Physics InternalAssessment Investigation between the height of water level, rangeof jet and discharge IntroductionIhave been always fascinated with water and the way it functions. Of course,back in time this wasn’t expressed through principles of hydraulics, butinstead with the fascination over dams, water purification plants, boats, andall sorts of water-based machinery. This was one of the most compelling factorsfor me to start a research in hydraulic engineering. I wanted to conduct aresearch on a rather simple model, find a way to then calculate more complexdetails in and even further connect it with various physical principles fromdistinct areas. In my research I will focus on an experimental model of acylinder with openings on various heights.
Furthermore, I will observe therange of jet of water at different heights and calculate the discharge at theorifices, taking into account the height of water level above the opening.Hence, the research question is: Doesheight of the water level affect the discharge through the opening?Theory Discharge,a term in hydrology, is used to describe the volumetric flow of water,transported through a cross-sectioned area.1Nonetheless,in order to fully comprehend the given task, I will continue to derive theequation for the discharge from a) Bernoulli’s equation and b) the applicationof the principles of horizontal throw.
Firstly, the Bernoulli principle states,that the speed of a fluid increases proportionally with the decrease in eitherpressure or potential energy of a fluid. Furthermore, this can be applied tothe form of so-called Bernoulli’s equation. The following form is derived fromthe principle of the conservation of energy. Where? is the density of the fluid, v is the velocity of water outflow, g is the gravitational constant, h is height above the opening and p being the pressure at that level.SinceI want to observe the relation between the water level at the top of the cylinderand the jet of water at the orifice, I will restructure the equation to suitthe principle of conversation of energy. Thus: Whereis the pressure at thedatum line, is the pressure at theopening, is the density of water, g is the gravitational constant, with v1 being the velocity of water at the datum line and v2 being the velocity ofstream at the opening.
Similarly, h1 and h2 represent theheight of the water above the opening and the height of the opening,respectively. Since the pressure is the same at both instances, we can crossout the and . The density of waterremains the same as well, meaning we can cross out all mentions of density aswell. After multiplying the equation by 2, we get the restructured form of:Thenext step requires us to neglect the speed with which the datum line changesand to substitute the difference between two heights with height between theopening and the datum line, denoted as h.That way, the velocity of jet of water passing through the orifice is equal to:Ifwe consider the definition of discharge as volumetric rate of water flowthrough a cross-sectioned area, we are able to form an equation such as: WhereQ is discharge, A is the area of the opening and v the velocity of water flow.
2Secondly,I will aim to theoretically apply the principle of a horizontal throw to mypractical example. As explained in the IB Physics course, the motion of aprojectile during horizontal throw can be deconstructed into two components,one facing downwards (y) and onefacing horizontally (x). Likewise,the jet of water can be described as such. In a horizontal projectile motion,the range of an object isWhereR is range, v is velocity, h isheight above the opening (in our case) and gis gravitational constant. Following that, the equation for time is alsoapplicable to the stream of water. Sinceit is clear that range R is equal tothe product of time t and velocity v, the equation of either, velocity ofan object or velocity of a stream is equal to:Thatway, I have proved the equation for discharge from two different points ofview. However, it is important to point out that these equations yield resultsfor what would be the ideal discharge.
My goal is to calculate the idealdischarge and try to measure the actual discharge, based on the gathered data.Hence, I will be able to compare the differences and asses them.Design and methodsMymain measuring equipment is a cylinder with a height of 50cm, a diameter (d1)of 70mm and with thickness of the cylinder wall of 5mm. The tube (cylinder) hasvertical centimetre markings.
Alongside the vertical, are openings with thediameter (d0) of 3,3mm, and the axis of these openings are locatedon heights 10 cm, 20 cm, 30 cm and 40 cm. With removal of each cap, we canregulate the outflow from individual holes. The medium used, is water.
Openingsare of same diameter on both sides of cylinder walls. Iconduct the experiment in a way, that I firstly pour water in the cylinder upto the height of 50 cm. After removing the plug, I start the stopwatch andmeasure a) range of jet and b) time in dependence of the water level (h) in the cylinder.
I finish mymeasurement when water level is 2.0 cm above the hole from which the jet flows,since past that point the shape of the jet is deformed. I repeat themeasurements for each opening.1 Haestad Methods, Inc. (2002). Computer applicationsin hydraulic engineering: Connecting theory to practice. Waterbury, CT:Haestad Press.2 Homer, D.
, -Jones, M. (2014). Physics: Course companion. Oxford: Oxford UniversityPress.