493 lines
17 KiB
C++
493 lines
17 KiB
C++
/**
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* @file RedlichKisterVPSSTP.h (see \ref thermoprops and class \link
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* Cantera::RedlichKisterVPSSTP RedlichKisterVPSSTP\endlink).
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*/
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/*
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* Copyright (2006) Sandia Corporation. Under the terms of
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* Contract DE-AC04-94AL85000 with Sandia Corporation, the
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* U.S. Government retains certain rights in this software.
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*/
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#ifndef CT_REDLICHKISTERVPSSTP_H
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#define CT_REDLICHKISTERVPSSTP_H
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#include "cantera/thermo/GibbsExcessVPSSTP.h"
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namespace Cantera
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{
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//! RedlichKisterVPSSTP is a derived class of GibbsExcessVPSSTP that employs the
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//! Redlich-Kister approximation for the excess Gibbs free energy
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/*!
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* RedlichKisterVPSSTP derives from class GibbsExcessVPSSTP which is derived
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* from VPStandardStateTP, and overloads the virtual methods defined there with
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* ones that use expressions appropriate for the Redlich Kister Excess Gibbs
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* free energy approximation.
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*
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* The independent unknowns are pressure, temperature, and mass fraction.
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*
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* ## Specification of Species Standard State Properties
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*
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* All species are defined to have standard states that depend upon both the
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* temperature and the pressure. The Redlich-Kister approximation assumes
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* symmetric standard states, where all of the standard state assume that the
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* species are in pure component states at the temperature and pressure of the
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* solution. I don't think it prevents, however, some species from being dilute
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* in the solution.
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*
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* ## Specification of Solution Thermodynamic Properties
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*
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* The molar excess Gibbs free energy is given by the following formula which is
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* a sum over interactions *i*. Each of the interactions are binary interactions
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* involving two of the species in the phase, denoted, *Ai* and *Bi*. This is
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* the generalization of the Redlich-Kister formulation for a phase that has
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* more than 2 species.
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*
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* \f[
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* G^E = \sum_{i} G^E_{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^E_{i} = n X_{Ai} X_{Bi} \sum_m \left( A^{i}_m {\left( X_{Ai} - X_{Bi} \right)}^m \right)
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* \f]
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*
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* and where we can break down the Gibbs free energy contributions into enthalpy and entropy contributions
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*
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* \f[
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* H^E_i = n X_{Ai} X_{Bi} \sum_m \left( H^{i}_m {\left( X_{Ai} - X_{Bi} \right)}^m \right)
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* \f]
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*
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* \f[
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* S^E_i = n X_{Ai} X_{Bi} \sum_m \left( S^{i}_m {\left( X_{Ai} - X_{Bi} \right)}^m \right)
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* \f]
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*
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* where n is the total moles in the solution. The activity of a species defined
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* in the phase is given by an excess Gibbs free energy formulation.
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*
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* \f[
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* a_k = \gamma_k X_k
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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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* R T \ln( \gamma_k )= \frac{d(n G^E)}{d(n_k)}\Bigg|_{n_i}
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* \f]
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*
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* Taking the derivatives results in the following expression
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* \f[
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* R T \ln( \gamma_k )= \sum_i \delta_{Ai,k} (1 - X_{Ai}) X_{Bi} \sum_m \left( A^{i}_m {\left( X_{Ai} - X_{Bi} \right)}^m \right)
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* + \sum_i \delta_{Ai,k} X_{Ai} X_{Bi} \sum_m \left( A^{i}_0 + A^{i}_m {\left( X_{Ai} - X_{Bi} \right)}^{m-1} (1 - X_{Ai} + X_{Bi}) \right)
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* \f]
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*
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* This object inherits from the class VPStandardStateTP. Therefore, the
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* specification and calculation of all standard state and reference state
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* values are handled at that level. Various functional forms for the standard
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* state are permissible. The chemical potential for species *k* is equal to
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*
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* \f[
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* \mu_k(T,P) = \mu^o_k(T, P) + R T \ln(\gamma_k X_k)
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* \f]
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*
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* The partial molar entropy for species *k* is given by the following relation,
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*
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* \f[
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* \tilde{s}_k(T,P) = s^o_k(T,P) - R \ln( \gamma_k X_k )
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* - R T \frac{d \ln(\gamma_k) }{dT}
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* \f]
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*
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* The partial molar enthalpy for species *k* is given by
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*
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* \f[
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* \tilde{h}_k(T,P) = h^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
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* \f]
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*
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* The partial molar volume for species *k* is
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*
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* \f[
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* \tilde V_k(T,P) = V^o_k(T,P) + R T \frac{d \ln(\gamma_k) }{dP}
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* \f]
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*
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* The partial molar Heat Capacity for species *k* is
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*
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* \f[
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* \tilde{C}_{p,k}(T,P) = C^o_{p,k}(T,P) - 2 R T \frac{d \ln( \gamma_k )}{dT}
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* - R T^2 \frac{d^2 \ln(\gamma_k) }{{dT}^2}
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* \f]
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*
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* ## %Application within Kinetics Managers
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*
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* \f$ C^a_k\f$ are defined such that \f$ a_k = C^a_k / C^s_k, \f$ where
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* \f$ C^s_k \f$ is a standard concentration defined below and \f$ a_k \f$ are
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* activities used in the thermodynamic functions. These activity (or
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* generalized) concentrations are used by kinetics manager classes to compute
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* the forward and reverse rates of elementary reactions. The activity
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* concentration,\f$ C^a_k \f$,is given by the following expression.
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*
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* \f[
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* C^a_k = C^s_k X_k = \frac{P}{R T} X_k
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* \f]
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*
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* The standard concentration for species *k* is independent of *k* and equal to
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*
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* \f[
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* C^s_k = C^s = \frac{P}{R T}
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* \f]
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*
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* For example, a bulk-phase binary gas reaction between species j and k,
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* producing a new gas species l would have the following equation for its rate
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* of progress variable, \f$ R^1 \f$, which has units of kmol m-3 s-1.
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*
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* \f[
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* R^1 = k^1 C_j^a C_k^a = k^1 (C^s a_j) (C^s a_k)
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* \f]
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* where
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* \f[
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* C_j^a = C^s a_j \mbox{\quad and \quad} C_k^a = C^s a_k
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* \f]
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*
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* \f$ C_j^a \f$ is the activity concentration of species j, and \f$ C_k^a \f$
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* is the activity concentration of species k. \f$ C^s \f$ is the standard
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* concentration. \f$ a_j \f$ is the activity of species j which is equal to the
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* mole fraction of j.
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*
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* The reverse rate constant can then be obtained from the law of microscopic
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* reversibility and the equilibrium expression for the system.
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*
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* \f[
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* \frac{a_j a_k}{ a_l} = K_a^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} )
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* \f]
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*
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* \f$ K_a^{o,1} \f$ is the dimensionless form of the equilibrium constant,
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* associated with the pressure dependent standard states \f$ \mu^o_l(T,P) \f$
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* and their associated activities, \f$ a_l \f$, repeated here:
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*
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* \f[
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* \mu_l(T,P) = \mu^o_l(T, P) + R T \log(a_l)
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* \f]
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*
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* We can switch over to expressing the equilibrium constant in terms of the
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* reference state chemical potentials
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*
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* \f[
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* K_a^{o,1} = \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} ) * \frac{P_{ref}}{P}
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* \f]
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*
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* The concentration equilibrium constant, \f$ K_c \f$, may be obtained by
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* changing over to activity concentrations. When this is done:
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*
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* \f[
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* \frac{C^a_j C^a_k}{ C^a_l} = C^o K_a^{o,1} = K_c^1 =
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* \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} ) * \frac{P_{ref}}{RT}
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* \f]
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*
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* %Kinetics managers will calculate the concentration equilibrium constant, \f$
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* K_c \f$, using the second and third part of the above expression as a
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* definition for the concentration equilibrium constant.
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*
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* For completeness, the pressure equilibrium constant may be obtained as well
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*
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* \f[
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* \frac{P_j P_k}{ P_l P_{ref}} = K_p^1 = \exp(\frac{\mu^{ref}_l - \mu^{ref}_j - \mu^{ref}_k}{R T} )
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* \f]
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*
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* \f$ K_p \f$ is the simplest form of the equilibrium constant for ideal gases.
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* However, it isn't necessarily the simplest form of the equilibrium constant
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* for other types of phases; \f$ K_c \f$ is used instead because it is
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* completely general.
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*
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* The reverse rate of progress may be written down as
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* \f[
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* R^{-1} = k^{-1} C_l^a = k^{-1} (C^o a_l)
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* \f]
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*
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* where we can use the concept of microscopic reversibility to write the
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* reverse rate constant in terms of the forward reate constant and the
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* concentration equilibrium constant, \f$ K_c \f$.
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*
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* \f[
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* k^{-1} = k^1 K^1_c
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* \f]
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*
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* \f$k^{-1} \f$ has units of s-1.
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*
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* @ingroup thermoprops
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*/
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class RedlichKisterVPSSTP : public GibbsExcessVPSSTP
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{
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public:
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//! Constructor
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/*!
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* This doesn't do much more than initialize constants with default values.
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*/
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RedlichKisterVPSSTP();
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//! Construct and initialize a RedlichKisterVPSSTP ThermoPhase object
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//! directly from an XML input file
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/*!
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* @param inputFile Name of the input file containing the phase XML data
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* to set up the object
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* @param id ID of the phase in the input file. Defaults to the
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* empty string.
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*/
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RedlichKisterVPSSTP(const std::string& inputFile, const std::string& id = "");
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//! Construct and initialize a RedlichKisterVPSSTP ThermoPhase object
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//! directly from an XML database
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/*!
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* @param phaseRef XML phase node containing the description of the phase
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* @param id id attribute containing the name of the phase.
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* (default is the empty string)
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*/
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RedlichKisterVPSSTP(XML_Node& phaseRef, const std::string& id = "");
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RedlichKisterVPSSTP(const RedlichKisterVPSSTP& b);
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RedlichKisterVPSSTP& operator=(const RedlichKisterVPSSTP& b);
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virtual ThermoPhase* duplMyselfAsThermoPhase() const;
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virtual std::string type() const {
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return "RedlichKister";
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}
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//! @name Molar Thermodynamic Properties
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//! @{
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virtual doublereal enthalpy_mole() const;
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virtual doublereal entropy_mole() const;
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virtual doublereal cp_mole() const;
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virtual doublereal cv_mole() const;
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/**
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* @}
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* @name Activities, Standard States, and Activity Concentrations
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*
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* The activity \f$a_k\f$ of a species in solution is
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* related to the chemical potential by \f[ \mu_k = \mu_k^0(T)
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* + \hat R T \log a_k. \f] The quantity \f$\mu_k^0(T,P)\f$ is
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* the chemical potential at unit activity, which depends only
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* on temperature and pressure.
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* @{
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*/
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virtual void getLnActivityCoefficients(doublereal* lnac) const;
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//@}
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/// @name Partial Molar Properties of the Solution
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//@{
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virtual void getChemPotentials(doublereal* mu) const;
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//! Returns an array of partial molar enthalpies for the species in the
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//! mixture.
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/*!
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* Units (J/kmol)
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*
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* For this phase, the partial molar enthalpies are equal to the standard
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* state enthalpies modified by the derivative of the molality-based
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* activity coefficient wrt temperature
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*
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* \f[
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* \bar h_k(T,P) = h^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
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* \f]
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*
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* @param hbar Vector of returned partial molar enthalpies
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* (length m_kk, units = J/kmol)
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*/
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virtual void getPartialMolarEnthalpies(doublereal* hbar) const;
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//! Returns an array of partial molar entropies for the species in the
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//! mixture.
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/*!
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* Units (J/kmol)
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*
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* For this phase, the partial molar enthalpies are equal to the standard
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* state enthalpies modified by the derivative of the activity coefficient
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* wrt temperature
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*
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* \f[
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* \bar s_k(T,P) = s^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
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* - R \ln( \gamma_k X_k)
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* - R T \frac{d \ln(\gamma_k) }{dT}
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* \f]
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*
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* @param sbar Vector of returned partial molar entropies
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* (length m_kk, units = J/kmol/K)
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*/
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virtual void getPartialMolarEntropies(doublereal* sbar) const;
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//! Returns an array of partial molar entropies for the species in the
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//! mixture.
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/*!
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* Units (J/kmol)
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*
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* For this phase, the partial molar enthalpies are equal to the standard
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* state enthalpies modified by the derivative of the activity coefficient
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* wrt temperature
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*
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* \f[
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* ???????????????
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* \bar s_k(T,P) = s^o_k(T,P) - R T^2 \frac{d \ln(\gamma_k)}{dT}
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* - R \ln( \gamma_k X_k)
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* - R T \frac{d \ln(\gamma_k) }{dT}
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* ???????????????
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* \f]
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*
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* @param cpbar Vector of returned partial molar heat capacities
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* (length m_kk, units = J/kmol/K)
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*/
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virtual void getPartialMolarCp(doublereal* cpbar) const;
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virtual void getPartialMolarVolumes(doublereal* vbar) const;
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//! Get the array of temperature second derivatives of the log activity
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//! coefficients
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/*!
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* units = 1/Kelvin
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*
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* @param d2lnActCoeffdT2 Output vector of temperature 2nd derivatives of
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* the log Activity Coefficients. length = m_kk
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*/
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virtual void getd2lnActCoeffdT2(doublereal* d2lnActCoeffdT2) const;
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virtual void getdlnActCoeffdT(doublereal* dlnActCoeffdT) const;
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/// @}
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/// @name Initialization
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/// The following methods are used in the process of constructing
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/// the phase and setting its parameters from a specification in an
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/// input file. They are not normally used in application programs.
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/// To see how they are used, see importPhase().
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virtual void initThermo();
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virtual void initThermoXML(XML_Node& phaseNode, const std::string& id);
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//! @}
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//! @name Derivatives of Thermodynamic Variables needed for Applications
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//! @{
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virtual void getdlnActCoeffds(const doublereal dTds, const doublereal* const dXds, doublereal* dlnActCoeffds) const;
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virtual void getdlnActCoeffdlnX_diag(doublereal* dlnActCoeffdlnX_diag) const;
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virtual void getdlnActCoeffdlnN_diag(doublereal* dlnActCoeffdlnN_diag) const;
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virtual void getdlnActCoeffdlnN(const size_t ld, doublereal* const dlnActCoeffdlnN);
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//@}
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private:
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//! Process an XML node called "binaryNeutralSpeciesParameters"
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/*!
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* This node contains all of the parameters necessary to describe the
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* Redlich-Kister model for a particular binary interaction. This function
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* reads the XML file and writes the coefficients it finds to an internal
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* data structures.
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*
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* @param xmlBinarySpecies Reference to the XML_Node named
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* "binaryNeutralSpeciesParameters" containing the binary interaction
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*/
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void readXMLBinarySpecies(XML_Node& xmlBinarySpecies);
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//! Resize internal arrays within the object that depend upon the number
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//! of binary Redlich-Kister interaction terms
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/*!
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* @param num Number of binary Redlich-Kister interaction terms
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*/
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void resizeNumInteractions(const size_t num);
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//! Initialize lengths of local variables after all species have been
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//! identified.
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void initLengths();
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//! Update the activity coefficients
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/*!
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* This function will be called to update the internally stored natural
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* logarithm of the activity coefficients
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*/
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void s_update_lnActCoeff() const;
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//! Update the derivative of the log of the activity coefficients wrt T
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/*!
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* This function will be called to update the internally stored derivative
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* of the natural logarithm of the activity coefficients wrt temperature.
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*/
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void s_update_dlnActCoeff_dT() const;
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//! Internal routine that calculates the derivative of the activity
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//! coefficients wrt the mole fractions.
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/*!
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* This routine calculates the the derivative of the activity coefficients
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* wrt to mole fraction with all other mole fractions held constant. This is
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* strictly not permitted. However, if the resulting matrix is multiplied by
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* a permissible deltaX vector then everything is ok.
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*
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* This is the natural way to handle concentration derivatives in this
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* routine.
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*/
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void s_update_dlnActCoeff_dX_() const;
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//! Internal routine that calculates the total derivative of the activity
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//! coefficients with respect to the log of the mole fractions.
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/*!
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* This function will be called to update the internally stored vector of
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* the total derivatives (i.e. not assuming other mole fractions are
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* constant) of the natural logarithm of the activity coefficients with
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* respect to the log of the mole fraction.
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*/
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void s_update_dlnActCoeff_dlnX_diag() const;
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public:
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//! Utility routine that calculates a literature expression
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/*!
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* @param VintOut Output contribution to the voltage corresponding to
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* nonideal term
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* @param voltsOut Output contribution to the voltage corresponding to
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* nonideal term and mf term
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*/
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void Vint(double& VintOut, double& voltsOut);
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protected:
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//! number of binary interaction expressions
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size_t numBinaryInteractions_;
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//! vector of species indices representing species A in the interaction
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/*!
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* Each Redlich-Kister excess Gibbs free energy term involves two species,
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* A and B. This vector identifies species A.
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*/
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std::vector<size_t> m_pSpecies_A_ij;
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//! vector of species indices representing species B in the interaction
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/*!
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* Each Redlich-Kister excess Gibbs free energy term involves two species,
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* A and B. This vector identifies species B.
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*/
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std::vector<size_t> m_pSpecies_B_ij;
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//! Vector of the length of the polynomial for the interaction.
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std::vector<size_t> m_N_ij;
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//! Enthalpy term for the binary mole fraction interaction of the excess
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//! Gibbs free energy expression
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mutable std::vector< vector_fp> m_HE_m_ij;
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//! Entropy term for the binary mole fraction interaction of the excess
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//! Gibbs free energy expression
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mutable std::vector< vector_fp> m_SE_m_ij;
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//! form of the RedlichKister interaction expression. Currently there is
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//! only one form.
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int formRedlichKister_;
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//! form of the temperature dependence of the Redlich-Kister interaction
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//! expression. Currently there is only one form -> constant wrt
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//! temperature.
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int formTempModel_;
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//! Two dimensional array of derivatives of activity coefficients wrt mole
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//! fractions
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mutable Array2D dlnActCoeff_dX_;
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};
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}
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#endif
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