Doxygen update -
Update of Header files
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Harry Moffat
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3 changed files with 282 additions and 41 deletions
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/**
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* @file HMWSoln.cpp
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* Member functions of Pitzer activity coefficient implementation.
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* Definitions for the %HMWSoln ThermoPhase object, which models concentrated
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* electrolyte solutions
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* (see \ref thermoprops and \link Cantera::HMWSoln HMWSoln \endlink) .
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*
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* Class %HMWSoln represents a concentrated liquid electrolyte phase which
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* obeys the Pitzer formulation for nonideality using molality-based
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* standard states.
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*/
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/*
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* Copywrite (2006) Sandia Corporation. Under the terms of
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@ -1,6 +1,12 @@
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/**
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* @file HMWSoln.h
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* Header file for Pitzer activity coefficient implementation
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* Headers for the %HMWSoln ThermoPhase object, which models concentrated
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* electrolyte solutions
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* (see \ref thermoprops and \link Cantera::HMWSoln HMWSoln \endlink) .
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*
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* Class %HMWSoln represents a concentrated liquid electrolyte phase which
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* obeys the Pitzer formulation for nonideality using molality-based
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* standard states.
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*/
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/*
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* Copywrite (2006) Sandia Corporation. Under the terms of
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@ -297,8 +303,140 @@ namespace Cantera {
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* category. A neutral solute species is put into the "nonpolarNeutral" category by default.
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*
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*
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* <H3> Specification of the Excess Gibbs Free Energy </H3>
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*
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* Pitzer's formulation may best be represented as a specification of the excess gibbs
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* free energy, \f$ G^{ex} \f$, defined as the deviation of the total gibbs free energy from
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* that of an ideal molal solution.
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* \f[
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* G = G^{id} + G^{ex}
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* \f]
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*
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* The ideal molal solution contribution, not equal to an ideal solution contribution
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* and in fact containing a singularity at the zero solvent mole fraction limit, is
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* given below.
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* \f[
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* G^{id} = n_o \mu^o_o + \sum_{k\ne o} n_k \mu_k^{\triangle}
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* + \tilde{M}_o n_o ( RT (\sum{m_i(\ln(m_i)-1)}))
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* \f]
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*
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* From the excess Gibbs free energy formulation, the activity coefficient expression
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* and the osmotic coefficient expression for the solvent may be defined, by
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* taking the appropriate derivatives. Using this approach garranties that the
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* entire system will obey the Gibbs-Duhem relations.
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*
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* Pitzer employs the following general expression for the excess Gibbs free energy
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*
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* \f[
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* \begin{array}{cclc}
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* \frac{G^{ex}}{\tilde{M}_o n_o RT} &= &
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* \left( \frac{4AI}{3b} \right) \ln(1 + b \sqrt{I})
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* + 2 \sum_c \sum_a m_c m_a B_{ca}
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* + \sum_c \sum_a m_c m_a Z C_{ca}
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* \\&&
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* + \sum_{c < c'} \sum m_c m_{c'} \left[ 2 \Phi_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
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* + \sum_{a < a'} \sum m_a m_{a'} \left[ 2 \Phi_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
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* \\&&
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* + 2 \sum_n \sum_c m_n m_c \lambda_{nc} + 2 \sum_n \sum_a m_n m_a \lambda_{na}
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* + 2 \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
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* + \sum_n m^2_n \lambda_{nn}
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* \end{array}
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* \f]
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*
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* <I>a</I> is a subscribt over all anions, <I>c</I> is a subscript extending over all
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* cations, and <I>i</I> is a subscrit that extends over all anions and cations.
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* <I>n</I> is a subscript that extends only over neutral solute molecules.
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* The second line contains cross terms where cations affect cations and/or cation/anion pairs,
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* and anions affect anions or cation/anion pairs. Note part of the coefficients,
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* \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ stem from the theory
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* of unsymmetrical mixing of electrolytes with different charges. This
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* theory depends on the total ionic stregnth of the solution, and therefore,
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* \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ will depend on <I>I</I>, the
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* ionic strength. \f$ B_{ca}\f$ is a strong function of the total ionic strength, <I>I</I>,
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* of the electrolyte. The rest of the coefficients are assumed to be independent of the
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* molalities or ionic strengths. However, all coefficients are potentially functions
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* of the temperature and pressure of the solution.
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*
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* <I>A</I> is the Debye-Huckel constant. It's specification is described in its own
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* section below.
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*
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* \f$ I\f$ is the ionic strength of the solution, and is given by:
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*
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* \f[
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* I = \frac{1}{2} \sum_k{m_k z_k^2}
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* \f]
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*
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* In contrast to several other Debye-Huckel implementations (see \ref DebyeHuckel), the
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* parameter \f$ b\f$ in the above equation is a constant that
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* doesn not vary with respect to ion idenity. This is an important simplification
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* as it avoids troubles with satisfaction of the Gibbs-Duhem analysis.
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*
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* The function \f$ Z \f$ is given by
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*
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* \f[
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* Z = \sum_i m_i \left| z_i \right|
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* \f]
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*
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* The value of \f$ B_{ca}\f$ is given by the following function
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*
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* \f[
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* B_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} g(\alpha_1 \sqrt{I})
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* + \beta^{(2)}_{ca} g(\alpha_2 \sqrt{I})
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* \f]
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*
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* where
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*
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* \f[
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* g(x) = 2 \frac{(1 - (1 + x)\exp[-x])}{x^2}
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* \f]
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*
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* The formulation for \f$ B_{ca}\f$ combined with the formulation of the
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* Debye-Huckel term in the eqn. for the excess Gibbs free energy stems
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* essentially from an empirical fit to the ionic strength dependent data
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* based over a wide sampling of binary electroyte systems. \f$ C_{ca} \f$,
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* \f$ \lambda_{nc} \f$, \f$ \lambda_{na} \f$, \f$ \lambda_{nn} \f$,
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* \f$ \Psi_{c{c'}a} \f$, \f$ \Psi_{a{a'}c} \f$ are experimentally derived
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* coefficients that may have pressure and/or temperature dependencies.
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* The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ formulations are
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* slightly more complicated. \f$ b \f$ is a univeral
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* constant defined to be equal to \f$ 1.2 kg^{1/2} gmol^{-1/2} \f$. The exponential
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* coefficient \f$ \alpha_1 \f$ is usually fixed at \f$ \alpha_1 = 2.0 kg^{1/2} gmol^{-1/2}\f$
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* except for 2-2 electrolytes, while other parameters were fit to experimental
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* data. For 2-2 electrolytes, \f$ \alpha_1 = 1.4 kg^{1/2} gmol^{-1/2}\f$
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* is used in combination with either \f$ \alpha_2 = 12 kg^{1/2} gmol^{-1/2}\f$
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* or \f$ \alpha_2 = k A_\psi \f$, where k is a constant. For electrolytes other
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* than 2-2 electrolytes the \f$ \beta^{(2)}_{ca} g(\alpha_2 \sqrt{I}) \f$ term
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* is not used in the fitting procedure; it is only used for divalent metal
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* solfates and other high-valence electrolytes which exhibit significant
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* association at low ionic strengths.
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*
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* The \f$ \beta^{(0)}_{ca} \f$, \f$ \beta^{(1)}_{ca} \f$, \f$ \beta^{(2)}_{ca} \f$,
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* and \f$ C_{ca}\f$ binary coefficients are referred to as ion-interaction or
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* Pitzer parameters. These Pitzer parameters may vary with temperature and pressure
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* but they do not depend on the ionic strength. Their values and temperature
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* derivatives of their values have been tabulated for a range of electrolytes
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*
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* The \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$ contributions, which
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* capture cation-cation and anion-anion interactions, also have an
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* ionic strength dependence.
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*
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* Ternary contributions \f$ \Psi_{c{c'}a} \f$ and \f$ \Psi_{a{a'}c} \f$
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* have been measured also for some systems. The success of the Pitzer
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* method lies in its ability to model nonlinear activity coefficients
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* of complex multicomponent systems with just binary and minor
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* ternary contributions, which can be independently measured in
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* binary or ternary subsystems.
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*
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*
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* <H3> Multicomponent Activity Coefficients for Solutes </H3>
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*
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* The formulas for activity coefficients of solutes may be obtained by taking the
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* following derivative of the excess Gibbs Free Energy formulation described above:
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*
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* \f[
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* \ln(\gamma_k^\triangle) = \frac{d\left( \frac{G^{ex}}{M_o n_o RT} \right)}{d(m_k)}\Bigg|_{n_i}
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* \f]
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*
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* In the formulas below the following conventions are used. The subscript <I>M</I> refers
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* to a particular cation. The subscript X refers to a particular anion, whose
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* activity is being currently evaluated. the subscript <I>a</I> refers to a summation
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@ -335,16 +473,11 @@ namespace Cantera {
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* + \sum_{a < a'} \sum m_a m_{a'} \Phi'_{a{a'}}
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* \f]
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*
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* where \f$ I\f$ is the ionic strength
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* We have employed the definition of \f$ A_{\phi} \f$, also used by Pitzer
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* which is equal to
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*
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* \f[
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* I = \frac{1}{2} \sum_k{m_k z_k^2}
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* \f]
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*
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* and the function \f$ Z \f$ is given by
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*
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* \f[
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* Z = \sum_i m_i \left| z_i \right|
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* A_{\phi} = \frac{A}{3}
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* \f]
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*
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* In the above formulas, \f$ \Phi'_{c{c'}} \f$ and \f$ \Phi'_{a{a'}} \f$ are the
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@ -387,51 +520,147 @@ namespace Cantera {
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*
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* The result is the following
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*
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* \f[
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* \phi - 1 =
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* \f[
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* \begin{array}{ccclc}
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* \phi - 1 &= &
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* \frac{2}{\sum_{i \ne 0} m_i}
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* \left[
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* \begin{array}{c}
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* \bigg[ &
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* - A_{\phi} \frac{I^{3/2}}{1 + b \sqrt{I}}
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* + \sum_c \sum_a m_c m_a \left( B^{\phi}_{ca} + Z C_{ca}\right)
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* \\
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* + \sum_c \sum_a m_c m_a \left( B^{\phi}_{ca} + Z C_{ca}\right)
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* \\&&&
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* + \sum_{c < c'} \sum m_c m_{c'} \left[ \Phi^{\phi}_{c{c'}} + \sum_a m_a \Psi_{c{c'}a} \right]
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* + \sum_{a < a'} \sum m_a m_{a'} \left[ \Phi^{\phi}_{a{a'}} + \sum_c m_c \Psi_{a{a'}c} \right]
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* \\
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* \\&&&
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* + \sum_n \sum_c m_n m_c \lambda_{nc} + \sum_n \sum_a m_n m_a \lambda_{na}
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* + \sum_{n < n'} \sum m_n m_{n'} \lambda_{n{n'}}
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* + \frac{1}{2} \left( \sum_n m^2_n \lambda_{nn}\right)
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* \end{array}
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* \right]
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* + \frac{1}{2} \left( \sum_n m^2_n \lambda_{nn}\right)
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* \bigg]
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* \end{array}
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* \f]
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*
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* It can be shown that the expression
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*
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* \f[
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* B^{\phi}_{ca} = \beta^{(0)}_{ca} + \beta^{(1)}_{ca} \exp{(- \alpha_1 \sqrt{I})}
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* + \beta^{(2)}_{ca} \exp{(- \alpha_2 \sqrt{I})}
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* \f]
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*
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* is consistent with the expression \f$ B_{ca}\f$ in the \f$ G^{ex}\f$ expression
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* after carrying out the derivative wrt \f$ m_M\f$.
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*
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* Also taking into account that \f$ \Phi_{c{c'}} \f$ and
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* \f$ \Phi_{a{a'}} \f$ has an ionic strength dependence
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*
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* \f[
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* \Phi^{\phi}_{c{c'}} = \Phi_{c{c'}} + I \frac{d\Phi_{c{c'}}}{dI}
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* \f]
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* \f[
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* \Phi^{\phi}_{a{a'}} = \Phi_{a{a'}} + I \frac{d\Phi_{a{a'}}}{dI}
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* \f]
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*
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*
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* <H3> Temperature and Pressure Dependence of the Pitzer Parameters </H3>
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*
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* In general most of the coefficients introduced in the previous section may
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* have a temperature and pressure dependence. The temperature and pressure
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* dependence of these coefficients strongly influence the value of the
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* excess Enthalpy and excess Volumes of Pitzer solutions. Therefore, these
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* are readily measurable quantities.
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* HMWSoln provides several
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* different methods for putting these dependencies into the coefficients.
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* HMWSoln has an implementation described by Silverter and Pitzer (1977),
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* which was used to fit experimental data for NaCl over an extensive range,
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* below the critical temperature of water.
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* They found a temperature funcdtional form for fitting the 3 following
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* coefficients that describe the Pitzer parameterization for a single salt
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* to be adequate to describe how the excess gibbs free energy values for
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* the binary salt changes with respect to temperature.
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* The following functional form
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* was used to fit the temperature dependence of the Pitzer Coefficients.
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*
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* \f[
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* \beta^{(0)} = q_1 + q_2 \left( \frac{1}{T} - \frac{1}{T_r}\right)
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* + q_3 \ln \left( \frac{T}{T_r} \right)
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* + q_4 \left( T - T_r \right)
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+ + q_5 \left( T^2 - T_r^2 \right)
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* \f]
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* \f[
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* \beta^{(1)} = q_6 + q_9 \left( T - T_r \right) + q_{10} \left( T^2 - T_r^2 \right)
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* \f]
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* \f[
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* C^{\phi} = q_{11} ++ q_{12} \left( \frac{1}{T} - \frac{1}{T_r}\right)
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* + q_{13} \ln \left( \frac{T}{T_r} \right) + + q_{14} \left( T - T_r \right)
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* \f]
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*
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* In later papers, Pitzer has added additional temperature dependencies
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* to all of the other remaining second and third order virial coefficients.
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* Some of these dependencies are justified and motivated by theory. Therefore,
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* a formalism wherein all of the coefficients in the base theory have
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* temperature dependencies associated with them has been implemented into the
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* %HMWSoln object.
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*
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* <H3> Example of the specification of Paramters for the Activity Coefficients </H3>
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*
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* An example is given below.
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*
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* An example <TT> activityCoefficients </TT> XML block for this formulation is supplied below
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*
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* * @code
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* <activityCoefficients model="Beta_ij">
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* <!-- A_Debye units = sqrt(kg/gmol) -->
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* <A_Debye> 1.172576 </A_Debye>
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* <!-- B_Debye units = sqrt(kg/gmol)/m -->
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* <B_Debye> 3.28640E9 </B_Debye>
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* <ionicRadius default="3.042843" units="Angstroms">
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* </ionicRadius>
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* <DHBetaMatrix>
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* H+:Cl-:0.27
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* Na+:Cl-:0.15
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* Na+:OH-:0.06
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* </DHBetaMatrix>
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* <stoichIsMods>
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* NaCl(aq):-1.0
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* </stoichIsMods>
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* <electrolyteSpeciesType>
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* H+:chargedSpecies
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* NaCl(aq):weakAcidAssociated
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* </electrolyteSpeciesType>
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* </activityCoefficients>
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* @code
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<activityCoefficients model="Pitzer" TempModel="complex1">
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<!-- Pitzer Coefficients
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These coefficients are from Pitzer's main
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paper, in his book.
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-->
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<A_Debye model="water" />
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<ionicRadius default="3.042843" units="Angstroms">
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</ionicRadius>
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<binarySaltParameters cation="Na+" anion="Cl-">
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<beta0> 0.0765, 0.008946, -3.3158E-6,
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-777.03, -4.4706
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</beta0>
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<beta1> 0.2664, 6.1608E-5, 1.0715E-6 </beta1>
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<beta2> 0.0 </beta2>
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<Cphi> 0.00127, -4.655E-5, 0.0,
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33.317, 0.09421
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</Cphi>
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<Alpha1> 2.0 </Alpha1>
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</binarySaltParameters>
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<binarySaltParameters cation="H+" anion="Cl-">
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<beta0> 0.1775, 0.0, 0.0, 0.0, 0.0</beta0>
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<beta1> 0.2945, 0.0, 0.0 </beta1>
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<beta2> 0.0 </beta2>
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<Cphi> 0.0008, 0.0, 0.0, 0.0, 0.0 </Cphi>
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<Alpha1> 2.0 </Alpha1>
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</binarySaltParameters>
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<binarySaltParameters cation="Na+" anion="OH-">
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<beta0> 0.0864, 0.0, 0.0, 0.0, 0.0 </beta0>
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<beta1> 0.253, 0.0, 0.0 </beta1>
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<beta2> 0.0 </beta2>
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<Cphi> 0.0044, 0.0, 0.0, 0.0, 0.0 </Cphi>
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<Alpha1> 2.0 </Alpha1>
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</binarySaltParameters>
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<thetaAnion anion1="Cl-" anion2="OH-">
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<Theta> -0.05 </Theta>
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</thetaAnion>
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<psiCommonCation cation="Na+" anion1="Cl-" anion2="OH-">
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<Theta> -0.05 </Theta>
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<Psi> -0.006 </Psi>
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</psiCommonCation>
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<thetaCation cation1="Na+" cation2="H+">
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<Theta> 0.036 </Theta>
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</thetaCation>
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<psiCommonAnion anion="Cl-" cation1="Na+" cation2="H+">
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<Theta> 0.036 </Theta>
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<Psi> -0.004 </Psi>
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</psiCommonAnion>
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</activityCoefficients>
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* @endcode
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*
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*
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@ -1,5 +1,11 @@
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/**
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* @file HMWSoln_input.cpp
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* Definitions for the %HMWSoln ThermoPhase object, which models concentrated
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* electrolyte solutions
|
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* (see \ref thermoprops and \link Cantera::HMWSoln HMWSoln \endlink) .
|
||||
*
|
||||
* This file contains definitions for reading in the interaction terms
|
||||
* in the formulation.
|
||||
*/
|
||||
/*
|
||||
* Copywrite (2006) Sandia Corporation. Under the terms of
|
||||
|
|
|
|||
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Add table
Reference in a new issue