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PENG ROBINSONEOS

TEAM:

-Marianella LucesIzarra

-Soath Karime Parra

-Javier Zozaya Pineda

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Introduction The Peng-Robinson (PR) has become the most popular equation

of state for natural gas systems in the petroleum industry.

Place and date of discovery: during the decade of the 1970s,D.Peng was a PhD student of Prof. D.B. Robinson at the Universityof Calgary (Canada). In 1976 they developed The Peng andRobinson (PR) EOS, specifically focused on natural gas systems.

Mean contribution: initially was found to calculate the volume of

100% methane gas as a function of pressure and temperature.This equation expresses fluid properties in terms of the criticalproperties and acentric factor of each species involved.Nowadays it is applied to the representation of properties ofmixtures as well.

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Preceding Correlations Van der Waals (1873):

Is the simplest EOS and this equation is capable to describe the

continuity between the liquid phase and vapor phase, to use it onlyneeds the knowledge of Tc and Pc, which are used to calculate theconstants a and b, as show the following equations:

Where:

a = Attraction parameter and b = Repulsion Parameter

2v

a

bv

TR

P

C

C

P

TRa

22

421875,0

c

c

P

TRb

125,0

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Preceding Correlations Redlich and Kwong(RK) (1949) :

RK modified the term of force attractionfrom the Van der Waals equation

for a term dependent on the temperature:

5,0)( Tbvv

a

bv

TRP

c

c

P

TRa

5,22

42747,0

c

c

P

TR

b

08664,0

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Preceding Correlations SoaveRedlichKwong (SRK) (1972) :

SRK replaced the term dependent on the temperature of the equation of

RK by an expression (T,

)function of the temperature and of theacentric factor:

)(

)(

bvv

Ta

bv

TR

P

c

cc

P

TR

a

22

42747,0

c

c

P

TRb

08664,0

)()( TaaTa cc

25,0

11

cT

TkT

2176,0574,1480,0 wwk

= is the acentric factor

of each pure substance ,

this parameter measures

the sphericity of the

molecules

k = is a constant

characteristic of each

substance

(T) = dimensionless

parameter dependently

of T.

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Equation Studied Peng and Robinson (PR) (1976):

PR improved the prediction of Lparticularly closed to the criticalregion, treating the dimensionless factor for the attraction

parameter as a function of acentric factor and reduced

temperature:

bvbbvva

bv

TRP

)(

c

c

P

TR

a

22

45724,0

25,0

11

cT

TkT

22699,0574,13746,0 wwk

c

c

P

RTb 07780,0

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Deriving the Equation PR departed from the equation of state of SRK improving liquid

density values and accuracy for vapor pressures and equilibrium

ratios.

Base equation of SRK:

PR proposed the need for an improvement in the ability of the

equation of the state to predict liquid densities, particularly close

to the critical region, as illustrated the following equation

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Deriving the EquationAs VdW showed to determine the values of the two constants a and b

PR solved these derivatives regardless of the type of substance and set an

universal gas compressibility factor Z= 0.307 , and the parameters a and b

can be calculated in a traditional way using in the following equations:

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Deriving the EquationImportant: At temperatures others than critical the values of a and b,

can be calculated using:

Where (Tr, )will

be equal to 1 at

critical temperature

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Deriving the EquationPR used the fugacity of each of the phases to determine the balance

conditions of each phase, through the equation:

The fugacity in each phase was introduced to develop a thermodynamic

equilibrium. The fugacity is a measure of the potential for transfer of the

components between phases ,where the phase with the lower componentfugacity accepts the component from the phase with a higher component

fugacity. Therefore, the condition of the thermodynamic equilibrium can be

expressed mathematically using the Newtonsmethod by:

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Deriving the EquationWith a convergence criterion of L - V 10 -4 kPa about two to fouriterations were required to obtain the value of at each temperature.

For all substances examined by PR the relationship between and Tc canbe described by the following equation:

Where k is a constant characteristic of each substance:

and is the acentric factor of each pure substance

)1(1 2121 rTk

226992.054226.137464.0 k

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Deriving the Equation Rearranging the PR Equation of State

It can be expressed as a cubic equation, in a more practical form in terms

of the compressibility factor and replacing the molar volume with ZRT/P,as following:

Where:

For pure components:

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Deriving the EquationAnd for mixtures:

This Form of the equations produces one or three real roots depending

upon the number of phases in the system. In the two phases region the

largest root is for the compressibility factor of de vapor, while the smallest

positive root corresponds to the liquid.

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Considerations Used The PR parameters are expressed in function of the critical properties and acentric

factor ().

The equation shows accuracy values for Z and L at critical conditions.

The mixing rules did not have to use more than one binary interaction parameter,

which must be independent of P, T and composition. The mixture parameters used are

defined for the following mixing rules:

Where ij is an emp ir ical ly determined binary interaction co eff icient characterizing the binary formed b y

comp onent i and j .

The equations had to be applicable to calculate all the properties of the fluids for

natural gas processes .

i j

ijTTjiT

TTT

i j

TjiT

j

jj

ji

jiij

ij

aayya

aaija

ayyabyb

)1()(

))(1(

2/1

2/1

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Modifications Nikos et al. in 1986 proposed a generalized correlation for

generating the binary interaction coefficient, .The authors correlated

these coefficients with system pressure, temperature, and theacentric factor

Where i refers to the principal component, N2, CO2, or CH4, and j

refers to the other hydrocarbon component of the binary.

The acentric-factor-dependent coefficients K0 K1 and K2, are

determined for each set of binaries.

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Modifications PR-VT, Stryjek-Vera (1986) proposed an improvement in the

reproduction of vapor pressures of a pure component by the PR EOS, by

replacing the K term in the equation

with the following expression:

0.7 < Tr < 1

Tr

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ModificationsCOMPONENT K

1

Nitrogen

Carbon dioxide

Water

Methane

Ethane

Propane

Butane

Pentane

Hexane

Heptane

Octane

Nonane

Decane

Undecane

Dodecane

Tridecane

Tetradecane

Pentadecane

Hexadecane

Heptadecane

Octadecane

0.01996

0.04285

0.06635

0.00159

0.02669

0.03136

0.03443

0.03946

0.05104

0.04648

0.04464

0.04104

0.04510

0.02919

0.05426

0.04157

0.02686

0.01892

0.02665

0.04048

0.08291

For components with a reduced

temperature greater than 0.7,

the optimum values of K1 for

compounds of industrial interest

are tabulated below:

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Modifications Ahmed (1991) To use the Peng-Robinson equation of state to predict

the phase and volumetric behavior of mixtures, one must be able to

provide the critical pressure, the critical temperature, and the acentric

factor for each component in the mixture. But the procedure isinadequate for calculating the parameters a, b, and of the equation for

the C7+ fraction.

parameter D as defined by the ratio of the molecular weight to the specific

gravity of the heptanes-plus fraction:

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ModificationsFor parameters a and b of C7+, the following generalized correlation

is proposed

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Solutions of the Equation Cubic Equation Generalized:

Schmidt and Wenzel (1980) demonstrated that all the cubic equations of

state can be expressed as:

Liquid

Vapor

Other

22wbubvv

a

bv

TR

P

EOS u w

VdW 0 0

RK 1 0

SRK 1 0

PR 2 -1

Heyen 1-w f(w,b

)

Kubic f(w) u2/4

Patel-Teja 1-w f(w)

Schmidt

-Wenzel

1-w f(w)

Yu-Lu f(w) u-3

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Solutions of the Equation Algorithm of the Equation

Input Values

Composition (Xi,Yi)

Temperature and Pressure of operations(T,P)

Acentric factor()

Critical Pressure and Critical Temperature (Tc,Pc)

Constant of gas (R)

Calculate the parameters , b and for each component in the system.

Calculate the mixture parameters and bmfor the gas and liquid phase.

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Solutions of the Equation Calculate the coefficients A and B for the gas and liquid phase.

Solve cubic equation for the compressibility factor of the gas and liquidphase.

Obtain the root of cubic equation with the coefficients A and B

Obtained Z you can calculate Volume or density.

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Application in the Oil

Industry The Peng-Robinson EOS has become the most popular equation of

state for natural gas systems in the petroleum industry.

A slightly better performance around critical conditions makes thePR EOS somewhat better suited to gas/condensate systems.

PR obtains better liquid densities than SRK.

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Thank You!