ch 1 fluid mechanics

Fluid Mechanics

Intoduction 1.1

Flow in pipes 1.2

Stresses

In fluid mechanics it is convenient to define a force per unit area

(=F/A), called a stress (same units as pressure).

• Normal stress acts perpendicular to the surface (F=normal force).

Tensile causes elongation Compressive causes shrinkage

(Pressure is the most important

example of a compressive stress)

F F F F A A

A

Fnormalnormal

Flow in pipes 1.3

Stresses

• Shear stress acts tangentially to the surface (F=tangential or shear

force). F

F A

A

Fshearshear

A fluid is defined as a substance that deforms continuously

when acted on by a shearing stress of any magnitude.

Intoduction 1.4

Introduction to Fluid Mechanics

• Fluid Mechanics is concerned with the behavior of fluids at rest and in

motion

• Distinction between solids and fluids:

– According to our experience: A solid is “hard” and not easily

deformed. A fluid is “soft” and deforms easily.

– Fluid is a substance that alters its shape in response to any force

however small, that tends to flow or to conform to the outline of its

container, and that includes gases and liquids and mixtures of solids

and liquids capable of flow.

– A fluid is defined as a substance that deforms continuously when

acted on by a shearing stress of any magnitude.

Intoduction 1.5

Course Organization

Textbook: deNevers “Fluid Mechanics for Chemical Engineers”

Introduction (Chapter 1) / Dimensions, Units Fluid statics: Fluid is at rest Fluid mechanics Fluid dynamics: Fluid is moving

Fluid statics (Chapter 2): Pressure, measurement of pressure, hydrostatic

forces, buoyancy

Fluid dynamics (Chapters 3-5, 7): Mass, energy and momentum balances

Applications in Engineering (Chapters 6, 9, 11, 12): Flow in pipes,

turbomachines, flow over immersed bodies, flow through porous media

Dimensional analysis and modeling (Chapter 13)

Motivation for Studying Fluid Mechanics

• Fluid Mechanics is omnipresent – Aerodynamics

– Bioengineering and biological systems

– Combustion

– Energy generation

– Geology

– Hydraulics and Hydrology

– Hydrodynamics

– Meteorology

– Ocean and Coastal Engineering

– Water Resources

– …numerous other examples…

• Fluid Mechanics is beautiful

Aerodynamics

Bioengineering

Energy generation

Geology

River Hydraulics

Hydraulic Structures

Hydrodynamics

Meteorology

Water Resources

Fluid Mechanics is Beautiful

Intoduction 1.17

TOPIC 1

Introduction (de Nevers 1.1-1.5.2, 1.8-1.10)

Intoduction 1.18

Dimensions and Units

In fluid mechanics we must describe various fluid characteristics in

terms of certain basic quantities such as length, time and mass

• A dimension is the measure by which a physical variable is expressed

qualitatively, i.e. length is a dimension associated with distance, width,

height, displacement.

Basic dimensions: Length, L

(or primary quantities) Time, T

Mass, M

Temperature, Q

We can derive any secondary quantity from the primary quantities

i.e. Force = (mass) x (acceleration) : F = M L T-2

• A unit is a particular way of attaching a number to the qualitative

dimension: Systems of units can vary from country to country, but

dimensions do not

Intoduction 1.19

Dimensions and Units

Primary

Dimension SI Unit

British

Gravitational

(BG) Unit

English

Engineering

(EE) Unit

Mass [M] Kilogram (kg) Slug Pound-mass

(lbm)

Length [L] Meter (m) Foot (ft) Foot (ft)

Time [T] Second (s) Second (s) Second (s)

Temperature [Q] Kelvin (K) Rankine (°R) Rankine (°R)

Force [F] Newton

(1N=1 kg.m/s2) Pound (lb) Pound-force (lbf)

Conversion factors are available in the textbook inside of front cover.

Intoduction 1.20

Units of Force: Newton’s Law F=m.g

• SI system: Base dimensions are Length, Time, Mass, Temperature

A Newton is the force which when applied to a mass of 1 kg

produces an acceleration of 1 m/s2.

Newton is a derived unit: 1N = (1Kg).(1m/s2)

• BG system: Base dimensions are Length, Force, Time, Temperature

A slug is the mass which produces an acceleration of 1 ft/s2 when

a force of 1lb is applied on it:

Slug is a derived unit: 1slug=(1lb) (s2)/(ft)

• EE system: Base dimensions are Length, Time, Mass, Force and

Temperature

The pound-force (lbf) is defined as the force which accelerates

1pound-mass (lbm), 32.174 ft/s2.

Intoduction 1.21

Units of Force – EE system

To make Newton’s law dimensionally consistent we must include a

dimensional proportionality constant:

cg

gmF

where

2

f

mc

)s)(lb(

)ft)(lb(1740.32g

Intoduction 1.22

Example: Newton’s Law

• An astronaut weighs 730N in Houston, TX, where the local

acceleration of gravity is g=9.792 m/s2. What is the mass of the

astronaut? What is his weight on the moon, where g=1.67 m/s2?

• Redo the same problem in EE units. In EE units the astronaut weighs

164.1lbf, gHouston=32.13 ft/s2 and gmoon=5.48 ft/s2.

Intoduction 1.23

Dimensional Homogeneity

• All theoretically derived equations are dimensionally homogeneous:

dimensions of the left side of the equation must be the same as those

on the right side.

– Some empirical formulas used in engineering practice are not

dimensionally homogeneous

• All equations must use consistent units: each term must have the

same units. Answers will be incorrect if the units in the equation are

not consistent. Always chose the system of units prior to solving the

problem

Intoduction 1.24

Properties of Fluids

Fundamental approach: Study the behavior of individual molecules

when trying to describe the behavior of fluids

Engineering approach: Characterization of the behavior by considering

the average, or macroscopic, value of the quantity of interest, where the

average is evaluated over a small volume containing a large number of

molecules

Treat the fluid as a CONTINUUM: Assume that all the fluid

characteristics vary continuously throughout the fluid

Intoduction 1.25

Measures of Fluid Mass and Weight

• Density of a fluid, r (rho), is the amount of mass per unit volume of a

substance: r = m / V

– For liquids, weak function of temperature and pressure

– For gases: strong function of T and P

from ideal gas law: r = P M/R T

where R = universal gas constant, M=mol. weight

R= 8.314 J/(g-mole K)=0.08314 (liter bar)/(g-mole K)=

0.08206 (liter atm)/(g-mole K)=1.987 (cal)/(g-mole K)=

10.73 (psia ft3)/(lb-mole °R)=0.7302 (atm ft3)/(lb-mole °R)

)T,P(rr

(1.1)

Intoduction 1.26

Measures of Fluid Mass and Weight

• Specific volume: u = 1 / r

• Specific weight is the amount of weight per unit volume of a substance:

g = w / V = r g

• Specific Gravity (independent of system of units)

C4@OH2

SGr

r

Flow in pipes 6.27

Forces acting on a fluid

The forces acting on a fluid are divided into two groups:

• Body forces act without physical contact. They act on every mass

element of the body and are proportional to its total mass. Examples

are gravity and electromagnetic forces

• Surface forces require physical contact (i.e. surface contact) with

surroundings for transmission. Pressure and stresses are surface

forces.

28

Definition of Pressure

Pressure is defined as the amount of force exerted on a unit

area of a substance:

Pam

N

area

forceP

2

29

Direction of fluid pressure on boundaries

Furnace duct Pipe or tube

Heat exchanger

Dam

Pressure is a Normal Force

(acts perpendicular to surfaces)

It is also called a Surface Force

30

Absolute and Gauge Pressure

• Absolute pressure: The pressure of a fluid is

expressed relative to that of vacuum (=0)

• Gauge pressure: Pressure expressed as the

difference between the pressure of the fluid and

that of the surrounding atmosphere.

Usual pressure guages record guage pressure.

To calculate absolute pressure:

gaugeatmabs PPP

31

Absolute & Gauge Pressure: Schematic

+

- +

+

32

Units for Pressure

Unit Definition or

Relationship

1 pascal (Pa) 1 kg m-1 s-2

1 bar 1 x 105 Pa

1 atmosphere

(atm)

101,325 Pa

1 torr 1 / 760 atm

760 mm Hg 1 atm

14.696 pounds

per sq. in. (psi)

1 atm

Flow in pipes 6.33

Shear Flow

NO-SLIP CONDITION: The fluid “sticks” to the solid boundaries.

The velocity of the fluid touching each plate is the same as that of the

plate (Vo for the top plate, 0 for the bottom plate).

The velocity profile is a straight line: The velocity varies uniformly from 0

to Vo

Vo

= F/A

A

yo

Fluid

x

y

F

= F/A

o

o

o

o

y

V

dy

dV and y

y

V)y(V

Flow in pipes 6.34

Shear Flow

The force, F is proportional to the velocity Vo, the area in contact with

the fluid, A and inversely proportional to the gap, yo:

y

V AF

o

o

Recall, shear stress, = F / A

y

V

o

o

In the limit of small deformations the ratio Vo/yo can be replaced by the

velocity gradient dV/dy:

dy

dVg

Rate of shearing strain

or shear rate:

g

or

dy

dV

Flow in pipes 6.35

Newton’s law of Viscosity

Newton’s law of viscosity

dy

dV

[=N/m2 . s=Pa . s]: Viscosity

n = /r : Kinematic viscosity [=m2/s]

Newtonian fluids: Fluids which obey Newton’s law: Shearing stress is

linearly related to the rate of shearing strain.

(6.6)

The viscosity of a fluid measures its resistance to flow under an

applied shear stress.

Flow in pipes 6.36

True or False?

• When a fluid is subjected to a steady shear stress, it will reach a state of equilibrium in which no further motion occurs

• Pressure and shear stress have identical units

• When a flowing fluid is in contact with a solid surface, the velocity of the portion of the fluid in direct contact with the surface is always equal to zero.

• Newton’s law of viscosity relates shear stress to rate of shearing strain

Flow in pipes 6.37

Example: Shear stress

The space between two plates, as shown in the figure, is filled with

water. Find the shear stress and the force necessary to move the

upper plate at a constant velocity of 10 m/s. The gap width is

yo=0.1 mm and the area A is 0.2 m2. The viscosity of water is 0.001

Pa.s.

Vo F

A

yo Water

= F/A

Flow in pipes 6.38

Non-Newtonian fluids

Non-Newtonian fluids: Fluids which do not obey Newton’s law: Shearing

stress is not linearly related to the rate of shearing strain.

•Bingham plastics

•Shear thinning

•Shear thickening

The study of these materials is the subject of rheology

Flow in pipes 6.39

Typical Viscosity Values (Pa-s)

Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept.

• Asphalt Binder……..

• Polymer Melt ………

• Molasses …………..

• Liquid Honey ………

• Glycerol ……………

• Olive Oil ……………

• Water ………………

• Air ………………….

100,000

1,000

100

10

1

0.01

0.001

0.00001

The range of viscosity values one encounters

suggests the use of logarithmic scales when

plotting data.

40

http://www.uaeu.ac.ae/

General Flow Classifications


Shear Rate (1/sec)

She

ar S

tres

s (P

a)

Dilatant

Newtonian

Pseudoplastic

An example of dilatancy is

wet sand on the beach.

Most materials used

industrially are

pseudoplastic.

41


General Flow Classifications


Shear Rate (1/sec)

She

ar S

tres

s (P

a)

Dilatant

Newtonian

Pseudoplastic

An example of dilatancy is

wet sand on the beach.

Most materials used

industrially are

pseudoplastic.

42


Bingham Fluid


τ, P

a

g /s

Ideal Yield Stress (Bingham Yield)

43



Some Interesting Phenomena

• shear thinning

• shear thickening

• yield stress

• viscoelastic effects

– Weissenberg effect

– Fluid memory

– Die Swell

– Tubeless Syphon

44



World’s Longest Running Laboratory Experiment – The Pitch Drop Experiment

• Pitch – derivative of tar (a dark sticky substance obtained from tar and

used in the building trades, especially for waterproofing roofs)

– @room temperature feels solid and can be shattered with a blow

of a hammer

– This experiment shows that in fact at room temperature pitch is a

fluid!

45



World’s Longest Running Laboratory

Experiment – The Pitch Drop Experiment

1927 – Prof Parnell in Univ. of Queensland

Australia heated a sample of pitch and poured it

into a glass funnel with a sealed stem. Three years

where allowed for it to settle, after which the stem

was cut.

Examine the viscosity of the pitch by the speed at

which it flows from a funnel into a jar.

Only eigth drops has fallen in 80 years.

The viscosity is approximated as 100 billion times

that of water.

46


Weissenberg Effect (Rod Climbing

Effect)


The fluid does not flow outward

when stirred at high speeds

Ms. Garcia-Rodrigues is s t u d y i n g R h e o l o g y a t U . o f Wisconsin-Madison, USA. The fluid shown is a 2% aqueous polyacrylamide s o l u t i o n , a n d t h e r o t a t i o n a l s p e e d i s n o m i n a l l y 0 . 5 H z .

47


Weissenberg Effect

Basim Abu-Jdayil - Chemical & Petroleum Eng. Dept. 48


Fluid Memory


Conserve their shape over

time periods or seconds or

minutes

Elastic like rubber

Can bounce or partially

retract

Example: clay

49


Die Swell


– as a polymer exits a die, the diameter of liquid stream

increases by up to an order of magnitude

– caused by relaxation of extended polymer coils, as stress is

reduced from high flow producing stresses present within the

die to low stresses, associated with the extruded stream

moving through ambient air

50


Tubeless Siphon


When the siphon tube is lifted out of

the fluid, the Newtonian liquid (N)

stops flowing; the macromolecular

fluid (P) continues to be siphoned.

51


../Desktop/Rheology/tubeless_syphon_s.avi

Application of Rheology


• Polymer melts, solutions including composites • Rubber • Lubricants • Paints • Printing inks • Paper and pulp • Food • Biological fluids • Concrete and Clay • Cosmetics • Pharmaceuticals • Smart materials – ER-fluids

52


Importance of Rheology


Measuring rheological properties helps us bridge the

gap between molecular structure and product performance.

Rheology is important in industrial processing and design of a wide

range of complex chemical products.

53


Yield Stress


Rheological modifiers are used to control the

yield behavior of fluids.

Why modify the yield behavior?

* to avoid sedimentation and increase the shelf live

* to reduce flow under gravity

* to stabilize a fluid against vibration

54



Emulsion Yield Stress

0 20 40 60 80 100 120 140 160 -20

shear rate [1/s}

0

5

10

15

20

25

30

str

ess [P

a]

yield stress determined in a stress ramp

Requirements are:

minimum yield stress (10 Pa) to retard sedimentation low shear viscosity of 500 Pas to impede flocculation shear thinning to 2 Pas at 10'000 s-1 to allow fast application on the hair

Formulation of a conditioning shampoo

55


Toothpaste


Toothpaste consists of solid particles

suspended in an aqueous solution of

various polymers.

A good toothpaste should be a Bingham

fluid, so that it can easily be squeezed out

of the tube but will not drip off the

toothbrush the way water or honey would.

56


Drilling Mud



Thixotropy


The thixotropy characterizes the time dependence of

reversible structure changes in complex fluids. The

control of thixotropy is important to control:

sagging and leveling and the related gloss

of paints and coatings, etc..

process conditions for example to avoid

structure build up in pipes at low pumping

rates i.e. rest periods, etc.... 58


Paints


A good paint should be a

thixotropic with a yield stress fluid,

So in the can it will be very

viscous and the pigment will not

settle to the bottom,

But when it is stirred, it will

become less viscous and can easily

be brushed onto a surface.

In addition, the brushing should

temporarily reduce the viscosity so

that the paint will flow sideways

and fill in the brush marks (called

leveling in the paint industry); then,

as it stands, its viscosity should

increase, so that it will not form

drops and run down the wall.

59



Viscosity ranges of coatings

The two coatings show the same consistancy after formulation,

but they exhibit very different application performance

0.1 1 10 100 1000 10000 100000 10

-3

10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

HSV MSV LSV

Roling Brushing Spraying

Mixing Pumping Consistency Appearence

Leveling Sagging Sedimentation

At Rest Processing Performance

Vis

cosity h

[P

as]

shear rate g [1/s]

60


Engine Oil


Motor oil, or engine

oil, is an oil used for

lubrication of various

internal combustion

engines

Base oil is Newtonian

Polymers are added to

give eth engine oil the

shear thinning behavior

61


http://en.wikipedia.org/wiki/Oil

http://en.wikipedia.org/wiki/Lubrication

http://en.wikipedia.org/wiki/Internal_combustion_engine

http://en.wikipedia.org/wiki/Internal_combustion_engine

Rheopectic Behavior



Shear Thickening


They filled a pool with a mix of cornstarch and water

made on a concrete mixer truck. It becomes a non-

newtonian fluid. When stress is applied to the liquid it

exhibits properties of a solid.

63

../../../../Desktop/Rheology/A pool filled with non-newtonian fluid.flv


Flow in pipes 6.64

Effect of temperature on viscosity

Viscosity is very sensitive to temperature

• The viscosity of gases increases with temperature:

n

oo T

T

ST

T C 3/2

o

Power-law

Sutherland equation

• The viscosity of liquids decreases with temperature:

-B/Te D

ch 1 fluid mechanics

Documents

fluid mechanics intoduction

examples fluid mechanics

denevers fluid mechanics

units fluid statics

fluid statics chapter

various fluid characteristics

compressive stress f

surface f