chapter 3 thermodynamics properties of fliuds
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Chapter 3 Thermodynamics Properties
of Fluids
Chemical Engineering Thermodynamics
3.1 Property Relations for Homogeneous Phases
3.2 Residual Properties
3.3 Residual Properties by Equations of State
Chapter Outline
SYSTEM
Process design and analysis
PROPERTY CALCULATION
IN OUT
To find volume, V, we are completely prepared, whether it is ideal gas, non-ideal gas or liquid.
But for other properties such as H, S, U, we have no idea what is the pressure and volume functions especially for non-ideal gas.
As such, we need to develop just such relations so that calculations can be carried out.
We have numerically calculated the main thermodynamics properties such as P, V, T.Now we are going to calculate enthalpy, H and entropy, S from PVT and heat-capacity data using 1st and 2nd law property relations.
And develop generalized correlations which provide estimates of property values in the absence of experimental information.
3.1 Property Relations forHomogeneous Phases
Primary thermodynamic properties are given as: P, V, T, U and S.
Other thermodynamic properties arise only by definition using the relation of these primary properties:Enthalpy: H ≡ U + PVHelmholtz Energy: A ≡ U – TSGibbs Energy: G ≡ H – TS
PdVTdSdU
VdPTdSdH
SdTPdVdA
SdTVdPdG
Fundamental property relations which are general equations for a homogenous fluid of constant composition:
VS S
P
V
T
PS S
V
P
T
TV V
S
T
P
TP P
S
T
V
They are all useful for evaluation ofthermodynamic properties from experimental data.
The thermodynamic properties can be expressed by equation for each of these equations which called as Maxwell’s equations:
Heat capacity at constant pressure
PP
CT
H
Other expression for heat capacity
PP T
ST
T
H
Hence, combine these two equation gives:
T
C
T
S P
P
Enthalpy and Entropy as Functions of T and P
From Maxwell’s equations…
TP P
S
T
V
From fundamental property relations,the corresponding derivative for enthalpy is
VP
ST
P
H
TT
As a result, heat capacity at constant pressure:
PT T
VTV
P
H
The functional relations chosen here for H and S are: H = H ( T , P ) and S = S ( T, P ) Whence,
dPT
VTVdTCdH
PP
dPT
V
T
dTCdS
PP
These are general equations relating the properties of homogeneous fluids of constant composition to constant temperature and pressure.
The pressure dependence of the internal energy is obtained by:
VP
VP
P
H
P
U
TTT
Replaced by previous equation gives:
TPT P
VP
T
VT
P
U
Internal Energy as a Function of P
Internal energy is given by equation U = H - PV
TP
H
The Ideal-Gas State
dTCdH igP
ig P
dPR
T
dTCdS ig
Pig
where superscript “ig” denotes an ideal-gas value.
Alternative Forms for Liquids
VP
S
T
VT
P
H
T
1 VTPP
U
T
At critical point the volume itself is verysmall, as are ß and κ. Thus at most conditions pressure has little effect on the properties of liquid. Hence for incompressible fluid:
VdPTdTCdH P 1
(Example 6.1)
VdPTdT
CdSP
Internal Energy and Entropy as Functions of T and V
Temperature and volume often serve as more convenient independent variables than do temperature and pressure.
These are general equations relating the internal energy, U and entropy, S of homogeneous fluids of constant composition to temperature and volume.
dVPT
PTdTCdU
VV
dVT
P
T
dTCdS
VV
VT
P
dVPTdTCdU V
dVT
dTCdS V
A change of state a constant volume becomes:
Then the previous equations above become:
(Example 6.2)
A fundamental property relation, follows fromthe mathematical identity:
dTRT
GdG
RTRT
Gd
2
1
After algebraic reduction:
dTRT
HdP
RT
V
RT
Gd
2
The Gibbs Energy as a Generating Function
When G/RT is known as a function of T and P, V/RT and H / RT follow by simple differentiation.
TP
RTG
RT
V
PT
RTGT
RT
H
The remaining properties are given by defining equations. In particular,
RT
G
RT
H
R
S
RT
PV
RT
H
RT
U
The Gibbs energy when given as a function of T and P serves as a generating function for the other thermodynamic properties, and represents complete property information .
where; GR = Residual Gibbs energy G = Actual Gibbs energy Gig = Ideal-gas values of Gibbs energy
For residual volume:
3.2 Residual Properties
igR GGG
PRT
VVVV igR
Since V = ZRT/ P,
1 ZP
RTV R
The definition for generic residual property is:
where M is molar value of any extensive thermodynamic property eg. V , U , H , S , or G.
igR MMM
dTRT
HdP
RT
V
RT
Gd
RRR
2
Fundamental property relation for residual properties applies to fluids of constant composition.
T
RR
P
RTG
RT
V
P
RR
T
RTGT
RT
H
Useful restricted forms are:
P
RR
dPRT
V
RT
G0
constTdPRT
V
RT
Gd
RR
Thus the residual Gibbs energy serves as a generating function for the other residual properties and here a direct link with experiment does exist.
Integration from zero pressure to arbitrary pressure P yields:
where at the lower limit GR/RT is equal to zero because the zero-pressure state is an ideal-gas state.
TP
dPZ
RTG P
R
constant 10
Differentiation with respect to temperature in accord to
give:
P
P
R
P
dP
T
ZT
RT
H0
P
RR
T
RTGT
RT
H
Defining equation for Gibbs energy is:
RT
G
RT
H
R
S RRR
The residual entropy is therefore:
The residual entropy is found:
P P
P
R
P
dPZ
P
dP
T
ZT
R
S0 0
1
RRR TSHG
The compressibility factor is defined asZ = PV/RT; values of Z and (∂Z/∂T)p
therefore come from experimental PVT data and the two integrals are evaluated by numerical or graphical method.
Alternatively the two integrals are evaluated analytically when Z is expressed as function of T and P by volume- explicit equation of state .
Thus, given PVT data or an appropriate equation of state, HR and SR can be evaluated and all other residual properties.
H = Hig + HR S = Sig + SR
Thus H and S follow from corresponding ideal-gas and residual properties by simple addition. Since:
T
T
igP
igig dTCHH0
0
and
0
0ln
0 PP
RTdT
CSS TT
ig
P
igig
Substitution into the preceding equations gives:
RT
T
igP
ig HdTCHH 0
0
RTT
ig
P
ig SPP
RTdT
CSS 0
0ln
0
(Example 6.3)
These two equation also may expressed alternatively to include the mean heat capacities:
R
H
igP
ig HTTCHH 00
R
S
ig
P
ig SPP
RTT
CSS 00
0lnln
Residual properties have validity for both gases and liquids.
Since the equations of thermodynamics which derive from the first and second laws do not permit calculation of absolute values for enthalpy and entropy and since in practice only relative values are needed the reference-state conditions T0 and P0
are selected for convenience and values are assigned to igH0 and igS0 arbitrarily.
Compressibility factor, Z is given:
Then,RT
BP
RT
G R
3.3 Residual Properties by Equations of State
RTBP
Z 1
By Equation,
2
,
1
T
B
dT
dB
TR
PT
T
RTGT
RT
H
xP
RR
or
dT
dB
T
B
R
P
RT
H R
Substitution of both equations,
dT
dB
R
P
RT
S R
Equation PV = ZRT can be written in the alternative form,
Differentiation at constant T gives:
RTZPV
Combination both gives:
Z
dZd
P
dP
dZZdRTdP
Upon substitution for dP /P ,
ZZd
ZRT
G R
ln110
10
Z
dTZ
TRTH R
00 1lnd
Zd
TZ
TZRS R
21 CBZ Using the 3-term virial equation:
ZCBRT
G R
ln2
32 2
2
2
1 dT
dC
T
C
dT
dB
T
BT
RT
H R
2
2
1ln
dT
dC
T
C
dT
dB
T
BTZ
RT
S R
The End
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