Post on 25-Feb-2016
description
Jeffery Lewins (MIT ‘56-’59
Some lessons fro early student research
(Mistakes I have made)
Just Three of My Mistakes
• The undergraduate paper• Reactor kinetics ‘generation time’• Adjoint equations and ‘importance’
Reactor Kinetics Definitons• Neutron Production rate P• Neutron Removal rate R• Neutron Lifetime 1/R• Neutron Generation time 1/P• k effective P/R• k excess (P-R)/P• reactivity (P-R)/P• Delayed neutron production fraction
l Λkeff
kex
ρβ
One group of delayed neutrons using the lifetime
dndt=(kex −βkeff)
ln + lc→ ωn
dcdt=βkeffln−lc→ ωc
ω = 12
kex −βkeffl
−l⎛⎝⎜
⎞⎠⎟ 1± 1+
4lkex / lkex −βkeff
l−l
⎛⎝⎜
⎞⎠⎟2
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
1/Removal rate
dndt=(ρ −β)Λ
n + lc→ ωn
dcdt=βΛn−lc→ ωc
ω = 12
ρ −βΛ
−l⎛⎝⎜
⎞⎠⎟ 1± 1+
4lρ / Λρ −βΛ
−l⎛⎝⎜
⎞⎠⎟2
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
Using the Generation time: 1/production rate
The search for exact solutions with varying .
• Time varying reactivity especially ramp and oscillations
ρ(t)
The search for exact solutions with varying .
• Time varying reactivity especially ramp and oscillations
• Step change: converging series solution with infinite radius of convergence (the exponential)
ρ(t)
The search for exact solutions with varying .
• Time varying reactivity especially ramp and oscillations
• Step change: converging series solution with infinite radius of convergence (the exponential)
• Ramp:the second order (or 1+Ithorder) does not converge!
ρ(t)
• Various elegant approximations but not ‘exact’
• Various elegant approximations but not ‘exact’
• Henri Smets and the Legendre transform solution for ramp and oscillatory reactivities requiring 1+I independent inverse integral transformations
• Various elegant approximations but not ‘exact’
• Henri Smets and the Legendre transform solution for ramp and oscillatory reactivities requiring 1+I independent inverse integral transformations
• Thought:If it is there in transform space surely it must be there in real space?
• Various elegant approximations but not ‘exact’
• Henri Smets and the Legendre transform solution for ramp and oscillatory reactivities requiring 1+I independent inverse integral transformations
• Thought:If it is there in transform space surely it must be there in real space?
• Second thought: How about 1+I simultaneous first order equations?
• Various elegant approximations but not ‘exact’
• Henri Smets and the Legendre transform solution for ramp and oscillatory reactivities requiring 1+I independent inverse integral transformations
• Thought:If it is there in transform space surely it must be there in real space?
• Second thought: How about 1+I simultaneous first order equations?
• It works!! Finite radius of convergence so solve for 1+I Dirac distributions and step out as far as wanted
Exact ramp reactivity solution
Exact oscillating reactivity solution
• Generation time: The time for one neutron to produce neutrons
• Reproduction time: The time for one neutron to produce one neutronn
Variational theory: deriving the adjoint equation from the
“Conservation of Importance’
Importance, the adjoit equation cummutation
and the detector distribution H
Critica:
(Ussachev)
volume of phase-space
Mψ =0,M*ψ + =0
ψ +Mψ dV =
°∫ ψ M *ψ +d
°∫ V
Importance, the adjoit equation cummutation
and the detector distribution H
Critica:
(Ussachev)
Source-free
Time dependent
(Lewins)
Mψ =0,M*ψ + =0
ψ +Mψ dV =
°∫ ψ M *ψ +dV
°∫
∂ψ∂t
= Mψ ,−∂ψ +
∂t= M *ψ +
Importance, the adjoit equation cummutation
and the detector distribution H
Critica:
(Ussachev)
Source-free
Time dependent
(Lewins)
Steady state
With source (Selengut)
Mψ =0,M*ψ + =0
ψ +Mψ dV =
°∫ ψ M *ψ +dV
°∫
∂ψ∂t
= Mψ ,−∂ψ +
∂t= M *ψ +
ψ +S⎡⎣ ⎤⎦d
3V =°∫ ψ H[ ]d
°∫ V
Mψ + S=0 =M*ψ + + H
Importance, the adjoit equation cummutation
and the detector distribution H
Critica:
(Ussachev)
Source-free
Time dependent
(Lewins)
Steady state
With source
The works
Mψ =0,M*ψ + =0
ψ +Mψ dV =
°∫ ψ M *ψ +d
°∫ V
∂ψ∂t
= Mψ ,−∂ψ +
∂t= M *ψ +
ψ +Mψ⎡⎣ ⎤⎦dVdt =
°∫ti
t f∫ ψ M *ψ +⎡⎣ ⎤⎦d°∫ Vdt
ti
t f
∫
ψ +SrdVdt
°∫ti
t f
∫ = ψ HdVdt°∫ti
t f
∫
Mψ + S=0 =M*ψ + + H
Variational Approximation
Lagrangian for the question of interest
L = ψH +ψ + Mψ +S[ ]dV°∫ = ψHd 3V
°∫
First-order error dL(δψ ) = δψ M *ψ + + H⎡⎣ ⎤⎦dV°∫ = 0
dψ =ψ~
−ψNatural boundary conditions
Second-order error d
2L = δψ +Mδψ dV°∫ = δψ M *δψ +dV
°∫10%,10% gives 1%
Problem: Non-natural boundary conditions
Natural BC: Outer boundaries
ψ + =0,∇ψ + = 0,etc
ψ =0,∇ψ = 0,etc
Then sources commute ψHdV
°∫ = ψ +SdV°∫
Non-natural bc for ?ψ
Can non-natural bcs be represented through Dirac distributions as sources?
Or does it? What about non-naturtal bcs?
ψ +t f SddV° ,t∫∫ + ψ +ψ bc (s)dAS“∫∫ dt
= ψ HdV + ψ +bc (s)ψ dAdtS“∫∫° ,t∫∫
ψ +Mδψ
ψ +Mψ It does not commute!
commutes
Solution: write the non-natural bcs as Dirac distributions in the source S. so that is normal. ?Ho Expectw?ψ
?? ψ +t f SddV° ,t∫∫ dt = ψ HdVdt
° ,t∫∫
ψ bc (s) = δ s (r = s)ψ bc (s) = Sbc (r)ψ bc
+ (s) = δ s (r = s)ψ bc+ (s) = Hbc (r
only
Desired relationship
Sources?
Write the non-natural boundary conditions as Dirac distributions ?
Develop a Dirac notation that has to be integrated normal to the boundary surface.
Try it on a simple heat conduction problem to see if it works in two dimensions?
dn(s)
T =T0
∇T = 0 ∇T = 0
T =0