Worked problem

Batch Reactor: Time to Reach a Target Conversion

Find the batch time for 90% conversion of a first-order reaction (k = 0.5 per minute) and compare it with ideal CSTR and PFR residence times.

Chemical & Process · Last reviewed October 17, 2026

Given

The first-order, irreversible, constant-density reaction A → B from the CSTR and PFR problem in this section, with k = 0.5 min⁻¹. A well-mixed batch reactor is charged with A and run until the conversion of A is 90%. Charging, heating, discharging and cleaning take 25 minutes per batch.

Find

The reaction time, the residence times that an ideal CSTR and an ideal PFR would need for the same conversion, and the number of batches per day.

Solution

t(batch) = −ln(1 − X) ÷ k

Batch reaction time: t = −ln(1 − 0.90) ÷ 0.5 = ln(10) ÷ 0.5 = 2.303 ÷ 0.5 = 4.61 min.
PFR: the space time needed is the same, τ = ln(1 ÷ (1 − X)) ÷ k = 4.61 min. A plug-flow reactor behaves like a batch reactor moving in space.
CSTR: τ = X ÷ [k × (1 − X)] = 0.90 ÷ (0.5 × 0.10) = 18.0 min, which is 3.9 times longer.
Cycle time per batch = 4.61 + 25 = 29.6 min, so a reactor can run 1,440 ÷ 29.6 = 48 batches per day.

Answer

batch 4.61 min  |  PFR 4.61 min  |  CSTR 18.0 min  |  48 batches per day

Reactor	Time for 90% conversion	Relative
Batch (reaction time)	4.61 min	1.0
Ideal PFR (space time)	4.61 min	1.0
Ideal CSTR (space time)	18.0 min	3.9

What it shows

The batch and the PFR need the same time, because every element of fluid follows the same concentration history.
The CSTR is much slower for high conversions, because the whole tank is always at the low outlet concentration.
For a batch reactor the reaction is only 16% of the cycle time. Turnaround time often matters as much as kinetics, so the batch size and the turnaround time are both part of the design.

Common mistakes

Using the CSTR formula for a batch reactor, or the reverse.
Comparing reaction time with residence time without remembering that a batch also needs charging and discharging time.
Using a first-order formula for a reaction that is not first order.
Using percent conversion (90) where the formula needs a fraction (0.90).

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

Questions

Why does the PFR match the batch?

In a PFR, a plug of fluid moves through the tube without mixing with its neighbors, so over the residence time it behaves like a small batch reactor.

When is a batch reactor a good choice?

For small volumes, many products in one vessel, long reaction times and cases where flexibility matters more than throughput.

How does the volume follow?

For a CSTR or PFR the volume is the flow rate times the space time. For a batch reactor the volume is the batch size, and the production rate is the batch size divided by the cycle time.

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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