Rep rinted from May 1986, Vol. 108, Journal of Heat T ransfer

Use of a Cubic Equation to Predict

P. O. 8iney1 Surface Tension and Spinodal

Wei-guo Oong2 Limits The very general Shamsundar-Murali cubic equation is used to interpolate p-v-T J. H. Lienhard data into the metastable and unstable regions. This y ields a spinodal line that closely Fellow ASME matches the homogeneous nucleation limit predicted by an improved kinetic theory. Only the pressure, the saturated liquid and vapor volumes, and the liquid com­ Heat Transfer and Phase-Change Laboratory, pressibility at saturation, as well as one compressed liquid data point. are needed to Mechanical Engineering Department , use the cubic equation for the interpolation process. The equation also yields an ac­ University 01 Houston , curate prediction of the temperature dependence of surf ace tension when it is Houston, TX 77004 substituted in van der Waals'surface tension formula. Thus, by capitalizing on the inherent relation among the p-v-T equation, the spinodal prediction, and the sur­ face tension - all three - it is possible to obtain each with high accuracy and minimal experimental data.

Introduction p Surface tension is intimately related to the metastable and unstable fluid states, and to the p-v-T equation that describes Pc these states, The aim of this study is to use this interrelation to assist in the development of means for estimating and predict­ ing both p-v-T and surface tension data, We therefore begin by reviewing the character of this relationship. Surface Tension and the Equation of Stale. It was shown in 1894 by van der Waals [1] that the temperature dependence of surface tension could be predicted precisely by the expression

~=f(Tr)= f'g ---k-[Pr,sa,(Vr-Vrj )- r PrdVrJ 1/2dvr (I) ao Vrj Vr Vrj where a is the surface tension ; Tn Pro and Vr are the reduced3 temperature, pressure, and volume: and where g and f denote O ~~~------~ saturated vapor and liquid values. The constant ao is a reference value of the surface tension which van der Waals showed how to evaluate in terms of molecular properties. (No one to date has managed to make accurate evaluations of ao.) Little has been done with equation (I) because its use re­ quires a full knowledge of p-v-T information throughout the metastable and unstable fluid regimes (see Fig. I). van der Waals used his own famous equation of state in equation (1) Fig. 1 Typical real,lIuid Isotherms and - without the aid of a computer - succeeded in making an approximation valid only near the critical point: · 16 I1m a=--a (1- T )3/ 2 (2) On Locating the Spinodal Line. A knowledge of the loca­ Tr-I V6 0 r tion of the liquid and /or vapor spinodal lines can be par­ Recent studies (see, e.g., [2]) of the variation of a with Tr near ticularly helpful in the process of developing the p-v-T equa­ the critical point suggest that tion of sta te that is needed to complete the integration o f equa­ tion (1). In 1981 , Lienhard and Karimi [3] provi ded m olecular lim acx(1 _ T )1.280r 1.29 (3) 4 Tr-l r arguments that ha wed that the liquid spinodal limit could be predicted quite accurately by h omogeneo us nucleation theory. gives a more plausible temperature dependence for real fluids They also howed that this was not true fo r the vapor spinodal than equation (2). It is also known that a form of equation (2) limit {3. 4] . Vapor spinodal li nes lie nowhere ncar the limit of with an exponent of 1119, or 1. 22, represents a wide variety of homogeneous nucleation for vapors. fluids pretty well at lower temperatures. In the course of their work, Lienhard and K rimi used the conventional homogeneous nucleation expression I PreSe nt address: Mec hanical Engineering Department, Prairie Vi ew A. and . nu leation events -Gb M. University, Prairie View, TX 77445. } =e (4) 20n leave from the Thermal Power Engineering Re search Inslitute, Xi-an, molecule co llisions People's Republic of China. 3 A "reduced" properly is one divided by its thermodynamic critical value. Contributed by the Heat Transfer Division and presented at the 23rd National Heat Transfer Conference, Denver, CO, August 1985 . Manuscript received by 4The spinodaiiimit is the locus of points at which (ap/au)T is zero (see Fig. the Heat Transfer Division November 8, 1984. 1).

Journal of Heat Transfer MAY 1986, Vol. 108/405 where5 closely to the homogeneous nucleation pressures given by j=probability of nucleati ng a bubble in a given collision (5) equation (8) with j= 10- 5 and with kTc used as the and characteristic energy. -' he interpolations did not come close to equation (8) when the conventional energy kT was used. (The Gb= Wkcn / (kTor k Tc) (6) minimal valu~ ofj= 10- 5 was not purely empirical. Molecular and (see e.g. [5]) arguments in [3] fixed it within an order of magnitude or so.) It is important to note that when these arguments are ap­ (7) plied to the vapor spinodal [3J they show that the limit of homogeneous nucleation o f droplets is far from the vapor Notice that in equation (6) we suggest that the Gibbs spinodal limit. number Gb should be a ratio of the critical work required to form a nucleus to either kT or kTc' The conventional nuclea­ A New Form of Cubic i'A:Juation for Fitting p-v-T tion theory is based on the average kinetic energy of the sur­ Data. Shamsundar and M urali [9, 10] have recently made ef­ rounding molecules, which is on the order of kT. However, it fective use of the following general form of cubic equation in was noted in [3] that the energy required to separate molecules fitting indivi dual isotherms from one another is on the order of k T . T hi s seemed to be an c ~ = 1- (V-VI) (v-vrn ) (v- vg ) equally plausible candidate for the characteristic energy of the (11) .0 'at (v+b) (v+c) (v+d) system. When equations (4), (6), and (7) are combined, we obtain The quantities P ,al' Vj ' Vrn , vg , b, c, and d all vary with the following expression for the homogeneous nucleation temperature. This form has the advantage that it automati­ pressure P corresponding with a given temperature Tsp cally satisfies critical point criteria, but it need not be tied to them. It is also pre factored to simplify fitting the constants. 167ru3 - In j = 2 2 (8) Three of the coefficients in equation (II) are the known or p ] (I - 3(kT k Tc )[Psat (Tsp) - vllvg) temperature-dependent properties: P sal , VI' and vg • Thus the most straightforward use of the equation is one in which it is where vI and Vg are to be evaluated at Tsp- Two issues re­ fit to one isotherm at a time. mained: identifying the value of j that will give the spinodal Murali simplified equation (II) by setting c = d and thereby limit, and deciding whether to use kT or k Tc' reducing the number of unknown coefficients to three. These Lienhard and Karimi next curve-fit cubic equations to the three coefficients were determined by imposing the following well-documented stable equilibrium states of water, constrain­ three conditions on the equation: the ideal gas limit at low ing them to satisfy the "Gibbs-Maxwell " requirement that pressures, the Gibbs-Maxwell condition, and the measured r~ isothermal compressibility of saturated liquid. He thus J vdp=O (9) evaluated the coefficients of the equation directly (rather than I statistically) using very few data. which stipulates that the two regions between an isotherm and The condition under which equation (11) reduces to the a horizontal li ne connecting j and g must be equal to one ideal gas law, by the way, is another in area (e.g., Area A in Fig. 1 equals area B). They chose the Himpan form (8) of cubic equation RT --=b+c+d+vl+vm+vg (12) A a P sa t Pr=--- (10) vr-b (vr-c) (vr +d) This kind of temperature-by-temperature application of equation (11) yielded far higher accuracies [10) than any ex­ and evaluated all five constants using least squares fit. They isting cubic equation, particularly in the liquid range. The re-evaluated the constants for each of many isotherms. equation also performs extremely well in the stable There were two weaknesses in this curve-fit procedure. The superheated vapor range. The key to this success is, of course, first is that equation (10) was not forced to fit the ideal gas law the fact that the coefficients do not have to obey any predeter­ precisely at high temperatures. T he second is that the H impan mined dependence on temperature. form turns out to be slightly restrictive. We remedy these features subsequently. An nJustrative Application of tbe Preceding Ideas. Infor­ Equation (10) with the five statistically fitted constants gave mation of the kind \ e have been describing can be used to ex­ interpolated liquid spinodal pressures that showed some pand existing knowledge. Kari mi , for example, developed a numerical data scatter. These pressures corresponded very new fu ndamental equation [11 ] for water, based in part on his interpolations. When this fundamental equation was used to 5 Actually. it is more common (see Skripov [5. 6] and Avedi sian [7]) to use J generate p-v-T data fo r use in equation (I), the resulting instead ofj. J is equal toj multiplied be{ the rate of molecular collisions per cubi c values of u/uo lay within 4.3 percent of measured values over centimeter. For water, J is about 103 times j in these units. all but the lowest range of saturation temperatures. This was

---- Nomenclature

A, a, b, c, d undetermined constants in the various cubic equations u, Uo = surface tension, undetermined reference Gb Gibbs number (see equation (6» value of u j nucleation probability, equation (5) w = the P itzer acentric factor, J j expressed as a rate per unit volume - 1 -IOgIOPr.sat (Tr = 0.7) k Boltzmann's constant p pressure Superscripts and Subscripts T temperature c a property at the critical point specific volume, saturated liquid volume, j, g saturated liquid or vapor properties saturated vapor volume r a "reduced" property (Xr=XIX c) v at the root of a cubic p-V- T equation be­ sat a property at a saturation condition tween vf and Vg sp a property at a spinodal point

406/ Vol. 108, MAY 1986 Transactions of the ASME 2 c 0 O r------~~~ Ol '­ a: ~ T, =0.53 T, =0.7 T,=0.8 T,= 0.9 (I) -10 (I) 0 predicted conventi on al hydrocarbons. For the straight-chain sion theory to verify p-v-T information, to our knowledge, hydrocarbons, we used the da ta o f Starling [13] as curve-fitted and it d isplays the potential interaction of (J and p -v-T by Reynolds [14], where we could . (Ho wever , in a few cases information. the Starling data for liquids disagreed with API data [15] or Present Objectives. The a vailability of Sh amsundar and the data of Vargafti k [16] .) These were the only substances fo r Murali's new method fo r interpolating isotherms gives means which we co uld readily obtain reliable p - v- T and surface t n­ for substantia ll y impro ing upon the work in (3] and for re­ sion data [15 , 16, 17J over broad ranges o f temperature. In all opening tbe two questions: (1) "Is k Tc a better characteristic cases, the high-pressure p-u-T data point used in the curve fit nergy tha n k T?" (2) "What minimum value of j gives the was [he value at Pr = 5. spinodal limit when it is used in eq uation (8)?" We al 0 10 ked at ammonia, argon, benzene, carbon di­ We therefore address these matters usi ng van der Waals' oxide, methane, hydrogen, oxygen , and nitrogen for which we surface tension prediction as a hitherto little-used validity did not have complete data over large ranges. In these cases check. curve fi ts were only made at one temperature each. The data sources for these cases were [16, J7, 18] . Isothermal Curve Fits The fl uids considered here reso lve roughly into two clas ifi cations: those for which we are confident o f the ac­ We altered two o f M urali ' s assumptions: We did not use the curacy (water. oxygen, hydrogen, nitrogen, butane, heptane, assumption that e = d. This increased the number of unknown pentane, and propane) and those for which we found some coefficiems by one, but stood to improve th accuracy of the level of unresolvable disagreement in the relevam properties resulting eq uation. The second alteration dealt with the (ammonia, argon, benzene, carbon d iOXide, ethane, hexane, restrictive form of the denominator of equalion (L1 ) (v + b) methane, and octane). In t he subsequent discussions we take • (v + c) (v + d). In t his form the equation is restrictive if b, e, care to base our conclusions only on the results obtained in the and d are to be real. T hi implication was relaxed by writing former fluids. 2 the denominator in the form (v +a) ( v +/ v +g) where/ = b + c Figure 2 show the resulting errors o f the cubic interpola­ = and g be (9J . This form allows b and e to be complex without tions for several typical fluids. These plot~ of error in the USi ng complex numbers explicitly. p redict d pressure, at selected values o f Tr in the liquid range, To complete the curve fit we needed one more condition reflect a very severe test of the curve fits. They consistently than M urali did since we chose not to set c = d. For this we show err rs substantiall y less than one percent, at reduced selected a high-pressure liquid point. We identified the temp ratures below 0.9. pressure that made the isotherm best fi t the available data for stable liquid and vapor stales by trying several pressures until Liquid Spinodal Limits we found the one that worked best. For water this pressure proved to be about 800 bars, or a reduced pressure o f about Figure 3 compares the spinodal limit of water as predicted 3.7, although substantially higher or lower values worked by our cu bic equation with the predicted spinodal limit based a lmost as well. Data from the IFC Formulation for Scientific on equation (8). The NBS surface tension recommendation Use (12) were used to do this. [19] was used in this calculation. (The value of j used in Fig. 3

Journal of Heat Transfer MAY 1986, Vol . 108/407 0.5 0. 6 0.7 0 .8 0 .9 1.0 0.8 0 . 9 1.0 0 . 9 1.0 0 .8 0 .9 1. 0

- ~ r7~ J '-/1 '7 iT - 1 - - -2 4 C -2 • J / f ./ / - 5 " _ - ___ -- ~ - 818H j =10 I .;. 1 c..~ 0 I. .., / / .. / ,,' /' \ -4 -5 ' ...... : . ..- .- . - - . -.- -­ '" - 4 ~ .- ,' ., /i ., ' ~. / j=3(10) -6 ; :: r ,it' ' (,ii / /' ;/:: .. -8 ~ ! , cf.. .. I :;0, ~"""'"uw.H"",UI 2 • O II CO -1 0 ~ 2 2 -1 0 I I ~ ~f' t :::'~:C: ;" 'M • N2 [J CsHs a: .: .II"'~-." - 1 2 I ' I 1 . , .1, ~ - 1 2 ... H2 o NH3 _ f-' 3, 2- .... 1 / • ~'_:~':'::'Ir ' o CH4 .;. Ar -1.A r ' 1f I _IIU'"''• -_ ,4' 5 - , 5 - " " ' ,1-1-'--' --'---'--- '---' 0 .5 0. 6 0 .7 0. 5 0.6 0 .7 0 .6 0 . 7 0.8 0 . 6 0.7 0.8 0 .9 1.0 0 .4 0 .5 1. 0 Reduced temperat ure, Tr ° Fig, 4 Comparison 01 the spinodal limit 01 several lIulds as predicted Reduced temperature, Tr by homogeneous nucleation theory and by the general cubic equation Fig. 5 Comparison 01 the values of v'=TnUl for which equations (8) and (11) agree exaclly

was 2 x 10- 5 . We return to the question of specifyin g) below.) bO The agreement is very good when kTc is used in p l ac ~ of kT!n b 7 the equation, except at such low temperatures and hIgh hqwd c· 0 tensile stresses that both theories are being pushed to the edge 'a; o f their limits of applicability. The choice between k Tand k Tc c: 6 ~ lAPS correl a tion of data Q) makes little difference in the region of positive pre sure, and predicted u s ing cubic this is the only region in which nucleation exper iments have Q) equation of state and () ever been made for large). At lower pressures the two diverge ca 5 equation (1) very strongly. ... -~ Our sugges ti on that k Tbe replaced with kTc is largely based (/) on this kind o f extrapolation. This kind of demonstration was 4 -0 made less conclusi ely (with the less flexible Himpan equa­ Q) tion) in [3]. The present evidence is very c.ompe.lI ing. indeed. C,) c: 3 Several other such comparisons are given In FIg. 4 for Q) 'C butane, heptane, hexane, and propane. (Since urface tensi?n C Q) data were not available for butane and propane over tile en ti re Cl Q) 2 j rang of temperature, the missing values had to be filled in 'C with the help of equation (1 ) in these cases.) In each case, we Q) have used a limiting value of j that bes t fits the extrapolation. ... ::I 1 Thesej's do not all match the value of 2 x 10 - 5 used for waler. -ca L. The four flu ids selected for display in Fig. 4 were chosen Q) because they embrace a wide range of j values. By the same Cl E 0 token, the four flu ids shown in Fig, 2 were selected because 0,) 0.4 0.5 0 .6 0.7 O.B 0.9 they typified the error o f th many fluids thal have been fitted. ~ The results of an inverse kind of calculation are shown in Reduced t emperature. Tr Fig. 5. Val ues of -J -In(j), which is in versely proportional to Fig. 6 Predicted and measured temperature dependence of the sur· the pressure difference between saturation and the liquid face tension 01 water spinodal line, were calculated at each point using equation (8) , with kTc. and the pressure difference predicted by equation (1). available. Furthermore, we suggest that the middle Figure 5 strongly suggests that a "best value" of) for the temperature range in Fig. 5 is the most reliable. We have spinodal limit - if one truly exists - is one slight ly in excess of averaged the ordinate values of the more reliable data in Fig. 5 5, giving water double weight, to obtain the recommendation 10- , in preference [ 0 10 - 5 which was previously suggested (3). It is clear that these ) limits are fairly sen 'itive to the ac­ that curacy of the data upon which they are based. Thus, in choos­ ) ,pinodal == 3 X 10-) ing the appropriate limiting value, one must be guided strong­ 5 This is just a little higher than the values of (l or 2) x 10 - , ly by water and the other very weLl- docwnented fluids. used previously. However this must be accompanied by the One must also consider whelher or not these ) alues were warning that we might eventually have to admit some obtained in regimes in which the cubic equation is truly ve ry depen d e :~ce of the limiting ) on T, and the fluid. (Of course accurate. Figure 2 makes it clear that the general cubic equa­ 3 X 10-5 is an approximation that one would only wa nt to us e tion interpolations begin to lose precision at very high if better information about j were u navailable.) temperatures - typically before T, == 0.9. It also becomes ap­ Notice, too, that replacing kTwith k Tc was a pretty revolu­ parent in the subsequent section that, although it interpolates tionary suggestion. The modifica ti on o f ) by even so much as stable properties very accurately at low temperatures. equa­ an order of magnitude, on the other hand, is far less important tion (11) probably fails to represent metastable and unstable because most calculations based on) are very insensitive to its properties with very high accuracy at low temperatures. ThL is value. evident in its failure to predict the temperature dependence of surface tension wi th high accuracy below T, == 0.5. Prediction of Surface Tension We accordingly restrict the plots in Fig. 5 to the range 0.5 < T, < 0.85, or to a smaller range in whi ch reliable data are The acid test of any p -u-Tequation that purports to predict

4081 Vol. 108, MAY 1986 Transactions of the ASME 10 10

O b 8 "­ bO 8 b ~ 6 ..... u b 6 Q) 4 "0 -0 /------­ CIJ 4 u 2 . ~ ...... Q) U Q "0 2 0 Q) c: ... 0. ... -2 '\ 0 0... I: ... -4 Cubic equation \...,\ Q) \ ... - 2 Karimi & Lienhard's \ 0 equation ... c: -6 I ... Q) Q) - 4 0 Uncertainty of Hexane ... -8 Q) experimental data .... a. I: - 6 -10 Solid symbols denote the fluids tor b 1.6 which abundant consistent data are >-" a w.lI able C 3 3 b· o Open symbols d e n o te oth er fluid s 0 n '" 1.4 Le ast squares Ii! through solid '"c: 7 2 d' points Q) ~ 1.2 6 C Q) '" 0 ;;; 1.0 5 0 "0 ...'" " :J 4 Heptane 4 " 0. 8 ~" '" 0 • ",0 C 6H 1 4 0 3 3 '" 0 .6 - '" • CsH'2 £\. CeHe Q) " 0 co, 0 7 2 .~" 0.4 • C3H,!I c: '" .. C .. H,O ¢. NH:J Q) " H "'C 6 "E ~N, ..,. C2 e c: 0 0.2 Q) . 0, o C H4 Q 5 0 + H, -9- A, Q) 0 u -0.3 -0. 2 - 0 .1 0 0 . 1 0.2 0.3 0.4 0.5 0. 6 4 4 ...Q) Pitzer acentric factor, w :J 3 3 Fig. 10 Corresponding states correlallon of the surface tension lead Octane I ~... constant Q) 2 2 a. E Figure 7 shows the error in the predicted temperature Q) 1 1 I­ dependence of surface tension, for water. The prediction is 0 0 very nearly within the reported accuracy of the NBS surface 0.5 0.7 0 .8 0. 9 1 .0 0.6 tension data [19) for Tr ~ 0.5 . The accuracy of Karimi's Reduced temperature, Tr prediction is also included for comparison. Figure 8 includes a comparable set of curves for heptane, Fig. 8 Predicted and measured temperature dependence of the sur· face tensIon of three fluIds hexane, and octane, the only fluids besides water for whi ch convincing surface tension data were available over a wide range of temperature [161 . These curves again how that the metastable and uns table properties is whether or not it cor­ cubic interpolations give very good predictions of the rectly predicts the temperature dependence of surface tension temperature dependence or surface tension hen they are when it is used in van der Waals' equation (I). We have sub­ substituted in equation (1 ). Figure 9 shows the percent error in jected our cubic equations for water to this test at each a/ ao for these fluids . Once again the results are very accurate temperature, and the results are shown in Fig. 6. for Tr ::;0.85. Figure 6 makes it quite clear that, except at the very lowest While equation (I) only predicts a/ ao , we would like to be temperatures, this prediction has been extraordinarily suc­ able to predict ao as well . To make the comparisons in Figs. 6 cessful. Nevertheless, it is this evidence that suggests that, and 8, it was necessary to calculate the average value of ao for while the cubic fi ts the low-temperature stable points with each substance, based on the surface tension data. We also great accuracy, it is probably less accurate than we would wish computed some values of ao at single points for fluids for in the metastable-unstable range. Of course, this observation which reliable p-u-Tand a data were not available over ranges is based on water - the only substance for which we had full of temperature. data below Tr = 0.5, but one which is also known for its In 1955, Brock and Bi rd [20] showed that the appropriate strange behavior at low temperatures. corresponding states nondimensionalizati on of a was

Journal of Heat Transfer MAY 1986, Vol. 108 /409

-­ - ­ ---­ 2 ao / pc / 3(kTJ 1/3 . Figure 10 presents the correlation of our ao very helpful counsel in all aspects of this work. This work has values as a function of the Pitzer factor w using this non­ received support from the Electric Power Research Institute dimensionalization.6 The points based upon the data in which with Dr. G. Srikantiah as Project Manager, and the University we have high confidence are presented as solid symbols. They of HOll ton Energy Laboratory. define the following correlation References I ao/p//3(kTJ / 3= 1.08 -0.65w (13) I van der Waals, J. D ., "Thermodynamische Theorie der Kapillaritat unter with a correlation coefficient of 0.995. The remaining data are Voraussetzung st etiger Dichte3'nderung," Zeit. Phys. Chemie, Vol. 13, 1894, somewhat more widely scattered, but they do not significantly pp. 657-725; also published as "The T hermodynamic Theory of Capillary alter the correlation. Under the Hpothesis of a Continuous Variation of Densily," tT. by J. S. Others, startulg with Hakim et al. [23], have formed cor­ Rowlinson, J. Stat. Phys., Vol. 20, 0.2, 1979, pp. 197-244. 2 Fisk, S., and Widom, B., "Structure and Free Energy of the Interface responding states correlations for a that include express ions Between Fluid Phases in Equilibrium Near the Critical Point," 1. Chern. Phys., for ao. These can be very useful, but they are normally based Vol. 50, . 0.8, 1969, p. 3219. on assumed forms of the temperature dependence of a/ao , 3 Lienhard, J. H., and Karimi, A., "Homogeneous Nucleation and the that differ from that given by van def Waals' integral. Yet, Spinodal Line," ASME JOURNAL OF HEAT TRANSFER, Vol. 102 , No.3, 1980, pp. 457-460. even though lhese ao expressions might also be linear in w (as is 4 Lienhard, J. H., and Karimi , A., "Corresponding States Correlations of true in [23)), they do not and should not match equation (13). the Extreme Liquid Superheat and Vapor Subcooling," ASME JOURNAL OF Equation (13) gives lhe lead constant specifically for the van HEAT TRANSF ER, Vol. 100, No.3, 1978, pp. 492-495. der Waals integral. 5 Sk ripov, V. P., Metastable Liquids, Wiley, New York, 1974. One can thus predict surface tension with acceptable ac­ 6 Sk ripov, V. P. , Sinitsin, E. N., Pavlov, P. A., Ermakov, G. V. , Muratav. G. N., Bulaoov, N. V., and Baidakov, V. G ., Th ermophysical Properties oj curacy for many applications, using p-u-T data alone, with Liquids ill the Metastable State [in Russianl, Atomizdat, USSR, 1980. the help of equations (I) and (13). As a matter of academic in­ 7 Avedisian, C. T ., "The Homogeneous Nuclealion Limits of Liquids," 1. terest, we can predict a dimensionless (]o fo r the van der Waals Chern. Phys. Ref. Data (in press ). equations (for which w = - 0.302) . Th value is 1.276. 8 Himpan, J., "Die definitive Form der neuen thermischcn Zustan­ dgleichung nest ihren Stoffkonstanten von uber 100 verschiedenen Stoffen," Monatschejte jur Chemie, Vol. 86, 1955, pp. 259-268. Conclusions 9 Shamsundar, N., private communication, 1984. 10 Murali, C. S., "Improved Cubic Equations of State for Polar and Non­ It appears possibl to interpolate p-v-T data with great polar Fluids," MSME thesis, Dept. of Meeh. Engr., University of Houston, accuracy and a minimum of experimental data, using equation 1983. (1J). The accuracy of such predictions has proven to be best 11 Karimi, A ., and Lienhard, J. H., " A Fundamental Equation Representing (for the 16 fluids studied) in the range 0. 5Gases, 2nd ed., Hemisphere, Washington, D.C., 1975. 5 The lead constant ao in equation (I ) is given by equation 17 Jaspar, J. J ., "The Surface Tension of Pure liquid Compounds," J. (13). Phys. and Chern . Ref. Data, Vol. I, No.4, 1972, pp, 841-1010. 18 Heat Exchanger Design Handbook, Vol. 5, Hemisphere. Washington, 6 The Shamsundar-Murali cubic equation has only been D .C., 1983, Section 5.5. 1. used for indi vidual isotherms here. We strongly recommend 19 The International Associalion for the Properties of Steam, "Release of that the problem of developi ng general corresponding states Surface Tension of Water Substance, " National Bureau of Standards, Dec . 1976. correlations to represent the temperature dependence of its 20 Brock, J. R., and Bird. R. B., "Surface Tension and the Principle of Cor­ coefficients be undertaken in the future. responding States ." AlehE Jour., Vol. I, 1955, p. 174. 21 Pitzer, K. S., Lippman, D. Z. , Curl, R. r ., Huggins, C. M. , and Peterson, Acknowledgments D. E. , "The Volumetric and Thermodynamic Properties of Fluids. II. Com­ pressi bility FacIO r, Vapor Pressure and Entropy of Vaporization," J. Am. We are particularly grateful to Prof. N. Shamsundar for his Chem. Soc., Vol. 77, 1955, pp. 3433-3440. 22 Lienhard, J. H., Shamsundar, N., and Biney, P . O., "Spinodal Lines and Equations of State - A Review," 1. Nuclear Engr. and Design (in press). 6The Pitzer acentric factor [21], w'" - I - log I0 Pr.sat (Tr : 0.7), is probably 23 Hakim, D. 1.. Stei nberg, D., and SLiel, L. I. , "Generalized Relalionship the best choice of molecular parameter available for correcting Corresponding for the SlIrface Tension of Polar Fluids," Ind. Eng. Chem. FUlldam., Vol. 10, Stales correlations. See, e.g., [221 for a further discussion of recent applications. No. I, 1971, pp. 174-175.

410 1 Vol. 108, MAY 1986 Transactions of the ASME