1. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Unit 14 Fluid

2. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Density  Look 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: 14-1

3. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Density Calculations – WS 63 #2  The typical air in a room has a density of 1.29 kg/ m 3.  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? 14-2

4. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Continuity Equation – WS 65 #3  Consider the piping system below.  The black circles represent atoms of a fluid moving through the pipe.  The area on the left (A 1 ) is twice as large as the area on the right (A 2 ).  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 (V 2 ) must be twice that as the speed on the left (V 1 ).  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? 14-3

5. © 2001-2005 Shannon W. Helzer. All Rights Reserved. WS 65 #4  A 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. 14-4 ABCD

6. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Pressure and Bernoulli’s Principle – WS 66  Pressure – a force per unit area.  The unit of pressure is the Pascal (Pa) and is equal to a N/m 2.  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: 14-5

7. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Effects of Surface Area on Pressure  A 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 m 2, and the area of the man’s shoe is 0.8m 2.  What are the pressures exerted on the ground by each of the people? 14-6

8. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 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? 14-7

9. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Pressure  Which of the identical objects shown below will exert the most pressure on the ground below it?  Why? 14-8

10. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Pressure and Density in Fluids  The 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/m 3. 14-9

11. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Pressure at Various Depths  Suppose 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. 14-10

12. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 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. 14-11

13. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Bernoulli’s Equation – WS 66  Water 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 (y 2 ) 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. 14-12

14. © 2001-2005 Shannon W. Helzer. All Rights Reserved. WS 64 #5  At 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. 14-13 A B

15. © 2001-2005 Shannon W. Helzer. All Rights Reserved. A Tie Race  Consider 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. 14-14

16. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Bernoulli’s Principle & the Wing  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 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. 14-15

17. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Laminar Flow – WS 67  Consider 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. 14-16

18. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Turbulent Flow – WS 65  Consider 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. 14-17

19. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Pressure Difference – WS 66 #1  When 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. 14-18

20. © 2001-2005 Shannon W. Helzer. All Rights Reserved. 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. 14-19

21. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Closed Fluid System  An 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. 14-20

22. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Work Done in Closed Fluid Systems  In 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. 14-21

23. © 2001-2005 Shannon W. Helzer. All Rights Reserved. Work Done in an Open Fluid System  One 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. 14-22

24. © 2001-2005 Shannon W. Helzer. All Rights Reserved. This presentation was brought to you by Where we are committed to Excellence In Mathematics And Science Educational Services.

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