Development of EoS for Vapours & Gases

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Models for Highly Bountiful Phase…..…..

Behaviour of Vapour

= interatomic potential, Joules.• r = separation of molecules, nm (mean Free path).• r = equivalent “hard sphere” radius of molecule (overlap of

electron clouds).• At high T, high p, collisions in the repulsive part of – positive

deviations from constancy.• At low T, moderate p, collisions in the attractive portion of –

negative deviations from constancy.

P – v- T Relation

• Greatest need for EoS of saturated and superheated steam.

Tv

aRTpv

•R and a are constants.

•The is called as Rankine’s Equation of state, 1849.

n

T

Tcc

0

0

• The specific volume of A vapour:

v = f (p,T)

dTT

fdp

p

fdv

pT

bT

Tc

p

RTv

n

0

0

P – v- T Relation

• Callender’s Characteristic Equation for saturated and superheated vapours.

bcp

RTv

•R and b are constants.

•c is a function of temperature and it is called as co-aggregation volume.

n

T

Tcc

0

0

• The specific volume of A vapour:

v = f (p,T)

dTT

fdp

p

fdv

pT

bT

Tc

p

RTv

n

0

0

Pressure Volume Diagram

bT

Tc

p

RTv

n

0

0

Van der Waals EOS

• One of the oldest but most extensively used of the EOS of non ideal gases

• Any EOS model must reproduce graphs such as that of the previous

RTbvv

ap

2

•a, b are the Van der Waals constants for the particular gas;

•for water: a = 0.5658 J-m3/mole2; b = 3.049x10-5 m3/mole,

abavvRTpbpv 23

p

RTvideal

JO H A N N E S D . V A N D E R W A A L SThe equation of state for gases and liquids

Nobel Lecture, December 12, 1910

I intend to discuss in sequence:

(1) the broad outlines of my equation of state and how I arrived at it;

(2) what my attitude was and still is to that equation;

(3) how in the last four years I have sought to account for the discrepancies which remained between the experimental results and this equation;

(4) how I have also sought to explain the behaviour of binary and ternary mixtures by means of the equation of state.

Van der Waals EOS

RTbvv

ap

2

•a, b are the Van der Waals constants for the particular gas;

•for water: a = 0.5658 J-m3/mole2; b = 3.049x10-5 m3/mole,

Van der Waals CoefficientsGas a (Pa m3) B (m3/mol)

Helium 3.46 x 10-3 23.71 x 10-6

Neon 2.12 x 10-2 17.10 x 10-6

Hydrogen 2.45 x 10-2 26.61 x 10-6

Carbon dioxide 3.96 x 10-1 42.69 x 10-6

Water vapor 5.47 x 10-1 30.52 x 10-6

abavvRTpbpv 23 RTpvideal

Van der Waals Isotherms

Isotherms of Real Gases

Improved Cubic Equations of State

The constants a, b, c, Ao, Bo varies with substance

Compressibility Factor

• The deviation from ideal gas behaviour can also be expressed by compressibility factor, Z.

• The ratio of volume of real gas, Vreal to the ideal volume of that gas, Vperfect calculated by ideal gas equation is known as compressibility factor.

RT

pvZ real

p

RTvideal

ideal

real

v

vZ

• Compact description of non-ideality: the compressibility factor,

• Z 1 as p 0 (ideality)

• Z < 1 at low T, moderate p (point A)

• Z > 1 at high p, high T (point B)

Generalized Compressibility Chart

Reduced Pressure pR = p/pc

Reduced Temperature TR = T/Tc

VdW EOS & Compressibility

• a represents the attractive part of the potential; with b = 0, the VdW EOS gives a smaller v for the same T than the ideal gas

• b represents the repulsive portion of the potential; with a = 0, the VdW EOS gives a larger v for the same T than the ideal gas

• The VdW EOS is easily expressed in the forms p(T,v) or T(p,v).

• For the v(T,p) form, or, equivalently, Z(p,T):

vRT

a

vbRT

pvZ real

1

1

The ideal gas equation of state may be written several ways.

What More Happens at System Boundary during Change of State

The Happenings Which are our Benefits!!!

Global Wind Patterns : The Simple Resource

The Ancient Green Method for Better Living

Traditional Egyptian architecture in Ancient Egypt as demonstrated on the Pharonic house of Neb- Ammun, Egypt,

19th Dynasty, c.1300 BC.

Persian: بادگیر bâdgir: bâd "wind" + gir "catcher“

Arabic: ملقف malqaf

Eastern Arabia:بارجيل barjeel

An ancient Idea for Better LivingWindcatcher (Bagdir)

Evolution of Wind Turbines

• Wind is a clean, safe, renewable form of energy.• Although the use of wind power in sailing vessels appeared in

antiquity, the widespread use of wind power for grinding grain and pumping water was delayed until – the 7th century in Persia, – the 12th century in England, and – the 15th century in Holland.

• 17th century, Leibniz proposed using windmills and waterwheels together to pump water from mines in the Harz Mountains.

• Dutch settlers brought Dutch mills to America in the 18th century. • This led to the development of a multiblade wind turbine that was

used to pump water for livestock.• Wind turbines were used in Denmark in 1890 to generate electric

power. • Early in the 20th century American farms began to use wind

turbines to drive electricity generators for charging storage batteries.

The Modern Green Idea for Better Living : Wind Power

What happens When there is a change in state?

• Any of these happenings is/are useful for engineering world?

• Does it consume any resource?

• How to recognize these Happenings?

• Thermal In-equilibrium

• Mechanical In-equilibrium

• Chemical In-equilibrium

• Any combinations of above.

• These are happenings or actions or path functions or interactions.

• Present only during a change of state.

• What action is work transfer?

• What action is Heat transfer?

• What action is Mass transfer?

• How to differentiate?

Mechanical Work Tranfer• Work is a mechanical concept given by the expression:

• F is a force and s is a displacement

• Work is a scalar product

• Force components along the displacement vector only can do work

• Force components perpendicular to the displacement vector cannot do work.

• This relationship will be useful to find work required to raise a weight, to stretch a wire or to move a charged particle through a magnetic field.

sdFW

.

2

1

21 . sdFW