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.

$${\displaystyle f_{p}={\frac {150}{Gr_{p}}}+1.75}$${\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 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}={\frac {\rho v_{s}D_{p}}{(1-\epsilon )\mu }}={\frac {Re}{(1-\epsilon )}},}
• ${\displaystyle Gr_{p}}${\displaystyle Gr_{p}} is the modified Reynolds number,
• ${\displaystyle f_{p}}${\displaystyle f_{p}} is the packed bed friction factor,
• ${\displaystyle \Delta p}${\displaystyle \Delta p} is the pressure drop across the bed,
• ${\displaystyle L}${\displaystyle L} is the length of the bed (not the column),
• ${\displaystyle D_{p}}${\displaystyle D_{p}} is the equivalent spherical diameter of the packing,
• ${\displaystyle \rho }${\displaystyle \rho } is the density of fluid,
• ${\displaystyle \mu }${\displaystyle \mu } is the dynamic viscosity of the fluid,
• ${\displaystyle v_{s}}${\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),
• ${\displaystyle \epsilon }${\displaystyle \epsilon } is the void fraction (porosity) of the bed, and
• ${\displaystyle Re}${\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:

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

• Equations
• Chemical process engineering
• Fluid dynamics

• Articles with short description
• Short description matches Wikidata