chapter 5 co hoc chat long revised

Chapter 5 Fluid Mechanics5.1 Some Basic Concepts5.2 Pressure5.3 Variation of Pressure with Depth5.4 Pressure Measurements 5.5 Pascal’s Principle5.6 Buoyant Forces and Archimedes’s Princi

TẬP ĐOÀN DẦU KHÍ VIỆT NAM

TRƯỜNG ĐẠI HỌC DẦU KHÍ VIỆT NAM

General Physics I

Lecturer : Assoc Prof Pham Hong

Chapter 5 Fluid Mechanics

5.1 Some Basic Concepts5.2 Pressure

5.3 Variation of Pressure with Depth5.4 Pressure Measurements

5.5 Pascal’s Principle

5.6 Buoyant Forces and Archimedes’s Principle5.7 Streamlines and the Equation of Continuity5.8 Bernoulli’s Equation

5.9 Surface Tension5.10 Wettability

5.11 Vapor Pressure

Learning outcome

•Identify Pascal’s principle.

•For a hydraulic lift, apply the relationship between the input area and displacement and the output area and displacement•Describe Archimedes’ principle.

•Apply the relationship between the buoyant force on a body and the mass of the fluid displaced by the body.

•For a floating body, relate the buoyant force to the gravitational force.

•For a floating body, relate the gravitational force to the mass of the fluid displaced by the body.

•Distinguish between apparent weight and actual weight.•Calculate the apparent weight of a body that is fully orpartially submerged.

Learning outcome

•Apply the equation of continuity to relate the cross-sectional area and flow speed at one point in a tube to those quantities at a different point.

•Apply Bernoulli’s equation to relate the total energy density at one point on a streamline to the value at another point.

•Identify that Bernoulli's equation is a statement of theconservation of energy.

•Explain the term Surface Tension and describe the methods to measure surface tension.

•Explain the term Wettability and describe the contact angle method.

•Explain the term Vapor Pressure and Viscosity

5.1 Some Basic Concepts

Density & Specific Gravity

The mass density  of a substance is the mass of the substance divided by the volume it occupies:



5.1 Some Basic Concepts

A bottle has a mass of 35.00 g when empty and 98.44 g when filled with water When filled with another fluid, the mass is 88.78 g What is the specific gravity of this other fluid?

Take the ratio of the density of the fluid to that of water, noting that the same volume is used for both liquids.



5.1 Some Basic Concepts

5.1 Some Basic Concepts

The three (common) states or phases of matter

1 Solid: Has a definite volume & shape Maintains

it’s shape & size (approximately), even under large forces.

2 Liquid: Has a definite volume, but not a definite

shape It takes the shape of it’s container.

3 Gas: Has neither a definite volume nor a definite

5.1 Some Basic Concepts

Fluids: Have the ability to flow

A fluid is a collection of molecules that are

randomly arranged & held together by weak

cohesive forces & by forces exerted by the walls of a container.

Both liquids & gases are fluids

Fluids

5.2 PressureAny fluid can exert a force perpendicular to its surface

on the walls of its container The force is described in terms of the pressure it exerts, or force per unit area:

Units: N/m2 or Pa (1 Pascal*)

1 atm = 1.013 x 105 Pa (One atmosphere is the

pressure exerted on us every day by the earth’s atmosphere)

AFp 

Đơn vị áp suất

14.504 x10−5

5.2 PressureConsider a solid object submerged

in a STATIC fluid as in the figure

The pressure P of the fluid at the

level to which the object has been submerged is the ratio of the force (due to the fluid surrounding it in all directions) to the area

At a particular point, P has the



5.3 Variation of Pressure with Depth

Experimental Fact: Pressure depends

on depth.

See figure If a static fluid is in a container, all

portions of the fluid must be in static equilibrium.All points at the same depth must be at the

same pressure

Otherwise, the fluid would not be static.

Consider the darker region, which is a sample of liquid with a cylindrical shape

It has a cross-sectional area A

Extends from depth d to d + h below the

The liquid has a density ρ

Assume the density is the same throughout the fluid This means it is an incompressible

5.3 Variation of Pressure with Depth

There are three external forces acting on the darker region These are:

The downward force on the top, P0AUpward force on the bottom, PA

Gravity acting downward, Mg

The mass M can be found from the density:

The net force on the dark region must be zero:

 ∑Fy = PA – P0A – Mg = 0Solving for the pressure gives

P = P0 + ρghgh

So, the pressure P at a depth h below a point in the liquid at

M V Ah

5.3 Variation of Pressure with Depth

Earth’s atmosphere:  A fluid

But doesn’t have a fixed top “surface”!

Change in height h above Earth’s surface:  Change in pressure: P = ρghgh

Sea level: P0  1.013  105 N/m2

= 101.3 kPa  1 atm Old units: 1 bar = 1.00  105 N/m2

Note: Cause of pressure at any height:Weight of air above that height!

Atmospheric Pressure

5.3 Variation of Pressure with DepthA barometer compares the

pressure due to the atmosphere to the

pressure due to a column of fluid, typically mercury The mercury column has a vacuum above it, so the only pressure is due to the mercury itself

5.3 Variation of Pressure with Depth

Gauge Pressure

Since atmospheric pressure acts uniformly in all directions, we don’t usually notice it Therefore, if you want to, say, add air to your tires to the

manufacturer’s specification, you are not interested in the total pressure What you are interested in is the gauge pressure –how much more pressure is there in the tire than in the atmosphere?

Pg 

5.4 Pressure Measurements

A possible method of measuring the pressure in a fluid is to submerge a measuring device in the fluid A common device is shown in the lower figure It is an evacuated

cylinder with a piston connected to an ideal spring It is first calibrated with a known force

After it is submerged, the force due to the fluid presses on the top of the

piston & compresses the spring.

The force the fluid exerts on the piston is then measured Knowing the area A, the pressure can then be found

5.4 Pressure Measurements

Manometer (Direct reading gages)

A gauge pressure measurement is positive when the unknown

pressure exceeds atmospheric pressure (A), and is negative when the unknown pressure is less than atmospheric pressure (B)

5.4 Pressure Measurements

In a well-type manometer, the cross-sectional area of one leg (the well) is much larger than the other leg When pressure is applied to the well, the fluid lowers only slightly compared to the fluid rise in the other leg

5.4 Pressure Measurements

Typical pressure sensor functional blocks

All except diaphragms provide a fairly large displacement that is useful in mechanical gauges and for electrical

sensors that require a significant movement

5.4 Pressure Measurements

The basic pressure sensing element can be configured as a C-shaped Bourdon

5.4 Pressure Measurements

C-shaped Bourdon tube

5.5 Pascal’s Principle

“If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount”

Car lift in a service station A large output force can be

applied by means of a small input force Volume of liquid

pushed down on left must equal volume pushed up on right.

A xA x

1212

5.5 Pascal’s Principle

5.6 Buoyant Forces and Archimedes’s PrincipleThis is an object submerged in a fluid There is a

net force on the object because the pressures at the top and bottom of it are different.

The buoyant force is found to be the upward force on the same volume of water:

5.6 Buoyant Forces and Archimedes’s PrincipleThe net force on the object is then the

difference between the buoyant force and the gravitational force

5.6 Buoyant Forces and Archimedes’s PrincipleIf the object’s density is less than that of water, there will

be an upward net force on it, and it will rise until it is partially out of the water.

5.6 Buoyant Forces and Archimedes’s PrincipleFor a floating object, the fraction that is submerged

is given by the ratio of the object’s density to that of the fluid.

5.6 Buoyant Forces and Archimedes’s Principle

This principle also works in the air; this is why hot-air and helium balloons rise.

A geologist finds that a Moon rock whose mass is 9.28 kg has an apparent mass of 6.18 kg when submerged in water What is the density of the rock?

The difference in the actual mass and the apparent mass is the mass of the water displaced by the rock The mass of the water displaced is the volume of the rock times the density of water, and the volume of the rock is the mass of the rock divided by its

density Combining these relationships yields an expression for the density of the rock.

rockactualapparentwater rockwater

rockrockwater

A crane lifts the 18,000-kg steel hull of a ship out of the water Determine the tension in the crane’s cable when the hull is submerged in the water

When the hull is submerged, both the buoyant force and the tension force act upward on the hull, and so their sum is equal to the weight of the hull The buoyant force is the weight of the water displaced.

A 5.25-kg piece of wood floats on water What minimum mass of lead, hung from the wood by a string, will cause it to sink? (the SG of wood and Pb are known)

For the combination to just barely sink, the total weight of the wood and lead must be equal to the total buoyant force on the wood and the lead.



5.6 Buoyant Forces and Archimedes’s Principle

5.7 Streamlines and the Equation of ContinuityWe will deal with laminar flow.

The mass flow rate is the mass that passes a given point per unit time The flow rates at any two points must be equal, as long as no fluid is being added or taken away.

This gives us the equation of continuity:

5.7 Streamlines and the Equation of ContinuityIf the density doesn’t change – typical for liquids

– this simplifies to Where the pipe is wider, the flow is slower.

5.8 Bernoulli’s EquationA fluid can also change its height By looking at the work done as it moves, we find:

This is Bernoulli’s equation One thing it tells us is that as the speed goes up, the

pressure goes down.

5.8 Bernoulli’s Equation

Proof of Bernoulli’s Equation

Work has to be done to make the fluid flow

Change in kinetic energy

Change in potential energy

P1 1 1  2 2 2 ( 1  2)

Where γ=ρghgOther form:

5.8 Bernoulli’s Equation

5.8 Bernoulli’s Equation

Venturi Flow Meter

5.8 Bernoulli’s Equation

Pitot tube

5.8 Bernoulli’s Equation

Net force on wing?

½ Aρgh(v22 – v12)

ρghair = 1.29 kg/m3

5.8 Bernoulli’s Equation

Curve Ball

5.8 Bernoulli’s Equation

As air passes at top of tube,

the pressure decreases and fluid is drawn upthe tube.

5.8 Bernoulli’s Equation

Water drains out of the bottom of a cooler at 3 m/s, what is the depth of the water above the valve?

45.9 cm

Various intermolecular forces draw the liquid particles

together Along the surface, the particles are pulled toward the rest of the liquid, as shown in the picture to the right

Surface tension (denoted with

the Greek variable gamma γ) is

defined as the ratio of the

surface force F to the length d

along which the force acts:

γ= F / d

The higher the attraction forces (intermolecular forces), the higher the

5.9 Surface Tension

Surface tension has the dimension of force per unit length, or of energy per unit area The two are

equivalent—but when referring to energy per unit of area, people use the term surface energy—which is a more general term in the sense that it applies

also to solid and not just liquids.

Unit of the Surface tension are N/m, J/ m2 , D/cm

5.9 Surface Tension

Soap filmForce =2Lγ

Force =mg

5.9 Surface Tension

Surface Tensions of Pure Liquids at 293 K

5.9 Surface Tension

The meniscus is the curve in the upper surface of a liquid close

to the surface of the container, caused by surface tension It can be either convex or concave A convex meniscus occurs when

the molecules have a stronger attraction to each other (cohesion) than to the material of the container (adhesion) Conversely, a

concave meniscus occurs when the molecules of the liquid attract those of the container's, causing the surface of the liquid to cave downwards

5.9 Surface Tension

If a capillary tube of inside radius =r

immersed in a liquid that wet its surface, the liquid continues to rise in the tube due to the surface tension, until the upward movement is just balanced by the downward force of gravity due to the weight of the liquid.

The total upward force around the inside circumference of the tube is

θ= the contact angle between the surface of



For water the angle Ө is insignificant, i.e the liquid

wets the capillary wall so that cos Ө = unity

The downward force of gravity is given by

gh

W = W + 4πRg

The method is simple and measures the detachment force

(the surface tension multiplied by the periphery 2*2R)

The Ring Method (du Nouy 1919)

5.9 Surface Tension

5.10 Wettability

According to the nature of the liquid and the solid, a drop of liquid placed on a solid surface will adhere to it or no That is the wettability between liquids and solids

When the forces of adhesion are greater than the forces of cohesion, the liquid tends to wet the surface and vice versa.

Place a drop of a liquid on a smooth surface of a solid According to the wettability, the drop will make a certain angle of contact with the solid

A contact angle is lower than 90°, the solid is called wettable

A contact angle is wider than 90°, the solid is named non-wettable

A contact angle equal to zero indicates complete wettability

5.10 Wettability

5.11 Vapor Pressure

Vapor Pressure: The pressure exerted on the surface of

a liquid by the vapor that is in equilibrium with the liquid is called as “vapor pressure”

Once equilibrium between a liquid and vapor is reached, the number of molecules per unit volume in a vapor does not change with time Hence, the vapor pressure over the liquid remains constant at a given temperature.

Vapor Pressure is independent of the volume of the container

Vapor pressure increases with the increase in

5.11 Vapor Pressure

5.12 Viscosity

Defined as “resistance to flow” of a fluid.

The dynamic viscosity (η) of a fluid is a measure of the resistance ) of a fluid is a measure of the resistance

it offers to relative shearing motion.

Viscosity (η) is defined as the ratio of shear stress (τ)to shear rate

(u/h) η) of a fluid is a measure of the resistance = F/ [A×(u/h)]

η) of a fluid is a measure of the resistance = τ /(u/h) kg/m.s or N.s/m² or Pa.s

5.12 Viscosity

The viscosity of liquids decreases with increase the temperature.

The viscosity of gases increases with the increase the temperature

5.12 Viscosity

For Newtonian fluids, shear stress linearly vary with the

shear rate as shown in Figure Viscosity is constant for this kind of fluid.

τ = η) of a fluid is a measure of the resistance (u/h)

5.12 Viscosity

5.12 Viscosity

Key words of the chapterPressure; Pascal’s Principle; Buoyant Forces;

Archimedes’s Principle; Streamlines; Equation of Continuity; Bernoulli’s Equation; Surface Tension; Wettability; Vapor Pressure; Contact angle; Dynamic Viscosity; Kinematic Viscosity

•Density: • Pressure:

• Atmospheric pressure:• Gauge pressure:

• Pressure with depth:

Archimedes’ principle: An object completely immersed in a fluid experiences an upward buoyant force equal in

magnitude to the weight of fluid displaced by the object

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