Ergun equation

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
$$f_p = \frac {150}{Gr_p}+1.75$$

where:


 * $$f_p = \frac{\Delta p}{L} \frac{D_p}{\rho v_s^2} \left(\frac{\epsilon^3}{1-\epsilon}\right),$$
 * $$Gr_p = \frac{\rho v_s D_p}{(1-\epsilon)\mu} = \frac{Re}{(1-\epsilon)},$$
 * $$Gr_p$$ is the modified Reynolds number,
 * $$f_p$$ is the packed bed friction factor,
 * $$\Delta p$$ 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),
 * $$\epsilon$$ 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:

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

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: $$k = \frac{D_p^2}{150} ~\frac{\epsilon^3}{(1-\epsilon)^2},$$ and $$k_1 = \frac{D_p}{1.75} ~\frac{\epsilon^3}{1-\epsilon}.$$

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