MargulesVPSSTP.h

Go to the documentation of this file.

/**
 *  @file  MargulesVPSSTP.h (see \ref thermoprops and class \link
 *      Cantera::MargulesVPSSTP MargulesVPSSTP\endlink).
 */

// This file is part of Cantera. See License.txt in the top-level directory or
// at http://www.cantera.org/license.txt for license and copyright information.

#ifndef CT_MARGULESVPSSTP_H
#define CT_MARGULESVPSSTP_H

#include "GibbsExcessVPSSTP.h"

namespace Cantera
{

//! MargulesVPSSTP is a derived class of GibbsExcessVPSSTP that employs the
//! Margules approximation for the excess Gibbs free energy
/*!
 * MargulesVPSSTP derives from class GibbsExcessVPSSTP which is derived from
 * VPStandardStateTP, and overloads the virtual methods defined there with ones
 * that use expressions appropriate for the Margules Excess Gibbs free energy
 * approximation.
 *
 * The independent unknowns are pressure, temperature, and mass fraction.
 *
 * ## Specification of Species Standard State Properties
 *
 * All species are defined to have standard states that depend upon both the
 * temperature and the pressure. The Margules approximation assumes symmetric
 * standard states, where all of the standard state assume that the species are
 * in pure component states at the temperature and pressure of the solution.  I
 * don't think it prevents, however, some species from being dilute in the
 * solution.
 *
 * ## Specification of Solution Thermodynamic Properties
 *
 * The molar excess Gibbs free energy is given by the following formula which is
 * a sum over interactions *i*. Each of the interactions are binary interactions
 * involving two of the species in the phase, denoted, *Ai* and *Bi*. This is
 * the generalization of the Margules formulation for a phase that has more than
 * 2 species.
 *
 * \f[
 *     G^E = \sum_i \left(  H_{Ei} - T S_{Ei} \right)
 * \f]
 * \f[
 *    H^E_i = n X_{Ai} X_{Bi} \left( h_{o,i} +  h_{1,i} X_{Bi} \right)
 * \f]
 * \f[
 *    S^E_i = n X_{Ai} X_{Bi} \left( s_{o,i} +  s_{1,i} X_{Bi} \right)
 * \f]
 *
 * where n is the total moles in the solution.
 *
 * The activity of a species defined in the phase is given by an excess Gibbs
 * free energy formulation.
 *
 * \f[
 *      a_k = \gamma_k  X_k
 * \f]
 *
 * where
 *
 * \f[
 *      R T \ln( \gamma_k )= \frac{d(n G^E)}{d(n_k)}\Bigg|_{n_i}
 * \f]
 *
 * Taking the derivatives results in the following expression
 *
 * \f[
 *      R T \ln( \gamma_k )= \sum_i \left( \left( \delta_{Ai,k} X_{Bi} + \delta_{Bi,k} X_{Ai}  - X_{Ai} X_{Bi} \right)
 *       \left( g^E_{o,i} +  g^E_{1,i} X_{Bi} \right) +
 *       \left( \delta_{Bi,k} - X_{Bi} \right)      X_{Ai} X_{Bi}  g^E_{1,i} \right)
 * \f]
 * where
 * \f$  g^E_{o,i} =  h_{o,i} - T s_{o,i} \f$ and
 * \f$ g^E_{1,i} =  h_{1,i} - T s_{1,i} \f$ and where
 * \f$ X_k \f$ is the mole fraction of species *k*.
 *
 * This object inherits from the class VPStandardStateTP. Therefore, the
 * specification and calculation of all standard state and reference state
 * values are handled at that level. Various functional forms for the standard
 * state are permissible. The chemical potential for species *k* is equal to
 *
 * \f[
 *      \mu_k(T,P) = \mu^o_k(T, P) + R T \ln(\gamma_k X_k)
 * \f]
 *
 * The partial molar entropy for species *k* is given by
 *
 * \f[
 *       \tilde{s}_k(T,P) =  s^o_k(T,P)  - R \ln( \gamma_k X_k )
 *              - R T \frac{d \ln(\gamma_k) }{dT}
 * \f]
 *
 * The partial molar enthalpy for species *k* is given by
 *
 * \f[
 *      \tilde{h}_k(T,P) = h^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
 * \f]
 *
 * The partial molar volume for species *k* is
 *
 * \f[
 *        \tilde V_k(T,P)  = V^o_k(T,P)  + R T \frac{d \ln(\gamma_k) }{dP}
 * \f]
 *
 * The partial molar Heat Capacity for species *k* is
 *
 * \f[
 *      \tilde{C}_{p,k}(T,P) = C^o_{p,k}(T,P)   - 2 R T \frac{d \ln( \gamma_k )}{dT}
 *              - R T^2 \frac{d^2 \ln(\gamma_k) }{{dT}^2}
 * \f]
 *
 * ## %Application within Kinetics Managers
 *
 * \f$ C^a_k\f$ are defined such that \f$ a_k = C^a_k / C^s_k, \f$ where
 * \f$ C^s_k \f$ is a standard concentration defined below and \f$ a_k \f$ are
 * activities used in the thermodynamic functions.  These activity (or
 * generalized) concentrations are used by kinetics manager classes to compute
 * the forward and reverse rates of elementary reactions. The activity
 * concentration,\f$  C^a_k \f$,is given by the following expression.
 *
 * \f[
 *      C^a_k = C^s_k  X_k  = \frac{P}{R T} X_k
 * \f]
 *
 * The standard concentration for species *k* is independent of *k* and equal to
 *
 * \f[
 *     C^s_k =  C^s = \frac{P}{R T}
 * \f]
 *
 * For example, a bulk-phase binary gas reaction between species j and k,
 * producing a new gas species l would have the following equation for its rate
 * of progress variable, \f$ R^1 \f$, which has units of kmol m-3 s-1.
 *
 * \f[
 *    R^1 = k^1 C_j^a C_k^a =  k^1 (C^s a_j) (C^s a_k)
 * \f]
 * where
 * \f[
 *    C_j^a = C^s a_j \mbox{\quad and \quad} C_k^a = C^s a_k
 * \f]
 *
 * \f$ C_j^a \f$ is the activity concentration of species j, and \f$ C_k^a \f$
 * is the activity concentration of species k. \f$ C^s \f$ is the standard
 * concentration. \f$ a_j \f$ is the activity of species j which is equal to the
 * mole fraction of j.
 *
 * The reverse rate constant can then be obtained from the law of microscopic
 * reversibility and the equilibrium expression for the system.
 *
 * \f[
 *       \frac{a_j a_k}{ a_l} = K_a^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} )
 * \f]
 *
 * \f$  K_a^{o,1} \f$ is the dimensionless form of the equilibrium constant,
 * associated with the pressure dependent standard states \f$ \mu^o_l(T,P) \f$
 * and their associated activities, \f$ a_l \f$, repeated here:
 *
 * \f[
 *      \mu_l(T,P) = \mu^o_l(T, P) + R T \log(a_l)
 * \f]
 *
 * We can switch over to expressing the equilibrium constant in terms of the
 * reference state chemical potentials
 *
 * \f[
 *     K_a^{o,1} = \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} ) * \frac{P_{ref}}{P}
 * \f]
 *
 * The concentration equilibrium constant, \f$ K_c \f$, may be obtained by
 * changing over to activity concentrations. When this is done:
 *
 * \f[
 *       \frac{C^a_j C^a_k}{ C^a_l} = C^o K_a^{o,1} = K_c^1 =
 *           \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} ) * \frac{P_{ref}}{RT}
 * \f]
 *
 * %Kinetics managers will calculate the concentration equilibrium constant, \f$
 * K_c \f$, using the second and third part of the above expression as a
 * definition for the concentration equilibrium constant.
 *
 * For completeness, the pressure equilibrium constant may be obtained as well
 *
 * \f[
 *     \frac{P_j P_k}{ P_l P_{ref}} = K_p^1 = \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} )
 * \f]
 *
 * \f$ K_p \f$ is the simplest form of the equilibrium constant for ideal gases.
 * However, it isn't necessarily the simplest form of the equilibrium constant
 * for other types of phases; \f$ K_c \f$ is used instead because it is
 * completely general.
 *
 * The reverse rate of progress may be written down as
 * \f[
 *    R^{-1} = k^{-1} C_l^a =  k^{-1} (C^o a_l)
 * \f]
 *
 * where we can use the concept of microscopic reversibility to write the
 * reverse rate constant in terms of the forward reate constant and the
 * concentration equilibrium constant, \f$ K_c \f$.
 *
 * \f[
 *    k^{-1} =  k^1 K^1_c
 * \f]
 *
 * \f$k^{-1} \f$ has units of s-1.
 *
 * @ingroup thermoprops
 */
class MargulesVPSSTP : public GibbsExcessVPSSTP
{
public:
    MargulesVPSSTP();

    //! Construct and initialize a MargulesVPSSTP ThermoPhase object directly
    //! from an XML input file
    /*!
     * Working constructors
     *
     * The two constructors below are the normal way the phase initializes
     * itself. They are shells that call the routine initThermo(), with a
     * reference to the XML database to get the info for the phase.
     *
     * @param inputFile Name of the input file containing the phase XML data
     *                  to set up the object
     * @param id        ID of the phase in the input file. Defaults to the
     *                  empty string.
     */
    MargulesVPSSTP(const std::string& inputFile, const std::string& id = "");

    //! Construct and initialize a MargulesVPSSTP ThermoPhase object directly
    //! from an XML database
    /*!
     *  @param phaseRef XML phase node containing the description of the phase
     *  @param id     id attribute containing the name of the phase.
     *                (default is the empty string)
     */
    MargulesVPSSTP(XML_Node& phaseRef, const std::string& id = "");

    virtual std::string type() const {
        return "Margules";
    }

    //! @name  Molar Thermodynamic Properties
    //! @{

    virtual doublereal enthalpy_mole() const;
    virtual doublereal entropy_mole() const;
    virtual doublereal cp_mole() const;
    virtual doublereal cv_mole() const;

    /**
     * @}
     * @name Activities, Standard States, and Activity Concentrations
     *
     * The activity \f$a_k\f$ of a species in solution is related to the
     * chemical potential by \f[ \mu_k = \mu_k^0(T) + \hat R T \log a_k. \f] The
     * quantity \f$\mu_k^0(T,P)\f$ is the chemical potential at unit activity,
     * which depends only on temperature and pressure.
     * @{
     */

    virtual void getLnActivityCoefficients(doublereal* lnac) const;

    //@}
    /// @name  Partial Molar Properties of the Solution
    //@{

    virtual void getChemPotentials(doublereal* mu) const;

    //! Returns an array of partial molar enthalpies for the species in the
    //! mixture.
    /*!
     * Units (J/kmol)
     *
     * For this phase, the partial molar enthalpies are equal to the standard
     * state enthalpies modified by the derivative of the molality-based
     * activity coefficient wrt temperature
     *
     * \f[
     *   \bar h_k(T,P) = h^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
     * \f]
     *
     * @param hbar  Vector of returned partial molar enthalpies
     *              (length m_kk, units = J/kmol)
     */
    virtual void getPartialMolarEnthalpies(doublereal* hbar) const;

    //! Returns an array of partial molar entropies for the species in the
    //! mixture.
    /*!
     * Units (J/kmol)
     *
     * For this phase, the partial molar enthalpies are equal to the standard
     * state enthalpies modified by the derivative of the activity coefficient
     * wrt temperature
     *
     * \f[
     *   \bar s_k(T,P) = s^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
     *                              - R \ln( \gamma_k X_k)
     *                              - R T \frac{d \ln(\gamma_k) }{dT}
     * \f]
     *
     * @param sbar  Vector of returned partial molar entropies
     *              (length m_kk, units = J/kmol/K)
     */
    virtual void getPartialMolarEntropies(doublereal* sbar) const;

    //! Returns an array of partial molar entropies for the species in the
    //! mixture.
    /*!
     * Units (J/kmol)
     *
     * For this phase, the partial molar enthalpies are equal to the standard
     * state enthalpies modified by the derivative of the activity coefficient
     * wrt temperature
     *
     *  \f[
     *   ???????????????
     *   \bar s_k(T,P) = s^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
     *                              - R \ln( \gamma_k X_k)
     *                              - R T \frac{d \ln(\gamma_k) }{dT}
     *   ???????????????
     *  \f]
     *
     * @param cpbar  Vector of returned partial molar heat capacities
     *              (length m_kk, units = J/kmol/K)
     */
    virtual void getPartialMolarCp(doublereal* cpbar) const;

    virtual void getPartialMolarVolumes(doublereal* vbar) const;

    //! Get the array of temperature second derivatives of the log activity
    //! coefficients
    /*!
     *  units = 1/Kelvin
     *
     * @param d2lnActCoeffdT2  Output vector of temperature 2nd derivatives of
     *                         the log Activity Coefficients. length = m_kk
     */
    virtual void getd2lnActCoeffdT2(doublereal* d2lnActCoeffdT2) const;

    virtual void getdlnActCoeffdT(doublereal* dlnActCoeffdT) const;

    /// @}
    /// @name Initialization 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()
    /// @{

    virtual void initThermo();
    virtual void initThermoXML(XML_Node& phaseNode, const std::string& id);

    //! Add a binary species interaction with the specified parameters
    /*!
     * @param speciesA    name of the first species
     * @param speciesB    name of the second species
     * @param h0         first excess enthalpy coefficient
     * @param h1         second excess enthalpy coefficient
     * @param s0         first excess entropy coefficient
     * @param s1         second excess entropy coefficient
     * @param vh0        first enthalpy coefficient for excess volume
     * @param vh1        second enthalpy coefficient for excess volume
     * @param vs0        first entropy coefficient for excess volume
     * @param vs1        second entropy coefficient for excess volume
     */
    void addBinaryInteraction(const std::string& speciesA,
        const std::string& speciesB, double h0, double h1, double s0, double s1,
        double vh0, double vh1, double vs0, double vs1);

    //! @}
    //! @name  Derivatives of Thermodynamic Variables needed for Applications
    //! @{

    virtual void getdlnActCoeffds(const doublereal dTds, const doublereal* const dXds, doublereal* dlnActCoeffds) const;
    virtual void getdlnActCoeffdlnX_diag(doublereal* dlnActCoeffdlnX_diag) const;
    virtual void getdlnActCoeffdlnN_diag(doublereal* dlnActCoeffdlnN_diag) const;
    virtual void getdlnActCoeffdlnN(const size_t ld, doublereal* const dlnActCoeffdlnN);

    //@}

private:
    //! Process an XML node called "binaryNeutralSpeciesParameters"
    /*!
     * This node contains all of the parameters necessary to describe the
     * Margules model for a particular binary interaction. This function reads
     * the XML file and writes the coefficients it finds to an internal data
     * structures.
     *
     * @param xmlBinarySpecies  Reference to the XML_Node named "binaryNeutralSpeciesParameters"
     *                          containing the binary interaction
     */
    void readXMLBinarySpecies(XML_Node& xmlBinarySpecies);

    //! Initialize lengths of local variables after all species have been
    //! identified.
    void initLengths();

    //! Update the activity coefficients
    /*!
     * This function will be called to update the internally stored natural
     * logarithm of the activity coefficients
     */
    void s_update_lnActCoeff() const;

    //! Update the derivative of the log of the activity coefficients wrt T
    /*!
     * This function will be called to update the internally stored derivative
     * of the natural logarithm of the activity coefficients wrt temperature.
     */
    void s_update_dlnActCoeff_dT() const;

    //! Update the derivative of the log of the activity coefficients wrt
    //! log(mole fraction)
    /*!
     * This function will be called to update the internally stored derivative
     * of the natural logarithm of the activity coefficients wrt logarithm of
     * the mole fractions.
     */
    void s_update_dlnActCoeff_dlnX_diag() const;

    //! Update the derivative of the log of the activity coefficients wrt
    //! log(moles) - diagonal only
    /*!
     * This function will be called to update the internally stored diagonal
     * entries for the derivative of the natural logarithm of the activity
     * coefficients wrt logarithm of the moles.
     */
    void s_update_dlnActCoeff_dlnN_diag() const;

    //! Update the derivative of the log of the activity coefficients wrt
    //! log(moles_m)
    /*!
     * This function will be called to update the internally stored derivative
     * of the natural logarithm of the activity coefficients wrt logarithm of
     * the mole number of species
     */
    void s_update_dlnActCoeff_dlnN() const;

protected:
    //! number of binary interaction expressions
    size_t numBinaryInteractions_;

    //! Enthalpy term for the binary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_HE_b_ij;

    //! Enthalpy term for the ternary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_HE_c_ij;

    //! Entropy term for the binary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_SE_b_ij;

    //! Entropy term for the ternary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_SE_c_ij;

    //! Enthalpy term for the binary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_VHE_b_ij;

    //! Enthalpy term for the ternary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_VHE_c_ij;

    //! Entropy term for the binary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_VSE_b_ij;

    //! Entropy term for the ternary mole fraction interaction of the
    //! excess Gibbs free energy expression
    mutable vector_fp m_VSE_c_ij;

    //! vector of species indices representing species A in the interaction
    /*!
     *  Each Margules excess Gibbs free energy term involves two species, A and
     *  B. This vector identifies species A.
     */
    std::vector<size_t> m_pSpecies_A_ij;

    //! vector of species indices representing species B in the interaction
    /*!
     *  Each Margules excess Gibbs free energy term involves two species, A and
     *  B. This vector identifies species B.
     */
    std::vector<size_t> m_pSpecies_B_ij;

    //! form of the Margules interaction expression
    /*!
     *  Currently there is only one form.
     */
    int formMargules_;

    //! form of the temperature dependence of the Margules interaction expression
    /*!
     *  Currently there is only one form -> constant wrt temperature.
     */
    int formTempModel_;
};

}

#endif