L6-Corresponding States - EOS

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Termodinamica de Termodinamica de HidrocarburosHidrocarburos

Generalized Phase Generalized Phase Equilibria ModelsEquilibria Models

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Generalized Phase Equilibria ModelsGeneralized Phase Equilibria Models

The Principle of Corresponding States. The Principle of Corresponding States. Correlations and Models.Correlations and Models.Extension of Corresponding States to Mixtures.Extension of Corresponding States to Mixtures.Phase equilibrium. Phase rule.Phase equilibrium. Phase rule.Thermodynamic Properties of Homogeneous and Thermodynamic Properties of Homogeneous and Heterogeneous Systems. Heterogeneous Systems. Phase Equilibrium: Vapor-Liquid-Equilibrium Phase Equilibrium: Vapor-Liquid-Equilibrium (VLE), Liquid-Liquid Equilibrium (LLE), Solid-(VLE), Liquid-Liquid Equilibrium (LLE), Solid-Liquid-Equilibrium (SLE). Liquid-Equilibrium (SLE). Phase Equilibrium Models: Single Components. Phase Equilibrium Models: Single Components. Reduced Equations of State (EOS). Reduced Equations of State (EOS). Multicomponents. Mixing Rules.Multicomponents. Mixing Rules.

What’s on it ?What’s on it ?

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Types of VLE Computations: Dew Point and Types of VLE Computations: Dew Point and Bubble Point Calculations. Multiphase Flash. Bubble Point Calculations. Multiphase Flash. Low Pressure Phase Equilibria Computations Low Pressure Phase Equilibria Computations (Surface Separators).(Surface Separators).Ideal Systems.Ideal Systems.K-value correlations. Empirical methods to K-value correlations. Empirical methods to determine equilibrium ratios (K-values).determine equilibrium ratios (K-values).

What’s on it?What’s on it?

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Learning ObjectivesLearning Objectives

After completing this module you will After completing this module you will be able to:be able to: Understand the Principle of Understand the Principle of

Corresponding States.Corresponding States. Calculate the compressibility factor using Calculate the compressibility factor using

different correlations and models.different correlations and models. Understand phase equilibrium.Understand phase equilibrium. Determine the number of variables Determine the number of variables

required to define a system in equilibrium required to define a system in equilibrium (Phase Rule).(Phase Rule).

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Learning ObjectivesLearning Objectives

After completing this module you After completing this module you will be able to:will be able to: Evaluate energy relationships using Evaluate energy relationships using

the First and Second Law of the First and Second Law of Thermodynamics.Thermodynamics.

Evaluate dew and bubble points given Evaluate dew and bubble points given pressure or temperature as pressure or temperature as independent variables.independent variables.

Evaluate flash separation processes.Evaluate flash separation processes.

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The Principle of The Principle of Corresponding StatesCorresponding States

““All fluids when compared at All fluids when compared at the same reduced temperature the same reduced temperature and reduced pressure, have and reduced pressure, have approximately the same approximately the same compressibility factor, and all compressibility factor, and all deviate from ideal gas behavior deviate from ideal gas behavior to about the same degree” to about the same degree”

The Principle of Corresponding states (POC) The Principle of Corresponding states (POC) originated with single component fluids. originated with single component fluids.

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Material properties are usually Material properties are usually expressed in terms of reduced expressed in terms of reduced parameters such as:parameters such as:

Reduced Temperature: Reduced Temperature:

Typical Reduced Typical Reduced ParametersParameters

cr TTT /

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Reduced Pressure:Reduced Pressure:

Reduced Molar Volume:Reduced Molar Volume:

cr PPP /

cr VVV ~/~~

Typical Reduced ParametersTypical Reduced Parameters

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Reduced ParametersReduced Parameters

Usually Usually TTrr and and PPrr VVrr obtained as obtained as a function of a function of TTrr and and PPrr

These are called two-parameter These are called two-parameter Corresponding States modelsCorresponding States modelsThree-parameter corresponding Three-parameter corresponding states models improve predictions states models improve predictions but third parameter is not but third parameter is not VVrr (not (not independent variable)independent variable)

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This third parameter is called the This third parameter is called the acentric factor.acentric factor.It takes into account the non-It takes into account the non-spherical nature of moleculesspherical nature of moleculesPeng Robinson and the Soave Peng Robinson and the Soave Redlich Kwong equations of state Redlich Kwong equations of state (EOS) are examples of three (EOS) are examples of three parameter corresponding states parameter corresponding states models.models.

Generalized Generalized Corresponding States Corresponding States

Three-ParameterThree-Parameter

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The acentric factor The acentric factor is is tabulated for most pure tabulated for most pure components and is defined ascomponents and is defined as

Acentric FactorAcentric Factor

7.0log1 rT

satrP

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Acentric Factor DefinitionAcentric Factor Definition1/T r

log(P rSat )

-1

1.431.0

Slope = -2.3 ( Ar, Kr, Xe)

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Compressibility Factor Compressibility Factor ChartsCharts

Following the POC only one Following the POC only one compressibility factor chart compressibility factor chart can be used to determine can be used to determine volumetric properties of any volumetric properties of any pure fluid by using its reduced pure fluid by using its reduced properties. The shape of this properties. The shape of this chart is in general.chart is in general.

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Compressibility Factor Compressibility Factor ChartsCharts

T r

P r

Z

1

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Corresponding States Corresponding States Correlations & ModelsCorrelations & Models

The objective is then to find a model The objective is then to find a model (models) to predict the Z factor. (models) to predict the Z factor. Ideal gas behavior is described from the Ideal gas behavior is described from the ideal gas Equation of State (EOS) with a ideal gas Equation of State (EOS) with a compressibility factor of 1.compressibility factor of 1.

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1RTVP id~

ZRT

VP

~

Equations of State for Equations of State for GasesGases

Ideal gasIdeal gas

Real gas Real gas

ZZVP

RTRT

VP

id

1~

~Z is the ratio of the “real molar volume” Z is the ratio of the “real molar volume” over the “ideal molar volume” over the “ideal molar volume” of a substance measured at the same of a substance measured at the same pressure and temperature.pressure and temperature.

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Virial Equations of StateVirial Equations of State

Pressure Virial EquationPressure Virial Equation

Density Virial EquationDensity Virial Equation

...'''1~

32 PDPCPBRT

VP

...~~~1~

32 VD

VC

VB

RTVP

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Evaluation of the Evaluation of the Compressibility Factor (Z-Compressibility Factor (Z-

factor)factor)The simplest correlation for The simplest correlation for the compressibility factor is the compressibility factor is expressed in terms of the expressed in terms of the second virial coefficient.second virial coefficient.

r

r

c

c

TP

RTBP

RTBPZ

11

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factor)factor)And the term And the term (BP(BPcc/RT/RTcc)) is is determined using Pitzer’s determined using Pitzer’s Correlation as follows,Correlation as follows,

withwith

10 BBRTBP

c

c

610 42200830 .

rT..B 24

1 01721390 .rT

.B

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factor)factor)Therefore the compressibility Therefore the compressibility factor is expressed as:factor is expressed as:

Note the use of reduced Note the use of reduced properties and the acentric factorproperties and the acentric factor

r

r

r

r

TPB

TPBZ 101

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factor)factor)Virial expansion is not Virial expansion is not adequate for high pressures (P adequate for high pressures (P > 500 psia)> 500 psia)Cubic Equations of State are Cubic Equations of State are more appropriatemore appropriate

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Johannes Diderik van der Johannes Diderik van der Waals Waals

The Nobel Prize in The Nobel Prize in Physics 1910 Physics 1910 "for his work on "for his work on the equation of the equation of state for gases state for gases and liquids" and liquids" Amsterdam Amsterdam University University Amsterdam, the Amsterdam, the Netherlands Netherlands 1837 - 19231837 - 1923

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Cubic Equations of StateCubic Equations of State

The most well-known and older The most well-known and older EOS is the Van der Waals EOS is the Van der Waals equation, which isequation, which is

2~~ Va

bVRTP

(two-parameter corresponding states)(two-parameter corresponding states)

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Probable the most widely used Probable the most widely used EOS in the gas and petroleum EOS in the gas and petroleum industry is the Peng-Robinson industry is the Peng-Robinson EOSEOS

VbVbVVa

bVRTP ~)~()~(~~

(three-parameter corresponding states)(three-parameter corresponding states)

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Mathematical Foundation Mathematical Foundation of Cubic Equations of of Cubic Equations of

StateStateThe critical point for a single The critical point for a single component in a component in a (PV)(PV) diagram diagram is an inflection point on the is an inflection point on the critical isotherm (critical isotherm (TcTc) ) The critical point for a single The critical point for a single component on a component on a (PT)(PT) diagram diagram is a maximum in is a maximum in PP and and TT

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PV Phase BehaviorPV Phase BehaviorPressure-Pressure-volume volume behavior behavior indicating indicating isotherms isotherms for a pure for a pure component component systemsystem

Pres

sur e

MolarVolume

Tc

T2

T1

P1v

L

2 -Phases

CP

V

L

V

Pres

sur e

MolarVolume

Tc

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Application of Constraints at Application of Constraints at the Critical Point – Van der the Critical Point – Van der

Waals EOS ExampleWaals EOS Example

2~~ Va

bVRTP

32 ~2

)~(~ Va

bVRT

VP

T

432

2

~6

)~(2

~ Va

bVRT

VP

T

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Application of Constraints Application of Constraints at the Critical Pointat the Critical Point

These two derivatives must These two derivatives must vanish at the critical pointvanish at the critical point

32 ~2

)~(0~

cc

c

TT Va

bVRT

VP

c

32 ~2

)~(0~

cc

c

TT Va

bVRT

VP

c

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Evaluation of Evaluation of aa and and bb ParametersParameters

Parameter Parameter bb is found from is found from

Next Next aa is … is …

3

~ ~

)~(32

)~( ~2~

)~(32

~2

)~(32

33

32c

cc

c

c

c

cc

c

c

cc

c

VbVbV

RTbV

RT

VaV

bVRT

Va

bVRT

8

~9~8

~9

3/~~2

~~)~(2 2

3

2

33

2cc

c

cc

cc

ccc

c

c VRTV

VRT

VV

VRTVbV

RTa

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Determination of a and b Determination of a and b ParametersParameters

The critical compressibility The critical compressibility factor isfactor is

Replace Replace aa and and bb in VdW EOS in VdW EOS and evaluate the critical and evaluate the critical compressibilitycompressibility

cc

c ZRT

VP

~

c

c

c

c

c

c

c

cc

cc

cc V

RTVRT

VRT

VVRT

VVRTP ~8

3~8

9~2

3~8

~93~~ 2

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Critical Compressibility Critical Compressibility FactorFactor

And the critical compressibility And the critical compressibility calculated from VdW EOS iscalculated from VdW EOS is

A unique value for all substancesA unique value for all substances

375.083~

cc

c ZRT

VP

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Parameters a and b as a Parameters a and b as a Function of Tc and PcFunction of Tc and Pc

Using this we can express the Using this we can express the aa and and bb constants as constants as

c

c

c

cc

c

cccc

PRT

PRTZ

PRTRTVRTa

6427

83

89

89

8

~9 22

c

c

c

cc

c

cc

PRT

PRTZ

PRTVb

883

333

~

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The The ZZ factor is then evaluated factor is then evaluated as as

RTVa

bVV

RTVPZ ~~

~~

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Solution of Cubic Solution of Cubic EquationsEquations

To solve cubic equations there To solve cubic equations there are analytical techniques and are analytical techniques and free software in the WEBfree software in the WEB

http://www.uni-koeln.de/math-http://www.uni-koeln.de/math-nat-fak/phchem/deiters/nat-fak/phchem/deiters/quartic/quartic.htmlquartic/quartic.html

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Solutions of Cubic Solutions of Cubic EquationsEquations

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Tartaglia: the solver of cubic equations

http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/tartalia.html



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Cubic Equation Solver Cubic Equation Solver

http://www.1728.com/cubic.htmhttp://www.1728.com/cubic.htm

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Single vs Multicomponent Single vs Multicomponent Mixtures Mixtures

So far have evaluated Z-So far have evaluated Z-factors for single components factors for single components onlyonlyFor hydrocarbon mixtures we For hydrocarbon mixtures we use the SAME models with use the SAME models with mixing rulesmixing rules

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Example Problem Example Problem Mr. Jones wants to use some 30 liter cans Mr. Jones wants to use some 30 liter cans to ship ethane from College Station to to ship ethane from College Station to Conroe. He would like to fill each of Conroe. He would like to fill each of these cylinders with 10 kg of ethane, but these cylinders with 10 kg of ethane, but he does not know the pressure at which he does not know the pressure at which he needs to fill these tanks or if the walls he needs to fill these tanks or if the walls of the tanks will be able to withstand of the tanks will be able to withstand that kind of pressure. The shipping that kind of pressure. The shipping should be done at an average should be done at an average temperature of 25 temperature of 25 ooC. C.

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Example Problem Example Problem Use three different methods:Use three different methods:

Ideal gas EOSIdeal gas EOS

ZZ factor compressibility correlations factor compressibility correlations given in classgiven in classZZ factor charts using chart given in class factor charts using chart given in class notes (you will do this one)notes (you will do this one)Z factor from a cubic EOS (you will do Z factor from a cubic EOS (you will do this)this)Z factor using properties evaluated from Z factor using properties evaluated from NIST website (you will do this)NIST website (you will do this)

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Example Problem Example Problem

The Critical properties and The Critical properties and acentric factor for ethane are: acentric factor for ethane are:

Mw = 30 g / molMw = 30 g / molTc = 305.5 ºKTc = 305.5 ºKPc = 48.8 barPc = 48.8 barw = 0.098w = 0.098

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(a) Ideal Gas EOS(a) Ideal Gas EOS

…………………….. .. (about 4,000 psia)(about 4,000 psia)

) K.(

mol Kbar cm.

g/mol g,

MwmRTnRTPV 25152731483

3000010 3

bar.cm

barcm,

.MwVmRTP 424275

0003010262738

3

36

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(b) Z-factor Correlations(b) Z-factor Correlations

To use Pitzer correlation we To use Pitzer correlation we must calculate the reduced must calculate the reduced temperature which istemperature which is

9755.05.305

15.27325

c

r TTT

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Example ProblemExample ProblemThe correlation isThe correlation is

(A)(A)

we also know that we also know that

(B)(B)

r

r

c

c

TP

RTBPZ

1

rr PPZ 177181.0

)2515.273)(14.83)(000,10()30)(000,30)(8.48(

mRTVMwPP

nRTPVZ cr

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Example ProblemExample Problem

with with

The acentric factor for ethane The acentric factor for ethane can be obtained from the can be obtained from the tables provided with tables provided with properties for pure properties for pure components.components.

10 BBRTBP

c

c

35610

975504220083042200830 6161

0 ....

T..B ..r

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andand

0519.0

9755.00172139.00172139.0 2.42.4

1 rT

B

3612.00519.0098.03561.010

BB

RTBP

c

c

9755.03612.011 r

r

r

c

c PTP

RTBPZ

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Combining the 2 equations A Combining the 2 equations A and B, we have and B, we have

and and rPZ 37027.01

rPZ 177181.0

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Equating these two equations (A) & Equating these two equations (A) & (B) we solve for the reduced (B) we solve for the reduced pressure, and then for the pressure pressure, and then for the pressure P.P.

There is a substantial difference There is a substantial difference from the ideal gas model!from the ideal gas model!

bar...PPP.P

cr

r

12898488263182631

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(c)(c), , (d) (d) and and (f)(f) are part of your are part of your homework assignmenthomework assignment

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Extension of Extension of Corresponding States to Corresponding States to

MixturesMixturesZZ factor charts (all built from factor charts (all built from EOS) are also used for EOS) are also used for multicomponent systems in this multicomponent systems in this case the coordinates used are case the coordinates used are “pseudo-reduced properties”“pseudo-reduced properties”For a mixture you can use the For a mixture you can use the same charts as for a pure same charts as for a pure component.component.

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Pseudoreduced PropertiesPseudoreduced PropertiesFor mixtures the same type of charts For mixtures the same type of charts apply but using “pseudoreduced apply but using “pseudoreduced properties” which are defined similarly properties” which are defined similarly as the ratio of pressure (or as the ratio of pressure (or temperature) with “pseudoreduced temperature) with “pseudoreduced critical pressure" (or temperature). critical pressure" (or temperature). These pseudocritical properties are an These pseudocritical properties are an average of the critical properties of average of the critical properties of the components in the mixture. the components in the mixture. Charts for mixtures can also be used Charts for mixtures can also be used for single component fluids.for single component fluids.

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CompressibiliCompressibility factor Z as ty factor Z as a function or a function or

pseudoreducepseudoreduced pressured pressure

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Pseudocritical Properties Pseudocritical Properties of Natural Gasesof Natural Gases

Pseudoreduced PressurePseudoreduced Pressure

Pseudoreduced TemperaturePseudoreduced Temperature

pcpr P

PP

pcpr T

TT

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If only the specific gravity of If only the specific gravity of the gases is known then the gases is known then charts/equations are available charts/equations are available to estimate these to estimate these pseudocritical properties (SPE pseudocritical properties (SPE paper 26668)paper 26668)

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Naturally the degree of Naturally the degree of accuracy is reduced accuracy is reduced substantially. We well see substantially. We well see methods when compositional methods when compositional information is available, in this information is available, in this case: case: cii

N

ipc PyP

c

1

cii

N

ipc TyT

c

1

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Once Once ZZ is evaluated you can is evaluated you can find the gas density asfind the gas density as

3/ ftlbmVM

g

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So far we were just determining So far we were just determining properties either for a “gas” or a properties either for a “gas” or a highly compressed fluid (liquid like highly compressed fluid (liquid like density) in the SINGLE PHASE density) in the SINGLE PHASE REGION.REGION.

Notice that Z- factor charts DO Notice that Z- factor charts DO NOT HELP AT ALL IN NOT HELP AT ALL IN

DETERMINING PROPERTIES OF DETERMINING PROPERTIES OF GAS AND LIQUID COEXISTING GAS AND LIQUID COEXISTING

PHASES.PHASES.

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Z-factor Z-factor chart for chart for

low low reduced reduced

pressurespressures

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SPE Paper 26668SPE Paper 26668

Real Gas EquationReal Gas EquationEvaluation of Ppc & Tpc when Evaluation of Ppc & Tpc when composition is knowncomposition is knownEvaluation of Ppc & Tpc when Evaluation of Ppc & Tpc when gas specific gravity is knowngas specific gravity is knownCorrections for NCorrections for N22, CO, CO22, H, H22SS

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Pseudocritical PropertiesPseudocritical Properties

When all gas compositions are knownWhen all gas compositions are known

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Pseudocritical PropertiesPseudocritical PropertiesCompositions known including Compositions known including corrections for Hcorrections for H22S, NS, N22 and CO and CO22

andand

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Pseudocritical PropertiesPseudocritical PropertiesCompositions known including Compositions known including corrections for Hcorrections for H22S, NS, N22 and CO and CO22

AndAndIn this orderIn this order

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Coefficients in Equation 4Coefficients in Equation 4

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Pseudocritical PropertiesPseudocritical Properties

Specific gas gravity & Specific gas gravity & composition of impurities is composition of impurities is known known

i = Hi = H22S, NS, N22 and CO and CO22- 3 major impurities)- 3 major impurities)

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Coefficients in Equation 6Coefficients in Equation 6

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RR22 - Statistical Evaluation of - Statistical Evaluation of Goodness of Fit (How well Goodness of Fit (How well

you fit the data)you fit the data)2

n

1ii )YY(SSR

2

n

1iii )YY(SSE

SYYSSRR ionDeterminat of tCoefficien 2

SSESSRSSY

sum of squares of (predicted – average)

sum of errors squared(true value – predicted value)

When R2 is close to 1 SSE is close to zero! sum of errors squared is nearly zero

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Appreciate Estimates…Appreciate Estimates…

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A Faster Procedure (but less A Faster Procedure (but less accurate)accurate)

Given gas specific gravity read Given gas specific gravity read JJ Given compositions of NGiven compositions of N22, CO, CO22, , HH22S correct S correct JJ using charts using charts

JJ = = JJ + sum(Correction_i) + sum(Correction_i)Note the corrections are negativeNote the corrections are negative

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A Faster Procedure (but less A Faster Procedure (but less accurate)accurate)

Given gas specific gravity read Given gas specific gravity read KK Given compositions of NGiven compositions of N22, CO, CO22, , HH22S correct S correct KK using charts using charts

KK = = KK + sum(Correction_i) + sum(Correction_i)Note the corrections are negativeNote the corrections are negative

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The Chart for JThe Chart for J

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The Corrections for JThe Corrections for J

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The Chart & Corrections The Chart & Corrections for Kfor K

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Procedure Procedure

Determine Tpc and PpcDetermine Tpc and PpcGiven gas compositionGiven gas compositionGiven gas gravity & compositions of Given gas gravity & compositions of impuritiesimpurities

At given p and T determine Tpr At given p and T determine Tpr and Pprand PprDetermine z for the gas requestedDetermine z for the gas requested

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Evaluate z-factor…Evaluate z-factor…

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Exercise in ClassExercise in Class

Determine the density of an Determine the density of an acid gas at 3,500 psia and 220 acid gas at 3,500 psia and 220 ooF F The gas contains 5% COThe gas contains 5% CO22 and and 15% H15% H22S and has a specific S and has a specific gravity of 1.4gravity of 1.4

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Questions & AssignmentQuestions & Assignment

What errors could you have in What errors could you have in your predictions if the corrections your predictions if the corrections for impurities are neglected?for impurities are neglected?How good are these new How good are these new correlations compared with the correlations compared with the existing ones?existing ones?READ the paper SPE 26668 for READ the paper SPE 26668 for additional details additional details

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L6-Corresponding States - EOS