Fluids 1

Fluid Mechanics

CHE211

Fall 2014

Grading Based on

Homework: 10% (5-6)

Quiz :10 % (3-4)

Mid-Term: 30% (TBA)

Final: 50%

Lectures: Monday, 10:30am-12:20 pm

Wednesday , 10:30-11:20 am

Tutorials : Wednesday, 11:30 am-12:20 pm.

Text books and Materials Desire to learn

1. De Nevers N, Grahn R. Fluid mechanics for chemical engineers:

McGraw-Hill; 2004.

2. Wilkes JO. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD: Pearson Education; 2006.

3. Fox RW, McDonald AT, Pritchard PJ. Introduction to fluid

mechanics. Vol 5: John Wiley & Sons New York; 1998.

Outline GENERAL INTRODUCTION

What is a fluid?; Shear force and Shear stress; Shear strain and Rate of shear; Examples of fluids.

Fluid Mechanics - Statics and Dynamics; Why study fluid mechanics (practical significance)?

Continuum hypothesis; flow field; flow field variables (velocity, pressure, density, velocity gradient, shear-stress, etc.); steady and unsteady flows; compressible and incompressible flows;

.

2. FLUID STATICS

Principle of hydrostatic equilibrium; Pressure variation throughout a static fluid; Applications of fluid statics - Barometric equation; manometers; Hydrostatic equilibrium in a centrifugal field;

3. INTRODUCTION TO FLUID FLOW (Basic Concepts)

Influence of solid boundaries on fluid flow; No-slip boundary condition; Boundary layer; Boundary layer in flow over a thin plate; Boundary layer in pipe flow.

Definition of viscosity (Newton's law of viscosity); Dependence of viscosity on temperature and pressure; Kinematic viscosity; Ideal fluid and potential flow; Introduction to non-Newtonian fluids.

Laminar and turbulent flows (Reynolds experiment); Reynolds number; Generalized Reynolds number; Turbulence in boundary layers.

4. BASIC EQUATIONS OF FLUID FLOW

Microscopic versus Macroscopic balances

Macroscopic Mass Balance; average velocity.

Macroscopic Momentum Balance; Momentum correction factor.

Macroscopic Mechanical Energy Balance for ideal frictionless fluids (Bernoulli equation); Bernoulli equation for viscous fluids; Kinetic energy correction factor; Pump work in Bernoulli equation.

5. FLOW OF INCOMPRESSIBLE FLUIDS IN CLOSED CHANNELS

Shear-stress distribution in a cylindrical tube; Friction factor.

Laminar flow in pipes; velocity distribution for Newtonian fluids; Maximum velocity; Average velocity; Hagen-Poiseuille Law; Reynolds number-friction factor law; Velocity distribution for non-Newtonian fluids.

Turbulent flow in pipes; Velocity distribution for Newtonian fluids; Reynolds number-friction factor law for smooth tubes; Friction factors for rough pipes; Colebrook's equation; Reynolds number-friction factor law for non-Newtonian fluids (Dodge-Metzner equation).

Flow in non-circular pipes; Concept of hydraulic diameter; Friction factor relations.

Friction loss in sudden expansion and sudden contraction of cross section; Effect of fittings and valves on friction loss.

6. FLOW PAST IMMERSED BODIES

Flow past single objects; Skin (wall) drag and Form drag; Definition of Drag coefficient; Variation of Drag coefficient with Reynolds number; Stokes' regime (Stokes' law), Intermediate regime and Newton's regime.

Free settling of particles through fluids; Terminal velocity; criteria for settling regimes (Stokes, intermediate or Newton's regimes).

Flow through bed of solids; Hydraulic diameter; Ergun equation; Blake-Kozeny equation (Darcy's Law); Burke-Plummer equation; Filtration.

Fluidization; conditions for fluidization; Minimum fluidization velocity; Types of fluidization (particulate fluidization versus bubbling fluidization); Expansion of fluidized beds - particulate fluidization and bubbling fluidization; Applications of fluidization.

7. METERING OF FLUIDS

Flow rate measurement; Venturi-meter - application of Bernoulli equation, Venturi-coefficient, Pressure recovery; Orifice-meter - application of Bernoulli equation, Orifice coefficient, Pressure recovery; Pitot tube.

Course Focus

6

Fundamental Concepts

Fluid Statics

Conservation of Mass, Momentum and Energy

Incompressible Fluids In Closed Channels

Flow Past Immersed Bodies

Dimensional Analysis

Metering of Fluids

GENERAL INTRODUCTION


What is a fluid?

Shear force and Shear stress; Rate of shear; Examples of fluids.

Fluid Mechanics

Statics and Dynamics; Why study fluid mechanics (practical significance)?

Dimensions and Units Flow field variables (velocity, pressure, density, velocity gradient, shear-stress,

etc.);

Continuum hypothesis

Principles involved

conservation of mass conservation of momentum (Newton's second law of motion), conservation of energy.

Comparison of tensile, compressive

and shear stresses

Tensile, Shear and Compressive forces

and stresses

T=

Tensile stress

=

shear stress

P =

Compressive stress

(Pressure)

Fluid Definition

Solids can permanently resist very large shear forces

(steel)

A fluid is a substance that deforms continuously when

undergoes a shear stress of any magnitude Fluids do

not sustain shearing or tensile stresses like solids they

can only hold compression stresses from all sides

(water)

Materials between solids and fluids can resist small

amount of shear permanently but not the large amount

(peanut butter)

Fluid Definition Cont

Fluids are either liquids or gases

Liquid: A state of matter in which the molecules

are relatively free to change their positions with

respect to each other but restricted by

cohesive forces so as to maintain a relatively

fixed volume

Gas: a state of matter in which the molecules

are practically unrestricted by cohesive forces. A

gas has neither definite shape nor volume.


What is a fluid?

Shear force and Shear stress; Shear strain and Rate of shear; Examples of fluids.

Fluid Mechanics

Definition, Statics and Dynamics; Why study fluid mechanics (practical significance)?


etc.);


Principles involved

- conservation of mass conservation of momentum (Newton's second law of motion), conservation of energy.

Fluid Mechanics

Mechanics: the branch of applied

mathematics that deals with the motion and

equilibrium of bodies and the action of forces,

and includes kinematics, dynamics, and statics.

Fluid mechanics: Its a branch of mechanics that only deals with fluids.

In Fluid Mechanics we can consider three

branches:

Fluid Statics: mechanics of fluids at rest

Kinematics: deals with velocities and

streamlines considering forces or energy

Fluid Dynamics: deals with the relations

between velocities and accelerations and

forces exerted by or upon fluids in

motion

Application

Fluid Mechanics is applied in:

Mechanical Engineering Aerodynamics

Chemical Engineering Combustion

Energy generation

Separation

Flow in pipes

Civil Engineering Hydraulics

Hydrology

Bioengineering

Hydrodynamics

Geology

Meteorology

Dimensions and Units

Two primary sets of units are used:

SI (System International) units

English units

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.

More on Dimensions

To remember units of a slug :

F=ma => m = F / a

[m] = [F] / [a] = lb / (ft / sec2) = lb*sec2 / ft

1 lb is the force of gravity acting on (or

weight of ) a platinum standard whose

mass is 0.453592 kg

Dimensions and Units cont

1 Newton Force required to accelerate a 1 kg of mass to 1 m/s2

1 slug is the mass that accelerates at 1 ft/s2 when acted upon by a force of 1 lb

To remember units of a Newton use F=ma

(Newtons 2nd Law)

[F] = [m][a]= kg*m/s2 = N

Weight

Weight (W) of object mass m is defined as

W = mg ; g is gravitational acceleration

g = 9.81 m/s2 in SI units

g = 32.2 ft/sec2 in English units

To convert units we can always consider:

1= lbf/lbf = =32.2 (ft.lbm/lbf.s2)

See back of front cover of textbook for conversion tables between SI and English units

Flow field variables and Some

physical properties of fluids Velocity

velocity gradient:

du/dy

22

( , , )

( , , )

( , , )x y z x y z

x y z

i j k

v v v v v i v j v k

Density, specific gravity and specific weight

Density (mass per unit volume): m

V

[ ][ ]

[ ]( )

m

V

kg

min SI units3Units of density:

For liquids, weak function of temperature and pressure

For gases: strong function of T and P

from ideal gas law: = P M/R T

where R = universal gas constant

Quetion: Prove the above relation for an ideal gas.

water

liquid

water

liquid

water

liquid

g

gS

Specific weight (weight per unit volume): g

Units of specific weight:

[ ] [ ][ ] ( ) gkg

m

m

s

N

min SI units3 2 3

Specific Gravity (independent of system of units)

COH

SG

4@2

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

Specific Gravity in Degrees Baume or

Degree API

For liquids heavier than water,

For liquids lighter than water,

Baumedeg145

145sg

sg

145145Baumedeg

Baumedeg130

140sg

130sg

140Baumedeg

Specific Gravity in Degrees Baume or Degree

API

The API has developed a scale that is slightly different from the Baum scale for liquids lighter than water.

The formulas are

APIdeg5.131

5.141sg

5.131sg

5.141APIdeg

Pressure Forces applied by static fluids always

perpendicular to the objects surface area)

Pressure

Pressure is a scalar, not a vector and is defined as the force applied over the surface area

2m

NPa

A

FP

Other Pressure units:

Psi = lbf/in2

Bar=100000 pa

atm=101325 pa

Absolute and Gage Pressure

When making calculations involving pressure in a fluid, you

must make the measurements relative to some reference

pressure.

Normally the reference pressure is that of the atmosphere,

and the resulting measured pressure is called gage pressure.

Pressure measured relative to a perfect vacuum is called

absolute pressure.

Absolute and Gage Pressure

A simple equation relates the two pressure-measuring

systems:

Pabs=Pgage + Patm

where

pabs = Absolute pressure

pgage = Gage pressure

patm = Atmospheric pressure

The comparison between absolute and gage pressures.

Example

Express a pressure of 225 kPa(abs) as a gage pressure.

The local atmospheric pressure is 101 kPa(abs).

Solving algebraically for pgage gives:

Pabs=Pgage + Patm

Pgage = Pabs - Patm

Pgage =225 Kpa(abs)- 101 Kpa (abs)= 124 Kpa (gage)

Example

Express a pressure of 75.2 kPa as a gage pressure. The local

atmospheric pressure is 103.4 kPa.

Notice that the result is negative. This can also be read 28.2

kPa below atmospheric pressure or 28.2 kPa vacuum.

Pgage = Pabs - Patm

Pgage =75.2Kpa 103.4 Kpa = -28.2 Kpa

Compressibility

All fluids compress if pressure increases resulting in an

increase in density

Compressibility is the change in volume due to a

change in pressure

A good measure of compressibility is the bulk modulus

(It is inversely proportional to compressibility)

Edp

d

1

( )specific volume

p is pressure

Compressibility

From previous expression we may write

For water at 15 psia and 68 degrees Farenheit,

From above expression, increasing pressure by 1000 psi will compress the water by only 1/320 (0.3%) of its original volume

( ) ( )

final initial

initial

final initialp p

E

E psi 320 000,

This result shows that water is incompressible .

In reality, no fluid is incompressible, but this is a good approximation for certain fluids

Viscosity ( ) Viscosity can be thought as the resistance against flow

Representative of internal friction in fluids

Internal friction forces in flowing fluids result from cohesion and momentum interchange between molecules.

Viscosity is important,

in determining amount of fluids that can be transported in a pipeline during a specific period of time

determining energy losses associated with transport of fluids in ducts, channels and pipes

Moving plate

Fixed plate

Y

x

yForce F

Velocity V(y=0)=0

Velocity V(y=Y)=U

Two plates with a small gap Y and large width

Fluid

Conditions:

No-slip conditions :Because of viscosity, at boundaries (walls) particles

of fluid adhere to the walls, and so the fluid velocity is zero relative to

the wall

The velocity profile can be drawn as follows:

y

u y( )

Y

U

u yU

Yy( )

Experimentally has been approved that for most of the fluids , The velocity profile

is linear and can be expressed linearly as follows:

du

dy

U

Y

u y velocity profile( ) ( )

And empirically, FAU

Y

More specifically, FAU

Y ; is coefficient of vis itycos

Shear stress induced by is F F

A

U

Y

Thus, shear stress is

du

dy

In general we may use previous expression to find shear stress at a point

inside a moving fluid. dy/dy is called strain rate or shear rate.

Note that if fluid is at rest shear stress is zero because

du

dy 0

Newtons equation of viscosity

- viscosity (coeff. of viscosity) Pa.s

Kinematic viscosity

Units of Kinematic viscosity: m2/s, Stoke

***** See Conversion factor table

Newtons Law of Viscosity can be used to evaluate the shear stress

exerted by a moving fluid onto the fluids boundaries.

at boundarydu

dyat boundary

Note is direction normal to the boundary y

Viscosity of a fluid depends on temperature: In liquids, viscosity decreases with increasing temperature (i.e.

cohesion decreases with increasing temperature)

In gases, viscosity increases with increasing temperature (i.e.

molecular interchange between layers increases with temperature

setting up strong internal shear)

Viscosity behavior of fluids

Newtonian.

Shear thickening.

Shear thinning.

Thixotropic.

Bingham plastic

Magnetorheological fluid

Newtonian Fluids

These are fluids which obey Newtons law viscosity which says the shear stress is linearly

related to the velocity gradient. Most

common fluids such as water, gasoline, fall into

this category.

Non-Newtonian Fluids

These are fluids which do not obey Newtons law of viscosity. Polymer solution and

emulsions

(ii) Pseudoplastic

The dynamic viscosity, of the fluid decreases as the rate of shear increases (e.g. milk, clay, colloidal solutions, sewage, sludge, cement). If n 1in Eq.(1.1), it is called dilatant.

(iv) Thioxotropic Substances

The dynamic viscosity decreases with the time for which shearing forces are

applied e.g. thioxotropic jelly paints.

(v) Rheopectic Materials

The dynamic viscosity of these materials increases with the time for which

shearing forces are applied.

(vi) Viscoelastic Materials

These behave in manner similar to Newtonian fluids under time-invariant

conditions but if shear stress changes suddenly behave as plastic.

Problem 1. 11 (streeter)

A 3.0-in.diameter shaft slides at 0.4 ft/sec through a

6-in.-long sleeve with radial clearance of 0.01 in.

when a 10.0-lb force is applied. Determine the

viscosity of fluid between shaft and sleeve.


What is a fluid?


Fluid Mechanics



etc.);


Principles involved

- conservation of mass conservation of momentum (Newton's second law of motion), conservation of energy.

Continuum Assumption

A fluid particle is a volume large enough to contain a sufficient

number of molecules of the fluid to give an average value for

any property that is continuous in space, independent of the

number of molecules.

A good way to determine if the continuum model is acceptable

is to compare a characteristic length L of the flow region with

the mean free path of molecules,

Mean free path ( ) Average distance a molecule travels

before it collides with another molecule.

Knudsen number: Kn = / L

- mean free path L - characteristic length

Continuum Assumption

For continuum assumption: Kn

For air duct:

49

Characteristic scales for standard air:

-> mean free path, (sea level) ~ 10-7 m

Characteristic length (L):

-> Diameter of the duct (D) = 1 inch (25.4mm)

Kn = 10-7/(0.0254) = 3.937x10-6 < 0.001

(Continuum and non-slip fluid flow)

For micro-channel:

50

Characteristic scales for standard air:

-> mean free path, (sea level) ~ 10-7 m

Characteristic length (L):

-> Width of the micro-channel = 1m = 10-6m

Kn = 10-7/(10-6) = 0.1

L

Microscopic:

Study the behavior of molecules

Macroscopic:

Continuum assumption

Navier-Stokes Equation

51

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


What is a fluid?


Fluid Mechanics



etc.);


Principles involved

Conservation of mass Conservation of momentum (Newton's second law of motion), Conservation of energy.

Principles involved

Conservation of mass

dm/dt=0

Conservation of momentum (Newton's

second law of motion)

F=ma

Conservation of energy

Energy is not produced or destroyed and only is changed from one to another form

Fluids 1

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