Bernoulli

bernoulli lab report tABLE OF CONTENTs TITLE PAGE …………………………………………..............………………………............. 1 ABSTARCT …………………………………………...……....………………………............... 3 introduction ……………………………………………...………………………............ 4 objective …………………………………………………....……………………….............. 6 Theory ………………………………….…………………….……………………….............. 6 Apparatus …………………………………….……………..………………………............ 10 Procedure …………………………………………………...………………………........... 11 Results ……………………………………………………….………………………............. 12 discussion …………………………………………………...………………………............ 14 conclusion …………………………………………………………………………............. 15 reference ……………………………………………………………………………............ 16 ABSTRACT This experiment is carried out to examine in depth on the validity of Bernoulli’s theorem when applied to the steady flow of water in tapered duct and to measure the flow rates and both static and total pressure heads in a rigid convergent or divergent tube of known geometry for a range of steady flow rates. The relation among the pressure, velocity and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady or laminar is indicated in Bernoulli’s theorem. The Bernoulli’s Apparatus Test Equipment is used in this, in order to demonstrate the Bernoulli’s theorem. The reading shown by manometer h* is the sum of the pressure and velocity heads and the reading in manometer hi measured the pressure head only. The time to collect 10L water in the tank was measured. Lastly, the flow rate and total velocity was calculated by using both Bernoulli and Continuity equation and the difference in velocity for both equations was also calculated from the data of the results. Based on the results taken, it has been analyzed that the velocity of the fluid is increased when it is flowing from a wider to a narrower tube as the pressure is lower at constrictions and the pressure increased as the cross-sectional area increases. introduction The flow of a fluid has to conform with a number of scientific principles in particular the conservation of mass and the conservation of energy. The first of these when applied to a liquid flowing through a conduit requires that for steady flow the velocity will be inversely proportional to the flow area. The second requires that if the velocity increases then the pressure must decrease. Bernoulli's Principle is a physical principle formulated that states that "as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases”. Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. Bernoulli’s Principle can be demonstrated by the Bernoulli equation. The Bernoulli equation is an approximate relation between pressure, velocity, and elevation. While the Continuity equation relates the speed of a fluid that moving through a pipe to the cross sectional area of the pipe. It says that as a radius of the pipe decreases the speed of fluid flow must increase and vice-versa. However, Bernoulli’s Principle can only be applied under certain conditions. The conditions to which Bernoulli’s equation applies are the fluid must be frictionless (inviscid) and of constant density; the flow must be steady, and the relation holds in general for single streamlines. In general, frictional effects are always important very close to solid wall (boundary layers) and directly downstream of bodies (wakes). Thus, the Bernoulli approximation is typically useful in flow regions outside of boundary layers and wakes, where the fluid motion is governed by the combined effects of pressure and gravity forces. Bernoulli's principle can be explained in terms of the law of conservation of energy. As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. At the same time, the work done by corresponding volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. Since the speed is greater in the narrower pipe, the kinetic energy of that volume is greater. Then, by the law of conservation of energy, this increase in kinetic energy must be balanced by a decrease in the pressure-volume product, or, since the volumes are equal, by a decrease in pressure. Figure 1.0(Convergent Flow) The converging-diverging nozzle apparatus is used to show the validity of Bernoulli’s equation. It is also used to show the validity of the continuity equation where the fluid flows is relatively incompressible. The data taken will show the presence of fluid energy losses, often attributed to friction and the turbulence and eddy currents associated with a separation of flow from the conduit walls. Figure 1.2(Divergent flow) objective The objectives of this Bernoulli’s Theorem experiment are; To investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapered duct. To measure flow rates, static and total pressure heads in a rigid convergent or divergent tube of known geometry for a range of steady flow rates Theory Bernoulli’s principle theorem states that an increase in the speed of the fluid, occurs at same time, whether with a decrease in pressure or potential energy of the fluid. This only occurs for invincible flow. The classical venture is the main apparatus of Bernoulli’s principle, which made from pure acrylic pipe of shifting roundabout cross section area, known as a venture as shown in fig (2.0). The static pressure can be measured by a series of wall tapping along the convergent duct, while a total head tube is provided to traverse along the center lie of the test section. The characteristics of the flow through both converging and departing are studied by hydraulics bench, since this test can be used to approve those conditions in which Bernoulli’s principle might be applied or in those condition in which Bernoulli’s principle is not enough to describe the fluid behavior. Figure 2.0(Venture) Bernoulli’s principle conforms to the principle of conservation of energy. The sum of all form of mechanical energy along the streamline is the same as the point of the flowing fluid. For steady flows each particles slide along its path, and its velocity vector is everywhere tangent to the path. The lines that are tangent to the velocity vector throughout the flow feed are called the streamlines. The fluid particles main properties are the pressure and weight. If the velocity increase, means that the fluid is moving horizontally along a streamline and this occurs at the lowest pressure, means Z1=Z2 will be canceled and the pressure heads h1= P1/and h2 = P2/can be measured from a common chance datum so that Bernoulli’s Equation simplifies as written. The Bernoulli equation: kinetic energy + potential energy + flow energy = constant. But if the velocity decrease, the fluid is moving vertically along a streamline and this occurs at the highest pressure, means Z1=Z2 (elevation head) and the P/(pressure head) and the velocity V2/2 (which represent the vertical distance needed for the fluid to fall freely), all must be calculated according to the equation: This can be demonstrated when the fluid is moving from a region of high pressure to a region of low pressure, the velocity of fluid might be high or low. Friction is negligible along the streamline through the venture tube. (4) P = fluid static pressure at the cross section. = density of the flowing fluid. g = acceleration due to gravity. V = mean velocity of fluid flow at the cross section. Allowance for friction losses and conversion of the pressure, P1 and P2 into static pressure heads, h1 and h2 yields: For dynamic pressure head when the mass flow rate is constant in closed system The velocity, Wmeas was calculated from the dynamic pressure: apparatus Bernoulli equipment Description: The equipment is designed and fabricated to demonstrate the Bernoulli’s theorem. It consists of a test section made of acrylic. It had convergent and divergent sections. Pressure tapings are provided at different locations in convergent and divergent section. Present set-up is self-contained water re-circulating unit, provided with a sump tank, centrifugal pump etc. An arrangement is done to conduct the experiment on different flow rates. Figure 3.0 (Bernoulli equipment) (1) 6-fold water pressure gauge (pressure distribution in venturi tube) (2) Venturi tube with 6 measurement points (3) Hose connection, water supply (4) Unions (5) Air bleed screw (6) Discharge valve (7) Gland nut (8) Probe for measuring overall pressure (9) Adjustable feet Stopwatch Description: The stopwatch that used for the timing to the flow measurement of the water Figure 3.1(stopwatch) procedure The pump switch was switched on. The discharge valve was adjusted to high measurable flow rate to fill the manometer tube to make sure the air bubble inside the manometer is free. The water level was reduced by using air bleed screw. The first flow rate was started by controlling the control valve. After the level stabilizes, gently slide the hypodermic tube connected to manometer H. So that its end reaches the cross section of the venturi tube at A. The readings from manometer H and A was taken. The step 5 was repeated for cross section (B, C, D, E and F) After that, the water flow rate is filled into the measuring tube. The time was taken by using the stopwatch. The step 4 to 7 was repeated with two other flow. The velocity was calculated using the Bernoulli’s and continuity equation. The difference between two calculated velocities was determined. Results Hdyn = Htot – Hstat. = 238.93 – 173 = 65.93m H1 H2 H3 H4 H5 H6 [mm ws] [mm ws] [mm ws] [mm ws] [mm ws] [mm ws] H stat 173 165 50 123 138 144 H total 238.9 238.9 238.9 238.9 238.9 238.9 H dyn (mm) 65.9 73.9 188.9 115.9 100.9 94.9 Wmeas = dyn = = 35.9842 m/s Wcal. = = = 0.705m/s Time ,s (10L) = 78 sec H1 H2 H3 H4 H5 H6 [mm ws] [mm ws] [mm ws] [mm ws] [mm ws] [mm ws] W mea. 35.98 38.10 60.91 47.71 44.52 43.17 W calc. 0.70 1.02 2.82 1.40 1.06 0.70 Discussion The goal of the experiment is to find out legitimacy of the Bernoulli’s mathematical statement when connected to the relentless flow of water in a decreased pipe. Bernoulli's Principle is essentially a work energy conservation principle which states that for an ideal fluid or for situations where effects of viscosity are neglected, with no work being performed on the fluid, total energy remains constant. This principle is a simplification of Bernoulli's equation, which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path. The total head for convergent flow is decreased from h1 to h6 while the total head value for divergent flow is the lowest at h1 and the highest at h6. However it is to be noted that there might have been some human and apparatus related errors unintentionally done in the experiment process which might have given us some deviated results from the actual results. However, the results can be improved if some precautions are taken during the experiment for example the eyes level must be placed parallel to the scale when manometer readings are taken. Besides that, the valve is also needed to be controlled slowly to stabilize the water level in the manometer. The human reaction error while noting the time using a stop watch can be avoided by using light gates to give out highly accurate results for the time measured. Bernoulli’s theorem has several applications in everyday lives. In certain problems in fluid flows when given the velocities at two points of the streamline and pressure at one point, the unknown is the pressure of the fluid at the other point. In such cases (if they satisfy the required condition for Bernoulli's Equation) Bernoulli's Equation can be used to find the unknown pressure. One such example is the flow through a converging nozzle. CONCLUSION From the experiment conducted, there are different cross-sections for each tube H1, H2, H3, H4, H5, and H6. These differences resulted in varieties of value obtained for stagnation head (H) and pressure head (hi). By using Bernoulli equation to calculate the velocity, it can be said that the velocity of fluid increase as the fluid is flowing from a wider to narrower tube and the velocity decrease in the opposite direction. This also indicates that the pressure of fluid decreases as the velocity increases. Therefore, the Bernoulli’s principle is proven. The Bernoulli equation forms the basis for solving a wide variety of fluid flow problems such as jets issuing from an orifice, flows associated with pumps and also turbines. Bernoulli’s equation is also useful in demonstration of aerodynamic properties such as drag and lift. From the data and results calculated, we can conclude that the Bernoulli equation is valid for flow as it obeys the equation and the objectives are successfully achieved. References http://documents.tips/documents/bernoullis-theorem-experiment.html http://documents.tips/documents/bernoullis-55f8286f6e7ec.html http://www.dummies.com/how-to/content/use-bernoullis-equation-to-calculate-pressure-diff.html Prentice Hall (2012). Fluid Mechanics for Chemical Engineers. Oxford publications. Pg.34-39. Douglas, J.F.Gasiorek, J.M. and Swaffield, J.A.(1999) Fluid Mechanics, 3rdedition. Longmans Singapore Publisher, Pg. 99-101 Giles, R.V., Evett, J.B. and Cheng Liu, Schaumm’s (2010)Outline Series Theory and Problemsof Fluid Mechanics and Hydraulic, McGraw Hill Intl. Pg.559-620 19