# Minimum fluidization velocity¶

For a bed of particles, the minimum fluidization velocity is the gas velocity at which the drag force of the upward moving gas equals the weight of the particles. As discussed in Chapter 3 of the Kunnii and Levenspiel book, the minimum fluidization velocity ($$u_{mf}$$) can be calculated from the equation shown below. This formula is based on the Ergun pressure drop equation for a bed of particles.

$\frac{1.75}{\epsilon_{mf}^3 \phi} \left( \frac{d_p u_{mf} \rho_g}{\mu} \right)^2 + \frac{150(1-\epsilon_{mf})}{\epsilon_{mf}^3 \phi^2} \left( \frac{d_p u_{mf} \rho_g}{\mu} \right) = \frac{d_p^3 \rho_g (\rho_s - \rho_g) g}{\mu^2}$

The above equation can be written in terms of the Reynolds and Archimedes numbers as follows

$\frac{1.75}{\epsilon_{mf}^3 \phi} Re_{p, mf}^2 + \frac{150 (1 - \epsilon_{mf})}{\epsilon_{mf}^3 \phi^2} Re_{p, mf} = Ar$

where

$Ar = \frac{d_p^3 \rho_g (\rho_s - \rho_g) g}{\mu^2}$
$Re_{p,mf} = \frac{d_p u_{mf} \rho_g}{\mu}$

Kunii and Levenspiel further simplify the equation to the following form

$K_1 Re_{p,mf}^2 + K_2 Re_{p,mf} = Ar$

where

$K_1 = \frac{1.75}{\epsilon_{mf}^3 \phi}$
$K_2 = \frac{150(1-\epsilon_{mf})}{\epsilon_{mf}^3 \phi^2}$

Solving for the Reynolds number provides

$Re_{p,mf} = \left( a^2 + b Ar \right)^{1/2} - a$

where

$a = \frac{K_2}{2 K_1}$
$b = \frac{1}{K_1}$

Finally, the minimum fluidization velocity can be calculated from the above Reynolds number as

$u_{mf} = \frac{Re_{p,mf} \mu}{d_p \rho_g}$

For very small particles where Re < 20, the above equation can be simplified to

$u_{mf} = \frac{d_p^2 (\rho_s - \rho_g) g}{150 \mu} \frac{\epsilon_{mf}^3 \phi^2}{1 - \epsilon_{mf}}$

and for large particles where Re > 1000, the following equation can be used

$u_{mf}^2 = \frac{d_p (\rho_s - \rho_g) g}{175 \rho_g} \epsilon_{mf}^3 \phi$

When void fraction and sphericity are not known, values for $$a$$ and $$b$$ from Table 4 in Chapter 3 of Kunii and Levenspiel can be used to estimate $$u_{mf}$$.

## Nomenclature¶

$$a$$, $$b$$ - dimensionless constants (-)
$$Ar$$ - Archimedes number (-)
$$d_p$$ - Particle diameter (m)
$$\epsilon_{mf}$$ - Bed void fraction at minimum fluidizing conditions (-)
$$g$$ - Acceleration due to gravity, 9.81 m/s²
$$K_1$$, $$K_2$$ - dimensionless constants (-)
$$\mu$$ - Gas viscosity (kg/(m s))
$$\phi$$ - Sphericity of a particle (-)
$$\rho_g$$ - Gas density (kg/m³)
$$\rho_s$$ - Solid particle density (kg/m³)

## Source code¶

chemics.minimum_fluidization_velocity.umf_coeff(dp, mu, rhog, rhos, coeff='wenyu')[source]

Determine minimum fluidization velocity using experimental coefficients from Wen and Yu, Richardson, Saxena and Vogel, Babu, Grace, and Chitester. This approach can be used when bed void fraction and particle sphericity are not known. Refer to Equation 25 and Table 4 in Chapter 3 of Kunii and Levenspiel 1.

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

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

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

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

• coeff (string) – Keyword to determine which coefficients to use for umf calculation. Valid options are ‘wenyu’, ‘rich’, ‘sax’, ‘babu’, ‘grace’, and ‘chit’. Default coefficients are set to ‘wenyu’.

Returns

umf (float) – Minimum fluidization velocity [m/s]

Example

>>> umf_coeff(0.0005, 3.6e-5, 0.44, 2500, 'rich')
0.1192


References

1

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

chemics.minimum_fluidization_velocity.umf_ergun(dp, ep, mu, phi, rhog, rhos)[source]

Determine minimum fluidization velocity from particle and gas properties. This approach is based on the Ergun pressure drop equation for a bed of particles. Refer to Equations 18 and 19 in Chapter 3 of Kunii and Levenspiel 2,

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

• ep (float) – Void fraction of the bed [-]

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

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

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

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

Returns

umf (float) – Minimum fluidization velocity [m/s]

Example

>>> umf_ergun(0.0005, 0.46, 3.6e-5, 0.86, 0.44, 2500)
0.1488


References

2

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

chemics.minimum_fluidization_velocity.umf_reynolds(dp, ep, mu, phi, re, rhog, rhos)[source]

Calculate minimum fluidization velocity for very small particles where Reynolds number < 20 and for very large particles where Reynolds number > 1000. See Equations 21 and 22 in Chapter 3 of Kunii and Levenspiel 3.

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

• ep (float) – Void fraction [-]

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

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

• re (float) – Reynolds number where Re < 20 or Re > 1000 [-]

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

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

Returns

umf (float) – Minimum fluidization velocity [m/s]

Example

For small Reynolds number where Re = 19

>>> umf_reynolds(0.0005, 0.46, 3.6e-5, 0.86, 19, 0.44, 2500)
0.1513


For large Reynolds number where Re = 1001

>>> umf_reynolds(0.0005, 0.46, 3.6e-5, 0.86, 1001, 0.44, 2500)
1.1545


References

3

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