Pressurized Water Reactor (PWR)

Transcription

1 Pressurized Water Reactor (PWR) Models: Involve neutron transport, thermal-hydraulics, chemistry Inherently multi-scale, multi-physics CRUD Measurements: Consist of low resolution images at limited number of locations.

2 3-D Neutron Transport Equations: Pressurized Water Reactor (PWR) Challenges: Linear in the state but function of 7 independent variables: Very large number of inputs or parameters; e.g., 1,; Parameter selection critical. ORNL Code: Denovo Codes can take hours to days to run.

3 Pressurized Water Reactor (PWR) Thermo-Hydraulic Model: Mass, momentum and energy balance for fluid Note: Similar equations for gas Challenges: Nonlinear coupled PDE with nonphysical parameters due to closure relations; CASL code: COBRA (CTF) COBRA is a sub-channel code, which cannot be resolved between pins. Codes can take minutes to days to run.

4 Microscopic Cross-Sections Cross-Sections: Probability that a neutron-nuclei reaction will occur is characterized by nuclear cross-sections. Assume target is sufficiently thin so no shielding. Rate of Reactions: R = IN A Note: is microscopic cross-section (cm 2 ) Flux I neutrons cm 2 s Microscopic Cross-Sections: Related to types of reactions. f : Fission s: Scatter t: Total N A nuclei cm 2 Reference: J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis, John Wiley and Sons, 1976.

5 Macroscopic Cross-Sections Macroscopic Cross-Sections: Accounts for shielding Total Reaction Rate per Unit Area: dr = t IdN A = t INdx I Strategy: Equate reaction rate to decrease in intensity di(x) = [I(x + dx) I(x)] ) di dx = = t INdx N ti(x) ) I(x) =I e N tx Total Macroscopic Cross-Section: t N t Frequency: With which reactions occur v t cm s 1 cm = 1 s

6 Neutron Flux in Reactor Neutron Density: N(r, t) N(r, t)dr 3 : Expected number of neutrons in dr 3 about r at time t N(r, E, t)dr 3 de: Expected number neutrons in dr 3 with energies E in de r dr 3 Reaction Rate Density: Interaction frequency v where v is neutron speed F (r, E, t)dr 3 = v N(r, E, t)dedr 3 Angular Neutron Density: n(r, E, ˆ,t)dr 3 dedˆ : Expected number of neutrons in dr 3 about r with energy E about de, moving in direction ˆ in solid angle ˆ at time t Angular Neutron Flux: '(r, E, ˆ,t)=vn(r, E, ˆ,t)

7 Neutron Transport Equation Conservation Law: n(r, ˆ,t)dr 3 = gain in V loss in V V Gain Mechanisms: (1) Neutron sources (fission) (2) Neutrons entering V (3) Neutrons of different E, ˆ that change to E, ˆ due to scattering collision Loss Mechanisms: (4) Neutrons leaving V (5) Neutrons suffering a collision

8 Terms: (1) Z V Neutron Transport Equation S(r, E, ˆ,t)dr 3 dedˆ where S is a source term (5) Rate at which neutrons collide at point r is f t = v t (r, E)n(r, E, ˆ,t) ) (5) = h R V v P t (r, E)n(r, E, ˆ,t)dr i 3 dedˆ (2), (4) Consider the rate at which neutrons leak out of surface j(r, E, ˆ,t) ds = v ˆ n(r, E, ˆ,t) ds so leakage is (4) - (2) = applez applez = applez = S V V ds v ˆ n(r, E, ˆ,t) dedˆ r v ˆ n(r, E, ˆ,t)dr 3 dedˆ v ˆ rn(r, E, ˆ,t)dr 3 dedˆ

9 Terms: Scattering cross-sections Neutron Transport Equation Consider first a beam of neutrons of incident intensity I, all of energy E, hitting a thin target of surface atomic density N A E E Note: Microscopic differential scattering cross-section is proportionality constant Rate cm 2 = s(e! E)IN A de when neutron scatters from energy E to final energy E in range E to E+dE Microscopic scattering cross-section: s(e )= Z 1 de s (E! E) Macroscopic scattering cross-section: s (E! E) N s (E! E)

10 Neutron Transport Equation Terms: Scattering cross-sections Consider how change in direction ˆ ˆ s(ˆ )= Z 4 dˆ s (ˆ! ˆ ) (3) Rate at which neutrons scatter from E, ˆ to E, ˆ is applez v s (E! E, ˆ! ˆ )n(r, E, ˆ,t)dr 3 V dedˆ To incorporate contributions from any E, ˆ, we integrate to obtain applez Z Z 1 (3) = dr 3 dˆ de v s (E! E, ˆ! ˆ )n(r, E, ˆ,t) dedˆ V 4

11 Neutron Transport Equation Conservation Law: Z = + v ˆ rn + v t n(r, E, ˆ,t) Z 1 Z de dˆ v s (E! E, ˆ! ˆ )n(r, E, ˆ,t) 4 S(r, E, ˆ,t) dr 3 dedˆ Since this must hold for any control volume, it + v ˆ rn + v t n(r, E, ˆ,t) = Z 4 Z 1 dˆ de v s (E! E, ˆ! ˆ )n(r, E, ˆ,t)+S(r, E, ˆ,t) Angular Flux: '(r, E, ˆ,t)=vn(r, E, ˆ!,t) + ˆ r' + t (r, E)'(r, E, ˆ,t) Z Z 1 = dˆ de s (E! E, ˆ! ˆ )'(r, E, ˆ,t)+S(r, E, ˆ,t) 4

12 Neutron Transport Equation Plane Symmetry: Flux depends only on x x = cos Z dˆ = 4 Z d sin ˆ = cos 1 µ x Then so ˆ r' = @' + + t'(x, E,,t) Z Z 1 d sin de s (E! E,! )'(x, E,,t)+S(x, E,,t) 1-D Neutron Transport Equation: + + t'(x, E, µ, t) = Z 1 1 Z 1 dµ de s (E! E,µ! µ)'(x, E,µ,t)+S(x, E, µ, t)