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Properties of Fluids for Fluid Mechanics. P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi. Basic Steps to Design…………. Continuum Hypothesis.

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Properties of Fluids for Fluid Mechanics

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Properties of fluids for fluid mechanics l.jpg

Properties of Fluids for Fluid Mechanics

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Basic Steps to Design………….


Continuum hypothesis l.jpg

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 l.jpg

Static Fluid

For a static fluid

Shear Stress should be zero.

For A generalized Three dimensional fluid Element, Many forms of shear stress

is possible.


One dimensional fluid element l.jpg

One dimensional Fluid Element

+Y

u=U

u=0

+X

+


Fluid statics l.jpg

tyx

tzx

txy

tzy

tyz

txz

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.

Y

X

Z

  • 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


Stress tensor l.jpg

Y

tyx

tzx

tyy

tzy

txy

txx

tyz

X

tzz

txz

txz

Z

Stress Tensor


Slide7 l.jpg

First Law of Pascal

Proof ?


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Simple Non-trivial Shape of A Fluid Element


Fluid statics for power generation l.jpg

Fluid Statics for Power Generation

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Steps for Design of Flow Devices………….


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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


Slide19 l.jpg

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 l.jpg

Pressure Variation for a Uniform-Density Fluid


Reading material l.jpg

Reading Material

  • Fluid Mechanics – Frank M White, McGraw Hill International Editions.

  • Introduction to Fluid Mechanics – Fox & McDOnald, John Wiley & Sons, Inc.

  • Fluid Mechanics – V L Streeter, E Benjamin Wylie & K W Bedfore, WCB McGraw Hill.

  • Fluid Mechanics – P K Kundu & I M Cohen, Elsevier Inc.


Pressure variation for compressible fluids l.jpg

Pressure Variation for Compressible Fluids

Basic equation for pressure variation with elevation

Pressure variation equation can be integrated for γ(p,z) known.

For example, here we solve for the pressure in the atmosphere assuming ρ(p,T) given from ideal gas law, T(z) known, and g ≠ g(z).


Draft required to establish air flow l.jpg

Draft Required to Establish Air Flow

Flue gas out

Air in


Natural draft l.jpg

Natural Draft

Zref,,pref

pA = pref +Dp

Hchimney

Tgas

Tatm

B

A


Slide28 l.jpg

Zref,,pref

pA = pref +Dp

Hchimney

Tgas

Tatm

B

A


Slide29 l.jpg

Pressure variations in Troposphere:

Linear increase towards earth surface

Tref & pref are known at Zref.

a : Adiabatic Lapse rate : 6.5 K/km


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Reference condition:

At Zref : T=Tref & p = pref


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Pressure at A:

Pressure variation inside chimney differs from atmospheric pressure.

The variation of chimney pressure depends on temperature variation along

Chimney.

Temperature variation along chimney depends on rate of cooling of hot gas

Due to natural convection.

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

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

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


Natural draft32 l.jpg

Natural Draft

  • Natural Draft across the furnace,

  • Dpnat = 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.


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Pressure Measurement

Another Application of Fluid Statics


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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 l.jpg

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.


Slide36 l.jpg

  • 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 l.jpg

Enlarged Leg

Inverted U Tube

U Tube

Two Fluid

Inclined Tube

Manometers


U tube or differential manometer l.jpg

U-tube or differential manometer

Right Limb fluid statics :

Left Limb fluid statics :

Point 3 and 2 are at the same elevation and same fluid


Slide39 l.jpg

Gauge Pressure:


Absolute gauge vacuum pressures l.jpg

Absolute, Gauge & Vacuum Pressures

System Pressure

Gauge Pressure

Absolute Pressure

Atmospheric Pressure

Absolute zero pressure


Slide41 l.jpg

Absolute, Gauge & Vacuum Pressures

Atmospheric Pressure

Vacuum Pressure

System Pressure

Absolute Pressure

Absolute zero pressure


Stress tensor for a static fluid l.jpg

Y

tyy

txx

X

tzz

Z

Stress Tensor for A Static Fluid

0

0

0

0

0

0


An important property of a fluid under motion l.jpg

An important Property of A Fluid under Motion


Slide44 l.jpg

Shear stress(t): Tangential force on per unit area of contact

between solid & fluid


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Elasticity (Compressibility)

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

  • The amount of deformation is called elasticity.


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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 l.jpg

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.


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