Fm 2

Properties of Fluids for Fluid Mechanics

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Basic Steps to Design………….

Continuum Hypothesis

• In this course, the assumption is made that the fluid behaves as a continuum, i.e., the number of molecules within the smallest region of interest (a point) are sufficient that all fluid properties are point functions (single valued at a point).

• For example:• Consider definition of density ρ of a fluid

• δV* = limiting volume below which molecular variations may be important and above which macroscopic variations may be important.

Static Fluid

For a static fluid

Shear Stress should be zero.

For A generalized Three dimensional fluid Element, Many forms of shear stressis possible.

One dimensional Fluid Element

+X

+

+Y

u=0

u=U

Fluid Statics

• Pressure : For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion.

• What is the significance of Diagonal Elements?

• Vectorial significance : Normal stresses.

•Physical Significance : ?

•For the general case, the stress on a fluid element or at a point is a tensor

X

Y

Z

xy

xz

yz

yx

zx

zy

Stress Tensor

X

Y

Z

yy

zz

yz

yx

zx

zy xx

xy

xzxz

First Law of Pascal

Proof ?

Simple Non-trivial Shape of A Fluid Element

Fluid Statics for Power Generation

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Steps for Design of Flow Devices………….

Pressure Variation with Elevation

• For a static fluid, pressure varies only with elevation within the fluid.

• This can be shown by consideration of equilibrium of forces on a fluid element

•Basic Differential Equation:

Newton's law (momentum principle) applied to a static fluid

ΣF = ma = 0 for a static fluid

i.e., ΣFx = ΣFy = ΣFz = 0

1st order Taylor series estimate for pressure variation over dz

For a static fluid, the pressure only varies with elevation z and is constant in horizontal xy planes.

• The basic equation for pressure variation with elevation can be integrated depending on

• whether ρ = constant i.e., the fluid is incompressible (liquid or low-speed gas)

• or ρ = ρ(z), or compressible (high-speed gas) since g is constant.

Pressure Variation for a Uniform-Density Fluid

Draft Required to Establish Air Flow

Air in

Flue as out

Natural Draft

Hchimney

TatmTgas

A B

pA = pref +p

)(zT

dz

R

pgdp

Zref

A

ref

Z

Z airairrefA zT

dz

R

pgpp

)(

B

ref

Z

Z gasgasrefB zT

dz

R

pgpp

)(

Hchimney

TatmTgas

A B

pA = pref +p

)(zT

dz

R

pgdp

Zref,,pref

Pressure variations in Troposphere:

)( ZZTT refref

Linear increase towards earth surface

Tref & pref are known at Zref.)(zT

dz

R

g

p

dp

Adiabatic Lapse rate : 6.5 K/km

)( ZZT

dz

R

g

p

dp

refrefatm

tconsZZTR

gp refref

atm

tan)(lnln

Reference condition:At Zref : T=Tref & p = pref

tconsTR

gp ref

atmref tanlnln

refatm

refrefrefatm

TR

gpZZT

R

gp lnln)(lnln

ref

refref

atmref T

ZZT

R

g

p

p )(lnln

atmR

g

ref

refref

ref T

ZZT

p

p

)(

Pressure at A:atmR

g

ref

Arefref

ref

A

T

ZZT

p

p

)(

Pressure variation inside chimney differs from atmospheric pressure.

The variation of chimney pressure depends on temperature variation alongChimney.

Temperature variation along chimney depends on rate of cooling of hot gasDue to natural convection.

Using principles of Heat transfer, one can calculate, Tgas(Z).

If this is also linear: T = Tref,gas + (Zref-Z).

Lapse rate of gas, gas is obtained from heat transfer analysis.

atmR

g

gasref

Brefgasgasref

ref

B

T

ZZT

p

p

,

, )(

Natural Draft

• Natural Draft across the furnace, pnat = pA – pB

The difference in pressure will drive the exhaust.

•Natural draft establishes the furnace breathing by –Continuous exhalation of flue gas–Continuous inhalation of fresh air.

•The amount of flow is limited by the strength of the draft.

Pressure Measurement

Pressure Measurement

Pressure is an important variable in fluid mechanics and many instruments have been devised for its measurement.

Many devices are based on hydrostatics such as barometers and manometers, i.e., determine pressure through measurement of a column (or columns) of a liquid using the pressure variation with elevation equation for an incompressible fluid.

PRESSURE

• Force exerted on a unit area : Measured in kPa

• Atmospheric pressure at sea level is 1 atm, 76.0 mm Hg, 101 kPa

• In outer space the pressure is essentially zero. The pressure in a vacuum is called absolute zero.

• All pressures referenced with respect to this zero pressure are termed absolute pressures.

• Many pressure-measuring devices measure not absolute pressure but only difference in pressure. This type of pressure reading is called gage pressure.

• Whenever atmospheric pressure is used as a reference, the possibility exists that the pressure thus measured can be either positive or negative.

• Negative gage pressure are also termed as vacuum pressures.

Manometers

U Tube

Inverted U Tube

Enlarged Leg

Two FluidInclined Tube

Absolute, Gauge & Vacuum Pressures

System Pressure

Atmospheric Pressure

Gauge Pressure

Absolute Pressure

Absolute zero pressure

Absolute, Gauge & Vacuum Pressures

System Pressure

Atmospheric Pressure

Vacuum Pressure

Absolute Pressure

Absolute zero pressure

An important Property of A Fluid

y

tu

tan

Shear stress(: Tangential force on per unit area of contact between solid & fluid

Elasticity (Compressibility)

• Increasing/decreasing pressure corresponds to contraction/expansion of a fluid.

• The amount of deformation is called elasticity.

Surface Tension

• Two non-mixing fluids (e.g., a liquid and a gas) will form an interface.

• The molecules below the interface act on each other with forces equal in all directions, whereas the molecules near the surface act on each other with increased forces due to the absence of neighbors.

• That is, the interface acts like a stretched membrane, e.g.

Vapour Pressure

• When the pressure of a liquid falls below the vapor pressure it evaporates, i.e., changes to a gas.

• If the pressure drop is due to temperature effects alone, the process is called boiling.

• If the pressure drop is due to fluid velocity, the process is called cavitation.

• Cavitation is common in regions of high velocity, i.e., low p such as on turbine blades and marine propellers.