Worked problem

CSTR vs PFR: Conversion for a First-Order Reaction

Compare the conversion of a first-order reaction (k = 0.5 per minute) in a CSTR and a PFR with the same 4-minute residence time.

Chemical & Process · Last reviewed

Given

An irreversible first-order liquid-phase reaction A → B with rate constant k = 0.5 min⁻¹. Constant density. Both reactors have a space time (residence time) τ = 4 min.

Find

The conversion of A in an ideal CSTR and in an ideal PFR.

Solution

  1. Damköhler number: kτ = 0.5 × 4 = 2.
  2. CSTR (perfectly mixed): X = kτ ÷ (1 + kτ) = 2 ÷ 3 = 0.667 (66.7%).
  3. PFR (plug flow): X = 1 − e^(−kτ) = 1 − e^(−2) = 1 − 0.1353 = 0.865 (86.5%).

Answer

CSTR: 66.7% | PFR: 86.5%

What it shows

For a positive-order reaction the PFR gives more conversion than a CSTR of the same volume, because the CSTR runs at the low outlet concentration everywhere and so at the slowest rate. To reach 86.5% in a CSTR, kτ would have to be 6.4, so the CSTR volume would be 3.2 times larger.

Common mistakes

  • Using the PFR formula for a stirred tank, or the reverse.
  • Applying these formulas to a reaction that is not first order or where density changes with conversion (gas-phase reactions with a change in moles).
  • Comparing reactors of different volume or flow rate without using space time.

Try a variation: a longer residence time

  1. With τ = 8 min, kτ = 4. CSTR: X = 4 ÷ 5 = 0.80. PFR: X = 1 − e⁻⁴ = 0.982.
  2. Doubling the volume lifted the CSTR from 66.7% to 80%, and the PFR from 86.5% to 98.2%.

Two CSTRs in series

Split the original 4-minute total residence time between two equal tanks of 2 minutes each. For each tank kτ = 1.

X = 1 − (1 ÷ (1 + kτ))² = 1 − (1 ÷ 2)² = 0.75

Two tanks in series give 75% conversion in the same total volume, between the single CSTR (66.7%) and the PFR (86.5%). More tanks in series approach plug-flow behavior.

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Questions

When is a CSTR the better choice?

When good mixing, temperature control or handling of solids matters more than volume efficiency, and for reactions where a low concentration is wanted for selectivity.

How is the CSTR volume 3.2 times larger?

For X = 0.865, kτ = X ÷ (1 − X) = 6.4. Against kτ = 2 for the PFR, the space time and volume ratio is 6.4 ÷ 2 = 3.2.

Why do CSTRs in series approach PFR performance?

Each tank works at a lower concentration than the one before it, so the series follows the concentration profile of a plug-flow reactor more closely as the number of tanks grows.

What is the Damköhler number?

The ratio of the reaction rate to the flow rate. For a first-order reaction it is kτ. A large value means the reaction runs close to completion.

How does reaction order change the comparison?

For reactions of positive order the PFR needs less volume for a given conversion. For some reactions, such as autocatalytic ones, a CSTR can need less volume at low conversion.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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