© 2001-2005 shannon w. helzer. all rights reserved. unit 14 fluid

© 2001-2005 Shannon W. Helzer. All Rights Reserved.

Unit 14Fluid


DensityLook at the two blocks below.Both blocks are the same size (have the same volume).However, one is Lead and the other is copper.An internal look at these blocks reveals that there are many more atoms in the lead block than there are in the copper block.As a result, we say that the lead block is more dense than the copper block.The formula for calculating density is as follows:

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m

V


Density Calculations – WS 63 #2

The typical air in a room has a density of 1.29 kg/ m3.Suppose your classroom has the dimensions of 3.5 m by 4 m by 2.5 m. Calculate the volume of your classroom. What is the mass of the air in your classroom?

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m

V

V lwh


Continuity Equation – WS 65 #3Consider the piping system below.The black circles represent atoms of a fluid moving through the pipe.The area on the left (A1) is twice as large as the area on the right (A2).However, the amount of fluid per unit time flowing through this pipe is the same.As a result, the speed of the fluid flowing through the right pipe segment (V2) must be twice that as the speed on the left (V1).The mathematical relationship used to demonstrate this fact is known as the continuity equation.

A certain pipe has a radius of 0.25 m at point A (left) and a radius of 0.12 m at point B (right).If the fluid in the pipe is flowing at 5.2 m/s at point A, then how fast is it flowing at point B?

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2211 vAvA


WS 65 #4A piping system has a velocity of 14.8 m/s and a radius of 0.10 m at point C. The radius at points A, B, and D are 0.50 m, 0.30 m, and 0.50 m respectively. Calculate the velocities of the fluid flowing through the pipes at points B and D. What do you think the velocity of the fluid at A will be? Why Calculate the fluid velocity at A in order to prove or disprove your prediction.

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A B C D

2211 vAvA


Pressure and Bernoulli’s Principle – WS 66Pressure – a force per unit area.

The unit of pressure is the Pascal (Pa) and is equal to a N/m2.

Bernoulli’s Principle – if the velocity of a fluid is high, then the pressure is low. Conversely, if the velocity of a fluid is low, then the pressure is high.

When the pressure at a certain depth in a liquid is desired, we can use the following formula:

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A

FP

ygP


Effects of Surface Area on PressureA man and a woman have made a footprint impression in the concrete by standing on one foot. They both weigh the same amount.Which person, man or woman, exerted the most pressure on the ground? The following problem is similar to WS 66 #1.A man and a woman both weigh 750.0 N.The woman’s shoe has a surface area that is 0.5 m2, and the area of the man’s shoe is 0.8m2.What are the pressures exerted on the ground by each of the people?

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ManWoman ManWoman


Pressure

A trunk full of valuable treasure falls over as shown.In which instance did the trunk exert the most pressure on the ground?Why?

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Pressure

Which of the identical objects shown below will exert the most pressure on the ground below it?

Why?

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Pressure and Density in FluidsThe pressure experienced at any depth in fluid is constant everywhere at that same depth.The formula for calculating that pressure is given below.WS 66 #3 - What is the pressure at the bottom of a swimming pool that is 2.25 m deep?The density () of water is 1000.0 kg/m3.

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ygP


Pressure at Various DepthsSuppose you had a coffee can full of water.If there were three holes in the can, then from which hole would the water flow the farthest? Why?If we were to measure the fluid pressure at the top of the fluid, then we would see that it would be “low.”As we move the gauge deeper, we would observe and increase in the pressure. However, if we were to move the gauge from left to right at the same depth, we would see that there would be no change in pressure.Pressure is constant at any given depth in a fluid that is opened to the atmosphere.The same behavior is observed in oddly shaped containers.

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Bernoulli’s Principle

Bernoulli’s Principle – if the velocity of a fluid is high, then the pressure is low. Conversely, if the velocity of a fluid is low, then the pressure is high.The speed of the fluid coming into the pipe from the left is slow; therefore, the pressure is high.The speed of the fluid leaving the pipe is fast; therefore, the pressure is low.

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Bernoulli’s Equation – WS 66Water enters a house through the basement at a speed of 0.60 m/s through a pipe that is 4.2 cm in diameter. When the water enters the house, it is under a pressure of 303.9 kPa. The water is pumped up to a height (y2) of 5.2 m and out a faucet that is 2.7 cm in diameter.Use the continuity equation to determine how fast the water is going when it leaves the faucet. Use Bernoulli’s equation to determine the pressure of the water when it leaves the faucet.

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2 21 1 1 2 2 2

1 1

2 2P v gy P v gy

2211 vAvA


WS 64 #5At point A on the pipe to the left, the water’s speed is 4.8 m/s pressure is 52.0 kPa.The water drops down 14.8 m to point B where the pipe’s cross sectional area is twice that at point A. Calculate the velocity of the water at point B. Calculate the pressure at point B.

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A

B22

2212

11 2

1

2

1gyvPgyvP


A Tie RaceConsider the two ball’s on the track below. If they race and tie, then what can you tell me about the speed of the orange ball when compared to the yellow ball?Why?The orange ball traveled further; therefore, it had to go faster in order to reach the end at the same time as the yellow ball.A similar effect helps to partially explain how a wing produces lift enabling an airplane to fly.

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Bernoulli’s Principle & the WingBernoulli’s Principle – if the velocity of a fluid is high, then the pressure is low. Conversely, if the velocity of a fluid is low, then the pressure is high.The distance across the top of the wing is farther; therefore, the top molecule must go faster in order to reach the rear wing at the same time as the bottom molecule.The air above the wing moves faster; therefore, the downward pressure acting on the top of the wing is less than the upward pressure acting on the bottom of the wing.Recall, P = F/A: therefore, F = PA. This means that the upward force on the wing is greater than the downward force on the wing.This difference in force results in the generation of lift which enables a plane to fly.

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UPF

DOWNF


Laminar Flow – WS 67Consider the piping system below.Depending on the fluid and piping material properties, the fluid may move easily and smoothly through the reduction in the pipe diameter.We can replace the molecules with flow lines that represent the paths of the different layers of fluid.In the case of Laminar flow, the flow lines would look like those shown below.

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Turbulent Flow – WS 65Consider the piping system below.Depending on the fluid and piping material properties, the fluid may find it hard to move and may move roughly through the reduction in the pipe diameter.We can replace the molecules with flow lines that represent the paths of the different layers of fluid.In the case of Turbulent flow, the flow lines would look like those shown below.

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Pressure Difference – WS 66 #1When the pressure in two chambers is uniform, no fluid flows from one chamber to the other.However, when there is a pressure difference, fluid moves from one chamber to the other.In a fluid system, work is done when a pressure Difference causes liquids to move. Notice how the pressure is constant everywhere after the pressure has equalized after the valve is opened.

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Open Fluid System

An open fluid system is one in which the fluid is not retained and is not recirculated.When you flush a toilet, the fluid drains into a septic tank or a municipal sewage system.The toilet retention tank is then refilled from a well or a municipal water source.The water from the septic tank does not return to the municipal water source.As a result, a toilet is an open fluid system.

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Closed Fluid SystemAn closed fluid system is one in which the fluid is retained and is recirculated.In isolated rest areas, the toilets use a special oil instead of water.When you flush the toilet, the fluid drains into a septic tank.The oil is clean because the solids and urine automatically settle to the bottom.The oil is able to flow over a baffle (left side).It is pumped from the left side of the tank back up to the toilet retention tank.This toilet is an example of a closed fluid system because the fluid is retained and recirculated.

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Work Done in Closed Fluid SystemsIn a closed fluid system, the pressure throughout the system is constant as long as there is no change in volume. When the pump in a hydraulic cylinder is turned on, it moves a volume of fluid from one side of the piston to the other.When fluid is pumped from one side of the piston, a low pressure is established on that side.When fluid is pumped to the other side of the piston, a high pressure is established on that side.As a result, the pressure difference causes the piston to move in order to reestablish a uniform pressure throughout the system.The work done is calculated using the following equation.V is the change in volume of the system.The final system pressure is equal to the initial system pressure.

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W P V


Work Done in an Open Fluid SystemOne example of an open fluid system is shown below.The fluid must be pumped from the reservoir to the storage tank.Both the storage tank and the reservoir a under the influence of atmospheric pressure.In order to move the fluid up to the storage tank, the pump must be able to overcome the pressure associated with the weight of the fluid moved.This pressure may be calculated using the following equation where w is the weight density of the fluid and h is the height to which the fluid is pumped.

The work done in an open fluid system may be calculated using the equation below where V is the volume of fluid moved.

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W P V

wP h


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© 2001-2005 shannon w. helzer. all rights reserved. unit 14 fluid

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