Properties of Materials & Equation of State

Mathematical Description of Material Behaviour…..

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Second Level Thermodynamic Analysis

•If a solid is heated, strains and stresses develop.•Conversely if body is strained rapidly, then heat is generated inside the body.•The changes undergone by this system can be characterized by some functions.•What kind functions can be used for correct recognition of these changes.

The Functional Relation for Description of A System

• What kind of Functional Relation?

• Assume that variables p, V, T are functionally related.

• Say F(p, V, T) = Constant.

• Assume that each variable can be explicitly “solved” from this functional relation in terms of two other variables, which are allowed to vary freely.

• p to obtain an expression of the form p = g(V, T), where V and T are chosen as free variables.

• Any function of p, V, T can be expressed as a function of any pair of free variables of your choice.

• F(p, V, T) = F(g(V, T), V, T) is expressed as a function of a pair of free variables V and T.

Functions of Several Variables

• Develop a function using these variables : F(x,y,z,…)

• If F(x,y,z,……) = Constant.

• This is called as Pfaffian function.

• Pfaffian function is denoted as F(.) and called as Point Function.

A total change in point function is expressed as:

0,,,

dzz

Fdy

y

Fdx

x

FdF

yxxzzy

Define functions M,N & P such that:

dzzyxPdyzyxNdxzyxMdF ),,(),,(),,(

As F(x,y,z) is a point function, differentiation is independent of order.

zyzxx

N

y

M

,,

This is called as pfaffian differential equation.

Properties of A Point Function

zxyx y

P

z

N

,,

yxzy z

P

x

P

,,

vp p

K

v

I

The necessary and sufficient condition for g to be accepted as a property is.

Creation of New Property Variable in Thermodynamics

dvvpKdpvpId ),(),(

If we develop an equation for change in a new characteristic of any thermodynamic system as

The variable can provide a functional relationship

0),,(),,(),,( dvpLdvvpKdpvpId

vpgvgvvpgvgp g

I

p

L

p

L

g

K

p

K

v

I

,,,,,,

;;

(p,v, )= Constant

Definition of A Thermodynamic Property

• Any Macroscopic variable, which can be written as a point function can be used as a thermodynamic property

• Thermodynamic properties are so related that F(.) is constant.

• Every substance is represented as F(.) in Mathematical (Caratheodory) Thermodynamics.

• This is shows a surface connectivity of Property of a substance.

p-v-T Diagram of solid Phase

Top View : V – T Diagram

Front View : p – V Diagram

Volume

Temperature

Pressure

Mathematical description of A Substance

Identification of Phase of A Substance:

β ≈ 10−3/K for liquids β ≈ 10−5/K for solids

General Behaviour of Solids

• Incompressible Substance.• Change in volume is infinitesimally small.• Huge increase in temperature or pressure required for a

finite change in volume/area/length.• In an ideal (Hookean) solid, finite increase in pressure

(stress) produces constant deformation (strain) at constant temperature.

• Thermal Expansion of Solids• As the thermal energy in a solid increases, the mean

separation of the atoms increases because the force curve is anharmonic.

• This causes the solid to expand.• Linear, superficial and cubical Expansion coefficient.

EOS for Solids

• The volume of A solid:

V = f (p,T) & p = g (V,T)

dTT

fdp

p

fdV

pT

Bulk modulus :

TV

gVB

Coefficient of volume or cubical expansion.

V

Tf

p

dTT

gdV

V

gdp

pT

Universal Equation of State for Solids

where

and

V0 is the volume of solid and B0 is bulk modulus at reference pressure .

RTTBXXX

BXTp 0002

0 1exp12

,

3

1

00

TV

VX

12

3

0

0 p

B

Constants of EoS

Parameter Gold Nacl Xenon

B0 (1010 Pa) 16.6 2.35 0.302

0(10-5 K-1) 4.25 12.0 60.0

(B/p)0 5.5 – 6.5 5.35 7.8

TR, K 300 298 60

A common equation of state for Solid

TpCpCTCTCCVm 542

321Vm = molar volume T = temperature p = pressure C1, C2, C3, C4, C5 = empirical constants

The empirical constants are all positive and specific to each substance.

For constant pressure processes, this equation is often shortened to

20 1 TBTAVV mm

Vmo = molar volume at 00CA, B = empirical constants

p-V-T Diagram of crystalline solid Phase

Volume

Pressure

Temperature