12-1 Lesson 12 Objectives Time dependent solutions Time dependent solutions Derivation of point kinetics equation Derivation of point kinetics equation Derivation of diffusion theory from transport theory (1D) Derivation of diffusion theory from transport theory (1D) Simple view of 1 st order perturbation theory Simple view of 1 st order perturbation theory

12-2 Time dependent BE Moving back to a previous slide (Lecture 2), the last time we had time dependence the BE looked like this: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:Since I am too lazy to write all that, I will shorten it to:

12-3 Time dependent BE (2) (I took out the fixed source since time-dependent problems usually involve reactors) (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)We will look at 3 techniques for attacking this equation (given that we can readily solve both the forward and adjoint STATIC BE)

12-4 Time dependent BE (3) It gets a little more complicated in practice because we have to account for the delayed neutrons:It gets a little more complicated in practice because we have to account for the delayed neutrons:

12-5 Method 1: Explicit For the explicit approach, we employ a forward finite difference for the time derivative:For the explicit approach, we employ a forward finite difference for the time derivative: and solve for the future values:and solve for the future values:

12-6 Method 1: Explicit (2)

12-7 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)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 termNotice the velocity term And it has the added advantage (shared by several methods) that you can determine the bracketed term (production rate – loss rate) BEFORE you pick the time stepAnd it has the added advantage (shared by several methods) that you can determine the bracketed term (production rate – loss rate) BEFORE you pick the time step Therefore, for example, you can restrict the timestep so that no value increases more than a certain percentageTherefore, for example, you can restrict the timestep so that no value increases more than a certain percentage

12-8 Method 2: Implicit For the implicit approach, we employ a backward finite difference for the time derivatives on Slide 12-4 to get: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 solutionThis uses the standard static solution I did not include the precursor solution since it is usually much easier (and explicit works fine)I did not include the precursor solution since it is usually much easier (and explicit works fine)

12-9 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 fluxThe 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 stepWe 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 shapeEvery 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)This usually allows for extremely large times steps (compared to the other two methods)

12-10 Derivation of point kinetics equation The transport equation can be reduced to the time (only) dependent point kinetics equationThe transport equation can be reduced to the time (only) dependent point kinetics equation The equation itself can be solved implicitly or explicitly (or using fancier time-dependent solutions like Runge-Kutta)The equation itself can be solved implicitly or explicitly (or using fancier time-dependent solutions like Runge-Kutta) It is the DEFINITION of the parameters from the current flux that is of prime importance in this derivationIt is the DEFINITION of the parameters from the current flux that is of prime importance in this derivation

12-11 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):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):

12-12 Derivation of point kinetics equation (3) Yielding:Yielding:(static)(time-dependent)

12-13 Derivation of point kinetics equation (4) This can be cleaned up a bit. The scattering terms can be reduced to:This can be cleaned up a bit. The scattering terms can be reduced to:

12-14 Derivation of point kinetics equation (5) And the fission term can be reduced to:And the fission term can be reduced to:

12-15 Derivation of point kinetics equation (6) Leaving us with (slightly rearranged):Leaving us with (slightly rearranged):(static)(time-dependent)

12-16 Derivation of point kinetics equation (7) Substituting the static into the time-dependent, the latter becomes:Substituting the static into the time-dependent, the latter becomes: Finally, if we define:Finally, if we define:

12-17 Derivation of point kinetics equation (8) this becomes:this becomes: if we define reactivity as:if we define reactivity as:

12-18 Derivation of diffusion theory (1D) Avoiding the much more complicated 3D form, we begin with the 1D slab BE with first order Legendre scattering:Avoiding the much more complicated 3D form, we begin with the 1D slab BE with first order Legendre scattering: First we integrate the entire equation using :First we integrate the entire equation using :

12-19 Derivation of diffusion theory (2) Or:Or: Next we integrate the entire equation usingNext we integrate the entire equation using

12-20 Derivation of diffusion theory (3) To get:To get: We get rid of the remaining integral by assuming that the second LEGENDRE moment of the flux is zero:We get rid of the remaining integral by assuming that the second LEGENDRE moment of the flux is zero:

12-21 Derivation of diffusion theory (4) Substituting this leaves us with the coupled equations:Substituting this leaves us with the coupled equations: The second equation can be solved for to get:The second equation can be solved for to get:

12-22 Derivation of diffusion theory (5) Substituting this into the first equation gives us the familiar diffusion equation:Substituting this into the first equation gives us the familiar diffusion equation: