Bernoulli Equation

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NO. SECTIONS PAGE 1 Abstract/summary 2 2 Introduction 3 3 Aims/objectives 4 4 Theory 4-7 5 Apparatus 8 6 Experimental procedure 9-10 7 Results 11-13 8 Sample of calculations 14-16 9 Discussions 17-18 10 Conclusions 19 11 Recommendations 19 12 References 20 13 Appendices 21-22 1

Transcript of Bernoulli Equation

NO.SECTIONSPAGE

1Abstract/summary2

2Introduction3

3Aims/objectives4

4Theory4-7

5Apparatus8

6Experimental procedure9-10

7Results11-13

8Sample of calculations14-16

9Discussions17-18

10Conclusions19

11Recommendations19

12References20

13Appendices21-22

1.0) ABSTRACT

The main objectives in this experiment is to investigate the validity of Bernoulli equation when applied to a steady flow of water in tapered duct and to measure the flow rate and both static and total pressure heads in a rigid convergent or divergent tube known geometry range of steady flow rates. In this experiment,the apparatus used is Bernoullis Theorem Demonstration Apparatus and this apparatus contains of many parts,which are venturi meter,pad of monometer tubes,pump,water tank equipped with valves water controller and water hosts and tubes.The experiment is done after the level of pressure for all the tubes are maintained.Then,the valve is opened to make the flow rate different for three times. The reading of pressure different was taken for each tubes A,B,C,D,E, and F. The time to collect 3L of water in the tank was recorded. Lastly, the flow rate, velocity, dynamic head, and total head were calculated using the readings we got from the experiment and from the data given for both convergent and divergent. Based on the results taken,it has been analysed that the velocity of convergent flow is increasing, whereas the velocity of divergent flow is the opposite, whereby the velocity decreased, since the water flow from a narrow area to a wider area. Therefore,bernoullis principle is valid for a steady flow in rigid convergent and divergent tube of known geometry for a range of steady flow rates and all the datas are as well calculated.The experiment was completed and successfully conducted.

2.0) INTRODUCTION

This lab was designed to helped student to verify the validity of Bernoullis equation for fluid flow and demonstrating the relationship between pressure head and kinetic head. Bernoulli's Principle is a physical principle formulated that states that "as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. Bernoullis Principle can be demonstrated by the Bernoulli equation. The Bernoulli equation is an approximate relation between pressure, velocity, and elevation. However, Bernoullis Principle can only be applied under certain conditions. The conditions to which Bernoullis equation applies are the fluid must be frictionless (inviscid) and of constant density; the flow must be steady, and the relation holds in general for single streamlines. The objectives of this experiment was to verify experimentally the validity of Bernoullis equation for fluid flow, to verify Bernoulli's equation by demonstrating the relationship between pressure head and kinetic head and to measure flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. Performing these objectives helped me to understand the Bernoulli equation and relationship between pressure head and kinetic head by calculated the flow rate, velocity, dynamic head, and total head using the readings we got from the experiment.Bernoulli's principle can be explained in terms of the law of conservation ofenergy. As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. At the same time, the work done by corresponding volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. Since the speed is greater in the narrower pipe, the kinetic energy of that volume is greater. Then, by the law of conservation of energy, this increase in kinetic energy must bebalanced bya decrease in thepressure-volumeproduct, or,since the volumesareequal,by a decrease in pressure. The Bernoulli equation:Kinetic energy + potential energy + flow energy = constant

3.0) OBJECTIVESIn this experiment, there are three objectives need to be achieved at the end of the experiment which is:1. To verify experimentally the validity of Bernoullis equation for fluid flow2. To verify Bernoulli's equation by demonstrating the relationship between pressure head and kinetic head.3. To measure flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates.

4.0) THEORYThe well-known Bernoulli equation is derived under the following assumptions: The liquid is incompressible. The liquid is non-viscous. The flow is steady and the velocity of the liquid is less than the critical velocity forthe liquid. There is no loss of energy due to friction.

Then, it is expressed with the following equation:

Where (in SI units):p= fluid static pressure at the cross section in N/m2= density of the flowing fluid in kg/m3g= acceleration due to gravity in m/s2 (its value is 9.81 m/s2=9810 mm/s2)v= mean velocity of fluid flow at the cross section in m/sz= elevation head of the centre ofthe cross section with respect to a datum z=0h*= total (stagnation) head in m

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 isconstant (based on the assumptions given) (Munson, 1998).

In this experimental, the centre line of the entire cross sections we are considering lie on the same horizontal plane (which we may choose as the datum, z=0), and thus, all the zvalues are zeros so thatthe above equation reduces to:

Pressure head is a term used in fluid mechanics to represent the internal energy of a fluid due to the pressure exerted on its container. It may also be called static pressure head orsimply static head (but not static head pressure). It is mathematically expressed as:

Where: is pressure head (Length, typically in units ofm);p is fluid pressure(Forceper unit Area, often ask kPa units); and is the specific weight(Weightper unit volume, typically Nm3units) is the density of the fluid (Massper unit volume ,typically kgm3)gis acceleration due to gravity (rate of change of velocity, given in ms2)

Venture Meter The venture 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 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 (Sylvia, 2003).

Assume incompressible flow and no frictional losses, from Bernoullis Equation

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

Ideally,

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,

Based on Douglas (1995), in metering practice, this non-ideality is accounted by insertion of an experimentally determined discharged coefficient,Cd that is termed as the coefficient of discharge. With Z1 = Z2 in this apparatus, the discharge coefficient is determined as below:

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

FACULTY OF CHEMICAL ENGINEERINGNovember 9, 2012

5.0) APPATARUS1. Bernoullis Theorem demonstration unit: Venturi Manometer Baseboard Discharge valve Connections Hydraulic bench Water tank 2. Stopwatch 3. Water

6.0) PROCEDUREGeneral Start-up ProceduresThe Bernoullis Theorem Demonstration (Model: FM 24) is supplied ready for use and only requires connection to the Hydraulic Bench (Model : FM110 ) as follows:1. The clear acrylic test section is ensured installed with the converging section upstream. The unions tightened and checked. To dismantle the test suction, the total pressure probe was withdrawn fully before the couplings being released.2. The apparatus located on the flat top of the bench.3. A spirit level attached to the baseboard and the unit was level on the top of the bench by adjusting the feet.4. Water filled into the volumetric tank of the hydraulic bench until approximately 90% full.5. The flexible inlet tube connected using the quick release coupling in the bed channel.6. The flexible hose connected to the outlet and it directed into the channel.7. The outlet flow control partially opened at the Bernoullis Theorem Demonstration Unit.8. The bench flow control valve, V1 fully closed then the pump switched on.9. V1 was gradually opened and water allowed filling until all air has been expelled from the system.10. All the trapped bubbles checked in the glass tube or plastic transfer tube. To remove air bubbles, the air were bleed out using a pen or screw driver to press the air bleed valve at the top right side of the manometer board.11. Water flowing into the venturi and discharge into the collection tank of hydraulic bench.12. The water flow rate proceeds to increased it. When all the water flow was steady and there were no trapped bubbles, the discharge valve closed to reduce the flow to the maximum measurable flow rate.13. Water level in the manometer was in different heights.14. V1 and outlet control valve adjusted to obtain the flow through the test section and the static pressure profile observed along the converging and diverging sections is indicated on its respective manometers. The total head pressure along the venture tube being 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 rate measured by using the volumetric tank with a stop watch.

General Shut Down Procedures1. Water supply valve and venturi discharge valve closed.2. The water supply pump closed.3. Water drain off from the unit when not in use.Bernoullis Theorem DemonstrationObjectives: To demonstrate Bernoullis TheoremProcedures:1. The general start-up procedures was performed 2. All manometers checked that properly connected to the corresponding pressure taps and were air bubble free.3. The discharge valve adjusted to high measureable flow rate.4. After the level stabilized, the water flow rate measured using volumetric method.5. The hypodermic tube (total head measuring) connected to manometer #H was gently slide , so that its end reached the cross section of the venturi tube at #A. After some time the readings from manometer #H and #A noted down. The reading shown by manometer #H was the sum of the static head and velocity heads, i.e the total (or stagnation) head (h*), because the hypodermic tube was held against the flow of the fluid forcing it to a stop (zero velocity). The reading in manometer #A measures just the pressure head (hi) because it was connect to the venturi tube pressure tap, which does not obstruct the flow, thus measuring the flow static pressure.6. Step 5 repeated for other cross sections (#B,#C,#D,#E,#F).7. Step 3 to 6 repeated with three other decreasing flow rates by regulating the venturi discharge valve.8. The velocity , ViB calculated using the Bernoullis equation where ;

9. The velocity , Vic using continuity equation where ;

10. The difference between two calculated velocities was determined.7.0) RESULTS

Experiment 1Volume (L)3

Average Time (sec)10.94

Flow Rate (LPM)16.67

Cross sectionUsing Bernoulli equationUsing Continuity equationdifference

#h* = hH(mm)hi(mm)ViB = (m/s)Ai = (m2)Vic =

(m/s)ViB - Vic(m/s)

A1951700.7005.31 x 10-40.5160.184

B1861650.6413.66 x 10-40.7490.108

C1831560.7282.01 x 10-41.3630.635

D1791141.1293.14 x 10-40.8730.256

E1741300.9293.80 x 10-40.7210.208

F1711430.7415.31 x 10-40.5160.225

Experiment 2Volume (L)3

Average Time (sec)9.53

Flow Rate (LPM)18.89

Cross sectionUsing Bernoulli equationUsing Continuity equationdifference

#h* = hH(mm)hi(mm)ViB = (m/s)Ai = (m2)Vic =

(m/s)ViB - Vic(m/s)

A2402000.8865.31 x 10-40.5930.293

B2331880.9403.66 x 10-40.8610.079

C2301681.1032.01 x 10-41.5670.464

D224741.7043.14 x 10-41.0030.701

E2241341.2993.80 x 10-40.8290.470

F2161401.2215.31 x 10-40.5930.628

Experiment 3Volume (L)3

Average Time (sec)7.53

Flow Rate (LPM)23.03

Cross sectionUsing Bernoulli equationUsing Continuity equationdifference

#h* = hH(mm)hi(mm)ViB = (m/s)Ai = (m2)Vic =

(m/s)ViB - Vic(m/s)

A2452020.9195.31 x 10-40.7500.169

B2351550.9903.66 x 10-41.0870.097

C2321621.1722.01 x 10-41.9800.808

D225561.8213.14 x 10-41.2680.553

E2181781.3293.80 x 10-41.0470.282

F2111561.0395.31 x 10-40.7500.289

8.0) CALCULATIONS

Experiment 1:Flow rate of water =

Sample Calculation (cross section A):

Bernoulli equation:ViB = ViB = ViB = 0.700 m/s

Continuity equation:Ai = Ai = Ai = 5.31 x 10-4 m2Vic = Vic = Vic = 0.516 m/s

Therefore, the difference is = ViB - Vic = 0.700 m/s - 0.516 m/s = 0.184 m/s

Experiment 2:Flow rate of water =

Sample Calculation (cross section A):

Bernoulli equation:ViB = ViB = ViB = 0.886 m/s

Continuity equation:Ai = Ai = Ai = 5.31 x 10-4 m2Vic = Vic = Vic = 0.593 m/s

Therefore, the difference is = ViB - Vic = 0.886 m/s - 0.593 m/s = 0.293 m/s

Experiment 3:Flow rate of water =

Sample Calculation (cross section A):

Bernoulli equation:ViB = ViB = ViB = 0.919 m/s

Continuity equation:Ai = Ai = Ai = 5.31 x 10-4 m2Vic = Vic = Vic = 0.593 m/s

Therefore, the difference is = ViB - Vic = 0.919 m/s - 0.593 m/s = 0.293 m/s

9.0) DISCUSSION

The objectives of this experiment, is to demonstrate the Bernoullis Theorem using Bernoullis Theorem Demonstration unit (model: FM24), to verify experimentally the validity of Bernoullis equation for fluid flow, to verify Bernoulli's equation by demonstrating the relationship between pressure head and kinetic head, and to measure flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates.To achieve the objectives of this experiment, Bernoullis theorem demonstration apparatus along with the hydraulic bench were used. This instrument was combined with a venturi meter and the pad of manometer tubes which indicate the pressure of hA to hF. A venturi is basically a converging-diverging section (like an hourglass), typicallyplaced between tubeorductsectionswithfixedcross-sectionalarea.Theflowratesthrough the venturi meter can be related to pressure measurements by using Bernoullis equation. From the result obtained through this experiment, it is been observed that when the pressure difference increase, the flow rates of the water increase and thus the velocities also increase for both convergent and divergent flow. The result show a rise at each manometer tubes when the pressure difference increases. As fluid flows from a wider pipe to a narrower one, the velocity of the flowing fluid increases. This is shown in all the result stables, where the velocity of water that flows in the tapered duct increases as the duct area decreases, regardless of the pressure difference and type of flow of each result taken. From the analysis of the results, it can be concluded that the velocity of waterdecrease as the water flow rate decrease.From the analysis of the results, it can be concluded that the velocity of water decrease as the water flow rate decrease. For instance, the velocity difference at cross section A for water flow rate of 23.03 LPM is bigger than velocity difference at cross section A for water flow rate of 16.67 LPM and 18.89 LPM. It also can be concluded that the diameter of the tube will affect the differences in velocity as a bigger tube will cause the differences in velocity become bigger while the smaller tube cause the velocity differences between ViB and Vic to be smaller.

There must be some parallax and zero error occurs when taking the measurement of each data. One of them is, the observer must have not read the level of static head properly. This will affect the results of the experiment. Moreover, the eyes are not perpendicular to the water level on the manometer. Therefore, there are some minor effects on the calculations due to the errors and this can be seen from the result obtained which there is few value calculatedget negative values for ViB - Vic. The eyes must be perpendicular to the water level on the manometer so that the parallax error can be reduced in this experiment. Next, the valve and bleed screw should regulate smoothly to reduce the errors. Another error is the bubbles left in the manometer. The bubbles must be removed in the manometer before start the experiment so that it will not affect the results of the experiment. Last but not least, the valve should be controlled slowly to maintain the pressure difference and make sure there is no leakage along the tube to avoid the water flowing out.Therefore, it can be concluded that the Bernoullis equation is valid when applied to steady flow of water in tapered duct and absolute velocity values increase along the same channel. Although the experiment proof that the Bernoullis equation is valid for both flow butthevaluesobtainmightbeslightlydifferfromtheactualvalue.Thisisbecausethereis some error maybe happen during the experiment is done. While taking the reading of the manometer, there might be possibility that the eye position of the readers is not parallel to the scale. Thus, this error will contribute to the different in the values obtained. Other than that, the readers must take the accurate reading from the manometers. In order to get the accurate value, the water level must be let to be really stable. Thus, a patient is needed in order to run this experiment successfully because sometimes the way the experiment is conduct may influence the result of the experiment.

10.0) CONCLUSION

The results show the reading of each manometer tubes increase when the pressure difference increases. From the result obtained, we can conclude that the Bernoullis equation is valid for convergent and divergent flow asboth of it does obey the equation. For both flow, as the pressure difference increase, the time taken for 3L water collected increase and the flow rates of the water also increase. Thus, as the velocity of the same channel increase, the total head pressure also increase for both convergent and divergent flow. The conservation of mass results in what's called the continuity equation.Through the results,it can be proved that when the area of the tube increases,the velocity that have been calculated by using continuity equation will decreases.It means that the area and the velocity are inversely proportional to each other.

11.0) RECOMENDATION

1) Make sure the trap bubbles must be removing first before start running the experiment.2)Repeat the experiment for several times toget the average values in order to getmoreaccurateresults.3) The valve must be control carefully to maintain the constant values of the pressure difference as it is quite difficult to control.4) The eye position ofthe observer must beparallel tothe water meniscus whentakingthe reading at the manometers to avoid parallax error.5) The time keeper must be alert with the rising of water volume to avoid errorandmustbe onlya personwho taking the time.VI.6) The leakage of water inthe instrument must be avoided.

12) REFERENCES

1) B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals ofFluid Mechanics 3rd ed., 1998, Wileyand Sons, New York

2) Sylvia et al (2003). Bernoulli Theorem Assignment Help.Retrieved November 20, 2013 fromhttp://www.transtutors.com/physics-homework-help/fluid-mechanics/bernoullis- theorem.aspx3) Douglas. J.F., Gasiorek. J.M. and Swaffield, Fluid Mechanics,3rd edition, (1995), Longmans Singapore Publisher.

4) Johnson et al (2000). Bernoulli Principle.Retrieved November 22, 2013 from http://en.wikipedia.org/wiki/Bernoulli's_principle5) Giles, R.V., Evett, J.B. and Cheng Lui, Schaumms OutlineSeries Theory and Problems of Fluid Mechanics andHydraulic,(1994), McGraw-Hill intl

13) APPENDIX

Bernoullis Theorem demonstration unit

Flowrate of water

Water Tank

Baseboard22