Equations of State in Engineering and Research - ACS Publications

19 The Nonanalytic Equation of State for Pure Fluids Applied to Propane

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ROBERT D. GOODWIN Thermophysical Properties Division, Center for Mechanical Engineering Process Technology, National Bureau of Standards, Boulder, CO 80303

and

An isochoric equation is designed for computing thermo dynamic functions of fluids. It has its origin on the liquidvapor coexistence boundary, and it yields a maximum in isochoric specific heats at the critical point. Its basic structure is similar to that of the Beattie-Bridgeman equa tion. With only five least-squares coefficients, it describes a P(ρ,Τ) surface free of irregularities. A modified function in the equation is presented, for the problem of behavior in the limit of low densities, especially as required for integra tion of the thermodynamic equation of state, to obtain the change of internal energy along isotherms. Recently derived vapor pressures for propane at low temperatures also have been introduced. Constants are reported for all equations, as needed for computations on propane.

A

n i s o c h o r i c e q u a t i o n has b e e n d e v e l o p e d for c o m p u t i n g t h e r m o d y n a m i c ^ f u n c t i o n s of p u r e

fluids.

It has its o r i g i n o n a g i v e n l i q u i d - v a p o r

coexistence b o u n d a r y , a n d it is s t r u c t u r e d to be consistent w i t h the k n o w n b e h a v i o r of specific heats, e s p e c i a l l y a b o u t the c r i t i c a l p o i n t . T h e n u m b e r of

adjustable, least-squares coefficients

i r r e g u l a r i t i e s i n t h e c a l c u l a t e d F(p,T) perature-dependent

has b e e n m i n i m i z e d to

avoid

surface b y u s i n g selected, t e m

f u n c t i o n s w h i c h are q u a l i t a t i v e l y consistent

isochores a n d specific heats o v e r the entire surface.

with

Several nonlinear

p a r a m e t e r s a p p e a r i n these f u n c t i o n s . A p p r o x i m a t e l y f o u r t e e n a d d i t i o n a l constants a p p e a r i n a u x i l i a r y equations, n a m e l y the v a p o r - p r e s s u r e a n d orthobaric-densities P(p,T)

equations,

w h i c h provide

the

boundary

equation-of-state surface. This chapter not subject to U.S. copyright. Published 1979 American Chemical Society In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

for

the

346

EQUATIONS

OF

STATE

T h e r a n g e of v a l i d i t y of the present e q u a t i o n of state is f o r a l l

fluid

states at densities to the t r i p l e - p o i n t l i q u i d d e n s i t y , t e m p e r a t u r e s f r o m the t r i p l e - p o i n t to infinity, a n d pressures to at least 700 b a r . V a p o r pressures of the s o l i d at t e m p e r a t u r e s b e l o w the t r i p l e p o i n t are r e p l a c e d b y a n e x t r a p o l a t i o n of the v a p o r - p r e s s u r e c u r v e to absolute z e r o t e m p e r a t u r e , a n d a c o m p l e t e l y n e w t y p e of f o r m u l a t i o n is p r e s e n t e d for densities of saturated v a p o r f r o m absolute z e r o to the c r i t i c a l - p o i n t t e m p e r a t u r e . S o m e c o m m e n t s o n a c c u r a c y are a p p r o p r i a t e i n this i n t r o d u c t i o n . T h e single m o t i v a t i o n for the present w o r k is the fact t h a t e x p e r i m e n t a l d a t a almost i n v a r i a b l y are of m u c h l o w e r absolute a c c u r a c y t h a n

PpT Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch019

r e q u i r e d for d e r i v e d t h e r m o d y n a m i c p r o p e r t i e s , s u c h as specific heats. W i t h a n excessive n u m b e r of adjustable, least-squares coefficients, o b t a i n s a P(p,T)

one

surface w i t h i r r e g u l a r i t i e s i n the d e r i v a t i v e s . A t the

c r i t i c a l p o i n t , e.g., the d e r i v a t i v e s dp/dT a n d dp/dP b e c o m e infinite, s u c h t h a t t h e slightest inconsistencies i n t e m p e r a t u r e a n d / o r pressure scales b e t w e e n different l a b o r a t o r i e s m u s t y i e l d gross d e v i a t i o n s of densities f r o m a n y c a l c u l a t e d surface. F o r c o m p r e s s e d l i q u i d at l o w t e m p e r a t u r e s the d e r i v a t i v e dP/dp

becomes extremely large, such that very

small

i r r e g u l a r i t i e s i n the e x p e r i m e n t a l densities y i e l d d e v i a t i o n s of pressure g r e a t l y e x c e e d i n g the a c c u r a c y of pressure measurements. I n this d o m a i n , a n y e q u a t i o n of state s h o u l d b e u s e d o n l y to

find

p(P,T).

W e h a v e a t t e m p t e d to d e p a r t f r o m the fruitless p r o c e d u r e of u s i n g m o r e least-squares coefficients to o b t a i n a best p o s s i b l e " f i t " of a mass of PpT d a t a , w h i c h do not h a v e t h e a c c u r a c y n e e d e d to define a n e q u a t i o n of state consistent w i t h the k n o w n b e h a v i o r of specific heats. T h e present t y p e of e q u a t i o n of state is m o r e h i g h l y c o n s t r a i n e d t h a n a n y p r e v i o u s l y k n o w n , a n d therefore serves as a r e l i a b l e s m o o t h i n g a n d i n t e r p o l a t i o n f u n c t i o n b y w h i c h means the i n a c c u r a c i e s a n d inconsistencies of v a r i o u s e x p e r i m e n t a l d a t a m a y be o b s e r v e d a n d i n t e r c o m p a r e d .

W e p r e f e r to

r e p l a c e the q u e s t i o n " H o w a c c u r a t e is the e q u a t i o n of state?" b y

the

q u e s t i o n " H o w a c c u r a t e a n d consistent are the e x p e r i m e n t a l d a t a u s e d h e r e ? " I n present w o r k , the a n s w e r is that, g e n e r a l l y , densities are w i t h i n a f e w tenths of one p e r c e n t o v e r the entire fluid d o m a i n ( e x c e p t for the c r i t i c a l r e g i o n w h e r e t h e y are m u c h l a r g e r , b u t are w i t h i n e x p e r i m e n t a l u n c e r t a i n t i e s ) (see

Tables II and I I I ) .

A n a d v a n t a g e of this e q u a t i o n is t h a t specific heat d a t a n e e d not b e i n c l u d e d i n the least-squares d e t e r m i n a t i o n of coefficients

i n order

to

o b t a i n a c c e p t a b l e agreement w i t h those d a t a . T h e present e q u a t i o n thus m a y b e u s e d to estimate specific heats f o r some d o m a i n s of t h e

P(p,T)

surface i n t h e absence of d a t a other t h a n those for i d e a l gas states. D i s a d v a n t a g e s of this e q u a t i o n i n c l u d e : ( a ) l e n g t h y t i m e of d e v e l o p ment

for

each

substance;

(b)

it cannot

be

integrated analytically

( n u m e r i c a l integrations are p e r f o r m e d for e a c h t h e r m o d y n a m i c v a l u e ) ; a n d ( c ) i t is not defined i n s i d e the l i q u i d - v a p o r coexistence envelope.

In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.


19.

GOODWIN

Nonanalytic

100

Equation

200

347

of State

300

400

TEMPERATURE,

500

Κ

National Bureau of Standards (U.S.), Interagency Report

Figure 1.

Density-temperature

diagram of propane (5)

I n this c h a p t e r , w e s h a l l at first g i v e a f u l l d e s c r i p t i o n of t h e l o g i c a l development

of t h e e q u a t i o n of state, necessarily r e p e a t i n g some p u b

l i s h e d m a t e r i a l (1,2,3,4,5).

P a r a m e t e r s a n d coefficients f o r p r o p a n e a r e

g i v e n i n t h e text, b u t t h e a u x i l i a r y equations ( v a p o r pressures a n d o r t h o b a r i c densities ) are p r e s e n t e d at t h e e n d o f this c h a p t e r . I n t h e second p a r t of this r e p o r t w e d e s c r i b e a l t e r n a t i v e f u n c t i o n s for use i n t h e e q u a t i o n of state, a n d c o m m e n t o n t h e i r m e r i t s . F i n a l l y , w e resolve t h e l o n g - s t a n d i n g p r o b l e m of b e h a v i o r of this e q u a t i o n of state i n t h e l i m i t of l o w densities, w h i c h arises i n the i n t e g r a t i o n of t h e t h e r m o d y n a m i c e q u a t i o n of state, to o b t a i n t h e c h a n g e of i n t e r n a l energy as a f u n c t i o n of d e n s i t y a l o n g isotherms. T h e d e n s i t y - t e m p e r a t u r e d i a g r a m f o r p r o p a n e is g i v e n b y F i g u r e 1. T h e u p p e r , l e f t - h a n d c o r n e r of F i g u r e 1 gives the f r e e z i n g l i q u i d l i n e . S y m b o l s a n d u n i t s are g i v e n i n A p p e n d i x A , a n d fixed-point values u s e d f o r p r o p a n e are i n T a b l e I. Table I.

F i x e d Points Used f o r Propane

Triple Point Temperature ( K ) Pressure (bar) Density (mol/L) vapor liquid

85.47 1.6609 · 1 0 " 2.3373 · 1 0 16.620

Boiling

9

1 0

Point

231.0679 1.01325 0.05479 13.1687

Critical

Point

369.80 42.3974 4.96 4.96

In Equations of State in Engineering and Chemical Research; Chao, K., et al.; American Advances in Chemistry; American Chemical Society: Snniptv \ \U Washington, DC, 1979.

348

EQUATIONS

Developing

the Equation

of

OF

State

E x p e r i m e n t a l specific heats, C (p,T), v

are k n o w n t o increase a p p a r

e n t l y w i t h o u t l i m i t o n t h e close a p p r o a c h t o t h e c r i t i c a l p o i n t . nonanalytic behavior

STATE

influences

a f a r greater p o r t i o n o f t h e

surface t h a n g e n e r a l l y is a p p r e c i a t e d .

This P(p,T)

T h e thermodynamic relation b e

t w e e n specific heats a n d the e q u a t i o n o f state a l o n g isotherms i s


(1)

F i g u r e 2 shows b e h a v i o r o f t h e c u r v a t u r e s o f isochores as i n d i c a t e d b y R o w l i n s o n (6) f o r consistency w i t h E q u a t i o n 1. F i g u r e 3 shows t h e zero slope a n d c u r v a t u r e o f the c r i t i c a l i s o t h e r m at t h e c r i t i c a l p o i n t , w h i c h a r e n e e d e d f o r t h e r m o d y n a m i c

consistency

w i t h the relations ( 6 ) ,

C ( ,T) P

P

=

+Τ •

C (P,T) V

W(P,T)

[c

=

p

•

(dP/dT)*/(dP/d )/p P

2

(dP/d )/c y P

v

2

(2) (3)

I n o u r e x p e r i e n c e , a necessary b u t insufficient c o n d i t i o n for a w e l l - b e h a v e d c r i t i c a l i s o t h e r m is that, at t h e c r i t i c a l p o i n t , t h e slope o f t h e c r i t i c a l i s o c h o r e f r o m t h e e q u a t i o n o f state b e e q u a l t o t h e slope o f t h e v a p o r pressure e q u a t i o n , dP/dT = dP /dT. a

T h i s c o n s t r a i n t a l w a y s is a p p l i e d i n

t h e f o l l o w i n g w o r k v i a t h e least-squares p r o g r a m ( 7 ) . T o o b t a i n isochores w h o s e c u r v a t u r e s b e c o m e v e r y l a r g e , a p p r o a c h i n g the c r i t i c a l d e n s i t y a l o n g t h e c r i t i c a l i s o t h e r m , as r e q u i r e d b y E q u a t i o n 1, w e h a v e d e s i g n e d a n infinite c u r v a t u r e f o r isochores a t a n o r i g i n , 0(p),

Journal of Research of the National Bureau of Standards

Figure 2.

The locus of isochore inflection points (I)


19.

GOODWIN

Nonanalytic

Equation

349

of State


Τ \ Journal of Research of the National Bureau of Standards

Figure 3.

Behavior of the critical isotherm (1)

i n s i d e t h e coexistence envelope, (see F i g u r e 4 ) . T h e f u n c t i o n g i v i n g this c u r v a t u r e s h a l l h a v e a d e n s i t y - d e p e n d e n t coefficient w i t h a root a t t h e c r i t i c a l d e n s i t y , s u c h that t h e v e r y l a r g e c u r v a t u r e s w i l l c h a n g e s i g n w h e n integrating E q u a t i o n 1 through the critical density, a n d the critical isochore w i l l b e c h a r a c t e r i z e d b y d^/dT

2

= 0 at t h e c r i t i c a l p o i n t .

Β

\\ \\ C

D

j

A

TEMPERATURE Journal of Research of the National Bureau of Standards

Figure 4.

Behavior of the locus, Θ(ρ) (1)


350

EQUATIONS

OF

STATE

W i t h t h e a b o v e objectives i n m i n d , w e c o n s t r a i n t h e e q u a t i o n t o t h e l i q u i d - v a p o r coexistence t e m p e r a t u r e , Τ (ρ),

F o r a n y d e n s i t y , t h e coexistence

is o b t a i n e d b y i t e r a t i o n f r o m equations f o r t h e o r t h o -

σ

b a r i c densities.

boundary.

T h u s t h e v a p o r pressure, Ρ [Τ (ρ)], σ

is a f u n c t i o n o f

σ

density. B y s u b t r a c t i o n Ρ _

Ρ

Σ

(

Ρ

)

=

R*

P

. [Τ -

Τ (ρ)] σ

+

P R*T 2

Q

· f(p,T)

O n l y t w o temperature-dependent functions ( i n addition to

(4) pR*T)

are n e e d e d to d e s c r i b e t h e s i g m o i d shape of isochores a t p < ρ < 2 · p c

c


(see F i g u r e 2 ) · Φ(ρ,Τ)

f(p,T)=B(p)

(5)

+ C(p) · *(p,T)

w h e r e B(p) a n d C(p) a r e p o l y n o m i a l coefficients t o b e d e v e l o p e d

from

PpT d a t a b y least squares. The function Φ(ρ,Τ)

EEEX"

2

· l n [Τ/Τ ( )]

(6)

σ Ρ

(see F i g u r e 5 ) has t h e v a l u a b l e p r o p e r t y that θ Φ / θ Γ 2

o n t h e coexistence b o u n d a r y at Τ = Τ (ρ). σ

2

= 0 everywhere

Its w e a k , n e g a t i v e c u r v a t u r e

at v e r y h i g h t e m p e r a t u r e s corresponds to t h e d e c l i n e o f v i r i a l coefficients t o w a r d zero at these t e m p e r a t u r e s .

Journal of Research of the National Bureau of Standards

Figure 5.

Behavior of the function,

Φ(ρ,Τ) (1)



19.

GOODWIN

Nonanalytic

Equation

of State

351

i Journal of Research of the National Bureau of Standards

Figure 6.

Behavior of the function,

T h e f u n c t i o n Ψ(ρ,Τ)

Φ(ρ,Τ) (I)

(see F i g u r e 6 ) y i e l d s a m a x i m u m i n

C (p,T) v

at t h e c r i t i c a l p o i n t v i a E q u a t i o n s 1 a n d 4. Its o r i g i n is ·βχρ[-α·

θ(ρ)=Τ (ρ) σ

f(p)] (7)

f (P) — |p — i | V ( p t — D » w h e r e p is the r e d u c e d d e n s i t y at t h e l i q u i d t r i p l e p o i n t .

T h e function

t

== ( 1 — θ (ρ) /Τ) is a n a r g u m e n t f o r E q u a t i o n 9 b e l o w .

ω(ρ,Τ)

As *

m u s t b e zero at coexistence, i t is defined as the difference Ψ ( ρ , Τ ) = φ{ρ,Τ) w h e r e ψσ(ρ) is o b t a i n e d f r o m ψ( ;Τ) Ρ

φ(ρ,Τ)

= δ · exp [c · (1 -

-ψσ(ρ)

(8)

merely b y replacing Τ w i t h χ)] +

[1 -

ω + ω · 1η ( ω ) ]

Τ (ρ), σ

(9)

T h e p a r a m e t e r , 0 < δ < 1, i n E q u a t i o n 9 is f o r r e l a t i v e w e i g h t i n g of t h e a n a l y t i c a n d n o n a n a l y t i c parts. T a b l e I I s u m m a r i z e s t h e PpT d a t a u s e d here, as d e t a i l e d i n R e f . 5. E a r l i e r c o m p i l a t i o n s o n p r o p a n e are f o u n d i n Refs. 8 a n d 9. P a r a m e t e r s a n d coefficients f o r E q u a t i o n 1, d e v e l o p e d as d e s c r i b e d i n R e f s . 3 a n d 5 are B(p) C(p)

s

^ Β + B ί

d

α = 1,

· (

2

P

-

·ρ+ B

1) · (P -

y = 0.07,

s

·p + B 2

(10)

3

4

δ=2/3,

0.2555 5013 087

B =

B

=

0.8275 9684 245

d —

Β

= -0.2818

Ά

·p

2) • exp ( - γ · p )

Β, = 2

4

4

e=2 0.0961 8894 561 -0.3681

8676 338

2856 341


(11)

352

EQUATIONS

OF

STATE


PROPANE

ο

ο

-1 0

1

2

3

P/R

c

National Bureau of Standards (U.S.), Interagency Report

Figure 7.

Behavior of coefficients B(p), C(p) for propane (5)

T h e b e h a v i o r of B(p)

a n d C(p)

f o r p r o p a n e is s h o w n i n F i g u r e 7. T h e

n u m b e r of PpT d a t a u s e d h e r e for a d j u s t i n g t h e e q u a t i o n of state is 843, w i t h different least-squares w e i g h t i n g s t h a n i n R e f s . 3 a n d 5.

Overall

d e v i a t i o n s , w i t h e q u a l w e i g h t i n g f o r a l l p o i n t s , are 2.07 b a r f o r t h e m e a n of absolute pressure d e v i a t i o n s a n d 0 . 3 4 % f o r t h e r m s of r e l a t i v e density deviations. W e a l w a y s e x a m i n e the b e h a v i o r of t h e c a l c u l a t e d c r i t i c a l i s o t h e r m i n m i n u t e d e t a i l near t h e c r i t i c a l d e n s i t y ( ± 1 0 % ) . S m a l l adjustments i n values of p a r a m e t e r s ( i n c l u d i n g t h e c r i t i c a l d e n s i t y ) m a y b e m a d e to e l i m i n a t e a n y n e g a t i v e slopes (dP/dp) , Tc

as d e s c r i b e d i n Ref. 3.


19.

GOODWIN

Nonanalytic

Equation

353

of State

T a b l e I I . S u m m a r y o f PpT D a t a f o r P r o p a n e Deviations

a

Range of the


Authors

à(mc >l/L)

G o o d w i n (5) B e a t t i e (12) C h e r n e y (18) D a w s o n (14) D e s c h n e r (15) D i t t m a r (16) E l y (17) R e a m e r (18) T o m l i n s o n (19)

0.30 1.0 --10.0 0.04- - 2.6 0.02- - 0.07 0.03- - 9.5 7.3 --13.4 11.5 --14.8 0.02- -13.0 10.3 --12.0

Δά/d (rms

Data

Τ (Κ)

Ρ (bar)

NP

290--700 370--548 323--398 243--348 303--609 273-413 166-322 311--511 277--327

6 - 17 23 - 310 11 - 50 1.8 0.91.Ο 142 ΙΟ - 1035 2 . 5 - 428 1.0- 690 20 - 137

42 110 25 18 236 336 222 306 40

%)

Mean ΔΡ (bar)

0.04 1.39 0.37 0.14 1.93 0.41 0.06 0.42 0.07

0.004 1.00 0.038 0.002 1.04 3.02 2.54 1.42 0.94

* Using Equations 7 and 8 for B(p), C(/>).

6

—

PROPANE Τ = T c

1

ι

COΟ

r-

Û. CM

1—

• CM

H

J

0.5

1.5

1.0

P'Pr. National Bureau of Standards (U.S.), Interagency Report

Figure 8.

Behavior of d P / d T on the critical isotherm (5) 2

2




W

CO

Ο

CO

δ

Η

W

£t

CO Ol

19.

GOODWIN

Nonanalytic

Equation

355

of State

T h e n o n a n a l y t i c c h a r a c t e r of E q u a t i o n 1 is seen c l e a r l y i n F i g u r e 8, w h i c h is a l o g a r i t h m i c p l o t of absolute v a l u e s of 2

isochore

a l o n g the c r i t i c a l i s o t h e r m . T h e v a l u e s of d P/dT

\d P/dT \ 2

2

at ρ
2, a n d i n some cases to isochores h a v i n g d e r i v a t i v e s d P / d T 3

w h i c h are

3

i r r e g u l a r . S t a r t i n g a n y n e w w o r k w i t h E q u a t i o n 13, therefore, c o u l d b e advantageous. Table I I I .

Deviations with Alternate Equations Equations

Authors G o o d w i n (5) B e a t t i e (12) C h e r n e y (13) D a w s o n (14) D e s c h n e r (15) D i t t m a r (16) E l y (17) R e a m e r (18) T o m l i n s o n (19)

7 and 9

Coefficients

forB(p),

C(p)

Equations

10 and 9

Ad/d (rms %)

Mean Δ Ρ (bar)

Ad/d (rms %)

Mean Δ Ρ (bar)

0.05 1.07 0.27 0.17 1.70 0.40 0.06 0.54 0.07

0.003 1.13 0.03 0.002 0.82 3.03 2.60 1.76 1.04

0.19 1.14 0.55 0.22 1.80 0.35 0.06 0.43 0.07

0.017 0.94 0.075 0.002 0.92 2.86 2.26 1.60 1.00


19.

GOODWIN

Behavior

Nonanalytic

at Very

Low

Equation

357

of State

Densities

P l a c i n g E q u a t i o n 4 i n the t h e r m o d y n a m i c e q u a t i o n of state y i e l d s a l e a d i n g t e r m as f o l l o w s

ΔΕ =

-1)

[(Ζ ( ) σ Ρ

-RT*(p)/

w h e r e ρ is the r e d u c e d density. A t ρ < 1, Ζ

σ

(15)

+ . · ·] 'dp

P

is the c o m p r e s s i b i l i t y f a c t o r


for s a t u r a t e d v a p o r , Ζ ( )=Ρ (ρ)/[ρ·ΙΡ σ Ρ

·Τ (ρ)]

σ

σ

A n i n i t i a l p r o b l e m w i t h E q u a t i o n 15 is loss of significant figures as —> 1. T h e m o r e serious p r o b l e m is that i f i n d e p e n d e n t equations are

Ζ

σ

u s e d for the v a p o r pressures, Ρ [Τ (ρ)], σ

densities, ρσ(Τ), Ζ

σ

a n d for t h e s a t u r a t e d v a p o r

σ

w e find that i n the l i m i t ρ - » 0 ( a n d h e n c e Τ

σ

-»

0),

m a y a p p r o a c h values f r o m z e r o to infinity, d e p e n d i n g o n the f o r m u l a

tions f o r Ρ

and ρ.

σ

σ

T h i s difficulty arises o n l y at e x t r e m e l y l o w densities, because u n d e r these c o n d i t i o n s the t e m p e r a t u r e d i m i n i s h e s m e r e l y as a l o g a r i t h m i c f u n c t i o n of d e n s i t y , 1/T ~

ln(l/p).

O u r n u m e r i c a l integrations for t h e r m o

d y n a m i c p r o p e r t i e s w e r e p e r f o r m e d s u c h t h a t t h e e q u a t i o n of state w a s not c a l l e d at these e x t r e m e l y l o w densities ( 4 , 5 ) . A s o l u t i o n for the p r o b l e m s m e n t i o n e d a b o v e is to r e p l a c e

conven

t i o n a l f o r m u l a t i o n s of the saturated v a p o r densities b y a f o r m u l a t i o n of the c o m p r e s s i b i l i t y factor, Ζ ( Τ ) , for s a t u r a t e d v a p o r σ

(see

Saturated

V a p o r D e n s i t i e s ). B y u s i n g this n e w f o r m u l a t i o n for substitutions, E q u a t i o n 15 c a n be t r a n s f o r m e d to

AE =

where Z Ζ

σ

=

[(Z

-

c

1) · (Z /Z ) a

c

- RT

C

· ί(χ ) σ

+ . . . ] ·

(16)

dp

is the v a l u e of the c o m p r e s s i b i l i t y f a c t o r at the c r i t i c a l p o i n t ,

Ζ [ Τ ( ρ ) ] is the c o m p r e s s i b i l i t y factor for saturated v a p o r , χ σ

Ta/T , c

Ρ

Equations of State in Engineering and Research - ACS Publications

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