h2o.hh

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#ifndef DUMUX_H2O_HH
#define DUMUX_H2O_HH
 
#include <cmath>
#include <cassert>
 
#include <dumux/nonlinear/findscalarroot.hh>
#include <dumux/material/idealgas.hh>
#include <dumux/common/exceptions.hh>
 
#include "iapws/common.hh"
#include "iapws/region1.hh"
#include "iapws/region2.hh"
#include "iapws/region4.hh"
 
#include <dumux/material/components/base.hh>
#include <dumux/material/components/liquid.hh>
#include <dumux/material/components/gas.hh>
 
namespace Dumux {
namespace Components {
 
template <class Scalar>
class H2O
: public Components::Base<Scalar, H2O<Scalar> >
, public Components::Liquid<Scalar, H2O<Scalar> >
, public Components::Gas<Scalar, H2O<Scalar> >
{
 
    using Common = IAPWS::Common<Scalar>;
    using Region1 = IAPWS::Region1<Scalar>;
    using Region2 = IAPWS::Region2<Scalar>;
    using Region4 = IAPWS::Region4<Scalar>;
 
    // specific gas constant of water
    static constexpr Scalar Rs = Common::Rs;
 
public:
    static std::string name()
    { return "H2O"; }
 
    static constexpr Scalar molarMass()
    { return Common::molarMass; }
 
    static constexpr Scalar acentricFactor()
    { return Common::acentricFactor; }
 
    static constexpr Scalar criticalTemperature()
    { return Common::criticalTemperature; }
 
    static constexpr Scalar criticalPressure()
    { return Common::criticalPressure; }
 
    static constexpr Scalar criticalMolarVolume()
    { return Common::criticalMolarVolume; }
 
    static constexpr Scalar tripleTemperature()
    { return Common::tripleTemperature; }
 
    static constexpr Scalar triplePressure()
    { return Common::triplePressure; }
 
    static Scalar vaporPressure(Scalar T)
    {
        using std::min;
        using std::max;
        T = min(T, criticalTemperature());
        T = max(T,tripleTemperature());
 
        return Region4::saturationPressure(T);
    }
 
    static Scalar vaporTemperature(Scalar pressure)
    {
        using std::min;
        using std::max;
        pressure = min(pressure, criticalPressure());
        pressure = max(pressure, triplePressure());
 
        return Region4::vaporTemperature(pressure);
    }
 
    static const Scalar gasEnthalpy(Scalar temperature,
                                    Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Enthalpy");
 
        // regularization
        if (pressure < triplePressure() - 100) {
            // We assume an ideal gas for low pressures to avoid the
            // 0/0 for the gas enthalpy at very low pressures. The
            // enthalpy of an ideal gas does not exhibit any
            // dependence on pressure, so we can just return the
            // specific enthalpy at the point of regularization, i.e.
            // the triple pressure - 100Pa
            return enthalpyRegion2_(temperature, triplePressure() - 100);
        }
        Scalar pv = vaporPressure(temperature);
        if (pressure > pv) {
            // the pressure is too high, in this case we use the slope
            // of the enthalpy at the vapor pressure to regularize
            Scalar dh_dp =
                Rs*temperature*
                Region2::tau(temperature)*
                Region2::dPi_dp(pv)*
                Region2::ddGamma_dTaudPi(temperature, pv);
 
            return
                enthalpyRegion2_(temperature, pv) +
                (pressure - pv)*dh_dp;
        }
 
        return enthalpyRegion2_(temperature, pressure);
    }
 
    static const Scalar liquidEnthalpy(Scalar temperature,
                                       Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Enthalpy");
 
        // regularization
        Scalar pv = vaporPressure(temperature);
        if (pressure < pv) {
            // the pressure is too low, in this case we use the slope
            // of the enthalpy at the vapor pressure to regularize
            Scalar dh_dp =
                Rs * temperature*
                Region1::tau(temperature)*
                Region1::dPi_dp(pv)*
                Region1::ddGamma_dTaudPi(temperature, pv);
 
            return
                enthalpyRegion1_(temperature, pv) +
                (pressure - pv)*dh_dp;
        }
 
        return enthalpyRegion1_(temperature, pressure);
    }
 
    static const Scalar gasHeatCapacity(Scalar temperature,
                                        Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Heat capacity");
 
        // regularization
        if (pressure < triplePressure() - 100) {
            return heatCap_p_Region2_(temperature, triplePressure() - 100);
        }
        Scalar pv = vaporPressure(temperature);
        if (pressure > pv) {
            // the pressure is too high, in this case we use the heat
            // cap at the vapor pressure to regularize
            return heatCap_p_Region2_(temperature, pv);
        }
        return heatCap_p_Region2_(temperature, pressure);
    }
 
    static const Scalar liquidHeatCapacity(Scalar temperature,
                                           Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Heat capacity");
 
        // regularization
        Scalar pv = vaporPressure(temperature);
        if (pressure < pv) {
            // the pressure is too low, in this case we use the heat cap at the vapor pressure to regularize
            return heatCap_p_Region1_(temperature, pv);
        }
 
        return heatCap_p_Region1_(temperature, pressure);
    }
 
    static const Scalar liquidInternalEnergy(Scalar temperature,
                                             Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Internal energy");
 
        // regularization
        Scalar pv = vaporPressure(temperature);
        if (pressure < pv) {
            // the pressure is too low, in this case we use the slope
            // of the internal energy at the vapor pressure to
            // regularize
 
            /*
            // calculate the partial derivative of the internal energy
            // to the pressure at the vapor pressure.
            Scalar tau = Region1::tau(temperature);
            Scalar dGamma_dPi = Region1::dGamma_dPi(temperature, pv);
            Scalar ddGamma_dTaudPi = Region1::ddGamma_dTaudPi(temperature, pv);
            Scalar ddGamma_ddPi = Region1::ddGamma_ddPi(temperature, pv);
            Scalar pi = Region1::pi(pv);
            Scalar dPi_dp = Region1::dPi_dp(pv);
            Scalar du_dp =
                Rs*temperature*
                (tau*dPi_dp*ddGamma_dTaudPi + dPi_dp*dPi_dp*dGamma_dPi + pi*dPi_dp*ddGamma_ddPi);
            */
 
            // use a straight line for extrapolation. use forward
            // differences to calculate the partial derivative to the
            // pressure at the vapor pressure
            static const Scalar eps = 1e-7;
            Scalar uv = internalEnergyRegion1_(temperature, pv);
            Scalar uvPEps = internalEnergyRegion1_(temperature, pv + eps);
            Scalar du_dp = (uvPEps - uv)/eps;
            return uv + du_dp*(pressure - pv);
        }
 
        return internalEnergyRegion1_(temperature, pressure);
    }
 
    static Scalar gasInternalEnergy(Scalar temperature, Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Internal energy");
 
        // regularization
        if (pressure < triplePressure() - 100) {
            // We assume an ideal gas for low pressures to avoid the
            // 0/0 for the internal energy of gas at very low
            // pressures. The enthalpy of an ideal gas does not
            // exhibit any dependence on pressure, so we can just
            // return the specific enthalpy at the point of
            // regularization, i.e.  the triple pressure - 100Pa, and
            // subtract the work required to change the volume for an
            // ideal gas.
            return
                enthalpyRegion2_(temperature, triplePressure() - 100)
                -
                Rs*temperature; // = p*v   for an ideal gas!
        }
        Scalar pv = vaporPressure(temperature);
        if (pressure > pv) {
            // the pressure is too high, in this case we use the slope
            // of the internal energy at the vapor pressure to
            // regularize
 
            /*
            // calculate the partial derivative of the internal energy
            // to the pressure at the vapor pressure.
            Scalar tau = Region2::tau(temperature);
            Scalar dGamma_dPi = Region2::dGamma_dPi(temperature, pv);
            Scalar ddGamma_dTaudPi = Region2::ddGamma_dTaudPi(temperature, pv);
            Scalar ddGamma_ddPi = Region2::ddGamma_ddPi(temperature, pv);
            Scalar pi = Region2::pi(pv);
            Scalar dPi_dp = Region2::dPi_dp(pv);
            Scalar du_dp =
                Rs*temperature*
                (tau*dPi_dp*ddGamma_dTaudPi + dPi_dp*dPi_dp*dGamma_dPi + pi*dPi_dp*ddGamma_ddPi);
 
            // use a straight line for extrapolation
            Scalar uv = internalEnergyRegion2_(temperature, pv);
            return uv + du_dp*(pressure - pv);
            */
 
            // use a straight line for extrapolation. use backward
            // differences to calculate the partial derivative to the
            // pressure at the vapor pressure
            static const Scalar eps = 1e-7;
            Scalar uv = internalEnergyRegion2_(temperature, pv);
            Scalar uvMEps = internalEnergyRegion2_(temperature, pv - eps);
            Scalar du_dp = (uv - uvMEps)/eps;
            return uv + du_dp*(pressure - pv);
        }
 
        return internalEnergyRegion2_(temperature, pressure);
    }
 
    static const Scalar liquidHeatCapacityConstVolume(Scalar temperature,
                                                      Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Heat capacity for a constant volume");
 
        // regularization
        Scalar pv = vaporPressure(temperature);
        if (pressure < pv) {
            // the pressure is too low, in this case we use the heat cap at the vapor pressure to regularize
 
            return heatCap_v_Region1_(temperature, pv);
        }
 
        return heatCap_v_Region1_(temperature, pressure);
    }
 
    static Scalar gasHeatCapacityConstVolume(Scalar temperature, Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Heat capacity for a constant volume");
 
        // regularization
        if (pressure < triplePressure() - 100) {
            return
                heatCap_v_Region2_(temperature, triplePressure() - 100);
        }
        Scalar pv = vaporPressure(temperature);
        if (pressure > pv) {
            return heatCap_v_Region2_(temperature, pv);
        }
 
        return heatCap_v_Region2_(temperature, pressure);
    }
 
    static constexpr bool gasIsCompressible()
    { return true; }
 
    static constexpr bool liquidIsCompressible()
    { return true; }
 
    static Scalar gasDensity(Scalar temperature, Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Density");
 
        // regularization
        if (pressure < triplePressure() - 100) {
            // We assume an ideal gas for low pressures to avoid the
            // 0/0 for the internal energy and enthalpy.
            Scalar rho0IAPWS = 1.0/volumeRegion2_(temperature,
                                                  triplePressure() - 100);
            Scalar rho0Id = IdealGas<Scalar>::density(molarMass(),
                                                      temperature,
                                                      triplePressure() - 100);
            return
                rho0IAPWS/rho0Id *
                IdealGas<Scalar>::density(molarMass(),
                                          temperature,
                                          pressure);
        }
        Scalar pv = vaporPressure(temperature);
        if (pressure > pv) {
            // the pressure is too high, in this case we use the slope
            // of the density energy at the vapor pressure to
            // regularize
 
            // calculate the partial derivative of the specific volume
            // to the pressure at the vapor pressure.
            const Scalar eps = pv*1e-8;
            Scalar v0 = volumeRegion2_(temperature, pv);
            Scalar v1 = volumeRegion2_(temperature, pv + eps);
            Scalar dv_dp = (v1 - v0)/eps;
            /*
            Scalar pi = Region2::pi(pv);
            Scalar dp_dPi = Region2::dp_dPi(pv);
            Scalar dGamma_dPi = Region2::dGamma_dPi(temperature, pv);
            Scalar ddGamma_ddPi = Region2::ddGamma_ddPi(temperature, pv);
 
            Scalar RT = Rs*temperature;
            Scalar dv_dp =
                RT/(dp_dPi*pv)
                *
                (dGamma_dPi + pi*ddGamma_ddPi - v0*dp_dPi/RT);
            */
 
            // calculate the partial derivative of the density to the
            // pressure at vapor pressure
            Scalar drho_dp = - 1/(v0*v0)*dv_dp;
 
            // use a straight line for extrapolation
            return 1.0/v0 + (pressure - pv)*drho_dp;
        }
 
        return 1.0/volumeRegion2_(temperature, pressure);
    }
 
    static Scalar gasMolarDensity(Scalar temperature, Scalar pressure)
    { return gasDensity(temperature, pressure)/molarMass(); }
 
    static constexpr bool gasIsIdeal()
    { return false; }
 
    static Scalar gasPressure(Scalar temperature, Scalar density)
    {
        // We use the newton method for this. For the initial value we
        // assume steam to be an ideal gas
        Scalar pressure = IdealGas<Scalar>::pressure(temperature, density/molarMass());
        Scalar eps = pressure*1e-7;
 
        Scalar deltaP = pressure*2;
        using std::abs;
        for (int i = 0; i < 5 && abs(pressure*1e-9) < abs(deltaP); ++i)
        {
            const Scalar f = gasDensity(temperature, pressure) - density;
 
            Scalar df_dp;
            df_dp = gasDensity(temperature, pressure + eps);
            df_dp -= gasDensity(temperature, pressure - eps);
            df_dp /= 2*eps;
 
            deltaP = - f/df_dp;
 
            pressure += deltaP;
        }
 
        return pressure;
    }
 
    static Scalar liquidDensity(Scalar temperature, Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Density");
 
        // regularization
        Scalar pv = vaporPressure(temperature);
        if (pressure < pv) {
            // the pressure is too low, in this case we use the slope
            // of the density at the vapor pressure to regularize
 
            // calculate the partial derivative of the specific volume
            // to the pressure at the vapor pressure.
            const Scalar eps = pv*1e-8;
            Scalar v0 = volumeRegion1_(temperature, pv);
            Scalar v1 = volumeRegion1_(temperature, pv + eps);
            Scalar dv_dp = (v1 - v0)/eps;
 
            /*
            Scalar v0 = volumeRegion1_(temperature, pv);
            Scalar pi = Region1::pi(pv);
            Scalar dp_dPi = Region1::dp_dPi(pv);
            Scalar dGamma_dPi = Region1::dGamma_dPi(temperature, pv);
            Scalar ddGamma_ddPi = Region1::ddGamma_ddPi(temperature, pv);
 
            Scalar RT = Rs*temperature;
            Scalar dv_dp =
                RT/(dp_dPi*pv)
                *
                (dGamma_dPi + pi*ddGamma_ddPi - v0*dp_dPi/RT);
            */
 
            // calculate the partial derivative of the density to the
            // pressure at vapor pressure
            Scalar drho_dp = - 1/(v0*v0)*dv_dp;
 
            // use a straight line for extrapolation
            return 1.0/v0 + (pressure - pv)*drho_dp;
        }
 
        return 1/volumeRegion1_(temperature, pressure);
    }
 
    static Scalar liquidMolarDensity(Scalar temperature, Scalar pressure)
    { return liquidDensity(temperature, pressure)/molarMass(); }
 
    static Scalar liquidPressure(Scalar temperature, Scalar density)
    {
        // We use Brent's method for this, with a pressure range including the surroundings of the
        // vapor pressure line
        Scalar minPressure = vaporPressure(temperature)/1.11;
        Scalar maxPressure = 100e6;
        const auto residualFunction = [&] (const Scalar pressure) {
            return liquidDensity(temperature, pressure) - density;
        };
        try
        {
            return findScalarRootBrent(minPressure, maxPressure, residualFunction);
        }
        catch (const NumericalProblem& e)
        {
            DUNE_THROW(NumericalProblem,
                    "searched for pressure(T=" << temperature << ",rho=" << density
                    <<") in [" << minPressure << ", " << maxPressure << "]: "
                    << e.what());
        }
    }
 
    static Scalar gasViscosity(Scalar temperature, Scalar pressure)
    {
        Region2::checkValidityRange(temperature, pressure, "Viscosity");
 
        Scalar rho = gasDensity(temperature, pressure);
        return Common::viscosity(temperature, rho);
    }
 
 
    static Scalar liquidViscosity(Scalar temperature, Scalar pressure)
    {
        Region1::checkValidityRange(temperature, pressure, "Viscosity");
 
        Scalar rho = liquidDensity(temperature, pressure);
        return Common::viscosity(temperature, rho);
    }
 
    static Scalar liquidThermalConductivity( Scalar temperature,  Scalar pressure)
    {
        // Thermal conductivity of water is empirically fit.
        // Evaluating that fitting-function outside the area of validity does not make sense.
        if ( !(   (pressure <= 400e6 && (273.15 <= temperature) && (temperature <= 398.15))
               || (pressure <= 200e6 && (398.15 <  temperature) && (temperature <=  523.15))
               || (pressure <= 150e6 && (523.15 <  temperature) && (temperature <=  673.15))
               || (pressure <= 100e6 && (673.15 <  temperature) && (temperature <=  1073.15)) ))
        {
            DUNE_THROW(NumericalProblem,
                       "Evaluating the IAPWS fit function for thermal conductivity outside range of applicability."
                       "(T=" << temperature << ", p=" << pressure << ")");
        }
 
        Scalar rho = liquidDensity(temperature, pressure);
        return Common::thermalConductivityIAPWS(temperature, rho);
    }
 
    static Scalar gasThermalConductivity(const Scalar temperature, const Scalar pressure)
    {
        // Thermal conductivity of water is empirically fit.
        // Evaluating that fitting-function outside the area of validity does not make sense.
        if ( !(   (pressure <= 400e6 && (273.15 <= temperature) && (temperature <= 398.15))
               || (pressure <= 200e6 && (398.15 <  temperature) && (temperature <= 523.15))
               || (pressure <= 150e6 && (523.15 <  temperature) && (temperature <= 673.15))
               || (pressure <= 100e6 && (673.15 <  temperature) && (temperature <= 1073.15)) ))
        {
            DUNE_THROW(NumericalProblem,
                       "Evaluating the IAPWS fit function for thermal conductivity outside range of applicability."
                       "(T=" << temperature << ", p=" << pressure << ")");
        }
 
        Scalar rho = gasDensity(temperature, pressure);
        return Common::thermalConductivityIAPWS(temperature, rho);
    }
 
private:
    // the unregularized specific enthalpy for liquid water
    static constexpr Scalar enthalpyRegion1_(Scalar temperature, Scalar pressure)
    {
        return
            Region1::tau(temperature) *
            Region1::dGamma_dTau(temperature, pressure) *
            Rs*temperature;
    }
 
    // the unregularized specific isobaric heat capacity
    static constexpr Scalar heatCap_p_Region1_(Scalar temperature, Scalar pressure)
    {
        return
            - Region1::tau(temperature) * Region1::tau(temperature) *
            Region1::ddGamma_ddTau(temperature, pressure) *
            Rs;
    }
 
    // the unregularized specific isochoric heat capacity
    static Scalar heatCap_v_Region1_(Scalar temperature, Scalar pressure)
    {
        Scalar tau = Region1::tau(temperature);
        Scalar num = Region1::dGamma_dPi(temperature, pressure) - tau * Region1::ddGamma_dTaudPi(temperature, pressure);
        Scalar diff = num * num / Region1::ddGamma_ddPi(temperature, pressure);
 
        return
            - tau * tau *
            Region1::ddGamma_ddTau(temperature, pressure) * Rs +
            diff;
    }
 
    // the unregularized specific internal energy for liquid water
    static constexpr Scalar internalEnergyRegion1_(Scalar temperature, Scalar pressure)
    {
        return
            Rs * temperature *
            ( Region1::tau(temperature)*Region1::dGamma_dTau(temperature, pressure) -
              Region1::pi(pressure)*Region1::dGamma_dPi(temperature, pressure));
    }
 
    // the unregularized specific volume for liquid water
    static constexpr Scalar volumeRegion1_(Scalar temperature, Scalar pressure)
    {
        return
            Region1::pi(pressure)*
            Region1::dGamma_dPi(temperature, pressure) *
            Rs * temperature / pressure;
    }
 
    // the unregularized specific enthalpy for steam
    static constexpr Scalar enthalpyRegion2_(Scalar temperature, Scalar pressure)
    {
        return
            Region2::tau(temperature) *
            Region2::dGamma_dTau(temperature, pressure) *
            Rs*temperature;
    }
 
    // the unregularized specific internal energy for steam
    static constexpr Scalar internalEnergyRegion2_(Scalar temperature, Scalar pressure)
    {
        return
            Rs * temperature *
            ( Region2::tau(temperature)*Region2::dGamma_dTau(temperature, pressure) -
              Region2::pi(pressure)*Region2::dGamma_dPi(temperature, pressure));
    }
 
    // the unregularized specific isobaric heat capacity
    static constexpr Scalar heatCap_p_Region2_(Scalar temperature, Scalar pressure)
    {
        return
            - Region2::tau(temperature) * Region2::tau(temperature) *
            Region2::ddGamma_ddTau(temperature, pressure) *
            Rs;
    }
 
    // the unregularized specific isochoric heat capacity
    static Scalar heatCap_v_Region2_(Scalar temperature, Scalar pressure)
    {
        Scalar tau = Region2::tau(temperature);
        Scalar pi = Region2::pi(pressure);
        Scalar num = 1 + pi * Region2::dGamma_dPi(temperature, pressure) + tau * pi * Region2::ddGamma_dTaudPi(temperature, pressure);
        Scalar diff = num * num / (1 - pi * pi * Region2::ddGamma_ddPi(temperature, pressure));
        return
            - tau * tau *
            Region2::ddGamma_ddTau(temperature, pressure) * Rs
            - diff;
    }
 
    // the unregularized specific volume for steam
    static constexpr Scalar volumeRegion2_(Scalar temperature, Scalar pressure)
    {
        return
            Region2::pi(pressure)*
            Region2::dGamma_dPi(temperature, pressure) *
            Rs * temperature / pressure;
    }
};
 
template <class Scalar>
struct IsAqueous<H2O<Scalar>> : public std::true_type {};
 
} // end namespace Components
 
namespace FluidSystems::Detail {
// viscosity according to Reid, R.C.
template <class Scalar>
Scalar viscosityReid_(Scalar temperature)
{
    constexpr Scalar tc = 647.3;
    using std::max;
    const Scalar tr = max(temperature/tc, 1e-8);
 
    const Scalar fp0 = 1.0 + 0.221*(0.96 + 0.1*(tr - 0.7));
    constexpr Scalar xi = 3.334e-3;
    using std::pow;
    using std::exp;
    const Scalar eta_xi = (0.807*pow(tr, 0.618) - 0.357*exp((-0.449)*tr)
                           + 0.34*exp((-4.058)*tr) + 0.018)*fp0;
 
    return 1.0e-7*eta_xi/xi;
}
 
// viscosity according to Nagel, T. et al.
template <class Scalar>
Scalar viscosityNagel_(Scalar temperature)
{
    constexpr Scalar a1 = -4.4189440e-6;
    constexpr Scalar a2 = 4.6876380e-8;
    constexpr Scalar a3 = -5.3894310e-12;
    constexpr Scalar a4 = 3.2028560e-16;
    constexpr Scalar a5 = 4.9191790e-22;
 
    return a1 + a2*temperature + a3*temperature*temperature
              + a4*temperature*temperature*temperature
              + a5*temperature*temperature*temperature*temperature;
}
} // end FluidSystems::Detail
namespace FluidSystems {
template <class Scalar>
Scalar h2oGasViscosityInMixture(Scalar temperature, Scalar pressure)
{
    if (temperature < 480.0)
        return Detail::viscosityReid_(temperature);
 
    else if (temperature > 500.0)
        return Detail::viscosityNagel_(temperature);
 
    else // interpolate
    {
        const Scalar op = (500.0 - temperature)/20.0;
 
        return op*Detail::viscosityReid_(temperature)
               + (1.0 - op)*Detail::viscosityNagel_(temperature);
    }
}
} // end namespace FluidSystems
} // end namespace Dumux
 
#endif