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Chemical Engineering Science, 1970, Vol. 25, pp. 160% 1609. Pergamon Press. Printed in Great Britain.

Abstract The results of tracer experiments can depend on the methods of injection and measurementof the tracer when the fluid velocity is not uniform through the injection and measurement planes.Some simple examples show that incorrect residence-time distributions will be derived if this fact isnot considered. For laminar flow in a tube the results obtamedwilfdiffer widely, depending on thetechniques used.TRACER experiments can give usefufinformationon the residence-time -&tribution of fluid in avessel. For example, if fluid flows past the tracerinjection point and past the measuring point ata uniform velocity, then the output from a pulseinjection gives directly the residence-time distri-bution (r.t.d.) of fluid in the vessel. If, however,the velocity profile at these locations is not flat,then the different ways of injecting and measur-ing the tracer will give different tracer curves,only one of which is identical to the r.t.d. Ignor-ance of this fact has led to conflicting equationsfor the r.t.d. in the literature for seeminglyidentical situations.

flowing fluid (G. I. Taylors technique), orexposing a light sensitive fluid to a pulse of lightdoes this.

As for measuring techniques we again havetwo possibilities:

Measurement A: the mixing cup reading. Con-ceptually, the outflow is collected at differenttimes in containers and the average concentrationin each is measured.

Here we consider this question of differingresponses when the fluid velocity is not uniformat tracer injection and/or past the measuringpoint, their relationship, and the commonly usedtechniques which lead to these different tracercurves.

Measurement B: tracer is monitored withoutdisturbance as it flows past a measuring plane.It is usually supposed that it is the average con-centration in the cross-sectional plane which ismeasured. This measurement is approachedwhen using concentration probes within thevessel, or photocells or geiger counters at thewall, to monitor the fluid as it flows past themeasuring plane.

When the fluid velocity is not uniform throughthe cross-section at the injection point there aretwo ways of injecting a pulse of tracer.

Znjection A: tracer is injected in quantitiesproportional to the velocity of fluid at each par-ticular radial position. A rapid turbulent squirtof tracer may approximate to this method ofinjection. A step change, such as switching fluids,is also equivalent to this.

Four combinations of techniques are possible:A-A, B-A, A-B and B-B . The combinationB-B has frequently been used.

Injection B: tracer is injected uniformlyacross the cross-section. Replacing a slice of

Let us illustrate the difference in outputobtained with these techniques, using the simplestof examples, shown in Fig. 1. This flow-systemconsists of two sections, of volumes VI and V,,through which a constant-density fluid flows atrates o1 through VI, and u2 through V,. The flowin each section is plug-flow. We shall use forour numerical illustrations the values V, = V, =2m3, ul 3 3m3 hr-, and v2 = lm3 hr-. Thesefigures ensure that the mean residence-time of

The interpretation of residence-time experimentsOCTAVE LEVENSPlELt and J. C. R. TURNER

Department of Chemical Engineering, University of Cambridge, Cambridge, England(Received 10 February 1970)

tPresent address: Department of Chemical Engineering, Oregon State University, Corvallis, Oregon 97331, U.S.A.1605

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Fig. 1.fluid within the flow-system (as shown withinthe dotted line, neglecting the volume of pipe-work, of course) is (2 + 2)/(3 + 1) = 1 hr. We shallnow examine the results which would be obtainedfrom four possible tracer experiments on thissystem.Technique B-B

(i) Put equal quantities, Q, of tracer in eachentry stream in time 6 t, Fig. 2. Though thisprocedure might seem improbable in this case,it does correspond to the replacement of a sliceof flowing fluid when flows V, and vz are not inseparate pipes. Analyse at the exit from thevessels in the manner shown (e.g. a light beamshone through both exit streams). The tracer,Q, entering V, will be spread over a volumev16 and will take r/l/u, hr to reach the exit. Theanalyser will see an average concentration ofQ/2v,8t for a time 6 t. Similar expressions holdfor the tracer passing through V,.

The resultant ct diagram is also shown inFig. 2.The usual expression for the mean residence-time from a ct vs. t plot is

r, = I,; tc,dt/ j-w ctdt.0

In this case2&+2xQf=3 6e &+Q ,

*Fi ;;; j

Technique B-B 2l3 Ihr 2hrTimeFig. 2.

0. LEVENSPIEL and J . C. R. TURNERwhich is not the mean residence-time of thefluid. (Note that one can in this case obtain amaterial balance on the tracer entering andleaving the system if the two peaks in Fig. 2 aremultiplied by different flow-velocities: if the twovessels gave overlapping peaks (e.g. each vesselhad a well-mixed section) real difficulties wouldarise.)

The ct diagram gives this wrong answerbecause there is involved in it both the non-uniform tagging of the entering fluid and thenon-uniform weighting of the analysis of the twoexit streams.Technique B-A

(ii) Change case (i) by analysing after the twoexit streams have joined (assume completemixing at the join). We then have Fig. 3.

Now the analyser will see the tracer enteringVi, after time I/l/v,, as a pulse of concentrationQ/v& . vl/(ul + v2) lasting a time 6t. Compare thect diagram of Fig. 3 with Fig. 2.

It is clear from Fig. 3 that r, = 13, which iscorrect for the tracer, if not for the fluid. Thereis non-uniform tagging of the fluid, but a correctweighting of the two exit streams in the analysis.Technique A-B(iii) Inject the tracer in the inlet stream beforeit splits to enter the two vessels: analyse as incase (i). We then have Fig. 4.The concentrations of tracer (not the amounts)entering V, and V, are now the same and equalto 2 QJ( v1 + v,)6t, and a peak of this concentra-tion, lasting for at, is seen by the analyser at timeV,lv, in half of its field-hence the ct diagramin Fig. 4.

It is clear from Fig. 4 that & = 14. In this case

Fig. 3.1606

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Technique A-BFig. 4.

Time

the fluid is uniformly tagged but the weightingof the exit streams at analysis is incorrect.Technique A-A

(iv) Injection as in case (iii), analysis as incase (ii).Now the exit peak from V, will be analysedas a peak of concentration 2Q/(v1 + Q) .ul/(v, + u2), lasting for at, and at time V,/V,-hence the ct diagram of Fig. 5.

In this case ?, is given by2 3Q 2Q

fc =3x7+7 =

;Q+f 1.

This answer is correct for the residence-timeof the fluid, since we have uniformly tagged thefluid with tracer and have correctly weightedthe outlet streams in analysing for the tracer.Of these four cases only one, that using tech-nique A-A, gives directly the r.t.d. of the fluidin the vessel, or even gives the correct meanresidence time of fluid in the vessel. Now thelarger the difference in flow velocities at theinjection and measurement points the morethe tracer curves for techniques A-B, B-A andB-B will differ from the true r.t.d.Application to laminar fl ow with negligiblediflision

For a physical situation, where the tracercurves differ greatly according to the techniqueused, consider laminar flow in a cylindricalpipe of length 1, where 1 is short enough so thatmolecular diffusion does not significantly enterthe picture. In this case calculations give thefollowing dimensionless tracer curves and meanresidence time.

The interpretation of residence-time experimentspl>f~F~, pisi

Technique A-d 2/3 Ihr 2hrTimeFig. 5.Technique A-A

e, = 1C = l/283 for 8 > 3.

Technique A-B and B-Ae, =coC = l/283 for 8 > 4.

Although Techniques A-B and B-A give thesame results in this example and in that con-sidered earlier (both of which involve segregatedflows) they would not in general be expectedto do so.Technique B-B

e, =a2c = 1128 for 8 > 4.

These curves are shown in Fig. 6 and illustratewhat may well be the maximum expected varia-tion of tracer curves in the flow of a single fluidwith no reverse flow (diffusion or recycle flow)past the injection or measuring points.Appl ication to the tanks-in-ser ies model

For this model we always visualise that weinject and measure in a portion of well mixedfluid, hence only one set of equations, corre-sponding to technique A-A , apply. Thus thecalculated tracer curves give the correct r.t.d.directly.Appl ication to the dispersion model

This model visualises a uniform velocity pro-file with superimposed longitudinal dispersion,even though the actual velocity profile may bequite non-uniform. Because a diffusional processallows fluid to move upstream and pass the

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0. LEVENSPIEL and J. C. R. TURNER

Laminar flowin 0 tube

I 2 3 48

Fig. 6.injection and measurement points more thanone time, this introduces additional complica-tions and subcases, each leading to a differenttracer curve. Further work is being done on thisproblem.

directly; hence care is needed to obtain meaning-ful answers from tracer experimentation.Acknowledgment- 0. L. is grateful for the award of a SeniorFulbright Scholarship for the period during which this workwas undertaken.

Discuss ionIt is possible that some cases of poor materialbalance reported in experiments may have arisen

from the points considered above, rather thanfrom loss of tracer by decomposition, adsorptionand holdback, or experimental error.The general conclusion of this note is thatwhere there is a distribution of velocities in theregion of tracer injection or tracer measurementthen how one introduces and measures the tracerwill influence the shape of the tracer curve. Onlyone of these curves gives the correct r.t.d.

NOTATIONconcentration of tracer, kg mol m-3, mea-sured at time tcV/Q, a dimensionless concentrationtracer added, kg moltime, hrmean residence-time, calculated from ctplot, hrvolume flow rate, m3 hr-vessel volume, m3t/(true mean residence-time)dimensionless mean residence-time, cal-culated from C plot

RbumC- Les resultats des experiences traceuses peuvent dependre des mtthodes dinjection et demesure du traceur quand la velocite du fluide nest pas igale a travers les surfaces dinjection et demesure. Quelques exemples t&s simples demontrent que les distributions fausses de temps deresidence sont d&i&es si ce fait est ignore. Lecoulement laminaire dans un tube donne des resultatsqui seraient differents Iun de Iautre selon les techniques employees.1608

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The interpretation of residence-time experimentsZusammenfassung- Die Ergebnisse von Tracer Versuchen kiinnen von de Methoden der Einsprit-zung und der Messung des Tracers abhangen wenn die Fliissigkeitsgeschwindigkeit in den Bereichender Einspritzung und der Messung nicht gleichmiissig ist. Aus einigen einfachen Beispielen ist ersicht-lich, dass bei Nichtbeachtung dieser Tatsache ungenaue Verweilzeitverteilungen erhalten werden.Fur Laminarstromung in einem Rohr werden die erhaltenen Resultate, je nach den verwendetenMethoden, weitgehend von einander abweichen.

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