Engineering guide
This module calculates the pressure drop in flow through packed beds according to Section L1.6 of the VDI-Wärmeatlas (VDI Heat Atlas, 12th edition, 2019). Fixed beds of particles, spheres, pellets or random packings are core components of many process apparatus: catalytic fixed bed reactors, adsorbers, regenerators, filter and dryer beds. Anyone who wants to calculate the pressure drop of a packed bed thereby directly determines the required fan or pump power and the mechanical load on the bed and its support grids.
The calculation relies on established approaches such as the Ergun equation and its generalizations: the pressure gradient is composed of a viscous, laminar contribution and an inertia-driven, turbulent contribution. The governing parameters are the porosity (void fraction) of the bed, the characteristic particle diameter – for non-spherical particles the Sauter diameter or the sphericity – as well as the superficial velocity and the fluid properties. Via the model selection (V33), several calculation approaches are available in the module.
Since the porosity enters the denominator with the third power, the pressure drop reacts extremely sensitively to packing density and wall channeling – a key point when transferring laboratory measurements to the industrial apparatus.
Scope
- Pressure drop in flow through beds
Typical workflow
Characterize the bed
Particle diameter (for size distributions the Sauter diameter), particle shape or sphericity and the porosity of the bed are defined. If the porosity is unknown, it is determined from bulk density and solid density.
Enter operating conditions and fluid properties
Superficial velocity or mass flow rate, density and dynamic viscosity of the fluid at operating pressure and temperature form the input data; for gases the pressure dependence of the density must be observed.
Choose the calculation model
The model selection defines the correlation approach, e.g. the Ergun equation or a drag-coefficient-based formulation with the bed Reynolds number; the models differ in their range of validity and their constants.
Check the flow regime
From the Reynolds number based on the particle diameter it is assessed whether the flow is in the creeping, transition or inertia-dominated regime; this determines which term of the pressure drop expression dominates.
Calculate the pressure gradient and total pressure drop
The pressure gradient is composed of the laminar and turbulent contributions and integrated over the bed height; for compressible media with a larger pressure drop, the calculation proceeds section by section with updated gas density.
Frequently asked questions
In which range is the Ergun equation valid?
The classical Ergun constants 150 and 1.75 were determined for random beds of roughly equal-sized particles with porosities around 0.35 to 0.55 and apply over a wide Reynolds number range from laminar into the turbulent regime. For strongly deviating porosities, very broad particle size distributions or ordered packings, the constants can deviate significantly – adapted models or measured data are then preferable.
Which diameter should be used for non-spherical particles?
The Sauter diameter, i.e. the diameter of the sphere with the same volume-to-surface ratio as the particle collective, combined with the sphericity where appropriate. If a sieve or median diameter is used instead, the calculated pressure drop can deviate considerably from the real one, because the specific surface area is captured incorrectly.
Why does the pressure drop react so sensitively to the porosity?
The laminar term contains (1−ε)²/ε³, the turbulent term (1−ε)/ε³. Reducing the porosity from 0.40 to 0.36 already increases the pressure drop by around 50 %. Compaction during filling, settling in operation or fines in the voids therefore have a strong effect and should be covered by tolerance considerations.
What is the wall channeling effect and when must it be considered?
At the vessel wall the packing is looser, so the fluid preferentially flows there. At small ratios of vessel to particle diameter (below about 10) this lowers the effective pressure drop and worsens the flow distribution; the correlations for infinitely extended beds are then only approximately valid.
Worked example
Air at 20 °C and about 1 bar flows through a fixed bed of spherical catalyst particles. This worked example calculates the pressure drop of the bed using the Ergun equation.
Given
| Particle diameter d | 5.0 mm |
|---|---|
| Porosity ε | 0.40 |
| Bed height L | 0.5 m |
| Superficial velocity u | 1.0 m/s |
| Density of air ρ | 1.19 kg/m³ |
| Dynamic viscosity of air η | 18.2 · 10⁻⁶ Pa·s |
Solution
Ergun equation
The pressure gradient is composed of a viscous and an inertial contribution:
Δp/L = 150 · (1−ε)²/ε³ · η·u/d² + 1.75 · (1−ε)/ε³ · ρ·u²/d
Laminar (viscous) contribution
150 · (0.60)²/(0.40)³ · 18.2·10−6 · 1.0 / (0.005)²
= 150 · 0.36/0.064 · 0.728 Pa/m ≈ 614 Pa/m
Turbulent (inertial) contribution
1.75 · 0.60/0.064 · 1.19 · 1.0² / 0.005
= 16.41 · 238 Pa/m ≈ 3,905 Pa/m
The inertial contribution clearly dominates – the flow lies in the inertia-controlled regime.
Total pressure drop
Δp/L = 614 + 3,905 ≈ 4,519 Pa/m
Δp = 4,519 Pa/m · 0.5 m ≈ 2,260 Pa ≈ 22.6 mbar
Result
| Pressure gradient Δp/L | ≈ 4,519 Pa/m |
|---|---|
| Pressure drop Δp (0.5 m bed height) | ≈ 2,260 Pa ≈ 22.6 mbar |
Technical content source: LV-Solved-Problems-Content 628d313247e7