Stefan Maxwell Diffusion Coefficients can be solved for given ion conductivity, mobility ratios, and self diffusion coeffs. More...

#include <LiquidTranInteraction.h>

Inheritance diagram for LTI_StefanMaxwell_PPN:

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Collaboration diagram for LTI_StefanMaxwell_PPN:

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Public Member Functions
	LTI_StefanMaxwell_PPN (TransportPropertyType tp_ind=TP_UNKNOWN)

void	setParameters (LiquidTransportParams &trParam)
	Copy constructor. More...

doublereal	getMixTransProp (doublereal valueSpecies, doublereal weightSpecies=0)
	Return the mixture transport property value. More...

doublereal	getMixTransProp (std::vector< LTPspecies * > LTPptrs)

void	getMatrixTransProp (DenseMatrix &mat, doublereal *speciesValues=0)
	Return the matrix of binary interaction parameters. More...

Public Member Functions inherited from LiquidTranInteraction
	LiquidTranInteraction (TransportPropertyType tp_ind=TP_UNKNOWN)
	Constructor. More...

	LiquidTranInteraction (const LiquidTranInteraction &right)

LiquidTranInteraction &	operator= (const LiquidTranInteraction &right)

virtual void	init (const XML_Node &compModelNode=XML_Node(), thermo_t *thermo=0)
	initialize LiquidTranInteraction objects with thermo and XML node More...

Protected Types
typedef std::vector< LTPspecies * >	LTPvector

Protected Attributes
doublereal	m_ionCondMix

LiquidTranInteraction *	m_ionCondMixModel

std::vector< LTPspecies * >	m_ionCondSpecies

DenseMatrix	m_mobRatMix

std::vector < LiquidTranInteraction * >	m_mobRatMixModel

std::vector< LTPvector >	m_mobRatSpecies

std::vector < LiquidTranInteraction * >	m_selfDiffMixModel

vector_fp	m_selfDiffMix

std::vector< LTPvector >	m_selfDiffSpecies

Protected Attributes inherited from LiquidTranInteraction
LiquidTranMixingModel	m_model
	Model for species interaction effects Takes enum LiquidTranMixingModel. More...

TransportPropertyType	m_property
	enum indicating what property this is (i.e viscosity) More...

thermo_t *	m_thermo
	pointer to thermo object to get current temperature More...

std::vector< DenseMatrix * >	m_Aij
	Matrix of interaction coefficients for polynomial in molefraction*weight of speciesA (no temperature dependence, dimensionless) More...

std::vector< DenseMatrix * >	m_Bij
	Matrix of interaction coefficients for polynomial in molefraction*weight of speciesA (linear temperature dependence, units 1/K) More...

DenseMatrix	m_Eij
	Matrix of interactions (in energy units, 1/RT temperature dependence) More...

std::vector< DenseMatrix * >	m_Hij
	Matrix of interaction coefficients for polynomial in molefraction*weight of speciesA (in energy units, 1/RT temperature dependence) More...

std::vector< DenseMatrix * >	m_Sij
	Matrix of interaction coefficients for polynomial in molefraction*weight of speciesA (in entropy units, divided by R) More...

DenseMatrix	m_Dij
	Matrix of interactions. More...

Detailed Description

Stefan Maxwell Diffusion Coefficients can be solved for given ion conductivity, mobility ratios, and self diffusion coeffs.

This class is only valid for a common anion mixture of two salts with cations of equal charge. Hence the name _PPN.

This class requres you specify

1 - ion conductivity

2 - mobility ratio of the two cations (set all other ratios to zero)

3 - Self diffusion coefficients of the cations (set others to zero) is used to calculate the "mutual diffusion coefficient". The approximation needed to do so requires the cations have equal charge.

We than calculate the Stefan Maxwell Diffusion Coefficients by

\[ \frac{1}{D_{12}} = (1-\epsilon X_A)(1+\epsilon X_B) \frac{\nu_- + \nu_+}{\nu_-\nu_+^2D} + \frac{z_-z_+ F^2}{\kappa V R T} \]

\[ \frac{1}{D_{12}} = -\epsilon X_B(1-\epsilon X_A) \frac{\nu_- + \nu_+}{\nu_-^2\nu_+D} - \frac{z_-z_+ F^2}{\kappa V R T} \]

\[ \frac{1}{D_{23}} = \epsilon X_A(1+\epsilon X_B) \frac{\nu_- + \nu_+}{\nu_-^2\nu_+D} - \frac{z_-z_+ F^2}{\kappa V R T} \]

where F is Faraday's constant, RT is the gas constant times the tempurature, and V is the molar volume (basis is moles of ions) that is calculated by the thermophase member. X_A and X_B are the mole fractions of the salts composed of cation(1) and cation(2), respectively, that share a common anion(3). \(\nu_{+,-}\) are the stoichiometric coefficients in the dissociation reaction of the salts to the ions with charges of \(z_{+,-}\). Assuming that the cations have equal charge, the "mutual diffusion coefficient" is calculated using the cation self diffusion coefficients.

\[ \frac{1}{\nu_-\nu_+D} = \left(1+\frac{\partial \gamma_B}{\partial N_B} \right)\frac{X_A}{D_2^*}+\left(1+\frac{\partial \gamma_A}{\partial N_A} \right)\frac{X_B}{D_1^*} \]

where the self diffusion coefficients, \(D_i^*\), are temperature and composition parameterized inputs and the derivative of the activity coefficient, \(\frac{\partial \gamma_B}{\partial N_B}\), is calculated by the thermophase member using the excess enthalpy and entropy upon mixing.

Finally, the deviation of the transferrence numbers from ideality, \(\epsilon\), is calculated from the mobility ratio of the cations.

\[ \epsilon = \frac{1-b_2/b_1}{X_A+X_Bb_2/b_1} \]

Where \(b_i\) are the mobilities of the two cations. Everywhere, cation 1 corresponds with salt A and cation 2 with salt B.

Sample input for this method is

*    <transport model="Liquid">
*       <speciesDiffusivity>
*          <compositionDependence model="stefanMaxwell_PPN">
*          </compositionDependence>
*       </speciesDiffusivity>
*    </transport>
*

Definition at line 488 of file LiquidTranInteraction.h.