Momentum principal and jet impacts
University of Birmingham Mechanical Engineering Experiments and Statistics The Short Laboratory Report Momentum principle and jet impacts Introduction Water turbines are widely used throughout the world to generate power, they were developed in the 19th century and were widely used for industrial power prior to electrical grids. This is applying the principle of conservation of linear momentum theorem. When a jet of fluid strike a flat or curved surface it will produce a force. The fluid will not rebound from the plate and moves over the surface tangentially.
This report will examine the application of the momentum principle to fluid flows by analysing the behaviour of water jets and their impact on different surface. It will show the force produced by a jet of water as it strikes a flat plate, curve surface and then compare this to the momentum flow rate in the jet. The report is divided into four main sections. It will first consider different results that were taken in the experiment and then required calculation were made for both surface. It will then go on to describe the difference between the experimental and theoretical results.
The third part compares forces in a graph. Finally some conclusion will be provided as to explain the errors and difference between the theoretical and experimental result. Objectives This experiment demonstrates the principle of conservation of linear momentum by measuring the force generated on a flat and curved surface due to an impinging water jet and comparing these forces with those that would be expected from an application of the momentum principle to the control volume that surrounds the water jet. Theoretical consideration
Mechanical work can be produced by using pressure of moving fluid at high velocity. As an example jet of water from nozzle can produce force when it strikes a plane of the surface of plate. If the jet of water impinges on the flat plate which deflects the water flow through 90 degrees the force on the plate will be: F = ? A = ? Qu (equation 1) If the jet of water impinges on the curved plate that deflect the water flow through 180 degrees it can again be shown that: F = 2? A = 2? Qu (equation 2) Where, Q is the discharge ? is the density of water (=1000 kg/ ) u is the velocity
A is the cross sectional area of water jet Some data required for calculation such as nozzle diameter d? , the vertical distance z between the end of the nozzle and the plate and the jockey weight were given as shown in table 1. There is a difference between the velocity at the plate u and the velocity at the nozzle un. The velocity at the plate u is somewhat smaller than the velocity at the nozzle-exit un due to the deceleration caused by gravity. To determine the velocity at the plate u the expression for vertical motion in a gravitational field were used: u? = ? – 2gz (equation 3)
Where un = Q/A (equation 4) The other thing that were considered was the force exerted on the plate can be determined from taking moments about the spring balance pivot which gives: 0. 15F = mgx (equation 5) Where, F is the force exerted on the plate m is the mass of the jockey weight g is the acceleration due to the gravity x is the position of the jockey weight The experiment consist of a nozzle from which a vertical jet of water impinges ,a set of impact surfaces (flat or curved) where the water jet impinges, an outlet at the base of the apparatus that directs the flow to a catch-tank for measuring the flow rate. A plate is attached to a lever arm which carries a jockey weight and is restrained by a spring scale. The lever arm is initially horizontal with the jockey weight at its zero position and the water jet turned off. The flat surface deflects the flow through 90 degrees and the curved surface through 180 degrees. Procedure A flat plate was installed on a lever arm connected to a spring scale. The spring scale was adjusted via the thumb screw until the lever arm become horizontal when the distance of the jockey weight on the lever arm is zero and the jet turned off.
Then the self-priming pump was turned on, the jockey weight was moved to the right and the lever arm was no longer horizontal due to the momentum caused by moving the jockey weight. Water was admitted into the nozzle by adjusting the bench valve (counter clockwise to open and clockwise to close) and the flow rate was increased until the lever arm become again horizontal and static equilibrium reached and then readings of the position of the jockey weight were made. The flow rate was measured by closing the outlet and stopwatch methods were used with the catch-tank. For every 0. 025 metres to the right along the lever arm was recorded how long it took to reach 10 litres of water. A total of 8 different position were recorded for gradually increasing flow rates as shown in table 2. 1, such that the jockey weight was moved to the right in every 0. 025 metres. After 8 stages some data needed to calculate the force measured with the spring balance system were calculated for both plates (flat and curved) for all stages such as, the velocity at the nozzle-exit un, and the velocity at the plate u, as shown in table 2. 1 and 2. 2. Observation and results
The velocity at the nozzle was obtained by dividing the flow rate Q by the cross-sectional area of the nozzle as shown in equation 4. This obtained value is the theoretical velocity of the fluid exiting the nozzle. This velocity was then used to obtain the velocity at the plate u as show in equation 3. Data analysis Table 1: Known values used to determine theoretical and actual values of force exerted by a jet of water Nozzle diameter (m) Nozzle area A(m? ) Z (m) m (kg) 0. 01 7. 85×10^-5 0. 035 0. 6 Table 2. 1: theoretical and experimental values for the flat plate x (m)
Two points on the graph were picked and the difference between the y coordinates was divides by the difference between the x coordinates. The obtained value is the gradient of the graph. The gradient of the flat plate graph is 1. 2 and of the curved plate graph is 1. 08. The expected gradient was to be 1 but the theoretical and experimental force have a percentage of error between them. Conclusion Theoretically, the calculated force should be the same as the measured force. However, this cannot be achieved experimentally due to the errors made during the experiment.
Most of this errors is due to some data obtained from the experiment was not exactly correct. The stopwatch may not have been stopped at the exact moment, the jockey weight could not be in the exact position, the friction could have reduce the water flow, and the lever arm could not be exactly horizontal. It can be noticed that the force acting on the curved plate is bigger than the force acting on the flat plate, so, if a force act on both plates the discharge and the velocity of the flow for the curved plate will be less than the flat plate.