CSTR

The continuous stirred-tank reactor operates perfectly mixed at steady state. Learn its design equation, first- and second-order sizing, CSTRs in series, residence time, mixing and agitation, jacket cooling, and steady-state multiplicity and stability.

A CSTR is a tank with continuous feed and product withdrawal and a strong agitator. Because it's perfectly mixed, the whole tank is at the outlet composition and temperature — easy to control, but conversion per volume is lower than in a plug-flow reactor.

CSTRs are the workhorse for liquid-phase reactions, polymerisation, fermentation and wastewater treatment.

An analogy

A CSTR is like a busy restaurant kitchen at steady state: orders come in and meals go out continuously, and at any instant the kitchen looks the same.

Perfect mixing

Feed is instantly blended; outlet composition = tank composition. The reaction therefore runs at the lowest (exit) concentration — for positive-order reactions that means lower rate than a PFR.

Sizing

Combine the design equation with the rate law at exit conditions. First order: X = Da/(1 + Da). Second order gives a quadratic. Tanks in series recover efficiency — 3–5 tanks approach PFR behaviour.

Agitation and heat transfer

Impellers (Rushton turbines, pitched blades, anchors for viscous fluids) provide mixing; power P = N_p ρ N³ D⁵. Jackets, internal coils or external loops remove reaction heat; heat-transfer area scales with V^(2/3) while heat generation scales with V — a scale-up challenge.

Multiplicity and stability

For exothermic reactions, heat generation (S-shaped vs T) and removal (straight line) can intersect up to three times: low (quench) and high (ignited) stable states and an unstable middle one. Control strategies and start-up procedures must manage this, and runaway protection is essential.

Formulas

• Design equation: V = F_A0 X / (−r_A), or τ = C_A0 X / (−r_A)
• First order: X = kτ / (1 + kτ)
• Second order: kC_A0τ = X / (1 − X)²
• N equal tanks, first order: X = 1 − 1/(1 + kτ_i)^N
• Heat removal: Q = UA (T − T_c)

Worked example: Size a CSTR

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

Find: τ and V.

• τ = X/(k(1 − X)) = 0.8/(0.2 × 0.2) = 20 min
• V = 10 × 20 = 200 L

Answer: 200 L

Worked example: Two tanks in series

Given: Two 100 L tanks, same feed and k

Find: Overall conversion.

• τ_i = 10 min → kτ_i = 2
• X = 1 − 1/3² = 0.889

Answer: 88.9 % (vs 80 % for one 200 L tank)

Common mistakes

• Wrong: Assuming conversion scales linearly with volume. — Right: Diminishing returns: X = kτ/(1 + kτ).
• Wrong: Scaling up without checking heat removal. — Right: Area/volume falls on scale-up — plan extra cooling.

Applications

• Polymerisation: Continuous PVC, polystyrene and emulsion polymer reactors.
• Fermentation & biotech: Chemostats and continuous bioreactors.
• Wastewater: Activated-sludge aeration tanks behave like CSTRs.

Key takeaways

• Perfectly mixed; operates at exit conditions.
• First order X = kτ/(1 + kτ).
• Tanks in series approach PFR.
• Watch heat removal and multiple steady states.

Going further

• Agitator scale-up: Constant power/volume or tip speed rules; CFD for mixing time.
• Adiabatic CSTR: Energy balance gives T = T₀ + ΔT_ad X.
• Bifurcation analysis: Maps steady states vs parameters for safe operating windows.