pressurized water reactor (pwr) - nc state universityrsmith/ma573_f15/neutron_transport.pdf ·...
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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.
Pressurized Water Reactor (PWR)
3-D Neutron Transport Equations:
Challenges:
• Linear in the state but function of 7 independent variables:
• Very large number of inputs or parameters; e.g., 100,000; Parameter selection critical.
• ORNL Code: Denovo
• Codes can take hours to days to run.
Pressurized Water Reactor (PWR) Thermo-Hydraulic Model: Mass, momentum and energy balance for fluid
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.
Note: Similar equations for gas
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:
• Microscopic Cross-Sections: Related to types of reactions.
Flux I
NA
neutrons
cm
2 · s
nuclei
cm2R = �INA
Note: � is microscopic cross-section (cm
2)
�f : Fission
�s: Scatter�t: Total
Reference: J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis, John Wiley and Sons, 1976.
Macroscopic Cross-Sections
Macroscopic Cross-Sections: Accounts for shielding
Total Reaction Rate per Unit Area:
dR = �tIdNA
= �tINdx
Strategy: Equate reaction rate to decrease in intensity
�dI(x) = �[I(x+ dx)� I(x)]
= �tINdx
) dI
dx
= �N�tI(x)
) I(x) = I0e�N�tx
Total Macroscopic Cross-Section: ⌃t ⌘ N�t
Frequency: With which reactions occur v⌃tcm
s
1
cm=
1
s
I0
Neutron Flux in Reactor
Neutron Density: N(r, t)
• N(r, t)dr3: Expected number of neutrons in dr3 about r at time t
Reaction Rate Density: Interaction frequency
dr3
r
• N(r, E, t)dr3dE: Expected number neutrons in dr3 with energies E in dE
Angular Neutron Density:
• n(r, E, ⌦̂, t)dr3dEd⌦̂: Expected number of neutrons in dr3 about r
with energy E about dE, moving in direction ⌦̂
in solid angle ⌦̂ at time t
Angular Neutron Flux: '(r, E, ⌦̂, t) = vn(r, E, ⌦̂, t)
F (r, E, t)dr3 = v⌃N(r, E, t)dEdr3
v⌃ where v is neutron speed
Neutron Transport Equation Conservation Law:
Gain Mechanisms: (1) Neutron sources (fission)
(2) Neutrons entering V
(3) Neutrons of different E0, ⌦̂0that change to E, ⌦̂ due to scattering collision
Loss Mechanisms: (4) Neutrons leaving V
(5) Neutrons suffering a collision
@
@t
Z
Vn(r, E, ˆ⌦, t)dr3
�= gain in V � loss in V
Neutron Transport Equation Terms:
(1)
Z
VS(r, E, ⌦̂, t)dr3dEd⌦̂ where S is a source term
(5) Rate at which neutrons collide at point r is
) (5) =hR
V vP
t(r, E)n(r, E, ⌦̂, t)dr3idEd⌦̂
j(r, E, ⌦̂, t) · dS = v⌦̂n(r, E, ⌦̂, t) · dSso leakage is
=
Z
Vr · v⌦̂n(r, E, ⌦̂, t)dr3
�dEd⌦̂
=
Z
Vv⌦̂ ·rn(r, E, ⌦̂, t)dr3
�dEd⌦̂
(2), (4) Consider the rate at which neutrons leak out of surface
(4) - (2) =
Z
SdS · v⌦̂n(r, E, ⌦̂, t)
�dEd⌦̂
ft = v⌃t(r, E)n(r, E, ⌦̂, t)
Neutron Transport Equation Terms: Scattering cross-sections
• Consider first a beam of neutrons of incident intensity I, all of energy E’, hitting a thin target of surface atomic density NA
E0 E
Note: Microscopic differential scattering cross-section is proportionality constant
Rate
cm2= �s(E
0 ! E)INAdE
Microscopic scattering cross-section:
�s(E0) =
Z 1
0dE�s(E
0 ! E)
when neutron scatters from energy E0to final energy E in range E to E+dE
Macroscopic scattering cross-section:
⌃s(E0 ! E) ⌘ N�s(E
0 ! E)
Neutron Transport Equation Terms: Scattering cross-sections
• Consider how change in direction
⌦̂0 ⌦̂�s(⌦̂
0) =
Z
4⇡d⌦̂�s(⌦̂
0 ! ⌦̂)
(3) Rate at which neutrons scatter from E0, ⌦̂0to E, ⌦̂ is
Z
Vv0⌃s(E
0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)dr3�dEd⌦̂
To incorporate contributions from any E0, ⌦̂0, we integrate to obtain
(3) =
Z
Vdr3
Z
4⇡d⌦̂0
Z 1
0dE0v0⌃s(E
0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)
�dEd⌦̂
Neutron Transport Equation Conservation Law:
Since this must hold for any control volume, it follows that @n
@t+ v⌦̂ ·rn+ v⌃tn(r, E, ⌦̂, t)
0 =
Z
V
@n
@t+ v⌦̂ ·rn+ v⌃tn(r, E, ⌦̂, t)
Angular Flux: '(r, E, ⌦̂, t) = vn(r, E, !̂, t)
1
v
@'
@t+ ⌦̂ ·r'+ ⌃t(r, E)'(r, E, ⌦̂, t)
�Z 1
0dE0
Z
4⇡d⌦̂0v0⌃s(E
0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t)�S(r, E, ⌦̂, t)
�dr3dEd⌦̂
=
Z
4⇡d⌦̂0
Z 1
0dE0⌃s(E
0 ! E, ⌦̂0 ! ⌦̂)'(r, E0, ⌦̂0, t) + S(r, E, ⌦̂, t)
=
Z
4⇡d⌦̂0
Z 1
0dE0v0⌃s(E
0 ! E, ⌦̂0 ! ⌦̂)n(r, E0, ⌦̂0, t) + S(r, E, ⌦̂, t)
Neutron Transport Equation Plane Symmetry: Flux depends only on x
x
⌦̂
✓ = cos
�1 µ• ⌦
x
= cos ✓
•Z
4⇡d⌦̂0 =
Z ⇡
0d✓0 sin ✓0
Then ˆ
⌦ ·r' = ⌦
x
@'
@x
= cos ✓
@'
@x
so 1
v
@'
@t
+ cos ✓
@'
@x
+ ⌃t'(x,E, ✓, t)
=
Z ⇡
0d✓
0 sin ✓0Z 1
0dE
0⌃s(E0 ! E, ✓
0 ! ✓)'(x,E0, ✓
0, t) + S(x,E, ✓, t)
1-D Neutron Transport Equation: 1
v
@'
@t
+ µ
@'
@x
+ ⌃t'(x,E, µ, t)
=
Z 1
�1dµ
0Z 1
0dE
0⌃s(E0 ! E, µ
0 ! µ)'(x,E0, µ
0, t) + S(x,E, µ, t)