# Source code for fluids.saltation

    """Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
    Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com>

    Permission is hereby granted, free of charge, to any person obtaining a copy
    of this software and associated documentation files (the "Software"), to deal
    in the Software without restriction, including without limitation the rights
    to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
    copies of the Software, and to permit persons to whom the Software is
    furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included in all
    copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
    AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
    OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
    SOFTWARE.

    This module contains correlations for calculating the saltation velocity of
    entrained particles.

    For reporting bugs, adding feature requests, or submitting pull requests,
    please use the `GitHub issue tracker <https://github.com/CalebBell/fluids/>`_
    or contact the author at Caleb.Andrew.Bell@gmail.com.

    .. contents:: :local:

    Correlations
    ------------
    .. autofunction :: Rizk
    .. autofunction :: Matsumoto_1974
    .. autofunction :: Matsumoto_1975
    .. autofunction :: Matsumoto_1977
    .. autofunction :: Schade
    .. autofunction :: Weber_saltation
    .. autofunction :: Geldart_Ling

    """

    from math import sqrt

    from fluids.constants import g, pi

    __all__ = ['Rizk', 'Matsumoto_1974', 'Matsumoto_1975', 'Matsumoto_1977',
    'Schade', 'Weber_saltation', 'Geldart_Ling']

    def Rizk(mp, dp, rhog, D):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_ as described in [2]_ and many others.

        .. math::
            \mu=\left(\frac{1}{10^{1440d_p+1.96}}\right)\left(Fr_s\right)^{1100d_p+2.5}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.

        Examples
        --------
        Example is from [3]_.

        >>> Rizk(mp=0.25, dp=100E-6, rhog=1.2, D=.078)
        9.8833092829357

        References
        ----------
        .. [1] Rizk, F. "Pneumatic conveying at optimal operation conditions and a
           solution of Bath's equation." Proceedings of Pneumotransport 3,
           paper D4. BHRA Fluid Engineering, Cranfield, England (1973)
        .. [2] Klinzing, G. E., F. Rizk, R. Marcus, and L. S. Leung. Pneumatic
           Conveying of Solids: A Theoretical and Practical Approach.
           Springer, 2013.
        .. [3] Rhodes, Martin J. Introduction to Particle Technology. Wiley, 2013.
        '''
        alpha = 1440.0*dp + 1.96
        beta = 1100.0*dp + 2.5
        term1 = 0.1**alpha
        Frs_sorta = 1.0/sqrt(g*D)
        expression1 = term1*Frs_sorta**beta
        expression2 = mp/rhog/(pi/4*D*D)
        return (expression2/expression1)**(1./(1. + beta))

    def Matsumoto_1974(mp, rhop, dp, rhog, D, Vterminal=1):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_. Also described in [2]_.

        .. math::
            \mu = 0.448 \left(\frac{\rho_p}{\rho_f}\right)^{0.50}\left(\frac{Fr_p}
            {10}\right)^{-1.75}\left(\frac{Fr_s}{10}\right)^{3}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhop : float
            Particle density, [kg/m^3]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]
        Vterminal : float
            Terminal velocity of particle settling in gas, [m/s]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.
        Result looks high, something may be wrong.
        For particles > 0.3 mm.

        Examples
        --------
        >>> Matsumoto_1974(mp=1., rhop=1000., dp=1E-3, rhog=1.2, D=0.1, Vterminal=5.24)
        19.583617317317895

        References
        ----------
        .. [1] Matsumoto, Shigeru, Michio Kara, Shozaburo Saito, and Siro Maeda.
           "Minimum Transport Velocity for Horizontal Pneumatic Conveying."
           Journal of Chemical Engineering of Japan 7, no. 6 (1974): 425-30.
           doi:10.1252/jcej.7.425.
        .. [2] Jones, Peter J., and L. S. Leung. "A Comparison of Correlations for
           Saltation Velocity in Horizontal Pneumatic Conveying." Industrial &
           Engineering Chemistry Process Design and Development 17, no. 4
           (October 1, 1978): 571-75. doi:10.1021/i260068a031
        '''
        A = pi/4*D**2
        Frp = Vterminal/sqrt(g*dp)
        Frs_sorta = 1./sqrt(g*D)
        expression1 = 0.448*sqrt(rhop/rhog)*(Frp/10.)**-1.75*(Frs_sorta/10.)**3
        expression2 = mp/rhog/A
        return (expression2/expression1)**(1/4.)

    def Matsumoto_1975(mp, rhop, dp, rhog, D, Vterminal=1):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_. Also described in [2]_.

        .. math::
            \mu = 1.11 \left(\frac{\rho_p}{\rho_f}\right)^{0.55}\left(\frac{Fr_p}
            {10}\right)^{-2.3}\left(\frac{Fr_s}{10}\right)^{3}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhop : float
            Particle density, [kg/m^3]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]
        Vterminal : float
            Terminal velocity of particle settling in gas, [m/s]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.
        Result looks high, something may be wrong.
        For particles > 0.3 mm.

        Examples
        --------
        >>> Matsumoto_1975(mp=1., rhop=1000., dp=1E-3, rhog=1.2, D=0.1, Vterminal=5.24)
        18.04523091703009

        References
        ----------
        .. [1] Matsumoto, Shigeru, Shundo Harada, Shozaburo Saito, and Siro Maeda.
           "Saltation Velocity for Horizontal Pneumatic Conveying." Journal of
           Chemical Engineering of Japan 8, no. 4 (1975): 331-33.
           doi:10.1252/jcej.8.331.
        .. [2] Jones, Peter J., and L. S. Leung. "A Comparison of Correlations for
           Saltation Velocity in Horizontal Pneumatic Conveying." Industrial &
           Engineering Chemistry Process Design and Development 17, no. 4
           (October 1, 1978): 571-75. doi:10.1021/i260068a031
        '''
        A = pi/4*D**2
        Frp = Vterminal/sqrt(g*dp)
        Frs_sorta = 1./sqrt(g*D)
        expression1 = 1.11*(rhop/rhog)**0.55*(Frp/10.)**-2.3*(Frs_sorta/10.)**3
        expression2 = mp/rhog/A
        return (expression2/expression1)**(1/4.)

    def Matsumoto_1977(mp, rhop, dp, rhog, D, Vterminal=1):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_ and reproduced in [2]_, [3]_, and [4]_.

        First equation is used if third equation yields d* higher than dp.
        Otherwise, use equation 2.

        .. math::
            \mu = 5560\left(\frac{d_p}{D}\right)^{1.43}\left(\frac{Fr_s}{10}\right)^4

        .. math::
            \mu = 0.373 \left(\frac{\rho_p}{\rho_f}\right)^{1.06}\left(\frac{Fr_p}
            {10}\right)^{-3.7}\left(\frac{Fr_s}{10}\right)^{3.61}

        .. math::
            \frac{d_p^*}{D} = 1.39\left(\frac{\rho_p}{\rho_f}\right)^{-0.74}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhop : float
            Particle density, [kg/m^3]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]
        Vterminal : float
            Terminal velocity of particle settling in gas, [m/s]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearanged to be explicit in terms of saltation velocity
        internally.r

        Examples
        --------
        Example is only a self-test.

        Course routine, terminal velocity input is from example in [2].

        >>> Matsumoto_1977(mp=1., rhop=1000., dp=1E-3, rhog=1.2, D=0.1, Vterminal=5.24)
        16.64284834446686

        References
        ----------
        .. [1] Matsumoto, Shigeru, Makoto Kikuta, and Siro Maeda. "Effect of
           Particle Size on the Minimum Transport Velocity for Horizontal Pneumatic
           Conveying of Solids." Journal of Chemical Engineering of Japan 10,
           no. 4 (1977): 273-79. doi:10.1252/jcej.10.273.
        .. [2] Klinzing, G. E., F. Rizk, R. Marcus, and L. S. Leung. Pneumatic
           Conveying of Solids: A Theoretical and Practical Approach.
           Springer, 2013.
        .. [3] Gomes, L. M., and A. L. Amarante Mesquita. "On the Prediction of
           Pickup and Saltation Velocities in Pneumatic Conveying." Brazilian
           Journal of Chemical Engineering 31, no. 1 (March 2014): 35-46.
           doi:10.1590/S0104-66322014000100005
        .. [4] Rabinovich, Evgeny, and Haim Kalman. "Threshold Velocities of
           Particle-Fluid Flows in Horizontal Pipes and Ducts: Literature Review."
           Reviews in Chemical Engineering 27, no. 5-6 (January 1, 2011).
           doi:10.1515/REVCE.2011.011.
        '''
        limit = 1.39*D*(rhop/rhog)**-0.74
        A = pi/4*D**2
        if limit < dp:
            # Coarse routine
            Frp = Vterminal/sqrt(g*dp)
            Frs_sorta = 1./sqrt(g*D)
            expression1 = 0.373*(rhop/rhog)**1.06*(Frp/10.)**-3.7*(Frs_sorta/10.)**3.61
            expression2 = mp/rhog/A
            return (expression2/expression1)**(1/4.61)
        else:
            Frs_sorta = 1./sqrt(g*D)
            expression1 = 5560*(dp/D)**1.43*(Frs_sorta/10.)**4
            expression2 = mp/rhog/A
            return (expression2/expression1)**(0.2)

    def Schade(mp, rhop, dp, rhog, D):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_ as described in [2]_, [3]_, [4]_, and [5]_.

        .. math::
            Fr_s = \mu^{0.11}\left(\frac{D}{d_p}\right)^{0.025}\left(\frac{\rho_p}
            {\rho_f}\right)^{0.34}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhop : float
            Particle density, [kg/m^3]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.

        Examples
        --------
        >>> Schade(mp=1., rhop=1000., dp=1E-3, rhog=1.2, D=0.1)
        13.697415809497912

        References
        ----------
        .. [1] Schade, B., Zum Ubergang Sprung-Strahnen-forderung bei der
           Horizontalen Pneumatischen Feststoffordrung. Dissertation, University of
           Karlsruche (1987)
        .. [2] Rabinovich, Evgeny, and Haim Kalman. "Threshold Velocities of
           Particle-Fluid Flows in Horizontal Pipes and Ducts: Literature Review."
           Reviews in Chemical Engineering 27, no. 5-6 (January 1, 2011).
           doi:10.1515/REVCE.2011.011.
        .. [3] Setia, G., S. S. Mallick, R. Pan, and P. W. Wypych. "Modeling
           Minimum Transport Boundary for Fluidized Dense-Phase Pneumatic Conveying
           Systems." Powder Technology 277 (June 2015): 244-51.
           doi:10.1016/j.powtec.2015.02.050.
        .. [4] Bansal, A., S. S. Mallick, and P. W. Wypych. "Investigating
           Straight-Pipe Pneumatic Conveying Characteristics for Fluidized
           Dense-Phase Pneumatic Conveying." Particulate Science and Technology
           31, no. 4 (July 4, 2013): 348-56. doi:10.1080/02726351.2012.732677.
        .. [5] Gomes, L. M., and A. L. Amarante Mesquita. "On the Prediction of
           Pickup and Saltation Velocities in Pneumatic Conveying." Brazilian
           Journal of Chemical Engineering 31, no. 1 (March 2014): 35-46.
           doi:10.1590/S0104-66322014000100005
        '''
        B = (D/dp)**0.025*(rhop/rhog)**0.34
        A = sqrt(g*D)
        C = mp/(rhog*pi/4*D**2)
        return (C**0.11*B*A)**(1/1.11)

    def Weber_saltation(mp, rhop, dp, rhog, D, Vterminal=4):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_ as described in [2]_, [3]_, [4]_, and [5]_.

        If Vterminal is under 3 m/s, use equation 1; otherwise, equation 2.

        .. math::
            Fr_s = \left(7 + \frac{8}{3}V_{terminal}\right)\mu^{0.25}
            \left(\frac{d_p}{D}\right)^{0.1}

        .. math::
            Fr_s = 15\mu^{0.25}\left(\frac{d_p}{D}\right)^{0.1}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhop : float
            Particle density, [kg/m^3]
        dp : float
            Particle diameter, [m]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]
        Vterminal : float
            Terminal velocity of particle settling in gas, [m/s]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.

        Examples
        --------
        Examples are only a self-test.

        >>> Weber_saltation(mp=1, rhop=1000., dp=1E-3, rhog=1.2, D=0.1, Vterminal=4)
        15.227445436331474

        References
        ----------
        .. [1] Weber, M. 1981. Principles of hydraulic and pneumatic conveying in
           pipes. Bulk Solids Handling 1: 57-63.
        .. [2] Rabinovich, Evgeny, and Haim Kalman. "Threshold Velocities of
           Particle-Fluid Flows in Horizontal Pipes and Ducts: Literature Review."
           Reviews in Chemical Engineering 27, no. 5-6 (January 1, 2011).
           doi:10.1515/REVCE.2011.011.
        .. [3] Setia, G., S. S. Mallick, R. Pan, and P. W. Wypych. "Modeling
           Minimum Transport Boundary for Fluidized Dense-Phase Pneumatic Conveying
           Systems." Powder Technology 277 (June 2015): 244-51.
           doi:10.1016/j.powtec.2015.02.050.
        .. [4] Bansal, A., S. S. Mallick, and P. W. Wypych. "Investigating
           Straight-Pipe Pneumatic Conveying Characteristics for Fluidized
           Dense-Phase Pneumatic Conveying." Particulate Science and Technology
           31, no. 4 (July 4, 2013): 348-56. doi:10.1080/02726351.2012.732677.
        .. [5] Gomes, L. M., and A. L. Amarante Mesquita. "On the Prediction of
           Pickup and Saltation Velocities in Pneumatic Conveying." Brazilian
           Journal of Chemical Engineering 31, no. 1 (March 2014): 35-46.
           doi:10.1590/S0104-66322014000100005
        '''
        if Vterminal <= 3:
            term1 = (7 + 8/3.*Vterminal)*(dp/D)**0.1
        else:
            term1 = 15.*(dp/D)**0.1
        term2 = 1./sqrt(g*D)
        term3 = mp/rhog/(pi/4*D**2)
        return (term1/term2*sqrt(sqrt(term3)))**(1/1.25)

    def Geldart_Ling(mp, rhog, D, mug):
        r'''Calculates saltation velocity of the gas for pneumatic conveying,
        according to [1]_ as described in [2]_ and [3]_.

        if Gs/D < 47000, use equation 1, otherwise use equation 2.

        .. math::
            V_{salt} = 1.5G_s^{0.465}D^{-0.01} \mu^{0.055}\rho_f^{-0.42}

        .. math::
            V_{salt} = 8.7G_s^{0.302}D^{0.153} \mu^{0.055}\rho_f^{-0.42}

        .. math::
            Fr_s = 15\mu^{0.25}\left(\frac{d_p}{D}\right)^{0.1}

        .. math::
            Fr_s = \frac{V_{salt}}{\sqrt{gD}}

        .. math::
            \mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}

        .. math::
            G_s = \frac{m_p}{A}

        Parameters
        ----------
        mp : float
            Solid mass flow rate, [kg/s]
        rhog : float
            Gas density, [kg/m^3]
        D : float
            Diameter of pipe, [m]
        mug : float
            Gas viscosity, [Pa*s]

        Returns
        -------
        V : float
            Saltation velocity of gas, [m/s]

        Notes
        -----
        Model is rearranged to be explicit in terms of saltation velocity
        internally.

        Examples
        --------
        >>> Geldart_Ling(1., 1.2, 0.1, 2E-5)
        7.467495862402707

        References
        ----------
        .. [1] Weber, M. 1981. Principles of hydraulic and pneumatic conveying in
           pipes. Bulk Solids Handling 1: 57-63.
        .. [2] Rabinovich, Evgeny, and Haim Kalman. "Threshold Velocities of
           Particle-Fluid Flows in Horizontal Pipes and Ducts: Literature Review."
           Reviews in Chemical Engineering 27, no. 5-6 (January 1, 2011).
           doi:10.1515/REVCE.2011.011.
        .. [3] Gomes, L. M., and A. L. Amarante Mesquita. "On the Prediction of
           Pickup and Saltation Velocities in Pneumatic Conveying." Brazilian
           Journal of Chemical Engineering 31, no. 1 (March 2014): 35-46.
           doi:10.1590/S0104-66322014000100005
        '''
        Gs = mp/(0.25*pi*D*D)
        if Gs/D <= 47000.0:
            return 1.5*Gs**0.465*D**-0.01*mug**0.055*rhog**-0.42
        else:
            return 8.7*Gs**0.302*D**0.153*mug**0.055*rhog**-0.42