Corresponding States Metastable

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Corresponding States Correlation of Saturated andMetastable PropertiesWEI-GUO DONG and JOHN H . LIENHARD

Heat Transfer and Pltase-Chonge Laboratory Mechanical Engineering Departmerlt University of HoustonHouston Texas 77004A corresponding states correlation is developed for the p v T properties of saturated liquids. is similar in form

to a previous correlation; however it is more accurate in the low pressure, low Pitzer factor range, owing to theincorporation of new information about low Pitzer factor behavior. The problem of creating correspond.ing statescorrelations of the spinodal lines and the metastable states is also discussed in the light of new equation of stateinformation. A partial preliminary correlation works particularly well for non-polar molecules in the range of fairly smallpositive Pitzer factors.On a etabli une correlation du type etats correspondan ts pour les proprietes p v T des liquides satures; elle estde forme semblable t celie d'une correlation anterieure, mais elle est plu s precise it basses pressions et pour de faiblesvaleurs du facteur de Pitzer, it cause de I' incorporation de renseignements nouv eaux sur Ie comportement pour de faiblesvaleurs du facteurs de Pitzer. On discute aussi e probJem e de l'etablissement de correlations de type etats correspondantspour les lignes spinodales et pour les etats metastables, it la lumiere de nouvelles informations sur les equations d'etat.Vne correlation partielle preliminaire fonctionne particulierement bien, dans Ie cas de molecules non polaires, dans Iedomaine des facteurs de Pitzer positifs et assez petits.

The first corresponding states correlations of thesaturation pressures were presented by Pitzer et al.(1955) and by Lydersen et al. (1955). They wrote the Lawof Corresponding States2 in the form:T - al = f (p molecular parameter) . . . . . . . . . . . . . (I)

For the molecular parameter, Pitzer used the Pitzer factor, W W I 10gl o[P,. sal(T, = 0.7)] . . . . . . . . . . . . . . (2)

and Lydersen et al. used the critical compressibility, eSeveral investigators subsequently made improved correlations of the vapor pressure curve usi ng w, which givessuperior results. The most accurate was Lee's and Kesler's(1975):In (P,.sa) = 5.92714 - 6.09648 / T, 1. 28862 In (T,)

important leverage in corresponding states correlations, thatwe use subsequently.Nobody has, to our knowledge, attempted to create acorresponding states correlation of properties within thesa t lration (or coexistence) dome. However, Biney et al.( 1985) recen tly improved on earlier work by Eberhart(1976) and Karimi an d Lienhard (1980) in making accuratepredictions of liquid spinodal lines. (The spinodal line, orlocus of points at which ap / a v)r = 0 in the liquid phase,is illustrated in Figure I.) Based on the earlier work ofKarimi and Lienhard (1980), Biney et al. represented spinodal points as limiting homogeneous nucleation points,

P+ 0.169347T,6 w[15.2518 - 15.6875 / T,-13.4721 n T, 0.43577T,6] (3)

In 1976 , Lienhard showed that the Lydersen et al. correlation naturally extrapolated to include the van der Waal ssaturation curve as Z, approached the van der Waals Zc of3/8. This was part of a demonstration that the van der Waalsfluid was a member of the family of real fluids at the appropriate value of Z, . Peck redid this demonstration in 1982using w in place of Z, and obtained even stronger corroboration of Lienhard s result. He found that, as w approachedthe van der Waals value of 0 .302, all reduced thennodynamic properties approached van der Waals values, except where the Law of Corresponding States is known to failas a consequence of low temperature quantum effects (see,e.g . Watkinson and Lielmezs (1967).)More recently, Shamsundar and Lienhard (1983) notedthat mercury , with an w close to 0 .302, is very wellrepresented by van der Waals' equation. The verifiable connection of van der Waals' equation to real fluids provides

IOn leave from the Thermal Power Engineering Research Institute,Xi-an, People's Republic of China.Conventional notation is listed in the No menclature section and nolexplained in context.

T. p cPc 1\ >, . . \ saturated liqUIdI vap o r spinodal

~

This negative pressure can be reachedby depressuriz ing a l iquid isothermal lybelow i ts saturat ion pressure

I , V and v areI used in predic.t inghomogeneous nucleation

v

{

Figure I - Typical real-fluid isotherms.THE CANADIAN JOURNAL OF CH i MICAL ENGINEERING, VOLUME 64, FEBRUARY 198658


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0.8

0.7

0.6

0 .2o. ,0. 05

0.0 1.005

0.5C1oH22 C1sH32 C19H40C11 H ., ,, C16H340.4 C12H2 8

0.3 Present workLee's and Kesler's equation

o 0 . 1 0.2 0 .3 0.4 0.5 0.6 0.7 0.8 0.9Pitzer factor, W

a typical real f luidis o therml o cus 01saturat ed l iquidand vspor s tat esl iquid and vaporspinodal lin es

1.0 .

1.0 P,

~

':lQ)a.EQ)lJQ)J

:llJQ)a:

z 1.2

cr:-.>a. 0.8N(;0 0.6'

nonpolar data : t > \ O0 .1 w 0 .35

' .oo . ~ ~ . ~

0.2 0.4 0.6 0.8 1.0 1.2 1.4igure 3 - Schematic representation of the metastable andunstable regimes on Z-coordinates.given by:

- I n ) = 2 2 (4)3 kTc)[Psa, T,p) - p] (I - vJwhere the group kTc replaces the conventional group, kT, asKarimi and Lienhard recommend. They showed that thenucleation probability,), normally ranges between 10-4 and10-5

Biney et a!. also showed that when the very generalShamsundar-MuraliP v - T equation of state (see Murali,1984 or Lienhard et al., 1985)P v - Y ; - ) v - vm v-vg )- = I - (5)PSI (v +b) v +c) v +d)

was fitted to real fluids, the results represented the metastable and unstable states - as well as the stable ones with high accuracy. They verified this by substituting theresults in van der Waals' (1894) surface tension predictionand accurately obtaining the temperature dependence of thesurface tension.Our present objective is to take a new look at the correlation of stable, metastable, and unstable data on andwithin the saturation dome. We wish to reconstruct Lee'sand Kesler's correlation to include the van der Waals satur

ation curve. Furthermore, armed with better information

>...0 0.4(/)(/)

a.E 0.20(.)

Figure 4 about the spinodal and metastable states , we seek to includethem in the correlation.CorrelationsA. CORRELATION OF SATURATED P - v - T DATA

The present vapor-pressure correlation is shown inFigure 2. The sources of p, T)sa w, Tc, and Pc data used todevelop this curve were: Reid et al. (1977), Reynolds(1979), Vargaftik (1975), and the IFC formulation of theproperties of water (1968). Data for the van der Waalsequation calculated by Shamsundar and Lienhard (1983) arealso included in the figure. (For clarity, Figure 2 only displays roughly every other isobar used in the correlation.)An equation similar to Lee's and Kesler'S was fitted to634 data points; however we found it possible to use twofewer terms than they did. The result, which fits all the datawith an rms accuracy in T, of : :: 0.42 percent , is

InP,.sa, = 5.37270(1 - I / T,)+ w(7.49408 - 11.18177T/ +3.68769T,6+ 17.92998InT,) . . . . . . . . .. . . . , . . . . . . . . . . . (6)

Figure 2 - Correlation of vapor-pressuredata for real fluids and the van dcr Waalsfluid.

C2H6 (0.100) o'" C6H'4 (0 . 308) CsH'2 (0.258) - C,H'D (0 . 205)

' ' k 0 C3Ha (0 .155) "H20 (0 .348) o ~ , '0faired curve through ~Reduced pressure, Pr

Z-chart correlation of the vapor-pressure data.

C7H'6 (0 .352)02 (0 . 022)H2 ( -0 .218)N (0 .037)A, (- 0 .003)van der Waalsf luid ( -0.302)

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, FEBRUARY 1986 159


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0.8 - - - r - - - - . - - - - . - - . - - ~ - - - - - - - - - - - - - - - - -I -a: Interpolated data M :> C2' HSa 0.6N i" CSHI4. CSH 20.4 . C H,o

C3Ha0.2 o H2O

0

o van der Waalsfluid

_ .. - lalred c urv e throughnonpolar dat a0.1 lC w E 0 .35

0 4 L ~ _ - L _ ~ _ ~ _ L - ~ - - ~ ~ ~ ~ ~-1.0 -0 .8 -0.6 -0.4 -0 .2 0 0.2 0.4 0.6 0.8 1.0Reduced pressure, Pr

Figure S - Z-chart correlation of the vapor and liquid spinodallines for various substances .

I -a:>a

N5U.,

0.8

0 .60.4

0.2

Int erpo lat ed -dalaM: C2HeIi' CeH,.

CSH 2C4 H lO

spinodal linefrom Fig. 5isotherms lor0.1;;( w;c 0.35

o

''0-

0p

0.2 0.4 0.6 0.8 1.0Reduced pressure , Pr

Figure 6 - Z-chart correlation of metastable / unstable isothermsfor various substances.The important feature of Equation (6) is that it is based on

data in a larger range of w than Equation (3) was, includingthose substances for which w extends well below zero.These include helium w = -0.391), cesium w =-0.267), hydrogen w = -0.218), mercury w = -0.21),potassium w = -0.11), and sodium w = .078), inaddition to the van der Waals fluid (which was given doubleweight in the curve fit.) We have excluded data for heliumand hydrogen below Tr = 0.6 where we expect to incur thefailures of the Law of Corresponding States that areinevitable when the behavior of these fluids must be dictatedby quantum statistics.Equation (6) gives pr,, , Tr= I = I and it gives Equation(2) at Tr = 0.7. It has not been forced to give a temperatureindependent Riedel factor at the critical point.

The choice between Equations (3) and (6) is a toss-up athigh wand T except in that Equation (6) is simpler. However, Equation (6) is far superior at low values of wand T.B. CORRESPONDING STATES CORRELATIONS OF METASTABLEPROPERTIES

Figure 3 is a schematic diagram that shows how thep - v - T surface (Figure 1) maps into generalized compressibility coordinates: Z = Z(p Tr) The various regimesare indicated on it for the readers convenience .Figures 4 and 5 show the vapor pressure and spinodal line

I - 1.0. . . . - - .a:>c. 0.5 _N

>- -0.5:0'iiiC.,N5U2>-.D


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Figures 4,5 and 6 thus suggest that it is feasible to createa Z-chart in the metastable and unstable regimes, particularly for a restrictive range of w s - just as is done in thestable ranges. Figures 7 and 8 are such Z-charts. However,Figure 7 is only based on the Biney et al. curve fits for water(with w = 0 .348) which while not the most general case- is the best documented and which best satisfied all oftheir tests for validity. Water is slightly displaced from thecenter of the data scatter in Figures 4, 5, and 6. ThusFigure 7 merely displays the qualitative features of a largerPr range - beyond the range where Biney et al. claimedgood accuracy.The curves in Figure 8, on the other hand, are limited tothe ranges in which pentane (with w = 0.258) represents theremaining data well. They are "best-fit" curves through thedata for the simple-substances and, while they do not strictlyrepresent pentane , they never stray far from it. We recommend Figure 8 as the corresponding states correlation for theregion within the saturation dome. It represents at least allof the nonpolar substances that we present within 0 .02 inZ, in the ranges:

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Corresponding States Metastable