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. ButterworthHeinemann, 2nd edition, 1991.

chemics.terminal_velocity.
ut_ganser
(dp, mu, phi, rhog, rhos)[source]¶ Estimate terminal velocity of a nonspherical 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 nonspherical particle [m/s]
Example
>>> ut_ganser(0.00016, 1.8e5, 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, 143152, 1993.
 3
R.P. Chhabra, L. Agarwal, and N.K. Sinha. Drag on nonspherical particles: An evaluation of available methods. Powder Technology, 101, 288295, 1999.
 4
Heping Cui and John R. Grace. Fluidization of biomass particles: A review of experimental multiphase flow aspects. Chemical Engineering Science, 62, 4555, 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, 1.8e5, 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.