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 <D 0.5 OJ critical data. Thus: oi " N2.2 ::~~741 ~l • ~n der /e6 8Tr __3 ~ 04 substonceWoo Is 7 (4) 3 . Pr = 3v ­ 1 v2 ' /'­ W"-0302 6 r r > . 6 '-"'I~0 ;­ 03 ,1' where Pr = plpc' etc. The dimensionless van der Waals <t / 7 C equation suggests the Law of Corresponding States ­ c:i 6 o that one equation of state, written in reduced coordi­ c: "" Q2 .J/ nates, should describe all fluids. Today, we know that the Law of Corresponding o./ States should be written as: 0.1 /c1 o Pr = f (T" V r ' primary molecular parameter, other OjJO molecular parameters) (5) o 0.1 0.2 03 0.4 0.5 0 .6 1- Tr - The strongest influences in eq. (5) are those of Tr and Vr' Indeed the need for any further parameters was Fig. 2. The use of the Pitzer factor to complete a Correspond­ 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.