n2.hh

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/*!
 * \file
 * \ingroup Components
 * \brief Properties of pure molecular nitrogen \f$N_2\f$.
 */
#ifndef DUMUX_N2_HH
#define DUMUX_N2_HH

#include <dumux/material/idealgas.hh>

#include <cmath>

#include <dumux/material/components/base.hh>
#include <dumux/material/components/gas.hh>

namespace Dumux {
namespace Components {

/*!
 * \ingroup Componentss
 * \brief Properties of pure molecular nitrogen \f$N_2\f$.
 *
 * \tparam Scalar The type used for scalar values
 */
template <class Scalar>
class N2
: public Components::Base<Scalar, N2<Scalar> >
, public Components::Gas<Scalar, N2<Scalar> >
{
    using IdealGas = Dumux::IdealGas<Scalar>;

public:
    /*!
     * \brief A human readable name for nitrogen.
     */
    static std::string name()
    { return "N2"; }

    /*!
     * \brief The molar mass in \f$\mathrm{[kg/mol]}\f$ of molecular nitrogen.
     */
    static constexpr Scalar molarMass()
    { return 28.0134e-3;}

    /*!
     * \brief Returns the critical temperature \f$\mathrm{[K]}\f$ of molecular nitrogen
     */
    static Scalar criticalTemperature()
    { return 126.192; /* [K] */ }

    /*!
     * \brief Returns the critical pressure \f$\mathrm{[Pa]}\f$ of molecular nitrogen.
     */
    static Scalar criticalPressure()
    { return 3.39858e6; /* [N/m^2] */ }

    /*!
     * \brief Returns the temperature \f$\mathrm{[K]}\f$ at molecular nitrogen's triple point.
     */
    static Scalar tripleTemperature()
    { return 63.151; /* [K] */ }

    /*!
     * \brief Returns the pressure \f$\mathrm{[Pa]}\f$ at molecular nitrogen's triple point.
     */
    static Scalar triplePressure()
    { return 12.523e3; /* [N/m^2] */ }

    /*!
     * \brief The vapor pressure in \f$\mathrm{[Pa]}\f$ of pure molecular nitrogen
     *        at a given temperature.
     *
     * \param T temperature of component in \f$\mathrm{[K]}\f$
     *
     * Taken from:
     *
     * R. Span, E.W. Lemmon, et al. (2000 ,pp. 1361-1433) \cite span2000
     */
    static Scalar vaporPressure(Scalar T)
    {
        if (T > criticalTemperature())
            return criticalPressure();
        if (T < tripleTemperature())
            return 0; // N2 is solid: We don't take sublimation into
                      // account

        // note: this is the ancillary equation given on page 1368
        using std::sqrt;
        Scalar sigma = Scalar(1.0) - T/criticalTemperature();
        Scalar sqrtSigma = sqrt(sigma);
        const Scalar N1 = -6.12445284;
        const Scalar N2 = 1.26327220;
        const Scalar N3 = -0.765910082;
        const Scalar N4 = -1.77570564;

        using std::exp;
        return
            criticalPressure() *
            exp(criticalTemperature()/T*
                     (sigma*(N1 +
                             sqrtSigma*N2 +
                             sigma*(sqrtSigma*N3 +
                                    sigma*sigma*sigma*N4))));
    }

    /*!
     * \brief The density \f$\mathrm{[kg/m^3]}\f$ of \f$N_2\f$ gas at a given pressure and temperature.
     *
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param pressure pressure of component in \f$\mathrm{[Pa]}\f$
     */
    static Scalar gasDensity(Scalar temperature, Scalar pressure)
    {
        // Assume an ideal gas
        return IdealGas::density(molarMass(), temperature, pressure);
    }

    /*!
     *  \brief The molar density of \f$N_2\f$ gas in \f$\mathrm{[mol/m^3]}\f$ at a given pressure and temperature.
     *
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param pressure pressure of component in \f$\mathrm{[Pa]}\f$
     *
     */
    static Scalar gasMolarDensity(Scalar temperature, Scalar pressure)
    { return IdealGas::molarDensity(temperature, pressure); }

    /*!
     * \brief Returns true if the gas phase is assumed to be compressible
     */
    static constexpr bool gasIsCompressible()
    { return true; }

    /*!
     * \brief Returns true if the gas phase is assumed to be ideal
     */
    static constexpr bool gasIsIdeal()
    { return true; }

    /*!
     * \brief The pressure of gaseous \f$N_2\f$ in \f$\mathrm{[Pa]}\f$ at a given density and temperature.
     *
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param density density of component in \f$\mathrm{[kg/m^3]}\f$
     */
    static Scalar gasPressure(Scalar temperature, Scalar density)
    {
        // Assume an ideal gas
        return IdealGas::pressure(temperature, density/molarMass());
    }

    /*!
     * \brief Specific enthalpy \f$\mathrm{[J/kg]}\f$ of pure nitrogen gas.
     *
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param pressure pressure of component in \f$\mathrm{[Pa]}\f$
     */
    static const Scalar gasEnthalpy(Scalar temperature,
                                    Scalar pressure)
    {
        return gasHeatCapacity(temperature, pressure) * temperature;
    }

    /*!
     * \brief Specific enthalpy \f$\mathrm{[J/kg]}\f$ of pure nitrogen gas.
     *
     *        Definition of enthalpy: \f$h= u + pv = u + p / \rho\f$.
     *
     *        Rearranging for internal energy yields: \f$u = h - pv\f$.
     *
     *        Exploiting the Ideal Gas assumption (\f$pv = R_{\textnormal{specific}} T\f$)gives: \f$u = h - R / M T \f$.
     *
     *        The universal gas constant can only be used in the case of molar formulations.
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param pressure pressure of component in \f$\mathrm{[Pa]}\f$
     */
    static const Scalar gasInternalEnergy(Scalar temperature,
                                          Scalar pressure)
    {
        return
            gasEnthalpy(temperature, pressure) -
            1/molarMass()* // conversion from [J/(mol K)] to [J/(kg K)]
            IdealGas::R*temperature; // = pressure * spec. volume for an ideal gas
    }

    /*!
     * \brief Specific isobaric heat capacity \f$\mathrm{[J/(kg*K)]}\f$ of pure
     *        nitrogen gas.
     *
     * This is equivalent to the partial derivative of the specific
     * enthalpy to the temperature.
     *
     * See: R. Reid, et al. (1987, pp 154, 657, 665) \cite reid1987
     */
    static const Scalar gasHeatCapacity(Scalar T,
                                        Scalar pressure)
    {
        // method of Joback
        const Scalar cpVapA = 31.15;
        const Scalar cpVapB = -0.01357;
        const Scalar cpVapC = 2.680e-5;
        const Scalar cpVapD = -1.168e-8;

        return
            1/molarMass()* // conversion from [J/(mol K)] to [J/(kg K)]
            (cpVapA + T*
              (cpVapB/2 + T*
                (cpVapC/3 + T*
                  (cpVapD/4))));
    }

    /*!
     * \brief The dynamic viscosity \f$\mathrm{[Pa*s]}\f$ of \f$N_2\f$ at a given pressure and temperature.
     *
     * \param temperature temperature of component in \f$\mathrm{[K]}\f$
     * \param pressure pressure of component in \f$\mathrm{[Pa]}\f$
     *
     * See:
     *
     * See: R. Reid, et al.: The Properties of Gases and Liquids,
     * 4th edition (1987, pp 396-397) \cite reid1987 <BR>
     * 5th edition (2001, pp 9.7-9.8 (omega and V_c taken from p. A.19)) \cite poling2001
     *
     */
    static Scalar gasViscosity(Scalar temperature, Scalar pressure)
    {
        const Scalar Tc = criticalTemperature();
        const Scalar Vc = 90.1; // critical specific volume [cm^3/mol]
        const Scalar omega = 0.037; // accentric factor
        const Scalar M = molarMass() * 1e3; // molar mas [g/mol]
        const Scalar dipole = 0.0; // dipole moment [debye]

        using std::sqrt;
        Scalar mu_r4 = 131.3 * dipole / sqrt(Vc * Tc);
        mu_r4 *= mu_r4;
        mu_r4 *= mu_r4;

        using std::pow;
        using std::exp;
        Scalar Fc = 1 - 0.2756*omega + 0.059035*mu_r4;
        Scalar Tstar = 1.2593 * temperature/Tc;
        Scalar Omega_v =
            1.16145*pow(Tstar, -0.14874) +
            0.52487*exp(- 0.77320*Tstar) +
            2.16178*exp(- 2.43787*Tstar);
        Scalar mu = 40.785*Fc*sqrt(M*temperature)/(pow(Vc, 2./3)*Omega_v);

        // convertion from micro poise to Pa s
        return mu/1e6 / 10;
    }

    /*!
     * \brief Thermal conductivity \f$\mathrm{[[W/(m*K)]}\f$ of nitrogen.
     *
     * Isobaric Properties for Nitrogen and Oxygen in: NIST Standard
     * Reference Database Number 69, Eds. P.J. Linstrom and
     * W.G. Mallard evaluated at p=.1 MPa, does not
     * change dramatically with p and can be interpolated linearly with temperature
     *
     * \param temperature absolute temperature in \f$\mathrm{[K]}\f$
     * \param pressure of the phase in \f$\mathrm{[Pa]}\f$
     */
    static Scalar gasThermalConductivity(Scalar temperature, Scalar pressure)
    {
        return 6.525e-5 * (temperature - 273.15) + 0.024031;
    }
};

} // end namespace Components

} // end namespace Dumux

#endif