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Lesson 12 Objectives. Time dependent solutions Derivation of point kinetics equation Derivation of diffusion theory from transport theory (1D) Simple view of 1 st order perturbation theory. Time dependent BE.
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Lesson 12 Objectives • Time dependent solutions • Derivation of point kinetics equation • Derivation of diffusion theory from transport theory (1D) • Simple view of 1st order perturbation theory
Time dependent BE • Moving back to a previous slide (Lecture 2), the last time we had time dependence the BE looked like this: • Since I am too lazy to write all that, I will shorten it to:
Time dependent BE (2) • (I took out the fixed source since time-dependent problems usually involve reactors) • We will look at 3 techniques for attacking this equation (given that we can readily solve both the forward and adjoint STATIC BE)
Time dependent BE (3) • It gets a little more complicated in practice because we have to account for the delayed neutrons:
Method 1: Explicit • For the explicit approach, we employ a forward finite difference for the time derivative: • and solve for the future values:
Method 1: Explicit (2) • This is a standard forward differencing, which tends to require VERY small time steps, although it always converges to the right solution (if you have the patience) • Notice the velocity term • And it has the added advantage (shared by several methods) that you can determine the derivatives BEFORE you pick the time step • Therefore, for example, you can restrict the time step size so that the final value increases no more than a certain percentage • and solve for the future flux:
Method 2: Implicit • For the implicit approach, we employ a backward finite difference for the time derivatives on Slide 12-4 to get: • This uses the standard static solution strategy—it just adds a bit to the total cross section (on the left side) and source term (on the right side) • I did not include the precursor solution since it is usually much easier (and explicit works fine)
Method 3: Quasi-static • The quasi-static approach rests on an assumption that the ABSOLUTE MAGNITUDE of the flux changes much more rapidly than the SHAPE (in space, direction, and energy) of the flux • We employ the point kinetics equation to solve for the rapidly-changing flux magnitude over a fairly large time step • Every so often we use the current cross sections (which have changed due to temperature and material movement) in a new STATIC flux calculation to get the flux shape • This usually allows for extremely large times steps (compared to the other two methods)
Derivation of point kinetics equation • The transport equation can be reduced to the time (only) dependent point kinetics equation • In addition to the equation itself, it is the DEFINITION of the parameters from the current flux that is of prime importance in this derivation
Derivation of point kinetics equation (2) • To get there, I am going to integrate the static and transient equations, returning to the brack-et notation for integration over all variables (except time):
Derivation of point kinetics equation (3) • Yielding: • (static) • (time-dependent)
Derivation of point kinetics equation (4) • This can be cleaned up a bit. The scattering terms can be reduced to:
Derivation of point kinetics equation (5) • And the fission term can be reduced to:
Derivation of point kinetics equation (6) • Leaving us with (slightly rearranged): • (static) • (time-dependent) • If you get this derivation on a test, START HERE.
Derivation of point kinetics equation (7) • Substituting the static into the time-dependent, the latter becomes: • Finally, if we define:
Derivation of point kinetics equation (8) • this becomes: • if we define reactivity as:
Derivation of diffusion theory (1D) • As an illustration of the much more complicated 3D form, we begin with the 1D slab BE with first order Legendre scattering and isotropic source: • First we integrate the entire equation using :
Derivation of diffusion theory (2) • Or: • Next we integrate the entire equation using
Derivation of diffusion theory (3) • Since integrals of odd powers go to 0, weget: • We get rid of the remaining integral by assuming that the second LEGENDRE moment of the flux is zero:
Derivation of diffusion theory (4) • Substituting this leaves us with the coupled equations: • The second equation can be solved for to get: • (This BTW is called “Fick’s law”.)
Derivation of diffusion theory (5) • Substituting this into the first equation gives us the familiar diffusion equation: