Post-Refinement Multiscale Method

Post-Refinement methods are one-way coupling methods from the nodal diffusion solver back to the higher order transport solver, as shown in Figure 3.2. These methods impose the core level solution by assigning either an albedo type boundary condition or a fixed boundary source problem. These boundary assignments have a fundamentally different solution technique which must be applied. The albedo boundary condition provides the ratio of the incoming angular flux to the outgoing angular flux. With these boundary conditions, the problem becomes an eigenvalue problem and a new eigenvalue is determined which represents the scaling of the fission source for the local domain. The fixed boundary source problem is not an eigenvalue problem. In this method the angular flux at the boundary is specified and the fission source is scaled with the eigenvalue from the global calculation. Both methods have advantages and disadvantages in the solution technique and implementation.

The solution procedure for the post-refinement method begins with a full core nodal diffusion calculation. Once the diffusion calculation has converged to the given tolerance, the surface fluxes and net currents are extracted at the boundary of the multiscale region. In the albedo method, the ratio of the incoming and outgoing partial currents are used for all angles, and are assigned to all fine groups which the coarse group solution represents. For the incoming angular flux, the surface flux and

Start

Solve CMFD

Update Eigenvalue

Nodal Update?

Deterimine Incoming Current Solve Corner Point Balance

Nodal Sweep Update Fluxes and Net Current

Update CMFD Parameters Converged? Setup Linear System

Calculate Boundary Conditions Run Transport Solver Scale Currents and Fluxes

Finish CMFD Kernel (Global) Nodal Kernel (Local) Multistep Method (Multiscale)

Figure 3.2: Flow Chart of the Post-Refinement Method

net current are used to project angular and energy shapes using one of the expansion methods mentioned above. Finally, the local problem is solved, and the pin power shape is normalized and projected onto the global solution. There are no changes made to the global solution except the shape of the pin power distribution.

The projection of the pin powers onto the global solution is one of the main shortfalls of the post-refinement method because it is completely dependent on the global solution. For example, if the assembly power is off by two percent, then the pin powers in that assembly will also be off by two percent. The post-refinement method does have some benefits as well. First of all, the post-refinement method could be implemented into any code system with only minor modifications. The second is that the post-refinement method can allow the user to specify a buffer region where the local solution may not be accurate. The buffer region only serves to decrease the impact of the boundary condition on the solution. When the pin power projection is determined, only the solution outside the buffer region is used.

3.2.1

Post-Refinement Fixed Source Implementation

The post-refinement methods use the global CMFD solver to fully converge the global FMFD linear system. The net current and surface flux from the global solution are calculated on the boundary of the multiscale region. The incoming angular flux at the boundary of the multiscale region is defined using the P1 angular distribution. The energy projection is performed using a predefined flux distribution obtained from the energy shape of the surface fluxes of the lattice calculation.

During the fixed boundary source calculation the fission source is scaled by the eigenvalue calculated from the global solution. Since the fixed source problem is not an eigenvalue problem, the acceleration scheme is slightly different. The CMFD solution is rewritten to have boundary sources instead of albedo boundary conditions. Since an eigenvalue iteration does not need to be performed, the fission and scattering sources can be moved to the left hand side of the equation, leaving only the boundary source on the right hand side of the system of equations.

(M − λglobalF) φ = C0Jin (3.11)

Here λglobal is the eigenvalue from the global calculations and must be used to scale the fission source. There are physical cases where the a steady state solution does not exist for the fixed source transport equation. This occurs when the production of neutrons through fission is greater than or equal to the loss of neutrons through absorption and leakage. This occurs when the eigenvalue of the scaled transport equation is greater than or equal to one. Equation (3.12) shows the eigenvalue problem for k. Since only the largest eigenvalue is required, the spectral radius of the matrix is taken to determine k.  M − 1 k(λglobalF)  φ = 0 (3.12a) k = ρ  M−1F kglobal  (3.12b) Since the maximum eigenvalue of M−1F is, by definition, the eigenvalue of the sub- domain, k can be written as the ratio of the local high order and global low order eigenvalue.

k = ksubdomain kglobal

(3.13) Although it is possible for this ratio to be greater than one, the leakage from the subdomain for most multiscale cases will make this ratio substancially smaller than 1.

The matrix M is slightly modified to handle the fact that the boundary currents are on the source side of the equations. The nonlinear correction factors from the CMFD are still added into the migration matrix to preserve the higher order net current.

3.2.2

Post-Refinement Albedo Boundary Implementation

The net current and surface flux from the global FMFD solution are combined to calculate the ratio of the incoming and outgoing partial currents.

β = φ + 2J net_{· n}

φ − 2Jnet_{· n} (3.14)

The albedos are applied to the local boundary and then a local eigenvalue calcula- tion is performed. The local eigenvalue calculation is identical to the global, except the extent of the domain is expanded. The albedo is applied uniformly for all fine energy groups contained in the coarse energy groups.