Serco Assurance. Resonance Theory and Transport Theory in WIMSD J L Hutton

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3 Serco Assurance Resonance Theory and Transport Theory in WIMSD J L Hutton 2 March 2004

4 Outline of Talk Resonance Treatment Outline of problem - pin cell geometry U 238 cross section Simple non-mathematical ideas More rigorous treatment Neutron Transport Theory Homogeneous - B1 Collision probabilities - PIJ Sn - DSN

5 DISCRETISATION OF GEOMETRY Infinite Homogeneous Problem - No geometry subdivision required. Heterogeneous Problem - Minimum subdivision: one calculation "mesh" per material region - In practice, material regions are normally subdivided into several meshes of about one transport mean free path in size (~ 1 cm in H O) 2

6 GEOMETRY OPTIONS HOMOGENEOUS SLAB REGULAR PINCELL ARRAY CLUSTER (PRESSURE TUBE) MULTICELL + choice of boundary conditions Typical cluster - subdivision into ~ 30 meshes Computing time and storage vary as : Number of groups x Number of meshes

7 PIN CELL GEOMETRY A "Pin Cell" consists essentially of a cylindrical fissile region surrounded by clad and coolant. Infinite arrays are generated, but leakage can be introduced. Isolated cylinders can be obtained by adopting a "free boundary. Fuel Can Water A WIMS Pin Cell Examples: Regular lattice benchmarking Safe number of CAGR pins in water

8 FLUX DISTRIBUTION WITHIN A FLUX CELL φ PIN CAN COOLANT φ FAST φthermal PIN CENTRE RADIUS

9 U238 σt

10 POSITIONS OF WIMS LIBRARY GROUP BOUNDARIES AND PRINCIPAL RESONANCES

11 RESONANCE TREATMENT (Non-mathematical) HOMOGENEOUS MIXTURE OF ABSORBER Σa(E) AND SCATTERER Σs (constant) φ ABSORPTION RATE ~ EFFECTIVE Σ a = φ Σ a φ Σ a φ If Σs is very large, flux depression is small and Σ a E Σ a Σ a = infinite dilution limit

12 RESONANCE TREATMENT (Non-mathematical) Σ a Infinite Dilution Limit As pin radius, this effect 0 Σ p = Σ s (potential scattering) ISOLATED PINS Neutrons pour in from the moderator as well as slowing down within the pin and tend to flatten the flux depression. 0, source completely swamps the absorptions Use Σ Σ p s d - where d is pin diameter.

13 RESONANCE TREATMENT (Non-mathematical) ARRAY OF PINS The source of neutrons from the moderator at the resonance energy is reduced by absorptions in neighbouring pins. Multiply 1/d effect by 0 as pins are compacted to solid. 1 as pins are widely separated γ γ Use Σ p Σ s + -- d 1 γ is known as the DANCOFF FACTOR pin spacing

14 RESONANCE TREATMENT HOMOGENEOUS φ ( Σ + Σ ) Σ p a p Σ p φ Σ du + Σ p a Resonance Integral I φ = = = = φσ a du Σ Σ a p Σ du + Σ p a Σ + Σ Σ p a a du Σ + Σ Σ + Σ p a p a u I Σ p Σ a I = -- = φ I I u Σ p

15 RESONANCE TREATMENT HETEROGENEOUS (isolated pin) 2 region model (f=fuel, m=moderator) but V Σ φ V Σ P + V Σ P f f f f p ff m m mf (1) V Σ P = V Σ P = V Σ ( 1 P ) m m mf f f fm f f ff (2) Σ φ Σ P + Σ ( 1 P ) f f p ff f ff (3)

16 RESONANCE TREATMENT Rational Approximation P ff = Σ f a Σ f + -- l (a=bell Factor) (4) I = Σ φ = a f Σ p Σ P + ( 1 P ) du a Σ ff ff f Σ Σ a p a Σ + -- f l a Σ -- a l a Σ + -- f l = du= Homog I with σp σe= σp+ Σ a Σ + -- a p l du a Σ + Σ + -- a p l a Nl

17 RESONANCE TREATMENT BELL FACTOR P ff = Σ Σ + a -- l where a~1.16 Error in 1-P ff for cylinder 10% 0-10% Σ l

18 RESONANCE TREATMENT DANCOFF FACTOR a σ σ e = p Nl for isolated rod p (homogeneous value) for packed array a --- Nl Require factor on to allow for geometry varying from 1 forisolated rod to 0 for packing Dancoff Factor ~ probability of collision in moderator before next fuel collision for neutrons leaving the fuel γ σ σ e = p a γ + a ( 1 γ ) N l

19 RESONANCE TREATMENT CORRECTIONS MULTIPLE ABSORBERS I Σ a u I Σ p LAMBDA VALUES (Finite Resonance Width) λσ p used instead of σ p where λ is the effectiveness of a scatterer relative to Hydrogen (λ depends on α, energy loss/collision and resonance width)

20 GRAPH OF λ(1-group) AGAINST ATOMIC WEIGHT 180,0 0,26 190,0 0,25 200,0 1,20 0,24 210,0 0,23 220,0 0,22 230,0 1,00 0,21 240,0 0,2 0,00 0,80 0,60 0,40 0,20 0,00 0,0 20,0 40,0 60,0 80,0 100,0 120,0 140,0 160,0 180,0 200,0 220,0 240,0

21 f(p) CORRECTION TO GROUP REMOVAL φ φ 1 Gp Average φ φ 2 Decreasing E Downscatter from the group = f (p) Σ r φ For heavy nuclides that remove neutrons only from the bottom of the group, Removals =Σ r φ 2 For Hydrogen f(p)~1.0 and f(p)= φ 2 / φ ave

22 f(p) CORRECTION TO GROUP REMOVAL In a well moderated system: φ φ φ 1 2 Gp Average φ φ ave < φ 2 and f(p) > 1 Decreasing E

23 NEUTRON TRANSPORT THEORY METHODS: Homogeneous - Leakage Effects (B1) Differential Transport - DSN Integral Transport (Collision Probabilities) - PERSEUS, PIJ, PRIZE.

24 Analytic Solutions Homogeneous - 1D - equation is µ φ + Σφ = ( E E) φ de dµ t Σ s z λχ E υ E φ de dµ Separation of variables φ = F, z + ( ) Σ ( ) f ( E µ ) φ = ωφ

25 Analytic Solution Solution has the form (,, ) φ = e ωz ϕ E µ ω Spectrum independent of position Spatial variation independent of energy

26 Bn Solutions General solution of Homogeneous transport equation - Used in WIMS in CRITIC module Solutions of form (,, ) φ ibz ϕ µ = e E B

27 Bn Solution Using Spherical Harmonics for flux gives following solution of transport equation Σ l = A g Σ k k + A g Σ g ϕ ϕ λ χ υ ϕ g k lk g gg g l 0 0 g fg g g

28 Bn Solution Where l + ϕ ( µ ) lm = ϕ ( µ ) g 2 1 P g lm lm 2 s 2l + 1 Σ Ω Ω = Σ Ω Ω P. gg gg l lm 2 ( ) l ( )

29 Bn Solution The variable A is given by A lk = 1 P ( ) P( ) g l k µ µ Σ g + ib d µ µ

30 Simplified Solution Only P1 terms - Only equations in scalar and first moment of flux Φ = ϕ 0, J = iϕ 1 g g g g This gives the following equations

31 Simplified Solution 2 Equations are Σ g 0 Σ + gg B { } Φ Σ Φ Σ Φ ( Σ Σ ) = + λχ υ 3α 2 1 g g gg J g = 2 B B Φ 0 g g g g g fg g g g g + Σ ( ) 1 3α Σ Σ g g gg B ( ) 1 3α Σ Σ g g gg J g g g g g g 1 Σ J 1 gg g g g

32 Simplified Solution Note new form of diffusion coefficient D g = 3α Σ 1 Σ 1 g g gg

33 LEAKAGE OPTIONS in CHAIN 14 Homogeneous solutions based on: Diagonal Transport Corrected Flux Solution B1 Flux Solution Diffusion Coefficients based on: Benoist 3-region model Transport cross sections Ariadne method

34 NUMERICAL INTEGRATION OF COLLISION PROBABILITIES PIJ

35 COLLISION PROBABILITIES Integral Transport Equation Σ ( r, E) φ ( r, E) = P ( r r, E) ψ ( r, E) dr r Reaction rate at position r Integrate over all positions r First flight probability that neutrons produced at energy E and position r will have next collision at position r Neutron source at position and energy E r

36 COLLISION PROBABILITIES (Continued) ψ ( r, E) = Σ ( r, E E) s E χ ( E) νσ ( r, E' ) φ ( r, E' ) de k f Neutron source at position r and Energy E Integrate over all energies E Neutron source at r due to scattering Neutron source at Energy E due to fission at position r and energy E

37 COLLISION PROBABILITIES (Continued) In group form V j Σ j g φ j g = g V P i ij h Σ ( h g) φ i i i h where the 's are average fluxes in regions i and j. + i g g V P Si i ij Reciprocity and Conservation Σi Vi Pij =Σj Vj Pji P and ij = 1 j Can also show that P si = 4Σ V i i S P is = 4Σ i V i ( 1 P ii ) S where S is the surface for a single region

38 NUMERICAL INTEGRATION OF COLLISION PROBABILITIES dx Σ j R j i Σ i R i B τ ij C j Integrate from x = - to + φ = 0 to 2π θ = 0 to π/2 π 2 1 P ij = dθ dx dφ 1 e 2πV i Σ i P ij 0 φ 2π 0 ( Σ i R i ) sin θ τ ij e sinθ 2π 1 = dφ dx{ki 2π V i Σ i 3 ( τ ij ) Ki 3 ( τ ij + Σ i R i ) Ki 3 ( τ ij + Σ j R j ) 0 1 e Σ j R j sinθ 2 sin θ +Ki 3 ( τ ij + Σ i R i + Σ j R j )}

39 NUMERICAL INTEGRATION OF COLLISION PROBABILITIES Note Symmetry Σ i V i P ij = Σ j V j P ji Collision Probability form of Transport Equation: Σφ space Pψ

40 ASSUMPTIONS IN COLLISION PROBABILITY METHODS Basic Assumptions: - isotropic scattering - isotropic flux - flat source in each region Consequence of flat source assumption: φ source Flat source Absorber Flat source moves neutrons closer to absorber, which increases absorptions and reduces k.

41 PIJ Geometry

42 ANNULI IN PIJ GEOMETRY In PIJ, meshes in annuli cannot be defined. Extra annuli must be defined as indicated above. nregion defines the number of PIJ regions. (If there are no azimuthal subdivisions of rods or annuli, nregion is the number of annuli plus the number of different rodsubs.)

43 Boundary Conditions Explicit Boundaries a boundary can be dealt with explicitly by PIJ Track is reflected back into problem as from a mirror repeat process at number of reflections until no loss results from terminating track

44 Sn Methods DSN

45 Boltzmann Equation Ω ϕ +Σ ϕ = S m m m m

46 Boltzmann Equation for Cylinders η cosφ r sinφ η + Σ N r E r (, ω, φ, ) φ = S η = sin ω

47 Scalar flux Sum over discrete directions φ = w ϕ m m m

48 Discrete Ordinates Set of points on the unit sphere Place triangular set of points on octant of unit sphere S N method of order N - N/2 levels - ith level has N/2-i+1 points

49 S 2 Lowest possible approximation One direction per octant 8 equations (3D) 4 equations (2D)

50 S 4 3 directions per octant 24 equations (3D) 12 equations (2D)

51 S 8 10 directions per octant 80 equations (3D) 40 equations (2D)

52 Weights Weights proportional to angle subtended on unit sphere w m m = Ω 4π w m = 1 m

53 Odd Moment Condition For isotropic flux net current is zero J = w Ω ϕ = 0 m m m w Ω = 0 m m m m

54 Scattering Source S Ω = Σ, µ ϕ Ω dω ( ) ( ) ( ) 0 g s g g g could expand angular flux... ϕ N + l ( Ω ) = ϕ lm Y lm ( Ω ) l = 0 m = l Y lm (, ) ( l m) ( + )! l m P m θψ = im l θ ψ! ( cos ) exp( )

55 Linearly Anisotropic Scattering S = Σ φde + 3Ω Σ JdE m s0 m s1 Use transport corrected cross sections - scattering assumed isotropic - correction for non isotropic effects

56 WIMSD DSN OPTION GEOMETRY OPTIONS: HOMOGENEOUS SLAB ANNULAR SPHERICAL + black or white boundary SOLVES spatial mesh neutron balance equations by differential transport method of k-infinity and neutron flux by mesh and group. NOT DIRECTLY APPLICABLE to complex geometries (eg. fuel clusters) without preliminary smearing of pincells into annuli.

57 S N v Diffusion Theory S N in principle more accurate than diffusion theory Difficult to calculate accurate transport cross sections in smeared geometries Restricted to simple geometries