Residence-Time Distribution — Mean Time and Tanks-in-Series Number: 23-Chem-A4 Chemical Reactor Engineering, May 2017, Q5

Question 5: Residence-Time Distribution — Mean Time and Tanks-in-Series Number (25 marks: a 18, b 7)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Pulse-tracer effluent response (equally spaced in time, $\Delta t=10$ s, endpoints zero).

$t$ (s)	$C$ (g/cm³)
0, 10, 20, 30, 40, 50	0, 0.2, 0.4, 0.7, 1.5, 1.8
60, 70, 80, 90, 100, 110	1.2, 0.8, 0.4, 0.3, 0.1, 0

Find. (a) The mean residence time $\bar t$; (b) the equivalent number of equal stirred tanks $N$.

Figure 5.1 — Pulse-tracer effluent curve. Its first moment gives $\bar t\approx51$ s; its spread gives $N\approx7$ tanks.

Approach. The mean residence time is the first moment of the pulse response and the tanks-in-series number follows from its normalized variance — both are discrete sums over the tabulated curve.

(a) Areas under the response. The zeroth and first moments (trapezoidal, endpoints zero) are $$\int C\,dt\;\propto\;\sum C_i=7.4,\qquad \int tC\,dt\;\propto\;\sum t_iC_i=378\ (\text{g}\cdot\text{s}/\text{cm}^3).$$
(a) Mean residence time. $$\bar t=\frac{\sum t_iC_i}{\sum C_i}=\frac{378}{7.4}=\boxed{51.1\ \text{s}.}$$
(b) Variance of the distribution. The second moment gives $$\sigma^2=\frac{\sum t_i^2C_i}{\sum C_i}-\bar t^{\,2}=\frac{21{,}940}{7.4}-(51.1)^2=2964.9-2609.2=\boxed{355.6\ \text{s}^2.}$$
(b) Tanks-in-series number. For the tanks-in-series model the dimensionless variance is $\sigma_\theta^2=\sigma^2/\bar t^{\,2}=1/N$, so $$N=\frac{\bar t^{\,2}}{\sigma^2}=\frac{2609.2}{355.6}=7.34\;\approx\;\boxed{7\ \text{tanks}.}$$

Quantity	Value
Mean residence time $\bar t$	51.1 s
Variance $\sigma^2$	355.6 s²
Dimensionless variance $\sigma_\theta^2$	0.136
Tanks-in-series number $N$	7.34 → 7 tanks

Concept & Theory

Topic. Residence-time distribution from a pulse tracer; moments and the tanks-in-series model of non-ideal flow.

Key relations. Exit-age distribution $E(t)=C/\int C\,dt$; mean $\bar t=\int tE\,dt$; variance $\sigma^2=\int(t-\bar t)^2E\,dt$; tanks-in-series $\sigma_\theta^2=\sigma^2/\bar t^{\,2}=1/N$.

Why this works. A pulse input labels each fluid element by the time it spends inside; the normalized response is the residence-time distribution, whose first moment is the mean holding time and whose relative spread measures how far the vessel departs from plug flow. A wide distribution (large $\sigma_\theta^2$) means few effective tanks; a sharp one ($\sigma_\theta^2\to0$, $N\to\infty$) approaches plug flow. Here $N\approx7$ indicates a moderately well-mixed but not ideal reactor.

Common pitfall. Forgetting to subtract $\bar t^{\,2}$ when computing the variance, or leaving $\sigma^2$ dimensional — $N$ requires the dimensionless variance $\sigma^2/\bar t^{\,2}$.

Source. Levenspiel, Chemical Reaction Engineering, Ch. 11–14 (RTD, tanks-in-series and dispersion models); Fogler, Ch. 16–18.

Question 5 of 5: Residence-Time Distribution — Mean Time and Tanks-in-Series Number

Question 5: Residence-Time Distribution — Mean Time and Tanks-in-Series Number (25 marks: a 18, b 7)