1. VOLUMETRIC PROPERTIES OF PURE FLUIDS
CHAPTER 3
VOLUMETRIC PROPERTIES OF PURE FLUIDS

2. Objectives Introduce the concept of a pure substance.
Discuss the physics of phase-change processes.
Illustrate the P-v, T-v, and P-T property diagrams and P-v-T surfaces of pure substances.
Demonstrate the procedures for determining thermodynamic properties of pure substances from tables of property data.
Describe the hypothetical substance “ideal gas” and the ideal-gas equation of state.
Apply the ideal-gas equation of state in the solution of typical problems.
Introduce the compressibility factor, which accounts for the deviation of real gases from ideal-gas behavior.
Present some of the best-known equations of state.

3. Pure Fluids/ Substance
A substance that has a fixed chemical composition throughout is called a Pure Substance.
Pure Substance:
- N2, O2, gaseous Air
-A mixture of liquid and gaseous water is a pure substance, but a mixture of liquid and gaseous air is not.

4. PHASES OF A PURE SUBSTANCE
The molecules in a SOLID are kept at their positions by the large springlike inter-molecular forces.
In a solid, the attractive and repulsive forces between the molecules tend to maintain them at relatively constant distances from each other.
The arrangement of atoms in different phases: (a) molecules are at relatively fixed positions in a solid, (b) groups of molecules move about each other in the LIQUID phase, and (c) molecules move about at random in the GAS phase.

5. Phase-Change Processes of Pure Substance
Compressed liquid or a subcooled liquid: A liquid that is not about to vaporize.
Saturated liquid: A liquid that is about to vaporize.
Saturated vapor: A vapor that is about to condense.
Saturated liquid-vapor mixture: the liquid and vapor phases coexist in equilibrium.
Superheated vapor: A vapor that is not about to condense

6. T-v diagram for the heating process of water at constant pressure.
2-1

7. Phase-Change Processes of Pure Substance
Saturated temperature, Tsat: At a given pressure, the temperature at which a pure substance changes phase.
Saturated pressure, Psat: At a given temperature, the pressure at which a pure substance changes phase.
Latent heat: the amount of energy absorbed or released during a phase-change process.
Latent heat of fusion: the amount of energy absorbed during melting.
Latent heat of vaporization: the amount of energy absorbed during vaporization.

8. Phase changes process
Macromedia 3.3

9. PVT BEHAVIOR OF PURE SUBSTANCES/ FLUIDS
Critical point – highest combination of pressure and temperature where the fluid exist in liq-vap equilibrium
The 2-C line, also known as vaporization curve is where liquid-vapor is in equilibrium
Triple point, three phases exist in equilibrium (F=0)
The 2-3 line, also known as fusion curve is where solid-liquid is in equilibrium
The 1-2 line, also known as sublimation curve is where solid-vapor is in equilibrium

10. Critical point becomes peak of the curve
PV diagram
Boundaries in PT diagram becomes region when illustrate with PV diagram
Critical point becomes peak of the curve
Triple point becomes horizontal line
Macromedia 3.4

11. Compressed liquid region
Saturated liquid line at boiling temperature
Superheated vapor region
Saturated vapor line at condensation temperature
T >Tc, the line do not cross the boundary
Isotherms in sub-cooled/ compressed lliquid region are steep, because liquid volumes change little with large changes in pressure

12. SINGLE PHASE REGION
At single phase regions in PV diagram, there is a relation connecting P,V and T.
This relation known as PVT equation of state; f(P,V,T)=0
If V is considered as a function of T and P, then V=V(T,P)

13. The combination will yield;
Because the isotherms on the left side of PV diagram are very steep, both β and κ are small.
Because of that, the liquid is known as incompressible fluid, where both constants are equal to zero
However, this is just idealization, and in incompressible fluid, no equation of state exist, since V is independent of T and P
If, we still want to calculate, for liquids, β is positive, and κ is positive as well. Integration of 3.4 yield
Check out Ex 3.1

14. Virial Equation of State
The coefficients a(T), b(T), c(T), and so on, that are functions of temperature alone are called virial coefficients.
Ideal-Gas Temperature; Universal Gas Constant

15. Figure 3.4: Plot of PV vs. P for 4 gaseous at triple-point temperature of water. The limiting value of PV as P0 is the same for all of the gaseous.
PV* = a = f(T)
Limiting value (asterisk)
This properties of gaseous is the basis for establishing an absolute temperature scale. The simplest procedure to define Kelvin scale:
(PV)* = a ≡ RT
(PV)*t = R x K 3.
(3.8)
(3.9)

16. In the limit, P0, molecules separated by finite distance
Volumes becomes negligible compare with the total volume
of the gas, and intermolecular forces approach zero
These condition define an IDEAL GAS state & Eq. 3.9 establishes
the ideal-gas temperature scale
The proportionality constant R = universal gas constant
Through the use of conversion factors, R may be expresses in various units.
Commonly used values are given as above table.

17. 2 Forms of the Virial Equation
Auxiliary thermodynamic property =
= Compressibility factor
Z = 1 + B’P+C’P2 +D’P3 +…
Z= 1 + B/V +C/V2 + D/V3 +…
Virial expansion = Eq 3.11 & Eq 3.12
B’, C’, D’ …, B, C, D = virial coefficient
B’ = 2nd virial coefficient
C’ = 3rd virial coefficient…etc
(3.10)
(3.11)
(3.12)

18. Eq 3.11 & Eq 3.12

19. THE IDEAL GAS
Because the Eq 3.12 arise on account of molecular interactions, the virial coefficients B,C…etc = 0 were no such interaction to exist
Phase Rule = Internal energy of a real gas is a function of pressure as well as of temperature.
Macromedia 3.6

20. THE IDEAL GAS
Equation of state: Any equation that relates the pressure, temperature, and specific volume of a substance.
The simplest and best-known equation of state for substances in the gas phase is the ideal-gas equation of state. This equation predicts the P-v-T behavior of a gas quite accurately within some properly selected region.
R: gas constant
M: molar mass (kg/kmol)
Ru: universal gas constant
Different substances have different gas constants.
Ideal gas equation of state
U = U(T) (Ideal gas) (3.15)

21. Implied Property Relations for an Ideal Gas (f(T) only)
Heat Capacity for constant volume, Cv
Eq 2.11 applied to an Ideal Gas
(3.1)
(3.17)

22. Implied Property Relations for an Ideal Gas (f(T) only)
Heat Capacity for constant pressure, Cp
Useful relation between Cv & Cp
(3.18)
(3.19)
NOTE: This equation does not imply that Cp and Cv are themselves
constant for an ideal gas, but only that they vary with temperature in such a way
that their differences is equal to R

23. For any ∆ of state of an ideal gas, Eq 3.16 and Eq.3.18 lead to:
T1 & T2
a---b = Constant volume process
a---c & a---d ≠ constant volume
The graph show that the ∆U=

24. Equation for Process Calculation for Ideal Gas
Working equation of dQ and dW depend on which pair of these variables
is selected as independent
With P=RT/V,
With V=RT/P and Cv given by Eq 3.19, dQ & dW written as Eq.3.24 & Eq.3.25
With T=PV/R, the work is simply dW=-PdV, and with Cv given by Eq.3.19,

25. Isothermal Process
Q=-W
Isobaric Process
Isochoric Process

26. Adiabatic Process; Constant Heat Capacities
Also,

27. (3.31)
IMPORTANT: Eq 3.30 are restricted in application to ideal gases with
Constant Heat capacities undergoing mechanically reversible adiabatic
expansion or compression
For ideal gas, the WORK of any adiabatic closed-system process is given by:
(3.32)
Because RT1 = P1V1 and RT2 = P2V2,
(3.33)

28. Elimination of V2 from Eq 3.33 by Eq 3.30c, valid for
mechanically reversible process, lead to:
(3.34)
Polytropic Process

29. Polytropic Process Isobaric process Isothermal process
Adiabatic process
Isochoric process

30. APPLICATION OF VIRIAL EQ.
All isotherms originate at Z=1 for P=0

31. (3.38)
Applied to vapors at sub critical temperature up to their saturation pressure
When the virial equation is truncated to 3 terms, the appropriate form is:
(3.40)

32. CUBIC EQ. OF STATE 1. The Van der Waals Equation of state (3.41)
2. A Generic Cubic Equation of State
(3.42)
3. Determination of Eq-of State Parameters

33. 4. Theorem of Corresponding State: Acentric Factor
(3.48)
NOTE: All fluid having the same value of , when compared at the same Tr
And Pr have about the same value of Z, and all deviate from ideal gas behavior
To about he same degree
5. Vapor& Vapor-like Root of the Generic Cubic Eq of state
6. Liquid & Liquid-like Root of the Generic Cubic Eq of state

34. GENERALIZE CORRELATION FOR GASES

35. GENERALIZE CORRELATION FOR GASES
Pitzer Correlation for the Compressibility Factor
Pitzer Correlation for the 2nd Virial Coefficient
Correlations for the 3rd Virial Coefficient
Condition of Approximate Vlidity of the Ideal-Gas Equation

36. GENERALIZE CORRELATION FOR LIQUID

37. Summary Pure substance Phases of a pure substance
Phase-change processes of pure substances
Compressed liquid, Saturated liquid, Saturated vapor, Superheated vapor
Saturation temperature and Saturation pressure
Property diagrams for phase change processes
The T-v diagram, The P-v diagram, The P-T diagram, The P-v-T surface
The ideal gas equation of state
Is water vapor an ideal gas?
Compressibility factor
Other equations of state

38. Thank you
Prepared by,
NMJ