diff --git a/Cantera/src/thermo/HMWSoln.h b/Cantera/src/thermo/HMWSoln.h index 1c49592a5..72489634e 100644 --- a/Cantera/src/thermo/HMWSoln.h +++ b/Cantera/src/thermo/HMWSoln.h @@ -657,11 +657,76 @@ namespace Cantera { * the beta0 block that fits the COMPLEX1 temperature * dependence given above is * - * @code - q0, q1, q2, q3, q4 + * @code + * + q0, q1, q2, q3, q4 + <\binarySaltParameters> * @endcode * - *

Mixing Parameters

+ *

Like-Charged Binary Ion Parameters and the Mixing Parameters

+ * + * 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_{i{j}} = \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, 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 + * 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 thetaCation and + * thetaAnion , which + * has the attribute cation1 , cation2 and + * anion1 , anion2 respectively + * to identify the interaction. No temperature or + * pressure dependence of this parameter is currently allowed. + * An example of the block is biven below + * + * @code + + 0.036 + + @endcode * * *

Ternary Pitzer Parameters

@@ -671,7 +736,7 @@ namespace Cantera { * * *

Example of the Specification of Parameters for the Activity - * Coefficients

+ * Coefficients * * An example is given below. *