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## Ch. 15 Fluids

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Fluids

- Substances that flow readily, usually from high pressure to low pressure.
- They take the shape of the container rather than retain their shape.
- We refer both liquids and gases as fluids.

Density

- Is the mass of a substance divided by the volume of that substance.

Sample Problem: What is the volume of my 2.16 g titanium wedding band if the density is 4.50g/ cm3?

Pressure

- is defined as force per unit of area.

Each book exerts the same force, but which one exerts a greater pressure?

Units of Pressure

- English: lb / ft2, lb / in2, or p.s.i.
- Metric: N/m2 or Pascal (Pa)

Sample Problem: If an airplane window that is 40 cm by 60 cm feels a pressure of 40,000 Pa. How much force is acting on the window?

F = (40,000)(.24)

F = 9600 N

Atmospheric Pressure

The pressure of the earth’s atmosphere pushing down on you.

Earth's atmosphere is pressing against each square inch of you with a force of 1 kilogram per square centimeter (14.7 pounds per square inch).

Sample ProblemCalculate the net force on an airplane window if cabin pressure is 90% of the pressure at sea level, and the external pressure is only 50% of that at sea level. Assume the window is 0.43 m tall and 0.30 m wide.

Since pressure acts equally in all directions, gauge pressure is used to deal with this.

- If you have a flat tire, is there a pressure in the tire or is the pressure zero?
- Yes, it is equal to atmospheric pressure.

Gauge Pressure (Pg)

- Is the pressure inside an inflated object, that is above the atmospheric pressure.
- Pg = P – Patm
- If a tire is inflated to 35 psi, that means it is 35 psi above the atmospheric pressure. So the pressure inside the tire is actually 35 psi + 14.7 psi.

Pressure increases with depth.

The deeper the body of water or you go, the more water there is on top of you per square inch.

The pressure of a liquid is the same at any given depth regardless the shape of the container.

Absolute Pressure

- Absolute pressure is obtained by adding the atmospheric pressure to the hydrostatic pressure.
- pabs = patm + rgh
- p2 = p1 + rgh

Pascal’s Principle

Pressure applied to a fluid in a closed container is transmitted equally

When the area of the 2 “pistons” are the same the input force is the same as the output force.

In order to maintain equality, the force has to be increased by the same factor.

Distance each piston moved is different.

F1d1 = F2d2

Sample problem: The small piston on a hydraulic lift has an area of 0.20 m2. A car weighing 1.2 x 104 N sits on the rack with a large piston. If the area of the large piston is 0.9 m2, how much force must be applied to the small piston to support the car?

S: F1=(1.2 x 104)(0.2)/(0.9)

S: F1 = 2666.7 N

The distance the 2nd piston moves is equal to the reciprocal of the factor of the ↑ or ↓ in area times the distance the 1st piston moves.

Buoyancy

- The apparent loss of weight of an object that is submerged in a fluid.

Buoyant Force

Is the upward force equal to the weight of the fluid displaced.

Pressure is greater at the bottom of the object because the depth is greater.

Buoyant force is due to pressure difference between top and bottom of object.

- Pressure difference does not depend on depth

If denser than the fluid, it sinks.

- If less dense than the fluid, it will float.
- If the same density, it will neither float nor sink.

Archimedes Principles

The buoyant force is equal to the weightof the liquid displaced. Immersion: all or part under water.Submersion: completely under water

An object completely submerged always displaces a volume equal to its own.

Sample problem: Assume a wooden raft has 80.0% of the density of water. The dimensions of the raft are 6.0 meters long by 3.0 meters wide by 0.10 meter tall. How much of the raft rises above the level of the water when it floats? (density of water is 1000 kg/m3)

Principle of Floatation

- A floating object will displace a weight of fluid equals to its own weight.

Specific Gravity

Is the ratio of any substance’s density to the density of water.

Fluid Flow Continuity

- The volume per unit time of a liquid flowing in a pipe is constant throughout the pipe.
- We can say this because liquids are not compressible, so mass conservation is also volume conservation for a liquid.

Volume of a fluid moving past a point can be calculated by:

V = Avt

V: volume of fluid (m3)

A: cross sectional areas at a point in the pipe (m2)

v: speed of fluid flow at a point in the pipe (m/s)

t: time (s)

Equation for Continuity

r1A1v1 = r2A2v2

- r1, r2: weight density of fluids at point 1 and 2
- A1, A2: cross sectional areas at points 1 and 2
- v1, v2: speed of fluid flow at points 1 and 2

Use for gases.

For an incompressible fluid (A liquid)

A1v1 = A2v2

- A1, A2: cross sectional areas at points 1 and 2
- v1, v2: speed of fluid flow at points 1 and 2

Sample problem: A pipe of diameter 6.0 cm has fluid flowing through it at 1.6 m/s. How fast is the fluid flowing in an area of the pipe in which the diameter is 3.0 cm? How much water per second flows through the pipe?

Sample problem: The water in a canal flows 0.10 m/s where the canal is 12 meters deep and 10 meters across. If the depth of the canal is reduced to 1.5 meters at an area where the canal narrows to 5.0 meters, how fast will the water be moving through this narrower region?

Bernoulli’s Principle

The sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any one location in the fluid is equal to the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any other location in the fluid for a non-viscous incompressible fluid in streamline flow.

Bernoulli’s Principle

- The pressure in a fluid decreases as the speed of the fluid increases.

Bernoulli’s Theorem

- p + r g h + ½ rv2 = Constant
- p : pressure (Pa)
- r : density of fluid (kg/m3)
- g: gravitational acceleration constant (9.8 m/s2)
- h: height above lowest point (m)
- v: speed of fluid flow at a point in the pipe (m/s)

fluid flows smoothly

- fluid flows without any swirls
- fluid flows everywhere through the pipe

fluid has the same density everywhere (it is "incompressible" like water)

Sample Problem: Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out of a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle? If the pressure in the hose is 350 kPa, what is the pressure in the nozzle?

Torricelli’s Law

- Deals with the speed of a fluid as it flows through a hole in a container.

P1 + r g h1 + ½ rv12 = P2 + r g h2 + ½ rv22

Patm + r g h1 + 0 = Patm + 0 + ½ rv22

- g h1 = ½ rv22

Sample Problem: An above-ground swimming pool has a hole of radius 0.10 cm in the side 2.0 meters below the surface of the water. How fast is the water flowing out of the hole? How much water flows out each second?

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