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

One group of delayed neutrons using the lifetime 1/Removal rate

Using the Generation time: 1/production rate

The search for exact solutions with varying . Time varying reactivity especially ramp and oscillations

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)

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!

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 neutron

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

Importance, the adjoit equation cummutation and the detector distribution H Critica: (Ussachev) Source-free Time dependent (Lewins)

Importance, the adjoit equation cummutation and the detector distribution H Critica: (Ussachev) Source-free Time dependent (Lewins) Steady state With source (Selengut)

Importance, the adjoit equation cummutation and the detector distribution H Critica: (Ussachev) Source-free Time dependent (Lewins) Steady state With source The works

Variational Approximation Lagrangian for the question of interest Natural boundary conditions First-order error Second-order error 10%,10% gives 1%

Problem: Non-natural boundary conditions Natural BC: Outer boundaries Then sources commute Non-natural bc for ? Can non-natural bcs be represented through Dirac distributions as sources?

Or does it? What about non-naturtal bcs? ?? It does not commute! commutes only Solution: write the non-natural bcs as Dirac distributions in the source S. so that is normal. ?Ho Expectw? 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?