pengrobinson.hh

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#ifndef DUMUX_PENG_ROBINSON_HH
#define DUMUX_PENG_ROBINSON_HH
 
#include <dumux/material/fluidstates/temperatureoverlay.hh>
#include <dumux/material/idealgas.hh>
#include <dumux/common/exceptions.hh>
#include <dumux/common/math.hh>
#include <dumux/common/tabulated2dfunction.hh>
 
#include <iostream>
 
namespace Dumux {
 
template <class Scalar>
class PengRobinson
{
    PengRobinson()
    { }
 
public:
    static void  init(Scalar aMin, Scalar aMax, int na,
                      Scalar bMin, Scalar bMax, int nb)
    {
        // resize the tabulation for the critical points
        criticalTemperature_.resize(aMin, aMax, na, bMin, bMax, nb);
        criticalPressure_.resize(aMin, aMax, na, bMin, bMax, nb);
        criticalMolarVolume_.resize(aMin, aMax, na, bMin, bMax, nb);
 
        Scalar VmCrit = 18e-6;
        Scalar pCrit = 1e7;
        Scalar TCrit = 273;
        for (int i = 0; i < na; ++i) {
            Scalar a = criticalTemperature_.iToX(i);
            for (int j = 0; j < nb; ++j) {
                Scalar b = criticalTemperature_.jToY(j);
 
                findCriticalPoint_(TCrit, pCrit, VmCrit,  a, b);
 
                criticalTemperature_.setSamplePoint(i, j, TCrit);
                criticalPressure_.setSamplePoint(i, j, pCrit);
                criticalMolarVolume_.setSamplePoint(i, j, VmCrit);
            }
        }
    };
 
    template <class Params>
    static Scalar computeVaporPressure(const Params &params, Scalar T)
    {
        using Component = typename Params::Component;
        if (T >= Component::criticalTemperature())
            return Component::criticalPressure();
 
        // initial guess of the vapor pressure
        Scalar Vm[3];
        const Scalar eps = Component::criticalPressure()*1e-10;
 
        // use the Ambrose-Walton method to get an initial guess of
        // the vapor pressure
        Scalar pVap = ambroseWalton_(params, T);
 
        // Newton-Raphson method
        for (int i = 0; i < 5; ++i) {
            // calculate the molar densities
            assert(molarVolumes(Vm, params, T, pVap) == 3);
 
            Scalar f = fugacityDifference_(params, T, pVap, Vm[0], Vm[2]);
            Scalar df_dp =
                fugacityDifference_(params, T, pVap  + eps, Vm[0], Vm[2])
                -
                fugacityDifference_(params, T, pVap - eps, Vm[0], Vm[2]);
            df_dp /= 2*eps;
 
            Scalar delta = f/df_dp;
            pVap = pVap - delta;
            using std::abs;
            if (abs(delta/pVap) < 1e-10)
                break;
        }
 
        return pVap;
    }
 
    template <class FluidState, class Params>
    static Scalar computeMolarVolume(const FluidState &fs,
                                     Params &params,
                                     int phaseIdx,
                                     bool isGasPhase)
    {
        Scalar Vm = 0;
 
        Scalar T = fs.temperature(phaseIdx);
        Scalar p = fs.pressure(phaseIdx);
 
        Scalar a = params.a(phaseIdx); // "attractive factor"
        Scalar b = params.b(phaseIdx); // "co-volume"
 
        Scalar RT = Constants<Scalar>::R*T;
        Scalar Astar = a*p/(RT*RT);
        Scalar Bstar = b*p/RT;
 
        Scalar a1 = 1.0;
        Scalar a2 = - (1 - Bstar);
        Scalar a3 = Astar - Bstar*(3*Bstar + 2);
        Scalar a4 = Bstar*(- Astar + Bstar*(1 + Bstar));
 
        // ignore the first two results if the smallest
        // compressibility factor is <= 0.0. (this means that if we
        // would get negative molar volumes for the liquid phase, we
        // consider the liquid phase non-existant.)
        Scalar Z[3] = {0.0,0.0,0.0};
        int numSol = invertCubicPolynomial(Z, a1, a2, a3, a4);
        if (numSol == 3) {
            // the EOS has three intersections with the pressure,
            // i.e. the molar volume of gas is the largest one and the
            // molar volume of liquid is the smallest one
            if (isGasPhase)
                Vm = Z[2]*RT/p;
            else
                Vm = Z[0]*RT/p;
        }
        else if (numSol == 1) {
            // the EOS only has one intersection with the pressure,
            // for the other phase, we take the extremum of the EOS
            // with the largest distance from the intersection.
            Scalar VmCubic = Z[0]*RT/p;
 
            if (T > criticalTemperature_(a, b)) {
                // if the EOS does not exhibit any extrema, the fluid
                // is critical...
                Vm = VmCubic;
                handleCriticalFluid_(Vm, fs, params, phaseIdx, isGasPhase);
            }
            else {
                // find the extrema (if they are present)
                Scalar Vmin, Vmax, pmin, pmax;
                using std::min;
                using std::max;
                if (findExtrema_(Vmin, Vmax,
                                 pmin, pmax,
                                 a, b, T))
                {
                    if (isGasPhase)
                        Vm = max(Vmax, VmCubic);
                    else
                        Vm = min(Vmin, VmCubic);
                }
                else
                    Vm = VmCubic;
            }
        }
 
        using std::isfinite;
        assert(isfinite(Vm) && Vm > 0);
        return Vm;
    }
 
    template <class Params>
    static Scalar computeFugacityCoeffient(const Params &params)
    {
        Scalar T = params.temperature();
        Scalar p = params.pressure();
        Scalar Vm = params.molarVolume();
 
        Scalar RT = Constants<Scalar>::R*T;
        Scalar Z = p*Vm/RT;
        Scalar Bstar = p*params.b() / RT;
        using std::sqrt;
        Scalar tmp =
            (Vm + params.b()*(1 + sqrt(2))) /
            (Vm + params.b()*(1 - sqrt(2)));
        Scalar expo = - params.a()/(RT * 2 * params.b() * sqrt(2));
        using std::exp;
        using std::pow;
        Scalar fugCoeff =
            exp(Z - 1) / (Z - Bstar) *
            pow(tmp, expo);
 
        return fugCoeff;
    }
 
    template <class Params>
    static Scalar computeFugacity(const Params &params)
    { return params.pressure()*computeFugacityCoeff(params); }
 
protected:
    template <class FluidState, class Params>
    static void handleCriticalFluid_(Scalar &Vm,
                                     const FluidState &fs,
                                     const Params &params,
                                     int phaseIdx,
                                     bool isGasPhase)
    {
        Scalar Vcrit = criticalMolarVolume_(params.a(phaseIdx), params.b(phaseIdx));
 
        using std::max;
        using std::min;
        if (isGasPhase)
            Vm = max(Vm, Vcrit);
        else
            Vm = min(Vm, Vcrit);
    }
 
    static void findCriticalPoint_(Scalar &Tcrit,
                                   Scalar &pcrit,
                                   Scalar &Vcrit,
                                   Scalar a,
                                   Scalar b)
    {
        Scalar minVm;
        Scalar maxVm;
 
        Scalar minP(0);
        Scalar maxP(1e100);
 
        // first, we need to find an isotherm where the EOS exhibits
        // a maximum and a minimum
        Scalar Tmin = 250; // [K]
        for (int i = 0; i < 30; ++i) {
            bool hasExtrema = findExtrema_(minVm, maxVm, minP, maxP, a, b, Tmin);
            if (hasExtrema)
                break;
            Tmin /= 2;
        }
 
        Scalar T = Tmin;
 
        // Newton's method: Start at minimum temperature and minimize
        // the "gap" between the extrema of the EOS
        for (int i = 0; i < 25; ++i) {
            // calculate function we would like to minimize
            Scalar f = maxVm - minVm;
 
            // check if we're converged
            if (f < 1e-10) {
                Tcrit = T;
                pcrit = (minP + maxP)/2;
                Vcrit = (maxVm + minVm)/2;
                return;
            }
 
            // backward differences. Forward differences are not
            // robust, since the isotherm could be critical if an
            // epsilon was added to the temperature. (this is case
            // rarely happens, though)
            const Scalar eps = - 1e-8;
            bool DUNE_UNUSED hasExtrema = findExtrema_(minVm, maxVm, minP, maxP, a, b, T + eps);
            assert(hasExtrema);
            Scalar fStar = maxVm - minVm;
 
            // derivative of the difference between the maximum's
            // molar volume and the minimum's molar volume regarding
            // temperature
            Scalar fPrime = (fStar - f)/eps;
 
            // update value for the current iteration
            Scalar delta = f/fPrime;
            if (delta > 0)
                delta = -10;
 
            // line search (we have to make sure that both extrema
            // still exist after the update)
            for (int j = 0; ; ++j) {
                if (j >= 20) {
                    DUNE_THROW(NumericalProblem,
                               "Could not determine the critical point for a=" << a << ", b=" << b);
                }
 
                if (findExtrema_(minVm, maxVm, minP, maxP, a, b, T - delta)) {
                    // if the isotherm for T - delta exhibits two
                    // extrema the update is finished
                    T -= delta;
                    break;
                }
                else
                    delta /= 2;
            }
        }
    }
 
    // find the two molar volumes where the EOS exhibits extrema and
    // which are larger than the covolume of the phase
    static bool findExtrema_(Scalar &Vmin,
                             Scalar &Vmax,
                             Scalar &pMin,
                             Scalar &pMax,
                             Scalar a,
                             Scalar b,
                             Scalar T)
    {
        Scalar u = 2;
        Scalar w = -1;
 
        Scalar RT = Constants<Scalar>::R*T;
 
        // calculate coefficients of the 4th order polynominal in
        // monomial basis
        Scalar a1 = RT;
        Scalar a2 = 2*RT*u*b - 2*a;
        Scalar a3 = 2*RT*w*b*b + RT*u*u*b*b  + 4*a*b - u*a*b;
        Scalar a4 = 2*RT*u*w*b*b*b + 2*u*a*b*b - 2*a*b*b;
        Scalar a5 = RT*w*w*b*b*b*b - u*a*b*b*b;
 
        // Newton method to find first root
 
        // if the values which we got on Vmin and Vmax are usefull, we
        // will reuse them as initial value, else we will start 10%
        // above the covolume
        Scalar V = b*1.1;
        Scalar delta = 1.0;
        using std::abs;
        for (int i = 0; abs(delta) > 1e-9; ++i) {
            Scalar f = a5 + V*(a4 + V*(a3 + V*(a2 + V*a1)));
            Scalar fPrime = a4 + V*(2*a3 + V*(3*a2 + V*4*a1));
 
            if (abs(fPrime) < 1e-20) {
                // give up if the derivative is zero
                Vmin = 0;
                Vmax = 0;
                return false;
            }
 
 
            delta = f/fPrime;
            V -= delta;
 
            if (i > 20) {
                // give up after 20 iterations...
                Vmin = 0;
                Vmax = 0;
                return false;
            }
        }
 
        // polynomial division
        Scalar b1 = a1;
        Scalar b2 = a2 + V*b1;
        Scalar b3 = a3 + V*b2;
        Scalar b4 = a4 + V*b3;
 
        // invert resulting cubic polynomial analytically
        Scalar allV[4];
        allV[0] = V;
        int numSol = 1 + invertCubicPolynomial(allV + 1, b1, b2, b3, b4);
 
        // sort all roots of the derivative
        std::sort(allV + 0, allV + numSol);
 
        // check whether the result is physical
        if (allV[numSol - 2] < b) {
            // the second largest extremum is smaller than the phase's
            // covolume which is physically impossible
            Vmin = Vmax = 0;
            return false;
        }
 
        // it seems that everything is okay...
        Vmin = allV[numSol - 2];
        Vmax = allV[numSol - 1];
        return true;
    }
 
    template <class Params>
    static Scalar ambroseWalton_(const Params &params, Scalar T)
    {
        using Component = typename Params::Component;
 
        Scalar Tr = T / Component::criticalTemperature();
        Scalar tau = 1 - Tr;
        Scalar omega = Component::acentricFactor();
        using std::sqrt;
        using std::pow;
        Scalar f0 = (tau*(-5.97616 + sqrt(tau)*(1.29874 - tau*0.60394)) - 1.06841*pow(tau, 5))/Tr;
        Scalar f1 = (tau*(-5.03365 + sqrt(tau)*(1.11505 - tau*5.41217)) - 7.46628*pow(tau, 5))/Tr;
        Scalar f2 = (tau*(-0.64771 + sqrt(tau)*(2.41539 - tau*4.26979)) + 3.25259*pow(tau, 5))/Tr;
        using std::exp;
        return Component::criticalPressure()*exp(f0 + omega * (f1 + omega*f2));
    }
 
    template <class Params>
    static Scalar fugacityDifference_(const Params &params,
                                      Scalar T,
                                      Scalar p,
                                      Scalar VmLiquid,
                                      Scalar VmGas)
    { return fugacity(params, T, p, VmLiquid) - fugacity(params, T, p, VmGas); }
 
    static Tabulated2DFunction<Scalar> criticalTemperature_;
    static Tabulated2DFunction<Scalar> criticalPressure_;
    static Tabulated2DFunction<Scalar> criticalMolarVolume_;
};
 
template <class Scalar>
Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalTemperature_;
 
template <class Scalar>
Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalPressure_;
 
template <class Scalar>
Tabulated2DFunction<Scalar> PengRobinson<Scalar>::criticalMolarVolume_;
 
} // end namespace
 
#endif