J. G . Eberhart
2730
and H. C. Schnyders
Application of the Mechanical Stability Condition to the Predictio of the Limit of Superheat for Normal Alkanes, Ether, and Water’ J. G. EberharP and H. C. Schnyders Chemicai Engineering Division, Argonne Nationai Laboratory. Argonne, lliinois 60439
(Received March 29, 1973)
Pubiication costs assisted by Argonne Nationa: Laboratory
If a liquid is in contact only with immiscible solid or liquid phases which it wets completely, then the liquid can be heated isobaricaliy until it reaches its limit of superheat where it vaporizes explosively. The liquid phase can likewise be “stretched” isothermally to this same limit, which is sometimes called the tensile strength of the liquid. A kinetic limit of superheat can be predicted from homogeneous nucleation theory. A thermodynamic limit of superheat can also be predicted from the liquid-phase spinoda1 which is that curve in PVT space which separates metastable states from unstable states aY densities iarger than the critica! density. Along this spinodal the equation of state satisfies the conditions (aP/ 8v)~ = 0 and ( P P / ~ V F )> T 0. Liquid spinodals for the normal alkanes from methane through nonane and for ether were derived from a variety of equations of state, including those of van der Waals, Berthelot, a modified Bertbelot equation, and a generalized van der Waals equation based on scaled particle theory for rigid, convex molecules. Good agreement is obtained between experimental limits of superheat for these liquids and the liquid-phase spinodals calculated from the generalized van der Waals equation of state for rigid, cylindrical molecules with a size based on the molecular structure which they represent. Our estimate of the limit of superheat of water, based on significant structure theory. is 305” a t atmospheric pressure. This exceeds superheatings observed thus far by 25”. Spinodals describing the limits of supersaturation of vapors and supercooling of liquids are briefly discussed.
Introduction The saturation or vapor pressure curve for a fluid represents the states where liquid and vapor are in two-phase equilibrium and the chemical potentials of the two phases, p1 and p”, are equal. This curve separates the equilibrium, one-phase. liquid field, where 111 < ,uL, from the equilibrium, one-phase, vapor field, where f i b < p i . A variety of experiments show that it is not only possible to superheat a liquid to pressures and temperatures in the vapor field, but that there is a well-defined limit to the extent of this metastable liquid state which is called the limit of superheat. This limit is approached experimentally either by heating a liquid a t constant pressure P to its upper temperature limit T I , or by “stretching” a liquid a t constant temperature T to its lower pressure limit, PI. The same limiting pressure-temperature relationship is defined by either experiment. Heating a liquid isobarically to its limit of superheat can be accomplished with an apparatus of a type first used by Mooreza and Wakashima and Taka?a2b and later modified by other i n v e s t i g a t o r ~ . 3 -This ~ ~ apparatus is essentially a vertical glass tube which is filled with a liquid heating medium which is immiscible with the liquid to be superheated. The tube is surrounded by a heater which produces an upward-increasing temperature gradient in the medium. A droplet of the liquid to be superheated 1s introduced in the bottom of the tube and, since the heating medium is more dense, the droplet rises in the tube, and its temperature increases as it moves upward. If the superheated droplet completely wets the liquid heating medium and also any solid impurities or motes which may be present in the droplet, then the droplet can be raised in temperature to its limit of superheat, T I ,and vaporizes explosively with a sharp “ping.” The Journal of Physical Chemistry, Voi. 77, No. 23, 7973
With a stabilized temperature gradient, within the heating medium, a series of small droplets of the same substance sent up the tube will explode a t essentially the same level in the tube, and the limit of superheat can be determined by moving the junction of a thermocouple to that level. The precision obtained in these measurements is of the order of 0.5-1.0”. Complete wetting of t,he liquid medium by the droplet permits the homogeneous nucleation of vapor bubbles within the body of the droplet, rather than heterogeneous nucleation at the superheated droplet-heating medium interface.2a.13~14The majority of the experiments performed to date have been with hydrocarbons in a mediuin of either sulfuric acid, glycerin, or ethylene glycol. Results which are independent of medium provide a strong indication that nucleation is homogeneous and t,hat maximum superheating has been achieved. A variety of techniques have also been developed for the isothermal stretching of liquids into these metastable states. These methods have been recently reviewed by Hayward.15 Typically the experiments are done in glass tubes near room temperature. At these temperatures metastable liquids can sustain large tensile stress or negative pressure. Lower pressure limits measured isothermally in the negative pressure regime are usually referred to as the tensile strength of the liquid. Experiments performed with the superheated liquid in direct contact- with glass give more erratic results than those described above, presumably because of the difficulty of obtaining complete wetting. Recently ApfelQ.10.16devised an acoustical technique for producing negative pressures in systems comprised of liquid droplets in a liquid heating medium, and has thus extended the range of these more reliable measurements of the limit of superheat. The scientific literature on superheating suggests two approaches to the prediction of the limit of superheat of a
Prediction of the Limit of Superheat for N o r m a l Alkanes, Ether, and Water
liquid. One approach is that of homogeneous nucleation t h e ~ r y . l ~ -Here ~ O the probability of bubble nucleation is considered and an expression is derived for the nucleation rate as a function of temperature T and pressure P.From a knowledge of the vapor pressure, the density, and the surface tension, the temperature a t which the nucleation rate becomes significant, TI, can be calculated. Because the nucleation rate increases very rapidly with temperature near the experimental limit of superheat (several orders of magnitude per degree a t 1 atm), T1 is relatively insensitive to the effects of variance in the volume of the droplet and its heating rate. Homogeneous nucleation theory has been used with notable success by a number of authors2-’l to predict the limit of superheat of various hydrocarbons. The second approach that has been suggested for the prediction of the limit of superheat is based on the mechanical stability condition of classical thermodynamics.21-24 According to this analysis the stable < pv) and the metastable (pl > pv) liquid states are those which satisfy the mechanical stability condition ( ~ F / ~ V 0, are located between the minimum and the maximum of the van der Waals loop of the isotherm. For these stales density fluctuations continue to grow rather than damp out. The states between the isotherm maximum and the saturated vapor represent metastable, supersaturated vapor states where (JP/dV)T < 0. These states are similar to the metastable liquid in their response to fluctuations in density. The locus of all the maxima and all the minima in the fluid isotherms is the spinodal, which separates the meta. stable and unstable fluid states. The condition (aP/dV)T = 0 is satisfied along the entire length of the spinodal which has two branches. The locus of the minima of all the isotherms, along which (d2P/dV2)T > 0, provides the upper bound t o the limit of superheat for the liquid as a function of ambient pressure, while the locus of the maxima, with ( a 2 P / a V 2 . ’
