CSTR vs PFR — Reactor Sizing, Design Equations & Damköhler Number

⚗️ CSTR vs PFR — Key Differences

The two most important continuous flow reactors in chemical engineering are the CSTR (Continuous Stirred Tank Reactor) and PFR (Plug Flow Reactor). Understanding their fundamental differences is essential for reactor selection and sizing.

🔵 CSTR

→ Contents perfectly mixed at all times
→ Exit concentration = reactor concentration
→ Operates at exit (lowest) concentration
→ Concentration is uniform throughout
→ Residence time distribution is exponential
→ Easy temperature control
→ Good for reactions needing dilution of feed

🔷 PFR

→ No mixing in axial direction
→ Concentration varies along tube length
→ Operates at all concentrations from inlet to outlet
→ Narrower residence time distribution
→ More complex temperature control
→ Smaller volume than CSTR for same conversion
→ Better for reactions with high activation energy

🎯

Key insight: Because a CSTR operates entirely at the exit (lowest) concentration, its reaction rate is always lower than a PFR for positive-order reactions. This means a CSTR always requires more volume than a PFR to achieve the same conversion — for the same reaction rate law.

Batch Reactor (BR) — Brief Comparison

Property	CSTR	PFR	Batch
Flow type	Continuous	Continuous	Non-flow
Mixing	Perfect (backmixed)	No axial mixing	Perfect
Operation	Steady state	Steady state	Unsteady state
Design variable	Volume (V)	Volume (V)	Time (t)
Scale-up	Moderate difficulty	Easier	Difficult
Equivalent to	1 stage (CSTR)	Infinite CSTRs in series	PFR (time-based)

📐 Design Equations — Mole Balance

Both reactor design equations come from a mole balance on the reacting species A. For a reaction A → Products with rate −r_A:

CSTR Design Equation

// CSTR Mole Balance (steady state, perfect mixing)

F_A0 − F_A + r_A × V = 0

V_CSTR = F_A0 × X_A / (−r_A)exit

F_A0 = Molar feed rate of A (mol/s) | X_A = Conversion | (−r_A)exit = Rate evaluated at EXIT conditions (lowest concentration) | V = Reactor volume (m³)

PFR Design Equation

// PFR Mole Balance (differential form)

dF_A / dV = r_A

// PFR Integral Form

V_PFR = F_A0 × ∫₀ˣᴬ dX_A / (−r_A)

Rate (−r_A) varies along reactor length — must integrate from X=0 to X=X_A. For analytical rate laws, this can be solved exactly. Otherwise, numerical integration is used.

Design Equations for Common Rate Laws

Reaction Order	Rate Law	V_CSTR	V_PFR
Zero order	−r_A = k	F_A0 × X / k	F_A0 × X / k (same)
First order	−r_A = k×C_A	C_A0×X / (k×(1−X))	C_A0×(−ln(1−X)) / k
Second order	−r_A = k×C_A²	C_A0×X / (k×C_A0²×(1−X)²)	X / (k×C_A0×(1−X))

✅

Important distinction: The CSTR volume is simply (F_A0 × X) / (−r_A) evaluated at the exit — one algebraic calculation. The PFR volume requires integration. For first-order reactions, the integral has a clean analytical solution: V_PFR = (F_A0/k) × [−ln(1−X)].

🔢 Damköhler Number (Da)

The Damköhler number is a dimensionless number that compares the reaction rate to the transport (flow) rate. It is one of the most useful quick-sizing tools in reactor engineering:

// Damköhler Number — General Definition

Da = (reaction rate at inlet) / (convective transport rate)

Da = −r_A0 × V / F_A0 = −r_A0 × τ / C_A0

τ = space time = V/v₀ | r_A0 = initial reaction rate | F_A0 = inlet molar flowrate

Da for Specific Reactor Types

// Da for First-Order CSTR

Da₁(CSTR) = k × τ = X_A / (1 − X_A)

// Da for First-Order PFR

Da₁(PFR) = k × τ = −ln(1 − X_A)

Da₁ = first-order Damköhler number. Subscript 1 denotes first-order reaction.

Damköhler Number as a Design Guide

Da Value	Physical Meaning	Expected Conversion
Da ≪ 1 (Da ~ 0.1)	Reaction much slower than flow	X_A ≈ 10% (low conversion)
Da = 1	Reaction rate ≈ transport rate	X_A ≈ 50% (CSTR), 63% (PFR)
Da ≫ 1 (Da ~ 10)	Reaction much faster than flow	X_A ≈ 90%+ (high conversion)

🎯

GATE tip: For a first-order reaction in a CSTR, the conversion can be directly written as X_A = Da / (1 + Da) = kτ / (1 + kτ). For a PFR, X_A = 1 − e^(−Da) = 1 − e^(−kτ). Memorising these two expressions solves the majority of GATE reactor problems.

⏱ Space Time, Space Velocity & Residence Time

These three related quantities are frequently confused — understanding each clearly is essential for reactor design and GATE problems.

// Space Time (τ)

τ = V / v₀

V = reactor volume (m³) | v₀ = volumetric flowrate at inlet conditions (m³/s) | Units: time (s, min, h). This is NOT the same as residence time for a PFR with variable density.

// Space Velocity (SV)

SV = 1/τ = v₀/V

Units: time⁻¹ (s⁻¹, h⁻¹). Common in gas-phase catalytic reactors (GHSV = Gas Hourly Space Velocity). Higher SV means shorter contact time.

// Mean Residence Time (t̄)

t̄ = V / v_avg

v_avg = average volumetric flowrate inside reactor. Equal to τ only for constant-density (liquid phase) reactions. For gas-phase reactions, t̄ ≠ τ.

Parameter	Symbol	Formula	When τ = t̄?
Space time	τ	V/v₀	Always defined this way
Mean residence time	t̄	V/v_avg	Only when density is constant
Space velocity	SV	1/τ	—
CSTR residence time	t̄_CSTR	V/v_exit	When v₀ = v_exit

📊 Levenspiel Plot — Graphical Reactor Sizing

The Levenspiel plot (1/−r_A vs X_A) is the most powerful graphical tool for reactor sizing and comparison. It works for any rate law — no analytical integration needed.

// Levenspiel Plot Interpretation

V_CSTR/F_A0 = X_A / (−r_A)exit → RECTANGLE area under the curve

V_PFR/F_A0 = ∫dX/(−r_A) → AREA under the curve

On a Levenspiel plot: CSTR volume = rectangular area (always larger for positive-order reactions). PFR volume = area under the curve (always smaller). The difference is the "wasted" volume due to CSTR backmixing.

// LEVENSPIEL PLOT — CSTR vs PFR VOLUME COMPARISON

💡

Exception — autocatalytic reactions: For autocatalytic reactions (rate passes through a maximum), the Levenspiel plot has a minimum. In this case, a CSTR can be more efficient than a PFR at low conversions — the optimal design uses a CSTR followed by a PFR.

🔗 Reactors in Series & Parallel

CSTRs in Series

Multiple CSTRs in series approach the performance of a PFR. Each CSTR operates at a progressively lower concentration, reducing the volume penalty of backmixing:

Feed

C_A0, X=0

→

CSTR 1

X₁

→

CSTR 2

X₂

→

CSTR N

X_A (final)

→

N→∞

≡ PFR

// N Equal CSTRs in Series — First Order

X_A = 1 − 1/(1 + kτ_i)^N

τ_i = V_i/v₀ = space time of each individual CSTR | N = number of CSTRs | As N→∞, performance approaches PFR

PFRs in Parallel

✅

PFRs in parallel: Identical PFRs in parallel give the same conversion as a single PFR with the same total volume, as long as the flow is split equally. This is commonly used when a single large tube is impractical (pressure drop, heat transfer limitations).

🎯 When to Use CSTR vs PFR

🌡️

Highly Exothermic Reactions

CSTR's backmixing moderates temperature — feed is diluted into a large, well-mixed volume, preventing hot spots.

Use CSTR

⚡

High Activation Energy Reactions

PFR maintains high temperature and concentration — benefits reactions that are strongly favoured at high T and high C.

Use PFR

🧫

Biological / Enzymatic Reactions

Michaelis-Menten kinetics with inhibition at high substrate concentrations — CSTR's dilution suppresses inhibition.

Use CSTR

🔥

Gas Phase, High T, High P

Tubular reactors (PFR) are structurally better for high pressure containment. Common in ammonia synthesis, methanol.

Use PFR

🔄

Autocatalytic Reactions

Rate is highest at intermediate conversion — CSTR first (to reach high rate), then PFR to push to final conversion.

CSTR then PFR

🎯

High Selectivity Required

For series reactions A→R→S where R is desired, PFR's concentration profile prevents over-reaction to S.

Use PFR

🧮 Worked Example — First Order Liquid Phase Reaction

Problem: A first-order liquid phase reaction A → B with k = 0.25 min⁻¹. Feed flowrate v₀ = 10 L/min, C_A0 = 2 mol/L. Find the reactor volume required for 80% conversion in (a) a CSTR and (b) a PFR.

Given

Parameter	Value	Unit
Rate constant (k)	0.25	min⁻¹
Volumetric flowrate (v₀)	10	L/min
Inlet concentration (C_A0)	2	mol/L
Target conversion (X_A)	0.80	—
Molar feed rate (F_A0 = C_A0 × v₀)	20	mol/min

(a) CSTR Volume

// CSTR — First Order

V_CSTR = F_A0 × X / (k × C_A0 × (1−X))

V_CSTR = 20 × 0.80 / (0.25 × 2 × (1−0.80))

V_CSTR = 16 / (0.25 × 2 × 0.20) = 16 / 0.10 = 160 L

Space time τ = V/v₀ = 160/10 = 16 min | Da = kτ = 0.25 × 16 = 4.0 | Check: X = Da/(1+Da) = 4/5 = 0.80 ✅

(b) PFR Volume

// PFR — First Order

V_PFR = (F_A0 / (k × C_A0)) × (−ln(1−X))

V_PFR = (20 / (0.25 × 2)) × (−ln(1−0.80))

V_PFR = 40 × (−ln(0.20)) = 40 × 1.609 = 64.4 L

Space time τ = V/v₀ = 64.4/10 = 6.44 min | Da = kτ = 0.25 × 6.44 = 1.61 | Check: X = 1−e^(−1.61) = 1−0.20 = 0.80 ✅

Comparison Summary

Parameter	CSTR	PFR	Ratio
Volume required	160 L	64.4 L	2.49×
Space time (τ)	16.0 min	6.44 min	2.49×
Damköhler number (Da)	4.0	1.61	—
Conversion achieved	80%	80%	Same
Capital cost implication	Higher (2.5× volume)	Lower	—

🎯

Conclusion: For this first-order reaction at 80% conversion, the CSTR requires 2.49× more volume than the PFR. This ratio increases at higher conversions. For 90% conversion, the ratio would be V_CSTR/V_PFR = 9/2.30 ≈ 3.9×, showing how the penalty grows sharply at high conversion.

CSTR vs PFR — When to Use Which Reactor & How to Size Them

⚗️ CSTR vs PFR — Key Differences

🔵 CSTR

🔷 PFR

Batch Reactor (BR) — Brief Comparison

📐 Design Equations — Mole Balance

CSTR Design Equation

PFR Design Equation

Design Equations for Common Rate Laws

🔢 Damköhler Number (Da)

Da for Specific Reactor Types

Damköhler Number as a Design Guide

⏱ Space Time, Space Velocity & Residence Time

📊 Levenspiel Plot — Graphical Reactor Sizing

// LEVENSPIEL PLOT — CSTR vs PFR VOLUME COMPARISON

🔗 Reactors in Series & Parallel

CSTRs in Series

PFRs in Parallel

🎯 When to Use CSTR vs PFR

Highly Exothermic Reactions

High Activation Energy Reactions

Biological / Enzymatic Reactions

Gas Phase, High T, High P

Autocatalytic Reactions

High Selectivity Required

🧮 Worked Example — First Order Liquid Phase Reaction

Given

(a) CSTR Volume

(b) PFR Volume

Comparison Summary

Practice Reactor Design — GATE Style Problems