Advances in Heat Transfer (eBook)

IV Analytical Modeling

A ANALYSIS OF BUBBLE DYNAMICS

As can be seen in Fig. 21, since they are in a highly superheat liquid and exposed to a negative pressure field (e.g., −15 bar), the cavitation bubbles start to grow rapidly. The bubbles are filled with vapor whose content changes with time due to evaporation/condensation kinetics. Bubble growth is slow relative to the speed of sound in vapor. Thus a low Mach number model, the extended Rayleigh equation, Eq. (1), may be used to describe the liquid and vapor motion during bubble growth [86,87,103]. As noted previously, the pressure inside the bubble is essentially uniform, although the temperature is not, and an incompressible model for the liquid adjacent to the interface may be used. The bubbles grow until increasing pressure in the liquid during the compression phase of the acoustic cycle arrests bubble growth and causes them to contract. At the beginning of bubble collapse, the velocity of the vapor/liquid interface is still small relative to the local speed of sound in vapor and the low Mach number model remains valid.

To model the latter stages of bubble implosion (and bubble rebound), a high Mach number, hydrodynamic shock code (i.e., HYDRO code) model, based on the full set of fluid dynamics conservation equations must be employed. This model can be derived using the “bubble-in-cell” model, which has been successfully used in many prior studies of bubble dynamics [86,87,104]. According to this scheme the entire region is divided into three zones, namely: (1) ≤r≤R(t) – the gas/vapor zone, where r is the radial coordinate and R(t) the bubble radius; (2) (t)≤r≤Rℓ(t) – the liquid boundary layer zone, filled with a compressible liquid, whose external radius ℓ(t) is larger than the bubble radii, but much smaller than the characteristic size of the acoustic chamber; (3) ℓ(t)≤r≤Rw – the liquid zone between the outer part of the cell (i.e., boundary layer zone) and the acoustic chamber wall (Rw). In this model, the radial velocity of the cell, ˙ℓ(t), is always small with respect to the speed of sound in the liquid. Therefore, the dynamics of ℓ(t) can be described by the generalized Rayleigh–Plesset equation, Eq. (1). Hence, it is necessary to conduct detailed calculations only in zones (1) and (2).

During bubble expansion, evaporation takes place keeping the vapor in the bubble almost at its saturation state. This leads to an almost constant vapor pressure. During the contraction stage, the vapor starts to condense. It is important to note that during bubble collapse non-equilibrium condensation takes place due to the thermal inertia. The faster the bubble implosion is, the farther the vapor is from thermodynamic equilibrium with the liquid, and higher the pressure and temperature of the vapor will be. When the temperature on the vapor side of the interface reaches the thermodynamic critical temperature, condensation ceases. From this moment on the bubble contracts as if it were filled with a “non-condensable vapor.” This phenomenon has been previously modeled and discussed by Akhatov et al. [104]. Hence, careful modeling of the evaporation/condensation process is important because it controls the amount of vapor, which is left in a collapsing bubble and will be involved in the subsequent compression process.

During implosion of the bubble, along with an almost adiabatic compression in the bubble’s interior, a shock wave, converging to the bubble’s center, may be generated. This shock wave ultimately compresses and heats a very small central part of the bubble. Vapor molecules in this region begin to dissociate, ionize and create a two-component, two-temperature fluid of ions and electrons, giving rise to plasma interactions.

The compression rate of the plasma is very high and the ion temperature increases rapidly. In contrast, as will be discussed in more detail subsequently, the electrons, which when ionization occurs, have essentially the same initial velocity as the ions, but much less mass, do not have enough time to exchange energy with the ions, and therefore stay relatively “cold” and thus have negligible impact on the energy and momentum of the plasma. In particular, the well-known energy loss mechanisms [105] associated with the electrons (i.e., Bremsstrahlung, line losses, recombination losses, etc.) are relatively small compared to those in laser-induced inertia confinement experiments [138]. Finally, the ion temperature and density reach conditions that are suitable for D/D thermonuclear fusion in a very small zone near the center of the collapsed bubble during a very short time interval. The neutron yield due to these thermonuclear fusion reactions may be calculated using the predicted, local, instantaneous thermal-hydraulic conditions and a suitable neutron kinetics model [105], Eq. (89).

It should be noted here that a realistic equation of state for the fluid, which accounts for the impact of vapor dissociation and ionization and liquid dissociation on the thermodynamic properties, is crucial in the theoretical predictions of experimentally observed sonofusion phenomena. Thus, we begin a discussion of the analytical model with a detailed description of the equation of state.

B EQUATIONS OF STATE

1 Equations of State for the Low Mach Number Period of Bubble Dynamics

During the low Mach number period of bubble dynamics, when the velocity of the interface is much less than the local speed of sound in vapor, the vapor density and pressure are not very high, and the vapor parameters satisfy the ideal gas equation of state

v=cvTv,pv=ρvRvTv,γv=(cv+Rv)cv

where v,pv,Tv,εv,cv,Rvandγv are the density, pressure, temperature, internal energy, heat capacity, gas constant and adiabatic exponent of the vapor, respectively.

The liquid in the region around the bubble during the low Mach number period can be treated as nearly incompressible (i.e., linearly compressible), because its density does not deviate significantly from its initial value. In this case the liquid pressure, ℓ, is calculated from the equations of liquid motion and the liquid internal energy is

ℓ=cℓTℓ

where ℓ,cℓ are temperature and heat capacity of the liquid, respectively.

According to thermodynamic matching conditions for the internal energies of the vapor and liquid on the saturation line, the following correction for vapor internal energy can be used [103]

v(ρvS(T),T)=hℓv(T)+pS(T)(1ρvS(T)-1ρℓS(T))+εℓ(ρℓS(T),T)

where v(ρvS(T),T),εℓ(ρℓS(T),T),ρvS(T)andρℓS(T) are the values of the internal energy and density of vapor and liquid on the saturation line, respectively; S(T) is the saturation pressure and ℓv(T) the latent heat of vaporization.

2 Equations of State for the High Mach Number Period

During the high Mach number period, when the velocity of the interface is comparable to, or higher than, the local speed of sound in vapor, the vapor density and pressure during a bubble implosion may be very high, and the ideal gas equation of state, Eq. (20), is no longer valid. The liquid around a bubble is also highly compressed and heated, and the incompressible (or linearly compressible) approximation is no longer valid.

To describe the thermodynamic properties of the liquid, vapor and the condensed matter (i.e., supercritical) phases, the Mie–Grüneisen equation of state can be used [106,107]. In this model, the pressure and the internal energy of a fluid are treated as the sum of potential, or cold, (p,εp), and thermal, or hot, (T,εT), components.

=pp+pT,ε=eρ-v22=εp+εT+εc

The cold, or potential components (p) quantity the short range atomic forces and are responsible for the elastic properties of the fluid and the hot, thermal components (T) quantify the effect of the oscillations of the atoms in a fluid [103,108–112]. The so-called chemical component (c) quantifies the chemical binding energy.

The thermal components of pressure and internal energy can be written as

T=ρΓ(ρ)εT,εT=c¯vT

where it may be assumed that the Grüneisen coefficient, ,depends only on fluid density , and the mean heat capacity, ¯v, is a constant.

In general, the internal energy of a fluid is the sum of the energy of the translational motion of the particles (i.e., molecules, atoms, ions and electrons), the rotational and vibrational energies of the molecules, the...