staggered/freeflow/maxwellstefanslaw.hh

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#ifndef DUMUX_DISCRETIZATION_STAGGERED_MAXWELL_STEFAN_LAW_HH
#define DUMUX_DISCRETIZATION_STAGGERED_MAXWELL_STEFAN_LAW_HH
 
#include <dune/common/float_cmp.hh>
 
#include <dumux/common/math.hh>
#include <dumux/common/properties.hh>
#include <dumux/common/parameters.hh>
#include <dumux/discretization/method.hh>
#include <dumux/flux/fluxvariablescaching.hh>
#include <dumux/flux/referencesystemformulation.hh>
 
namespace Dumux {
 
// forward declaration
template <class TypeTag, DiscretizationMethod discMethod, ReferenceSystemFormulation referenceSystem>
class MaxwellStefansLawImplementation;
 
template <class TypeTag, ReferenceSystemFormulation referenceSystem>
class MaxwellStefansLawImplementation<TypeTag, DiscretizationMethod::staggered, referenceSystem>
{
    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
    using Problem = GetPropType<TypeTag, Properties::Problem>;
    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using SubControlVolumeFace = typename FVElementGeometry::SubControlVolumeFace;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
    using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
    using VolumeVariables = GetPropType<TypeTag, Properties::VolumeVariables>;
    using CellCenterPrimaryVariables = GetPropType<TypeTag, Properties::CellCenterPrimaryVariables>;
    using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
    using FluidSystem = GetPropType<TypeTag, Properties::FluidSystem>;
 
    using ModelTraits = GetPropType<TypeTag, Properties::ModelTraits>;
 
    static const int numComponents = ModelTraits::numFluidComponents();
    static constexpr bool useMoles = getPropValue<TypeTag, Properties::UseMoles>();
 
    using ReducedComponentVector = Dune::FieldVector<Scalar, numComponents-1>;
    using ReducedComponentMatrix = Dune::FieldMatrix<Scalar, numComponents-1, numComponents-1>;
 
    static_assert(ModelTraits::numFluidPhases() == 1, "Only one phase allowed supported!");
 
    static_assert(referenceSystem == ReferenceSystemFormulation::massAveraged, "only the mass averaged reference system is supported for the Maxwell-Stefan formulation");
 
public:
    // state the discretization method this implementation belongs to
    static const DiscretizationMethod discMethod = DiscretizationMethod::staggered;
    //return the reference system
    static constexpr ReferenceSystemFormulation referenceSystemFormulation()
    { return referenceSystem; }
 
    using Cache = FluxVariablesCaching::EmptyDiffusionCache;
    using CacheFiller = FluxVariablesCaching::EmptyCacheFiller;
 
    static CellCenterPrimaryVariables flux(const Problem& problem,
                                           const Element& element,
                                           const FVElementGeometry& fvGeometry,
                                           const ElementVolumeVariables& elemVolVars,
                                           const SubControlVolumeFace& scvf)
    {
        //this is to calculate the maxwellStefan diffusion in a multicomponent system.
        //see: Multicomponent Mass Transfer. R. Taylor u. R. Krishna. J. Wiley & Sons, New York 1993
        CellCenterPrimaryVariables componentFlux(0.0);
        ReducedComponentVector moleFracInside(0.0);
        ReducedComponentVector moleFracOutside(0.0);
        ReducedComponentVector reducedFlux(0.0);
        ReducedComponentMatrix reducedDiffusionMatrixInside(0.0);
        ReducedComponentMatrix reducedDiffusionMatrixOutside(0.0);
 
        // get inside/outside volume variables
        const auto& insideVolVars = elemVolVars[scvf.insideScvIdx()];
        const auto& outsideVolVars = elemVolVars[scvf.outsideScvIdx()];
        const auto rhoInside = insideVolVars.density();
        const auto rhoOutside = outsideVolVars.density();
 
        //to implement outflow boundaries correctly we need to loop over all components but the main component as only for the transported ones we implement the outflow boundary. diffusion then is 0.
        for(int compIdx = 0; compIdx < numComponents; ++compIdx)
        {
            if(compIdx == FluidSystem::getMainComponent(0))
                continue;
 
            // get equation index
            const auto eqIdx = Indices::conti0EqIdx + compIdx;
            if(scvf.boundary())
            {
               const auto bcTypes = problem.boundaryTypes(element, scvf);
                 if((bcTypes.isOutflow(eqIdx)) || (bcTypes.isSymmetry()))
                    return componentFlux;
            }
        }
 
        //calculate the mole fraction vectors
        for (int compIdx = 0; compIdx < numComponents-1; compIdx++)
        {
            //calculate x_inside
            const auto xInside = insideVolVars.moleFraction(compIdx);
            //calculate outside molefraction with the respective transmissibility
            const auto xOutside = outsideVolVars.moleFraction(compIdx);
 
            moleFracInside[compIdx] = xInside;
            moleFracOutside[compIdx] = xOutside;
        }
 
        //now we have to do the tpfa: J_i = J_j which leads to: tij(xi -xj) = -rho Bi^-1 omegai(x*-xi) with x* = (omegai Bi^-1 + omegaj Bj^-1)^-1 (xi omegai Bi^-1 + xj omegaj Bj^-1) with i inside and j outside
        reducedDiffusionMatrixInside = setupMSMatrix_(problem, fvGeometry, insideVolVars, scvf);
 
        reducedDiffusionMatrixOutside = setupMSMatrix_(problem, fvGeometry, outsideVolVars, scvf);
 
        const auto insideScvIdx = scvf.insideScvIdx();
        const auto& insideScv = fvGeometry.scv(insideScvIdx);
        const auto outsideScvIdx = scvf.outsideScvIdx();
        const auto& outsideScv = fvGeometry.scv(outsideScvIdx);
 
        const Scalar insideDistance = (insideScv.dofPosition() - scvf.ipGlobal()).two_norm();
 
        const Scalar omegai = calculateOmega_(insideDistance, insideVolVars.extrusionFactor());
 
        //if on boundary
        if (scvf.boundary())
        {
            moleFracOutside -= moleFracInside;
            reducedDiffusionMatrixInside.solve(reducedFlux, moleFracOutside);
            reducedFlux *= omegai*rhoInside;
        }
        else
        {
            const Scalar outsideDistance = (outsideScv.dofPosition() - scvf.ipGlobal()).two_norm();
            const Scalar omegaj = calculateOmega_(outsideDistance, outsideVolVars.extrusionFactor());
 
            reducedDiffusionMatrixInside.invert();
            reducedDiffusionMatrixOutside.invert();
            reducedDiffusionMatrixInside *= omegai*rhoInside;
            reducedDiffusionMatrixOutside *= omegaj*rhoOutside;
 
            //in the helpervector we store the values for x*
            ReducedComponentVector helperVector(0.0);
            ReducedComponentVector gradientVectori(0.0);
            ReducedComponentVector gradientVectorj(0.0);
 
            reducedDiffusionMatrixInside.mv(moleFracInside, gradientVectori);
            reducedDiffusionMatrixOutside.mv(moleFracOutside, gradientVectorj);
 
            auto gradientVectorij = (gradientVectori + gradientVectorj);
 
            //add the two matrixes to each other
            reducedDiffusionMatrixOutside += reducedDiffusionMatrixInside;
 
            reducedDiffusionMatrixOutside.solve(helperVector, gradientVectorij);
 
            //Bi^-1 omegai rho(x*-xi)
            helperVector -=moleFracInside;
            reducedDiffusionMatrixInside.mv(helperVector, reducedFlux);
        }
 
        reducedFlux *= -scvf.area();
 
        for (int compIdx = 0; compIdx < numComponents-1; compIdx++)
        {
            componentFlux[compIdx] = reducedFlux[compIdx];
            componentFlux[numComponents-1] -= reducedFlux[compIdx];
        }
 
        return componentFlux ;
    }
 
private:
    static Scalar calculateOmega_(const Scalar distance,
                                  const Scalar extrusionFactor)
    {
        Scalar omega = 1/distance;
        omega *= extrusionFactor;
 
        return omega;
    }
 
    static ReducedComponentMatrix setupMSMatrix_(const Problem& problem,
                                                const FVElementGeometry& fvGeometry,
                                                const VolumeVariables& volVars,
                                                const SubControlVolumeFace& scvf)
    {
        ReducedComponentMatrix reducedDiffusionMatrix(0.0);
 
        for (int compIIdx = 0; compIIdx < numComponents-1; compIIdx++)
        {
            const auto xi = volVars.moleFraction(compIIdx);
            const auto avgMolarMass = volVars.averageMolarMass(0);
            const auto Mn = FluidSystem::molarMass(numComponents-1);
            const Scalar tin = volVars.effectiveDiffusivity(compIIdx, numComponents-1);
 
            // set the entries of the diffusion matrix of the diagonal
            reducedDiffusionMatrix[compIIdx][compIIdx] +=  xi*avgMolarMass/(tin*Mn);
 
            for (int compJIdx = 0; compJIdx < numComponents; compJIdx++)
            {
                // we don't want to calculate e.g. water in water diffusion
                if (compJIdx == compIIdx)
                    continue;
 
                const auto xj = volVars.moleFraction(compJIdx);
                const auto Mi = FluidSystem::molarMass(compIIdx);
                const auto Mj = FluidSystem::molarMass(compJIdx);
                const Scalar tij = volVars.effectiveDiffusivity(compIIdx, compJIdx);
                reducedDiffusionMatrix[compIIdx][compIIdx] +=  xj*avgMolarMass/(tij*Mi);
                if (compJIdx < numComponents-1)
                    reducedDiffusionMatrix[compIIdx][compJIdx] += xi*(avgMolarMass/(tin*Mn) - avgMolarMass/(tij*Mj));
            }
        }
        return reducedDiffusionMatrix;
    }
};
} // end namespace
 
#endif