Flow through packed and fluidised beds

In short

A packed bed is a fixed column of solid particles through which a fluid is forced; a fluidised bed is the same column at a velocity high enough for the drag on the particles to support their weight, so the bed behaves like a boiling liquid. The design question in both cases is pressure drop versus velocity, which fixes blower/pump power, and the velocity window between minimum fluidisation and particle carry-over. The topic carries heavy weight in GATE-CH (Mechanical Operations and Fluid Mechanics) and in university Unit Operations papers, and is used daily in catalytic reactors, FCC units, AFBC/CFBC boilers, fluid-bed dryers and granulators, adsorbers, ion-exchange columns and absorption towers. Interviewers at Reliance, IOCL, BPCL, Tata Chemicals, UPL, Thermax, Cipla, Sun Pharma and GMM Pfaudler ask for the Ergun equation, u_mf and why bed pressure drop stays flat after fluidisation.

Explained

Geometry of a bed of particles

• Voidage (porosity) ε = void volume / total bed volume. Loose random packing of spheres ε ≈ 0.40-0.45; dense packing ≈ 0.36; crushed irregular solids 0.45-0.55; Raschig rings and other tower packings 0.6-0.8.
• Bulk density ρ_b = (1 − ε) ρ_s, so ε = 1 − ρ_b/ρ_s — the easiest lab measurement (weigh a known bed volume).
• Sphericity (shape factor) φ_s = surface area of a sphere of equal volume / actual particle surface area. Sphere = 1.0; rounded sand 0.8-0.86; angular/crushed solids 0.6-0.7; flakes and cylinders lower.
• Equivalent diameter: for non-spheres take d_p = d_v, the diameter of the volume-equivalent sphere, and carry φ_s separately in the equation — folding the shape factor into the diameter as well counts it twice. For a size distribution use the Sauter (surface-volume) mean, because drag depends on surface area per unit volume.
• Specific surface of the bed a = 6(1 − ε)/(φ_s d_p) m² per m³ of bed — the quantity that actually appears in the derivation.
• Hydraulic radius model: the bed is treated as a bundle of tortuous channels of hydraulic radius r_H = ε/a, giving an interstitial velocity u/ε. This is the route to Kozeny-Carman.

Pressure drop through a fixed packed bed

• Laminar (creeping) flow, Re_p < 10 — Blake-Kozeny / Kozeny-Carman: ΔP ∝ μ u, and ΔP ∝ 1/d_p². Doubling the particle size cuts pressure drop four-fold.
• Fully turbulent, Re_p > 1000 — Burke-Plummer: ΔP ∝ ρ u², independent of viscosity.
• Ergun equation adds the two and covers the whole range (the examined form). Viscous term dominates for fine particles/viscous liquids, inertial term for gases and coarse packing.
• Strong sensitivity to ε: the ε³ in the denominator means a drop from ε = 0.45 to 0.40 raises ΔP by roughly 55-70% (about 70% on the viscous term, 55% on the inertial). Hence careful, uniform packing; avoid fines segregating at the wall.
• Wall effect: use D/d_p > 10 (preferably > 20), else channelling along the wall lowers ΔP and ruins contacting; apply a wall correction below that.
• ΔP is independent of flow direction in a fixed bed; for upflow the measured ΔP includes the hydrostatic head, so always state whether it is total or frictional.
• Limitations of Ergun: assumes uniform spheres of one size, isothermal incompressible flow, no bed movement; errors of ±25% are normal for irregular or sticky solids, and for gases at high ΔP the density change must be handled by writing the equation in differential form.

Onset of fluidisation

• Raise the upflow velocity through a bed of solids: ΔP rises linearly (fixed bed, Ergun) until it equals the buoyant weight of the bed per unit area. That point is minimum fluidisation, velocity u_mf, voidage ε_mf (typically 0.4-0.5, slightly above the packed value).
• Beyond u_mf, ΔP stays essentially constant at W_buoyant/A however much the velocity rises, because the bed simply expands. This flat line is the signature of fluidisation and is the basis of the standard exam plot.
• Hysteresis / overshoot: on the first upward run a slightly higher ΔP peak appears because interlocked particles must be broken apart; on the way down the curve falls along the fluidised line and then along a fixed-bed line of higher voidage. u_mf is therefore read from the decreasing branch.
• Calculating u_mf: equate Ergun ΔP to the buoyant weight. For Re_mf < 20 only the viscous term survives (simple closed form); for Re_mf > 1000 only the inertial term; in between solve the quadratic. Alternatively use the Wen-Yu correlation in terms of the Archimedes number — generally the more reliable of the two for real solids.
• ε_mf is often not measured; Leva's and Baeyens' empirical correlations avoid it, and many designers simply take ε_mf ≈ 0.4-0.45 for sand-like solids.
• u_mf rises with particle size squared (fine range) and rises with solid density; it falls as gas temperature rises because μ increases and ρ decreases — a hot FBC needs a lower u_mf than its cold start-up test shows.

Regimes above u_mf, and Geldart classification

• Particulate (homogeneous) fluidisation: smooth expansion, no bubbles — typical of liquid-solid systems; bed expansion follows Richardson-Zaki, u/u_t = εⁿ with n ≈ 4.65 in the Stokes range falling to ≈ 2.4 at high Re.
• Aggregative (bubbling) fluidisation: gas in excess of u_mf passes as bubbles through a dense phase close to ε_mf (two-phase theory) — typical of gas-solid systems. Bubbles give excellent solids mixing but bypass the solids, which lowers gas-solid conversion.
• Slugging: in tall, narrow beds bubbles grow until their diameter exceeds about 0.6 × column diameter and the bed rises and collapses in slugs — severe pressure fluctuation, poor contacting.
• Turbulent and fast fluidisation: at still higher velocity bubbles lose identity; with solids recirculation this becomes the circulating fluidised bed (CFB). Above the terminal velocity of most particles the bed becomes pneumatic transport.
• Channelling (gas cuts preferential paths, bed partly dead) and spouting (a central jet) are maldistribution faults, caused by cohesive fines, wet solids or a poor distributor.
• Geldart groups: C (cohesive, < ~30 µm — channels, hard to fluidise, needs stirring/vibration), A (aerateable, ~30-100 µm, e.g. FCC catalyst — expands between u_mf and the minimum bubbling velocity u_mb), B (sand-like, ~100-800 µm — bubbles from u_mf itself, u_mb = u_mf), D (coarse/spoutable, > ~1 mm — large bubbles, spouting; used in FBC burning crushed coal).
• Elutriation and TDH: fines with u_t below the gas velocity are carried out; freeboard height above the bed is set at or above the transport disengaging height so splashed particles fall back, and cyclones/bag filters capture the rest.

Design elements of a fluidised bed

• Distributor: perforated or bubble-cap plate, sintered plate or nozzle grid. Its own pressure drop must be a substantial fraction of the bed ΔP (commonly ≳ 30%, often 0.1-0.5 of bed ΔP) so gas distributes evenly; it must also prevent solids draining back (sifting).
• Bed expansion: solids volume is conserved, so L(1 − ε) = L_mf(1 − ε_mf) — the standard route from voidage to bed height.
• Operating window: between u_mf and the carry-over velocity, with u_t/u_mf ratios of roughly 9 (coarse, Newton-range particles) to 90 (fine, Stokes-range particles); gas velocities are typically 2-10 × u_mf in practice.
• Heat transfer: bed-to-surface coefficients of several hundred W/m²K and near-isothermal beds are why FBC boilers burn high-ash Indian coal at 850-900 °C, below the NOx-forming and ash-fusion range, with limestone injected in-bed for SO₂ capture.
• Internals: baffles and tube banks break bubbles and limit slugging; attrition of catalyst and erosion of in-bed tubes are the chief maintenance issues.
• Indian practice and statute: AFBC and CFBC boilers are registered and inspected under the Indian Boiler Regulations, 2026 (in force from October 2026, superseding the 1950 Regulations), framed under the Boilers Act, 2025 (in force from 1 May 2025, replacing the Boilers Act, 1923); stack emissions follow CPCB/MoEFCC norms notified under the Environment (Protection) Act, 1986 and the Air Act, 1981; dust exposure and explosion control in pharmaceutical fluid-bed dryers come under the Occupational Safety, Health and Working Conditions Code, 2020, in force from 21 November 2025, which replaced the Factories Act, 1948. Vessels are built to the Indian unfired-pressure-vessel code where pressure-rated. Fluid-bed dryer/granulator qualification in pharma follows Schedule M of the Drugs and Cosmetics Rules and CDSCO GMP requirements.

Packed versus fluidised: choosing one

Key points

• Voidage ε - fraction of bed volume that is void; ΔP varies roughly as (1 − ε)/ε³, so small packing errors matter a lot.
• Sphericity φ_s - corrects a non-spherical particle to an equivalent sphere; always ≤ 1.
• Specific surface a = 6(1 − ε)/(φ_s d_p) - the drag-generating area per unit bed volume.
• Ergun equation - viscous + inertial pressure drop valid over the whole Re range for fixed beds.
• Kozeny-Carman - laminar limit, ΔP ∝ μu/d_p²; Burke-Plummer - turbulent limit, ΔP ∝ ρu².
• Particle Reynolds number Re_p = d_p u ρ/μ (superficial u); laminar below ~10, turbulent above ~1000.
• u_mf - velocity at which bed ΔP equals buoyant bed weight; found by equating Ergun to that weight, or by Wen-Yu.
• ΔP at fluidisation = L_mf(1 − ε_mf)(ρ_s − ρ)g - independent of velocity and of particle size.
• Hysteresis peak - first-run overshoot from interlocked particles; read u_mf from the decreasing branch.
• Bed expansion - L(1 − ε) is constant; solids volume does not change.
• Richardson-Zaki u/u_t = εⁿ - expansion law for smooth liquid (particulate) fluidisation.
• Terminal velocity u_t - upper bound; above it particles are entrained (pneumatic conveying).
• Geldart A/B/C/D - aerateable, sand-like, cohesive, spoutable; decides whether a solid will fluidise at all.
• Bubbling versus particulate - gas-solid beds bubble, liquid-solid beds expand smoothly.
• Slugging - bubble diameter > ~0.6D in tall narrow beds; avoid by limiting L/D and adding internals.
• Distributor ΔP - must be a large fraction of bed ΔP to prevent channelling and back-sifting.
• TDH and freeboard - space above the bed that lets splashed solids fall back; cyclones catch the rest.
• D/d_p > 10 - minimum column-to-particle ratio to avoid wall channelling in packed beds.

Formulas and examples

Example 1 - Ergun pressure drop in a catalyst bed. Spheres d_p = 2 mm, φ_s = 1, ε = 0.40, bed L = 1.2 m, column D = 0.6 m; air at 20 °C, ρ = 1.2 kg/m³, μ = 1.8×10⁻⁵ Pa·s, superficial u = 0.5 m/s.

1. Re_p = d_p u ρ/μ = (0.002 × 0.5 × 1.2)/1.8×10⁻⁵ = 66.7 → transition, both terms needed.
2. Viscous term = 150 × 1.8×10⁻⁵ × 0.5 × (0.6)² / (0.002² × 0.40³) = 4.86×10⁻⁴ / 2.56×10⁻⁷ = 1,898 Pa/m.
3. Inertial term = 1.75 × 1.2 × 0.5² × 0.6 / (0.002 × 0.064) = 0.315 / 1.28×10⁻⁴ = 2,461 Pa/m.
4. ΔP/L = 4,359 Pa/m → ΔP = 4,359 × 1.2 = 5.23 kPa (≈ 533 mm water gauge).
5. A = (π/4)(0.6)² = 0.2827 m², Q = 0.1414 m³/s. Blower power at η = 0.60: P = 0.1414 × 5,231/0.60 = 1.23 kW. At 8,000 h/yr and ₹8/kWh the fan alone costs about ₹0.79 lakh a year — which is why the inertial term, and hence velocity, is cut first.

Example 2 - u_mf of sand in air, two methods. Sand d_p = 300 µm, ρ_s = 2,600 kg/m³, ε_mf = 0.40, φ_s = 1; air as above.

1. Laminar Ergun form: d_p²(ρ_s − ρ)g/(150μ) = (9×10⁻⁸ × 2,598.8 × 9.81)/(2.7×10⁻³) = 2.294×10⁻³/2.7×10⁻³ = 0.850 m/s.
2. × ε_mf³/(1 − ε_mf) = 0.064/0.60 = 0.1067 → u_mf = 0.091 m/s. Check Re_mf = 0.0003 × 0.091 × 1.2/1.8×10⁻⁵ = 1.8 (< 20, so the laminar form is legitimate).
3. Wen-Yu: Ar = (2.7×10⁻¹¹ × 1.2 × 2,598.8 × 9.81)/(3.24×10⁻¹⁰) = 8.26×10⁻⁷/3.24×10⁻¹⁰ = 2,550.
4. Re_mf = √(1,135.7 + 0.0408×2,550) − 33.7 = √1,239.7 − 33.7 = 35.21 − 33.70 = 1.51 → u_mf = 1.51 × 1.8×10⁻⁵/(3×10⁻⁴ × 1.2) = 0.076 m/s.
5. The two differ by ~20% purely through the assumed ε_mf — report the range 7.6-9.1 cm/s and design the blower on the higher figure.

Example 3 - fluidised bed ΔP and expansion. Same sand, L_mf = 0.5 m, A = 0.2827 m².

1. ΔP = L_mf(1 − ε_mf)(ρ_s − ρ)g = 0.5 × 0.60 × 2,598.8 × 9.81 = 7.65 kPa — constant for all u > u_mf.
2. Solids mass = L_mf(1 − ε_mf)ρ_s A = 0.5 × 0.6 × 2,600 × 0.2827 = 220.5 kg; check ΔP = W(1 − ρ/ρ_s)g/A = 220.5 × 0.99954 × 9.81/0.2827 = 7.65 kPa ✓.
3. At ε = 0.55: L = L_mf(1 − ε_mf)/(1 − ε) = 0.5 × 0.60/0.45 = 0.667 m, i.e. 33% expansion — set freeboard above this.

Example 4 - upper limit of the operating window. Terminal velocity of the same sand in air.

1. Try Stokes: u_t = 2.294×10⁻³/(18 × 1.8×10⁻⁵) = 7.08 m/s → Re = 142, far outside Stokes' range, so reject.
2. Allen (intermediate) law: u_t = 0.153 × 9.81^0.71 × (3×10⁻⁴)^1.14 × 2,598.8^0.71 / (1.2^0.29 × (1.8×10⁻⁵)^0.43) = 0.01982/9.61×10⁻³ = 2.06 m/s. Check Re = 41 — inside 2-500 ✓.
3. u_t/u_mf = 2.06/0.076 ≈ 27, a wide window; an operating velocity of 0.3-0.6 m/s (4-8 × u_mf) gives vigorous bubbling with little carry-over of the main cut, though fines below ~80 µm will elutriate and need a cyclone.

Example 5 - mixed sizes. Mass fractions 0.30 at 1.0 mm, 0.50 at 0.50 mm, 0.20 at 0.20 mm. Σ(x_i/d_pi) = 0.30/1.0 + 0.50/0.50 + 0.20/0.20 = 2.30 mm⁻¹ → d_sm = 0.435 mm. Using the mass-weighted arithmetic mean (0.59 mm) instead would under-predict Ergun's viscous ΔP by about 45%.

Where it is used

• Refining and petrochemicals: FCC riser-regenerator units at Jamnagar, Vadinar, Panipat and Mathura; fixed-bed hydrotreaters, reformers and sulphur-guard beds where Ergun-based ΔP sets the recycle compressor duty.
• Power and captive steam: AFBC and CFBC boilers from Thermax, BHEL, ISGEC and Cethar burning high-ash Indian coal, washery rejects, rice husk and bagasse, with in-bed limestone for SO₂ capture.
• Pharma and food: fluid-bed dryers, top-spray granulators and Wurster coaters (Cipla, Sun Pharma, Dr Reddy's; equipment from ACG, GMM Pfaudler, Spray Engineering) — air velocity and distributor plate selection decide drying rate and fines loss.
• Metallurgy and minerals: fluid-bed roasting of zinc and pyrite concentrates (Hindustan Zinc), alumina calciners, DRI and fluid-bed coolers.
• Water and effluent treatment: packed sand/anthracite filters, activated-carbon adsorbers, ion-exchange and fluidised-bed biological reactors in CETPs; backwash velocity is a direct u_mf-to-u_t calculation.
• Job roles: process design engineer (ΔP and blower sizing), reactor/catalyst engineer, utilities and boiler engineer, production engineer troubleshooting channelling and distributor plugging, EPC proposals engineer (Technip, L&T, Toyo, Thyssenkrupp).

Interview questions

• Q: Why does the pressure drop across a fluidised bed stay constant above u_mf? A: Because the drag force must only balance the buoyant weight of the solids, which is fixed; extra gas is accommodated by bed expansion and bubbles, not by higher ΔP.
• Q: What are the two terms of the Ergun equation physically? A: A viscous (surface-friction) term proportional to μu that dominates in laminar flow, and an inertial (form-drag/kinetic-loss) term proportional to ρu² that dominates in turbulent flow.
• Q: How would you find u_mf for a solid you have never handled? A: Measure it: run a bed in a transparent column, plot ΔP against velocity on the decreasing branch and take the break point; cross-check with Wen-Yu using measured ρ_s and Sauter mean diameter.
• Q: Why read u_mf from the decreasing branch of the ΔP curve? A: The first upward run shows a spurious overshoot because interlocked particles have to be broken apart; the descending curve reflects a settled, reproducible bed.
• Q: Why is bed ΔP independent of particle size at fluidisation, but strongly size-dependent in a fixed bed? A: At fluidisation ΔP equals solids weight per unit area, which depends only on bed mass; in a fixed bed ΔP depends on surface area for drag, which scales with 1/d_p.
• Q: What is channelling and what causes it? A: Gas finding preferential paths and leaving part of the bed unfluidised, caused by cohesive fines (Geldart C), damp or sticky solids, or a distributor whose ΔP is too low relative to the bed.
• Q: Why must the distributor plate have appreciable pressure drop? A: Its resistance forces even gas distribution across the plate and stops solids sifting back; a common rule is at least about a third of the bed ΔP.
• Q: Which Geldart group is FCC catalyst and why is that chosen? A: Group A (roughly 60-80 µm); it fluidises smoothly, expands before bubbling and circulates readily between riser and regenerator.
• Q: Fixed bed or fluidised bed for a strongly exothermic catalytic reaction? A: Fluidised, because the vigorous solids mixing keeps the bed nearly isothermal and avoids hot spots and runaway; the cost is bubble bypassing and catalyst attrition.
• Q: What happens if you raise gas velocity above the terminal velocity of the particles? A: The bed is carried out of the vessel — pneumatic transport; with a cyclone and standpipe this becomes a deliberate circulating fluidised bed.
• Q: How does bed height affect u_mf and ΔP? A: u_mf is independent of bed height; ΔP is directly proportional to it.
• Q: Why does u_mf fall when a fluidised bed is heated to operating temperature? A: Gas viscosity rises and density falls; in the viscosity-controlled (fine particle) regime the higher μ means less gas velocity is needed to generate the same drag.

Common mistakes

• Using the interstitial velocity u/ε in the Ergun equation — it is written for the superficial velocity.
• Forgetting the sphericity for crushed or flaky solids, or applying φ_s twice (once in d_p and again in the equation).
• Taking an arithmetic mean size for a mixture instead of the Sauter mean, which under-predicts surface area and ΔP.
• Using the Stokes law for u_t without checking Re — for sand in air it is wrong by a factor of three.
• Claiming that bed ΔP keeps rising after fluidisation, or that it depends on the gas velocity above u_mf.
• Mixing units: mm for d_p with SI elsewhere, bar with Pa, poise with Pa·s (1 cP = 10⁻³ Pa·s).
• Ignoring the hydrostatic/static head in an upflow liquid bed, so the reported ΔP is the total rather than the frictional loss.
• Assuming any powder will fluidise — Geldart C solids channel and need a stirrer, vibration or granulation first.

Related topics

Filtration, sedimentation and fluidisation · Size reduction and screening · Conveying and storage of solids · Fluid flow and pumps in process plants · Dimensional analysis and similitude · Heterogeneous catalytic reactor design