1. Lecture 11 Critical point and real gases

2. Phase diagrams of water and CO2
H2O
CO2
Phase transition from solid to gas is called sublimation
If T >Tcritlical and p > pcritical fluid is called a supercritical fluid

3. Specific volumes of saturated water (v’) and steam/vapor (v’’)
TC = oC and
pC = bar
At the critical point, molar/specific volumes (or densities) of the
saturated liquid and vapor are the same.

4. Isotherms of water in p,v-diagram
Liduid
Gas

5. Critical values for some gases
pC. bar
tC. oC
vC. dm3/kg He
2.29
-267.9
14.49 H2
12.96
-239.9
32.26 O2
50.35
-118.8
2.326 N2
33.98
-146.9
3.215
Air
37.60
-140.6
2.830
H2O
221.1
374.2
3.18
CO2
73.6
31.0
2.156
NH3 (ammoniac)
112.95
132.4
4.255
CH4 (methane)
46.30
-82.5
6.173

6. A supercritical power plant
A non-supercitical power plant
pfeed water < pC
A supercitical power plant
pfeed water > pC
To the turbine
To the turbine
Superheated steam
190bar. 540oC
Supercritical steam
240bar. 540oC
Liquid water
200bar. 202oC
Liquid water
250bar. 202oC
From the
feed water tank
From the
feed water tank
Suprcritical boilers are once-through boilers with no
steam drums, for example a Benson boiler

7. Specific heat capacities at the critical point
t. oC
p. bar
cp’. specific heat of
saturated water. kJ/kgK
cp’’. specific heat of
saturated steam/vapor. kJ/kgK 10
4.193
1.860 50
4.181
1.899
100
1.013
4.216
2.028
150
4.760
4.310
2.314
200
15.549
4.497
2.843
250
39.776
4.867
3.772
300
85.927
5.762
5.863
330
128.63
7.219
9.361
350
165.35
10.11
17.15
360
186.75
14.58
25.12
370
210.54
43.17
76.92
374.15
221.20 

8. Ideal gas versus real gases
Molecules are point masses.
There is no interaction between molecules.
The model is usually valid when densities are low.
Real gas
Molecules take a finite volume => the volume cannot be smaller than the total
molecular volume of the gas.
Molecules interact with each other.
Gases do not obey the ideal gas equation of state when gas temperatures and
pressures begin to ”approach” the critical point.

9. Some equations of state for real gases
wan der Waals equation
a and b are constants which have different values for a given gas.
Redlich-Kwong equation
a and b are constants which have different values for a given gas.
Other real gas equations
- Beattie Bridgeman (is used for water)
- The virial equation of state

10. Calculating the constants a and b
wan der Waals equation
Redlich-Kwong equation
TC critical temperature and pC critical pressure

11. Calculating the enthalpy change using van der Waals equation
Total differential of the internal energy
=>
=>
=> the enthalpy becomes h = u +pvm [J/mol]
cv is the specific heat capacity at constant volume [J/molK]
The dependence of the enthalpy on the pressure

12. Example 1: van der Waals vs. Ideal gas
Evaluate the accuracy of the van der Waals and ideal gas equations for water
vapor at 300oC.
TC = 374.2oC pC = 221.1bar
wan der Waals equation
Ideal gas equation

13. Example 1: van der Waals vs. Ideal gas results
bar
vm van der Waals
m3/mol
vm. ideal
vm real. m3/mol
0.40
1.33
2.65
5.28
10.52
14.42
19.59
24.72
29.81
34.86
38.63
43.62
47.34
52.27
55.94
59.59
64.43
69.23
72.81
75.19

14. Principle of the corresponding state and the compression factor
The compression factor Z is defined for all gases as follows:
where
Z(pr . Tr) is a universal function that can be used for all gases.
The accuracy of Z should be Z  6%

15. Nelson-Obert Generalized Compressibility Chart for Gases
If Z  1 => gas is an ideal gas
If Z differs from 1 => gas is a real gas

16. Example 2: Gas density
Calculate the density of methane at 6bar and 17oC using the generalized compressibility
chart and the ideal gas equation.
For methane TC = 191.1K and pC = 46.4bar
From the Generalized Compressibility Chart Z  0.99
=>
The ideal gas equation

17. Example 2: Gas density

18. Example 2: Gas density
At 100bar and -23oC. the corresponding densities for CH4 would be
The Generalized Compressibility Chart
=>
The ideal gas equation

19. Example 3: Throttling valve
What is the temperture of N2 after throttling?
Throttling valve
N bar. -10oC
N2 . 70bar. t2?
The energy balance over the valve
w1  w2 => h1 = h2

20. Example 3: Throttling valve
The change of the enthalpy
where
and dh = =>
where
 is the volumetric expansion coefficient

21. Example 3: Throttling valve
For N2 PC = 33.6 bar and TC = K
=> Z1  and
Note: dfg = f’g +fg’

22. Example 3: Throttling valve
The term Z(T.p)/ Z(T) may be approximated from the Nelson-Obert chart as follows
=>
After integration

23. Example 3: Throttling valve
Tr1 Z
Tr1’’
Z1 and Tr1 represent the values before the valve

24. Example 3: Throttling valve
Numerical values
on the basis of the compressibility chart
cpm (70bar K)  33.7J/molK
=>
=> t2 = -18.5oC . for an ideal gas T = 0oC

25. Temperature of the superheated steam after throttling
Intial state 10 bar, 300oC
Final state 4 bar, ?
The temperature after throttling
can be read from the h,s diagram
t  290oC

26. Main course content Enthalpy Entropy Chemical equilibrium
Electrochemical systems
Thermodynamics of solutions
Thermodynamics of humid air
Real gases