PengRobinson.h

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//! @file PengRobinson.h
 
// This file is part of Cantera. See License.txt in the top-level directory or
// at https://cantera.org/license.txt for license and copyright information.
 
#ifndef CT_PENGROBINSON_H
#define CT_PENGROBINSON_H
 
#include "MixtureFugacityTP.h"
#include "cantera/base/Array.h"
 
namespace Cantera
{
/**
 * Implementation of a multi-species Peng-Robinson equation of state
 *
 * @ingroup thermoprops
 */
class PengRobinson : public MixtureFugacityTP
{
public:
 
    //! Construct and initialize a PengRobinson object directly from an
    //! input file
    /*!
     * @param infile    Name of the input file containing the phase YAML data.
     *                  If blank, an empty phase will be created.
     * @param id        ID of the phase in the input file. If empty, the
     *                  first phase definition in the input file will be used.
     */
    explicit PengRobinson(const string& infile="", const string& id="");
 
    string type() const override {
        return "Peng-Robinson";
    }
 
    //! @name Molar Thermodynamic properties
    //! @{
 
    double cp_mole() const override;
    double cv_mole() const override;
 
    //! @}
    //! @name Mechanical Properties
    //! @{
 
    //! Return the thermodynamic pressure (Pa).
    /*!
     * Since the mass density, temperature, and mass fractions are stored,
     * this method uses these values to implement the
     * mechanical equation of state @f$ P(T, \rho, Y_1, \dots, Y_K) @f$.
     *
     * @f[
     *    P = \frac{RT}{v-b_{mix}}
     *        - \frac{\left(\alpha a\right)_{mix}}{v^2 + 2b_{mix}v - b_{mix}^2}
     * @f]
     *
     * where:
     *
     * @f[
     *    \alpha = \left[ 1 + \kappa \left(1-T_r^{0.5}\right)\right]^2
     * @f]
     *
     *  and
     *
     * @f[
     *    \kappa = \left(0.37464 + 1.54226\omega - 0.26992\omega^2\right),
     *        \qquad \qquad \text{For } \omega <= 0.491 \\
     *
     *    \kappa = \left(0.379642 + 1.487503\omega - 0.164423\omega^2 + 0.016667\omega^3 \right),
     *        \qquad \text{For } \omega > 0.491
     * @f]
     *
     * Coefficients @f$ a_{mix}, b_{mix} @f$ and @f$ (a \alpha)_{mix} @f$ are calculated as
     *
     * @f[
     *    a_{mix} = \sum_i \sum_j X_i X_j a_{i, j} = \sum_i \sum_j X_i X_j \sqrt{a_i a_j}
     * @f]
     *
     * @f[
     *    b_{mix} = \sum_i X_i b_i
     * @f]
     *
     * @f[
     *   {a \alpha}_{mix} = \sum_i \sum_j X_i X_j {a \alpha}_{i, j}
     *       = \sum_i \sum_j X_i X_j \sqrt{a_i a_j} \sqrt{\alpha_i \alpha_j}
     * @f]
     */
    double pressure() const override;
 
    //! @}
 
    //! Returns the standard concentration @f$ C^0_k @f$, which is used to
    //! normalize the generalized concentration.
    /*!
     * This is defined as the concentration by which the generalized
     * concentration is normalized to produce the activity.
     * The ideal gas mixture is considered as the standard or reference state here.
     * Since the activity for an ideal gas mixture is simply the mole fraction,
     * for an ideal gas,  @f$ C^0_k = P/\hat R T @f$.
     *
     * @param k Optional parameter indicating the species. The default is to
     *          assume this refers to species 0.
     * @return
     *   Returns the standard Concentration in units of m^3 / kmol.
     */
    double standardConcentration(size_t k=0) const override;
 
    //! Get the array of non-dimensional activity coefficients at the current
    //! solution temperature, pressure, and solution concentration.
    /*!
     * For all objects with the Mixture Fugacity approximation, we define the
     * standard state as an ideal gas at the current temperature and pressure of
     * the solution. The activities are based on this standard state.
     *
     * @param ac Output vector of activity coefficients. Length: m_kk.
     */
    void getActivityCoefficients(double* ac) const override;
 
    //! @name  Partial Molar Properties of the Solution
    //! @{
 
    void getChemPotentials(double* mu) const override;
    void getPartialMolarEnthalpies(double* hbar) const override;
    void getPartialMolarEntropies(double* sbar) const override;
    void getPartialMolarIntEnergies(double* ubar) const override;
    //! Calculate species-specific molar specific heats
    /*!
     *  This function is currently not implemented for Peng-Robinson phase.
     */
    void getPartialMolarCp(double* cpbar) const override;
    void getPartialMolarVolumes(double* vbar) const override;
    //! @}
 
    //! Calculate species-specific critical temperature
    /*!
     *  The temperature dependent parameter in P-R EoS is calculated as
     * @f[ T_{crit} = (0.0778 a)/(0.4572 b R) @f]
     *  Units: Kelvin
     *
     * @param a    species-specific coefficients used in P-R EoS
     * @param b    species-specific coefficients used in P-R EoS
     */
    double speciesCritTemperature(double a, double b) const;
 
    //! @name Initialization Methods - For Internal use
    //!
    //! The following methods are used in the process of constructing
    //! the phase and setting its parameters from a specification in an
    //! input file. They are not normally used in application programs.
    //! To see how they are used, see importPhase().
    //! @{
 
    bool addSpecies(shared_ptr<Species> spec) override;
    void initThermo() override;
    void getSpeciesParameters(const string& name, AnyMap& speciesNode) const override;
 
    //! Set the pure fluid interaction parameters for a species
    /*!
     *  @param species   Name of the species
     *  @param a         @f$ a @f$ parameter in the Peng-Robinson model [Pa-m^6/kmol^2]
     *  @param b         @f$ a @f$ parameter in the Peng-Robinson model [m^3/kmol]
     *  @param w         acentric factor
     */
    void setSpeciesCoeffs(const string& species, double a, double b, double w);
 
    //! Set values for the interaction parameter between two species
    /*!
     *  @param species_i   Name of one species
     *  @param species_j   Name of the other species
     *  @param a           @f$ a @f$ parameter in the Peng-Robinson model [Pa-m^6/kmol^2]
     */
    void setBinaryCoeffs(const string& species_i, const string& species_j, double a);
    //! @}
 
protected:
    // Special functions inherited from MixtureFugacityTP
    double sresid() const override;
    double hresid() const override;
 
    double liquidVolEst(double T, double& pres) const override;
    double densityCalc(double T, double pressure, int phase, double rhoguess) override;
 
    double densSpinodalLiquid() const override;
    double densSpinodalGas() const override;
    double dpdVCalc(double T, double molarVol, double& presCalc) const override;
 
    // Special functions not inherited from MixtureFugacityTP
 
    //! Calculate temperature derivative @f$ d(a \alpha)/dT @f$
    /*!
     *  These are stored internally.
     */
    double daAlpha_dT() const;
 
    //! Calculate second derivative @f$ d^2(a \alpha)/dT^2 @f$
    /*!
     *  These are stored internally.
     */
    double d2aAlpha_dT2() const;
 
public:
 
    double isothermalCompressibility() const override;
    double thermalExpansionCoeff() const override;
    double soundSpeed() const override;
 
    //! Calculate @f$ dp/dV @f$ and @f$ dp/dT @f$ at the current conditions
    /*!
     *  These are stored internally.
     */
    void calculatePressureDerivatives() const;
 
    //! Update the @f$ a @f$, @f$ b @f$, and @f$ \alpha @f$ parameters
    /*!
     *  The @f$ a @f$ and the @f$ b @f$ parameters depend on the mole fraction and the
     *  parameter @f$ \alpha @f$ depends on the temperature. This function updates
     *  the internal numbers based on the state of the object.
     */
    void updateMixingExpressions() override;
 
    //! Calculate the @f$ a @f$, @f$ b @f$, and @f$ \alpha @f$ parameters given the temperature
    /*!
     * This function doesn't change the internal state of the object, so it is a
     * const function.  It does use the stored mole fractions in the object.
     *
     * @param aCalc (output)  Returns the a value
     * @param bCalc (output)  Returns the b value.
     * @param aAlpha (output) Returns the (a*alpha) value.
     */
    void calculateAB(double& aCalc, double& bCalc, double& aAlpha) const;
 
    void calcCriticalConditions(double& pc, double& tc, double& vc) const override;
 
    //! Prepare variables and call the function to solve the cubic equation of state
    int solveCubic(double T, double pres, double a, double b, double aAlpha,
                   double Vroot[3]) const;
protected:
    //! Value of @f$ b @f$ in the equation of state
    /*!
     *  `m_b` is a function of the mole fractions and species-specific b values.
     */
    double m_b = 0.0;
 
    //! Value of @f$ a @f$ in the equation of state
    /*!
     *  `m_a` depends only on the mole fractions.
     */
    double m_a = 0.0;
 
    //! Value of @f$ a \alpha @f$ in the equation of state
    /*!
     *  `m_aAlpha_mix` is a function of the temperature and the mole fractions.
     */
    double m_aAlpha_mix = 0.0;
 
    // Vectors required to store species-specific a_coeff, b_coeff, alpha, kappa
    // and other derivatives. Length = m_kk.
    vector<double> m_b_coeffs;
    vector<double> m_kappa;
    vector<double> m_acentric; //!< acentric factor for each species, length #m_kk
    mutable vector<double> m_dalphadT;
    mutable vector<double> m_d2alphadT2;
    vector<double> m_alpha;
 
    // Matrices for Binary coefficients a_{i,j} and {a*alpha}_{i.j} are saved in an
    // array form. Size = (m_kk, m_kk).
    Array2D m_a_coeffs;
    Array2D m_aAlpha_binary;
 
    //! Explicitly-specified binary interaction parameters, to enable serialization
    map<string, map<string, double>> m_binaryParameters;
 
    int m_NSolns = 0;
 
    double m_Vroot[3] = {0.0, 0.0, 0.0};
 
    //! Temporary storage - length = m_kk.
    mutable vector<double> m_pp;
 
    // Partial molar volumes of the species
    mutable vector<double> m_partialMolarVolumes;
 
    //! The derivative of the pressure with respect to the volume
    /*!
     * Calculated at the current conditions. temperature and mole number kept
     * constant
     */
    mutable double m_dpdV = 0.0;
 
    //! The derivative of the pressure with respect to the temperature
    /*!
     *  Calculated at the current conditions. Total volume and mole number kept
     *  constant
     */
    mutable double m_dpdT = 0.0;
 
    //! Vector of derivatives of pressure with respect to mole number
    /*!
     *  Calculated at the current conditions. Total volume, temperature and
     *  other mole number kept constant
     */
    mutable vector<double> m_dpdni;
 
    enum class CoeffSource { EoS, CritProps, Database };
    //! For each species, specifies the source of the a, b, and omega coefficients
    vector<CoeffSource> m_coeffSource;
 
private:
    //! Omega constant: a0 (= omega_a) used in Peng-Robinson equation of state
    /*!
     *  This value is calculated by solving P-R cubic equation at the critical point.
     */
    static const double omega_a;
 
    //! Omega constant: b0 (= omega_b) used in Peng-Robinson equation of state
    /*!
     *  This value is calculated by solving P-R cubic equation at the critical point.
     */
    static const double omega_b;
 
    //! Omega constant for the critical molar volume
    static const double omega_vc;
};
}
 
#endif