Department of Mechanical EngineeringME 322 – Mechanical Engineering

Thermodynamics

Lecture 6

Thermodynamic DiagramsPhase ChangeDetermination of Properties with Equations of State (EOS)

Thermodynamic Diagrams

The P-v Diagram The P-T Diagram

Phase Diagram

The P-v-T Surface To view the T-v diagram look at the top view of

the P-v-T surface

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Thermodynamic Diagrams

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The P-v DiagramThe P-T (Phase) Diagram

Projections of the P-v-T 3D surface

The State PostulateTwo independent, intensive properties fix the thermodynamic state of a simple substance

Typical Phase Diagram

This state is fixed by T and P which are independent in the single phase. Once the state is identified, all thermodynamic

properties of the state are known.

,T P

1/

,p v

vuhs

c c

Defines

4

Phase Change

5

A thought experiment ... boiling water

T

v

14.696 psiaatmP P

atmP P

atmP P

atmP P

72 FT

212 FT

T-v Diagram

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T

v

critical point

L

cT

cP

V

GSC

Sat

urat

ed L

iqui

d

Saturated Vapor

L + V

P-v Diagram

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P

v

critical point

L

cT

cP

VG

SC

Sat

urat

ed L

iqui

d

Saturated Vapor

L + V

Thermodynamic Nomenclature

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Phase Thermodynamic NameLiquid Compressed liquid

Vapor Superheated vapor

Liquid + Vapor Wet or Saturated Mixture

Gas Gas

Supercritical Supercritical

Saturation Property NomenclatureSaturated liquid properties are signified with a subscript fSaturated vapor properties are signified with a subscript gSaturated solid properties are signified with a subscript i

saturated liquid specific volume

saturated vapor specific enthalpyf

g

v

h

For example ...

Quality – A New Property

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p

vvfv gv

p,T are NOT independent in the wet region. However, p,v and T,v are independent.

But, what if you are trying to find v? To do this, I need another independent, intensive property to fix the state in the wet region. The property we need is the quality, x, of the mixture.

Quality – A New PropertyConsider the total volume of a wet mixture …

f gf g

m mv v v

m m

gV

fV

f gV V V

f gV VVvm m m

f f g gm v m vv

m m

Define ... gmx

m quality

Therefore ... 1fmx

m

Substituting ... 1 f gv x v xv

Significance: p,x and T,x are independent, intensive properties in the wet region!

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Quality – A New Property

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1

1

1

1

f g f fg

f g f fg

f g f fg

f g f fg

v x v xv v xv

u x u xu u xu

h x h xh h xh

s x s xs s xs

Quality expressions are valid for other internal energy, enthalpy, and entropy too!

A little algebra ...

1

f g f f g f g f f fg

fg g f

v x v xv v xv xv v x v v v xv

v v v

specific volume of vaporization

Department of Mechanical EngineeringME 322 – Mechanical Engineering

Thermodynamics

Property Models

The Incompressible Substance ModelThe Ideal Gas ModelThe Real Fluid Model

Department of Mechanical EngineeringME 322 – Mechanical Engineering

Thermodynamics

The Incompressible Substance Model

v

u u T

constant Liquids and

Solids

The Incompressible Substance Model

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As seen in the reading (Section 3.9.1),

vdu c dT

If we know the value of c (or the variation of c with T), the above equation, du = cdT can be integrated between any two states to determine the change in internal energy. Once the change in internal energy is known, the change in enthalpy can also be found!

It can also be shown (see Section 3.9.1) that,

p vp

h duc c cT dT

(specific heat)

Department of Mechanical EngineeringME 322 – Mechanical Engineering

Thermodynamics

The Ideal Gas Model

pv RTu u T

GasesT >> Tc

and P << Pc

The Ideal Gas Model

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As seen in the reading (Section 3.9.2),

vdu c dT

Since we are dealing with an ideal gas, pv = RT. Therefore,

h u pv h u RT h h T

This leads to the following conclusion (section 3.9.2),

pdh c dT

It can also be shown (section 3.9.2) that,

p vc c R

The Ideal Gas Model

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v p p vdu c dT dh c dT c c R

Significance:

These equations allow us to determine internal energy and enthalpy changes for ideal gases. In order to integrate the du and dh equations, we need to determine one of the specific heats (the other can be found with the third equation).

With ideal gases, the specific heat dependence on temperature may be stronger compared to incompressible substances. Exceptions: The heat capacities are constant for the inert gases. For the inert gases, cp = 5R/2

Department of Mechanical EngineeringME 322 – Mechanical Engineering

Thermodynamics

The Real Fluid Model

,p p v TTrue for the

complete P-v-T surface

The Real Fluid Model• Theoretical extension of the Ideal Gas EOS

– Clausius, van der Waals, Beattie-Bridgeman, Redlich-Kwong (Section 3.9.4)

• Theory cannot fully predict correct fluid behavior– Example: The van der Waals EOS is not valid

in the liquid phase!• Modern EOSs include theoretically

significant terms, but also have empirical terms to make up the deficiency in theory

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The Modified Benedict-Webb-Rubin EOS

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2 2

5 8 941 2 3 6 72 2 3 2

13 15 1612 1410 114 5 6 2 7

/ /17 18 19 20 221 22

8 2 9 2 3 2 3 5 2

1 1

1 1 1

1 1 v v

N N NNpv RT N T N T N N T Nv T T v T T

N N NN NN T Nv T v v T T v T

N N N N NN Ne ev T T v T v T T v T

2 2 2

2

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/ / /25 26 27 28 2924

7 2 3 9 2 4 11 2 3

/30 31 32

13 2 3 4

v v v

v

T

N N N N NNe e ev T T v T T v T T

N N Nev T T T

A high-accuracy EOS