A Manual for the

MECHANICS of FLUIDS LABORATORY

William S. JannaDepartment of Mechanical Engineering

Memphis State University

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©1997 William S. Janna

All Rights Reserved.No part of this manual may be reproduced, stored in a retrieval

system, or transcribed in any form or by any means—electronic, magnetic,mechanical, photocopying, recording, or otherwise—

without the prior written consent of William S. Janna

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TABLE OF CONTENTS

Item Page

Report Writing ................................................................................................................. 4Cleanliness and Safety .................................................................................................... 6Experiment 1 Density and Surface Tension..................................................... 7Experiment 2 Viscosity ......................................................................................... 9Experiment 3 Center of Pressure on a Submerged Plane Surface ............. 10Experiment 4 Measurement of Differential Pressure .................................. 12Experiment 5 Impact of a Jet of Water ............................................................ 14Experiment 6 Critical Reynolds Number in Pipe Flow............................... 16Experiment 7 Fluid Meters ................................................................................ 18Experiment 8 Pipe Flow ..................................................................................... 22Experiment 9 Pressure Distribution About a Circular Cylinder................ 24Experiment 10 Drag Force Determination ....................................................... 27Experiment 11 Analysis of an Airfoil................................................................ 28Experiment 12 Open Channel Flow—Sluice Gate ......................................... 30Experiment 13 Open Channel Flow Over a Weir .......................................... 32Experiment 14 Open Channel Flow—Hydraulic Jump ................................ 34Experiment 15 Open Channel Flow Over a Hump........................................ 36Experiment 16 Measurement of Velocity and Calibration of

a Meter for Compressible Flow ............................. 39Experiment 17 Measurement of Fan Horsepower ......................................... 44Experiment 18 Measurement of Pump Performance .................................... 46Appendix ......................................................................................................................... 50

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REPORT WRITING

All reports in the Fluid MechanicsLaboratory require a formal laboratory reportunless specified otherwise. The report should bewritten in such a way that anyone can duplicatethe performed experiment and find the sameresults as the originator. The reports should besimple and clearly written. Reports are due oneweek after the experiment was performed, unlessspecified otherwise.

The report should communicate several ideasto the reader. First the report should be neatlydone. The experimenter is in effect trying toconvince the reader that the experiment wasperformed in a straightforward manner withgreat care and with full attention to detail. Apoorly written report might instead lead thereader to think that just as little care went intoperforming the experiment. Second, the reportshould be well organized. The reader should beable to easily follow each step discussed in thetext. Third, the report should contain accurateresults. This will require checking and recheckingthe calculations until accuracy can be guaranteed.Fourth, the report should be free of spelling andgrammatical errors. The following format, shownin Figure R.1, is to be used for formal LaboratoryReports:

Title Page–The title page should show the titleand number of the experiment, the date theexperiment was performed, experimenter'sname and experimenter's partners' names.

Table of Contents –Each page of the report mustbe numbered for this section.

Object –The object is a clear concise statementexplaining the purpose of the experiment.This is one of the most important parts of thelaboratory report because everythingincluded in the report must somehow relate tothe stated object. The object can be as short asone sentence and it is usually written in thepast tense.

Theory –The theory section should contain acomplete analytical development of allimportant equations pertinent to theexperiment, and how these equations are usedin the reduction of data. The theory sectionshould be written textbook-style.

Procedure – The procedure section should containa schematic drawing of the experimentalsetup including all equipment used in a partslist with manufacturer serial numbers, if any.Show the function of each part whennecessary for clarity. Outline exactly step-

Bibliography

Calibration Curves

Original Data Sheet(Sample Calculation)

Appendix Title Page

Discussion & Conclusion(Interpretation)

Results (Tablesand Graphs)

Procedure (Drawingsand Instructions)

Theory(Textbook Style)

Object(Past Tense)

Table of ContentsEach page numbered

Experiment NumberExperiment Title

Your Name

Due Date

Partners’ Names

FIGURE R.1. Format for formal reports.

by-step how the experiment was performed incase someone desires to duplicate it. If itcannot be duplicated, the experiment showsnothing.

Results – The results section should contain aformal analysis of the data with tables,graphs, etc. Any presentation of data whichserves the purpose of clearly showing theoutcome of the experiment is sufficient.

Discussion and Conclusion – This section shouldgive an interpretation of the resultsexplaining how the object of the experimentwas accomplished. If any analyticalexpression is to be verified, calculate % error†

and account for the sources. Discuss thisexperiment with respect to its faults as well

†% error–An analysis expressing how favorably theempirical data approximate theoretical information.There are many ways to find % error, but one method isintroduced here for consistency. Take the differencebetween the empirical and theoretical results and divideby the theoretical result. Multiplying by 100% gives the% error. You may compose your own error analysis aslong as your method is clearly defined.

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as its strong points. Suggest extensions of theexperiment and improvements. Alsorecommend any changes necessary to betteraccomplish the object. Each experiment write-up contains anumber of questions. These are to be answeredor discussed in the Discussion and Conclusionssection.

Appendix(1) Original data sheet.(2) Show how data were used by a samplecalculation.(3) Calibration curves of instrument whichwere used in the performance of theexperiment. Include manufacturer of theinstrument, model and serial numbers.Calibration curves will usually be suppliedby the instructor.(4) Bibliography listing all references used.

Short Form Report FormatOften the experiment requires not a formal

report but an informal report. An informal reportincludes the Title Page, Object, Procedure,Results, and Conclusions. Other portions may beadded at the discretion of the instructor or thewriter. Another alternative report form consistsof a Title Page, an Introduction (made up ofshortened versions of Object, Theory, andProcedure) Results, and Conclusion andDiscussion. This form might be used when adetailed theory section would be too long.

GraphsIn many instances, it is necessary to compose a

plot in order to graphically present the results.Graphs must be drawn neatly following a specificformat. Figure R.2 shows an acceptable graphprepared using a computer. There are manycomputer programs that have graphingcapabilities. Nevertheless an acceptably drawngraph has several features of note. These featuresare summarized next to Figure R.2.

Features of note

• Border is drawn about the entire graph.• Axis labels defined with symbols and

units.• Grid drawn using major axis divisions.• Each line is identified using a legend.• Data points are identified with a

symbol: “ ´” on the Qac line to denotedata points obtained by experiment.

• The line representing the theoreticalresults has no data points represented.

• Nothing is drawn freehand.• Title is descriptive, rather than

something like Q vs ∆h.

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

Qth

Qac

Q

∆ hhead loss in m

flow

rat

e

in m

3 /s

FIGURE R.2. Theoretical and actual volume flow ratethrough a venturi meter as a function of head loss.

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CLEANLINESS AND SAFETY

CleanlinessThere are “housekeeping” rules that the user

of the laboratory should be aware of and abideby. Equipment in the lab is delicate and eachpiece is used extensively for 2 or 3 weeks persemester. During the remaining time, eachapparatus just sits there, literally collecting dust.University housekeeping staff are not required toclean and maintain the equipment. Instead, thereare college technicians who will work on theequipment when it needs repair, and when theyare notified that a piece of equipment needsattention. It is important, however, that theequipment stay clean, so that dust will notaccumulate too badly.

The Fluid Mechanics Laboratory containsequipment that uses water or air as the workingfluid. In some cases, performing an experimentwill inevitably allow water to get on theequipment and/or the floor. If no one cleaned uptheir working area after performing anexperiment, the lab would not be a comfortable orsafe place to work in. No student appreciateswalking up to and working with a piece ofequipment that another student or group ofstudents has left in a mess.

Consequently, students are required to cleanup their area at the conclusion of the performanceof an experiment. Cleanup will include removalof spilled water (or any liquid), and wiping thetable top on which the equipment is mounted (ifappropriate). The lab should always be as cleanor cleaner than it was when you entered. Cleaning

the lab is your responsibility as a user of theequipment. This is an act of courtesy that studentswho follow you will appreciate, and that youwill appreciate when you work with theequipment.

SafetyThe layout of the equipment and storage

cabinets in the Fluid Mechanics Lab involvesresolving a variety of conflicting problems. Theseinclude traffic flow, emergency facilities,environmental safeguards, exit door locations,etc. The goal is to implement safety requirementswithout impeding egress, but still allowingadequate work space and necessary informalcommunication opportunities.

Distance between adjacent pieces ofequipment is determined by locations of floordrains, and by the need to allow enough spacearound the apparatus of interest. Immediateaccess to the Safety Cabinet is also considered.Emergency facilities such as showers, eye washfountains, spill kits, fire blankets and the likeare not found in the lab. We do not work withhazardous materials and such safety facilitiesare not necessary. However, waste materials aregenerated and they should be disposed ofproperly.

Every effort has been made to create apositive, clean, safety conscious atmosphere.Students are encouraged to handle equipmentsafely and to be aware of, and avoid beingvictims of, hazardous situations.

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EXPERIMENT 1

FLUID PROPERTIES: DENSITY AND SURFACE TENSION

There are several properties simpleNewtonian fluids have. They are basicproperties which cannot be calculated for everyfluid, and therefore they must be measured.These properties are important in makingcalculations regarding fluid systems. Measuringfluid properties, density and viscosity, is theobject of this experiment.

Part I: Density Measurement.

EquipmentGraduated cylinder or beakerLiquid whose properties are to be

measuredHydrometer cylinderScale

The density of the test fluid is to be found byweighing a known volume of the liquid using thegraduated cylinder or beaker and the scale. Thebeaker is weighed empty. The beaker is thenfilled to a certain volume according to thegraduations on it and weighed again. Thedifference in weight divided by the volume givesthe weight per unit volume of the liquid. Byappropriate conversion, the liquid density iscalculated. The mass per unit volume, or thedensity, is thus measured in a direct way.

A second method of finding density involvesmeasuring buoyant force exerted on a submergedobject. The difference between the weight of anobject in air and the weight of the object in liquidis known as the buoyant force (see Figure 1.1).

W1

W2

FIGURE 1.1. Measuring the buoyant force on anobject with a hanging weight.

Referring to Figure 1.1, the buoyant force B isfound as

B = W1 - W2

The buoyant force is equal to the differencebetween the weight of the object in air and theweight of the object while submerged. Dividingthis difference by the volume displaced gives theweight per unit volume from which density can becalculated.

Questions1. Are the results of all the density

measurements in agreement?2. How does the buoyant force vary with

depth of the submerged object? Why?

Part II: Surface Tension Measurement

EquipmentSurface tension meterBeakerTest fluid

Surface tension is defined as the energyrequired to pull molecules of liquid from beneaththe surface to the surface to form a new area. It istherefore an energy per unit area (F⋅L/L2 = F/L).A surface tension meter is used to measure thisenergy per unit area and give its value directly. Aschematic of the surface tension meter is given inFigure 1.2.

The platinum-iridium ring is attached to abalance rod (lever arm) which in turn is attached

to a stainless steel torsion wire. One end of thiswire is fixed and the other is rotated. As the wireis placed under torsion, the rod lifts the ringslowly out of the liquid. The proper technique isto lower the test fluid container as the ring islifted so that the ring remains horizontal. Theforce required to break the ring free from theliquid surface is related to the surface tension ofthe liquid. As the ring breaks free, the gage atthe front of the meter reads directly in the unitsindicated (dynes/cm) for the given ring. Thisreading is called the apparent surface tension andmust be corrected for the ring used in order toobtain the actual surface tension for the liquid.The correction factor F can be calculated with thefollowing equation

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FIGURE 1.2. A schematic of thesurface tension meter.

torsion wire

test liquid

platinumiridium ring

clampbalance rod

F = 0.725 + √0.000 403 3(σa/ρ) + 0.045 34 - 1.679(r/R)

where F is the correction factor, σa is theapparent surface tension read from the dial(dyne/cm), ρ is the density of the liquid (g/cm3),and (r/R) for the ring is found on the ringcontainer. The actual surface tension for theliquid is given by

σ = Fσa

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EXPERIMENT 2

FLUID PROPERTIES: VISCOSITY

One of the properties of homogeneous liquidsis their resistance to motion. A measure of thisresistance is known as viscosity. It can bemeasured in different, standardized methods ortests. In this experiment, viscosity will bemeasured with a falling sphere viscometer.

The Falling Sphere Viscometer When an object falls through a fluid medium,

the object reaches a constant final speed orterminal velocity. If this terminal velocity issufficiently low, then the various forces acting onthe object can be described with exact expressions.The forces acting on a sphere, for example, that isfalling at terminal velocity through a liquid are:

Weight - Buoyancy - Drag = 0

ρsg 43 πR3 - ρg

43 πR3 - 6πµVR = 0

where ρs and ρ are density of the sphere andliquid respectively, V is the sphere’s terminalvelocity, R is the radius of the sphere and µ isthe viscosity of the liquid. In solving thepreceding equation, the viscosity of the liquid canbe determined. The above expression for drag isvalid only if the following equation is valid:

ρVD

µ < 1

where D is the sphere diameter. Once theviscosity of the liquid is found, the above ratioshould be calculated to be certain that themathematical model gives an accuratedescription of a sphere falling through theliquid.

EquipmentHydrometer cylinderScaleStopwatchSeveral small spheres with weight and

diameter to be measuredTest liquid

Drop a sphere into the cylinder liquid andrecord the time it takes for the sphere to fall acertain measured distance. The distance dividedby the measured time gives the terminal velocityof the sphere. Repeat the measurement and

average the results. With the terminal velocityof this and of other spheres measured and known,the absolute and kinematic viscosity of the liquidcan be calculated. The temperature of the testliquid should also be recorded. Use at least threedifferent spheres. (Note that if the density ofthe liquid is unknown, it can be obtained from anygroup who has completed or is taking data onExperiment 1.)

Questions1. Should the terminal velocity of two

different size spheres be the same?2. Does a larger sphere have a higher

terminal velocity?3. Should the viscosity found for two different

size spheres be the same? Why or why not?4. If different size spheres give different

results for the viscosity, what are the errorsources? Calculate the % error and accountfor all known error sources.

5. What are the shortcomings of this method?6. Why should temperature be recorded.7. Can this method be used for gases?8. Can this method be used for opaque liquids?9. Can this method be used for something like

peanut butter, or grease or flour dough?Why or why not?

d

V

FIGURE 2.1. Terminal velocity measurement (V =d/t ime) .

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EXPERIMENT 3

CENTER OF PRESSURE ON A SUBMERGEDPLANE SURFACE

Submerged surfaces are found in manyengineering applications. Dams, weirs and watergates are familiar examples of submergedsurfaces used to control the flow of water. Fromthe design viewpoint, it is important to have aworking knowledge of the forces that act onsubmerged surfaces.

A plane surface located beneath the surfaceof a liquid is subjected to a pressure due to theheight of liquid above it, as shown in Figure 3.1.Increasing pressure varies linearly withincreasing depth resulting in a pressuredistribution that acts on the submerged surface.The analysis of this situation involvesdetermining a force which is equivalent to thepressure, and finding the location of this force.

F

yF

FIGURE 3.1. Pressure distribution on a submergedplane surface and the equivalent force.

For this case, it can be shown that theequivalent force is:

F = ρgycA (3.1)

in which ρ is the liquid density, yc is the distancefrom the free surface of the liquid to the centroidof the plane, and A is the area of the plane incontact with liquid. Further, the location of thisforce yF below the free surface is

yF = Ix x

ycA + yc (3.2)

in which Ixx is the second area moment of theplane about its centroid. The experimental

verification of these equations for force anddistance is the subject of this experiment.Center of Pressure Measurement

Equipment Center of Pressure Apparatus

Weights

Figure 3.2 gives a schematic of the apparatusused in this experiment. The torus and balancearm are placed on top of the tank. Note that thepivot point for the balance arm is the point ofcontact between the rod and the top of the tank.The zeroing weight is adjusted to level thebalance arm. Water is then added to apredetermined depth. Weights are placed on theweight hanger to re-level the balance arm. Theamount of needed weight and depth of water arethen recorded. The procedure is then repeated forfour other depths. (Remember to record thedistance from the pivot point to the free surfacefor each case.)

From the depth measurement, the equivalentforce and its location are calculated usingEquations 3.1 and 3.2. Summing moments about thepivot allows for a comparison between thetheoretical and actual force exerted. Referring toFigure 3.2, we have

F = W L

(y + yF) (3.3)

where y is the distance from the pivot point tothe free surface, yF is the distance from the freesurface to the line of action of the force F, and L isthe distance from the pivot point to the line ofaction of the weight W. Note that both curvedsurfaces of the torus are circular with centers atthe pivot point. For the report, compare the forceobtained with Equation 3.1 to that obtained withEquation 3.3. When using Equation 3.3, it will benecessary to use Equation 3.2 for yF.

Questions1. In summing moments, why isn't the buoyant

force taken into account?2. Why isn’t the weight of the torus and the

balance arm taken into account?

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L

F

y

h

w

yF

Ri

Ro

zeroing weight

pivot point(point of contact)

level

torus

weighthanger

FIGURE 3.2. A schematic of the center of pressure apparatus.

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EXPERIMENT 4

MEASUREMENT OF DIFFERENTIAL PRESSURE

Pressure can be measured in several ways.Bourdon tube gages, manometers, and transducersare a few of the devices available. Each of theseinstruments actually measures a difference inpressure; that is, measures a difference betweenthe desired reading and some reference pressure,usually atmospheric. The measurement ofdifferential pressure with manometers is thesubject of this experiment.

Manometry A manometer is a device used to measure a

pressure difference and display the reading interms of height of a column of liquid. The heightis related to the pressure difference by thehydrostatic equation.

Figure 4.1 shows a U-tube manometerconnected to two pressure vessels. The manometerreading is ∆h and the manometer fluid hasdensity ρm. One pressure vessel contains a fluid ofdensity ρ1 while the other vessel contains a fluidof density ρ2. The pressure difference can be foundby applying the hydrostatic equation to eachlimb of the manometer. For the left leg,

p1

p2

z2

z1

pA pA

h

m

1

2

FIGURE 4.1. A U-tube manometer connected totwo pressure vessels.

p1 + ρ1gz1 = pA

Likewise for the right leg,

p2 + ρ2gz2 + ρmg∆h = pA

Equating these expressions and solving for thepressure difference gives

p1 - p2 = ρ2gz2 + ρ1gz1 + ρmg∆h

If the fluids above the manometer liquid are bothgases, then ρ1 and ρ2 are small compared to ρµ.

The above equation then becomes

p1 - p2 = ρmg∆h

Figure 4.2 is a schematic of the apparatusused in this experiment. It consists of three U-tubemanometers, a well-type manometer, a U-tube/inclined manometer and a differentialpressure gage. There are two tanks (actually, twocapped pieces of pipe) to which each manometerand the gage are connected. The tanks have bleedvalves attached and the tanks are connectedwith plastic tubing to a squeeze bulb. The bulblines also contain valves. With both bleed valvesclosed and with both bulb line valves open, thebulb is squeezed to pump air from the low pressuretank to the high pressure tank. The bulb issqueezed until any of the manometers reaches itsmaximum reading. Now both valves are closedand the liquid levels are allowed to settle ineach manometer. The ∆h readings are allrecorded. Next, one or both bleed valves areopened slightly to release some air into or out of atank. The liquid levels are again allowed tosettle and the ∆h readings are recorded. Theprocedure is to be repeated until 5 different sets ofreadings are obtained. For each set of readings,convert all readings into psi or Pa units, calculatethe average value and the standard deviation.Before beginning, be sure to zero each manometerand the gage.

Questions1. Manometers 1, 2 and 3 are U-tube types and

each contains a different liquid. Manometer4 is a well-type manometer. Is there anadvantage to using this one over a U-tubetype?

2. Manometer 5 is a combined U/tube/inclinedmanometer. What is the advantage of thistype?

3. Note that some of the manometers use aliquid which has a specific gravitydifferent from 1.00, yet the reading is ininches of water. Explain how this ispossible.

4. What advantages or disadvantages doesthe gage have over the manometers?

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5. Is a low value of the standard deviationexpected? Why?

6. What does a low standard deviationimply?

7. In your opinion, which device gives themost accurate reading. What led you to thisconclusion?

U-tube manometers Well-typemanometer

U-tube/inclinedmanometer

Gage

Bleed valvesLow pressure tankHigh pressure tank

FIGURE 4.2. A schematic of the apparatus used in this experiment.

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EXPERIMENT 5

IMPACT OF A JET OF WATER

A jet of fluid striking a stationary objectexerts a force on that object. This force can bemeasured when the object is connected to a springbalance or scale. The force can then be related tothe velocity of the jet of fluid and in turn to therate of flow. The force developed by a jet streamof water is the subject of this experiment.

Impact of a Jet of Liquid

EquipmentJet Impact ApparatusObject plates

Figure 5.1 is a schematic of the device used inthis experiment. The device consists of a tankwithin a tank. The interior tank is supported on apivot and has a lever arm attached to it. Aswater enters this inner tank, the lever arm willreach a balance point. At this time, a stopwatchis started and a weight is placed on the weighthanger (e.g., 10 lbf). When enough water hasentered the tank (10 lbf), the lever arm willagain balance. The stopwatch is stopped. Theelapsed time divided into the weight of watercollected gives the weight or mass flow rate ofwater through the system (lbf/sec, for example).

The outer tank acts as a support for the tabletop as well as a sump tank. Water is pumped fromthe outer tank to the apparatus resting on thetable top. As shown in Figure 5.1, the impactapparatus contains a nozzle that produces a highvelocity jet of water. The jet is aimed at an object(such as a flat plate or hemisphere). The forceexerted on the plate causes the balance arm towhich the plate is attached to deflect. A weightis moved on the arm until the arm balances. Asummation of moments about the pivot point ofthe arm allows for calculating the force exertedby the jet.

Water is fed through the nozzle by means ofa centrifugal pump. The nozzle emits the water ina jet stream whose diameter is constant. After thewater strikes the object, the water is channeled to

the weighing tank inside to obtain the weight ormass flow rate.

The variables involved in this experimentare listed and their measurements are describedbelow:1. Mass rate of flow–measured with the

weighing tank inside the sump tank. Thevolume flow rate is obtained by dividingmass flow rate by density: Q = m/ρ.

2. Velocity of jet–obtained by dividing volumeflow rate by jet area: V = Q/A. The jet iscylindrical in shape with a diameter of 0.375in.

3. Resultant force—found experimentally bysummation of moments about the pivot pointof the balance arm. The theoretical resultantforce is found by use of an equation derived byapplying the momentum equation to a controlvolume about the plate.

Impact Force Analysis The total force exerted by the jet equals the

rate of momentum loss experienced by the jet afterit impacts the object. For a flat plate, the forceequation is:

F = ρQ2

A(flat plate)

For a hemisphere,

F = 2ρQ2

A(hemisphere)

For a cone whose included half angle is α,

F = ρQ2

A (1 + cos α) (cone)

For your report, derive the appropriateequation for each object you use. Compose a graphwith volume flow rate on the horizontal axis,and on the vertical axis, plot the actual andtheoretical force. Use care in choosing theincrements for each axis.

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flat plate

pivot

balancingweight lever arm with

flat plate attached

water jet

nozzle

drain

weigh tank

plug

sump tank

motor pump

weight hanger

flow controlvalve

tank pivot

FIGURE 5.1. A schematic of the jet impact apparatus.

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EXPERIMENT 6

CRITICAL REYNOLDS NUMBER IN PIPE FLOW

The Reynolds number is a dimensionless ratioof inertia forces to viscous forces and is used inidentifying certain characteristics of fluid flow.The Reynolds number is extremely important inmodeling pipe flow. It can be used to determinethe type of flow occurring: laminar or turbulent.Under laminar conditions the velocitydistribution of the fluid within the pipe isessentially parabolic and can be derived from theequation of motion. When turbulent flow exists,the velocity profile is “flatter” than in thelaminar case because the mixing effect which ischaracteristic of turbulent flow helps to moreevenly distribute the kinetic energy of the fluidover most of the cross section.

In most engineering texts, a Reynolds numberof 2 100 is usually accepted as the value attransition; that is, the value of the Reynoldsnumber between laminar and turbulent flowregimes. This is done for the sake of convenience.In this experiment, however, we will see thattransition exists over a range of Reynolds numbersand not at an individual point.

The Reynolds number that exists anywhere inthe transition region is called the criticalReynolds number. Finding the critical Reynoldsnumber for the transition range that exists in pipeflow is the subject of this experiment.

Critical Reynolds Number Measurement

EquipmentCritical Reynolds Number Determination

Apparatus

Figure 6.1 is a schematic of the apparatusused in this experiment. The constant head tankprovides a controllable, constant flow throughthe transparent tube. The flow valve in the tubeitself is an on/off valve, not used to control theflow rate. Instead, the flow rate through the tubeis varied with the rotameter valve at A. Thehead tank is filled with water and the overflowtube maintains a constant head of water. Theliquid is then allowed to flow through one of thetransparent tubes at a very low flow rate. Thevalve at B controls the flow of dye; it is openedand dye is then injected into the pipe with thewater. The dye injector tube is not to be placed inthe pipe entrance as it could affect the results.Establish laminar flow by starting with a verylow flow rate of water and of dye. The injected

dye will flow downstream in a threadlikepattern for very low flow rates. Once steady stateis achieved, the rotameter valve is openedslightly to increase the water flow rate. Thevalve at B is opened further if necessary to allowmore dye to enter the tube. This procedure ofincreasing flow rate of water and of dye (ifnecessary) is repeated throughout theexperiment.

Establish laminar flow in one of the tubes.Then slowly increase the flow rate and observewhat happens to the dye. Its pattern maychange, yet the flow might still appear to belaminar. This is the beginning of transition.Continue increasing the flow rate and againobserve the behavior of the dye. Eventually, thedye will mix with the water in a way that willbe recognized as turbulent flow. This point is theend of transition. Transition thus will exist over arange of flow rates. Record the flow rates at keypoints in the experiment. Also record thetemperature of the water.

The object of this procedure is to determinethe range of Reynolds numbers over whichtransition occurs. Given the tube size, theReynolds number can be calculated with:

Re = VDν

where V (= Q/A) is the average velocity ofliquid in the pipe, D is the hydraulic diameter ofthe pipe, and ν is the kinematic viscosity of theliquid.

The hydraulic diameter is calculated fromits definition:

D = 4 x Area

Wetted Perimeter

For a circular pipe flowing full, the hydraulicdiameter equals the inside diameter of the pipe.For a square section, the hydraulic diameter willequal the length of one side (show that this isthe case). The experiment is to be performed forboth round tubes and the square tube. With goodtechnique and great care, it is possible for thetransition Reynolds number to encompass thetraditionally accepted value of 2 100.

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Questions1. Can a similar procedure be followed for

gases?2. Is the Reynolds number obtained at

transition dependent on tube size or shape?3. Can this method work for opaque liquids?

drilled partitions

dye reservoir

on/off valverotameter

A

to drain

inlet totank

overflowto drain

B

transparent tube

FIGURE 6.1. The critical Reynolds number determination apparatus.

18

EXPERIMENT 7

FLUID METERS IN INCOMPRESSIBLE FLOW

There are many different meters used in pipeflow: the turbine type meter, the rotameter, theorifice meter, the venturi meter, the elbow meterand the nozzle meter are only a few. Each meterworks by its ability to alter a certain physicalcharacteristic of the flowing fluid and thenallows this alteration to be measured. Themeasured alteration is then related to the flowrate. A procedure of analyzing meters todetermine their useful features is the subject ofthis experiment.

The Venturi Meter The venturi meter is constructed as shown in

Figure 7.1. It contains a constriction known as thethroat. When fluid flows through theconstriction, it must experience an increase invelocity over the upstream value. The velocityincrease is accompanied by a decrease in staticpressure at the throat. The difference betweenupstream and throat static pressures is thenmeasured and related to the flow rate. Thegreater the flow rate, the greater the pressuredrop ∆p. So the pressure difference ∆h (= ∆p/ρg)can be found as a function of the flow rate.

12

h

FIGURE 7.1. A schematic of the Venturi meter.

Using the hydrostatic equation applied tothe air-over-liquid manometer of Figure 7.1, thepressure drop and the head loss are related by(after simplification):

p1 - p2

ρg = ∆h

By combining the continuity equation,

Q = A1V1 = A2V2

with the Bernoulli equation,

p1

ρ + V1

2

2 = p2

ρ + V2

2

2

and substituting from the hydrostatic equation, itcan be shown after simplification that thevolume flow rate through the venturi meter isgiven by

Qth = A2 √2g∆h1 - (D2

4/D14) (7.1)

The preceding equation represents the theoreticalvolume flow rate through the venturi meter.Notice that is was derived from the Bernoulliequation which does not take frictional effectsinto account.

In the venturi meter, there exists smallpressure losses due to viscous (or frictional)effects. Thus for any pressure difference, theactual flow rate will be somewhat less than thetheoretical value obtained with Equation 7.1above. For any ∆h, it is possible to define acoefficient of discharge Cv as

Cv = Qac

Qth

For each and every measured actual flow ratethrough the venturi meter, it is possible tocalculate a theoretical volume flow rate, aReynolds number, and a discharge coefficient.The Reynolds number is given by

Re = V2D2

ν (7.2)

where V2 is the velocity at the throat of themeter (= Qac/A2).

The Orifice Meter andNozzle-Type Meter

The orifice and nozzle-type meters consist ofa throttling device (an orifice plate or bushing,respectively) placed into the flow. (See Figures7.2 and 7.3). The throttling device creates ameasurable pressure difference from its upstreamto its downstream side. The measured pressuredifference is then related to the flow rate. Likethe venturi meter, the pressure difference varieswith flow rate. Applying Bernoulli’s equation topoints 1 and 2 of either meter (Figure 7.2 or Figure7.3) yields the same theoretical equation as thatfor the venturi meter, namely, Equation 7.1. Forany pressure difference, there will be twoassociated flow rates for these meters: thetheoretical flow rate (Equation 7.1), and the

19

actual flow rate (measured in the laboratory).The ratio of actual to theoretical flow rate leadsto the definition of a discharge coefficient: Co forthe orifice meter and Cn for the nozzle.

1 2

h

FIGURE 7.2. Cross sectional view of the orificemeter.

1 2

h

FIGURE 7.3. Cross sectional view of the nozzle-type meter, and a typical nozzle.

For each and every measured actual flowrate through the orifice or nozzle-type meters, itis possible to calculate a theoretical volume flowrate, a Reynolds number and a dischargecoefficient. The Reynolds number is given byEquation 7.2.

The Turbine-Type Meter The turbine-type flow meter consists of a

section of pipe into which a small “turbine” hasbeen placed. As the fluid travels through thepipe, the turbine spins at an angular velocitythat is proportional to the flow rate. After acertain number of revolutions, a magnetic pickupsends an electrical pulse to a preamplifier whichin turn sends the pulse to a digital totalizer. Thetotalizer totals the pulses and translates theminto a digital readout which gives the totalvolume of liquid that travels through the pipeand/or the instantaneous volume flow rate.Figure 7.4 is a schematic of the turbine type flowmeter.

rotor supportedon bearings(not shown)

turbine rotorrotational speedproportional to

flow rate

to receiver

flowstraighteners

FIGURE 7.4. A schematic of a turbine-type flowmeter.

The Rotameter (Variable Area Meter) The variable area meter consists of a tapered

metering tube and a float which is free to moveinside. The tube is mounted vertically with theinlet at the bottom. Fluid entering the bottomraises the float until the forces of buoyancy, dragand gravity are balanced. As the float rises theannular flow area around the float increases.Flow rate is indicated by the float position readagainst the graduated scale which is etched onthe metering tube. The reading is made usually atthe widest part of the float. Figure 7.5 is a sketchof a rotameter.

tapered, graduatedtransparent tube

freelysuspendedfloat

inlet

outlet

FIGURE 7.5. A schematic of the rotameter and itsoperation.

Rotameters are usually manufactured withone of three types of graduated scales:1. % of maximum flow–a factor to convert scale

reading to flow rate is given or determined forthe meter. A variety of fluids can be usedwith the meter and the only variable

20

encountered in using it is the scale factor. Thescale factor will vary from fluid to fluid.

2. Diameter-ratio type–the ratio of crosssectional diameter of the tube to thediameter of the float is etched at variouslocations on the tube itself. Such a scalerequires a calibration curve to use the meter.

3. Direct reading–the scale reading shows theactual flow rate for a specific fluid in theunits indicated on the meter itself. If thistype of meter is used for another kind of fluid,then a scale factor must be applied to thereadings.

Experimental Procedure

EquipmentFluid Meters ApparatusStopwatch

The fluid meters apparatus is shownschematically in Figure 7.6. It consists of acentrifugal pump, which draws water from asump tank, and delivers the water to the circuitcontaining the flow meters. For nine valvepositions (the valve downstream of the pump),record the pressure differences in eachmanometer. For each valve position, measure theactual flow rate by diverting the flow to thevolumetric measuring tank and recording the timerequired to fill the tank to a predeterminedvolume. Use the readings on the side of the tankitself. For the rotameter, record the position ofthe float and/or the reading of flow rate givendirectly on the meter. For the turbine meter,record the flow reading on the output device.

Note that the venturi meter has twomanometers attached to it. The “inner”manometer is used to calibrate the meter; that is,to obtain ∆h readings used in Equation 7.1. The“outer” manometer is placed such that it readsthe overall pressure drop in the line due to thepresence of the meter and its attachment fittings.We refer to this pressure loss as ∆H (distinctlydifferent from ∆h). This loss is also a function offlow rate. The manometers on the turbine-typeand variable area meters also give the incurredloss for each respective meter. Thus readings of∆H vs Qac are obtainable. In order to use theseparameters to give dimensionless ratios, pressurecoefficient and Reynolds number are used. TheReynolds number is given in Equation 7.2. Thepressure coefficient is defined as

Cp = g∆HV2/2 (7.3)

All velocities are based on actual flow rate andpipe diameter.

The amount of work associated with thelaboratory report is great; therefore an informalgroup report is required rather than individualreports. The write-up should consist of anIntroduction (to include a procedure and aderivation of Equation 7.1), a Discussion andConclusions section, and the following graphs:

1. On the same set of axes, plot Qac vs ∆h andQth vs ∆h with flow rate on the verticalaxis for the venturi meter.

2. On the same set of axes, plot Qac vs ∆h andQth vs ∆h with flow rate on the verticalaxis for the orifice meter.

3. Plot Qac vs Qth for the turbine type meter.4. Plot Qac vs Qth for the rotameter.5. Plot Cv vs Re on a log-log grid for the

venturi meter.6. Plot Co vs Re on a log-log grid for the orifice

meter.7. Plot ∆H vs Qac for all meters on the same set

of axes with flow rate on the vertical axis.8. Plot Cp vs Re for all meters on the same set

of axes (log-log grid) with Cp vertical axis.

Questions1. Referring to Figure 7.2, recall that

Bernoulli's equation was applied to points 1and 2 where the pressure differencemeasurement is made. The theoreticalequation, however, refers to the throat areafor point 2 (the orifice hole diameter)which is not where the pressuremeasurement was made. Explain thisdiscrepancy and how it is accounted for inthe equation formulation.

2. Which meter in your opinion is the best oneto use?

3. Which meter incurs the smallest pressureloss? Is this necessarily the one that shouldalways be used?

4. Which is the most accurate meter?5. What is the difference between precision

and accuracy?

21

orifice meter

venturi meter

manometer

valve

turbine-type meter

rotameter

sump tank

volumetricmeasuringtank

return

motor pump

FIGURE 7.6. A schematic of the Fluid Meters Apparatus. (Orifice and Venturi meters: upstreamdiameter is 1.025 inches; throat diameter is 0.625 inches.)

22

EXPERIMENT 8

PIPE FLOW

Experiments in pipe flow where the presenceof frictional forces must be taken into account areuseful aids in studying the behavior of travelingfluids. Fluids are usually transported throughpipes from location to location by pumps. Thefrictional losses within the pipes cause pressuredrops. These pressure drops must be known todetermine pump requirements. Thus a study ofpressure losses due to friction has a usefulapplication. The study of pressure losses in pipeflow is the subject of this experiment.

Pipe Flow

EquipmentPipe Flow Test Rig

Figure 8.1 is a schematic of the pipe flow testrig. The rig contains a sump tank which is used asa water reservoir from which a centrifugal pumpdischarges water to the pipe circuit. The circuititself consists of four different diameter lines anda return line all made of drawn copper tubing. Thecircuit contains valves for directing andregulating the flow to make up various series andparallel piping combinations. The circuit hasprovision for measuring pressure loss through theuse of static pressure taps (manometer board notshown in schematic). Finally, because the circuitalso contains a rotameter, the measured pressurelosses can be obtained as a function of flow rate.

As functions of the flow rate, measure thepressure losses in inches of water for (as specifiedby the instructor):

1. 1 in. copper tube 5. 1 in. 90 T-joint2. 3/4-in. copper tube 6. 1 in. 90 elbow (ell)3. 1/2-in copper tube 7. 1 in. gate valve4. 3/8 in copper tube 8. 3/4-in gate valve

• The instructor will specify which of thepressure loss measurements are to be taken.

• Open and close the appropriate valves on theapparatus to obtain the desired flow path.

• Use the valve closest to the pump on itsdownstream side to vary the volume flowrate.

• With the pump on, record the assignedpressure drops and the actual volume flowrate from the rotameter.

• Using the valve closest to the pump, changethe volume flow rate and again record thepressure drops and the new flow rate value.

• Repeat this procedure until 9 differentvolume flow rates and corresponding pressuredrop data have been recorded.

With pressure loss data in terms of ∆h, thefriction factor can be calculated with

f = 2g∆h

V 2(L/D)

It is customary to graph the friction factor as afunction of the Reynolds number:

Re = VDν

The f vs Re graph, called a Moody Diagram istraditionally drawn on a log-log grid. The graphalso contains a third variable known as theroughness coefficient ε/D. For this experimentthe roughness factor ε is that for drawn tubing.

Where fittings are concerned, the lossincurred by the fluid is expressed in terms of a losscoefficient K. The loss coefficient for any fittingcan be calculated with

K = ∆ h

V2/2g

where ∆h is the pressure (or head) loss across thefitting. Values of K as a function of Qac are to beobtained in this experiment.

For the report, calculate friction factor f andgraph it as a function of Reynolds number Re foritems 1 through 4 above as appropriate. Compareto a Moody diagram. Also calculate the losscoefficient for items 5 through 8 above asappropriate, and determine if the loss coefficientK varies with flow rate or Reynolds number.Compare your K values to published ones.

Note that gate valves can have a number ofopen positions. For purposes of comparison it isoften convenient to use full, half or one-quarteropen.

23

valve

static pressure tapmotor

pump

tank

rotameter

FIGURE 8.1. Schematic of the pipe friction apparatus.

24

EXPERIMENT 9

PRESSURE DISTRIBUTION ABOUT A CIRCULAR CYLINDER

In many engineering applications, it may benecessary to examine the phenomena occurringwhen an object is inserted into a flow of fluid. Thewings of an airplane in flight, for example, maybe analyzed by considering the wings stationarywith air moving past them. Certain forces areexerted on the wing by the flowing fluid thattend to lift the wing (called the lift force) and topush the wing in the direction of the flow (dragforce). Objects other than wings that aresymmetrical with respect to the fluid approachdirection, such as a circular cylinder, willexperience no lift, only drag.

Drag and lift forces are caused by thepressure differences exerted on the stationaryobject by the flowing fluid. Skin friction betweenthe fluid and the object contributes to the dragforce but in many cases can be neglected. Themeasurement of the pressure distribution existingaround a stationary cylinder in an air stream tofind the drag force is the object of thisexperiment.

Consider a circular cylinder immersed in auniform flow. The streamlines about the cylinderare shown in Figure 9.1. The fluid exerts pressureon the front half of the cylinder in an amountthat is greater than that exerted on the rearhalf. The difference in pressure multiplied by theprojected frontal area of the cylinder gives thedrag force due to pressure (also known as formdrag). Because this drag is due primarily to apressure difference, measurement of the pressuredistribution about the cylinder allows for findingthe drag force experimentally. A typical pressuredistribution is given in Figure 9.2. Shown in

Figure 9.2a is the cylinder with lines andarrowheads. The length of the line at any pointon the cylinder surface is proportional to thepressure at that point. The direction of thearrowhead indicates that the pressure at therespective point is greater than the free streampressure (pointing toward the center of thecylinder) or less than the free stream pressure(pointing away). Note the existence of aseparation point and a separation region (orwake). The pressure in the back flow region isnearly the same as the pressure at the point ofseparation. The general result is a net drag forceequal to the sum of the forces due to pressureacting on the front half (+) and on the rear half(-) of the cylinder. To find the drag force, it isnecessary to sum the components of pressure ateach point in the flow direction. Figure 9.2b is agraph of the same data as that in Figure 9.2aexcept that 9.2b is on a linear grid.

FreestreamVelocity V

StagnationStreamline Wake

FIGURE 9.1. Streamlines of flow about a circularcylinder.

separationpoint

separationpoint

0 30 60 90 120 150 180

p

(a) Polar Coordinate Graph (b) Linear Graph

FIGURE 9.2. Pressure distribution around a circular cylinder placed in a uniform flow.

25

Pressure Measurement

EquipmentA Wind TunnelA Right Circular Cylinder with Pressure

Taps

Figure 9.3 is a schematic of a wind tunnel. Itconsists of a nozzle, a test section, a diffuser and afan. Flow enters the nozzle and passes throughflow straighteners and screens. The flow isdirected through a test section whose walls aremade of a transparent material, usuallyPlexiglas or glass. An object is placed in the testsection for observation. Downstream of the testsection is the diffuser followed by the fan. In thetunnel that is used in this experiment, the testsection is rectangular and the fan housing iscircular. Thus one function of the diffuser is togradually lead the flow from a rectangularsection to a circular one.

Figure 9.4 is a schematic of the side view ofthe circular cylinder. The cylinder is placed inthe test section of the wind tunnel which isoperated at a preselected velocity. The pressuretap labeled as #1 is placed at 0° directly facingthe approach flow. The pressure taps areattached to a manometer board. Only the first 18taps are connected because the expected profile issymmetric about the 0° line. The manometers willprovide readings of pressure at 10° intervalsabout half the cylinder. For two differentapproach velocities, measure and record thepressure distribution about the circular cylinder.

Plot the pressure distribution on polar coordinategraph paper for both cases. Also graph pressuredifference (pressure at the point of interest minusthe free stream pressure) as a function of angle θon linear graph paper. Next, graph ∆p cosθ vs θ(horizontal axis) on linear paper and determinethe area under the curve by any convenientmethod (counting squares or a numericaltechnique).

The drag force can be calculated byintegrating the flow-direction-component of eachpressure over the area of the cylinder:

Df = 2RL ∫0

π

∆p cosθdθ

The above expression states that the drag force istwice the cylinder radius (2R) times the cylinderlength (L) times the area under the curve of ∆pcosθ vs θ.

Drag data are usually expressed as dragcoefficient CD vs Reynolds number Re. The dragcoefficient is defined as

CD = Df

ρV2A/2

The Reynolds number is

Re = ρVD

µ

inlet flowstraighteners

nozzle

test sectiondiffuser fan

FIGURE 9.3. A schematic of the wind tunnel used in this experiment.

26

where V is the free stream velocity (upstream ofthe cylinder), A is the projected frontal area ofthe cylinder (2RL), D is the cylinder diameter, ρis the air density and µ is the air viscosity.Compare the results to those found in texts.

static pressuretaps attach tomanometers

60

0

30

90120

150

180

FIGURE 9.4. Schematic of the experimentalapparatus used in this experiment.

27

EXPERIMENT 10

DRAG FORCE DETERMINATION

An object placed in a uniform flow is actedupon by various forces. The resultant of theseforces can be resolved into two force components,parallel and perpendicular to the main flowdirection. The component acting parallel to theflow is known as the drag force. It is a function ofa skin friction effect and an adverse pressuregradient. The component perpendicular to theflow direction is the lift force and is caused by apressure distribution which results in a lowerpressure acting over the top surface of the objectthan at the bottom. If the object is symmetricwith respect to the flow direction, then the liftforce will be zero and only a drag force will exist.Measurement of the drag force acting on an objectimmersed in the uniform flow of a fluid is thesubject of this experiment.

EquipmentSubsonic Wind TunnelObjects

A description of a subsonic wind tunnel isgiven in Experiment 9 and is shown schematicallyin Figure 9.3. The fan at the end of the tunneldraws in air at the inlet. An object is mounted on astand that is pre calibrated to read lift and dragforces exerted by the fluid on the object. Aschematic of the test section is shown in Figure10.1. The velocity of the flow at the test section isalso pre calibrated. The air velocity past theobject can be controlled by changing the angle ofthe inlet vanes located within the fan housing.Thus air velocity, lift force and drag force areread directly from the tunnel instrumentation.

There are a number of objects that areavailable for use in the wind tunnel. Theseinclude a disk, a smooth surfaced sphere, a roughsurface sphere, a hemisphere facing upstream,and a hemisphere facing downstream. Forwhichever is assigned, measure drag on the objectas a function of velocity.

Data on drag vs velocity are usually graphedin dimensionless terms. The drag force Df iscustomarily expressed in terms of the dragcoefficient CD (a ratio of drag force to kineticenergy):

CD = Df

ρV2A/2

in which ρ is the fluid density, V is the freestream velocity, and A is the projected frontalarea of the object. Traditionally, the dragcoefficient is graphed as a function of theReynolds number, which is defined as

Re = VDν

where D is a characteristic length of the objectand ν is the kinematic viscosity of the fluid. Foreach object assigned, graph drag coefficient vsReynolds number and compare your results tothose published in texts. Use log-log paper ifappropriate.

Questions1. How does the mounting piece affect the

readings?2. How do you plan to correct for its effect, if

necessary?

drag forcemeasurement

lift forcemeasurement

uniform flow mounting stand

object

FIGURE 10.1. Schematic of an object mounted inthe test section of the wind tunnel.

28

EXPERIMENT 11

ANALYSIS OF AN AIRFOIL

A wing placed in the uniform flow of anairstream will experience lift and drag forces.Each of these forces is due to a pressuredifference. The lift force is due to the pressuredifference that exists between the lower andupper surfaces. This phenomena is illustrated inFigure 11.1. As indicated the airfoil is immersedin a uniform flow. If pressure could be measured atselected locations on the surface of the wing andthe results graphed, the profile in Figure 11.1would result. Each pressure measurement isrepresented by a line with an arrowhead. Thelength of each line is proportional to themagnitude of the pressure at the point. Thedirection of the arrow (toward the horizontalaxis or away from it) represents whether thepressure at the point is less than or greater thanthe free stream pressure measured far upstream ofthe wing.

stagnationpoint

negative pressuregradient on uppersurface

positive pressureon lower surface

Cp

pressurecoefficient

stagnationpoint

c

chord, c

FIGURE 11.1. Streamlines of flow about a wingand the resultant pressure distribution.

Lift and Drag Measurements for a Wing

EquipmentWind Tunnel (See Figure 9.3)Wing with Pressure TapsWing for Attachment to Lift & Drag Instruments (See Figure 11.2)

Experiment I Mount the wing with pressure taps in the

tunnel and attach the tube ends to manometers.Select a wind speed and record the pressuredistribution for a selected angle of attack (asassigned by the instructor). Plot pressure vs chordlength as in Figure 11.1, showing the verticalcomponent of each pressure acting on the uppersurface and on the lower surface. Determinewhere separation occurs for each case.

Mount the second wing on the lift and dragbalance (Figure 11.2). For the same wind speedand angle of attack, measure lift and drag exertedon the wing.

drag forcemeasurement

lift forcemeasurement

uniform flow mounting stand

cdrag

lift

FIGURE 11.2. Schematic of lift and dragmeasurement in a test section.

The wing with pressure taps providedpressure at selected points on the surface of thewing. Use the data obtained and sum thehorizontal component of each pressure to obtainthe drag force. Compare to the results obtainedwith the other wing. Use the data obtained andsum the vertical component of each pressure toobtain the lift force. Compare the resultsobtained with the other wing. Calculate %errors.

29

Experiment II For a number of wings, lift and drag data

vary only slightly with Reynolds number andtherefore if lift and drag coefficients are graphedas a function of Reynolds number, the results arenot that meaningful. A more significantrepresentation of the results is given in what isknown as a polar diagram for the wing. A polardiagram is a graph on a linear grid of liftcoefficient (vertical axis) as a function of dragcoefficient. Each data point on the graphcorresponds to a different angle of attack, allmeasured at one velocity (Reynolds number).Referring to Figure 11.2 (which is theexperimental setup here), the angle of attack α ismeasured from a line parallel to the chord c to aline that is parallel to the free stream velocity.If so instructed, obtain lift force, drag force andangle of attack data using a pre selected velocity.Allow the angle of attack to vary from a negativeangle to the stall point and beyond. Obtain data

at no less than 9 angles of attack. Use the data toproduce a polar diagram.

Analysis Lift and drag data are usually expressed in

dimensionless terms using lift coefficient and dragcoefficient. The lift coefficient is defined as

CL = Lf

ρV2A/2

where Lf is the lift force, ρ is the fluid density, Vis the free stream velocity far upstream of thewing, and A is the area of the wing when seenfrom a top view perpendicular to the chordlength c. The drag coefficient is defined as

CD = Df

ρV2A/2

in which Df is the drag force.

30

EXPERIMENT 12

OPEN CHANNEL FLOW—SLUICE GATE

Liquid motion in a duct where a surface of thefluid is exposed to the atmosphere is called openchannel flow. In the laboratory, open channelflow experiments can be used to simulate flow in ariver, in a spillway, in a drainage canal or in asewer. Such modeled flows can include flow overbumps or through dams, flow through a venturiflume or under a partially raised gate (a sluicegate). The last example, flow under a sluice gate,is the subject of this experiment.

Flow Through a Sluice Gate

EquipmentOpen Channel Flow ApparatusSluice Gate Model

Figure 12.1 shows a schematic of the sideview of the sluice gate. Flow upstream of the gatehas a depth ho while downstream the depth is h.The objective of the analysis is to formulate anequation to relate the volume flow rate through(or under) the gate to the upstream anddownstream depths.

h

direction ofmovement

hand cranksluice gate

patm

patm

ho

FIGURE 12.1. Schematic of flow under a sluicegate .

The flow rate through the gate is maintained atnearly a constant value. For various raisedpositions of the sluice gate, different liquidheights ho and h will result. Applying theBernoulli equation to flow about the gate gives

p0

ρg +

V02

2g + h0 =

pρg

+ V2

2g + h

Pressures at the free surface are both equal toatmospheric pressure, so they cancel. Rearranginggives

h0 = V2

2g -

V02

2g +h

In terms of flow rate, the velocities are written as

V0 = QA

= Q

bh0

V = Qb h

where b is the channel width at the gate.Substituting into the Bernoulli Equation andsimplifying gives

h0 = Q2

2gb2

1

h 2 - 1

h 02 + h

Dividing by h0,

1 = Q2

2gb2h0

1

h 2 - 1

h 02 +

hh 0

Rearranging further,

1 -

hh 0

= Q2

2gb2h2h0

1 -

h 2

h 02

Multiplying both sides by h2/h02, and continuing

to simplify, we finally obtain

h 2 / h 02

1 + h/h0 =

Q2

2gb2h03

here Q is the theoretical volume flow rate. Theright hand side of this equation is recognized as1/2 of the upstream Froude number. So bymeasuring the depth of liquid before and afterthe sluice gate, the theoretical flow rate can becalculated with the above equation. Thetheoretical flow rate can then be compared to theactual flow rate obtained by measurements usingthe orifice meters.

For 9 different raised positions of the sluicegate, measure the upstream and downstreamdepths and calculate the actual flow rate. Inaddition, calculate the upstream Froude numberfor each case and determine its value formaximum flow conditions. Graph h/h0 (vertical

31

axis) versus (Q2/b2h03g). Determine h/h0

corresponding to maximum flow. Note that h/h0varies from 0 to 1.

Figure 12.2 is a sketch of the open channelflow apparatus. It consists of a sump tank with apump/motor combination on each side. Each pumpdraws in water from the sump tank anddischarges it through the discharge line tocalibrated orifice meters and then to the headtank. Each orifice meter is connected to its ownmanometer. Use of the calibration curve(provided by the instructor) allows for findingthe actual flow rate into the channel. The headtank and flow channel have sides made ofPlexiglas. Water flows downstream in thechannel past the object of interest (in this case asluice gate) and then is routed back to the sumptank.

Questions1. For the required report, derive the sluice

gate equation in detail.2. What if it was assumed that V0 << V, and

so V0 could be canceled from the Bernoulliequation? Derive the resulting equationwhich will contain h/h0 and the upstreamFroude number.

2. What is the significance of Froude number?3. What is the significance of the Reynolds

number in this or in any open channel flowsituation?

4. Where are sluice gates found?5. What are they used for?

sump tankpump/motor

pump dischargepipe

valve

head tank

sluice gate

orificemeter

hand crank

flow channel

FIGURE 12.2. Schematic of the open channel flow apparatus.

32

EXPERIMENT 13

OPEN CHANNEL FLOW OVER A WEIR

Flow meters used in pipes introduce anobstruction into the flow which results in ameasurable pressure drop that in turn is related tothe volume flow rate. In an open channel, flowrate can be measured similarly by introducing anobstruction into the flow. A simple obstruction,called a weir, consists of a vertical plateextending the entire width of the channel. Theplate may have an opening, usually rectangular,trapezoidal, or triangular. Other configurationsexist and all are about equally effective. The useof a weir to measure flow rate in an open channelis the subject of this experiment.

Flow Over a Weir

EquipmentOpen Channel Flow Apparatus (See Figure 12.1)Several Weirs

The open channel flow apparatus allows forthe insertion of a weir and measurement of liquiddepths. The channel is fed by two centrifugalpumps. Each pump has a discharge line whichcontains an orifice meter attached to amanometer. The pressure drop reading from themanometers and a calibration curve provide themeans for determining the actual flow rate intothe channel.

Figure 13.1 is a sketch of the side andupstream view of a 90 degree (included angle) V-notch weir. Analysis of this weir is presentedhere for illustrative purposes. Note thatupstream depth measurements are made from thelowest point of the weir over which liquid flows.

This is the case for the analysis of allconventional weirs. A coordinate system isimposed whose origin is at the intersection of thefree surface and a vertical line extending upwardfrom the vertex of the V-notch. We select anelement that is dy thick and extends the entirewidth of the flow cross section. The velocity ofthe liquid through this element is found byapplying Bernoulli's equation:

pa

ρ + Vo

2

2 + gh = pa

ρ + V2

2 + g(h - y)

Note that in pipe flow, pressure remained in theequation when analyzing any of the differentialpressure meters (orifice or venturi meters). In openchannel flows, the pressure terms representsatmospheric pressure and cancel from theBernoulli equation. The liquid height istherefore the only measurement required here.From the above equation, assuming Vo negligible:

V = √2gy (13.1)

Equation 13.1 is the starting point in the analysisof all weirs. The incremental flow rate of liquidthrough layer dy is:

dQ = 2Vxdy = √2gy(2x)dy

From the geometry of the V-notch and withrespect to the coordinate axes, we have y = h - x.

pa

pa

Vo

Vh

y

dy

x

x axis

y axis

FIGURE 13.1. Side and upstream views of a 90° V-notch weir.

33

Therefore,

Q = ∫ 0

h

(2√2g)y1/2(h - y)d y

Integration gives

Qth = 8

15 √2g h5/2 =Ch5/2 (13.2)

where C is a constant. The above equationrepresents the ideal or theoretical flow rate ofliquid over the V-notch weir. The actualdischarge rate is somewhat less due to frictionaland other dissipative effects. As with pipemeters, we introduce a discharge coefficientdefined as:

C' = Qac

Qth

The equation that relates the actual volume flowrate to the upstream height then is

Qac = C'Ch5/2

It is convenient to combine the effects of theconstant C and the coefficient C’ into a singlecoefficient Cvn for the V-notch weir. Thus wereformulate the previous two equations to obtain:

Cvn ≈ Qac

Qth(13.3)

Qac = Cvnh5/2 (13.4)

Each type of weir will have its own coefficient. Calibrate each of the weirs assigned by the

instructor for 7 different upstream heightmeasurements. Use the flow rate chart providedwith the open channel flow apparatus to obtainthe actual flow rate. Derive an appropriateequation for each weir used (similar to Equation13.4) above. Determine the coefficient applicablefor each weir tested. List the assumptions madein each derivation. Discuss the validity of eachassumption, pointing out where they break down.Graph upstream height vs actual and theoreticalvolume flow rates. Plot the coefficient ofdischarge (as defined in Equation 13.3) as afunction of the upstream Froude number.

FIGURE 13.2. Other types of weirs–semicircular, contracted and suppressed.

34

EXPERIMENT 14

OPEN CHANNEL FLOW—HYDRAULIC JUMP

When spillways or other similar openchannels are opened by the lifting of a gate,liquid passing below the gate has a high velocityand an associated high kinetic energy. Due to theerosive properties of a high velocity fluid, itmay be desirable to convert the high kineticenergy (e.g. high velocity) to a high potentialenergy (e.g., a deeper stream). The problem thenbecomes one of rapidly varying the liquid depthover a short channel length. Rapidly varied flowof this type produces what is known as ahydraulic jump.

Consider a horizontal, rectangular openchannel of width b, in which a hydraulic jumphas developed. Figure 14.1 shows a side view of ahydraulic jump. Figure 14.1 also shows the depthof liquid upstream of the jump to be h1, and adownstream depth of h2. Pressure distributionsupstream and downstream of the jump are drawnin as well. Because the jump occurs over a veryshort distance, frictional effects can be neglected.A force balance would therefore include onlypressure forces. Applying the momentum equationin the flow direction gives:

p1A1 - p2A2 = ρQ(V2 - V1)

Pressure in the above equation represents thepressure that exists at the centroid of the crosssection. Thus p = ρg(h/2). With a rectangularcross section of width b (A = bh), the aboveequation becomes

h1g

2 (h1b) - h2g

2 (h2b) = Q(V2 - V1)

From continuity, A1V1 = A2V2 = Q. Combining andrearranging,

h1

2 - h22

2 = Q 2

gb2

1

h 2 -

1 h 1

Simplifying,

h22 + h2h1 - 2

Q2

gb2h1 = 0

Solving for the downstream height yields onephysically (nonnegative) possible solution:

h2 = - h1

2 + √2Q 2

gb2h1 +

h12

4

from which the downstream height can be found.By applying Bernoulli’s Equation along the freesurface, the energy lost irreversibly can becalculated as

Lost Energy = E = g(h2 - h1)3

4h 2h 1

and the rate of energy loss is

d Wd t

= ρQE

The above equations are adequate to properlydescribe a hydraulic jump.

Hydraulic Jump Measurements

EquipmentOpen Channel Flow Apparatus (Figure 12.1)

The channel can be used in either ahorizontal or a sloping configuration. The devicecontains two pumps which discharge waterthrough calibrated orifice meters connected tomanometers. The device also contains on thechannel bottom two forward facing brass tubes.Each tube is connected to a vertical Plexiglastube. The height of the water in either of thesetubes represents the energy level at therespective tube location. The difference in heightis the actual lost energy (E) for the jump ofinterest.

FIGURE 14.1. Schematic of ahydraulic jump in an openchannel.

h1

V2

V1

p1

p2

h2

35

Develop a hydraulic jump in the channel;record upstream and downstream heights,manometer readings (from which the actualvolume flow rate is obtained) and the lost energyE. By varying the flow rate, upstream height,downstream height and/or the channel slope,record measurements on different jumps. Derivethe applicable equations in detail and substituteappropriate values to verify the predicteddownstream height and lost energy. In otherwords, the downstream height of each jump is tobe measured and compared to the downstreamheight calculated with Equation 14.1. The sameis to be done for the rate of energy loss (Equation14.2).

Analysis Data on a hydraulic jump is usually specified

in two ways both of which will be required forthe report. Select any of the jumps you havemeasurements for and construct a momentumdiagram . A momentum diagram is a graph ofliquid depth on the vertical axis vs momentum onthe horizontal axis. The momentum of the flow isgiven by:

M = 2Q2

gb2h +

h2

4

Another significant graph of hydraulic jumpdata is of depth ratio h2/h1 (vertical axis) as afunction of the upstream Froude number, Fr1 (=Q2/gb2h1

3 ). Construct such a graph for any of thejumps for which you have taken measurements.

36

EXPERIMENT 15

OPEN CHANNEL FLOW OVER A HUMP

Flow over a hump in an open channel is aproblem that can be successfully modeled in orderto make predictions about the behavior of thefluid. This experiment involves makingappropriate measurements for such a system, andrelating flow rate to critical depth. The flowrate, critical depth, and specific energy aredetermined theoretically and experimentally.

TheoryFlow in a channel is modeled in terms of a

parameter called the specific energy head (or justspecific energy) of the flow, E. The specificenergy head is defined as

E = h + Q2

2gh2b2 (15.1)

where h is the depth of the flow, Q is the volumeflow rate, g is gravity, and b is the channelwidth. The dimension of the specific energy headis L (ft or m).

Figure 15.1 is a sketch of flow over a hump,with flow from left to right. Shown is the chan-nel bed and the hump. Upstream of the hump(subscript 1 notation), the flow is subcritical;downstream (subscript 2) the flow is super-critical. Just at the highest point of the hump,the flow is critical (subscript c). Also shown inthe figure is the total energy line, which weassume is parallel to the flow channel bed; i.e.,the total energy remains a constant in the flow.

Upstream of the hump, the total specificenergy head of the flow is denoted as E1, and thedepth of the liquid is h1, as shown graphically inFigure 15.1. At any location z on the hump beforezc, the energy head is E, and the depth is h. Atthis same height z downstream of zc, the liquiddepth is h’, but the energy head is still E. At thehighest point of the hump zc, the energy head isEc and the liquid depth is hc. The total specificenergy head and the liquid depth anywhere arerelated according to Equation 15.1.

E1 h1

z

Ehhc

h2

h'

Ec

zc

total energy line

hump

channel bed

flow direction

FIGURE 15.1. Flow over ahump in an openchannel.

We can illustrate the relationship betweenthese parameters graphically by drawing aspecific energy head diagram, as illustrated inFigure 15.2. This graph has flow depth on thevertical axis and specific energy head on thehorizontal axis. The condition of the flow isrepresented by the solid line with arrowsshowing how the flow changes from subcritical tosupercritical. At the location on the hump wherethe height is z, the energy head is E. We draw avertical line at this value of the specific energyhead; it will intersect the line at h (upstream)and h’ (downstream).

E1, E2

h1

z

E

h

hc

h2

h'

Ec

zc

specific energy head E

dep

th h

supercritical

subcritical

FIGURE 15.2. Specific energy diagram.

37

At any upstream (of the hump) location, sayh1, we see that the corresponding specific energyhead is E1. The vertical line that locates E1 alsolocates the energy E2 which is downstream of thehump. A vertical line drawn at E1 intersects theline at h1 and h2, which are the upstream anddownstream liquid heights, respectively. Notethat the minimum specific energy head is at thehighest point of the hump zc, and the energyhead there is Ec.

As water flows over the hump, the initialspecific energy head E1 is reduced to a value E byan amount equal to the height of the hump. So atany location along the hump, the specific energyhead is E1 - z, where z is the elevation above thechannel bed. At the point where the flow iscritical, the critical depth hc is given by

hc =

Q2

b2g

1/3 =

2Ec

3 (15.2)

Flow Over a Hump

EquipmentOpen Channel Flow Apparatus (Figure

12.1)Installed hump

The open channel flow apparatus is describedin Experiment 12 and illustrated schematically inFigure 12.1. Adjust the channel so that it ishorizontal. Make every effort to minimizeleakage of water past the sides of the hump.Start both pumps and adjust the valves to give asmooth water surface profile over the hump. Forone set of conditions, take readings from themanometers to determine the volume flow rateover the hump.

The open channel flow apparatus has adepth gage attached. It will be necessary to

measure the water depth at certain specificlocations on or about the hump. These locationsare shown in Figure 15.3 (dimensions are in feet).There are 8 water depths to be measured. So forone flow rate, two manometer readings and 8water depths will be recorded. Gather data forthe assigned number of flow rates.

ResultsAlthough the data taken in this experiment

seem simple, the calculations required to reducethe data appropriately can occupy much time.With the data obtained:

Determine the flow rate using the manometerreadings. This value will be referred to as theactual flow rate QAC (subscript AC will referto an actual value, while TH refers to atheoretical value).

Calculate the flow rate using a rearrangedform of Equation 15.2. This value will bereferred to as the theoretical flow rate QTH.Compare the two flow rates and find % error.

Use Equation 15.2 to find the value of thecritical depth using QAC. Compare this valueto the measured value, and find % error.

Calculate the theoretical and actual valuesof the minimum energy Ec using Equation 15.2.Compare the results.

Calculate the actual specific energy head EACat each measurement station using Equation15.1. Determine also the total energy headHAC (= EAC + z) for all readings.

Compose a chart using the column and rowheadings shown in Table 15.1.

flow direction

0.313 0.313 0.3130.313

0.276

2

hump

1

1 2 3 4 5 6 7 8

FIGURE 15.3. Water depthmeasurement locationsfor flow over a hump.(Dimensions in feet.)

38

TABLE 15.1. Data reduction table for flow over a hump.

Station 1 2 3 4 5 6 7 8Depth of flow hAC in ftSpecific energy head EAC in ftHeight of hump above channel bed zin ftTotal energy head HAC in ft

Construct a graph of the flow configuration.On the horizontal axis, plot distancedownstream, and plot depth on the verticalaxis. On this set of axes, plot (a) the totalenergy line (HAC); (b) the water surfaceprofile; and, (c) the elevation z. Show datapoints on the graph.

Construct a specific energy head diagramsimilar to that of Figure 15.2. Show thetheoretical results (based on QTH), and showthe actual data points.

Derive Equation 2.

Questions1. What is the value of the Froude number (a)

upstream of the hump, (b) at the highestpoint of the hump, and (c) downstream of thehump?

2. Is the Froude number used in finding thecritical depth in Equation 15.2?

3. What equations is used to develop theexpression for specific energy head (Equation15.1)?

4. How is the second term in Equation 15.1 (i.e.;Q2/2gh2b2) related to the Froude number?

5. Is the total energy line (HAC) a constant as weassumed with reference to Figure 15.1, or doesit change?

39

EXPERIMENT 16

MEASUREMENT OF VELOCITYAND

CALIBRATION OF A METER FOR COMPRESSIBLE FLOW

The objective of this experiment is todetermine a calibration curve for a meter placedin a pipe that is conveying air. The meters ofinterest are an orifice meter and a venturi meter.These meters are calibrated in this experiment byusing a pitot-static tube to measure the velocity,from which the flow rate is calculated.

Pitot Static TubeWhen a fluid flows through a pipe, it exerts

pressure that is made up of static and dynamiccomponents. The static pressure is indicated by ameasuring device moving with the flow or thatcauses no velocity change in the flow. Usually, tomeasure static pressure, a small holeperpendicular to the flow is drilled through thecontainer wall and connected to a manometer (orpressure gage) as indicated in Figure 16.1.

The dynamic pressure is due to the movementof the fluid. The dynamic pressure and the staticpressure together make up the total or stagnationpressure. The stagnation pressure can be measuredin the flow with a pitot tube. The pitot tube is anopen ended tube facing the flow directly. Figure16.1 gives a sketch of the measurement ofstagnation pressure.

static pressuremeasurement

stagnation pressuremeasurement

pitot tube

flow

hh

FIGURE 16.1. Measurement of static andstagnation pressures.

The pitot-static tube combines the effects ofstatic and stagnation pressure measurement intoone device. Figure 16.2 is a schematic of the pitot-static tube. It consists of a tube within a tubewhich is placed in the duct facing upstream. Thepressure tap that faces the flow directly gives ameasurement of the stagnation pressure, while

the tap that is perpendicular to the flow givesthe static pressure.

When the pitot-static tube is immersed in theflow of a fluid, the pressure difference(stagnation minus static) can be read directlyusing a manometer and connecting the pressuretaps to each leg. Applying the Bernoulli equationbetween the two pressure taps yields:

A

A

section A-Aenlarged

four to eight holesequally spaced

manometerconnections

flow direction

FIGURE 16.2. Schematic of a pitot-static tube.

p1

ρg +

V12

2g + z1 =

p2

ρg +

V22

2g + z2

where state “1” as the stagnation state (whichwill be changed to subscript “t”), and state “2” asthe static state (no subscript). Elevationdifferences are negligible, and at the point wherestagnation pressure is measured, the velocity iszero. The Bernoulli equation thus reduces to:

ptρg

= pρg

+ V2

2g

Next, we rearrange the preceding equation andsolve for velocity

V = √2(p t - p)ρ

A manometer connected to the pitot-static tubewould provide head loss readings ∆h given by

40

∆h = pt - p

ρg

where density is that of the flowing fluid. Sovelocity in terms of head loss is

V = √2g∆h

Note that this equation applies only toincompressible flows. Compressibility effects arenot accounted for. Furthermore, ∆h is the headloss in terms of the flowing fluid and not in termsof the reading on the manometer.

For flow in a duct, manometer readings are tobe taken at a number of locations within the crosssection of the flow. The velocity profile is thenplotted using the results. Velocities at specificpoints are then determined from these profiles.The objective here is to obtain data, graph avelocity profile and then determine the averagevelocity.

Average VelocityThe average velocity is related to the flow

rate through a duct as

V = QA

where Q is the volume flow rate and A is thecross sectional area of the duct. We can dividethe flow area into five equal areas, as shown inFigure 16.3. The velocity is to be obtained atthose locations labeled in the figure. The chosenpositions divide the cross section into five equalconcentric areas. The flow rate through each arealabeled from 1 to 5 is found as

Q1 = A1V1 Q2 = A2V2

Q3 = A3V3 Q4 = A4V4

Q5 = A5V5

0.316 R

R

0.837 R0.949 R0.707 R0.548 R

FIGURE 16.3. Five positions within the crosssection where velocity is to be determined.

The total flow rate through the entire crosssection is the sum of these:

Qtotal = ∑1

5

Qi = A1V1 + A2V2 + A3V3 + A4V4

+ A5V5

or Qtotal = A1 (V1 + V2 + V3 + V4 + V5)

The total area Atotal is 5A1 and so

V = Qtotal

Atotal =

(Atotal/5)(V1 + V2 + V3 + V4 + V5)Atotal

The average velocity then becomes

V = (V1 + V2 + V3 + V4 + V5)

5

The importance of the five chosen radialpositions for measuring V1 through V5 is nowevident.

Velocity MeasurementsEquipment

Axial flow fan apparatusPitot-static tubeManometer

The fan of the apparatus is used to move airthrough the system at a rate that is small enoughto allow the air to be considered incompressible.While the fan is on, make velocity profilemeasurements at a selected location within theduct at a cross section that is several diametersdownstream of the fan. Repeat thesemeasurements at different fan speed settings sothat 9 velocity profiles will result. Use thevelocity profiles to determine the averagevelocity and the flow rate.

Questions1. Why is it appropriate to take velocity

measurements at several diametersdownstream of the fan?

2. Suppose the duct were divided into 6 equalareas and measurements taken at selectpositions in the cross section. Should theaverage velocity using 6 equal areas be thesame as the average velocity using 5 or 4equal areas?

41

Incompressible Flow Through a MeterIncompressible flow through a venturi and an

orifice meter was discussed in Experiment 9. Forour purposes here, we merely re-state theequations for convenience. For an air over liquidmanometer, the theoretical equation for bothmeters is

Qth = A2 √2g∆h(1 - D2

4/D14)

Now for any pressure drop ∆hi, there are twocorresponding flow rates: Qac and Qth. The ratio ofthese flow rates is the venturi dischargecoefficient Cv, defined as

Cv = Qac

Qth = 0.985

for turbulent flow. The orifice dischargecoefficient can be expressed in terms of the Stolzequation:

Co = 0.595 9 + 0.031 2β 2.1 - 0.184β 8 +

+ 0.002 9β 2.5

106

Re β 0.75

+ 0.09L1

β 4

1 - β 4 - L2 (0.003 37β 3)

where Re = ρVoDo

µ = 4ρQac

πDoµ β = Do

D1

L1 = 0 for corner tapsL1 = 1/D1 for flange tapsL1 = 1 for 1D & 12D taps

and if L1 ≥ 0.433 3, the coefficient of the

β 4

1 - β 4

term becomes 0.039.

L2 = 0 for corner tapsL2 = 1/D1 for flange tapsL2 = 0.5 - E/D1 for 1D & 12D taps

E = orifice plate thickness

Compressible Flow Through a MeterWhen a compressible fluid (vapor or gas)

flows through a meter, compressibility effectsmust be accounted for. This is done by introductionof a compressibility factor which can bedetermined analytically for some meters(venturi). For an orifice meter, on the other hand,the compressibility factor must be measured.

The equations and formulation developedthus far were for incompressible flow through ameter. For compressible flows, the derivation is

somewhat different. When the fluid flowsthrough a meter and encounters a change in area,the velocity changes as does the pressure. Whenpressure changes, the density of the fluid changesand this effect must be accounted for in order toobtain accurate results. To account forcompressibility, we will rewrite the descriptiveequations.

Venturi MeterConsider isentropic, subsonic, steady flow of

an ideal gas through a venturi meter. Thecontinuity equation is

ρ1A1V1 = ρ2A2V2 = ·misentropic = ·ms

where section 1 is upstream of the meter, andsection 2 is at the throat. Neglecting changes inpotential energy (negligible compared to changesin enthalpy), the energy equation is

h1 + V1

2

2 = h2 + V2

2

2

The enthalpy change can be found by assumingthat the compressible fluid is ideal:

h1 - h2 = Cp(T1 - T2)

Combining these equations and rearranging gives

CpT1 + ·ms

2

2ρ12A1

2 = CpT2 + ·ms

2

2ρ22A2

2

or

·ms2

1

ρ22A2

2 - 1

ρ12A1

2 = 2Cp(T1 - T2)

= 2CpT1

1 -

T2

T1

If we assume an isentropic compression processthrough the meter, then we can write

p2

p1 =

T2

T1

γγ - 1

where γ is the ratio of specific heats (γ = Cp/Cv).Also, recall that for an ideal gas,

Cp = R γ

γ - 1

Substituting, rearranging and simplifying, we get

42

·ms2

ρ22A2

2

1 -

ρ22A2

2

ρ12A1

2 = 2 R γ

γ - 1 T1

1 -

p2

p1

γ - 1γ

For an ideal gas, we write ρ = p/RT. Substitutingfor the RT1 term in the preceding equation yields

·ms2

A22 = 2ρ2

2 γγ - 1

p1

ρ1

1 - (p2/p1) (γ - 1)/γ 1 - (ρ2

2A22/ρ1

2A12)

For an isentropic process, we can also write

p1

ρ1γ =

p2

ρ2γ

or ρ2 =

p2

p1

1/γ ρ1

from which we obtain

ρ22 =

p2

p1

2/γ ρ1

2

Substituting into the mass flow equation, we getafter considerable manipulation Equation 16.1 ofTable 16.1, which summarizes the results.

Thus for compressible flow through a venturimeter, the measurements needed are p1, p2, T1,the venturi dimensions, and the fluid properties.By introducing the venturi discharge coefficientCv, the actual flow rate through the meter isdetermined to be

·mac = Cv ·ms

Combining this result with Equation 16.1 givesEquation 16.2 of Table 16.1.

It would be convenient if we could re-writeEquation 16.2 in such a way that thecompressibility effects could be consolidated intoone term. We attempt this by using the flow rateequation for the incompressible case multipliedby another coefficient called the compressibilityfactor Y; we therefore write

·mac = CvYρ1A2 √2(p1 - p2)ρ1(1 - D2

4/D14)

We now set the preceding equation equal toEquation 16.2 and solve for Y. We obtain Equation16.3 of the table.

The ratio of specific heats γ will be known fora given compressible fluid, and so Equation 16.3

could be plotted as compressibility factor Y versuspressure ratio p2/p1 for various values of D2/D1.The advantage of using this approach is that apressure drop term appears just as with theincompressible case, which is convenient if amanometer is used to measure pressure. Moreover,the compressibility effect has been isolated intoone factor Y.

Orifice MeterThe equations and formulation of an analysis

for an orifice meter is the same as that for theventuri meter. The difference is in the evaluationof the compressibility factor. For an orifice meterthe compressibility factor is much lower thanthat for a venturi meter. The compressibilityfactor for an orifice meter cannot be derived, butinstead must be measured. Results of such testshave yielded the Buckingham equation, Equation16.4 of Table 16.1, which is valid for mostmanometer connection systems.

Calibration of a MeterFigures 16.4 and 16.5 show how the apparatus

is set up. An axial flow fan is attached to theshaft of a DC motor. The rotational speed of themotor, and hence the volume flow rate of air, iscontrollable. The fan moves air through a ductinto which a pitot-static tube is attached. Thepitot static tube is movable so that the velocityat any radial location can be measured. An orificeor a venturi meter can be placed in the ductsystem.

The pitot static tube has pressure taps whichare to be connected to a manometer. Likewise eachmeter also has pressure taps, and these will beconnected to a separate manometer.

A meter for calibration will be assigned bythe instructor. For the experiment, makemeasurements of velocity using the pitot-statictube to obtain a velocity profile. Draw thevelocity profile to scale. Obtain data from thevelocity profile and determine a volume flowrate.

For one velocity profile, measure the pressuredrop associated with the meter. Graph volumeflow rate as a function of head loss ∆h obtainedfrom the meter, with ∆h on the horizontal axis.Determine the value of the compressibility factorexperimentally and again using the appropriateequation (Equation 16.3 or 16.4) for each datapoint. A minimum of 9 data points should beobtained. Compare the results of bothcalculations for Y.

43

TABLE 16.1. Summary of equations for compressible flow through a venturi or an orifice meter.

·ms = A2

2p1ρ1 (p2/p1)2/γ [γ/(γ - 1)] [1 - (p2/p1)(γ - 1)/γ ]

1 - (p2/p1)2/γ (D24/D1

4)

1/2

(16.1)

·mac = Cv A2

2p1ρ1 (p2/p1)2/γ [γ/(γ - 1)] [1 - (p2/p1)(γ - 1)/γ ]

1 - (p2/p1)2/γ (D24/D1

4)

1/2

(16.2)

Y = √γγ - 1

[(p2/p1)2/γ - (p2/p1)(γ + 1)/γ](1 - D24/D1

4)[1 - (D2

4/D14)(p2/p1)2/γ](1 - p2/p1) (venturi meter) (16.3)

Y = 1 - (0.41 + 0.35β 4) (1 - p2/p1)

γ (orifice meter) (16.4)

roundedinlet

outlet ductaxial flowfan

motor

manometerconnections

pitot-statictube

venturi meter

FIGURE 16.4. Experimental setup for calibrating a venturi meter.

roundedinlet

outlet ductaxial flowfan

motor

orifice plate

manometerconnections

pitot-statictube

FIGURE 16.5. Experimental setup for calibrating an orifice meter.

44

EXPERIMENT 17

MEASUREMENT OF FAN HORSEPOWER

The objective of this experiment is to measureperformance characteristics of an axial flow fan,and display the results graphically.

Figure 17.1 shows a schematic of theapparatus used in this experiment. A DC motorrotates an axial flow fan which moves airthrough a duct. The sketch shows a venturi meterused in the outlet duct to measure flow rate.However, an orifice meter or a pitot-static tubecan be used instead. (See Experiment 16.) Thecontrol volume from section 1 to 2 includes all thefluid inside. The inlet is labeled as section 1, andhas an area (indicated by the dotted line) so hugethat the velocity at 1 is negligible compared tothe velocity at 2. The pressure at 1 equalsatmospheric pressure. The fan thus acceleratesthe flow from a velocity of 0 to a velocity weidentify as V2. The continuity equation is

m· 1 = m· 2

The energy equation is

0 = - dWd t

+ m· 1

h1 +

V 12

2 - m· 2

h2 +

V22

2

where dW/dt is the power input from the fan tothe air, which is what we are solving for. Bysubstituting the enthalpy terms according to thedefinition (h = u + pv), the preceding equationbecomes

d Wd t

= m· (u1 - u2)

+ m·

p1

ρ + V1

2

2 -

p2

ρ + V2

2

2

Assuming ideal gas behavior, we have

u1 - u2 = Cv(T1 - T2)

With a fan, however, we assume an isothermal

process, so that T1 ≈ T2 and ρ1 ≈ ρ2 = ρ. With m· =ρA V (evaluated at the outlet, section 2), theequation for power becomes

d Wd t

= A2V2

p1 +

ρV12

2 -

p2 +

ρV22

2

Recall that in this analysis, we set up our controlvolume so that the inlet velocity V1 = 0; actuallyV1 << V2. Thus

p1 +

ρV12

2 -

p2 +

ρV22

2 ≈

p1 - p2 -

ρV22

2

in which p1 is atmospheric pressure, and p2 ismeasured at section 2. The quantity in brackets inthe previous equation is the change in totalpressure ∆pt. Thus, the power is

d Wd t

= ∆ptQ (17.1)

This is the power imparted to the air from thefan.

Data AcquisitionThe motor controller is used to set the

rotational speed of the fan, which in turn controlsthe volume flow rate of air through the duct. Onthe side of the motor is a shaft (a torque arm)that extends outward normal to the axis ofrotation. When the fan rotates, the motor tends torotate in the opposite direction. A weight can be

roundedinlet

outlet ductaxial flowfan

motor

manometerconnections

venturi meter

1 2

FIGURE 17.1. Schematic of setup of fan horsepower experiment.

45

placed on the torque arm to reposition the motorto its balanced position. The product of weightand torque arm length gives the torque input frommotor to fan.

A tachometer is used to measure therotational speed of the motor. The product oftorque and rotational speed gives the power inputto the fan:

dWa

d t = Tω (17.2)

This is the power delivered to the fan from themotor.

The efficiency of the fan can now becalculated using Equations 1 and 2:

η = d W / d td W a/d t

(17.3)

Thus for one setting of the motor controller, thefollowing readings should be obtained:

1. An appropriate reading for the flow meter.2. Weight needed to balance the motor, and its

position on the torque arm.3. Rotational speed of the fan and motor.4. The static pressure at section 2.

With these data, the following parameterscan be calculated, again for each setting of themotor controller:1. Outlet velocity at section 2: V2 = Q/A2.2. The power using Equation 17.1.3. The input power using Equation 17.2.4. The efficiency using Equation 17.3.

Presentation of ResultsOn the horizontal axis, plot volume flow

rate. On the vertical axis, graph the power usingEquation 1, and Equation 2, both on the same set ofaxes. Also, again on the same set of axes, graphtotal pressure ∆pt as a function of flow rate. On aseparate graph, plot efficiency versus flow rate(horizontal axis).

46

EXPERIMENT 18

MEASUREMENT OF PUMP PERFORMANCE

The objective of this experiment is to performa test of a centrifugal pump and display theresults in the form of what is known as aperformance map.

Figure 18.1 is a schematic of the pump andpiping system used in this experiment. The pumpcontains an impeller within its housing. Theimpeller is attached to the shaft of the motorand the motor is mounted so that it is free torotate, within limits. As the motor rotates andthe impeller moves liquid through the pump, themotor housing tends to rotate in the oppositedirection from that of the impeller. A calibratedmeasurement system gives a readout of the torqueexerted by the motor on the impeller.

The rotational speed of the motor is obtainedwith a tachometer. The product of rotationalspeed and torque is the input power to theimpeller from the motor.

Gages in the inlet and outlet lines about thepump give the corresponding pressures in gagepressure units. The gages are located at knownheights from a reference plane.

After moving through the system, the wateris discharged into an open channel containing aV-notch weir. The weir is calibrated to providethe volume flow rate through the system.

The valve in the outlet line is used to controlthe volume flow rate. As far as the pump isconcerned, the resistance offered by the valvesimulates a piping system with a controllablefriction loss. Thus for any valve position, thefollowing data can be obtained: torque, rotationalspeed, inlet pressure, outlet pressure, and volumeflow rate. These parameters are summarized inTable 18.1.

TABLE 18.1. Pump testing parameters.

Raw Data

Parameter Symbol Dimensions

torque T F·L

rotational speed ω 1/T

inlet pressure p1 F/L2

outlet pressure p2 F/L2

volume flow rate Q L3/T

The parameters used to characterize thepump are calculated with the raw data obtainedfrom the test (listed above) and are as follows:input power to the pump, the total headdifference as outlet minus inlet, the powerimparted to the liquid, and the efficiency. Theseparameters are summarized in Table 18.2. Theseparameters must be expressed in a consistent set ofunits.

TABLE 18.2. Pump characterization parameters.

Reduced Data

Parameter Symbol Dimensions

input power dWa/dt F·L/T

total head diff ∆H L

power to liquid dW/dt F·L/T

efficiency η —

The raw data are manipulated to obtain thereduced data which in turn are used tocharacterize the performance of the pump. Theinput power to the pump from the motor is theproduct of torque and rotational speed:

- dWa

d t = Tω (18.1)

where the negative sign is added as a matter ofconvention. The total head at section 1, wherethe inlet pressure is measured (see Figure 18.1), isdefined as

H1 = p1

ρg +

V12

2g + z1

where ρ is the liquid density and V1 (= Q/A) isthe velocity in the inlet line. Similarly, thetotal head at position 2 where the outlet pressureis measured is

H2 = p2

ρg +

V22

2g + z2

The total head difference is given by

47

∆H = H2 - H1 = p2

ρg +

V22

2g + z2

-

p1

ρg +

V12

2g + z1

The dimension of the head H is L (ft or m). Thepower imparted to the liquid is calculated withthe steady flow energy equation applied fromsection 1 to 2:

- d Wd t

= m· g

p2

ρg +

V22

2g + z2

-

p1

ρg +

V12

2g + z1

In terms of total head H, we have

- d Wd t

= m· g (H2 - H1) = m· g ∆H (18.2)

The efficiency is determined with

η = d W / d td W a/d t

(18.3)

Experimental MethodThe experimental technique used in obtaining

data depends on the desired method of expressingperformance characteristics. For this experiment,data are taken on only one impeller-casing-motorcombination. One data point is first taken at acertain valve setting and at a preselectedrotational speed. The valve setting would then bechanged and the speed control on the motor (notshown in Figure 18.1) is adjusted if necessary sothat the rotational speed remains constant, andthe next set of data are obtained. This procedureis continued until 6 data points are obtained forone rotational speed.

Next, the rotational speed is changed andthe procedure is repeated. Four rotational speedsshould be used, and at least 6 data points perrotational speed should be obtained.

•sump tank

inlet

valve

valve

pumpmotor

control paneland gages

v-notch weir

return

1-1/2 nominalschedule 40PVC pipe

1 nominalschedule 40PVC pipe

pressuretap

pressuretap

•

z1

z2

motorshaft

FIGURE 18.1. Centrifugal pump testing setup.

48

Performance MapA performance map is to be drawn to

summarize the performance of the pump over itsoperating range. The performance map is a graphif the total head ∆H versus flow rate Q(horizontal axis). Four lines, corresponding to thefour pre-selected rotational speeds, would bedrawn. Each line has 6 data points, and theefficiency at each point is calculated. Lines ofequal efficiency are then drawn, and the resultinggraph is known as a performance map. Figure 18.2is an example of a performance map.

0 200 400 600 8000

10

20

30

40

Volume flow rate in gallons per minute

Tot

al h

ead

in ft

3600 rpm

1760

2700

900

85%

80%

75%

75%

65%

65%

Efficiency in %

FIGURE 18.2. Example of a performance map ofone impeller-casing-motor combinationobtained at four different rotational speeds.

Dimensionless GraphsTo illustrate the importance of

dimensionless parameters, it is prudent to use thedata obtained in this experiment and produce adimensionless graph.

A dimensional analysis can be performed forpumps to determine which dimensionless groupsare important. With regard to the flow of anincompressible fluid through a pump, we wish torelate three variables introduced thus far to theflow parameters. The three variables of interesthere are the efficiency η, the energy transfer rateg∆H, and the power dW/dt. These threeparameters are assumed to be functions of fluidproperties density ρ and viscosity µ, volume flowrate through the machine Q, rotational speed ω,and a characteristic dimension (usually impellerdiameter) D. We therefore write three functionaldependencies:

η = f1(ρ, µ, Q, ω, D )

g∆H = f2(ρ, µ, Q, ω, D)

d Wd t

= f3(ρ, µ, Q, ω, D)

Performing a dimensional analysis gives thefollowing results:

η = f1

ρωD2

µ , Q

ωD3

g∆Hω2D2 = f2

ρωD2

µ , Q

ωD3

d W / d t ρω3D5 = f3

ρωD2

µ , Q

ωD3

where

g∆Hω2D2 = energy transfer coefficient

QωD3 = volumetric flow coefficient

ρωD2

µ = rotational Reynolds number

d W / d t ρω3D5 = power coefficient

Experiments conducted with pumps show that therotational Reynolds number (ρωD2/µ) has asmaller effect on the dependent variables thandoes the flow coefficient. So for incompressibleflow through pumps, the preceding equationsreduce to

η ≈ f1

Q

ωD3 (18.4)

g∆Hω2D2 ≈ f2

Q

ωD3 (18.5)

d W / d t ρω3D5 ≈ f3

Q

ωD3 (18.6)

For this experiment, construct a graph ofefficiency, energy transfer coefficient, and powercoefficient all as functions of the volumetric flowcoefficient. Three different graphs can be drawn,or all lines can be placed on the same set of axes.

49

Specific SpeedA dimensionless group known as specific

speed can also be derived. Specific speed is foundby combining head coefficient and flowcoefficient in order to eliminate characteristiclength D:

ωss =

Q

ωD3

1/2

ω2D2

g∆H

3/4

or ωss = ωQ1/2

(g∆H)3/4 [dimensionless]

Exponents other than 1/2 and 3/4 could be used (toeliminate D), but 1/2 and 3/4 are customarilyselected for modeling pumps. Another definitionfor specific speed is given by

ωs = ωQ1/2

∆H3/4

rpm =

rpm(gpm)1/2

ft3/4

in which the rotational speed ω is expressed inrpm, volume flow rate Q is in gpm, total head ∆His in ft of liquid, and specific speed ωs isarbitrarily assigned the unit of rpm. The equationfor specific speed ωss is dimensionless whereasωs is not.

The specific speed of a pump can becalculated at any operating point, butcustomarily specific speed for a pump isdetermined only at its maximum efficiency. Forthe pump of this experiment, calculate itsspecific speed using both equations.

50

Appendix

Calibration Curves

Orifice plates—open channel flow apparatus .................................... 51

V-notch weir—turbomachinery experiments...................................... 52

51

0.01

0.1

1

0.01 0.1 1 10

volu

me

flow

rat

e in

ft3 /s

manometer deflection in ft of water

large orifice

small orifice

FIGURE A.1. Calibration curve for the open channel flow device.

52

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

volume flow rate in liters/min

heig

ht r

ead

ing

in m

m

FIGURE A.2. Calibration curve for the V-notch weir, turbomachinery experiments.