Description

Pressure drop of a single-phase fluid flowing through a fixed bed of particles, by the Ergun equation. The equation adds a viscous term, linear in velocity, to an inertial term, quadratic in velocity, and so covers the laminar, transitional and turbulent regimes with one expression. It applies to packed columns, catalytic reactors, adsorbers, granular filters and beds at the onset of fluidization.

Ergun equation

Calculation methods

• Pressure drop: pressure drop and pressure gradient of a bed for a given flow, with the superficial velocity, the modified Reynolds number and the share of the viscous term.
• Flow for a given pressure drop: flow that a bed passes under a given pressure drop.
• Minimum fluidization: superficial velocity and flow at which the bed's pressure drop equals its buoyant weight, with the bed pressure drop at that point.

Model

$\frac{\Delta P}{L} = \frac{150 \mu (1 - \varepsilon)^2}{\phi^2 d_p^2 \varepsilon^3} u + \frac{1.75 \rho (1 - \varepsilon)}{\phi d_p \varepsilon^3} u^2 \qquad u = \frac{Q}{A}$

$Re' = \frac{\rho u \phi d_p}{\mu (1 - \varepsilon)}$

where $\Delta P$ is the pressure drop across a bed of length $L$ and cross-sectional area $A$, $Q$ the volumetric flow, $u$ the superficial velocity, $\varepsilon$ the bed porosity, $d_p$ the volume-equivalent particle diameter, $\phi$ the particle sphericity ($\phi = 1$ for spheres), and $\rho$ and $\mu$ the fluid density and viscosity. The viscous term governs for $Re' < 10$ and the inertial term for $Re' > 1000$.

At minimum fluidization the pressure drop equals the buoyant weight of the bed per unit area:

$\Delta P_{mf} = (1 - \varepsilon_{mf}) (\rho_p - \rho) g L_{mf}$

$\frac{1.75}{\varepsilon_{mf}^3 \phi} Re_{mf}^2 + \frac{150 (1 - \varepsilon_{mf})}{\varepsilon_{mf}^3 \phi^2} Re_{mf} = Ar \qquad Ar = \frac{d_p^3 \rho (\rho_p - \rho) g}{\mu^2} \qquad Re_{mf} = \frac{\rho u_{mf} d_p}{\mu}$

The correlation of Wen and Yu, $Re_{mf} = \sqrt{33.7^2 + 0.0408 Ar} - 33.7$, which requires neither $\varepsilon_{mf}$ nor $\phi$, is shown in the calculation steps for comparison.

Range of application

• Incompressible flow. For a gas, the result holds while the pressure drop is small relative to the absolute pressure (roughly below 10 %); otherwise the bed should be divided into segments evaluated at their mean density.
• The porosity has the strongest influence on the result: $\Delta P$ varies approximately with $(1 - \varepsilon)^2 / \varepsilon^3$, so an error of 0.02 in $\varepsilon$ changes the pressure drop by about 20 %. Randomly packed spheres have $\varepsilon \approx 0.36$ to $0.40$.
• Wall effects are neglected; they become significant when the bed diameter is less than about ten particle diameters.
• Default values: in the first two methods, a bed of 3 mm spheres in air at 20 °C and 1 atm; in the minimum fluidization method, round sand ($\phi = 0.86$, 2650 kg/m³) in air.