A laboratory fixed bed should drop well under 1–5 kPa: a 20 mm bed of 1 mm pellets at 100 mL/min costs about 7 Pa, the same bed of 100 µm powder about 630 Pa, and halving particle size multiplies the viscous term by four. Above 10 kPa, suspect fines, a blocked frit or over-tight packing before blaming the catalyst.
Pressure drop through a packed bed follows the Ergun equation: ΔP/L = 150·μ·(1−ε)2·u / (ε3dp2) + 1.75·ρ·(1−ε)·u2 / (ε3dp), with ε the bed voidage, u the superficial velocity and dp the particle diameter. The first term is viscous, the second inertial.
At lab flows the particle Reynolds number Rep = ρu·dp/μ stays below 3, and the inertial term contributes under 5 %. Darcy's law with a 1/dp2 scaling is then accurate enough for design, which matters because it makes the sensitivity obvious: halve the particle size and the pressure drop quadruples.
| dp | Rep | ΔP/L | ΔP over 20 mm bed | Viscous share | Practical note |
|---|---|---|---|---|---|
| 2 mm | 2.8 | 83 Pa/m | 1.7 Pa | 95 % | Below transducer resolution; diluent or inert packing |
| 1 mm | 1.4 | 324 Pa/m | 6.5 Pa | 97 % | Typical pellet; measure at higher flow to resolve |
| 0.5 mm | 0.7 | 1.3 kPa/m | 26 Pa | 99 % | Common compromise for kinetics |
| 0.2 mm | 0.28 | 7.9 kPa/m | 158 Pa | 99.5 % | Powder; fines migration becomes likely |
| 0.1 mm | 0.14 | 31.6 kPa/m | 632 Pa | 99.7 % | Fluidised or entrained at higher flow |
Conditions: 10 mm internal diameter, 20 mm bed, ε = 0.40, nitrogen at 25 °C and 1 bar, 100 mL/min (u = 0.021 m/s), μ = 1.76 × 10−5 Pa·s. Scale linearly with bed length, quadratically with diameter for the same flow, and with μ(T) when the reactor runs hot: nitrogen viscosity rises about 40 % between 25 and 300 °C, raising the viscous term proportionally.
Fines are the first suspect. A charge containing 5 % of particles at dp/10 raises the effective pressure drop by 30–60 %, because the small fraction occupies the voids between large pellets and blocks the paths that carried most of the flow. Sieve to 20 % either side of the nominal size and keep the fines out.
The second suspect is packing density. Voidage of loose spheres sits near 0.42; tapping the column down to 0.35 increases (1−ε)2/ε3 by a factor of 2.2, so the same bed doubles its pressure drop without any change in particle size. Fill in increments, tap lightly and once, and record the settled bed height.
Condensation and coking come next. A bed at 30 bar holding condensable products develops liquid bridges in the cold zone below the heated section, and the pressure drop climbs over hours before it appears in the product analysis. Watch ΔP against time at constant flow; a monotonic rise is deposition, a step change is a plug.
Smaller particles are attractive because they shorten the diffusion path inside the pellet, where the concentration actually falls. External film transfer is rarely the limit at lab scale: with a bed-averaged rate of 0.3 mol/m3·s, dp of 1 mm, kc of 0.06 m/s and a bulk concentration of 10 mol/m3, the Mears criterion gives 5 × 10−4, three orders of magnitude below the 0.15 threshold.
Intraparticle transport is different. With an effective diffusivity of 1 × 10−7 m2/s, typical of Knudsen-limited mesopores, the Weisz–Prater modulus reaches 0.08 at 1 mm and 0.75 at 3 mm: measurable internal gradients in the larger pellets. The 0.3–1 mm range is the usual compromise.
Use a differential transducer across the bed, not two absolute gauges subtracted. At 30 bar, gauges accurate to 0.1 % of full scale each carry 30 kPa of uncertainty, which is a thousand times the 30 Pa signal and will show a negative pressure drop on a good day. A differential transducer rated for line pressure with a 1–10 kPa span resolves the signal directly.
Log the baseline with an empty column and with the bed at process temperature. Gas viscosity changes with temperature and the reading must be corrected before comparison across runs; a cold bed measured at room temperature and a hot bed at 300 °C differ by 40 % at the same flow. State flow in mL/min at a defined temperature and pressure, or in GHSV, and never in "rotameter units".
Under 1–5 kPa for a lab-scale bed; atmospheric systems tolerate 5–10 kPa, while a 30 bar rig should stay under 1 % of line pressure, about 30 kPa. A reading an order of magnitude above the calculated Ergun value indicates a mechanical problem, not a kinetic one.
Yes, through intraparticle diffusion. Below about 0.3 mm the effectiveness factor approaches 1 and the measured rate is intrinsic; at 3 mm it can fall to 0.3–0.6, which is why rate comparisons across papers with different sieve fractions often disagree.
Deposition. Coke, sulfur or condensed product narrows the voids, and the rise is usually monotonic over hours. A sudden step points to a dislodged plug of fines or a collapsed glass-wool support rather than gradual fouling.
Only above roughly 100 Pa. Water columns resolve about 10 Pa with care, which covers 0.2 mm powders but not 1 mm pellets, and they cannot be used on a pressurised system without a condensing trap.
