BERNOULLI’S EXPERIMENT
ABSTRACT
Two fluid properties, the Bernoulli principle and the Venturi effect, were examined and analyzed in this experiment. A venturi meter was used to measure the fluid flow rate, velocity, and pressure of water (the fluid being examined). The results showed that the fluid pressure decreases as the fluid traveled through a smaller cross sectional area. The minimum fluid pressure (in effect, maximum fluid velocity) was observed in the smallest cross sectional area at position D (diameter 16 mm). The decrease in fluid pressure was compensated by an increase in the fluid velocity. With these two measured phenomena, both the Bernoulli principle and the Venturi effect were observed to happen in the experiment. However, water inside the venturi meter did not completely behave as an ideal fluid. Some discrepancies were observed between the theoretical and experimental values of the piezometric height.
INTRODUCTION
This experiment is about testing a fluid inside a venturi meter and observing the behavior whether it follows Bernoulli principle and Venturi effect. The Bernoulli’s experiment has the following aims:
For a fluid to be ideal, it has to strictly follow the Bernoulli’s principle which states that a fluid’s pressure is inversely proportional to its velocity. This inverse proportionality, which includes the fluid density in the equation, is explained in theory and in detail in the background section. The Venturi effect is a phenomenon in which the pressure of a moving fluid decreases as it approaches a tube with smaller area. In effect, following Bernoulli’s principle, the fluid will travel faster through this smaller area. The Venturi effect is further discussed in terms of its practical applications.
The background section presents the theory behind this experiment, which includes the mathematical model of Bernoulli’s principle and Venturi effect. The results section presents the tabulated measurements from the venturi meter of the piezometric height and the fluid flow rate. Included in the table are the fluid velocities at different points across the venturi meter and the theoretical values of the piezometric height. The discussion section presents the analysis regarding the acquired results. The analysis will be based upon compliance with theory and possible applications of the observed phenomena. The summary provides a brief overview of the whole experiment including the conclusions derived from the results.
BACKGROUND
Bernoulli’s equation relates the fluid density, fluid pressure, fluid velocity, and potential head of an ideal fluid. The following equation is Bernoulli’s equation:
Z+v22g+pρg=constant=H=total head m
Where:
Z=potential head
v22g=kinetic head
pρg=kinetic head
The above equation is a manifestation of the conservation of mechanical energy of an ideal fluid integrated in a connected system. The total energy of the system must be constant unless some energy is lost due to friction or other external (or internal) elements. Incorporating this concept, Bernoulli’s equation can be expanded to:
Z1+v122g+p1ρg=Zn+vn22g+pnρg 1
Figure 1. Venturi Meter Diagram
The above diagram shows that the piezometric head at any point in the venturi meter is equal to the sum of the potential head, and the pressure head. Rearranging the equation:
p1ρg+Z1-pnρg+Zn=vn22g-v122g
The venturi meter in this experiment is horizontal, thus Z1=Z2==Zn, so the equation simplifies to:
p1-pnρg=v12-vn22g (2)
These equations mean that as the inlet and outlet diameters of a venturi meter are the same, then the inlet and outlet velocities must be the same:
Eq. 1 predicts that for an ideal fluid the first and last piezometric heights (h1 and h11) must be the same. This behavior will be examined in the experiment.
Eq. 2 predicts that for an ideal fluid any change in velocity must be matched by a change in piezometric head – when the venturi is horizontal, this simplifies to pressure and velocity energy being mutually interchangeable. Eq. 2 is only applicable when the path of the fluid is horizontal. This observation, often called the Bernoulli Effect, has endless applications in engineering, science, medicine, etc. (It even explains how footballers and cricketers make balls swerve). The Bernoulli effect is also examined in the experiment.
As shown in equation (2), the fluid pressure and the fluid velocity have an inverse proportionality relationship. This means that an increase in the fluid velocity will create a decrease in the fluid pressure in the horizontal motion of a fluid .
RESULTS
The results of the experiment are compiled and tabulated in table 1. The ‘hn experimental’ and the ‘Time (s)’ values are the readings measured from the venturi meter. The flow rates Q are computed by dividing the mass (6 kg) by the product of the fluid density and the time (the third and final time reading for each test run is used). The formula is:
Q=Massρ×Time
The velocity of the fluid is computed by dividing the fluid flow rate Q by the cross-sectional area A:
vn=QA
Figure 2,3 and 4 displays the graphs of the piezometric heights against the positions across the venturi meter (A to K) for the test runs 1, 2, and 3, respectively.
Figure 2. Test 1 hn vs. Position
Figure 3. Test 2 hn vs. Position
Figure 4. Test 3 hn vs. Position
DISCUSSION
The tabular results showed that the piezometric height tends to decrease in positions where the cross-sectional area of the venturi meter is decreasing. The minimum piezometric height occurred at the minimum cross-sectional area at position D. The fluid travelling away from this position displayed eventual increase of piezometric height values. Since the piezometric height is directly related to the fluid pressure following the formula p=ρgh, this means that the pressure decreases as the fluid passes through smaller cross-sectional area. This fluid behavior is the Venturi effect. The graphs provide a visual display of this behavior. The lowest points in the graph of hn occur at position D where the cross-sectional area is minimum. All the test runs showed similar behavior in terms of pressure and cross-sectional area. Furthermore, the relative values of the piezometric heads are decreasing as the flow rate is increasing (from test 1 to test 3).
Consequently, the velocity of the fluid tends to increase as the cross-sectional area is decreasing. This phenomenon is observed in all the test runs. This is consistent with both fluid properties Venturi effect and Bernoulli Effect. When the fluid pressure decreases on small cross-sectional areas, the fluid compensates by increasing the fluid velocity. Hence, mechanical energy is conserved in the system.
Water as a fluid in this experiment does not behave as an ideal fluid. This is because the measured piezometric heights on the first and last positions are not equal to each other (should be equal for ideal fluids). Moreover, the measured piezometric heights are slightly deviated from the theoretical values. This is caused primarily by the non-ideal fluid behavior of water; also, mechanical losses on the system like friction on the inside walls of the venturi meter could also cause discrepancies of the measured values from the theoretical ones.
The Venturi effect can be utilized in applications like pumping water. In addition to water pump engine with sufficient power capacity, a carefully designed pipe system can maximize the efficiency of water volume delivered to a certain location. In locations where there is high fluid pressure buildup, smaller pipes can be used to ease down the fluid pressure. In effect, the fluid velocity will increase in these locations, ensuring that the water would continuously travel with the desired rate. However, the pipe size cannot be too small in which the water volume delivered becomes less than the target value. Proper trade off of pipe size, engine power, fluid velocity and pressure should be considered in the design process of the water pump application.
The Bernoulli Effect can be used in real world applications like aircraft design. Airlift is generated by the Bernoulli Effect in which the wings of aircrafts are built specifically to use the fluid around it (in this case, air) to push it upwards. When an aircraft is moving before takeoff, the winds that hit the wings have a much higher velocity on the upper surface of the wing compared to the winds on the lower surface. This is done by making the upper surface curved while the lower surface is flat. The difference in wind velocities would create a pressure difference on the top and the bottom of wing: the top will be low pressure winds, while the bottom will be high pressure winds. Because of this pressure difference, the winds on the bottom will force to move up in order even out the pressure. The upward force of the wing is what creates the lift for the aircraft during takeoff.
SUMMARY
The Bernoulli experiment exhibited the two fluid properties Bernoulli Effect and Venturi effect. As the fluid travels across a path with small cross-sectional area, the fluid pressure decreases and the fluid velocity increases. The inverse proportionality of the pressure and velocity is consistent with the Bernoulli Effect; the behavior of the fluid when approaching small cross-sectional areas is consistent with the Venturi effect. The measured piezometric heights are slightly deviated from the theoretical values, but still the trends of proportionality are consistent across the three test runs.
REFERENCES
Blinder, SM 2016, The Venturi Effect, viewed 25 April 2016, <http://demonstrations.wolfram.com/TheVenturiEffect/>.
Romero, D 2007, Bernoulli Effect, viewed 25 April 2016, <http://scienceworld.wolfram.com/physics/BernoulliEffect.html>.
Solken, W 2008, Pipe Flow Measurement - Venturi Flow Meter, viewed 25 April 2016, <http://www.wermac.org/specials/venturiflowmeter.html>.
The Columbia Electronic Encyclopedia 2012, Bernoulli's principle, viewed 25 April 2016, <http://www.infoplease.com/encyclopedia/science/bernoulli-principle.html>.
The Engineering Toolbox 2003, Orifice, Nozzle and Venturi Flow Rate Meters, viewed 25 April 2016, <http://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html>.
