# 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

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

where ${\displaystyle f_{p}}$ and ${\displaystyle Gr_{p}}$ are defined as

${\displaystyle f_{p}={\frac {\Delta p}{L}}{\frac {D_{p}}{\rho v_{s}^{2}}}\left({\frac {\epsilon ^{3}}{1-\epsilon }}\right)}$ and ${\displaystyle Gr_{p}={\frac {\rho v_{s}D_{p}}{(1-\epsilon )\mu }}}$

where: ${\displaystyle Gr_{p}}$ is the modified Reynolds number,

fp 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), and
${\displaystyle \epsilon }$ is the void fraction of the bed (bed porosity at any time).

## Extension

The extension of the Ergun equation to fluidized beds is discussed by Akgiray and Saatçı (2001). 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.