Detailed Info on EOS_2

7/28/2019 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.

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