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.

[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.