1. The Transport Equation (cont’d)
Lesson 2 Objectives
The Transport Equation (cont’d)
Particle distributions
Interaction rates
Boundary crossings
Derivation of the Boltzmann Equation (twice)
Initial conditions and boundary conditions

2. Particle distributions
A basic concept we will use is the particle distribution:
By (arbitrary) convention, we use the angular flux:

3. Particle distributions (2)
Angular flux is more useful for two types of events:
Interaction rates
Boundary crossings
Since the individual terms of the B.E. involve these two types of events, the angular flux is the primary unknown of the equation.

4. Interaction rates

5. Boundary crossings
Consider an element of surface area dA not perpendicular to :
Create a volume element by projecting dA backwards along the direction a distance vdt

6. Boundary crossings (2)

7. Boundary crossings (3)
Note that the total crossing rate (regardless of direction) can be found by integrating over :

8. Boundary crossings (4)
This gives us the “net current”
We are also sometimes interested in “partial currents”, which count particles crossing the surface in positive and negative directions (where positive is defined by YOU by which way points):

9. Derivation of Boltzmann Equation (Eulerian)
Particle balance on the subset of particles occupying a fixed dEdWdxdydzdt:

10. Term#1: Increase of particles
For this, we will use a Cartesian (dx,dy,dz) volume element: dz dy dx
Obviously:
differs for the 6 different faces

11. Term#1: Increase of particles (2)
And the increase in the number of particles in the phase space element in the time interval is:

12. Term#2: Particles “born”
For now, we will combine all sources (fixed source, scattering, fission) into one term:

13. Term#3: Particles streaming out
Each of the dimensions (x,y,z) has a positive and a negative face
Using the boundary crossing rates from before, this is:

14. Term#4: Particle colliding
Using the interaction rates from before, this is simply:

15. Putting it together
Combining all the terms and dividing by dxdydzdtdWdE gives us:

16. Putting it together (2)
Taking the limits as and and converting to angular flux (=vN) gives us:

17. Putting it together (3)
We can simplify (i.e., obscure) this by recalling that the gradient operator, is defined as:
which further allows us to write:

18. Putting it together (4)
And our final form of the Boltzmann Eqn. comes from substituting this to give us:
where I have simplified the notation by using:

19. Derivation of Boltzmann Equation (Lagrangian)
Now that we have successfully derived the Boltzmann equation, let’s do it again…..
This time let’s use a Lagrangian grid, that moves with the particles
With this approach, instead of letting all of the variables (x,y,z,E,W,t) define the state of the particle, we characterize the state in terms of particle incoming parameters at the boundary plus a SINGLE parameter that takes care of changes that have occurred since that initial state
For this derivation, we will let DISTANCE, s, from the boundary (in the direction of travel of the particle) be the single parameter that we use.

20. Lagrangian state description
In this alternative viewpoint, we have:
Problem boundary s

21. Lagrangian Derivation of BE (2)
We have replaced one 7D PDE with seven 1D ODEs
The advantages of this form are:
The seven are easy to solve (I solved 6 of them on the previous slide!)
This is the form of the equation that integral transport methods and Monte Carlo methods begin with
It is much easier to understand curvilinear geometries with this form
It is much easier to derive the adjoint equation with this form

22. Lagrangian Derivation of BE (3)
Now our question is how does the flux change as the particle moves from s to s+ds?
s+ds s
Problem boundary

23. Lagrangian Derivation of BE (4)
Beams are particularly easy to solve (We have been doing it since NE301!)
The particle flux is depleted by any interaction, with the probability of interaction per unit path given by the total cross section
Therefore the total probability of interaction is total cross section times ds
The gains per unit path are given by the source term times the unit path, so the balance equation is:
or:

24. Lagrangian Derivation of BE (5)
This equation will be our starting point for the integral transport equation in Chapter 5.
For now, except for the derivative term we can jump to the “deterministic” equation by simply substituting the dependencies:
If the derivatives exist, we can use the chain rule:

25. Lagrangian Derivation of BE (6)
Plugging in the derivatives (the constants that were buried in the integrals of Slide 2-20) gets us to the familiar:
This is the equation from Slide 2-16, which leads to the same final form as before (Slide 2-18)

26. Homework Problems
(2-1) Repeat the Lagrangian derivation using time, t, as the parameter instead of distance, s, along the direction of travel.
(2-2) How would the equation look for a charged particle with stopping power (i.e., energy loss per unit distance) of S(E)?
(2-3) Fermi developed his “age theory” by assuming that neutron scattering was a continuous process (instead of happening instantaneously at each collision). Using xsi (average lethargy gain per collision), show that S(E)=E*xsi*sigma-s.