Introduction
The experiment was conducted to examine the validity of Bernoulli’s Principle as applied to the flow of water in a tapering circular duct. It employed the use of Bernoulli’s Theorem Demonstration Apparatus, hydraulics bench and a stopwatch to gather relevant data. The volume of water was measured and its flow was timed using the stopwatch. The two values were used to calculate the flow rate of the fluid. Additionally, the manometer readings were collected and used to compute the velocity of the fluid at each tapping position. The values were used to calculate the theoretical velocity head at each tapping position. The calculated results were compared with the measured result so as to determine the validity of Bernoulli’s Theorem.
Literature Review
Bernoulli principle has been applied in many areas of physic to achieve various significant inventions and innovations in equal measure. In most cases, Bernoulli principle is used to investigate the relationship between speed and pressure in a flowing fluid. The principle explores the association that is existing between the speed of the fluid and the pressure or potential energy. Bernoulli’s Theorem holds that; for an in-viscid flow of a non-conducting fluid, there is an inverse proportionality between the speed of the fluid and the pressure (Castro-Orgaz and Chanson 2009). That is to say that an increase in speed of the fluid results in a decrease in the fluid's pressure or potential energy. On the other hand, a decrease in the speed of the fluid results in a simultaneous increase in pressure.
Bernoulli's Theorem is applicable in numerous types of fluid flow which generate numerous types of Bernoulli's equation. It is important to note that different types of flow exhibit different forms of Bernoulli’s equation. For incompressible flows, the simplest form of Bernoulli’s equation is valid. On the other hand, compressible flows require the application of more sophisticated forms of Bernoulli’s equation. It is imperative to note that incompressible flows exist in lower Mach numbers while compressible flows exist at higher Mach numbers.
The principle borrows from the principle of conservation of energy. All energy within a flowing fluid remains the same but can be converted from one form to another without destruction or creation (Rouse 2011). It is imperative to point out that the sum of all types of energy in a fluid that exhibits a steady-state flow along a streamline remains the same at all points within the streamline.
v22+gz+pρ=Constant
V is the velocity of the fluid which represents the kinetic energy exhibited by the fluid as it flows. G and z represent the gravitational acceleration and elevation of the point respectively and are used to determine the potential energy in the fluid. On the other hand, p and ρ represent the pressure and density respectively.
This principle implies that the sum of internal energy, potential energy and kinetic energy remains the same within the streamline in a steady state flow (Castro-Orgaz and Chanson 2009). As a result, an increase in the velocity of the fluid means an increase in kinetic energy and the fluid's dynamic pressure alike. The increase in fluid's velocity transpires with a concurrent decline in the sum of its internal energy, potential energy and static pressure. In case fluid is flowing out of a streamline, the overall total of all the different types of energy should be the equivalent on all streamlines since the streamline the summation of gravitational potential and pressure (energy per unit volume) is equal everywhere.
Methodology
The following tools and equipment were used; hydraulics bench, stopwatch and Bernoulli's Theorem Demonstration Apparatus.
Procedure
The Bernoulli apparatus was levelled on the Hydraulics Bench. The manometer tubes were carefully filled with water to remove air pockets from the system and to ensure that all the connecting pipes are devoid of air. The inlet feed and the flow control valves were adjusted to achieve a combination of system pressure and flow rate which will output the biggest convenient difference between the lowest and the highest manometer levels. The volumetric ad the stop watch were used to measure the flow rate of the fluid. Three readings were taken so as to get an average flow. The probe was inserted into the end of the parallel duct and moved underneath the manometer opening. The scale reading of the manometer level and the corresponding probe level were recorded.
Results and Discussion
Flow rate
Manometer reading
Results
It is imperative to note that if a small amount of fluid is flowing in a horizontal manner to a region of low pressure for a region of high pressure within a streamline section, then the fluid exhibit more pressure behind as compared to the pressure in front (White and Corfield 2006). This principle presents an overall force on the fluid’s volume, thus accelerating the fluid along the streamline section. There are only two factors that fluid particle are subject to; their weight and pressure. The increase in speed exhibited by a fluid which flows in both as streamline and horizontal section is a result of the fluid's movement from a region of high pressure to a region of low pressure. On the other hand, a decrease in the speed of fluid in the same condition is caused by the fluid's movement from a low-pressure region to a high-pressure region. As a result, within a fluid flowing in a horizontal manner, the highest velocities of the fluid are recorded where the fluid's pressure is lowest while the lowest velocities are recorded where the fluid's pressure is the highest.
Conclusion
According to the results obtained, Bernoulli’s principle holds. The measured results and the calculated results exhibited a slight deviation given the various errors exhibited during the experiment. It is imperative to point out that the calculated result and the measured results exhibited a similar pattern.
References
Castro-Orgaz, O. and Chanson, H., 2009. Bernoulli theorem, minimum specific energy, and
water wave celerity in open-channel flow. Journal of irrigation and drainage
engineering, 135(6), pp.773-778.
Rouse, H., 2011. Elementary mechanics of fluids. Courier Corporation.
White, F.M. and Corfield, I., 2006. Viscous fluid flow (Vol. 3). New York: McGraw-Hill.
