NBSIR 80-1631

D t Reference^

^^y^catfons

analy¥s of liquid volume and liquid

MASS FRACTIONS AT COEXISTENCE

FOR PURE FLUIDS

Lambert John Van Poolen, Ph. D.t

AlllDS D3710T

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

I- Guest Worker; Professor of Engineering

Calvin College

Grand Rapids, Michigan 49506

QC

100

.056

80-1631

ly^u May 1980
7
NBSIR 80-1631

no

ANALYSIS OF LIQUID VOLUME AND LIQUID

MASS FRACTIONS AT COEXISTENCE

FOR PURE FLUIDS

Lambert John Van Poolen, Ph. D.+

Thermophysical Properties Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1980

f Guest Worker: Professor of Engineering

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

Calvin College

Grand Rapids, Michigan 49506

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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CONTENTS

Page

1. INTRODUCTION 1 2. MATHEMATICAL DESCRIPTION OF X, AND X, 2 LV LM

3. ANALYSIS OF X,_y AND X^^^ 4

3.1 Maxima and Minima Analysis for T-j-p < T < T^ 4

3.2 Maxima and Minima Analysis as T Goes to T^ 17

3.3 Analysis Along and X^ as T Goes to T^ 22

3.4 Maxima and Minima Analysis as T Goes to T-j-p 36

4. CORRELATING COEXISTENCE - CRITICAL POINT DATA USING THE LIQUID

VOLUME FRACTION 37

5. EXPERIMENTAL APPROACH TO SIMULTANEOUS DETERMINATION OF COEXISTENCE DATA 51 6. SUMMARY 55

7. ACKNOWLEDGMENTS 55 8. REFERENCES 56 APPENDIX A. Symbols and Units 58
LIST OF FIGURES

Page

Figure 1. vs T for CH^ at constant Pj 5

Figure 2. vs T for CH^ at constant p-j- 6

Figure 3. p-j- vs T for CH^ at constant X|^^ 7

Figure 4. p-j- vs T for CH^ at constant X^^j^ 8

Figure 5. X|^^ vs Py for CH^ at constant T 9

Figure 6. Xj^^^ vs Py for CH^ at constant T TO

Figure 7. Verification of Xj^^ - Xj^^^ structure 15

Figure 8. X^^ First Possibility 23

Figure 9. X|^^ Second Possibility 24

Figure 10. Data for X|^^ Possibility Analysis 26

Figure 11. Implications of X^ > 1/2 34

Figure 12. Percent Deviation Plot of

p^^ + p^ for Ethylene 45

Figure 13. 3o Confidence Limit Band on p^ + p^ of Ethylene 50

Figure 14. X^^ vs py

to Find

p^^ and p^ for Ammonia 52

Table 1.

Table 2.

Table 3.

Tabl e 4.

Table 5.

Table 6.

Table 7. Table 8.

Table 9.

Table 10

Table 11

Table 12 Table 13 Table 14

LIST OF TABLES

and for Ethylene

3py

Effect of f, B,., and, B-, on lim and lim (p‘ + p').

hrh.im

Analysis of for (B^^^ = -B^^)

Check on Inequalities for Ethylene

^LV c ^

Xfv c

''S T for Ammonia

Experimental Ethylene Data

X|^y ^ and + p^) Comparison for Ethylene

Critical Properties of Ethylene

3 (mol /dm ) Comparisons for Ethylene

3 p^ (mol /dm ) Comparisons for Ethylene

3 Comparison of Coexistence Densities (g/cm ) 3 Comparison of p Values (g/cm ) ^ 3 Comparison of p^ Values (g/cm )

27

32

33

36

39

40

42

43

46

47

47

53

54

54

i V
ANALYSIS OF LIQUID VOLUME AND LIQUID MASS FRACTIQNS AT CQEXISTEMCE FQR PURE FLUIDS

Lambert John Van Poolen

An analysis of the behavior of liquid volume and liquid mass fractions at coexistence for pure fluids is made. Scaled equations for the saturation liquid

and vapor densities are analyzed and relationships between various exponents and

among constant coefficients are presented. Inequalities which exist among the

saturation densities and thetr derivatives are developed. A procedure to correlate saturation data with the critical point is applied to ethylene. An experimental procedure to determine, simultaneously, saturated liquid and vapor

densities at a given temperature is presented.

Key words: Coexistence densities; critical density; critical point; liquid mass fraction; liquid volume fraction; phase equilibria; pure fluids.

1 . INTRODUCTIQN

A complete description of a pure fluid at liquid-vapor coexistence from a P-V-T

perspective is given by the intensive variables:

P, vapor pressure,

T, vapor temperature,

p^, saturated liquid density,

p , saturated vapor density,

and the following extensive variables:

m^, m^, masses of the two phases (m^^ + m^ = m^ = total mass),

V„ , V , volumes of the two phases (V„ + V = V-^ = total volume).

Of the four intensive variables only one is independent, i.e., there is one degree of

freedom as described by the Gibbs phase rule. For the purpose of this work, T is assumed

the independent variable.

An analysis of the extensive variables reveals that any two are independent variables.

Hence, to completely describe the pure fluid at coexistence it is sufficient to specify one intensive variable and two extensive variables, i.e., the overall system has three degrees

of freedom.

It is not obvious that useful information about the coexistence state is to be gained

by considering the extensive variables along with the intensive ones. However, an investigation of these extensive variables when normalized by the total volume and/or total mass has proven interesting.

Consider the mass balance at coexistence,

, (

1

)

normalized by V^, utilizing the fact that
= p V ( 2 ) V V

with the result that

(V^' “ Pi<V''t> * (3)

Now define:

overall or total density,

X| y

s liquid volume fraction,

and since

^ - \V =

the overall density is

Pf = Pj^X^y + P^(l - X^y) , (4)

or rewriting,

^LV

~ ~

*^v^

*

Now, p^ = P£ " Pji^^^’ therefore, eq. (5) describes a variable field having

two degrees of freedom - one from the intensive variable set (T) and one from the extensive

variable set = Py). Another extensive variable, perhaps Vy, would also have to be

explicitly known to completely describe a system at coexistence. However, eq. (5) does

provide a connection between the intensive and extensive variables and it is that

connection, X^^y, which provides the interest.

The liquid mass fraction:

^LM

""

is found directly from the liquid volume fraction, i.e..

MLM

= (VjPt)/(V^ y) (P£/Pt^ ^LV• (6)

The two variable fields to be explored are then (from (5)):

.PjJ) = 0, i.e.. \v = (Pt - f>v)^(Ot

- Pv^

and from (6 ) and (5)

,

,Py,T) = 0, i.e.. \m = (Pj^/Py)(Py - P^ )/(P, -

P ) (7)

2. MATHEMATICAL DESCRIPTION OF X^y AND X,,, LM For f(X,

y

5 ^ J’ ^ ~ 0 we expect:

3T 3Py

(8) 3T 3px

Pt \v T

= - 1

2
and from eq. (5) the derivatives are:

3X LV

3T

(p; - p;)

(P£ - Py) LV (P^ - (9)

3X LV

3p, no)

9p-|

W

- p; + X|_^,(p; - p;; ( 11 ) LV

Indeed eqs. (9), (10), and (11) together satisfy eq. (8) - the chain rule. It can also be shown that for:

3X dX

LV

LV 3p, '‘Ot ^

dT ( 12 )

that:

3T3p, 3P-J.3T

indicating that dXj^y is an exact differential and that X, ^

is a thermodynamic

LV property.

For » P j

> T)

MLM 3T 3Py

31 3Px Pj *'T Y LM \m

and from eq. (7) the deri

w

LM

- p£p;

3T Pj

' ^ j(p£ - Py)

= - 1

+ X, (p^p; - p;p„)

-

f>v>

(13)

(14)

3X LM

3pi T

" ^v'

(15)

3p,

3T

2

P-rP

' X, ,,P, (PpP' - P^P

V LM^T ' £ V £ 1 TM PrP£ V

(16)

Indeed eqs. (14), (15), and (16) satisfy eq. (13). It can also be shown that for:

3
3X dX

LM LM 3p.

dT (17)

that:

2

a'^x LM

2 a^xLM

aiap. ap^ai

indicating that dX^^^^ is an exact differential and that X|^|^ is a thermodynamic

property.

3. ANALYSIS OF X^_^ AND Xj_^

3.1 Maxima and Minima Analysis for T-j-p < T < T

The fields f(X|^^, Pj, T) = 0, and Pj, T) = 0, are plotted from smoothed

methane data presented by lUPAC [1]. The results are shown in figs. 1-6.

For the maximum in X|^^, shown in fig. 1, eq. (9) can be set to zero, i.e..

axj_v aT

- X 0 .

Designating the subscript "MAX" for the point where the maximum occurs,

- p' \v,MAX " (p^ - p^) ^ ° *

Now there is a corresponding p

^LV,MAX

" ^*^T,MAX

‘

T,MAX

such that:

(18)

or, ....

*^T,MAX (p[ - Py)

(p,p; p;p^)

(19)

Later on (see figs. 8, 9, and 10), when dealing directly with the critical liquid

volume fraction, X^y

^ (py = p^), it is shown that it is unlikely that a maximum occurs

along the curve describing it for T < T^. The implication is that if a maximum does occur, the Xj^^ curve is turning towards zero and hence the Py is a p^ (refer to fig. 1).

We can then conclude that Py = p^ < p^ and hence:

\v,c > \v,MAX •

4
mLT) m

N0I13VUJ 3IAiniOA QinDi! -

5

Figure

1.

X,

vs T

for

CH,

at

constant

p
Locus

of

min

pts

IT)

If) L»

cn

M

NOliOVUd SSVIAI QinDH -

6

Figure

2.

X,

^

vs T

for

CH,

at

constant

p
7

Figure

3.

p-p

vs T

for

CH,

at

constant X
8

Figure

4.

vs T

for

CHat

constant X
p^- DENSITY, mol/cm^

Figure 5. vs py

for CH^ at constant T,

9
,

LIQUID

MASS

FRACTION

/>, DENSITY, mol/cm3

Figure 6. vs for CH^ at constant T.

10
The tendency towards zero for the p-j. = curves (or towards one for p^ =

p^^) is

literally true in the absence of gravity effects. Indeed Hohenberg, et al . [2] indicate

that within a very narrow range of temperature and overall density near the critical the

meniscus, under the effect of gravity, dissappears somewhere other than at the top or bottom of the test cell. For the minimum in shown in fig. 2, eq. (14) can be set to zero, i.e.,

3X.'LM 3T

- p£Pv (P£pi - p£Pv^

Pt(p, - P,) * P,(P, - P,)

= »

Using the subscript "MIM" for the point where the minimum occurs,

po r (pop‘) 1

^LM,MIN ^T,MIN [- piPviJ

Now the corresponding Pj is that for which

^LM,MIM - [

^^T,MIN ^vH

^T,MIN L >

Q. 1 Q.

(2l)

or. Vv

T,MIN (p„p‘ - p'p ) ^^£ ” ^v^ ^v

~ ^^v*^£” *^£^ v^^^^^£^v PnP £^v '•^£^v

In an argument similar to that for X

values of Pt are p/s and hence;

T,MIN £

^LM,MIN ^ \m,c ’

where X.,, is the critical liquid mass fraction (p^ = p^). Ln,c I c If p^ is substituted for p^ in eqs. (5) and (7), the critical liquid volume

fraction and critical liquid mass fraction are explicitly

£^v' ( 22 )

LV c

^ ^LV MAXshown, based on fig. 2, that

(23)

\v,c = (Pc - Pv)/(P£ - Pv^

'LM,c ^^£'^^c^^^c

“ “ *^v^

(24)

(25)

Now, X, „ > X,., at a given temperature by inspection of eqs. (24) and (25) since,

Lm^C LVjC

P£ > p^, therefore:

^LM,MIN ^ \m,c ^ \v,c ^ \v,MAX (26)

and

11
( 27 ) Pt,MIN ^ ^ Pj,MAX ’

where > p^.

Also, considering the value of at the temperature and p^ for which X|^^ is a

maximum, i.e.,

^LM,MAX " pTTi^ \v,MAX

Substituting for Pj eg. (19) and for X^^ eg. (18) the result is:

LM,MAX

P£py

(p£p; - p£Pv^

(28)

Also, consider the value of X|^^ at the temperature and p^ for which X^j^ is a minimum,

i .e.

,

IV, MIN T,MIN

LM,MIN

(form of eg. 6]

^LM,MIN Substitute for Pj eg. (22) and for X, eg. (21) to obtain

P£Pv

p£Pv - p£Pv 'TV, MIN

which is the same as X LM,MAX

liguid volume fraction at a density p„ = p

for the same temperature (see eg. (28)). Now X LV,MIN

T MTM > P..- Therefore: TjMIN ^c

^LV,MIN

^ ^LV,c

and XLM,MAX

is the liguid mass fraction at a density p^ = p-j. < p^ so

^LM,MAX ^ ^LM,c *

Combining egs. (26), (28), (29), (30), and (31) results in:

^LM,MIM

^ ^LM,c^ ^LM,MAX

" ^LV,MIN

^ ^LV,c

^ ^LV,MAX '

(29)

is the

(30)

(31)

(32)

An investigation of the individual terms in eg. (32) and their related eguations

indicates they are only functions of temperature. This is interesting since the field

describing the coexistence states has two degrees of freedom. Therefore, these are loci of

points, having only one degree of freedom. This reduction comes via the "zero" condition

on the first derivative. Also, some conclusions can be drawn about the derivatives.

12
LM

3T and

3 ^

, at

From

^T,MAX

eq. (6),

^T,MIN

^'iM 1 r '^v

3T

Pj Pj

P£ 3T P£ \V

Pf J

, at the same temperature.

3X

Now at a minimum, LM

3T p- = 0 and from eq. (33)

,

( 33 )

3X LV

'a 3T

and from,

T,MIN

^LV,MIN ’ (34)

Y = L y ^LV TM ’

obtai n:

(form of eq. 6'

3X,v fl MLM \m^£'

3T 1—

CL

II

1— Q. 3T

1— Q. 4 J

3X At a maximum, —

3X

LV

= 0, therefore from eq. (35):

'LM 3T

'T,MAX

^LM,MAX P£ °

Combining eqs. (34) and (36)

3X.'LV IV,MIN

^LM,MAX

3T Pf.MIN

3X LM

3T T,MAX

and as previously shown from eqs. (28) and (29):

\v,MIN " ^LM,MAX ’

therefore:

!iv

3T

3X LM

^T,MIN

3T

T,MAX

:35)

(36)

(37)

(38)

(39)

13
The methane data of lUPAC [1] was analyzed to check the validity of eqs. (27), (32), (38),

and (39). The results are shown in fig. 7 for one temperature, 344 K (160°F). The data

used is published at two degree intervals and the slight errors in the equality of

to — reflect the effect of using this large

^T,MIN ^T.MAX

temperture increment in the numerical calculation of the derivatives.

An examination of fig. 3 reveals a minimum in Pj versus T at constant Xj^^ < X^

(limit of critical liquid volume fraction).

Substituting eq. (5) into eq. (11) and setting the derivative to zero.

dX to ond

LV 3T

3p-| W = 0 =

(Pi P +

'LV

- Pv^

(p; - p;)

the expression for p^ at the minimum is:

T,ZER0

SLOPE

PyP)l

- K

-

which is the same as eq. (19). Thus these mini mums in fig. 3 correspond to the maximums in

fi g . 1

.

A similar analysis of the maximums in fig. 4 reveal that they correspond to the

minimums in fig. 2. An interesting inequality arises from the fact that the second derivative of

at the maximum shown in fig. 1 must be less than zero. The derivative of eq. (9) is calculated recognizing that at the point of consideration:

= 0 by definition, and X|^^ is given by eq. (18),

^T,MAX

with the result:

(p>;) - (- p;pi)

(p; - p;)(Pi - P,)

^T ,MAX

Now by inspection of figs. 3 or 4,

p;

> 0 , p; < 0

p; > 0 , p^ < 0 .

In terms of signs, eq. (40) is:

(40)

(41)

14
N0I13VUJ 3IAiniOA QinDH ''^^X

15

Figure

7.

Verification

of

X,.,

- X,

„

structure
3^X

LV ar

T,MAX

2 3^X

Therefore if LV 3T T,MAX

is to be less than zero,

> PilPv ( 42 )

This same conclusion can be reached from the mini mums in fig. 3. From the derivative

of eq. (16),

.2

3 p.

3T"

3X

= o" + —

^v 3T K

- * \v <p«! - • (43)

LV

How, as seen in an earlier analysis, at this minimum:

3X,Lv

r Pi

_L1

3T

= 0

and;

^LV " \v,MAX

' ^v^

*

After substitution of these in eq. (43);

(18)

^2

3

3T^

(p>p * <-fWi

(p; - p;)(Pt - P,T

(44)

LV,MAX

If this is a true minimum

3^p^

3T^

LV,MAX

aqai n:

« H „ I P„P( v^£

1 "n ' '^Jc.^v

should be greater than zero. This is true if

(42)

And since we have already indicated that a maximum in X, ,, exists for all TTn < T < T , LV I P c eq. (42) is true for this temperature range.

Consider the minimum in Xj^^^ as seen in fig. 2. The related derivative is given

by eq. (14).

If the minimum is truly so, the curve is concave upward and the second derivative is positive at that point.

16
We had at this minimum,

3X LM

3T

T,MIN

= 0, and from previous work eqs. (21) and (22)

apply.

Taking the derivative of eq. (14) and substituting eqs. (21) and (22) the result is:

2

9'^X LM

9T^

T,MIN

p,p^p][p; ^ 2p;p^(p;p^-p;p^)) + (p^p^p;p;)

(Po - Pw)(p;pJ - PoP?,)

(45)

rv'

Equation (45) in terms of signs using eq. (41) is;

2

9"^X LM

9T‘ T,MIN

2 9^X

If LM 9T

(^) ^ (-)

(+)(+)

is to be greater than zero. T,MIN

- p p„p"p' + 2p'p((p„'p - p'p„) Pvp£p;p; (46)

This is true for Tjp < T < T^ for which a minimum in X, occurs. (No doubt eq. (46) LM can also be found by considering the maximum in fig. 4. This was not done.)

3.2 Maxima and Minima Analysis as T Goes to T

To aid in this limit analysis, use is made of the following coexistence dome equations

based on those given by Green, et al . [3], and Sengers, et al . [4] with the exception that

the exponents are not assumed the same for each phase, i.e..

Pfi. ^c '^2H^ ®3£^ (47)

0 <t> 'll V V V

p - p

= Bn £ + Bo e + e ^v ^c Iv 2v 3v

where

,

B's are constants.

(48)

„,<(> =1.00

I' V

> 1.00

£ = T^ - T.

To analyze the limits of various mass and liquid fractions in eq. (32) it is helpful to develop the relationship between the exponents 0^ and

0^^ in eqs. (47) and (48).

17
To that end, form the critical liquid -1 volume fraction X,,, = (p - p )/(p„ - p ) LV,c c V a V using eqs. (47) and (48), i.e.:

B 4> 't' - e ''-B, c ''-B, e Iv 2v 3v

LV,c B^ B ^ \ 4- -'iv^ -^2v^

^ -^3v^

(49)

Now fig. 1 indicates the generally accepted fact that the liquid volume fraction in

the limit along the critical isochore is neither one (a liquid) nor zero (a vapor). A recent detailed critical point investigation by Moldover [5] reinforces such an opinion.

With that in mind, if may be assumed in eq. (49) that B^ > B^, and each term can be

divided by e obtaining:

- B. + TERMS (e

1 V

' EXP>0,

(50)

where "EXP" means the exponent of e. In the limit as T + T^ (e + 0)

jl" \v,c ' ’ '»' h > »V

C

(51)

Or if it is assumed that, B^ > B^, and each term of eq. (49) is divided by c the result

is:

, ^ TERMS (c^^^^^)

B^^ + TERMS

(52)

In the limit then as T - T (e -<• 0)

c lim X ^

= 0 for e > . T*Tc

(53)

Neither of these conclusions, eqs. (51) and (53), agree with the accepted fact that

0 < lim X,„ < 1 . T^T. LV,c

(54)

The only possible conclusion is:

Bj^ = By = B . (55)

In most of the current work with eqs. (47) and (48) the "B exponents" are assumed

equal. However, the behavior of the critical liquid volume fraction gives a rational

g reason for their equality. If this is true each term of eq. (49) can be divided by e obtai ni ng.

18
\v,c

- + TERMS

+ TERMS

(62)

In the limit, then, as T + T^ (e 0):

- B Iv "T *'-V.c ' - "c,LIM

•

c

(63)

We note that obviously B^^ nor B-^^ is zero leading again to limits of zero and one respectively. Also, Sengers, et al . [4] indicate that for first order "dome"

symmetry at the critical point.

HJl

= - B Iv

(64)

indicating also (see eq. (63)) that B^^^ and B^^ are of opposite algebraic sign.

Normally B^^^ is positive while B-j^ is negative (see eqs. (47) and (48)).

Also, the limit, T T^, in eq. (6), i.e..

1 im

T^T. LM,c

1 im

T^T.

1 im

T^T, IV,

c

results in:

\m,c ^LV,c ^c,LIM (65)

The critical isochores in figs. 1 and 2 can then be extended to the point

"X" = “Biy/(Bij^ - Bjy), i.e., to at T^ = 190.555 K for the methane.

The limits of the liquid mass and volume fractions found in eq. (32) can be analyzed.

Substituting the derivatives of eqs. (47) and (48) into eq . (18) and dividing each term by.

\v,MAX

- B B, + TERMS (e^^'’^°)

1

V

) [ +B (B^j^ - B^y) + TERMS (e^^''>°)

( 66 )

In the limit (e 0):

” ^ 1 V \v,MAX " (B^g^ " ^c,LIM

c

and for eq. (21), in the limit (e 0) , by inspection,

^LM,MIN

" ^LV.MAX

" ^c,LIM *

T-*-T T+T_

(67)

( 68 )

19
The results given in eqs. (67) and (68) allow the extension of the locus of maximum

and mini mug py^nts tg X at T on the figs. 1 aijd 2 and to p on figs. 3 and 4 at

that as T+T„,

c

1 im X,

Examining X^^ and Xj^^^ in the context of eq. (38) it can be seen by inspection

= X. .... . (69)

T^T

LV.MIN

1 im X T>T. LM.MAX

= 1 i m X T>T. LV.MAX c,LIM

•

c c c

Thus eq. (32) can be rewritten for the limit at the critical as:

1 im X T^T

= 1 im X, „ =1 im X, T^T.

LM.MiN j;"' "LM,c “lm.max "lv.min

= lim X, T*T.

Px'^P- f-T “i

(70)

" ^LV,c" ^LV.MAX ^c,LIM

c c Pf-Pc

Since all these "X's" collapse to a single value at the critical point it might be

expected that the derivatives of these curves at the points in eq. (32) would also go to a single value in the limit.

By inspection of figs. 1 and 2 and considering eq. (68) it is concluded that:

9X

1 im

T-T,

LV 3T

= 0 (71)

^T

" ^T,MAX

P-r'^Pand

1 im

Pt-p+

3X LM 3T

= 0 . (72)

^T

" ‘^T.MIN

Further, consideration of eq. (69) suggests that the following limits also be considered:

3X lim

T^^c

PT-P+

LV

3X 3T

•^T " ^T,MIN

and lim

Pt-p-c

LM

3T

•^T " ‘^T.MAX

The values of these derivatives can be found by combining eqs. (18), (19), (28), (36)

and (39) , to obtai n:

3X LM

3T

3X LV

^T

" *^T,MAX

3T

P>.'

^T

- ^T,MIN

(oJOv - 0>)l)

(73)

20
Substituting the derivatives from eqs. (47) and (48) into eq. (73) and dividing each term

by (- Be^“^) results in: nLM 3T

3X LV

‘^T.MAX 3T

B + TERMS (e^’^''^°) ^

Ot,MIN (®U-8lv> " terms (cETRX))

(74)

Noting that, B-j^^ B-j^ <0 and (Bi^-B-|y) >0, in the limit (e 0):

3X.

1 im

T^T

c ^T,MIN

'LV 3T

= +

" ^T,MIN

(75)

3X

1 im

^T,MAX ^*^0

LM

3T

^t"^t,max

(76)

So rather than uniformity in the derivatives at the critical point eqs. (71), (72),

(75) and (76) indicate different values depending on whether the approach to is from

"below" or "above" and whether the function is X|^^ or These results are sensible in

light of figs. 1 and 2. Also, since the maxima and minima of figs. 3 and 4 are shown to extend to the critical point, the limits.

1 im

T'Tc

Y -Y. TV ^c,LIM

3p-| W = 0 ^LV ^T,MAX

(77)

1 im

TTc

X, M ^X+ , LM c,LIM

9p-| W

= 0 (78)

^LM ^T,MIN

seem reasonable. Also by analogy to eqs. (75) and (76) as well as by inspection of figs. 3 and 4

,

1 im

^LV ^^c,LIM

3p^

FT

— ^ oo (79)

Xlv Pt,MIN

1 im

X, M "X- , LM c,LM

3p-|

FT

= + 00 (80)

^LM ^T,MAX

seem to be rational conclusions.

21
3.3 Analysis Along and as T Goes to

It appears that to obtain a derivative at the critical point, unambiguously, the

approach to the critical point must be along the critical isochore in figs. 1 and 2 and

along the line in figs. 3 and 4. Furthermore, the

^ function has been

utilized by this author [6] to both check the internal consistency of coexistence data and

critical density values and to predict from saturation data. (These procedures are discussed further later in this report.) A knowledge of the derivative is an obvious help

to these ends.

The ambiguity in the limit prompts reconsideration of the function map and the areas of applicability of eqs . (8) and (13), i.e., the chain rules.

Now f(X^^,p^,T) - 0, becomes, along the critical isochore, ^,T) = 0.

Likewise g(X|^j^^ ,p-j-,T) = 0 becomes 9(.(X|_f^ ^,,T) = 0. This reduction in dependent variables

is also true on the saturation boundaries, i.e., f(Xj^^,Pj,T) = ^^£(Pjj^»T) = 0 for X^^ = 1,

and f(X|^^ ,p-|.,T) = fy(Py.T) = 0 for X|^^ = 0. This would occur also for g(X|^|^,Py,T) = 0.

Therefore the chain rule only holds in the regions between the saturation boundaries

and the critical isochore for both the liquid volume and liquid mass fractions in figs. 1 and 2 respectively. This may be a reason for the ambiguity of the limits for the

derivatives when the critical point is approached from these regions. To avoid this

ambiguity then, approach the critical point along p^ (figs. 1 and 2) or X^ (figs. 3

and 4)

.

Along p^.

9X LV,c

3T

dX

LV,c

3X

LM,c

3T

Pt = Pc

Pt = Pc

dT

x;

dX

JJi^

dT

LV,c

X

' ^LM,c

(81)

independent of the derivatives and their limits found previously.

Figure 1 shows that X|^^ ^ has positive values virtually to the critical point.

The argument that X^^^ ^ does not have a maximum (i.e., X

' c

~ except perhaps at T^ is

as follows. There are only two possibilities for a maximum in X|^^ ^ in the interval

T^p < T < T^. The first is illustrated graphically in fig. 8.

The locus of X^^ in fig. 8, stops at point "a" for some T < Tc» But it has

been shown that eq. (18) describes, for all f < fc

’ ^LV MAX*

addition, eq. (67)

indicates that X^^ = X^ at T^. Therefore this case could not exist, i.e., point

"a" and X of fig. 8 must be the same point. This leads to the second possibility

shown graphically in fig. 9.

22
E

o

X

23

Figure

8.

First

Possibility.
E

oX

X

24

Figure

9.

Second

Possibility.
Here the locus of the zeros in is shown going into the critical point.

However, for a maximum and a minimum occurs in Now at these "zero slope

points" the value of p^ is given by eq. (19).

In order for the case of fig. 9 to exist, it would be necessary, at two different

temperatures, for the right hand side of eq. (19) to be the same (for p^ > p^ in the

example). This behavior in X|^y has never been found by this author even when working

with data very near the critical point. An example of data close to the critical point is shown in fig. 10 based on oxygen data of Weber [7] in a one-half degree interval before the

critical point. The departures from smooth curves can only be attributed to experimental error and certainly the points lie on the smooth curves "within" that error. The only data

which might cause the reader to suspect the phenomena suggested in fig. 9 is that at 154.560 K in fig. 10 (second to the last temperature). But this "upward" trend appears at densities above and below the critical point and can most likely be attributed to an experimental bias. (The critical isochore appears to approach a value of X^ very

near one-hal f .

)

To further amplify the idea that the case of fig. 9 does not occur, data based on "dome" equations similar to eqs. (47) and (48) are presented. The equations are those of

Douslin, et al . [8] for ethylene. They are:

p^ = 7.635 + 1.9695 e + .02669 - .2731 x 10"^ (82)

Py = 7.635 - 1 .9695 e + .01404 e + .6783 x 10"^ (83)

These were substituted into eq. (24) for X|^^ ^ and eq. (18) for X|^^ The results

are gi ven in table 1

.

Equations (82) and (83), based on theory and fit to actual data, result in an X^^

^ which has a positive slope and is monotonic to the critical point. Also X|^^ mAX’

^

positive slope, increases monotonical ly into the critical point. The behavior shown in

table 1 coincides with that of fig. 1 and also reinforces the result of the last section

that lim = ^c,LIM*

T-*-T„

’ c The conclusion is that the cases shown in figs. 8 and 9 do not occur. Therefore

^LV c

^ slope except perhaps at the critical point. That is, the

behavior of X, „ in fig. 8 is incorrect mathematically and there is no evidence for

L V , c the behavior of X^^^ ^ as shown in fig. 9. The result is as suggested in fig. 1, that

^LV c monotonical ly into X^ and, that the possibilities for X|^^ ^ are:

0 < lim X' ^ T-T^ ’ c

25
CO

26

Figure

10.

Data

for

Possibility

Analysis.
Table 1.

T (K)

0.1 K INTERVALS

280.05

280.15

280.25

280.35

280.45

280.55

280.65

280.75

280.85

280.95

281 .05 281 .15 281 .25 281 .35 281 .45 281 .55 281 .65 281 .75

281 .85

281 .95 282.05

282.15

282.25

0.01 K INTERVALS

282.26

282 .27 282.28

282.29

282.30

282.31

282.32

282.33

282.34

282.35 (T^)

\v,C ^LV.MAX

\v,c

.4911 .4914

.4916

.4919

.4922

.4924

.4927

.4930

.4933

.4935

.4938

.4942

.4945

.4948

.4951 .4955

.4959

.4962

.4966

.4971 .4976

.4981 .4988

.4989

.49896

.49904

.49913

.49923

.49933

.49944

.49957

.49972

.50000

for Ethylene

\v,MAX

.4750

.4757

.4764

.4771

.4779

.4786

.4794

.4802

.4810

.4818

.4827

.4835

.4844

.4853

.4863

.4873

.4883

.4894

.4906

.4918

.4932

.4947

.4966

.4968

.4971 .4973

.4975

.4978

.4981 .4984

.4988

.4992

.4999

27
Having established eq. (84), further analysis can be done by developing the derivative

^LV c terms of eqs. (47) and (48). For convenience set an assumption usually

made. (See eqs. (82) and (83) where Douslin, et al . [8] have 't'- = = .984.) The

X/ V equation is as follows:

^ (eu62v - - B) . TERMS

(B^j^ - B^^)^ + TERMS (e^^^^°)

and eq. (84) allows the limit of this derivative to be either zero, positive finite, or positive infinity.

Before the exponents are examined, the nature of the constant coefficient

- ®2«,® ^^) should be analyzed. The possibilities for it are either zero

or greater than zero based on eq. (85). If it is to be zero then:

'^211

( 86 ) ^Iv ‘"2v

Since it has been previously shown that B-|^ and B^^ have opposite algebraic signs,

eq. (86) would mean that and B^^ would also have opposite signs. Sengers,

et al . [4] in an extensive study of data near the critical point suggest that no conclusion

can be made that B^g^ = B^^ but indicate that when taking the difference Pg^ - the B

2g^

B

2 y terms are nearly equal in magnitude and are of the same sign. The eqs. (82) and (83)

and

of Douslin, et al., indicate that BZl .02669 and B^y = .01404, i.e., they are of the same

SI gn.

It is reasonable then to suggest that since no evidence exists for assuming eq. (86),

^®U^2v

" ^2!l^lv^

^ ° (87)

An examination of eq. (85) in the light of eq. (87) would result in the virtual

elimination of zero as a limit for the derivative. Also, since the exponent (-B + 4

> - 1) is not equal to zero itself the choice of the positive finite limit is also removed. The

only reasonable choice left is that:

lim X'y = + » (88)

T^T

c

The derivative

^ behaves as though the fluid were a "liquid" at the critical

point (see fig. 1) and eq. (75).

A check of eqs. (82) and (83) for ethylene indicates that indeed eq. (87) is satisfied, i.e., (1 .9695)( .01404)-(-l .9695)( .02669) > 0, affirming that:

28
The data of table 1 indicates that the X,., function comes into X , with a very small

LV,c c,LIM

slope and therefore must then turn upward virtually at T^.

An exception to eq. (87) was found in the recent paper of Haar, et al . [9] for

ammonia. His values are;

B = 2.117

= -1 .4097

= -1.1390

Indeed, B^^^ and B^^ have the same sign but

(2 .1 17)(-1 .4097) - (-2.117)(-1 .1390) < 0 ,

which would indicate lim X.'w = - Hov^ever the equations are based mainly

1*1

c on data by Cragoe, et al [10,11] which are at best some 30 K away from the critical point.

These equations were used only to estimate T^ and and were not used in the

determination of the published saturation data. To check the anrionia data for the slope X|^^ ^ at the critical point, this author fit

the published saturation data from 116 - 132°C to the function X^^

^ (T^ = 132.24°C). The

procedure is outlined in a previous paper [6]. The results for an unweighted least squares

fit is (Var (data) = .2547 x 10 ^):

\v c

" ^

from which:

^LV c

" (1.893 X 10"^) . (90)

Clearly the limit is; +®, which agrees with eq. (88).

The equation for the derivative of the critical liquid mass fraction, is developed from eqs. (47) and (48) assuming The result is:

^LM,c - [[^(B^rBlvH^U^lv) " Pc(*-e)^^U^2v-Vlv^

+ j^{26 (B2j^-B2y) + \ B^j^B^^

(91)

+ <t> |(B^^-B^^) B^^B^y - (B^y+B^j^) B^^B^y

[]

e

+ TERMS (e^^^^°) ]]

29
6*1 The term containing e would have the strongest influence since

e-1 -V- - .65

i^-B-l X - .35

ifr- 1 "V- 0.0.

6 1 The coefficient of e^" , i.e., y) is < 0 (assuming as before,

> 0 and < 0). Hence the only choice for the limit of

^ is:

lim x; T-T. LM,c

_ CO

( 92 )

This limit is also strongly suggested by the graph of fig. 2 for the critical isochore.

The derivative

^ behaves as though the fluid were a "vapor" at critical point

(see eg . (76) )

.

As previously stated, the liquid volume fraction derivative behaves as a "liquid"

derivative, hence there is some ambiguity at the critical point in the derivatives in

addition to the non-liquid, non-vapor ambiguous behavior of X|^^ ^ and X^j^ ^ themselves at the critical point.

From eq. (6)

Y ^LM,c ^c

LV,c

The derivative with respect to temperature is

V = Y' ^ ^LM,c P&^c

\v,c \m,c p 2 ^ Pjl

In the limit:

1 im

T^T

c ^LV,c

lim

T^T.

Y

TM,c

V,LIM lim

T^T,

Since Pi and

lim X T-T. LV,c

= .00

in the limit are -

+ 00

,

(93)

(94)

(95)

an indeterminate form. Thus the limit could be anything. The limit as given by eq. (75)

(+ “) is allowed by eq. (95).

The ambiguity in the limit of the derivatives of eqs. (77), (78), (79) and (80), is avoided by accessing the critical point along the lines X^^^ = X^ in fig. 3 and

X, = X , ,,, in fi g. 4

,

LM c,LIM ^

To that end, in eq. (11), substitute the derivatives from eqs. (47) and (48) and

eq. (63) for X^ resulting in:

30
9p-j W

_ TER«S(eEX'’>'’)

\v"^c,LIM

- Biv)

( 96 )

6-1 But the coefficient of is zero hence:

3p-j W

- + TERMS

^LV"^c,LIM

-Blv)

(97)

Earlier analysis indicates that it is reasonable to have (B-u^B^^ - B^^B^j^) > 0, and

since 4' and (B^^^ - B^^) are greater than zero the possibilities are:

3p,

- “_< lim

T-^T. 3T

< 0 \v~^c,LIM

(98)

where the actual value is then determined by the value of <t), i.e..

0 < 4 < 1 , lim

T^T.

3P-|

W

S GO \v ^c,LIM

(99)

9Pj

= 1 , lim

r*T^ ^LV ^c,LIM

finite , and ( 100 )

> 1 , lim

T^T.

9p-|

W

^LV'^c.LIM

( 101 )

There is much interest in the literature of today in the slope of the rectilinear

diameter, (p' + p')/2, especially at the critical point. Theoretical work by Green,

V et al . [3] and experimental work by Weiner, et al . [12] indicate the possibility of a slight "hook" in the rectilinear diameter, i.e., the hypothesis is that

1 im

T^T.

(p; * Pi)

= » GO

( 102 )

31
9Pj

31^ (103)

If in eq. (11), = 1/2, then:

p; ^ Pi

\v ='/2

’ ^

the slope of the recti inear diameter.

The slope of the rectilinear diameter may be obtained by using the derivatives of

eqs. (47) and (48) directly for and obtaining,

p'+p; = - B (B^j^+B^^)c^-’' - i>(B2£+B2v)e‘^"^ + TERMS (c^^'">°) (104)

The variables of interest in eqs. (97), (103), and (104) are B-^j^, B-j^, and the

exponent In eq . :i04), it is assumed that if (Bi£ + B^y) ^ 0 the ,B-1 dominates.

Tabl e 2 summarizes the effect of varying B^^, B^^ tTM “ ” B, /(B

:,LIM Iv ' U"®lv )) and ii.

Tabl e 2 Effect of . , and, B, on lim

t^t^

9Pj

w~ ^LV"^C,LIM

1 im

T^^c

(Pi ^ p;>

^C,LIM

9Pj

1 im T-T

c '^C,LIM

1 im

T-Tc

(Pi + p;)

(4, Bi

1 V

< ) <1 _ 00 (97)^ — 00 (104)

II = 1 -FINITE (97) _ 00 (104)

II >1 0 (97) — 00 (104)

(=1. ^iv

=

) <1 — OO (97) — 00 (97,103)

II = 1 -FINITE (97) -FINITE (97,103)

II >1 0 (97) 0 (97,103)

(4’ ^iv > 'l£ ) <1 “ oo (97) + 00 (104)

II = 1 -FINITE (97) + 00 (104)

II >1 0 (97) + oo (104)

Vhe number in parentheses refers to the related equation.

Uhen B-|^ = - B^^ (X^ = 1/2), the rectilinear diameter slope is found

9pf P£ + P' from eq. (97) as y -y i and ^ are equal (see eq. (103). These should

^'LV ^c,LIM = j

^

match the values obtained from eq. (104) where at first order symmetry, B-^^ = -B-|y,

eq. (104) reduces to: 32
Pi + TERMS (105) ^ " ^2v)

"

4--1

From the values in table 2 for lim

(pj|^ + p^) , at B-j^ =

are the following implications, given in table 3, for + B^^

the values in eq. (105) should match.

(based on eq. (97)) there

since, as said above.

Table 3. Analysis of ( 82 ^^ + 82^) for (B,j = -B,^)

4- lim (p' + p')

T^T^

c

^®

2S, ^2 v^

<1 - ” (97)^ >0 (105)

=1 -FINITE (97) >0 (105)

>1 0 (97) NO RESTRICTION (105)

2The number in parentheses refers to the related equation.

If first order symmetry is assumed (B-^^ = -B-|y), and if < 1, both generally assumed

to be valid, then:

(B

2 J, + B2 y) > 0 . (106)

We see that for Douslin's [ 8 ] eqs., (82) and (83), both and B^^ are positive so

the criterion is met. The and B^^ of Haar, et al . [9] for ammonia are both negative

but see the previous discussion regarding the validity of these values in light of the data

range from which they were derived.

Figure 11 indicates that if the rectilinear diameter is to have a slope in the limit

of + ”, the diameter must have the value of p^ two times, i.e., once at T^ (point "a") and

again at the critical point. This author does not know of any evidence in the literature

for such an occurrence. A reasonable hypothesis is then:

^c,LIM - 1/2 (107)

3p. An expression for the derivative y along X . (see fig. 4) can also be

c,Lin

developed. Utilizing eqs. (7), (16), (67), ( 68 ) and the derivatives of eqs. (47) and

(48),

33
n

34

Figure

11.

Implications

of

>

1/2.
3Pf W

23-1

2 ^4>-l

- 4>P^ e

+ TERMS

+ FINITE NO. + TERMS

(108)

Since, (B^^^ - B-|y) > 0, < 0, and the term dominates near the

critical point,

9Pi

1 im

T^T.

3T

= + 00 ^LM"^c,LIM

(109)

regardless of what X , is. ^ c,LIM

A brief analysis was made regarding insights into the rectilinear diameter in light of

eq. (108). No additional information was found beyond that already reported.

The inequalities of eqs. (42) and (46) were investigated in the limit as T + T^.

By use of the derivatives of eqs. (47) and (48) it can be shown that.

1 im

T^T.

= 1 im

T-T,

P„P ( 110 )

Equations (82) and (83) for ethylene were used to demonstrate the validity of eq. (42)

and eq. (110). The results are given in table 4. A similar limit analysis for the inequality of eq. (46) was attempted with no conclusive result. However, the computer generated results of table 4 indicate that, not only is eq. (46) valid but also most likely.

1 im T>T.

- - <‘>i)

= lim

T--T,

( 111 )

35
Table 4. Check on Inequalities for Ethylene

(PiPv-p;ppl >1

T (K) + 2p;pi(p;p„-p;p;^)

242.15 -.970E-4 -.420E-4 .540E-2 -.192E-2

247.15 -.127E-3 -.604E-4 .794E-2 -.285E-2

252.15 -.173E-3 -.907E-4 .121E-1 -.440E-2

257.15 -.252E-3 -.145E-4 .193E-1 -.723E-2

262.15 -.402E-3 -.252E-3 .332E-1 -.130E-1

267.15 -.737E-3 -.507E-3 .650E-1 -.269E-1

272.1 5 -.175E-2 -.133E-2 .161 -.726E-1

277.15 -.777E-2 -.653E-2 .718 -.369

282.15 -.127E+2 -.124E+2 .942E+3 -.724E+3

(The following is for an increasingly finer AT.)

282.25 -.622E+2 -.613E+2 .443E+4 -.358E+4

282.30 -.305E+3 -.303E+3 .210E+5 -.177E+5

282.33 -.250E+4 -.249E+4 •165E+6 -.145E+6

282.34 -.1232E+5 -.1228E+5 .794E+6 -.716E+6

and for T

c 282.35 -.40274E+24 -.40274E+24 .23480E+26 -.23477E+26

3.4 Maxima and Minima Analysis as T Goes to T^p

Fi gures 3 and 4 indicate that the following limits obtain at the triple point:

3Pj

1 im

T-Ttp

= 0 \v

(112)

3Pj

w~ ' 'tp

= - FINITE VALUE

\v

(113)

3p-|-

1 im T-T ' 'tp , X. .. -O

^ 0 or + FINITE (114)

36
1 im

T^T. 0 (115) 3T

TP 'LM

'LM

Hypothesize that eqs. (32) and (112) simultaneously hold at the triple point.

Equation (112) implies that - 0 and

py p^ yp

as T

-* Typ . The corresponding

^LM,MAX ^T,MAX

" ^v,TP

^LM,MAX

P£,TP fPv,TP ^,tp1

^v,TP L^£,TP ^,TPJ

0 .

But eq. (32) says that X|^n^ > X^^

^ and X^^^ ^ is usually about .3 or .4 at the triple

point (see fig. 1) so our original hypothesis is wrong.

Hypothesize that eqs. (32) and (115) simultaneously hold at the triple point.

Equation (115) implies that Xj^^^ + 1 and Py yp. The corresponding

\v,MIN ^T,MIN

" P£,TP

V = P£.TP ~ Py.TP

^

. LV.MIN P^^yp - Py^yp

But eq. (32) says that X^^

^

> X^^ While Xj^j^ ^ is very close to one, about .998, or

so, at the triple point it can never be one because p^ j yp (also see fig. 1). So this

hypothesis is also wrong. The conclusion is that eq. (32) does not hold at the triple

point.

Equations (112) and (114) would appear to be expressions for p^ at the triple point

and eqs. (113) and (115) appear to be values of p^ at the triple point. However, the

ambiguity is obvious. Again, as at the critical point, this ambiguity of definition of the

derivative may be due to the inapplicability of eqs. (8) and (13) along the saturation

lines as has been already mentioned. Thus, the only non-ambi guous limit for p^ and p^ at

the triple point can be found along the saturated vapor and saturated liquid lines

respectively as l’’^f-|-p*

In the light of the behavior of the liquid volume and liquid mass fractions, the

triple point exhibits singularities of the same sort as the critical point particularly as to ambiguity in the derivatives.

4. CORRELATIMG COEXISTENCE - CRITICAL POINT DATA USING THE LIQUID VOLUME FRACTION

An earlier paper by this author [6] outlined a method of testing whether or not the

saturation density data correlated with published critical point data. The technique is based on fig. 1 and its subsequent analysis. If, as the critical liquid volume fraction

37
approached T^, it went through a maximum, the used was too low, i.e., in reality it was

a Py. Or, if, as the critical liquid volume fraction approached T^, it increased beyond

one-half and kept increasing, the p^ used was too high, i.e., in reality it was a p^^.

In that earlier paper it was assumed that the critical liquid volume

fraction is one-half at the critical point. This assumption was based on the

work by Sengers, et al . [4]. The macroscopic approaches of this current paper, could only

indicate that the fraction concerned is equal to or less than one-half at the critical

point. The value one-half is, however, assumed for the work that follows. The one-half is

a reasonable answer to the ambiguity at the critical, i.e., the fraction is not zero (a

vapor) or one (a liquid). A further aspect of the value one-half is that it implies a first order symmetry = - B^^) at the critical point, i.e., from eq. (63):

h,LW - - - “lv>

= V2 for (B,, = - B,„) (116)

A look at the ratio of the derivatives p^ and p^ further emphasizes the nature of

first order symmetry. From eqs. (47) and (48) and (B^ =3^=6) the following ratio may

be calculated;

-1

-

ipJl -1 ^ ^

- B, 3e^‘^ - B, (j) e'^'' - B. i|i c'*''' Iv 2v V 3v V

(117)

Di viding by e3-1taking the lim T T^ (e 0), the result is

’U

*lv

and if (B^^ = - B-j^) (first order syrmetry)

,

(118)

Pfl lim ^ - 1 . (119)

T-T^ ^v

c

This ratio of minus one also agrees with the idea that at the critical point neither the

liquid or vapor is dominant thus carrying out the theme of ambiguity at the critical point.

As a further indication that X , is one-half, the reader may refer to fig. 1 for

c,LIM ’

methane and fig. 10 for oxygen. In both cases the graph indicates that the critical

isochore on a vs T plot comes to nearly one-half at the critical point. It is

assumed, in the analysis that follows, that X^ is one-half.

The behavior! al test for is applied below to argon data published

by Gosman, et al. [13]. The results are shov;n in table 5.

38
Table 5.

c ^ Argon

(p^ = 13.4123 mol/L pub)

T (K) \v.c

145.00 .4762

146.00 .4777

147.00 . .4790

148.00 .4795

149.00 .4781 150.00 .4693

150.860 (T ) c (.5000)

The published (13.4123 mol/L) is behaving more like a p^ since the

^ function turns down before T^ is reached. These same six data points were fit to a

functional form of

^ by this author. From this fit an estimate of p^ can be made.

(The details of this procedure are outlined later.)

The result from the unweighted least squares fit at a minimum variance of

the data is:

p^ = 13.6138 mol/L

0 = 0.0031

Pc

for an X , assumed to be one-half,

c ,LIM

This "correctly correlated" p^ is indeed higher than the published value as the "X"

analysis predicts.

A similar analysis was done on more recent data - that of ammonia by Haar,

et al . [9]. The results are shown in table 6. The second column of table 6 indicates that

the proposed Xj^^ ^ goes through a maximum and decreases towards zero. These data

(116 - 132°C) were then fit to an X|^^ ^ function with one-half at the critical point

(Haar's assumption). The result is:

39
Table 6.

c ^

T (°C) Xlv q (Pc = .23502 g/cm^ pub) \v,C 'Oc ' -23652

116 .4664 .4711 117 .4678 .4727

118 .4693 .4743

119 .4708 .4759

120 .4724 .4776

121 .4739 .4793

122 .4755 .4811 123 .4771 .4829

124 .4788 .4848

125 .4804 .4867

126 .4821 .4887

128 .4852 .4927

130 .4873 .4963

132 .4854 .4983

132.24 (TJ (.5000)? .5000

23652 g/cm '^from fit)

.23652 g/cm''

= .00004

Again the properly correlated is slightly higher than the published value as the

analysis would indicate. The results for

^ using this new are given in column

three of table 6. The values go monotonically towards .5000 which is expected by the

earlier analysis in this report. (This type of analysis had been done previously by this

author [6] for data published by ASHRAE-American Society of Heating, Refrigeration, and Air

Conditioning Engineers, Inc., 1969.)

The foregoing procedure for testing the correlation of saturation/critical data may be

useful to those whose task it is to evaluate thermodynamic property data.

Ethylene data are also analyzed. Whereas, for the argon and aimonia, smoothed

published data were used, the weighted experimental values of Douslin, et al . [8] are used

for ethylene. The intent of the following is to present a method for obtaining a correctly

correlated value for p^ directly from data. Arguments are presented to indicate the

statistical soundness of the approach.

An outline of the procedure (reported in a previous paper [6]), used to fit Xj^^ ^ to

the data follows.

Substituting eqs. (47) and (48) into eq. (24) for the critical liquid volume fraction,

results in a series of terms of e having exponents (not necessarily integers). For convenience, a truncated form of this series of terms for X, is assumed as follows:

Lv ,c 40
( 120 ) X,IV.c

' * <^('1'*]

where:

b,c are constants,

P,R are exponents (not necessarily integers).

and:

Combining eqs. (24) and (120) and doing some algebraic manipulation the result (with

Xc lim one-half) is:

where:

A = 2p^, B, C are constants.

The exponents P and R are varied until the variance of the data for a least squares

fit is minimized. The digital computer utilizes a Gauss Jordon pivoting technique

yielding, among other things, the best estimate of (A/2) and its standard deviation.

Whereas eq. (121) looks like the rectilinear diameter, it really is quite different as far as the data used in the fit is concerned. A normal fit of the recti inear diameter uses only values of (p^^ + P^)/2 while eq. (121) also utilizes the difference (p^ - p^) which is

inherently quite accurate (see Weber [7] for this opinion). Also eq. (121) is not an extrapolation to the critical point but (from eq. (120)) is a fit constrained to a particular value, = one-half, at the critical point.

Equation (120) was fit (via eq. (121)) to the fourteen weighted data points presented

by Douslin, et al . [8] for ethylene from 238.15 to 282.15 K (T^ = 282.35 K) . The data

used are given in table 7 below. In fitting the data, first one exponential term in

eq. (121) is utilized, and a minimum variation in the data is found. Then a second term is added in eq . (121) until a lower minimum in the variation of the data is found. Then the

"local" area of these exponents is searched by successive approximations to check on a further minimization of the variance. A more sophisticated approach could be utilized in

which the computer finds the "best fit" of exponents using a non-linear fit procedure.

However, the p^ found is certainly correct in the first four significant figures - well

within experimental accuracy. The results given in table 8, later presented, attest to the

"goodness" of fit.

The use of only one term resulted in a systematic rather than a random error distribution found when using two exponential terms. The result of the weighted least

squares fit is (for = one-half which Douslin assumes):

A + B(e)^(pj^ - py) + C(e)’^(p^ - p^) ( 121 )

41
= 15 .258577 + .011666548 e

(p^^ - p^)

+ .65778747 x 10"^ -088

or in

^ form:

\m c " -0^1666548 - .65778747 x 10"^

and with a p^ found to be:

p^

= 7.6292884 mol /dm^

0 = 0.0000270 mol/dm^

Table 7. Experimental

(from Dousl in

Ethylene Data

, et al . [8]

T (K) ••(3)

Weight

238.15 16.036 1.1130 1280

243.15 15.634 1 .2986 1280

248.15 15.196 1 .5141 1280

253 .15 14.723 1 .7672 1280

258.15 14.207 2.0677 1280

263.15 13.630 2.4321 1280

268.15 1 2 .966 2 .8905 1279

273.15 12.146 3.5025 1053

278.15 10.995 4.4471 775

279 .15 10.674 4.7246 725

280.15 10.290 5.071 485

281 .15 9.767 5.546 232

281 .65 9.387 5.911 128

282.15 8.780 6.489 33

282.35 7.635 7.635

3 This p^ is from a straight line extrapolation of the rectilinear diameter.

If the derivative of eq. (123) is taken as T - , a limit of + “ is obtained

is in accordance with previous findings (see eq . (88)). Further, in reference to

eq. (122), if, from scaling laws:

( 122 )

(123)

. This

42

( 124 )
the derivative (p^ + pj^) goes to - » which is predicted for « < 1 (.350 + .528 = .878 < 1) and = one-half. This is in accordance with the results previously predicted and

listed in table 2.

The critical temperature utilized is Douslin's value of 282.35 K. The p value

c calculated comes out slightly less than the value (7.635 mol/dm^) presented by Douslin.

An analysis of X, „ versus T using this value of 7.635, resulted in an X, ,, = .5002 at LV,C

282.15 K < T^. This indicates tha't a correlated value for p^ must be slightly lower.

The fit of eqs. (122) and (123) to the data is shown in table 8. The percent

differences are very small and indicate a random error which is statistically desirable.

The results of the weighted least squares fit for ethylene are given below. For,

Pv +Pji = A + B(e)'' (Pj^ - Py) + C(e)'^ (Pj^ - Py) , (121)

Table 8. \v,c + p^) Comparison for Ethylene

T (K) (Pj^+Pv) »daTA ^^il'^v^'CALC % Diff

^LV.c.DATA \v.c,CALC % Diff

(P, = 7 .6292884)

238.15 17.149000 17.151151 -.0125 .43666075 .43658868 +.0165

243.15 16.932600 16.929469 +.0185 .44161226 .44172145 -.0247

248.15 16.710100 16.709095 + .0060 .44695462 .44699133 - .0082

253.15 16.490200 16.491149 -.0058 .45246827 .45243163 +.0081

258.15 16 .274700 16.276070 - .0084 .45814737 .45809092 + .0123

263.15 16.062100 16.063816 -.0107 .46412170 .46404508 + .0165

268.15 15.856500 15.854547 +.0123 .47032786 .47042478 -.0206

273.15 15.648500 15.647636 +.0055 .47744414 .47749413 -.0105

278.15 15.442100 15.442076 +.0002 .48598610 .48598790 -.0004

279 .15 15.398600 15.400722 -.0138 .48823216 .48805379 + .0365

280.15 15.361000 15.358999 + .0130 .49018747 .49037917 -.0391

281 .15 15.313000 15.316183 -.0208 .49355328 .49317621 +.0764

281.65 15.298000 15.293720 +.0280 .49432923 .49494492 -.1246

282.15 15.269000 15.270265 - .0083 .49772519 .49744914 +.0555

4 3 Density units is mol /dm .

43
A = .15258577 X 10^^ = 2p^ ; Var(A) = .29216971x10'®

B = .11666548 X 10'^

; Var(B) = .82952522x10'^^

C = .65778747 x 10'®; Var(C) = .10371048 x 10'^^

Cov (A,B) = - .13428749 x 10'^

Cov (A,C) = + .13600825 x 10'^®

Cov (B,C) = - .91386470 x 10'^^

Correlation Coefficient (A,B) = - .86258752

Correlation Coefficient (A,C) = .78133331

Correlation Coefficient (B,C) = - .98527119

An analysis of each coefficient with its variance indicates that the value zero does

not lie within a 3o confidence limit; hence, each coefficient is statistically

significant. It is also desirable that the value (A = 2p^) be quite insensitive to the

values of the coefficients B and C. Indeed, the correlation coefficients indicate that A does not correlate highly (statistically speaking) with B or C. The coefficients B and C are more highly correlated with each other. This is to be expected since they are each

involved in similar type terms in eq. (121).

Another consideration is the sensitivity of p^ to changes in the exponents P and R.

To analyze this, each exponent was held at its value found for the best fit while the other

was varied until p^ could no longer be reproduced as 7.629 mol/dm within four significant

figures. The results are shown below:

R = 1.079,

P = .528

R = 1 .088

p^

= 7.62947 (mol /dm )

R = 1 .088,

R = 1.126,

P = .5306,

P = .528 ,

P = .5184,

p^

= 7.62929

p^

= 7.6285

p^ = 7.62949

p^

= 7.62929

p^

= 7.6285

(Best Fit)

(Best Fit)

The results indicate that p^ is relatively insensitive to the exponents P and R,

i.e., the critical density is not highly dependent on the functional form - a desirable

result

.

The fit, if statistically sound, should result in a randomly distributed error or percent error. Data given in table 8 indicate a random error. Also, fig. 12 gives error

data in a deviation plot for eq. (122). The figure indicates a relatively random error.

Also, the question arises concerning the sensitivity of the critical density to changes in the critical temperature. Moldover [5] and Sengers [14] report values of T^

44
eo0

45

Figure

12.

Percent

Deviation

Plot

of

p„+p

for

Ethylene.
for ethylene differing slightly from Douslin's value of 282.35 K. Each of these

temperatures was utilized in the

^ fit with no change in the critical density to four

significant figures. These results are summarized in table 9 below.

Table 9. Critical Properties of Ethylene

Source T- (K) (mol/dm") Method

Moldover [5] 282.344 7.650 Visual observation of critical

Douslin, et al . [8] 282.350 7.635 Graphical - T ; Straight line

extrapolation of rectilinear

diameter - p

c Sengers [14] 282.3452 7.634 Fit of single phase data to scaled

equation of state This report Any of 7.629 Fit of X|^y ^ vs T to Douslin's

above saturation data

Having found from the data and

^ correlation it is possible to obtain an

equation for

p^^ of the form:

= + B U

+ B

2£ * ^

where, following Douslin, et al . [8]:

B = .350 (found from a graph of (p^ - p^) vs c)

= 3 + 1/G ,

il) = 3 + 2/c

The equation resulting from a weighted least squares fit for

p^^ - p^ is:

(47)

p^ = 7.6292884 + 1.9752911 + .025699752 - .23107950 x 10"^ (125)

compared to Douslin's equation repeated below.

Pj^ = 7.635 + 1 .9695 + .02669 - .2731 x 10“^ (82)

The fit of eq. (125) compared to eq. (82) is slightly better for one-half of the fourteen

data points and slightly worse for the other half. The results are shown in table 10. The next problem is to obtain values of p^. Two approaches were utilized. The first

approach was to combine eqs. (123) for X|^y ^ and (125) for p^ and solve for p^, i.e.,

^v,CALC

"

^“^c ^LV,c^

The results for this calculation are shown in table 11. The overall fit is slightly better than that for Douslin's fit given in eq. (83).

46
3 Table 10. (mol/dm ) Comparisons for Ethylene

T (K) ^A.DATA •^il.CALC ^Jl.DATA'^il.CALC ^il,DATA"P)l,CALC

Equation (125) % Diff Equation (82)

238.15 16.036 16.037977 -.00198 -.0123 -.001

243 .15 15.634 15.630700 +.00330 + .0211 + .003

248.15 15.196 15.195190 +.00081 +.0053 + .001 253.15 14.723 14.724569 -.00157 -.0107 -.002

258.15 14.207 14.208639 -.00164 -.0115 -.002

263 .15 13.630 13.631159 -.00116 -.0085 -.001

268.15 12.966 12.963389 + .00261 +.0201 + .003 273.15 12.146 12.144977 +.00102 +.0084 + .001 278 .15 10.995 10.996848 -.00185 -.0168 -.001

279 .15 10.674 10.676468 -.00247 -.0231 -.002

280.15 10.290 10.287406 +.00259 + .0252 + .003

281 .15 9.767 9 .7651870 + .00181 +.0185 + .001 281 .65 9.387 9.3907125 -.00371 -.0395 -.005

282.15 8.780 8.7591117 +.02088 +.2378 + .018

Table 3 11. Py (mol/dm ) Comparisons for Ethylene

T (K) '^v.DATA *^v,CALC

• ^v,DATA'Pv,CALC '^v.DATA"^v,CALC

Equation (126) % Diff Equation (83)

238 .15 1 .1130 1 .1 133772 -.000377 -.0339 -.0001

243 .15 1 .2986 1 .2984076 +.000192 +.0148 +.0003

248.15 1 .5141 1 .5138464 +.000254 +.0168 +.0002

253 .15 1 .7672 1 .7667706 + .000429 + .0243 + .0002

258.15 2.0677 2.0675788 +.000121 +.0059 -.0002

263.15 2.4321 2.4326972 -.000597 -.0245 - .0008

268.15 2.8905 2.8909751 - .000475 -.0164 -.0003

273 .1 5 3.5025 3.5026075 -.000107 -.0031 +.0006

278 .15 4 .4471 4.4453302 + .001769 +.0398 +.0020

279 .1 5 4.7246 4.7243203 +.000280 + .0059 +.0001

280 .1

5

5.0710 5.0715327 -.000533 -.0105 -.0020

281 .15 5 .5460 5.5509046 -.004905 - .0884 -.0070

281 .65 5.9110 5.9031244 +.007876 + .1332 + .0040

282 .15 6 .4890 6.5109346 -.021935 - .3380 - .0280

47
The other approach is to utilize the found from assume first order symmetry,

and assume the exponents in the equation are the same as for

p^^ in eq. (125), and fit;

.350

p^ - 7.6292884 + 1.9752911 e

- R r*9®^ + R p ''-622 ^2v

^ + 63^ e 127)

The results are not as good as those obtained using eq. (126). The differences between

DATA CALC

order of magnitude higher than those in table 11. This

is not satisfactory. The equation obtained was:

p

= 7.6292884 - 1.9752911 + .16287155 x 10"^

+ .50890319 X 10"^

(128)

Equations (125) for p^ and (128) for p^ meet the criteria of eqs. (87) and (106).

The following procedure is suggested for obtaining the critical temperature, density,

and smooth saturation density values from experimental saturation data.

Step One: Plot p^ - p^ vs e on log-log paper to obtain a value for 8. Use in

addition, a computer analysis of this plot to obtain a best value of T^. For

this procedure see Ueber [7]. This utilizes the well known scaling law:

Pji - . (124)

Step Two: Utilize this T in a fit of the data to X, ^ finding the best value of

C

^ LV ,C Pc- step Three: Utilize the p^, T^, and B to fit the

p^^ data to an equation.

Step Four: Utilize the Xj^^ ^ and

p^^ equations to generate smooth data for the

saturation densities both p^ and p .

This procedure allows a rational integration of the critical and saturation

properties. The X|^^ ^ function is the integration tool.

One of the major problems left in coexistence data work is to obtain equations which

will fit the entire range from the triple to the critical point. The "dome" equations which

are used here are not thought to be valid over this entire range. However, the relative

flatness of the X|^^ ^ curves over this entire temperature range (see fig. 1) may be helpful

in solving this problem.

Now, from eq. (125) and p^ from eq. (126) can be utilized in a final statistical

analysis of the original fit of eq. (121). This test is to look at the shape and magnitude

of the 3o confidence band over the temperature range of interest. The standard deviation

with temperature of the fit can be found from the square root of the variation of

(Pj^ + p^). This variance, from well known statistical considerations, is;

48
3(Pn+Py)

= " 9A

^(P^+Py)

3B

3(Pj+P„)

3C

where from eq . (121 )

a(p£+py)

3A

3(Pj^+P^

9A

9(P£+Py)

“~9A

Var(A) +

9(p^'''Pv) 9B

3{P0+Py)

Cov(A.B) + Cov(A

Cov(A,B) + 3(P£+Py) 3(P^+Py)

9B

3(P£+Py)

Var(B) +

9C Cov(B

3(pjl+Py)

Cov(A,C) + Cov(B.C) + Var(C)

,C)j

,c)j

']

( 129 )

9(P£ + Py)

1 ,

9(p£ + py)

9B

e (P£ - Py) .

9C ^^5,

obtaining and from eqs. (125) and (126) respectively. The variances and covariances

are those previously given. The standard deviation of (p. + p ) at each temperature is

V multiplied by three and the result is plotted to show the 3o confidence limit band. The

results are shown in fig. 13.

The behavior of the band deviates from the so-called classical form in the

temperature range from about 238 - 250 K. The classical form is found from the

latter temperature to the critical. The graph indicates that p^ is found to four

significant figures at a high degree of confidence, i.e.,

7.62921 1 P^ < 7.62937 (mol/dm^)

within a 3o confidence band.

Finally an analysis of the effect of the data weights on the p^ value was made. The

3 unweighted fit of

^ resulted in a value of p^

= 7.629 (mol/dm ) to four significant

figures with a slightly higher variance of the data. All in all, the value of p^, found by

the fit of

^ to the saturation data, results in a statistically reliable number. The

foregoing analysis supports the use of the X|^^ ^ method to obtain a correlated p^ from

saturation data.

49
e>i V esj

jUip/|oui^Olx

50

Figure

13.

3a

Confidence

Limit

Band

on

p„

+p

of

Ethylene.
5. EXPERIMENTAL APPROACH TO SIMULTANEOUS DETERMINATION OF COEXISTENCE DATA

This section presents a method for obtaining saturation density data simultaneously.

Current experimental approaches obtain and in separate experiments. Usually the

pj^ is obtained directly while the p^ is found from the intersection of isochores with the

vapor pressure curve. See Kleinrahm, et al . [15] for a review of these techniques.

The method proposed here is based on the straight line plots found in fig. 5. Theoretically, all one needs to know is two pairs of liquid volume fraction - total density

data at the same temperature to establish a line. Simple extrapolation to X^^^ = 1 for

Pg^ and to Xj^y = 0 for p^ produces the saturation values from the same set of data. This

should enhance internal consistency.

Currently, in many cases, the coexistence data takes a secondary role in property

fitting to equations of state. For example, the procedure used by Haar, et al . [9] for

ammonia is to obtain the saturation data from a surface described by an equation of state explicit in Helmholtz free energy. The coexistence data (mainly those of Cragoe,

et al . [10,11] were used only to obtain estimates of the critical properties. Perhaps a more internally consistent set of p^, p^^^ data can provide the foundation for an equation of

state

.

To test the validity of using data such as that illustrated in fig. 5, liquid volume

fraction data is needed. Unfortunately, the thermodynamic property literature yields very

little total density-liquid volume fraction information. However, Cragoe, et al . [10,11]

in obtaining p. and p for ammonia did measure and publish such data. They did not use

A/ V them, however, in the manner suggested here. To test this idea, one needs saturated liquid

and vapor density values data at the same temperature. Unfortunately, this occurs at only one point in all the data taken by Cragoe and his co-workers. In their report on the

liquid densities [10] at T = - 46.43°C the values are:

p^

= .5370715 g/cm^, and

X^^^ = .7695038

and in the vapor density report [11] at T

= - 46.42°C the values of

Qj

= .0037965898 g/cm^, and

X^^ = .0047679271

,

are given. These two points determine a straight line. Then X|^^ = 0 is substituted to

obtain p , and X, = 1 is substituted to obtain p.. See fig. 14 for a sketch of the data

V LV X and procedure. (Because of the relative sizes of the numbers, they are not plotted

precisely on the linear scales.)

Table 12 gives some comparisons related to the data in fig. 14. The first column

gives the results from the straight line. The second column gives the experimental values

of Cragoe, et al . [10,11] while the third column gives the values of Haar, et al . [9]

linearly interpolated to -46.425°C.

51
Figure 14. vs p-j- to Find and p^ for Ammonia

52
3 Table 12. Comparison of Coexistence Densities (g/cm ) This Report - Cragoe, et al . [10,11] Haar, et al . [9]. Equation

Straight Line Method Experimental of State

p

= .0004717609 p^ = .0004726344 (T = -46.42°C) Py = .000462 (T = -46.425°C)

p^ = .69780 p^ = .69781 (T = -46.43°C) p^ = .69770 (T = -46.425°C)

Another approach which could be taken would be to utilize accurate data = 1)

along with < 1, Pj) data and obtain a straight line. This line could be extrapolated

to Xj^y = 0 to obtain a value for p^. This was done utilizing data presented by Cragoe,

et al . [10] in their report on liquid densities only.

The results are presented in table 13. Data is taken from Haar, et al . [9] at the

nearest even temperature for comparison. A p^ found at the same temperature from

the equations presented in Cragoe, et al . [11] when dealing only with p^. The values for

p^ in table 13 not only reflect the range of error involved in p^ determination but also

the wide disagreement as to what the p^ actually are. The straight line values for p^ are

at least correlated with the experimental

p^^ values.

If the p^ is specified along with a point {X|^^ < 1, Pj), p^ can be determined from

the straight line at X|^^ = 1. Here the results are even more surprising. For very small

X^^y (x .005) there is good agreement in the

p^^ values at X^y = 1. See table 14.

53
3 Table 13. Comparison of Values (g/cm )

T (°C)

Cragoe (Haar)

^£,DATA ^T,DATA

Cragoe [10]

\v,DATA

V

^v,STR ^ Line^v.CALC ^

Cragoe [11]

^v.PUB ^

Haar [9]

-49.96 (-50) .702099 .576546 .821076 3.8557 3.8243 3.8100

-40.14 (-40) .690255 .57690 .835630 6.2074 6.4057 6.440

-30.04 (-30) .677782 .540417 .797018 10.454 10.367 10.380

-14.94 (-15) .658592 .57627 .87464 18.977 19.706 19.670

-00.05 (0) .638590 .553245 .865617 35.023 34.467 34.580

+10.00 (10) .624637 .553035 .884468 48.793 48.594 48.680

+20.01 (20) .610273 .55291 .904958 67.099 66.896 67.010

+29.98 (30) .595295 .55275 .927413 91 .772 90.258 90.050

+39.94 (40) .579606 .53600 .923170 119.52 119.86 120.29

+49.94 (50) .562905 .495826 .877389 158.24 157.36 157.80

+60.22 (60) .544802 .536281 .98373 208.72 206.08 204.90

+80.58 (81) .504322 .449203 .882783 340.94 345.88 347.44

+91.24 (91) .479773 .44896 .929207 444.78 453.74 445.80

3 Table 14. Comparison of Values (g/cm )

T (°C)

Craqoe (Haar)

‘^v,DATA^^°^

Craqoe [11]

\v,DATA '^il.STR Line

Pjl.CALC

Craqoe [10] Haar [9]

-50.00 (-50) .390442 3.6784 .004687 .701877 .701997 .70202

-42.07 (-42) .585035 3.8702 .004748 .692422 .692486 .69231

-32.99 (-33) .905551 3.8694 .004356 .681302 .681370 .68122

-17.88 (-18) 1 .76246 7.5843 .008815 .662238 .662279 .66232

-03.09 (-3) 3.10183 6.9142 .005960 .642766 .642762 .64267

+02.98 (+3) 3.84438 3.9526 .000172 .634123 .634480 .63454

+24.05 (24) 7.57576 14.6599 .011873 .604227 .604249 .60448

+32.03 (32) 9.58130 16.3919 .011693 .592041 .592090 .59227

+46.32 (46) 14.2653 14.6496 .000693 .568489 .569091 .56967

The results shown in tables 12, 13, 14, may motivate experimentalists to obtain

coexistence data from (p^, X|^^) plots. Cragoe, although he did not use the above method,

did measure liquid volume fractions using sight lines determined by careful volumetric

calibration. (In evaluating the coexistence data of amnonia, Haar, et al . [9] concludes

that the data of Cragoe, et al . [10,11] is the best-indicating that accurate measurements

can be made of the liquid volume fraction.)

54
An even earlier effort than Cragoe's to determine saturation densities by measuring

liquid volumes is that of Young [16]. He utilized pairs of measured liquid volumes and

total mass values in calculations based on mass balances. The approach is similar in concept to the ideas expressed in figs. 5 and 14 of this paper. He did not utilize,

however, the reduced variable form, i.e., liquid volume fraction vs. total density nor did

he explore the implication of these as thermodynamic functions as is done in this report.

An examination of fig. 5, indicates that these lines should be significantly different

(statistically speaking) over a small range of temperature just below the critical. This

is in contrast to the (P vs T) isochores near the critical point which are extrapolated to the vapor pressure to find p^. When utilizing the vs py

plots to obtain coexistence

data it appears that the large changes in p. and p near the critical point may actually be

X/ V helpful, vvhereas up to this time these changes were a problem.

The goal then is to obtain a good measurement of the liquid level. Aside from the

visual method utilized by Cragoe, newer techniques such as time-domain ref lectometry [17]

or fiber optics [18] should be investigated.

6. SUMMARY

An analysis of the liquid volume and liquid mass fractions has been made over the

temperature range from the triple point to the critical point. Limits of the functions and

derivatives have been studied both at the triple and critical points. Also the slope of

the rectilinear diameter at the critical was studied in some detail. Conclusions were made

about the constants and exponents in the p^ and p^ coexistence dome equations. A procedure

for correlating saturation data with the critical point has been presented using ethylene

as an example. Finally, an experimental approach for obtaining p^^ and p^ simultaneously

is presented.

7. ACKNOWLEDGMENTS

I wish to thank the staff of the Thermophysical Properties Division of the National

Bureau of Standards, Boulder, Colorado for making office, digital computer, and data resources available during my tenure as a guest worker. Thanks also to those staff members

who took the time for stimulating discussions, especially Robert McCarty regarding the

statistical analysis of the

^ fit. Salary support was provided by Calvin College,

Grand Rapids, Michigan during my Sabbatical leave.

55
8. REFERENCES

[1] Methane, International Thermodynamic Tables of the Fluid State-5, 1976, lUPAC,

Chemical Data Series, No. 16, Pergamon Press, Oxford.

[2] Hohenberg, P. C. and Barmatz, M., 1972, Gravity Effects Near the Gas-Liquid Critical

Point, Phys. Rev. A, 6: 289-313.

[3] Green, M. S., Cooper, M. J. and Sengers, J. M. H. Levelt, 1971, Extended

Thermodynamic Scaling From a Generalized Parametric Form, Phys. Rev. Lett.,

492-495.

[4] Sengers, J. M. H. Levelt, Straub, J. and Vincenti-Missoni , M. J., 1971, Coexistence

Curves of CO^ , N

2 O and CCilF^ in the Critical Region, J. Chem. Phys. 5034-5050.

[5] Moldover, M. R., 1974, Visual Observation of the Critical Temperature and Density:

CO

2 and C

2 H4

, J. Chem. Phys., £l_: No. 5, 1766-1778.

[ 6 ] Van Poolen, Lambert John, 1977, Analysis of Property Data in the Critical Region

Using the Function-Liquid Volume Fraction, ASHRAE Transactions, 8 : Part II, Dec. [7] Weber, Lloyd A., 1970, Density and Compressibility of Oxygen in the Critical Region,

Phys. Rev. A, 2: No. 6 , 2379-2388.

[ 8 ] Douslin, D. R. and Harrison, R. H., 1976, Pressure, Volume, Temperature Relations of Ethylene, J. Chem. Thermodynamics, 301-330.

[9] Haar, L. and Gallagher, J. S., 1978, Thermodynamic Properties of Ammonia, J. Phys.

Chem. Ref. Data, ]_: No. 3, 635-792.

[10] Cragoe, C. S. and Harper, 3rd, D. R., 1921, Specific Volume of Liquid Ammonia,

Scientific Papers of the Bureau of Standards, Vol 17, 287-315.

[11] Cragoe, C. S., McKelvey, E. C. and O'Connor, G. F., 1922, Specific Volume of

Saturated Ammonia Vapor, Scientific Papers of the Bureau of Standards, Vol 18,

707-735 (dated 3/17/23).

[12] Weiner, J., Langley, K. H. and Ford, N. C., Jr., 1974, Experimental Evidence for a Departure from the Law of the Rectilinear Diameter, Phys. Rev. Lett., No. 16,

879-886.

[13] Gosman, A. L., McCarty, R. D. and Hust, J. G., 1969, Thermodynamic Properties of

Argon From the Triple Point to 300 K at Pressures to 1000 Atmospheres, NSRDS-NBS

Report No. 27.

[14] Sengers, J. M. H. Levelt, private conmunication, 1979.

[15] Kleinrahm, R. and Wagner, W., 1978, Examination of Density Measuring Procedures and

Design of Device for Precise Measurement of Boiling and Thawing Densities, Chem.

Ing. Tech., No. 10, 805 (synopsis)(in German).

[16] Young, S., 1910, The Vapor-Pressures, Specific Volumes, Heats of Vaporization, and

Critical Constants of Thirty Pure Substances, Scientific Proceedings, Royal Dublin

Society, 374-443, June 10.

56
[17] Cruz, J. E., Rogers, E. H. and Heister, A. E., 1973, Continuous Liquid Level

Measurements with Time-Domain Reflectometry, Paper H-4, Advances in Cryogenic

Engineering, Vol 18, Plenum Press, N. Y., 323-327.

[18] Elson, B. M., 1978, Fiber Optics Technology Investigated, Aviation Week and Space

Tech (Oct. 16).

57
APPENDIX A. Symbols and Units

A constant

B constant

b constant

C constant

c constant

Cov covariance

m mass

P pressure, EXPONENT

R exponent

T temperature

V vol ume Var variance

X 0 _< fraction _< 1

X' mass or volume fraction derivative with respect to temperature

3 exponent

exponent

exponent

p density

p' density derivative with respect to temperature

0 standard deviation

SUBSCRIPTS

c critical point

c,LIM critical point, limit

£ saturated liquid

LM liquid mass

LV liquid volume MAX point at which Xj^^ has a maximum

MIN point at which X^^^ has a minimum

SAT saturation

T total or overall

TP triple point

V saturated vapor

1

O related to constant modifying e term

2 related to constant modifying e term

3 related to constant modifying e term

SUPERSCRIPTS

- limit from below

+ limit from above

58
NBS*114A (REV. 9-781

U.S. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. BIBLIOGRAPHIC DATA

SHEET NBSIR 80-1631

4. TITLE AND SUBTITLE

Mo. t, $ Ac<5«$sioa No»

ANALYSIS OF LIQUID VOLUME AND LIQUID MASS

FRACTIONS AT COEXISTENCE FOR PURE FLUIDS

5, Publication Date May 1980

7. AUTHOR(S)

Lambert John Van Poolen

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

9 'Task/Woffc tfe« Mo.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fStreer. city, state, ZIP) 13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leaa factual aummary of moat alghlflcanl Information. If document includes a significant bibliography or

literature aurvey, mention it here.) An analysis of the bsHavlor of liquid voluma and liquid mass fractions at cosxistenca

for pure fluids is made. Scaled equations for the saturation liquid and vapor

densities are analyzed and relationships between various exponents and among constant

coefficients are presented. Inequalities which exist among the saturation densities

and their derivatives are developed. A procedure to correlate saturation data with

the critical point is applied to ethylene. An experimental procedure to determine,

simultaneously, saturated liquid and vapor densities at a given temperature is

presented.

17. KEY WORDS fs lx to twelve entries; alphabetical order; capitalize only the firat tetter of the first key word unless a proper name;

separated by semicolons)

Coexistence densities; critical density; critical point; liquid mass fraction; liquid

volume fraction; phase equilibria; pure fluids.

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