# 10. Depletion and Transmutation¶

OpenMC supports transport-coupled and transport-independent depletion, or burnup, calculations through the openmc.deplete Python module. OpenMC uses transmutation reaction rates to solve a set of transmutation equations that determine the evolution of nuclide densities within a material. The nuclide densities predicted at some future time are then used to determine updated reaction rates, and the process is repeated for as many timesteps as are requested.

The depletion module is designed such that the reaction rate solution (the transport “operator”) is completely isolated from the solution of the transmutation equations and the method used for advancing time.

openmc.deplete supports multiple time-integration methods for determining material compositions over time. Each method appears as a different class. For example, openmc.deplete.CECMIntegrator runs a depletion calculation using the CE/CM algorithm (deplete over a timestep using the middle-of-step reaction rates). An instance of TransportOperator is passed to one of these Integrator classes along with the timesteps and power level:

power = 1200.0e6  # watts
timesteps = [10.0, 10.0, 10.0]  # days
openmc.deplete.CECMIntegrator(op, timesteps, power, timestep_units='d').integrate()


The depletion problem is executed, and once it is done a depletion_results.h5 file is written. The results can be analyzed using the openmc.deplete.Results class. This class has methods that allow for easy retrieval of k-effective, nuclide concentrations, and reaction rates over time:

results = openmc.deplete.Results("depletion_results.h5")
time, keff = results.get_keff()


Note that the coupling between the reaction rate solver and the transmutation solver happens in-memory rather than by reading/writing files on disk. OpenMC has two categories of transport operators for obtaining transmutation reaction rates.

## 10.1. Transport-coupled depletion¶

This category of operator solves the transport equation to obtain transmutation reaction rates. At present, the openmc.deplete module offers a single transport-coupled operator, openmc.deplete.CoupledOperator (which uses the OpenMC transport solver), but in principle additional transport-coupled operator classes based on other transport codes could be implemented and no changes to the depletion solver itself would be needed. The openmc.deplete.CoupledOperator class requires a Model instance containing material, geometry, and settings information:

model = openmc.Model()
...

op = openmc.deplete.CoupledOperator(model)


Any material that contains a fissionable nuclide is depleted by default, but this can behavior can be changed with the Material.depletable attribute.

Important

The volume must be specified for each material that is depleted by setting the Material.volume attribute. This is necessary in order to calculate the proper normalization of tally results based on the source rate.

### 10.1.1. Fixed-Source Transmutation¶

When the power or power_density argument is used for one of the Integrator classes, it is assumed that OpenMC is running in k-eigenvalue mode, and normalization of tally results is performed based on energy deposition. It is also possible to run a fixed-source simulation and perform normalization based on a known source rate. First, as with all fixed-source calculations, we need to set the run mode:

settings.run_mode = 'fixed source'


Additionally, all materials that you wish to deplete need to be marked as such using the Material.depletable attribute:

mat = openmc.Material()
mat.depletable = True


When constructing the CoupledOperator, you should indicate that normalization of tally results will be done based on the source rate rather than a power or power density:

op = openmc.deplete.CoupledOperator(model, normalization_mode='source-rate')


Finally, when creating a depletion integrator, use the source_rates argument:

integrator = openmc.deplete.PredictorIntegrator(op, timesteps, sources_rates=...)


As with the power argument, you can provide a different source rate for each timestep in the calculation. A zero source rate for a given timestep will result in a decay-only step, where all reaction rates are zero.

### 10.1.2. Caveats¶

#### 10.1.2.1. Energy Deposition¶

The default energy deposition mode, "fission-q", instructs the CoupledOperator to normalize reaction rates using the product of fission reaction rates and fission Q values taken from the depletion chain. This approach does not consider indirect contributions to energy deposition, such as neutron heating and energy from secondary photons. In doing this, the energy deposited during a transport calculation will be lower than expected. This causes the reaction rates to be over-adjusted to hit the user-specific power, or power density, leading to an over-depletion of burnable materials.

There are some remedies. First, the fission Q values can be directly set in a variety of ways. This requires knowing what the total fission energy release should be, including indirect components. Some examples are provided below:

# use a dictionary of fission_q values
fission_q = {"U235": 202e+6}  # energy in eV

# create a Model object
model = openmc.Model(geometry, settings)

# create a modified chain and write it to a new file
chain = openmc.deplete.Chain.from_xml("chain.xml", fission_q)
chain.export_to_xml("chain_mod_q.xml")
op = openmc.deplete.CoupledOperator(model, "chain_mod_q.xml")

# alternatively, pass the modified fission Q directly to the operator
op = openmc.deplete.CoupledOperator(model, "chain.xml",
fission_q=fission_q)


A more complete way to model the energy deposition is to use the modified heating reactions described in Heating and Energy Deposition. These values can be used to normalize reaction rates instead of using the fission reaction rates with:

op = openmc.deplete.CoupledOperator(model, "chain.xml",
normalization_mode="energy-deposition")


These modified heating libraries can be generated by running the latest version of openmc.data.IncidentNeutron.from_njoy(), and will eventually be bundled into the distributed libraries.

#### 10.1.2.2. Local Spectra and Repeated Materials¶

It is not uncommon to explicitly create a single burnable material across many locations. From a pure transport perspective, there is nothing wrong with creating a single 3.5 wt.% enriched fuel fuel_3, and placing that fuel in every fuel pin in an assembly or even full core problem. This certainly expedites the model making process, but can pose issues with depletion. Under this setup, openmc.deplete will deplete a single fuel_3 material using a single set of reaction rates, and produce a single new composition for the next time step. This can be problematic if the same fuel_3 is used in very different regions of the problem.

As an example, consider a full-scale power reactor core with vacuum boundary conditions, and with fuel pins solely composed of the same fuel_3 material. The fuel pins towards the center of the problem will surely experience a more intense neutron flux and greater reaction rates than those towards the edge of the domain. This indicates that the fuel in the center should be at a more depleted state than periphery pins, at least for the fist depletion step. However, without any other instructions, OpenMC will deplete fuel_3 as a single material, and all of the fuel pins will have an identical composition at the next transport step.

This can be countered by instructing the operator to treat repeated instances of the same material as a unique material definition with:

op = openmc.deplete.CoupledOperator(model, chain_file,
diff_burnable_mats=True)


For our example problem, this would deplete fuel on the outer region of the problem with different reaction rates than those in the center. Materials will be depleted corresponding to their local neutron spectra, and have unique compositions at each transport step. The volume of the original fuel_3 material must represent the volume of all the fuel_3 in the problem. When creating the unique materials, this volume will be equally distributed across all material instances.

Note

This will increase the total memory usage and run time due to an increased number of tallies and material definitions.

## 10.2. Transport-independent depletion¶

Warning

This feature is still under heavy development and has yet to be rigorously verified. API changes and feature additions are possible and likely in the near future.

This category of operator uses pre-calculated one-group microscopic cross sections to obtain transmutation reaction rates. OpenMC provides the IndependentOperator for this method of calculation. While the one-group microscopic cross sections can be calculated using a transport solver, IndependentOperator is not directly coupled to any transport solver. The IndependentOperator class requires a openmc.Materials object, a MicroXS object, and a path to a depletion chain file:

# load in the microscopic cross sections
materials = openmc.Materials()
...

micro_xs = openmc.deplete.MicroXS.from_csv(micro_xs_path)
op = openmc.deplete.IndependentOperator(materials, micro_xs, chain_file)


Note

The same statements from Transport-coupled depletion about which materials are depleted and the requirement for depletable materials to have a specified volume also apply here.

An alternate constructor, from_nuclides(), accepts a volume and dictionary of nuclide concentrations in place of the openmc.Materials object:

nuclides = {'U234': 8.92e18,
'U235': 9.98e20,
'U238': 2.22e22,
'U236': 4.57e18,
'O16': 4.64e22,
'O17': 1.76e19}
volume = 0.5
op = openmc.deplete.IndependentOperator.from_nuclides(volume,
nuclides,
micro_xs,
chain_file,
nuc_units='atom/cm3')


A user can then define an integrator class as they would for a coupled transport-depletion calculation and follow the same steps from there.

Note

Ideally, one-group cross section data should be available for every reaction in the depletion chain. If a nuclide that has a reaction associated with it in the depletion chain is present in the nuclides parameter but not the cross section data, that reaction will not be simulated.

### 10.2.1. Generating Microscopic Cross Sections¶

Users can generate the one-group microscopic cross sections needed by IndependentOperator using the MicroXS class:

import openmc

model = openmc.Model.from_xml()

micro_xs = openmc.deplete.MicroXS.from_model(model,
model.materials[0],
chain_file)


The from_model() method will produce a MicroXS object with microscopic cross section data in units of barns, which is what IndependentOperator expects the units to be. The MicroXS class also includes functions to read in cross section data directly from a .csv file or from data arrays:

micro_xs = MicroXS.from_csv(micro_xs_path)

nuclides = ['U234', 'U235', 'U238']
reactions = ['fission', '(n,gamma)']
data = np.array([[0.1, 0.2],
[0.3, 0.4],
[0.01, 0.5]])
micro_xs = MicroXS.from_array(nuclides, reactions, data)


Important

Both from_csv() and from_array() assume the cross section values provided are in barns by defualt, but have no way of verifying this. Make sure your cross sections are in the correct units before passing to a IndependentOperator object.

### 10.2.2. Caveats¶

#### 10.2.2.1. Reaction Rate Normalization¶

The IndependentOperator class supports two methods for normalizing reaction rates:

Important

Make sure you set the correct parameter in the openmc.abc.Integrator class. Use the source_rates parameter when normalization_mode == source-rate, and use power or power_density when normalization_mode == fission-q.

1. source-rate normalization, which assumes the source_rate provided by the time integrator is a flux, and obtains the reaction rates by multiplying the cross-sections by the source-rate.

2. fission-q normalization, which uses the power or power_density provided by the time integrator to obtain reaction rates by computing a value for the flux based on this power. The general equation for the flux is

$\phi = \frac{P}{\sum\limits_i (Q_i \sigma^f_i N_i)}$

where $$P$$ is the power, $$Q_i$$ is the fission Q value for nuclide $$i$$, $$\sigma_i^f$$ is the microscopic fission cross section for nuclide $$i$$, and $$N_i$$ is the number of atoms of nuclide $$i$$. This equation makes the same assumptions and issues as discussed in Energy Deposition. Unfortunately, the proposed solution in that section does not apply here since we are decoupled from transport code. However, there is a method to converge to a more accurate value for flux by using substeps during time integration. This paper provides a good discussion of this method.

Warning

The accuracy of results when using fission-q is entirely dependent on your depletion chain. Make sure it has sufficient data to resolve the dynamics of your particular scenario.

#### 10.2.2.2. Multiple Materials¶

Running a depletion simulation with multiple materials using the source-rate normalization method treats each material as completely separate with respect to reaction rates. This can be useful for running many different cases of a particular scenario. However, running a depletion simulation with multiple materials using the fission-q normalization method treats each material as part of the same “reactor” due to how fission-q normalization accumulates energy values from each material to a single value. This behavior may change in the future.

#### 10.2.2.3. Time integration¶

The one-group microscopic cross sections passed to openmc.deplete.IndependentOperator are fixed values for the entire depletion simulation. This implicit assumption may produce inaccurate results for certain scenarios.