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 Concepts 5.2 Pressure

5.3 Variation of Pressure with Depth 5.4 Pressure Measurements

5.5 Pascal’s Principle 5.6 Buoyant Forces and Archimedes’s Principle 5.7 Streamlines and the Equation of Continuity 5.8 Bernoulli’s Equation

5.9 Surface Tension 5.10 Wettability

5.11 Vapor Pressure

Learning outcome

The students must be able to

•Distinguish fluids from solids.

•When mass is uniformly distributed, relate density to mass

and volume.

•Apply the relationship between hydrostatic pressure, force,

and the surface area over which that force acts.

•Apply the relationship between the hydrostatic pressure, fluid density, and the height above or below a reference level.

•Distinguish between total pressure (absolute pressure) and

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 or

partially submerged.

•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 Pressure

Any 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 10 5 Pa (One atmosphere is the

pressure exerted on us every day by the earth’s

atmosphere)

A F

p 

Đơn vị

áp

suất

Pascal (Pa)

Bar (bar)

Kgf/cm 2

(at)

Atmotsphe

re (atm)

mmH 2 0 mmHg

(Torr)

psi (lbs/in 2 )

5.2 Pressure

Consider 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

following properties:

1 It is the same in all directions

2 It is  to any surface of the

object

F P

A



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

surface

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, P0A Upward 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 = 0 Solving 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 Depth

A 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?

0

P P

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)

gh p

Gauge pressure is a measurement relative to atmospheric pressure and it varies with the barometric reading

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

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.

5.5 Pascal’s Principle

5.6 Buoyant Forces and Archimedes’s Principle

This 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 Principle

The net force on the object is then the difference between the buoyant force and the gravitational force

5.6 Buoyant Forces and Archimedes’s Principle

If 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 Principle

For 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.

rock actual apparent water rock water

rock

rock rock water

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.

weight buoyant wood Pb wood water Pb water

1 1

5.6 Buoyant Forces and Archimedes’s Principle

5.7 Streamlines and the Equation of Continuity

We 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 Continuity

If the density doesn’t change – typical for liquids – this simplifies to Where the pipe is wider, the flow is slower

5.8 Bernoulli’s Equation

A 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

dV P

P l

A P l

A

P1 1 1  2 2 2  ( 1  2)

2 2

2 1

2 1

5.8 Bernoulli’s Equation

5.8 Bernoulli’s Equation

) (

2

2 2

2 1

2 2 1

A A

pA v

5.8 Bernoulli’s Equation

h g

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

Atomizer

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

Carbon Tetrachloride 27.0 Methylene Iodide 50.8

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

g h

At Maximum height, the opposing

forces are in equilibrium

1

5.9 Surface Tension

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

temperature

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

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)