detailed info on eos_2
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Equations of State
Lecture I
Cubic Equations
Virial Equation
Mixing Rules
Lecture II
Complex Equations
Generalized Correlations
Learning Objective
Use appropriate equations of state to estimate densities of liquid andvapor phases of pure substances and mixtures.
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Phase Behavior of Pure Fluids
1. Critical Pressure2. Liquid
3. Solid
4. Solid/Liquid
5. Liquid/Vapor
6. Gas
7. Satd Liquid8. Satd Vapor
9. Vapor
10. Solid/Vapor
11. Critical Volume
12. Critical Temperature
2
Solid
3
4
5
7 8
10
11
6
9
12
Pressure
Volume
1
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Lsat
Vsat
Solid/VaporVapor
GasLiquid/VaporSolid
Liquid
PC
VC
TC
Pressure
Volume
May beideal gas
What is the shape of an isotherm in the gas or vapor region?
What is an isotherm?
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Ideal Gas Equation
All gases approach ideality as P 0
Ideal gases have NO interparticle interactions
Real gases HAVE interparticle interactions
PV = nRT
There is no fixed algorithm for identifying the region where the gas behavesideally, because the behavior of a gas depends on the chemical properties ofthe molecule
Thermodynamic properties (i.e. H, U) of ideal gases are special because of
the lack of interparticle interactions.
Residual properties allow us to determine the deviation from ideality by
MR= M - MID
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Hard Sphere Model Equation of State
Like ideal gas assumption, this model applies at low fluid density and hightemperature.
All spheres have the same diameter -
Compressibility factor A depends on fluid density -
VN/~
Is the number of spheres involume VN~
Density () is conventionally in terms of the packing fraction
packing fraction is the ratio of V of spheres to container V
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v
N
V
N
V
VAspheres
63
)2/(4~ 33
Hard Sphere Model Equation of State, contd
The packing fraction can be written without the numberdensity term
v is the specific volume (volume/mole)
Because there are voids between spheres, < 1
The maximum packing fraction occurs when spheres form a face-centeredcubic lattice
...74048.06
2max
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This maximum packing is for a solid phase
Hard Sphere Model Equation of State, contd
For a fluid composed of hard spheres
...494..03
2 max
The compressibility factor can be written in terms of the packingfraction
3
32
)1(
1Z For < 0.494
In the limit as 0, Z 1 which is for an ideal gas
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Real Gases
Simplest representation of real gas is the compressibility factor
1RT
PVZ
Walas fig 1.2b
Effects of temperature and pressure on the compressibility of nitrogen atseveral temperatures. From Walas fig 1.2b.
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Corresponding states representations, from Walas fig. 1.8a
Figure 1.8 Walas
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Corresponding states representations, from Walas fig. 1.8b
Figure 1.8 Walas
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Real gases have non-negligibleintermolecular forces
Attractive forces
Repulsive forces
Molecules may be:
electrically neutral and symmetrical, usually non-polar
electrically neutral and unsymmetrical, polar
molecules that associate due to residual valences
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Cubic Equations of State
In bringing particles together there is a point where attractive forces begin, thenfinally there are repulsions at close distances. Coefficients of the virial equation
account for the attractive forces between distant molecules.
Earliest cubic equationvan der Waals (1873)
1. Takes into account the real volume of the particles b ~ volume of a particle
2. Takes into account the real pressure exerted by particles subject to
interactions, which are predominantly attractive.Pid = Preal + P0 P0 is called the inner pressure.
P0 (1/V)2 V is the specific volume, or 1/
we define the constant a, P0 = a(1/V2)
RTbVaV
P
2
1
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RTbVV
aP 2
Van der Waals Equation of State
a is called the attraction parameter
b is the repulsion parameter b is the effective molecular volume ~4x the volume of the actualmolecules
Useful forms of the equation:
Explicit in pressure:2V
a
bV
RTP
Volume form: 023
P
abV
P
aV
P
RTbV
How manyroots are there?What do they
physicallyrepresent?
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Diagram of a cubic equation ofstate in the two-phase region.Shows metastable regions andtie line connecting volumes of
liquid and vapor phases inequilibrium. Areas FEDF andDCBD are equal (Maxwellsprinciple). Region EDC isphysically impossible for a puresubstance
Figure 1.7 in Walas
Cubic equations can represent the continuity ofphysical states
3 volume roots: highest value is the vapor volume, lowest volume isthe liquid volume, third one is physically meaningless
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Determination of parameters for cubicequations
At the critical point,
023
C
C
C
C
C
CC
P
abV
P
aV
P
RTbV
0P
V
T
32
2
CC
C
V
a
bV
RT
0P
V
2
2
T
43
62
CC
C
V
a
bV
RT
Solve these three simultaneously to obtain
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Determination of parameters for cubicequations, contd
C
CCC
P
TRVPa
64
273
222
C
CCP
RTVb 83
Solution of three equations yields a third parameter
C
CC
T
VP
R 3
8
This value of R is meaningless, however, it enables the calculation of ZC
375.0CZ
If the Law ofCorresponding States
truly held, ALL gaseswould have the same ZC.
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Other Cubic Equations
After Van der Waals, the next major improvement in cubic equations
was Redlich Kwong (1949).
)( bVVT
a
bV
RTP
Redlich-Kwong Parameters
C
C
P
TRa
5.2242748.0
C
C
P
RTb
0866.0
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Redlich Kwong, contd
vs.
VV
a
bV
RTP
)( bVV
Ta
bV
RTP
1.) a was found to be dependent on T, so T0.5 has a significance
2.) Increased role of molecular volume.
Redlich (1976): They had no particular theoretical basis for theirequation, so it is to be regarded as an arbitrary but inspiredempirical modification of its predecessors.
)( bVVT
a
bV
RT
P
RK equation has two
parameters, a and b
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Soave Redlich-Kwong (1972)
)( bVV
a
bV
RTP
25.02 )]1)(15613.055171.148508.0(1[ rT
C
C
P
TRa
2242748.0
C
C
PRTb 0866.0
The Soave Equation has beendesigned for hydrocarbon
gaseous phases.
is a function of Tr and
a/T1/2 of RK isreplaced with (T, )
For hydrogen: )15114.0(096.1 rT
e
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Acentric Factor
Third parameter in equations of state.
Acentric factor () was identified by Pitzer
Pitzer used the fact that the log of vapor pressure of a pure fluid isapproximately linear with the reciprocal of absolute temperature.
r
sat
r
T
d
Pda
1
log The slope of a plot of log Prsat vs 1/Tr
For simple fluids (Ar, Kr, Xe), this slope = -2.3
1.2 1.4
-1
-2
Slope = -2.3 (Ar, Kr, Xe)
Tr = 0.7
1/Tr
log
(Pr
sat)
Find the difference of other fluids
with the value of log Prsat
at Tr = 0.7Then = -1-log(Pr
sat)Tr = 0.7
According to Pitzer, measures the deviation of intermolecular potentialfunctions from that of simple, spherical molecules.
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Peng Robinson Equation
)()(
)(
bVbbVV
Ta
bV
RTP
C
C
P
TRa
2245724.0
C
C
P
RTb
07780.0
Parameters
307.0CZ
Close for non-polars
At temperatures other than the critical temperature
)26992.05422.137464.0)(1(1 25.05.0 rT
),()()( rC TTaTa
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Peng Robinson Equation, contd
Clear goals for the development of this equation
1) All parameters expressible in terms of PC, TC, and
2) Focus accuracy near the critical point
3) No more than 1 binary interaction parameter for mixing rules and theseshould be independent of T, P and composition
4) Applicable to natural gas processes for calculation of all fluid properties
PR equation is more accurate for liquid densities than RK
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Example calculation: Finding roots of cubic equation
Methyl chloride
Critical properties:
TC = 416.3 K
PC = 66.8 bar
Using volume form of the Van der Waals equation
023
P
abV
P
aV
P
RTbV
C
C
P
TRa
64
27 22
C
C
P
RTb
8
=7.566e+06 bar cm6/mol2
=64.766 cm3/mol
See excel calculation, use goal seek
For initial guesses, use b for theliquid root, andRT/Pfor the vapor
root.
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Finding roots of cubic equations, contd
Poling criteria is an algorithm to determine what phase correspondsto the root you have found.
Poling Criteria
Find =TP
V
V
1What is this called??
23
22
)(2
)(
bVaRTV
VbV = For Van der Waals equation
TbVbVabVRTV
bVV
/))(2()(
)(222
222
= For Redlich Kwong equation
For liquid root, < 0.005 atm-1
Vapor phasePP
39.0
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Virial Equation of State
First described by Thiesen in 1885
.. .12V
C
V
B
RT
PVZ
...1 2CB
.''12
PCPB
Notice that B = BAnd C = C
This equation can be derived from statistical mechanical analysis ofthe forces between molecules.
B is the second virial coefficient describing interactions between
pairs of molecules.C is the third virial coefficient describing interactions betweentriplets of molecules.
Virial coefficients are functions only of temperature.
RT
BB' ; 2
2
)(' RT
BC
C
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Virial Equation of State, contd
The truncated form of the Virial Equation
RT
BPPB
V
B
RT
PVZ 1'11
Is extremely convenient to use because of its mathematical simplicity.
According to Prausnitz (1957) it is often adequate when
P < 0.5T(yiPCi)/(yiTCi)
A rough estimate (Smith & Van Ness): B truncated equation up to 5
bar, C truncated equation up to 15 bar
The virial equation cannot be used for liquid phases, only gaseous.
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Complex Equations of State
Four main categories (see Walas section 1.6)
1. Equations specific for individual substances water, air, CO2.These are required to have great accuracy over a wide range ofconditions, which is achieved by the use of many constants.
2. Equations of a particular form with different constants for use withdifferent substances. Combining rules may extend these to
mixtures
3. Equations with universal parameters evaluated in terms of thereadily known properties of individual pure substances.
4. Equations that can be applied to mixtures (from #2 and 3) thatincorporate binary interaction parameters.
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...)1( 32 DCBRTP
Some key equations
Several equations were developed from the original virial equation
The Benedict-Webb-Rubin equation was an improvement over theBeattie-Bridgeman equation (1940).
4
2
03
2
000
2
200
T
bcRB
T
cRBaAbRTB
T
RcARTBRTP
One of the first successful complex equations was the Beattie-Bridgemanequation (1927)
With 5 parameters, this equation worked well below the critical point.
)exp()1( 2232
6
32
2
000
T
ca
abRTT
CARTBRTP
For more details onthese equations,including mixing rulesand parameters, seeWalas, Chapter I.
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Complex Equations of State, contd
A number of researchers determined constants for many compounds.
Nishiumi (1980) extended the BWR equation to work for water, polarcompounds and heavy hydrocarbons, and calculated (measured)parameters for 92 substances. Their equation had 15 parameters that are
functions of the acentric factor.
)exp()1( 2232
6
32
4
0
3
0
2
000
Tc
Tda
T
dabRT
T
E
T
D
T
CARTBRTP
Starling modified the BWR equation(1973), referred to commonly as theBWRS equation.
The BWR and BWRS equations have high degree of accuracy - for Tr as low as0.3 and for r as high as 3 - and are used widely by industry, including forcryogenic systems.
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Corresponding States Equations
Correlation of the deviations of PVT properties from those of particular
reference substances.
The first of these equations was developed by Pitzer (1955 1958) of thecompressibility as a polynomial in acentric factor:
)2(2)1()0( ZZZZ
Normally truncated at Z(1)
Z(0) is the compressibility of a simple fluid (e.g. argon) and the additionalterms account for deviation from simple fluid behavior.
Values for Z(0) and Z(1) were tabulated and can also be found in plots. Theequation with data for Z(0) and Z(1) in these forms are accurate abovereduced temperatures of 0.7.
Correlations for Z(0) and Z(1) have also been developed, especially by Leeand Kesler (1975).
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Lee Kesler Equation
)1()0()0()(
)(
)0( ZZZZZZ rr
3978.0)(r Acentric factor for octane, the largest hydrocarbon forwhich extensive data is available.
2223
4
52exp1
rrrrrrrr
rr
VVVT
c
V
D
V
C
V
B
T
VPZ
34
232
1
rrr Tb
T
b
TbbB
3
321
rr T
c
T
ccC
rT
d
dD 21
Constant Simple Fluids Reference Fluids
b1 0.1181193 0.202657 9
b2 0.2657 28 0.331511
b3 0.1547 9 0.027 665
b4 0.030323 0.203488
c1 0.02367 44 0.0313385
c2 0.01 86984 0.050361 8c3 0.0 0.016901
c4 0.0427 24 0.04157 7
d1 x 104 0.155488 0.487 36
d2 x 104 0.623689 0.07 40336
0.65392 1 .226
0.060167 0.037 54
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Lee Kesler Equation, contd
Strategy to determine Z
1. From equation for Z, at specified Tr and Pr, find Vr(0) and Vr
(1) by trial
2. Find Z(0) and Z(1) and solve for Z.
3. Note that V r is not the reduced volume. Rather, 1/Vr = Pr/ZTr
Check the literature for updates on this equation.
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Equations for Liquids
RK, SRK and Lee-Kesler are not very accurate for liquids, especially
compressed liquids.
Rackett equation (1970)
2857.0)1( rT
CC
sat ZVVTo find the molar volume ofsaturated liquids
Yamada and Gunn (1973) stated the average error of the Rackett eqn. is 2.4%.They developed an equation similar in form:
))'1(exp(' 7/2))1(1(7/2
r
T
C
sat TZVV r
Zc can be calculated in terms of the acentric factorZc = 0.29056 0.08775
In this equation, V and Tr are corresponding values of the specific volumeand the reduced temperature.
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Equations for Liquids, contd
))1(1( 7/2/ rTCCiCiisat ZPTxRV
For liquid mixtures, a similar equation was developed by Spencer & Dunn
(1973)
CiiC ZxZ
To find the liquid volume for compressed liquids, a chart method wasdeveloped by XXX and is described in Smith & Van Ness.
2
112
r
rVV
Known volume
Read from a chart
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Equations for Liquids, contd
Calculate the Tr and Pr for the state of your liquid and then readthe r off the left axis.
Fig. 3.17: Generalized density correlation for liquids.
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Comparison of Equations
For calculation of volumetric properties and gas phase fugacity calculations,both RK and B-truncated virial equation are useful at Pr < 0.5Tr.
PR is more accurate for liquid densities than RK, but generalizedcorrelations by Rackett or Yamada & Dunn are more accurate for liquids.
BWR and BWRS are very accurate but difficult to use. BWRS is moreaccurate above the critical point. As stated easrlier, the BWR and BWRSequations have high degree of accuracy - for T
ras low as 0.3 and for
ras
high as 3 - and are used widely by industry, including for cryogenic systems.
Lee-Kesler equation is claimed to be accurate in the range of 0.3 < Tr < 4 and0 < Pr < 10, however some difficulties may be encountered between0.93 < Tr < 1.
Recommendation estimate values with simple equations first prior tousing more complex equations, because mistakes can be detected usingthe estimated values.