diff --git a/Cantera/src/thermo/HMWSoln.cpp b/Cantera/src/thermo/HMWSoln.cpp index a5f7a09c3..6897e001d 100644 --- a/Cantera/src/thermo/HMWSoln.cpp +++ b/Cantera/src/thermo/HMWSoln.cpp @@ -940,7 +940,6 @@ namespace Cantera { } /* - * * getPartialMolarEntropies() (virtual, const) * * Returns an array of partial molar entropies of the species in the @@ -954,10 +953,14 @@ namespace Cantera { * Combining this with the expression H = G + TS yields: * * \f[ - * \bar s_k(T,P) = \hat s^0_k(T) - R log(M0 * molality[k] ac[k]) - * - R T^2 d log(ac[k]) / dT + * \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 reference-state pure-species entropies,\f$ \hat s^0_k(T) \f$, * at the reference pressure, \f$ P_{ref} \f$, are computed by the @@ -1011,7 +1014,7 @@ namespace Cantera { } } - /** + /* * getPartialMolarVolumes() (virtual, const) * * Returns an array of partial molar volumes of the species @@ -1053,7 +1056,16 @@ namespace Cantera { * enthalpy of the kth species in the solution at constant * P and composition (p. 220 Smith and Van Ness). * - * Cp = -T d2(chemPot_i)/dT2 + * \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] */ void HMWSoln::getPartialMolarCp(doublereal* cpbar) const { /* @@ -1086,7 +1098,7 @@ namespace Cantera { * in the Solution ------------------ */ - /** + /* * getStandardChemPotentials() (virtual, const) * * @@ -1115,7 +1127,7 @@ namespace Cantera { mu[0] = m_waterSS->gibbs_mole(); } - /** + /* * Get the nondimensional gibbs function for the species * standard states at the current T and P of the solution. * @@ -1138,7 +1150,7 @@ namespace Cantera { } } - /** + /* * * getPureGibbs() * @@ -1227,7 +1239,7 @@ namespace Cantera { sr[0] /= GasConstant; } - /** + /* * Get the nondimensional heat capacity at constant pressure * function for the species * standard states at the current T and P of the solution. @@ -1250,7 +1262,7 @@ namespace Cantera { cpr[0] /= GasConstant; } - /** + /* * Get the molar volumes of each species in their standard * states at the current * T and P of the solution. diff --git a/Cantera/src/thermo/HMWSoln.h b/Cantera/src/thermo/HMWSoln.h index ca700d0b8..f26a5382c 100644 --- a/Cantera/src/thermo/HMWSoln.h +++ b/Cantera/src/thermo/HMWSoln.h @@ -276,8 +276,8 @@ namespace Cantera { * * 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. + * 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 electrolyteSpeciesType XML block. * Note, this is not @@ -346,13 +346,15 @@ namespace Cantera { * a is a subscribt over all anions, c is a subscript extending over all * cations, and i is a subscrit that extends over all anions and cations. * n 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, + * 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, the - * ionic strength. \f$ B_{ca}\f$ is a strong function of the total ionic strength, I, + * ionic strength. \f$ B_{ca}\f$ is a strong function of the + * total ionic strength, 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. @@ -519,6 +521,14 @@ namespace Cantera { * \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[ @@ -688,7 +698,7 @@ namespace Cantera { * Dependent in general on temperature and pressure, it's ionic * strength dependence is ignored in Pitzer's approach. * \f$ \,^E\Theta_{ij}(I) \f$ accounts for the electrostatic - * unsymmetrical mixing effects and is depeendnet only on the + * 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. @@ -925,10 +935,68 @@ namespace Cantera { *

Temperature and Pressure Dependence of the Activity Coefficients

* * Temperature dependence of the activity coefficients leads to nonzero terms - * for the excess enthalpy of solution. + * 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(). * *
*

%Application within %Kinetics Managers

@@ -941,7 +1009,8 @@ namespace Cantera { * kinetic rate constant specified * as if all reactants were on a concentration basis. * - * For example, a bulk-phase binary reaction between liquid species j and k, producing + * For example, a bulk-phase binary reaction between liquid species + * j and k, producing * a new liquid species l would have the * following equation for its rate of progress variable, \f$ R^1 \f$, which has * units of kmol m-3 s-1. @@ -966,7 +1035,7 @@ namespace Cantera { * and the equilibrium expression for the system. * * \f[ - * \frac{a_j a_k}{ a_l} = K^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} ) + * \frac{a_j a_k}{ a_l} = K^{o,1} = \exp(\frac{\mu^o_l - \mu^o_j - \mu^o_k}{R T} ) * \f] * * \f$ K^{o,1} \f$ is the dimensionless form of the equilibrium constant. @@ -1591,12 +1660,14 @@ namespace Cantera { * 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} + * \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}) + * = h^{o}_o(T,P) + * + R T^2 (\sum_{k \neq o} m_k) \tilde{M_o} (\frac{d \phi}{dT}) * \f] * * @@ -1609,7 +1680,7 @@ namespace Cantera { //! Returns an array of partial molar entropies of the species in the //! solution. Units: J/kmol/K. /*! - * Maxwell's equations provide an insight in how to calculate this + * Maxwell's equations provide an answer for how calculate this * (p.215 Smith and Van Ness) * * d(chemPot_i)/dT = -sbar_i @@ -1620,11 +1691,13 @@ namespace Cantera { * temperature derivative of the activity coefficents. * * \f[ - * \bar s_k(T,P) = \hat s^0_k(T) - R log(M0 * molality[k]) + * \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_solvent(T,P) = \hat s^0_solvent(T) - * - R ((xmolSolvent - 1.0) / xmolSolvent) + * \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. @@ -1638,6 +1711,15 @@ namespace Cantera { * 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 speciar partial molar volumes. * Length = m_kk. units are m^3/kmol. */ @@ -1646,13 +1728,26 @@ namespace Cantera { //! 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; - //@} /// @name Properties of the Standard State of the Species @@ -1661,7 +1756,8 @@ namespace Cantera { //! Get the array of chemical potentials at unit activity for the species - //! at their standard states at the current T and P of the solution. + //! at their standard states at the current T and P + //! of the solution. /*! * These are the standard state chemical potentials \f$ \mu^0_k(T,P) * \f$. The values are evaluated at the current @@ -2992,12 +3088,12 @@ namespace Cantera { //! This function calculates the temperature derivative of the //! natural logarithm of the molality activity coefficients. /*! - * Private function does the work + * This is the private function. It does all of the direct work. */ void s_update_dlnMolalityActCoeff_dT() const; /** - * This function calcultes the temperature second derivative + * This function calculates the temperature second derivative * of the natural logarithm of the molality activity * coefficients. */ diff --git a/Cantera/src/thermo/IdealMolalSoln.h b/Cantera/src/thermo/IdealMolalSoln.h index 09b2c86ca..e0761c942 100644 --- a/Cantera/src/thermo/IdealMolalSoln.h +++ b/Cantera/src/thermo/IdealMolalSoln.h @@ -560,7 +560,7 @@ namespace Cantera { * standard state species entropies plus the ideal molal solution contribution. * * \f[ - * \bar{s}_k(T,P) = s^0_k(T) - R log( m_k ) + * \bar{s}_k(T,P) = s^0_k(T) - R \ln( \frac{m_k}{m^{\triangle}} ) * \f] * \f[ * \bar{s}_w(T,P) = s^0_w(T) - R ((X_w - 1.0) / X_w) @@ -615,8 +615,8 @@ namespace Cantera { // in the Solution -- //@{ + //! Get the standard state chemical potentials of the species. /*! - * Get the standard state chemical potentials of the species. * This is the array of chemical potentials at unit activity * \f$ \mu^0_k(T,P) \f$. * We define these here as the chemical potentials of the pure