FireFoam

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fireFoam

   Transient solver for fires and turbulent diffusion flames with reacting
   particle clouds, surface film and pyrolysis modelling.

1 Solution Strategy

The source code can be found in fireFoam.C


 
 
/*---------------------------------------------------------------------------*\
  =========                 |
  \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
   \\    /   O peration     |
    \\  /    A nd           | Copyright (C) 2011-2017 OpenFOAM Foundation
     \\/     M anipulation  |
-------------------------------------------------------------------------------
License
    This file is part of OpenFOAM.
 
    OpenFOAM is free software: you can redistribute it and/or modify it
    under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.
 
    OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
    ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
    for more details.
 
    You should have received a copy of the GNU General Public License
    along with OpenFOAM.  If not, see <http://www.gnu.org/licenses/>.
 
Application
    fireFoam
 
Group
    grpCombustionSolvers
 
Description
    Transient solver for fires and turbulent diffusion flames with reacting
    particle clouds, surface film and pyrolysis modelling.
 
\*---------------------------------------------------------------------------*/
 
#include "fvCFD.H"
#include "turbulentFluidThermoModel.H"
#include "basicReactingCloud.H"
#include "surfaceFilmModel.H"
#include "pyrolysisModelCollection.H"
#include "radiationModel.H"
#include "SLGThermo.H"
#include "solidChemistryModel.H"
#include "psiReactionThermo.H"
#include "CombustionModel.H"
#include "pimpleControl.H"
#include "fvOptions.H"
 
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
int main(int argc, char *argv[])
{
    argList::addNote
    (
        "Transient solver for fires and turbulent diffusion flames"
        " with reacting particle clouds, surface film and pyrolysis modelling."
    );
 
    #include "postProcess.H"
 
    #include "addCheckCaseOptions.H"
    #include "setRootCaseLists.H"
    #include "createTime.H"
    #include "createMesh.H"
    #include "createControl.H"
    #include "createFields.H"
    #include "createFieldRefs.H"
    #include "initContinuityErrs.H"
    #include "createTimeControls.H"
    #include "compressibleCourantNo.H"
    #include "setInitialDeltaT.H"
    #include "createRegionControls.H"
 
    turbulence->validate();
 
    // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
 
    Info<< "\nStarting time loop\n" << endl;
 
    while (runTime.run())
    {
        #include "readTimeControls.H"
        #include "compressibleCourantNo.H"
        #include "solidRegionDiffusionNo.H"
        #include "setMultiRegionDeltaT.H"
        #include "setDeltaT.H"
 
        ++runTime;
 
        Info<< "Time = " << runTime.timeName() << nl << endl;
 
        parcels.evolve();
 
        surfaceFilm.evolve();
 
        if(solvePyrolysisRegion)
        {
            pyrolysis.evolve();
        }
 
        if (solvePrimaryRegion)
        {
            #include "rhoEqn.H"
 
            // --- PIMPLE loop
            while (pimple.loop())
            {
                #include "UEqn.H"
                #include "YEEqn.H"
 
                // --- Pressure corrector loop
                while (pimple.correct())
                {
                    #include "pEqn.H"
                }
 
                if (pimple.turbCorr())
                {
                    turbulence->correct();
                }
            }
 
            rho = thermo.rho();
        }
 
        runTime.write();
 
        runTime.printExecutionTime(Info);
    }
 
    Info<< "End" << endl;
 
    return 0;
}
 
 
// ************************************************************************* //
 

2 Equations

2.1 Equations Parcels

The particles in fireFoam are treated in a Lagrangian frame of reference. This means that each particle (or parcel which is an ensemble of a number of particles with the same properties) is tracked individually throughout the computational domain. For each parcel a ordinary differential equation is solved which describes the evolution of its mass, velocity (after one integration more also the position in space) and its temperature. The evolution of the particle mass is required besides for its own momentum equation to calculate the mass source term in the spices and the pressure equation of the carrier phase. The particle momentum equation through which the particle velocity and after one integration also the particle position is obtained, is required to calculate the forces acting from the particles on the carrier phase required in the momentum equation of the fluid. The energy equation of the particles (the temperature is obtained by an equation of state) is required to compute the source term due to the particles in the energy equation of the carrier phase. Note that some additional informations can be also found in https://www.politesi.polimi.it/bitstream/10589/134603/7/2017_07_Ghasemi.pdf and http://www.tfd.chalmers.se/~hani/kurser/OS_CFD_2013/EmilLjungskog/reactingParcelFilmFoam_report.pdf.


2.1.1 Evolution of Mass

According to [1] (see also https://trs.jpl.nasa.gov/bitstream/handle/2014/22473/97-0966.pdf?sequence=1) the evaporating mass of a spherical droplet with the diameter  d_{p,k} and mass  m_{p,k} (the subscript k denotes the spices k since in OpenFOAM different spices can be tracked) can be expressed in the following form:




    \frac{ d m_{p,k}}{d t}   =  - \frac{Sh}{3 Sc} \frac{m_{p,k}}{\tau_{p,k}} H_M
(x)

 Sh,  Sc ,  \tau_{p,k} = \rho_{p,k} d_p^2 / (18 \mu) and  H_M are the Sherwood number, the Schmidt number, the particle relaxation time and the potential driving the evaporation, respectively.  \rho_{p,k} is the particle density and  \mu the dynamic viscosity of the carrier phase. The Sherwood number describes the ratio between the convective mass transfer and the diffusion (https://en.wikipedia.org/wiki/Sherwood_number). The Schmidt number is defined as the ratio between viscous diffusion rate and mass diffusion rate (https://en.wikipedia.org/wiki/Schmidt_number).


The source therm in the pressure and in the species equations of the carrier phase are calculated in src/lagrangian/intermediate/clouds/Templates/ReactingCloud/ReactingCloudI.H:



 
 
template<class CloudType>
inline Foam::tmp<Foam::DimensionedField<Foam::scalar, Foam::volMesh>>
Foam::ReactingCloud<CloudType>::Srho(const label i) const
{
    tmp<volScalarField::Internal> tRhoi
    (
        new volScalarField::Internal
        (
            IOobject
            (
                this->name() + ":rhoTrans",
                this->db().time().timeName(),
                this->db(),
                IOobject::NO_READ,
                IOobject::NO_WRITE,
                false
            ),
            this->mesh(),
            dimensionedScalar
            (
                rhoTrans_[0].dimensions()/dimTime/dimVolume, Zero
            )
        )
    );
 
    if (this->solution().coupled())
    {
        scalarField& rhoi = tRhoi.ref();
        rhoi = rhoTrans_[i]/(this->db().time().deltaTValue()*this->mesh().V());
    }
 
    return tRhoi;
}
 
 
template<class CloudType>
inline Foam::tmp<Foam::DimensionedField<Foam::scalar, Foam::volMesh>>
Foam::ReactingCloud<CloudType>::Srho() const
{
    tmp<volScalarField::Internal> trhoTrans
    (
        new volScalarField::Internal
        (
            IOobject
            (
                this->name() + ":rhoTrans",
                this->db().time().timeName(),
                this->db(),
                IOobject::NO_READ,
                IOobject::NO_WRITE,
                false
            ),
            this->mesh(),
            dimensionedScalar
            (
                rhoTrans_[0].dimensions()/dimTime/dimVolume, Zero
            )
        )
    );
 
    if (this->solution().coupled())
    {
        scalarField& sourceField = trhoTrans.ref();
        forAll(rhoTrans_, i)
        {
            sourceField += rhoTrans_[i];
        }
 
        sourceField /= this->db().time().deltaTValue()*this->mesh().V();
    }
 
    return trhoTrans;
}


The field rhoTrans_ is modified in the file src/lagrangian/intermediate/parcels/Templates/ReactingParcel/ReactingParcel.C:


 
template<class ParcelType>
template<class TrackCloudType>
void Foam::ReactingParcel<ParcelType>::calc
(
    TrackCloudType& cloud,
    trackingData& td,
    const scalar dt
)
{
    const auto& composition = cloud.composition();
 
 
    // Define local properties at beginning of time step
    // ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
    const scalar np0 = this->nParticle_;
    const scalar d0 = this->d_;
    const vector& U0 = this->U_;
    const scalar T0 = this->T_;
    const scalar mass0 = this->mass();
 
 
    // Calc surface values
    scalar Ts, rhos, mus, Prs, kappas;
    this->calcSurfaceValues(cloud, td, T0, Ts, rhos, mus, Prs, kappas);
    scalar Res = this->Re(rhos, U0, td.Uc(), d0, mus);
 
 
    // Sources
    // ~~~~~~~
 
    // Explicit momentum source for particle
    vector Su = Zero;
 
    // Linearised momentum source coefficient
    scalar Spu = 0.0;
 
    // Momentum transfer from the particle to the carrier phase
    vector dUTrans = Zero;
 
    // Explicit enthalpy source for particle
    scalar Sh = 0.0;
 
    // Linearised enthalpy source coefficient
    scalar Sph = 0.0;
 
    // Sensible enthalpy transfer from the particle to the carrier phase
    scalar dhsTrans = 0.0;
 
 
    // 1. Compute models that contribute to mass transfer - U, T held constant
    // ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
    // Phase change
    // ~~~~~~~~~~~~
 
    // Mass transfer due to phase change
    scalarField dMassPC(Y_.size(), 0.0);
 
    // Molar flux of species emitted from the particle (kmol/m^2/s)
    scalar Ne = 0.0;
 
    // Sum Ni*Cpi*Wi of emission species
    scalar NCpW = 0.0;
 
    // Surface concentrations of emitted species
    scalarField Cs(composition.carrier().species().size(), 0.0);
 
    // Calc mass and enthalpy transfer due to phase change
    calcPhaseChange
    (
        cloud,
        td,
        dt,
        Res,
        Prs,
        Ts,
        mus/rhos,
        d0,
        T0,
        mass0,
        0,
        1.0,
        Y_,
        dMassPC,
        Sh,
        Ne,
        NCpW,
        Cs
    );
 
 
    // 2. Update the parcel properties due to change in mass
    // ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
    scalarField dMass(dMassPC);
    scalar mass1 = updateMassFraction(mass0, dMass, Y_);
 
    this->Cp_ = composition.Cp(0, Y_, td.pc(), T0);
 
    // Update particle density or diameter
    if (cloud.constProps().constantVolume())
    {
        this->rho_ = mass1/this->volume();
    }
    else
    {
        this->d_ = cbrt(mass1/this->rho_*6.0/pi);
    }
 
    // Remove the particle when mass falls below minimum threshold
    if (np0*mass1 < cloud.constProps().minParcelMass())
    {
        td.keepParticle = false;
 
        if (cloud.solution().coupled())
        {
            scalar dm = np0*mass0;
 
            // Absorb parcel into carrier phase
            forAll(Y_, i)
            {
                scalar dmi = dm*Y_[i];
                label gid = composition.localToCarrierId(0, i);
                scalar hs = composition.carrier().Hs(gid, td.pc(), T0);
 
                cloud.rhoTrans(gid)[this->cell()] += dmi;
                cloud.hsTrans()[this->cell()] += dmi*hs;
            }
            cloud.UTrans()[this->cell()] += dm*U0;
 
            cloud.phaseChange().addToPhaseChangeMass(np0*mass1);
        }
 
        return;
    }
 
    // Correct surface values due to emitted species
    correctSurfaceValues(cloud, td, Ts, Cs, rhos, mus, Prs, kappas);
    Res = this->Re(rhos, U0, td.Uc(), this->d(), mus);
 
 
    // 3. Compute heat- and momentum transfers
    // ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
    // Heat transfer
    // ~~~~~~~~~~~~~
 
    // Calculate new particle temperature
    this->T_ =
        this->calcHeatTransfer
        (
            cloud,
            td,
            dt,
            Res,
            Prs,
            kappas,
            NCpW,
            Sh,
            dhsTrans,
            Sph
        );
 
    this->Cp_ = composition.Cp(0, Y_, td.pc(), T0);
 
 
    // Motion
    // ~~~~~~
 
    // Calculate new particle velocity
    this->U_ =
        this->calcVelocity(cloud, td, dt, Res, mus, mass1, Su, dUTrans, Spu);
 
 
    // 4. Accumulate carrier phase source terms
    // ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
    if (cloud.solution().coupled())
    {
        // Transfer mass lost to carrier mass, momentum and enthalpy sources
        forAll(dMass, i)
        {
            scalar dm = np0*dMass[i];
            label gid = composition.localToCarrierId(0, i);
            scalar hs = composition.carrier().Hs(gid, td.pc(), T0);
 
            cloud.rhoTrans(gid)[this->cell()] += dm;
            cloud.UTrans()[this->cell()] += dm*U0;
            cloud.hsTrans()[this->cell()] += dm*hs;
        }
 
        // Update momentum transfer
        cloud.UTrans()[this->cell()] += np0*dUTrans;
        cloud.UCoeff()[this->cell()] += np0*Spu;
 
        // Update sensible enthalpy transfer
        cloud.hsTrans()[this->cell()] += np0*dhsTrans;
        cloud.hsCoeff()[this->cell()] += np0*Sph;
 
        // Update radiation fields
        if (cloud.radiation())
        {
            const scalar ap = this->areaP();
            const scalar T4 = pow4(T0);
            cloud.radAreaP()[this->cell()] += dt*np0*ap;
            cloud.radT4()[this->cell()] += dt*np0*T4;
            cloud.radAreaPT4()[this->cell()] += dt*np0*ap*T4;
        }
    }
}
 


The evaporation model is called in the file src/lagrangian/intermediate/parcels/Templates/ReactingParcel/ReactingParcel.C:



 
template<class ParcelType>
template<class TrackCloudType>
void Foam::ReactingParcel<ParcelType>::calcPhaseChange
(
    TrackCloudType& cloud,
    trackingData& td,
    const scalar dt,
    const scalar Re,
    const scalar Pr,
    const scalar Ts,
    const scalar nus,
    const scalar d,
    const scalar T,
    const scalar mass,
    const label idPhase,
    const scalar YPhase,
    const scalarField& Y,
    scalarField& dMassPC,
    scalar& Sh,
    scalar& N,
    scalar& NCpW,
    scalarField& Cs
)
{
    const auto& composition = cloud.composition();
    auto& phaseChange = cloud.phaseChange();
 
    if (!phaseChange.active() || (YPhase < SMALL))
    {
        return;
    }
 
    scalarField X(composition.liquids().X(Y));
 
    scalar Tvap = phaseChange.Tvap(X);
 
    if (T < Tvap)
    {
        return;
    }
 
    const scalar TMax = phaseChange.TMax(td.pc(), X);
    const scalar Tdash = min(T, TMax);
    const scalar Tsdash = min(Ts, TMax);
 
    // Calculate mass transfer due to phase change
    phaseChange.calculate
    (
        dt,
        this->cell(),
        Re,
        Pr,
        d,
        nus,
        Tdash,
        Tsdash,
        td.pc(),
        td.Tc(),
        X,
        dMassPC
    );
 
    // Limit phase change mass by availability of each specie
    dMassPC = min(mass*YPhase*Y, dMassPC);
 
    const scalar dMassTot = sum(dMassPC);
 
    // Add to cumulative phase change mass
    phaseChange.addToPhaseChangeMass(this->nParticle_*dMassTot);
 
    forAll(dMassPC, i)
    {
        const label cid = composition.localToCarrierId(idPhase, i);
 
        const scalar dh = phaseChange.dh(cid, i, td.pc(), Tdash);
        Sh -= dMassPC[i]*dh/dt;
    }
 
 
    // Update molar emissions
    if (cloud.heatTransfer().BirdCorrection())
    {
        // Average molecular weight of carrier mix - assumes perfect gas
        const scalar Wc = td.rhoc()*RR*td.Tc()/td.pc();
 
        forAll(dMassPC, i)
        {
            const label cid = composition.localToCarrierId(idPhase, i);
 
            const scalar Cp = composition.carrier().Cp(cid, td.pc(), Tsdash);
            const scalar W = composition.carrier().W(cid);
            const scalar Ni = dMassPC[i]/(this->areaS(d)*dt*W);
 
            const scalar Dab =
                composition.liquids().properties()[i].D(td.pc(), Tsdash, Wc);
 
            // Molar flux of species coming from the particle (kmol/m^2/s)
            N += Ni;
 
            // Sum of Ni*Cpi*Wi of emission species
            NCpW += Ni*Cp*W;
 
            // Concentrations of emission species
            Cs[cid] += Ni*d/(2.0*Dab);
        }
    }
}

For the case the "liquidEvaporation" model is used the function calculate of this specific model is called in the file src/lagrangian/submodels/Reacting/PhaseChangeModel/LiquidEvaporation/LiquidEvaporation.C:



 
template<class CloudType>
void Foam::LiquidEvaporation<CloudType>::calculate
(
    const scalar dt,
    const label celli,
    const scalar Re,
    const scalar Pr,
    const scalar d,
    const scalar nu,
    const scalar T,
    const scalar Ts,
    const scalar pc,
    const scalar Tc,
    const scalarField& X,
    scalarField& dMassPC
) const
{
    // immediately evaporate mass that has reached critical condition
    if ((liquids_.Tc(X) - T) < SMALL)
    {
        if (debug)
        {
            WarningInFunction
                << "Parcel reached critical conditions: "
                << "evaporating all available mass" << endl;
        }
 
        forAll(activeLiquids_, i)
        {
            const label lid = liqToLiqMap_[i];
            dMassPC[lid] = GREAT;
        }
 
        return;
    }
 
    // construct carrier phase species volume fractions for cell, celli
    const scalarField Xc(calcXc(celli));
 
    // calculate mass transfer of each specie in liquid
    forAll(activeLiquids_, i)
    {
        const label gid = liqToCarrierMap_[i];
        const label lid = liqToLiqMap_[i];
 
        // vapour diffusivity [m2/s]
        const scalar Dab = liquids_.properties()[lid].D(pc, Ts);
 
        // saturation pressure for species i [pa]
        // - carrier phase pressure assumed equal to the liquid vapour pressure
        //   close to the surface
        // NOTE: if pSat > pc then particle is superheated
        // calculated evaporation rate will be greater than that of a particle
        // at boiling point, but this is not a boiling model
        const scalar pSat = liquids_.properties()[lid].pv(pc, T);
 
        // Schmidt number
        const scalar Sc = nu/(Dab + ROOTVSMALL);
 
        // Sherwood number
        const scalar Sh = this->Sh(Re, Sc);
 
        // mass transfer coefficient [m/s]
        const scalar kc = Sh*Dab/(d + ROOTVSMALL);
 
        // vapour concentration at surface [kmol/m3] at film temperature
        const scalar Cs = pSat/(RR*Ts);
 
        // vapour concentration in bulk gas [kmol/m3] at film temperature
        const scalar Cinf = Xc[gid]*pc/(RR*Ts);
 
        // molar flux of vapour [kmol/m2/s]
        const scalar Ni = max(kc*(Cs - Cinf), 0.0);
 
        // mass transfer [kg]
        dMassPC[lid] += Ni*pi*sqr(d)*liquids_.properties()[lid].W()*dt;
    }
}
 

2.1.2 Evolution of Momentum

2.1.3 Evolution of Energy

2.2 Equations Liquid Film

The equation describing the evolution of the liquid film can be found in http://www.ilasseurope.org/ICLASS/iclass2012_Heidelberg/Contributions/Paper-pdfs/Contribution1294_b.pdf and https://www.witpress.com/Secure/elibrary/papers/MPF11/MPF11020FU1.pdf. A description of the method used to derive the equation can be found in https://www.researchgate.net/profile/Petr_Vita/publication/313655371_Thin_Film_Fluid_Flow_Simulation_on_Rotating_Discs/links/58a1a592a6fdccf5e9707665/Thin-Film-Fluid-Flow-Simulation-on-Rotating-Discs.pdf.

2.2.1 Mass Conservation Film

The conservation of mass reads:



    \frac{ \partial (\rho \delta )}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {U}_{j} \delta \right)  =  S_{\delta}
(1)

 U represents the mean film velocity,  \delta the film height and  S_\delta = S_{\delta,imp} + S_{\delta,eva} the mass source term resulting from the impingement of the liquid droplets on the film surface  S_{\delta,imp}  and due to the evaporation of the film   S_{\delta,eva} . Note that the source term due to evaporation   S_{\delta,eva} will be inserted into the pressure equation for the gas phase and also into the equation for the spices transport. The source code of the continuity equation can be found in src/regionModels/surfaceFilmModels/kinematicSingleLayer:



 
void kinematicSingleLayer::solveContinuity()
{
    if (debug)
    {
        InfoInFunction << endl;
    }
 
    solve
    (
        fvm::ddt(deltaRho_)
      + fvc::div(phi_)
     ==
      - rhoSp_
    );
}

2.2.2 Momentum Conservation Film

The momentum conservation reads:



    \frac{ \partial (\rho \delta U)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho U_i {U}_{j} \delta \right)  = -\delta \frac{\partial p}{\partial x_i} + \tau  + S_{U\delta}
(2)

 p represents the pressure,  \tau the stress like components of the forces acting on the filme and  S_{U\delta} represents the contribution to the momentum by the impinging droplets. The pressure contribution is divided in three components: The contribution due to capillary effects  p_\sigma = -\sigma\frac{\partial \delta }{\partial x_i \partial x_i}, the hydrostatic contribution  p_\delta = - \rho x_i g_i \delta and the contribution of the pressure originated by the surrounding gas  p_g .  \sigma represents the surface tension. The stress like contribution is divided in the wall shear stress  \tau_w =- \mu \frac{ 3 U_i}{\delta} , the gravity force tangential to the wall  \tau_t = \rho \delta g_{ti} and the contribution of the contact line force  \tau_\theta .  g_{ti} is the component of the gravity parallel to the wall. Finally the momentum equation can be written as:



    \frac{ \partial (\rho \delta U)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho U_i {U}_{j} \delta \right)  = -\delta \frac{\partial}{\partial x_i} \left ( - \rho x_i g_i \delta  -\sigma\frac{\partial \delta }{\partial x_i \partial x_i} + p_g  \right ) + \rho \delta g_{ti} - \mu \frac{ 3 U_i}{\delta} +   \tau_\theta  + S_{U\delta}
(3)

The source code of the momentum equation can be found in src/regionModels/surfaceFilmModels/kinematicSingleLayer:



 
tmp<Foam::fvVectorMatrix> kinematicSingleLayer::solveMomentum
(
    const volScalarField& pu,
    const volScalarField& pp
)
{
    if (debug)
    {
        InfoInFunction << endl;
    }
 
    // Momentum
    tmp<fvVectorMatrix> tUEqn
    (
        fvm::ddt(deltaRho_, U_)
      + fvm::div(phi_, U_)
     ==
      - USp_
   // - fvm::SuSp(rhoSp_, U_)
      - rhoSp_*U_
      + forces_.correct(U_)
      + turbulence_->Su(U_)
    );
 
    fvVectorMatrix& UEqn = tUEqn.ref();
 
    UEqn.relax();
 
    if (momentumPredictor_)
    {
        solve
        (
            UEqn
         ==
            fvc::reconstruct
            (
              - fvc::interpolate(delta_)
              * (
                    regionMesh().magSf()
                  * (
                        fvc::snGrad(pu, "snGrad(p)")
                      + fvc::snGrad(pp, "snGrad(p)")*fvc::interpolate(delta_)
                      + fvc::snGrad(delta_)*fvc::interpolate(pp)
                    )
                  - fvc::flux(rho_*gTan())
                )
            )
        );
 
        // Remove any patch-normal components of velocity
        U_ -= nHat()*(nHat() & U_);
        U_.correctBoundaryConditions();
    }
 
    return tUEqn;
}


2.2.3 Energy Conservation Film

The energy equation reads:



    \frac{ \partial (\rho \delta h )}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {U}_{j} \delta h \right)  =  S_{h\delta}
(4)

 h represents the mean film enthalpy and  S_{h\delta} =  S_{h\delta,imp} + S_{h\delta,eva} represents the contribution of the impinging droplets   S_{h\delta,imp} and the evaporation of the film   S_{h\delta,eva} to the energy equation. Note that the energy required by the film (or a portion of it) to evaporate   S_{h\delta,eva} is cast as source term into the energy equation of the surrounding gas. The source code can be found in src/regionModels/surfaceFilmModels/thermoSingleLayer:


 
void thermoSingleLayer::solveEnergy()
{
    if (debug)
    {
        InfoInFunction << endl;
    }
 
 
    dimensionedScalar residualDeltaRho
    (
        "residualDeltaRho",
        deltaRho_.dimensions(),
        1e-10
    );
 
    solve
    (
        fvm::ddt(deltaRho_, hs_)
      + fvm::div(phi_, hs_)
     ==
      - hsSp_
      + q(hs_)
      + radiation_->Shs()
    );
 
    correctThermoFields();
 
    // Evaluate viscosity from user-model
    viscosity_->correct(pPrimary_, T_);
 
    // Update film wall and surface temperatures
    updateSurfaceTemperatures();
}

2.2.4 Film Thickness Equation

The last equation missing is the linear equation used to get the film thickness  \delta . In order to get a tighter coupling between the momentum equation and the continuity equation, the same strategy used to get a coupling between momentum and continuity equation in pressure based solvers (see https://openfoamwiki.net/index.php/ChtMultiRegionFoam and https://openfoamwiki.net/index.php/SimpleFoam and references therein ) is used: The discrete momentum equation is used to get a relation for the velocity at point P  U_p which is then inserted into the continuity equation (1). The most convenient way to illustrate the procedure, is to write down the momentum equation in its discrete form:



\rho  \bold {U_p} \delta = \rho  \left [ \bold { H[U] } \bold {\frac {1}{A_P}} -  \bold {\frac {1}{A_P}}\delta \nabla \left (  \underbrace{-\rho g_i x_i}_{P_p} \delta - \underbrace{\sigma \nabla \nabla \delta + p_g}_{P_u}   \right )  +  \bold {\frac {1}{A_P}}\rho \delta g_t   \right ] \delta
(5)

The operator  \bold H[U] contained the contribution of the of the neighbouring velocities and the source term to the velocity at cell centre P. The explicit and implicite source terms included into the  \bold H[U] are the contributions due to the viscose shear, the influence of the contact angle and the contribution of the particle impingement. In order that equation (5) can be inserted into the continuity equation the divergence has to be taken:



\rho \nabla  \cdot \left (   \bold {U_p} \delta \right ) = 
\rho \nabla \cdot  \left ( \left [ \bold {\frac { \bold { H[U] }}{A_P}} -  \bold {\frac {\delta}{A_P}} \nabla \left (  {P_p} \delta +{P_u}   \right )  +   {\frac {\delta}{\bold A_P}}\rho  g_t   \right ] \delta \right )
(6)

The above equation can be inserted into the continuity equation to obtain:



     \frac{ \partial (\rho \delta )}{\partial t}  + 
\rho \nabla \cdot  \left ( \left [ \bold {\frac { \bold { H[U] }}{A_P}}   - \frac {\delta^2}{A_P} \nabla   {P_p} -     \frac {\delta P_p}{A_P} \nabla   {\delta}  -  \bold {\frac {\delta}{A_P}} \nabla {P_u}  +   {\frac {\delta}{\bold A_P}}\rho  g_t   \right ] \delta \right ) = S_{\delta}
(7)

It is obvious that the above equation is non linear in  \delta . In order to be solved it has to be linearised and an iterative method has be applied to obtain an approximate solution. In order to obtain the final set of equations which are solved in this solver, a Picard or fixed point iteration is used (see e.g. https://hplgit.github.io/num-methods-for-PDEs/doc/pub/nonlin/pdf/nonlin-4print.pdf or http://web.engr.uky.edu/~acfd/me690-lctr-nts.pdf):



     \frac{ \partial (\rho \delta^{k+1} )}{\partial t}  + 
\rho \nabla \cdot  \left ( \left [ \bold {\frac { \bold { H[U] }}{A_P}}   - \frac {{\delta^k}^2}{A_P} \nabla   {P_p} -     \frac {\delta^k P_p}{A_P} \nabla   {\delta^{k+1}}  -   \frac {\delta^k}{\bold A_P} \nabla {P_u}  +   {\frac {\delta^k}{\bold A_P}}\rho  g_t   \right ] \delta^{k+1} \right ) = S_{\delta}
(8)

The superscript k denote the solution computed at the last iteration an the superscript k+1 denotes the solution of the current iteration which has to be computed by solving the linear system resulting from the discretization. After separating the linear and the still non linear terms of the above equation we obtain:



     \frac{ \partial (\rho \delta^{k+1} )}{\partial t}  + 
\rho \nabla \cdot  \left ( \left [ \bold {\frac { \bold { H[U] }}{A_P}}   - \frac {{\delta^k}^2}{A_P} \nabla   {P_p} -   \frac {\delta^k}{\bold A_P} \nabla {P_u}  +   {\frac {\delta^k}{\bold A_P}}\rho  g_t   \right ] \delta^{k+1} \right )  -   \nabla \cdot \left (  \frac {\delta^k\delta^{k+1} P_p}{A_P} \nabla   {\delta^{k+1}}  \right )     = S_{\delta}
(9)

Since the above equation still contains a non linear contribution it has remaining  \delta^{k+1} has to be approximated by the value at the previous iteration step  \delta^{k}  :



     \frac{ \partial (\rho \delta^{k+1} )}{\partial t}  + 
\rho \nabla \cdot  \left ( \left [ \bold {\frac { \bold { H[U] }}{A_P}}   - \frac {{\delta^k}^2}{A_P} \nabla   {P_p} -   \frac {\delta^k}{\bold A_P} \nabla {P_u}  +   {\frac {\delta^k}{\bold A_P}}\rho  g_t   \right ] \delta^{k+1} \right )  -   \nabla \cdot \left (  \frac {\delta^k\delta^{k} P_p}{A_P} \nabla   {\delta^{k+1}}  \right )     = S_{\delta}
(10)

which is the final form solved in the present solver. The source code can be found in src/regionModels/surfaceFilmModels/kinematicSingleLayer:


 
void kinematicSingleLayer::solveThickness
(
    const volScalarField& pu,
    const volScalarField& pp,
    const fvVectorMatrix& UEqn
)
{
    if (debug)
    {
        InfoInFunction << endl;
    }
 
    volScalarField rUA(1.0/UEqn.A());
    U_ = rUA*UEqn.H();
 
    surfaceScalarField deltarUAf(fvc::interpolate(delta_*rUA));
    surfaceScalarField rhof(fvc::interpolate(rho_));
 
    surfaceScalarField phiAdd
    (
        "phiAdd",
        regionMesh().magSf()
      * (
            fvc::snGrad(pu, "snGrad(p)")
          + fvc::snGrad(pp, "snGrad(p)")*fvc::interpolate(delta_)
        )
      - fvc::flux(rho_*gTan())
    );
    constrainFilmField(phiAdd, 0.0);
 
    surfaceScalarField phid
    (
        "phid",
        fvc::flux(U_*rho_) - deltarUAf*phiAdd*rhof
    );
    constrainFilmField(phid, 0.0);
 
    surfaceScalarField ddrhorUAppf
    (
        "deltaCoeff",
        fvc::interpolate(delta_)*deltarUAf*rhof*fvc::interpolate(pp)
    );
 
    regionMesh().setFluxRequired(delta_.name());
 
    for (int nonOrth=0; nonOrth<=nNonOrthCorr_; nonOrth++)
    {
        // Film thickness equation
        fvScalarMatrix deltaEqn
        (
            fvm::ddt(rho_, delta_)
          + fvm::div(phid, delta_)
          - fvm::laplacian(ddrhorUAppf, delta_)
         ==
          - rhoSp_
        );
 
        deltaEqn.solve();
 
        if (nonOrth == nNonOrthCorr_)
        {
            phiAdd +=
                fvc::interpolate(pp)
              * fvc::snGrad(delta_)
              * regionMesh().magSf();
 
            phi_ == deltaEqn.flux();
        }
    }
 
    // Bound film thickness by a minimum of zero
    delta_.max(0.0);
 
    // Update U field
    U_ -= fvc::reconstruct(deltarUAf*phiAdd);
 
    // Remove any patch-normal components of velocity
    U_ -= nHat()*(nHat() & U_);
 
    U_.correctBoundaryConditions();
 
    // Update film wall and surface velocities
    updateSurfaceVelocities();
 
    // Continuity check
    continuityCheck();
}
 


2.2.5 Solution Strategy

Equations (1), (3) and (4) represent a system of 3 non-linear partial differential equations with the unknowns  \delta ,  U_i and  h . In order to solve this system of equations, a splitting method is adopted: The equations are linearised by using quantities from the previous iteration and solved sequentially. The steps performed are the following:

1. Solve continuity equation (1). Here has to be mentioned that the quantity solved for is  \rho_\delta = \rho \delta which appears also in the momentum and energy equation.

2. Solve momentum equation (3)

3. Solve energy equation (4)

4. Solve film thickness equation: The film thickness equation is derived by inserting the discrete momentum equation into the continuity equation.

Steps 2-4 are solved a number of time specified by the user in order to achieve sufficient convergence of the equations. The source code can be found in src/regionModels/surfaceFilmModels/thermoSingleLayer



 
 
 void thermoSingleLayer::evolveRegion()
{
    if (debug)
    {
        InfoInFunction << endl;
    }
 
    // Solve continuity for deltaRho_
    solveContinuity();
 
    // Update sub-models to provide updated source contributions
    updateSubmodels();
 
    // Solve continuity for deltaRho_
    solveContinuity();
 
    for (int oCorr=1; oCorr<=nOuterCorr_; oCorr++)
    {
        // Explicit pressure source contribution
        tmp<volScalarField> tpu(this->pu());
 
        // Implicit pressure source coefficient
        tmp<volScalarField> tpp(this->pp());
 
        // Solve for momentum for U_
        tmp<fvVectorMatrix> UEqn = solveMomentum(tpu(), tpp());
 
        // Solve energy for hs_ - also updates thermo
        solveEnergy();
 
        // Film thickness correction loop
        for (int corr=1; corr<=nCorr_; corr++)
        {
            // Solve thickness for delta_
            solveThickness(tpu(), tpp(), UEqn());
        }
    }
 
    // Update deltaRho_ with new delta_
    deltaRho_ == delta_*rho_;
 
    // Update temperature using latest hs_
    T_ == T(hs_);
}
 

2.3 Equations Fluid

2.3.1 Mass conservation

The variable-density continuity equation is



\frac{\partial \rho}{\partial t} +   \frac{\partial {\rho u}_j}{\partial x_j} = S_{\delta,eva} + S_{p,eva}
(11)


The source terms in the mass conservation equations come from the evaporation of the liquid film   S_{\delta,eva} and from the liquid droplets   S_{p,eva} .

The source code can be found in rhoEqn.H:


 
 
{ 
    fvScalarMatrix rhoEqn
    (
        fvm::ddt(rho)
      + fvc::div(phi)
      ==
        parcels.Srho(rho)
      + surfaceFilm.Srho()
      + fvOptions(rho)
    );
 
    rhoEqn.solve();
 
    fvOptions.correct(rho);
}
 

2.3.2 Momentum conservation

The equation of motion are written for a moving frame of reference. They are however formulated for the absolute velocity (the derivation of the equations of motion can be found in https://openfoamwiki.net/index.php/See_the_MRF_development and also in https://diglib.tugraz.at/download.php?id=581303c7c91f9&location=browse. Some additional information can be found in https://pingpong.chalmers.se/public/pp/public_courses/course07056/published/1497955220499/resourceId/3711490/content/UploadedResources/HakanNilssonRotatingMachineryTrainingOFW11-1.pdf):




    \frac{ \partial (\rho {u}_i)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_{rj} u_i \right)  + \rho\epsilon_{ijk}\omega_i u_j= 

   - \frac{\partial p_{rgh}} {\partial{x_i}} - \frac{\partial \rho g_j x_j}{\partial x_i}  + \frac{\partial}{\partial x_j} \left( \tau_{ij} + \tau_{t_{ij}} \right) 
+ F_{pi}
(12)

 u represent the velocity,  u_r the relative veloicty,  g_i the gravitational acceleration,  p_{rgh} = p - \rho g_j x_j the pressure minus the hydrostatic pressure and  \tau_{ij}  and  \tau_{t_{ij}}  are the viscose and turbulent stresses. Note that since the relative velocity  u_r appears in the divergence term, the face flux  \phi appearing in the finite volume discretization of the momentum equation should be calculated with the relative velocity.  F_{pi} represents the force excreted by the particles on the fluid.


The source code can be found in Ueqn.H:


 
 
      MRF.correctBoundaryVelocity(U);
 
    fvVectorMatrix UEqn
    (
        fvm::ddt(rho, U) + fvm::div(phi, U)
      + MRF.DDt(rho, U)
      + turbulence->divDevRhoReff(U)
     ==
        parcels.SU(U)
      + fvOptions(rho, U)
    );
 
    UEqn.relax();
 
    fvOptions.constrain(UEqn);
 
    if (pimple.momentumPredictor())
    {
        solve
        (
            UEqn
          ==
            fvc::reconstruct
            (
                (
                  - ghf*fvc::snGrad(rho)
                  - fvc::snGrad(p_rgh)
                )*mesh.magSf()
            )
        );
 
        fvOptions.correct(U);
        K = 0.5*magSqr(U);
    }
 
 

2.3.3 Species conservation

In order to account for the chemical reactions occurring between different chemical species a conservation equation for each species k has to be solved:



   \frac{ \partial (\rho Y_k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j Y_k \right) =      \frac{\partial } {\partial{x_j} }\mu_{eff}\frac{\partial Y_k}{\partial x_j} + R_k + S_{Yk,\delta} + S_{Yk,p}
(13)

 R_k is the reaction rate of the species k and  S_{Yk,\delta} + S_{Yk,p} is the contribution to the spices k by the evaporation of the film and the particle respectively.

The source code can be found in YEEqn.H:


 
 
tmp<fv::convectionScheme<scalar>> mvConvection
(
    fv::convectionScheme<scalar>::New
    (
        mesh,
        fields,
        phi,
        mesh.divScheme("div(phi,Yi_h)")
    )
);
{
    combustion->correct();
    Qdot = combustion->Qdot();
    volScalarField Yt(0.0*Y[0]);
 
    forAll(Y, i)
    {
        if (i != inertIndex && composition.active(i))
        {
            volScalarField& Yi = Y[i];
 
            fvScalarMatrix YiEqn
            (
                fvm::ddt(rho, Yi)
              + mvConvection->fvmDiv(phi, Yi)
              - fvm::laplacian(turbulence->alphaEff(), Yi)
              ==
                parcels.SYi(i, Yi)
              + surfaceFilm.Srho(i)
              + combustion->R(Yi)
              + fvOptions(rho, Yi)
            );
 
            YiEqn.relax();
 
            fvOptions.constrain(YiEqn);
 
            YiEqn.solve(mesh.solver("Yi"));
 
            fvOptions.correct(Yi);
 
            Yi.max(0.0);
            Yt += Yi;
        }
    }
 
    Y[inertIndex] = scalar(1) - Yt;
    Y[inertIndex].max(0.0);
 
    radiation->correct();
 

2.3.4 Energy conservation

The derivation of the energy equation (without source terms from the evaporation of the particle  S_{hp,eva} and the liquid film  S_{h\delta,eva} ) can be found in: https://cfd.direct/openfoam/energy-equation/

The total energy of a fluid element can be seen as the sum of kinetic energy  k = 0.5 u_i u_i and internal energy  e . The rate of change of the kinetic energy within a fluid element is the work done on this fluid element by the viscous forces, the pressure and eternal volume forces like the gravity:



    \frac{ \partial (\rho k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j k \right) =     - \frac{\partial p u_j} {\partial{x_i} }- \rho g_j u_j  + 
\frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right)
(14)

The rate of change of the internal energy  e of a fluid element is the heat transferred to this fluid element by diffusion and turbulence   q_i + q_{ti}  plus the heat source term  r plus the heat source by radiation  Rad  :



    \frac{ \partial (\rho e)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j e \right) =     - \frac{\partial q_i} {\partial{x_i} } +  \rho r  + Rad + S_{h\delta,eva} + S_{hp,eva}
(15)

The change rate of the total energy is the sum of the above two equations:



    \frac{ \partial (\rho e)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j e \right) + \frac{ \partial (\rho k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j k \right)   =     - \frac{\partial ( q_i + q_{ti}) } {\partial{x_i} } +  \rho r  + Rad    - \frac{\partial p u_j} {\partial{x_i} }- \rho g_j u_j  + 
\frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right) +  S_{h\delta,eva} + S_{hp,eva}
(16)

Instead of the internal energy  e there is also the option to solve the equation for the enthalpy  h =  e + p/\rho :



    \frac{ \partial (\rho h)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j h \right) + \frac{ \partial (\rho k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j k \right)   =     - \frac{\partial  ( q_i + q_{ti})} {\partial{x_i} } +  \rho r  + Rad    + \frac{\partial p} {\partial{t} }- \rho g_j u_j  + 
\frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right)  + S_{h\delta,eva} + S_{hp,eva}
(17)

The source code can be found in YEEqn.H:


 
    volScalarField& he = thermo.he();
 
    fvScalarMatrix EEqn
    (
        fvm::ddt(rho, he) + mvConvection->fvmDiv(phi, he)
      + fvc::ddt(rho, K) + fvc::div(phi, K)
      + (
            he.name() == "e"
          ? fvc::div
            (
                fvc::absolute(phi/fvc::interpolate(rho), U),
                p,
                "div(phiv,p)"
            )
          : -dpdt
        )
      - fvm::laplacian(turbulence->alphaEff(), he)
     ==
        Qdot
      + radiation->Sh(thermo, he)
      + parcels.Sh(he)
      + surfaceFilm.Sh()
      + fvOptions(rho, he)
    );
 
    EEqn.relax();
 
    fvOptions.constrain(EEqn);
 
    EEqn.solve();
 
    fvOptions.correct(he);
 
    thermo.correct();
 
    Info<< "min/max(T) = "
        << min(T).value() << ", " << max(T).value() << endl;


2.3.5 Pressure equation

As most of the solvers of OpenFOAM also fireFoam is a pressure based solver. In this class of solvers, the discrete momentum equation (which contains also the pressure gradient) is used to obtain a semi discrete equation for the velocity at the cell centre. This velocity is cast into the continuity equation in order to obtain an equation for the pressure which satisfies the mass conservation and by construction also the momentum equation. It has also to be mentioned, that the pressure equation which is solved, is only valid for law Mach number flows, i.e., it is assumed that the density has only a weak dependency from the pressure. Large temperature variation are however allowed (see the difference to the pressure equation solved in https://openfoamwiki.net/index.php/ChtMultiRegionFoam).

Starting point for the derivation of the pressure equation is the discrete continuity equation after integrating over the control volume:


    \frac{ \partial \rho}{\partial t} V_P + \sum_f  \rho_f  v_f   \cdot S_f = S_{\delta,eva}V_P +  S_{p,eva}V_P
(18)

The subscript f denotes the the quantities are evaluated at the cell faces. The semi discrete momentum equation reads:



\bold{A_P U_P} - \bold {H[U]} = - \nabla p_{rgh,P} - g \cdot x \nabla \rho
(19)

and after rearranging with respect to the velocity at the cell centre   \bold U_P :



\bold{ U_P}  = \frac{ \bold {H[U]}}{\bold A_P}  - \frac{\nabla p_{rgh,P}}{\bold A_P} - \frac{ g \cdot x \nabla \rho}{\bold A_P}
(20)
 \bold H[U]   and    \bold A_P   are the contribution of the off diagonal elements of the matrix and the diagonal elements of the matrix, respectively.

The above equation can be inserted into the continuity equation:


    \frac{ \partial \rho}{\partial t} V_P + \sum_f  \rho_f  \left (  \frac{ \bold {H[U]}}{\bold A_P}  - \frac{\nabla p_{rgh,P}}{\bold A_P} - \frac{ g \cdot x \nabla \rho}{\bold A_P} \right )_f   \cdot S_f = S_{\delta,eva}V_P +  S_{p,eva}V_P
(21)

Since the above equation still contains the density a relation between the density and the pressure is required. In the most general form it can be expressed as:


\rho = \psi p =  \psi (   p_{ref} + p_{rgh} + \rho g_i  x_i )
(22)

 p_{ref} is the reference pressure.

If we insert equation (22) into equation (21) we get a non-linear equation for the pressure when evaluating the divergence (the second term in equation (21)). However, if we assume only a week dependence of the density from the pressure, the density at the previous iteration  \rho_f^* can be taken as a reasonable approximation of the density at the current iteration step  \rho_f . Since we can still have a strong variation of the density with time (in turbulent combustion the temperature is highly in-stationary and hence also the density varies considerably with time) the temporal derivatives have to be kept. The resulting equation reads:


    \frac{ \partial  \psi (   p_{ref} + p_{rgh} + \rho^* g_i  x_i )}{\partial t} V_P + \sum_f  \rho_f^*  \left (  \frac{ \bold {H[U]}}{\bold A_P}  - \frac{\nabla p_{rgh,P}}{\bold A_P} - \frac{ g \cdot x \nabla \rho^*}{\bold A_P} \right )_f   \cdot S_f = S_{\delta,eva}V_P +  S_{p,eva}V_P
(23)

The source code can be found in pEqn.H:


 
rho = thermo.rho();
 
volScalarField rAU(1.0/UEqn.A());
surfaceScalarField rhorAUf("rhorAUf", fvc::interpolate(rho*rAU));
volVectorField HbyA(constrainHbyA(rAU*UEqn.H(), U, p));
 
surfaceScalarField phig("phig", -rhorAUf*ghf*fvc::snGrad(rho)*mesh.magSf());
 
surfaceScalarField phiHbyA
(
    "phiHbyA",
    (
        fvc::flux(rho*HbyA)
      + MRF.zeroFilter(rhorAUf*fvc::ddtCorr(rho, U, phi))
    )
  + phig
);
 
MRF.makeRelative(fvc::interpolate(rho), phiHbyA);
 
// Update the pressure BCs to ensure flux consistency
constrainPressure(p_rgh, rho, U, phiHbyA, rhorAUf, MRF);
 
while (pimple.correctNonOrthogonal())
{
    fvScalarMatrix p_rghEqn
    (
        fvm::ddt(psi, p_rgh)
      + fvc::ddt(psi, rho)*gh
      + fvc::ddt(psi)*pRef
      + fvc::div(phiHbyA)
      - fvm::laplacian(rhorAUf, p_rgh)
     ==
        parcels.Srho()
      + surfaceFilm.Srho()
      + fvOptions(psi, p_rgh, rho.name())
    );
 
    p_rghEqn.solve(mesh.solver(p_rgh.select(pimple.finalInnerIter())));
 
    if (pimple.finalNonOrthogonalIter())
    {
        phi = phiHbyA + p_rghEqn.flux();
        U = HbyA + rAU*fvc::reconstruct((p_rghEqn.flux() + phig)/rhorAUf);
        U.correctBoundaryConditions();
        fvOptions.correct(U);
    }
}
 
p = p_rgh + rho*gh + pRef;
 
#include "rhoEqn.H"
#include "compressibleContinuityErrs.H"
 
K = 0.5*magSqr(U);
 
if (thermo.dpdt())
{
    dpdt = fvc::ddt(p);
}
 


2.4 Pyrolysis

3 References

  1. Miller, R. S., K. Harstad, and J. Bellan. "Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations." International Journal of Multiphase Flow 24.6 (1998): 1025-1055.