TheErgun equation, derived by theTurkishchemical engineerSabri Ergun in 1952, expresses the friction factor in apacked column as a function of the modifiedReynolds number.

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

where:

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

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

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

This arrangement of the Ergun equation makes clear its close relationship to the simplerKozeny-Carman equation, which describeslaminar 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 theDarcy–Forchheimer equation. Specifically, the Ergun equation gives the following permeability${\displaystyle k}$ and inertial permeability${\displaystyle k_1}$ from the Darcy-Forchheimer law:$${\displaystyle k=\frac{D_p^2}{150}\frac{{ϵ}^3}{(1-ϵ{)}^2},}$$and$${\displaystyle k_1=\frac{D_p}{1.75}\frac{{ϵ}^3}{1-ϵ}.}$$

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

• Hagen–Poiseuille equation
• Kozeny–Carman equation

• 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.

• Equations
• Chemical process engineering
• Fluid dynamics

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