Doxygen Update
-> added more formulas to the header
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Harry Moffat
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1 changed files with 85 additions and 10 deletions
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@ -263,8 +263,8 @@ namespace Cantera {
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* may not be be included in the
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* species solution vector.
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* - <B>cEST_strongAcidAssociated</B> Species which always breaksapart into charged species.
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* It may or may not be charged. Normally, these aren't included
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* in the speciation vector.
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* It may or may not be charged. Normally, these
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* aren't included in the speciation vector.
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* - <B>cEST_polarNeutral </B> Polar neutral species
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* - <B>cEST_nonpolarNeutral</B> Non poloar neutral species
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*
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@ -309,29 +309,104 @@ namespace Cantera {
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*
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* \f[
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* \ln(\gamma_M^\triangle) = -z_M^2(F) + \sum_a m_a \left( 2 B_{Ma} + Z C_{Ma} \right)
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* + z_M \left( \sum_a \sum_c m_a m_c C_{Ma} \right)
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* + z_M \left( \sum_a \sum_c m_a m_c C_{ca} \right)
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* + \sum_c m_c \left[ 2 \Phi_{Mc} + \sum_a m_a \psi_{Mca} \right]
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* + \sum_{a < a'} \sum m_a m_{a'} \psi_{Ma{a'}}
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* + 2 \sum_n m_n \lambda_{nM}
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* \f]
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*
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* The activity coefficient for a particular anion <I>X</I> is given by
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*
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* \f[
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* \ln(\gamma_X^\triangle) = -z_X^2(F) + \sum_a m_c \left( 2 B_{cX} + Z C_{cX} \right)
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* + \left|z_X \right| \left( \sum_a \sum_c m_a m_c C_{ca} \right)
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* + \sum_a m_a \left[ 2 \Phi_{Xa} + \sum_c m_c \psi_{cXa} \right]
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* + \sum_{c < c'} \sum m_c m_{c'} \psi_{c{c'}X}
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* + 2 \sum_n m_n \lambda_{nM}
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* \f]
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* where the function \f$ F \f$ is given by
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*
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*
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* \f[
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* F = - A_{\phi} \left[ \frac{\sqrt{I}}{1 + b \sqrt{I}}
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* + \frac{2}{b} \ln{\left(1 + b\sqrt{I}\right)} \right]
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* + \sum_a \sum_c m_a m_c B'_{ca}
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* + \sum_{c < c'} \sum m_c m_{c'} \Phi'_{c{c'}}
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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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*
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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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* \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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* ionic strength derivatives of \f$ \Phi_{c{c'}} \f$ and \f$ \Phi_{a{a'}} \f$,
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* respectively.
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*
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* The function \f$ B'_{MX} \f$ is defined as:
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*
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* \f[
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* B'_{MX} = \left( \frac{\beta^1_{MX} h(\alpha \sqrt{I})}{I} \right)
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* \left( \frac{\beta^2_{MX} h(\alpha \sqrt{I})}{I} \right)
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* \f]
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*
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* where \f$ h(x) \f$ is defined as
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*
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* \f[
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* h(x) = g'(x) \frac{x}{2} =
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* \frac{2\left(1 - \left(1 + x + \frac{x^2}{2} \right)\exp(-x) \right)}{x^2}
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* \f]
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*
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* The activity coefficient for neutral species <I>N</I> is given by
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*
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* \f[
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* \ln(\gamma_N^\triangle) = 2 \left( \sum_i m_i \lambda_{iN}\right)
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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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* \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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* <H3> Activity of the Water Solvent </H3>
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*
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* The activity for the solvent water,\f$ a_o \f$, is not independent and must be
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* determined from the Gibbs-Duhem relation.
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* determined either from the Gibbs-Duhem relation or from taking the appropriate derivative
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* of the same excess Gibbs free energy function as was used to formulate
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* the solvent activity coefficients. Pitzer's description follows the later approach to
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* derive a formula for the osmotic coefficient, \f$ \phi \f$.
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*
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* \f[
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* \ln(a_o) = \frac{X_o - 1.0}{X_o} + \frac{ 2 A_{Debye} \tilde{M}_o}{3} (I)^{3/2}
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* \phi - 1 = - \left( \frac{d\left(\frac{G^{ex}}{RT} \right)}{d(\tilde{M}_o n_o)} \right)
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* \frac{1}{\sum_{i \ne 0} m_i}
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* \f]
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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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* \frac{2}{\sum_{i \ne 0} m_i}
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* \left[
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* \begin{array}{c}
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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 < 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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* + \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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* \f]
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*
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*
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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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