1 derivation of the fugacity coefficient for the peng-robinson ... - fet

8/16/2019 1 Derivation of the Fugacity Coefficient for the Peng-Robinson ... - FET

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University of JordanChemical Engineering Department

Chemical Engineering Thermodynamics 905721

Dr. Ali Khalaf [email protected]

Exam I Solution20/11/2003

1 Derivation of the fugacity coefficient for the

Peng-Robinson EOSThe derivation is similar to the derivation given in class for the van der Waal’sEOS. We need to get the equation of state in its mathematical form combinedwith its associated mixing rules. One of the tricks in the question is that re-gardless of the complexity of the combining rules given, the derivation is doneonce. This is a result of the combining rules being composition independent.Consequently, we need only to derive the expression for the fugacity coefficientonce (not three times for every case).

You can solve this problem using a simple or a hard approach. The simpleapproach includes using the residual Helmholtz free energy and its relation tofugacity. The hard approach involves the brute force calculation of the deriv-atives required within the volume integral for fugacity. I am solving using the

residual Helmholtz free energy approach.

1.1 Obtain the EOS

The Peng-Robinson EOS is given as

P = RT

v − b −

a

v(v + b) + b(v − b) =

RT

v − b −

a

v2 + 2vb − b2 (1)

The mixing rules to be used are

a =mX

i=1

mX

j=1

xixjaij = 1

n2T

mX

i=1

mX

j=1

ninjaij (2)

b =

mX

i=1

mX

j=1

xixjbij = 1n2T

mX

i=1

mX

j=1

ninjbij (3)

With the cross parameters given by the appropriate combining rule. Noticethat the formulation in the expressions for the a and b parameters is similarwhich reduces the amount of derivatives to be evaluated.

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1.2 Convert the EOS to Density and Compressibility No-

tation

The equation of state to be used (regardless of its complexity) needs to be con-verted to the density (specific volume) and compressibility factor. Consequently,the Peng-Robinson EOS becomes

Z = 1

1 − bρ−

aρ

RT

1

1 + 2bρ− b2ρ2 (4)

1.3 Obtain an Expression for Helmholtz Free Energy

We have an equation of state that we can apply for a pure component to obtainthe residual Helmholtz free energy. At a constant temperature and volume wecan use

(a − aIG

)RT

=

ρZ 0

Z − 1ρ

dρ =

bρZ 0

Z − 1bρ

d(bρ) (5)

From the Peng-Robinson EOS

Z − 1 = 1

1 − bρ−

aρ

RT

1

1 + 2bρ− b2ρ2 −

1 − bρ1 − bρ

= bρ

1 − bρ−

aρ

RT

1

1 + 2bρ− b2ρ2 (6)

Substitute this expression into the residual Helmholtz free energy

(a−

aIG)

RT =

bρ

Z 0

1

1 − bρd(bρ) + a

bRT

bρ

Z 0

1

1 + 2bρ− b2ρ2 d(bρ)

The first term to the right hand side of the equal sign is easy to integrate.However, the second can be obtained from the tables of integration formulas as

Z 1

a0x2 + b0x + c0dx =

1

(b02 − 4a0c0)1/2 ln

¯̄¯̄¯

2a0x + b0 −¡

b02 − 4a0c0¢1/2

2a0x + b0 + (b02 − 4a0c0)1/2

¯̄¯̄¯ (7)

This result applies when the discriminator is negative i.e.,

4a0c0 − b02


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Substituting back and carrying out the first integral to get

(a− aIG)RT

= − ln(1 − bρ) − a√ 8bRT

ln"

1 + (1 + √ 2)bρ1 + (1 −

√ 2)bρ

# (9)

We can use the compressibility factor, and the A and B factors to reduce thisequation into dimensionless form. Notice that the A factor is not the Helmholtzfree energy to avoid confusion between symbols.

(a− aIG)RT

= Z − 1 − ln(Z −B) −A

√ 8B

ln

"Z + (1 +

√ 2)B

Z + (1 −√

2)B

# (10)

1.4 Obtain an Expression for the Fugacity Coefficient

Using the relationship between fugacity coeffi

cient and residual Helmholtz freeenergy we have

lnφi =

µ ∂

∂ ni

(A−AIG)RT

¶T,V,nj

− lnZ (11)

Consequently, we need to put the residual Helmholtz free energy into its exten-sive form by multiplying by the total number of moles

(A−AIG)RT

= −nT ln(1−bρ)−an2T √

8bRTnT

hln³

1 + (1 +√

2)bρ´− ln

³1 + (1 −

√ 2)bρ

´i(12)

Carry out the diff erentiation with respect to the number of moles of any arbi-trary component

µ∂ (A−AIG)/RT

∂ ni

¶T,V,nj

= − ln(1− bρ) + n(1− bρ)

µ∂ (bρ)∂ ni

¶ (13)

−an2T √

8bRTnT

(1 +

√ 2)³

∂ (bρ)∂ ni

´1 + (1 +

√ 2)bρ

−(1−

√ 2)³

∂ (bρ)∂ ni

´1 + (1 −

√ 2)bρ

− ln

"1 + (1 +

√ 2)bρ

1 + (1 −√

2)bρ

#³

∂ (an2T )∂ ni

´√

8bRTnT −

an2T √ 8RT

³∂ (nT b)∂ ni

´(nT b)2

This contribution can be evaluated if the derivatives with respect to thenumber of moles are obtain. The next section derives general expressions forthe quadratic mixing rules.

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1.5 Obtain Derivatives of the Mixing Rules

The following derivatives are required to be obtained from the mixing rules

µ∂ (n2T a)

∂ ni

¶nj

=

∂

"mPi=1

mPj=1

ninjaij

#

∂ ni

nj

=

∂

"mPi=1

ni

mPj=1

njaij

#

∂ ni

njµ∂ (n2T a)

∂ ni

¶nj

= 2mXk=1

nkaik. (14)

Also,

µ∂ (nT b)

∂ ni

¶nj

=

∂ " 1nT

mPi=1

mPj=1

ninjbij#

∂ ni

nj

= 1

nT

∂

"mPi=1

mPj=1

ninjbij

#

∂ ni

nj

−

1

n2T

mXi=1

mXj=1

ninjbij

= 1nT

∂ "m

Pi=1nim

Pj=1njbij#∂ ni

nj

−b

µ∂ (nT b)

∂ ni

¶nj

= 2mXk=1

xkbik − b. (15)

For a single summation:

µ∂ (nT b)

∂ ni

¶nj

=

∂ (mPi=1

nibi)

∂ ni

nj

= bi (16)

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1.6 Plug into the Fugacity Coefficient Expression

The derivatives are ready to be plugged into the fugacity coefficient expressionsas follows

lnφi = − lnZ − ln(1− bρ) +ρ

µ2

mPk=1

xkbik − b¶

(1 − bρ)

−ρa

µ2

mPk=1

xkbik − b¶

√ 8bRT

" (1 +

√ 2)

1 + (1 +√

2)bρ−

(1 −√

2)

1 + (1 −√

2)bρ

#

− ln

"1 + (1 +

√ 2)bρ

1 + (1 −√

2)bρ

#

2mPk=1

xkaik

√ 8bRT

−a

√ 8bRT

µ2

mPk=1

xkbik − b¶

b

Simplifying this equation further by taking a common factor in the last term toobtain

lnφi = − lnZ − ln(1− bρ) +ρ

µ2

mPk=1

xkbik − b¶

(1 − bρ)

−ρa

µ2

mPk=1

xkbik − b¶

√ 8bRT

" (1 +

√ 2)

1 + (1 +√

2)bρ−

(1 −√

2)

1 + (1 −√

2)bρ

#

−a

√ 8bRT ln"1 + (1 +

√ 2)bρ

1 + (1 −√ 2)bρ#

2

m

Pk=1xkaik

a − µ

2m

Pk=1xkbik − b¶b

To simplify the notation further define

bi =

Ã2

mXk=1

xkbik − b

! (17)

ai =mXk=1

xkaik (18)

From which

lnφi = − lnZ − ln(1 − bρ) + biρ

(1−bρ)

−ρabi√ 8bRT

" (1 +

√ 2)

1 + (1 +√

2)bρ−

(1−√

2)

1 + (1 −√

2)bρ

#

−a

√ 8bRT

ln

"1 + (1 +

√ 2)bρ

1 + (1 −√

2)bρ

#·2aia −

bi

b

¸

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There is another simplification that can be carried out

biρ

(1 − bρ) − ρabi√

8bRT

" (1 + √ 2)

1 + (1 +√

2)bρ− (1

−√ 2)1 + (1 −

√ 2)bρ

#

= bi

b

· bρ

(1 − bρ) −

ρab√

8bRT

µ 1

1 + 2bρ− b2ρ2

¶¸ =

bi

b (Z − 1)

Consequently,

lnφi = − lnZ − ln(1 − bρ) + biρ

(1− bρ) +

bi

b (Z − 1) (19)

−a

√ 8bRT

ln

"1 + (1 +

√ 2)bρ

1 + (1 −√

2)bρ

#·2aia −

bi

b

¸.

To have a more compact notation, define the variables in terms of the reducingvariables A, B, and Z as follows

bρ = B

Z ;

a

bRT =

A

B; ai

a =

Ai

A ; bi

b =

Bi

B .

Applying these transformations, we end up with the desired expression for thefugacity coefficient using the Peng-Robinson equation of state. This expression isderived for quadratic mixing rules for the co-volume and the energy parameters.It simplifies a little computationally if we use arithmetic averages for the co-volume.

lnφi =−

ln(Z −B)+

Bi

B (Z

−1)−

A

√ 8Bln"

Z + (1 +√

2)B

Z + (1 −√ 2)B#·

2AiA −

Bi

B ¸ . (20)where

B = bP

RT ; (21)

Bi =

µ2

mPk=1

xkbik − b¶P

RT (22)

b = 0.07780RT c

P c(23)

A = aP

(RT )2; (24)

Ai =

µ mPk=1

xkaik

¶P

(RT )2 ; (25)

a(T ) = 0.45724(RT c)2

P cα(T ); (26)

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p α(T ) = 1 + κÃ1−r T

T c! ; (27)κ = 0.37464 + 1.5422ω − 0.26992ω2. (28)

2 Algorithm for using the Fugacity Expression

The expression for the fugacity coefficient and its associated variables and equa-tions needs to be solved to obtain a value for the fugacity. This is not hardcomputationally. However, one of the main points encountered frequently is thesolution for the roots of a cubic equation. The provided notes are implementedin the Excel worksheet to obtain the roots of the cubic at any given temperatureand pressure.

3 Mixing Rules

Three mixing rules were used to solve the exam statement.

• Lorentz-Berthelot

bij = 1

2(bii + bjj), (29)

aij = (aiiajj )1/2

. (30)

• Waldman-Hagler

bij =

Ãb2ii + b

2

jj

2

!1/2, (31)

aij = (aiiajj)1/2

Ãbiibjj

b2ij

! . (32)

• MADAR-1

bij = 1

3

2XL=0

Ã0.25 (bii + bjj )

2

bL/3ii b

L/3jj

! 12−2L/3

(33)

aij = (aiiajj)1/2

Ãbiibjj

b2ij

!. (34)

4 Isobutane and Carbon Dioxide System

The exam asked to generate plots for the fugacity as a function of compositionand pressure for the system: isobutane and carbon dioxide. The required inputinformation are given below for the two components.

The fugacity coefficient as obtained from the Peng-Robinson equation of state was evaluated as a function of pressure and composition using Microsoft

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Table 1: Critical parameters, and the acentric factors of carbon dioxide and

i-butane. Component T c (K) P c (MPa) ω (-) Z c (-)Carbon dioxide 304.14 7.375 0.239 0.274145Isobutane 407.8 3.604 0.183 0.275296

Excel. The Excel worksheet is obtained and modified from the textbook of Elliot and Lira.

Figures 1 through 3 present the final plots of fugacity for isobutane andcarbon dioxide respectively as a function of composition and pressure. Figure1 is plotted using the Lorentz-Berthelot set of mixing rules, Figure 2 is plottedusing the Waldman-Hagler, while Figure 3 is plotted using the MADAR-1 setof mixing rules. Additionally, Figure 4 is plotted to show the eff ect of mixing

rules on the same graph at pressures of 1, 2, and 4 MPa.

4.1 Pure Component Limit

From the first three figures, it is evident that at the pure component limit, thefugacity of CO2 approaches the pressure of the system. This is to be expectedsince CO2 is supercritical at the temperature of the system. However, for theisobutane it seems that the fugacity is almost an order of magnitude lower thanthe system pressure. Isobutane being a liquid at the given temperature explainsthe low fugacity of at the pure component limit. Furthermore, there is a linearcomposition dependence at any given pressure as the pure component limit isapproached.

4.2 Influence of System Pressure

The pressure aff ects the fugacity of both components appreciably. However, theinfluence of the pressure may be divided into two main regions:

• At pressures below approximately 1 MPa, the fugacity is a linear functionof the composition in all the composition range.

• At pressures above approximately 1 MPa, the fugacity begins to show somesort of going through a maximum, rising again, going through a minimum,then it begins to increase again. This is due to the mixture being avapor approaching its dew pressure at compositions rich in CO2 that isbeing enriched with isobutane tending to lower the fugacity. Consequently,condensing the mixture to a liquid phase at isobutane rich areas.

At very low pressures, the ideal gas mixture limit is approached. This leadsto the fugacity of any component being equal to its partial pressure. Thishappens at a pressure of 0.001 MPa which is sufficiently low to guarantee theapplication of the ideal gas mixture limit. This pressure was given to check

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the results obtained from the Excel sheet. The ideal gas mixture and pure

component limits are useful safeguards for checking and debugging the codes

used.

4.3 Eff ect of Diff erent Mixing Rules

The figures provided; indicate that there is an eff ect of the mixing rules used.

There are minor diff erences between the Waldman-Hagler and MADAR-1 rules.

However, both rules feature some major diff erences with the Lorentz-Berthelot

rules. These diff erences increase close to regions where there is a liquid phase.

The Lorentz-Berthelot rules provides lower values of the fugacity compared to

the Waldman-Hagler and MADAR-1 rules. This may be explained by the over-

estimation of repulsive forces using the Lorentz-Berthelot rules.

The diff erences between the Lorentz-Berthelot rules and the two other rules

is quantitative and qualitative. The trends predicted and locations of phase

changes are contradictory among these rules. These diff erences increase withthe pressure i.e., increase when the Lorentz-Berthelot rules results are not as

good as the gas limits. The diff erence in the values of fugacity amounts to

threefold diff erence. Consider P = 4 MPa, the maximum fugacity of isobutane

predicted by the Lorentz-Berthelot rules is about 0.2 MPa at x = 0.15, while

that predicted by Waldman-Hagler and MADAR-1 rules is 0.57 MPa at x =

0.37! Additionally, the Lorentz-Berthelot rules show an almost constant value of

carbon dioxide fugacity over the region where minimum and maximum fugacities

in isobutane fugacity occurs. This is some sort of an inconsistency compared to

the inverted trends predicted by the Waldman-Hagler and MADAR-1 rules.

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Mole fraction of isobuatne

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

F u g a c

i t y o f i s o b u t a n e ( M P a )

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa

Mole fraction of isobutane

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

F u g a c i t y o f c a r b o n d i o x i d e ( M P a )

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa

Figure 1: Fugacities of isobuatne and carbon dioxide as a function of composition

and pressure using the Lorentz-Berthelot mixing rules.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

F u g a c


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


0

1

2

3

4

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa


and pressure using the Waldman-Hagler mixing rules.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

F u g a c


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


0

1

2

3

4

0.001 MPa

0.100 MPa

1.000 MP

2.000 MPa

4.000 MPa


and pressure using the MADAR-1 mixing rules.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

F u g a c i t y o f i s o b u t a n e ( M P a )

0.0

0.2

0.4

0.6

0.8L-B, 1 MPa

L-B, 2 MPa

L-B, 4 MPa

W-H, 1 MPa

W-H, 2 MPa

W-H, 4 MPa

MADAR-1, 1 MPa

MADAR-1, 2 MPa

MADAR-1, 4 MPa


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


0

1

2

3

4L-B, 1 MPa

L-B, 2 MPa

L-B, 4 MPa

W-H, 1 MPa

W-H, 2 MPa

W-H, 4 MPa

MADAR-1, 1 MPa

MADAR-1, 2 MPa

MADAR-1, 4 MPa

Figure 4: The eff ect of diff erent mixing rules on the fugacity of isobutane and

carbon dioxide at pressures of 1, 2, and 4 MPa.

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1 derivation of the fugacity coefficient for the peng-robinson ... - fet

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