Ergun Packed-Bed Pressure Drop Formula (Catalyst & Filter Columns)

Define the bed and velocity convention

This example estimates frictional pressure-loss magnitude for a stationary bed of spherical particles, with constant density, viscosity, porosity and superficial velocity along the bed. It is an empirical model, not a universal rating for every filter or reactor.

Use viscosity in Pa·s, density in kg/m³, superficial speed in m/s, particle diameter in metres, dimensionless void fraction and bed length in metres. Superficial velocity is volumetric flow divided by the empty column cross-section; do not substitute the larger interstitial velocity. Enter 3 mm as 0.003 m.

The 150-term is proportional to viscosity and speed; the 1.75-term is proportional to density and speed squared. The spreadsheet returns a nonnegative pressure-loss magnitude. Signed pressure gradients, hydrostatic head and entrance losses are separate quantities.

Calculate total pressure loss in kPa

E2 and E3 contain the unrounded viscous and inertial contributions in Pa; E4 adds them and converts Pa to kPa. With the initial inputs, these are 2812.5 Pa, 3281.25 Pa and 6.09375 kPa. Format cells to show fewer decimals without rounding intermediate calculations.

All six inputs must be numeric. Viscosity, density and diameter must be positive, porosity strictly between zero and one, and speed and length nonnegative. Zero speed or zero length is allowed as a mathematical zero-loss limit. These input checks do not certify physical applicability.

=IF(COUNT(B2:B7)<>6,"Enter six numeric SI inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B6>=1,B7<0),"Require mu,rho,diameter>0; speed,length>=0; 0<porosity<1",(150*B2*B4*B7*(1-B6)^2/(B5^2*B6^3)+1.75*B3*B4^2*B7*(1-B6)/(B5*B6^3))/1000))

Compare the two contributions

Changing only porosity from 0.40 to 0.32 gives about 7055.664063 Pa viscous loss and 7263.183594 Pa inertial loss: 14.318847656 kPa total. The increase is approximately 2.35-fold for these inputs.

Try doubling speed: the viscous term doubles and the inertial term quadruples. Doubling length doubles both terms. Halving particle diameter quadruples the viscous term and doubles the inertial term. These relationships hold with the other model inputs fixed.

=IF(COUNT(B2:B7)<>6,"Enter six numeric SI inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B6>=1,B7<0),"Require mu,rho,diameter>0; speed,length>=0; 0<porosity<1",150*B2*B4*B7*(1-B6)^2/(B5^2*B6^3)))

Name the Reynolds-number definition

E5 computes the modified particle Reynolds number Re*=rho×speed×diameter/[mu×(1−porosity)], initially 100. Some references instead use rho×speed×diameter/mu, so always state the definition.

For nonzero speed and length, dividing the inertial term by the viscous term gives 1.75×Re*/150. At Re*=100 their ratio is 1.166667: neither contribution should be discarded. Equality occurs at Re*=150/1.75, about 85.714. This is an algebraic comparison of Ergun terms, not a universal transition threshold.

=IF(COUNT(B2:B7)<>6,"Enter six numeric SI inputs",IF(OR(B2<=0,B3<=0,B4<0,B5<=0,B6<=0,B6>=1,B7<0),"Require mu,rho,diameter>0; speed,length>=0; 0<porosity<1",B3*B4*B5/(B2*(1-B6))))

Recognize when the simple workbook is insufficient

Nonspherical particles, wide size distributions, narrow columns and wall effects require appropriate characterization or another correlation. Near porosity zero or one, even mathematically finite results may be physically inappropriate for a packed bed.

For gases with substantial pressure variation, density and superficial velocity change along the bed. A constant-property calculation is then insufficient; a coupled or integrated pressure model is needed. This workbook also does not model bed expansion, fluidization or changing porosity during operation.

Further reading

• Fluids documentation — Ergun packed-bed correlation
• COMSOL — Packed Bed Reactor, pressure-dependent gas-flow model