1. Jeffery Lewins (MIT ‘56-’59
Some lessons fro early student research
(Mistakes I have made)

2. Just Three of My Mistakes
The undergraduate paper
Reactor kinetics ‘generation time’
Adjoint equations and ‘importance’

3. Reactor Kinetics Definitons
Neutron Production rate P
Neutron Removal rate R
Neutron Lifetime 1/R
Neutron Generation time /P
k effective P/R
k excess (P-R)/P
reactivity (P-R)/P
Delayed neutron production fraction

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

5. Using the Generation time: 1/production rate

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

7. 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)

8. 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!

9. Various elegant approximations but not ‘exact’

10. 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

11. 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?

12. 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?

13. 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

14. Exact ramp reactivity solution

15. Exact oscillating reactivity solution

16. Generation time:
The time for one
neutron to produce
neutrons
Reproduction time:
The time for one neutron to produce one neutron

17. Variational theory: deriving the adjoint equation from the “Conservation of Importance’

18. Importance, the adjoit equation cummutation
and the detector distribution H
Critica:
(Ussachev)
volume of phase-space

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

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

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

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

23. 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?

24. 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?

25. 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?