BERNOULLI’S EXPERIMENT
INTRODUCTION
This experiment aims to verify Bernoulli’s equation that relates the pressure and the velocity of a fluid. The aims of the experiment are:
A venturi meter is a device used in fluid measurements. It uses a converging section of a pipe in order to change fluid velocity or pressure. The venturi meter in this experiment is used several times to estimate flow rates and measure piezometric head values at different points along the venturi meter.
The fluid used in this experiment is water. Bernoulli’s equation is a mathematical model for ideal fluids. Because of this, the results of the experiment will show how close to ideal behavior does water behave in terms of complying with Bernoulli’s equation.
The measured data from the venturi meter are used to verify the existence of the venturi effect. The venturi effect is the tendency of a fluid to increase its velocity, and decrease its pressure correspondingly, whenever it encounters a smaller cross sectional area in its flow path. This behavior of fluids in streamlined paths is consistent with two things: the principle of continuity, and the conservation of mechanical energy. The conservation of mechanical energy follows the equation:
emech=pρ+12v2+gz
Where:
z=potential head
p=fluid pressure
v=fluid velocity
ρ=fluid density
In order to conserve mechanical energy, any increase in the velocity of the fluid is balanced out by a decrease in the pressure of the fluid. This relationship between fluid velocity and pressure is known as the Bernoulli principle .
BACKGROUND AND THEORY
The behavior of an ideal fluid of constant density flowing under steady state conditions may be predicted by Bernoulli’s equation stated in the following:
Z+v22g+pρg=constant=H=total head m
Where:
Z=potential head
v22g=kinetic head
pρg=kinetic head
For an ideal fluid in a connected system, Bernoulli’s Theorem postulates that the total energy must remain constant at all points if there is no energy loss (due to friction or other factors). Thus:
Z1+v122g+p1ρg=Zn+vn22g+pnρg 1
In the above diagram, it can be seen that the piezometric head at any point is the sum of potential and pressure heads and rearranging the equation yields:
p1ρg+Z1-pnρg+Zn=vn22g-v122g
The laboratory venturi 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.
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. 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).
PROCEDURE
The method will be explained in detail at the start of the laboratory and you will be required to run the experiment and collect data as a group. The general procedure is as follows:
Establish a steady flow and measure the flow rate (Q) using the weight bench. Take at least two readings (ideally 3 if time permits).
Measure the piezometric head values (h) for all eleven points along the venturi meter.
For the results, the following steps must be done:
Plot the observed values hn against the position along the venturi meter (to scale) – use the attached graph.
Using the observed value of flow rate (Q), calculate the velocity at each section (vn) – use the attached calculation sheet.
Calculate the theoretical values of hn as predicted by Bernoulli’s Theorem:
The hydrostatic pressure (p) in a fluid of depth (h) is given by:
p=ρgh
If we substitute values into Eq. 2 above and rearrange, we find that:
hn=h1=vn2-v122g
Assume that the density of water is 1000kgm3: e.g. a mass flow of 5kgs=5ls=0.005m3s.
RESULTS
All the measurements from the experimental runs are tabulated in a single table. Additionally, the calculations for the velocities and the theoretical hn are also included in the table. Excel was used to calculate the flow rates Q, the velocities vn and theoretical hn values faster and be displayed in a neat fashion. The theoretical piezometric height hn assumes that the fluid is ideal, thus the values are the same for both the inlet and outlet points. The reference position for each theoretical piezometric height computation is the measured hn and vn at position A. The final value of the flow rate for each of the runs is used for the calculation of the fluid velocity. The flow rate Q is computed by:
Q=Massρ∙Time
The table of results is presented in the following:
The graph of the hn against the positions A to K for the first test run is shown in the following:
It is observed both in the table and the graph that the values of h at the positions 1 and 11 (h1 and h11) are not equal. The piezometric head values dip at the positions D and E. The minimum value occurs at position D (hn=0.19 m). The maximum value occurs at position A (hn=2.73 m).
CONCLUSIONS
The piezometric head value measurements showed that the pressure changes in different parts of the venturi meter. Near the terminal positions (h1 and h11), the pressure values are at the highest. The piezometric head value is at its minimum on the position D, which is also the point of smallest cross sectional area (smallest diameter of 16 mm). Consequently, this is also the point of highest velocities (highest velocity achieved in test 1 with v4=4.64ms).
For an ideal fluid, the first and last piezometric heights must be equal. In the case of water in this experiment, this was not achieved because the last piezometric height is always smaller than the first piezometric height (this is true for all the three test runs). Nevertheless, it is consistent that the maximum piezometric height occurs at position A.
The Bernoulli Effect is observed in this experiment as shown in the tabulated results. When the piezometric height decreases along the positions of the venturi meter, the pressure also decreases. In effect, the velocity values across these positions increase. This inverse relation of the fluid pressure and the fluid velocity is consistent at any position in the venturi meter. Moreover, the Venturi effect is also observed particularly on the regions of decreasing cross sectional areas. The velocities are observed to be at the maximum at the position of smallest cross sectional area (position D).
The Bernoulli Effect can be applied as the guiding principle in airplane wing design and architecture . The wing of the airplane is designed such that the velocity of air passing through the top of the wing is much greater than the velocity of air passing through the bottom of the wing. With this velocity discrepancy (considering air as a fluid) would cause a change in pressure between the two regions, top and bottom, of the wing. The bottom region would be a high pressure region, while the top region would be a low pressure region. Since the tendency of fluids is to flow from high to low pressures, the air at the bottom of the wing will force itself to rise towards the low pressure region at the top of wing. This force would generate the lifting force for the airplane to be able to take off.
Moreover, the Venturi effect can be applied to water pumping applications. In cases where water needs to be pumped up to a higher altitude location, the pipe cross sectional areas must be designed accordingly such that the pumping capacity of the water pump engine would be sufficient for water to reach the target location. The cross sectional area should be small enough to maximize the benefit of increased fluid velocity, but it should also be large enough such that the amount of water delivered is still at a desirable level.
REFERENCES
Meyers, JM, Fletcher, DG & Dubrief, Y 2013, MAE 123 : Mechanical Engineering Laboratory II - Fluids, viewed 23 April 2016, <http://www.cems.uvm.edu/~jmmeyers/ME123/Lectures/ME123%20Lecture%202.pdf>.
1999, Bernoulli's Principle, viewed 23 April 2016, <http://theory.uwinnipeg.ca/mod_tech/node68.html>.
Reader-Harris, MJ 2011, Venturi Meters, viewed 23 April 2016, <http://www.thermopedia.com/content/1241/>.
Smid, T, Bernoulli's Principle and Airplane Aerodynamics, viewed 23 April 2016, <http://www.physicsmyths.org.uk/bernoulli.htm>.
