FLUID LABORATORY (BEC205) Bernoullis theoremGroup B2

Table of ContentNoTitlesPages

1.0 Introduction to Bernoulli Theorem 2

1.1Explanation of the Bernoullis equation.3

1.1.1formula of STEADY FLOW ENERGY EQUATION3

1.1.2Bernoullis equation has some restrictions in its applicability3

1.2THE CONCEPT OF BERNOULLIS THEOREM4

1.2.1THEORY4-5

2.0Objective6

2.1General Description6-7

2.2Equipment8

2.3Procedures9

2.3.1General Shut-down Procedures9-10

3.0Data and Analysis10

3.1Calculation10-11

3.2Discussion11

3.3Conclusion11

Appendix12

1.0 Introduction to Bernoulli Theorem : In fluid dynamics, Bernoulli's principle states that for an in viscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamics in 1738. Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential gh) is the same everywhere. Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline. Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.1.1 Explanation of the Bernoullis equation.From the study of physics, it has been established that energy cannot be created nor destroyed. Energy can only be transformed from one from to another. In an ideal fluid, where is no viscosity by H at one point in a steady flow, must be equal to the total energy at any other point in the path of flow provided no energy is added to the fluid or taken from it. H= z++Where H is the total head, z is the elevation head , is the velocity head ,is the pressure head.

1.1.1 formula of STEADY FLOW ENERGY EQUATION

1. Is a statement of the principle of conservation of energy along a streamline. 2. Is one of the most important/useful equations in fluid mechanics.

1.1.2 Bernoullis equation has some restrictions in its applicability:

Flow is steady. Density is constant (which also means the fluid is incompressible) Friction losses are negligible. The equation relates the states at two points along a single streamline, (not conditions on two different streamlines).

1.2 THE CONCEPT OF BERNOULLIS THEOREM:Bernoulli's principle, sometimes known as Bernoulli's equation, holds that for fluids in an ideal state, pressure and density are inversely related: in other words, a slow-moving fluid exerts more pressure than a fast-moving fluid. Since "fluid" in this context applies equally to liquids and gases, the principle has as many applications with regard to airflow as to the flow of liquids. One of the most dramatic everyday examples of Bernoulli's principle can be found in the airplane, which staysaloftdue to pressure differences on the surface of its wing; but the truth of the principle is also illustrated in something asmundaneas a showercurtainthat billowsinward.1.2.1 THEORY:Bernoulli's law indicates that, if an in viscid fluid is flowing along a pipe of varying cross section, then the pressure is lower at constrictions where the velocity is higher, and higher where the pipe opens out and the fluid stagnates. Many people find this situation paradoxical when they first encounter it (higher velocity, lower pressure). The well-known Bernoulli equation is derived under the following assumptions:1. fluid is incompressible ( densityis constant ); 2. flow is steady:3. flow is frictionless (= 0);4. along a streamline;Then, it is expressed with the following equation:Where (in SI units):p= fluid static pressure at the cross section inN/m2.= density of the flowing fluid inkg/m3g= acceleration due to gravity inm/s2(its value is9.81 m/s2=9810 mm/s2)v= mean velocity of fluid flow at the cross section inm/sz= elevation head of the center of the cross section with respect to a datumz=0h*= total (stagnation) head inmThe 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 headh*at any cross section is constant (based on the assumptions given above). 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 thezvalues are zeros so that the above equation reduces to:(This is the total head at a cross section).For our experiment, we denote the pressure head ashiand the total head ash*i, where i represents the cross section we are referring to.Bernoullis theorem is a consequence of the principle of conservation of energy, applied to ideal liquids in motion. The theorem states that: For the streamline flow of an ideal liquid, the total energy (sum of pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section, throughout the flow.

2.0 Objective To demonstrate Bernoullis Theorem:1. To investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapered duct.2. To measure flow rates and both static and total pressure heads in a rigid convergent or divergent tube of known geometry for a range of steady flow rates.2.1General Description The unit is mounted on a base board which is to be placed on top of the hydraulic Bench. This base board has four adjustable feet to level the apparatus.The main test section is an accurately machined acrylic venture of varying circular crass section. It is provided with a number of side hole pressure tappings, which are connected to the manometer tubes on the rig. The test section incorporates two unions, one at either end, to Facilitate reversal for convergent or divergent as illustrated in figure 1 and 2.

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 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 the monometer 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 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.

Figure 2 Top view of Bernoullis Theorem Demonstration Unit.

2.2Equipment :a. ventureThe venturi is made of transparent acrylic with the following specifications:Throat diameter : 16mmUpstream diameter : 26mmDesigned flow Rate : 20 LPM

b. ManometerThere are eight manometer tubes; each length 320mm, for statics pressure and total head measuring along the venture 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 venture meter.d. Discharge valveOne discharge valve is installed at the venture 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: 0.37KW, 50 LPM

2.3 Procedures: 1. The general Start-up Procedures in the above section is performed.2. All manometer tubing is checked properly so that they are connected to the corresponding pressure taps and are air-bubble free.3. The discharge valve is adjusted to a high measurable flow rate.4. The water flow rate is measured using volumetric method after the level stabilizes.5. The hypodermic tube (total head measuring) connected to manometer G is slide gently so that its end reaches the cross section of the Venturi tube at A. Wait for some time and the readings from manometer G 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. Step 5 is repeated for other cross sections (A,B, C, D, E,F).7. Step 3 to 6 is repeated with three other decreasing flow rates by regulating the venture discharge valve.8. The velocity is calculated, ViB using the Bernoullis equation where;

ViB = 9. The velocity, ViC is calculated using the continuity equation where;

ViC = 10. Determined the difference between two calculated velocities.

2.3.1 General Shut-down Procedures1. closes water supply valve and venture discharge valve.2. Turn off the water supply pump.3.Drain off water from the unit when not in use.

3.0 Data and AnalysisCross sectionUsing Bernoulli equationUsing continuity equation Difference

iDi (mm)h*=hG(mm)hi(mm)ViB= [2 x g x (h* h i)](m/s)Ai= Di2/4(m2)ViC=Qav/Ai(m/s)ViB-Vic(m/s)Qav

A26.01221200.1980.0005310.758-0.564.03

B21.61191180.1400.0003660.110 0.034.03

C16.01171150.1980.0002010.200-0.0024.03

D20.01151140.1400.0003140.1280.0124.03

E22.01111100.1400.0003800.1060.0344.03

F26.01081060.1980.0005310.0760.0724.03

3.1 Calculation:Water volume (V) = 5 litresTime(T) =2m04s =124sQ == 0.4or 4

ViBA = [2 x g x (h* h i)] =29.81122-120) =0.198(m/s)Where g is 9.81Ai = Di2/ 4 = (0.026)2 / 4Ai = 0.000531 m2.ViC=Qav/Ai (m/s) = = 0.076 m/s.ViB-Vic (m/s)

3.2Discussion:We can see that the difference of the velocity is not very large. The difference between the velocities is caused due to the difference in the cross sectional area of the water channels. The difference of the velocity is also caused due to the inner surface frictional losses. In our case, the air bubble inside the tube is fail to remove, it cause some error to the result taken. The smaller cross sectional area have a larger velocity value. Therefore, the water pressure in this area is higher than the others.

3.3Conclusion:By doing this lab experiment, we learned practically the theorem of Bernoulli. The theory that stated in Bernoulli theorem, we see practically in this demonstration experiment. This experiment also proves that the smallest section in area having a largest velocity which is stated in the Bernoullis theorem.

Appendix

11