DustyGasTransport.h

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/**
 * @file DustyGasTransport.h Headers for the DustyGasTransport object, which
 *   models transport properties in porous media using the dusty gas
 *   approximation (see \ref tranprops and \link Cantera::DustyGasTransport
 *   DustyGasTransport \endlink) .
 */
 
// This file is part of Cantera. See License.txt in the top-level directory or
// at https://cantera.org/license.txt for license and copyright information.
 
#ifndef CT_DUSTYGASTRAN_H
#define CT_DUSTYGASTRAN_H
 
// Cantera includes
#include "TransportBase.h"
#include "cantera/numerics/DenseMatrix.h"
 
namespace Cantera
{
//! Class DustyGasTransport implements the Dusty Gas model for transport in porous media.
/*!
 * As implemented here, only species transport is handled. The viscosity,
 * thermal conductivity, and thermal diffusion coefficients are not implemented.
 *
 * The dusty gas model includes the effects of Darcy's law. There is a net flux
 * of species due to a pressure gradient that is part of Darcy's law.
 *
 * The dusty gas model expresses the value of the molar flux of species
 * \f$ k \f$, \f$ J_k \f$ by the following formula.
 *
 * \f[
 *     \sum_{j \ne k}{\frac{X_j J_k - X_k J_j}{D^e_{kj}}} + \frac{J_k}{\mathcal{D}^{e}_{k,knud}} =
 *             - \nabla C_k  - \frac{C_k}{\mathcal{D}^{e}_{k,knud}} \frac{\kappa}{\mu} \nabla p
 * \f]
 *
 * \f$ j \f$ is a sum over all species in the gas.
 *
 * The effective Knudsen diffusion coefficients are given by the following form
 *
 * \f[
 *    \mathcal{D}^e_{k,knud} =  \frac{2}{3} \frac{r_{pore} \phi}{\tau} \left( \frac{8 R T}{\pi W_k}  \right)^{1/2}
 * \f]
 *
 * The effective knudsen diffusion coefficients take into account the effects of
 * collisions of gas-phase molecules with the wall.
 *
 * References for the Dusty Gas Model
 *
 * 1. H. Zhu, R. J. Kee, "Modeling Electrochemical Impedance Spectra in SOFC
 *    Button Cells with Internal Methane Reforming," J. Electrochem. Soc.,
 *    153(9) A1765-1772 (2006).
 * 2. H. Zhu, R. J. Kee, V. M. Janardhanan, O. Deutschmann, D. G. Goodwin, J.
 *    Electrochem. Soc., 152, A2427 (2005).
 * 3. E. A. Mason, A. P. Malinauskas," Gas Transport in Porous Media: the Dusty-
 *    Gas Model", American Elsevier, New York (1983).
 * 4. J. W. Veldsink, R. M. J. van Damme, G. F. Versteeg, W. P. M. van Swaaij,
 *    "The use of the dusty gas model for the description of mass transport with
 *    chemical reaction in porous media," Chemical Engineering Journal, 57, 115
 *    - 125 (1995).
 * @ingroup tranprops
 */
class DustyGasTransport : public Transport
{
public:
    //! default constructor
    /*!
     *  @param thermo   Pointer to the ThermoPhase object for this phase.
     *      Defaults to zero.
     */
    DustyGasTransport(ThermoPhase* thermo=0);
 
    //  overloaded base class methods
 
    virtual void setThermo(ThermoPhase& thermo);
 
    virtual std::string transportType() const {
        return "DustyGas";
    }
 
    virtual void getMultiDiffCoeffs(const size_t ld, doublereal* const d);
 
    //! Get the molar fluxes [kmol/m^2/s], given the thermodynamic state at two nearby points.
    /*!
     *   \f[
     *       J_k = - \sum_{j = 1, N} \left[D^{multi}_{kj}\right]^{-1} \left( \nabla C_j  + \frac{C_j}{\mathcal{D}^{knud}_j} \frac{\kappa}{\mu} \nabla p \right)
     *   \f]
     *
     * @param  state1  Array of temperature, density, and mass fractions for state 1.
     * @param  state2  Array of temperature, density, and mass fractions for state 2.
     * @param  delta   Distance from state 1 to state 2 (m).
     *
     * @param fluxes   Vector of species molar fluxes due to diffusional driving force
     */
    virtual void getMolarFluxes(const doublereal* const state1,
                                const doublereal* const state2, const doublereal delta,
                                doublereal* const fluxes);
 
    // new methods added in this class
 
    //! Set the porosity (dimensionless)
    /*!
     * @param porosity  Set the value of the porosity
     */
    void setPorosity(doublereal porosity);
 
    //! Set the tortuosity (dimensionless)
    /*!
     * Tortuosity is considered to be constant within the object
     *
     * @param tort  Value of the tortuosity
     */
    void setTortuosity(doublereal tort);
 
    //! Set the mean pore radius (m)
    /*!
     * @param rbar  Value of the pore radius ( m)
     */
    void setMeanPoreRadius(doublereal rbar);
 
    //! Set the mean particle diameter
    /*!
     * @param dbar  Set the mean particle diameter (m)
     */
    void setMeanParticleDiameter(doublereal dbar);
 
    //! Set the permeability of the media
    /*!
     * If not set, the value for close-packed spheres will be used by default.
     *
     * The value for close-packed spheres is given below, where p is the
     * porosity, t is the tortuosity, and d is the diameter of the sphere
     *
     * \f[
     *     \kappa = \frac{p^3 d^2}{72 t (1 - p)^2}
     * \f]
     *
     * @param B  set the permeability of the media (units = m^2)
     */
    void setPermeability(doublereal B);
 
    //! Return a reference to the transport manager used to compute the gas
    //! binary diffusion coefficients and the viscosity.
    /*!
     * @returns a reference to the gas transport object
     */
    Transport& gasTransport();
 
    //! Make the TransportFactory object a friend, because this object has
    //! restricted its instantiation to classes which are friends.
    friend class TransportFactory;
 
protected:
    //! Initialization routine called by TransportFactory
    /*!
     * The DustyGas model is a subordinate model to the gas phase transport
     * model. Here we set the gas phase models.
     *
     * This is a protected routine, so that initialization of the Model must
     * occur within Cantera's setup
     *
     * @param  phase  Pointer to the underlying ThermoPhase model for the gas phase
     * @param  gastr  Pointer to the underlying Transport model for transport in
     *     the gas phase.
     */
    void initialize(ThermoPhase* phase, Transport* gastr);
 
private:
    //! Update temperature-dependent quantities within the object
    /*!
     * The object keeps a value m_temp, which is the temperature at which
     * quantities were last evaluated at. If the temperature is changed, update
     * Booleans are set false, triggering recomputation.
     */
    void updateTransport_T();
 
    //! Update concentration-dependent quantities within the object
    /*!
     * The object keeps a value m_temp, which is the temperature at which
     * quantities were last evaluated at. If the temperature is changed, update
     * Booleans are set false, triggering recomputation.
     */
    void updateTransport_C();
 
    //! Private routine to update the dusty gas binary diffusion coefficients
    /*!
     * The dusty gas binary diffusion coefficients \f$  D^{dg}_{i,j} \f$ are
     * evaluated from the binary gas-phase diffusion coefficients \f$
     * D^{bin}_{i,j} \f$  using the following formula
     *
     * \f[
     *     D^{dg}_{i,j} =  \frac{\phi}{\tau} D^{bin}_{i,j}
     * \f]
     *
     * where \f$ \phi \f$ is the porosity of the media and \f$ \tau \f$ is the
     * tortuosity of the media.
     */
    void updateBinaryDiffCoeffs();
 
    //! Update the Multicomponent diffusion coefficients that are used in the
    //! approximation
    /*!
     * This routine updates the H matrix and then inverts it.
     */
    void updateMultiDiffCoeffs();
 
    //! Update the Knudsen diffusion coefficients
    /*!
     * The Knudsen diffusion coefficients are given by the following form
     *
     * \f[
     *     \mathcal{D}^{knud}_k =  \frac{2}{3} \frac{r_{pore} \phi}{\tau} \left( \frac{8 R T}{\pi W_k}  \right)^{1/2}
     * \f]
     */
    void updateKnudsenDiffCoeffs();
 
    //! Calculate the H matrix
    /*!
     * The multicomponent diffusion H matrix \f$  H_{k,l} \f$ is given by the following form
     *
     * \f[
     *    H_{k,l} = - \frac{X_k}{D_{k,l}}
     * \f]
     * \f[
     *    H_{k,k} = \frac{1}{\mathcal(D)^{knud}_{k}} + \sum_{j \ne k}^N{ \frac{X_j}{D_{k,j}} }
     * \f]
     */
    void eval_H_matrix();
 
    //! Local copy of the species molecular weights
    /*!
     *  units kg /kmol
     *  length = m_nsp;
     */
    vector_fp m_mw;
 
    //! binary diffusion coefficients
    DenseMatrix m_d;
 
    //! mole fractions
    vector_fp m_x;
 
    //! Knudsen diffusion coefficients. @see updateKnudsenDiffCoeffs()
    vector_fp m_dk;
 
    //! temperature
    doublereal m_temp;
 
    //! Multicomponent diffusion coefficients. @see eval_H_matrix()
    DenseMatrix m_multidiff;
 
    //! work space of size m_nsp;
    vector_fp m_spwork;
 
    //! work space of size m_nsp;
    vector_fp m_spwork2;
 
    //! Pressure Gradient
    doublereal m_gradP;
 
    //! Update-to-date variable for Knudsen diffusion coefficients
    bool m_knudsen_ok;
 
    //! Update-to-date variable for Binary diffusion coefficients
    bool m_bulk_ok;
 
    //! Porosity
    doublereal m_porosity;
 
    //! Tortuosity
    doublereal m_tortuosity;
 
    //! Pore radius (meter)
    doublereal m_pore_radius;
 
    //! Particle diameter
    /*!
     * The medium is assumed to consist of particles of size m_diam. units =  m
     */
    doublereal m_diam;
 
    //! Permeability of the media
    /*!
     * The permeability is the proportionality constant for Darcy's law which
     * relates discharge rate and viscosity to the applied pressure gradient.
     *
     * Below is Darcy's law, where \f$ \kappa \f$ is the permeability
     *
     * \f[
     *     v = \frac{\kappa}{\mu} \frac{\delta P}{\delta x}
     * \f]
     *
     * units are m2
     */
    doublereal m_perm;
 
    //! Pointer to the transport object for the gas phase
    std::unique_ptr<Transport> m_gastran;
};
}
#endif