Exercise: Calculating pressure drop in a packed bed

Problem Statement

Design a packed bed column that contains spherical solid packing material with diameter 22 mm, and a void fraction of 0.5. The column is designed to purify water containing rust particles (Iron Oxide, Fe₂O₃) at a rate of 3 kg/s. The filtration is performed at 26.85^oC, steady state, with an actual velocity (v') of 1 m/s. The pressure difference across the column is about 1.3 pascals.

Solution Strategy

We are asked to "design" a packed bed column which means that we are looking for our column's dimensions such as length and diameter. The Ergun equation for a packed bed column is used to find the pressure difference:

$△ P_{B e d} = - f_{p} . \frac{L}{D_{p}} . \frac{(1 - ϵ)}{ϵ^{3}} * ρ V_{s}^{2}$

Where the terms are defined as below:

Table 1.0
Symbol	Name	Meaning
fp	Friction Factor	Empirical constant accounting for flow regime and particle geometry.
L	Column Length/Height	N/A
Dp	Particle Diameter	Average diameter of the packing particles
ϵ	Bed Porosity (void fraction)	Fraction of the packed bed volume that is not occupied by solid packing
Vs	Superficial Velocity	The velocity of the fluid assuming the entire cross-section is open (it is related to the interstitial velocity or v' by Vs = ϵ.v')

We can rearrange the above equation and solve for the length directly, but we do not have any information about the friction factor. Hence, the correlations below are introduced:

$f_{p} = \frac{150}{R e_{p}} + \frac{7}{4}$

$R e_{p} = \frac{ρ V_{s} D_{p}}{(1 - ϵ) μ}$

Using the properties of water at 26.85^oC or 300K, and conversion of v' to Vs from table 1.0, we can solve for the Reynolds number in the packed bed, and lastly, calculate the friction factor.

The diameter of column can be found from the mass flow rate definition as follows:

$M a s s F l o w R a t e (M F R) = ρ . A . v^{'}$ ; where A is the column's cross-sectional area and defined as:

$A = \frac{π . D^{2}}{4}$ .

Combining the above 2 equations and solving for the diameter we get:

$D = 2 \sqrt{\frac{M F R}{π . ρ . v^{'}}}$

Solving

Assuming dilute solution, properties can be found for pure water at 300K:

ρ = 996.57 kg/m³, and μ = 8.53 *10^-4 Pa.s Source: Density, Viscosity $V_{s} = ϵ . v^{'} \Rightarrow V_{s} = 0.5 * 1 ␣ m / s = 0.5 m / s$

$⟹ R e_{p} = \frac{996.57 k g / m^{3} . 0.5 m / s . 22 * 1 0^{- 3} m}{0.5.8.53 * 1 0^{- 4} P a . s} \approx 25, 700 ⟹ f_{p} \approx 1.756$

Inputting the values found above into the Ergun equation yields us:

$L \approx 0.016 m m$

Lastly, solving for the diameter using the mass flow rate equation above gives:

$D \approx 62 m m$ , for a bed packed with solids.