Investigation of Bernoulli’s Principle Mechanical Engineering Principles

The Investigation of Bernoulli’s Principle

Introduction –

This lab explored the validity of Bernoulli’s principle. This was achieved by investigating the change in fluid pressure when the velocity of the fluid was increased. This procedure was then repeated at three different flow-rates. The validity of the principle and the experiment were then measured by comparing the theoretical and practical results.
 

Bernoulli’s Principle is a Fluid Dynamics Principle stating that an increase in the speed of a flowing fluid is accompanied simultaneously with a decrease of pressure or the decrease of the potential energy of the fluid.
 

The Bernoulli Equation is;

P+½ ²+ gh

This equation stays constant throughout the points of interest in the flow such that [1];

P₁+½ ₁²+y₁=P₂+½ ₂²+y₂=…=P_n+½ _n²+y_n

In this experiment the number that goes in the place of n shows the points in the tube with varying cross-sections from one to six.  P is the hydrostatic pressure, is the velocity of the water, is the density of the fluid (which in this case is 1000kg/m³) and y is the position of the fluid in the tube. In a positional flow the value of y is the same across all six points so the equation can be simplified to;

P₁+½ ₁²=P₂+½ ₂²=…=P_n+½ _n²

As only water is used as a fluid in this experiment due to the conservation of mass, the equation of continuity is;

A_{1 1}=A_{2 2}=…=A_{n n}

Where A is the cross-sectional Area of the flow of the fluid.

To calculate the hydrostatic pressure the manometer tube readings are used. The manometer readings measure the height (h) at each point of the flow. The cross-sectional area of the flow is calculated by using the diameters of the pipe from point one to six. The velocity for each of the points can be calculated by finding the flow-rate for the fluid (Q) through the experiment and then use the following equation;

Q=A

Description of the experiment

Equipment-

Figure 1

Figure 2

Procedure-

1. Firstly set up the experiment using the equipment shown above in figures 1 and 2 connecting the hydraulic bench to the test pipe and manometers. As shown in figure 2 the test pipe is tapped at various positions of varying cross-sectional areas.
2. Then turn on the pump and ensure it’s producing the required flow. This may have been set before the experiment started.
3. After setting up the equipment use the outlet valve (this controls the outlet of the flow from the test pipe into the hydraulic bench) to set the maximum flow-rate.
4. Take the first readings on the manometer at this rate. Take a note of the difference in heights between the manometers one to six.
5. After taking these readings use the outlet valve again to change the flow-rate and alter the height difference between manometers one and six (as shown above) to 50mm. Then take the second readings at this flow rate.
6. Finally alter the difference in heights 1-6 so that the value falls halfway between the first and second values taken.

Results-

Calculations-

After each test calculate the flow velocity (Q). Also calculate the velocity ( ) at each tapping point by using the cross sectional area (A) of the tube at each position and using the equation Q=A .

During the tests after the ideal flow has been achieved (maximum flow rate for test 1 is achieved by fully opening the outlet, or by gaining the desired h1-h5 differences in tests 2 and 3) close the ball valve in the hydraulic bench.

Measure the time (t) taken for the hydraulic bench to gain a volume (V) of 6 litres of water. Use a stopwatch to measure this. There is a volume measure on the bench as shown below which is used to measure this amount. Q=V/t is used where V is in m³.

You can use these measured and calculated values to then calculate the (hydro)static pressure (P) using the equation P= gh. You can also use the equation P+½ ²(where the density of water ( )is 1000 kg/m³) to find the total pressure. Then record all the values in tables as shown below; including all of the values, calculated and measured in the experiment.

Figure 1: Test 1- A table of results to show the gathered and calculated values retrieved from test 1

Figure 2: Test 2- A table of results to show the gathered and calculated values retrieved from test 2

Figure 3: Test 3- A table of results to show the gathered and calculated values retrieved from test 3

As described in the intro the total pressure at each point should be the same according to the equation P₁+½ ₁²=P₂+½ ₂²=…=P_n+½ _n²as stated above. To validate the experiment and the Bernoulli equation all of the Total pressures should be equal to each other from points A to F. 

The (hydro)static pressure values at points B to E should also be easy to calculate by using the calculated (hydro)static pressure of A and the velocities from points B to F. These can then be compared to the calculated values from the experiment. The calculated values are shown below by using the rearranged equation; P₁+½ ₁²-½ _n²=P_nwhere n is the point in the tube where the measurements are taken.

Discussion and conclusion

Figure 4: Test 1 comparison

Figure 5: Test 2 comparison

Figure 6: Test 3 comparison

In test 1, (Shown in Figure 4) the percentage difference between the calculated and experimental values in this test are nearly 50% for one of the readings. This is a high percentage error and counts as a discrepancy. This may be due to human error in the calculations or due to an error in the equipment.

In test 2 (Shown in Figure 5), again there is a larger percentage error for the point at E which means that there is a larger possibility of an error in the equipment. However the percentage error for the other points on the tube are very low meaning that the experiment was valid in this test as the calculated and experimental values were very similar.

In test 3 (Shown in Figure 6) the percentage error between the calculated and experimental values were even smaller and the (hydro)static pressures at each point were very similar to each other. This validates the experiment and the theory further.

In conclusion, through experimentation this lab has proven the Bernoulli equation is a valid theory. The pressure throughout a tube should stay constant regardless of the cross-sectional area of the tube as shown in this experiment. Therefore the aim of this experiment has been achieved and it was successful overall.

References:

• [1]-Young H., Freedman R., Lewis F., (2016) University Physics with Modern Physics, Global Edition, Fourteenth edition, Global edition. Harlow, UK: Pearson
• Keller F., Gettys W., Skove M., (1989) Physics Classical and Modern, Second Edition. USA: McGraw-Hill, Inc.
• Cengal Y., Boles M., (2007) Thermodynamics: an engineering approach, Sixth edition. Boston, UK: McGraw-Hill