demonstration of bernoulli's theorem

Thermal-Fluid Engineering Lab Semester 6 2010/2011

1.0 INTRODUCTION

Bernoulli’s Theorem Demonstration (Model: FM 24) apparatus consists of a classical Venturi made of clear acrylic. A series of wall tappings allow measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. These tappings are connected to a manometer bank incorporating a manifold with air bleed valve. Pressurization of the manometers is facilitated by a hand pump.

This unit has been designed to be used with a Hydraulics Bench for students to study the characteristics of flow through both converging and diverging sections. During the experiment, water is fed through a hose connector and students may control the flow rate of the water by adjusting a flow regulator valve at the outlet of the test section.

The venturi can be demonstrated as a means of flow measurement and the discharge coefficient can be determined. This test section can be used to demonstrate those circumstances to which Bernoulli’s Theorem may be applied as well as in other circumstances where the theorem is not sufficient to describe the fluid behavior.

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2.0 GENERAL DESCRIPTION

The unit is mounted on a base board which is to be placed on top of the Hydraulic Bench (Model: FM110). This base board has four adjustable feet to level the apparatus.

The main test section is an accurately machined acrylic venturi of varying circular cross section. It is provided with a number of side hole pressure tappings, which are connected to the manometer tubes on the rig. These tappings allow the measurement of static pressure head simultaneously at each of 6 sections. The tapping positions and the test section diameters are shown in Appendix A. The test section incorporates two unions, one at either end, to facilitate reversal for convergent or divergent testing as illustrated in Figure 1 and Figure 2.

Figure 1: Front View of Bernoulli’s Theorem Demonstration Unit (Model: FM24)

Figure 2: Top View of Bernoulli’s Theorem Demonstration Unit (Model: FM24)

A hypodermic tube, the total pressure head probe, is provided which may be positioned to read the total pressure head at any section of the duct. This total pressure head probe may be moved after slacking the gland nut; this nut should

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be re-tightened by hand after adjustment. An additional tapping is provided to facilitate setting up. All eight pressure tapings are connected to a bank of pressurized manometer tubes. Pressurization of the manometers is facilitated by connecting any hand pump to the inlet valve on the manometer manifold.

The unit is connected to the hydraulic bench using flexible hoses. The hoses and the connections are equipped with rapid action couplings. The flexible hose attached to the outlet pipe which should be directed to the volumetric measuring tank on the hydraulics bench. A flow control valve is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve and the bench supply control valve.

Please familiarize with the unit before operating the unit. The unit consists of the followings:

a) VenturiThe venturi meter is made of transparent acrylic with the following specifications:Throat diameter : 16 mmUpstream Diameter : 26 mmDesigned Flow Rate : 20 LPM

b) ManometerThere 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) BaseboardThe baseboard is epoxy coated and designed with 4 height adjustable stands to level the venturi meter.

d) Discharge valveOne discharge valve is installed at the venturi discharge section for flow rate control.

e) ConnectionsHose Connections are installed at both inlet and outlet.

f) Hydraulic BenchSump tank : 120 litresVolumetric tank : 100 litresCentrifugal pump : 0.37 kW, 50 LPM

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2.1 Parts Identification

Figure 3: Parts Identification Diagram

1. Manometer Tubes 6. Flow Control Valve

2. Test Section 7. Gland Nut

3. Water Inlet 8. Hypodermic Probe

4. Unions 9. Adjustable Feet

5. Air Bleed Screw

4

5

1

2

3

4

6

7

8

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3.0 SUMMARY OF THEORY

3.1 Derivation Using Streamline Coordinates

Euler’s equation for steady flow along a streamline is

(3.1)

If a fluid particle moves a distance, ds, along a streamline,

(3.2)

(3.3)

(3.4)

Thus, after multiplying Equation 3.1 by ds,

(3.5)

Integration of this equation gives:

(3.6)

The relation between pressure and density must be applied in this equation. For the special case of incompressible flow, ρ = constant, and Equation 3.6 becomes the Bernoulli’s Equation.

(3.7)

Restrictions:i. Steady flowii. Incompressible flowiii. Frictionless flowiv. Flow along a streamline

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3.2 Bernoulli’s 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:

(3.8)

Where,

p = Fluid static pressure at the cross sectionρ = 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 Bernoulli’s 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 centerline 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:

(3.9)

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.

3.3 Static, Stagnation and Dynamic Pressures

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The pressure, p, which we have used in deriving the Bernoulli’s 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.

(a) Wall Pressure Tapping

(b) Wall Pressure Tapping

Figure 4: Measurement of Static Pressure

In 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

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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

(3.10)

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, may be computed from

(3.11)

Therefore,

(3.12)

Equation 3.12 is a mathematical statement of stagnation pressure, valid for incompressible flow. The term ½ ρV² generally is the dynamic pressure. Solving the dynamic pressure gives:

(3.13)

Or

(3.14)

Thus, if the stagnation pressure and the static pressure could be measured at a point, Equation 3.14 would give the local flow speed.

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Figure 5: Measurement of Stagnation Pressure

(a) Total Head Tube Used with Wall Static Tap

(a) Pitot-Static Tube

Figure 6: 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.

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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, while 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).

Note:

(3.15)

where, u = fluid velocityc = sonic velocity

3.4 Pressure Varies Along the Pipe

A number of factors can cause for pressure to vary along the pipe such as:

▪ Friction from the pipe’s inner surface,▪ The diameter of the pipe; if it is small the pressure is lower

because the velocity is increased (Bernoulli’s Theory),▪ Density of the fluid in the pipe,▪ The height of the pipe at which the pipe stands or the height at

which the flow through i.e. gravity,▪ Turbulence of the fluid

3.5 Venturi Meter

The venturi meter consists of a venturi tube and differential pressure gauge. The venturi tube has a converging portion, a throat and a diverging portion as shown in the figure below. The function of the converging portion is to increase the velocity of the fluid and lower its static pressure. A pressure difference between inlet and throat is thus

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developed, which pressure difference is correlated with the rate of discharge. The diverging cone serves to change the area of the stream back to the entrance area and convert velocity head into pressure head.

Figure 7: The Venturi Tube

Assume incompressible flow and no frictional losses, from Bernoulli’s Equation

(3.16)

Use of the continuity Equation Q = A1V1 = A2V2, equation (1) becomes

(3.17)

Ideally,

(3.18)

However, in the case of real fluid flow, the flow rate will be expected to be less than that given by equation (3.18) because of frictional effects and consequent head loss between inlet and throat. Therefore,

(3.19)

In metering practice, this non-ideality is accounted by insertion of an experimentally determined discharge coefficient, Cd that is termed as the

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coefficient of discharge. With Z1 = Z2 in this apparatus, the discharge coefficient is determined as follow:

(3.20)

Discharge coefficient, Cd usually lies in the range between 0.9 and 0.99.

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4.0 EXPERIMENTAL PROCEDURES

4.1 General Start-up Procedures

The Bernoulli’s Theorem Demonstration (Model: FM 24) is supplied ready for use and only requires connection to the Hydraulic Bench (Model: FM 110) as follows:

1. Ensure that the clear acrylic test section is installed with the converging section upstream. Also check that the unions are tighten (hand tight only). If necessary to dismantle the test section then the total pressure probe must be withdrawn fully (but not pulled out of its guide in the downstream coupling) before releasing the couplings.

2. Locate the apparatus on the flat top of the bench.

3. Attach a spirit level to baseboard and level the unit on top of the bench by adjusting the feet.

4. Fill water into the volumetric tank of the hydraulic bench until approximately 90% full.

5. Connect the flexible inlet tube using the quick release coupling in the bed of the channel.

6. Connect a flexible hose to the outlet and make sure that it is directed into the channel.

7. Partially open the outlet flow control valve at the Bernoulli’s Theorem Demonstration unit.

8. Fully close the bench flow control valve, V1 then switch on the pump.

9. Gradually open V1 and allow the piping to fill with water until all air has been expelled from the system.

10. Also check for “Trapped Bubbles” in the glass tube or plastic transfer tube. You would need to remove them from the system for better accuracy.Note: To remove air bubbles, you will have to bleed the air out as follow:i. Get a pen or screw driver to press the air bleed valve at the top

right side of manometer board. ii. Press air bleed valve lightly to allow fluid and trapped air to

escape out. (Take care or you will wet yourself or the premise). Allow sufficient time for bleeding until all bubbles escape.

11. At this point, you will see water flowing into the venturi and discharge into the collection tank of hydraulic bench.

12. Proceed to increase the water flowrate. When the flow in the pipe is steady and there is no trapped bubble, start to close the discharge valve to reduce the flow to the maximum measurable flow rate.

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13. You will see that water level in the manometer tubes will begin to display different level of water heights. If the water level in the manometer board is too low where it is out of visible point, open V1 to increase the static pressure. If the water level is too high, open the outlet control valve to lower the static pressure.Note: The water level can be adjusted facilitate by the air bleed valve.

14. Adjust V1 and outlet control valve to obtain a flow through the test section and observe that the static pressure profile along the converging and diverging sections is indicated on its respective manometers. The total head pressure along the venture tube can be measured by traversing the hypodermic tube.Note:The manometer tube connected to the tapping adjacent to the outlet flow control valve is used as a datum when setting up equivalent conditions for flow through test section.

15. The actual flow of water can be measured using the volumetric tank with a stop watch.

4.2 Discharge Coefficient Determination

Objective: To determine the discharge coefficient of the venturi meter

Procedures:

1. Perform the General Start-up Procedures in Section 4.1.2. Withdraw the hypodermic tube from the test section. 3. Adjust the discharge valve to the maximum measurable flow rate of

the venturi. This is achieved when tube 1 and 3 give the maximum observable water head difference.Note: Refer to the venturi specification for the designed flow rate.

4. After the level stabilizes, measure the water flow rate using volumetric method and record the manometers reading.

5. Repeat step 4 with at least three decreasing flow rates by regulating the venturi discharge valve.

6. Obtain the actual flow rate, Qa from the volumetric flow measurement method.

7. Calculate the ideal flow rate, Qi from the head difference between h1

and h3 using Equation 3.18.

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8. Plot Qa Vs Qi and finally obtain the discharge coefficient, Cd which is the slope.

Results:

Volume (L)

Time (s)

Qa (LPM)

Water Head (mm)

hA hB hC hD hE hF

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4.3 Flow Rate Measurement with Venturi Meter

Objective: To measure flow rate with venturi meter

Procedures:

1. Perform the General Start-up Procedures in Section 4.1.2. Withdraw the hypodermic tube from the test section.3. Adjust the discharge valve to a high measurable flow rate.4. After the level stabilizes, measure the water flow rate using

volumetric method and record the manometers reading. 5. Repeat step 4 with three other decreasing flow rates by regulating the

venturi discharge valve.6. Calculate the venturi meter flow rate of each data by applying the

discharge coefficient obtained.7. Compare the volumetric flow rate with venturi meter flow rate.

Results:

Volume (L)

Time (s)

Qa (LPM)

Water Head (mm)hA hB hC hD hE hF

4.4 Bernoulli’s Theorem Demonstration

Objective: To demonstrate Bernoulli’s Theorem

Procedures:

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1. Perform the General Start-up Procedures in Section 4.1.2. Check that all manometer tubings are properly connected to the

corresponding pressure taps and are air-bubble free.3. Adjust the discharge valve to a high measurable flow rate.4. After the level stabilizes, measure the water flow rate using

volumetric method. 5. Gently slide the hypodermic tube (total head measuring) connected

to manometer #G, so that its end reaches the cross section of the Venturi tube at #A. Wait for some time and note down the readings from manometer #G and #A. The reading shown by manometer #G is the sum of the static head and velocity heads, i.e. the total (or stagnation) head (h*), because the hypodermic tube is held against the flow of fluid forcing it to a stop (zero velocity). The reading in manometer #A measures just the pressure head (hi) because it is connected to the Venturi tube pressure tap, which does not obstruct the flow, thus measuring the flow static pressure.

6. Repeat step 5 for other cross sections (#B, #C, #D, #E and #F).7. Repeat step 3 to 6 with three other decreasing flow rates by

regulating the venturi discharge valve.8. Calculate the velocity, ViB using the Bernoulli’s equation where;

9. Calculate the velocity, ViC using the continuity equation where ViC = Qav / Ai

10. Determined the difference between two calculated velocities.

Results:

Cross Section

Using Bernoulli equation Using Continuity equation

Difference

i h*=hG

(mm)hi

(mm)

ViB = √[2*g*(h* - hi )]

(m/s)

Ai = ¶ Di

2 / 4(m2)

ViC =Qav /

Ai

(m/s)

ViB-ViC

(m/s)

ABCDEF

* Please refer to Appendix C for Cross Section Diameter

4.5 General Shut-down Procedures

1. Close water supply valve and venturi discharge valve.

2. Turn off the water supply pump.

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3. Drain off water from the unit when not in use.

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5.0 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, indicates 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 procedures.

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6.0 REFERENCES

Applied Fluid Mechanics 5th Edition, Robert L. Mott, Prentice-Hall

Elementary Fluid Mechanics 7th Edition, Robert L. Street, Gary Z. Watters, John K. Vennard, John Wiley & Sons Inc.

Fluid mechanics 4th Edition, Reynold C. Binder

Fluid Mechanics with applications, Anthony Esposito, Prentice-Hall International Inc.

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APPENDIX C Venturi Meter Drawing

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demonstration of bernoulli's theorem

Documents

static tubeflowsmall holesstem

decreasing flow rates

air bleed valve

outlet control valve

total head measuring

total head tube

venturi discharge valve

static pressure probe