10 l
Nuclear Engineering and Design 95 (1986) 297-314 297 North-Holland, Amsterdam
SPINODAL LINES AND EQUATIONS OF STATE: A REVIEW
John H. LIENHARD, N, SHAMSUNDAR and PaulO, BINEY * Heat Transfer/ Phase-Change Laboratory, Mechanical Engineering Department, University of Houston, Houston, TX 77004, USA
The importance of knowing superheated liquid properties, and of locating the liquid spinodal line, is discussed, The measurement and prediction of the spinodal line, and the limits of isentropic pressure undershoot, are reviewed, Means are presented for formulating equations of state and fundamental equations to predict superheated liquid properties and spinodal limits, It is shown how the temperature dependence of surface tension can be used to verify p - v - T equations of state, or how this dependence can be predicted if the equation of state is known.
1. Scope methods for making simplified predictions of property information, which can be applied to the full range of Today's technology, with its emphasis on miniaturiz fluids - water, mercury, nitrogen, etc. [3-5]; and predic ing and intensifying thennal processes, steadily de tions of the depressurizations that might occur in ther mands higher heat fluxes and poses greater dangers of mohydraulic accidents. (See e.g. refs. [6,7].) sending liquids beyond their boiling points into the metastable, or superheated, state. This state poses the threat of serious thermohydraulic explosions. Yet we 2. The spinodal limit of liquid superheat know little about its thermal properties, and cannot predict process behavior after a liquid becomes super heated. Some of the practical situations that require a 2.1. The role of the equation of state in defining the spinodal line knowledge the limits of liquid superheat, and the physi cal properties of superheated liquids, include: - Thennohydraulic explosions as might occur in nuclear Fig. 1 clarifies what happens when a liquid is heated coolant line breaks, liquefied light-hydrocarbon spills, beyond its boiling point. It shows real isotherms of a or Kraft paper process boiler leaks. fluid on p-v coordinates. All states along an isotherm - Quenching, as occurs in heat treating metals, rewet are equilibrium states. When the slope of an isotherm is ting nuclear cores, cooling liquid metal ejecta from a positive, that equilibrium is unstable. When the slope is melted reactor core, or the diagnosis of boiling heat negative, the equilibrium is stable. The spinodal line transfer. connects the points where the isotherms have zero slope. - Predicting the behavior of liquids heated beyond By locating the liquid spinodal line, one specifies the their boiling points, in nucleate and transition boil absolute limit beyond which a liquid can never be ing. superheated. Another feature of this curve is that the - Estimating how much damage a thermohydraulic ex Gibbs potentials, gf and gg' must be equal in the plosion can do. saturated liquid and vapor states. Thus: Much of the work reviewed here was the result of previous inquiries supported by the Electric Power Re (1) search Institute. The results developed under EPRI support included: a fundamental equation for water in The last term vanishes giving the "Gibbs-Maxwell" the superheated liquid and subcooled vapor states (1,2]; relation, which requires that area A in fig. 1 equals area B:
* Present address: Mechanical Engineering Department, !,8Vdp = O. (2) Prairie View A&M University, Prairie View, TX 77446, USA f
0029-5493/86/$03.50 © Elsevier Science Publishers B.v. (N orth-Holland Physics Publishing Division) 298 1.H. Lienhard el 01. / Spinodal lines and equalions of slale
P - The primary molecular parameter is usually taken to Tsa! be Zc; the Riedel factor. ilp (see ref. [8)); or the T,p Pitzer acentric factor, w (see, e.g. ref. [9]): Pc ~ saturated liquid w == -1 - 10glO [ Pr.sa.( Tr = 0.7)].
Fig. 2 is a recent Corresponding States correlation [5] v, and Vg are used In predicting of the ratio vr/ vg. The use of the Pitzer factor here homogeneous nucleation brings data for very different fluids into alignment. T,p - Little has been done with secondary molecular parameters. Fig. 3 shows how data for molecules with -- saturated vapor high dipole moments deviate slightly from an other '" - T, .. wise successful correlation of burnout heat fluxes, based on the Law of Corresponding States [10]. Thus the Law sometimes needs correction when it is ap OI'I,! ~ v plied to polar molecules. An interesting corollary to the preceding results is that van der Waals' equation should accurately describe any real fluid with Zc' ilR, or w ('.qual to the van der Waals' value. We therefore ask if there is any such fluid. This negative pressure can be reached Fig. 4 shows the raw data that established fig. 2. by depressurizinQ a liquid isothermally { When they are cross-plotted against w (in the inset), all, below its saturation pressure including the van der Waals value, lie on the same Fig. 1. Typical real-gas isotherms. straight line. Furthermore, vr/vg for mercury (w= -0.21) closely approximates that of the van der Waals fluid whose w of - 0.302 is only slightly lower. For a long time, van der Waals' equation The importance of this is that - since van der Waals' RT a (3) p = u- b - v 2 ' provided the only theoretical knowledge of real fluid isotherms. Van der Waals argued, on the basis of molec °'1 0010 0 ular behavior, that there is an inherent continuity from o woler , w"0348 7 the liquid to the vapor states. An important feature of 6 o.mmonio. w" 0 2 u ~ °T O F-I . 53 this equation is that it can be nondimensionalized using not clarified until the mid 1950's. ing States correlation of vf jVg (from ref. [5]). 1. H. Lienhard el 01. / Spinodal lines and equations of slate 299 1.0 0.8 g(w)=0.46+1.07 w __-----.. 0.6 Symbol Drsal 0.4 0 0.01 6 0.05 + 0.10 x 0.20 0.2 0 0.30 0 0.40 ~0~.3~--~0~.2~---0~.~1 ----~----~~--~~~~--~L---~~--~--~ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pitzer acentric factor, w Fig. 3 .Illustration of the failure of polar fluids to conform to a Corresponding States correlation (from ref. [10]). equation yields a pair of spinodal lines and it is the The van der Waals spinodal lines are obtained by basis for the Law of Corresponding States - real-fluid setting (3pr/3ur)r= 0 and eliminating Tr between this spinodal lines should also obey the Law of Correspond result and the original equation. The result: ing states. Pr = 3 /u~ - 2/u;, (6) Pitzer factor! w 1.0 -0.3 -0.2 o 0.2 0.4 describes the liquid spinodal for ur < 1 and the vapor 0.08 ,-,----,--,---,-,----,.---, spinodal for u > 1. Hq r 0.5 005 "e 0.05 . ""' VI "'" 2.2. Relation of the spinodal line to the homogeneous ""'- d.. Redlich-Kwong 0.02 nucleation limit Nz ""' 0.1 We would like to use measurements of the limiting " 'o NH 3 0.01 "H20 liquid superheat to obtain the location of the spinodal 0.05 I--;;_.L--;:'-;__.L-0~ 0 .005 0 .2 OJ 0.4 line. But can that be done? Are the homogeneous nucleation limit and the spinodal line related? To bring a real liquid all the way up to the spinodal limit, one would have to do so without any disturbances or imper 0 .01 fections in the system. However, real liquids are made of molecules that constantly move. As the liquid tem 0.005 perature rises these motions provide the disturbances needed to upset liquid stability at a temperature less than the spinodal temperature. Frenkel (see e.g. ref. [11]) first calculated the least disturbance needed to create a minimum stable vapor 0.001 '----'-----'_ ...L _---'-_'----'-----'_-'-'~.L.L"---'--___' o 0.1 0.2 0.3 0.4 0 .5 0.6 bubble or the "potential barrier" to nucleation. He (! - 7;) calculated the difference in Gibbs function of the liquid Fig. 4. Construction of the correlation in fig. 2 illustrating the with and without an unstable vapor bubble in it and influence of the Pitzer factor. obtained the critical work, Wk crit ,. needed to create the - --- . ~ ------ 300 J.H. Lienhard el al. / Spinodal lines and equations of slale bubble: .. . . - R • . - - " . Wkcrit = 17TR 6u . (7) ' ~ " -- " ". .. -~ - - . . - ~ . - -- 0...... " The radius, Ro, is the well-known unstable equilibrium -::: --=_ - __ - ~ -- R~ .' .... . _ radius: 0 _ 2u R 0- (8) liquid in which a An unstable bubble Psat a' T,up - P bubble of radius nucleus displaces Ro will form. N liquid molecules, where T,up is the local temperature of the superheated each of which suffers liquid, and P is the pressure in the surrounding liquid. about 1013 collisions per sec. Wkcrit must now be compared with a characteristic energy of the superheated liquid. Two energies are Fig. 5. Model for calculating the maximum j. appropriate to this purpose: - The average level of molecular vibrational energy is on the order of kT, where k is Boltzmann's constant It was thus shown in ref. [3] that the limiting value of j and T is Tsup - the temperature of the superheated is 10- 5 . This limit, of course, is only an order of liquid. Conventional theories of homogeneous magnitude estimate. Fortunately, nucleation is only sen nucleation are based on this energy. sitive to large variations in j. (Later we suggest 2 X 10 - 5 - The energy required to separate two molecules from might be a better value.) one another. This is the depth of the "potential well" We now have a prediction for the limit of homoge and it is well known to be on the order of k~ . The neous nucleation. To find how close to the the spinodal use of k~ was introduced in ref. [3], and is discussed line it lies, we use the thermodynamic availability of below. spinodal liquid, with respect to liquid at the limit of The ratio of Wkcrit to kT (or k~) is the Gibbs number: homogeneous nucleation: Gb==Wkcri.lkT. (9) .1a I ~Pn = Gbmin(kT or kTc) , (13) What is the minimum Gb for which nucleation ab where Gb = -In(2 X 10-5) = 10.8, and the isobaric solutely must occur? The least possible value of R will min ° difference in availabilities is : give the minimum Wkcri , and Gb. To find it, we first define: .1a I~Pn = [.1h - T h .n..1s1:' n .. (14) j == probability of nucleating a bubble It is shown in ref. [3] that eq. (14) implies a .1T on in a given molecular collision. (10) the order of 1°C between the homogeneous nucleation and spinodal lines. Therefore the liquid spinodal line Now j must be 1 for Gb = 0, and we expect that: can be located with high accuracy using a homogeneous nucleation prediction. We make this prediction by com dj = _ dWkcrit j kT or kT = -dGb, (11) bining eqs. (7) and (8) in eq. (12). The final result, c which includes a curvature correction (see ref. [11)), so: (1 - VI/Vg)2 that cannot be obtained from eq. (7), (8) . nucleation events -Gb and (12), is : (12) J molecule collisions = e . 3 08= 1617u (15) The problem thus reduces to establishing j. To do 1 . 2 0 3(kT)[Psat(T,p)-p] (1-vl/Vg) this, consider a spherical region of undisturbed liquid in which the smallest critical bubble, with radius Ro, The terms T,p, v , and Vg are defined in fig. 1. The term might nucleate, and then count the rate at which colli r kT is kT,p in the conventional theory, but we show in sions can occur in this region (fig. 5). Nucleation must section 3 that it probably should be k~. occur if just one of these collisions triggers a critical When this argument is applied to the nucleation of nucleus wilhin one relaxation time (or about 10 colli liquid drops in a vapor - which is less dense - far fewer sions). This gives: nucleation events are needed in the limiting case of 5 Probability of nucleation = j:O; 10- . complete nucleation. Thus, j must be far smaller and J.H. Lienhard et al. / Spinodal lines and equations 0/ state 301 Gb far larger. This leads to large temperature dif tion [3] that these data should be almost the same as the ferences between homogeneous nucleation and the vapor spinodal line which is a thermodynamic variable. The spinodal. second point made clear in fig. 6 is that - as we anticipated - the vapor nucleation data do not in any 2.3. Experimental data for homogeneous nucleation at way conform to Corresponding States correlation. high superheat, or large values ofj In ref. [14] the following equation was fitted to the Skripov data: V. Skripov and his coworkers at the Ural Institute at Sverdlovsk have pushed the limit of liquid superheat iJ.Tr = 0.905 - Tr,s., + 0.095 T/s., ' (16) much further in the laboratory than anyone else. Much nus gives values slightly less than the true generalized of this work is summarized in two books [11,12]. Avedi spinodal value, because j for this set of Skripov's data sian [13] recently provided an extensive compendium of is a factor of 10 - 8 smaller than the spinodal j *. measured liquid superheat and related j values *. All the variables in eq. (15) are subject to corre Many techniques exist for creating high liquid super sponding states correlation. Forming such a correlation heats. The most effective has been Skripov's method of for the limit of pressure undershoot was the subject of pulse heating a fine wire filament. When the wire is ref. [5J. This result, based on k~ instead of kT, was: subjected to a known electrical pulse, its temperature rises rapidly and predictably within a few microseconds. iJ.Pr = Psp - Psa, As the temperature rises, a few isolated instances of = 112.82 + 224.42w (1 _ T )1.83. nucleation occur; but then - at a certain temperature (17) j-In(j) r ,Sp a complete blanket of vapor appears on the wire. It is at this point that no further temperature increase is possi See fig. 1 for notation and note that, while j "" 2 X 10- 5 ble in the liquid. gives the spinodal line, any smaller j gives a homoge Skripov reached different limiting temperatures de neous nucleation limit that does not reach the spinodal pending upon the rate at which he heated the liquid. line. However, as the heating rate rose, the temperature The Corresponding States correlation can also be approached an asymptote. The value of j at that limit written in terms of temperature. The result for the was about 10- 5 (Skripov reports some values of j 5 spinodal line can be approximated with high accuracy slightly above 10- , but only at heating rates for which in the form: a different and less reliable experimental technique was used.) iJ. Tr = 0.923 - 1', ,5.' + 0.077 T,:sat. (18) Fig. 6 shows those data of Skripov et aI. , available in 13 1976 [14], for j values on the order of 10- . The nus lies about 2% above the correlation given by eq. difference between his homogeneous nucleation temper (16), and reflects the increase of j from the value of ature and the saturation temperature at the same pres 10 - 13 (which was characteristic of Skripov's experi 5 sure, is plotted against T,. The notational distinction ments) to 2 X 10- . between fsp (liquid spinodal) and gsp (vapor spinodal) is introduced temporarily in fig. 6 because it includes the best data available for nucleation of droplets in 3. The development of equations of state and fundamen sub cooled vapor. The liquid and vapor spinodal limits tal equations to describe the metastable regions calculated from van der Waals' equations are also in cluded. We can now place the spinodal limit with reasonable Fig. 6 makes two things clear: One is that the accuracy. However thermodynamic properties are gen homogeneous nucleation limits for the 12 liquids do erally unavailable in the metastable and unstable ranges. conform to Corresponding States correlation. Further Such data are constantly needed in boiling and two more, the shape of the dashed correlating line through phase heat transfer work, but they are extremely hard to them is very similar in form and placement to the van come by, and little analysis has been done. der Waals prediction. This corroborates the demonstra It has been customary in boiling work to estimate * Actually, Skripov (and Avedisian as well) use J instead of J. * People unfamiliar with these calculations will understanda J is equal to j multiplied by the rate of molecular collisions bly be startled at the notion that a factor of 10 - 8 makes so per cm3 J is normally about 10 39 times j in these units. little difference in this range. 302 J.H Lienhard el at. / Spinodal lines and equations of Slale 27132 data 0.8 ----O-M for 12 f luids rf sp • -LlT for noz zles and shock tubes rg sp V> 5; 0.7 0' ~ --- - 6 T f o r diffusi0n cloud cho mbers , o.~ rgsp V> 'V> ~ 0-- 0.6 III III 0. 0. V> '" 0---'" ~ "? 0 .5 -0 0.1 to '"::l -0 ~ 0 0 0 2 0. 3 0.4 05 0.6 0 7 0.8 1.0 reduced satu ration tem perature, Trs Fig. 6. Corresponding States correlations of the limiting liquid superheat and vapor subcooling. thermodynamic properties in these regimes by extrapo "caloric" , or specific heat, equation of state to give lating them linearly in temperature. This works in say - enthalpies or entropies. A fundamental equation, slightly superheated liquids, but at higher superheats when it can be written, is more convenient than a set of such a simple strategy becomes impossible. (The specific equations of state. It provides all thermodynamic infor heat at constant pressure approaches infinity on spinodal mation through straightforward calculations. The lines, liquids become highly compressible at high super Helmholtz function form of the fundamental equation, heat, etc.) for example, can be obtained from the p-u-T and c~ equations: 3.1. The formulation of fundamental equations RT/ P,,, '" ()u, T = '"ref + j P d u' .The "fundamental" or "canonical" thermodynamic u equation is one from which all equations of state can be + iT c~(T ' )[T/ T' - 1] dT' derived and in which all thermodynamic properties are Trd embodied. Four common ones for a pure substance take -R(T- Trer ) , (20) the forms s = s(u, u), h = f(s, p), '" = .peT, u) and g = geT, p). The several equations of state are obtained where (Pref' T,.ef) is a reference point where the ideal from the derivatives of these equations with respect to gas equation is valid and the entropy is taken as zero. their independent variables. For example, the p-u-T The Steam Tables (see, e.g., ref. [15) or ref. [16» are and energy equations of state are obtained from: normally based on empirical fundamental equations in the Helmholtz form with scores of constants in them. p/T= (as/au)" and l / T= (as/ au) u' (19) Such equations permit one to evaluate any thermody One tends to associate the p-u-T equation with the namic property within four decimal places. But we term "equation of state"; but it gives incomplete ther seldom havtl anything nearing such complete property modynamic information and must be combined with a information for fluids other than water. Furthermore, J.H. Lienhard et al. / Spinodal lines and equations o[ state 303 such equations do not normally provide valid informa Karimi based his work on the KKHM equation, tion in the metastable regimes. (22) 3.2. Surface tension where Q is a function of p and T, obtained by fitting The property, surface tension, a, is important m the data to a suitable expression with many terms of the predicting the burnout heat flux - a crucial design type (T- a)m(p - b)n e-cp KKHM only used data variable in any energy system in which beat is removed from the stable regions in their fitting. Karimi improved by boiling. Unfortunately, it has not been measured the coefficients with the help of additional theoretical over wide ranges of temperature for many fluids. constraints and the meager existing data for metastable Furthermore, 0 is intimately related to the p-v-T fluids. equation of state. Van der Waals - well known for his The KKHM equation is the sum of an ideal gas remarkably simple and successful equation of state expression and a correction. No such features as changes also developed this remarkable and completely precise of phase, stable and metastable regions, etc., can be equation for predicting tbe surface tension, 0 , from a modelled by this ideal gas" reference function". Many knowledge of the p-v-T equation of state [17]: of Karimi's difficulties arose because he had to absorb all of the complexity of water in the correction term, Q. ", 1 [ 0= ao j ' . 5/2 Pr ,sa, (vr- vr,r) urf Vr 3.4. An improved strategy for creating a fundamental equation (21) A new and highly accurate equation of state was The integration of eq. (21) requires complete p-v-T developed: t1e National Bureau of Standards (NBS) by values in the metastable and unstable regimes. Thus Haar, Gallagher, and Kell in 1979 [18] . It, too, gives surface tension can be used to extend metastable prop pressure as the sum of a reference function and a erty information and vice versa. residual correction, but the reference function has a theoretical basis. It displays all real fluid features quali tatively, and yields errors less than 1 % in wide ranges of 3.3. Karimi's fundamental equation for water T and p. Thus, its residual correction is much smaller than in the KKHM equation. It must only improve the Karimi made the first attempt to create an accurate accuracy - it need not . provide all the features of rea! fundamental equation for water in the metastable regi fluid behavior by itself. Thus it can be fitted accurately mes. He and Lienhard first (1] made a relatively crude with fewer terms. modification of the Keenan-Keyes-Hill-Moore We are currently modifying the NBS equation in (KKHM) equation of state for water [15]. Karimi then much the same way as Karimi modified the older refined the curve-fit and produced a greatly improved KKHM equation. A key factor to doing this is the use equation [2]. of cubic equations to extrapolate p-v-T data. We turn Some of the important results predicted by Karimi's to this matter next. equation are shown in the modified p-v-T and T-s-p surfaces shown in figs. 7 and 8. Notice that all of the 3.5. Cubic equations of state properties exhibit continuous change as they pass into the "two-phase" regime. The stable, metastable, and Cubic equations of state, of which the van der Waals unstable regimes are ali shown, and the conventional equation is the prototype, have the smallest number of straight-line representations of mixtures are not in free parameters that can represent real data accurately. cluded. They are therefore much safer to use in estimating data Although Karimi's results point out the potential for than more complicated ones. They display major fea accurately predicting metastable and unstable behavior tures of physical behavior correctly in relatively large of real substances, he had to use different coefficients interpolations, with no need for smoothing. When such on either side of T= 150°C in his final equation. equations have enough inherent flexibility and are care Consequently the equation displayed inconsistencies and fully fitted, they are remarkably accurate at a very low discontinuities in the neighborhood of 150°C. cost of calculation. 304 J.H. Lienhard et al. / Spinodal lines and equations of state 30 \ - , "'--...... 20 'l::~~o V i 3 ~ " IV 0 ' ' ~__ ___ ~ _ 10 ~ 1"0 ~_ "' _ c..'" '1. E>O ~______:::E I /)~>(; ~ =-=cc I XA~~%oo~ 0. OJ 0 ~ 0.02 0.03 (/)" 3 (/) volume, v (m /kg) Q) 0. -10 saturated liquid-vapor line spinodal lines -20 isotherms -30 -40 -50 Fig. 7. The p-v-T surface for water based on Karimi's [2J fundamental equation. 3.5.1. A new form of cubic equation not always - fitted with the help of critical data: We have developed a new volume-cubic equation of state that includes most existing cubic equations of state r; aTr-A. P =--- (23) as particular cases. General cubic equations have been r r + b ( r + c / _ d2 ' developed by others, but their applicability has usually been restricted by tying some coefficients to certain where r'= CUr - I)Zc and b'= 1/ 0:, and - if one elects groups of substances. to use the critical data - a, c, and d become: We avoid this trap by making the procedure for a = (1 - b)J, c =(1- b)/2, and d 2 = c2(l- 4b). 0: is determining the coefficients as flexible as possible. When related to the Riedel factor, CiR, by: data are plentiful they can be used to fit the coefficients statistically, and the resulting equation will be very _dpr.sa'i =0:(1+;\)-;\. (24) aCCurate. This can also be done with less accuracy when CiR - dTr T,-t the data are less plentiful; but then the result can be used to back out unknown data - even critical data. A The new variable, r, greatly simplifies the relations major strength of this equation is that it can be written among the coefficients. It has the virtue of vanishing at in many forms - suiting the form to the number of data the critical point, which is the natural origin for equa to be used in determining its coefficients. tions of state. The first form is of the same type as most existing Eq. (23) includes many other familiar cubic equa cubic equations, with the coefficients normally - but tions as special cases. With Zc = i, Ci = 4 and ;\ = 0, we J, H. Lienhard et ai, / Spinodal lines and equations of state 305 400 ~ o denotes critical point /II)!''It IC') !cv I!<0 350 II Q ..... 0° 300 l i ~ .....::J ro ~ t Cl) a. 250 E Cl) I saturated vapor 200 1501, 5 4 6 7 8 Specified entropy, S (kJ/kg- °C) Fig. 8, The T - s - p surface for water based on Karimi's [2] fundamental equation. get the van der Waals equation; with Ze = t 3.5,2, Application 0/ the cubic equation to a substance with a = 3/ (2 - 21 /3 )) and ,\ = ~ we get the Redlich-Kwong a well-documented critical point equation; etc, Water is extremely well-documented and existing We have yet to say how to assign a value to the cubic equations have generally failed to represent it at coefficient ,\ . Often, ,\ is chosen so as to make the all well - particularly in the liquid state. We used Pc , equation better fit data away from the critical point, Te , and Ve from the IFC 1967 Formulation [16], but not once the other constants have been fitted. However it the published values of aR' Thus band ,\ were un might be unwise to rely this heavily on the critical point, knowns obtained by fitting the equation to saturation which is mathematically singular as well as being a lone data from the IFC skeleton table. Details of the fit are point. Indeed we have found that if, after using critical given in ref. [19]. The results - with very modest predic data for all but two constants, we fix both ,\ and a R tion errors - are reported in table 1. Saturated liquid using data away from the critical point, then aR is properties are better represented by the present cubic liable to disagree with the measured value. This suggests equation than by any other, it predicts superheated that using data from all over the p-v-T field might vapor data very well, and it only shows inaccuracies in yield a better all-around equation if the critical point is the compressed liquid range [19]. allowed to float. We next see how this works. ------ 306 J.H. Lienhard el al. j Spinodal lines and equal ions of slale Table 1 Comparison of predictions from eq. (23) with IFC 1967 [16] data for water T Psat SPsa. Uf SUr ug SUg hrg Sh fg (OF) (psia) (%) (ft 3jlb) (%) (ft 3j lb) (%) (Btujib) (%) 198 9.783 -3.9 0.Q1716 -3.6 39.595 3.3 988.8 0.4 250 30.41 -2.8 0.01733 -1.9 13.357 1.2 943.9 0.2 286 54.47 -0.7 0.Q1753 -1.2 7.9282 -0.2 917.3 0.3 .. 370 171.7 1.0 0.01810 0.7 2.6842 -2.3 849.7 0.4 436 335.2 2.6 0.01878 2.2 1.3181 -4.0 787.7 0.3 .. 498 682.7 3.8 0.Q1954 3.3 0.7721 -4.0 727.0 0.0 536 899.8 3.5 0.02848 4.1 0.5889 -3.8 665.1 -0.2 .. 586 1343 3.3 0.02203 4.4 0.3117 -2.3 579.9 -0.3 .. 636 1955 2.4 0.02483 3.2 0.1859 1.1 461.5 8.9 .. 678 2491 1.6 0.02879 -8.5 0.1230 3.6 34109 3.4 686 2798 0.8 0.03238 -2.6 0.0949 6.4 256.7 8.2 .. 698 2879 0.6 0.03371 - 3.5 0.0879 99 229.6 10.3 .. Data on these lines were not used in fitting the cubic. 3.5.3. Application of the cubic equation to substances with (about twice the measured value) agree remarkably well unknown critical data with the published data. Consider what this implies: When critical data are unknown, it is pointless to use The published Zc for mercury is close to the van der eq. (23), whose volume variable, r, is centered on the Waals value of i and the calculated value is closer still. critical point. One may then return to eq. (23) in its (We [4] recently presented other close agreements be dimensional form and use non-critical data to de tween the properties of mercury and the van der Waals termine the constants: fluid.) Thus mercury is very nearly a van der Waals fluid, and its reported critical data are probably accu RT a'T-x p=--- 2 . (23a) rate. v-b (v+c') _d'2 The predicted Tc and Pc for sodium are very nearly the same as the published values, so we are inclined to This was done for methane and water in ref. [19] with think these properties are known correctly. The mea equal success. Table 2, for example, shows the results surement of Vc is very difficult and often inaccurate. for methane. It is clear that such prediction is extremely The value of Zc = 0.178, based upon the reported v , is accurate. c much lower than for other fluids. However, our predict The liquid metals are natural substances for this ion - Zc = 0.307 - places sodium squarely among the application since, in most cases, tabulated data only go light elements. It is thus likely that our Vc and Zc values to a fraction of Tc. When the extrapolation is applied to are the more correct ones, and that the frequent sugges mercury [20] and sodium [21], the results (table 2) are tion that liquid metals are not subject to the Law of astonishing both in their accuracy and their implication. Corresponding Stales, is wrong. The critical point predictions for everything but vc.sodium Table 2 Extrapolated critical constants obtained from eq. (23) 3 Substance Source pc{MPa) ~(K) uc {m /kg) Zc Methane Pred. 4.600 190.52 0.006225 0.2914 Litr. 4.599 190.55 0.006233 0.2903 Mercury Pred. 152.9 1870 0.0001913 0.377 Litr. 153.0 1763 0.0001818 0.381 Sodium Pred. 34.09 2581 0.008411 0.307 Litr. 34.10 2573 0.004850 0.178 J. H. Lienhard et al. / Spinodal lines and equations of Slale 307 3.5.4. A second and very general form of cubic equation 6 r------~ If one or more of the constants in eq. (23) is allowed to vary with temperature its accuracy might be further improved. The obvious way to identify accurate forms ~ \-- Redlich ::l of temperature dependence is to fit the equation at Ul 4 Ul \ several temperatures, and then to form the functions by Q) \ correlating the resulting coefficients with temperature. C. ~ "0 Cubic ~ \ III To this end we turn to a second and far more general () equation , \ ::l form of cubic equation that includes eq. (23) as a "0 2 ~ \ III special case: a:: Data (IFe 1967) """ '\ \ p (v-vf)(v-vm)(v-vg ) - = 1 ------~--"::...- (25) P,o' ( V b) ( d ) o ~----~~----~~----~ + v + c) ( v + 0.375 0.400 0.425 0.450 Reduced volume The quantities Ps." Vf' Vm' Vg , b, c and d all vary with temperature. The advantage of this form is that it Fig. 9. Comparison of cubic equation, eq. (25), with IFC data automatically satisfies critical point criteria, but it need (16] for liquid water at high pressures. not be tied to them. Another nice feature is that it is prefactored to do away with the need for finding roots of the cubic in fitting the constants. Three of the coefficients in eq. (25) are the known that of Redlich [22]. Fuller's (23) cubic equation temperature-dependent properties: Psa" Vf and vg . Thus intended for water - predicts values outside the figure. the most straightforward use of the equation is one in Another such comparison is presented. for ethylene which isotherms are fitted one at a time. in fig. 10. The present cubic again outclasses the exist Actually, two of these constants can be fixed with ing cubic equations and other" simple" equations. The the ideal gas law limit at low pressure and the key to this success is, of course, the fact that the Gibbs-Maxwell equal-areas condition (eq. (2).) Then coefficients are free from having to obey any prede just two pieces of data will fit an equation that should termined dependence upon temperature. be quite accurate along a given isotherm. We emphasize liquid properties because they are so The isothermal compressibility of saturated liquid, hard to predict; however Murali [19] has shown that eq. and one compressed liquid point, have proven to give (25) is extremely accurate in predicting superheated the best results. It turns out that if the two missing vapor properties. pieces of data are reduced to just one by the seemingly But a most important use of the new equation of arbitrary assumption that c = d, the resulting equation is still very accurate. For a given temperature, the coefficients for the equation can be found using very 6 \ few data, and making a direct, not a statistical, calcu \ \ T = 2 50 K lation. 5 \ \ \ , \ yCubi C e quation The remaining problem associated with this kind of ~ ::l use of eq. (25) - that of generalizing the temperature Ul 4 ~ \\o..a Ul \ dependence of the coefficients - remains to be solved. ~ 0- \. \ 3 "0 \ '\, 0.. III 3.5.5. Applications of the cubic equation with temperature (J \ '\, ~ ::l "0 2 \ ' ~ dependent coefficients III /"\ Marl>, in '-...... The temperature-by-temperature application of eq. a:: :~:~ jl~=n \equatio n " (25) to water has yielded accuracies that are within the '-... tolerance of the IFC [16] Skeleton Tables in the subcriti \ OL-____L- ____L- ____~ __~ cal range of pressures. A comparison of IFC data with 0.40 0.43 0 .46 0.49 0 .52 our equation is given over a wider range of pressures for water in fig. 9. The only other cubic equation (actually a Reduced volume modified. cubic that was not intended for use with Fig. 10. Comparison of cubic equation, eq. (25), to data for water) close enough to the data to appear on this plot is liquid ethylene at high pressures. 308 J.H. Lienhard et a/. / Spinodal lines and equal ions oj Slale state is that of predicting properties in the metastable bO b 7 .------; regimes. Fig. 11 compares the liquid spinodal of the c cubic equation with the one predicted in ref. [3). The .S? en lAPS correlation of data c 6 agreement is very good except at such low temperatures Q) predicted using cubic and high liquid tensile stresses that both theories are Q) () being pushed to their limits of applicability. 5 ~ The nucleation limits shown in fig. 11 are both :J'" en obtained from eq. (15), using both kT and kTc: the use 4 o of kTc is far superior. The choice makes little difference Q) () in the positive pressure range, which is the only range in c Q) 3 which experiments have ever been made for large j's. u c Q) At lower pressures the two diverge very strongly. Our n Q) 2 recommendation to use kTc is largely based on the "0 compelling evidence derived from the cubic equation. Q) ~ :J The value of j used with the present cubic is 2 X 10-5 5 instead of 10- (as recommended in ref. (3)) - a minor Q)'" n alteration. The replacement of kT with k1'c is the E o I " Q) 0.4 0.5 0.6 0.7 0.8 0.9 1.0 revolutionary issue here. ~ Reduced temperature, Tr . 3.6. The prediction of surface tension Fig. 12. Predicted and measured temperature dependence of An acid test of any p-u-T equation that purports to surface tension. predict metastable and unstable properties is that is must correctly predict the temperature dependence of surface tension when it is used in van der Waals' surface tension eq. (22). The cubic equations for each isotherm have been subjected to this test * and the results are shown in fig. 12. Fig. 12 makes it quite clear that, except at the very lowest temperatures, the cubic equa tion passes this test wi th flying colors. 0 4. Depressurization of hot liquids I -2 Consider a liquid in a container, suddenly depres surized from an initial point, (Pi' Tj ), as SBOwn in fig. a -4 13. Such a system was studied both analytically and experimentally by Alamgir, Lienhard and Trela [24-26]. ~ 9QuQlion :J (/) When the liquid is initially not too hot, or when the (/) -6 Q) depressurization ends in nucleation well before the C. predic ted by spinodal limit, the isentropic depressurization lies close u Q) eQualion (15) <.> -8 with kT r epla ce d to an isothermal path. Fig. 14 shows a typical :J u by kTc pressure-time history for such a process from [25] . The Q) a: sharpness with which the depressurization ends is very -10 predicted conventional interesting. Experiments in ref. [26] showed that some homogeneous nuclea tion nucleation oeeured on the pipe wall before the point at theory, eQuation (15) -12 which depressurization abruptly reversed. Alamgir and Lienhard [25] therefore concluded that a real homoge neous nucleation limit was reached on the two-dimen -14 L1----~----~--~~--~~--~~--~ 0.4 0.5 0.6 0.7 0.8 0.9 1.0 sional pipe wall. Thus the homogeneous nucleation theory had to be Reduced temperature, Tr Fig. 11. Comparison of spinodal limit as predicted by homoge * We thank Mr. Wei-guo Dong for his help with these calcula neous nucleation theory and by the general cubic equation. tions. J.H. Lienhard et al. / Spinodal lines and equations oj state 309 _ _ isother mal polh and an 1) "" Gb/ cf>. Then they demonstrated that: __ ___ isenlropic depressurizo1ion palh __ saluroted liquid and vapor sl ales e - ~q, - J7T1)cf> erfc N ___ spinodal lines ( loci of lop/o v];O) 2:'(l- vr/ vg) (26) Pi p saturat ion and used eq. (15) in the form applicable to real (not ' ''~ia~ homogeneous) nucleation: " pressure undershoot Dnuclealion _____ -- Tsat at Pn (15a) to eliminate 1) from eq. (26). They then used eq. (26) to calculate cf> for observed nucleation events. The resulting values of cf> are correlated as a function Fig. 13. Depressurization of a hot liquid. of 2:' and temperature in fig. 15. This correlation brings together the data of many investigators and it only breaks down when 2: becomes so low that single iso reconstructed in a sort of " Flatland" form. Nucleation lated nucleation events can stop the depressurization. was taken to occur everywhere on the wall when the This occurs when 2:' is less than 4000 atm/ s. two-dimensional J was equal to 10 18 nucleation The correlation shown in fig . 15 can be summarized 2 12 events/ m s. This translates to j = 0.5661 X 10- which in the form : is very close to j for Skripov's pulse heating results. (27) The pressure undershoot, in this case, depends on the rate of depressurization, 2:' atm/ s (see fig. 14.) The for reason is that a two-dimensional array of bubbles will have more time to grow and completely fill the wall if 0.62 :s;; T,. :s;; 0.935, 0.004 :s;; 2:' :s;; 1.8 Matm/s. the rate is slower. which correlates all available blowdown data with a Alamgir and Lienhard [25] defined a " heterogeneity 10.4% rms error. 12 factor," The data in fig. 15 imply that j is 0.5661 X 10- _ . actual potential barrier This gives Gb = 28.2. By substituting Gb/ cf> for 1) in eq. cf> = potential barrier for homogeneous nucleation (15a) and using eq. (27), we recast the undershoot prediction in dimensional form: slope in superheated t liquid range,I'=O.l7xI06 asm 1000 (28) For the reader's convenience, this pressure undershoot o 'in is displayed in dimensional form for water, as a func 0. 0. tion of 2:' and the initial saturation temperature, Ti , in - 500 fig. 16. ~ - psat =4355 psia :::J '"Q) Ci. -Pn=260 psia 4.1. The damage-doing potential of a sudden depressuriza \ tion A most important practical problem is that of lime, I ms (arbitrary zero) answering the question: "A large pipe carrying nearly Fig. 14. A typical pressure-time history during depressuriza saturated water at high pressure is suddenly ruptured. tion in 453°F (233.9 ° C) liquid. How much damage can each pound of water do to the 310 J.H. Lienhard et al. / Spinodal lines and equations of slate -I -2 -3 + 1.27 cm 1.0. Pipe PT - I o 1.27cm 1.0. Pipe PT-2 ® 5.08 cm 1.0. Pipe o Edwards -4 f:, CANON shock lube \l Kenning and Thirunavukkarasu x Hooper el. 01. -5 * Fr iz el. 01. 181 Sozzi and Fedrick o Gallagher ~Q OAllemann el . ol. -6 £'" -7 10g10[0 .1058T r 28.46 ] ® -8 x -9 _IO'L-___L____l_ ___L____~__~_____L ____L_ ___L____l_ ___L__~ -030 -0.28 -0.25 -0.22 -0.20 -0.17 -0.15 -0.12 -0.10 -007 -0.05 -0.02 I og loT , 0) The I u n c I ion, I I (Tr ) : 4> I 12 ( I ' ) - 1. 75 + 1.27 cm 1.0 . Pipe PT - I -1.50 o 1.27 em 1.0 . Pipe PT-2 ® 5.08 cm 1.0. Pipe o Edwards ++, f:, CANON shock tube \l -1.25 \l Kenning and Thirunavukkarasu + x Hooper et. 01. * Friz el . 01. 181 Sozzi and Fedrick o Gallogher t+ '--'r -1.00 E ~ o Allemann el. 01 . :.=c:. -&-1 ~/ + ev ·2 ~ -0.75 0'0 Q .§ ~ ci 00" ~ ~ x Ci. u Q -0.50 0._ '"o 00 ® ® -0.25 *x x -000 x ® x I0910[1+14I,08] XX - 0.25 ® I' in Malm / s * -050~'--~----~--~--~----~--~--~----~--~--~--~ -4.00 -3.60 -3.20 -2.80 -2.40 -2.00 -1.60 -1.20 -0.80 -0.40 0 .00 0.40 10gloI' b) The lunclion, f2(I') : Fig. 15. Correlation of 45 A description of the exact calculation, and the evaluation of ,1 a for van der Waals fluids at their spinodal lines are given in ref. [4]. We have also com 40 puted availabilities of superheated water for EPR! using Karimi's fundamental equation. One of these calcula 35 tions involved challenging the often-used assumption that isentropic depressurizations can be treated as iso thenna\. Fig. 17 shows what the left-hand region of fig. 30 1 or 13 would look like plotted to scale. Notice that at E -0 Tr = 0.8855 (or T= 300°C) the assumption is not bad as long as the pressure stays positive; but at Tr = 0.9627 c;-c 25 (350°C) it breaks down much more quickly. ~ a. This information is shown in another way in fig. 18 which shows how the relation between p and T changes 0 20 .c "~ along isentropic paths, starting from various initial Q) -0c: saturation temperatures. => 15 Q) Fig. 19 shows how the availability of water above the ~ normal boiling point, varies with the temperature at '"Q) which it first crosses the saturated liquid line. As p n: 10 decrease, the availability diminishes, but the reduction is too little to improve safety noticeably. A very rapid 5 depressurization of 1.5 Matm/s brings water to nuclea tion at a point whose availability almost matches the saturation point from which depressurization began. 0 The point where nucleation occurs in fig. 19 is obtained 50 100 200 300 374.14 wi th the help of eq. (28). It is sobering that ,1 a is on the Initial water temperature, Ti (Oel Fig. 16. Prediction of pressure undershoot applied to water. 1.0 system in the resulting thermohydraulic explosion?" Two /// thermodynamic issues lurk in this important safety-re lated question. The first - that of saying how far below 0.8 1/ its saturation pressure the water must fall before an I explosion is initiated - is answered by eq. (28). 0.6 i' / The second issue is that of determining the thermo 0.0 ~ ..... 09627 0. / \ LliQU~~= dynamic a vailabili ty of the liquid a t this limi t. The II 0.4 ~SPinOdal CU". availability of a liquid with respect to its surroundings o.~ specifies the maximum" useful" work - actually damage Q) I . Is entropi c :; ~ palh in this case - that it can deliver to these surroundings. ------.------ 312 J.H. Lienhard el al. / Spinodal lines and equalions of slale 5 C:: 10.0 order of RT ' or about 10 ft-lbr/ lb for water. Ther 0. c m mohydraulic explosions can do a lot of damage. Q"'~ "''' \ I T =O.653r..-r-\ '\ reat /"" OJ ~ 5. Summary c.'" / a; / Ol This review clearly reflects the ongoing interests of c: r-; eo / Change along the .<:: liquid spinodal its authors, and new results are currently being devel o Tr "0.8082/ cur .... e. Q) 991 / oped. New papers still under review at this writing 5 1.0 / include the development of corresponding correlations '" I '"Q) of saturated and metastable p-v-T properties [28], 0. further description of the cubic equation development, o I '5. / and the prediction of surface tension from cubic equa ::: I C tions [29]. The important points of the present discus Q) sion are: .!!1 T :"O.9627 r99 '0 (1) There is a real need to be able to predict spinodal Q) o lines, homogeneous nucleation, and properties of '0 Q)" superheated liquids. a: I ! t I lO-1 1_2 10-1 (2) These issues present an inherent inter-relation of 10 purposes: Reduced isentropic temperature change; - A knowledge of metastable and unstable p-v-T T••, - T(s••t. Pn) behavior facilitates the extrapolation of single measurements of a over the full range of temper Fig. 18. The relation between pressure and temperature during atures. By the same token, a knowledge of aCT) three isentropic processes in water. provides an important check on any equation of state. - The ability to predict homogeneous nucleation and the ability to predict the liquid spinodal are U I- 1. 1 equivalent...... a: ..., 1.0 - Thus knowledge of the p-v-T surface includes () 0 knowledge of the liquid spinodal and the limit of 0 0 <0, ""00.."., """ 0 .8 de pr e ss ur IZ a t ton -J; homogeneous nucleation as well. rat e i s 1.5 Matm/s ;. (3) Eq. (15) predicts homogeneous nucleation in a 0 1- f variety of practical situations: 0.6 E lor saluraled I;Qu;d ~ - By replacing the left side with the -In) ap eo t propriate to a real situation, one can use it to II 0.4 handle real situations. 8 I'/ eo - If Vf, Vg and a are not available for eq. (15), I c: / correlations (17) and (18) can be used with good eo 0.2 ,t' '-' accuracy . ,,; .t/ ",(' (4) The vapor spinodal is unrelated to the limit of 0.0 .D 0.6 0.7 homogeneous nucleation of liquid droplets. ~ (5) Strong evidence suggests that homogeneous nuclea '0; Reduced saturation > -0.2 eo temperature. Trsal tion theories should be based on the characteristic '0 Q) energy, k'Fc, instead of kT. 0 (6) Metastable p-v-T data can be accurately predicted '0 -0.4 "Q) by fitting a general cubic equation to the known a: -0.5 saturation points, a known saturated liquid com Fig. 19. Availability of water, when it nucleates, in comparison pressibility, a third stable equilibrium point and the with saturated water at one atmosphere. The independent Gibbs-Maxwell condition. variable is the reduced temperature at which the liquid be (7) When more than just p-v-T data are needed, comes saturated during depressurization. Karimi's method can be used to develop a funda J. H. Lienhard el 01. / Spinodal lines and equations of slale 313 mental equation, but it should be altered to use a ratio of actual to ideal potential barrier to more realistic reference function. nucleation (8) Karimi's fundamental equation provides the best Helmholtz function information for metastable water, to date. w the Pitzer factor, (9) Eq. (28) predicts the pressure undershoot in the -1 - 10giO[ Pr,sal (Tr = 0.7)] rapid depressurization of hot water. It reflects the fact that sufficiently rapid depressurizations reach a General superscripts and subscripts true (two-dimensional) homogeneous nucleation limit. denotes a dummy variable of integration, or the dimensional forms of a, b, c and d c a property at the critical point Nomenclature f, g saturated liquid or vapor properties fsp, gsp identify liquid and vapor spinodals in a availability function, or fig. 6 a. b van der Waals constants (eq. (3». (also h.n. a homogeneous nucleation limit used in other equations of state as unde r a "reduced" property (e.g. Xr =X/Xc) termined constants.) ref a reference state c, d undetermined constants (which could be sat a property at a saturation condition complex conjugates) sp a property at a spinodal point specific heat at constant pressure, low pressure cp References g specific Gibbs function, or function of w defined in fig. 3 [1] A. Karimi and J.H. Lienhard, Toward a fundamental Gb Gibbs number (see eq. (9» equation for water in the metastable states, High Temp. h specific enthalpy High Press. 11 (1979) 511-517. [2] A. Karimi and J.H. Lienhard, A fundamental equation j, f nucleation probability (see eq. (10»; j representing water in the stable, metastable and unstable expressed as a rate per unit volume states, EPRI Report NP-3328 (Dec. 1983). k Boltzmann's constant [3] A. Karimi and J.H. Lienhard, Homogeneous nucleation m, n undetermined constants and the spinodal line, J. Heat Transf. 102 (1980) 457-460. p pressure [4] N. Shamsundar and J.H. Lienhard, Properties of the Q residual correction function for a funda satlUated and metastable van der Waals fluid, Can. J. mental equation Chem. Engr. 61 (J 983) 876-880. R ideal gas constant [5] J.H. Lienhard, Corresponding states correlations of the Ro critical radius of nucleation (see eq. (8» spinodal and homogeneous nucleation limits, J. Heat s specific entropy Transf. 104 (1982) 379-881. [6] Md. Alamgir and 1.H. Lienhard, Correlation of pressure T temperature undershoot during hot water depressurization, 1. Heat u specific internal thermal energy Transf. 103 (1981) 52-55. V specific volume [7J Md. Alamgir, c.Y. Kan and J.H. Lienhard, An experi 1J, 1Jr, Vg saturated liquid and vapor volume. (at the mental study of the rapid depressurization of hot water, J. saturated spinodal condition in eq. (15» Heat Transf. 102 (1980) 433-438. v at the root of a cubic p-v-T equation [8] L. Riedel, Eine neue universelle Dampfdruckformel, between 1Jr and Vg Chemie-Ing. Technik 25 (1954) 83. potential barrier to nucleation (see [9] K.S. Pitzer, D.Z. Lippman, RF. Curl, C.M. Huggins and eq. (7» D.E. Peterson, J. Am. Chern. Soc. 77 (1955) 3433. critical compressibility, Pcve/RTc [10J A. Sharan, R. Kaul and 1.H. Lienhard, A corresponding states correlation for the peak pool boiling heat flux, J. (aR + A)/(1 - A); the Riedel factor Heat Transf. 107 (1985) 392-397. 7J Gb/ [13J C.T. Avedisian, The homogeneous nucleation limits of [22J O. Redlich, On the thIee-parameter representation of liquids, J. Phys. Chern. Ref. Data, in press. equation of state, I.E.C. Fundamentals 14 (1975) 257. [14J J.H. Lienhard , Correlation for the limiting liquid super [23J G.G. Fuller, A Modified Redlich-Kwong-Soave equation heat, Chern. Engr. Sci. 31 (1976) 847-849; see also: J.H. capable of representing the liquid state, I.E.C. Fundamen Lienhard and A. Karimi , Homogeneous nucleation and tals 15 (1976) 254. the spinodal line, J. Heat Transf. lO3 (1981) 61-64. [24J J.H. Lienhard, Md. AJamgir and M. Trela, Early response [15J J.H. Keenan, F.G. Keyes, P.G. Hill and J.G. Moore, of hot water to sudden release from high pressure, J. Heat Steam Tables (John Wiley and Sons, New York, 1969). Transf. 100 (1978) 473-479. [16J Steam Tables, 4th Ed. (American Society of Mechanical [25J Md. Alamgir and J.H. Lienhard, Correlation of pressure Engineers, New York, 1979). undershoot during hot water depressurization, J. Heat [17] J.D. Van der Waals, Thennodynamische Theorie der Transf. 103 (1981) 52-55. Kapillaritiit unter Voraussetzung stetiger Dichteiinderung. [26J Md. AJamgir, c.Y. Kan and J.H. Lienhard, An experi Zeit. Phys. Chern. 13 (1894) 657 - 725 . mental study of the rapid depressurization of hot water, J. [18] L. Haar, J.S. Gallagher, G.S. Kell , Thermodynamic prop Heat Transf. lO2 (1980) 433-438. erties for fluid water, in: Water and Steam, Eds. J. Straub [27J J.H. Lienhard, Some generalizations on the stability of and K. Scheffler (Pergamon Press, New York, 1980) pp. liquid- gas-vapor systems, Int. J. Heat Mass Transf. 7 69-82. (1964) 813. [19J C.S. Murali, Improved cubic equations of state for polar [28J W-g. Dong and J.H. Lienhard, Corresponding states cor and non-polar fluids, MSME thesis, Dept. of Mech. Engr., relation of saturated and metastable properties, Canadian Univ. of Houston (1983). J. Chern. Engrg. 64 (1986) 158-161. [20J N.B. Vargafti.k, Tables of Thermophysical Properties of [29J P.O. Biney, W-g. Dong and J.H. Lienhard, Use of a cubic Liquids and Gases, 2nd Ed. (Hemisphere Publishing Co., equation to predict surface tension and spinodal limits, J. Washington, 1975). Heat Transf. 108 (1986) 405-410. [21J c.T. Ewing, J.P. Stone, J.R. Spann and R.R. Miller, High temperature properties of sodium, J. Chern. Engr. Data 1 (1966) 468.