Drag and terminal velocity (fluids.drag)

fluids.drag.drag_sphere(Re, Method=None, AvailableMethods=False)[source]

This function handles calculation of drag coefficient on spheres. Twenty methods are available, all requiring only the Reynolds number of the sphere. Most methods are valid from Re=0 to Re=200,000. A correlation will be automatically selected if none is specified. The full list of correlations valid for a given Reynolds number can be obtained with the AvailableMethods flag.

If no correlation is selected, the following rules are used:

  • If Re < 0.01, use Stoke’s solution.
  • If 0.01 <= Re < 0.1, linearly combine ‘Barati’ with Stokes’s solution such that at Re = 0.1 the solution is ‘Barati’, and at Re = 0.01 the solution is ‘Stokes’.
  • If 0.1 <= Re <= ~212963, use the ‘Barati’ solution.
  • If ~212963 < Re <= 1E6, use the ‘Barati_high’ solution.
  • For Re > 1E6, raises an exception; no valid results have been found.
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

methods : list, only returned if AvailableMethods == True

List of methods which can be used to calculate Cd with the given Re

Other Parameters:
 

Method : string, optional

A string of the function name to use, as in the dictionary drag_sphere_correlations

AvailableMethods : bool, optional

If True, function will consider which methods which can be used to calculate Cd with the given Re

Examples

>>> drag_sphere(200)
0.7682237950389874
fluids.drag.v_terminal(D, rhop, rho, mu, Method=None)[source]

Calculates terminal velocity of a falling sphere using any drag coefficient method supported by drag_sphere. The laminar solution for Re < 0.01 is first tried; if the resulting terminal velocity does not put it in the laminar regime, a numerical solution is used.

\[v_t = \sqrt{\frac{4 g d_p (\rho_p-\rho_f)}{3 C_D \rho_f }}\]
Parameters:

D : float

Diameter of the sphere, [m]

rhop : float

Particle density, [kg/m^3]

rho : float

Density of the surrounding fluid, [kg/m^3]

mu : float

Viscosity of the surrounding fluid [Pa*s]

Method : string, optional

A string of the function name to use, as in the dictionary drag_sphere_correlations

Returns:

v_t : float

Terminal velocity of falling sphere [m/s]

Notes

As there are no correlations implemented for Re > 1E6, an error will be raised if the numerical solver seeks a solution above that limit.

The laminar solution is given in [R301340] and is:

\[v_t = \frac{g d_p^2 (\rho_p - \rho_f)}{18 \mu_f}\]

References

[R301340](1, 2) Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.
[R302340]Rushton, Albert, Anthony S. Ward, and Richard G. Holdich. Solid-Liquid Filtration and Separation Technology. 1st edition. Weinheim ; New York: Wiley-VCH, 1996.

Examples

>>> v_terminal(D=70E-6, rhop=2600., rho=1000., mu=1E-3)
0.004142497244531304
fluids.drag.integrate_drag_sphere(D, rhop, rho, mu, t, V=0, Method=None, distance=False)[source]

Integrates the velocity and distance traveled by a particle moving at a speed which will converge to its terminal velocity.

Performs an integration of the following expression for acceleration:

\[a = \frac{g(\rho_p-\rho_f)}{\rho_p} - \frac{3C_D \rho_f u^2}{4D \rho_p}\]
Parameters:

D : float

Diameter of the sphere, [m]

rhop : float

Particle density, [kg/m^3]

rho : float

Density of the surrounding fluid, [kg/m^3]

mu : float

Viscosity of the surrounding fluid [Pa*s]

t : float

Time to integrate the particle to, [s]

V : float

Initial velocity of the particle, [m/s]

Method : string, optional

A string of the function name to use, as in the dictionary drag_sphere_correlations

distance : bool, optional

Whether or not to calculate the distance traveled and return it as well

Returns:

v : float

Velocity of falling sphere after time t [m/s]

x : float, returned only if distance == True

Distance traveled by the falling sphere in time t, [m]

Notes

This can be relatively slow as drag correlations can be complex.

Examples

>>> integrate_drag_sphere(D=0.001, rhop=2200., rho=1.2, mu=1.78E-5, t=0.5,
... V=30, distance=True)
(9.686465044063436, 7.829454643649386)
fluids.drag.Stokes(Re)[source]

Calculates drag coefficient of a smooth sphere using Stoke’s law.

\[C_D = 24/Re\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 0.3

References

[R303342]Rhodes, Martin J. Introduction to Particle Technology. Wiley, 2013.

Examples

>>> Stokes(0.1)
240.0
fluids.drag.Barati(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R304343].

\[C_D = 5.4856\times10^9\tanh(4.3774\times10^{-9}/Re) + 0.0709\tanh(700.6574/Re) + 0.3894\tanh(74.1539/Re) - 0.1198\tanh(7429.0843/Re) + 1.7174\tanh[9.9851/(Re+2.3384)] + 0.4744\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R304343](1, 2, 3) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

Maching example in [R304343], in a table of calculated values.

>>> Barati(200.)
0.7682237950389874
fluids.drag.Barati_high(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R305344].

\[C_D = 8\times 10^{-6}\left[(Re/6530)^2 + \tanh(Re) - 8\ln(Re)/\ln(10)\right] - 0.4119\exp(-2.08\times10^{43}/[Re + Re^2]^4) -2.1344\exp(-\{[\ln(Re^2 + 10.7563)/\ln(10)]^2 + 9.9867\}/Re) +0.1357\exp(-[(Re/1620)^2 + 10370]/Re) - 8.5\times 10^{-3}\{2\ln[\tanh(\tanh(Re))]/\ln(10) - 2825.7162\}/Re + 2.4795\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 1E6 This model is the wider-range model the authors developed. At sufficiently low diameters or Re values, drag is no longer a phenomena.

References

[R305344](1, 2, 3) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

Maching example in [R305344], in a table of calculated values.

>>> Barati_high(200.)
0.7730544082789523
fluids.drag.Rouse(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R306345] as described in [R307345].

\[C_D = \frac{24}{Re} + \frac{3}{Re^{0.5}} + 0.34\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R306345](1, 2) H. Rouse, Fluid Mechanics for Hydraulic Engineers, Dover, New York, N.Y., 1938
[R307345](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Rouse(200.)
0.6721320343559642
fluids.drag.Engelund_Hansen(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R308347] as described in [R309347].

\[C_D = \frac{24}{Re} + 1.5\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R308347](1, 2) F. Engelund, E. Hansen, Monograph on Sediment Transport in Alluvial Streams, Monograpsh Denmark Technical University, Hydraulic Lab, Denmark, 1967.
[R309347](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Engelund_Hansen(200.)
1.62
fluids.drag.Clift_Gauvin(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R310349] as described in [R311349].

\[C_D = \frac{24}{Re}(1 + 0.152Re^{0.677}) + \frac{0.417} {1 + 5070Re^{-0.94}}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R310349](1, 2) R. Clift, W.H. Gauvin, The motion of particles in turbulent gas streams, Proc. Chemeca, 70, 1970, pp. 14-28.
[R311349](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Clift_Gauvin(200.)
0.7905400398000133
fluids.drag.Morsi_Alexander(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R312351] as described in [R313351].

\[\begin{split}C_D = \left\{ \begin{array}{ll} \frac{24}{Re} & \mbox{if $Re < 0.1$}\\ \frac{22.73}{Re}+\frac{0.0903}{Re^2} + 3.69 & \mbox{if $0.1 < Re < 1$}\\ \frac{29.1667}{Re}-\frac{3.8889}{Re^2} + 1.2220 & \mbox{if $1 < Re < 10$}\\ \frac{46.5}{Re}-\frac{116.67}{Re^2} + 0.6167 & \mbox{if $10 < Re < 100$}\\ \frac{98.33}{Re}-\frac{2778}{Re^2} + 0.3644 & \mbox{if $100 < Re < 1000$}\\ \frac{148.62}{Re}-\frac{4.75\times10^4}{Re^2} + 0.3570 & \mbox{if $1000 < Re < 5000$}\\ \frac{-490.5460}{Re}+\frac{57.87\times10^4}{Re^2} + 0.46 & \mbox{if $5000 < Re < 10000$}\\ \frac{-1662.5}{Re}+\frac{5.4167\times10^6}{Re^2} + 0.5191 & \mbox{if $10000 < Re < 50000$}\end{array} \right.\end{split}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5. Original was reviewed, and confirmed to contain the cited equations.

References

[R312351](1, 2) Morsi, S. A., and A. J. Alexander. “An Investigation of Particle Trajectories in Two-Phase Flow Systems.” Journal of Fluid Mechanics 55, no. 02 (September 1972): 193-208. doi:10.1017/S0022112072001806.
[R313351](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Morsi_Alexander(200)
0.7866
fluids.drag.Graf(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R314353] as described in [R315353].

\[C_D = \frac{24}{Re} + \frac{7.3}{1+Re^{0.5}} + 0.25\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R314353](1, 2) W.H. Graf, Hydraulics of Sediment Transport, Water Resources Publications, Littleton, Colorado, 1984.
[R315353](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Graf(200.)
0.8520984424785725
fluids.drag.Flemmer_Banks(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R316355] as described in [R317355].

\[ \begin{align}\begin{aligned}C_D = \frac{24}{Re}10^E\\E = 0.383Re^{0.356}-0.207Re^{0.396} - \frac{0.143}{1+(\log_{10} Re)^2}\end{aligned}\end{align} \]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R316355](1, 2) Flemmer, R. L. C., and C. L. Banks. “On the Drag Coefficient of a Sphere.” Powder Technology 48, no. 3 (November 1986): 217-21. doi:10.1016/0032-5910(86)80044-4.
[R317355](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Flemmer_Banks(200.)
0.7849169609270039
fluids.drag.Khan_Richardson(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R318357] as described in [R319357].

\[C_D = (2.49Re^{-0.328} + 0.34Re^{0.067})^{3.18}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R318357](1, 2) Khan, A. R., and J. F. Richardson. “The Resistance to Motion of a Solid Sphere in a Fluid.” Chemical Engineering Communications 62, no. 1-6 (December 1, 1987): 135-50. doi:10.1080/00986448708912056.
[R319357](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Khan_Richardson(200.)
0.7747572379211097
fluids.drag.Swamee_Ojha(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R320359] as described in [R321359].

\[C_D = 0.5\left\{16\left[(\frac{24}{Re})^{1.6} + (\frac{130}{Re})^{0.72} \right]^{2.5}+ \left[\left(\frac{40000}{Re}\right)^2 + 1\right]^{-0.25} \right\}^{0.25}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 1.5E5

References

[R320359](1, 2) Swamee, P. and Ojha, C. (1991). “Drag Coefficient and Fall Velocity of nonspherical particles.” J. Hydraul. Eng., 117(5), 660-667.
[R321359](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Swamee_Ojha(200.)
0.8490012397545713
fluids.drag.Yen(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R322361] as described in [R323361].

\[C_D = \frac{24}{Re}\left(1 + 0.15\sqrt{Re} + 0.017Re\right) - \frac{0.208}{1+10^4Re^{-0.5}}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R322361](1, 2) B.C. Yen, Sediment Fall Velocity in Oscillating Flow, University of Virginia, Department of Civil Engineering, 1992.
[R323361](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Yen(200.)
0.7822647002187014
fluids.drag.Haider_Levenspiel(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R324363] as described in [R325363].

\[C_D=\frac{24}{Re}(1+0.1806Re^{0.6459})+\left(\frac{0.4251}{1 +\frac{6880.95}{Re}}\right)\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5 An improved version of this correlation is in Brown and Lawler.

References

[R324363](1, 2) Haider, A., and O. Levenspiel. “Drag Coefficient and Terminal Velocity of Spherical and Nonspherical Particles.” Powder Technology 58, no. 1 (May 1989): 63-70. doi:10.1016/0032-5910(89)80008-7.
[R325363](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Haider_Levenspiel(200.)
0.7959551680251666
fluids.drag.Cheng(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R326365] as described in [R327365].

\[C_D=\frac{24}{Re}(1+0.27Re)^{0.43}+0.47[1-\exp(-0.04Re^{0.38})]\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 2E5

References

[R326365](1, 2) Cheng, Nian-Sheng. “Comparison of Formulas for Drag Coefficient and Settling Velocity of Spherical Particles.” Powder Technology 189, no. 3 (February 13, 2009): 395-398. doi:10.1016/j.powtec.2008.07.006.
[R327365](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Cheng(200.)
0.7939143028294227
fluids.drag.Terfous(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R328367] as described in [R329367].

\[C_D = 2.689 + \frac{21.683}{Re} + \frac{0.131}{Re^2} - \frac{10.616}{Re^{0.1}} + \frac{12.216}{Re^{0.2}}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is 0.1 < Re <= 5E4

References

[R328367](1, 2) Terfous, A., A. Hazzab, and A. Ghenaim. “Predicting the Drag Coefficient and Settling Velocity of Spherical Particles.” Powder Technology 239 (May 2013): 12-20. doi:10.1016/j.powtec.2013.01.052.
[R329367](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Terfous(200.)
0.7814651149769638
fluids.drag.Mikhailov_Freire(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R330369] as described in [R331369].

\[C_D = \frac{3808[(1617933/2030) + (178861/1063)Re + (1219/1084)Re^2]} {681Re[(77531/422) + (13529/976)Re - (1/71154)Re^2]}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 118300

References

[R330369](1, 2) Mikhailov, M. D., and A. P. Silva Freire. “The Drag Coefficient of a Sphere: An Approximation Using Shanks Transform.” Powder Technology 237 (March 2013): 432-35. doi:10.1016/j.powtec.2012.12.033.
[R331369](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Mikhailov_Freire(200.)
0.7514111388018659
fluids.drag.Clift(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R332371] as described in [R333371].

\[\begin{split}C_D = \left\{ \begin{array}{ll} \frac{24}{Re} + \frac{3}{16} & \mbox{if $Re < 0.01$}\\ \frac{24}{Re}(1 + 0.1315Re^{0.82 - 0.05\log Re}) & \mbox{if $0.01 < Re < 20$}\\ \frac{24}{Re}(1 + 0.1935Re^{0.6305}) & \mbox{if $20 < Re < 260$}\\ 10^{[1.6435 - 1.1242\log Re + 0.1558[\log Re]^2} & \mbox{if $260 < Re < 1500$}\\ 10^{[-2.4571 + 2.5558\log Re - 0.9295[\log Re]^2 + 0.1049[\log Re]^3} & \mbox{if $1500 < Re < 12000$}\\ 10^{[-1.9181 + 0.6370\log Re - 0.0636[\log Re]^2} & \mbox{if $12000 < Re < 44000$}\\ 10^{[-4.3390 + 1.5809\log Re - 0.1546[\log Re]^2} & \mbox{if $44000 < Re < 338000$}\\ 9.78 - 5.3\log Re & \mbox{if $338000 < Re < 400000$}\\ 0.19\log Re - 0.49 & \mbox{if $400000 < Re < 1000000$}\end{array} \right.\end{split}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 1E6.

References

[R332371](1, 2) R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops, and Particles, Academic, New York, 1978.
[R333371](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Clift(200)
0.7756342422322543
fluids.drag.Ceylan(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R334373] as described in [R335373].

\[C_D = 1 - 0.5\exp(0.182) + 10.11Re^{-2/3}\exp(0.952Re^{-1/4}) - 0.03859Re^{-4/3}\exp(1.30Re^{-1/2}) + 0.037\times10^{-4}Re\exp(-0.125\times10^{-4}Re) -0.116\times10^{-10}Re^2\exp(-0.444\times10^{-5}Re)\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is 0.1 < Re <= 1E6 Original article reviewed.

References

[R334373](1, 2) Ceylan, Kadim, Ayşe Altunbaş, and Gudret Kelbaliyev. “A New Model for Estimation of Drag Force in the Flow of Newtonian Fluids around Rigid or Deformable Particles.” Powder Technology 119, no. 2-3 (September 24, 2001): 250-56. doi:10.1016/S0032-5910(01)00261-3.
[R335373](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Ceylan(200.)
0.7816735980280175
fluids.drag.Almedeij(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R336375] as described in [R337375].

\[ \begin{align}\begin{aligned}C_D = \left[\frac{1}{(\phi_1 + \phi_2)^{-1} + (\phi_3)^{-1}} + \phi_4\right]^{0.1}\\\phi_1 = (24Re^{-1})^{10} + (21Re^{-0.67})^{10} + (4Re^{-0.33})^{10} + 0.4^{10}\\\phi_2 = \left[(0.148Re^{0.11})^{-10} + (0.5)^{-10}\right]^{-1}\\\phi_3 = (1.57\times10^8Re^{-1.625})^{10}\\\phi_4 = \left[(6\times10^{-17}Re^{2.63})^{-10} + (0.2)^{-10}\right]^{-1}\end{aligned}\end{align} \]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 1E6. Original work has been reviewed.

References

[R336375](1, 2) Almedeij, Jaber. “Drag Coefficient of Flow around a Sphere: Matching Asymptotically the Wide Trend.” Powder Technology 186, no. 3 (September 10, 2008): 218-23. doi:10.1016/j.powtec.2007.12.006.
[R337375](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Almedeij(200.)
0.7114768646813396
fluids.drag.Morrison(Re)[source]

Calculates drag coefficient of a smooth sphere using the method in [R338377] as described in [R339377].

\[C_D = \frac{24}{Re} + \frac{2.6Re/5}{1 + \left(\frac{Re}{5}\right)^{1.52}} + \frac{0.411 \left(\frac{Re}{263000}\right)^{-7.94}}{1 + \left(\frac{Re}{263000}\right)^{-8}} + \frac{Re^{0.8}}{461000}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

Returns:

Cd : float

Drag coefficient [-]

Notes

Range is Re <= 1E6.

References

[R338377](1, 2) Morrison, Faith A. An Introduction to Fluid Mechanics. Cambridge University Press, 2013.
[R339377](1, 2) Barati, Reza, Seyed Ali Akbar Salehi Neyshabouri, and Goodarz Ahmadi. “Development of Empirical Models with High Accuracy for Estimation of Drag Coefficient of Flow around a Smooth Sphere: An Evolutionary Approach.” Powder Technology 257 (May 2014): 11-19. doi:10.1016/j.powtec.2014.02.045.

Examples

>>> Morrison(200.)
0.767731559965325
fluids.drag.Song_Xu(Re, sphericity=1.0, S=1.0)[source]

Calculates drag coefficient of a particle using the method in [R340379]. Developed with data for spheres, cubes, and cylinders. Claims 3.52% relative error for 0.001 < Re < 100 based on 336 tests data.

\[C_d = \frac{24}{Re\phi^{0.65}S^{0.3}}\left(1 + 0.35Re\right)^{0.44}\]
Parameters:

Re : float

Particle Reynolds number of the sphere using the surrounding fluid density and viscosity, [-]

sphericity : float, optional

Sphericity of the particle

S : float, optional

Ratio of equivalent sphere area and the projected area in the particle settling direction [-]

Returns:

Cd : float

Drag coefficient of particle [-]

Notes

Notable as its experimental data and analysis is included in their supporting material.

References

[R340379](1, 2) Song, Xianzhi, Zhengming Xu, Gensheng Li, Zhaoyu Pang, and Zhaopeng Zhu. “A New Model for Predicting Drag Coefficient and Settling Velocity of Spherical and Non-Spherical Particle in Newtonian Fluid.” Powder Technology 321 (November 2017): 242-50. doi:10.1016/j.powtec.2017.08.017.

Examples

>>> Song_Xu(30.)
2.3431335190092444