PFR

The plug-flow reactor has no axial mixing — fluid moves like a piston. Learn its differential design equation, first- and second-order sizing, packed-bed reactors and pressure drop, temperature profiles and hot spots, multitubular designs and recycle reactors.

In a long tube, each fluid element travels like a plug: concentration falls gradually along the length. This keeps the average concentration high, so PFRs achieve high conversion in less volume than CSTRs for most reactions.

Catalytic packed beds, cracking furnaces and many gas-phase reactors are PFRs.

An analogy

A PFR is like a queue of batch reactors on a conveyor belt: each slice reacts as it travels, never mixing with slices ahead or behind.

Plug flow

No mixing along the flow; perfect mixing across the section. Every fluid element spends the same time (τ) in the reactor, like a batch reactor in time — the design equations have the same form.

Packed-bed reactors

Catalyst pellets fill the tube; design uses catalyst mass W. Pressure drop through the bed (Ergun) matters for gas reactions: smaller pellets increase rate (effectiveness) but also ΔP.

Temperature management

Exothermic reactions create hot spots near the inlet where rates are highest. Multitubular reactors (thousands of narrow tubes in a coolant shell), interstage cooling between adiabatic beds, or dilution control temperature.

Non-ideal and advanced PFRs

Axial dispersion models and RTD measurements quantify deviations. Recycle reactors span CSTR–PFR behaviour; membrane reactors remove products to shift equilibrium; reverse-flow reactors trap heat for autothermal operation. Parametric sensitivity analysis prevents runaway.

Formulas

• Differential design: F_A0 dX/dV = −r_A
• First order: τ = −ln(1 − X)/k
• Second order (A + A): k C_A0 τ = X/(1 − X)
• Packed bed: F_A0 dX/dW = −r′_A (per catalyst mass)
• Ergun (per length): ΔP/L = 150μ(1−ε)²u/(ε³d_p²) + 1.75ρ(1−ε)u²/(ε³d_p)

Worked example: Size a PFR

Given: First order, k = 0.2 min⁻¹; v₀ = 10 L/min, X = 0.8

Find: τ and V.

• τ = ln(5)/0.2 = 8.05 min
• V = 80.5 L

Answer: ≈ 80 L (vs 200 L CSTR)

Worked example: Second-order PFR

Given: k = 0.5 L/mol·min, C_A0 = 2 mol/L; X = 0.9

Find: τ.

• kC_A0τ = 0.9/0.1 = 9 → τ = 9 min

Answer: 9 min

Common mistakes

• Wrong: Ignoring pressure drop in packed beds. — Right: Include Ergun ΔP — it changes gas concentration and conversion.
• Wrong: Designing only for average temperature. — Right: Check hot spots and parametric sensitivity.

Applications

• Steam cracking & reforming: Fired tubular reactors for ethylene and hydrogen.
• Catalytic synthesis: Multitubular reactors for ethylene oxide, phthalic anhydride.
• Flow chemistry: Pharma continuous-flow tubular reactors.

Key takeaways

• F_A0 dX/dV = −r_A; first order τ = −ln(1 − X)/k.
• Smaller than CSTR for positive orders.
• Packed beds: catalyst mass and Ergun ΔP.
• Manage hot spots with multitubular designs.

Going further

• Axial dispersion model: Peclet number quantifies deviation from plug flow.
• Membrane reactors: Selective removal of H₂ shifts dehydrogenation equilibria.
• Reverse-flow reactors: Periodic flow reversal traps heat for lean mixtures.