Lesson 6: Diffusion Theory (cf. Transport), Applications

Transcription

1 Lesson 6: Diffusion Theory (cf. Transport), Applications Transport Equation Diffusion Theory as Special Case Multi-zone Problems (Passive Media) Self-shielding Effects Diffusion Kernels Typical Values of L, D ( different moderators) Transport, Diffusion Theories.. 1

2 Diffusion, as Special Case of Transport (Ligou, Ch. 8) Integro-differential form of the Transport Equation (monoenergetic) Neutron balance in a cylindrical volume element angular flux emission rate Making a Taylor expansion of and for : Transport, Diffusion Theories.. 2

3 Emission Term The complexity comes from the emission term In addition to the true sources, one needs to consider n s which have been scattered in the right direction Scattering rate: Neutrons scattered along : angular distribution f is an item of nuclear data, just as σ s is, with Transport, Diffusion Theories.. 3

4 Transport Equation Effectively, Scattering ~ cosine of the angle of deflection (spherical symmetry of the nucleus) Thus, true volumic source term The integro-differential equation (monoenergetic, steady-state) is thus: Transport, Diffusion Theories.. 4

5 Planar Geometry (1-D) For demonstrating the passage to diffusion theory, sufficient to consider this simple case (Axis OX: common perpendicular to all the infinite, homog. plates ) All Ω corresponding to same θ are equivalent Angular fluxes depend on just x, θ : polar, azimuthal angles Noting Transport, Diffusion Theories.. 5

6 Simplified 1-D Transport Equation For the double integral to be evaluated, consider with One has Laboratory for Reactor Physics and Systems Behaviour with (integration over α not applicable to flux) One assumes (as is often the case) that the scattering anisotropy is linear, i.e. Transport, Diffusion Theories.. 6

7 Simplified 1-D Transport Equation (contd.) Laboratory for Reactor Physics and Systems Behaviour For calculating a, b consider and average cosine of θ 0 (measure of anisotropy) Substituting (3) in the identities, Using this expression, as also (4), in (2), Finally (1) becomes Transport, Diffusion Theories.. 7

8 Passage to Diffusion Theory Multiplying (5) by dµ and integrating over µ (-1, 1), (Once again, the neutron balance equation but with 2 unknowns) For obtaining a 2 nd equation, multiply (5) by µdµ and integrate over µ Transport, Diffusion Theories.. 8

9 Passage to Diffusion Theory (contd.) Laboratory for Reactor Physics and Systems Behaviour It is only here that one makes the assumption which leads to Fick s Law : ϕ(x,µ) A(x) + B(x)µ (8) (linear anisotropy of ϕ) Substituting (8) into the expressions for Φ and J, A = Φ(x)/ 2 and B = 3J(x)/ 2 The integral in (7) becomes Eq. (7) may thus be written as: which is Fick s Law! Hypothesis equivalent to linear anisotropy of ϕ (more stringent condition that anistropy of scattering) Transport, Diffusion Theories.. 9

10 Diffusion Equation Applications (Passive Media) Multizone problems, e.g. absorbing region in a diffusive medium Consider infinite, homogeneous medium (uniform source distribution: Q n/cm 3 -s) (result independent of diffusion equation) If Σ a 0, Φ (n s produced continuously, zero absorptions, leakage : Φ ) Consider, in planar geometry, a separate region within the medium with Σ a > Σ a Σ a still << Σ s Effectively, an infinite plate of thickness 2h, containing a supplementary absorber (e.g. boron), Σ s ~ same We have 2 different regions, with a non-uniform flux distribution Transport, Diffusion Theories.. 10

11 Two-Zone Example (contd.) Consider the absorbing plate between z = 0, h (symmetry: ± z) Diffusion equations: Solutions: (D = 1/ 3Σ t 1/3Σ s, i.e. D ~ same) with Transport, Diffusion Theories.. 11

12 Application of Boundary Conditions Condition at z = 0 : net current = 0 For z : Φ 0 Thus, Laboratory for Reactor Physics and Systems Behaviour (For z : Φ(z) Φ ) For A, A one needs to apply the conditions at z = h (interface) Continuity of Φ(z) and of J (z) (i.e. of dφ/dz) Transport, Diffusion Theories.. 12

13 Final Solution, Comments One finally obtains: For z h and for z h Comments: 1. For Σ a Σ a, Φ(z ) Φ (The flux is not affected absorption at infinite dilution) 2. For h 0, Φ is, once again, the value of the flux for all z 0 3. In general, Φ(z ) < Φ for z h (The flux is depressed) 4. Absorption in the absorbing region: with reduced, relative to Σ a Φ (self-shielding phenomenon) Transport, Diffusion Theories.. 13

14 Another Example Two contiguous zones with a planar source at the centre Diffusion equations: General solutions: Conditions: Transport, Diffusion Theories.. 14

15 Solution From (iii), From (iv), Thus, Comments: 1. Φ 1, Φ 2 S 2. Φ is continuous at interfaces, not dφ/dx - Condition D 1.(dΦ 1 /dx) = D 2.(dΦ 2 /dx) implies dφ 1 /dx dφ 2 /dx if D 1 D 2 Transport, Diffusion Theories.. 15

16 Kernels for the Diffusion Equation (infinite, homog. medium) Point Kernel (flux at due to a source of 1 n/s at r r 0 ) With a distributed source S( ) n/cm 3 -s r 0 Planar Kernel (flux at x due to a planar source of 1 n/cm 2 -s at x 0 ) With a distribution of planar sources, S(x 0 ) The concept of a kernel is useful in considering the diffusion of thermal neutrons as a process which follows slowing down (the latter providing the source distribution) Transport, Diffusion Theories.. 16

17 Typical Values of D, L ρ (g/cm 3 ) Σ a (cm -1 ) D (cm) L (cm) C H 2 O D 2 O Be Transport, Diffusion Theories.. 17

18 Comments Due to the significantly high Σ a value for H 2 O, L is quite small LWRs have relatively tight lattices (cf. CANDU, AGR, ) Slowing down is very efficient (next chapter ) For UO 2 (3% enr), Σ a = 0.52 cm -1, Σ t = 0.89 cm -1 (D = 0.37 cm) L = (0.72) 1/2 = 0.85 cm (cf. λ t = 1.12 cm) Thus, L ~ λ t Diffusion theory not valid for pure fuel For an explicit treatment of the heterogeneous unit cell, transport theory needed After homogenisation calculation (equivalent homog. mixture of fuel, moderator) L >> λ t (due to the dominating effect of the moderator) Transport, Diffusion Theories.. 18

19 Summary, Lesson 6 Integro-differential form of the transport equation (Boltzmann Equation) Diffusion theory (Fick s Law) as special case (linear anisotropy of ϕ) Multi-zone problems (passive media) Boundary conditions Self-shielding effects Diffusion kernels Typical values of L, D (λ t ) Comparison fuels, moderators Transport, Diffusion Theories.. 19