From aed4f06f141c52a0f532df3177066f59b2b297de Mon Sep 17 00:00:00 2001 From: Bryan Herman Date: Wed, 10 Sep 2014 22:25:47 -0400 Subject: [PATCH] added last toy problem --- docs/source/methods/cmfd.rst | 132 ++++++++++++++++++++++++++++++++++- 1 file changed, 131 insertions(+), 1 deletion(-) diff --git a/docs/source/methods/cmfd.rst b/docs/source/methods/cmfd.rst index 67967e933..39573c2fa 100644 --- a/docs/source/methods/cmfd.rst +++ b/docs/source/methods/cmfd.rst @@ -453,7 +453,7 @@ because no fission neutrons appear with energies in the thermal group. +--------------------------------------------------------------------------------------------+----------------+---------------------------+ +--------------------------------------------------------------------------------------------+----------------+---------------------------+ | tally | score | filter | - +--------------------------------------------------------------------------------------------+----------------+---------------------------+ + +============================================================================================+================+===========================+ | \ :math:`\left\langle\overline{\overline\phi}_{l,m,n}^g | flux | mesh, energy | | \Delta_l^u\Delta_m^v\Delta_n^w\right\rangle` | | | +--------------------------------------------------------------------------------------------+----------------+---------------------------+ @@ -523,6 +523,136 @@ weights of neutrons from the source bank on a given spatial and energy mesh. Once weight adjustment factors were calculated, each neutron's statistical weight in the source bank was modified according to its location and energy. +------------------- +Toy Problem Example +------------------- + +Before applying CMFD to a large reactor, a simple $1$-D slab toy problem was +analyzed to understand how CMFD works. Table :ref:`tab_1Dtoyinput` presents +data used to construct this problem. For MFD, the mesh was 2 cm over the +geometry with one energy group. A comparison of fission source convergence +using Shannon entropy is shown in figure :ref:`fig_1Dentropy`. The figure +illustrates that it takes about 150 FSGs (equivalent to batches) to converge +the source with standard MC without CMFD. For the case with CMFD, it is +activated at batch 11 and directly affects the fission source for batch 12. +Convergence of the fission source is almost immediately reached. + +.. _tab_1Dtoyinput: + +.. table:: Input data for 1-D slab toy problem + + +--------------------------------------------+-------------------+ + +--------------------------------------------+-------------------+ + | Slab Length | 200 cm | + +============================================+===================+ + | Homogeneous material of :math:`\rm UO_2` | 19 g/cc density | + +--------------------------------------------+-------------------+ + | U-235 weight percent | 0.21 | + +--------------------------------------------+-------------------+ + | U-238 weight percent | 0.68 | + +--------------------------------------------+-------------------+ + | O-16 weight percent | 0.11 | + +--------------------------------------------+-------------------+ + | Number of particles per | 4,000,000 | + +--------------------------------------------+-------------------+ + | Number of inactive | 400 | + +--------------------------------------------+-------------------+ + +| +| + +.. _fig_1Dentropy: + +.. figure:: ../_images/entropy_1Dslab.png + :scale: 10 + + Source convergence comparison for $1$-D slab toy problem + +To further show this convergence, source distributions were edited at various +batches and compared. Figures :ref:`fig_toy6` to :ref:`fig_toy200` compare the +OpenMC source distribution from the no CMFD case, the OpenMC source +distribution from the CMFD case and the CMFD source distribution for six +different batches. Figure :ref:`fig_toy6` compares the distributions at batch +6. Here, CMFD has not been activated yet, so it is just plotted at zero. As +expected, the OpenMC source distributions from the two cases are equal because +CMFD has not yet affected it. From this plot, one can also see that the source +distribution is very flat because the initial guess was uniform over space. +Because the dominance ratio is close to unity, the source will slowly converge +to a cosine-like shape. Results from batch 10 are presented in figure +:ref:`fig_toy10`. The same information is shown in this plot, but the source is +slightly more converged. It is plotted here to show how little the source +changes in four batches. Batch 11 is the first time a CMFD source is +calculated. Right away, it appears as a smooth cosine-like shape as shown in +figure :ref:`fig_toy11`. Thus, when CMFD is fed back, its source shape will +modify the OpenMC source bank to preserve this distribution on the CMFD mesh. +On the next batch, shown in :ref:`fig_toy12`, the OpenMC source and the CMFD +source from the CMFD case match, while the OpenMC source from the no CMFD case +lags behind. Two more batches are shown in figure :ref:`fig_toy40` and figure +:ref:`fig_toy200` to illustrate that all source distributions eventually line +up at batch 200. + +.. _fig_toy6: + +.. figure:: ../_images/statepoint6.png + :scale: 10 + + Source at FSG 6 + +| + +.. _fig_toy10: + +.. figure:: ../_images/statepoint10.png + :scale: 10 + + Source at FSG 10 + +| + +.. _fig_toy11: + +.. figure:: ../_images/statepoint11.png + :scale: 10 + + Source at FSG 11 + +| + +.. _fig_toy12: + +.. figure:: ../_images/statepoint12.png + :scale: 10 + + Source at FSG 12 + +| + +.. _fig_toy40: + +.. figure:: ../_images/statepoint40.png + :scale: 10 + + Source at FSG 40 + +| + +.. _fig_toy200: + +.. figure:: ../_images/statepoint200.png + :scale: 10 + + Source at FSG 200 + +| + +This simple illustration shows the power of NDA. Because of the nature of the +diffusion equation, it can propagate and dampen higher harmonics much faster +than the MC transport solution. It should be noted here that this is a very +simplified problem where the dominance ratio was increased by changing the size +of the slab. Also, the source distribution is very smooth and there was only +one homogeneous material. The problem becomes more difficult to solve when +expanding to more spatial dimensions and complex materials. + ---------- References ----------