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5.1. Bubble Nucleation and Onset of Nucleate Boiling

Vapor may form from a liquid (a) at a vapor-liquid interface away from surfaces, (b) in the bulk of the liquid due to density fluctuations, or (c) at a solid surface with pre-existing vapor or gas pockets. In each situation one can observe the departure from a stable or a metastable state of equilibrium. The first physical situation can occur at a planar interface when the liquid temperature is fractionally increased above the saturation temperature of the vapor at the vapor pressure in the gas or vapor region. Thus, the liquid "evaporates" into the vapor because its temperature is maintained at a temperature minimally higher than its vapor "saturation" temperature at the vapor system pressure. Evaporation is the term commonly used to describe such a situation which can also be described on a microscopic level as the imbalance between molecular fluxes at these two distinctly different temperatures. We consider this conceptual picture again, when condensation is later considered in Section 9.

When considering the other two situations of vapor formation, a vapor bubble or "nucleus" must be formed and be mechanically and thermally stable. Consider the simplest case of a spherical vapor bubble of pressure, tex2html_wrap_inline4924 , with a saturation temperature of tex2html_wrap_inline4926 in its liquid with pressure, tex2html_wrap_inline4928 , which corresponds to its saturation temperature tex2html_wrap_inline4283 . Mechanical equilibrium requires that

equation1892

where tex2html_wrap_inline4932 is the interfacial surface tension and r is the bubble radius of curvature. If the liquid is also in thermal equilibrium with the vapor, tex2html_wrap_inline4936 , which then implies tex2html_wrap_inline4938 . If one uses simple thermodynamics, combining equation 5.1 with the equality of local temperatures, one finds the needed liquid superheat tex2html_wrap_inline4940 for the vapor bubble to exist

equation1904

One can now relate the superheat required within the bulk of the liquid or at a solid-gas-liquid interface to the size of nuclei.

In a bulk liquid, thermal fluctuations always exist as a small but finite cluster of molecules can take on higher than macroscopic average energies (i.e., temperature). As the bulk liquid increases in its superheat these molecule clusters can take on "vapor-like" energies with increasing probability, and possibly form a stable "vapor nucleus." This process of vapor bubble nucleation is referred to as "homogeneous nucleation." One can use a thermodynamics approach to estimate the degree of liquid superheat necessary to form a stable vapor nucleus; e.g., tex2html_wrap_inline4942 for atmospheric pressures. However, a statistical mechanics approach (Blander 1975, Skripov, 1970) provides a more complete picture of the nucleation rate tex2html_wrap_inline4944 as

equation1919

where tex2html_wrap_inline4946 is the collision frequency

equation1925

k is the Boltzman's constant, h is Planck's constant and tex2html_wrap_inline4952 is liquid molecular density tex2html_wrap_inline4954 . tex2html_wrap_inline4956 is the free energy of formation for the vapor nucleus of radius, r, given by

equation1931

where tex2html_wrap_inline4932 is evaluated at tex2html_wrap_inline4926 and tex2html_wrap_inline4924 is the saturation pressure at tex2html_wrap_inline4926 . As the liquid superheat tex2html_wrap_inline4968 increases, the surface tension decreases and tex2html_wrap_inline4970 increases. Thus, at a particular tex2html_wrap_inline4926 the nucleation rate increases markedly and this corresponds to the "homogeneous nucleation" temperature, tex2html_wrap_inline4974 ; e.g., tex2html_wrap_inline4976 for water at atmospheric pressure, which corresponds to vapor nuclei radii of ;SPMlt;1 micron. Such a superheat value for the onset of vapor nucleation is far above experimental observations for water, under commercial applications, thus it is not the primary mode of vapor nucleation, under normal circumstances. Nevertheless, it must be considered as operating conditions change (e.g., pressure) especially for organic liquids.

Finally, consider the situation where a vapor/gas pocket exists near a solid surface in a liquid (Figure 5.3). Container surfaces can provide sites for vapor formation. This third method of vapor generation from pre-existing vapor nuclei is called "heterogeneous nucleation." Examples of such pre-existing nuclei include noncondensible gas bubbles held in an emulsion in the liquid pool or gas/vapor filled cracks or cavities on container surfaces (Figure 5.3). The latter example is probably the most common circumstance for vapor bubble nucleation. In fact, one could derive the required liquid superheat necessary for the case of an ideal cavity of known radius. One finds it is substantially lower than that needed for homogeneous nucleation, because the cavity radius is much larger. Thus, the bubble requires less superheat and associated pressure difference for thermal and mechanical stability. It is hypothesized that the maximum superheat occurs at the throat of any cavity where the radius is smallest for most aqueous fluids with large contact angles. (Note: the contact angle is the angle through the liquid between the solid-liquid interface and the liquid-vapor interface, and depends on liquid-surface chemistry; e.g., water and commercial steel tex2html_wrap_inline4980 ).

Only a small fraction of all cavities become effective sites for vapor nucleation, because one must consider the balance between the required superheat for a cavity of radius, tex2html_wrap_inline4982 , and the temperature gradient from the wall, tex2html_wrap_inline4984 , to the bulk liquid at saturation, tex2html_wrap_inline4283 ; as depicted by Hsu (1962) Figure 5.4 gives a conceptual picture of the model. As the heat flux at the wall is increased, the wall temperature, which is probably representative of the vapor bubble and local liquid temperature, exceeds the saturation temperature. The liquid will locally vaporize and the vapor nuclei in the cavity will grow toward the cavity throat at the heater surface. If one assumes that the liquid temperature gradient from the wall to the bulk is approximately linear, then the requirement for mechanical and thermal stability of a vapor nucleus at the cavity exit is that the whole bubble should be in a liquid region of boundary layer size, tex2html_wrap_inline4988 where the temperature is at least above a value of tex2html_wrap_inline4990 which satisfies the equilibrium condition of equation 5.2, for tex2html_wrap_inline4992 . If there is a sufficiently large array of cavity sizes this "onset of nucleation" will first occur when the liquid temperature profile is tangent to the line of thermal and mechanical equilibrium (Figure 5.4). One can algebraically eliminate the cavity radius, tex2html_wrap_inline4982 , from the two equations by equality of temperature and slope and find the relation between the heat flux, tex2html_wrap_inline4996 , at which the "onset of nucleate boiling", ONB, occurs

equation1960

where all properties are evaluated at tex2html_wrap_inline4998 . Now if there are no cavities at this size the heat flux must increase so that the superheat temperature increases to a point where a cavity first exists and the temperature profile intersects the equilibrium curve in Figure 5.4.

One should notice that this model only provides a stability line where the "onset of nucleate boiling" may first occur. To find the particular heat flux and superheat pair one must look for the intersection of this stability line with natural convection mode of heat transfer that would exist prior to boiling (Figure 5.5)

equation1975

equation1980

and Gr is the liquid Grashof number for tex2html_wrap_inline5002 and Pr is the liquid Prandtl number. For water at atmospheric pressure this model predicts an "onset of nucleate boiling" for a superheat less than tex2html_wrap_inline5006 C, which corresponds to a cavity size of about 50 microns. In practice the superheat may be as high as tex2html_wrap_inline5008 C for very smooth, clean metallic surfaces, which indicates larger cavities were not available on the surface. As a very rough guide (Brown, 1967) aqueous fluids seem to have active sites ;SPMlt; 10 microns, organic ;SPMlt; 5 and cryogens ;SPMlt; 1.5 microns.


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