What units does the Ergun equation use?

The classic Ergun equation is expressed in SI units: pressure drop in Pascals, bed length in meters, particle diameter in meters, fluid density in kg/m³, viscosity in Pa·s, and superficial velocity in m/s. However, the equation can be used in any consistent unit system as long as all inputs are converted properly. This calculator supports both SI and US customary units.

What is superficial velocity in a packed bed?

Superficial velocity is defined as the volumetric flow rate divided by the empty cross-sectional area of the bed, ignoring the presence of particles. It is not the actual fluid velocity in the pores, but it is the standard velocity used in correlations like the Ergun equation.

How accurate is the Ergun equation?

The Ergun equation is an empirical correlation that generally provides good estimates for pressure drop in randomly packed beds of roughly spherical particles over a wide range of Reynolds numbers. Accuracy decreases for highly non-spherical particles, very wide particle size distributions, or strongly non-Newtonian fluids. In such cases, experimental calibration or more advanced models may be needed.

Ergun Equation Packed Bed Pressure Drop Calculator

Q: What is the Ergun equation used for?

The Ergun equation is an empirical correlation used to estimate the pressure drop of a fluid flowing through a packed bed of particles. It combines viscous and inertial contributions and is valid over a wide range of Reynolds numbers for both laminar and turbulent flow in packed beds.

Compute pressure drop, pressure gradient, friction factor, Reynolds number, and flow regime for gas or liquid flow through packed beds using the Ergun equation.

1. Choose unit system

2. Enter packed bed and fluid data

Bed length L

Particle diameter d_p

Bed void fraction ε

Typical random packed spheres: 0.35–0.45

Bed cross-sectional area A

m²

If unknown, estimate from column diameter: A = πD²/4.

Fluid density ρ

kg/m³

Dynamic viscosity μ

Pa·s

Water at 20°C ≈ 0.001 Pa·s (1 cP).

Volumetric flow rate Q

m³/s

Enter Q or superficial velocity; the other will be computed.

Superficial velocity v

m/s

If left blank, it will be calculated from Q and A.

3. Results

Pressure drop

Total pressure drop across the packed bed:

–

Pressure gradient (ΔP/L):

–

Hydrodynamics

Superficial velocity v:: –
Reynolds number Re_p:: –
Flow regime:: –
Friction factor f:: –

Friction factor defined such that ΔP/L = 2 f (1−ε) ρ v² / (ε³ d_p).

Ergun equation for packed bed pressure drop

The Ergun equation is a widely used empirical correlation for estimating the pressure drop of a fluid flowing through a packed bed of particles. It combines viscous and inertial contributions and is valid over a broad range of Reynolds numbers for both laminar and turbulent flow.

Ergun equation (SI form)

\[ \frac{\Delta P}{L} = \frac{150 (1-\varepsilon)^2}{\varepsilon^3} \frac{\mu v}{d_p^2} + \frac{1.75 (1-\varepsilon)}{\varepsilon^3} \frac{\rho v^2}{d_p} \]

\(\Delta P\): pressure drop [Pa]
\(L\): bed length [m]
\(\varepsilon\): bed void fraction (porosity) [–]
\(\mu\): dynamic viscosity [Pa·s]
\(\rho\): fluid density [kg/m³]
\(v\): superficial velocity [m/s]
\(d_p\): particle diameter [m]

Reynolds number and friction factor in packed beds

For packed beds, a particle-based Reynolds number is commonly defined as:

\[ \mathrm{Re}_p = \frac{\rho v d_p}{\mu (1-\varepsilon)} \]

Using the Ergun equation, a Fanning-type friction factor \(f\) can be written as:

\[ \frac{\Delta P}{L} = 2 f \frac{(1-\varepsilon)}{\varepsilon^3} \frac{\rho v^2}{d_p} \quad\Rightarrow\quad f = \frac{1}{2} \left[ \frac{150}{\mathrm{Re}_p} + 1.75 \right] \]

This expression shows how the viscous term (150/Re_p) dominates at low Reynolds numbers (laminar flow), while the inertial term (1.75) dominates at high Reynolds numbers (turbulent flow).

Typical ranges and engineering guidance

Porosity ε: 0.35–0.45 for randomly packed spheres; can be lower for ordered packings or irregular particles.
Laminar regime: Re_p ≲ 10 – viscous term dominates, pressure drop ∝ v.
Transition: 10 ≲ Re_p ≲ 1000 – both viscous and inertial terms are important.
Turbulent regime: Re_p ≳ 1000 – inertial term dominates, pressure drop ∝ v².

How to use this Ergun equation calculator

Select the unit system. Choose SI (metric) or US/Imperial. All inputs and outputs will update accordingly.
Enter bed and particle data. Provide bed length, particle diameter, and porosity. If you know the column diameter, you can compute the area externally or use the hint above.
Enter fluid properties. Density and viscosity should match the operating temperature and pressure of your process.
Specify flow. Enter either volumetric flow rate or superficial velocity. The calculator will compute the missing one from the bed area.
Review results. The tool returns pressure drop, pressure gradient, Reynolds number, friction factor, and a qualitative flow regime.

Limitations of the Ergun equation

While the Ergun equation is very popular, keep in mind:

It is based on experiments with rigid, randomly packed, roughly spherical particles.
Accuracy decreases for highly non-spherical particles, very broad size distributions, or non-Newtonian fluids.
Wall effects can be significant when the column diameter is only a few particle diameters wide.
For fluidized beds, additional correlations and regime maps are required.

Frequently asked questions

Can I use the Ergun equation for gases and liquids?

Yes. The Ergun equation is valid for both gases and liquids as long as the fluid behaves approximately as a Newtonian fluid and the properties (ρ, μ) are evaluated at the operating conditions. For compressible gases with large pressure changes, you may need to integrate along the bed using an average density.

What if I only know pressure drop and want to find flow rate?

The Ergun equation is quadratic in superficial velocity, so you can rearrange it and solve for \(v\) given ΔP, L, ε, d_p, ρ, and μ. This calculator currently solves in the forward direction (from v to ΔP), but you can iteratively adjust Q or v until the computed ΔP matches your target.

How does this differ from the Carman–Kozeny equation?

The Carman–Kozeny equation describes laminar flow through packed beds and includes only the viscous term. The Ergun equation extends this by adding an inertial term, making it applicable over a much wider Reynolds number range that includes turbulent flow.