MolalityVPSSTP.h

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/**
 *  @file MolalityVPSSTP.h
 *   Header for intermediate ThermoPhase object for phases which
 *   employ molality based activity coefficient formulations
 *  (see @ref thermoprops
 * and class @link Cantera::MolalityVPSSTP MolalityVPSSTP@endlink).
 *
 * Header file for a derived class of ThermoPhase that handles
 * variable pressure standard state methods for calculating
 * thermodynamic properties that are further based upon activities
 * based on the molality scale.  These include most of the methods for
 * calculating liquid electrolyte thermodynamics.
 */
 
// 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_MOLALITYVPSSTP_H
#define CT_MOLALITYVPSSTP_H
 
#include "VPStandardStateTP.h"
 
namespace Cantera
{
 
//! Scale to be used for the output of single-ion activity coefficients is that
//! used by Pitzer.
/*!
 * This is the internal scale used within the code. One property is that the
 * activity coefficients for the cation and anion of a single salt will be
 * equal. This scale is the one presumed by the formulation of the single-ion
 * activity coefficients described in this report.
 *
 * Activity coefficients for species k may be altered between scales s1 to s2
 * using the following formula
 *
 * @f[
 *     \ln(\gamma_k^{s2}) = \ln(\gamma_k^{s1})
 *        + \frac{z_k}{z_j} \left( \ln(\gamma_j^{s2}) - \ln(\gamma_j^{s1}) \right)
 * @f]
 *
 * where j is any one species.
 */
const int PHSCALE_PITZER = 0;
 
//! Scale to be used for evaluation of single-ion activity coefficients is that
//! used by the NBS standard for evaluation of the pH variable.
/*!
 * This is not the internal scale used within the code.
 *
 * Activity coefficients for species k may be altered between scales s1 to s2
 * using the following formula
 *
 * @f[
 *     \ln(\gamma_k^{s2}) = \ln(\gamma_k^{s1})
 *        + \frac{z_k}{z_j} \left( \ln(\gamma_j^{s2}) - \ln(\gamma_j^{s1}) \right)
 * @f]
 *
 * where j is any one species. For the NBS scale, j is equal to the Cl- species
 * and
 *
 * @f[
 *     \ln(\gamma_{Cl-}^{s2}) = \frac{-A_{\phi} \sqrt{I}}{1.0 + 1.5 \sqrt{I}}
 * @f]
 *
 * This is the NBS pH scale, which is used in all conventional pH measurements.
 * and is based on the Bates-Guggenheim equations.
 */
const int PHSCALE_NBS = 1;
 
/**
 * MolalityVPSSTP is a derived class of ThermoPhase that handles variable
 * pressure standard state methods for calculating thermodynamic properties that
 * are further based on molality-scaled activities. This category incorporates
 * most of the methods for calculating liquid electrolyte thermodynamics that
 * have been developed since the 1970's.
 *
 * This class adds additional functions onto the ThermoPhase interface that
 * handle molality based standard states. The ThermoPhase class includes a
 * member function, ThermoPhase::activityConvention() that indicates which
 * convention the activities are based on. The default is to assume activities
 * are based on the molar convention. However, classes which derive from the
 * MolalityVPSSTP class return `cAC_CONVENTION_MOLALITY` from this member
 * function.
 *
 * The molality of a solute, @f$ m_i @f$, is defined as
 *
 * @f[
 *     m_i = \frac{n_i}{\tilde{M}_o n_o}
 * @f]
 * where
 * @f[
 *     \tilde{M}_o = \frac{M_o}{1000}
 * @f]
 *
 * where @f$ M_o @f$ is the molecular weight of the solvent. The molality has
 * units of gmol/kg. For the solute, the molality may be considered
 * as the amount of gmol's of solute per kg of solvent, a natural experimental
 * quantity.
 *
 * The formulas for calculating mole fractions if given the molalities of the
 * solutes is stated below. First calculate @f$ L^{sum} @f$, an intermediate
 * quantity.
 *
 * @f[
 *     L^{sum} = \frac{1}{\tilde{M}_o X_o} = \frac{1}{\tilde{M}_o} + \sum_{i\ne o} m_i
 * @f]
 * Then,
 * @f[
 *     X_o =   \frac{1}{\tilde{M}_o L^{sum}}
 * @f]
 * @f[
 *     X_i =   \frac{m_i}{L^{sum}}
 * @f]
 * where @f$ X_o @f$ is the mole fraction of solvent, and @f$ X_o @f$ is the
 * mole fraction of solute *i*. Thus, the molality scale and the mole fraction
 * scale offer a one-to-one mapping between each other, except in the limit of a
 * zero solvent mole fraction.
 *
 * The standard states for thermodynamic objects that derive from MolalityVPSSTP
 * are on the unit molality basis. Chemical potentials of the solutes, @f$ \mu_k
 * @f$, and the solvent, @f$ \mu_o @f$, which are based on the molality form,
 * have the following general format:
 *
 * @f[
 *    \mu_k = \mu^{\triangle}_k(T,P) + R T \ln(\gamma_k^{\triangle} \frac{m_k}{m^\triangle})
 * @f]
 * @f[
 *    \mu_o = \mu^o_o(T,P) + RT \ln(a_o)
 * @f]
 *
 * where @f$ \gamma_k^{\triangle} @f$ is the molality based activity coefficient
 * for species @f$ k @f$.
 *
 * The chemical potential of the solvent is thus expressed in a different format
 * than the chemical potential of the solutes. Additionally, the activity of the
 * solvent, @f$ a_o @f$, is further reexpressed in terms of an osmotic
 * coefficient, @f$ \phi @f$.
 * @f[
 *     \phi = \frac{- \ln(a_o)}{\tilde{M}_o \sum_{i \ne o} m_i}
 * @f]
 *
 * MolalityVPSSTP::osmoticCoefficient() returns the value of @f$ \phi @f$. Note
 * there are a few of definitions of the osmotic coefficient floating around. We
 * use the one defined in (Activity Coefficients in Electrolyte Solutions, K. S.
 * Pitzer CRC Press, Boca Raton, 1991, p. 85, Eqn. 28). This definition is most
 * clearly related to theoretical calculation.
 *
 * The molar-based activity coefficients @f$ \gamma_k @f$ may be calculated from
 * the molality-based activity coefficients, @f$ \gamma_k^\triangle @f$ by the
 * following formula.
 * @f[
 *     \gamma_k = \frac{\gamma_k^\triangle}{X_o}
 * @f]
 * For purposes of establishing a convention, the molar activity coefficient of
 * the solvent is set equal to the molality-based activity coefficient of the
 * solvent:
 * @f[
 *     \gamma_o = \gamma_o^\triangle
 * @f]
 *
 * The molality-based and molarity-based standard states may be related to one
 * another by the following formula.
 *
 * @f[
 *   \mu_k^\triangle(T,P) = \mu_k^o(T,P) + R T \ln(\tilde{M}_o m^\triangle)
 * @f]
 *
 * An important convention is followed in all routines that derive from
 * MolalityVPSSTP. Standard state thermodynamic functions and reference state
 * thermodynamic functions return the molality-based quantities. Also all
 * functions which return activities return the molality-based activities. The
 * reason for this convention has been discussed in supporting memos. However,
 * it's important because the term in the equation above is non-trivial. For
 * example it's equal to 2.38 kcal/gmol for water at 298 K.
 *
 * In order to prevent a singularity, this class includes the concept of a
 * minimum value for the solvent mole fraction. All calculations involving the
 * formulation of activity coefficients and other non-ideal solution behavior
 * adhere to this concept of a minimal value for the solvent mole fraction. This
 * makes sense because these solution behavior were all designed and measured
 * far away from the zero solvent singularity condition and are not applicable
 * in that limit.
 *
 * This objects add a layer that supports molality. It inherits from
 * VPStandardStateTP.
 *
 * All objects that derive from this are assumed to have molality based standard
 * states.
 *
 * Molality based activity coefficients are scaled according to the current pH
 * scale. See the Eq3/6 manual for details.
 *
 * Activity coefficients for species k may be altered between scales s1 to s2
 * using the following formula
 *
 * @f[
 *     \ln(\gamma_k^{s2}) = \ln(\gamma_k^{s1})
 *        + \frac{z_k}{z_j} \left( \ln(\gamma_j^{s2}) - \ln(\gamma_j^{s1}) \right)
 * @f]
 *
 * where j is any one species. For the NBS scale, j is equal to the Cl- species
 * and
 *
 * @f[
 *     \ln(\gamma_{Cl-}^{s2}) = \frac{-A_{\phi} \sqrt{I}}{1.0 + 1.5 \sqrt{I}}
 * @f]
 *
 * The Pitzer scale doesn't actually change anything. The pitzer scale is
 * defined as the raw unscaled activity coefficients produced by the underlying
 * objects.
 *
 * ### SetState Strategy
 *
 * The MolalityVPSSTP object does not have a setState strategy concerning the
 * molalities. It does not keep track of whether the molalities have changed.
 * It's strictly an interfacial layer that writes the current mole fractions to
 * the State object. When molalities are needed it recalculates the molalities
 * from the State object's mole fraction vector.
 *
 * @todo Make two solvent minimum fractions. One would be for calculation of the
 *       non-ideal factors. The other one would be for purposes of stoichiometry
 *       evaluation. the stoichiometry evaluation one would be a 1E-13 limit.
 *       Anything less would create problems with roundoff error.
 */
class MolalityVPSSTP : public VPStandardStateTP
{
public:
    //! Default Constructor
    /*!
     * This doesn't do much more than initialize constants with default values
     * for water at 25C. Water molecular weight comes from the default
     * elements definitions. It actually differs slightly from the IAPWS95 value of
     * 18.015268. However, density conservation and therefore element
     * conservation is the more important principle to follow.
     */
    MolalityVPSSTP();
 
    //! @name  Utilities
    //! @{
 
    //! String indicating the mechanical phase of the matter in this Phase.
    /*!
     * All derived phases from `MolalityVPSSTP` always represent liquids.
     */
    string phaseOfMatter() const override {
        return "liquid";
    }
 
    //! Set the pH scale, which determines the scale for single-ion activity
    //! coefficients.
    /*!
     * Single ion activity coefficients are not unique in terms of the
     * representing actual measurable quantities.
     *
     * @param pHscaleType  Integer representing the pHscale
     */
    void setpHScale(const int pHscaleType);
 
    //! Reports the pH scale, which determines the scale for single-ion activity
    //! coefficients.
    /*!
     * Single ion activity coefficients are not unique in terms of the
     * representing actual measurable quantities.
     *
     * @returns the pHscale type
     */
    int pHScale() const;
 
    //! @}
    //! @name Utilities for Solvent ID and Molality
    //! @{
 
    /**
     * Sets the minimum mole fraction in the molality formulation. Note the
     * molality formulation is singular in the limit that the solvent mole
     * fraction goes to zero. Numerically, how this limit is treated and
     * resolved is an ongoing issue within Cantera. The minimum mole fraction
     * must be in the range 0 to 0.9.
     *
     * @param xmolSolventMIN  Input double containing the minimum mole fraction
     */
    void setMoleFSolventMin(double xmolSolventMIN);
 
    //! Returns the minimum mole fraction in the molality formulation.
    double moleFSolventMin() const;
 
    //! Calculates the molality of all species and stores the result internally.
    /*!
     * We calculate the vector of molalities of the species in the phase and
     * store the result internally:
     * @f[
     *     m_i = \frac{X_i}{1000 * M_o * X_{o,p}}
     * @f]
     * where
     *    - @f$ M_o @f$ is the molecular weight of the solvent
     *    - @f$ X_o @f$ is the mole fraction of the solvent
     *    - @f$ X_i @f$ is the mole fraction of the solute.
     *    - @f$ X_{o,p} = \max (X_{o}^{min}, X_o) @f$
     *    - @f$ X_{o}^{min} @f$ = minimum mole fraction of solvent allowed
     *              in the denominator.
     */
    void calcMolalities() const;
 
    //! This function will return the molalities of the species.
    /*!
     * We calculate the vector of molalities of the species in the phase
     * @f[
     *     m_i = \frac{X_i}{1000 * M_o * X_{o,p}}
     * @f]
     * where
     *    - @f$ M_o @f$ is the molecular weight of the solvent
     *    - @f$ X_o @f$ is the mole fraction of the solvent
     *    - @f$ X_i @f$ is the mole fraction of the solute.
     *    - @f$ X_{o,p} = \max (X_{o}^{min}, X_o) @f$
     *    - @f$ X_{o}^{min} @f$ = minimum mole fraction of solvent allowed
     *              in the denominator.
     *
     * @param molal       Output vector of molalities. Length: m_kk.
     */
    void getMolalities(double* const molal) const;
 
    //! Set the molalities of the solutes in a phase
    /*!
     * Note, the entry for the solvent is not used. We are supplied with the
     * molalities of all of the solute species. We then calculate the mole
     * fractions of all species and update the ThermoPhase object.
     * @f[
     *     m_i = \frac{X_i}{M_o/1000 * X_{o,p}}
     * @f]
     * where
     *    -  @f$ M_o @f$ is the molecular weight of the solvent
     *    -  @f$ X_o @f$ is the mole fraction of the solvent
     *    -  @f$ X_i @f$ is the mole fraction of the solute.
     *    -  @f$ X_{o,p} = \max(X_o^{min}, X_o) @f$
     *    -  @f$ X_o^{min} @f$ = minimum mole fraction of solvent allowed
     *                     in the denominator.
     *
     * The formulas for calculating mole fractions are
     * @f[
     *     L^{sum} = \frac{1}{\tilde{M}_o X_o} = \frac{1}{\tilde{M}_o} + \sum_{i\ne o} m_i
     * @f]
     * Then,
     * @f[
     *     X_o =   \frac{1}{\tilde{M}_o L^{sum}}
     * @f]
     * @f[
     *     X_i =   \frac{m_i}{L^{sum}}
     * @f]
     * It is currently an error if the solvent mole fraction is attempted to be
     * set to a value lower than @f$ X_o^{min} @f$.
     *
     * @param molal   Input vector of molalities. Length: m_kk.
     */
    void setMolalities(const double* const molal);
 
    //! Set the molalities of a phase
    /*!
     * Set the molalities of the solutes in a phase. Note, the entry for the
     * solvent is not used.
     *
     * @param xMap  Composition Map containing the molalities.
     */
    void setMolalitiesByName(const Composition& xMap);
 
    //! Set the molalities of a phase
    /*!
     * Set the molalities of the solutes in a phase. Note, the entry for the
     * solvent is not used.
     *
     * @param name  String containing the information for a composition map.
     */
    void setMolalitiesByName(const string& name);
 
    //! @}
    //! @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 \ln 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.
    //! @{
 
    /**
     *  We set the convention to molality here.
     */
    int activityConvention() const override;
 
    void getActivityConcentrations(double* c) const override;
    double standardConcentration(size_t k=0) const override;
 
    //! Get the array of non-dimensional activities (molality based for this
    //! class and classes that derive from it) at the current solution
    //! temperature, pressure, and solution concentration.
    /*!
     * All standard state properties for molality-based phases are evaluated
     * consistent with the molality scale. Therefore, this function must return
     * molality-based activities.
     *
     * @f[
     *     a_i^\triangle = \gamma_k^{\triangle} \frac{m_k}{m^\triangle}
     * @f]
     *
     * This function must be implemented in derived classes.
     *
     * @param ac     Output vector of molality-based activities. Length: m_kk.
     */
    void getActivities(double* ac) const override;
 
    //! Get the array of non-dimensional activity coefficients at
    //! the current solution temperature, pressure, and solution concentration.
    /*!
     * These are mole-fraction based activity coefficients. In this
     * object, their calculation is based on translating the values
     * of the molality-based activity coefficients.
     *  See Denbigh p. 278 @cite denbigh1981 for a thorough discussion.
     *
     * The molar-based activity coefficients @f$ \gamma_k @f$ may be calculated
     * from the molality-based activity coefficients, @f$ \gamma_k^\triangle @f$
     * by the following formula.
     * @f[
     *     \gamma_k = \frac{\gamma_k^\triangle}{X_o}
     * @f]
     *
     * For purposes of establishing a convention, the molar activity coefficient of the
     * solvent is set equal to the molality-based activity coefficient of the
     * solvent:
     *
     * @f[
     *     \gamma_o = \gamma_o^\triangle
     * @f]
     *
     * Derived classes don't need to overload this function. This function is
     * handled at this level.
     *
     * @param ac  Output vector containing the mole-fraction based activity
     *            coefficients. length: m_kk.
     */
    void getActivityCoefficients(double* ac) const override;
 
    //! Get the array of non-dimensional molality based activity coefficients at
    //! the current solution temperature, pressure, and solution concentration.
    /*!
     * See Denbigh p. 278 @cite denbigh1981 for a thorough discussion. This method must
     * be overridden in classes which derive from MolalityVPSSTP. This function
     * takes over from the molar-based activity coefficient calculation,
     * getActivityCoefficients(), in derived classes.
     *
     * These molality based activity coefficients are scaled according to the
     * current pH scale. See the Eq3/6 manual for details.
     *
     * Activity coefficients for species k may be altered between scales s1 to
     * s2 using the following formula
     *
     * @f[
     *     \ln(\gamma_k^{s2}) = \ln(\gamma_k^{s1})
     *        + \frac{z_k}{z_j} \left( \ln(\gamma_j^{s2}) - \ln(\gamma_j^{s1}) \right)
     * @f]
     *
     * where j is any one species. For the NBS scale, j is equal to the Cl-
     * species and
     *
     * @f[
     *     \ln(\gamma_{Cl-}^{s2}) = \frac{-A_{\phi} \sqrt{I}}{1.0 + 1.5 \sqrt{I}}
     * @f]
     *
     * @param acMolality Output vector containing the molality based activity
     *                   coefficients. length: m_kk.
     */
    virtual void getMolalityActivityCoefficients(double* acMolality) const;
 
    //! Calculate the osmotic coefficient
    /*!
     * @f[
     *     \phi = \frac{- \ln(a_o)}{\tilde{M}_o \sum_{i \ne o} m_i}
     * @f]
     *
     * Note there are a few of definitions of the osmotic coefficient floating
     * around. We use the one defined in (Activity Coefficients in Electrolyte
     * Solutions, K. S. Pitzer CRC Press, Boca Raton, 1991, p. 85, Eqn. 28).
     * This definition is most clearly related to theoretical calculation.
     *
     *  units = dimensionless
     */
    virtual double osmoticCoefficient() 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().
    //! @{
 
    bool addSpecies(shared_ptr<Species> spec) override;
    void initThermo() override;
 
    //! @}
 
    //! Set the temperature (K), pressure (Pa), and molalities
    //!(gmol kg-1) of the solutes
    /*!
     * @param t           Temperature (K)
     * @param p           Pressure (Pa)
     * @param molalities  Input vector of molalities of the solutes.
     *                    Length: m_kk.
     */
    void setState_TPM(double t, double p, const double* const molalities);
 
    //! Set the temperature (K), pressure (Pa), and molalities.
    /*!
     * @param t           Temperature (K)
     * @param p           Pressure (Pa)
     * @param m           Composition containing the molalities
     */
    void setState_TPM(double t, double p, const Composition& m);
 
    //! Set the temperature (K), pressure (Pa), and molalities.
    /*!
     * @param t           Temperature (K)
     * @param p           Pressure (Pa)
     * @param m           String which gets translated into a composition
     *                    map for the molalities of the solutes.
     */
    void setState_TPM(double t, double p, const string& m);
 
    //! @copydoc ThermoPhase::setState
    /*!
     * Additionally uses the keys `molalities` or `M` to set the molalities.
     */
    void setState(const AnyMap& state) override;
 
    void getdlnActCoeffdlnN(const size_t ld, double* const dlnActCoeffdlnN) override {
        getdlnActCoeffdlnN_numderiv(ld, dlnActCoeffdlnN);
    }
 
    string report(bool show_thermo=true, double threshold=1e-14) const override;
 
protected:
    //! Get the array of unscaled non-dimensional molality based activity
    //! coefficients at the current solution temperature, pressure, and solution
    //! concentration.
    /*!
     * See Denbigh p. 278 @cite denbigh1981 for a thorough discussion. This method must
     * be overridden in classes which derive from MolalityVPSSTP. This function
     * takes over from the molar-based activity coefficient calculation,
     * getActivityCoefficients(), in derived classes.
     *
     * @param acMolality Output vector containing the molality based activity
     *                   coefficients. length: m_kk.
     */
    virtual void getUnscaledMolalityActivityCoefficients(double* acMolality) const;
 
    //! Apply the current phScale to a set of activity Coefficients or
    //! activities
    /*!
     * See the Eq3/6 Manual for a thorough discussion.
     *
     * @param acMolality input/Output vector containing the molality based
     *                   activity coefficients. length: m_kk.
     */
    virtual void applyphScale(double* acMolality) const;
 
private:
    //! Returns the index of the Cl- species.
    /*!
     * The Cl- species is special in the sense that its single ion molality-
     * based activity coefficient is used in the specification of the pH scale
     * for single ions. Therefore, we need to know what species index is Cl-. If
     * the species isn't in the species list then this routine returns -1, and
     * we can't use the NBS pH scale.
     *
     * Right now we use a restrictive interpretation. The species must be named
     * "Cl-". It must consist of exactly one Cl and one E atom.
     */
    size_t findCLMIndex() const;
 
protected:
    //! Scaling to be used for output of single-ion species activity
    //! coefficients.
    /*!
     * Index of the species to be used in the single-ion scaling law. This is
     * the identity of the Cl- species for the PHSCALE_NBS scaling. Either
     * PHSCALE_PITZER or PHSCALE_NBS
     */
    int m_pHScalingType = PHSCALE_PITZER;
 
    //! Index of the phScale species
    /*!
     * Index of the species to be used in the single-ion scaling law. This is
     * the identity of the Cl- species for the PHSCALE_NBS scaling
     */
    size_t m_indexCLM = npos;
 
    //! Molecular weight of the Solvent
    double m_weightSolvent = 18.01528;
 
    /**
     * In any molality implementation, it makes sense to have a minimum solvent
     * mole fraction requirement, since the implementation becomes singular in
     * the xmolSolvent=0 limit. The default is to set it to 0.01. We then modify
     * the molality definition to ensure that molal_solvent = 0 when
     * xmol_solvent = 0.
     */
    double m_xmolSolventMIN = 0.01;
 
    //! This is the multiplication factor that goes inside log expressions
    //! involving the molalities of species. It's equal to Wt_0 / 1000, where
    //! Wt_0 = weight of solvent (kg/kmol)
    double m_Mnaught = 18.01528E-3;
 
    //! Current value of the molalities of the species in the phase. Note this
    //! vector is a mutable quantity. units are (kg/kmol)
    mutable vector<double> m_molalities;
};
 
}
 
#endif