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cantera/include/cantera/thermo/HMWSoln.h

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/**
* @file HMWSoln.h
* Headers for the %HMWSoln ThermoPhase object, which models concentrated
* electrolyte solutions
* (see \ref thermoprops and \link Cantera::HMWSoln HMWSoln \endlink) .
*
* Class %HMWSoln represents a concentrated liquid electrolyte phase which
* obeys the Pitzer formulation for nonideality using molality-based
* standard states.
*/
/*
* Copyright (2006) Sandia Corporation. Under the terms of
* Contract DE-AC04-94AL85000 with Sandia Corporation, the
* U.S. Government retains certain rights in this software.
*/
#ifndef CT_HMWSOLN_H
#define CT_HMWSOLN_H
#include "MolalityVPSSTP.h"
#include "electrolytes.h"
namespace Cantera
{
/**
* Major Parameters:
* The form of the Pitzer expression refers to the
* form of the Gibbs free energy expression. The temperature
* dependence of the Pitzer coefficients are handled by
* another parameter.
*
* m_formPitzer = Form of the Pitzer expression
*
* PITZERFORM_BASE = 0
*
* Only one form is supported atm. This parameter is included for
* future expansion.
*
*/
#define PITZERFORM_BASE 0
/*!
* @name Temperature Dependence of the Pitzer Coefficients
*
* Note, the temperature dependence of the
* Gibbs free energy also depends on the temperature dependence
* of the standard state and the temperature dependence of the
* Debye-Huckel constant, which includes the dielectric constant
* and the density. Therefore, this expression defines only part
* of the temperature dependence for the mixture thermodynamic
* functions.
*
* PITZER_TEMP_CONSTANT
* All coefficients are considered constant wrt temperature
* PITZER_TEMP_LINEAR
* All coefficients are assumed to have a linear dependence
* wrt to temperature.
* PITZER_TEMP_COMPLEX1
* All coefficients are assumed to have a complex functional
* based dependence wrt temperature; See:
* (Silvester, Pitzer, J. Phys. Chem. 81, 19 1822 (1977)).
*
* beta0 = q0 + q3(1/T - 1/Tr) + q4(ln(T/Tr)) +
* q1(T - Tr) + q2(T**2 - Tr**2)
*
*/
//@{
#define PITZER_TEMP_CONSTANT 0
#define PITZER_TEMP_LINEAR 1
#define PITZER_TEMP_COMPLEX1 2
//@}
/*
* @name ways to calculate the value of A_Debye
*
* These defines determine the way A_Debye is calculated
*/
//@{
#define A_DEBYE_CONST 0
#define A_DEBYE_WATER 1
//@}
class WaterProps;
class PDSS_Water;
/**
* Class %HMWSoln represents a dilute or concentrated liquid electrolyte
* phase which obeys the Pitzer formulation for nonideality.
*
* As a prerequisite to the specification of thermodynamic quantities,
* The concentrations of the ionic species are assumed to obey the
* electroneutrality condition.
*
* <HR>
* <H2> Specification of Species Standard %State Properties </H2>
* <HR>
*
* The solvent is assumed to be liquid water. A real model for liquid
* water (IAPWS 1995 formulation) is used as its standard state.
* All standard state properties for the solvent are based on
* this real model for water, and involve function calls
* to the object that handles the real water model, #Cantera::WaterPropsIAPWS.
*
* The standard states for solutes are on the unit molality basis.
* Therefore, in the documentation below, the normal \f$ o \f$
* superscript is replaced with
* the \f$ \triangle \f$ symbol. The reference state symbol is now
* \f$ \triangle, ref \f$.
*
*
* It is assumed that the reference state thermodynamics may be
* obtained by a pointer to a populated species thermodynamic property
* manager class (see ThermoPhase::m_spthermo). How to relate pressure
* changes to the reference state thermodynamics is resolved at this level.
*
* For solutes that rely on ThermoPhase::m_spthermo, are assumed to
* have an incompressible standard state mechanical property.
* In other words, the molar volumes are independent of temperature
* and pressure.
*
* For these incompressible,
* standard states, the molar internal energy is
* independent of pressure. Since the thermodynamic properties
* are specified by giving the standard-state enthalpy, the
* term \f$ P_0 \hat v\f$ is subtracted from the specified molar
* enthalpy to compute the molar internal energy. The entropy is
* assumed to be independent of the pressure.
*
* The enthalpy function is given by the following relation.
*
* \f[
* \raggedright h^\triangle_k(T,P) = h^{\triangle,ref}_k(T)
* + \tilde{v}_k \left( P - P_{ref} \right)
* \f]
*
* For an incompressible,
* stoichiometric substance, the molar internal energy is
* independent of pressure. Since the thermodynamic properties
* are specified by giving the standard-state enthalpy, the
* term \f$ P_{ref} \tilde v\f$ is subtracted from the specified reference molar
* enthalpy to compute the molar internal energy.
*
* \f[
* u^\triangle_k(T,P) = h^{\triangle,ref}_k(T) - P_{ref} \tilde{v}_k
* \f]
*
*
* The solute standard state heat capacity and entropy are independent
* of pressure. The solute standard state gibbs free energy is obtained
* from the enthalpy and entropy functions.
*
* The vector Constituents::m_speciesSize[] is used to hold the
* base values of species sizes. These are defined as the
* molar volumes of species at infinite dilution at 300 K and 1 atm
* of water. m_speciesSize are calculated during the initialization of the
* %HMWSoln object and are then not touched.
*
* The current model assumes that an incompressible molar volume for
* all solutes. The molar volume for the water solvent, however,
* is obtained from a pure water equation of state, waterSS.
* Therefore, the water standard state varies with both T and P.
* It is an error to request standard state water properties at a T and P
* where the water phase is not a stable phase, i.e., beyond its
* spinodal curve.
*
* <HR>
* <H2> Specification of Solution Thermodynamic Properties </H2>
* <HR>
*
* 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$k\f$.
*
* Individual activity coefficients of ions can not be independently measured. Instead,
* only binary pairs forming electroneutral solutions can be measured. This problem
* leads to a redundancy in the evaluation of species standard state properties.
* The redundancy issue is resolved by setting the standard state chemical potential
* enthalpy, entropy, and volume for the hydrogen ion, H+, to zero, for every temperature
* and pressure. After this convention is applied, all other standard state
* properties of ionic species contain meaningfull information.
*
*
* <H3> Ionic Strength </H3>
*
* Most of the parameterizations within the model use the ionic strength
* as a key variable. The ionic strength, \f$ I\f$ is defined as follows
*
* \f[
* I = \frac{1}{2} \sum_k{m_k z_k^2}
* \f]
*
*
* \f$ m_k \f$ is the molality of the kth species. \f$ z_k \f$ is the charge
* of the kth species. Note, the ionic strength is a defined units quantity.
* The molality has defined units of gmol kg-1, and therefore the ionic
* strength has units of sqrt( gmol kg<SUP>-1</SUP>).
*
* In some instances, from some authors, a different
* formulation is used for the ionic strength in the equations below. The different
* formulation is due to the possibility of the existence of weak acids and how
* association wrt to the weak acid equilibrium relation affects the calculation
* of the activity coefficients via the assumed value of the ionic strength.
*
* If we are to assume that the association reaction doesn't have an effect
* on the ionic strength, then we will want to consider the associated weak
* acid as in effect being fully dissociated, when we calculate an effective
* value for the ionic strength. We will call this calculated value, the
* stoichiometric ionic strength, \f$ I_s \f$, putting a subscript s to denote
* it from the more straightforward calculation of \f$ I \f$.
*
* \f[
* I_s = \frac{1}{2} \sum_k{m_k^s z_k^2}
* \f]
*
* Here, \f$ m_k^s \f$ is the value of the molalities calculated assuming that
* all weak acid-base pairs are in their fully dissociated states. This calculation may
* be simplified by considering that the weakly associated acid may be made up of two
* charged species, k1 and k2, each with their own charges, obeying the following relationship:
*
* \f[
* z_k = z_{k1} + z_{k2}
* \f]
* Then, we may only need to specify one charge value, say, \f$ z_{k1}\f$,
* the cation charge number,
* in order to get both numbers, since we have already specified \f$ z_k \f$
* in the definition of original species.
* Then, the stoichiometric ionic strength may be calculated via the following formula.
*
* \f[
* I_s = \frac{1}{2} \left(\sum_{k,ions}{m_k z_k^2}+
* \sum_{k,weak_assoc}(m_k z_{k1}^2 + m_k z_{k2}^2) \right)
* \f]
*
* The specification of which species are weakly associated acids is made in the input
* file via the
* <TT> stoichIsMods </TT> XML block, where the charge for k1 is also specified.
* An example is given below:
*
* @code
* <stoichIsMods>
* NaCl(aq):-1.0
* </stoichIsMods>
* @endcode
*
*
* Because we need the concept of a weakly associated acid in order to calculated
* \f$ I_s \f$ we need to
* catalog all species in the phase. This is done using the following categories:
*
* - <B>cEST_solvent</B> : Solvent species (neutral)
* - <B>cEST_chargedSpecies</B> Charged species (charged)
* - <B>cEST_weakAcidAssociated</B> Species which can break apart into charged species.
* It may or may not be charged. These may or
* may not be be included in the
* species solution vector.
* - <B>cEST_strongAcidAssociated</B> Species which always breaksapart into charged species.
* It may or may not be charged. Normally, these
* aren't included in the speciation vector.
* - <B>cEST_polarNeutral </B> Polar neutral species
* - <B>cEST_nonpolarNeutral</B> Non poloar neutral species
*
* Polar and non-polar neutral species are differentiated, because some additions
* to the activity
* coefficient expressions distinguish between these two types of solutes.
* This is the so-called salt-out effect.
*
* The type of species is specified in the <TT>electrolyteSpeciesType</TT> XML block.
* Note, this is not
* considered a part of the specification of the standard state for the species,
* at this time. Therefore,
* this information is put under the <TT>activityCoefficient</TT> XML block. An example
* is given below
*
* @code
* <electrolyteSpeciesType>
* H2L(L):solvent
* H+:chargedSpecies
* NaOH(aq):weakAcidAssociated
* NaCl(aq):strongAcidAssociated
* NH3(aq):polarNeutral
* O2(aq):nonpolarNeutral
* </electrolyteSpeciesType>
* @endcode
*
*
* Much of the species electrolyte type information is infered from other information in the
* input file. For example, as species which is charged is given the "chargedSpecies" default
* category. A neutral solute species is put into the "nonpolarNeutral" category by default.
*
*
* <H3> Specification of the Excess Gibbs Free Energy </H3>
*
* Pitzer's formulation may best be represented as a specification of the excess gibbs
* free energy, \f$ G^{ex} \f$, defined as the deviation of the total gibbs free energy from
* that of an ideal molal solution.
* \f[
* G = G^{id} + G^{ex}
* \f]
*
* The ideal molal solution contribution, not equal to an ideal solution contribution
* and in fact containing a singularity at the zero solvent mole fraction limit, is
* given below.
* \f[
* G^{id} = n_o \mu^o_o + \sum_{k\ne o} n_k \mu_k^{\triangle}
* + \tilde{M}_o n_o ( RT (\sum{m_i(\ln(m_i)-1)}))
* \f]
*
* From the excess Gibbs free energy formulation, the activity coefficient expression
* and the osmotic coefficient expression for the solvent may be defined, by
* taking the appropriate derivatives. Using this approach garranties that the
* entire system will obey the Gibbs-Duhem relations.
*
* Pitzer employs the following general expression for the excess Gibbs free energy
*
* \f[
* \begin{array}{cclc}
* \frac{G^{ex}}{\tilde{M}_o n_o RT} &= &
* \left( \frac{4A_{Debye}I}{3b} \right) \ln(1 + b \sqrt{I})
* + 2 \sum_c \sum_a m_c m_a B_{ca}
* + \sum_c \sum_a m_c m_a Z C_{ca}
* \\&&
* + \sum_{c < c'} \sum m_c m_{c'} \left[ 2 \Phi_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
* + \sum_{a < a'} \sum m_a m_{a'} \left[ 2 \Phi_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
* \\&&
* + 2 \sum_n \sum_c m_n m_c \lambda_{nc} + 2 \sum_n \sum_a m_n m_a \lambda_{na}
* + 2 \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
* + \sum_n m^2_n \lambda_{nn}
* \end{array}
* \f]
*
* <I>a</I> is a subscript over all anions, <I>c</I> is a subscript extending over all
* cations, and <I>i</I> is a subscript that extends over all anions and cations.
* <I>n</I> is a subscript that extends only over neutral solute molecules.
* The second line contains cross terms where cations affect
* cations and/or cation/anion pairs,
* and anions affect anions or cation/anion pairs. Note part of the coefficients,
* \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ stem from the theory
* of unsymmetrical mixing of electrolytes with different charges. This
* theory depends on the total ionic stregnth of the solution, and therefore,
* \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ will depend on <I>I</I>, the
* ionic strength. \f$ B_{ca}\f$ is a strong function of the
* total ionic strength, <I>I</I>,
* of the electrolyte. The rest of the coefficients are assumed to be independent of the
* molalities or ionic strengths. However, all coefficients are potentially functions
* of the temperature and pressure of the solution.
*
* <I>A</I> is the Debye-Huckel constant. Its specification is described in its own
* section below.
*
* \f$ I\f$ is the ionic strength of the solution, and is given by:
*
* \f[
* I = \frac{1}{2} \sum_k{m_k z_k^2}
* \f]
*
* In contrast to several other Debye-Huckel implementations (see \ref DebyeHuckel), the
* parameter \f$ b\f$ in the above equation is a constant that
* does not vary with respect to ion identity. This is an important simplification
* as it avoids troubles with satisfaction of the Gibbs-Duhem analysis.
*
* The function \f$ Z \f$ is given by
*
* \f[
* Z = \sum_i m_i \left| z_i \right|
* \f]
*
* The value of \f$ B_{ca}\f$ is given by the following function
*
* \f[
* B_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} g(\alpha^{(1)}_{ca} \sqrt{I})
* + \beta^{(2)}_{ca} g(\alpha^{(2)}_{ca} \sqrt{I})
* \f]
*
* where
*
* \f[
* g(x) = 2 \frac{(1 - (1 + x)\exp[-x])}{x^2}
* \f]
*
* The formulation for \f$ B_{ca}\f$ combined with the formulation of the
* Debye-Huckel term in the eqn. for the excess Gibbs free energy stems
* essentially from an empirical fit to the ionic strength dependent data
* based over a wide sampling of binary electrolyte systems. \f$ C_{ca} \f$,
* \f$ \lambda_{nc} \f$, \f$ \lambda_{na} \f$, \f$ \lambda_{nn} \f$,
* \f$ \Psi_{c{c'}a} \f$, \f$ \Psi_{a{a'}c} \f$ are experimentally derived
* coefficients that may have pressure and/or temperature dependencies.
* The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ formulations are
* slightly more complicated. \f$ b \f$ is a universal
* constant defined to be equal to \f$ 1.2\ kg^{1/2}\ gmol^{-1/2} \f$. The exponential
* coefficient \f$ \alpha^{(1)}_{ca} \f$ is usually
* fixed at \f$ \alpha^{(1)}_{ca} = 2.0\ kg^{1/2} gmol^{-1/2}\f$
* except for 2-2 electrolytes, while other parameters were fit to experimental
* data. For 2-2 electrolytes, \f$ \alpha^{(1)}_{ca} = 1.4\ kg^{1/2}\ gmol^{-1/2}\f$
* is used in combination with either \f$ \alpha^{(2)}_{ca} = 12\ kg^{1/2}\ gmol^{-1/2}\f$
* or \f$ \alpha^{(2)}_{ca} = k A_\psi \f$, where <I>k</I> is a constant. For electrolytes other
* than 2-2 electrolytes the \f$ \beta^{(2)}_{ca} g(\alpha^{(2)}_{ca} \sqrt{I}) \f$ term
* is not used in the fitting procedure; it is only used for divalent metal
* solfates and other high-valence electrolytes which exhibit significant
* association at low ionic strengths.
*
* The \f$ \beta^{(0)}_{ca} \f$, \f$ \beta^{(1)}_{ca}\f$, \f$ \beta^{(2)}_{ca} \f$,
* and \f$ C_{ca} \f$ binary coefficients are referred to as ion-interaction or
* Pitzer parameters. These Pitzer parameters may vary with temperature and pressure
* but they do not depend on the ionic strength. Their values and temperature
* derivatives of their values have been tabulated for a range of electrolytes
*
* The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ contributions, which
* capture cation-cation and anion-anion interactions, also have an
* ionic strength dependence.
*
* Ternary contributions \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{a{a'}c} \f$
* have been measured also for some systems. The success of the Pitzer
* method lies in its ability to model nonlinear activity coefficients
* of complex multicomponent systems with just binary and minor
* ternary contributions, which can be independently measured in
* binary or ternary subsystems.
*
*
* <H3> Multicomponent Activity Coefficients for Solutes </H3>
*
* The formulas for activity coefficients of solutes may be obtained by taking the
* following derivative of the excess Gibbs Free Energy formulation described above:
*
* \f[
* \ln(\gamma_k^\triangle) = \frac{d\left( \frac{G^{ex}}{M_o n_o RT} \right)}{d(m_k)}\Bigg|_{n_i}
* \f]
*
* In the formulas below the following conventions are used. The subscript <I>M</I> refers
* to a particular cation. The subscript X refers to a particular anion, whose
* activity is being currently evaluated. the subscript <I>a</I> refers to a summation
* over all anions in the solution, while the subscript <I>c</I> refers to a summation
* over all cations in the solutions.
*
* The activity coefficient for a particular cation <I>M</I> is given by
*
* \f[
* \ln(\gamma_M^\triangle) = -z_M^2(F) + \sum_a m_a \left( 2 B_{Ma} + Z C_{Ma} \right)
* + z_M \left( \sum_a \sum_c m_a m_c C_{ca} \right)
* + \sum_c m_c \left[ 2 \Phi_{Mc} + \sum_a m_a \Psi_{Mca} \right]
* + \sum_{a < a'} \sum m_a m_{a'} \Psi_{Ma{a'}}
* + 2 \sum_n m_n \lambda_{nM}
* \f]
*
* The activity coefficient for a particular anion <I>X</I> is given by
*
* \f[
* \ln(\gamma_X^\triangle) = -z_X^2(F) + \sum_a m_c \left( 2 B_{cX} + Z C_{cX} \right)
* + \left|z_X \right| \left( \sum_a \sum_c m_a m_c C_{ca} \right)
* + \sum_a m_a \left[ 2 \Phi_{Xa} + \sum_c m_c \Psi_{cXa} \right]
* + \sum_{c < c'} \sum m_c m_{c'} \Psi_{c{c'}X}
* + 2 \sum_n m_n \lambda_{nM}
* \f]
* where the function \f$ F \f$ is given by
*
* \f[
* F = - A_{\phi} \left[ \frac{\sqrt{I}}{1 + b \sqrt{I}}
* + \frac{2}{b} \ln{\left(1 + b\sqrt{I}\right)} \right]
* + \sum_a \sum_c m_a m_c B'_{ca}
* + \sum_{c < c'} \sum m_c m_{c'} \Phi'_{c{c'}}
* + \sum_{a < a'} \sum m_a m_{a'} \Phi'_{a{a'}}
* \f]
*
* We have employed the definition of \f$ A_{\phi} \f$, also used by Pitzer
* which is equal to
*
* \f[
* A_{\phi} = \frac{A_{Debye}}{3}
* \f]
*
* In the above formulas, \f$ \Phi'_{c{c'}} \f$ and \f$ \Phi'_{a{a'}} \f$ are the
* ionic strength derivatives of \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$,
* respectively.
*
* The function \f$ B'_{MX} \f$ is defined as:
*
* \f[
* B'_{MX} = \left( \frac{\beta^{(1)}_{MX} h(\alpha^{(1)}_{MX} \sqrt{I})}{I} \right)
* \left( \frac{\beta^{(2)}_{MX} h(\alpha^{(2)}_{MX} \sqrt{I})}{I} \right)
* \f]
*
* where \f$ h(x) \f$ is defined as
*
* \f[
* h(x) = g'(x) \frac{x}{2} =
* \frac{2\left(1 - \left(1 + x + \frac{x^2}{2} \right)\exp(-x) \right)}{x^2}
* \f]
*
* The activity coefficient for neutral species <I>N</I> is given by
*
* \f[
* \ln(\gamma_N^\triangle) = 2 \left( \sum_i m_i \lambda_{iN}\right)
* \f]
*
*
* <H3> Activity of the Water Solvent </H3>
*
* The activity for the solvent water,\f$ a_o \f$, is not independent and must be
* determined either from the Gibbs-Duhem relation or from taking the appropriate derivative
* of the same excess Gibbs free energy function as was used to formulate
* the solvent activity coefficients. Pitzer's description follows the later approach to
* derive a formula for the osmotic coefficient, \f$ \phi \f$.
*
* \f[
* \phi - 1 = - \left( \frac{d\left(\frac{G^{ex}}{RT} \right)}{d(\tilde{M}_o n_o)} \right)
* \frac{1}{\sum_{i \ne 0} m_i}
* \f]
*
* The osmotic coefficient may be related to the water activity by the following relation:
*
* \f[
* \phi = - \frac{1}{\tilde{M}_o \sum_{i \neq o} m_i} \ln(a_o)
* = - \frac{n_o}{\sum_{i \neq o}n_i} \ln(a_o)
* \f]
*
*
* The result is the following
*
* \f[
* \begin{array}{ccclc}
* \phi - 1 &= &
* \frac{2}{\sum_{i \ne 0} m_i}
* \bigg[ &
* - A_{\phi} \frac{I^{3/2}}{1 + b \sqrt{I}}
* + \sum_c \sum_a m_c m_a \left( B^{\phi}_{ca} + Z C_{ca}\right)
* \\&&&
* + \sum_{c < c'} \sum m_c m_{c'} \left[ \Phi^{\phi}_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
* + \sum_{a < a'} \sum m_a m_{a'} \left[ \Phi^{\phi}_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
* \\&&&
* + \sum_n \sum_c m_n m_c \lambda_{nc} + \sum_n \sum_a m_n m_a \lambda_{na}
* + \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
* + \frac{1}{2} \left( \sum_n m^2_n \lambda_{nn}\right)
* \bigg]
* \end{array}
* \f]
*
* It can be shown that the expression
*
*
* \f[
* B^{\phi}_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} \exp{(- \alpha^{(1)}_{ca} \sqrt{I})}
* + \beta^{(2)}_{ca} \exp{(- \alpha^{(2)}_{ca} \sqrt{I} )}
* \f]
*
* is consistent with the expression \f$ B_{ca} \f$ in the \f$ G^{ex} \f$ expression
* after carrying out the derivative wrt \f$ m_M \f$.
*
* Also taking into account that \f$ {\Phi}_{c{c'}} \f$ and
* \f$ {\Phi}_{a{a'}} \f$ has an ionic strength dependence.
*
* \f[
* \Phi^{\phi}_{c{c'}} = {\Phi}_{c{c'}} + I \frac{d{\Phi}_{c{c'}}}{dI}
* \f]
*
* \f[
* \Phi^{\phi}_{a{a'}} = \Phi_{a{a'}} + I \frac{d\Phi_{a{a'}}}{dI}
* \f]
*
*
* <H3> Temperature and Pressure Dependence of the Pitzer Parameters </H3>
*
* In general most of the coefficients introduced in the previous section may
* have a temperature and pressure dependence. The temperature and pressure
* dependence of these coefficients strongly influence the value of the
* excess Enthalpy and excess Volumes of Pitzer solutions. Therefore, these
* are readily measurable quantities.
* HMWSoln provides several
* different methods for putting these dependencies into the coefficients.
* HMWSoln has an implementation described by Silverter and Pitzer (1977),
* which was used to fit experimental data for NaCl over an extensive range,
* below the critical temperature of water.
* They found a temperature funcdtional form for fitting the 3 following
* coefficients that describe the Pitzer parameterization for a single salt
* to be adequate to describe how the excess gibbs free energy values for
* the binary salt changes with respect to temperature.
* The following functional form
* was used to fit the temperature dependence of the Pitzer Coefficients
* for each cation - anion pair, M X.
*
* \f[
* \beta^{(0)}_{MX} = q^{b0}_0
* + q^{b0}_1 \left( T - T_r \right)
* + q^{b0}_2 \left( T^2 - T_r^2 \right)
* + q^{b0}_3 \left( \frac{1}{T} - \frac{1}{T_r}\right)
* + q^{b0}_4 \ln \left( \frac{T}{T_r} \right)
* \f]
* \f[
* \beta^{(1)}_{MX} = q^{b1}_0 + q^{b1}_1 \left( T - T_r \right)
* + q^{b1}_{2} \left( T^2 - T_r^2 \right)
* \f]
* \f[
* C^{\phi}_{MX} = q^{Cphi}_0
* + q^{Cphi}_1 \left( T - T_r \right)
* + q^{Cphi}_2 \left( T^2 - T_r^2 \right)
* + q^{Cphi}_3 \left( \frac{1}{T} - \frac{1}{T_r}\right)
* + q^{Cphi}_4 \ln \left( \frac{T}{T_r} \right)
* \f]
*
* where
*
* \f[
* C^{\phi}_{MX} = 2 {\left| z_M z_X \right|}^{1/2} C_{MX}
* \f]
*
* In later papers, Pitzer has added additional temperature dependencies
* to all of the other remaining second and third order virial coefficients.
* Some of these dependencies are justified and motivated by theory.
* Therefore,
* a formalism wherein all of the coefficients in the base theory have
* temperature dependencies associated with them has been implemented
* within the %HMWSoln object. Much of the formalism, however,
* has been unexercised.
*
* In the %HMWSoln object, the temperature dependence of the Pitzer
* parameters are specified in the following way.
*
* - PIZTER_TEMP_CONSTANT - string name "CONSTANT"
* - Assumes that all coefficients are independent of temperature
* and pressure
* - PIZTER_TEMP_COMPLEX1 - string name "COMPLEX" or "COMPLEX1"
* - Uses the full temperature dependence for the
* \f$\beta^{(0)}_{MX} \f$ (5 coeffs),
* the \f$\beta^{(1)}_{MX} \f$ (3 coeffs),
* and \f$ C^{\phi}_{MX} \f$ (5 coeffs) parameters described above.
* - PITZER_TEMP_LINEAR - string name "LINEAR"
* - Uses just the temperature dependence for the
* \f$\beta^{(0)}_{MX} \f$, the \f$\beta^{(1)}_{MX} \f$,
* and \f$ C^{\phi}_{MX} \f$ coefficients described above.
* There are 2 coefficients for each term.
*
* The temperature dependence is specified in an attributes field
* in the <TT> activityCoefficients </TT> XML block,
* called <TT> TempModel </TT>. Permissible values for that
* attribute are <TT> CONSTANT, COMPLEX1</TT>, and <TT> LINEAR.</TT>
*
* The specification of the binary interaction between a cation and
* an anion is given by the coefficients, \f$ B_{MX}\f$ and
* \f$ C_{MX}\f$
* The specification of \f$ B_{MX}\f$ is a function of
* \f$\beta^{(0)}_{MX} \f$, \f$\beta^{(1)}_{MX} \f$,
* \f$\beta^{(2)}_{MX} \f$, \f$\alpha^{(1)}_{MX} \f$, and
* \f$\alpha^{(2)}_{MX} \f$.
* \f$ C_{MX}\f$ is calculated from \f$C^{\phi}_{MX} \f$
* from the formula above.
* All of the underlying coeficients are specified in the
* XML element block <TT> binarySaltParameters </TT>, which
* has the attribute <TT> cation </TT> and <TT> anion </TT>
* to identify the interaction. XML elements named
* <TT> beta0, beta1, beta2, Cphi, Alpha1, Alpha2 </TT>
* within each <TT> binarySaltParameters </TT> block
* specify the parameters. Within each of these blocks
* multiple parameters describing temperature or pressure
* dependence are serially listed in the order that they
* appear in the equation in this document. An example of
* the <TT> beta0 </TT> block that fits the <TT> COMPLEX1 </TT> temperature
* dependence given above is
*
* @code
<binarySaltParameters cation="Na+" anion="OH-">
<beta0> q0, q1, q2, q3, q4 </beta0>
</binarySaltParameters>
@endcode
*
* The parameters for \f$ \beta^{(0)}\f$ fit the following equation:
*
* \f[
* \beta^{(0)} = q_0^{{\beta}0} + q_1^{{\beta}0} \left( T - T_r \right)
* + q_2^{{\beta}0} \left( T^2 - T_r^2 \right)
* + q_3^{{\beta}0} \left( \frac{1}{T} - \frac{1}{T_r} \right)
* + q_4^{{\beta}0} \ln \left( \frac{T}{T_r} \right)
* \f]
*
* This same COMPLEX1 </TT> temperature
* dependence given above is used for the following parameters:
* \f$ \beta^{(0)}_{MX} \f$, \f$ \beta^{(1)}_{MX} \f$,
* \f$ \beta^{(2)}_{MX} \f$, \f$ \Theta_{cc'} \f$, \f$\Theta_{aa'} \f$,
* \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{ca{a'}} \f$.
*
*
* <H3> Like-Charged Binary Ion Parameters and the Mixing Parameters </H3>
*
* The previous section contained the functions, \f$ \Phi_{c{c'}} \f$,
* \f$ \Phi_{a{a'}} \f$ and their derivatives wrt the
* ionic strength, \f$ \Phi'_{c{c'}} \f$ and \f$ \Phi'_{a{a'}} \f$.
* Part of these terms come from theory.
*
* Since like charged ions repel each other and are generally
* not near each other, the virial coefficients for same-charged ions
* are small. However, Pitzer doesn't ignore these in his
* formulation. Relatively larger and longer range terms between
* like-charged ions exist however, which appear only for
* unsymmetrical mixing of same-sign charged ions with different
* charges. \f$ \Phi_{ij} \f$, where \f$ ij \f$ is either \f$ a{a'} \f$
* or \f$ c{c'} \f$ is given by
*
*
*
* \f[
* {\Phi}_{ij} = \Theta_{ij} + \,^E \Theta_{ij}(I)
* \f]
*
*
* \f$ \Theta_{ij} \f$ is the small virial coefficient expansion term.
* Dependent in general on temperature and pressure, its ionic
* strength dependence is ignored in Pitzer's approach.
* \f$ \,^E\Theta_{ij}(I) \f$ accounts for the electrostatic
* unsymmetrical mixing effects and is dependent only on the
* charges of the ions i, j, the total ionic strength and on
* the dielectric constant and density of the solvent.
* This seems to be a relatively well-documented part of the theory.
* They theory below comes from Pitzer summation (Pitzer) in the
* appendix. It's also mentioned in Bethke's book (Bethke), and
* the equations are summarized in Harvie & Weare (1980).
* Within the code, \f$ \,^E\Theta_{ij}(I) \f$ is evaluated according
* to the algorithm described in Appendix B [Pitzer] as
*
* \f[
* \,^E\Theta_{ij}(I) = \left( \frac{z_i z_j}{4I} \right)
* \left( J(x_{ij}) - \frac{1}{2} J(x_{ii})
* - \frac{1}{2} J(x_{jj}) \right)
* \f]
*
* where \f$ x_{ij} = 6 z_i z_j A_{\phi} \sqrt{I} \f$ and
*
* \f[
* J(x) = \frac{1}{x} \int_0^{\infty}{\left( 1 + q +
* \frac{1}{2} q^2 - e^q \right) y^2 dy}
* \f]
*
* and \f$ q = - (\frac{x}{y}) e^{-y} \f$. \f$ J(x) \f$ is evaluated by
* numerical integration.
*
* The \f$ \Theta_{ij} \f$ term is a constant that is specified
* by the XML element <TT> thetaCation </TT> and
* <TT> thetaAnion </TT>, which
* has the attribute <TT> cation1 </TT>, <TT> cation2 </TT> and
* <TT> anion1 </TT>, <TT> anion2 </TT> respectively
* to identify the interaction. No temperature or
* pressure dependence of this parameter is currently allowed.
* An example of the block is presented below.
*
* @code
<thetaCation cation1="Na+" cation2="H+">
<Theta> 0.036 </Theta>
</thetaCation>
@endcode
*
*
* <H3> Ternary Pitzer Parameters </H3>
*
* The \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{ca{a'}} \f$ terms
* represent ternary interactions between two cations and
* an anion and two anions and a cation, respectively.
* In Pitzer's implementation these terms are usually small
* in absolute size. Currently these parameters do not have
* any dependence on temperature, pressure, or ionic strength.
*
* Their values are input using the XML element
* <TT> psiCommonCation </TT> and <TT> psiCommonAnion </TT>.
* The species id's are specified in attribute fields in
* the XML element. The fields <TT>cation</TT>,
* <TT> anion1</TT>, and <TT> anion2</TT>
* are used for <TT>psiCommonCation</TT>. The fields <TT> anion</TT>,
* <TT>cation1</TT> and <TT>cation2</TT> are used for
* <TT> psiCommonAnion</TT>. An example block is given below.
* The <TT> Theta </TT> field below is a duplicate of the
* <TT> thetaAnion </TT> field mentioned above. The two fields
* are input into the same block for convenience, and because
* their data are highly correlated, in practice.
* It is an error for the
* two blocks to specify different information about
* thetaAnion (or thetaCation) in different blocks. It's
* ok to specify duplicate but consistent information
* in multiple blocks.
*
* @code
<psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
<Theta> -0.05 </Theta>
<Psi> -0.006 </Psi>
</psiCommonCation>
@endcode
*
* <H3> Treatment of Neutral Species </H3>
*
* Binary virial-coefficient-like interactions between two neutral
* species may be specified in the \f$ \lambda_{mn} \f$ terms
* that appear in the formulas above.
* Currently these interactions are independent of temperature,
* pressure, and ionic strength. Also, currently, the neutrality
* of the species are not checked. Therefore, this interaction
* may involve charged species in the solution as well.
* The identity of the species is specified by the
* <TT>species1</TT> and <TT>species2</TT> attributes to the XML
* <TT>lambdaNeutral</TT> node. These terms are symmetrical;
* <TT>species1</TT> and <TT>species2</TT> may be reversed and
* the term will be the same. An example is given below.
*
* @code
<lambdaNeutral species1="CO2" species2="CH4">
<lambda> 0.05 </lambda>
</lambdaNeutral>
@endcode
*
* <H3> Example of the Specification of Parameters for the Activity
* Coefficients </H3>
*
* An example is given below.
*
* An example <TT> activityCoefficients </TT> XML block for this
* formulation is supplied below
*
* @verbatim
<activityCoefficients model="Pitzer" TempModel="complex1">
<!-- Pitzer Coefficients
These coefficients are from Pitzer's main
paper, in his book.
-->
<A_Debye model="water" />
<ionicRadius default="3.042843" units="Angstroms">
</ionicRadius>
<binarySaltParameters cation="Na+" anion="Cl-">
<beta0> 0.0765, 0.008946, -3.3158E-6,
-777.03, -4.4706
</beta0>
<beta1> 0.2664, 6.1608E-5, 1.0715E-6, 0.0, 0.0 </beta1>
<beta2> 0.0, 0.0, 0.0, 0.0, 0.0 </beta2>
<Cphi> 0.00127, -4.655E-5, 0.0,
33.317, 0.09421
</Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<binarySaltParameters cation="H+" anion="Cl-">
<beta0> 0.1775, 0.0, 0.0, 0.0, 0.0 </beta0>
<beta1> 0.2945, 0.0, 0.0, 0.0, 0.0 </beta1>
<beta2> 0.0, 0.0, 0.0, 0.0, 0.0 </beta2>
<Cphi> 0.0008, 0.0, 0.0, 0.0, 0.0 </Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<binarySaltParameters cation="Na+" anion="OH-">
<beta0> 0.0864, 0.0, 0.0, 0.0, 0.0 </beta0>
<beta1> 0.253, 0.0, 0.0 0.0, 0.0 </beta1>
<beta2> 0.0 0.0, 0.0, 0.0, 0.0 </beta2>
<Cphi> 0.0044, 0.0, 0.0, 0.0, 0.0 </Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<thetaAnion anion1="Cl-" anion2="OH-">
<Theta> -0.05, 0.0, 0.0, 0.0, 0.0 </Theta>
</thetaAnion>
<psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
<Theta> -0.05, 0.0, 0.0, 0.0, 0.0 </Theta>
<Psi> -0.006 </Psi>
</psiCommonCation>
<thetaCation cation1="Na+" cation2="H+">
<Theta> 0.036, 0.0, 0.0, 0.0, 0.0 </Theta>
</thetaCation>
<psiCommonAnion anion="Cl-" cation1="Na+" cation2="H+">
<Theta> 0.036, 0.0, 0.0, 0.0, 0.0 </Theta>
<Psi> -0.004 </Psi>
</psiCommonAnion>
</activityCoefficients>
@endverbatim
*
*
* <H3> Specification of the Debye-Huckel Constant </H3>
*
* In the equations above, the formula for \f$ A_{Debye} \f$
* is needed. The %HMWSoln object uses two methods for specifying these quantities.
* The default method is to assume that \f$ A_{Debye} \f$ is a constant, given
* in the initialization process, and stored in the
* member double, m_A_Debye. Optionally, a full water treatment may be employed that makes
* \f$ A_{Debye} \f$ a full function of <I>T</I> and <I>P</I> and creates nontrivial entries for
* the excess heat capacity, enthalpy, and excess volumes of solution.
*
* \f[
* A_{Debye} = \frac{F e B_{Debye}}{8 \pi \epsilon R T} {\left( C_o \tilde{M}_o \right)}^{1/2}
* \f]
* where
*
* \f[
* B_{Debye} = \frac{F} {{(\frac{\epsilon R T}{2})}^{1/2}}
* \f]
* Therefore:
* \f[
* A_{Debye} = \frac{1}{8 \pi}
* {\left(\frac{2 N_a \rho_o}{1000}\right)}^{1/2}
* {\left(\frac{N_a e^2}{\epsilon R T }\right)}^{3/2}
* \f]
*
* Units = sqrt(kg/gmol)
*
* where
* - \f$ N_a \f$ is Avrogadro's number
* - \f$ \rho_w \f$ is the density of water
* - \f$ e \f$ is the electronic charge
* - \f$ \epsilon = K \epsilon_o \f$ is the permitivity of water
* where \f$ K \f$ is the dielectric constant of water,
* and \f$ \epsilon_o \f$ is the permitivity of free space.
* - \f$ \rho_o \f$ is the density of the solvent in its standard state.
*
* Nominal value at 298 K and 1 atm = 1.172576 (kg/gmol)<SUP>1/2</SUP>
* based on:
* - \f$ \epsilon / \epsilon_0 \f$ = 78.54
* (water at 25C)
* - \f$ \epsilon_0 \f$= 8.854187817E-12 C<SUP>2</SUP> N<SUP>-1</SUP> m<SUP>-2</SUP>
* - e = 1.60217653E-19 C
* - F = 9.6485309E7 C kmol<SUP>-1</SUP>
* - R = 8.314472E3 kg m<SUP>2</SUP> s<SUP>-2</SUP> kmol<SUP>-1</SUP> K<SUP>-1</SUP>
* - T = 298.15 K
* - B_Debye = 3.28640E9 (kg/gmol)<SUP>1/2</SUP> m<SUP>-1</SUP>
* - \f$N_a\f$ = 6.0221415E26 kmol<SUP>-1</SUP>
*
* An example of a fixed value implementation is given below.
* @code
* <activityCoefficients model="Pitzer">
* <!-- A_Debye units = sqrt(kg/gmol) -->
* <A_Debye> 1.172576 </A_Debye>
* <!-- object description continues -->
* </activityCoefficients>
* @endcode
*
* An example of a variable value implementation within the %HMWSoln object is given below.
* The model attribute, "water", triggers the full implementation.
*
* @code
* <activityCoefficients model="Pitzer">
* <!-- A_Debye units = sqrt(kg/gmol) -->
* <A_Debye model="water" />
* <!-- object description continues -->
* </activityCoefficients>
* @endcode
*
*
* <H3> Temperature and Pressure Dependence of the Activity Coefficients </H3>
*
* Temperature dependence of the activity coefficients leads to nonzero terms
* for the excess enthalpy and entropy of solution. This means that the
* partial molar enthalpies, entropies, and heat capacities are all
* non-trivial to compute. The following formulas are used.
*
* The partial molar enthalpy, \f$ \bar s_k(T,P) \f$:
*
* \f[
* \bar h_k(T,P) = h^{\triangle}_k(T,P)
* - R T^2 \frac{d \ln(\gamma_k^\triangle)}{dT}
* \f]
* The solvent partial molar enthalpy is equal to
* \f[
* \bar h_o(T,P) = h^{o}_o(T,P) - R T^2 \frac{d \ln(a_o)}{dT}
* = h^{o}_o(T,P)
* + R T^2 (\sum_{k \neq o} m_k) \tilde{M_o} (\frac{d \phi}{dT})
* \f]
*
* The partial molar entropy, \f$ \bar s_k(T,P) \f$:
*
* \f[
* \bar s_k(T,P) = s^{\triangle}_k(T,P)
* - R \ln( \gamma^{\triangle}_k \frac{m_k}{m^{\triangle}}))
* - R T \frac{d \ln(\gamma^{\triangle}_k) }{dT}
* \f]
* \f[
* \bar s_o(T,P) = s^o_o(T,P) - R \ln(a_o)
* - R T \frac{d \ln(a_o)}{dT}
* \f]
*
* The partial molar heat capacity, \f$ C_{p,k}(T,P)\f$:
*
* \f[
* \bar C_{p,k}(T,P) = C^{\triangle}_{p,k}(T,P)
* - 2 R T \frac{d \ln( \gamma^{\triangle}_k)}{dT}
* - R T^2 \frac{d^2 \ln(\gamma^{\triangle}_k) }{{dT}^2}
* \f]
* \f[
* \bar C_{p,o}(T,P) = C^o_{p,o}(T,P)
* - 2 R T \frac{d \ln(a_o)}{dT}
* - R T^2 \frac{d^2 \ln(a_o)}{{dT}^2}
* \f]
*
* The pressure dependence of the activity coefficients leads to non-zero terms
* for the excess Volume of the solution.
* Therefore, the partial molar volumes are functions
* of the pressure derivatives of the activity coefficients.
* \f[
* \bar V_k(T,P) = V^{\triangle}_k(T,P)
* + R T \frac{d \ln(\gamma^{\triangle}_k) }{dP}
* \f]
* \f[
* \bar V_o(T,P) = V^o_o(T,P)
* + R T \frac{d \ln(a_o)}{dP}
* \f]
*
* The majority of work for these functions take place in the internal
* routines that calculate the first and second derivatives of the log
* of the activity coefficients wrt temperature,
* s_update_dlnMolalityActCoeff_dT(), s_update_d2lnMolalityActCoeff_dT2(),
* and the first
* derivative of the log activity coefficients wrt pressure,
* s_update_dlnMolalityActCoeff_dP().
*
* <HR>
* <H2> %Application within %Kinetics Managers </H2>
* <HR>
*
* For the time being, we have set the standard concentration for all solute
* species in
* this phase equal to the default concentration of the solvent at the system temperature
* and pressure multiplied by Mnaught (kg solvent / gmol solvent). The solvent
* standard concentration is just equal to its standard state concentration.
*
*
* This means that the
* kinetics operator essentially works on an generalized concentration basis (kmol / m3),
* with units for the kinetic rate constant specified
* as if all reactants (solvent or solute) are on a concentration basis (kmol /m3).
* The concentration will be modified by the activity coefficients.
*
* For example, a bulk-phase binary reaction between liquid solute species
* <I>j</I> and <I>k</I>, producing
* a new liquid solute species <I>l</I> would have the
* following equation for its rate of progress variable, \f$ R^1 \f$, which has
* units of kmol m-3 s-1.
*
* \f[
* R^1 = k^1 C_j^a C_k^a = k^1 (C^o_o \tilde{M}_o a_j) (C^o_o \tilde{M}_o a_k)
* \f]
*
* where
*
* \f[
* C_j^a = C^o_o \tilde{M}_o a_j \quad and \quad C_k^a = C^o_o \tilde{M}_o a_k
* \f]
*
* \f$ C_j^a \f$ is the activity concentration of species <I>j</I>, and
* \f$ C_k^a \f$ is the activity concentration of species <I>k</I>. \f$ C^o_o \f$
* is the concentration of water at 298 K and 1 atm. \f$ \tilde{M}_o \f$
* has units of kg solvent per gmol solvent and is equal to
*
* \f[
* \tilde{M}_o = \frac{M_o}{1000}
* \f]
*
* \f$ a_j \f$ is
* the activity of species <I>j</I> at the current temperature and pressure
* and concentration of the liquid phase is given by the molality based
* activity coefficient multiplied by the molality of the jth species.
*
* \f[
* a_j = \gamma_j^\triangle m_j = \gamma_j^\triangle \frac{n_j}{\tilde{M}_o n_o}
* \f]
*
* \f$k^1 \f$ has units of m<SUP>3</SUP> kmol<SUP>-1</SUP> s<SUP>-1</SUP>.
*
* Therefore the generalized activity concentration of a solute species has the following form
*
* \f[
* C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
* \f]
*
* The generalized activity concentration of the solvent has the same units, but it's a simpler form
*
* \f[
* C_o^a = C^o_o a_o
* \f]
*
* 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^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} )
* \f]
*
* \f$ K^{o,1} \f$ is the dimensionless form of the equilibrium constant.
*
* \f[
* R^{-1} = k^{-1} C_l^a = k^{-1} (C_o \tilde{M}_o a_l)
* \f]
*
* where
*
* \f[
* k^{-1} = k^1 K^{o,1} C_o \tilde{M}_o
* \f]
*
* \f$ k^{-1} \f$ has units of s<SUP>-1</SUP>.
*
* Note, this treatment may be modified in the future, as events dictate.
*
* <HR>
* <H2> Instantiation of the Class </H2>
* <HR>
*
* The constructor for this phase is now located in the default ThermoFactory
* for %Cantera. The following code snippet may be used to initialize the phase
* using the default construction technique within %Cantera.
*
* @code
* ThermoPhase *HMW = newPhase("HMW_NaCl.xml", "NaCl_electrolyte");
* @endcode
*
*
* A new %HMWSoln object may be created by the following code snippets:
*
* @code
* HMWSoln *HMW = new HMWSoln("HMW_NaCl.xml", "NaCl_electrolyte");
* @endcode
*
* or
*
* @code
* char iFile[80], file_ID[80];
* strcpy(iFile, "HMW_NaCl.xml");
* sprintf(file_ID,"%s#NaCl_electrolyte", iFile);
* XML_Node *xm = get_XML_NameID("phase", file_ID, 0);
* HMWSoln *dh = new HMWSoln(*xm);
* @endcode
*
* or by the following call to importPhase():
*
* @code
* char iFile[80], file_ID[80];
* strcpy(iFile, "HMW_NaCl.xml");
* sprintf(file_ID,"%s#NaCl_electrolyte", iFile);
* XML_Node *xm = get_XML_NameID("phase", file_ID, 0);
* HMWSoln dhphase;
* importPhase(*xm, &dhphase);
* @endcode
*
*
* <HR>
* <H2> XML Example </H2>
* <HR>
*
* The phase model name for this is called StoichSubstance. It must be supplied
* as the model attribute of the thermo XML element entry.
* Within the phase XML block,
* the density of the phase must be specified. An example of an XML file
* this phase is given below.
*
* @verbatim
<phase id="NaCl_electrolyte" dim="3">
<speciesArray datasrc="#species_waterSolution">
H2O(L) Na+ Cl- H+ OH-
</speciesArray>
<state>
<temperature units="K"> 300 </temperature>
<pressure units="Pa">101325.0</pressure>
<soluteMolalities>
Na+:3.0
Cl-:3.0
H+:1.0499E-8
OH-:1.3765E-6
</soluteMolalities>
</state>
<!-- thermo model identifies the inherited class
from ThermoPhase that will handle the thermodynamics.
-->
<thermo model="HMW">
<standardConc model="solvent_volume" />
<activityCoefficients model="Pitzer" TempModel="complex1">
<!-- Pitzer Coefficients
These coefficients are from Pitzer's main
paper, in his book.
-->
<A_Debye model="water" />
<ionicRadius default="3.042843" units="Angstroms">
</ionicRadius>
<binarySaltParameters cation="Na+" anion="Cl-">
<beta0> 0.0765, 0.008946, -3.3158E-6,
-777.03, -4.4706
</beta0>
<beta1> 0.2664, 6.1608E-5, 1.0715E-6 </beta1>
<beta2> 0.0 </beta2>
<Cphi> 0.00127, -4.655E-5, 0.0,
33.317, 0.09421
</Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<binarySaltParameters cation="H+" anion="Cl-">
<beta0> 0.1775, 0.0, 0.0, 0.0, 0.0</beta0>
<beta1> 0.2945, 0.0, 0.0 </beta1>
<beta2> 0.0 </beta2>
<Cphi> 0.0008, 0.0, 0.0, 0.0, 0.0 </Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<binarySaltParameters cation="Na+" anion="OH-">
<beta0> 0.0864, 0.0, 0.0, 0.0, 0.0 </beta0>
<beta1> 0.253, 0.0, 0.0 </beta1>
<beta2> 0.0 </beta2>
<Cphi> 0.0044, 0.0, 0.0, 0.0, 0.0 </Cphi>
<Alpha1> 2.0 </Alpha1>
</binarySaltParameters>
<thetaAnion anion1="Cl-" anion2="OH-">
<Theta> -0.05 </Theta>
</thetaAnion>
<psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
<Theta> -0.05 </Theta>
<Psi> -0.006 </Psi>
</psiCommonCation>
<thetaCation cation1="Na+" cation2="H+">
<Theta> 0.036 </Theta>
</thetaCation>
<psiCommonAnion anion="Cl-" cation1="Na+" cation2="H+">
<Theta> 0.036 </Theta>
<Psi> -0.004 </Psi>
</psiCommonAnion>
</activityCoefficients>
<solvent> H2O(L) </solvent>
</thermo>
<elementArray datasrc="elements.xml"> O H Na Cl </elementArray>
<kinetics model="none" >
</kinetics>
</phase>
@endverbatim
*
*
*
* @ingroup thermoprops
*
*/
class HMWSoln : public MolalityVPSSTP
{
public:
//! Default Constructor
HMWSoln();
//! Construct and initialize an HMWSoln ThermoPhase object
//! directly from an asci 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.
*/
HMWSoln(std::string inputFile, std::string id = "");
//! Construct and initialize an HMWSoln 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)
*/
HMWSoln(XML_Node& phaseRef, std::string id = "");
//! Copy Constructor
/*!
* Copy constructor for the object. Constructed
* object will be a clone of this object, but will
* also own all of its data.
* This is a wrapper around the assignment operator
*
* @param right Object to be copied.
*/
HMWSoln(const HMWSoln& right);
//! Assignment operator
/*!
* Assignment operator for the object. Constructed
* object will be a clone of this object, but will
* also own all of its data.
*
* @param right Object to be copied.
*/
HMWSoln& operator=(const HMWSoln& right);
//! This is a special constructor, used to replicate test problems
//! during the initial verification of the object
/*!
*
*
* test problems:
* 1 = NaCl problem - 5 species -
* the thermo is read in from an XML file
*
* speci molality charge
* Cl- 6.0954 6.0997E+00 -1
* H+ 1.0000E-08 2.1628E-09 1
* Na+ 6.0954E+00 6.0997E+00 1
* OH- 7.5982E-07 1.3977E-06 -1
* HMW_params____beta0MX__beta1MX__beta2MX__CphiMX_____alphaMX__thetaij
* 10
* 1 2 0.1775 0.2945 0.0 0.00080 2.0 0.0
* 1 3 0.0765 0.2664 0.0 0.00127 2.0 0.0
* 1 4 0.0 0.0 0.0 0.0 0.0 -0.050
* 2 3 0.0 0.0 0.0 0.0 0.0 0.036
* 2 4 0.0 0.0 0.0 0.0 0.0 0.0
* 3 4 0.0864 0.253 0.0 0.0044 2.0 0.0
* Triplet_interaction_parameters_psiaa'_or_psicc'
* 2
* 1 2 3 -0.004
* 1 3 4 -0.006
*
* @param testProb Hard -coded test problem to instantiate.
* Current valid values are 1.
*/
HMWSoln(int testProb);
//! Destructor.
virtual ~HMWSoln();
//! Duplicator from the ThermoPhase parent class
/*!
* Given a pointer to a ThermoPhase object, this function will
* duplicate the ThermoPhase object and all underlying structures.
* This is basically a wrapper around the copy constructor.
*
* @return returns a pointer to a ThermoPhase
*/
ThermoPhase* duplMyselfAsThermoPhase() const;
/**
* @name Utilities
* @{
*/
/**
* Equation of state type flag. The base class returns
* zero. Subclasses should define this to return a unique
* non-zero value. Constants defined for this purpose are
* listed in mix_defs.h.
*/
virtual int eosType() const;
/**
* @}
* @name Molar Thermodynamic Properties of the Solution --------------
* @{
*/
/// Molar enthalpy. Units: J/kmol.
/**
* Molar enthalpy of the solution. Units: J/kmol.
* (HKM -> Bump up to Parent object)
*/
virtual doublereal enthalpy_mole() const;
/**
* Excess molar enthalpy of the solution from
* the mixing process. Units: J/ kmol.
*
* Note this is kmol of the total solution.
*/
virtual doublereal relative_enthalpy() const;
/**
* Excess molar enthalpy of the solution from
* the mixing process on a molality basis.
* Units: J/ (kmol add salt).
*
* Note this is kmol of the guessed at salt composition
*/
virtual doublereal relative_molal_enthalpy() const;
/// Molar internal energy. Units: J/kmol.
/**
* Molar internal energy of the solution. Units: J/kmol.
* (HKM -> Bump up to Parent object)
*/
virtual doublereal intEnergy_mole() const;
/// Molar entropy. Units: J/kmol/K.
/**
* Molar entropy of the solution. Units: J/kmol/K.
* For an ideal, constant partial molar volume solution mixture with
* pure species phases which exhibit zero volume expansivity:
* \f[
* \hat s(T, P, X_k) = \sum_k X_k \hat s^0_k(T)
* - \hat R \sum_k X_k log(X_k)
* \f]
* The reference-state pure-species entropies
* \f$ \hat s^0_k(T,p_{ref}) \f$ are computed by the
* species thermodynamic
* property manager. The pure species entropies are independent of
* temperature since the volume expansivities are equal to zero.
* @see SpeciesThermo
*
* (HKM -> Bump up to Parent object)
*/
virtual doublereal entropy_mole() const;
/// Molar Gibbs function. Units: J/kmol.
/*
* (HKM -> Bump up to Parent object)
*/
virtual doublereal gibbs_mole() const;
/// Molar heat capacity at constant pressure. Units: J/kmol/K.
virtual doublereal cp_mole() const;
/// Molar heat capacity at constant volume. Units: J/kmol/K.
/*
* (HKM -> Bump up to Parent object)
*/
virtual doublereal cv_mole() const;
//@}
/// @name Mechanical Equation of State Properties ---------------------
//@{
/**
* In this equation of state implementation, the density is a
* function only of the mole fractions. Therefore, it can't be
* an independent variable. Instead, the pressure is used as the
* independent variable. Functions which try to set the thermodynamic
* state by calling setDensity() may cause an exception to be
* thrown.
*/
/**
* Pressure. Units: Pa.
* For this incompressible system, we return the internally stored
* independent value of the pressure.
*/
virtual doublereal pressure() const;
//! Set the internally stored pressure (Pa) at constant
//! temperature and composition
/*!
* This method sets the pressure within the object.
* The water model is a completely compressible model.
* Also, the dielectric constant is pressure dependent.
*
* @param p input Pressure (Pa)
*
* @todo Implement a variable pressure capability
*/
virtual void setPressure(doublereal p);
protected:
/**
* Calculate the density of the mixture using the partial
* molar volumes and mole fractions as input
*
* The formula for this is
*
* \f[
* \rho = \frac{\sum_k{X_k W_k}}{\sum_k{X_k V_k}}
* \f]
*
* where \f$X_k\f$ are the mole fractions, \f$W_k\f$ are
* the molecular weights, and \f$V_k\f$ are the pure species
* molar volumes.
*
* Note, the basis behind this formula is that in an ideal
* solution the partial molar volumes are equal to the pure
* species molar volumes. We have additionally specified
* in this class that the pure species molar volumes are
* independent of temperature and pressure.
*
* NOTE: This is a non-virtual function, which is not a
* member of the ThermoPhase base class.
*/
void calcDensity();
public:
//! Returns the current value of the density
/*!
* @return value of the density. Units: kg/m^3
*/
virtual doublereal density() const;
//! Set the internally stored density (kg/m^3) of the phase.
/*!
* Overwritten setDensity() function is necessary because of
* the underlying water model.
*
* @todo Now have a compressible ss equation for liquid water.
* Therefore, this phase is compressible. May still
* want to change the independent variable however.
*
* NOTE: This is an overwritten function from the State.h
* class
*
* @param rho Input density (kg/m^3).
*/
void setDensity(const doublereal rho);
//! Set the internally stored molar density (kmol/m^3) for the phase.
/**
* Overwritten setMolarDensity() function is necessary because of the
* underlying water model.
*
* This function will now throw an error condition if the input
* isn't exactly equal to the current molar density.
*
* NOTE: This is a virtual function overwritten from the State.h
* class
*
* @param conc Input molar density (kmol/m^3).
*/
void setMolarDensity(const doublereal conc);
//! Set the temperature (K)
/*!
* Overwritten setTemperature(double) from State.h. This
* function sets the temperature, and makes sure that
* the value propagates to underlying objects, such as
* the water standard state model.
*
* @todo Make State::setTemperature a virtual function
*
* @param temp Temperature in kelvin
*/
virtual void setTemperature(const doublereal temp);
//! Set the temperature (K) and pressure (Pa)
/*!
* Set the temperature and pressure.
*
* @param t Temperature (K)
* @param p Pressure (Pa)
*/
virtual void setState_TP(doublereal t, doublereal p);
/**
* The isothermal compressibility. Units: 1/Pa.
* The isothermal compressibility is defined as
* \f[
* \kappa_T = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T
* \f]
*/
virtual doublereal isothermalCompressibility() const;
/**
* The thermal expansion coefficient. Units: 1/K.
* The thermal expansion coefficient is defined as
*
* \f[
* \beta = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P
* \f]
*/
virtual doublereal thermalExpansionCoeff() const;
/**
* @}
* @name Potential Energy
*
* Species may have an additional potential energy due to the
* presence of external gravitation or electric fields. These
* methods allow specifying a potential energy for individual
* species.
* @{
*/
/**
* @}
* @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 the pressure.
* Activity is assumed to be molality-based here.
* @{
*/
//! This method returns an array of generalized activity concentrations
/*!
* The generalized activity concentrations, \f$ C_k^a\f$, are defined such that
* \f$ a_k = C^a_k / C^0_k, \f$ where \f$ C^0_k \f$
* is a standard concentration
* defined below. These generalized concentrations are used
* by kinetics manager classes to compute the forward and
* reverse rates of elementary reactions.
*
* The generalized activity concentration of a solute species has the following form
*
* \f[
* C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
* \f]
*
* The generalized activity concentration of the solvent has the same units, but it's a simpler form
*
* \f[
* C_o^a = C^o_o a_o
* \f]
*
*
* @param c Array of generalized concentrations. The
* units are kmol m-3 for both the solvent and the solute species
*/
virtual void getActivityConcentrations(doublereal* c) const;
//! Return the standard concentration for the kth species
/*!
* The standard concentration \f$ C^0_k \f$ used to normalize
* the activity (i.e., generalized) concentration for use
*
* We have set the standard concentration for all solute species in
* this phase equal to the default concentration of the solvent at the system temperature
* and pressure multiplied by Mnaught (kg solvent / gmol solvent). The solvent
* standard concentration is just equal to its standard state concentration.
*
* \f[
* C_j^0 = C^o_o \tilde{M}_o \quad and C_o^0 = C^o_o
* \f]
*
* The consequence of this is that the standard concentrations have unequal units
* between the solvent and the solute. However, both the solvent and the solute
* activity concentrations will have the same units of kmol kg<SUP>-3</SUP>.
*
* This means that the
* kinetics operator essentially works on an generalized concentration basis (kmol / m3),
* with units for the kinetic rate constant specified
* as if all reactants (solvent or solute) are on a concentration basis (kmol /m3).
* The concentration will be modified by the activity coefficients.
*
* For example, a bulk-phase binary reaction between liquid solute species
* <I>j</I> and <I>k</I>, producing
* a new liquid solute species <I>l</I> would have the
* following equation for its rate of progress variable, \f$ R^1 \f$, which has
* units of kmol m-3 s-1.
*
* \f[
* R^1 = k^1 C_j^a C_k^a = k^1 (C^o_o \tilde{M}_o a_j) (C^o_o \tilde{M}_o a_k)
* \f]
*
* where
*
* \f[
* C_j^a = C^o_o \tilde{M}_o a_j \quad and \quad C_k^a = C^o_o \tilde{M}_o a_k
* \f]
*
* \f$ C_j^a \f$ is the activity concentration of species <I>j</I>, and
* \f$ C_k^a \f$ is the activity concentration of species <I>k</I>. \f$ C^o_o \f$
* is the concentration of water at 298 K and 1 atm. \f$ \tilde{M}_o \f$
* has units of kg solvent per gmol solvent and is equal to
*
* \f[
* \tilde{M}_o = \frac{M_o}{1000}
* \f]
*
* \f$ a_j \f$ is
* the activity of species <I>j</I> at the current temperature and pressure
* and concentration of the liquid phase is given by the molality based
* activity coefficient multiplied by the molality of the jth species.
*
* \f[
* a_j = \gamma_j^\triangle m_j = \gamma_j^\triangle \frac{n_j}{\tilde{M}_o n_o}
* \f]
*
* \f$k^1 \f$ has units of m<SUP>3</SUP> kmol<SUP>-1</SUP> s<SUP>-1</SUP>.
*
* Therefore the generalized activity concentration of a solute species has the following form
*
* \f[
* C_j^a = C^o_o \frac{\gamma_j^\triangle n_j}{n_o}
* \f]
*
* The generalized activity concentration of the solvent has the same units, but it's a simpler form
*
* \f[
* C_o^a = C^o_o a_o
* \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<SUP>3</SUP> kmol<SUP>-1</SUP>.
*
* @param k Species index
*/
virtual doublereal standardConcentration(size_t k=0) const;
//! Returns the natural logarithm of the standard
//! concentration of the kth species
/*!
* @param k Species index
*/
virtual doublereal logStandardConc(size_t k=0) const;
//! Returns the units of the standard and generalized concentrations.
/*!
* Note they have the same units, as their
* ratio is defined to be equal to the activity of the kth
* species in the solution, which is unitless.
*
* This routine is used in print out applications where the
* units are needed. Usually, MKS units are assumed throughout
* the program and in the XML input files.
*
* The base %ThermoPhase class assigns the default quantities
* of (kmol/m3) for all species.
* Inherited classes are responsible for overriding the default
* values if necessary.
*
* @param uA Output vector containing the units
* uA[0] = kmol units - default = 1
* uA[1] = m units - default = -nDim(), the number of spatial
* dimensions in the Phase class.
* uA[2] = kg units - default = 0;
* uA[3] = Pa(pressure) units - default = 0;
* uA[4] = Temperature units - default = 0;
* uA[5] = time units - default = 0
* @param k species index. Defaults to 0.
* @param sizeUA output int containing the size of the vector.
* Currently, this is equal to 6.
*/
virtual void getUnitsStandardConc(double* uA, int k = 0,
int sizeUA = 6) const;
//! Get the array of non-dimensional activities at
//! the current solution temperature, pressure, and solution concentration.
/*!
*
* We resolve this function at this level by calling
* on the activityConcentration function. However,
* derived classes may want to override this default
* implementation.
*
* (note solvent is on molar scale).
*
* @param ac Output vector of activities. Length: m_kk.
*/
virtual void getActivities(doublereal* ac) const;
//@}
/// @name Partial Molar Properties of the Solution -----------------
//@{
//! Get the species chemical potentials. Units: J/kmol.
/*!
*
* This function returns a vector of chemical potentials of the
* species in solution.
*
* \f[
* \mu_k = \mu^{\triangle}_k(T,P) + R T ln(\gamma_k^{\triangle} m_k)
* \f]
*
* @param mu Output vector of species chemical
* potentials. Length: m_kk. Units: J/kmol
*/
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 coefficent wrt temperature
*
* \f[
* \bar h_k(T,P) = h^{\triangle}_k(T,P)
* - R T^2 \frac{d \ln(\gamma_k^\triangle)}{dT}
* \f]
* The solvent partial molar enthalpy is equal to
* \f[
* \bar h_o(T,P) = h^{o}_o(T,P) - R T^2 \frac{d \ln(a_o)}{dT}
* = h^{o}_o(T,P)
* + R T^2 (\sum_{k \neq o} m_k) \tilde{M_o} (\frac{d \phi}{dT})
* \f]
*
*
* @param hbar Output vector of species partial molar enthalpies.
* Length: m_kk. units are J/kmol.
*/
virtual void getPartialMolarEnthalpies(doublereal* hbar) const;
//! Returns an array of partial molar entropies of the species in the
//! solution. Units: J/kmol/K.
/*!
* Maxwell's equations provide an answer for how calculate this
* (p.215 Smith and Van Ness)
*
* d(chemPot_i)/dT = -sbar_i
*
* For this phase, the partial molar entropies are equal to the
* SS species entropies plus the ideal solution contribution
* plus complicated functions of the
* temperature derivative of the activity coefficents.
*
* \f[
* \bar s_k(T,P) = s^{\triangle}_k(T,P)
* - R \ln( \gamma^{\triangle}_k \frac{m_k}{m^{\triangle}}))
* - R T \frac{d \ln(\gamma^{\triangle}_k) }{dT}
* \f]
* \f[
* \bar s_o(T,P) = s^o_o(T,P) - R \ln(a_o)
* - R T \frac{d \ln(a_o)}{dT}
* \f]
*
* @param sbar Output vector of species partial molar entropies.
* Length = m_kk. units are J/kmol/K.
*/
virtual void getPartialMolarEntropies(doublereal* sbar) const;
//! Return an array of partial molar volumes for the
//! species in the mixture. Units: m^3/kmol.
/*!
* For this solution, the partial molar volumes are functions
* of the pressure derivatives of the activity coefficients.
*
* \f[
* \bar V_k(T,P) = V^{\triangle}_k(T,P)
* + R T \frac{d \ln(\gamma^{\triangle}_k) }{dP}
* \f]
* \f[
* \bar V_o(T,P) = V^o_o(T,P)
* + R T \frac{d \ln(a_o)}{dP}
* \f]
*
* @param vbar Output vector of species partial molar volumes.
* Length = m_kk. units are m^3/kmol.
*/
virtual void getPartialMolarVolumes(doublereal* vbar) const;
//! Return an array of partial molar heat capacities for the
//! species in the mixture. Units: J/kmol/K
/*!
* The following formulas are implemented within the code.
*
* \f[
* \bar C_{p,k}(T,P) = C^{\triangle}_{p,k}(T,P)
* - 2 R T \frac{d \ln( \gamma^{\triangle}_k)}{dT}
* - R T^2 \frac{d^2 \ln(\gamma^{\triangle}_k) }{{dT}^2}
* \f]
* \f[
* \bar C_{p,o}(T,P) = C^o_{p,o}(T,P)
* - 2 R T \frac{d \ln(a_o)}{dT}
* - R T^2 \frac{d^2 \ln(a_o)}{{dT}^2}
* \f]
*
*
* @param cpbar Output vector of species partial molar heat
* capacities at constant pressure.
* Length = m_kk. units are J/kmol/K.
*/
virtual void getPartialMolarCp(doublereal* cpbar) const;
//@}
protected:
//! Updates the standard state thermodynamic functions at the current T and P of the solution.
/*!
* @internal
*
* This function gets called for every call to a public function in this
* class. It checks to see whether the temperature or pressure has changed and
* thus whether the ss thermodynamics functions must be recalculated.
*
* @param pres Pressure at which to evaluate the standard states.
* The default, indicated by a -1.0, is to use the current pressure
*/
//virtual void _updateStandardStateThermo() const;
//@}
/// @name Thermodynamic Values for the Species Reference States ---
//@{
///////////////////////////////////////////////////////
//
// The methods below are not virtual, and should not
// be overloaded.
//
//////////////////////////////////////////////////////
/**
* @name Specific Properties
* @{
*/
/**
* @name Setting the State
*
* These methods set all or part of the thermodynamic
* state.
* @{
*/
//@}
/**
* @name Chemical Equilibrium
* Chemical equilibrium.
* @{
*/
public:
//!This method is used by the ChemEquil equilibrium solver.
/*!
* It sets the state such that the chemical potentials satisfy
* \f[ \frac{\mu_k}{\hat R T} = \sum_m A_{k,m}
* \left(\frac{\lambda_m} {\hat R T}\right) \f] where
* \f$ \lambda_m \f$ is the element potential of element m. The
* temperature is unchanged. Any phase (ideal or not) that
* implements this method can be equilibrated by ChemEquil.
*
* @param lambda_RT Input vector of dimensionless element potentials
* The length is equal to nElements().
*/
virtual void setToEquilState(const doublereal* lambda_RT) {
updateStandardStateThermo();
err("setToEquilState");
}
//@}
//! Set the equation of state parameters
/*!
* @internal
* The number and meaning of these depends on the subclass.
*
* @param n number of parameters
* @param c array of \a n coefficients
*/
virtual void setParameters(int n, doublereal* const c);
//! Get the equation of state parameters in a vector
/*!
* @internal
* The number and meaning of these depends on the subclass.
*
* @param n number of parameters
* @param c array of \a n coefficients
*/
virtual void getParameters(int& n, doublereal* const c) const;
//! Set equation of state parameter values from XML
//! entries.
/*!
* This method is called by function importPhase in
* file importCTML.cpp when processing a phase definition in
* an input file. It should be overloaded in subclasses to set
* any parameters that are specific to that particular phase
* model.
*
* @param eosdata An XML_Node object corresponding to
* the "thermo" entry for this phase in the input file.
*/
virtual void setParametersFromXML(const XML_Node& eosdata);
//---------------------------------------------------------
/// @name Critical state properties.
/// These methods are only implemented by some subclasses.
//@{
/// Critical temperature (K).
virtual doublereal critTemperature() const {
err("critTemperature");
return -1.0;
}
/// Critical pressure (Pa).
virtual doublereal critPressure() const {
err("critPressure");
return -1.0;
}
/// Critical density (kg/m3).
virtual doublereal critDensity() const {
err("critDensity");
return -1.0;
}
//@}
/// @name Saturation properties.
/// These methods are only implemented by subclasses that
/// implement full liquid-vapor equations of state.
///
virtual doublereal satTemperature(doublereal p) const {
err("satTemperature");
return -1.0;
}
//! Get the saturation pressure for a given temperature.
/*!
* Note the limitations of this function. Stability considerations
* concernting multiphase equilibrium are ignored in this
* calculation. Therefore, the call is made directly to the SS of
* water underneath. The object is put back into its original
* state at the end of the call.
*
* @todo This is probably not implemented correctly. The stability
* of the salt should be added into this calculation. The
* underlying water model may be called to get the stability
* of the pure water solution, if needed.
*
* @param T Temperature (kelvin)
*/
virtual doublereal satPressure(doublereal T) const;
virtual doublereal vaporFraction() const {
err("vaprFraction");
return -1.0;
}
virtual void setState_Tsat(doublereal t, doublereal x) {
err("setState_sat");
}
virtual void setState_Psat(doublereal p, doublereal x) {
err("setState_sat");
}
//@}
/*
* -------------- Utilities -------------------------------
*/
//! Initialization of a HMWSoln phase using an xml file
/*!
* This routine is a precursor to initThermo(XML_Node*)
* routine, which does most of the work.
*
* @param inputFile XML file containing the description of the
* phase
*
* @param id Optional parameter identifying the name of the
* phase. If none is given, the first XML
* phase element will be used.
*/
void constructPhaseFile(std::string inputFile, std::string id);
//! Import and initialize a HMWSoln phase
//! specification in an XML tree into the current object.
/*!
* Here we read an XML description of the phase.
* We import descriptions of the elements that make up the
* species in a phase.
* We import information about the species, including their
* reference state thermodynamic polynomials. We then freeze
* the state of the species.
*
* Then, we read the species molar volumes from the xml
* tree to finish the initialization.
*
* @param phaseNode This object must be the phase node of a
* complete XML tree
* description of the phase, including all of the
* species data. In other words while "phase" must
* point to an XML phase object, it must have
* sibling nodes "speciesData" that describe
* the species in the phase.
*
* @param id ID of the phase. If nonnull, a check is done
* to see if phaseNode is pointing to the phase
* with the correct id.
*/
void constructPhaseXML(XML_Node& phaseNode, std::string id);
//! Internal initialization required after all species have
//! been added
/*!
* @internal Initialize. This method is provided to allow
* subclasses to perform any initialization required after all
* species have been added. For example, it might be used to
* resize internal work arrays that must have an entry for
* each species. The base class implementation does nothing,
* and subclasses that do not require initialization do not
* need to overload this method. When importing a CTML phase
* description, this method is called just prior to returning
* from function importPhase.
*
* @see importCTML.cpp
*/
virtual void initThermo();
//! Initialize the phase parameters from an XML file.
/*!
* initThermoXML() (virtual from ThermoPhase)
*
* This gets called from importPhase(). It processes the XML file
* after the species are set up. This is the main routine for
* reading in activity coefficient parameters.
*
* @param phaseNode This object must be the phase node of a
* complete XML tree
* description of the phase, including all of the
* species data. In other words while "phase" must
* point to an XML phase object, it must have
* sibling nodes "speciesData" that describe
* the species in the phase.
* @param id ID of the phase. If nonnull, a check is done
* to see if phaseNode is pointing to the phase
* with the correct id.
*/
virtual void initThermoXML(XML_Node& phaseNode, std::string id);
//! Report the molar volume of species k
/*!
*
* units - \f$ m^3 kmol^-1 \f$
*
* @param k species index
*
* @deprecated
* The getPartialMolarVolumes() expression is more precise.
*/
double speciesMolarVolume(int k) const;
//! Value of the Debye Huckel constant as a function of temperature
//! and pressure.
/*!
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
*
* Units = sqrt(kg/gmol)
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
virtual double A_Debye_TP(double temperature = -1.0,
double pressure = -1.0) const;
//! Value of the derivative of the Debye Huckel constant with
//! respect to temperature as a function of temperature
//! and pressure.
/*!
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
*
* Units = sqrt(kg/gmol)
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
virtual double dA_DebyedT_TP(double temperature = -1.0,
double pressure = -1.0) const;
/**
* Value of the derivative of the Debye Huckel constant with
* respect to pressure, as a function of temperature
* and pressure.
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
*
* Units = sqrt(kg/gmol)
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
virtual double dA_DebyedP_TP(double temperature = -1.0,
double pressure = -1.0) const;
/**
* Return Pitzer's definition of A_L. This is basically the
* derivative of the A_phi multiplied by 4 R T**2
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
* dA_phidT = d(A_Debye)/dT / 3.0
* A_L = dA_phidT * (4 * R * T * T)
*
* Units = sqrt(kg/gmol) (RT)
*
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
double ADebye_L(double temperature = -1.0,
double pressure = -1.0) const;
/**
* Return Pitzer's definition of A_J. This is basically the
* temperature derivative of A_L, and the second derivative
* of A_phi
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
* dA_phidT = d(A_Debye)/dT / 3.0
* A_J = 2 A_L/T + 4 * R * T * T * d2(A_phi)/dT2
*
* Units = sqrt(kg/gmol) (R)
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
double ADebye_J(double temperature = -1.0,
double pressure = -1.0) const;
/**
* Return Pitzer's definition of A_V. This is the
* derivative wrt pressure of A_phi multiplied by - 4 R T
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
* dA_phidT = d(A_Debye)/dP / 3.0
* A_V = - dA_phidP * (4 * R * T)
*
* Units = sqrt(kg/gmol) (RT) / Pascal
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*
*/
double ADebye_V(double temperature = -1.0,
double pressure = -1.0) const;
//! Value of the 2nd derivative of the Debye Huckel constant with
//! respect to temperature as a function of temperature
//! and pressure.
/*!
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
*
* Units = sqrt(kg/gmol)
*
* @param temperature Temperature of the derivative calculation
* or -1 to indicate the current temperature
*
* @param pressure Pressure of the derivative calcualtion
* or -1 to indicate the current pressure
*/
virtual double d2A_DebyedT2_TP(double temperature = -1.0,
double pressure = -1.0) const;
//! Reports the ionic radius of the kth species
/*!
* @param k Species index
*/
double AionicRadius(int k = 0) const;
/**
*
* formPitzer():
*
* Returns the form of the Pitzer parameterization used
*/
int formPitzer() const {
return m_formPitzer;
}
/**
* Print out all of the input coefficients.
*/
void printCoeffs() const;
//! Get the array of unscaled non-dimensional molality based
//! activity coefficients at the current solution temperature,
//! pressure, and solution concentration.
/*!
* See Denbigh p. 278 for a thorough discussion. This class must be overwritten 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.
*/
void getUnscaledMolalityActivityCoefficients(doublereal* acMolality) const;
private:
//! Apply the current phScale to a set of activity Coefficients
/*!
* See the Eq3/6 Manual for a thorough discussion.
*
*/
void s_updateScaling_pHScaling() const;
//! Apply the current phScale to a set of derivatives of the activity Coefficients
//! wrt temperature
/*!
* See the Eq3/6 Manual for a thorough discussion of the need
*/
void s_updateScaling_pHScaling_dT() const;
//! Apply the current phScale to a set of 2nd derivatives of the activity Coefficients
//! wrt temperature
/*!
* See the Eq3/6 Manual for a thorough discussion of the need
*/
void s_updateScaling_pHScaling_dT2() const;
//! Apply the current phScale to a set of derivatives of the activity Coefficients
//! wrt pressure
/*!
* See the Eq3/6 Manual for a thorough discussion of the need
*/
void s_updateScaling_pHScaling_dP() const;
//! Calculate the Chlorine activity coefficient on the NBS scale
/*!
* We assume here that the m_IionicMolality variable is up to date.
*/
doublereal s_NBS_CLM_lnMolalityActCoeff() const;
//! Calculate the temperature derivative of the Chlorine activity coefficient
//! on the NBS scale
/*!
* We assume here that the m_IionicMolality variable is up to date.
*/
doublereal s_NBS_CLM_dlnMolalityActCoeff_dT() const;
//! Calculate the second temperature derivative of the Chlorine activity coefficient
//! on the NBS scale
/*!
* We assume here that the m_IionicMolality variable is up to date.
*/
doublereal s_NBS_CLM_d2lnMolalityActCoeff_dT2() const;
//! Calculate the pressure derivative of the Chlorine activity coefficient
/*!
* We assume here that the m_IionicMolality variable is up to date.
*/
doublereal s_NBS_CLM_dlnMolalityActCoeff_dP() const;
//@}
private:
/**
* This is the form of the Pitzer parameterization
* used in this model.
* The options are described at the top of this document,
* and in the general documentation.
* The list is repeated here:
*
* PITZERFORM_BASE = 0 (only one supported atm)
*
*/
int m_formPitzer;
/**
* This is the form of the temperature dependence of Pitzer
* parameterization used in the model.
*
* PITZER_TEMP_CONSTANT 0
* PITZER_TEMP_LINEAR 1
* PITZER_TEMP_COMPLEX1 2
*/
int m_formPitzerTemp;
/**
* Format for the generalized concentration:
*
* 0 = unity
* 1 = molar_volume
* 2 = solvent_volume (default)
*
* The generalized concentrations can have three different forms
* depending on the value of the member attribute m_formGC, which
* is supplied in the constructor.
* <TABLE>
* <TR><TD> m_formGC </TD><TD> GeneralizedConc </TD><TD> StandardConc </TD></TR>
* <TR><TD> 0 </TD><TD> X_k </TD><TD> 1.0 </TD></TR>
* <TR><TD> 1 </TD><TD> X_k / V_k </TD><TD> 1.0 / V_k </TD></TR>
* <TR><TD> 2 </TD><TD> X_k / V_N </TD><TD> 1.0 / V_N </TD></TR>
* </TABLE>
*
* The value and form of the generalized concentration will affect
* reaction rate constants involving species in this phase.
*
* (HKM Note: Using option #1 may lead to spurious results and
* has been included only with warnings. The reason is that it
* molar volumes of electrolytes may often be negative. The
* molar volume of H+ is defined to be zero too. Either options
* 0 or 2 are the appropriate choice. Option 0 leads to
* bulk reaction rate constants which have units of s-1.
* Option 2 leads to bulk reaction rate constants for
* bimolecular rxns which have units of m-3 kmol-1 s-1.)
*/
int m_formGC;
//! Vector containing the electrolyte species type
/*!
* The possible types are:
* - solvent
* - Charged Species
* - weakAcidAssociated
* - strongAcidAssociated
* - polarNeutral
* - nonpolarNeutral .
*/
vector_int m_electrolyteSpeciesType;
/**
* Species molar volumes \f$ m^3 kmol^-1 \f$
* -> m_speciesSize in Constituents.h
*/
//array_fp m_speciesMolarVolume;
/**
* a_k = Size of the ionic species in the DH formulation
* units = meters
*/
array_fp m_Aionic;
/**
* Current value of the ionic strength on the molality scale
* Associated Salts, if present in the mechanism,
* don't contribute to the value of the ionic strength
* in this version of the Ionic strength.
*/
mutable double m_IionicMolality;
/**
* Maximum value of the ionic strength allowed in the
* calculation of the activity coefficients.
*/
double m_maxIionicStrength;
/**
* Reference Temperature for the Pitzer formulations.
*/
double m_TempPitzerRef;
/**
* Stoichiometric ionic strength on the molality scale.
* This differs from m_IionicMolality in the sense that
* associated salts are treated as unassociated salts,
* when calculating the Ionic strength by this method.
*/
mutable double m_IionicMolalityStoich;
public:
/**
* Form of the constant outside the Debye-Huckel term
* called A. It's normally a function of temperature
* and pressure. However, it can be set from the
* input file in order to aid in numerical comparisons.
* Acceptable forms:
*
* A_DEBYE_CONST 0
* A_DEBYE_WATER 1
*
* The A_DEBYE_WATER form may be used for water solvents
* with needs to cover varying temperatures and pressures.
* Note, the dielectric constant of water is a relatively
* strong function of T, and its variability must be
* accounted for,
*/
int m_form_A_Debye;
private:
/**
* A_Debye -> this expression appears on the top of the
* ln actCoeff term in the general Debye-Huckel
* expression
* It depends on temperature. And, therefore,
* most be recalculated whenever T or P changes.
* This variable is a local copy of the calculation.
*
* A_Debye = (F e B_Debye) / (8 Pi epsilon R T)
*
* where B_Debye = F / sqrt(epsilon R T/2)
* (dw/1000)^(1/2)
*
* A_Debye = (1/ (8 Pi)) (2 Na * dw/1000)^(1/2)
* (e * e / (epsilon * kb * T))^(3/2)
*
* Units = sqrt(kg/gmol)
*
* Nominal value = 1.172576 sqrt(kg/gmol)
* based on:
* epsilon/epsilon_0 = 78.54
* (water at 25C)
* epsilon_0 = 8.854187817E-12 C2 N-1 m-2
* e = 1.60217653 E-19 C
* F = 9.6485309E7 C kmol-1
* R = 8.314472E3 kg m2 s-2 kmol-1 K-1
* T = 298.15 K
* B_Debye = 3.28640E9 sqrt(kg/gmol)/m
* dw = C_0 * M_0 (density of water) (kg/m3)
* = 1.0E3 at 25C
*/
mutable double m_A_Debye;
//! Water standard state calculator
/*!
* derived from the equation of state for water.
*/
PDSS* m_waterSS;
//! density of standard-state water
/*!
* internal temporary variable
*/
double m_densWaterSS;
/**
* Pointer to the water property calculator
*/
WaterProps* m_waterProps;
/**
* Vector containing the species reference exp(-G/RT) functions
* at T = m_tlast
*/
mutable vector_fp m_expg0_RT;
/**
* Vector of potential energies for the species.
*/
mutable vector_fp m_pe;
/**
* Temporary array used in equilibrium calculations
*/
mutable vector_fp m_pp;
/**
* vector of size m_kk, used as a temporary holding area.
*/
mutable vector_fp m_tmpV;
/**
* Stoichiometric species charge -> This is for calculations
* of the ionic strength which ignore ion-ion pairing into
* neutral molecules. The Stoichiometric species charge is the
* charge of one of the ion that would occur if the species broke
* into two charged ion pairs.
* NaCl -> m_speciesCharge_Stoich = -1;
* HSO4- -> H+ + SO42- = -2
* -> The other charge is calculated.
* For species that aren't ion pairs, its equal to the
* m_speciesCharge[] value.
*/
vector_fp m_speciesCharge_Stoich;
/**
* Array of 2D data used in the Pitzer/HMW formulation.
* Beta0_ij[i][j] is the value of the Beta0 coefficient
* for the ij salt. It will be nonzero iff i and j are
* both charged and have opposite sign. The array is also
* symmetric.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*/
mutable vector_fp m_Beta0MX_ij;
//! Derivative of Beta0_ij[i][j] wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta0MX_ij_L;
//! Derivative of Beta0_ij[i][j] wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta0MX_ij_LL;
//! Derivative of Beta0_ij[i][j] wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta0MX_ij_P;
//! Array of coefficients for Beta0, a variable in Pitzer's papers
/*!
* column index is counterIJ
* m_Beta0MX_ij_coeff.ptrColumn(counterIJ) is a double* containing
* the vector of coefficients for the counterIJ interaction.
*/
mutable Array2D m_Beta0MX_ij_coeff;
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Beta1_ij[i][j] is the value of the Beta1 coefficient
* for the ij salt. It will be nonzero iff i and j are
* both charged and have opposite sign. The array is also
* symmetric.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*/
mutable vector_fp m_Beta1MX_ij;
//! Derivative of Beta1_ij[i][j] wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta1MX_ij_L;
//! Derivative of Beta1_ij[i][j] wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta1MX_ij_LL;
//! Derivative of Beta1_ij[i][j] wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta1MX_ij_P;
//! Array of coefficients for Beta1, a variable in Pitzer's papers
/*!
* column index is counterIJ
* m_Beta1MX_ij_coeff.ptrColumn(counterIJ) is a double* containing
* the vector of coefficients for the counterIJ interaction.
*/
mutable Array2D m_Beta1MX_ij_coeff;
/**
* Array of 2D data used in the Pitzer/HMW formulation.
* Beta2_ij[i][j] is the value of the Beta2 coefficient
* for the ij salt. It will be nonzero iff i and j are
* both charged and have opposite sign, and i and j
* both have charges of 2 or more. The array is also
* symmetric.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*/
mutable vector_fp m_Beta2MX_ij;
//! Derivative of Beta2_ij[i][j] wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta2MX_ij_L;
//! Derivative of Beta2_ij[i][j] wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta2MX_ij_LL;
//! Derivative of Beta2_ij[i][j] wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Beta2MX_ij_P;
//! Array of coefficients for Beta2, a variable in Pitzer's papers
/*!
* column index is counterIJ
* m_Beta2MX_ij_coeff.ptrColumn(counterIJ) is a double* containing
* the vector of coefficients for the counterIJ interaction.
* This was added for the YMP database version of the code since it
* contains temperature-dependent parameters for some 2-2 electrolytes.
*/
mutable Array2D m_Beta2MX_ij_coeff;
/**
* Array of 2D data used in the Pitzer/HMW formulation.
* Alpha1MX_ij[i][j] is the value of the alpha1 coefficient
* for the ij interaction. It will be nonzero iff i and j are
* both charged and have opposite sign.
* It is symmetric wrt i, j.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*/
vector_fp m_Alpha1MX_ij;
/**
* Array of 2D data used in the Pitzer/HMW formulation.
* Alpha2MX_ij[i][j] is the value of the alpha2 coefficient
* for the ij interaction. It will be nonzero iff i and j are
* both charged and have opposite sign, and i and j
* both have charges of 2 or more, usually.
* It is symmetric wrt i, j.
* counterIJ, where counterIJ = m_counterIJ[i][j],
* is used to access this array.
*/
vector_fp m_Alpha2MX_ij;
/**
* Array of 2D data used in the Pitzer/HMW formulation.
* CphiMX_ij[i][j] is the value of the Cphi coefficient
* for the ij interaction. It will be nonzero iff i and j are
* both charged and have opposite sign, and i and j
* both have charges of 2 or more. The array is also
* symmetric.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*/
mutable vector_fp m_CphiMX_ij;
//! Derivative of Cphi_ij[i][j] wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CphiMX_ij_L;
//! Derivative of Cphi_ij[i][j] wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CphiMX_ij_LL;
//! Derivative of Cphi_ij[i][j] wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CphiMX_ij_P;
//! Array of coefficients for CphiMX, a parameter in the activity
//! coefficient formulation
/*!
* Column index is counterIJ
* m_CphiMX_ij_coeff.ptrColumn(counterIJ) is a double* containing
* the vector of coefficients for the counterIJ interaction.
*/
mutable Array2D m_CphiMX_ij_coeff;
//! Array of 2D data for Theta_ij[i][j] in the Pitzer/HMW formulation.
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Theta_ij[i][j] is the value of the theta coefficient
* for the ij interaction. It will be nonzero for charged
* ions with the same sign. It is symmetric.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*
* HKM Recent Pitzer papers have used a functional form
* for Theta_ij, which depends on the ionic strength.
*/
mutable vector_fp m_Theta_ij;
//! Derivative of Theta_ij[i][j] wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Theta_ij_L;
//! Derivative of Theta_ij[i][j] wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Theta_ij_LL;
//! Derivative of Theta_ij[i][j] wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Theta_ij_P;
//! Array of coefficients for Theta_ij[i][j] in the Pitzer/HMW formulation.
/*!
* Theta_ij[i][j] is the value of the theta coefficient
* for the ij interaction. It will be nonzero for charged
* ions with the same sign. It is symmetric.
* Column index is counterIJ.
* counterIJ where counterIJ = m_counterIJ[i][j]
* is used to access this array.
*
* m_Theta_ij_coeff.ptrColumn(counterIJ) is a double* containing
* the vector of coefficients for the counterIJ interaction.
*/
Array2D m_Theta_ij_coeff;
//! Array of 3D data used in the Pitzer/HMW formulation.
/*!
* Psi_ijk[n] is the value of the psi coefficient for the
* ijk interaction where
*
* n = k + j * m_kk + i * m_kk * m_kk;
*
* It is potentially nonzero everywhere.
* The first two coordinates are symmetric wrt cations,
* and the last two coordinates are symmetric wrt anions.
*/
mutable vector_fp m_Psi_ijk;
//! Derivitive of Psi_ijk[n] wrt T
/*!
* see m_Psi_ijk for reference on the indexing into this variable.
*/
mutable vector_fp m_Psi_ijk_L;
//! Derivitive of Psi_ijk[n] wrt TT
/*!
* see m_Psi_ijk for reference on the indexing into this variable.
*/
mutable vector_fp m_Psi_ijk_LL;
//! Derivitive of Psi_ijk[n] wrt P
/*!
* see m_Psi_ijk for reference on the indexing into this variable.
*/
mutable vector_fp m_Psi_ijk_P;
//! Array of coefficients for Psi_ijk[n] in the Pitzer/HMW formulation.
/*!
* Psi_ijk[n] is the value of the psi coefficient for the
* ijk interaction where
*
* n = k + j * m_kk + i * m_kk * m_kk;
*
* It is potentially nonzero everywhere.
* The first two coordinates are symmetric wrt cations,
* and the last two coordinates are symmetric wrt anions.
*
*
* m_Psi_ijk_coeff.ptrColumn(n) is a double* containing
* the vector of coefficients for the n interaction.
*/
Array2D m_Psi_ijk_coeff;
//! Lambda coefficient for the ij interaction
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Lambda_nj[n][j] represents the lambda coefficient for the
* ij interaction. This is a general interaction representing
* neutral species. The neutral species occupy the first
* index, i.e., n. The charged species occupy the j coordinate.
* neutral, neutral interactions are also included here.
*/
mutable Array2D m_Lambda_nj;
//! Derivative of Lambda_nj[i][j] wrt T. see m_Lambda_ij
mutable Array2D m_Lambda_nj_L;
//! Derivative of Lambda_nj[i][j] wrt TT
mutable Array2D m_Lambda_nj_LL;
//! Derivative of Lambda_nj[i][j] wrt P
mutable Array2D m_Lambda_nj_P;
//! Array of coefficients for Lambda_nj[i][j] in the Pitzer/HMW formulation.
/*!
* Lambda_ij[i][j] is the value of the theta coefficient
* for the ij interaction.
* Array of 2D data used in the Pitzer/HMW formulation.
* Lambda_ij[i][j] represents the lambda coefficient for the
* ij interaction. This is a general interaction representing
* neutral species. The neutral species occupy the first
* index, i.e., i. The charged species occupy the j coordinate.
* Neutral, neutral interactions are also included here.
*
* n = j + m_kk * i
*
* m_Lambda_ij_coeff.ptrColumn(n) is a double* containing
* the vector of coefficients for the (i,j) interaction.
*/
Array2D m_Lambda_nj_coeff;
//! Mu coefficient for the self-ternary neutral coefficient
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Mu_nnn[i] represents the Mu coefficient for the
* nnn interaction. This is a general interaction representing
* neutral species interacting with itself.
*/
mutable vector_fp m_Mu_nnn;
//! Mu coefficient temperature derivative for the self-ternary neutral coefficient
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Mu_nnn_L[i] represents the Mu coefficient temperature derivative for the
* nnn interaction. This is a general interaction representing
* neutral species interacting with itself.
*/
mutable vector_fp m_Mu_nnn_L;
//! Mu coefficient 2nd temperature derivative for the self-ternary neutral coefficient
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Mu_nnn_L[i] represents the Mu coefficient 2nd temperature derivative for the
* nnn interaction. This is a general interaction representing
* neutral species interacting with itself.
*/
mutable vector_fp m_Mu_nnn_LL;
//! Mu coefficient pressure derivative for the self-ternary neutral coefficient
/*!
* Array of 2D data used in the Pitzer/HMW formulation.
* Mu_nnn_L[i] represents the Mu coefficient pressure derivative for the
* nnn interaction. This is a general interaction representing
* neutral species interacting with itself.
*/
mutable vector_fp m_Mu_nnn_P;
//! Array of coefficients form_Mu_nnn term
/*!
*
*/
Array2D m_Mu_nnn_coeff;
//! Logarithm of the activity coefficients on the molality
//! scale.
/*!
* mutable because we change this if the composition
* or temperature or pressure changes.
*
* index is the species index
*/
mutable vector_fp m_lnActCoeffMolal_Scaled;
//! Logarithm of the activity coefficients on the molality
//! scale.
/*!
* mutable because we change this if the composition
* or temperature or pressure changes.
*
* index is the species index
*/
mutable vector_fp m_lnActCoeffMolal_Unscaled;
//! Derivative of the Logarithm of the activity coefficients on the molality
//! scale wrt T
/*!
* index is the species index
*/
mutable vector_fp m_dlnActCoeffMolaldT_Scaled;
//! Derivative of the Logarithm of the activity coefficients on the molality
//! scale wrt T
/*!
* index is the species index
*/
mutable vector_fp m_dlnActCoeffMolaldT_Unscaled;
//! Derivative of the Logarithm of the activity coefficients on the molality
//! scale wrt TT
/*!
* index is the species index
*/
mutable vector_fp m_d2lnActCoeffMolaldT2_Scaled;
//! Derivative of the Logarithm of the activity coefficients on the molality
//! scale wrt TT
/*!
* index is the species index
*/
mutable vector_fp m_d2lnActCoeffMolaldT2_Unscaled;
//! Derivative of the Logarithm of the activity coefficients on the
//! molality scale wrt P
/*!
* index is the species index
*/
mutable vector_fp m_dlnActCoeffMolaldP_Scaled;
//! Derivative of the Logarithm of the activity coefficients on the
//! molality scale wrt P
/*!
* index is the species index
*/
mutable vector_fp m_dlnActCoeffMolaldP_Unscaled;
/*
* -------- Temporary Variables Used in the Activity Coeff Calc
*/
//! Cropped and modified values of the molalities used in activity
//! coefficient calculations
mutable vector_fp m_molalitiesCropped;
//! Boolean indicating whether the molalities are cropped or are modified
mutable bool m_molalitiesAreCropped;
//! a counter variable for keeping track of symmetric binary
//! interactions amongst the solute species.
/*!
* n = m_kk*i + j
* m_CounterIJ[n] = counterIJ
*/
mutable array_int m_CounterIJ;
/**
* This is elambda, MEC
*/
mutable double elambda[17];
/**
* This is elambda1, MEC
*/
mutable double elambda1[17];
/**
* Various temporary arrays used in the calculation of
* the Pitzer activity coefficents.
* The subscript, L, denotes the same quantity's derivative
* wrt temperature
*/
//! This is the value of g(x) in Pitzer's papers
/*!
* vector index is counterIJ
*/
mutable vector_fp m_gfunc_IJ;
//! This is the value of g2(x2) in Pitzer's papers
/*!
* vector index is counterIJ
*/
mutable vector_fp m_g2func_IJ;
//! hfunc, was called gprime in Pitzer's paper. However,
//! it's not the derivative of gfunc(x), so I renamed it.
/*!
* vector index is counterIJ
*/
mutable vector_fp m_hfunc_IJ;
//! hfunc2, was called gprime in Pitzer's paper. However,
//! it's not the derivative of gfunc(x), so I renamed it.
/*!
* vector index is counterIJ
*/
mutable vector_fp m_h2func_IJ;
//! Intermediate variable called BMX in Pitzer's paper
//! This is the basic cation - anion interaction
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BMX_IJ;
//! Derivative of BMX_IJ wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BMX_IJ_L;
//! Derivative of BMX_IJ wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BMX_IJ_LL;
//! Derivative of BMX_IJ wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BMX_IJ_P;
//! Intermediate variable called BprimeMX in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BprimeMX_IJ;
//! Derivative of BprimeMX wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BprimeMX_IJ_L;
//! Derivative of BprimeMX wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BprimeMX_IJ_LL;
//! Derivative of BprimeMX wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BprimeMX_IJ_P;
//! Intermediate variable called BphiMX in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BphiMX_IJ;
//! Derivative of BphiMX_IJ wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BphiMX_IJ_L;
//! Derivative of BphiMX_IJ wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BphiMX_IJ_LL;
//! Derivative of BphiMX_IJ wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_BphiMX_IJ_P;
//! Intermediate variable called Phi in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Phi_IJ;
//! Derivative of m_Phi_IJ wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Phi_IJ_L;
//! Derivative of m_Phi_IJ wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Phi_IJ_LL;
//! Derivative of m_Phi_IJ wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Phi_IJ_P;
//! Intermediate variable called Phiprime in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_Phiprime_IJ;
//! Intermediate variable called PhiPhi in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_PhiPhi_IJ;
//! Derivative of m_PhiPhi_IJ wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_PhiPhi_IJ_L;
//! Derivative of m_PhiPhi_IJ wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_PhiPhi_IJ_LL;
//! Derivative of m_PhiPhi_IJ wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_PhiPhi_IJ_P;
//! Intermediate variable called CMX in Pitzer's paper
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CMX_IJ;
//! Derivative of m_CMX_IJ wrt T
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CMX_IJ_L;
//! Derivative of m_CMX_IJ wrt TT
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CMX_IJ_LL;
//! Derivative of m_CMX_IJ wrt P
/*!
* vector index is counterIJ
*/
mutable vector_fp m_CMX_IJ_P;
//! Intermediate storage of the activity coefficient itself
/*!
* vector index is the species index
*/
mutable vector_fp m_gamma_tmp;
//! Logarithm of the molal activity coefficients
/*!
* Normally these are all one. However, stability schemes will change that
*/
mutable vector_fp IMS_lnActCoeffMolal_;
//! IMS Cutoff type
int IMS_typeCutoff_;
//! value of the solute mole fraction that centers the cutoff polynomials
//! for the cutoff =1 process;
doublereal IMS_X_o_cutoff_;
//! gamma_o value for the cutoff process at the zero solvent point
doublereal IMS_gamma_o_min_;
//! gamma_k minimun for the cutoff process at the zero solvent point
doublereal IMS_gamma_k_min_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_cCut_;
//! Parameter in the polyExp cutoff treatment
/*!
* This is the slope of the f function at the zero solvent point
* Default value is 0.6
*/
doublereal IMS_slopefCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_dfCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_efCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_afCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_bfCut_;
//! Parameter in the polyExp cutoff treatment
/*!
* This is the slope of the g function at the zero solvent point
* Default value is 0.0
*/
doublereal IMS_slopegCut_;
//! Parameter in the polyExp cutoff treatment having to do with
//! rate of exp decay
doublereal IMS_dgCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_egCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_agCut_;
//! Parameter in the polyExp cutoff treatment having to do with rate of exp decay
doublereal IMS_bgCut_;
//! value of the solvent mole fraction that centers the cutoff polynomials
//! for the cutoff =1 process;
doublereal MC_X_o_cutoff_;
//! gamma_o value for the cutoff process at the zero solvent point
doublereal MC_X_o_min_;
//! Parameter in the Molality Exp cutoff treatment
/*!
* This is the slope of the p function at the zero solvent point
* Default value is 0.0
*/
doublereal MC_slopepCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal MC_dpCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal MC_epCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal MC_apCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal MC_bpCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal MC_cpCut_;
//! Parameter in the Molality Exp cutoff treatment
doublereal CROP_ln_gamma_o_min;
//! Parameter in the Molality Exp cutoff treatment
doublereal CROP_ln_gamma_o_max;
//! Parameter in the Molality Exp cutoff treatment
doublereal CROP_ln_gamma_k_min;
//! Parameter in the Molality Exp cutoff treatment
doublereal CROP_ln_gamma_k_max;
//! This is a boolean-type vector indicating whether
//! a species's activity coefficient is in the cropped regime
/*!
*
* 0 = Not in cropped regime
* 1 = In a transition regime where it is altered but there
* still may be a temperature or pressure dependence
* 2 = In a cropped regime where there is no temperature
* or pressure dependence
*/
mutable std::vector<int> CROP_speciesCropped_;
//! Local error routine
/*!
* @param msg print out a message and error exit
*/
doublereal err(std::string msg) const;
//! Initialize all of the species - dependent lengths in the object
void initLengths();
//! 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(doublereal* acMolality) const;
private:
/*
* This function will be called to update the internally stored
* natural logarithm of the molality activity coefficients
*/
void s_update_lnMolalityActCoeff() const;
//! This function calculates the temperature derivative of the
//! natural logarithm of the molality activity coefficients.
/*!
* This is the private function. It does all of the direct work.
*/
void s_update_dlnMolalityActCoeff_dT() const;
/**
* This function calculates the temperature second derivative
* of the natural logarithm of the molality activity
* coefficients.
*/
void s_update_d2lnMolalityActCoeff_dT2() const;
/**
* This function calculates the pressure derivative of the
* natural logarithm of the molality activity coefficients.
*/
void s_update_dlnMolalityActCoeff_dP() const;
//! This function will be called to update the internally stored
//! natural logarithm of the molality activity coefficients
/*
* Normally they are all one. However, sometimes they are not,
* due to stability schemes
*
* gamma_k_molar = gamma_k_molal / Xmol_solvent
*
* gamma_o_molar = gamma_o_molal
*/
void s_updateIMS_lnMolalityActCoeff() const;
private:
/**
* This function does the main pitzer coefficient
* calculation
*/
void s_updatePitzer_lnMolalityActCoeff() const;
//! Calculates the temperature derivative of the
//! natural logarithm of the molality activity coefficients.
/*!
* Public function makes sure that all dependent data is
* up to date, before calling a private function
*/
void s_updatePitzer_dlnMolalityActCoeff_dT() const;
/**
* This function calculates the temperature second derivative
* of the natural logarithm of the molality activity
* coefficients.
*/
void s_updatePitzer_d2lnMolalityActCoeff_dT2() const;
//! Calculates the Pressure derivative of the
//! natural logarithm of the molality activity coefficients.
/*!
* Public function makes sure that all dependent data is
* up to date, before calling a private function
*/
void s_updatePitzer_dlnMolalityActCoeff_dP() const;
//! Calculates the Pitzer coefficients' dependence on the temperature.
/*!
* It will also calculate the temperature
* derivatives of the coefficients, as they are important
* in the calculation of the latent heats and the
* heat capacities of the mixtures.
*
* @param doDerivs If >= 1, then the routine will calculate
* the first derivative. If >= 2, the
* routine will calculate the first and second
* temperature derivative.
* default = 2
*/
void s_updatePitzer_CoeffWRTemp(int doDerivs = 2) const;
//! Calculate the lambda interactions.
/*!
*
* Calculate E-lambda terms for charge combinations of like sign,
* using method of Pitzer (1975).
*
* @param is Ionic strength
*/
void calc_lambdas(double is) const;
/**
* Calculate etheta and etheta_prime
*
* This interaction will be nonzero for species with the
* same charge. this routine is not to be called for
* neutral species; it core dumps or error exits.
*
* MEC implementation routine.
*
* @param z1 charge of the first molecule
* @param z2 charge of the second molecule
* @param etheta return pointer containing etheta
* @param etheta_prime Return pointer containing etheta_prime.
*
* This routine uses the internal variables,
* elambda[] and elambda1[].
*
* There is no prohibition against calling
*
*/
void calc_thetas(int z1, int z2,
double* etheta, double* etheta_prime) const;
//! Set up a counter variable for keeping track of symmetric binary
//! interactactions amongst the solute species.
/*!
* The purpose of this is to squeeze the ij parameters into a
* compressed single counter.
*
* n = m_kk*i + j
* m_Counter[n] = counter
*/
void counterIJ_setup() const;
//! Calculate the cropped molalities
/*!
* This is an internal routine that calculates values
* of m_molalitiesCropped from m_molalities
*/
void calcMolalitiesCropped() const;
//! Process an XML node called "binarySaltParameters"
/*!
* This node contains all of the parameters necessary to describe
* the Pitzer model for that particular binary salt.
* This function reads the XML file and writes the coefficients
* it finds to an internal data structures.
*
* @param BinSalt reference to the XML_Node named binarySaltParameters
* containing the
* anion - cation interaction
*/
void readXMLBinarySalt(XML_Node& BinSalt);
//! Process an XML node called "thetaAnion"
/*!
* This node contains all of the parameters necessary to describe
* the binary interactions between two anions.
*
* @param BinSalt reference to the XML_Node named thetaAnion
* containing the
* anion - anion interaction
*/
void readXMLThetaAnion(XML_Node& BinSalt);
//! Process an XML node called "thetaCation"
/*!
* This node contains all of the parameters necessary to describe
* the binary interactions between two cations.
*
* @param BinSalt reference to the XML_Node named thetaCation
* containing the
* cation - cation interaction
*/
void readXMLThetaCation(XML_Node& BinSalt);
//! Process an XML node called "psiCommonAnion"
/*!
* This node contains all of the parameters necessary to describe
* the ternary interactions between one anion and two cations.
*
* @param BinSalt reference to the XML_Node named psiCommonAnion
* containing the
* anion - cation1 - cation2 interaction
*/
void readXMLPsiCommonAnion(XML_Node& BinSalt);
//! Process an XML node called "psiCommonCation"
/*!
* This node contains all of the parameters necessary to describe
* the ternary interactions between one cation and two anions.
*
* @param BinSalt reference to the XML_Node named psiCommonCation
* containing the
* cation - anion1 - anion2 interaction
*/
void readXMLPsiCommonCation(XML_Node& BinSalt);
//! Process an XML node called "lambdaNeutral"
/*!
* This node contains all of the parameters necessary to describe
* the binary interactions between one neutral species and
* any other species (neutral or otherwise) in the mechanism.
*
* @param BinSalt reference to the XML_Node named lambdaNeutral
* containing multiple
* Neutral - species interactions
*/
void readXMLLambdaNeutral(XML_Node& BinSalt);
//! Process an XML node called "MunnnNeutral"
/*!
* This node contains all of the parameters necessary to describe
* the self-ternary interactions for one neutral species.
*
* @param BinSalt reference to the XML_Node named Munnn
* containing the self-ternary interaction
*/
void readXMLMunnnNeutral(XML_Node& BinSalt);
//! Process an XML node called "zetaCation"
/*!
* This node contains all of the parameters necessary to describe
* the ternary interactions between one neutral, one cation, and one anion.
*
* @param BinSalt reference to the XML_Node named psiCommonCation
* containing the
* neutral - cation - anion interaction
*/
void readXMLZetaCation(const XML_Node& BinSalt);
//! Process an XML node called "croppingCoefficients"
//! for the cropping coefficients values
/*!
* @param acNode Activity Coefficient XML Node
*/
void readXMLCroppingCoefficients(const XML_Node& acNode);
//! Precalculate the IMS Cutoff parameters for typeCutoff = 2
void calcIMSCutoffParams_();
//! Calculate molality cut-off parameters
void calcMCCutoffParams_();
//! Utility function to assign an integer value from a string
//! for the ElectrolyteSpeciesType field.
/*!
* @param estString string name of the electrolyte species type
*/
static int interp_est(std::string estString);
public:
/*!
* Turn on copious debug printing when this
* is true and DEBUG_MODE is turned on.
*/
mutable int m_debugCalc;
//! Return int specifying the amount of debug printing
/*!
* This will return 0 if DEBUG_MODE is not turned on
*/
int debugPrinting();
};
}
#endif
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