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MAE 3130-Fluid Mechanics

Spring 2015

Laboratory report 1:

Bernoulli`s equation

Name: Breno Oliveira Student ID: 104714828

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Table of Contents 1-Abstract3 2-Introduction.4 3-Results and Discussion..6 4-Conclusion..16 5-References..17 6-Appendix.18

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1-Abstract

In this experiment was concluded that we can experimentally validate the Bernoulli equation by the comparison of the results application of the equations. The experiment consists essentially in a three laminar and three non-laminar flows of water passing through a Venturi`s tube, whose total pressure was measured at six different sections through a direct and an indirect method. According to Bernoulli's Equation, the total pressure in all points would be constant if in a steady, inviscid and incompressible flow. Experimentally noticed that the indirect total pressure was higher than the direct due to systematic errors. In the section F, the divergent nozzle, there is a greater error due to a wrong estimation of the velocity in that spot. In addition, the computational simulation was an important tool for the understanding of the water behavior in different flow conditions and provided results that reinforced the arguments presented in the experimental discussion.

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2-Introduction 2.1-Theory 2.1.1-Bernoulli`s equation

In 1738, the physicist and mathematician Daniel Bernoulli changed the Science of fluids by publishing his book Hydrodynamics. In this book, appeared an equation, after called Bernoullis equation, which is a powerful tool in fluid mechanics and overused in aerodynamics theories.

Bernoulli's principle can be derived from the principle of conservation of energy. It states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. The Bernoulli`s equation used in this experiment is:

+1

2V2 + z = constant along streamline = (1)

Where p=pressure, = density, V= velocity, =specific weight and z= height. This equation assumes four conditions for its derivation:

Inviscid flow(viscous effects are considered negligible),

steady flow(no changes with time at a given location in the flow field),

incompressible flow(no changes in the density) and

equation`s application along a streamline. 2.1.2-Reynold`s number

In fluid mechanics, the Reynolds number (Re) is a dimensionless quantity that is used to help predict similar flow patterns in different fluid flow situations. This number characterizes different flow regimes within a similar fluid, such as laminar or flow.

Laminar flow occurs at low Reynolds number (Re2100) and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The equation for the Reynold`s number is:

=

(2)

Where =density, V=velocity, D=the smallest diameter in the test section and = viscosity.

http://en.wikipedia.org/wiki/Conservation_of_energyhttp://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlineshttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/Potential_energyhttp://en.wikipedia.org/wiki/Internal_energyhttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Dimensionless_quantityhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Vortex

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2.2-Objective

The objective of this experiment was determine if Bernoullis equation accurately models steady water flow through a converging diverging duct by applying the Bernoullis principle in different flow conditions. Also, analyze the water behavior in different flow conditions through computational simulation. 2.3-Methods Was used two different methods: experimental and computational. 2.3.1-Experimental method: The equipments used in this experiment were:

Hydraulic bench

Bernoulli accessory

Graduated cylinder

Stop watch The procedure were to calculate the total pressure of 6 points in the tube which has a form a

Venturi tube, as showed in the Figure 1, by two methods: direct and indirect.

Figure 1- Schematic of test section detailing manometer tap positions and associated cross-section diameters. 2.3.1.1-Direct method

Were collected the heights of the water in the manometer connected to the total pressure head probe, which moved along the test section centerline. The total pressure of each point was calculated directly through the stagnation pressure of the probe using the following equation:

= = (3)

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Where, p=density, g= gravity, and h= height. 2.3.1.2-Indirect method

Were measured the heights of the manometers that are attached to the top of the 6 points in the tube. The total pressure in this case can be obtained by an simplification of the Bernoulli`s equation assuming that all the points have the same height (1 = 2).Thus, the total pressure was indirectly calculated with the stagnation pressure related to the manometers plus the dynamic pressure of the point by the equation:

= + 1

22 (4)

Where V=velocity. The relationship between the flow rate (Q), which was measured in this experiment, and the area (A) obtain V:

=

(5)

The flow rate was measured with a cylinder and stopwatch by collecting the volume (ml) in a

certain amount of time(s). The equation used is:

=

(6)

Where v=volume and t=time.

2.3.2-Computational

The computational method was a simulation of a Venturi`s tube (showed in the Figure 14 in the appendix) with a flow rate fixed of 10ml/s. The program used was the COMSOL Multiphysics. Three studies in different conditions were done as described in the table 1.

Table 1: Computational test matrix.

3-Results and Discussion 3.1Experimental Data

The experiment was repeated six times with different flow rates. According with the theory, the flow`s behavior depends on its velocity which is related with the flow rate. The flow rates measured and their corresponding Reynolds number are showed in the table 2.

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Table 2-Reynold`s number relative to the flow rate

Flow rate(ml/s) Reynoldss number

7.2 817.6926248

10.4 1181.111569

11.8 1340.107357

34.4 3906.753652

45 5110.578905

50 5678.421005

Observing this table, it is noticeable that there is three laminar and three non-laminar flows,

once a laminar flow have a Reynoldss number less than 2100 according with the theory. The following graphics are demonstrations of the direct and indirect total pressure head versus position for flow rates that configure a laminar flow.

Figure 2- total pressure head versus position in laminar flow

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2540

2560

2580

2600

2620

2640

A B C D E F

Pre

ssu

re(P

a)

Tapping position

7.2ml/s(laminar)

Indirect totalpressure

Direct totalpressure

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Figure 3- total pressure head versus position in laminar flow

Figure 4- total pressure head versus position in laminar flow

In regards to Figures 2-4, the direct total pressure remained constant at all points. This fact is

expected since the Bernoulli`s principle states that the total pressure is constant along the streamline. Simultaneously, the indirect total pressure was not constant and higher than the direct total pressure contradicting the theory. The reasons for these results are the errors in the theory assumptions and in the experiment .

Instead of being constant, the indirect total pressure curve has a considerable decline in the last point. This circumstance is explained by a not proper assumption made in order to calculate the velocity at this spot. According to the equation 5, the velocity is the flow rate divided by the section area. However, the fluid accelerates as de diameter reduces and it does not loose acceleration enough when the diameter of the tube expands. Thus, the velocity used to calculate the dynamic pressure at this point was lower than the reality. Consequently, the total pressure obtained was lower than the expected once the static pressure reduces its value conforming the dynamic pressure raises. A proof of this statement

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2450

2500

2550

2600

2650

A B C D E F

Pre

ssu

re(P

a)

Tapping position

10.4ml/s(laminar)

indirect totalpressure

Direct totalpressure

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2450

2500

2550

2600

2650

A B C D E F

Pre

ssu

re(P

a)

Tapping position

11.8ml/s(laminar)

Indirecttotalpressure

Direct totalpressure

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is the velocity contours at Figures 9 and 10, where the right edge has higher velocity than the left edge even that they have the same diameter.

In this experiment, the operation of the instrument is a possible error source. A bad control of the time that the water filled the cylinder in order to measure the flow rate is an example of that. In addition, as the indirect total pressure is higher than the direct total pressure the three flow rates, demonstrated in Figures 2-4, it can be assumed that there is a systematic error in the measurements. One possible error is the probe not aligning in the fluids direction. Consequently, it can produce a nonsymmetrical fluid. Furthermore, the measurements can have errors in the reading of the manometers.

The following graphs are demonstrations of the direct and indirect total pressure head versus position for flow rates that configure a non-laminar flow.

Figure 5- total pressure head versus position in non-laminar flow

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2350

2400

2450

2500

2550

2600

2650

A B C D E F

Pre

ssu

re(P

a)

Tapping position

34.4ml/s(Non-laminar)

Indirect totalpressure

Direct totalpressure

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Figure 6- total pressure head versus position in non-laminar flow

Figure 7- total pressure head versus position in non-laminar flow

In these graphs the total pressure is not constant in the direct nor the indirect methods. It can

be explained by the fact that the flow rates measured configures a non-laminar flow, what characterize an unsteady flow. Consequently, the total pressure is not expected to be constant along the tapping positions.

In addition, the indirect and direct total pressure of the non-laminar flow measurements are more distant from each other than the in laminar flow measurements. The unsteady flow condition of the non-laminar flow is the also the reason of this fact. 3.1-Computatiotal Data:

Were made three different studies as described in the table 1.

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2350

2400

2450

2500

2550

2600

2650

A B C D E F

Pre

ssu

re(P

a)

Tapping position

45ml/s(Non-laminar)

Indirect totalpressureDirect totalpressure

2200

2250

2300

2350

2400

2450

2500

2550

2600

2650

A B C D E F

Pre

ssu

re(P

a)

Tapping position

50ml/s(Non-laminar)

IndirecttotalpressureDirect totalpressure

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The first comparison between the studies showed in table 1, is the velocity versus position data of the no slip and slip cases . A 3-D cut line was created in the simulated tube as showed in Figure 15 in the Appendix. The Figure 8 provides the velocity along the cut line for no slip and slip conditions.

Figure 8: Velocity x Position for slip and no slip wall conditions.

The profiles of the no slip and slip condition graphs are similar. In both circumstances, the

highest velocity was in the smallest diameter of the tube and the lowest velocity was in the biggest diameter. This is expected since the velocity is calculated by the equation 5 in the theory section. Additionally, the slip condition velocity is smaller than the no slip condition velocity. The reason is the presence of friction on the walls resulting in the loss of the Kinect energy of the fluid reducing its velocity.

The Figures 9 and 10 provide the velocity contours for the no slip and slip cases respectively.

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Figure 9: velocity contour for no slip wall condition.

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Figure 10-velocity contour for slip wall condition.

In both graphics, the velocity starts low and gets higher as the diameter of the pipe decreases

agreeing with the theory related with the equation 5.Altoguh the right and left edges have the same diameter, the right region has a higher velocity. It is due the fact that the fluid, earlier accelerated as the diameter reduces, does not decelerate enough as the diameter expands.

The comparison between the two graphs evidence that the slip condition has lower velocity than the no slip condition. The explanation for it is the friction generated in the walls, which reduces the fluid velocity. In addition, it is noticeable that, due to the friction in the walls, the fluid have lower velocities as closer to the boundaries it is.

To reinforce this statement the Figures 11 and 12 shows the streamlines of no slip and slip condition respectively.

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Figure 11- Streamlines of no slip wall condition.

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Figure 12-Streamlines of slip wall condition.

In this simulation, the streamlines of the no slip condition are spreader than the slip condition

streamlines, which are concentrated in the middle of the pipe, once there is friction force preventing movements near to the boundaries. It evident that there is no streamlines close to the tube walls.

Finally, the last result is the graph of velocity versus position as a function of time provided by the Figure 13.

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Figure 13-Velocity versus position as a function of time.

The lines of the graph were generated in each five seconds of one minute. The study presented

in this graph is a transient flow. Similarly, to the graph of the Figure 8, the velocity increases as closer to the lowest diameter of the tube it is. As time goes, the lines have higher velocity and diminish with less intensity. For example, the ten seconds line(red line) have a velocity of 0.002 m/s at the position 0.14, while the fifty seconds line(yellow line) have the same velocity at the position 0.16. 4-Conclusion

In conclusion, the experiment shows that the Bernoullis equation have a good accuracy in the steady water flow`s modeling in the Venturi tube. The decline of the total pressure in the last tapping point due to the calculation of a lower velocity proves the Bernoullis principle because the static pressure reduced due to a higher velocity of the fluid. Also, the experiment shows that the total pressure is not constant along the streamline in an unsteady flow, reinforcing the Bernoulli`s theory.

The experiment error sources such as the reading of the manometers or the cylinder, the operation measurements can be avoided with two improvements in the system. First, a better precision of the manometers and the cylinder, and second, a better way to measure the flow rate of the system, for example, the usage of a flow meter with a good precision. Moreover, the computational experiment provided a good opportunity to the analyses of the water behavior in different flow conditions. The velocity versus position graph and the velocity contours for no slip a slip conditions demonstrated the effect of the friction in the fluid. The friction reduces the

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average velocity of the fluid and impedes the appearance of streamlines near the boundaries. Finally, the velocity versus position as function of time graph was an important tool for the understanding of the water behavior in a transient flow.

With the results, it is possible to prove the importance and the truth of the Bernoullis principle, and to understand why the theoretical values and the experimental values are different.

5-References Munson, Bruce R., and Theodore H. Okiishi, Wade W. Huebsch, Alric P. Rothmayer. Fundamentals of fluid Mechanics. 7th ed. Wiley.

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6-Appendix The following Figures are demonstrations of the computational simulation.

Figure 14- Simulation of Venturi tube

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Figure 15- 3-D cut line along the tube