Pneumatic conveying (fluids.saltation)

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 or contact the author at Caleb.Andrew.Bell@gmail.com.

Correlations

fluids.saltation.Rizk(mp, dp, rhog, D)

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Returns

Vfloat

Saltation velocity of gas, [m/s]

Notes

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

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.

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)

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhopfloat

Particle density, [kg/m^3]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Vterminalfloat

Terminal velocity of particle settling in gas, [m/s]

Returns

Vfloat

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

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

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)

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhopfloat

Particle density, [kg/m^3]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Vterminalfloat

Terminal velocity of particle settling in gas, [m/s]

Returns

Vfloat

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

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

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)

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.

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

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$Fr_p = \frac{V_{terminal}}{\sqrt{gd_p}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhopfloat

Particle density, [kg/m^3]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Vterminalfloat

Terminal velocity of particle settling in gas, [m/s]

Returns

Vfloat

Saltation velocity of gas, [m/s]

Notes

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

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.

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)

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhopfloat

Particle density, [kg/m^3]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Returns

Vfloat

Saltation velocity of gas, [m/s]

Notes

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

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

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)

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.

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhopfloat

Particle density, [kg/m^3]

dpfloat

Particle diameter, [m]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

Vterminalfloat

Terminal velocity of particle settling in gas, [m/s]

Returns

Vfloat

Saltation velocity of gas, [m/s]

Notes

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

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

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)

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.

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

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

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

$$Fr_s = \frac{V_{salt}}{\sqrt{gD}}$$

$$\mu = \frac{m_p}{\frac{\pi}{4}D^2V \rho_f}$$

$$G_s = \frac{m_p}{A}$$

Parameters

mpfloat

Solid mass flow rate, [kg/s]

rhogfloat

Gas density, [kg/m^3]

Dfloat

Diameter of pipe, [m]

mugfloat

Gas viscosity, [Pa*s]

Returns

Vfloat

Saltation velocity of gas, [m/s]

Notes

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

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

Examples

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