# Pneumatic conveying (fluids.saltation)¶

fluids.saltation.Rizk(mp, dp, rhog, D)[source]

Calculates saltation velocity of the gas for pneumatic conveying, according to [1] as described in [2] and many others.

\begin{align}\begin{aligned}\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}\end{aligned}\end{align}
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] V : float Saltation velocity of gas, [m/s]

Notes

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

References

 [1] (1, 2) 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] (1, 2) Klinzing, G. E., F. Rizk, R. Marcus, and L. S. Leung. Pneumatic Conveying of Solids: A Theoretical and Practical Approach. Springer, 2013.
 [3] (1, 2) Rhodes, Martin J. Introduction to Particle Technology. Wiley, 2013.

Examples

Example is from [3].

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

fluids.saltation.Matsumoto_1974(mp, rhop, dp, rhog, D, Vterminal=1)[source]

Calculates saltation velocity of the gas for pneumatic conveying, according to [1]. Also described in [2].

\begin{align}\begin{aligned}\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}\end{aligned}\end{align}
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] 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.

References

 [1] (1, 2) 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] (1, 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

Examples

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

fluids.saltation.Matsumoto_1975(mp, rhop, dp, rhog, D, Vterminal=1)[source]

Calculates saltation velocity of the gas for pneumatic conveying, according to [1]. Also described in [2].

\begin{align}\begin{aligned}\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}\end{aligned}\end{align}
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] 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.

References

 [1] (1, 2) 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] (1, 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

Examples

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

fluids.saltation.Matsumoto_1977(mp, rhop, dp, rhog, D, Vterminal=1)[source]

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.

\begin{align}\begin{aligned}\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}\end{aligned}\end{align}
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] V : float Saltation velocity of gas, [m/s]

Notes

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

References

 [1] (1, 2) 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] (1, 2) Klinzing, G. E., F. Rizk, R. Marcus, and L. S. Leung. Pneumatic Conveying of Solids: A Theoretical and Practical Approach. Springer, 2013.
 [3] (1, 2) 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] (1, 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.

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

fluids.saltation.Schade(mp, rhop, dp, rhog, D)[source]

Calculates saltation velocity of the gas for pneumatic conveying, according to [1] as described in [2], [3], [4], and [5].

\begin{align}\begin{aligned}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}\end{aligned}\end{align}
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] V : float Saltation velocity of gas, [m/s]

Notes

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

References

 [1] (1, 2) Schade, B., Zum Ubergang Sprung-Strahnen-forderung bei der Horizontalen Pneumatischen Feststoffordrung. Dissertation, University of Karlsruche (1987)
 [2] (1, 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] (1, 2) 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] (1, 2) 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] (1, 2) 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

Examples

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

fluids.saltation.Weber_saltation(mp, rhop, dp, rhog, D, Vterminal=4)[source]

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.

\begin{align}\begin{aligned}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}\end{aligned}\end{align}
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] V : float Saltation velocity of gas, [m/s]

Notes

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

References

 [1] (1, 2) Weber, M. 1981. Principles of hydraulic and pneumatic conveying in pipes. Bulk Solids Handling 1: 57-63.
 [2] (1, 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] (1, 2) 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] (1, 2) 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] (1, 2) 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

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

fluids.saltation.Geldart_Ling(mp, rhog, D, mug)[source]

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.

\begin{align}\begin{aligned}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}\end{aligned}\end{align}
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] V : float Saltation velocity of gas, [m/s]

Notes

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

References

 [1] (1, 2) Weber, M. 1981. Principles of hydraulic and pneumatic conveying in pipes. Bulk Solids Handling 1: 57-63.
 [2] (1, 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] (1, 2) 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

Examples

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