A Physical Introduction to Fluid Mechanics - Research Paper Example

EXPERIMENT 1: IMPACT OF A JET EXPERIMENT 4: BERNOULLI’S THEOREM DEMONSTRATION Name Professor Institution Course Date Experiment 1 - Impact of a Jet Aim The impact of a jet experiment was aimed at determining the values obtained through the experiment of the force directed towards a variety of shape targets and then subjecting this to comparison with the values in theory and therefore obtain a validation of the force values in theory. Theory In the past decades scientists have discovered several ways of utilization of the force that can be exerted by a fluid jet on a surface that diverts the flow (Nicholas, 1990). For instance, the use of the rotating has been used in making of flour. Other than that, several turbines are still in use is in the second and at times in the first stages turbine running on steam. People dealing with fire use the moving energy possessed by a jet to avail water at a level that is above the nozzle so as to be able to extinguish fires in buildings that are much taller. Jets of fluids are also utilized in the metal industry for debarring as well as cutting operations. Several applications involving fluid jets play a very important part in the technological development. A model in theory for the force required in holding the surface of impact in a stationery position is attained through the application of several expressions of momentum and continuity in an integral form. The model details are dependent on the nature of the fluid that leaves the surface of the impact. That is whether it is symmetrical in nature in relation to the vertical axis or it is symmetrical in nature in relation to the horizontal axis. Fluid motions are often caused by the force effects of the surrounding of the fluid. For a general situation the flux of the momentum in a fluid jet is give as; (m x Uo) Where: is the flow rate of the mass (Kgs) is the velocity of the jet at the upper stream of the vane. Following deflection through an angle, the flux of the momentum is (m x u1cos) which is in the x-axis direction. The force experienced by the fluid is then given as (m x u1cos) – (m x u); this force also acts in the direction of the x-axis. Therefore the force F in the x-axis direction is found to be: (1) For the case of a flat surface target whereby =90 and therefore then the equation (1) changes to: In the case of a hemispherical target surface that is a cup, where by the =180 and therefore cos  = -1. This makes the equation (1) to change to: F=m x (Uo-U) Furthermore if reduction in speed happens to a rate that is negligible such that (Uo=U1) then the equation (1) takes the form In this king of experiment it is much difficult to directly measure the upper stream vane value. However the initial velocity u with which the fluid exits the nozzle is possibly determined. The velocity at entry is very minimal due to declamation resulting from the force of gravity and its calculation can be made from the expression of Bernoulli relationship which is; Considering the target surfaces where; Z=0 Po=P Zo=s the above Bernoulli expression yields to: Where; The distance from the nozzle exit to the valve surface For purposes of making calculations of the force exerted on the valve as a result of the jet, the moment about the pivot is taken into consideration for the of the beam. The direction of the fluid that is deflected is largely dependent on the nature of the target solid. The rate of momentum change in fluids is caused by a change in the direction of flow of fluid in relation to the deflection. The force of hydrostatic causes the surface of the target of the fluid flow to be in motion. Force is exerted by a fluid jet in the straight line direction and is expressed as follows; fluid density (kg/m3) fluid volumetric rate (m3/s) velocity = Q/A (m/s) The angle of fluid impact against the target surface (Theoretically, the impact angle for flat shaped targets is 90o, 180o) Therefore forces that are exerted in the vertical direction by the fluid jet on flat shaped target Fy is expressed as; And for hemispherical shaped surface, force exerted by the jet fluid in the vertical direction is expressed as is Fy is expressed as; However, the ideal jet force is similar in magnitude to the product gravitational acceleration and weight of the pan. Apparatus The apparatus used in this experimental performance included the following; A jet impact equipment Bench of hydraulics Stop watch Method The impact of a jet experiment used the following apparatus in the setup method installed as follows; Apparatus: - An equipment for jet impact, bench of hydraulics and a stop watch The supply of water was made to the jet apparatus that was situated in a closed loop by a pump. The determination of the flow rate was done using a stop watch and the weighing tank. The water was released and directed in an upwards direction, through a nozzle into the air. The following surface targets were obtained; flat plate, hemispherical cup as well as the cone surface. Every object was mounted on a horizontal lever that was placed on top of the water jet and where impacts were made. The force experienced by the object was determined through using weights that were hung at various lever positions. The apparatus were stand on the hydraulic bench with the drainage pipe on top of the whole that lead to tank used for weighing. The bench was then connected to the horse inlet pipe with the use of a horse clip for the purpose of securing connection. The flat target surface was fitted onto the apparatus and held in place through tightening of a screw. While ding this care was taken not to drop the plastic surface. The cover plate was then fitted on top of the flat stem plate and it was held in position at a level lower than the beam and then screwed to tighten it. The weighing beam was set at a position coinciding with the datum and then the jokey was first set to bring the groove of the datum to zero. The switch on the pump of the bench was opened to allow water into the apparatus. The drain pipe was checked to confirm whether it was correctly installed. The supply valve was then fully opened and the jokey weight slid on the side of the beam until the tally retuned for recording and measurement. The jockey was moved in an inside direction by about which aided in the reduction of the rate of flow up to a point when it was approximately equal to the level of the beam. The beam was then set at exactly the same position as was indicated by the tally. The entire procedure was repeated again with various surface targets and the data recoded. Results The following results were collected from observation and recording of experimental data values shown; The measurement of the nozzle diameter size was done and recorded as and the nozzle cross sectional are area was then obtained with the use of the formula; For the case of the flat surface target, the following data vales were observed and recorded and the result of values indicated in a tabulated style as shown; Flat target Pan (kg) (L) Vol (m^3) Time (s) Flow Rate Q(m^3/s) Q^2(m^3/s)^2 Actual force (N) Theoretical force (N) Theoretical mass 0.05 5 0.005 30.56 1.70E-04 2.89E-08 0.49 0.624 0.0636 0.1 5 0.005 23.31 2.15E-04 4.60E-08 0.9981 1.785 0.182 0.15 5 0.005 17.87 2.80E-04 7.83E-08 1.962 2.874 0.292 0.2 5 0.005 14.44 3.46E-04 1.20E-07 2.943 3.67 0.374 0.25 5 0.005 13.84 3.61E-04 1.31E-07 3.924 4.405 0.449 0.3 5 0.005 12.71 3.93E-04 1.55E-07 4.905 6.371 0.649 Hemispherical Target Mass on pan (kg) Vol (L) Vol (m^3) Time (s) Flow Rate Q(m^3/s) Q^2(m^3/s)^2 Actual force (N) Theoretical force (N) Theoretical mass 0.05 5 0.005 40.47 1.23548*10^4 1.53E-08 0.981 0.8028 0.0818 0.1 5 0.005 29.19 1.71292*10^4 2.93E-08 1.962 1.3641 0.139 0.15 5 0.005 24.16 2.06954*10^4 4.28E-08 2.943 2.0863 0.2126 0.2 5 0.005 21.12 2.36742*10^4 5.60E-08 3.924 2.5595 0.2609 0.25 5 0.005 19.6 2.55102*10^4 6.51E-08 4.905 3.0452 0.3104 0.3 5 0.005 16.5 3.0303*10^4 9.18E-08 5.8836 3.6551 0.3725 Discussion It was determined from the analysis of the graph of the recorded data that there was a relationship between the experimentally obtained data and the ones known in theory. These differences were found to have resulted from the availability of errors as well as unfavorable experimental environments and surroundings. The graphical information could not be relied upon following the fact that they were not as per the expectations. This was caused by number factors which involved the aspect of human error, changes in the experimental conditions as well as discrepancies in the instrument and equipment of measurement. The human errors were taken to have contributed significantly in affecting the unreliability of the results obtained. These were cited to operations which consisted of activities like the changes in the datum which involved the use of naked human eye. The eye sight is likely to bring about errors to do with the readings parallax. Errors were also committed to a worrying level in the course of the operations involving operation of the stop watch while at the same time monitoring the flow of water. The errors in the experiment were several and therefore it is very important to come up with ways and procedures of reducing the amount of the factors that facilitated the binging about of mistakes and errors as well as strategizing on improvement measures to focus on accuracies on the experimental data process and also the appropriate operations on equipment and used in the measurement process. It is also advisable to do the reading and data recording of values from observation for several numbers of times then computing the average values which are more accurate. The use of mathematical expression of in the calculation of figures as well as values for use in, making decisions in the course of the experiment decisions and analysis is expected to be performed with a lot of precision and accuracy. Another factor that contributed to errors in this experiment for this experiment was the functionality of the stop watch that was used in timing and time data collection. It was such that is displayed valued that were already rounded and approximated time readings. This brought about several mistakes. Conclusion The experimental comparison of the theoretically known values of the forces responsible for casing the different kind of behavior of fluid jet and the experimentally determined force values was done and the results were analyzed which were able to clearly demonstrate the relationship. It was established that the two values were slightly different and this slight difference was attributed to the experimentation errors involved in the process of conducting the experiment. Experiment 4 – Bernoulli’s Theorem demonstration Aim The Bernoulli’s theorem demonstration experiment was aimed at investigation of the validity of the principle of Bernoulli with regard to incompressible steady water flow in both diverging and converging water flows. Theory The various forms of mechanical energy find their relationship with one another on the basis of the analysis of the Bernoulli’s theorem expression. The mechanical forms of energy which kinetic energy; the pressure energy as well as the potential energy is summed up at any one particular point during the flow to be equal to a constant value. The Bernoulli’s theorem in this case explains that for an incompressible fluid flowing through along a streamline the energy that is possessed by the fluid is conserved. The Bernoulli’s theorem can be mathematically described as; …………………………………………………. (I) Where; P = is the static pressure (Pa) p = is the density of fluid (Kg/m3) V = is the velocity of the fluid (m/s) Z = is the fluid vertical elevation (m) g = is the gravitational acceleration (9.81 m/s2) The Bernoulli’s expression is actually a sum or combination of heads that are equivalent to a constant the constant disappears when comparison analysis are being performed using the Bernoulli’s equation for different points of fluid flow along the streamline which could otherwise be termed as state one and state two. The relationship between the static pressure and the total pressure that is possessed by an incompressible fluid flowing at any one point in time is that the total pressure is the sum of the static pressure and the dynamic pressure where by a Pitot tube is used in the measurement of the total pressure since it brings about deceleration of the flowing fluid and it then takes the measurement of the pressure at that particular point at this point the velocity of the fluid is zero. Figure 1: Schematic illustration of total and static pressure measurement The flow of fluid through a convergent section is different from the fluid flow through a divergent section. This is because at a converging point, the pipe’s cross sectional area gradually reduces and therefore leading to an increase in velocity. The increase in velocity in turn causes a decrease in the fluid pressure as is governed by the principle of conservation of energy. However, for the case of divergent fluid flow there is turbulence that is experienced at points with a higher rate of flow. This in turn subjects the fluid to a swirling motion and the streamlines are absent in this kind of fluid flow. Apparatus Stop watch Measuring cylinder Bench of hydraulics Pressure manometers Venturi apparatus Method The method used in this experiment involved first setting up the apparatus which consisted of preparing the hydraulic bench and while ensuring that the closure of the rig valves. The rig valve was then opened at half turn before the bench was started through switching on the on/off switch. This was followed by slow and gradual opening of the bench valve which allowed for the filling of the Venturi apparatus with water. At this point in time, the rig valves were confirmed to be open enough which enabled the letting out of excess water which was a preventive measure enforced to prevent the bursting of the horse pipe. The entire head probe was then entirely inserted with its tip present at first manometer known as manometer one. After the process of starting the apparatus the followed the installation of bleeding manometers and the flow rates were set appropriately before the reading process commenced. The reading and recording of the data values was done and the values were presented as experimental results for evaluation and analysis. Results Time (Sec) Static head (mm) Total head (mm) Flow rate (m3/s) Diameter (mm) Area(m2) Dynamic head (mm) T1 h1 275 ho1 278 0.000749520 25 19.75 3 12.64 h2 218 ho2 278 0.000115759 13.9 152.64 60 h3 163 ho3 278 0.000159895 11.8 110 15 h4 104 ho4 276 0.000196639 10.7 90.45 172 h5 30 ho5 273 0.000242650 10 79 243 h6 116 ho6 163 0.000293327 25 19.75 47 h7 114 T2 h1 148 ho1 148 0 25 19.75 0 32.3 h2 138 ho2 147 0.000044371 13.9 152.64 9 h3 127 ho3 146 0.000067507 11.8 110 19 h4 120 ho4 146 0.000075957 10.7 90.45 26 h5 107 ho5 146 0.000099516 10 79 39 h6 119 ho6 140 0.000214342 25 19.75 21 h7 118 T3 h1 140 ho1 139 0.000000721 25 19.75 1 36.52 h2 132 ho2 138 0.000033446 13.9 152.64 6 h3 126 ho3 138 0.000048206 11.8 110 14 h4 120 ho4 138 0.000059456 10.7 90.45 18 h5 111 ho5 138 0.000077897 10 79 27 h6 120 ho6 132 0.00008655 25 19.75 12 h7 118 Discussion Following the calculations and analysis of the results it was determined that indeed the total head of a fluid is equal to the sum of the dynamic and the static head. The three heads that are involved include; the static or the pressure head, the kinetic or the dynamic head as well as the potential head which is denoted by z. The sum of all the above mentioned three heads is referred to as the total head and denoted as (ho) The reason as to why the above three quantities are referred to as heads is because all of their units of measurements are in meters . The equation (I) holds for the analysis performed at any given point along the streamline fluid flow. Conclusion The experimentation demonstration of the Bernoulli’s theorem was successful since the Bernoulli’s theorem was clearly demonstrated where by the experimental value of head were computed and compared with the theoretical values for the liquid water. It was also experimentally demonstrated that the overall head is equivalent to the sum of the dynamic and the static heads as is expressed in the Bernoulli’s equation. References Alexander, JS 2000, A Physical Introduction to Fluid Mechanics, John Wiley, New York. John, JB 2000, Practical Fluid Mechanics for Engineering Applications, Marcel Dekker Incorporated, New Jersey. Nicholas, PC 1990, Practical Fluid Mechanics for Engineers and Scientists, Technomic Pub. Co., London. Read More