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.