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