Applications of the Peng-Robinson Equation of State - American

8 Applications of the Peng-Robinson Equation of State DONALD B. ROBINSON, DING-YU PENG, and HENG-JOO NG

Downloaded by UNIV OF ARIZONA on November 28, 2012 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch008

University of Alberta, Edmonton, Alberta, Canada

The recently proposed Peng-Robinson equation of state (1) incorporates the best features of the Soave (2) treatment of the Redlich-Kwong (3) equation into a new model which has some signi ficant advantages over earlier two-parameter equation of state models. The purpose of this contribution is to indicate how the equation performs when it is used for calculating fluid thermo dynamic properties for systems of industrial interest. The flexibility of the equation and the generality of the situations for which it can be used to give answers of acceptable reliability at a reasonable cost are illustrated through example calculations of vapor pressure, density, vapor-liquid equilibrium, critical properties, three phase L L G equilibrium for systems containing water, and HLG, HL L G, and HL L equilibrium in hydrate forming systems. The equation shows its best advantages in any situation involving liquid density calculations and in situations near the critical region, but it is usually better than other two para meter models in all regions. In developing the new equation certain criteria were estab lished at the outset. In order for the equation to be acceptable the following conditions had to be met. 1

1

1

2

2

1

2

a. The equation was to be a two-constant equation not higher than cubic in volume. b. The model should result in significantly improved perfor mance in the vicinity of the critical point, in particular with Z and liquid density calculations. c. The constants in the equation should be expressable in terms of P , T and ω. d. The mixing rules for evaluating mixture constants should not contain more than one fitted binary interaction para meter, and if possible this parameter should be temper ature, pressure, and composition independent. c

c

c

200

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

8.

Peng-Robinson Equation of State

ROBINSON E T A L .

e.

201

The equation should be s u f f i c i e n t l y general i n i t s a p p l i c a b i l i t y so that the one equation could be used t o handle a l l f l u i d property c a l c u l a t i o n s f o r systems normally encountered i n the p r o d u c t i o n , t r a n s p o r t a t i o n , or p r o c e s s i n g or n a t u r a l gas, condensate, or other r e l a t e d hydrocarbon mixtures.


The Peng-Robinson Equation The d e t a i l s of the development of the Peng-Robinson (PR) equat i o n are given i n the o r i g i n a l paper ( 1 ) . The f i n a l r e s u l t s are summarized here f o r convenience. The equation has the form:

p

=

_RT

a(T)

v - b

v ( v + b) + b(v - b)

In t h i s equation: RT b = 0.07780 a(T) = a ( T ) c

c • a(T ,a>) R

2 2 R T a(T ) = 0.45724 „ c P c C

Z = 0.307 c a^(T ,co) = 1 + K ( 1 - T S R

R

K = 0.37464 + 1.54226a) - 0.26992a)

2

The values of b and a ( T ) are obtained by equating the f i r s t and second d e r i v a t i v e of pressure w i t h respect t o volume t o zero along the c r i t i c a l isotherm at the c r i t i c a l p o i n t , s o l v i n g the two equations simultaneously f o r a and b, and then u s i n g the equation of s t a t e t o s o l v e f o r Z . The v a l u e of K f o r each pure component was obtained by l i n e a r i z i n g the or2(T ,a)) V S T ^ r e l a t i o n s h i p between the normal b o i l i n g point and the c r i t i c a l p o i n t . This i s s l i g h t l y d i f f e r e n t that the method of Soave (2) which assumed a l i n e a r r e l a t i o n s h i p from T = 1.0 to T = 0.7. The i n f l u e n c e of a) and K was obtained by a l e a s t squares f i t of the data f o r a l l components of i n t e r e s t . c

c

R

R

r

R


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P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

TABLE 1.

Comparison of Vapor Pressure P r e d i c t i o n s (Reprinted w i t h Permission from Ind. Eng. Chem. Fundamentals, 15, 293 (1976a). Copyright by The American Chemical S o c i e t y . ) R e l a t i v e E r r o r , %. No. of Data Points


Component

AAD SRK

BIAS PR

Methane

28

Ethane

27

0 .70

Propane

31

Isobutane n-Butane

RMS PR

SRK

PR

0 .47

0.38

1. 57

0. 77

0.34

-0 .10

-0.34

0.95

0.38

0 .98

0.36

0 .87

0.31

1. 10

0.42

27

1 .06

0.32

0 .82

0.16

1. 18

0.34

28

0 .75

0.37

0 .47

-0.22

0.86

0.42

Isopentane

15

0 .46

0.54

0 .17

-0.53

0.,49

0. 60

n-Pentane

30

0 .92

0.58

0 .50

-0.29

1.02

0. 66

n-Hexane

29

1 .55

0.90

1 .31

0.37

1. 75

1.06

n-Heptane

18

1 .51

0.79

1 .48

0.63

1.88

1. 04

n-Octane

16

1 .99

1.04

1 .97

1.02

2. 24

1. 26

Nitrogen

17

0 .56

0.31

0 .00

-0.02

0.75

0.37

Carbon Dioxide

30

0 .53

0.62

0 .50

-0.49

0.63

0.71

Hydrogen S u l f i d e

30

0 .66

0.96

0 .34

0.42

1.00

1. 48

1 .01

0.60

0 .68

0.11

1.19

0.73

Average, %

1 .44

SRK

0.66

N l a

AAD = ± = L

I d.i

1

BIAS

1=1

RMS

d. = e r r o r f o r each p o i n t l


8.

ROBINSON ET AL.


203

The mixing r u l e s which are recommended are as f o l l o w s : b

= m

a

m


a.

(1 - 6. .) a.' in i

In the equation f o r a y , i s a f i t t e d parameter p r e f e r a b l y obtained by o p t i m i z i n g the p r e d i c t i o n of b i n a r y bubble p o i n t p r e s sures over a reasonable range of temperature and composition. Applications Vapor Pressure. One of the important c o n s i d e r a t i o n s f o r any equation o f s t a t e that i s t o be used f o r v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s i s whether or not i t can a c c u r a t e l y p r e d i c t the vapor pressure of pure substances. Table 1 shows a comparison between the PR equation and the Soave-Redlich-Kwong (SRK) equation f o r p r e d i c t i n g the vapor pressure of ten p a r a f f i n hydrocarbons and three commonly encountered non-hydrocarbons. I t w i l l be noted that the use of the PR equation has improved the RMS r e l a t i v e e r r o r by a f a c t o r of about 40 percent over the SRK p r e d i c t i o n s u s i n g the same data. F u r t h e r , i t i s seen that the SRK p r e d i c t i o n s are biased high i n every case but one and that PR p r e d i c t i o n s are evenly s p l i t between p o s i t i v e and negative departures w i t h a r e s u l t i n g o v e r a l l p o s i t i v e b i a s that i s only 16 percent of the value obtained u s i n g the SRK equation. D e n s i t i e s . I t i s w e l l known that the SRK equation tends to p r e d i c t l i q u i d m o l a l volumes that are too h i g h , and that t h i s i s p a r t i c u l a r l y n o t i c e a b l e i n the v i c i n i t y of the c r i t i c a l p o i n t . The f a c t t h a t the PR equation gives a u n i v e r s a l c r i t i c a l c o m p r e s s i b i l i t y f a c t o r of 0.307 compared t o 0.333 f o r the SRK equation has improved the a b i l i t y of the PR equation t o p r e d i c t l i q u i d d e n s i t i e s . F i g u r e 1 i l l u s t r a t e s the performance of the two equations f o r p r e d i c t i n g the m o l a l volume of s a t u r a t e d l i q u i d s and vapors f o r pure n-pentane. A t reduced temperatures above about 0.8, the average e r r o r i n l i q u i d d e n s i t y has been reduced by a f a c t o r o f about 4 by using the new equation. At lower reduced temperatures, the p r e d i c t i o n s by the new equation are b e t t e r by a f a c t o r of about 2. Both equations g i v e acceptable p r e d i c t i o n s of the vapor d e n s i t y . The a b i l i t y o f the new equation t o p r e d i c t the s p e c i f i c volume of s a t u r a t e d l i q u i d s i n multicomponent systems i s c l e a r l y i l l u s t r a t e d i n F i g u r e s 2 and 3. F i g u r e 2 shows the c a l c u l a t e d l i q u i d volume percent i n the r e t r o g r a d e r e g i o n f o r a 6 component p a r a f f i n hydrocarbon mixture c o n t a i n i n g components from methane through n-decane. F i g u r e 3 shows the same k i n d o f i n f o r m a t i o n f o r a 9 component system


P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY


204




ROBINSON ET AL.

0

5

10 VOLUME

15

20

25

PERCENT LIQUID

Figure 3. Volumetric behavior of nine-component condensate-type fluid containing hydrocarbons, nitrogen, carbon dioxide, and hydrogen sulfide at 250°F


206

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

• YARBOROUGH, 1972 I —PR PREDICTION

O


Applications of the Peng-Robinson Equation of State - American

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