lesson 6: diffusion theory (cf. transport), applications · 2010. 6. 23. · transport, diffusion...

Transport, Diffusion Theories.. 1

Laboratory for Reactor Physics and Systems Behaviour

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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)



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Diffusion, as Special Case of Transport (Ligou, Ch. 8)

Integro-differential form of the Transport Equation (monoenergetic) • Neutron balance in a cylindrical volume element

Making a Taylor expansion of and for :

angular flux emission rate



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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 :

f is an item of nuclear data, just as σs is, with

angular distribution



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Transport Equation

Effectively, • Scattering ~ cosine of the angle of deflection (spherical symmetry of the nucleus)

Thus,

The integro-differential equation (monoenergetic, steady-state) is thus:

true volumic source term



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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 corresponding to same θ are equivalent

→ Angular fluxes depend on just x, θ :

Noting

polar, azimuthal angles

€

Ω



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Simplified 1-D Transport Equation

For the double integral to be evaluated, consider with One has

with

(integration over α’ not applicable to flux)

One assumes (as is often the case) that the scattering anisotropy is linear, i.e.



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Simplified 1-D Transport Equation (contd.)

For calculating a , b consider

and

Substituting (3) in the identities,

Using this expression, as also (4), in (2),

Finally (1) becomes

average cosine of θ0 (measure of anisotropy)



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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 2nd equation, multiply (5) by µdµ and integrate over µ



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Passage to Diffusion Theory (contd.) It is only here that one makes the assumption which leads to Fick’s Law : … (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)

€

ϕ(x,µ) ≅ A(x) + B(x)µ



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Diffusion Equation Applications (Passive Media)

Multizone problems, e.g. absorbing region in a diffusive medium • Consider infinite, homogeneous medium (uniform source distribution: Q n/cm3-s)

(result independent of diffusion equation)

• If Σa → 0 , Φ → ∞ (n’s produced continuously, zero absorptions, leakage : Φ↑)

Consider, in planar geometry, a separate region within the medium with Σ’a > Σa • Σ’a still << Σs • Effectively, an infinite plate of thickness 2h, containing a supplementary absorber

(e.g. boron), Σs ~ same

⇒ We have 2 different regions, with a non-uniform flux distribution



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Two-Zone Example (contd.)

Consider the absorbing plate between z = 0, h (symmetry: ± z)

Diffusion equations:

(D = 1/ 3Σt ≈ 1/3Σs , i.e. D ~ same)

Solutions:

with



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Application of Boundary Conditions

Condition at z = 0 : net current = 0

For z → ∞ : Φ ≠0

Thus,

(For z → ∞ : Φ(z) → Φ∞)

For A , A’ one needs to apply the conditions at z = h (interface)

→ Continuity of Φ(z) and of (i.e. of dΦ/dz)

€

J (z)



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Final Solution, Comments

One finally obtains:

For z ≤ h …

and for z ≥ h …

Comments:

1. For Σ’a → Σa , 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’aΦ∞ (self-shielding phenomenon)



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Another Example

Two contiguous zones with a planar source at the centre

Diffusion equations:

General solutions:

Conditions:



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Solution From (iii),

From (iv),

Thus,

Comments:

1. Φ1 , Φ2 ∝ S

2. Φ is continuous at interfaces, not dΦ/dx - Condition D1.(dΦ1/dx) = D2.(dΦ2/dx) implies dΦ1/dx ≠ dΦ2/dx if D1 ≠ D2



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Kernels for the Diffusion Equation (infinite, homog. medium) Point Kernel

(flux at due to a source of 1 n/s at )

With a distributed source S( ) n/cm3-s

Planar Kernel (flux at x due to a planar source of 1 n/cm2-s at x0)

With a distribution of planar sources, S(x0)

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)

€

r

€

r 0

€

r 0



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Typical Values of D , L

ρ (g/cm3) Σa (cm-1) D (cm) L (cm)

C 1.6 3.8.10-4 0.87 47.5

H2O 1.0 0.018 0.14 2.76

D2O 1.1 7.10-5 0.80 107

Be 1.85 0.0011 0.49 21



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Comments

Due to the significantly high Σa value for H2O, L is quite small • LWRs have relatively tight lattices (cf. CANDU, AGR, …) • Slowing down is very efficient (next chapter…)

For UO2 (3% enr) , Σa = 0.52 cm-1 , Σt = 0.89 cm-1 (D = 0.37 cm) → L = (0.72)1/2 = 0.85 cm (cf. λt = 1.12 cm)

Thus, L ~ λt → 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 >> λt (due to the dominating effect of the moderator)



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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 (λt )

Comparison fuels, moderators

lesson 6: diffusion theory (cf. transport), applications · 2010. 6. 23. · transport, diffusion...

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