CAVITATION AND BUBBLE DYNAMICS BRENNAN PDF

Particular attention will be given to the process of the creation of vapor bubbles in a liquid. In doing so we will attempt to meld together several overlapping areas of research activity. First, there are the studies of the fundamental physics of nucleation as epitomized by the books of Frenkel and Skripov These deal largely with very pure liquids and clean environments in order to isolate the behavior of pure liquids. On the other hand, most engineering systems are impure or contaminated in ways that have important effects on the process of nucleation. The later part of the chapter will deal with the physics of nucleation in such engineering environments.

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Particular attention will be given to the process of the creation of vapor bubbles in a liquid. In doing so we will attempt to meld together several overlapping areas of research activity. First, there are the studies of the fundamental physics of nucleation as epitomized by the books of Frenkel and Skripov These deal largely with very pure liquids and clean environments in order to isolate the behavior of pure liquids.

On the other hand, most engineering systems are impure or contaminated in ways that have important effects on the process of nucleation. The later part of the chapter will deal with the physics of nucleation in such engineering environments.

This engineering knowledge tends to be divided into two somewhat separate fields of interest, cavitation and boiling. Of course, from a basic physical point of view, there is little difference between the two processes, and we shall attempt to review the two processes of nucleation simultaneously. The differences in the two processes occur because of the different complicating factors that occur in a cavitating flow on the one hand and in the temperature gradients and wall effects that occur in boiling on the other hand.

The last sections of this first chapter will dwell on some of these complicating factors. Though simple kinetic theory understanding of the gaseous state is sufficient for our purposes, it is necessary to dwell somewhat longer on the nature of the liquid state. In doing so we shall follow Frenkel , though it should also be noted that modern studies are usually couched in terms of statistical mechanics for example, Carey Figure 1. Our discussion will begin with typical phase diagrams, which, though idealized, are relevant to many practical substances.

The triple point is that point in the phase diagram at which the solid, liquid, and vapor states coexist; that is to say the substance has three alternative stable states. Thermodynamically it is defined by the fact that the chemical potentials of the two coexisting phases must be equal.

The line joining the maxima in the theoretical isotherms is called the vapor spinodal line; the line joining the minima is called the liquid spinodal line. Clearly both spinodals end at the critical point.

The two regions between the spinodal lines and the saturated or binodal lines are of particular interest because the conditions represented by the theoretical isotherm within these regions can be realized in practice under certain special conditions.

If, for example, a pure liquid at the state A Figure 1. If sufficient numbers of nucleation sites of sufficient size are present and this needs further discussion later the liquid will become vapor as the state moves horizontally from B to C, and at pressure below the vapor pressure the state will come to equilibrium in the gaseous region at a point such as E.

A liquid at a point such as D is said to be in tension, the pressure difference between B and D being the magnitude of the tension. The first and most obvious difference between the saturated liquid and saturated vapor states is that the density of the liquid remains relatively constant and similar to that of the solid except close to the critical point. On the other hand the density of the vapor is different by at least 2 and up to 5 or more orders of magnitude, changing radically with temperature.

Since it will also be important in later discussions, a plot of the ratio of the saturated liquid density to the saturated vapor density is included as Figure 1. Second, an examination of the measured specific heat of the saturated liquid reveals that this is of the same order as the specific heat of the solid except at high temperature close to the critical point. The above two features of liquids imply that the thermal motion of the liquid molecules is similar to that of the solid and involves small amplitude vibrations about a quasi-equilibrium position within the liquid.

Thus the arrangement of the molecules has greater similarity with a solid than with a gas. One needs to stress this similarity with a solid to counteract the tendency to think of the liquid state as more akin to the gaseous state than to the solid state because in many observed processes it possesses a dominant fluidity rather than a dominant elasticity.

Indeed, it is of interest in this regard to point out that solids also possess fluidity in addition to elasticity. At high temperatures, particularly above 0. When the strain rate is high, this creep occurs due to the nonisotropic propagation of dislocations this behavior is not like that of a Newtonian liquid and cannot be characterized by a simple viscosity.

At low strain rates, high-temperature creep occurs due simply to the isotropic migration of molecules within the crystal lattice due to the thermal agitation.

This kind of creep, which is known as diffusion creep, is analogous to the fluidity observed in most liquids and can be characterized by a simple Newtonian viscosity.

Following this we may ask whether the liquid state possesses an elasticity even though such elasticity may be dominated by the fluidity of the liquid in many physical processes. Then if the typical time, t, associated with the applied force is small compared with tm, the substance will not be capable of permanent deformation during that process and will exhibit elasticity rather than fluidity.

Thus, though the conclusion is overly simplistic, one can characterize a solid as having a large tm and a liquid as having a small tm relative to the order of magnitude of the typical time, t, of the applied force. The observation time, t, becomes important when the phenomenon is controlled by stochastic events such as the diffusion of vacancies in diffusion creep.

In many cases the process of nucleation is also controlled by such stochastic events, so the observation time will play a significant role in determining this process. Over a longer period of time there is a greater probability that vacancies will coalesce to form a finite vapor pocket leading to nucleation.

Conversely, it is also possible to visualize that a liquid could be placed in a state of tension negative pressure for a significant period of time before a vapor bubble would form in it. Such a scenario was visualized many years ago. In , Berthelot subjected purified water to tensions of up to 50 atmospheres before it yielded. This ability of liquids to withstand tension is very similar to the more familiar property exhibited by solids and is a manifestation of the elasticity of a liquid.

Consider two molecules separated by a variable distance, s. Equilibrium occurs at the separation, xo, typically of the order of m. In practice solids do not reach these limits the rupture stress is usually about times less because of stress concentrations; that is to say, the actual stress encountered at certain points can achieve the large values quoted above at certain points even when the overall or globally averaged stress is still times smaller.

In liquids the large theoretical values of the tensile strength defy all practical experience; this discrepancy must be addressed. It is valuable to continue the above calculation one further step Frenkel This is in agreement with the order of magnitude of the latent heat of vaporization measured for many liquids. Taking a typical molecules per m3, this implies that TC is given by equating the kinetic energy of the thermal motions per unit volume, or 1.

Consequently we find that this simplistic model presents a dilemma because though it correctly predicts the order of magnitude of the latent heat of vaporization and the critical temperature, it fails dismally to predict the tensile strength that a liquid can withstand. One must conclude that unlike the latent heat and critical temperature, the tensile strength is determined by weaknesses at points within the liquid. Such weaknesses are probably ephemeral and difficult to quantify, since they could be caused by minute impurities.

This difficulty and the dependence on the time of application of the tension greatly complicate any theoretical evaluation of the tensile strength. The process of rupturing a liquid by decrease in pressure at roughly constant liquid temperature is often called cavitation. A liquid at constant pressure may be subjected to a temperature, T, in excess of the normal saturation temperature, TS. The process of rupturing a liquid by increasing the temperature at roughly constant pressure is often called boiling.

Though the basic mechanics of cavitation and boiling must clearly be similar, it is important to differentiate between the thermodynamic paths that precede the formation of vapor. There are differences in the practical manifestations of the two paths because, although it is fairly easy to cause uniform changes in pressure in a body of liquid, it is very difficult to uniformly change the temperature.

Note that the critical values of the tension and superheat may be related when the magnitudes of these quantities are small. By the Clausius-Clapeyron relation, It is important to emphasize that Equation 1. The thermal motions within the liquid form temporary, microscopic voids that can constitute the nuclei necessary for rupture and growth to macroscopic bubbles.

This is termed homogeneous nucleation. In practical engineering situations it is much commoner to find that the major weaknesses occur at the boundary between the liquid and the solid wall of the container or between the liquid and small particles suspended in the liquid. When rupture occurs at such sites, it is termed heterogeneous nucleation.

In the following sections we briefly review the theory of homogeneous nucleation and some of the experimental results conducted in very clean systems that can be compared with the theory. In covering the subject of homogeneous nucleation, it is important to remember that the classical treatment using the kinetic theory of liquids allows only weaknesses of one type: the ephemeral voids that happen to occur because of the thermal motions of the molecules. In any real system several other types of weakness are possible.

First, it is possible that nucleation might occur at the junction of the liquid and a solid boundary. Kinetic theories have also been developed to cover such heterogeneous nucleation and allow evaluation of whether the chance that this will occur is larger or smaller than the chance of homogeneous nucleation.

It is important to remember that heterogeneous nucleation could also occur on very small, sub-micron sized contaminant particles in the liquid; experimentally this would be hard to distinguish from homogeneous nucleation. Another important form of weaknesses are micron-sized bubbles microbubbles of contaminant gas, which could be present in crevices within the solid boundary or within suspended particles or could simply be freely suspended within the liquid.

In water, microbubbles of air seem to persist almost indefinitely and are almost impossible to remove completely. As we discuss later, they seem to resist being dissolved completely, perhaps because of contamination of the interface.

While it may be possible to remove most of these nuclei from a small research laboratory sample, their presence dominates most engineering applications. In liquids other than water, the kinds of contamination which can occur in practice have not received the same attention. Another important form of contamination is cosmic radiation.

A collision between a high energy particle and a molecule of the liquid can deposit sufficient energy to initiate nucleation when it would otherwise have little chance of occurring. Such, of course, is the principal of the bubble chamber Skripov While this subject is beyond the scope of this text, it is important to bear in mind that naturally occurring cosmic radiation could be a factor in promoting nucleation in all of the circumstances considered here.

The modern theory of homogeneous nucleation is due to Volmer and Weber , Farkas , Becker and Doring , Zeldovich , and others. For reviews of the subject, the reader is referred to the books of Frenkel and Skripov , to the recent text by Carey and to the reviews by Blake , Bernath , Cole , Blander and Katz , and Lienhard and Karimi We present here a brief and simplified version of homogeneous nucleation theory, omitting many of the detailed thermodynamical issues; for more detail the reader is referred to the above literature.

In a pure liquid, surface tension is the macroscopic manifestation of the intermolecular forces that tend to hold molecules together and prevent the formation of large holes.

The liquid pressure, p, exterior to a bubble of radius, R, will be related to the interior pressure, pB, by In this and the section which follow it is assumed that the concept of surface tension or, rather, surface energy can be extended down to bubbles or vacancies a few intermolecular distances in size. Such an approximation is surprisingly accurate Skripov If the temperature, T, is uniform and the bubble contains only vapor, then the interior pressure pB will be the saturated vapor pressure pV T.

This would then yield a probability that the liquid would rupture under a given tension during the available time. This is clearly in accord with the estimate of the tensile strength outlined in section 1. Equation 1. The second expression we need to identify is that giving the increment of energy that must be deposited in the body of the pure liquid in order to create a nucleus or microbubble of the critical size, RC. Assuming that the critical nucleus is in thermodynamic equilibrium with its surroundings after its creation, then the increment of energy that must be deposited consists of two parts.

First, energy must be deposited to account for that stored in the surface of the bubble. But, in addition, the liquid has to be displaced outward in order to create the bubble, and this implies work done on or by the system.

Thus the net energy, WCR, that must be deposited to form the bubble is For more detailed considerations the reader is referred to the works of Skripov and many others.

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