1. 공정 열역학 Chapter 3. Volumetric Properties of Pure Fluids – Part 2
고려대학교 화공생명공학과
강정원

2. Equations for Process Calculation for Ideal Gases
From the first law of thermodynamics
Combine ideal gas law
T, V
T, P
P,V

3. DO NOT memorize the equation ! Memorize the sequences of derivation !
Equations for Process Calculation for Ideal Gases - (1) Isothermal Process
T= const.
DO NOT memorize the equation !
Memorize the sequences of derivation !

4. Equations for Process Calculation for Ideal Gases - (2) Isobaric Process
P= const.

5. Equations for Process Calculation for Ideal Gases - (3) Isochoric Process
V= const.

6. Equations for Process Calculation for Ideal Gases - (4) Adiabatic Process
dQ=0
integration
integration
integration

7. Equations for Process Calculation for Ideal Gases - (4) Adiabatic Process
These equations are restricted
to const. heat capacities and
reversible, adiabatic processes

8. Equations for Process Calculation for Ideal Gases - (4) Adiabatic Process
Other expression for WORK calculation
Use P instead of T1 or T2

9. Equations for Process Calculation for Ideal Gases - (4) Adiabatic Process
Values of Cp/Cv
Monatomic gases : 1.67 (He, Ne, Kr,…)
Diatomic gases : 1.4 (H2, N2, O2, …)
Simple polyatomic gases : 1.3 (CO2, SO2, NH3, CH4,…)

10. Equations for Process Calculation for Ideal Gases - (5) Polytropic Process
Polytropic : “Turning many ways”
isobaric
isothermal
adiabatic
isochoric P
Can be used to represent a in-between processes V

11. constant heat capacity
Equations for Process Calculation for Ideal Gases - (5) Polytropic Process
Using ideal gas equation,
constant heat capacity

12. QUIZ – 20 min
Obtain the following expressions

13. Irreversible Processes
Equations in (1)-(5)
Only valid for mechanically reversible, closed system for ideal gases
Properties changes (dU, dH) are also same regardless of the process  State Properties
Heat and Work amount depends on the nature of process (reversible/irreversible, closed/open) Path function
For irreversible processes, the following procedures are commonly employed
W is determined
For reversible process
Efficiency (h)
is multiplied

14. 3.4 Application of the Virial Equations
Infinite Series
Only useful for engineering purpose when convergence is rapid
Two or three terms
Derivatives of compressibility
Truncation equation
Compressibility-factor graph for methane

15. Virial Equation for Engineering Purpose
Truncated two terms
Truncated three terms
Extended Virial equation
(Benedict-Webb-Rubin Equation)
similar accuracy
but convenient
more accurate
eight parameters

16. 3.5 Cubic Equations of State
PVT behavior of liquid & vapor over wide range
Equation must not be so complex
Polynomial equation  at least cubic equation
Cubic EOS are simplest eqn. P P V V
real behavior
cubic EOS representation

17. The van der Waals Equation of State
J. D. Van der Waals
First proposed in 1873
Won Novel Prize in 1910

18. The van der Waals Equation of State
T > Tc ; monotonically decreasing function
T = Tc ; Critical Region
Saturated liquid and vapor
T < Tc ; Liquid region
T < Tc ; Unrealistic Behavior
T < Tc ; Vapor region
T < Tc ; Real behavior : Metastable region

19. The van der Waals Equation of State
Molar volume cannot be smaller than b
At T=Tc and T>Tc, only single root exists
At T<Tc, three roots exist
Smaller : liquid-like volume
Middle : no significance
Larger : vapor-like volume b

20. Limitation of Van der Waals EOS
Inaccurate critical point prediction
Inaccurate vapor pressure prediction
a parameters are not optimized to fit vapor pressure
Only historical interest
For all fluids
Zc = 0.24 to 0.29 for real fluid
(mainly hydrocarbons)

21. Cubic Equation of State - History
Improvement over Van der Waals EOS
Redlich and Kwong (1949)
Slightly different volume dependence in attractive term
Improved critical compressibility (0.3333) and Second Virial Coefficient
Vapor pressure and liquid density are still inaccurate
Wison (1964)
First introduced T dependency a parameter
Acentric factor

22. Cubic Equation of State - History
Soave (1972)
More refined temperature dependency of a parameter
Improved vapor-pressure calculation
Also called Soave-Redlich-Kwong (SRK) EOS
Peng-Ronbinson (1976)

23. Cubic Equation of State - History
Equation of State with more parameters
Scmidt and Wenzel (1980)
Hamens and Knapp (1980)
Patel and Teja (1982)
Results can be improved …
But meaning of the third parameter is not clear…
It is not easy to write a meaningful mixing rule for the third parameter

24. Cubic Equation of State - History
a(T) law improvement
Stryjek-Vera (1986)
Soave (1984)
Carrier (1988)
Mathias and Copeman (1983)
Numerous functional forms have been proposed …

25. Cubic Equation of State - History
Review : J. O. Valderrama, Ind. Eng. Chem. Res., 42,1603 (2003)

26. A Generic Cubic Equation of State
Several hundred cubic equation of state have been proposed
Most EOS can be reduced to a generic form;

27. A Generic Cubic Equation of State

28. Determination of Equation-of-State Parameters
How to determine a and b parameters ?
An EOS should represent PVT behavior of pure fluid
Two conditions at critical point
At critical point ; T=Tc, P=Pc, V=Vc
93-94 page derivation (Van der Waals EOS)

29. Determination of Equation-of-State Parameters
For other EOS (RK, SRK, PR), similar equation can be obtained

30. Determination of Equation-of-State Parameters
For better representation of vapor pressure, a parameters are assumed to be temperature dependent

31. Theorem of Corresponding State: Acentric Factor
a(T) law can be fitted using vapor pressure data
Is there any method not using vapor pressure ?
 Corresponding state theorem  acentric factor (w )

32. Theorem of Corresponding State: Acentric Factor
Reduced Properties : How far from critical point ?
Theorem of Corresponding State (Simplest Form)
All fluids, when compared at the same reduced temperature and pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree.
Only valid for simple fluid (argon, krypton and xenon)
Systematic deviations are observed

33. Theorem of Corresponding State: Acentric Factor
Introduced by K. S. Pitzer and coworkers
Plot of log (Prsat) vs. (1/Tr)
(1/Tr)
S=-2.3 (for Ar, Kr, Xe)
log(1/Prsat)
Slope varies depending on the chemical species

34. Theorem of Corresponding State: Acentric Factor
Basis : w = 0 for Ar, Kr, Xe
This parameter represent the degree how far from simple gases
C1 :0.012/C2:0.100/C3:0.152/C4:0.200/…./C10:0.492
(1/0.7)
(1/Tr) -1
S=-2.3 (for Ar, Kr, Xe)
log(1/Prsat)
Slope varies depending on the chemical species

35. Theorem of Corresponding State: Acentric Factor
Three-Parameter Theorem of Corresponding State
All fluids having the same value of w, when compared at the same Tr and Pr, have about the same value of Z, and all deviate from ideal gas behavior to about the same degree.
Use of acentric factor in Cubic EOS

36. How to Solve Cubic EOS ? Solution Technique
Root formula for Cubic equation
Iterative calculation (반복계산)
Newton-Raphson iteration or Secant iteration
Successive Substitution
Pseudo-Root method

37. Reformulation
SRK EOS
PR EOS

38. Solution Method (1) Root formula for cubic EOS
Three Root case
Single Root Case

39. Solution Method (2) Newton-Raphson Iteration
Initialization
For gas phase : Ideal Gas Root
For liquid phase : Hard Sphere Volume(0.9b), Rackett equation , ….

40. Solution Method (3) Successive Substitution
Vapor or Vapor-Like Roots
Start with V=RT/P
Liquid and Liquid-Like Roots
Start with V=b

41. 3.6 Generalized Correlations for Gases
Pitzer correlations for the Compressibility factor
Z0 and Z1 are given as a generalized function of Tr and Pr
Lee-Kesler Method (1975)
Given by chart (Z0 and Z1)
Basis : Benedict/Webb/Rubin Equation
For quantum gases (hydrogen, helium, neon)
Small molecules
Do not conform corresponding behavior
Use effective critical parameters ( )
Errors
2-3 percent for nonpolar or slightly polar small molecules (mainly hydrocarbons)
Larger errors for polar/associating components, large molecules

42. 3.6 Generalized Correlations for Gases
Pitzer correlations for the Second Virial Coefficient
Not recommended
Not accurate for highly polar or associating molecules

43. 3.7 Generalized Correlation for Liquids
Cubic EOS
Not accurate for liquid phase molar volume (5-10 % error)
Lee-Kesler correlation
Accurate liquid volume but not accurate for polar species
Saturated molar volume
Rackett equation (1978)

44. Homework 문제풀이 (기한 1주) 프로그램 작성 프로그램 (기한 2주) 리포트 작성제출 마감일은 조교가 공지함.
3.3, 3.8, 3.10, 3.20, 3.27, 3.64, 3.68,3.76
프로그램 작성 프로그램 (기한 2주)
PR 또는 SRK 상태방정식을 이용하여 기체 및 액체의 해를 구하는 프로그램을 작성하라.
Successive Substitution / Newton Raphson / Root Formula 중 한가지를 사용한다.
FORTRAN / C / MATLAB 중 어느것을 이용해도 무방
예제 또는 연습문제에 나타나 있는 기체/액체 몰 부피 자료를 비교하여 에러를 계산하라.
에러를 계산 데이터를 Lee-Kesler Equation의 자료와 비교하여 에러를 계산하라.
리포트 작성제출 마감일은 조교가 공지함.