Drop model criticisms

Qu W, Mudawar I (2002) Prediction and measurement of incipient boiling heat flux in micro-channel heat sinks. Int J Heat Mass Transfer 45 3933-3945 Qu W, Mudawar I (2004) Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks. Int J Heat Mass Transfer 47 2045-2059 Quiben JM, Thome JR (2007a) Flow pattern based two-phase pressure drop model for horizontal tubes. Part I. Diabatic and adiabatic experimental study. Int. J. Heat and Fluid Flow. 28(5) 1049-1059... 

Fig. 9. Critical supersaturation for homogeneous gas phase nucleation of n-butylbenzene vapor as a function of temperature. The predictions of the drop model (dashed curve) are in good agreement with experiment (solid curve) although a systematic difference is apparent.

It is clear from Table 1 and Fig. 9 that classical nucleation theory, based on the drop model, does not yield complete agreement with experiment. Before proceeding to the various attempts to improve on classical nucleation theory, it is desirable to understand why it fails. There are two basic types of criticisms, which we will now discuss. 

Here a is the microscopic surface free energy of the cluster, ai the area of a cluster of i molecules, and t and c constants. [Note that if x is, as for spherical particles, for suitable values of r and c Eq. (48) looks like either the classical drop model or the drop model modified to include rotation, translation, etc. Therefore, Eq. (48) is, in a sense, a generalization of the classical drop model.] The Fisher drop model, Eq. (48), was originally developed to describe properties of gases very near the critical point, x and r can be obtained from critical-point indices and are found to be x = and t = 2.333. Hamill showed that c could be obtained from the density of the gas and cr from the second virial coefficient. Using the same equation for the free energy of a... 

Now Eq. (52) for a droplet containing an ion is subject to all of the criticisms of the drop model itself. It neglects rotation and translation of the drop. It is based on macroscopic continuum thermodynamics and so there is no reason to expect that it should apply to small drops. It contains no consideration of the structure of a small drop. In addition, it does not consider that the ion itself may perturb the configurations of the molecules in the drop. 

Mercury electrodeposition is a model system for experimental studies of electrochemical phase formation. On the one hand, the product obtained is a liquid drop, corresponding very well with the liquid drop model of classical nucleation theory. Besides, electron transfer is fast [61] and therefore the growth of nuclei is controlled by mass transport to the electrode surface [44]. On the other hand, the properties of the mercuryjaqueous solution interface have been the object of study for over a century and hence are fairly well understood. The high overpotential for proton reduction onto both mercury and vitreous carbon favor the study of the process over a wide range of overpotentials. In spite of the complications introduced by the equilibrium between the Hg +, Hg2 " ", and Hg species, this system offers an excellent opportunity to verily the fundamental postulates of the electrochemical nucleation theory. In fact, the dependence of the nucleation rate on the oxidation state of the electrodepositing species is fiiUy consistent with theory critical nuclei appear with similar sizes and onto similar number densities of active sites... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

In the following sections, the flow patterns, void fraction and slip ratio, and local phase, velocity, and shear distributions in various flow patterns, along with measuring instruments and available flow models, will be discussed. They will be followed by the pressure drop of two-phase flow in tubes, in rod bundles, and in flow restrictions. The final section deals with the critical flow and unsteady two-phase flow that are essential in reactor loss-of-coolant accident analyses. 

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