1. Chapter Three: Part Two

4. Superheated Vapor
In the superheated water Table A-6, T and P are the independent properties. The value of temperature to the right of the pressure is the saturation temperature for that pressure. 4

7. Compressed Liquid 7

8. Approximation of Compressed Liquid as
Saturated Liquid 8

10. How to Choose the Right Table
The correct table to use to find the thermodynamic properties of a real substance can always be determined by:
comparing the known state properties to the properties in the saturation region.
Given the temperature or pressure and one other property from the group v, u, h, and s, the following procedure is used. If
We have subcooled liquid, use Table A-7 (use A-7E for English system of units) If
We have saturated liquid, use Table A-4 or A-5 (use A-4E or A-5E for English system)
We have saturated vapor-liquid mixture, use Table A-4 or A-5
(use A-4E or A-5E for English system)
We have saturated vapor, use Table A-4 or A-5 (use A-4E or A-5E for English system)
We have superheated vapor, use Table A-6 (use A-6E for English system)
Where y is v, u, h, or s 10

12. 120.23
Sat. vapor-liquid mixture

13. 120.23
Sat. vapor-liquid mixture
232.1
0.535

14. 120.23
Sat. vapor-liquid mixture
395.6
232.1
0.535
Superheated vapor
---

15. 120.23
Sat. vapor-liquid mixture
395.6
232.1
0.535
Superheated vapor
---
313.90
Subcooled liquid
120.23
Sat. vapor-liquid mixture
395.6
232.1
0.535
Superheated vapor
---
313.90
Subcooled liquid
Sat. liquid
731.27
172.96

16. Equations of State
The relationship among the state variables, temperature, pressure, and specific volume is called the equation of state. We now consider the equation of state for the vapor or gaseous phase of simple compressible substances.
Boyle's Law
At constant temperature, the volume of a given quantity of gas is inversely proportional to its pressure : V α 1/P
Charles' Law
At constant pressure, the volume of a given quantity of gas is directly proportional to the absolute temperature : V α T (in Kelvin)
Ideal Gas
where R is the constant of proportionality and is called the gas constant and takes on a different value for each gas. If a gas obeys this relation, it is called an ideal gas. We often write this equation as 16

17. The ideal gas equation of state may be written several ways.
The gas constant for ideal gases is related to the universal gas constant valid for all substances through the molar mass (or molecular weight). Let Ru be the universal gas constant. Then,
The mass, m, is related to the moles, N, of substance through the molecular weight or molar mass, M, see Table A-1. The molar mass is the ratio of mass to moles and has the same value regardless of the system of units.
Since 1 kmol = 1000 gmol or 1000 gram-mole and 1 kg = 1000 g, 1 kmol of air has a mass of kg or 28,970 grams.
The ideal gas equation of state may be written several ways. 17

18. P = absolute pressure in MPa, or kPa
Here
P = absolute pressure in MPa, or kPa
= molar specific volume in m3/kmol
T = absolute temperature in K
Ru = kJ/(kmolK)
Some values of the universal gas constant are
Universal Gas Constant, Ru
8.314 kJ/(kmolK)
8.314 kPam3/(kmolK)
1.986 Btu/(lbmolR)
1545 ftlbf/(lbmolR)
10.73 psiaft3/(lbmolR) 18

19. 1. Intermolecular forces are small.
The ideal gas equation of state can be derived from basic principles if one assumes
1. Intermolecular forces are small.
2. Volume occupied by the particles is small.
Example
Determine the particular gas constant for air and hydrogen. 19

20. 20

21. Compressibility Factor (Z)
The ideal gas equation of state is used when:
the pressure is small compared to the critical pressure or
the temperature is at least twice the critical temperature and the pressure is less than 10 times the critical pressure.
The critical point is the state where there is an instantaneous
change from the liquid phase to the vapor phase for a
substance. Critical point data are given in Table A-1.
To understand the above criteria and to determine how much
the ideal gas equation of state deviates from the actual gas
behavior, we introduce the compressibility factor Z as
follows. or 21

22. Figure 3-51 gives a comparison of Z factors for various gases.
For an ideal gas Z = 1, and the deviation of Z from unity measures the deviation of a gas from ideality.
The compressibility factor is expressed as a function of the reduced pressure (PR) and the reduced temperature (TR).
The Z factor is approximately the same for all gases at the same reduced temperature and reduced pressure, which are defined as
where Pcr and Tcr are the critical pressure and temperature, respectively.
The critical constant data for various substances are given in Table A-1 (Table A-1E for English System of Units).
Figure 3-51 gives a comparison of Z factors for various gases. 22

23. Figure 3-51
When either P or T is unknown, Z can be determined from the compressibility chart (Figure A-34a and A-34b) with the help of the pseudo-reduced specific volume, defined as: 23

24. Behavior of Gases
At very low pressures (PR <<1), the gases behave as an ideal gas
regardless of temperature.
At high temperatures (TR > 2), ideal-gas behavior can be assumed
with good accuracy regardless of pressure (except when PR >>1).
The deviation of a gas from ideal-gas behavior is greatest in the vicinity
of the critical point. 24

25. Examples
Example 1: Given that Vr =0.9 and Tr=1.05, Calculate P for a gas that has critical pressure of 1.2 MPa using the generalized compressibility chart.
Example 2: Given that Vr =1.2 and Pr=0.7, Calculate T for a gas that has critical
temperature of 300 K using the generalized compressibility chart. 25

26. Van der Waals Equation of State
Accounts for intermolecular forces
Volume occupied by molecules
where 26

27. Van der Waals: Example
A 3.27-m3 tank contains 100 kg of nitrogen at 225 K.
Determine the pressure in the tank using the van der Waals equation.
Properties: The gas constant, molar mass, critical pressure, and critical
temperature of nitrogen are (Table A-1)
R = kPa·m3/kg·K, M = kg/kmol,
Tcr = K, Pcr = 3.39 Mpa

28. Van der Waals
Then,

29. 29

30. The specific heat ratio k is defined as:
The change in internal energy (du) and enthaply (dh) from state one to state two can be calculated using the following expressions:
du= CvdT
and
dh= CpdT
The specific heat ratio k is defined as: 30

31. Some relations for ideal gas
Show that
and 31

32. Example
Calculate the change in specific internal energy (Δu) and the change in specific enthalpy (Δh) of nitrogen gas when temperature increases from 100 to 300 °C. Assume that Cv and Cp do not vary with temperature.

33. End of Chapter Three. 33