1. LECTURES IN THERMODYNAMICS Claus Borgnakke CHAPTER 2
For the 8th Edition of:
Fundamentals of Thermodynamics
Claus Borgnakke, Richard Sonntag
John Wiley & Sons, 2013

2. Chapter 2 Phase Boundaries and The P-v-T Surface The Two Phase States
The Liquid and Solid States
The Superheated Vapor States
The Ideal gas States
Compressibility Factor
Equations Of State (EOS)
Computerized Tables

3. Phase Boundaries and The P-v-T Surface
Measure P-v-T and observe phase for water.
Water at 1 atm (101 kPa), room temperature is liquid.
Heat the water, state moves from (a) towards (b) .
At (b) it is boiling with co-existing liquid and vapor.
Continued heating: more vapor, less liquid until no more liquid.
Further heating gives warm vapor at (c).
If water is cooled it will change state from (a) towards (e) and it
will start to freeze at (d).
Further cooling gives solid water (ice) with no liquid at (e).
If process is repeated at different pressures we can indicate the
phases in a P-T diagram, where the points for the phase changes
are connected with red curves.
On these lines points are pairs of saturated (P, T) a b c d e

4. Phase Boundaries and The P-v-T Surface
Complete phase diagram (P, T) separates the solid (S), the liquid (L) and the vapor (V) phases. Sublimation line (S above, V below) Fusion line (S left, L right) Vaporization curve (L above, V below) On each curve: P = Psat(T) or T = Tsat(P) One point on each curve is a pair (Psat, Tsat) State a: Superheated vapor (T > Tsat for same P) Expanded vapor (P < Psat for same T) State b: Compressed liquid (P > Psat for same T) Subcooled liquid (T < Tsat for same P)
Vaporization curve

5. Phase Boundaries and The P-v-T Surface
The phase diagram for water Notice the P axis must be log(P) to fit the large range.
The phase diagram for CO2

6. Phase Boundaries Concept Questions

7. The P-v-T surfaces
Show the (T, v) combinations following constant P processes for water
Small v’s are liquid
Large v’s are vapor (gas)
Liquid and vapor regions
are separated by a 2-phase
region of mixtures with a
combination of liquid and
vapor.
The borders are the
Saturated liquid line and
Saturated vapor line
They meet at critical point
vapor
liquid

8. The P-v-T surface for water
The total P-v-T surface shows all the different regions
Small v’s are liquid or solid
Large v’s are vapor (gas)
There are 3 different 2-phase
regions:
Solid-Vapor
Solid-Liquid
Liquid-Vapor
Look at this diagram from the right
along the volume axis. You see
the P-T phase diagram.
Look from the top down, you see
the T-v diagram.
Look from the left along the T axis
you see a P-v diagram partly
obscured by the solid surface.
We will use these 2-D figures/projections (for simplicity)

9. The P-v-T surface for most substances other than water
Water has a higher v for solid than liquid
so the solid is lighter than the liquid
Most other substances behaves opposite
Notice also that the fusion line separating
the solid and liquid region slopes positive
Recall the phase diagram for CO2
All the two-phase surfaces are plane in
the v direction. When you look from the
left along the v axis the planes show up
as curves:
Sublimation, Fusion and Vaporization lines

10. The B Section Tables for Water
Saturated two-phase liquid + vapor region
Table B.1.1 Gives listing of Tentry Psat, vf, vfg, vg, ….
Table B.1.2 same but with Pentry Tsat, …
vf vfg vg Psat as function of T

11. The Tables for Water
Table B.1.3 superheated vapor along a constant P curve: Tentry, v, …
Table B.1.4 compressed liquid
Subheading P (Tsat)
TABLE B.1.3
Superheated
Vapor v u h s

12. The Tables for Water
The saturated solid-vapor table and the sublimation line
Table B.1.5 Gives listing of Tentry Psat, vi, vig, vg, ….

13. The Tables for Water

14. The Two Phase States

15. The Tables for Water

16. The Tables for Water
Table B.5.1
T Psat vf vfg vg uf ufg ug

17. The Tables for Water
Table F.10.1
T Psat vf vfg vg uf ufg ug

18. The Liquid and Solid States
Main characteristic: Incompressible, weak function of T, meaning
v ≈ constant = 1/ρ = v(T)
Solid: v = v(T) ≈ vi Liquid: v = v(T) ≈ vf
If P > Psat compressed solid If P > Psat compressed liquid
Figures illustrate properties for water

19. The Superheated Vapor States
You have a superheated vapor when v is larger
than the saturated vapor value v > vg
The region above the critical point is called
dense fluid.

20. The Superheated Vapor States

21. The Superheated Vapor States

22. The Superheated Vapor States
Example 2.4
The red circles indicates
the two states used for the
interpolation

23. The Superheated Vapor States
Example 2.4E
The red circles indicates
the two states used for the
interpolation

24. The Superheated Vapor States
Example 2.5

25. The Superheated Vapor States
Example 2.5 continued
From B.1.3: Locate state 200 C, v = 0.4 m3/kg
500 kPa, 200 C: v = m3/kg; 600 kPa, 200 C: v = m3/kg

26. The P-v-T Surface Concept Questions

27. The Superheated Vapor States and The Ideal Gas States
Constant v, T and P curves are shown below.
When extended far out in the superheated vapor region they become simple (math) curves.
P = BT for constant v; Pv = C for constant T; T = Av for constant P
B = Bo / v C = Co T A = Ao P
Unified equation: Pv = RT Ideal gas law, R ideal gas constant
R = R /M R : universal gas constant
M: molecular mass L V
L + V S

28. Ideal Gas Law P v = R T P V = m R T = n R T
A different scaling can be used
P v = R T
P V = m R T = n R T
Here n is the number of moles and M is the molecular mass.
n = m / M
From ideal gas law: 1 mol occupies same V at a given (P, T) regardless of which substance it is.
Compare a mass at state 1 with same mass at a state 2:
m R = P2V2 T2 = P1V1 T1
Differentiate the ideal gas law with time for constant state (P, T ) to get:

29. Ideal Gas Law
Example 2.6

30. Ideal Gas Law
Example 2.7 Since substance not given R is calculated from universal gas constant, recall 1 kN m = 1 kJ.

31. Ideal Gas Law
Example 2.7E Since substance not given R is calculated from universal gas constant.

32. Ideal Gas Law
Example 2.8 Differentiate the ideal gas law with time at constant ( P, T ) to get:

33. Ideal Gas Law
When do you have an ideal gas? For very large v, so molecular distances are large (very small force between molecules) To get a sense look at a T-v diagram for water. If T is higher you can allow smaller v’s and still have an ideal gas.

34. The Compressibility Factor
The deviation from the ideal gas law can be expressed by a factor Z so
P v = Z RT
For an ideal gas Z = 1 and for states approaching liquid Z becomes very small. For very high pressures Z can actually be higher than 1. A chart for nitrogen gives a general shape of a compressibility chart.

35. The Compressibility Factor
The compressibility chart shows Z plotted as an
average chart in reduced properties. The scaling
is done with the critical point properties:
Pr = P/Pc
Tr = T/Tc
The chart is plotted using an equation of state
called Lee Kesler.

36. The Compressibility Factor
Example 2.11

37. The Compressibility Factor
Example 2.11 continued

38. The Ideal Gas and Compressibility Concept Questions

39. Equations of State P = RT v – b – a v2 + cbv + db2
The P-v-T surface is approximated in an equation of state. Several of these are extensions of the ideal gas law as
P = RT v – b – a v2 + cbv + db2
Here a and b are scaled to the critical properties
a = ao Pc x2 ; b = bo x ; x = RTc/Pc
c and d are model constants shown in Tabel. D.1 including a correction with the accentric factor, ω

40. Computerized Tables CATT3
CATT3 program with water selected. Shows the T-s or the P-v (log-log) diagrams. L + V two-phase region purple, superheated vapor red and liquid, dense fluid blue, green/aqua State selected (100 oC, 0.1 MPa) shown by crosshair lines.