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.

$${\displaystyle f_{p}={\frac {150}{Gr_{p}}}+1.75}$$

where:

• ${\displaystyle f_{p}={\frac {\Delta p}{L}}{\frac {D_{p}}{\rho v_{s}^{2}}}\left({\frac {\epsilon ^{3}}{1-\epsilon }}\right),}$
• ${\displaystyle Gr_{p}={\frac {\rho v_{s}D_{p}}{(1-\epsilon )\mu }}={\frac {Re}{(1-\epsilon )}},}$
• ${\displaystyle Gr_{p}}$ is the modified Reynolds number,
  
• ${\displaystyle f_{p}}$ is the packed bed friction factor,
  
• ${\displaystyle \Delta p}$ is the pressure 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 the density of fluid,
  
• ${\displaystyle \mu }$ is the dynamic viscosity of the fluid,
  
• ${\displaystyle v_{s}}$ is the superficial velocity (i.e. the velocity that the fluid would have through the empty tube at the same volumetric flow rate),
  
• ${\displaystyle \epsilon }$ is the void fraction (porosity) of the bed, and
  
• ${\displaystyle Re}$ is the particle Reynolds Number (based on superficial velocity^[1])..

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

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

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 ${\displaystyle k}$ and inertial permeability ${\displaystyle k_{1}}$ from the Darcy-Forchheimer law: $${\displaystyle k={\frac {D_{p}^{2}}{150}}~{\frac {\epsilon ^{3}}{(1-\epsilon )^{2}}},}$$ and $${\displaystyle k_{1}={\frac {D_{p}}{1.75}}~{\frac {\epsilon ^{3}}{1-\epsilon }}.}$$

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

• 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.