Ergun equation

Template:Short description The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number.

Equation

$$f_p=\frac{150}{G{r}_p}+1.75$$

where:

• $f_p=\frac{\mathrm{\Delta }p}{L}\frac{D_p}{\rho v_s^2}\left(\frac{{ϵ}^3}{1-ϵ}\right),$
• $G{r}_p=\frac{\rho v_s{D}_p}{(1-ϵ)\mu }=\frac{Re}{(1-ϵ)},$
• $G{r}_p$ is the modified Reynolds number,
• $f_p$ is the packed bed friction factor,
• $\mathrm{\Delta }p$ is the pressure drop across the bed,
• $L$ is the length of the bed (not the column),
• $D_p$ is the equivalent spherical diameter of the packing,
• $\rho$ is the density of fluid,
• $\mu$ is the dynamic viscosity of the fluid,
• $v_s$ is the superficial velocity (i.e. the velocity that the fluid would have through the empty tube at the same volumetric flow rate),
• $ϵ$ is the void fraction (porosity) of the bed, and
• $Re$ is the particle Reynolds Number (based on superficial velocity)..

Extension

To calculate the pressure drop in a given reactor, the following equation may be deduced:

$$\mathrm{\Delta }p=\frac{150\mu L}{D_p^2}\frac{(1-ϵ{)}^2}{{ϵ}^3}{v}_s+\frac{1.75L\rho }{D_p}\frac{(1-ϵ)}{{ϵ}^3}{v}_s|v_s|.$$

This arrangement of the Ergun equation makes clear its close relationship to the simpler Kozeny-Carman equation, which describes laminar flow of fluids across packed beds via the first term on the right hand side. On the continuum level, the second-order velocity term demonstrates that the Ergun equation also includes the pressure drop due to inertia, as described by the Darcy–Forchheimer equation. Specifically, the Ergun equation gives the following permeability $k$ and inertial permeability $k_1$ from the Darcy-Forchheimer law: $$k=\frac{D_p^2}{150}\frac{{ϵ}^3}{(1-ϵ{)}^2},$$ and $$k_1=\frac{D_p}{1.75}\frac{{ϵ}^3}{1-ϵ}.$$

The extension of the Ergun equation to fluidized beds, where the solid particles flow with the fluid, is discussed by Akgiray and Saatçı (2001).

See also

• Hagen–Poiseuille equation
• Kozeny–Carman equation

References

• Ergun, Sabri. "Fluid flow through packed columns." Chem. Eng. Prog. 48 (1952).
• Ö. Akgiray and A. M. Saatçı, Water Science and Technology: Water Supply, Vol:1, Issue:2, pp. 65–72, 2001.