# Multiphase Density Calculation Correction for Soot in OpenFOAM 5.x:

## Multiphase soot density treatment in OpenFOAM 5.x

This article details some of the theory and a lot of the implementation details involved in correcting the gas phase density calculation when soot is included via the Two Equation Soot Model. There is also some general interest knowledge to be gained about how to work the the thermodynamic classes in OpenFOAM (OF). Unfortunately a relatively deep knowledge of C++ features, especially templating and (virtual) inheritance, will be necessary to understand some of it and I will also be jumping right into the OF speak with their custom data types and runtime type instantiation stuff so a familiarity with OF is assumed too.

The specific models I'm referring to here can be found in the OpenFOAM_5.x_Libraries repository of my github account if you are interested in checking it out. I have given some cursory instructions there on installing and using it but, as with most OF packages, the onus for configuring and implementing it into a model falls on the user.

### Overview and background

With the Two Equation soot model the soot is essentially treated as a gas phase species and we use the graphite Janaf data to represent it within the OF thermodynamic database. This makes it much easier to use the devolatilization and (most of the) thermodynamic/transport machinery that is built in to OF. One significant drawback to this approach is in the calculation of the gas phase density.

The density of the gas phase mixture (at least with the thermo classes used in coalChemistryFoam and its derivatives) is calculated as $\rho = \psi \cdot P$ in the psiThermo::rho() function.

Here, $\psi$ is the fluid compressibility field and $P$ is the pressure field. The pressure field is calculated with the Poisson equation but $\psi$ is determined from a mass weighted average of the individual species compressibilities ($\psi_i$). This is inappropriate for our soot species because physically it is small discrete particles of soot and not individual carbon atoms floating in the gas mixture as the use of the graphite Janaf chemistry data implies.

The proper multiphase treatment is to calculate the gas density as

$$\rho = (1 - V_s) \cdot (\psi_{*} \cdot P) + (V_s) \cdot (\rho_{soot}) \tag{1}$$

Where $V_s$ is the soot volume fraction, $\psi_*$ is the fluid compressibility calculated as before but neglecting the contribution of the soot species. The density of coal soot, $\rho_{soot}$, is empirically known to be $2000 \left[kg/m^3\right]$.

### Relevant classes and functions in OpenFOAM

The thermo class used in coalChemistryFoam, i.e. what the main thermo the solver points at, is an object of the class

hePsiThermo<psiReactionThermo,
SpecieMixture<reactingMixture<gasHthermoPhysics>>
>

where gasHthermoPhysics is a typedef for another class

sutherlandTransport<species::thermo<janafThermo<perfectgas<specie>>,
sensibleEnthalpy
>
>

Obviously that really becomes confusing quickly but fortunately there are only a few functions that are relevant to what we are trying to accomplish. Foremost is

• psiThermo::rho()

This is the same function as mentioned above. It calculates the density volScalarField based on the multiplication of stored values of inherited member variables psi_ and p_ (that psi_ being the total mixture compressibility field).

• multiComponentMixture::cellMixture(celli)

multiComponentMixture is inherited by the reactingMixture class. The function calculates the thermodynamic properties for a cell based on the current mass fractions of species in that cell.

      mixture_ = Y_[0][celli]*speciesData_[0];

for (label n=1; n<Y_.size(); n++)
{
mixture_ += Y_[n][celli]*speciesData_[n];
}

Here speciesData_ is a list of gasHThermoPhysics objects, one object entry per species. For example, to find the molecular weight of $CH_4$, if it were the second species in the list one could use speciesData_.W(1). The overloaded += operator is used in coordination with the species mass fractions, Y_, to calculate mixture quantities. Importantly to us, the compressibility $\psi$, is one of these averaged quantities. We will therefore need to modify this function to exclude soot when taking the mass weighted average.

• multiComponentMixture::patchFaceMixture(patchi, facei)

Same as multiComponentMixture::cellMixture(celli) but determines the mixture thermo properties on a boundary face, not at a cell center. Everything that we do later to modify cellMixture needs to be done in this function as well.

• hePsiThermo::calculate()

This function loops over all cells and calls multiComponentMixture::cellMixture(celli) on each of them. Once the mixture properties are calculated for a cell it sets the main thermodynamic fields of that cell based on the mixture, these include the temperature, compressibility, dynamic viscosity and thermal diffusivity.

It subsequently calls multiComponentMixture::patchFaceMixture(celli) for each boundary face to similarly calculate and set the thermodynamic properties on the boundaries.

It is in this function that we will need to differentiate between the standard cellMixture and a sootCellMixture function, which excludes soot from the mixture, when setting the values for the psi_ field.

## Source code modifications

It turns out that all of the needed modifications can be performed at the level of hePsiThermo which is particularly convenient since that is one of the classes that can be specified at run-time in the thermophysicalProperties dictionary file. Assuming the appropriate modifications can be made and compiled in a library with a renamed class, here it is called sootHePsiThermo, we can just select that class in the dictionary and avoid further changes to the solver source code. Unfortunately as described below there is an additional complication that is resolved with a solver modification but at least major changes to the compile time solver models are avoided.

The overall plan is as follows:

1. Copy hePsiThermo.H and hePsiThermo.C to a user source directory and rename the files and class sootHePsiThermo
2. Write, within the new class, a sootCellMixture(celli) function that excludes soot from consideration
3. Modify the calculate() function to use the new sootCellMixture(celli) function when setting global field psi_
4. Add sootVolume_ field and updateSootVolume() function to class
5. Create a function sootHePsiThermo::rho() that overrides psiThermo::rho() and uses equation 1 to calculate the density. This is possible because sootHePsiThermo is derived from psiThermo.
6. Modify solver to downcast the thermo pointer, allowing us direct access
to the functions we just wrote.

A lot of the code I will reproduce in the article, but whatever is not provided here you can find in my GitHub repository within the sootReactionThermo library.

#### 1. Copy and rename the class

This is a standard first step in any modifications or model development in OF. We don't want to work on the main installation source code so copy it to your own 'User' folder first. Copy the class from the thermophysicalModels/basic/psiThermo/ directory by copying hePsiThermo.H and hePsiThermo.C to a user source directory. You will also need to copy the source file in which the new type will be created in the runtime selection table. That file is named psiReactionThermos.C and is located at thermophysicalModels/reactionThermo/psiReactionThermo/. Then change the file names (I used 'sootHePsiThermo') and use the Unix sed command from the terminal to change all 'hePsiThermo' to 'sootHePsiThermo' e.g.

user\$ sed -i -e "s/hePsiThermo/sootHePsiThermo/g" *

Be sure not to change any other names, it is easy to get carried away and accidentally change 'psiReactionThermo' to 'sootPsiReactionThermo'for instance.

You will also need to copy a Make/ directory if you want to use wmake, which I would recommend. I took it from the thermophysicalModels/reactionThermo directory. The only file that you will be compiling (i.e. the only listing in the Make/files file) is sootPsiThermos.C, the Make/options file should be okay as is but if, when compiling, you get an error just be sure to add whichever library/header file it complains about missing to it.

#### 2. Writing sootCellMixture function

As mentioned above the built in version of this function, multicomponentMixture::cellMixture(celli), needs to be modified to exclude the soot species from consideration. Here is the implementation of the new function

template<class BasicPsiThermo, class MixtureType>
typename MixtureType::thermoType
Foam::sootHePsiThermo<BasicPsiThermo, MixtureType>::sootCellMixture
(
const label celli
) const
{

// Get list of mass fractions
const PtrList<volScalarField>& Y_ = MixtureType::Y();

const scalar Ysoot = MixtureType::Y("SOOT")[celli];
const label sootIdx = MixtureType::species()["SOOT"];

// find a good index to start from, not at the soot idx
label startIdx = 0;
if (sootIdx == 0)
{
startIdx = 1;
}

// Start the mixture averaging with first species, not at soot
typename MixtureType::thermoType mixture =
(Y_[startIdx][celli]/(1.0 - Ysoot))*MixtureType::speciesData()[startIdx];

// Loop through all other species and get the average
// properties, excluding soot and whatever startIdx is
for (label n=(startIdx + 1); n<Y_.size(); n++)
{
if (n != sootIdx && n != startIdx)
{
// adjusted Y to renomalize mass fractions with soot excluded
const scalar adjY = Y_[n][celli]/(1.0 - Ysoot);

mixture += adjY * MixtureType::speciesData()[n];
}
}

return mixture;
}

Fortunately we have the MixtureType template argument here that allows us to refer to anything we need from the mixture classes so this can almost be directly copied from the main OpenFOAM implementation in the multiComponentMixture class.

Hopefully the changes here are pretty clear, we are just excluding soot from the calculation of the mixture thermo dynamic properties. I think that there are two potential points of confusion.

• We don't know before hand the position of soot within the speciesData() list. It's therefore possible that its at position 0. You can see how that is handled by checking to be sure sootidx isn't 0 and skipping it if it is.

• It is important to renormalize the mass fractions after soot has been removed from consideration since the soot mass fraction can be quite significant. That is accomplished with all of the adjY[i] = Y[i]/(1.0 - Ysoot) business.

For brevity I won't show the code here but as I mentioned earlier you will need to create another function (mine is named sootPatchFaceMixture(patchi,facei)) that does for the boundary faces what sootCellMixture(celli) does for the cell centers (i.e. exclude soot from the calculation). The necessary code changes are entirely analogous.

#### 3. Modify calculate() to use our new sootCellMixture(celli)

Now that we can calculate important thermodynamic variables in a cell while excluding soot we need to modify this function to set the main them member volScalarField psi_ accordingly. Here is the modified implementation which utilized sootCellMixture(celli) to calculate $\psi$.

template<class BasicPsiThermo, class MixtureType>
void Foam::sootHePsiThermo<BasicPsiThermo, MixtureType>::calculate()
{
const scalarField& hCells = this->he_;
const scalarField& pCells = this->p_;

scalarField& TCells = this->T_.primitiveFieldRef();
scalarField& psiCells = this->psi_.primitiveFieldRef();
scalarField& muCells = this->mu_.primitiveFieldRef();
scalarField& alphaCells = this->alpha_.primitiveFieldRef();

forAll(TCells, celli)
{
// Use the sootCellMixture function to exclude
// soot from the calculation
const typename MixtureType::thermoType sootMixture_ =
this->sootCellMixture(celli);

const typename MixtureType::thermoType& mixture_ =
this->cellMixture(celli);

TCells[celli] = mixture_.THE
(
hCells[celli],
pCells[celli],
TCells[celli]
);

psiCells[celli] = sootMixture_.psi(pCells[celli], TCells[celli]);

muCells[celli] = mixture_.mu(pCells[celli], TCells[celli]);
alphaCells[celli] = mixture_.alphah(pCells[celli], TCells[celli]);
}

volScalarField::Boundary& pBf =
this->p_.boundaryFieldRef();

volScalarField::Boundary& TBf =
this->T_.boundaryFieldRef();

volScalarField::Boundary& psiBf =
this->psi_.boundaryFieldRef();

volScalarField::Boundary& heBf =
this->he().boundaryFieldRef();

volScalarField::Boundary& muBf =
this->mu_.boundaryFieldRef();

volScalarField::Boundary& alphaBf =
this->alpha_.boundaryFieldRef();

forAll(this->T_.boundaryField(), patchi)
{
fvPatchScalarField& pp = pBf[patchi];
fvPatchScalarField& pT = TBf[patchi];
fvPatchScalarField& ppsi = psiBf[patchi];
fvPatchScalarField& phe = heBf[patchi];
fvPatchScalarField& pmu = muBf[patchi];
fvPatchScalarField& palpha = alphaBf[patchi];

if (pT.fixesValue())
{
forAll(pT, facei)
{
const typename MixtureType::thermoType& mixture_ =
this->patchFaceMixture(patchi, facei);

const typename MixtureType::thermoType& sootMixture_ =
this->sootPatchFaceMixture(patchi, facei);

phe[facei] = mixture_.HE(pp[facei], pT[facei]);

ppsi[facei] = sootMixture_.psi(pp[facei], pT[facei]);
pmu[facei] = mixture_.mu(pp[facei], pT[facei]);
palpha[facei] = mixture_.alphah(pp[facei], pT[facei]);
}
}
else
{
forAll(pT, facei)
{
const typename MixtureType::thermoType& mixture_ =
this->patchFaceMixture(patchi, facei);

const typename MixtureType::thermoType& sootMixture_ =
this->sootPatchFaceMixture(patchi, facei);

pT[facei] = mixture_.THE(phe[facei], pp[facei], pT[facei]);

ppsi[facei] = sootMixture_.psi(pp[facei], pT[facei]);
pmu[facei] = mixture_.mu(pp[facei], pT[facei]);
palpha[facei] = mixture_.alphah(pp[facei], pT[facei]);
}
}
}
}

#### 4. Add a sootVolume_ field and updateSootVolume() function to the sootHePsiThermo class

In order to calculate the density in accordance with Equation 1 we need to first determine the soot volume fraction field. Since this field is of interest in its own right another valid approach might be to create the field within the solver and then just pass it to this class to make the density calculation. Here I have made it a member of the class and just added an access function for it.

You can see the addition of the field on my github there is nothing too special about it. For convenience I also added a soot density member variable, sootDensity_, that is hard-coded to $2000 \, [kg/m^3]$.

The function updateSootVolume() is taken with only minor modifications from the 'greyMeanSolidAbsoprtionEmission' radiation absorption/emission model. I thought it was a little complicated when I first saw it but I now think it is the best way to calculate volume fraction of a species. The basic idea is to first calculate something like the mixture specific volume

$$\nu_{mix} = \sum_{i}^{species} Y_i / \rho_i(T,P)$$

Where $Y_i$ is a species mass fraction and the species density $\rho_i$ is calculated with the ideal gas law for the individual species. We first sum $Y_i / \rho_i$, note that those terms individually have units $[V_i / \text{unit mass}\,]$. Summing them yields the total volume of all gas species per unit mass ($\dot{=} \, V_{mix} / \text{unit mass}$). Then taking $\nu_{soot} = Y_{soot} / \rho_{soot}$ we can calculate the volume fraction as $\nu_{soot} / \nu_{mix}$ which has units of $V_{soot} / V_{mix}$. It is important here to use the known density of soot, 2000 $[kg/m^3]$, rather than the ideal gas law prediction based on Janaf thermodynamic data or this entire exercise would be in vain.

A problem that remains in this approach is the question of what to do with the boundary values. The density field is a volScalarField and it therefore needs to have boundary conditions defined and we will in turn need boundary values for the soot volume fraction to calculate it. I think it should be possible to calculate boundary face values since the p_, T_ and Y fields are also volScalarFields but for now I am going to assume zero soot volume fraction at the boundaries (the last line in the function). At any rate if you do nothing the uninitialized boundary values will cause drastic density fluctuations and crash the simulation immediately.

template<class BasicPsiThermo, class MixtureType>
void Foam::sootHePsiThermo<BasicPsiThermo, MixtureType>::updateSootVolume()
{

// Hardcoded soot density
scalar sootDensity(2000.0); // kg\m^3 from dasgupta thesis

// To be the sum overall species of [m^3_species / kg_total]
scalarField specificVolumeSum =
scalarField(this->sootVolume_.size(), 0.0);

// As we iterate we will grab the SOOT species index
label sootIdx(-1);

// Pointer to the mixture for this thermo
basicSpecieMixture& mixture_ = this->composition();

forAll(this->Y(), specieI)
{
const scalarField& Yi = mixture_.Y()[specieI];
const word specieName = mixture_.Y()[specieI].name();

if (specieName == "SOOT")
{
sootIdx = specieI;
specificVolumeSum += Yi/sootDensity;
}
else
{
// loop through cells for non-constant density
forAll(specificVolumeSum, celli)
{
specificVolumeSum[celli] += Yi[celli]/
mixture_.rho(specieI, this->p_[celli], this->T_[celli]);
}
}

}// end loop through species

// now find and set the soot volume fraction as
// [V_soot/kg_total] / [V_total/kg_total]
this->sootVolume_.primitiveFieldRef() =
(this->Y()[sootIdx]/sootDensity) / (specificVolumeSum);

// Set this to 0 so that psi_* P_ is used for the boundary density field.
this->sootVolume_.boundaryFieldRef() = 0.0;
}

#### 5. Create the sootHePsiThermo::rho() function

Finally we are ready to actually write a new density calculation function. All of the work is already done and we can just write out the density function as described in equation 1 using the member variables we added earlier.

template<class BasicPsiThermo, class MixtureType>
Foam::tmp<Foam::volScalarField>
Foam::sootHePsiThermo<BasicPsiThermo, MixtureType>::rho() const
{

return (1.0 - this->sootVolume_)*(this->p_*this->psi_) +
(this->sootVolume_) * this->sootDensity_;
}

#### 6. Modify the solver to allow access to new class

As mentioned above I thought this would be a good approach because the use of sootHePsiThermo rather than hePsiThermo is specified at runtime from the 'thermophysicalProperties' dictionary. That means that no solver modifications are necessary. Unfortunately I was mistaken because the thermo class within the solver is not created directly but instantiated within the combusition class. The combustion class then passes an upcasted reference to the solver, here is the code from the coalChemistryFoam createFields.H file

// In solver createFields.C file

Info<< "Creating combustion model\n" << endl;

autoPtr<combustionModels::psiCombustionModel> combustion
(
combustionModels::psiCombustionModel::New(mesh)
);

psiReactionThermo& thermo = combustion->thermo();

Examining the thermo class in a little more detail we can see why this is possible. First here is the class again, as shown at the top of this document I have replace the specific template parameters used with more general names (those used for the template parameters in the source) for brevity.

hePsiThermo<BasicThermo,MixtureType>

You can look at the documentaton of the classes and discover that while hePsiThermo does not directly inherit from its template parameters, BasicThermo and MixtureType, it does inherit from heThermo<BasicThermo,MixtureType>. And heThermo<BasicThermo,MixtureType> inherits from both BasicThermo and MixtureType. So in the specific case where we BasicThermo is psiReactionThermo we know that hePsiThermo indirectly inherits from psiReactionThermo and it can therefore be upcast as implied in the solver code snippet above. I think they actually do a dynamic_cast on the original thermo reference within the combustion model to upcast and then just pass that member reference variable here.

The problem with the upcast to psiReactionThermo is that all of the functions we just wrote in hePsiThermo are now inaccessible (you can't use a base class reference to access derived class functions when those functions they aren't present in the base class, and even then only if they are virtual functions). So the options are to either create a new combustion model too, with only the type of the thermo reference changed. Or to just use a downcast within the solver to change the reference to psiReactionThermo to a reference to sootHePsiThermo. The second option is used here.

Here is my new version of the createFields file for the SootCoalFoam solver (one of the solvers that is available on my
GitHub) in which the thermo pointer is downcast using dynamic_cast

// Modified solver createField.C file within SootCoalFoam

autoPtr<combustionModels::psiCombustionModel> combustion
(
combustionModels::psiCombustionModel::New(mesh)
);

psiReactionThermo& baseThermo = combustion->thermo();

// Downcast baseThermo to thermo
// changes type from psiReactionThermo to sootHePsiThermo
// which enables us to use the functions in sootHePsiThermo
sootHePsiThermo<
psiReactionThermo,
SpecieMixture< reactingMixture< gasHThermoPhysics > >
> &thermo =
dynamic_cast<
sootHePsiThermo<
psiReactionThermo,
SpecieMixture<reactingMixture<gasHThermoPhysics > >
> &>
(baseThermo);

It is really messy, maybe some typdefs would be helpful to understand here but all we are doing is telling the pointer that it now points to type

hePsiThermo<psiReactionThermo,
SpecieMixture<reactingMixture<gasHthermoPhysics>>
>

rather than just psiReactionThermo

I assume that the creators of OpenFOAM wrote it that way for a reason and I'm a little afraid that having the thermo pointer like that will cause a problem but so far in my testing I have not encountered any problems.

You will need to link to the new library we created with the sootHePsiThermo class before compiling the solver. You will also need to include some header files for the additional classes needed in the downcast like SpecieMixture, gasHthermoPhysics and reactingMixture. And finally you will also need to adjust your thermoPhysicalProperties file within your case to use sootHePsiThermo rather than hePsiThermo or you will get a bad cast exception.