Effux Time Equations

Efflux Time from Tanks with Exit Pipesand Fittings*

DAVID B. VAN DONGENTechnology Development Department, Cultor Food Science, Inc., Ardsley NY, USA.E-mail: [email protected]

EDWARD C. ROCHE, JR.Dept. of Chemical Engineering, Chemistry, and Environmental Science,New Jersey Institute of Technology, Newark NJ, USA

The efflux time experiment is simple to construct, simple to operate, and very useful for teachingthe fundamentals of fluid flow, friction loss, and data analysis in an engineering laboratory coursethat addresses fluid mechanics. Students have the opportunity to study the effect of various criticalparameters and dimensions on the time to drain a tank, analyze the data, and draw meaningfulconclusions.

INTRODUCTION

THE DEVELOPMENT of cost-effective teachingtools is an important issue for all institutions ofhigher learning particularly in light of recent cost-cutting initiatives. Departmental budgets leavelittle room to maintain or even construct anyapparatus that could be used to demonstrate andteach abstract concepts. These abstract conceptswill simply remain in the student's mind only asmemorized abstract equations and Greek mathe-matical symbols unless students can see theseconcepts in action, and be able to understand thephysical significance of the equations and symbols.Add to this the pressure on students to carry afull course load, work full-time to pay tuition,compete for a limited number of opportunitiesafter graduation, and the onerous task of com-pleting the senior-level chemical-engineering unit-operations laboratory can become an unpleasantexperience for all.

There are simple solutions. While instructorsare lecturing at the university level, it is importantfor them to continually bring students `back toreality'. That is, instructors need to demonstratethe simple fact that abstract concepts do indeedhave physical significance that can be measuredand observed. (The authors constantly remindstudents that employers do not sell abstractconcepts and mathematical equations; ratherthey sell tangible products and valuable services.Employers want students who can relate conceptsto the physical world.) One way to demonstratethe physical significance of abstract concepts isthrough the simple efflux time experiment asdescribed below.

Even though the efflux time experiment at NJITis part of the fluid mechanics experiment repertoirein the senior-level chemical engineering laboratory,it does have broader appeal to other engineeringdisciplines. Mechanical and civil engineeringstudents also study fluid mechanics, usually witha laboratory course. This particular cost-effectiveexperiment is suitable for all three engineeringdisciplines. Furthermore, many schools are inte-grating simple engineering experiments into lower-level and introductory engineering courses toprovide the students an initial experience to thefields of engineering and its application, prior tohaving students make a firm selection of theirpreferred field of study. The efflux time experimentis one that should be considered.

APPARATUS

The efflux time experimental apparatus involvesa supported tank with some means of gauging ormonitoring liquid level (preferably a simple closed-end Plexiglas tube with a 1-inch diameter nipple atthe discharge), a 1-inch diameter quick-openingvalve (such as a knife gate or ball valve), and aseries of pre-cut threaded end-connection exitpipes (and/or tubing lengths, of various lengthsand diameters) with reducing couplings and adap-ters to fit the exit pipes to the quick-action valve.All parts can be obtained off the shelf from anindustrial supply house such as McMaster-CarrSupply Co. or W. W. Grainger, Inc. A simpleschematic diagram is shown in Fig. 1.

Students will also need a method of monitor-ing liquid level as a function of real time. Thiscan be done with a simple stopwatch, or with adifferential-pressure gauge connected to a trans-mitter with a local read-out on a strip-chart* Accepted 16 January 1999.

206

Int. J. Engng Ed. Vol. 15, No. 3, pp. 206±212, 1999 0949-149X/91 $3.00+0.00Printed in Great Britain. # 1999 TEMPUS Publications.

recorder. If a stopwatch is used, then the tank or alevel gauge should be marked at regular intervals.Several chronometers can also be ganged togethersuch that they can all be started simultaneously,and then stopped individually as the liquid levelfalls past the interval marks on the level gauge.

Lastly, there should be an easy method of fillingthe tank with fixed piping, a garden hose, or abucket and funnel. In addition a method of collect-ing the discharge or directing it to a sewer drainmust be provided for.

PROCEDURE

The experiment is simple to operate:

1. Attach the desired exit pipe to the quick-opening valve.

2. Fill the tank to a level slightly above the highestmark on the level scale.

3. Reset the timers and start them all simul-taneously.

4. Open the quick-action valve.5. Stop the first timer when the liquid level falls to

the first (highest) mark on the level scale.6. Stop a timer as the liquid level falls past pre-

selected uniformly spaced marks on the levelscale.

7. Record the times.8. After the liquid has drained out, close the valve,

and refill the tank.9. Reset the timers, and repeat the experiment as

needed.

The tank is always filled to a level several inchesabove the top mark to provide enough liquid to

hydraulically fill the exit pipe, and to accelerate theflowing liquid to an initial quasi-steady state.

THEORY

The theory describing the efflux time of a tankhas been derived by Crosby [1] and by Bird,Stewart, and Lightfoot [2], and further extendedto systems with the installed fittings by Hanesian[3]. Under conditions of turbulent flow in the exitpipe, the efflux time, t, can be related to the heightof the liquid relative to the bottom of the exit pipe,H, by:

t � k1�H 3=7i ÿH 3=7� �1�

where

k1 � 7

3

r

r0

� �20:0791 � Le � �1=4

21=4 � g � �1=4 � r5=40

" #4=7

�2�

This constant contains the relevant physicalproperties of the liquid and critical dimensionsof the apparatus with the total equivalent lengthof the exit pipe and fittings represented by theterm Le. (See nomenclature section for definitionof terms.) The underlying key goal of this experi-ment is to teach the student that an abstract termsuch as Le (that is `equivalent' length, not actuallength) does have physical significance, a termthat can be indirectly measured and observedthrough proper data analysis. Furthermore, theanalysis of a set of data, from different runswhere the length of pipe has been changed, canclearly demonstrate the concept of `entranceeffect'. Students can calculate the equivalentlength pipe that would have the same pressuredrop or flow resistance that is caused by theentrance effect.

The constants and exponents appearing in theseequations are a direct result of approximating theFanning friction factor for flow through a pipe bythe Blassius relation:

f � 0:0791

Re1=4�3�

If we ignore the change in friction factor andsimply use a value that represents the average forthe flow regime and pipe roughness, then:

t � k2�H 1=2i ÿH 1=2� �4�

where:

k2 � 2r

r0

� �2favg � Le

g � r0

� �1=2

�5�

In the case of laminar flow in the exit pipe, effluxtime is related to liquid height by:

t � k3 lnH i

H

� ��6�

Fig. 1. Schematic diagram of apparatus.

Efflux Time from Tanks with Exit Pipes and Fittings 207

where:

k1 � 8 � Le � � � r 2

g � � � r40

�7�

Generalized modelIn the most general sense where the Fanning

friction factor is locally approximated by amodified Blassius relation:

f � k4

Ren �8�

the efflux time is related to liquid level by:

t � k5�H mi ÿH m� �9�

where:

k5 � 1

m

r

r0

� �2k4 � Le � �n

2n � g � �n � rn�10

" #1=�2ÿn��10�

and:

m � 1ÿ n

2ÿ n�11�

For example, the Fanning friction factor forflow of water, at room temperature, through a

1/2-inch, Schedule 40 steel pipe in the Reynoldsnumber range of 40,000 to 80,000 can beapproximated by:

fex � 0:016

Re0:0742�12�

and the appropriate value for m is:

mex � 0:4807 �13�The friction factor is easily estimated for a widerange of flow conditions and pipe roughnessfactors using the model presented by Churchill [4].

TYPICAL EXPERIMENTAL RESULTS

Typical results for one run are shown in Fig. 2.As expected, the height and apparent rate ofefflux decreases with time. It is difficult toassign an exact zero time for the run, but, aswe shall see below, this is not necessary. The timerecorded when the levels falls to the highest markneed not be accepted as time zero as the error inmarking this time is the same as marking any ofthe time intervals. Furthermore, the slope of theplotted data is important for this analysis, not the

Fig. 2. Typical experimental data.

D. B. Van Dongen and E. C. Roche208

intercept. Thus the students do not have to adjustall efflux times relative to the time on the firsttimer.

Data regression for one runFrom the run in Fig. 2, it was found that the

Reynolds number changed from about 40,000 to80,000. For this example, the data are plotted asefflux time versus the height raised to the 0.4807power as shown in Fig. 3. This data should be fit tothe best straight line where the intercept defines thebest value for time zero, and the slope is the bestvalue for k5 from which the equivalent length, Le,is found.

Data set for varying lengthThe experiments were repeated for a series of

pipe lengths. The slopes of the lines were found bylinear regression. For each pipe length, an equiva-lent length is found. Figure 4 shows a plot of theequivalent length versus the actual pipe length, notaccounting for the length of the valve and fittings.We should expect to be able to conclude two factsfrom analyzing this plot after performing a linear

regression. First, the intercept of the plot shouldprovide the equivalent length of all valves, fittings,and contractions as flow enters the pipe from thetank. Second, the slope of the line should be unity.That is, every foot of additional exit pipe shouldincrease the overall equivalent length by one foot.Figure 4 does not show this.

There are several explanations for this, and itwould be a useful exercise to allow the students toperform a sensitivity analysis of their data to findout which variables have the greatest impact on thefinal results. In this case, it is found that smallchanges in the tank and pipe diameters, and in therelative roughness factor for the pipe being used,can have a profound impact on the results.Students should go back to the experimentalapparatus to confirm the dimensions of the equip-ment, and to inspect the condition of the pipelengths. Depending on the age of the equipmentand the quality of water that had been used overthe years that the equipment was in use, there maybe scale build-up in the lengths of pipe. A smallamount of scale can in fact make a rough pipesmoother as scale would tend to collect in crevicesand so reduce the relative roughness of the pipe.

Fig. 3. Typical data analysis for turbulent flow.


The use of a spreadsheet to analyze experimentaldata obviously simplifies and eases the process ofdoing a sensitivity analysis.

Data set for laminar-flow runFigure 5 shows typical data from a run where

the flow was in the laminar regime. Laminar flowis easily obtained by attaching a long length of1/4- or 1/8-inch diameter plastic or metal tubing.(We have found that we can use old, discardedGC columns where the ends have been cut offand the packing blown out.) The results presentedin Fig. 5 predict an equivalent length of 33.7 ftfor a 6-m long, 3/16-inch diameter, coiled tube,including the fittings, reducers, and quick-actionvalve.

EXTENSION OF THE EXPERIMENT

Several other experiments can be set up thatwould allow the student to investigate the impactof other parameters (physical dimensions or fluidproperties). Students can do a series of runs toexamine the effect of pipe or tank diameter, or exitvalve position. Experiments can be repeated where

the thermophysical properties of the liquid arevaried (for example, heating the liquid to reduceviscosity, adding a solute such as a biodegradablethickening agent to increase viscosity, etc.).

The apparatus can also be modified to allow thestudents to study the effect of surface roughness.For example, a series of pipes can be prepared withvarying roughness factors. Conventional stainlesssteel pipe can be electropolished to provide pipewith essentially a smooth bore. Other pipes canhave fine threads cut into the inner surface on alathe to produce pipes with a known, very largeroughness factor.

The students also have an opportunity to deter-mine if the flow through the exit pipes is fullydeveloped, turbulent flow. They can also postulateand test various methods for ensuring that the flowat the entrance of the exit pipe (after all fittings) isfully developed.

We must not discount the possibility of pocketsof air or vapor under some high flow-rate situa-tions. Immediately downstream of a contraction orfitting, there will exist a vena contracta that maycontain trapped air or vapor. The presence of airor vapor may be due to inadequate flushing duringthe time when liquid is accelerating, or there may

Fig. 4. Total equivalent length vs. exit-pipe length.


be a small leak of air into that zone from outside.The presence of this zone would call for adifferent analysis of the data. Indeed, becauseour raw data (time required to attain a particularlevel) seems to be relatively insensitive to thelength of the exit pipe, it is quite possible, inour experimental set-up, that air or water vaporis present somewhere in the exit pipe or fittings.An experimental apparatus that uses some typeof transparent pipe, such as glass or Plexiglas,would verify this situation.

Still another factor that can be examined is thesiphon effect. That is, students can determine theeffect of the level of the outlet of the pipe. This canbe examined by using a length of garden hosecoiled up so that the experiment can be repeatedafter raising or lowering the outlet of the hose.Students can determine if centrifugal forces havean impact on the drain time by varying thediameter of the coil of hose. They can also examinethe effect of the direction of the discharge from thehose. The analysis above assumes that the liquidflows are unidirectional. If the angle of thedischarge is changed, even pointing it straight upas in a fountain, the effect of kinetic energychanges can be observed.

In all of the above parametric studies, thestudents should estimate, or predict, the results inadvance. Theoretical analysis may suggest thatchanges in pipe roughness, for example, wouldalter the efflux time by only a fraction of asecond, not enough difference to be clearlyobserved with manually operated chronometers.Several repeat runs for a particular set-up wouldallow students to perform statistical analysis ofexperimental data, a skill that is gaining ever moreprominence and importance in industry.

CONCLUSION

One of the most important lessons that studentscan learn from running this experiment is this: themethod of data analysis depends on the experi-mental set-up and conditions. Our experience hasshown that students are quick to fit any data(usually numerical results without units) to astraight line, calculate the slope and intercept ofthat best-fit line, report those numbers (usuallywithout units), and finally walk away thinking thatthey had successfully completed the experiment

Fig. 5. Typical data analysis for laminar flow.


and data analysis simply because they indeed didget a computer-generated `best fit' of a straight linefrom the data. This experiment teaches the studentto carefully consider what data can fit a straightline, allow him or her to assign and appreciate thephysical significance to the slope and intercept,and then draw meaningful conclusions. Thesimplicity of the apparatus and its operationallow students to concentrate more on the dataanalysis, and less on actual time-consumingphysical testing. Thus several runs can beperformed in a reasonable time period so that thestudents have a lot of `real-world' data to workwith.

AcknowledgmentÐThe authors wish to thank the seniorchemical engineering students at NJIT for conducting theexperiments and providing the data used in this report.

NOMENCLATURE

d pipe inner diameterf Fanning friction factorg acceleration of gravityI liquid level in the tankk constant with collected termsLe equivalent lengthm constant �1ÿ n�=�2ÿ n�n exponent in friction-factor correlation with

Reynolds numberr tank inner radiusr0 pipe inner radiusRe Reynolds number � dv�=�t timev fluid velocity in pipe� fluid viscosity� fluid density

REFERENCES

1. E. J. Crosby, Experiments in Transport Phenomena, Wiley, New York, 1961.2. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960.3. D. Hanesian, Chemical Engineering Laboratory Manual, NJIT, Newark, 1984.4. S. W. Churchill, Chem. Eng., Nov. 7, 1977, pp. 91±92.

Edward C. Roche Jr. is an emeritus professor from New Jersey Institute of Technology,with 31 years of service. He has a Sc.D. from Stevens Institute of Technology in ChemicalEngineering. Industrial experience includes ten years of continuous employment with theExxon Research and Engineering Company as a process design engineer, plus more thantwenty years of active process design experience as a consulting engineer. Other academicappointments as a visiting professor are: University of Kuwait, US Military Academy, andCanterbury University (NZ).

David B. Van Dongen is a Senior Development Engineer with Cultor Food Science, Inc., amanufacturer of food ingredients. He is responsible for pilot-scale development of processesfor the manufacture of new products. He has a Ph.D. from the University of Massachusettsat Amherst in Chemical Engineering, and has over fifteen years of experience in processdevelopment in the specialty-chemicals industry. He taught at the New Jersey Institute ofTechnology for a few years as an adjunct professor where he and several faculty memberscollaborated on many projects directed at presenting the unit-operations laboratory as an``industrial'' experience for the students.


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