Physics Internal

Assessment

Investigation between the height of water level, range

of jet and discharge

Introduction

I

have been always fascinated with water and the way it functions. Of course,

back in time this wasn’t expressed through principles of hydraulics, but

instead with the fascination over dams, water purification plants, boats, and

all sorts of water-based machinery. This was one of the most compelling factors

for me to start a research in hydraulic engineering. I wanted to conduct a

research on a rather simple model, find a way to then calculate more complex

details in and even further connect it with various physical principles from

distinct areas. In my research I will focus on an experimental model of a

cylinder with openings on various heights. Furthermore, I will observe the

range of jet of water at different heights and calculate the discharge at the

orifices, taking into account the height of water level above the opening.

Hence, the research question is: Does

height 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.1

Nonetheless,

in order to fully comprehend the given task, I will continue to derive the

equation for the discharge from a) Bernoulli’s equation and b) the application

of the principles of horizontal throw. Firstly, the Bernoulli principle states,

that the speed of a fluid increases proportionally with the decrease in either

pressure or potential energy of a fluid. Furthermore, this can be applied to

the form of so-called Bernoulli’s equation. The following form is derived from

the 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.

Since

I want to observe the relation between the water level at the top of the cylinder

and the jet of water at the orifice, I will restructure the equation to suit

the principle of conversation of energy. Thus:

Where

is the pressure at the

datum line, is the pressure at the

opening, 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 of

stream at the opening. Similarly, h1 and h2 represent the

height of the water above the opening and the height of the opening,

respectively. Since the pressure is the same at both instances, we can cross

out the and . The density of water

remains the same as well, meaning we can cross out all mentions of density as

well. After multiplying the equation by 2, we get the restructured form of:

The

next step requires us to neglect the speed with which the datum line changes

and to substitute the difference between two heights with height between the

opening and the datum line, denoted as h.

That way, the velocity of jet of water passing through the orifice is equal to:

If

we consider the definition of discharge as volumetric rate of water flow

through a cross-sectioned area, we are able to form an equation such as:

Where

Q is discharge, A is the area of the opening and v the velocity of water flow.2

Secondly,

I will aim to theoretically apply the principle of a horizontal throw to my

practical example. As explained in the IB Physics course, the motion of a

projectile during horizontal throw can be deconstructed into two components,

one facing downwards (y) and one

facing horizontally (x). Likewise,

the jet of water can be described as such. In a horizontal projectile motion,

the range of an object is

Where

R is range, v is velocity, h is

height above the opening (in our case) and g

is gravitational constant. Following that, the equation for time is also

applicable to the stream of water.

Since

it is clear that range R is equal to

the product of time t and velocity v, the equation of either, velocity of

an object or velocity of a stream is equal to:

That

way, I have proved the equation for discharge from two different points of

view. However, it is important to point out that these equations yield results

for what would be the ideal discharge. My goal is to calculate the ideal

discharge 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 methods

My

main 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) has

vertical centimetre markings. Alongside the vertical, are openings with the

diameter (d0) of 3,3mm, and the axis of these openings are located

on heights 10 cm, 20 cm, 30 cm and 40 cm. With removal of each cap, we can

regulate the outflow from individual holes. The medium used, is water. Openings

are of same diameter on both sides of cylinder walls.

I

conduct the experiment in a way, that I firstly pour water in the cylinder up

to the height of 50 cm. After removing the plug, I start the stopwatch and

measure a) range of jet and b) time in dependence of the water level (h) in the cylinder. I finish my

measurement 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 the

measurements for each opening.

1 Haestad Methods, Inc. (2002). Computer applications

in hydraulic engineering: Connecting theory to practice. Waterbury, CT:

Haestad Press.

2 Homer, D., &

Bowen-Jones, M. (2014). Physics: Course companion. Oxford: Oxford University

Press.