Reactor sizing (CSTR & PFR): practice problems with step-by-step solutions

Free Reactor sizing (CSTR & PFR) practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: CSTR first order, PFR first order, conversion from volume.

Free practice problemEasy
V = ?, k = 0.08 min⁻¹Q = 80 L/minX = 50%

A first-order liquid-phase reaction A → B with k = 0.08 min⁻¹ is carried out in a continuous stirred-tank reactor (CSTR) fed at 80 L/min. The density is constant and the reactor is at steady state. Find the reactor volume V in litres for 50% conversion of A.

Q = 80 L/mink = 0.08 min⁻¹X = 50%
L
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What you practise

  1. EasyCSTR first order
  2. MediumPFR first order
  3. Hardconversion from volume

Key formulas

Space time
τ = V/Q
CSTR, first order
τ = X/(k·(1 − X)), CA = CA0/(1 + kτ)
PFR, first order
τ = ln(1/(1 − X))/k, CA = CA0·e−kτ
Conversion from volume
CSTR: X = kτ/(1 + kτ), PFR: X = 1 − e−kτ
N equal CSTRs in series
X = 1 − 1/(1 + kτi)N

Learn Reactor sizing (CSTR & PFR) from scratch

Before this: Reaction kinetics

1What reactor sizing is

In industry, reactions usually run in continuous reactors: feed flows in all the time and product flows out all the time. The question is how big the reactor must be to convert enough of the feed.

Two ideal reactors cover most exam questions:

  • The CSTR (continuous stirred-tank reactor): a tank with a stirrer. It is perfectly mixed, so the whole tank has the same concentration as the outlet.
  • The PFR (plug-flow reactor): a long tube. Each slice of fluid moves through without mixing with the slices before and after it, like a row of tiny batch reactors.
QCA0CAV
QCA0CAV

Here the reaction is first order, −rA = k·CA, the density stays constant and the reactor runs at steady state.

2Space time and conversion

The space time τ is the reactor volume divided by the volumetric feed rate. It is the average time the fluid stays inside:

τ = V / Q

A 200 L reactor fed at 25 L/min has τ = 200/25 = 8 min.

The conversion X is the fraction of A that has reacted: CA = CA0·(1 − X). 75% conversion means a quarter of A is left.

For a first-order reaction everything depends on one number, the product k·τ. A larger reactor, a slower feed or a faster reaction all raise kτ and so the conversion.

Self-check A 200 L reactor is fed at 25 L/min. What is the space time τ?
  1. 0.125 min
  2. 225 min
  3. 5000 min
  4. 8 min

Answer: 8 min. τ = V/Q = 200 L / 25 L/min = 8 min. Multiplying (5000) or inverting the ratio (0.125) gives the wrong units: the result must be a time.

3The design equations

A CSTR reacts everywhere at the low outlet concentration. A mole balance on A gives:

CSTR: τ = X / (k·(1 − X)), CA = CA0 / (1 + kτ), X = kτ / (1 + kτ)

In a PFR each slice behaves like a batch reactor, so the first-order decay from kinetics applies with t = τ:

PFR: τ = ln(1/(1 − X)) / k, CA = CA0·e−kτ, X = 1 − e−kτ

Then the volume is V = Q·τ.

For the same conversion a CSTR always needs more volume than a PFR, because it reacts at the lowest concentration. The gap grows quickly at high conversion.

Several equal CSTRs in series act more and more like a PFR. Each tank divides the concentration by (1 + kτi), so after N tanks X = 1 − 1/(1 + kτi)N.

Self-check For the same feed, rate constant and conversion, which ideal reactor needs the larger volume?
  1. Both need the same volume
  2. It depends on the feed concentration
  3. The CSTR
  4. The PFR

Answer: The CSTR. A CSTR runs entirely at the low outlet concentration, so the rate is low everywhere. In a PFR most of the fluid sees higher concentrations and reacts faster.

4The method step by step

  1. Decide which reactor it is: stirred tank (CSTR) or tube (PFR). For tanks in series, treat one tank at a time.
  2. Write the conversion as a decimal: 85% is X = 0.85.
  3. Find τ: from V/Q if the volume is given, or from the design equation if the conversion is given.
  4. Use the matching design equation for the unknown: τ, V = Q·τ, CA or X.
  5. Check: the CSTR result needs more volume (or gives less conversion) than a PFR of the same size.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A first-order liquid-phase reaction A → B with k = 0.15 min⁻¹ runs in a CSTR fed at 25 L/min. The density is constant and the reactor is at steady state. Find the reactor volume V for 75% conversion of A.

1

Draw the reactor

Q = 25 L/minCA0X = 0.75V = ?

It is a CSTR, so the whole tank works at the outlet concentration CA0·(1 − X).

2

Space time

τ = X / (k·(1 − X)) = 0.75 / (0.15 · 0.25) = 20 min
3

Volume

V = Q · τ = 25 · 20 = 500 L
4

Check the answer

Put it back: kτ = 0.15 · 20 = 3, so X = 3/(1 + 3) = 0.75. Correct.

A PFR would need τ = ln 4/0.15 = 9.24 min, so V = 231 L. The CSTR is about 2.2 times larger, as expected.

Answer

V = 500 L (space time 20 min).

6Common mistakes

  • Using the PFR equation for a CSTR, or the other way round.
  • Using the volume V in the equations instead of the space time τ = V/Q.
  • Forgetting the (1 − X) in the CSTR equation.
  • Using log₁₀ in the PFR equation. It is the natural logarithm ln.
  • Giving τ in minutes when the volume in litres is asked: multiply by Q.
  • For tanks in series, treating them as one tank of the total volume.
Self-check A 250 L CSTR is fed at 25 L/min, with k = 0.15 min⁻¹ (first order). What conversion does it reach?
  1. 150%
  2. 60%
  3. 77.7%
  4. 97.4%

Answer: 60%. τ = 10 min, so kτ = 1.5 and X = 1.5/2.5 = 60%. 77.7% is the PFR result 1 − e⁻¹·⁵, and 97.4% comes from using V = 250 instead of τ.

7How exam questions look

  • Easy: a CSTR with first-order kinetics. Find τ, the volume or the outlet concentration.
  • Medium: the same for a PFR, or the ratio VCSTR/VPFR for a given conversion.
  • Hard: the conversion from a given volume: one CSTR, one PFR, or two equal CSTRs in series.

Series example: two 400 L CSTRs fed at 10 L/min, k = 0.05 min⁻¹. Each tank has kτ = 0.05 · 40 = 2, so C/CA0 = 1/3² = 1/9 and X = 88.9%. One 800 L tank would give only 80%.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
V = 800 L, k = 0.4 min⁻¹Q = 20 L/minCA0​ = 2.5 mol/LCA​ = ?

A liquid-phase first-order reaction A → B (k = 0.4 min⁻¹) runs in an 800 L continuous stirred-tank reactor (CSTR). The feed enters at 20 L/min with CA0 = 2.5 mol/L. The density is constant and the reactor is at steady state. Find the outlet concentration CA.

V = 800 LQ = 20 L/mink = 0.4 min⁻¹CA0 = 2.5 mol/L

Find CA (mol/L)

Show hints
  1. Space time: τ = V/Q = 40 min.
  2. CSTR, first order: CA = CA0/(1 + kτ).
  3. kτ = 16.
Show the answer and the step-by-step solution

Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.

Space time

τ = V/Q = 800/20 = 40 min

Outlet concentration

CA = 2.5/(1 + 0.4·40) = 0.1471 mol/L

Answer

CA = 0.1471 mol/L

Conversion: X = 94.12%.

Example 2Medium
Q = 20 L/minCA0​ = 2.5 mol/LCA​ = ?V = 200 L, k = 0.25 min⁻¹

A liquid-phase first-order reaction A → B (k = 0.25 min⁻¹) runs in a 200 L plug-flow reactor (PFR). The feed enters at 20 L/min with CA0 = 2.5 mol/L. The density is constant and the reactor is at steady state. Find the outlet concentration CA.

V = 200 LQ = 20 L/mink = 0.25 min⁻¹CA0 = 2.5 mol/L

Find CA (mol/L)

Show hints
  1. Space time: τ = V/Q = 10 min.
  2. PFR, first order: CA = CA0·e^(−kτ).
  3. kτ = 2.5.
Show the answer and the step-by-step solution

Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.

Space time

τ = V/Q = 200/20 = 10 min

Outlet concentration

CA = 2.5·e−0.25·10 = 0.2052 mol/L

Answer

CA = 0.2052 mol/L

Conversion: X = 91.79%.

Example 3Hard
Q = 25 L/minX = ?V = 50 L

A first-order liquid-phase reaction (k = 0.5 min⁻¹) runs in a 50 L plug-flow reactor (PFR) fed at 25 L/min. The density is constant and the reactor is at steady state. Find the conversion X in percent.

V = 50 LQ = 25 L/mink = 0.5 min⁻¹

Find X (%)

Show hints
  1. Space time per reactor: τ = V/Q = 2 min, so kτ = 1.
  2. PFR: X = 1 − e^(−kτ).
  3. Give X as a percentage.
Show the answer and the step-by-step solution

Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.

Space time

τ = V/Q = 50/25 = 2 min, kτ = 0.5·2 = 1

Conversion

X = 1 − e−kτ = 1 − e−1 = 63.21%

Answer

X = 63.21%

Questions and answers

What is Reactor sizing (CSTR & PFR)?

In industry, reactions usually run in continuous reactors: feed flows in all the time and product flows out all the time. The question is how big the reactor must be to convert enough of the feed.

Two ideal reactors cover most exam questions:

How do I solve Reactor sizing (CSTR & PFR) problems step by step?

Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.

What are common mistakes in Reactor sizing (CSTR & PFR)?
  • CA0·e−kτ is the PFR (or batch) result. A CSTR reacts at the outlet concentration: CA = CA0/(1 + kτ).
  • Use the space time τ = V/Q, not the volume, in the rate expression.
  • CA0/(1 + kτ) is the CSTR result. In a PFR the concentration decays exponentially: CA = CA0·e−kτ.
What do the Reactor sizing (CSTR & PFR) exercises cover?
  • Easy: CSTR first order
  • Medium: PFR first order
  • Hard: conversion from volume

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Learn first

Reaction kinetics

Calculators

CSTR and PFR volume calculator

Courses with this topic

Chemical Reaction EngineeringFE Chemical exam practice

More topics in Chemical

Mass balanceEnergy balanceIdeal gas lawReaction kineticsHeat exchangers (LMTD)

Printable worksheet

Reactor sizing (CSTR & PFR) worksheet with answers
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