiplying Media (Reactors)

Transcription

1 Lesson 9: Multiplying Media (Reactors) Laboratory for Reactor Physics and Systems Behaviour Multiplication Factors Reactor Equation for a Bare, Homogeneous Reactor Geometrical, Material Buckling Spherical, Slab Reactors Cylindrical Reactor Absolute Value of Flux Maximum- to-average Flux Ratio Comments on Criticality Condition Multiplying Media.. 1

2 Media Containing Fuel Neutron propagation considered in passive media till now Characterised by Σ a, Σ t (or Σ s ), External neutron source(s) For a medium containing fissile material, one has a source per unit volume: arising from fissions Here, Q f is not a part of the data provided, but rather is dependent on the flux, i.e. on the sought solution itself The neutronics description yields one or several homogeneous equations Stationary solution exists only if a certain criticality condition is satisfied by the geometrical and material characteristics of the system Multiplying Media.. 2

3 Media Containing Fuel (contd.) For the critical state: productions = absorptions + leakage Condition ~ independent of the flux (all three reaction rates vary in about the same proportions) We will first consider a reactor which is homogeneous and bare Isolated zone, without a reflector or a blanket (i.e. just the core ) Treatment will first be one-group Monoenergetic neutrons (e.g. thermal with appropriately averaged cross-sections) Later will be considered Multizone reactors (e.g. with a reflector, i.e. an outer zone of pure moderator) Multigroup theory (basis used, in most practical cases, for numerical calculations) Certain aspects of the heterogeneity of the reactor lattice (heterogeneous unit-cell: fuel/moderator/ ) Multiplying Media.. 3

4 Multiplication Factors For the infinite medium, in general Multiplying Media.. 4

5 Multiplication Factors (contd.) η c fuel multiplication factor f utilisation factor (for the homogeneous mixture) Laboratory for Reactor Physics and Systems Behaviour For a heterogeneous unit-cell, one needs to consider that the flux is depressed within the fuel region Multiplying Media.. 5

6 System of Finite Size For the infinite medium: k = η c. f This corresponds to k = η c. f. ε. p, with ε = p = 1 All the neutrons have the same energy There are neither resonance absorptions, nor fast fissions For the finite system, one has the effective multiplication factor Multiplying Media.. 6

7 System of Finite Size (contd.) Laboratory for Reactor Physics and Systems Behaviour Three cases to be considered: k eff = 1 critical reactor (self-sustaining chain reaction, constant flux) k eff > 1 supercritical reactor (divergent reaction, increasing flux) k eff < 1 subcritical reactor (convergent reaction, decreasing flux) k eff most important single parameter for the functioning of a reactor Important to note: k eff < k < η c < ν Multiplying Media.. 7

8 Equation for the Critical Reactor Laboratory for Reactor Physics and Systems Behaviour One uses the 1-group diffusion equation for the stationary case (critical reactor) Thereby: Thus, with One-group Reactor Equation Eqn. homogeneous Can be solved for a given geometry, Condition to be satisfied can be identified Simplest systems homogeneous spherical reactor, slab reactor Material Buckling (depends only material properties) Multiplying Media.. 8

9 Spherical Reactor One has (positive sign, cf. passive medium with point source), i.e. For the finite system: 1. Φ at ρ = 0 C = 0 2. Φ = 0 at ρ = R+d = R+ 0.71λ t, i.e. sin {B m (R+d)} = 0 Multiplying Media.. 9

10 Spherical Reactor (contd.) From the condition sin {B m (R+d)} = 0 : For the critical reactor, only i = 1 is valid Smallest eigenvalue Fundamental Mode Other solutions: higher harmonics Exist only in a subcritical system E.g. near the external source B m = B i = iπ R + d for i = 1, 2, Critical condition for the spherical reactor is thus: Multiplying Media.. 10

11 Spherical Reactor (contd.2) The flux distribution is: For a given medium (specific values of criticality condition Critical radius: ), R is determined by the Critical mass: Conversely, if the size (R) is fixed, the material properties need to be identified which yield the appropriate (material buckling) E.g. adjust the enrichment, i.e. k Multiplying Media.. 11

12 Comments For a bare homogeneous reactor, the criticality condition demands an eigenvalue search for the reactor equation: Eigenvalues need to go to zero at the extrapolated outer surface The square of the smallest eigenvalue: B 2 (geometrical buckling) Criticality condition: Flux distribution: proportional to the eigenfunction corresponding to B 2 Absolute level of the flux not yet known A constant ( A, in the example considered) remains undetermined, depends on the power (determined, in turn, by technological constraints ) allows us to determine the criticality condition and the spatial distribution of the flux, but does not fix the latter s absolute value Multiplying Media.. 12

13 Slab Reactor System infinite in the X, Y directions Height H Flux, function only of z Reactor Equation: General solution: Flux, symmetric relative to plane z = 0 ( dφ/dz = 0 at z = 0 ) C = 0 Multiplying Media.. 13

14 Slab Reactor (contd.) Thus, Condition Φ = 0 at yields as eigenvalues: with Square of the smallest eigenvalue ( i = 0 ) (geometrical buckling) Flux distribution: Multiplying Media.. 14

15 Cylindical Reactor For a cylindrical reactor of height H and radius a, Critical Reactor Equation: With the asumption, Multiplying Media.. 15

16 Cylindical Reactor (contd.) Multiplying Media.. 16

17 Cylindical Reactor (contd.2) Comments: The criticality condition implies: (i) If H is so small that, even a does not suffice System remains subcritical (ii) Similarly, if a is too small, Even with H, Multiplying Media.. 17

18 Cylindical Reactor (contd.3) (iii) There are minimal values for H, a (iv) Many combinations of H, a for the critical state - Shaded area corresponds to supercritical states (e.g. power reactor at start-of-cycle ) Multiplying Media.. 18

19 Absolute Flux value For a spherical reactor, if one neglects d extrapolated radius Constant A is determined by the reactor power watts joules/fission Multiplying Media.. 19

20 Absolute Flux value (contd.) For a slab reactor with Per cm 2 of the slab, watts/cm 2 For a cylindrical reactor, one can show: Multiplying Media.. 20

21 ( Φ max / Φ ) Ratio The flux distribution determines the power distribution Reactor homogeneous constant Σ f The maximum-to-average ratio, same for flux, power For bare, homogeneous reactors, flux always maximum at centre, varies strongly (goin to zero at extrapolated surface) In practice, one has a reflector and/ or several different zones in the reactor, which render the flux distribution flatter Multiplying Media.. 21

22 ( Φ max / Φ ) Ratio (Examples) Laboratory for Reactor Physics and Systems Behaviour Spherical Reactor Slab Reactor Multiplying Media.. 22

23 Comments on Criticality Condition Laboratory for Reactor Physics and Systems Behaviour (a) Rewriting above equation: (1) Previously, one had : (2) From (1) (criticality condition): (3) Comparing (2), (3): Multiplying Media.. 23

24 Comments on Criticality Condition (contd.) (b) In general, for given values of k, B 2 : (system either subctritical or supercritical) One may nevertheless use the formalism of a critical system (stationary flux) - We consider a fictitious medium in which the number of n s produced per fission For this medium, yields (criticality condition) For the actual system, one needs to search for the modification which leads to. - Change in B 2 or in k. For a bare homogeneous reactor, the search is direct For a multizone system (heterogeneous layout), one needs to adopt an iterative approach - The dimensions (or the material characteristics) are varied until. Multiplying Media.. 24

25 Comments on Criticality Condition (contd.) (c) We have Laboratory for Reactor Physics and Systems Behaviour Comparing this with: Leakage term DB 2 is like a supplementary Σ a One may consider a reactor of finite dimensions (with a geometrical buckling of B 2 ), as though it were an infinite medium with a poisoning of (However, reactor equation still needs to be solved for obtaining B 2 ) Multiplying Media.. 25

26 Summary, Lesson 9 Multiplying media, multiplication factors 1-group, diffusion equation (Reactor Eq.) Material and geometrical buckling Bare homogeneous reactors (sphere, slab, cylinder, etc.) Absolute flux and reactor power Maximum- to-average flux ratio Comments on criticality condition Non-leakage probability, criticality search, leakage as absorptions (DB 2 ) Multiplying Media.. 26