Impulse Response of Two and Three Continuous Stirred-Tank Reactors in Series: Exact and Approximate Solutions

Consider two or three continuous stirred-tank reactors (CSTRs) in series. If the feed to the first reactor is an impulse tracer, then the outlet concentration from the last reactor is given in the Laplace domain by
, where is the number of CSTRs in series (in this Demonstration, 2 or 3) and is the residence time of the CSTR.
An exact solution can be found by using the Mathematica built-in function InverseLaplaceTransform. This solution is shown in blue. A very accurate approximate solution was derived by B. J. McCoy [1] using lower-order moments (the first three moments) and fitting the solution to a Poisson function. This approximate solution is shown by the dashed red curve.
If the CSTRs are identical, then the approximate solution is identical to the exact solution, as can be seen from the first two snapshots. A good approximation to the exact solution is also obtained for the two CSTRs in series cases when the residence times and have very different values.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] B. J. McCoy, "Approximate Polynomial Expansion Method for Inverting Laplace Transforms of Impulse Responses," Chemical Engineering Communications, 52(1–3), 1987 pp. 93–103.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+