Pulse-Tracer Test on a Fouled Polymerization Reactor: 23-Chem-A4 Chemical Reactor Engineering, Undated paper, Q3

Question 3: Pulse-Tracer Test on a Fouled Polymerization Reactor (25 marks: a 22, b 3)

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, $M=100{,}000$ units; water flow $Q=1000$ gal/h $=16.67$ gal/min; original volume 500 gal; effluent $C(t)$ from 0 to 100 min.

$t$ (min)	0	1	2	3	4	5	10	15	20
$C$ (units/gal)	0	205	225	222	215	205	165	138	111
$t$ (min)	25	30	40	50	60	70	80	90	100
$C$ (units/gal)	92	76	53	35	24	16	10	4	0

Find. (a) The active (swept) reactor volume; (b) the volume of polymer deposited.

Figure 3 — Effluent response to the inlet pulse: a sharp rise and a long exponential-like tail, close to a stirred tank. The dashed line marks the mean residence time, 23.2 min.

Approach. For a closed vessel the mean residence time is the first moment of the exit-age distribution, $t_m=\int tE\,dt$ with $E=C/\int C\,dt$ (Fogler Ch. 16). At steady flow the volume actually swept by fluid is $V=Qt_m$, and the original volume minus this active volume is the deposit. The tracer balance $Q\int C\,dt=M$ checks that the data are complete.

Tracer balance (data check). Trapezoidal rule over the unequal intervals: $\int_0^{100}C\,dt=6002$ units·min/gal. Then $Q\int C\,dt=16.67\times6002=100{,}030$ units, or 100.0% of the 100,000 injected. All the tracer is accounted for, so the tail is complete and no tracer is held in a slowly exchanging zone.
First moment. $\int_0^{100}tC\,dt=139{,}456$ units·min²/gal (trapezoidal rule on $tC$).
Mean residence time. $$t_m=\frac{\int tC\,dt}{\int C\,dt}=\frac{139{,}456}{6002}=23.2\ \text{min}.$$ Using the injected amount instead, $\int C\,dt=M/Q=6000$, gives the same result to three figures.
(a) Volume available for reaction. $$V_{active}=Q\,t_m=\left(\frac{1000}{60}\ \frac{\text{gal}}{\text{min}}\right)(23.2\ \text{min})=\boxed{387\ \text{gal}}$$
(b) Polymer build-up. $$V_{polymer}=V_{original}-V_{active}=500-387=\boxed{113\ \text{gal}}$$ That is 22.6% of the original volume, about 427 L in US gallons. At a typical solid-polymer density of 1.0–1.1 kg/L this is roughly 430–470 kg of deposit.

Assumption

All of the lost volume is treated as solid polymer, not stagnant liquid. The complete tracer recovery supports this: a stagnant pocket that slowly exchanged tracer would stretch the tail beyond 100 min. Gallons are taken as US gallons, as the problem gives no other basis.

The shape of the curve is also informative. Its variance is $\sigma^2=\int t^2C\,dt/\int C\,dt-t_m^2=396\ \text{min}^2$, which gives a tanks-in-series number $N=t_m^2/\sigma^2=1.4$. The vessel is still close to well mixed, so the deposit has shrunk the vessel rather than creating channels.

Quantity	Result
Tracer recovered	100,030 units (100.0%)
Mean residence time $t_m$	23.2 min
(a) Active volume	387 gal
(b) Polymer built up	113 gal (22.6% of 500 gal)

Concept & Theory

Topic. Residence-time distribution (RTD) from a pulse input; diagnosing lost reactor volume.

Key relations. $E(t)=C(t)/\int_0^\infty C\,dt$; $t_m=\int_0^\infty tE(t)\,dt$; $V=Qt_m$; tracer balance $M=Q\int_0^\infty C\,dt$; variance $\sigma^2=\int(t-t_m)^2E\,dt$.

Why this works. Every tracer molecule samples the flow paths open to the fluid, so the average time it spends inside is the open volume divided by the flow rate. That result holds whatever the mixing pattern. Solid polymer takes up space without holding tracer, so it shortens $t_m$ in proportion to the volume it occupies.

Common pitfall. Taking the peak time (2 min) as the residence time, integrating with equal time steps when the table switches from 1 min to 5 min and then 10 min intervals, or forgetting to convert 1000 gal/h to gal/min before multiplying by a time in minutes.

Source. Fogler, Elements of Chemical Reaction Engineering, Ch. 16 (pulse input, mean residence time, reactor diagnostics); Levenspiel, Chemical Reaction Engineering, Ch. 11.

Question 3 of 5: Pulse-Tracer Test on a Fouled Polymerization Reactor

Question 3: Pulse-Tracer Test on a Fouled Polymerization Reactor (25 marks: a 22, b 3)