Tank Training Exercise - Experiment 3 | EAS 361, Lab Reports of Fluid Mechanics

Download Tank Training Exercise - Experiment 3 | EAS 361 and more Lab Reports Fluid Mechanics in PDF only on Docsity! Tank Draining Exercise EAS 361, Fall 2008 Engineering of Everyday Things Gerald Recktenwald Portland State University gerry@me.pdx.edu eet.cecs.pdx.edu Engineering of Everyday Things – Fall 2008 Participant Code: 1 Apparatus Figure 1 shows the equipment for this laboratory exercise. The key components are 1. Two cylindrical tanks, one with straight walls (single diameter) and another with a step-shaped wall (two diameters). 2. A pressure transducer mounted on the side of each tank, and at a distance H from the base. Not shown in the sketches are 3. A digital camera and tripod used to capture images of the free jet of water. 4. A power supply to provide electrical energy to the transducer. 5. A data acquisition device (DAQ) for digitizing the transducer output. 6. A computer to record and display the digitized output of the transducer h 1 H L 1 p 1 catch basin Vj Vs h 2 H L 2 p 2 catch basin Vj Vs Figure 1: Apparatus for the tank-draining experiment. 2 Learning Objectives After completing this lab exercise students will be able to 1. Apply mass conservation to a control volume with a time-varying mass. 2. Apply the Bernoulli and Energy Equations to compute the velocity of a free jet emerging from a hole in a tank; 3. Determine the pressure at depth in a tank with a small hole in its side; 4. Use digital photographs to measure geometric features, specifically the trajectory of the jet. 5 3.4 Pressure versus Depth 1 2 3 4 V j z Figure 2: Streamlines used to determine pressure at tank side wall during tank draining. In a stationary liquid, the pressure at depth h is p = γh, where γ is the specific weight of the liquid. Can that formula be applied when water is flowing out through the hole in the side of the tank? Figure 2 depicts two streamlines that converge at the hole at Station 3 in the side of the tank. Station 1 is at the free surface. Station 2 is adjacent to the pressure transducer attached to the tank wall. Station 4 is on the streamline immediately outside of the tank. The following analysis shows that under some very reasonable assump- tions, the pressure at Station 2 is determined only by the depth of water in the tank. Write the Bernoulli Equation for the streamline from Station 1 to Station 3. p1 γ + V 21 2g + z1 = p3 γ + V 23 2g + z3 (6) Since Station 1 is a free surface: p1 = 0, V1 = 0. Choose z3 = 0. =⇒ p3 = γz1 − 1 2 ρV 23 (7) Write the Bernoulli Equation for the streamline from Station 2 to Station 3. p2 γ + V 22 2g + z2 = p3 γ + V 23 2g + z3 (8) Since Station 2 is adjacent to the wall, V2 = 0. Also z1 = z3 = 0. =⇒ p2 = p3 + 1 2 ρV 23 (9) Combine Equation (7) and Equation (9) to obtain a formula for p2 as a function of z1. p2 = ( γz1 − 1 2 ρV 23 ) + 1 2 ρV 23 Therefore, p2 = γz1 (10) Your Summary: Based on the preceding analysis, list two conditions under which p = γh can be used to measure the instantaneous tank depth with the pressure transducer attached to the side wall of the tank. 1. 2. STOP Before continuing, show your lab manual to the instructor. It’s impor- tant at this point to make sure you are on the right track. Instructor Approval 6 3.5 Trajectory of a Free Jet Figure 3 depicts the trajectory of the jet of water emanating from the hole in the side of the tank. The goal of the following exercise is to obtain a formula for Ljet as a function of the velocity Vj of the fluid as it leaves the hole. The water jet is assumed to retain the same diameter as it flows. The water is assumed to have a purely horizontal velocity component Vj when it leaves the hole at xp = 0, yp = H. Vj Vy = –gt yp xp Vj Ljet H Figure 3: Arc of the water jet. The path of the fluid is assumed to be the same as that of a rigid particle falling under the influence of only gravity, and having an initial horizontal velocity Vj . 3.5.1 Jet Shape Determined by Equations for Particle Motion Consider the trajectory of a particle of water emerging from the hole in the side of the tank. Rearranging Newton’s law of motion F = ma as a = F/m, and writing the differential form of the acceleration in the x and y direction gives. d2xp dt2 = 0 d2xp dt2 = −g Integrate the equations of motion with the initial conditions (t = 0) dxp dt = Vj , xp = 0, dyp dt = 0, yp = 0. The particle crosses the yp = 0 plane when 12gt 2 0 = H or at time t0 = √ 2H/g. At time t0 the particle has travelled a horizontal distance Ljet = Vjt0. Therefore, Ljet = Vj √ 2H g (11) 3.5.2 Analytical Model of Jet Velocity Apply the Bernoulli Equation between Station 1 and Station 4 in Figure (2) on page 5. p1 γ + V 21 2g + z1 = p4 γ + V 24 2g + z4 7 At the free surface, p1 = 0, and V1 = 0. At the jet exit, p2 = 0. Making these assumptions the equation becomes z1 = V 24 2g + z4 or V 24 2g = z1 − z4. (12) Since Vj is the same as V4, we finally get Vj = V4 = √ 2gh. (13) Substituting Equation (13) into Equation (11) gives Ljet = 2 √ Hh (14) 3.6 Study Questions 1. What physical effect(s) could cause the jet to not follow the trajectory of a rigid particle? 2. Suppose you had a series of measurements of Ljet and h at different times. After making a plot of Ljet versus h, how would you make a quantitative (i.e., numerical) comparison between the data in the plot and Equation (14)? STOP Before continuing, show your lab manual to the instructor. It’s impor- tant at this point to make sure you are on the right track. Instructor Approval 10 2 2 3 4 5 6 7 8 1 1 3 Figure 5: Annotated screen shot of the tankDrainGUI program. 11 4.3 Automated Analysis of Jet and Transducer Data Start Matlab and run the tankDrainGUI program by typing >> tankDrainGUI at the command prompt. Use the following steps to analyze the data. The numbers of the items in the list correspond to the large numbers in circles in Figure 5. 1. Enter (t, Ljet) data in the table in the upper left quadrant of the GUI. Alternatively, if you have already saved some (t, Ljet) data from previous work with tankDrainGUI, then load it by selecting File→Open Jet Data from menu at the top of the GUI window. 2. Data entered in the table is automatically plotted in the figure pane in the upper right quadrant of the GUI. You can directly edit data in the first two columns of the table. Row and table-level actions are achieved with the three buttons at the top of the table. Clear Table replaces all values in the table with zeros. Add Row inserts a new row at the bottom of the table. Delete Row removes the row that is currently selected. A row is selected by clicking on a cell in the t or L columns. Note that data in the h column is calculated by the program and cannot be edited manually. 3. Load pressure versus time data by selecting File→Open Pressure Data from the menu at the top of the GUI window. Locate the file that was saved by the LABVIEW VI. After it is loaded, the (t, p) data is displayed in the figure pane in the lower right quadrant of the GUI. 4. The t axis scaling for the Ljet(t) and p(t) plots are unlikely to be the same. Click the Set Axes Equal button to use the same scale for t in both plots. 5. After Ljet(t) and p(t) have been loaded, click the Transfer Jet Data button. This causes the t values from the Ljet(t) data set to be transfered to the p(t) plot. Those t values are represented by short vertical lines that cross the p(t) data. For each of the t values in the Ljet(t) data set, a new pressure value, call it p̂, is obtained by interpolating in the measured pressure data set. The interpolated p̂ values are taken as the pressure (and hence tank depth) for each of the Ljet(t) values. Note that when the pressure is given in units of head, the pressure value is h, the depth of water measured from the location of the pressure transducer. 6. Click the plot L vs. h button to combine the jet length and pressure measurements in a single plot in the lower left quadrant of the GUI. 7. Click the Add Curve fit button to compute and plot a curve fit of the form Ljet = c √ h. The curve fit to the data appears as a dashed red line. The curve fit coefficient c is displayed on the plot. 8. Click the Export Data button at the bottom of the data table in the upper left quadrant of the GUI. This opens a dialog box for specifying the file name and location of the data file. The file consists of the three columns of data shown in the upper left quadrant of the GUI. 12 4.4 Qualitative Analysis of Tank Draining Data On the axes to the right, sketch the relationship of Ljet versus h. Use an arrow to indicate the direction that the data on the curve follows as time increases during the tank draining experiment. Ljet h 4.5 Anticipation of Results for the Step-shaped Tank On the axes to the right, sketch the relationship of Ljet versus t for the step-shaped tank. Put a circle on the curve to identify the time where the free surface passes the change in area of the tank. Ljet t On the axes to the right, sketch the relationship of pressure transducer output versus t for the step- shaped tank. The pressure recorded by the trans- ducer is proportional to the depth of water in the tank so the vertical axis is labeled h. Put a circle on the curve to identify the time where the free surface passes the change in area of the tank. h t On the axes to the right, sketch the relationship of Ljet versus h for the step-shaped tank. Use an arrow to indicate the direction that the data on the curve follows as time increases during the tank draining experiment. Add a dashed curve to indicate the Ljet(h) relationship for the straight tank. Ljet h STOP Before continuing, show your lab manual to the instructor. It’s impor- tant at this point to make sure you are on the right track. Instructor Approval