PETE 310 Lectures # 32 to 34

PETE 310PETE 310Lectures # 32 to 34Lectures # 32 to 34

Cubic Equations of StateCubic Equations of State

……Last LecturesLast Lectures

Instructional ObjectivesInstructional Objectives Know the data needed in the EOS to Know the data needed in the EOS to

evaluate fluid propertiesevaluate fluid properties Know how to use the EOS for single and Know how to use the EOS for single and

multicomponent systemsmulticomponent systems Evaluate the volume (density, or z-factor) Evaluate the volume (density, or z-factor)

roots from a cubic equation of state forroots from a cubic equation of state for

Gas phase (when two phases exist)Gas phase (when two phases exist) Liquid Phase (when two phases exist)Liquid Phase (when two phases exist) Single phase when only one phase existsSingle phase when only one phase exists

Equations of State (EOS)Equations of State (EOS)

Single Component SystemsSingle Component Systems

Equations of State (EOS) are Equations of State (EOS) are mathematical relations between mathematical relations between pressure (P) temperature (T), and molar pressure (P) temperature (T), and molar volume (V). volume (V).

Multicomponent SystemsMulticomponent Systems

For multicomponent mixtures in For multicomponent mixtures in addition to (P, T & V) , the overall molar addition to (P, T & V) , the overall molar composition and a set of mixing rules composition and a set of mixing rules are needed.are needed.

Uses of Equations of State Uses of Equations of State (EOS)(EOS)

Evaluation of gas injection processes Evaluation of gas injection processes (miscible and immiscible)(miscible and immiscible)

Evaluation of properties of a reservoir Evaluation of properties of a reservoir oil (liquid) coexisting with a gas cap oil (liquid) coexisting with a gas cap (gas)(gas)

Simulation of volatile and gas Simulation of volatile and gas condensate production through condensate production through constant volume depletion evaluations constant volume depletion evaluations

Recombination tests using separator Recombination tests using separator oil and gas streamsoil and gas streams


One of the most used EOS’ is the One of the most used EOS’ is the Peng-Robinson EOS (1975). This is a Peng-Robinson EOS (1975). This is a three-parameter corresponding three-parameter corresponding states model.states model.

)()( bVbbVV

a

bV

RTP

attrrep PPP

PV Phase BehaviorPV Phase Behavior

Pressure-Pressure-volume volume behavior behavior indicating indicating isotherms for isotherms for a pure a pure component component systemsystem

Pre

ssu

r e

Mo lar V olume

Tc

T2

T1

P1v

L

2 - P has es

CP

V

L

V

Pre

ssu

r e

Mo lar V olume

Tc


The critical point conditions are The critical point conditions are used to determine the EOS used to determine the EOS parametersparameters

0

0

2

2

c

c

T

T

V

P

V

P


Solving these two equations Solving these two equations simultaneously for the Peng-simultaneously for the Peng-Robinson EOS providesRobinson EOS provides

c

ca P

TRa

22

c

cb P

RTb andand


Where

and

with

07780.0

45724.0

b

a

211 rTm

22699.054226.137464.0 m

EOS for a Pure ComponentEOS for a Pure Component

-10

0

12

A1

A2

Pre

ssu

re

Mo la r V olum e

T2

T1P1

v

L

2 - P hases

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P

-10

12

A1

A2

Pre

ssu

re

112

A1

A2

Pre

ssu

re

Mo la r V olum e

T2

T1P1

v

L

2 - P has

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P


PR equation can be expressed PR equation can be expressed as a cubic polynomial in V, as a cubic polynomial in V, density, or Z. density, or Z.

RT

bPB

RT

PaA

2

3 2

2

2 3

( 1)

( 3 2 )

( ) 0

Z B Z

A B B Z

AB B B

withwith


When working with mixtures When working with mixtures (a(a) and (b) are evaluated ) and (b) are evaluated using a set of mixing rulesusing a set of mixing rules

The most common mixing The most common mixing rules are:rules are: Quadratic for a Quadratic for a

Linear for bLinear for b

Quadratic MR for aQuadratic MR for a

where the kij’s are called interaction where the kij’s are called interaction parameters and by definitionparameters and by definition

0.5

1 1

1Nc Nc

i j i j i j i jmi j

a x x a a k

0

ij ji

ii

k k

k

Linear MR for bLinear MR for b

1

Nc

m i ii

b x b

ExampleExample

For a three-component For a three-component mixture (Nc = 3) the attraction mixture (Nc = 3) the attraction (a) and the repulsion constant (a) and the repulsion constant (b) are given by(b) are given by

1

0.5 0.5

1 2 1 2 1 2 12 2 3 2 3 2 3 23

0.5 2 21 3 1 3 1 3 13 1 1 2 2 2

23 3 3

2 (1 ) 2 (1 )

2 (1 )

ma x x a a k x x a a k

x x a a k x a x a

x a

1 1 2 2 3 3

mb x b x b x b


The constants a and b can be The constants a and b can be evaluated usingevaluated using

Overall compositions Overall compositions zzii with with ii = = 1, 2…Nc1, 2…Nc

Liquid compositions Liquid compositions xxii with with ii = = 1, 2…Nc1, 2…Nc

Vapor compositions Vapor compositions yyii with with ii = = 1, 2…Nc1, 2…Nc


The cubic expression for a mixture The cubic expression for a mixture is then evaluated usingis then evaluated using

2 mm

m m

a P b PA B

RTRT

Analytical Solution of Analytical Solution of Cubic EquationsCubic Equations

The cubic EOS can be arranged The cubic EOS can be arranged into a polynomial and be into a polynomial and be solved analytically as follows. solved analytically as follows.

3 2

2

2 3

( 1)

( 3 2 )

( ) 0

Z B Z

A B B Z

AB B B


Let’s write the polynomial in Let’s write the polynomial in the following waythe following way

Note:Note: “x” could be either the “x” could be either the molar volume, or the density, or molar volume, or the density, or the z-factorthe z-factor

3 2 31 2 0x a x a x a


When the equation is When the equation is expressed in terms of the z expressed in terms of the z factor, the coefficients factor, the coefficients aa11 to to aa33 are:are:

1

22

2 33

( 1)

( 3 2 )

( )

a B

a A B B

a AB B B

Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation

AnalyticallyAnalytically2

2 1

31 2 3 1

3 23

3 23

3

9

9 27 2

54

a aQ

a a a aR

S R Q R

T R Q R

LetLet


AnalyticallyAnalytically

1 1

2 1

3 1

1

31 1 1

32 3 21 1 1

32 3 2

x S T a

x S T a i S T

x S T a i S T

The solutions are,The solutions are,



If If aa11, , aa22 and and aa33 are real and if are real and if D = D = QQ33 + R + R22 is the discriminant, then is the discriminant, then

One root is real and two complex One root is real and two complex conjugate if conjugate if D > 0D > 0;;

All roots are real and at least two All roots are real and at least two are equal if are equal if D = 0D = 0; ;

All roots are real and unequal if All roots are real and unequal if D D < 0< 0..



wherewhere

1 1

2 1

3 1

1 12 cos

3 3

1 1If 0 2 cos 120

3 3

1 12 cos 240

3 3

x Q a

D x Q a

x Q a

3cos

R

Q



where where xx11, , xx22 and and xx33 are the three are the three roots.roots.

1 2 3 1

1 2 2 3 3 1 2

1 2 3 3

x x x a

x x x x x x a

x x x a



The range of solutions that are The range of solutions that are used for the engineer are used for the engineer are those for positive volumes and those for positive volumes and pressures, we are not pressures, we are not concerned about imaginary concerned about imaginary numbers.numbers.

Solutions of a Cubic Solutions of a Cubic PolynomialPolynomial

From the shape of the polynomial we are only

interested in the first

quadrant.

Solutions of a Cubic Solutions of a Cubic PolynomialPolynomial

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 contains contains Fortran codes to solve the Fortran codes to solve the roots of polynomials up to fifth roots of polynomials up to fifth degree.degree.

Web site to download Fortran Web site to download Fortran source codes to solve polynomials source codes to solve polynomials

up to fifth degreeup to fifth degree

EOS for a Pure ComponentEOS for a Pure Component

-10

0

12

A1

A2

Pre

ssu

re

Mo la r V olum e

T2

T1P1

v

L

2 - P hases

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P

-10

12

A1

A2

Pre

ssu

re

112

A1

A2

Pre

ssu

re

Mo la r V olum e

T2

T1P1

v

L

2 - P has

CP

V

L

V

1

2

3

4

7 6

5

0

TV~P

Parameters needed to Parameters needed to solve EOSsolve EOS

Tc, Pc, (acentric factor for some Tc, Pc, (acentric factor for some equations I.e Peng Robinson)equations I.e Peng Robinson)

Compositions (when dealing Compositions (when dealing with mixtures)with mixtures) Specify P and T Specify P and T determine Vm determine Vm

Specify P and Vm Specify P and Vm determine T determine T

Specify T and Vm Specify T and Vm determine P determine P

Tartaglia: the solver of cubic equations

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

Cubic Equation Solver Cubic Equation Solver

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


Phase equilibrium for a single Phase equilibrium for a single component at a given component at a given temperature can be graphically temperature can be graphically determined by selecting the determined by selecting the saturation pressure such that saturation pressure such that the areas above and below the the areas above and below the loop are equal, these are known loop are equal, these are known as the van der Waals loops.as the van der Waals loops.

Two-phase VLETwo-phase VLE

The phase equilibria equations The phase equilibria equations are expressed in terms of the are expressed in terms of the equilibrium ratios, the “K-equilibrium ratios, the “K-values”. values”.

ˆ

ˆ

li i

i vi i

yK

x

Dew Point CalculationsDew Point Calculations

Equilibrium is always stated Equilibrium is always stated as:as:

(i = 1, 2, 3 ,…Nc) (i = 1, 2, 3 ,…Nc)

with the following material with the following material balance constrainsbalance constrains

ˆ ˆl vi i i ix P y P

1 1 1

1, 1, 1Nc Nc Nc

i i ii i i

x y z


At the dew-pointAt the dew-point

ˆ ˆl vi i i i

i i i

x z

x K z

(i = 1, 2, 3 ,…Nc)(i = 1, 2, 3 ,…Nc)


Rearranging, we obtain the Rearranging, we obtain the Dew-Point objective functionDew-Point objective function

1

1 0Nc

i

i i

z

K

Bubble Point Equilibrium Bubble Point Equilibrium CalculationsCalculations

For a Bubble-pointFor a Bubble-point

1

1 0Nc

i ii

z K

Flash Equilibrium Flash Equilibrium CalculationsCalculations

Flash calculations are the work-Flash calculations are the work-horse of any compositional horse of any compositional reservoir simulation package. reservoir simulation package.

The objective is to find the The objective is to find the ffvv in in a VL mixture at a specified T a VL mixture at a specified T and P such thatand P such that

1

( 1)0

1 ( 1)

cNi i

i v i

z K

f K

Evaluation of Fugacity Evaluation of Fugacity Coefficients and K-values from Coefficients and K-values from

an EOSan EOSThe general expression to The general expression to

evaluate the fugacity evaluate the fugacity coefficient for component “coefficient for component “ii” ” isis

fixedT

P

ivi dP

P

RTVRT

0

ˆln

The final expression to The final expression to evaluate the fugacity evaluate the fugacity coefficient using an EOS is.coefficient using an EOS is.

vvtv

inT

v

V

vi ZRTdV

V

RT

n

PRT

tvj

i

vt

lnˆln,

Evaluation of Fugacity Evaluation of Fugacity Coefficients and K-values from Coefficients and K-values from

an EOSan EOS

Documents

PETE 310 Lectures # 32 to 34