Bernoulli Distribution

Bernoulli Distribution. In our everyday life, Bernoulli's principle can be used to explain the work of carburetors, venturi meters, Pitot tubes, perfume sprayer bottles and insecticide spray guns. The Bernoulli's principle is also used to calculate the leakage of the water tank.

The Application of Bernoulli's Principle

a.  Flow from a Tank Hole

A drum that is filled with water to a height h1. There is a hole in the tank which is located at a height h2. measured from the bottom of the tank. What is the speed of water flowing from that hole? The speed of the liquid in the tank (v₁) is very small compared to the speed of the fluid through the hole (v₂). Thus we can assume that v₁ = 0. Both the water surface and the hole are affected by the atmospheric pressure, therefore P1 = P2 = P0. Based on equation

P₁ + ½ pv₁² + ρgh₁ = P₂ + ½ ρv₂² ρgh₂ 

we have:

 ρgh₁ = ½ ρv₂² - ρgh₂

Example
A cylindrical container having a large cross-sectional area is filled with water to a height of 170 cm. There is a narrow hole at a height of 125 cm from the bottom of the container. If g = 10m/s². calculate (a) the escape speed of water through the hole, and (b) the distance from the container where the water stream strikes the floor.

b. The Venturi meter

A venturi pipe or venturi meter can be used to measure the flow speed of a fluid in a pipe. There are two types of venturi meter; venturi meter without manometer and venturi meter with the manometer.

1) Venturimeter without Manometer

There is no difference in height in the flow of fluid which is going be measured so the Bernoulli's equation, in this case, will be in the form of:

P₁ + ½ pv₁² = P₂ + ½ ρv₂²     or

P₁ - P₂ = ½ ρ (v₂² - pv₁² )   

c. Perfume Sprayer

When we press the sprayer button of a perfume bottle, a rubber ball is compressed and the air is forced to come out through a narrow hole above a cylinder tube which has enough length to reach the perfume. The air which is moving faster causes the air pressure at the upper part of the tube to decrease, as a result the atmospheric pressure forces the liquid to rise to the upper part of the tube. Finally, a fast moving air spray blows the perfume liquid, and the perfume is flowed out as a fine mist.

d.  Insecticide Sprayer

Basically, the principle of an insecticide sprayer is the same with a perfume sprayer works by pressing the sprayer button, meanwhile, an insecticide sprayer works by pushing the pump-rod.

e.  The airplane wing

The wing is shaped is such a way that air flows more rapidly over the top of the wing than along the bottom. As a result, the pressure on top of the wing is smaller that the pressure on the bottom, therefore the net upward force (lift) is generated.

Exercise 6.2

1. A fluid through a pipe having a cross-sectional area of 3 cm², with a volume flow rate of 15 cm³/s. Calculate: (a) the amount of fluid flowing in one hour, and  (b) the flow speed of the fluid.
2. A tube having different cross-sectional areas is placed in a horizontal position. The radius of each portion 2 cm and 1 cm. If the flow speed through the wider portion is 20 m/s. find out the flow speed at the narrow portion.
3. The volume flow rate of water flowing through a hole is 100 cm³/s. The hole is 10 m deep from the water surface in a large container. If there is an additional pressure of 3 x 10⁴ N/m² on the surface, calculate: (a) the flow's speed of water through the hole, and (b) the volume flow rate of water through the hole.
4. When the air flows through a Pitot tube, the difference in height between mercury columns in manometer is 2 cm. Determine the flow speed of the air.  and ρ_air = 1,29 kg/m³ and ρ_mercury = 13,6 kg/m³)
5. An airplane with a mass of 500 kg has a total area of wings A = 50 m². The air flows through the upper part of the wing a speed of 40 m/s ( ρ = 1.29 kg/m³). Find out the flow speed of air below the wing. ( g = 10m/s²)

Multiple Choice

1)  A pipe of a barometer is replaced with another pipe having twice the cross-sectional area the previous pipe. If the atmospheric pressure is 1 atm, the height of the mercury column is.....

a.  19 cm

b.  38 cm

c.  76 cm

d.  114 cm

e.  152 cm

2)  The water level at a dam is 5 m. If the lenght of the dam wall is 40 m, the force required by the wall to hold the water is ..........

a.  1.00 x 10⁴ N

b.  2.50 x 10⁵ N

c.  5.00 x 10⁶ N

d.  2.50 x 10⁷ N

e.  3.00 x 10⁸ N

3)  The pressure of a liquid depends on ......

a.  The density of the object

b.  The density and volume of the object

c.  The density and depth of the liquid

d.  The volume and depth of the liquid

e.  The density, volume and depth of the liquid

4)  The pressure applied to an enclosed fluid is transmitted equally to every part of the fluid. That statement is....

a.  The principal law of hydrostatic

b.  Archimedes' principle

c.  Pascal's law

d.  Boyle's law

e.  Hooke's law

5)  The volume flow rate is defined as the amount of fluid which flows through a section of a tube in every unit if ......

a.  volume

b.  area

c.  length

d.  time

e.  mass

6)  Water flows through a pipe with a cross-sectional area of 10 m². The time required of fully fill a container with a volume of 1 m3 is five minutes. The flow speed of the water is ..... m/s

a.  0.1

b.  0.33

c.  1/300

d.  3.33

e.  6.66

7)  Water flows into a container with a constant volume flow rate of 0.5 liters/s. If te volume of the container is 1 m³, it will be full in.....

a.  33.3

b.  3.33

c.  333

d.  0.33

e.  0.033

8)  The following devices work based on Bernoulli's principle, except ......

a.  airplane wing design

b.  hydraulic jack

c.  perfume sprayer

d.  venturi tube

e.  pitot tube

9)  A container is 20 m high. Two  holes are made in the container which is located 2 m from the surface and 2 m from the bottom of the container. The ratio between the distances where the water stream from hole 1 and hole 2 strike the floor is ......

a.   1 : 1

b.   2 : 1

c.   1 : 2

d.   3 : 2

e.   2 : 3

10)  There is a hole in a container. The hole is h₁ deep from the surface and h₂ high from the bottom of the container. The water strikes the floor at a maximum horizontal distance if the ratio between h₁ : h₂ is ......

a.   1 : 2

b. √2 : 1

c. 2 : 1

d. 1 : √2

e. 1 : 1 

Activity

Observing Capillarity and Mass Density Of Fluid

Tools :

kerosene lamp, kerosene, water, lighter

Procedure:

1. Prepare a kerosene lamp
2. Fill the lamp with kerosene. Make sure that the wick is not immersed.
3. Add some water in a way that the wick is then immersed
4. Light it on.

Question 

1. Can you turn on the lamp? Can water be used as fuel for the lamp?
2. Can the lamp be turned on if the wick is not wet yet by the kerosene?
3. Explain your finding.

You Need To know

We are familiar with egg; we may even consume it every day. Do you know how to differentiate between a good and fresh egg and the rotten one?

It is very easy and simple. We can do it by putting the eggs in a basin of water. A good egg will sink to the bottom of the basin because it has a fresh yolk and album en. The density of a fresh egg is higher than the density of water (ρ_aegg = ρ_mwater). Meanwhile, the yolk and the albumen of a rotten egg are dehydrated, thus its density becomes smaller than the density of water (ρ_egg < ρ_water). Therefore, if we put a rotten egg into the water, the egg will float.

Case Study

Mini Submarines (Mini-Sub)

A mini submarine is a specially designed submarine having different size and shapes. Inside its hull, there is a round-shaped cabin made of very thick metals. A crew man can look outside through a window. It is also equipped with very strong lamps, the sunlight could not reach the bottom of the sea. A mini-submarine is very useful for repairing the offshore oil drilling instruments, investigating the wrecked ships, and finding some valuable minerals that might exist in the seabed. A mini-sub is equipped with an arm which can be moved and controlled from the hull. The arm is able to hold or take some objects from the sea floor.

Discussion Material

A driver cannot work a very deep place below the surface without using additional equipment to protect his body. Why is that so? Is it related to the hydrostatic pressure?

Bernoulli Principle fluid

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