The test section, which is provided with a number of hole-sided pressure tapings, connected to the manometers housed on the rig, is indeed an accurately machined clear acrylic duct of varying circular cross section. The tapings allow the measurement of static pressure head simultaneously.
A flow control valve is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve, and the bench supply control valve.
Consider a system whereby Chamber A is under pressure and is connected to Chamber B, which is as well under pressure. The pressure in Chamber A is static pressure of 689.48 kPa. The pressure at some point, x along the connecting tube consists of a velocity pressure of 68.95 kPa exerted 10 psi exerted in a direction parallel to the line of flow, plus the unused static pressure of 90 psi, and operates equally in all directions. As the fluid enters chamber B, it is slowed down, and its velocity is changed back to pressure. The force required to absorb its inertia equals the force required to start the fluid moving originally, so that the static pressure in chamber B is equal to that in chamber A.
From the above illustration, Bernoulli’s principle relates much with incompressible flow. Below is a common form of Bernoulli’s equation, where it is valid at any arbitrary point along a streamline when gravity is constant.
...............(1)
where:
is the fluid flow speed at a point on a streamline,
is the ,
is the of the point above a reference plane, with the positive z-direction pointing upward — so in the direction opposite to the gravitational acceleration,
is the at the point, and
is the of the fluid at all points in the fluid.
If equation (1) is multiplied with fluid density, ρ, it can be rewritten as the followings;
...........(2)
Or
........(3)
where:
is ,
is the or (the sum of the elevation z and the and
is the total pressure (the sum of the static pressure p and dynamic pressure q).
The above equations suggest there is a flow speed at which pressure
is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids, when the pressure becomes too low, occur. The above equations use a linear relationship between flow speed squared and pressure.
Generally in many applications of Bernoulli’s equations, it is common to neglect the values of ρ g z term, since the change is so small compared to other values. Thus, the previous expression can be simplified as the following;
.......(3)
where p0 is called total pressure, and q is dynamic pressure, whereas p usually refers as static pressure. Thus,
Total pressure = static pressure + dynamic pressure.......(4)
However, a few assumptions are taken into account in order to achieve the objectives of experiment, which are as the followings:
- The fluid involved is incompressible
- The flow is steady
- The flow is frictionless
APPARATUS
- Venturi meter
- Pad of monometer tubes
- Pump
- Stopwatch
- Water
- Water tank equipped with valves water controller
- Water hosts and tubes
EXPERIMENTAL PROCEDURE
- The test section tube is set to be converging in the direction of flow.
- The pump switch is opened. The flow control valve is then opened and the bench valve is adjusted to allow the flow through the manometer.
-
The air bleed screw is opened and the cap is removed from the adjacent air valve until the same level of water in manometer is reached. The bench valve is adjusted until the h1 – h5 head difference of 50mm water is obtained.
- The ball valve is closed and the time taken to accumulate a known volume of 3L fluid in the tank is measured to determine the volume flow rate.
-
The whole process is repeated using Δ (h1 – h5) 100 and 150 mm water.
- Next, the experiment is repeated for divergent test section tube.
RESULTS
Convergent Flow
Pressure difference = 50 mm water
Volume (m3) = 0.003
Time (s) = 46
Flow rate (m3/s) = 6.522x10-5
Pressure difference = 100 mm water
Volume (m3) = 0.003
Time (s) = 31
Flow rate (m3/s) = 9.677x10-5
Pressure difference = 150 mm water
Volume (m3) = 0.003
Time (s) = 25
Flow rate (m3/s) = 1.200x10-4
Divergent Flow
Pressure difference = 50mm water
Volume (m3) = 0.003
Time (s) = 30
Flow rate (m3/s) = 1.000x10-4
Pressure difference = 100 mm water
Volume (m3) = 0.003
Time (s) = 23
Flow rate (m3/s) = 1.304x10-4
Pressure difference = 150 mm water
Volume (m3) = 0.003
Time (s) = 20
Flow rate (m3/s) = 1.500x10-4
Figure 1 Graph of Total Head versus Pressure Head for Convergent Flow
Figure 2 Graph of Total Head versus Pressure Head for Divergent Flow
SAMPLE CALCULATIONS
Divergent Flow
Pressure difference = h1 - h5 = 100 mm water
Flow rate = 0.003/23
= 1.304× 10-4 m3/s
Velocity, v = Flow rate
Area into duct
= 1.304 × 10-4m3/s
490.9 x 10-6 m2
= 0.2657 m/s
Dynamic head = v2
2g
= (0.2657 m/s) 2
2 x 9.81m/s2
= 0.0036 m
Total head = Static head + Dynamic head
= (0.0036 + 1175x10-3) m
= 0.1786 m
DISCUSSION
Referring back to the objectives of the experiment, which are to investigate the validity of the Bernoulli’s equation when applied to the steady flow of water in a tapered duct as well as to measure the flow rate and both static and total pressure heads in a rigid convergent and divergent tube of known geometry for a range of steady flow rates.
As fluid flows from a wider pipe to a narrower one, the velocity of the flowing fluid increases. This is shown in all the results tables, where the velocity of water that flows in the tapered duct increases as the duct area decreases, regardless of the pressure difference and type of flow of each result taken.
From the analysis of the results, we can conclude that for both type of flow, be it convergent or divergent, the velocity increases as the pressure difference increases. For instance, the velocities at pressure head h5 at pressure difference of 50 millimetres, 100 millimetres and 150 millimetres for convergent flow are 0.8308 m/s, 1.5290 m/s and 1.2740 m/s respectively, which are increasing. The same goes to divergent flow, whereby the velocities are decreasing when the pressure difference between h1 and h5 is increased. Note that for divergent flow, the water flows form pressure head h5 to h1, which is from narrow tube to wider tube.
Next, the total head value for convergent flow is calculated to be the highest at pressure head h1 and the lowest at pressure head h5, whereas the total head for divergent flow is in a different case where it is calculated to be the highest at pressure head h5 and the lowest at pressure head h1.
There must be some error or weaknesses when taking the measurement of each data. One of them is, the observer must have not read the level of static head properly, where the eyes are not perpendicular to the water level on the manometer. Therefore, there are some minor effects on the calculations due to the errors.
CONCLUSION
From the experiment conducted, the total head pressure increases for both convergent and divergent flow. This is exactly following the Bernoulli’s principle for a steady flow of water and the velocity is increasing along the same channel.
The second objectives, where the flow rates and both static and total head pressures in a rigid convergent / divergent of known geometry for a range of steady flow rates are to be calculated, are also achieved through the experiment.
RECOMMENDATION
- Repeat the experiment several times to get the average value.
- Make sure the bubbles are fully removed and not left in the manometer.
- The eye of the observer should be parallel to the water level on the manometer.
- The valve should be controlled slowly to maintain the pressure difference.
- The valve and bleed screw should regulate smoothly to reduce the errors
- Make sure there is no leakage along the tube to avoid the water flowing out
REFERENCES
-
B.R. Munson, D.F. Young, and T.H. Okiishi, Fundamentals of Fluid Mechanics, 3rd ed., 1998, Wiley
and Sons, New York.
-
Douglas. J.F., Gasiorek. J.M. and Swaffield, Fluid Mechanics, 3rd edition, (1995), Longmans Singapore Publisher.
-
Giles, R.V., Evett, J.B. and Cheng Lui, Schaumm’s Outline Series Theory and Problems of Fluid Mechanics and Hydraulic, (1994), McGraw-Hill intl.
APPENDICES