Reactor Conversion Calculator (CSTR, PFR, batch, first & second order)

Understanding the CSTR and PFR conversion calculator — concept and principle

The two ideal reactor types of chemical reaction engineering, the continuous stirred-tank reactor (CSTR) and the plug-flow reactor (PFR), reach different conversions with the same rate constant and residence time. The calculator compares them for first- and second-order reactions, handles a chain of equal CSTRs in series, and reports the Damköhler number. A batch reactor run for time τ gives the same conversion as a PFR with residence time τ.

In the pipe the reactant is used up gradually; the stirred tank runs everywhere at the low outlet concentration.

How it is calculated

First order, PFR or batch: X = 1 − e^(−kτ)
First order, N equal CSTRs: X = 1 − (1 + kτ/N)^(−N)
Second order, PFR or batch: X = 1 − 1/(1 + k C_A0 τ)
Second order, CSTR: each tank solves k τᵢ C² + C − C_in = 0, so C = [−1 + √(1 + 4 k τᵢ C_in)] ÷ (2 k τᵢ), with τᵢ = τ/N
Damköhler number Da = kτ (first order) or k C_A0 τ (second order)

τ is the total residence time (volume ÷ volumetric flow); k must use the same time unit, and for second order k is in L/(mol·time).

The principle

In a PFR every fluid element travels like a small batch, so the concentration falls gradually along the tube and the reaction runs at high rates near the inlet. A CSTR is perfectly mixed: the whole tank sits at the outlet concentration, which is low, so the reaction runs at the slowest rate throughout. For positive-order reactions this makes a single CSTR less effective. Splitting the volume into several tanks in series lets the early tanks run at higher concentrations; as N grows the series approaches PFR behaviour. Da compares the reaction rate with the flow rate through the reactor.

Worked example

Inputs: k = 0.1, τ = 20, first order, C_A0 = 1 mol/L, 1 CSTR.
Da = 0.1 × 20 = 2.
PFR: X = 1 − e^(−2) = 1 − 0.1353 = 86.47%.
CSTR: X = 1 − 1/(1 + 2) = 66.67%.

Each point is the calculator re-run with only “Rate constant k” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Rate constant k	PFR / batch conversion
0.025	39.35%
0.05	63.21%
0.075	77.69%
0.1	86.47%
0.125	91.79%
0.15	95.02%
0.2	98.17%

How to read the result

For the same volume the CSTR here converts 20 points less. With two equal tanks the conversion rises to 75%, with three to 78.4%, closing in on the PFR value. Da much larger than 1 means high conversion; Da near or below 1 means much of the feed leaves unreacted. To reach a target conversion, a CSTR needs a larger volume than a PFR, and the difference grows as the target approaches 100%.

Common mistakes

Mixing time units, such as k per minute with τ in seconds.
Using a first-order k for a second-order reaction; the units and results differ.
Ignoring volume change in gas-phase reactions, which these constant-density formulas do not include.

Keep in mind: These are ideal, isothermal, constant-density models. Real reactors have imperfect mixing, dead zones, heat effects and side reactions; reactor design and scale-up require kinetic data and review by qualified engineers.

Frequently asked questions

When is a CSTR still the better choice?

When good temperature control, easy operation or low concentration is needed, for example in strongly exothermic or selectivity-sensitive reactions.

Is batch conversion really the same as PFR?

Yes for constant density: time in a batch reactor plays the role of residence time in a plug-flow tube.

What is the residence time?

τ = V/Q, the reactor volume divided by the volumetric feed rate.

Sources: O. Levenspiel, Chemical Reaction Engineering, 3rd edition, Wiley · H. S. Fogler, Elements of Chemical Reaction Engineering, Pearson

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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