Physics with the decrease in either pressure or

 

 

 

Physics Internal
Assessment

 

Investigation between the height of water level, range
of jet and discharge

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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.