Fluid Mechanics Report t4j5w

1.

Introduction 1.1

Bernoulli’s Theorem

Bernoulli’s theorem states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously whether with a decrease in pressure or the potential energy of the fluid. It is named after the Dutch-Swiss mathematician Daniel Bernoulli. The apparatus used in this experiment consist of a classical Venturi which is made of clear acrylic. There is a series of wall tapping allow measurement of the static pressure distribution along the converging duct and a total head tube is provided to traverse along the centre line of the test section. These tapping are connected to a manometer bank incorporating a manifold with air bleed valve. The unit has been designed to be used with a Hydraulics Bench to study the characteristics of the flow through both converging and diverging sections. 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. (2) (5) Bernuolli’s principle complies with the principles of conservation of energy. In a steady flow, at all points of the streamline of a flowing fluid is the same as the sum of all forms of mechanical energy along the streamline. It can be simplified as constant practices of the sum of potential energy as well as kinetic energy. Fluid particles’ core properties are the pressure and weight. If a fluid is moving horizontally along a streamline, an increase in speed can be explained due to the fluid that moves from a region of high pressure to a region of low pressure and so with the inverse condition where the speed is decreased. In the case of a fluid that moves horizontally, the highest speed occurs at the lowest pressure, whereas the lowest speed is present at the most highest pressure. (1) Bernuolli’s theorem usually relates to Bernuolli’s equation. The equation is express as below:

Where, P = fluid static pressure at the cross section p = density of the flowing fluid g = acceleration due to gravity v = mean velocity of fluid flow at the cross section z = elevation head of the center at the cross section with respect to a datum h*= total (stagnation) head

1.2

Equation Derivation pg1

Reduction of Bernoulli’s equation to find the flow rate In the experiment 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 the equation reduces to:

For the experiment, the pressure head is denoted as hi and the total head as h*i where the i represents the cross sections at different tapping points. 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 the changes in speed and pressure along a streamline for such a process. If the static pressure is P at a point in the flow where the speed is v, then the stagnation pressure P0, where the stagnation speed v0 is zero, maybe computed from

Therefore,

Thus, if the stagnation pressure and the static pressure could be measured at a point, it would give the local flow speed. If the velocity of the stream at A is v, a particle moving from A to the mouth of the tube B will be brought to rest so that v0 at B is zero. By Bernoulli’s equation, Total energy per unit weight at A = Total energy per unit weight at B

Since v0 = 0, thus P0 will be greater than P. So,

Therefore,

Thus, velocity at A is

.

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Theoretically, the measured velocity, velocity is given by

, Pitot tubes may require calibration. The true

, where C is the coefficient of the instrument and h is the

difference of the head measured in of the fluid flowing. (3) (4)

2.

3.

Objective •

To understand the Bernoulli’s Theorem and the characteristics of the flow through both converging and diverging sections

•

To determine the flow rate and pressure of the water in the venturi meter

•

To calculate the discharge coefficient of the venturi meter

•

To measure the percentage error between theoretical and experimental data

•

To compare the difference in between the calculation of using Bernoulli equation and Continuity equation

Equipment

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Figure 1: Bernoulli’s Theorem Demonstration Unit

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4.

Procedures: Experiment_1: Discharge Coefficient Determination i.

The equipment of the venturi meter has been set up as shown in Figure 1.

ii.

The hypodermic tube is withdrawn from the test section.

iii.

The discharge valve is adjusted to the maximum measurable flow rate of the venturi and will only be achieved when tube 1 and 3 give the maximum observable water head difference.

iv.

The water flow rate is measured using volumetric method and the manometers reading is recorded after the level stabilizes.

v.

Step 4 is repeated with at least three decreasing flow rates by regulating the venturi discharge valve

vi.

The actual flow rate, Qa is obtained from the volumetric flow measurement method.

vii.

The ideal flow rate, Qi is calculated from the head difference between h1 and h3 using the equation provided.

viii.

A graph of Qa VS Qi is plotted and the discharge coefficient, Cd which is the slope is obtained. Experiment_2: Flow rate measurement with venturi meter

i.

The equipment of the venturi meter has been set up as shown in Figure 1.

ii.

The hypodermic tube is withdrawn from the test section.

iii.

The discharge valve is adjusted to a high measurable flow rate.

iv.

The water flow rate is measured using the volumetric method and the manometer reading is recorded after the level stabilizes.

v.

Step 4 is repeated with three other decreasing flow rates by regulating the venturi discharge valve.

vi.

The venturi meter flow rate of each data is calculated by applying the discharge coefficient obtained.

vii.

The volumetric flow rate with venturi meter flow rate is compared. Experiment_3: Bernoulli’s Theorem Demonstration

i.

The equipment of the venturi meter has been set up as shown in Figure 1.

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ii.

All the manometer tubings are checked to ensure that they are properly connected to the corresponding pressure taps and are air-bubble free.

iii.

The discharge valve is adjusted to a high measurable flow rate.

iv.

The water flow rate is measured using volumetric method after the level stabilizes.

v.

The hypodermic tube connected to manometer #H is gently slide so that the end reaches the cross section of the Venturi tube at #A. The readings from manometer #H and #A is noted down after some time. Manometer #H reading shows the sum of the static head and velocity heads as the hypodermic tube is held against the flow of fluid forcing it to a stop. Manometer #A reading shows the pressure head because it is connected to the Venturi tube pressure tap, which does not obstruct the flow, thus, measuring the flow static pressure.

vi.

Step 5 is repeated for other cross section (#B, #C, #D, #E, #F)

vii.

Step 3 to 6 is repeated with three other decreasing flow rates by regulating the venturi discharge valve.

viii.

The velocity, ViB is calculated using the Bernoulli’s equation

ix.

The velocity, ViC is calculated using the continuity equation

x.

5.

The difference between two calculated velocities is determined.

Results 5.1

Experiment_1: Discharge Coefficient Determination

Data Analysis: Table 1 Throat Diameter, D3 (mm) Inlet Diameter, D3 (mm) Throat Area, At (m2) Inlet Area, Ai (m2) g (m/s2) ρ (kg/m3)

16.0 26.0 2.011×10-4 5.309×10-4 9.81 1000

Table 2: Flow rate and the height of each manometer Qav (LPM) 11.8977 11.4921 10.6421

hA (mm) 250 240 230

hB (mm) 241 233 225

hC (mm) 192 195 191

hD (mm) 224 219 214

hE (mm) 232 225 218

hF (mm) 239 230 223

hA - hC (m) 0.058 0.045 0.039

Qi (LPM) 13.9100 12.2500 11.4000 pg6

7.3233

220

217

195

Volume of water for Qav = 13L – 10L = 10 L

208

211

214

0.025

9.1309

Time taken to collect 10L water for the first Qav = 50.43 s

For the first Qav = 10L / (50.43 /60)min = 11.8977 LPM ≈ 11.90 LPM For Qav=11.8977LPM, Qi = [ At [ √ 2 × g × ( hA – hC ) ] ] / √ 1 – (At/Ai )2 = [ 2.011 × 10-4 m2[ √ 2 × 35316 m/min2 × ( 0.058m ) ] ] / √ 1 – (2.011 × 10-4 m2/5.309×10-4m2 )2 = 0.01391 m3/min × 1000L/1m3 = 13.91LPM Graph 1: Qav vs Qi

Cd = Gradient = (11.8977 – 7.3233)LPM / (13.9100 - 9.1309)LPM = 0.9572 5.2

Experiment_2: Flow rate measurement with venturi meter Table 3: Flow rate and the height of each manometer pg7

Qav (LPM) 11.8977 11.4921 10.6421 7.3233

hA (mm) 250 240 230 220

hB (mm) 241 233 225 217

hC (mm) 192 195 191 195

hD (mm) 224 219 214 208

hE (mm) 232 225 218 211

hF (mm) 239 230 223 214

Table 4: Theoretical and practical flow rate with percentage error

Cd

Qav (LPM) 11.8977 11.4921 10.6421 7.3233 = 0.9572

Qa

hA - hC (m) 0.058 0.045 0.039 0.025

Qi (LPM) 13.9100 12.2500 11.4000 9.1309

Calculated Flow Rate (LPM) 13.3147 11.7257 10.9121 8.7401

Error (%) 10.60 1.99 2.47 16.21

= Cd × Q i = 0.9572× 13.9100 LPM = 13.3147 LPM

Error = |Qav – Qa| / Qa × 100% = | 11.8977 LPM – 13.3147 LPM | / 13.3147 LPM × 100% = 10.60% 5.3

Experiment_3: Bernoulli’s Theorem Demonstration Table 5: Flow rate used (Liter/Min)

Volume (L) Average Time (min) Flow Rate (LPM)

10 L 0.52 min 19.2308 LPM

Table 6: Velocity calculated by using Bernoulli’s equation and Continuity equation Cross Section # A B C D E F

Using Bernoulli equation h*=hh (mm)

hi (mm)

290 286 279 270 269 266

252 238 145 207 222 236

ViB = √[2*g*(h*hi)] 0.8635 0.9704 1.6214 1.1118 0.9603 0.7672

Using Continuity equation Ai = π Di2 / 4 Vic = Qav (m) / Ai (m/s) 5.3093×10-4 0.6037 3.6644×10-4 0.8747 -5 2.0106×10 1.5941 -4 3.1416×10 1.0202 3.8013×10-4 0.8432 -4 5.3093×10 0.6037

Difference ViB - ViC (m/s) 0.2598 0.0957 0.0273 0.0916 0.1171 0.1635

For cross section #A, 19.2308 LPM = 19.2308 × 1/1000 m3 × 1/60 s pg8

= 3.2051 × 10-4 m3/s ViB = √[2*g*(h*-hi)] = √[2*9.81 m/s2*(0.290m - 0.252m)] = 0.8635 m/s ViC = Qav / Ai = 3.2051 × 10-4 m3/ s / 5.3093×10-4m2 = 0.6037 m/s ViB - ViC = 0.8635 m/s – 0.6037m/s = 0.2598 m/s

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6.

Discussion The aim of this experiment is to study the characteristics of flow through both converging and diverging sections. During the experiment, water is fed through a hose connector and we need to 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 has been determined. Our result of the experiment is obeying the Bernoulli’s law which states that if a nonviscous 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. (3) However, there are some restrictions of using Bernoulli’s theorem which include: •

Steady flow

•

Incompressible flow

•

Frictionless flow

•

Flow along a streamline (5)

From the analysis of the results of experiment 1 (Table 2), we can conclude that the height of water in manometer tube C (hc) is always the lowest even though we test it with different flow rate (Qav). Moreover, the water level of manometer tube from hA to hC will decrease whereas; the water level of manometer tube from hC to hF will increase. This is because of the converging and diverging section of the venturi tube. There is a convergent flow from A to C due to the diameter of the tube is narrowing while there will be a divergent flow from C to F because of the tube is widening. Thus, according to Bernoulli’s theorem, as fluid flows from a wider pipe to a narrower one, the velocity of the flowing fluid increases and the pressure will decrease. Besides, the difference in height of hA - hC is increasing while the flow rate (Qav) is increasing. This has indirectly shown that the faster the flow rate, the bigger the pressure difference between manometer tube A and C. In our experiment, the highest flow rate (Qav) that we used is 11.8977liter/min, so we obtain the biggest difference which is 0.058m between manometer tube A and C. The flow rate (Qi) in experiment 1 is calculated by using the equation below:

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Where

is equal to (hA - hC).

Therefore, Qi can be known as theoretical flow rate as it is calculated by using the formula. However, in the case of real fluid flow, the flow rate will be expected to be less than that given by equation because of frictional effects and consequent head loss between the inlet and throat. Therefore, this non-ideality is ed by insertion of an experimentally determined discharge coefficient, Cd that is termed as the coefficient of discharge. A graph of Qav VS Qi (Graph 1) is plotted and the discharge coefficient, Cd which is 0.9572 is obtained. For the results in experiment 2 (Table 4), the discharge coefficient is determined as follow: Cd= (Qa/Qi).Thus, Qa is being calculated by multiplying the Cd with Qi and the Qa can be known as theoretical data as it is calculated by using an equation whereas Qav is a practical data. Therefore, there must be some error if compare between practical and theoretical data as shown in Table 4. The biggest error calculated by using our collected data is 16.21% which is acceptable due to the parallax error of the readers. The most common weakness is the observers have not read the water level properly where the eyes are not perpendicular to the water level on the manometer. Thus, there are some small effects on the calculations of our data. The flow rate that we used for experiment 3 was 19.2308 liter/min which is also shown in Table 5. For the result in Table 6, the height of water level for cross section C is the lowest, 145mm but the velocity is the highest which is 1.6214m/s for Bernoulli’s equation and 1.5941m/s for Continuity equation. This is because the diameter of C is the smallest among all of the cross section, thus the velocity will be the highest, therefore the pressure will be the lowest and hence the reading of the manometer will be the lowest. In addition, for the difference of ViB - ViC (whereby the ViB is calculated by using the Bernoulli’s equation and ViC is calculated by Continuity equation in Table 6), the reason of the presence of the differences is because we are using two different equations to calculate the velocity of the fluid. For an ideal answer, the V iB should equal to ViC as both of them are the same velocity which is interrelated to the flow rate that we used. Due to the different equation being used, the variables that consist in the equation will be different. For an example, we measure the difference of water level in each manometer to determine the V iB in Bernoulli’s equation, whereas we use the Qav to determine the ViC for Continuity equation. So, the difference of water level and the Qav are the different variables that we measured practically and all those figures are used for calculating the velocity. That is why there might be some errors when taking the measurement of each data, causing the difference between the value ViB and ViC.

7.

Conclusion pg11

In conclusion, the result collected from the experiment is according to the Bernoulli’s Theorem which is the highest speed is the one at the lowest pressure, whereas the lowest speed is present at the most highest pressure. Due to the highest pressure of the water, it causes the reading of manometer become the highest. This principle complies with the principle of conservation of energy which it is the sum if all forms of mechanical energy along the streamline. During the converging section which is from manometer A to C, the diameter of the venturi meter at this section is narrowing which means that the diameter of the tube is decreasing. Furthermore, section C has the smallest diameter among all the cross section area. Thus, it can create the highest velocity and hence it has the lowest pressure. This has indirectly caused the reading of manometer C remain the lowest among all manometer tubes. Meanwhile, the diverging section is begin from manometer C to manometer F. Therefore, the reading of the height of manometer from C to F is increasing due to the pressure increased because of the lowering of velocity along C to F. All the data collected and calculated are obeyed the Bernoulli’s Theorem, even the discharge coefficient, 0.9572 that we obtained from the slope of the graph is also within the acceptable range of 0.9 to 0.99. However, there will be some errors when measuring the reading of each data due to the parallax error and valve regulation problem. Due to the unstable reading of water level of manometer, we believe that this has affected our result. Thus, some errors and differences between the calculated flow rate and practical flow rate Qav have formed as we are conducting the experiment manually. If we repeat the experiment by several times in order to get the average value, we strongly believe that the errors can be minimized. 8.

9.

Precautions and Recommendation •

The eye of the observer should be parallel to the water level to avoid parallax error.

•

Repeat the experiment several times to get the average value.

•

Make sure the bubbles are fully removed.

•

The valve should be controlled slowly to maintain the pressure difference.

•

The valve should be regulated smoothly to reduce errors.

•

Always wear protective clothing, shoes and goggles throughout the laboratory session.

References pg12

1. Bernoulli’s Theorem.(2010).Retrieved on February 6,2010 from

http://www.idoub.com/doc/39165346/Bernoulli-s-Theorem-Distribution-Experiment 2. Bernoulli’s Theorem .(2010).Retrieved on February 6,2010 from

http://www.solution.com.my/pdf/FM24%28A4%29.pdf 3. Bernoulli’s Theorem .(2001).Retrieved on February 6,2010 from

http://www.ceet.niu.edu/faculty/kostic/bernoulli.html 4. John F.Douglas. (2001). Fluid Mechanics (4th ed.). Pearson Education

Limited(pg.188) 5. Lab Manual: Bernoulli’s Theorem demonstration Unit

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