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# -*- coding: utf-8 -*- 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.'''
'Schade', 'Weber_saltation', 'Geldart_Ling']
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}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
\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 rearanged 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. '''
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}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}
\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. 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 '''
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}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}
\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. 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 '''
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 yeilds 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
\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}
\frac{d_p^*}{D} = 1.39\left(\frac{\rho_p}{\rho_f}\right)^{-0.74}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}
\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.
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. ''' # Corase routine else:
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}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
\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 rearanged 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 '''
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}
Fr_s = 15\mu^{0.25}\left(\frac{d_p}{D}\right)^{0.1}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
\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.
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 ''' else:
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}
V_{salt} = 8.7G_s^{0.302}D^{0.153} \mu^{0.055}\rho_f^{-0.42}
Fr_s = 15\mu^{0.25}\left(\frac{d_p}{D}\right)^{0.1}
Fr_s = \frac{V_{salt}}{\sqrt{gD}}
\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}
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 rearanged 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 ''' else:
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