Tortuosity.h

/**
 * @file TortuosityBruggeman.h
 *  Class to compute the increase in diffusive path length in porous media
 *  assuming the Bruggeman exponent relation
 */

/*
 * Copyright (2005) Sandia Corporation. Under the terms of
 * Contract DE-AC04-94AL85000 with Sandia Corporation, the
 * U.S. Government retains certain rights in this software.
 */

#ifndef CT_TORTUOSITY_H
#define CT_TORTUOSITY_H

namespace Cantera
{

//! Specific Class to handle tortuosity corrections for diffusive transport
//! in porous media using the Bruggeman exponent
/*!
 * Class to compute the increase in diffusive path length associated with
 * tortuous path diffusion through, for example, porous media.
 * This base class implementation relates tortuosity to volume fraction
 * through a power-law relationship that goes back to Bruggeman.  The
 * exponent is referred to as the Bruggeman exponent.
 *
 * Note that the total diffusional flux is generally written as
 *
 * \f[
 *   \frac{ \phi C_T D_i \nabla X_i }{ \tau^2 }
 * \f]
 *
 * where \f$ \phi \f$ is the volume fraction of the transported phase,
 * \f$ \tau \f$ is referred to as the tortuosity.  (Other variables are
 * \f$ C_T \f$, the total concentration, \f$ D_i \f$, the diffusion
 * coefficient, and \f$ X_i \f$, the mole fraction with Fickian
 * transport assumed.)
 *
 * The tortuosity comes into play in conjunction the the
 */
class Tortuosity
{

public:
    //! Default constructor uses Bruggeman exponent of 1.5
    Tortuosity(double setPower = 1.5) : expBrug_(setPower) {
    }

    //! The tortuosity factor models the effective increase in the
    //! diffusive transport length.
    /**
     * This method returns \f$ 1/\tau^2 \f$ in the description of the
     * flux \f$ \phi C_T D_i \nabla X_i / \tau^2 \f$.
     */
    virtual double tortuosityFactor(double porosity) {
        return pow(porosity, expBrug_ - 1.0);
    }

    //! The McMillan number is the ratio of the flux-like
    //! variable to the value it would have without porous flow.
    /**
     * The McMillan number combines the effect of tortuosity
     * and volume fraction of the transported phase.  The net flux
     * observed is then the product of the McMillan number and the
     * non-porous transport rate.  For a conductivity in a non-porous
     * media, \f$ \kappa_0 \f$, the conductivity in the porous media
     * would be \f$ \kappa = (\rm McMillan) \kappa_0 \f$.
     */
    virtual double McMillan(double porosity) {
        return pow(porosity, expBrug_);
    }

protected:
    //! Bruggeman exponent: power to which the tortuosity depends on the volume fraction
    double expBrug_ ;

};



/** This class implements transport coefficient corrections
 * appropriate for porous media where percolation theory applies.
 * It is derived from the Tortuosity class.
 */
class TortuosityPercolation : public  Tortuosity
{

public:
    //! Default constructor uses Bruggeman exponent of 1.5
    TortuosityPercolation(double percolationThreshold = 0.4, double conductivityExponent = 2.0) : percolationThreshold_(percolationThreshold), conductivityExponent_(conductivityExponent)  {
    }

    //! The tortuosity factor models the effective increase in the
    //! diffusive transport length.
    /**
     * This method returns \f$ 1/\tau^2 \f$ in the description of the
     * flux \f$ \phi C_T D_i \nabla X_i / \tau^2 \f$.
     */
    double tortuosityFactor(double porosity) {
        return McMillan(porosity) / porosity;
    }

    //! The McMillan number is the ratio of the flux-like
    //! variable to the value it would have without porous flow.
    /**
     * The McMillan number combines the effect of tortuosity
     * and volume fraction of the transported phase.  The net flux
     * observed is then the product of the McMillan number and the
     * non-porous transport rate.  For a conductivity in a non-porous
     * media, \f$ \kappa_0 \f$, the conductivity in the porous media
     * would be \f$ \kappa = (\rm McMillan) \kappa_0 \f$.
     */
    double McMillan(double porosity) {
        return pow((porosity - percolationThreshold_)
                   / (1.0 - percolationThreshold_),
                   conductivityExponent_);
    }

protected:
    //! Critical volume fraction / site density for percolation
    double percolationThreshold_;
    //! Conductivity exponent
    /**
     * The McMillan number (ratio of effective conductivity
     * to non-porous conductivity) is
     * \f[ \kappa/\kappa_0 = ( \phi - \phi_c )^\mu \f]
     * where \f$ \mu \f$ is the conductivity exponent (typical
     * values range from 1.6 to 2.0) and \f$ \phi_c \f$
     * is the percolation threshold.
     */
    double conductivityExponent_;
};



/** This class implements transport coefficient corrections
 * appropriate for porous media with a dispersed phase.
 * This model goes back to Maxwell.  The formula for the
 * conductivity is expressed in terms of the volume fraction
 * of the continuous phase, \f$ \phi \f$, and the relative
 * conductivities of the dispersed and continuous phases,
 * \f$ r = \kappa_d / \kappa_0 \f$.  For dilute particle
 * suspensions the effective conductivity is
 * \f[
 *    \kappa / \kappa_0 = 1 + 3 ( 1 - \phi ) ( r - 1 ) / ( r + 2 )
 *                        + O(\phi^2)
 * \f]
 * The class is derived from the Tortuosity class.
 */
class TortuosityMaxwell : public Tortuosity
{

public:
    //! Default constructor uses Bruggeman exponent of 1.5
    TortuosityMaxwell(double relativeConductivites = 0.0) : relativeConductivites_(relativeConductivites)  {
    }

    //! The tortuosity factor models the effective increase in the
    //! diffusive transport length.
    /**
     * This method returns \f$ 1/\tau^2 \f$ in the description of the
     * flux \f$ \phi C_T D_i \nabla X_i / \tau^2 \f$.
     */
    double tortuosityFactor(double porosity) {
        return McMillan(porosity) / porosity;
    }

    //! The McMillan number is the ratio of the flux-like
    //! variable to the value it would have without porous flow.
    /**
     * The McMillan number combines the effect of tortuosity
     * and volume fraction of the transported phase.  The net flux
     * observed is then the product of the McMillan number and the
     * non-porous transport rate.  For a conductivity in a non-porous
     * media, \f$ \kappa_0 \f$, the conductivity in the porous media
     * would be \f$ \kappa = (\rm McMillan) \kappa_0 \f$.
     */
    double McMillan(double porosity) {
        return 1 + 3 * (1.0 - porosity) * (relativeConductivites_ - 1.0) / (relativeConductivites_ + 2);
    }

protected:
    //! Relative  conductivities of the dispersed and continuous phases,
    //! \code{relativeConductivites_}\f$  = \kappa_d / \kappa_0 \f$.
    double relativeConductivites_;

};

}
#endif