the advanced chemical engineering thermodynamics the...
TRANSCRIPT
The Advanced Chemical
Engineering Thermodynamics
The thermodynamics properties of
fluids (II)
Q&A_-10- 11/17/2005(10)
Ji-Sheng Chang
Property relations
� The residual Gibbs free energy
� The definition of residual properties
� MR(T,P) = M(T,P) - M
IG(T,P)
� R: Residual property
� IG : Ideal gas behavior property
� MR(T,P) = M(T,P) - M
IG(T,P),
M are the properties as V, G, H, S, �
Property relations
� The residual properties
� VR(T,P) = V(T,P) - V
IG(T,P) = V - RT/P;
Basic thermodynamics relationship for calculations
� GR(T,P) = G(T,P) - G
IG(T,P);
More important in traditional Chemical Engineering Thermodynamics
� HR(T,P) = H(T,P) - H
IG(T,P)
Property relations
� Deriving the residual properties in terms of the PVT properties.
� From the generating function of Gibbs free energy, V
R/RT= [∂(GR
/RT)/∂P]T,
� Then d(GR/RT) = (V
R/RT)dP,
� to integrate from P=0 to P at constant T.
Property relations
� The integrated result in terms of PVT property as the formula (6.45)
T) (constant ,)1(0∫ −=P
R
P
dPZ
RT
G
T) (constant ,)1
(0∫ −=P
R
dPPRT
V
RT
G
Property relations
� The residual enthalpy, HR/RT= - T [∂(GR/RT)/∂T]P, The integrated result in terms of PVT property as the formula (6.46)
T) (constant ,0∫
∂∂
−=P
P
R
P
dP
T
ZT
RT
H
Property relations
� In previous section S = H/T - G/T, then the residual entropy SR/R = HR/RT - GR/RT The result in terms of PVT property as the formula (6.48)
T) (constant ,)1(0 0∫ ∫ −−
∂
∂−=
P P
P
R
P
dPZ
P
dP
T
ZT
RT
S
Property relations
� Using the independent property of T and V
� More commonly as the form P=P(V,T) of the PVT equations of state, especially the cubic equations of state.
� P = ZRT/V, P = ZρRT; as ρ = 1/V
� dP=RTd(Zρ)=RT(Zdρ+ρdZ) at constant T
Property relations
� dP/P=dρ/ρ+dZ/Z,
� as the formula (6.58)
� the alternative form.
T) (constant ,ln1)1(0
ZZd
ZRT
GR
−−+−= ∫ρ
ρρ
T) (constant ,ln1)1( ZZV
dVZ
RT
G VR
−−+−= ∫∞
T) (constant ,)1(0∫ −=P
R
P
dPZ
RT
G
Property relations
� Calculating the residual properties � The PVT data,
Using graphical integration method to calculate the residual Gibbs free energy from steam data.
T) (constant ,1
0∫
−=P
R
dPP
Z
RT
G
Phase diagram
0 200 400 600 800 1000T/K
0
100
200
300
400
500
P/b
ar
Solid Liquid
Gas
PC,TC
1. PT phase diagram of water
2. The Solid/Liquid equilibrium curve
for water
for most component
200OC path
Critical properties
PC=220.55 bar
TC=647.1 K
TTri=273.16 K
Vapor/Liquid equilibrium curve
� The Antoine equation
� The constant of Antoine
equation for water
� A=16.3872
� B=3056.96
� C=217.625
CCT/
BA/kPaPln
sat
+°−=
Property
� Critical properties of water
� PC=220.55 bar
� TC=647.1 K
� PTri=0.611 kPa
� TTri=273.16 K
� Tn=373.15 K at P= 1.01325 bar = 1 atm
Property relations
� Problems
� Calculate the residual Gibbs
energy of the 200°C superheat steam at 10 bar, using the PVT
data from steam tables
Properties
P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P
1 0.01 218350 3933575 0.999951 -0.00493
50 0.5 4356.0 78473.34 0.997432 -0.00514
100 1 2172.3 39133.98 0.994822 -0.00518
200 2 1080.4 19463.41 0.989555 -0.00522
300 3 716.35 12905.05 0.984174 -0.00528
400 4 534.26 9624.694 0.978674 -0.00533
500 5 424.96 7655.654 0.973069 -0.00539
600 6 352.04 6342.001 0.967317 -0.00545
700 7 299.92 5403.059 0.961455 -0.00551
800 8 260.79 4698.132 0.955446 -0.00557
Properties
P/kPa P/bar V/cm3g-1 V/cm3mol-1 Z (Z-1)/P
900 9 230.32 4149.215 0.949291 -0.00563
1000 10 205.92 3709.649 0.943027 -0.0057
1100 11 185.92 3349.349 0.936579 -0.00577
1200 12 169.23 3048.678 0.930002 -0.00583
1300 13 155.09 2793.946 0.923321 -0.0059
1400 14 142.94 2575.064 0.916447 -0.00597
1500 15 132.38 2384.826 0.909367 -0.00604
1550 15.5 127.61 2298.894 0.90582 -0.00608
1554.9 15.55 127.20 2291.508 0.905764 -0.00606
Residual Gibbs free energy
properties
0 4 8 12 16 20P [ bar ]
-0.008
-0.006
-0.004
-0.002
0
(Z-1
)/P
[
ba
r-1
]
GR/RT of superheat steam
200OC
Example 1
� Calculating the residual properties
� The PVT equations of state
� Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the carbon dioxide at the states of temperature of 25°C and pressure of 1 bar.
Deriving the
formulas
� The equations of residual Gibbs free energy
� Three type of the two term virial equation
� The residual Gibbs free energy for a gas at each P and T
Deriving the
formulas
� Second virial coefficient
� From PVT data
� From generalized
correlation equations
� Based molecular
thermodynamic, it is integrated from
potential function
Example 2
� Calculating the residual properties
� The PVT equations of state
� Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the methane at the states of temperature of 25°C and pressure of 1 bar.
Example 3
� Calculating the residual properties
� The PVT equations of state
� (?) Using the two term virial equation of states, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the benzene at the states of temperature of 25°C and pressure of 1 bar.
Theorem of corresponding states
� Two-parameter theorem of corresponding states
� Given T and P
� Finding Tc, Pc
� Calculate Tr, Pr
� Using Tr, Pr to finding the reduced properties from Table E.1 to E.16
Theorem of corresponding states
� Three-parameter theorem of corresponding states
� Given T and P
� Finding Tc, Pc, ω
� Calculate Tr, Pr
� Using Tr, Pr, ω to finding the reduced properties from Table E.1 to
E.16
Theorem of corresponding states
� Based on the corresponding states principles
� The Lee/Kesler generalized correlation tables, Appendix E at text books of pages from 695 to 711.
� The properties of some pure species tables, Appendix B at text books of pages from 679 to 682.
Example 4
� Application of the theorem of corresponding states to estimate the
thermodynamics properties of the
pure components.
� At temperature of 25°C and pressure of 1 bar, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the carbon dioxide.
Example 5
� Application of the theorem of corresponding states to estimate the
thermodynamics properties of the
pure components.
� At temperature of 25°C and pressure of 1 bar, estimate the molar volume, residual enthalpy, residual entropy, and residual Gibbs free energy of the methane.