Lecture 21

22.05 Reactor Physics Part Twenty-One Extension of Group Theory to Reactors of Multiple Region Two Energy Groups 1 1.

“The basic approximation which leads to one-group theory is that Φ(r, E) has a separable energy dependence ψk(E) throughout each given material composition k. This approximation is very questionable at interfaces between regions k and l, since the form of ψk(E) will not in general match that of ψ1(E) at such boundary. We tried to minimize this problem by matching Rk(rc)ψk(E) and R1(rc)ψ1(E) in an energy-integral sense. But if ψk(E) and ψ1(E) are quite different (as will be in the case if k is core material an l is reflector material), there are bound to be significant mismatches over particular subranges of the overall energy range 0 to ∞. For example, at an interface between core and reflector material, the net leakage of high-energy neutrons, which are originally created by fission in the core, is from the core to the reflector, whereas the net leakage of low-energy neutrons, created in abundance by the superior moderating power of the reflector material, is in the opposite direction. A one-group model cannot describe this process. Depending on the sign of n k ⋅ D k ∇R k (r ) = n k ⋅ D1∇R1 (r ) , either a net number of neutrons having the energy distribution ψk(E) leak from k to l per second or a net number having the distribution ψ1(E) leak from l to k.

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Two-group theory represents an attempt to improve the accuracy with which the flux can be described near such interfaces. The basic idea is to split the spectrum functions ψk(E) into two parts, ψ1k (E) (for energy group one, extending from a “cut-point” energy Ec to ∞) and ψ k2 (E) (for energy group two, extending from 0 to Ec) and to associate separate spatial functions R1k (r ) and R k2 (r ) with neutrons belonging to each of these two groups. Continuity of flux and current across an interface is then required in an integral sense, individually, for the ranges 0 to Ec and Ec to ∞. Thus the two-group model permits a net leakage rate of group-one neutrons in one direction across an interface and a net leakage rate of group-two neutrons in the opposite direction. In the interior portions of a given region, where it is expected that the separable form Φ(r, E) = Rk(r)ψk(E) will be valid over the entire energy range, the two-group approximation will yield the result that R1k (r ) and R k2 (r ) have the same spatial shape, so that R1k (r )ψ1k (E ) and R k2 (r )ψ k2 (E) for the two energy ranges fit together to form a single function of the form R k (r )ψ k (E) . 1

Material in this section follows that of Henry pp. 162-165. Portions that are verbatim are indicated by quotations.

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Mathematically the assumptions of two-group theory may be summarized as follows: The scalar flux function may be written, for r in region k, as: ⎧⎪R k (r )ψ k (E) for E < E < ∞ 1 c Φ (r, E) = ⎨ 1k ⎪⎩R 2 (r )ψ k2 (E) for E < E c

Where ψ1k (E) and ψ k2 (E) are the parts of ψ k (E) .” (Henry, pp. 162-163)

2.

Derivation of Two-Group Equations a)

Spectrum Equations: The starting point is again the continuous energy diffusion equation and we follow our now standard procedure. (i)

Write out the continuous energy diffusion equation − ∇ ⋅ D(r, E ) ∇Φ(r, E ) + Σ t (r, E )Φ (r, E ) ⎤ ∞⎡ = ∫0 ⎢∑ χ j ( E )ν j Σ jf (r, E ′) + Σ s (r, E ′ → E )⎥Φ (r, E ′)dE ′ ⎣ j ⎦

(ii)

Modify this equation for a homogeneous medium (i.e., material parameters are not position dependent.) − D( E ) ∇ 2 Φ (r, E ) + Σ t ( E )Φ (r, E ) ∞⎡1 ⎤ = ∫0 ⎢ χ ( E )νΣ f ( E ′) + Σ s ( E ′ → E )⎥Φ (r, E ′)dE ′ ⎣λ ⎦

(iii)

Add superscripts k to denote the various regions of the reactor. We again chose k=2 as to describe two regions: Fuel/moderator and reflector. (Note: The regions are NOT the groups. The word “group” in group theory refers to the division of the neutron energies.)

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− ∇ ⋅ D k (E ) ∇Φ (r, E ) + Σ kt (E )Φ (r, E ) ∞ ⎡1 ⎤ = ∫ ⎢ χ k (E )νΣ fk (E′) + Σ sk (E′ → E )⎥Φ (r, E′)dE′ 0 λ ⎣ ⎦

(k = 1, 2, 3, .. ., K )

(4.9.1) (iv)

Approximate the leakage term in each region as:

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− ∇ ⋅ D k ( E ) ∇Φ (r , E ) = D k ( E ) Bmk Φ (r , E ) and rewrite the diffusion equation for each region as:

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⎛ k ⎞ k k 2 ⎜ D (E) Bm + Σ t (E) ⎟ Φ (r, E) ⎝ ⎠ ∞ ⎡1 ⎤ = ∫ ⎢ χ k (E)νΣ fk (E′) + Σsk (E′ → E)⎥Φ (r, E′)dE′ 0 ⎣λ ⎦

for r in the interior of region k; k = 1, 2. (v)

Substitute the two group definition of the scalar flux ⎧ R1k (r )ψ 1k ( E ) for Ec < E < ∞ Φ (r, E ) = ⎨ k k ⎩ R2 (r )ψ 2 ( E ) for E < Ec

into the above relation and obtain two spectral equations for each region. For region 1, these would be

[D (E) (B ) + Σ (E )]R (r)ψ (E ) 1 2 m

1

∞

1 t

1 1

1 1

[

]

= ∫E χ ′( E )νΣ1f ( E ′) + Σ1s ( E ′ → E ) R11 (r )ψ 11 ( E ′)dE ′ c

and

[D (E) (B ) + Σ (E)]R (r)ψ (E) 1 2 m

1

Ec

1 t

1 2

[

1 2

]

= ∫0 χ ′( E )νΣ1f ( E ′) + Σ1s ( E ′ → E ) R21 (r )ψ 21 ( E ′)dE ′

(Note: The difficult part of these equations is the notation. Superscripts denote regions; subscripts denote energy groups.)

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The equations for region 2 are similar. These four equations are solved for the spectral functions to obtain ψ1(E) and ψ2(E). b.

Spatial Equations: We now have the energy distribution of the scalar neutron flux and can obtain two group cross-sections. Hence, we can proceed with the spatial analysis. We now quote again from Henry’s Nuclear Reactor analysis (p. 163):

ƒ

“The scalar flux function may be written, for r in region k, as: ⎧ R k (r )ψ 1k ( E ) for Ec < E < ∞ Φ (r, E ) = ⎨ 1k k ⎩ R2 (r )ψ 2 ( E ) for E < Ec

(4.11.1)

Where ψ1k (E) and ψ k2 (E) are parts of ψ k (E) found as described above.

ƒ

Boundary conditions, for k and l indicating any two adjacent compositions and rc being a point on the interface separating them, require that ∞

k

∞

k

l

l

∫ E c dE R1 (rc )Ψ1 (E) =∫ E c dE R1(rc )Ψ1 (E) Ec

∫0

Ec dE R l2 (rc )Ψ2l (E) 0

dE R k2 (rc )Ψ2k (E) = ∫

∞

[ ⋅ D (E)∇[R

∫ E c dE nk ⋅ D Ec

∫0

dE n k

k

] (r )Ψ (E)] = ∫

(E)∇ R1k (rc )Ψ1k (E) = ∫

k

k 2 c

k 2

∞ dE Ec

[ ] ⋅ D (E)∇[R (r )Ψ (E) ]

n k ⋅ Dl (E)∇ R1l (rc )Ψ1l (E)

Ec dE 0

nk

l

l 2 c

As with the one group case, it simplifies the algebra if, after having found the shape in energy of ψ1k (E) and ψ k2 (E) for the material of region k, we renormalize these two segments ψ k (E) so that

∫ ∫

∞ Ec

Ec 0

Ψ1k ( E ) dE = 1 (k = 1,2,...., K ), Ψ2k ( E ) dE = 1 (k = 1,2,...., K ).

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l 2

Then the continuity equations show that R1k (rc ) = R1l (rc ) and R k2 (rc ) = R l2 (rc ) for points rc on the interface between regions k and l. Thus, as in the one-group case, the superscript k on the functions R are superfluous, and we shall change notation by replacing the various R1k (r ) by the single, everywhere-continuous function Φ1 (r ) and similarly

replacing R k2 (r ) by Φ 2 (r ) . The functions Φ1 (r ) and Φ 2 (r ) are called the two-group fluxes. Note that they are not fluxes per unit energy (in the way that Φ (r, E) = v(E)n(r, E) is). Physically Φ1 (r ) is the two-group approximation to the number of neutrons per cc having energies in the ∞ range Ec to ∞ ⎛⎜ ∫ n(r, E )dE ⎞⎟ multiplied by the average “fast” speed. ⎝ Ec ⎠ ∞

v1 =

∫E c ν(E)n (r, E)dE ∞

∫E c n (r, E)dE

And Φ 2 (r ) is the two-group approximation to the number in the range 0 to Ec multiplied by the average thermal speed. v2 =

E

∫0 c ν(E)n (r, E)dE E

∫0 c n (r, E)dE

(These interpretations are somewhat ambiguous since (4.11.1), on which they are based, is an approximation which cannot be rigorously correct.) To find differential equations for the two-group fluxes Φ1 (r ) and Φ 2 (r ) we substitute the approximation (4.11.1) into (4.9.1) which is the homogenized diffusion equation for two regions. This will give us different results for E > Ec and E < Ec. - D k (E)ψ1k (E )∇ 2Φ1 (r ) + Σ kt (E)ψ1k (E)Φ1 (r ) ∞ ⎡1 ⎤ = ∫ ⎢ χ k (E)νΣ fk (E′)ψ1k (E′) + Σsk (E′ → E)ψ1k (E′)⎥dE′Φ1 (r ) Ec λ ⎣ ⎦

+∫

Ec

0

⎡1 k ⎤ k k ⎢⎣ λ χ (E)νΣ f (E′)ψ 2 (E′)⎥⎦dE′Φ 2 (r )

for E > E c

(4.11.4) and

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- D k ( E )ψ 2k ( E )∇ 2Φ 2 (r ) + Σ tk ( E )ψ 2k ( E )Φ 2 (r ) ∞ ⎡1 ⎤ = ∫E ⎢ χ k ( E )νΣ kf ( E ′)ψ 1k ( E ′) + Σ ks ( E ′ → E )ψ 1k ( E ′)⎥ dE ′Φ1 (r ) c ⎦ ⎣λ

Ec ⎡ 1 ⎤ + ∫0 ⎢ χ k ( E )νΣ kf ( E ′)ψ 2k ( E ′) + Σ ks ( E ′ → E )ψ 2k ( E ′)⎥ dE ′Φ 2 (r ) ⎦ ⎣λ

for E ≤ Ec

(4.11.5) Where on physical grounds we have omitted from (4.11.4) the scattering from E′ < E c to E > E c (For thermal reactors the cut point Ec will always be such that χ k (E) is zero for E < Ec; hence no fission terms will appear in (4.11.5). We retain them for possible application to fast reactors, for which Ec will be much higher. There is no solution Φ1 (r ), Φ 2 (r ) that will satisfy (4.11.4) and (4.11.5) at all energies since the form (4.11.1) is not sufficiently general. However, we can force equality of the right- and left-hand sides in an integral sense and in that way find equations which, when solved, will give us Φ1 (r ) and Φ 2 (r ) . Accordingly, we shall integrate (4.11.4) from Ec to ∞ and (4.11.5) from 0 to Ec and require that the resultant equations be valid at all r. To simplify the result, we first define a set of “two-group constants”: ∞ Ec

D1k ≡ ∫ D(E) Ψ1k (E) dE,

Ec D(E) Ψ2k (E) dE, 0

D k2 ≡ ∫

∞ Σ kt1 ≡ ∫ Σ kt (E)Ψ1k (E) dE, Ec k χ1

∞ Ec

≡ ∫ χ k (E) dE,

Ec k Σ t (E )Ψ2k 0

Σ kt 2 ≡ ∫

Ec k χ (E) dE 0

χ k2 ≡ ∫

∞ νΣ fk1 ≡ ∫ νΣfk (E) Ψ1k (E ) dE, Ec ∞

∞

k

Ec νΣfk (E) Ψ2k (E) dE, 0

νΣ fk2 ≡ ∫

k Σ11 ≡

∫Ec dE, ∫E c dE′ Σs (E′ → E)ψ1 (E′),

Σ k21 ≡

∫0

Ec

(E) dE

k

∞ Ec

dE, ∫ dE′ Σsk ( E′ → E )ψ1k ( E′)

Σ k22 ≡

Ec

∫0

Ec dE′ Σsk ( E′ → E )ψ k2 ( E′) 0

dE, ∫

k k k k Σ1k ≡ Σ kt1 − Σ11 , Σ 2 ≡ Σ t 2 − Σ 22

Using these definitions, integrating (4.11.4) from Ec to ∞. and integrating (4.11.5) from 0 to Ec, we get

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− D1k ∇ 2Φ1 (r ) + Σ1k Φ1 (r ) =

− D2k ∇ 2Φ 2 (r ) + Σ 2k Φ 2 (r ) =

1

λ 1

λ

χ1k [νΣ kf 1Φ1 (r ) + νΣ kf 2Φ 2 (r )] k χ 2k [νΣ kf 1Φ1 (r ) + νΣ kf 2Φ 2 (r )] + Σ 21 Φ1 (r )

(4.11.7) The boundary conditions become: 1. Φ1 (r ) , Φ 2 (r ) must be continuous everywhere, 2. n ⋅ D1∇Φ1 (r ) and n ⋅ D 2∇Φ 2 (r ) must be continuous across interfaces separating different material compositions, 3. Φ1 (r ) = Φ 2 (r ) on the outer boundary of the reactor. The above equations (4.11.7) are the “two-group diffusion equations.” They are the standard workhorses of thermal-reactor design. It can be proved that a unique, positive solution corresponding to a most-positive real eigenvalue λ always exists for the two-group equations. Thus, a physically acceptable solution for the group fluxes Φ1 and Φ 2 can always be found, and from this the two-group approximation Φ1 (r )ψ1k (E) ; Φ 2 (r )ψ k2 (E) for the scalar flux Φ (r, E) can be constructed throughout each material composition k.” (Henry, pp. 163-165)

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