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# -*- coding: utf-8 -*- 

'''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.''' 

 

from __future__ import division 

from scipy.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} 

 

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. 

''' 

alpha = 1440*dp + 1.96 

beta = 1100*dp + 2.5 

term1 = 1./10**alpha 

Frs_sorta = 1/(g*D)**0.5 

expression1 = term1*Frs_sorta**beta 

expression2 = mp/rhog/(pi/4*D**2) 

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} 

 

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 

''' 

A = pi/4*D**2 

Frp = Vterminal/(g*dp)**0.5 

Frs_sorta = 1./(g*D)**0.5 

expression1 = 0.448*(rhop/rhog)**0.5*(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} 

 

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 

''' 

A = pi/4*D**2 

Frp = Vterminal/(g*dp)**0.5 

Frs_sorta = 1./(g*D)**0.5 

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 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. 

''' 

limit = 1.39*D*(rhop/rhog)**-0.74 

A = pi/4*D**2 

if limit < dp: 

# Corase routine 

Frp = Vterminal/(g*dp)**0.5 

Frs_sorta = 1./(g*D)**0.5 

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./(g*D)**0.5 

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} 

 

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 

''' 

B = (D/dp)**0.025*(rhop/rhog)**0.34 

A = (g*D)**0.5 

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} 

 

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 

''' 

if Vterminal <= 3: 

term1 = (7 + 8/3.*Vterminal)*(dp/D)**0.1 

else: 

term1 = 15.*(dp/D)**0.1 

term2 = 1./(g*D)**0.5 

term3 = mp/rhog/(pi/4*D**2) 

return (term1/term2*term3**0.25)**(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} 

 

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 

''' 

Gs = mp/(pi/4*D**2) 

if Gs/D <= 47000: 

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