Terminal velocity

An individual particle can be carried by a stream of gas when the gas velocity exceeds the termnial velocity \(u_t\) of the particle. However, in fluidized bed reactors, entrainment of particles out of the bed may require a gas velocity many times higher than the terminal velocity.

chemics.terminal_velocity.ut(cd, dp, rhog, rhos)[source]

Calculate terminal velocity of a single particle based on Equation 28 on page 80 in the Kunii and Levenspiel book 1 where \(C_D\) is an experimentally determined drag coefficient.

\[u_t = \left( \frac{4 d_p\, (\rho_s - \rho_g) g}{3 \rho_g\, C_D} \right)^{1/2}\]
Parameters
  • cd (float) – Drag coefficient [-]

  • dp (float) – Diameter of particle [m]

  • rhog (float) – Density of gas [kg/m^3]

  • rhos (float) – Density of solid [kg/m^3]

Returns

ut (float) – Terminal velocity [m/s]

Example

>>> ut(11.6867, 0.00016, 1.2, 2600)
0.6227

References

1

Daizo Kunii and Octave Levenspiel. Fluidization Engineering. Butterworth-Heinemann, 2nd edition, 1991.

chemics.terminal_velocity.ut_ganser(dp, mu, phi, rhog, rhos)[source]

Estimate terminal velocity of a non-spherical particle based on the Ganser drag coefficient 2. According to the Chhabra paper 3, the Ganser drag correlation is applicable for sphericity values from 0.09 to 1.

\[ \begin{align}\begin{aligned}C_d &= \frac{24}{Re\, K_1} \left( 1 + 0.1118 (Re\, K_1 K_2)^{0.6567} \right) + \frac{0.4305 K_2}{1 + \frac{3305}{Re\, K_1 K_2}}\\K_1 &= \left( \frac{1}{3} + \frac{2}{3}\phi \right)\\K_2 &= 10^{1.8148 (-log\, \phi)^{0.5743}}\end{aligned}\end{align} \]

where K₁ is Stokes’ shape factor and K₂ is Newton’s shape factor. The Cui 2007 and Chhabra 1999 papers leave out the \(-2.25*d_v/D\) term in the shape factor equations.

Parameters
  • dp (float) – Diameter of the particle [m]

  • mu (float) – Viscosity of gas [kg/(m s)]

  • phi (float) – Sphericity of the particle [-]

  • rhog (float) – Density of the gas [kg/m³]

  • rhos (float) – Density of the particle [kg/m³]

Returns

ut (float) – Terminal velocity of non-spherical particle [m/s]

Example

>>> ut_ganser(0.00016, 1.8e-5, 0.67, 1.2, 2600)
0.6230

Note

Drag coefficient is referenced from Equation 18 in Ganser, Equation 6 in Chhabra, and as Equation 2 in Cui 4.

References

2

Gary H. Ganser. A rational approach to drag prediction of spherical and nonsperical particles. Powder Technology, 77, 143-152, 1993.

3

R.P. Chhabra, L. Agarwal, and N.K. Sinha. Drag on non-spherical particles: An evaluation of available methods. Powder Technology, 101, 288-295, 1999.

4

Heping Cui and John R. Grace. Fluidization of biomass particles: A review of experimental multiphase flow aspects. Chemical Engineering Science, 62, 45-55, 2007.

chemics.terminal_velocity.ut_haider(dp, mu, phi, rhog, rhos)[source]

Calculate terminal velocity of a particle as discussed in the Haider and Levenspiel 5. Valid for particle sphericities of 0.5 to 1. Particle diameter should be an equivalent spherical diameter, such as the diameter of a sphere having same volume as the particle.

To determine the terminal velocity for a range of particle sphericities, Haider and Levenspiel first define two dimensionless quantities

\[d_{*} = d_p \left[ \frac{g\, \rho_g (\rho_s - \rho_g)}{\mu^2} \right]^{1/3} \ u_* = \left[ \frac{18}{d{_*}^2} + \frac{2.3348 - 1.7439\, \phi}{d{_*}^{0.5}} \right]^{-1}\]

where \(0.5 \leq \phi \leq 1\) and particle diameter \(d_p\) is an equivalent spherical diameter, the diameter of a sphere having the same volume as the particle. The relationship between \(u_*\) and \(u_t\) is given by

\[u_* = u_t \left[ \frac{\rho{_g}^2}{g\, \mu\, (\rho_s - \rho_g)} \right]^{1/3}\]

The terminal velocity of the particle can finally be determined by rearranging the above equation such that

\[u_t = u_* \left[ \frac{g\, \mu\, (\rho_s - \rho_g)}{\rho{_g}^2} \right]^{1/3}\]
Parameters
  • dp (float) – Diameter of particle [m]

  • mu (float) – Viscosity of gas [kg/(m s)]

  • phi (float) – Sphericity of particle [-]

  • rhog (float) – Density of gas [kg/m^3]

  • rhos (float) – Density of particle [kg/m^3]

Returns

ut (float) – Terminal velocity of a particle [m/s]

Example

>>> ut_haider(0.00016, 1.8e-5, 0.67, 1.2, 2600)
0.8857

References

5

A. Haider and O. Levenspiel. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technology, 58:63–70, 1989.