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- 1. A Manual for the MECHANICS of FLUIDS LABORATORY removable glass cover dye reservoir valve manifold with injectors dye water flow water inlet to drain plan view profile view grid on surface beneath glass manifold with injectors grid on surface beneath glass circular disk circular disk William S. Janna Department of Mechanical Engineering The University of Memphis
- 2. 2 ©2012 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.
- 3. 3 TABLE OF CONTENTS Item Page Course Learning Outcomes, Cleanliness and Safety................................................4 Code of Student Conduct ...............................................................................................5 Statistical Treatment of Experimental.........................................................................6 Report Writing...............................................................................................................16 Experiment 1 Density and Surface Tension...................................................18 Experiment 2 Viscosity.......................................................................................20 Experiment 3 Center of Pressure on a Submerged Plane Surface.............21 Experiment 4 Impact of a Jet of Water ............................................................23 Experiment 5 Critical Reynolds Number in Pipe Flow...............................26 Experiment 6 Fluid Meters................................................................................28 Experiment 7 Pipe Flow .....................................................................................32 Experiment 8 Pressure Distribution About a Circular Cylinder................34 Experiment 9 Drag Force Determination .......................................................37 Experiment 10 Analysis of an Airfoil................................................................38 Experiment 11 Open Channel Flow—Sluice Gate .........................................40 Experiment 12 Open Channel Flow Over a Weir ..........................................42 Experiment 13 Open Channel Flow—Hydraulic Jump ................................44 Experiment 14 Measurement of Pump Performance....................................46 Experiment 15 Measurement of Velocity and Calibration of a Meter for Compressible Flow.............................50 Experiment 16 Measurement of Fan Horsepower .........................................55 Experiment 17 External Laminar Flows Over Immersed Bodies................57 Experiment 18 Series-Parallel Pump Performance ........................................59 Experiment 19 Design of Experiments: Calibration of an Elbow Meter.....63 Experiment 20 Design of Experiments: Measurement of Force on a Conical Object ...........................................................65 Appendix .........................................................................................................................67
- 4. 4 Course Learning Outcomes The Fluid Mechanics Laboratory experiments are set up so that experiments can be performed to complement the theoretical information taught in the fluid mechanics lecture course. Thus topical areas have been identified and labeled as Course Learning Outcomes (CLOs). The CLOs in the MECH 3335 Laboratory are as follows: TABLE 1. Course Learning Outcomes 1. Identify safe operating practices and requirements for laboratory experiments 2. Measure fluid properties 3. Measure hydrostatic forces on a submerged body 4. Use flow meters to measure flow rate in a pipe 5. Measure pressure loss due to friction for pipe flow 6. Measure drag/lift forces on objects in a flow, or measure flow rate over a weir 7. Design and conduct an experiment, as well as analyze and interpret data 8. Function effectively as a member of a team Cleanliness There are “housekeeping” rules that the user of the laboratory should be aware of and abide by. Equipment in the lab is delicate and each piece is used extensively for 2 or 3 weeks per semester. During the remaining time, each apparatus just sits there, literally collecting dust. University housekeeping staff are not required to clean and maintain the equipment. Instead, there are college technicians who will work on the equipment when it needs repair, and when they are notified that a piece of equipment needs attention. It is important, however, that the equipment stay clean, so that dust will not accumulate too heavily. The Fluid Mechanics Laboratory contains equipment that uses water or air as the working fluid. In some cases, performing an experiment will inevitably allow water to get on the equipment and/or the floor. If no one cleaned up their working area after performing an experiment, the lab would not be a comfortable or safe place to work in. No student appreciates walking up to and working with a piece of equipment that another student or group of students has left in a mess. Consequently, students are required to clean up their area at the conclusion of the performance of an experiment. Cleanup will include removal of spilled water (or any liquid), and wiping the table top on which the equipment is mounted (if appropriate). The lab should always be as clean or cleaner than it was when you entered. Cleaning the lab is your responsibility as a user of the equipment. This is an act of courtesy that students who follow you will appreciate, and that you will appreciate when you work with the equipment. Safety The layout of the equipment and storage cabinets in the Fluid Mechanics Lab involves resolving a variety of conflicting problems. These include traffic flow, emergency facilities, environmental safeguards, exit door locations, unused equipment stored in the lab, etc. The goal is to implement safety requirements without impeding egress, but still allowing adequate work space and necessary informal communication opportunities. Distance between adjacent pieces of equipment is determined by locations of water supply valves, floor drains, electrical outlets, and by the need to allow enough space around the apparatus of interest. Immediate access to the Safety Cabinet and the Fire Extinguisher is also considered. We do not work with hazardous materials and safety facilities such as showers, eye wash fountains, spill kits, fire blankets, etc., are not necessary. Safety Procedures. There are five exit doors in this lab, two of which lead to other labs. One exit has a double door and leads directly to the hallway on the first floor of the Engineering Building. Another exit is a single door that also leads to the hallway. The fifth exit leads directly outside to the parking lot. In case of fire, the doors to the hallway should be closed, and the lab should be exited to the parking lot. There is a safety cabinet attached to the wall of the lab adjacent to the double doors. In case of personal injury, the appropriate item should be taken from the supply cabinet and used in the recommended fashion. If the injury is serious enough to require professional medical attention, the student(s) should contact the Civil Engineering Department in EN 104, Extension 2746. Every effort has been made to create a positive, clean, safety conscious atmosphere. Students are encouraged to handle equipment safely and to be aware of, and avoid being victims of, hazardous situations.
- 5. 5 THE CODE OF STUDENT CONDUCT Taken from The University of Memphis 1998–1999 Student Handbook Institution Policy Statement The University of Memphis students are citizens of the state, local, and national governments, and of the academic community. They are, therefore, expected to conduct themselves as law abiding members of each community at all times. Admission to the University carries with it special privileges and imposes special responsibilities apart from those rights and duties enjoyed by non-students. In recognition of this special relationship that exists between the institution and the academic community which it seeks to serve, the Tennessee Board of Regents has, as a matter of public record, instructed “the presidents of the universities and colleges under its jurisdiction to take such action as may be necessary to maintain campus conditions…and to preserve the integrity of the institution and its educational environment.” The following regulations (known as the Code of Student Conduct) have been developed by a committee made up of faculty, students, and staff utilizing input from all facets of the University Community in order to provide a secure and stimulating atmosphere in which individual and academic pursuits may flourish. Students are, however, subject to all national, state and local laws and ordinances. If a student’s violation of such laws or ordinances also adversely affects the University’s pursuit of its educational objectives, the University may enforce its own regulations regardless of any proceeding instituted by other authorities. Additionally, violations of any section of the Code may subject a student to disciplinary measures by the University whether or not such conduct is simultaneously violative of state, local or national laws. The term “academic misconduct” includes, but is not limited to, all acts of cheating and plagiarism. The term “cheating” includes, but is not limited to: a. use of any unauthorized assistance in taking quizzes, tests, or examinations; b. dependence upon the aid of sources beyond those authorized by the instructor in writing papers, preparing reports, solving problems, or carrying out other assignments; c. the acquisition, without permission, of tests or other academic material before such material is revealed or distributed by the instructor; d. the misrepresentation of papers, reports, assignments or other materials as the product of a student’s sole independent effort, for the purpose of affecting the student’s grade, credit, or status in the University; e. failing to abide by the instructions of the proctor concerning test-taking procedures; examples include, but are not limited to, talking, laughing, failure to take a seat assignment, failing to adhere to starting and stopping times, or other disruptive activity; f. influencing, or attempting to influence, any University official, faculty member, graduate student or employee possessing academic grading and/or evaluation authority or responsibility for maintenance of academic records, through the use of bribery, threats, or any other means or coercion in order to affect a student’s grade or evaluation; g. any forgery, alteration, unauthorized possession, or misuse of University documents pertaining to academic records, including, but not limited to, late or retroactive change of course application forms (otherwise known as “drop slips”) and late or retroactive withdrawal application forms. Alteration or misuse of University documents pertaining to academic records by means of computer resources or other equipment is also included within this definition of “cheating.” The term “plagiarism” includes, but is not limited to, the use, by paraphrase or direct quotation, of the published or unpublished work of another person without full or clear acknowledgment. It also includes the unacknowledged use of materials prepared by another person or agency engaged in the selling of term papers or other academic materials. Course Policy Academic misconduct (acts of cheating and of plagiarism) will not be tolerated. The Student Handbook is quite specific regarding the course of action to be taken by an instructor in cases where academic misconduct may be an issue.
- 6. 6 Statistical Treatment of Experimental Data Introduction This laboratory course concerns making measurements in various fluid situations and geometries, and relating results of those measurements to derived equations. The objective is to determine how well the derived equations describe the physical phenomena we are modeling. In doing so, we will need to make physical measurements, and it is essential that we learn how to practice good techniques in making scientific observations and in obtaining measurements. We are making quantitative estimates of physical phenomena under controlled conditions. Measurements There are certain primary desirable characteristics involved when making these physical measurements. We wish that our measurements would be: a ) Observer-independent, b) Consistent, and c) Quantitative So when reporting a measurements, we will be stating a number. Furthermore, we will have to add a dimension because a physical value without a unit has no significance. In reporting measurements, a question arises as to how should we report data; i.e., how many significant digits should we include? Which physical quantity is associated with the measurement, and how precise should it or could it be? It is prudent to scrutinize the claimed or implied accuracy of a measurement. Performing experiments In the course of performing an experiment, we first would develop a set of questions or a hypothesis, or put forth the theory. We then identify the system variables to be measured or controlled. The apparatus would have to be developed and the equipment set up in a particular way. An experimental protocol, or procedure, is established and data are taken. Several features of this process are important. We want accuracy in our measurements, but increased accuracy generally corresponds to an increase in cost. We want the experiments to be reproducible, and we seek to minimize errors. Of course we want to address all safety issues and regulations. After we run the experiment, and obtain data, we would analyze the results, draw conclusions, and report the results. Comments on Performing Experiments • Keep in mind the fundamental state of questions or hypotheses. • Make sure the experiment design will answer the right questions. • Use estimation as a reality check, but do not let it affect objectivity. • Consider all possible safety issues. • Design for repeatability and the appropriate level of accuracy. Error & Uncertainty—Definitions The fluid mechanics laboratory is designed to provide the students with experiments that verify the descriptive equations we derive to model physical phenomena. The laboratory experience involves making measurements of depth, area, and flow rate among other things. In the following paragraphs, we will examine our measurement methods and define terms that apply. These terms include error, uncertainty, accuracy, and precision. Error. The error E is the difference between a TRUE value, x, and a MEASURED value, xi: E x xi= − (1) There is no error-free measurement. All measurements contain some error. How error is defined and used is important. The significance of a measurement cannot be judged unless the associated error has been reliably estimated. In Equation 1, because the true value of x is unknown, the error E is unknown as well. This is always the case. The best we can hope for is to obtain the estimate of a likely error, which is called an uncertainty. For multiple measurements of the same quantity, a mean value, x, (also called a nominal value) can be calculated. Hence, the error becomes: E x x= − However, because x is unknown, E is still unknown.
- 7. 7 Uncertainty. The uncertainty, ∆x, is an estimate of E as a possible range of errors: ∆x E≈ (2) For example, suppose we measure a velocity and report the result as V = 110 m/s ± 5 m/s The value of ± 5 m/s is defined as the uncertainty. Alternatively, suppose we report the results as V = 110 m/s ± 4.5% The value of ± 4.5% is defined as the relative uncertainty. It is common to hear someone speak of “experimental errors,” when the correct terminology should be “uncertainty.” Both terms are used in everyday language, but it should be remembered that the uncertainty is defined as an estimate of errors. Accuracy. Accuracy is a measure (or an estimate) of the maximum deviation of measured values, xi, from the TRUE value, x: accuracy estimate of x xi= −max (3) Again, because the true value x is unknown, then the value of the maximum deviation is unknown. The accuracy, then, is only an estimate of the worst error. It is usually expressed as a percentage; e.g., “accurate to within 5%.” Accuracy and Precision. As mentioned, accuracy is a measure (or an estimate) of the maximum deviation of measured values from the true value. So a question like: “Are the measured values accurate?” can be reformulated as “Are the measured values close to the true value?” Accuracy was defined in Equation 3 as accuracy estimate of x xi= −max (3) Precision, on the other hand, is a measure (or an estimate) of the consistency (or repeatability). Thus it is the maximum deviation of a reading (measurement), xi, from its mean value, x : precision estimate of x xi= −max (4) Note the difference between accuracy and precision. Regarding the definition of precision, there is no true value identified, only the mean value (or average) of a number of repeated measurements of the same quantity. Precision is a characteristic of the measurement. In everyday language we often conclude that “accuracy” and “precision” are the same, but in error analysis there is a difference. So a question like: “Are the measured values precise?” can be reformulated as “Are the measured values close to each other?” As an illustration of the concepts of accuracy and precision, consider the dart board shown in the accompanying figures. Let us assume that the blue darts show the measurements taken, and that the bullseye represents the value to be measured. When all measurements are clustered about the bullseye, then we have very accurate and, therefore, precise results (Figure 1a). When all measurements are clustered together but not near the bullseye, then we have very precise but not accurate results (Figure 1b). When all measurements are not clustered together and not near the bullseye, but their nominal value or average is the bullseye, then we have accurate (on average) but not precise results (Figure 1c). When all measurements are not clustered together and not near the bullseye, and their average is the not at the bullseye, then we have neither accurate nor precise results (Figure 1d). We conclude that accuracy refers to the correctness of the measurements, while precision refers to their consistency. Classification of Errors Random error. A random error is one that arises from a random source. Suppose for example that a measurement is made many thousands of times using different instruments and/or observers and/or samples. We would expect to have random errors affecting the measurement in either direction (±) roughly the same number of times. Such errors can occur in any scenario: • Electrical noise in a circuit generally produces a voltage error that may be positive or negative by a small amount.
- 8. 8 1 18 4 13 6 10 15 2 173 7 16 8 11 14 9 12 5 FIGURE 1a. Accurate and Precise 1 18 4 13 6 10 15 2 173 7 16 8 11 14 9 12 5 FIGURE 1b. Precise but not Accurate. 1 18 4 13 6 10 15 2 173 7 16 8 11 14 9 12 5 FIGURE 1c. Precise but not Accurate. 1 18 4 13 6 10 15 2 173 7 16 8 11 14 9 12 5 FIGURE 1d. Neither Precise nor Accurate. • By counting the total number of pennies in a large container, one may occasionally pick up two and count only one (or vice versa). The question arises as to how can we reduce random errors? There are no random error free measurements. So random errors cannot be eliminated, but their magnitude can be reduced. On average, random errors tend to cancel out. Systematic Error. A systematic error is one that is consistent; that is, it happens systematically. Typically, human components of measurement systems are often responsible for systematic errors. For example, systematic errors are common in reading of a pressure indicated by an inclined manometer. Consider an experiment involving dropping a ball from a given height. We wish to measure the time it takes for the ball to move from where it is dropped to when it hits the ground. We might repeat this experiment several times. However, the person using the stopwatch may consistently have a tendency to wait until the ball bounces before the watch is stopped. As a result, the time measurement might be systematically too long. Systematic measurements can be anticipated and/or measured, and then corrected. This can be done even after the measurements are made. The question arises as to how can we reduce systematic errors? This can be done in several ways: 1. Calibrate the instruments being used by checking with a known standard. The standard can be what is referred to as: a) a primary standard obtained from the “National Institute of standards and technology” (NIST— formerly the National Bureau of Standards); or b) a secondary standard (with a higher accuracy instrument); or c) A known input source. 2. Make several measurements of a certain quantity under varying test conditions, such as different observers and/or samples and/or instruments. 3. Check the apparatus. 4. Check the effects of external conditions 5. Check the coherence of results. A repeatability test using the same instrument is one way of gaining confidence, but a far more
- 9. 9 reliable way is to use an entirely different method to measure the desired quantity. Uncertainty Analysis Determining Uncertainty. When we state a measurement that we have taken, we should also state an estimate of the error, or the uncertainty. As a rule of thumb, we use a 95% relative uncertainty, or stated otherwise, we use a 95% confidence interval. Suppose for example, that we report the height of a desk to be 38 inches ± 1 inch. This suggests that we are 95% sure that the desk is between 37 and 39 inches tall. When reporting relative uncertainty, we generally restrict the result to having one or two significant figures. When reporting uncertainty in a measurement using units, we use the same number of significant figures as the measured value. Examples are shown in Table 1: TABLE 1. Examples of relative and absolute uncertainty. Relative uncertainty Uncertainty in units 3.45 cm ± 8.5% 5.23 cm ± 0.143 cm 6.4 N ± 2.0% 2.5 m/s ± 0.082 m/s 2.3 psi ± 0.1900% 9.25 in ± 0.2 in 9.2 m/s ± 8.598% 3.2 N ± 0.1873 N The previous tables shows uncertainty in measurements, but to determine uncertainty is usually difficult. However, because we are using a 95% confidence interval, we can obtain an estimage. The estimate of uncertainty depends on the measurement type: single sample measurements, measurements of dependent variables, or multi variable measurements. Single-sample measurements. Single-sample measurements are those in which the uncertainties cannot be reduced by repetition. As long as the test conditions are the same (i.e., same sample, same instrument and same observer), the measurements (for fixed variables) are single- sample measurements, regardless of how many times the reading is repeated. Single-sample uncertainty. It is often simple to identify the uncertainty of an individual measurement. It is necessary to consider the limit of the “scale readability,” and the limit associated with applying the measurement tool to the case of interest. Measurement Of Function Of More Than One Independent Variables. In many cases, several different quantities are measured in order to calculate another quantity—a dependent variable. For example, the measurement of the surface area of a rectangle is calculated using both its measured length and its measured width. Such a situation involves a propagation of uncertainties. Consider some measuring device that has as its smallest scale division δx. The smallest scale division limits our ability to measure something with any more accuracy than δx/2. The ruler of Figure 2a, as an example, has 1/4 inch as its smallest scale division. The diameter of the circle is between 4 and 4 1/4 inches. So we would correctly report that D = 41/8 ± 1/8 in. This is the correct reported measurement for Figure 2a and Figure 2b, even though the circles are of different diameters. We can “guesstimate” the correct measurement, but we cannot report something more accurately than our measuring apparatus will display. This does not mean that the two circles have the same diameter, merely that we cannot measure the diameters with a greater accuracy than the ruler we use will allow. 0 1 2 3 4 5 6 (a) 0 1 2 3 4 5 6 (b) FIGURE 2. A ruler used to measure the diameter of a circle. The ruler depicted in the figure could be any arbitrary instrument with finite resolution. The uncertainty due to the resolution of any instrument is one half of the smallest increment
- 10. 10 displayed. This is the most likely single sample uncertainty. It is also the most optimistic because reporting this values assumes that all other sources of uncertainty have been removed. Multi-Sample Measurements. Multi-sample measurements involve a significant number of data points collected from enough experiments so that the reliability of the results can be assured by a statistical analysis. In other words, the measurement of a significant number of data points of the same quantity (for fixed system variables) under varying test conditions (i.e., different samples and/or different instruments) will allow the uncertainties to be reduced by the sheer number of observations. Uncertainty In Measurement of a Function of Independent Variables. The concern in this measurement is in the propagation of uncertainties. In most experiments, several quantities are measured in order to calculate a desired quantity. For example, to estimate the gravitational constant by dropping a ball from a known height, the approximate equation would be: g L t = 2 2 (5) Now suppose we measured: L = 50.00 ± 0.01 m and t = 3.1 ± 0.5 s. Based on the equation, we have: g L t = = × = 2 2 50 00 3 1 10 42 2 2. . . m/s We now wish to estimate the uncertainty ∆g in our calculation of g. Obviously, the uncertainty ∆g will depend on the uncertainties in the measurements of L and t. Let us examine the “worst cases.” These may be calculated as: gmin . . .= × = 2 49 99 3 6 7 72 2m/s and gmax . . .= × = 2 50 01 2 6 14 82 2m/s The confidence interval around g then is: 7 7 14 82 2 . .m/s m/s≤ ≤g (6) Now it is unlikely for all single-sample uncertainties in a system to simultaneously be the worst possible. Some average or “norm” of the uncertainties must instead be used in estimating a combined uncertainty for the calculation of g. Uncertainty In Multi-Sample Measurements. When a set of readings is taken in which the values vary slightly from each other, the experimenter is usually concerned with the mean of all readings. If each reading is denoted by xi and there are n readings, then the arithmetic mean value is given by: x x n i i n = ∑ =1 (7) Deviation. The deviation of each reading is defined by: d x xi i= − (8) The arithmetic mean deviation is defined as: d n di i n = ∑ = = 1 0 1 Note that the arithmetic mean deviation is zero: Standard Deviation. The standard deviation is given by: σ = −∑ − = ( )x x n i i n 2 1 1 (9) Due to random errors, experimental data is dispersed in what is referred to as a bell distribution, known also as a Gaussian or Normal Distribution, and depicted in Figure 3. xi f(xi ) FIGURE 3. Gaussian or Normal Distribution. The Gaussian or Normal Distribution is what we use to describe the distribution followed by random errors. A graph of this distribution is
- 11. 11 often referred to as the “bell” curve as it looks like the outline of a bell. The peak of the distribution occurs at the mean of the random variable, and the standard deviation is a common measure for how “fat” this bell curve is. Equation 10 is called the Probability Density Function for any continuous random variable x. f x e x x ( ) ( ) = − − 1 2 2 22 σ π σ (10) The mean and the standard deviation are all the information necessary to completely describe any normally-distributed random variable. Integrating under the curve of Figure 3 over various limits gives some interesting results. • Integrating under the curve of the normal distribution from negative to positive infinity, the area is 1.0 (i.e., 100 %). Thus the probability for a reading to fall in the range of ±∞ is 100%. • Integrating over a range within ± σ from the mean value, the resulting value is 0.6826. The probability for a reading to fall in the range of ± σ is about 68%. • Integrating over a range within ± 2σ from the mean value, the resulting value is 0.954. The probability for a reading to fall in the range of ± 2σ is about 95%. • Integrating over a range within ± 3σ from the mean value, the resulting value is 0.997. The probability for a reading to fall in the range of ± 3σ is about 99%. TABLE 2. Probability for Gaussian Distribution (tabulated in any statistics book) Probability ± value of the mean 50% 0.6754σ 68.3% σ 86.6% 1.5σ 95.4% 2σ 99.7% 3σ Estimating Uncertainty. We can now use the probability function to help in determining the accuracy of data obtained in an experiment. We use the uncertainty level of 95%, which means that we have a 95% confidence interval. In other words, if we state that the uncertainty is ∆x, we suggest that we are 95% sure that any reading xi will be within the range of ± ∆x of the mean. Thus, the probability of a sample chosen at random of being within the range ± 2σ of the mean is about 95%. Uncertainty then is defined as twice the standard deviation: ∆x ≈ 2σ Example 1. The manufacturer of a particular alloy claims a modulus of elasticity of 40 ± 2 kPa. How is that to be interpreted? Solution: The general rule of thumb is that ± 2 kPa would represent a 95% confidence interval. That is, if we randomly select many samples of this manufacturer’s alloy we should find that 95% of the samples meet the stated limit of 40 ± 2 kPa. Now it is possible that we can find a sample that has a modulus of elasticity of 37 kPa; however, it means that it is very unlikely. Example 2 If we assume that variations in the product follow a normal distribution, and that the modulus of elasticity is within the range 40 ± 2 kPa, then what is the standard deviation, σ? Solution: The uncertainty ≈ 95% of confidence interval ≈ 2σ. Thus ± 2 kPa = ± 2σ So σ = 1 kPa Example 3. Assuming that the modulus of elasticity is 40 ± 2 kPa, estimate the probability of finding a sample from this population with a modulus of elasticity less than or equal to 37 kPa. Solution: With σ = 1 kPa, we are seeking the value of the integral under the bell shaped curve, over the range of -∞ to – 3σ. Thus, the probability that the modulus of elasticity is less than 37 kPa is: P(E < 37 kPa) = 100 - 99.7 2 = 0.15% Statistically Based Rejection of “Bad” Data– Chauvenet’s Criterion Occasionally, when a sample of n measurements of a variable is obtained, there may be one or more that appear to differ markedly from the others. If some extraneous
- 12. 12 influence or mistake in experimental technique can be identified, these “bad data” or “wild points” can simply be discarded. More difficult is the common situation in which no explanation is readily available. In such situations, the experimenter may be tempted to discard the values on the basis that something must surely have gone wrong. However, this temptation must be resisted, since such data may be significant either in terms of the phenomena being studied or in detecting flaws in the experimental technique. On the other hand, one does not want an erroneous value to bias the results. In this case, a statistical criterion must be used to identify points that can be considered for rejection. There is no other justifiable method to “throw away” data points. One method that has gained wide acceptance is Chauvenet’s criterion; this technique defines an acceptable scatter, in a statistical sense, around the mean value from a given sample of n measurements. The criterion states that all data points should be retained that fall within a band around the mean that corresponds to a probability of 1-1/(2n). In other words, data points can be considered for rejection only if the probability of obtaining their deviation from the mean is less than 1/(2n). This is illustrated in Figure 4. xi f(xi ) Probability 1 - 1/(2n) Reject data Reject data FIGURE 4. Rejection of “bad” data. The probability 1-1/(2n) for retention of data distributed about the mean can be related to a maximum deviation dmax away from the mean by using a Gaussian probability table. For the given probability, the non dimensional maximum deviation τmax can be determined from the table, where τmax = |(xi – –x )|max sx = dmax sx and sx is the precision index of the sample. All measurements that deviate from the mean by more than dmax/sx can be rejected. A new mean value and a new precision index can then be calculated from the remaining measurements. No further application of the criterion to the sample is allowed. Using Chauvenet’s criterion, we say that the values xi which are outside of the range x C± σ (11) are clearly errors and should be discarded for the analysis. Such values are called outliers. The constant C may be obtained from Table 3. Note that Chauvenet’s criterion may be applied only once to a given sample of readings. The methodology for identifying and discarding outlier(s) is a follows: 1. After running an experiment, sort the outcomes from lowest to highest value. The suspect outliers will then be at the top and/or the bottom of the list. 2. Calculate the mean value and the standard deviation. 3. Using Chauvenet’s criterion, discard outliers. 4. Recalculate the mean value and the standard deviation of the smaller sample and stop. Do not repeat the process; Chauvenet’s criterion may be applied only once. TABLE 3. Constants to use in Chauvenet’s criterion, Equation 11. Number, n dmax sx = C 3 1.38 4 1.54 5 1.65 6 1.73 7 1.80 8 1.87 9 1.91 10 1.96 15 2.13 20 2.24 25 2.33 50 2.57 100 2.81 300 3.14 500 3.29 1,000 3.48 Example 4. Consider an experiment in which we measure the mass of ten individual “identical” objects. The scale readings (in grams) are as shown in Table 4.
- 13. 13 By visual examination of the results, we might conclude that the 4.85 g reading is too high compared to the others, and so it represents an error in the measurement. We might tend to disregard it. However, what if the reading was 2.50 or 2.51 g? We use Chauvenet’s criterion to determine if any of the readings can be discarded. TABLE 4. Data obtained in a series of experiments. Number, n reading in g 1 2.41 2 2.42 3 2.43 4 2.43 5 2.44 6 2.44 7 2.45 8 2.46 9 2.47 10 4.85 We apply the methodology described earlier. The results of the calculations are shown in Table 5: 1. Values in the table are already sorted. Column 1 shows the reading number, and there are 10 readings of mass, as indicated in column 2. 2. We calculate the mean and standard deviation. The data in column 2 are added to obtain a total of 26.8. Dividing this value by 10 readings gives 2.68, which is the mean value of all the readings: m– = 2.68 g In column 3, we show the square of the difference between each reading and the mean value. Thus in row 1, we calculate (x– – x1)2 = (2.68 – 2.41)2 = 0.0729 We repeat this calculation for every data point. We then add these to obtain the value 5.235 shown in the second to last row of column 3. This value is then divided by (n –1) = 9 data points, and the square root is taken. The result is 0.763, which is the standard deviation, as defined earlier in Equation 9: σ = −∑ − = ( )x x n i i n 2 1 1 = 0.763 (9) 3. Next, we apply Chauvenet’s criterion; for 10 data points, n = 10 and Table 3 reads C = 1.96. We calculate Cσ = 1.96(0.763) = 1.50. The range of “acceptable” values then is 2.68 ± 1.50, or: m– – Cσ ≤ mi ≤ m– + Cσ 1.18 g ≤ m– ≤ 4.18 g Any values outside the range of 1.18 and 4.18 are outliers and should be discarded. 4. Thus for the data of the example, the 4.85 value is an outlier and may be discarded. All other points are valid. The last two columns show the results of calculations made without data point #10. The mean becomes 2.44, and the standard deviation is 0.019 (compare to 2.68, and 0.763, respectively).
- 14. 14 TABLE 5. Calculations summary for the data of Table 4. Number, n reading in g (x– – xi)2 remove #10 (x– – xi)2 1 2.41 0.0729 2.41 0.000835 2 2.42 0.0676 2.42 0.000357 3 2.43 0.0625 2.43 0.000079 4 2.43 0.0625 2.43 0.000079 5 2.44 0.0576 2.44 0.000001 6 2.44 0.0576 2.44 0.000001 7 2.45 0.0529 2.45 0.000123 8 2.46 0.0484 2.46 0.000446 9 2.47 0.0441 2.47 0.000968 10 4.85 4.7089 ∑= 26.8 5.235 21.95 0.002889 2.68 0.763 2.44 0.019 f(∂T,∂x
- 15. 16 REPORT WRITING All reports in the Fluid Mechanics Laboratory require a formal laboratory report unless specified otherwise. The report should be written in such a way that anyone can duplicate the performed experiment and find the same results as the originator. The reports should be simple and clearly written. Reports are due one week after the experiment was performed, unless specified otherwise. The report should communicate several ideas to the reader. First the report should be neatly done. The experimenter is in effect trying to convince the reader that the experiment was performed in a straightforward manner with great care and with full attention to detail. A poorly written report might instead lead the reader to think that just as little care went into performing the experiment. Second, the report should be well organized. The reader should be able to easily follow each step discussed in the text. Third, the report should contain accurate results. This will require checking and rechecking the calculations until accuracy can be guaranteed. Fourth, the report should be free of spelling and grammatical errors. The following format, shown in Figure R.1, is to be used for formal Laboratory Reports: Title Page–The title page should show the title and number of the experiment, the date the experiment was performed, experimenter's name and experimenter's partners' names, all spelled correctly. Table of Contents –Each page of the report must be numbered for this section. Object –The object is a clear concise statement explaining the purpose of the experiment. This is one of the most important parts of the laboratory report because everything included in the report must somehow relate to the stated object. The object can be as short as one sentence. Theory –The theory section should contain a complete analytical development of all important equations pertinent to the experiment, and how these equations are used in the reduction of data. The theory section should be written textbook-style. Procedure – The procedure section should contain a schematic drawing of the experimental setup including all equipment used in a parts list with manufacturer serial numbers, if any. Show the function of each part when necessary for clarity. Outline exactly step- Bibliography Calibration Curves Original Data Sheet (Sample Calculation) Appendix Title Page Discussion & Conclusion (Interpretation) Results (Tables and Graphs) Procedure (Drawings and Instructions) Theory (Textbook Style) Object (Past Tense) Table of Contents Each page numbered Experiment Number Experiment Title Your Name Due Date Partners’ Names FIGURE R.1. Format for formal reports. by-step how the experiment was performed in case someone desires to duplicate it. If it cannot be duplicated, the experiment shows nothing. Results – The results section should contain a formal analysis of the data with tables, graphs, etc. Any presentation of data which serves the purpose of clearly showing the outcome of the experiment is sufficient. Discussion and Conclusion – This section should give an interpretation of the results explaining how the object of the experiment was accomplished. If any analytical expression is to be verified, calculate % error† and account for the sources. Discuss this †% error–An analysis expressing how favorably the empirical data approximate theoretical information. There are many ways to find % error, but one method is introduced here for consistency. Take the difference between the empirical and theoretical results and divide by the theoretical result. Multiplying by 100% gives the % error. You may compose your own error analysis as long as your method is clearly defined.
- 16. 16 experiment with respect to its faults as well as its strong points. Suggest extensions of the experiment and improvements. Also recommend any changes necessary to better accomplish the object. Each experiment write-up contains a number of questions. These are to be answered or discussed in the Discussion and Conclusions section. Appendix (1) Original data sheet. (2) Show how data were used by a sample calculation. (3) Calibration curves of instrument which were used in the performance of the experiment. Include manufacturer of the instrument, model and serial numbers. Calibration curves will usually be supplied by the instructor. (4) Bibliography listing all references used. Short Form Report Format Often the experiment requires not a formal report but an informal report. An informal report includes the Title Page, Object, Procedure, Results, and Conclusions. Other portions may be added at the discretion of the instructor or the writer. Another alternative report form consists of a Title Page, an Introduction (made up of shortened versions of Object, Theory, and Procedure) Results, and Conclusion and Discussion. This form might be used when a detailed theory section would be too long. Graphs In many instances, it is necessary to compose a plot in order to graphically present the results. Graphs must be drawn neatly following a specific format. Figure R.2 shows an acceptable graph prepared using a computer. There are many computer programs that have graphing capabilities. Nevertheless an acceptably drawn graph has several features of note. These features are 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 denote data points obtained by experiment. • The line representing the theoretical results 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 Q th Q ac Q ∆ hhead loss in m flowrateinm 3 /s FIGURE R.2. Theoretical and actual volume flow rate through a venturi meter as a function of head loss.
- 17. 17 EXPERIMENT 1 FLUID PROPERTIES: DENSITY AND SURFACE TENSION There are several properties simple Newtonian fluids have. They are basic properties which cannot be calculated for every fluid, and therefore they must be measured. These properties are important in making calculations regarding fluid systems. Measuring fluid properties, density and surface tension, is the object of this experiment. Part I: Density Measurement. Equipment Graduated cylinder or beaker Liquid whose properties are to be measured Hydrometer cylinder Scale Method 1. The density of the test fluid is to be found by weighing a known volume of the liquid using the graduated cylinder or beaker and the scale. The beaker is weighed empty. The beaker is then filled to a certain volume according to the graduations on it and weighed again. The difference in weight divided by the volume gives the weight per unit volume of the liquid. By appropriate conversion, the liquid density is calculated. The mass per unit volume, or the density, is thus measured in a direct way. Method 2. A second method of finding density involves measuring buoyant force exerted on a submerged object. The difference between the weight of an object in air and the weight of the object in liquid is known as the buoyant force (see Figure 1.1). W1 W2 FIGURE 1.1. Measuring the buoyant force on an object with a hanging weight. Referring to Figure 1.1, the buoyant force B is found as B = W1 - W2 The buoyant force is equal to the difference between the weight of the object in air and the weight of the object while submerged. Dividing this difference by the volume displaced gives the weight per unit volume from which density can be calculated. Method 3. A third method of making a density measurement involves the use of a calibrated hydrometer cylinder. The cylinder is submerged in the liquid and the density is read directly on the calibrated portion of the cylinder itself. Experiment Measure density using the methods assigned by the instructor. Compare results of all measurements. Questions 1. Are the results of all the density measurements in agreement? 2. How does the buoyant force vary with depth of the submerged object? Why? 3. In your opinion, which method yielded the “most accurate” results? 4. Are the results precise? 5. What is the mean of the values you obtained? 6. What is the standard deviation of the results? 7. Using Chauvenent’s rule, can any of the measurements be discarded?
- 18. 18 Part II: Surface Tension Measurement Equipment Surface tension meter Beaker Test fluid Surface tension is defined as the energy required to pull molecules of liquid from beneath the surface to the surface to form a new area. It is therefore an energy per unit area (F⋅L/L2 = F/L). A surface tension meter is used to measure this energy per unit area and give its value directly. A schematic of the surface tension meter is given in Figure 1.2. The platinum-iridium ring is attached to a balance rod (lever arm) which in turn is attached to a stainless steel torsion wire. One end of this wire is fixed and the other is rotated. As the wire is placed under torsion, the rod lifts the ring slowly out of the liquid. The proper technique is to lower the test fluid container as the ring is lifted so that the ring remains horizontal. The force required to break the ring free from the liquid surface is related to the surface tension of the liquid. As the ring breaks free, the gage at the front of the meter reads directly in the units indicated (dynes/cm) for the given ring. This reading is called the apparent surface tension and must be corrected for the ring used in order to obtain the actual surface tension for the liquid. The correction factor F can be calculated with the following equation F = 0.725 + √0.000 403 3(σa/ρ) + 0.045 34 - 1.679(r/R) where F is the correction factor, σa is the apparent 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 ring container. The actual surface tension for the liquid is given by σ = Fσa Experiment Measure the surface tension of the liquid assigned. Each member of your group should make a measurement to become familiar with the apparatus. Are all measurements in agreement? FIGURE 1.2. A schematic of the surface tension meter. torsion wire test liquid platinum iridium ring clamp balance rod
- 19. 19 EXPERIMENT 2 FLUID PROPERTIES: VISCOSITY One of the properties of homogeneous liquids is their resistance to motion. A measure of this resistance is known as viscosity. It can be measured in different, standardized methods or tests. In this experiment, viscosity will be measured with a falling sphere viscometer. The Falling Sphere Viscometer When an object falls through a fluid medium, the object reaches a constant final speed or terminal velocity. If this terminal velocity is sufficiently low, then the various forces acting on the object can be described with exact expressions. The forces acting on a sphere, for example, that is falling at terminal velocity through a liquid are: Weight - Buoyancy - Drag = 0 ρsg 4 3 πR3 - ρg 4 3 πR3 - 6πµVR = 0 where ρs and ρ are density of the sphere and liquid respectively, V is the sphere’s terminal velocity, R is the radius of the sphere and µ is the viscosity of the liquid. In solving the preceding equation, the viscosity of the liquid can be determined. The above expression for drag is valid only if the following equation is valid: ρVD µ < 1 where D is the sphere diameter. Once the viscosity of the liquid is found, the above ratio should be calculated to be certain that the mathematical model gives an accurate description of a sphere falling through the liquid. Equipment Cylinder filled with test liquid Scale Stopwatch Several small spheres with weight and diameter to be measured Drop a sphere into the cylinder liquid and record the time it takes for the sphere to fall a certain measured distance. The distance divided by the measured time gives the terminal velocity of the sphere. Repeat the measurement and average the results. With the terminal velocity of this and of other spheres measured and known, the absolute and kinematic viscosity of the liquid can be calculated. The temperature of the test liquid should also be recorded. Use at least three different spheres. (Note that if the density of the liquid is unknown, it can be obtained from any group who has completed or is taking data on Experiment 1.) d V FIGURE 2.1. Terminal velocity measurement (V = d/time). Questions 1. 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. What are the shortcomings of this method? 5. Why should temperature be recorded? 6. Can this method be used for gases? 7. Can this method be used for opaque liquids? 8. Can this method be used for something like peanut butter, or grease or flour dough? Why or why not? 9. Perform an error analysis for one of the data points. That is, determine the error associated with all the measurements, and provide an error band about the mean value.
- 20. 20 EXPERIMENT 3 CENTER OF PRESSURE ON A SUBMERGED PLANE SURFACE Submerged surfaces are found in many engineering applications. Dams, weirs and water gates are familiar examples of submerged plane surfaces. It is important to have a working knowledge of the forces that act on submerged surfaces. A plane surface located beneath the surface of a liquid is subjected to a pressure due to the height of liquid above it, as shown in Figure 3.1. Pressure increases linearly with increasing depth resulting in a pressure distribution that acts on the submerged surface. The analysis of this situation involves determining a force which is equivalent to the pressure, and finding the line of action of this force. F yF FIGURE 3.1. Pressure distribution on a submerged plane surface and the equivalent force. For this case, it can be shown that the equivalent force is: F = ρgycA (3.1) in which ρ is the liquid density, yc is the distance from the free surface of the liquid to the centroid of the plane, and A is the area of the plane in contact with liquid. Further, the location of this force yF below the free surface is yF = Ix x ycA + yc (3.2) in which Ixx is the second area moment of the plane about its centroid. The experimental verification of these equations for force and distance is the subject of this experiment. Figure 3.2a is a sketch of an apparatus that we use to illustrate the concepts behind this experiment. The apparatus consists of one-fourth of a torus, consisting of a solid piece of material. The torus is attached to a lever arm, which is free to rotate (within limits) about a pivot point. The torus has inside and outside radii, Ri and Ro respectively, and it is constructed such that the center of these radii is at the pivot point of the lever arm. The torus is now submerged in a liquid, and there will exist an unbalanced force F that is exerted on the plane of dimensions h x w. In order to bring the torus and lever arm back to their balanced position, a weight Wmust be added to the weight hanger. The force and its line of action can be found with Equations 3.1 and 3.2. Consider next the apparatus sketched in Figure 3.2b. It is quite similar to that in Figure 3.2a, in that it consists of a torus attached to a lever arm. In this case, however, the torus is hollow, and can be filled with liquid. If the depth of the liquid is equal to that in Figure 3.2a, (as measured from the bottom of the torus), then the forces in both cases will be equal in magnitude but opposite in direction. Moreover, the distance from the free surface of the liquid to the line of action of both forces will also be equal. Thus, there is an equivalence between the two devices. Center of Pressure Measurement Equipment Center of Pressure Apparatus (Figure 3.2b) Weights The torus and balance arm are located on a pivot rod. Note that the pivot point for the balance arm is the point of contact between the rod and the torus. Place the weight hanger on the apparatus, and add water into the trim tank (not shown in the figure) to bring the submerged plane back to the vertical position. Start by adding 20 g to the weight hanger. Then pour water into the torus until the submerged plan is brought back to the vertical position. Record the weight and the liquid depth. Repeat this procedure for 4 more weights. (Remember to record the distance from the pivot point to the free surface for each case.) From the depth measurement, the equivalent force and its location are calculated using Equations 3.1 and 3.2. Summing moments about the pivot allows for a comparison between the theoretical and actual force exerted. Referring to Figure 3.2b, we have
- 21. 21 F = W L (y + yF) (3.3) where y is the distance from the pivot point to the free surface, yF is the distance from the free surface to the line of action of the force F, and L is the distance from the pivot point to the line of action of the weight W. Recalling that both curved surfaces of the torus are circular with centers at the pivot point, we see that the forces acting on the curved surfaces have a zero moment arm. For the report, compare the force obtained with Equation 3.1 to that obtained with Equation 3.3. When using Equation 3.3, it will be necessary to use Equation 3.2 for yF. Questions 1. In summing moments, why isn't the buoyant force taken into account in Figure 3.2a? 2. Why isn’t the weight of the torus and the balance arm taken into account? weight hanger L Ri F y h w yF Ro torus FIGURE 3.2a weight hanger L F y h w yF Ri Ro torus FIGURE 3.2b. A schematic of the center of pressure apparatus.
- 22. 22 EXPERIMENT 4 IMPACT OF A JET OF WATER A jet of fluid striking a stationary object exerts a force on that object. This force can be measured when the object is connected to a spring balance or scale. The force can then be related to the velocity of the jet of fluid and in turn to the rate of flow. The force developed by a jet stream of water is the subject of this experiment. Impact of a Jet of Liquid Equipment Jet Impact Apparatus Object plates Figure 4.1 is a schematic of the device used in this experiment. The device consists of a catch basin within a sump tank. A pump moves water from the sump tank to the impact apparatus, after which the water drains to the catch basin. The plug is used to allow water to accumulate in the catch basin. On the side of the sump tank is a sight glass (not shown in Figure 4.1) showing the water depth in the catch basin. When flow rate is to be measured, water is allowed to accumulate in the catch basin, and a stopwatch is used to measure the time required for the water volume to reach a pre-determined volume, using the sight glass as an indicator. In other words, we use the stopwatch to measure the time required for a certain volume of water to accumulate in the catch basin. The sump tank acts as a support for the table top which supports the impact apparatus. As shown in Figure 4.1, the impact apparatus contains a nozzle that produces a high velocity jet of water. The jet is aimed at an object (such as a flat plate or hemisphere). The force exerted on the plate causes the balance arm to which the plate is attached to deflect. A weight is moved on the arm until the arm balances. A summation of moments about the pivot point of the arm allows for calculating the force exerted by the jet. Water is fed through the nozzle by means of a pump. The nozzle emits the water in a jet stream whose diameter is constant. After the water strikes the object, the water is channeled to the catch basin to obtain the volume flow rate. The variables involved in this experiment are listed and their measurements are described below: 1. Volume rate of flow–measured with the catch basin (to obtain volume) and a stopwatch (to obtain time). The volume flow rate is obtained by dividing volume by time: Q = V/t. 2. Velocity of jet–obtained by dividing volume flow rate by jet area: V = Q/A. The jet is cylindrical in shape. 3. Resultant force—found experimentally by summation of moments about the pivot point of the balance arm. The theoretical resultant force is found by use of an equation derived by applying the momentum equation to a control volume about the plate. Impact Force Analysis (Theoretical Force) The total force exerted by the jet equals the rate of momentum loss experienced by the jet after it impacts the object. For a flat plate, the force equation 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) These equations are easily derivable from the momentum equation applied to a control volume about the object.
- 23. 23 flat plate pivot balancing weight lever arm with flat plate attached water jet nozzle drain sump tank flow control valve motorpump plug catch basin FIGURE 4.1. A schematic of the jet impact apparatus. Procedure I. Figure 4.2 shows a sketch of the lever arm in the impact experiment. The impact object should be in place and the thumbscrew on the spring should be used to zero the lever arm. This is done without any water flow. (Units of the scales in the figures are arbitrary.) II. The pump is now turned on and a water jet hits the impact object, which will deflect the lever arm causing it to rotate slightly counterclockwise. The balancing weight is moved from the zero position to the position required to re-balance the lever arm (in this case identified as “3” in Figure 4.3). The spring is left untouched. Only the balancing weight is moved in order to re- balance the lever arm. III. During the time that the water jet impacts the object, the time required to calculate volume flow rate is measured.
- 24. 24 40 1 2 3 5 Fs Fo Fw dw1 do ds O FIGURE 4.2. Lever arm in zero position without any water flow. 40 1 2 3 5 Fs Fo Fw F dw2 do ds O water jet FIGURE 4.3. Lever arm in zero position when the water jet is on. Nomenclature SYMBOL FORCE DISTANCE Fs spring force ds Fw balancing weight dw Fo impact object do F exerted by water jet do Analysis (Actual Force as Measured) Summing moments about point O in Figure 4.2 gives the following equation for the lever arm: Fsds + Fodo + Fwdw1 = 0 (4.1) Summing moments about point O gives the following equation for the lever arm in Figure 4.3: Fsds + Fodo – Fdo + Fwdw2 = 0 (4.2) Now we compare Equations 4.1 and 4.2. We can identify parameters that appear in both equations that are constants. These are Fsds and Fodo. We rearrange Equation 4.1 to solve for the sum of these force-distance products: Fsds + Fodo = – Fwdw1 (4.3) Likewise, Equation 4.2 gives Fsds + Fodo = + Fdo – Fwdw2 (4.4) Subtracting Equation 4.4 from 4.3, we get 0 = – Fwdw1 – Fdo + Fwdw2 The force we are seeking is that exerted by the water jet F; rearranging gives Fdo = – Fwdw1 + Fwdw2 = Fw(dw2 – dw1) or F = Fw(dw2 – dw1) do (4.5) Thus, the force exerted by the water equals the weight of what we have called the balancing weight times a ratio of distances. The distance (dw2 – dw1) is just the difference in readings of the position of the balancing weight. The distance do is the distance from the pivot to the location of the impact object. For your report, derive the appropriate equation for each object you are assigned to use. Compose a graph with volume flow rate on the horizontal axis, and on the vertical axis, plot the actual and theoretical force. Use care in choosing the increments for each axis.
- 25. 25 EXPERIMENT 5 CRITICAL REYNOLDS NUMBER IN PIPE FLOW The Reynolds number is a dimensionless ratio of inertia forces to viscous forces and is used in identifying certain characteristics of fluid flow. The Reynolds number is extremely important in modeling pipe flow. It can be used to determine the type of flow occurring: laminar or turbulent. Under laminar conditions the velocity distribution of the fluid within the pipe is essentially parabolic and can be derived from the equation of motion. When turbulent flow exists, the velocity profile is “flatter” than in the laminar case because the mixing effect which is characteristic of turbulent flow helps to more evenly distribute the kinetic energy of the fluid over most of the cross section. In most engineering texts, a Reynolds number of 2 100 is usually accepted as the value at transition; that is, the value of the Reynolds number between laminar and turbulent flow regimes. This is done for the sake of convenience. In this experiment, however, we will see that transition exists over a range of Reynolds numbers and not at an individual point. The Reynolds number that exists anywhere in the transition region is called the critical Reynolds number. Finding the critical Reynolds number for the transition range that exists in pipe flow is the subject of this experiment. Critical Reynolds Number Measurement Equipment Critical Reynolds Number Determination Apparatus Figure 5.1 is a schematic of the apparatus used in this experiment. The constant head tank provides a controllable, constant flow through the transparent tube. The flow valve in the tube itself is an on/off valve, not used to control the flow rate. Instead, the flow rate through the tube is varied with the rotameter valve at A. The head tank is filled with water and the overflow tube maintains a constant head of water. The liquid is then allowed to flow through one of the transparent tubes at a very low flow rate. The valve at B controls the flow of dye; it is opened and dye is then injected into the pipe with the water. The dye injector tube is not to be placed in the pipe entrance as it could affect the results. Establish laminar flow by starting with a very low flow rate of water and of dye. The injected dye will flow downstream in a threadlike pattern for very low flow rates. Once steady state is achieved, the rotameter valve is opened slightly to increase the water flow rate. The valve at B is opened further if necessary to allow more dye to enter the tube. This procedure of increasing flow rate of water and of dye (if necessary) is repeated throughout the experiment. Establish laminar flow in one of the tubes. Then slowly increase the flow rate and observe what happens to the dye. Its pattern may change, yet the flow might still appear to be laminar. This is the beginning of transition. Continue increasing the flow rate and again observe the behavior of the dye. Eventually, the dye will mix with the water in a way that will be recognized as turbulent flow. This point is the end of transition. Transition thus will exist over a range of flow rates. Record the flow rates at key points in the experiment. Also record the temperature of the water. The object of this procedure is to determine the range of Reynolds numbers over which transition occurs. Given the tube size, the Reynolds number can be calculated with: Re = VD ν where V (= Q/A) is the average velocity of liquid in the pipe, D is the hydraulic diameter of the pipe, and ν is the kinematic viscosity of the liquid. The hydraulic diameter is calculated from its definition: D = 4 x Area Wetted Perimeter For a circular pipe flowing full, the hydraulic diameter equals the inside diameter of the pipe. For a square section, the hydraulic diameter will equal the length of one side (show that this is the case). The experiment is to be performed for both round tubes and the square tube. With good technique and great care, it is possible for the transition Reynolds number to encompass the traditionally accepted value of 2 100.
- 26. 26 Questions 1. 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 valve rotameter A to drain inlet to tank overflow to drain B transparent tube FIGURE 5.1. The critical Reynolds number determination apparatus.
- 27. 27 EXPERIMENT 6 FLUID METERS IN INCOMPRESSIBLE FLOW There are many different meters used in pipe flow: the turbine type meter, the rotameter, the orifice meter, the venturi meter, the elbow meter and the nozzle meter are only a few. Each meter works by its ability to alter a certain physical characteristic of the flowing fluid and then allows this alteration to be measured. The measured alteration is then related to the flow rate. A procedure of analyzing meters to determine their useful features is the subject of this experiment. The Venturi Meter The venturi meter is constructed as shown in Figure 6.1. It contains a constriction known as the throat. When fluid flows through the constriction, it must experience an increase in velocity over the upstream value. The velocity increase is accompanied by a decrease in static pressure at the throat. The difference between upstream and throat static pressures is then measured and related to the flow rate. The greater the flow rate, the greater the pressure drop ∆p. So the pressure difference ∆h (= ∆p/ρg) can be found as a function of the flow rate. 1 2 h FIGURE 6.1. A schematic of the Venturi meter. Using the hydrostatic equation applied to the air-over-liquid manometer of Figure 6.1, the pressure 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, it can be shown after simplification that the volume flow rate through the venturi meter is given by Qth = A2 √ 2g∆h 1 - (D2 4/D1 4) (6.1) The preceding equation represents the theoretical volume flow rate through the venturi meter. Notice that is was derived from the Bernoulli equation which does not take frictional effects into account. In the venturi meter, there exists small pressure losses due to viscous (or frictional) effects. Thus for any pressure difference, the actual flow rate will be somewhat less than the theoretical value obtained with Equation 6.1 above. For any ∆h, it is possible to define a coefficient of discharge Cv as Cv = Qac Qth For each and every measured actual flow rate through the venturi meter, it is possible to calculate a theoretical volume flow rate, a Reynolds number, and a discharge coefficient. The Reynolds number is given by Re = V2D2 ν (6.2) where V2 is the velocity at the throat of the meter (= Qac/A2). The Orifice Meter and Nozzle-Type Meter The orifice and nozzle-type meters consist of a throttling device (an orifice plate or bushing, respectively) placed into the flow. (See Figures 6.2 and 6.3). The throttling device creates a measurable pressure difference from its upstream to its downstream side. The measured pressure difference is then related to the flow rate. Like the venturi meter, the pressure difference varies with flow rate. Applying Bernoulli’s equation to points 1 and 2 of either meter (Figure 6.2 or Figure 6.3) yields the same theoretical equation as that for the venturi meter, namely, Equation 6.1. For any pressure difference, there will be two associated flow rates for these meters: the theoretical flow rate (Equation 6.1), and the
- 28. 28 actual flow rate (measured in the laboratory). The ratio of actual to theoretical flow rate leads to the definition of a discharge coefficient: Co for the orifice meter and Cn for the nozzle. 1 2 h FIGURE 6.2. Cross sectional view of the orifice meter. 1 2 h FIGURE 6.3. Cross sectional view of the nozzle- type meter, and a typical nozzle. For each and every measured actual flow rate through the orifice or nozzle-type meters, it is possible to calculate a theoretical volume flow rate, a Reynolds number and a discharge coefficient. The Reynolds number is given by Equation 6.2. The Turbine-Type Meter The turbine-type flow meter consists of a section of pipe into which a small “turbine” has been placed. As the fluid travels through the pipe, the turbine spins at an angular velocity that is proportional to the flow rate. After a certain number of revolutions, a magnetic pickup sends an electrical pulse to a preamplifier which in turn sends the pulse to a digital totalizer. The totalizer totals the pulses and translates them into a digital readout which gives the total volume of liquid that travels through the pipe and/or the instantaneous volume flow rate. Figure 6.4 is a schematic of the turbine type flow meter. rotor supported on bearings (not shown) turbine rotor rotational speed proportional to flow rate to receiver flow straighteners FIGURE 6.4. A schematic of a turbine-type flow meter. The Rotameter (Variable Area Meter) The variable area meter consists of a tapered metering tube and a float which is free to move inside. The tube is mounted vertically with the inlet at the bottom. Fluid entering the bottom raises the float until the forces of buoyancy, drag and gravity are balanced. As the float rises the annular flow area around the float increases. Flow rate is indicated by the float position read against the graduated scale which is etched on the metering tube. The reading is made usually at the widest part of the float. Figure 6.5 is a sketch of a rotameter. tapered, graduated transparent tube freely suspended float inlet outlet FIGURE 6.5. A schematic of the rotameter and its operation. Rotameters are usually manufactured with one of three types of graduated scales: 1. % of maximum flow–a factor to convert scale reading to flow rate is given or determined for the meter. A variety of fluids can be used with the meter and the only variable
- 29. 29 encountered in using it is the scale factor. The scale factor will vary from fluid to fluid. 2. Diameter-ratio type–the ratio of cross sectional diameter of the tube to the diameter of the float is etched at various locations on the tube itself. Such a scale requires a calibration curve to use the meter. 3. Direct reading–the scale reading shows the actual flow rate for a specific fluid in the units indicated on the meter itself. If this type of meter is used for another kind of fluid, then a scale factor must be applied to the readings. Experimental Procedure Equipment Fluid Meters Apparatus Stopwatch The fluid meters apparatus is shown schematically in Figure 6.6. It consists of a centrifugal pump, which draws water from a sump tank, and delivers the water to the circuit containing the flow meters. For nine valve positions (the valve downstream of the pump), record the pressure differences in each manometer. For each valve position, measure the actual flow rate by diverting the flow to the volumetric measuring tank and recording the time required to fill the tank to a predetermined volume. Use the readings on the side of the tank itself. For the rotameter, record the position of the float and/or the reading of flow rate given directly on the meter. For the turbine meter, record the flow reading on the output device. Note that the venturi meter has two manometers attached to it. The “inner” manometer is used to calibrate the meter; that is, to obtain ∆h readings used in Equation 6.1. The “outer” manometer is placed such that it reads the overall pressure drop in the line due to the presence of the meter and its attachment fittings. We refer to this pressure loss as ∆H (distinctly different from ∆h). This loss is also a function of flow rate. The manometers on the turbine-type and variable area meters also give the incurred loss for each respective meter. Thus readings of ∆H vs Qac are obtainable. In order to use these parameters to give dimensionless ratios, pressure coefficient and Reynolds number are used. The Reynolds number is given in Equation 6.2. The pressure coefficient is defined as Cp = g∆H V2/2 (6.3) All velocities are based on actual flow rate and pipe diameter. The amount of work associated with the laboratory report is great; therefore an informal group report is required rather than individual reports. The write-up should consist of an Introduction (to include a procedure and a derivation of Equation 6.1), a Discussion and Conclusions section, and the following graphs: 1. On the same set of axes, plot Qac vs ∆h and Qth vs ∆h with flow rate on the vertical axis for the venturi meter. 2. On the same set of axes, plot Qac vs ∆h and Qth vs ∆h with flow rate on the vertical axis 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. Questions 1. Referring to Figure 6.2, recall that Bernoulli's equation was applied to points 1 and 2 where the pressure difference measurement is made. The theoretical equation, however, refers to the throat area for point 2 (the orifice hole diameter) which is not where the pressure measurement was made. Explain this discrepancy and how it is accounted for in the equation formulation. 2. Which meter in your opinion is the best one to use? 3. Which meter incurs the smallest pressure loss? Is this necessarily the one that should always be used? 4. Which is the most accurate meter? 5. What is the difference between precision and accuracy? Air Over Liquid Manometry Each corresponding pair of pressure taps on the apparatus is attached to an air over liquid (water, in this case), inverted U-tube manometer. Use of the manometers can lead to some difficulties that may need attention. Figure 6.7 is a sketch of one manometer. The left and right limbs are attached to pressure taps,
- 30. 30 denoted as p1 and p2. Accordingly, when the system is operated, the liquid will rise in each limb and reach an equilibrium point. The pressure difference will appear as a difference in height of the water columns. That is, the pressure difference is given by: p1 – p2 = ρg∆h where ρ is that of the liquid, and ∆h is read directly on the manometer. In some cases, the liquid levels are at places beyond where we would like them to be. To alleviate this problem, the air release valve may be opened (slowly) to let air out or in. When this occurs, the two levels will still have the same ∆h reading, but located at a different place on the manometer. Sometimes, air bubbles will appear within the liquid. The apparatus used has water with a small amount of liquid soap dissolved to reduce the surface tension of the water. However, if the presence of bubbles persists, the pump should be cycled on and off several times, and this should solve the problem. air liquid air release valve p1 p2 ∆h FIGURE 6.7. Air over liquid manometer. orifice meter venturi meter manometer valve turbine-type meter rotameter sump tank volumetric measuring tank return pump motor FIGURE 6.6. A schematic of the Fluid Meters Apparatus. (Orifice and Venturi meters: upstream diameter is 1.025 inches; throat diameter is 0.625 inches.)
- 31. 31 EXPERIMENT 7 PIPE FLOW Experiments in pipe flow where the presence of frictional forces must be taken into account are useful aids in studying the behavior of traveling fluids. Fluids are usually transported through pipes from location to location by pumps. The frictional losses within the pipes cause pressure drops. These pressure drops must be known to determine pump requirements. Thus a study of pressure losses due to friction has a useful application. The study of pressure losses in pipe flow is the subject of this experiment. Pipe Flow Equipment Pipe Flow Test Rig Figure 7.1 is a schematic of the pipe flow test rig. The rig contains a sump tank which is used as a water reservoir from which one or two centrifugal pumps discharge water to the pipe circuit. The circuit itself consists of six different diameter lines and a return line all made of PVC pipe. The circuit contains ball valves for directing and regulating the flow, and can be used to make up various series and parallel piping combinations. The circuit has provision for measuring pressure loss through the use of static pressure taps (manometer board and pressure taps are not shown in the schematic). The six lines are 1/2, 3/4, 1, 11/4, 11/2, and 2 inch schedule 80 pipe. The topmost line is a return line, and it is made of 2 inch pipe as well. The apparatus contains two flow meters. The 1/2 line contains a flow meter which is used only for that pipe. The other flow meter in the return line is for all other flows through the system. Because the circuit contains flow meters, the measured pressure losses can be obtained as a function of flow rate. As functions of the flow rate, measure the pressure losses in inches of water for whatever combination of flows and minor losses specified by the instructor • The instructor will specify which of the pressure loss measurements are to be taken. • Open and close the appropriate valves on the apparatus to obtain the desired flow path. • Use the valve closest to the pump(s) on its downstream side to vary the volume flow rate. • With the pump on, record the assigned pressure drops and the actual volume flow rate from the flow meter. • Using the valve closest to the pump, change the volume flow rate and again record the pressure drops and the new flow rate. • Repeat this procedure until 9 different volume flow rates and corresponding pressure drop data have been recorded. With pressure loss data in terms of ∆h, the friction factor can be calculated with f = 2g∆h V2(L/D) It is customary to graph the friction factor as a function of the Reynolds number: Re = VD ν The f vs Re graph, called a Moody Diagram, is traditionally drawn on a log-log grid. The graph also contains a third variable known as the roughness coefficient ε/D. For this experiment the roughness factor ε is that for smooth walled tubing. Where fittings are concerned, the loss incurred by the fluid is expressed in terms of a loss coefficient K. The loss coefficient for any fitting can be calculated with K = ∆ h V2/2g where ∆h is the pressure (or head) loss across the fitting. Values of K as a function of the flow rate are to be obtained in this experiment. For the report, calculate friction factor f and graph it as a function of Reynolds number Re for the assigned pipe(s). Compare to a Moody diagram. Also calculate the loss coefficient for the fitting(s) assigned, and determine if the loss coefficient K varies with flow rate or Reynolds number. Compare your K values to published ones.
- 32. 32 flow meter Q2 open ball valve closed ball valve union fitting flow direction pumps Q1 Q2 1/2 nominal 3/4 1 1 1/4 1 1/2 2 FIGURE 7.1. Schematic of the pipe friction apparatus (not to scale).
- 33. 33 EXPERIMENT 8 PRESSURE DISTRIBUTION ABOUT A CIRCULAR CYLINDER In many engineering applications, it may be necessary to examine the phenomena occurring when an object is inserted into a flow of fluid. The wings of an airplane in flight, for example, may be analyzed by considering the wings stationary with air moving past them. Certain forces are exerted on the wing by the flowing fluid that tend to lift the wing (called the lift force) and to push the wing in the direction of the flow (drag force). Objects other than wings that are symmetrical with respect to the fluid approach direction, such as a circular cylinder, will experience no lift, only drag. Drag and lift forces are caused by the pressure differences exerted on the stationary object by the flowing fluid. Skin friction between the fluid and the object contributes to the drag force but in many cases can be neglected. The measurement of the pressure distribution existing around a stationary cylinder in an air stream to find the drag force is the object of this experiment. Consider a circular cylinder immersed in a uniform flow. The streamlines about the cylinder are shown in Figure 8.1. The fluid exerts pressure on the front half of the cylinder in an amount that is greater than that exerted on the rear half. The difference in pressure multiplied by the projected frontal area of the cylinder gives the drag force due to pressure (also known as form drag). Because this drag is due primarily to a pressure difference, measurement of the pressure distribution about the cylinder allows for finding the drag force experimentally. A typical pressure distribution is given in Figure 8.2. Shown in Figure 8.2a is the cylinder with lines and arrowheads. The length of the line at any point on the cylinder surface is proportional to the pressure at that point. The direction of the arrowhead indicates that the pressure at the respective point is greater than the free stream pressure (pointing toward the center of the cylinder) or less than the free stream pressure (pointing away). Note the existence of a separation point and a separation region (or wake). The pressure in the back flow region is nearly the same as the pressure at the point of separation. The general result is a net drag force equal to the sum of the forces due to pressure acting on the front half (+) and on the rear half (- ) of the cylinder. To find the drag force, it is necessary to sum the components of pressure at each point in the flow direction. Figure 8.2b is a graph of the same data as that in Figure 8.2a except that 8.2b is on a linear grid. Freestream Velocity V Stagnation Streamline Wake FIGURE 8.1. Streamlines of flow about a circular cylinder. separation point separation point 0 30 60 90 120 150 180 p (a) Polar Coordinate Graph (b) Linear Graph FIGURE 8.2. Pressure distribution around a circular cylinder placed in a uniform flow.
- 34. 34 Pressure Measurement Equipment A Wind Tunnel A Right Circular Cylinder with Pressure Taps Figure 8.3 is a schematic of a wind tunnel. It consists of a nozzle, a test section, a diffuser and a fan. Flow enters the nozzle and passes through flow straighteners and screens. The flow is directed through a test section whose walls are made of a transparent material, usually Plexiglas or glass. An object is placed in the test section for observation. Downstream of the test section is the diffuser followed by the fan. In the tunnel that is used in this experiment, the test section is rectangular and the fan housing is circular. Thus one function of the diffuser is to gradually lead the flow from a rectangular section to a circular one. Figure 8.4 is a schematic of the side view of the circular cylinder. The cylinder is placed in the test section of the wind tunnel which is operated at a preselected velocity. The pressure tap labeled as #1 is placed at 0° directly facing the approach flow. The pressure taps are attached to a manometer board. Only the first 18 taps are connected because the expected profile is symmetric about the 0° line. The manometers will provide readings of pressure at 10° intervals about half the cylinder. For two different approach velocities, measure and record the pressure distribution about the circular cylinder. Plot the pressure distribution on polar coordinate graph paper for both cases. Also graph pressure difference (pressure at the point of interest minus the free stream pressure) as a function of angle θ on linear graph paper. Next, graph ∆p cosθ vs θ (horizontal axis) on linear paper and determine the area under the curve by any convenient method (counting squares or a numerical technique). The drag force can be calculated by integrating the flow-direction-component of each pressure over the area of the cylinder: Df = 2RL ∫ 0 π ∆p cosθdθ The above expression states that the drag force is twice the cylinder radius (2R) times the cylinder length (L) times the area under the curve of ∆p cosθ vs θ. Drag data are usually expressed as drag coefficient CD vs Reynolds number Re. The drag coefficient is defined as CD = Df ρV2A/2 The Reynolds number is Re = ρVD µ inlet flow straighteners nozzle test section diffuser fan FIGURE 8.3. A schematic of the wind tunnel used in this experiment.
- 35. 35 where V is the free stream velocity (upstream of the cylinder), A is the projected frontal area of the cylinder (2RL), D is the cylinder diameter, ρ is the air density and µ is the air viscosity. Compare the results to those found in texts. static pressure taps attach to manometers 60 0 30 90 120 150 180 FIGURE 8.4. Schematic of the experimental apparatus used in this experiment.
- 36. 36 EXPERIMENT 9 DRAG FORCE DETERMINATION An object placed in a uniform flow is acted upon by various forces. The resultant of these forces can be resolved into two force components, parallel and perpendicular to the main flow direction. The component acting parallel to the flow is known as the drag force. It is a function of a skin friction effect and an adverse pressure gradient. The component perpendicular to the flow direction is the lift force and is caused by a pressure distribution which results in a lower pressure acting over the top surface of the object than at the bottom. If the object is symmetric with respect to the flow direction, then the lift force will be zero and only a drag force will exist. Measurement of the drag force acting on an object immersed in the uniform flow of a fluid is the subject of this experiment. Equipment Subsonic Wind Tunnel Objects A description of a subsonic wind tunnel is given in Experiment 8 and is shown schematically in Figure 8.3. The fan at the end of the tunnel draws in air at the inlet. An object is mounted on a stand that is pre calibrated to read lift and drag forces exerted by the fluid on the object. A schematic of the test section is shown in Figure 9.1. The velocity of the flow at the test section is also pre calibrated. The air velocity past the object can be controlled by changing the rotational speed of the fan. Thus air velocity, lift force and drag force are read directly from the tunnel instrumentation. There are a number of objects that are available for use in the wind tunnel. These include a disk, a smooth surfaced sphere, a rough surface sphere, a hemisphere facing upstream, and a hemisphere facing downstream. For whichever is assigned, measure drag on the object as a function of velocity. Data on drag vs velocity are usually graphed in dimensionless terms. The drag force Df is customarily expressed in terms of the drag coefficient CD (a ratio of drag force to kinetic energy): CD = Df ρV2A/2 in which ρ is the fluid density, V is the free stream velocity, and A is the projected frontal area of the object. Traditionally, the drag coefficient is graphed as a function of the Reynolds number, which is defined as Re = VD ν where D is a characteristic length of the object and ν is the kinematic viscosity of the fluid. For each object assigned, graph drag coefficient vs Reynolds number and compare your results to those published in texts. Use log-log paper if appropriate. Questions 1. How does the mounting piece affect the readings? 2. How do you plan to correct for its effect, if necessary? drag force measurement lift force measurement uniform flow mounting stand object FIGURE 9.1. Schematic of an object mounted in the test section of the wind tunnel.
- 37. 37 EXPERIMENT 10 ANALYSIS OF AN AIRFOIL A wing placed in the uniform flow of an airstream will experience lift and drag forces. Each of these forces is due to a pressure difference. The lift force is due to the pressure difference that exists between the lower and upper surfaces. This phenomena is illustrated in Figure 10.1. As indicated the airfoil is immersed in a uniform flow. If pressure could be measured at selected locations on the surface of the wing and the results graphed, the profile in Figure 10.1 would result. Each pressure measurement is represented by a line with an arrowhead. The length of each line is proportional to the magnitude of the pressure at the point. The direction of the arrow (toward the horizontal axis or away from it) represents whether the pressure at the point is less than or greater than the free stream pressure measured far upstream of the wing. stagnation point negative pressure gradient on upper surface positive pressure on lower surface Cp pressure coefficient stagnation point c chord, c FIGURE 10.1. Streamlines of flow about a wing and the resultant pressure distribution. Lift and Drag Measurements for a Wing Equipment Wind Tunnel (See Figure 8.3) Wing with Pressure Taps Wing for Attachment to Lift & Drag Instruments (See Figure 10.2) Experiment For a number of wings, lift and drag data vary only slightly with Reynolds number and therefore if lift and drag coefficients are graphed as a function of Reynolds number, the results are not that meaningful. A more significant representation of the results is given in what is known as a polar diagram for the wing. A polar diagram is a graph on a linear grid of lift coefficient (vertical axis) as a function of drag coefficient. Each data point on the graph corresponds to a different angle of attack, all measured at one velocity (Reynolds number). Referring to Figure 10.2 (which is the experimental setup here), the angle of attack α is measured from a line parallel to the chord c to a line that is parallel to the free stream velocity. Obtain lift force, drag force and angle of attack data using a pre selected velocity. Allow the angle of attack to vary from a negative angle to the stall point and beyond. Obtain data at no less than 9 angles of attack. Use the data to produce a polar diagram. Analysis Lift and drag data are usually expressed in dimensionless terms using lift coefficient and drag coefficient. The lift coefficient is defined as CL = Lf ρV2A/2 where Lf is the lift force, ρ is the fluid density, V is the free stream velocity far upstream of the wing, and A is the area of the wing when seen from a top view perpendicular to the chord length c. The drag coefficient is defined as CD = Df ρV2A/2 in which Df is the drag force.
- 38. 38 drag force measurement lift force measurement uniform flow mounting stand c drag lift FIGURE 10.2. Schematic of lift and drag measurement in a test section.
- 39. 39 EXPERIMENT 11 OPEN CHANNEL FLOW—SLUICE GATE Liquid motion in a duct where a surface of the fluid is exposed to the atmosphere is called open channel flow. In the laboratory, open channel flow experiments can be used to simulate flow in a river, in a spillway, in a drainage canal or in a sewer. Such modeled flows can include flow over bumps or through dams, flow through a venturi flume or under a partially raised gate (a sluice gate). The last example, flow under a sluice gate, is the subject of this experiment. Flow Through a Sluice Gate Equipment Open Channel Flow Apparatus Sluice Gate Model Figure 11.1 is a sketch of the flow pattern under a sluice gate. Upstream of the gate, the velocity is V0, and the liquid height is h0. The gate is a distance h1 above the bottom of the channel, and downstream, the liquid height is h2. The channel width is b. The objective of this experiment is to make measurements for a number of gate positions and flow rates, and to determine whether the equations we derive for a sluice gate are accurate in their description of the resulting flows. h0 h1 h2 V0 sluice gate FIGURE 11.1. Schematic of flow under a sluice gate. Theory The continuity equation applied about the sluice gate is V0h0b = V2h2b (11.1) Under real conditions, h2 is somewhat less than h1. We therefore introduce a contraction coefficient Cc defined as Cc = h2 h1 Substituting into Equation 11.1, canceling the channel width b, and solving for V0, we get V0 = V2 Cch1 h0 (11.2) The Bernoulli Equation applied about the gate is V0 2 2g + h0 = V2 2 2g + h2 (11.3) Substituting for V0 from Equation 11.2, we obtain V2 2Cc 2h1 2/h0 2 2g + h0 = V2 2 2g + Cch1 Rearranging and solving for V2, we have V2 = 2g h0 – Cch1 1 – Cc 2h1 2/h0 2 1/2 Factoring h0 and noting the relationship between the numerator and denominator, the preceding equation becomes V2 = 2gh0 1 + Cch1/h0 1/2 (11.4) The flow rate is the product of area and velocity. At section 2, the flow rate may be written as Q = V2h2b = V2Cch1b Substituting from Equation 11.4 yields Q = Cch1b 2gh0 1 + Cch1/h0 1/2 (11.5) Introducing a discharge coefficient Cs, we write Q = Csb √2gh0 (11.6) Comparison with Equation 11.5 gives Cs = Cc √1 + Cch1/h0 (11.7)
- 40. 40 We see that the contraction coefficient and the discharge coefficient depend only on the upstream height and the gate height. Procedure Set up the open channel flow apparatus (Figure 11.2) to obtain flow under a sluice gate. For nine (if possible) different flow rate/gate position combinations, record upstream height, gate height, downstream height, and volume flow rate. Calculations Calculate the contraction coefficient, the discharge coefficient (Equation 11.7), and the expected volume flow rate (Equation 11.6). Compare the calculated flow rate (theoretical) with the measured (actual) value. Include in Your Report • Detailed derivation of the equations • Graph of contraction coefficient as a function of the ratio h1/h0 • Graph of discharge coefficient as a function of the ratio h1/h0 • Graph of flow rates (actual and theoretical) as a function of the ratio h1/h0 • Discussion of your results sump tank pump/motor pump discharge pipe valve head tank sluice gate turbine meter flow channel FIGURE 11.2. Schematic of the open channel flow apparatus.
- 41. 41 EXPERIMENT 12 OPEN CHANNEL FLOW OVER A WEIR Flow meters used in pipes introduce an obstruction into the flow which results in a measurable pressure drop that in turn is related to the volume flow rate. In an open channel, flow rate can be measured similarly by introducing an obstruction into the flow. A simple obstruction, called a weir, consists of a vertical plate extending the entire width of the channel. The plate may have an opening, usually rectangular, trapezoidal, or triangular. Other configurations exist and all are about equally effective. The use of a weir to measure flow rate in an open channel is the subject of this experiment. Flow Over a Weir Equipment Open Channel Flow Apparatus (See Figure 11.2) Several Weirs The open channel flow apparatus allows for the insertion of a weir and measurement of liquid depths. The channel is fed by two centrifugal pumps. Each pump has a discharge line which contains an turbine meter with digital readout, which provide the means of determining the actual flow rate into the channel. Figure 12.1 is a sketch of the side and upstream view of a 90 degree (included angle) V- notch weir. Analysis of this weir is presented here for illustrative purposes. Note that upstream depth measurements are made from the lowest point of the weir over which liquid flows. This is the case for the analysis of all conventional weirs. A coordinate system is imposed whose origin is at the intersection of the free surface and a vertical line extending upward from the vertex of the V-notch. We select an element that is dy thick and extends the entire width of the flow cross section. The velocity of the liquid through this element is found by applying Bernoulli's equation: pa ρ + Vo 2 2 + gh = pa ρ + V2 2 + g(h - y) Note that in pipe flow, pressure remained in the equation when analyzing any of the differential pressure meters (orifice or venturi meters). In open channel flows, the pressure terms represents atmospheric pressure and cancel from the Bernoulli equation. The liquid height is therefore the only measurement required here. From the above equation, assuming Vo negligible: V = √2gy (12.1) Equation 12.1 is the starting point in the analysis of all weirs. The incremental flow rate of liquid through layer dy is: dQ = 2Vxdy = √2gy(2x)dy From the geometry of the V-notch and with respect to the coordinate axes, we have y = h - x. pa pa Vo V h y dy x x axis y axis FIGURE 12.1. Side and upstream views of a 90° V-notch weir.
- 42. 42 Therefore, Q = ∫ 0 h (2√2g)y1/2(h - y)d y Integration gives Qth = 8 15 √2g h5/2 =Ch5/2 (12.2) where C is a constant. The above equation represents the ideal or theoretical flow rate of liquid over the V-notch weir. The actual discharge rate is somewhat less due to frictional and other dissipative effects. As with pipe meters, we introduce a discharge coefficient defined as: C' = Qac Qth The equation that relates the actual volume flow rate to the upstream height then is Qac = C'Ch5/2 It is convenient to combine the effects of the constant C and the coefficient C’ into a single coefficient Cvn for the V-notch weir. Thus we reformulate the previous two equations to obtain: Cvn ≈ Qac Qth (12.3) Qac = Cvnh5/2 (12.4) Each type of weir will have its own coefficient. Calibrate each of the weirs assigned by the instructor for 7 different upstream height measurements. Derive an appropriate equation for each weir used (similar to Equation 12.4) above. Determine the coefficient applicable for each weir tested. List the assumptions made in each derivation. Discuss the validity of each assumption, pointing out where they break down. Graph upstream height vs actual and theoretical volume flow rates. Plot the coefficient of discharge (as defined in Equation 12.3) as a function of the upstream Froude number. FIGURE 12.2. Other types of weirs–semicircular, contracted and suppressed, respectively.

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