TOPIC 5: BERNOULLIS THEOREM DEMOSTRATION4.0 THEORYBernoullis Law Bernoulli's law states that if a non-viscous fluid is flowing along a pipe of varying cross section, then the pressure is lower at constrictions where the velocity is higher, and the pressure is higher where the pipe opens out and the fluid stagnate. Many people find this situation paradoxical when they first encounter it (higher velocity, lower pressure). This is expressed with the following equation:

Where,P= Fluid static pressure at the cross section= Density of the flowing fluidg= Acceleration due to gravityv= Mean velocity of fluid flow at the cross sectionz= Elevation head of the center at the cross section with respect to a datumh*= Total (stagnation) head

The terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv ), and elevation head (z), respectively. The sum of these terms is known as the total head (h*). According to the Bernoullis theorem of fluid flow through a pipe, the total head h* at any cross section is constant. In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones. In our experimental setup, the centreline of all the cross sections we are considering lie on the same horizontal plane (which we may choose as the datum, z = 0, and thus, all the z values are zeros so that the above equation reduces to:

This represents the total head at a cross section. For the experiments, the pressure head is denoted as hi and the total head as h*i, where i represents the cross sections at different tapping points.

Static, Stagnation and Dynamic PressuresThe pressure, p, which we have used in deriving the Bernoullis equation, Equation 3.7, is the thermodynamic pressure; it is commonly called the static pressure. The static pressure is that pressure which would be measured by an instrument moving with the flow. However, such a measurement is rather difficult to make in a practical situation. As we know, there was no pressure variation normal to straight streamlines. This fact makes it possible to measure the static pressure in a flowing fluid using a wall pressure tapping, placed in a region where the flow streamlines are straight, as shown in Figure 4 (a). The pressure tap is a small hole, drilled carefully in the wall, with its axis perpendicular to the surface. If the hole is perpendicular to the duct wall and free from burrs, accurate measurements of static pressure can be made by connecting the tap to a suitable pressure measuring instrument.

Figure 5.1: Measurement of Static PressureIn a fluid stream far from a wall, or where streamlines are curved, accurate static pressure measurements can be made by careful use of a static pressure probe, shown in Figure 4 (b). Such probes must be designed so that the measuring holes are place correctly with respect to the probe tip and stem to avoid erroneous results. In use, the measuring section must be aligned with the local flow direction.Static pressure probes or any variety of forms are available commercially in sizes as small as 1.5 mm (1/16 in.) in diameter. The stagnation pressure is obtained when a flowing fluid is decelerated to zero speed by a frictionless process. In incompressible flow, the Bernoulli Equation can be used to relate changes in speed and pressure along a streamline for such a process. Neglecting elevation differences, Equation 3.7 becomes

If the static pressure is p at a point in the flow where the speed is v, then the stagnation pressure, Po, where the stagnation speed, Vo, is zero, therefore,

Equation 3.11 is a mathematical statement of stagnation pressure, valid for incompressible flow. The term V generally is the dynamic pressure. Solving the dynamic pressure gives:

Thus, if the stagnation pressure and the static pressure could be measured at a point, Equation 3.13 would give the local flow speed.

Figure 5.2: Measurement of Stagnation Pressure

(a) Total Head Tube Used with Wall Static Tap

(b) Pitot-Static TubeFigure 5.3: Simultaneous Measurement of Stagnation and Static Pressures

Stagnation pressure is measured in the laboratory using a probe with a hole that faces directly upstream as shown in Figure 5. Such a probe is called a stagnation pressure probe (hypodermic probe) or Pitot (pronounced pea-toe) tube. Again, the measuring section must be aligned with the local flow direction.

We have seen that static pressure at a point can be measured with a static pressure tap or probe (Figure 4). If we know the stagnation pressure at the same point, then the flow speed could be computed from Equation 3.14. Two possible experimental setups are shown in Figure 6.

In Figure 6(a), the static pressure corresponding to point A is read from the wall static pressure tap. The stagnation pressure is measured directly at A by the total head tube, as shown. (The stem of the total head tube is placed downstream from the measurement location to minimize disturbance of the local flow)

Two probes often are combined, as in the Pitot-static tube shown in Figure 6(b). The inner tube is used to measure the stagnation pressure at point B, whereas the static pressure at C is sensed using the tapping on the wall. In flow fields where the static pressure variation in the streamwise direction is small, the Pitot-static tube may be used to infer the speed at point B in the flow by assuming pB =pC and using Equation 3.14. (Note that when pB pC, this procedure will give erroneous results)

Remember that the Bernoulli equation applies only for incompressible flow (Mach number, M 0.3).

5.2 APPARATUS

The unit consists of the followings: a) Venturi: The venturi meter is made of transparent acrylic with the following specifications: Throat diameter : 16 mm Upstream Diameter : 26 mm Designed Flow Rate : 20 LPM

b) Manometer: There are eight manometer tubes; each length 320 mm, for static pressure and total head measuring along the venturi meter. The manometer tubes are connected to an air bleed screw for air release as well as tubes pressurization. c) Baseboard: The baseboard is epoxy coated and designed with 4 height adjustable stands to level the venturi meter. d) Discharge valve: One discharge valve is installed at the venturi discharge section for flow rate control. e) Connections: Hose Connections are installed at both inlet and outlet.

5.3 MAINTENANCE AND SAFETY PRECAUTIONS

1.It is important to drain all water from the apparatus when not in use. The apparatus should be stored properly to prevent damage.2.Any manometer tube, which does not fill with water or slow fill, indicated that tapping or connection of the manometer is blocked. To remove the obstacle, disconnect the flexible connection tube and blow through.3.The apparatus should not be exposed to any shock and stresses.4.Always wear protective clothing, shoes, helmet and goggles throughout the laboratory session.5.Always run the experiment after fully understand the unit and procedure.

5.4 EXPERIMENTAL PROCEDURES:5.4.1 Experiment 1: Discharge Coefficient DeterminationObjectiveTo determine the discharge coefficient of the venture meter.

Procedures1. The hypodermic tube is withdrawn from the test section.2. The discharge valve is adjusted to the maximum measureable flow rate of the venture.3. The water flow rate is measured using volumetric method and the manometers reading are recorded.4. The steps are repeated for a few data collection.5. From results, the discharge coefficient, Cd is determined.6. The results of actual flow rate and ideal flow rate are compared.

5.4.2 Experiment 2: Bernoullis Theorem DemonstrationObjectiveTo demonstrate Bernoullis theorem.

Procedures1. The discharge valve is adjusted to a high measureable flow rate.2. After the level stabilizes, the water flow rate is measured using volumetric method.3. The hypodermic tube (total head measuring) connected to manometer #G is gently slide, so that its end reaches the cross section of the Venturi tube at #A. Wait for some time and the readings from the manometers are noted down.4. Steps 1 to 3 are repeated with different flow rates.5. The difference between two calculated velocities are determined using Bernoullis equation and using continuity equation.6. The comparison between the two velocities are discussed.

B) RESULTS, DISCUSSION, CONCLUSION, OPEN ENDED QUESTIONS, REFERENCESName: Mohammad Iskandar Zulkarnain b. Roslan Matric number: 42188

6.0 RESULTS6.0.1 Discharge Coefficient DeterminationIn this experiment, the volume of water used is 5L. The actual flow rate, Qa can be determined asQa = Actual flow rate, Qa for each time:Qa1= 5L / 33.22s= 0.1505 L/sQa2= 5L / 35.66s= 0.1402 L/sQa3= 5L / 38.28s= 0.1306 L/sQa4= 5L / 40.11s= 0.1247 L/s

Qa (L/s)hahbhchdhehfTime (s)

Water head (mm)

0.150523122920422022523033.22

0.140224123821222723023635.66

0.130624424021323023423838.28

0.124725124621823724124540.11

Data analysis:Throat diameter, D3 (mm)= 16Upstream diameter, D1 (mm)= 26Throat area, At (m2)= 2.011 x 10-4Upstream diameter, A (m2)= 5.309 x 10-4g (m/s2)= 9.81 (kg/m3)= 1000

The difference between the maximum value and minimum value of water head is ha hc, as observed in the table above.(ha hc)1= 231 mm 206 mm= 25 mm(ha - hc)2= 241 mm 218 mm= 23 mm(ha hc)3= 244 mm 223 mm= 21 mm(ha hc)4= 251 mm 233 mm= 18 mm

Actual flow rate, Qi is calculated by using equation (3).

And since Z1 = Z2 in this apparatus, this makes Z1 Z2 = 0For (ha hc)1 = 27 (10-3) m,(Qi)1= 2.011(10-4) [ = 1.5815 (10-4) m3/s= 1.5815 (10-4) m3/s (1000L/1m3)= 0.1582 L/s

For (ha hc)2 = 23 (10-3) m,(Qi)2= 2.011(10-4) [ = 1.4597 (10-4) m3/s= 1.4597 (10-4) m3/s (1000L/1m3)= 0.1459 L/sFor (ha hc)3 = 21 (10-3) m,(Qi)3= 2.011(10-4) [ = 1.3948 (10-4) m3/s= 1.3948 (10-4) m3/s (1000L/1m3)= 0.1395 L/sFor (ha hc)3 = 18 (10-3) m,(Qi)3= 2.011(10-4) [ = 1.2913 (10-4) m3/s= 1.2913 (10-4) m3/s (1000L/1m3)= 0.1291 L/s

ha - hc (10-3 m)

Ideal Flow Rate, Qi (L/s)Actual Flow Rate, Qa (L/s)

270.15820.1505

290.14590.1402

270.13950.1306

270.12910.1247

Graph 1: Graph of Qa against Qi

6.0.2 Bernoullis Theorem Demonstration6.0.2.1 Sub-experiment 1Volume= 5LTime= 24.47s Cross SectionUsing Bernoullis EquationUsing Continuity EquationDifference

ih*=hG (mm)hi (mm)ViB (m/s)Ai (x10-4 m2)ViC (m/s)V (m/s)Percentage error (%)

ha2852790.34315.309 0.26970.073421.39

hb2802750.31323.6640.3908-0.077624.77

hc2782241.02932.011 0.71210.317230.82

hd2732450.80463.142 0.45580.348843.35

he2712500.64193.801 0.37670.265241.31

hf2682560.48525.3090.26970.215544.49

SAMPLE CALCULATIONSArea of Tapping Points A to F.Area= d2Aa= 2= 5.309 E-4 m2

Velocity of water by Bernoullis equation, ViBViB= ViB(a)= = 0.3431 m/s

Velocity of water by Continuity equation, ViCViC=

Average ideal flow rate, Qi,avgQi,avg= [ (0.1582+0.1459+0.1395+0.1291) L/s ] / 4= 0.1432 L/s= 0.1432 L/s [1m3/1000L]= 1.432 E-4 m3/sViC(a)= [1.432 E-4 m3/s] / 5.309E-4 m2= 0.2697 m/sDifference in velocities of Bernoullis equation and Continuity equation V = ViB - ViCVa= 0.3431 m/s 0.2697 m/s= 0.0734 m/sPercentage error, PEa = x 100% = 21.39%

6.0.2.2 Sub-experiment 2Volume= 5LTime= 10.35s Cross SectionUsing Bernoullis EquationUsing Continuity EquationDifference

ih*=hG (mm)hi (mm)ViB (m/s)Ai (x10-4 m2)ViC (m/s)V (m/s)Percentage error (%)

ha2742640.44295.309 0.26970.173239.11

hb2712520.61063.6640.39080.219835.99

hc2691971.18852.011 0.71210.476440.08

hd2642290.82873.142 0.45580.372944.99

he2642350.75433.801 0.37670.377650.06

hf2602410.61065.3090.26970.340955.83

SAMPLE CALCULATIONSVelocity of water by Bernoullis equation, ViBViB= ViB(a)= = 0.4429 m/s

Velocity of water by Continuity equation, ViCViC=

Average ideal flow rate, Qi,avgQi,avg= [ (0.1582+0.1459+0.1395+0.1291) L/s ] / 4= 0.1432 L/s= 0.1432 L/s [1m3/1000L]= 1.432 E-4 m3/sViC(a)= [1.432 E-4 m3/s] / 5.309E-4 m2= 0.2697 m/sDifference in velocities of Bernoullis equation and Continuity equation V = ViB - ViCVa= 0.4429 m/s 0.2697 m/s= 0.1732 m/sPercentage error, PEa = x 100% = 39.11%

6.0.2.3 Sub-experiment 3Volume= 5LTime= s Cross SectionUsing Bernoullis EquationUsing Continuity EquationDifference

ih*=hG (mm)hi (mm)ViB (m/s)Ai (x10-4 m2)ViC (m/s)V (m/s)Percentage error (%)

ha2182130.31325.309 0.26970.043513.89

hb2171910.71423.6640.39080.323445.28

hc2151221.35082.011 0.71210.638747.28

hd2111630.97043.142 0.45580.514653.03

he2101730.85203.801 0.37670.475355.79

hf2091840.70045.3090.26970.430761.49

SAMPLE CALCULATIONSVelocity of water by Bernoullis equation, ViBViB= ViB(a)= = 0.3132 m/s

Velocity of water by Continuity equation, ViCViC=

Average ideal flow rate, Qi,avgQi,avg= [ (0.1582+0.1459+0.1395+0.1291) L/s ] / 4= 0.1432 L/s= 0.1432 L/s [1m3/1000L]= 1.432 E-4 m3/sViC(a)= [1.432 E-4 m3/s] / 5.309E-4 m2= 0.2697 m/sDifference in velocities of Bernoullis equation and Continuity equation V = ViB - ViCVa= 0.3132 m/s 0.2697 m/s= 0.0435 m/sPercentage error, PEa = x 100% = 13.89%

7.0 DISCUSSIONS1. From the graph of actual flow rate, Qa against ideal flow rate, Qi on the first experiment (Discharge Coefficient Determination), it shows that the discharge coefficient, Cd is 0.9177, which is in between the ideal Cd values which are from 0.9 to 0.99. 2. From the second experiment (Bernoullis Theorem Demonstration), there are too many significant errors such as the large values of percentage errors and the significant difference in values between the velocities calculated using Bernoullis equation and Continuity equation. One of the reason behind this is that the apparatus is leaked when we conducted the experiment. This leakage of water contributes to errors obtained in calculations. Besides, the manometer reading fluctuates frequently during the experiment. This makes the data reading inconsistent and not accurate, thus the errors are significant.

8.0 CONCLUSIONSFrom the first experiment, the discharge coefficient is determined to be 0.9177, based on the graph plotted. This value is in the range of standard discharge coefficient value, which is in between 0.9 to 0.99.For the second experiment, Bernoullis theorem can be demonstrated and the data are obtained. However, from the calculations, the errors are quite significant as most of the percentage errors are more than 20%. These are due to the errors and technical problems encountered during the experiment. In order to reduce the errors, the apparatus should not leak and no air bubbles must present on the tube.

9.0 OPEN-ENDED QUESTIONS1. One of the differences between the stagnation pressure and dynamic pressure is that stagnation pressure is the summation of static pressure and dynamic pressure of a fluid. Therefore, stagnation pressure indicates both static and dynamic pressure at a point of a fluid.The equation of stagnation pressure, Pstag = P + V2/g whereby, as indicated just now, P is static pressure and V2/g is dynamic pressure, and the unit is given in kilopascal (kPa). By definition, stagnation pressure is the static pressure at the stagnation point, where V = 0.

2. There are a few instruments that can be used to measure the airflow rate in a duct, such as pitot tube (stagnation tube), orifice plate and radial turbine flowmeter. Pitot tube is one of the most conventional device to measure the flow of fluid by measuring the pressure at the stagnation point. The velocity is then calculated using the formula V = This device is can be easily found, but the drawback of using pitot tube is that it is restricted to point measuring only, as it only measures the pressure at a point. This is impractical to measure the pressure at different points, especially when it involves a lot of points.Next, the orifice plate can also be used to measure the airflow rate in a duct. The fluid flow is measured through the difference in pressure from the upstream side to the downstream side of a moderately congested pipe. The advantages of orifice plate are it is simple, relatively inexpensive and is universal in nature. However, the orifice plate is not suitable for low fluid flow rate, as it will cause inaccurate calculations.

Orifice plate

Besides, a radial turbine flow meter can be used to measure the airflow rate in a duct. The working principle of this flow meter is that it is inserted in through the side of a pipe and the flow across the turbine blades will generate a measurement which is related to general flow in the pipe. The advantage of radial turbine flow meter is it works very well in smaller cross-sectional area pipes or ducts. Nevertheless, this device is not commonly found and is relatively expensive.

Radial turbine flow meterI would choose pitot tube to measure the airflow rate in a duct, because it is easy to handle, inexpensive and less calculations need to be done.10.0 REFERENCES1.Cengel, Y.A., Cimbala, J.M. (2014). Fluid Mechanics: Fundamentals and Applications (3rd Edition in SI Unit). McGraw-Hill Education.2.Roberson, J. A. (2013). Engineering Fluid Mechanics (Tenth Edition). John Wiley & Sons, Inc.3.White, F. M. (2008). Fluid Mechanics (Sixth Edition). McGraw-Hill.4.http://www.engineeringtoolbox.com/flow-meters-d_493.html5.http://web.mst.edu/~cottrell/ME240/Resources/Fluid_Flow/Fluid_flow.pdf