CE 321 INTRODUCTION TO FLUID MECHANICS Fall 2009
LABORATORY 3: THE BERNOULLI EQUATION
OBJECTIVES To investigate the validity of Bernoulli's Equation as applied to the flow of water in a tapering horizontal tube to determine if the total pressure head remains constant along the length of the tube as the equation predicts.
To determine if the variations in static pressure head along the length of the tube can be predicted with Bernoulli’s equation
APPROACH Establish a constant flow rate (Q) through the tube and measure it. Use a pitot probe and static probe to measure the total pressure head h Tm and static pressure head h Sm at six locations along the length of the tube. The values of h Tm will show if total pressure head remains constant along the length of the tube as required by the Bernoulli Equation. Using the flow rate and cross sectional area of the tube, calculate the velocity head h Vc at each location. Use Bernoulli’s Equation, h Tm and h Vc to predict the variations
in static pressure head h St expected along the tube. Compare the calculated and measured values of static pressure head to determine if the variations in fluid pressure along the length of the tube can be predicted with Bernoulli’s Equation.
EQUIPMENT
Hydraulic bench with Bernoulli apparatus, stop watch
THEORY Considering flow at any two positions on the central streamline of the tube (Fig. 1), Bernoulli's equation may be written as
V 2 p V 2 p 1 + 1 + z = 2 + 2 + z (1) 2g γ 1 2g γ 2
1 Bernoulli’s equation indicates that the sum of the velocity head (V 2/2g), pressure head (p/ γ), and elevation (z) are constant along the central streamline. As explained in the
Appendix 1, Eq. 1 can be simplified for this apparatus. The tube is horizontal so z 1=z 2, and the pressure heads h 1= p 1/γ and h 2 = p 2/γ can be measured from a common arbitrary datum so that Bernoulli’s Equation simplifies to V 2 V 2 1 + h = 2 + h (2) 2g 1 2g 2
Note that the sum on either side of the equal sign is the total pressure head h T. If
Bernoulli’s Eq. is correct, the total pressure head (h T)has the same value at all locations on the central streamline. This is not obvious since the narrowing tube causes the velocity to vary along the length of the tube. For Bernoulli’s Eq. to be true the pressure head must decrease as much as the velocity head increases in moving from point 1 to point 2.
The pitot probe you will use measures the total pressure head h T of the fluid a short distance upstream of the probe’s tip. The probe’s tip must be positioned about 3mm downstream from point where the measurement is desired and oriented so that it opens directly into the flow (Fig.1).
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Fig.1. Test section and manometer tubes
The piezometers installed along the side of the tube measure static pressure head h Sm.
PROCEDURE
A. Familiarize yourself with the apparatus.
B. Level the apparatus.
C. Carefully fill the manometer tubes with water to flush all air pockets from the system and ensure all connecting pipes are free of air. With the pump on and the flow control valve closed, the top of the water columns in the manometer tubes should all be at the same elevation. For best results, this level should be set at approximately 150 mm. To lower the levels, attach the hand pump to the inlet
valve and increase the air pressure (p oa ) in the chamber at the top of the
manometer tubes. To raise the levels, lower p oa by releasing air at the inlet valve.
3 D. Make sure that all of the manometer tubes have their zero readings at the same elevation. If not, get help from your TA.
E. The pitot probe can be positioned by loosening the gland nut and carefully moving the probe to the desired position. Before a reading is taken, the gland nut should be re-tightened by hand.
F. Carefully adjust the bench supply control valve and the Bernoulli apparatus flow control valve to provide the combination of flow rate and system pressure
(p oa ) which will give the largest convenient difference between the highest and lowest manometer levels.
G. For a given valve setting (flow rate), do the following:
1. Measure volume and time so that you can determine flow rate. 2. With the pitot probe removed entirely from the tube’s test section,
3. Record each piezometer’s manometer reading. This is h Sm . 4. Move the tip of the pitot probe to a position about 3mm downstream of each piezometer tap and record the reading of the pitot probe manometer
at each location. This is h Tm . 5. Repeat your measurements of volume and time to show that the flow rate has remained constant while you were making your measurements. 6. Repeat steps 1 to 4 for each of the three flow rates. 7. When finished, drain the apparatus. Make sure the pitot probe is completely inserted into the tube and the gland nut is tightened before leaving.
4 ANALYSIS
A. Determine the three flow rates. • Use the volume-time data.
2 B. Calculate the velocity head h vc (Va 2/ g ) at each cross section. • Use each flow rate and the six cross-sectional areas of the tube to determine
the average fluid velocity (V a=Q/A) at each measurement position along the length of the tube. The tube diameters at measurement sections are in Fig.3 Appendix2
C. Calculate the best estimate of the true value of h T in the tube.
• Use the value of h Tm at point A in the converging section.
D. Predict the variation of static pressure head h St along the length of the tube.
• Use Bernoulli’s Equation to predict h St .
• Use the measured values of total pressure head h Tm .
• Use the calculated values of velocity head h Vc .
E. Graph the values as a function of position along the tube. • Make one graph for each flow rate. • Make each graph the same size. • Use identical scales on all three graphs.
• Show the measured values of total pressure head h Tm , static pressure head h Sm ,
and calculated values of velocity head h Vc on each graph .
• Show the theoretical variation of h T. • Label the graphs so that the different heads are clearly identifiable.
F. Consider what the measurements show happened in the converging section of the tube. • Look at the graphs and tables.
5 • Describe how measured total pressure head h Tm varied with distance in this region.
• Compare the variation of h Tm with the theoretical variation of h Tt that Bernoulli’s Equation predicts. • Comment on whether this suggests that the assumptions made in deriving Bernoulli’s Equation are satisfied in this part of the tube. • Describe how measured static pressure head varied with distance in this region.
• Compare the variation of measured static pressure head h Sm with the theoretical
variation h ST that you predicted using Bernoulli’s Equation. • Comment on what this comparison suggests about the ability to predict fluid pressure in the converging section of the tube using Bernoulli’s Equation if you know only the flow rate and the geometry of the tube.
G. Consider what the measurements show happened in the diverging section of the tube. • Look at the graphs and tables.
• Describe how measured total pressure head h Tm varied with distance in this region.
• Compare the variation of h Tm with the theoretical variation of h Tt that Bernoulli’s Equation predicts. • Considering the assumptions made to derive Bernoulli’s Equation, offer an explanation for why the variation in total pressure head that you measured agrees or disagrees with what Bernoulli’s Equation predicts.
• Describe how measured static pressure head h Sm varied with distance in this region. Where are the lowest static pressure heads?
• Compare the variation of measured static pressure head h Sm with the theoretical
variation h ST that you predicted using Bernoulli’s Equation.
H. Consider what the measurements show happened as the flow rate increased.
6 • Describe how the increase in flow influenced the total pressure head in the converging region. • Describe how the increase in flow influenced the total pressure head in the diverging region. • Describe how the increase in flow influenced the measured static pressure head in the converging region. I. Comment on the apparent validity of Bernoulli’s Equation for convergent and divergent flows in general
7 APPENDIX1
JUSTIFICATION FOR EQ.2
Some justification for the simplifications made to obtain Eq. 2 are required. First, in Eq.
1 both elevations (z 1,z 2) must be measured from a common horizontal datum for elevation. In setting up the apparatus for this experiment, the tube is leveled so that the central streamline is also horizontal. Thus the elevation(z) is forced to remain constant
(z 1 = z 2) so there is no need to determine its actual value in order to evaluate whether Bernoulli’s equation correctly describes conditions along the streamline where the velocity is changing. Second, the pressures in Eq. 1 may be measured as indicated above (p= γh) by employing a special pressure datum (Fig. 1) that is selected to simplify
measurement. The absolute pressure of the datum is p da =p 0a +γ∆ z. Pressures p 1=γh1 and p 2=γh2 are the portion of the absolute pressure of the water at locations 1 and 2 that is in excess of the datum pressure p da . To show this we need to first determine the
absolute pressure at 1 and then subtract the pressure datum p da . We obtain the absolute pressure p 1a by analyzing the manometer column at 1 (Fig. 1) which shows that
p1a = p0a + γ (h1 + ∆z) (3)
Subtracting the datum pressure (p da =p 0a +γ∆ z) leads to the desired result for pressure
p1 = p1a − pda = [ p0a + γ (h1 + ∆z)] −[( p0a + γ∆z)] = γh1 (4)
Thus the pressure head at 1 based on the special pressure datum is p 1 = h (5) γ 1
The same approach works at point 2 or any other point on the central streamline. Note
8 that the pressures that you are working with (p 1=γh1 and p 2=γh2) are neither absolute pressures nor gage pressures because we have selected a special datum. We can pick any pressure datum that we want with Bernoulli’s equation so long as we use it all of the time, because, as Eq. 1 shows, we are only concerned with how the pressure varies from one point to the other.
9 APPENDIX2 DESCRIPTION OF THE BERNOULLI APPARATUS Fig. 2 shows the location of each of the main elements of the apparatus. The test section (5) is a machined tube of varying circular cross section provided with pressure tappings so that static pressures can be measured simultaneously at each of 6 locations. Fig. 3 provides the location and dimension of each cross section. A pitot probe (7) is provided which can be positioned to read the total head at any section of the tube. The probe can be moved after loosening the gland nut (6); this nut should be re-tightened by hand. (To prevent damage, the probe should be fully inserted during storage.) All eight pressure tappings are connected to a bank of pressurized manometer tubes (3). The pressure in the air chamber (2) can be increased by connecting a hand pump to the inlet valve (4). Pressure can be reduced by depressing the valve (4). The inlet pipe (1) should be connected directly to the bench supply. A flexible hose attached to the outlet pipe should cause water flowing from the apparatus to empty directly into the volumetric measuring tank. The apparatus should be leveled with the adjustable feet (8). Flow rate and pressure in the apparatus can be varied by adjusting the flow control valve (9) and the bench supply control valve. Opening the flow control valve will increase flow rate through the tube and decrease fluid pressure throughout the tube. Whereas, opening the bench supply control valve will increase flow rate and fluid pressure throughout the tube. By use of the two valves, it is possible to vary flow rate and pressure independently. 1. Inlet pipe 2. Air chamber 3. Manometer tubes 4. Inlet valve 5. Test section 6. Gland nut 7. Pitot probe 8. Adjustable feet 9. Flow control valve
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Fig.2. Different elements of Bernoulli apparatus
Fig.3. Measurement section locations and dimensions in mm
11 APPENDIX 3 CHECK LIST FOR BERNOULLI EXPERIMENT
DATA APPEAR TO BE REASONABLY ACCURATE
1. Your plots show smooth variations of head with distance. 2. You employed a method of measurement that is expected to give a flow rate with an uncertainty less than 5%.
In the Discussion and Results Section make sure of the following 1.Your observations are correct, clearly stated, and supported by the data you present. 2. Your logical arguments are correct and clearly stated. 3. Your graphs are in the form requested • Graph values as a function of position along the tube. • Make one graph for each flow rate. • Make each graph the same size. • Use identical scales on all three graphs.
Show the measured values of total pressure head h Tm , static pressure head hSm , and
calculated values of velocity head h Vc on each graph .
• Show the theoretical variation of h T. • Label the graphs so that the different heads are clearly identifiable. 4. You described what happened in the converging Section:
• Described how measured total pressure head h Tm varied with distance in this region.
• Compared the variation of h Tm with the theoretical variation of h Tt that Bernoulli’s Equation predicts. • Commented on whether this comparison suggests that the assumptions made in deriving Bernoulli’s Equation are satisfied in this part of the tube. • Described how measured static pressure head varied with distance in this region.
12 • Compared the variation of measured static pressure head h Sm with the theoretical
variation h St that you predicted using Bernoulli’s Equation. • Commented on what this comparison suggests about the ability to predict fluid pressure in the converging section of the tube using Bernoulli’s Equation if you know only the flow rate and the geometry of the tube.
5. You described what happened in the diverging section of the tube:
• Described how measured total pressure head h Tm varied with distance in this
region Compared the variation of h Tm with the theoretical variation of h Tt that Bernoulli’s Equation predicts • Considering the assumptions made to derive Bernoulli’s Equation, offer an explanation for why the variation in total pressure head that you measured agrees or disagrees with what Bernoulli’s Equation predicts.
• Described how measured static pressure head h Sm varied with distance in this region. Where are the lowest static pressure heads?
• Compared the variation of measured static pressure head h Sm with the theoretical
variation h St that you predicted using Bernoulli’s equation 6. You described what happened as the flow rate increased: • Described how the increase in flow influenced the total pressure head in the converging region. • Described how the increase in flow influenced the total pressure head in the diverging region. • Described how the increase in flow influenced the measured static pressure head in the converging section.
7. You commented on the apparent validity of Bernoulli’s Equation for convergent and divergent flows in general
Make sure the important results and conclusions are restated in the Conclusions Section and summarized in the Abstract
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