SMJM 1023 ENGINEERING MATHEMATICS 2PROJECT

• MUBAREK KURTGROUP X+Y=0

Water Flow from Plastic Bottles through Pin-Hole T

INTRODUCTION• We performed an experiment on elementary hydrodynamics.

The basic system is a cylindrical bottle from which water flow through a pin-hole located at the bottom of its lateral surface. We measured the speed of the water leaving the pin-hole, as a function of both the time and current level of water still inside the bottle.• We use the concept of conservation of energy that lead to a

differential equation and Toricelli`s Law .

OBJECTIVESTo investigate the relationship between the speed of the flow and the height of the level of water .

To apply the Torricelli’s law in this problem cases.

To apply differential equation in our daily life.

To increase our knowledge on differential equation by solving the problem cases given.

PROBLEM

SOLVING

Construct a differential equation for the speed of the flow and height of the level of water, and find its solution.

water exiting the hole:

Conservation of energy:

Relate the velocity of the fluid leaving the hole to the height of the water in the Tank :

The speed of the fluid is related to the height of the water :

Recall that the volume of the water in the tank, V (t) is related to the height of fluid h(t) by

where A > 0 is a constant, the cross-sectional area of the tank. Thus, we can simplify as follows:

where k is a constant that depends on the size and shape of the cylinder and its hole:

The height h(t) of water in the tank at time t satisfies the following differential equation:

 

Integration of the formula would give the time and height:

Thus the decreasement of the height over time is:

, In order the empty the tank, the

Where is the time taken for the tank to empty, is the initial height of the water in the

container.

Sketch of the graph of the solutions

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25 Graph of solution

Data for the level and speed of water leaving the pin-hole through the range X

TIME/s Height/m X/m Speed of water/ms-1

0 0.135 0 020 0.128 0.0230 0.322240 0.123 0.0225 0.315260 0.122 0.0220 0.308280 0.118 0.0210 0.2942100 0.106 0.0205 0.2871

EXPERIMENT VS THEORY

0 20 40 60 80 100 120 14000.020.040.060.080.1

0.120.140.160.180.2

f(x) = 2.02470862470862E-05 x² − 0.00402470862470862 x + 0.200006993006993

Experimental Vs Theoretical

REASON FOR THE SMALL HOLE

With the bigger cross sectional area on the top, there will be a very high atmospheric pressure act on the surface of the water. The pressure is transmitted throughout the liquid equally. As the high atmospheric pressure pushing the water to the bottom, the water will run through the small hole with high velocity, inflicting greater distance of X. If bigger hole is made, the velocity would not be as high as the small one.

What will happen if the change the bottle to rectangular form?

NOTHING CHANGE BUT IT DEPENDS ON THE HEIGHT OF THE WATER IN THE CONTAINER

BECAUSE HEIGHT INFLUENCES THE VELOCITY OF THE WATER FLOW

IF H=60M, WHAT IS MAX SPEED OF WATER EMERGE?

CONCLUSIONThe experiment values is different with theoretical values because of some error .Height is directly proportional to the speed of water emerge.

THE END