Equations of State - American Chemical Society

19 Optimal Temperature-Dependent Parameters for the Redlich-Kwong Equation of State

Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

Randall W. Morris and Edward A. Turek Amoco Production Company, Tulsa, OK 74102

Redlich-Kwong "a" and "b" parameters for several light hydrocarbons, carbon dioxide, nitrogen, and hydrogen sulfide have been optimized at many temperatures using accurate correlations of PVT data available in the l i t erature. The regressed parameter values have been fit as functions of temperature for convenient use in cal culations. The accuracy of volumetric calculations using the new parameters is compared to that obtained using the original Redlich-Kwong equation and the Soave and Yarborough modifications. The new parameters sig nificantly improve the accuracy of volumetric calcula tions using the Redlich-Kwong equation. One a p p l i c a t i o n o f e q u a t i o n s o f s t a t e i n the p e t r o l e u m i n d u s t r y i s i n the n u m e r i c a l modeling o f p e t r o l e u m r e s e r v o i r s . S i n c e t h e s e mathe m a t i c a l models a r e v e r y complex, a r e l a t i v e l y s i m p l e e q u a t i o n o f s t a t e i s needed f o r computer s o l u t i o n o f the model t o be p r a c t i c a l . The Redlich-Kwong e q u a t i o n (1.) i s a common form o f e q u a t i o n o f s t a t e used i n such s i t u a t i o n s . The e q u a t i o n , RT

a

(1)

Ρ = V-b

5

T°* V(V+b)

r e l a t e s t h e p r e s s u r e , molar volume, and temperature o f a f l u i d u s i n g the two p a r a m e t e r s , " a " and "b". F o r a m i x t u r e , t h e v a l u e s o f t h e s e parameters a r e determined by combining t h e parameters o f t h e i n d i v i d u a l components. Thus, i n g e n e r a l , m i x t u r e c a l c u l a t i o n s w i l l y i e l d the b e s t r e s u l t s when t h e i n d i v i d u a l component parameters p r o v i d e t h e best f i t s t o t h e p r o p e r t i e s o f t h o s e components. A c c u r a t e c o r r e l a t i o n s o f PVT d a t a a r e a v a i l a b l e from IUPAC ( 2 - 4 ) f o r carbon d i o x i d e , n i t r o g e n , and methane, t h e N a t i o n a l Bureau o f Standards (5-8) f o r ethane, propane, i s o b u t a n e , and normal butane, and t h e Texas A&M U n i v e r s i t y Thermodynamics R e s e a r c h C e n t e r (9) f o r hydrogen s u l f i d e . We have used d a t a from t h e s e c o r r e l a t i o n s t o e v a l uate t h e a c c u r a c y o f the Redlich-Kwong e q u a t i o n o f s t a t e and t o

0097-6156/86/0300-0389$06.00/0 © 1986 American Chemical Society

In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

390

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

d e v e l o p new parameters. S i n c e the q u a l i t a t i v e r e s u l t s were the same f o r a l l s u b s t a n c e s s t u d i e d , a d e t a i l e d a n a l y s i s of the r e s u l t s w i l l be p r e s e n t e d o n l y f o r carbon d i o x i d e .


Background In the o r i g i n a l development of the Redlich-Kwong e q u a t i o n of s t a t e , the parameters " a " and "b" were s p e c i f i e d to be c o n s t a n t f o r a g i v e n fluid. For a pure s u b s t a n c e , they were d e f i n e d i n terms of the c r i t i c a l temperature and c r i t i c a l p r e s s u r e of the s u b s t a n c e . These v a l u e s o f the parameters f o r carbon d i o x i d e are r e p r e s e n t e d by the h o r i z o n t a l l i n e s i n F i g u r e s 1 and 2, and w i l l be r e f e r r e d to as the " s t a n d a r d " Redlich-Kwong parameters i n the f o l l o w i n g d i s c u s s i o n . E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g these parameters a r e shown i n F i g u r e s 3 and 4. Note the l a r g e e r r o r s i n the c a l c u l a t i o n of l i q u i d volumes. In the Soave m o d i f i c a t i o n (10,11 ) of the Redlich-Kwong e q u a t i o n , the " a " parameter i s r e p l a c e d w i t h a f u n c t i o n of temperature which i s a d j u s t e d to f i t vapor p r e s s u r e s . The "b" parameter r e t a i n s i t s s t a n dard v a l u e . T h i s m o d i f i c a t i o n i s i n t e n d e d to improve phase e q u i l i brium c a l c u l a t i o n s and r e s u l t s i n l i t t l e improvement i n the v o l u metric c a l c u l a t i o n s . F i g u r e 1 shows the temperature dependence of the Soave " a " parameter c o n s i s t e n t w i t h E q u a t i o n 1. F i g u r e s 5 and 6 show the e r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g the Soave parameters. Z u d k e v i t c h and J o f f e (_12) a l l o w e d both parameters to v a r y w i t h temperature below the c r i t i c a l temperature, d e t e r m i n i n g t h e i r v a l u e s from the p r e s s u r e , l i q u i d d e n s i t y , and f u g a c i t y c o e f f i c i e n t at s a t u r a t i o n f o r a substance at a g i v e n temperature. They used the v a l u e s of the parameters determined at the c r i t i c a l temperature f o r temperat u r e s above the c r i t i c a l . Yarborough ( O ) a p p l i e d t h i s method u s i n g g e n e r a l i z e d c o r r e l a t i o n s to c a l c u l a t e vapor p r e s s u r e , s a t u r a t e d l i q u i d d e n s i t y , and f u g a c i t y c o e f f i c i e n t . The r e s u l t i n g c o r r e l a t i o n e x p r e s s e s the e q u a t i o n of s t a t e parameters i n d i m e n s i o n l e s s form as f u n c t i o n s o n l y of reduced temperature and a c e n t r i c f a c t o r . V a l u e s of t h e s e parameters f o r C O 2 a r e i n c l u d e d i n F i g u r e s 1 and 2 f o r compari s o n to the s t a n d a r d and Soave parameters. F i g u r e s 7 and 8 show the e r r o r s i n v o l u m e t r i c c a l c u l a t i o n s f o r C O 2 u s i n g these parameters. Note t h a t the a c c u r a c y of the l i q u i d volume c a l c u l a t i o n s i s improved considerably. New

Parameters

While the Yarborough parameters o f f e r s u b s t a n t i a l improvement o v e r the s t a n d a r d and Soave parameters at temperatures below the c r i t i c a l t e m p e r a t u r e , a f u r t h e r improvement i s p o s s i b l e by a l l o w i n g the parame t e r s t o v a r y w i t h temperature above the c r i t i c a l a l s o . We have done t h i s f o r s e v e r a l s u b s t a n c e s , l i s t e d above, which a r e commonly encount e r e d at s u p e r c r i t i c a l temperatures. V a l u e s of the parameters were d e t e r m i n e d by o p t i m i z i n g the f i t of the Redlich-Kwong e q u a t i o n to single-phase volumetric data. We a l s o determined new v a l u e s of the parameters at s u b c r i t i c a l temperatures u s i n g both s i n g l e - p h a s e v o l u m e t r i c d a t a and s a t u r a t i o n d a t a . O p t i m a l v a l u e s of the parameters were determined a t many temperatures. For each temperature, 197 pressure-volume d a t a p o i n t s were


MORRIS A N D TUREK

Parameters for the Redlich-Kwong

T = T

360000·

Equation

Legend:

C

S t a n d a r d Redlich-Kwong _ Soave Redlich-Kwong

_


Yarborough

-100

0

100

200

300

400

500

600

700

800

Temperature, *F Figure

1.

C O 2 " a " Parameter f o r t h e Redlich-Kwong

Λ

Equation

= Τ

Legend: Standard Redlich-Kwong Yarborough New Parameters . -100

0

1

100

200

1 300

1

1

400

500

600

700

800

e

Temperature, F F i g u r e 2.

C O 2 "b

,f

Parameter f o r t h e Redlich-Kwong

Equation



EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

10-H - 1 0 0

1 0

h 100

1

1

1

1

1

1

|

J

1

200

300

400

500

600

700

800

900

1 1000

1 1100

1 1200

Temperature, °F F i g u r e 3. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h S t a n d a r d Parameters

3

Volume, ft /lbmol F i g u r e 4. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h S t a n d a r d Parameters



MORRIS A N D TUREK

10-Η -100

1 0

h 100


1

1

1

1

1

200

300

400

500

600

1 700

r

800

Equation

1 900

1 1000

1 1100

1 1200

e

Temperature, F F i g u r e 5. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Soave Parameters

F i g u r e 6. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Soave Parameters



394


100

200

300

400

500

600

700

600

800

1000

1100

1200

Temperature, °F Figure 7 . E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Yarborough Parameters

Γ

T=T

C

3

Volume, ft /lbmol F i g u r e 8 . E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h Yarborough Parameters



19.


MORRIS A N D TUREK

Equation

395

o b t a i n e d from t h e c o r r e l a t i o n s . The d i s t r i b u t i o n o f these p o i n t s was l i n e a r w i t h r e s p e c t t o l o g - l o g o f volume i n o r d e r t o g i v e t h e d e s i r e d d i s t r i b u t i o n o f l i q u i d and vapor volumes. Any o f t h e 197 p o i n t s t h a t f e l l i n t h e two-phase r e g i o n f o r t h e g i v e n temperature were e l i m i nated. F o r temperatures below t h e c r i t i c a l temperature, the s a t u r a t i o n p r e s s u r e and s a t u r a t e d l i q u i d and vapor volumes were a l s o used r e s u l t i n g i n a maximum o f 200 d a t a p o i n t s a t a g i v e n temperature. V o l u m e t r i c d a t a were weighted a c c o r d i n g t o t h e d e v i a t i o n o f t h e Ζ f a c t o r from 1, s i n c e t h e f i t t o d a t a e x h i b i t i n g i d e a l gas b e h a v i o r was found t o be i n s e n s i t i v e t o t h e v a l u e s o f t h e e q u a t i o n o f s t a t e parameters. W e i g h t i n g s o f t h e s a t u r a t i o n p r o p e r t y d e v i a t i o n s were a d j u s t e d by t r i a L t o g i v e a r e a s o n a b l e b a l a n c e w i t h t h e v o l u m e t r i c data. The f i n a l o b j e c t i v e f u n c t i o n minimized a t each temperature was 2

F = Σ[(Ζ-1) Δν]

2

2

+ (16ΔΡ

) + (25Δν

sat where the Δ q u a n t i t i e s a r e f r a c t i o n a l

ΔΧ = (Χ

Λ

calc

Λ

)

2

+ (2Δν ^

sat,l

- X

)

2

(2)

sat,ν

deviations, i . e . , )/X exp exp

f o r a g i v e n q u a n t i t y X. Parameters were c o n s t r a i n e d d u r i n g t h e r e g r e s s i o n t o be c o n s i s t e n t w i t h t h e e x p e r i m e n t a l c r i t i c a l tempera t u r e , i . e . , parameter v a l u e s y i e l d i n g t h r e e r o o t s i n t h e e q u a t i o n o f s t a t e above t h e e x p e r i m e n t a l c r i t i c a l temperature o r o n l y one r o o t below t h e e x p e r i m e n t a l c r i t i c a l temperature were r e j e c t e d . The o p t i m i z e d parameter v a l u e s were f i t as f u n c t i o n s o f tempera t u r e f o r each s u b s t a n c e . Since petroleum r e s e r v o i r a p p l i c a t i o n s of e q u a t i o n s o f s t a t e a r e g e n e r a l l y a t a c o n s t a n t temperature, t h e p a r ameters must be determined o n l y once f o r a l a r g e number o f c a l c u l a tions. F o r t h i s r e a s o n , t h e f u n c t i o n a l forms used t o f i t t h e o p t i mized parameter v a l u e s c o u l d be made as complex as n e c e s s a r y f o r the d e s i r e d a c c u r a c y w i t h o u t a d v e r s e l y a f f e c t i n g computing t i m e s . I t s h o u l d a l s o be noted t h a t t h e method used t o d e t e r m i n e these parame t e r s r e s u l t s i n a d i s c o n t i n u i t y i n t h e temperature d e r i v a t i v e s o f t h e parameters a t t h e c r i t i c a l t e m p e r a t u r e . T h i s i s not a problem f o r c o n s t a n t temperature a p p l i c a t i o n s but may need t o be c o n s i d e r e d f o r c a l c u l a t i o n s r e q u i r i n g t h e temperature d e r i v a t i v e s o f t h e e q u a t i o n o f s t a t e p a r a m e t e r s , e.g., e n t h a l p y c a l c u l a t i o n s . The f i t s t o t h e o p t i mized parameters a r e d e s c r i b e d i n t h e Appendix. The new parameters f o r C O 2 a r e compared t o t h e s t a n d a r d v a l u e s , and t h e Soave and Yarborough v a l u e s i n F i g u r e s 1 and 2. E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s u s i n g t h e new parameters a r e shown i n F i g u r e s 9 and 10. Note t h a t t h e l a r g e e r r o r s i n c a l c u l a t e d volumes a r e now c o n f i n e d t o t h e near p r o x i m i t y o f t h e c r i t i c a l p o i n t where they a r e u n a v o i d a b l e w i t h t h e Redlich-Kwong e q u a t i o n o f s t a t e . The i n c l u s i o n o f v o l u m e t r i c d a t a i n t h e parameter r e g r e s s i o n s a t s u b c r i t i c a l temperatures s t i l l a l l o w s a r e a s o n a b l e f i t t o t h e s a t u r a t i o n p r e s s u r e s as shown i n F i g u r e 11. S a t u r a t i o n p r e s s u r e s f o r C O 2 c a l c u l a t e d u s i n g t h e new parameters show an average e r r o r o f about 0.3%. Improvements i n t h e a c c u r a c y o f v o l u m e t r i c c a l c u l a t i o n s f o r t h e o t h e r s u b s t a n c e s s t u d i e d a r e s i m i l a r t o those o b t a i n e d f o r C 0 2 E r r o r s i n v o l u m e t r i c c a l c u l a t i o n s a t s e l e c t e d c o n d i t i o n s f o r a l l sub s t a n c e s s t u d i e d a r e shown i n T a b l e s I through V I I I a l o n g w i t h average e r r o r s over t h e e n t i r e range o f p r e s s u r e s and volumes c o n s i d e r e d .


396

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

r

T=T

C

1

H Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch019

ι m

\ in J 1 V \}f/ Ύ

Y - 1 0 0

0

1

j-

Î

1 • 100

„

=

! 200

1 300

I 400

1 500

1

1

600

700

1• BOO

E Z 3 0.2 to 0.5% WM\ 0.5 to 2.0% i H 2.0 to 5.0% H i 5.0 to 10.0% • ϋ Over 10.0%

! 000

1

1

1000

1100

1 1200

e

Temperature, F F i g u r e 9. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h New Parameters

3

Volume, ft /lbmol F i g u r e 10. E r r o r i n M o l a r Volume C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n w i t h New Parameters


19.

MORRIS A N D TUREK


Equation


Legend: S t a n d a r d Redlich-Kwong _ Suave Redlich-Kwong

\^

_

V

Yarborough

\

New j ) a r a meters

\^

\

-4H

0

Û

20

40

60

80

100

Temperature, *F F i g u r e 11. E r r o r i n Vapor P r e s s u r e C a l c u l a t e d f o r C O 2 u s i n g t h e Redlich-Kwong E q u a t i o n


397

398

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

Table I .

Error

i n C a l c u l a t e d Volume f o r Carbon

Dioxide

r

r

Standard Parameters

Soave Parameters

Yarborough Parameters

New Parameters

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50 2.00 2.00 2.00

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

16.68 11.66 5.91 0.10 12.92 3.89 0.59 0.16 2.79 0.89 3.38 1.28 0.75 2.67 4.88

11.49 8.65 4.90 0.42 16.46 4.39 0.49 11.47 4.50 0.70 4.14 5.34 0.71 2.91 5.03

0.17 1.20 3.33 0.61 14.70 6.09 0.93 0.39 4.84 1.00 3.64 0.22 0.78 2.70 3.95

0.27 1.72 3.96 1.07 2.13 4.52 0.27 0.38 3.84 0.41 1.07 0.37 0.34 0.91 0.03

3.15

4.10

2.19

1.02

Τ


Percent

ρ

Average

Table I I .

Percent

Error in

C a l c u l a t e d Volume f o r N i t r o g e n

r

r

Standard Parameters

Soave Parameters


New Parameters

2.00 2.00 2.00 3.00 3.00 3.00

0.5 2.0 10.0 0.5 2.0 10.0

0.32 1.06 2.78 0.30 1.10 3.83

0.26 1.20 1.16 0.20 0.71 0.79

0.32 1.08 2.95 0.30 1.12 3.92

0.29 0.72 0.56 0.04 0.09 0.14

1.85

0.88

1.91

0.20

Τ

ρ

Average

Table I I I .

Percent

Error in

C a l c u l a t e d Volume f o r Methane

r

r

Standard Parameters

Soave Parameters


New Parameters

1.15 1.15 1.15 1.50 1.50 1.50 2.00 2.00 2.00 3.00 3.00

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0

0.35 7.56 0.38 0.23 0.31 0.63 0.21 0.36 0.62 0.21 0.70

0.27 8.55 0.50 0.08 1.88 1.87 0.22 1.35 2.39 0.20 0.80

0.29 7.08 0.37 0.21 0.30 0.04 0.20 0.38 1.01 0.21 0.73

1.01 0.63 0.22 0.65 1.62 0.81 0.28 0.56 0.21 0.02 0.01

1.12

1.72

1.11

0.75

Τ

ρ

Average


19.

MORRIS A N D TUREK

T a b l e IV.


τI r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

ρ r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average

T a b l e V. τ 1

r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

ρ r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average

Table VI. τ 1

r 0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15

ρ r r 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Average


Percent

Equation

399

E r r o r i n C a l c u l a t e d Volume f o r Ethane

Standard Parameters 11.11 6.83 1.96 0.17 11.24 1.44 0.46 2.90 1.10 0.57 1.67 0.71

Soave Parameters 9.17 5.74 1.60 0.06 12.73 1.65 0.05 8.58 1.87 0.29 2.58 2.81

Yarborough Parameters 1.53 0.76 3.82 0.33 11.66 2.04 0.56 2.73 1.66 0.61 1.77 0.30

New Parameters 0.31 2.34 5.13 0.59 2.48 4.87 0.07 0.40 3.95 0.49 1.13 0.61

2.45

3.14

1.53

1.03

Percent

E r r o r i n C a l c u l a t e d Volume f o r Propane

Standard Parameters 13.72 8.88 3.32 0.29 12.01 1.98 0.54 1.12 1.24 0.49 2.85 1.56

Soave Parameters 10.37 6.97 2.69 0.06 14.37 2.31 0.22 9.40 2.40 0.70 2.83 3.27

Yarborough Parameters 1.33 0.78 3.77 0.59 13.09 3.28 0.74 0.98 2.46 0.55 3.01 0.66

New Parameters 0.41 2.31 5.08 0.68 2.93 4.98 0.17 0.60 2.62 0.31 1.72 1.03

3.65

4.24

2.00

1.33

Percent

E r r o r i n C a l c u l a t e d Volume f o r Isobutane

Standard Parameters 14.88 9.65 3.55 0.60 11.81 1.85 0.91 0.11 0.91

Soave Parameters 10.68 7.25 2.76 0.17 14.69 2.26 0.00 9.67 2.31

Yarborough Parameters 0.77 1.29 4.34 0.99 13.21 3.53 1.17 0.27 2.48

New Parameters 0.37 2.32 5.26 0.24 4.11 4.10 0.76 0.67 0.55

4.42

4.82

2.20

1.64


E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

400

Table


τ 1

VII.

Percent

r

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15

r

VIII.

r

0.85 0.85 0.85 1.05 1.05 1.05 1.15 1.15 1.15 1.50 1.50 1.50

Parameters

Butane

Yarborough

New

Parameters

Parameters

0.37 2.19 5.00 0.66 3.75 4.35 0.37 0.69 1.22

5.07

5.58

2.44

1.59

Error

Volume

in Calculated

Standard

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Normal

1.31 0.68 3.69 0.78 14.60 4.69 0.96 0.32 3.58

Soave

Parameters

r

for

11.68 8.28 3.77 0.12 16.21 3.28 0.29 11.04 3.36

Percent

ρ

XT

Volume

16.28 10.92 4.65 0.34 13.06 2.84 0.68 0.50 1.85

Average

τ 1

Calculated Soave

Parameters

0.5 2.0 10.0 0.5 2.0 10.0 0.5 2.0 10.0

Table

in

Standard

ρ

Jr

Error

Parameters

for

Hydrogen

Sulfide

Yarborough

New

Parameters

Parameters

9.75 6.28 2.34 0.28 11.22 2.20 0.08 6.42 2.14 0.24 0.19 1.46

8.05 5.32 2.02 0.49 12.58 2.40 0.40 11.85 2.86 0.58 3.89 4.85

0.68 0.98 3.29 0.14 11.53 2.69 0.16 6.23 2.59 0.27 0.27 1.80

0.39 2.06 4.37 0.63 3.23 2.79 0.07 0.68 3.64 0.11 0.20 2.84

2.87

3.79

1.79

0.93

Average

Conclusion The

accuracy

tion

of

state

parameters

to

temperature. in

the

made

volumetric

was

improved

vary By

without for

with of

the

s a c r i f i c i n g

developed

convenient

in use

calculations

the this

by

both

saturation new

in

using

significantly

temperature

u t i l i z i n g

development

parameters ture

of

above

data

parameters, f i t work

to

vapor

were

the

f i t

Redlich-Kwong

allowing and

as

well

these

the

"a"

below

the

as

The

functions

of

tempera

Nomenclature =

parameter

for

equation

1,

psia

f t

b

=

parameter

for

equation

1,

ft /lbmol

6

°R

0

,

5

/lbmol

data

were

optimized

calculations.

a

"b"

c r i t i c a l

volumetric

improvements

pressures. as

equa

and

2

3


19.


b*

MORRIS A N D TUREK

= limiting


v a l u e f o r parameter

F

= objective function

ΔΡ

= fractional

ΔΤ

= (T/T ) - 1 c

Δν

= fractional

f o r parameter

i n molar volume

= pressure, psia

P^

= reduced p r e s s u r e , P/P

calculation

pressure, psia

R

= gas c o n s t a n t , p s i a

Τ

= t e m p e r a t u r e , °R = critical

optimization

deviation

= critical

c

3

i n pressure c a l c u l a t i o n

P^

401

b, f t / l b m o l

deviation

Ρ

T

Equation

c

3

f t / l b m o l °R

t e m p e r a t u r e , °R

= r e d u c e d t e m p e r a t u r e , T/T^ V

= molar volume,

Ζ

= compressibility

3

ft /lbmol factor,

PV/RT

Subscripts c = c r i t i c a l point calc = calculated exp = experimental 1 = liquid r = reduced sat = saturation ν = vapor

Appendix The o p t i m i z e d Redlich-Kwong " a " and "b" parameters d e t e r m i n e d i n t h i s work were f i t as f u n c t i o n s o f t e m p e r a t u r e . The f u n c t i o n s used i n t h e s e f i t s a r e e m p i r i c a l and no p h y s i c a l s i g n i f i c a n c e s h o u l d be a t t a c h e d to t h e i r forms. S i n c e the f i t s were d e v e l o p e d s e p a r a t e l y over a p e r i o d o f time, some v a r i a t i o n i n f u n c t i o n a l form may be seen. For c o n v e n i e n c e , a l l the f u n c t i o n s a r e e x p r e s s e d i n terms o f t h e v a r i a b l e Δ Τ , d e f i n e d as

ΔΤ = (T/T ) - 1 c

i . e . , the d i m e n s i o n l e s s d i s p l a c e m e n t from the c r i t i c a l t e m p e r a t u r e . C r i t i c a l temperatures and ranges o f f i t s o f the parameters a r e shown i n T a b l e IX.



402

T a b l e IX.

Critical


Substance Carbon D i o x i d e Nitrogen Methane Ethane Propane Isobutane Normal Butane Hydrogen S u l f i d e

Temperatures and Ranges o f Parameter

Critical

temperature, 547.578 227.16 343.00 549.594 665.730 734.130 765.288 672.354

Fits

Range of f i t , °F -63 t o 800 -75 to 650 -75 to 650 -200 t o 600 -200 to 600 -200 to 600 -200 to 600 -90 t o 600

R

The e q u a t i o n o f s t a t e parameters were f i t as d i m e n s i o n a l q u a n t i ties. Parameter " a i s i n p s i a f t °R° · / l b m o l . Parameter "b" i s in f t / l b m o l . S e p a r a t e f i t s were r e q u i r e d f o r s u b c r i t i c a l and s u p e r c r i t i c a l parameters i n o r d e r t o f i t the " s p i k e " i n the parameter v a l u e s a t the c r i t i c a l temperature. In o r d e r t o i n s u r e the p r o p e r b e h a v i o r at temperatures slightly above the c r i t i c a l , a check must be a p p l i e d t o the c a l c u l a t e d v a l u e s of the parameters. The q u a n t i t y "b*", the minimum a l l o w a b l e v a l u e o f the s u p e r c r i t i c a l "b" parameter i s c a l c u l a t e d from the " a " parameter at t h e same temperature w i t h the f o l l o w i n g e q u a t i o n : ff

6

5

2

3

b* = b i a / [ T

1.5

(3)

2

1.8886163E-02 0.99974

bi b

where:

(l+b AT)]

c

2

I f ΔΤ i s l e s s than 0.004 o r i f the c a l c u l a t e d v a l u e o f the "b" param e t e r i s l e s s than "b*", "b*" must be used as the v a l u e o f the "b" p a r a m e t e r . T h i s check may be o m i t t e d f o r n i t r o g e n and methane s i n c e t h e i r f i t s do not extend down to t h e i r c r i t i c a l t e m p e r a t u r e s . The parameter f i t s f o r a l l s u b s t a n c e s s t u d i e d a r e g i v e n below.

Carbon D i o x i d e Parameters, a

= aoAT

b

= b AT

ao ai a a a as a6 a as

= = = = = = = = =

0

2

3

4

7

3

3

+ aiAT

2

+ biAT

2

AT = 0

+ a AT • a

+ a (a -AT)

a e

3

+ b AT + b

+ b (b -AT)

b 6

3

2

1.1323818E+05 -1.9341561E+05 -2.0363902E+05 -5.2494786E+07 5.2775697E+07 1.6289565E-04 -7.9739155E-05 4.5206914E-05 8.4787834E-09

2

4

4

5

5

+ a /(a -AT) 7

8

+ b /(bg-AT) 7

bo = -2.3973118E+00 bi

b b b b be by bg 2

3

4

5

= = = = = = = =

-1.8705039E+00 -7.0399570E-01 4.6945137E-01 -1.9580259E-01 3.3775759E-06 3.1643771E-01 1.3103149E-11 1.2868684E-09


MORRIS A N D T U R E K

19.


Equation

403

Carbon D i o x i d e Parameters, AT ^ 0 a

= aoAT + a i A T

b

= b AT

ao ai a2 a3 a as ae

= = = = = = =

3

+ biAT

0

4


3

2

+

2

a2AT

+

a

3

+ a (AT-a )

+ b AT + b

3

+ b (AT-b )

2

-1.2327970E+04 -3.9510605E+03 1.5753912E+04 2.9883762E+05 1.2638522E-02 -1.2669980E-01 -6.9947616E+00

b bi b b b b b 0

2

3

4

5

6

4

5

4

5

= 4.9137761E-02 = -1.5082045E-01 = 1.9277928E-01 = 4.0844576E-01 = 1.9737045E-04 = -3.8637696E-02 = -1.7936190E+00

N i t r o g e n and Methane Parameters = aoAT

3

b

= b AT

3

ao ai a2 a a as

= = = = = =

Nitrogen -1.6347508E+01 1.2161843E+03 -2.2892243E+04 1.2959794E+05 -4.8072688E+04 -7.5945730E-01

Methane 5.7694890E+03 -3.2063240E+04 3.0940267E+04 1.6229084E+05 -4.6636347E+00 1.1820010E-01

bo bi b b b b

= = = = = =

-2.8577341E-04 3.8658153E-03 -1.7061211E-02 5.2704092E-01 -1.2361065E-01 -1.2508721E+00

3.0568302E-03 -1.5876295E-02 2.2593112E-02 4.8294387E-01 -4.0107835E-04 8.0512599E-02

0

3

4

2

3

4

5

+ aiAT

2

+ biAT

2

& ΔΊ

a

+

+ a

3

+ a /(AT-as)

+ b AT + b

3

+ b /(AT-b )

2

2

4

4

5

Ethane, Propane, I s o b u t a n e , and Normal Butane Parameters, ΔΤ = 0 a

= aiAT

b

= biAT

ai a a a as a6 a7 as a9

= =

2

3

4

= = = = = =

5

5

+

a2AT

+ b AT 2

4

4

+

a AT

3

+ b AT

3

Ethane 0.0 0.0 6.0408944E+04 -3.3524012E+05 -2.4440624E+05 -5.3770245E+06 5.8143088E+06 4.5467943E-05 -9.6079783E-04

3

3

+ a AT

2

+ b AT

2

4

3 9

+ a AT + a

6

+ a ( a g- Δ Τ )

+ b AT + b

6

+ b (b -AT)

5

4

Propane 1.3807587E+05 0.0 0.0 -6.8075273E+05 -5.1847288E+05 -4.1533054E+08 4.1613493E+08 3.2664789E-05 -2.5461817E-05

5

7

7

Isobutane 3.6060539E+05 0.0 0.0 -8.9042232E+05 -7.4744128E+05 1.1845554E+06 3.6937924E+04 1.7804777E-04 -1.8159933E-01

b 9

8

Normal Butane 4.0124834E+05 0.0 0.0 -9.9038190E+05 -8.2677046E+05 1.2222223E+06 6.7757452E+04 1.1836024E-04 -1.3348188E-01


404

bi

b b b b b b? b bg 2

3

4

5

6

8

E Q U A T I O N S O F STATE: T H E O R I E S A N D

= = = = = = = =

0.0 0.0 -3.8152843E-•01 -7.6440305E-•01 -4.9300521E-•01 8.3401715E-•01 -2.5418456E--01 2.8828651E-•05 7.9453607E-•02

0.0 2.8826091E-01 0.0 -7.2744096E-01 -5.4966388E-01 -7.0736939E+02 7.0819594E+02 3.7042340E-05 -2.4660292E-05

APPLICATIONS

0.0 0.0 -5.0388863E-01 -1.2016256E+00 -7.5953251E-01 -4.9144994E+02 4.9249286E+02 5.0910144E-05 -5.1232547E-05

0.0 0.0 -5.0788946E-01 -1.2234509E+00 -7.8285224E-01 -4.8981625E+02 4.9086550E+02 6.0889312E-05 -5.1127477E-05


Ethane, Propane, I s o b u t a n e , and Normal Butane P a r a m e t e r s , AT = 0 a

= aiAT

b

= biAT

ai a a3 a as ae a as ag

= = = = = = = = =

Ethane 0.0 0.0 0.0 -2.9168739E+04 6.6705082E+04 4.6578248E+05 1.2357826E+00 4.3979460E-02 -3.2057354E+00

Propane 0.0 1.2332546E+06 0.0 -1.7413558E+06 1.2077341E+06 -5.4364187E+08 5.4425460E+08 1.5085713E-02 -1.3187677E-04

Isobutane 0.0 0.0 3.7513525E+06 -4.1274194E+06 1.4272780E+06 1.2027989E+06 2.5675094E+02 4.7010838E-02 -2.1024705E+00

Normal Butane 0.0 5.2547626E+06 0.0 -4.3472555E+06 2.4222201E+06 -3.2439274E+07 3.3361438E+07 1.3513673E-02 -3.6386297E-03

= = = = = = = =

2.3099616E-01 -2.9478506E-01 0.0 0.0 1.9013417E-01 6.4692946E-01 1.1226280E-03 1.5821016E-02 -1.0175663E+00

0.0 1.3021642E+00 0.0 -1.8915504E+00 1.4616927E+00 -1.1210554E+01 1.1849368E+01 9.8746681E-03 -6.5869382E-03

0.0 0.0 3.3538827E+00 -3.6241228E+00 1.4281118E+00 1.1243492E+00 1.3058917E-05 6.0297320E-02 -3.3703921E+00

0.0 1.1231303E+01 -7.6112989E+00 0.0 1.0382612E+00 1.1002003E+00 5.5989194E-04 3.5280578E-02 -1.7457633E+00

2

4

7

bi

b b b b be by b b 2

3

4

5

8

9

5

+ a AT

4

+ b AT

4

2

5

+ a AT

3

+ b AT

3

3

2

+ a AT

2

+ b AT

2

+ a AT + a6 + a ( a i + A T )

4

3

5

7

+ b AT + b

4

5

6

+ b (b +AT) 7

a 9

3

b 9

8

Hydrogen S u l f i d e P a r a m e t e r s , AT ύ 0 a

= aiAT

3

b

= biAT

3

ai a a a as ae a 2

3

4

7

= = = = = = =

+ a AT

2

+ b AT

2

2

+ a AT + a

+ a (a -AT)

a 7

4

+ b AT + b

+ b (b -AT)

b 7

4

3

2

2.7553548E+05 -1.5271335E+05 -2.2153932E+05 -7.8818355E+09 7.8822161E+09 3.7436399E-04 -9.8433542E-07

3

bi b b b b b b 2

3

4

5

6

7

= = = = = = =

5

5

6

6

7.5061764E-02 -2.6513067E-01 -2.7365804E-01 -7.1403337E+03 7.1407259E+03 2.4591169E-04 -1.4554375E-06


19.


MORRIS AND TUREK

Equation

405

Hydrogen S u l f i d e Parameters, AT ^ 0 a

=

aiAT

3

b

=

biAT

3

ai

=

+ a AT

2

+ b AT

2

2

+ a AT + a

4

+ a (a +AT)

+ b AT + b

4

+ b (b +AT)

3

2

3

8.9992497E+04

= = = a7 = 4


as ae

1. 2.

3.

4.

5.

6.

7.

8.

9.

10. 11. 12. 13.

6

2

3

4.1713206E+05 5.4872735E+02 3.4032689E-02 -1.1257065E+00

Literature

5

6

1.6468196E-01 bi = b = •-2.7842604E-01 1.7715814E-01 b = b = •-1.8220125E+02 b = 1.8259959E+02 b = 8.3692118E-03 b = •-9.1864978E-05

*2 = -2.1018269E+05 *3 = 9.3022318E+04 a

5

4

5

6

7

Cited

R e d l i c h , O.; Kwong, J. N. S. Chemical Reviews 1949, 44, 233-244 Angus, S.; Armstrong, B.; de Reuck, Κ. M. " I n t e r n a t i o n a l T h e r modynamic T a b l e s of the F l u i d S t a t e - 3 , Carbon D i o x i d e " ; P e r gamon: O x f o r d , 1976 Angus, S.; de Reuck, Κ. M.; Armstrong, B. " I n t e r n a t i o n a l Ther modynamic T a b l e s of the F l u i d S t a t e - 6 , N i t r o g e n " ; Pergamon: O x f o r d , 1979 Angus, S.; Armstrong, B.; de Reuck, Κ. M. " I n t e r n a t i o n a l Ther modynamic T a b l e s of the F l u i d S t a t e - 5 , Methane"; Pergamon: O x f o r d , 1978 Goodwin, R. D.; Roder, H. M.; S t r a t y , G. C. "Thermophysical P r o p e r t i e s of Ethane from 90 to 600 Κ a t P r e s s u r e s to 700 Bar"; N a t i o n a l Bureau of Standards T e c h n i c a l Note 684, N a t i o n a l Bureau of S t a n d a r d s : Washington, D.C., 1976 Goodwin, R. D; Haynes, W. M. " T h e r m o p h y s i c a l P r o p e r t i e s of P r o pane from 85 t o 700 Κ a t P r e s s u r e s to 70 MPa"; N a t i o n a l Bureau of Standards Monograph 170, N a t i o n a l Bureau of S t a n d a r d s : Wash i n g t o n , D.C., 1982 Goodwin, R. D; Haynes, W. M. " T h e r m o p h y s i c a l P r o p e r t i e s of I s o butane from 114 to 700 Κ at P r e s s u r e s to 70 MPa"; National Bureau of Standards T e c h n i c a l Note 1051, N a t i o n a l Bureau of Standards: Washington, D.C., 1982 Haynes, W. M.; Goodwin, R. D. " T h e r m o p h y s i c a l P r o p e r t i e s of Normal Butane from 135 to 700 Κ at P r e s s u r e s to 70 MPa"; N a t i o n a l Bureau of Standards Monograph 169, N a t i o n a l Bureau of Standards: Washington, D.C., 1982 Chen, S. S.; K r e g l e w s k i , A. " S e l e c t e d V a l u e s of P r o p e r t i e s of Chemical Compounds"; Thermodynamics Research C e n t e r Data P r o ject: Texas A&M U n i v e r s i t y , 1976; T a b l e s 14-2-(1.01)-j, 1 4 - 2 - i , 14-2-k, and 14-2-kb Soave, G. Chem. Eng. Sci. 1972, 27, 1197-1203 G r a b o s k i , M. S; Daubert, T. E. Ind. Eng. Chem. P r o c e s s D e s . Dev. 1978, 17, 448-454 Z u d k e v i t c h , D.; J o f f e , J . AIChE J . 1970, 16, 112-119 Yarborough, L. In " E q u a t i o n s of S t a t e i n E n g i n e e r i n g and R e s e a r c h " ; Chao, K. C.; Robinson, R. L., Eds.; ADVANCES IN CHEMISTRY SERIES No. 182, American Chemical S o c i e t y : Wash i n g t o n , D.C., 1979; pp. 385-439

RECEIVED November 8,

1985


Equations of State - American Chemical Society

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