##### Аннотация

*Introduction*. Cavitation is the most significant factor that affects liquid food products during ultrasound treatment. Ultrasonic treatment intensifies diffusion, dissolution, and chemical interactions. However, no physical model has yet been developed to unambiguously define the interaction between ultrasonic cavities and structural particles of liquid food media. Physical models used to describe ultrasonic interactions in liquid food media are diverse and, sometimes, contradictory. The research objective was to study ultrasonic devices in order to improve their operating modes and increase reliability.

*Study objects and methods*. The present research featured ultrasonic field generated in water by the cylindrical emitter, the intensity of flexural ultrasonic waves and their damping rate at various distances from the emitter.

*Results and discussion*. The paper offers a review of available publications on the theory of acoustic cavitation in various media. The experimental studies featured the distribution of cavities in the ultrasound field of rod vibrating systems in water. The research revealed the erosion capacity of ultrasonic waves generated by the cylindrical emitter. The article also contains a theoretical analysis of the cavitation damage to aluminum foil in water and the erosive effect of cavitation on highly rigid materials of ultrasonic vibration systems. The obtained results were illustrated by semi-graphical dependences.

*Conclusion*. The present research made it possible to assess the energy capabilities of cavities generated by ultrasonic field at different distances from the ultrasonic emitter. The size of the contact spot and the penetration depth can serve as a criterion for the erosion of the surface of the ultrasonic emitter.

##### Ключевые слова

Ultrasound, cavitation, aqueous medium, foil screen, erosion, oscillatory system##### ВВЕДЕНИЕ

Ultrasound technologies in the food and processing industries have been the focus of numerous studies. As a result, information about ultrasound treatment of food media is quite abundant. Cavitation is the main process of ultrasound treatment of liquids. Ultrasonic waves accelerate diffusion, chemical reactions, and dissolution processes in liquid foods. The ultrasonic acceleration of chemical interactions results from free ions formed during ultrasonic cavitation.

Contemporary food science knows no universally accepted theory that would unambiguously describe the physical nature of the interaction of ultrasonic cavities and structural particles during food processing. The existing physical models are diverse and often contradictory. They give a very vague explanation of the processes that occur in the ultrasonic field. According to some models, cavities have high pressures and temperatures. According to others, cavitation creates microcavities of deep vacuum and cryogenic temperatures. Both destroy microparticles in liquid media. For instance, ultrasonic cavities break milk fat globules into smaller fragments [1–5]. Cavitation is the formation of microcavities in a liquid medium, which are called bubbles, voids, or just cavities. These cavities are filled with vapor phase, gas, or a mixture of both. Cavities are formed in local zones of the liquid phase, where the pressure drops to a critical level, which usually coincides with the saturation pressure. Hydrodynamic cavities develop in a flow of fluid; acoustic cavities develop as a result of acoustic treatment [3].

Physical models of cavitation can be contradictory because experimental studies of cavitation are quite complex. To be used in food industry, ultrasonic cavitation requires optimal operating modes of ultrasonic devices. Reliability and safety of ultrasonic devices remain an important issue of food science [6–8].

**Cavitation theories.** The ability of ultrasound to
accelerate food processing was established in the early
XX century. For example, such phenomena as the
acceleration of water-fat emulsion or finely dispersed
suspensions are quite old. In 1960s, ultrasonic treatment
began to be used in chocolate production: it provided
a more effective mixing, emulsification, and better
dispersion. However, these methods found little practical
use because of their economic inexpediency. Ultrasound
methods and technical means used to be imperfect and
expensive [9, 10].

The latest technology of controlled, focused, and highly intensive ultrasonic field had low energy costs, which made it economically feasible. As the technology gained more popularity, it revealed some new opportunities. For example, ultrasound treatment raised the intensity of extraction in cognac production by hundreds of times. Similar beneficial effects were observed in other processes, e.g. extraction of vegetable oils.

However, high-intensity ultrasound technologies erode the device surfaces that come in direct contact with the product. As a result, the processed product might contain some components that are not part of the formulation. Studies of the interaction between the product and the surface of the ultrasonic cavitator (oscillatory system) can help reduce or eliminate erosion [11].

To find out more about various theories of collapse and waves of a single cavity, see [3–5]. The results of these researches made theoretical studies as reliable as actual practical experiments in cavitation.

In its most general form, the equation of motion of the outer wall of the cavity in spherical coordinates looks like this:

where *t* – time, s; *r* – current radius, m; *ρ* – liquid
density, kg/m^{3}; *p* – hydrostatic pressure, Pa.
The continuity equation has the following form:

The vortex-free motion of the surface can be
considered as a potential with potential *φ*, i.e.:

Provided the bubble surface is *r* = *R*, *u* = *U*, then:

Equations (3) and (4) produce the following result:

Integration of (1) from *r* to *r* = ∞ results in:

The condition of incompressibility of liquid means
that *ρ* = *ρ*_{0} = const; therefore, (6) is:

Inserting (5) in (7) at *r* = *R*, we get the following equation for an empty cavity:

For cavity surface *r* = *R*, *U* = *dR*/*dt*; thus, equation (8) looks like this:

where *P*(*R*) is cavity surface pressure.

The resulting equation describes the motion of
the cavity surface depending on the regularity of the
pressure change *P*(*R*).

Lamb and Rayleigh obtained a solution to equation
(9) assuming the simplest boundary conditions: *P*(*R*) = 0,
*P*_{∞} = *P*_{0}), i.e. pressure at a sufficient distance equals
hydrostatic pressure:

where *R*_{m} is maximal cavity radius early during collapse.

Provided *U* = *dR*_{dt}, we get Rayleigh’s equation:

where τ is time of cavity collapse, s.

In the middle of the XX century, Nolting and Nepiras obtained an equation for cavity pulsations based on the Laplace surface tension forces and the change in the gas volume in the cavity during adiabatic expansion and compression [5]. They used harmonic pressure fluctuations as a boundary condition at a sufficient distance from the cavity. The equation is:

where *P*_{s} – steam pressure in the cavity; *ω* = 2*πf*, *f* – oscillation frequency; *σ* – Laplace
surface tension; *γ* – adiabatic exponent (4/3), *ρ*_{0} – density of liquid unaffected by fluctuations, *P*_{m} – maximal pressure in the cavity attained at its minimal radius.

The numerical solutions proposed in [6, 7] describe the changes in the radius of the cavity. The speed of movement of the cavity surface approaches the speed of sound in a liquid medium. It is the boundary condition for the application of the abovementioned equations. The continuity equation under these conditions has the following form:

In the considered approximation *u*_{2}/*c*_{2} << 1, whereas:

where *U* – the speed of movement of the cavity surface during collapse and pulsation; *с*_{0} – speed of sound in the liquid unaffected by fluctuations.

Flynn supplemented this equation with a term that takes into account the viscosity of the liquid and pressure fluctuations by a harmonic law. As a result, he obtained the following equation:

Kirkwood and Bethe introduced the concept of
specific enthalpy *h* and kinetic enthalpy:

Thus, they obtained an equation that describes cavity pulsations and takes into account the compression and surface tension of the liquid, as well as the polytropic (adiabatic) character of vapor expansion in the cavity:

where *H* – free enthalpy on the sphere surface; *A* and *n* – constants for water (*А* = 3.001×10^{8} Pa = 3.001×10^{5} MPa; *n* = 7).

An analysis of the equations of Nolting-Nepiras, Herring-Flynn, and Kirkwood-Bethe showed the similarity of the results, which diverged only at high ultrasound frequencies and in the case of a long collapse time [6].

**The theory of shock waves during cavitation.**
Many researchers performed visual observations and
proved that the erosion of the surfaces within the
cavitation field occurs due to the high local pressures
that take up the form of shock waves when cavities
collapse [12, 13].

The analysis of the propagation of spherical shock
waves consists in determining function *G* (*R*, *t*) on the cavity surface with radius *R* and calculating the time
it takes the waves to appear at a distance *r* from their source:

After necessary transformations, equation (18) looks like this:

where the speed of the cavity surface *U* is determined by
the following transcendental equation:

The pressure along the shock front can be calculated using the formula given in [16]:

where *В* – constant for water (3.0×10^{5} MPa); constants *А* and *n* (see Eq. (17)).

Formula (20) makes it possible to analyze the conditions of cavitation effect on the surface of oscillatory system, which creates ultrasonic waves in liquid.

**Theory and practice of erosion studies for solid
materials in cavitation field.** Erosion of material
surfaces in cavitation field occurs following the
destruction of bonds in the crystal lattice of the object.
Cavitation erosion is determined by the decrease in the
mass of the object in the cavitation field, or the state
of the photosensitive layer on the glass plate surface.
Another variant is to measure the total area of the
holes formed in the aluminum foil under the effect of
cavitation in a certain period of time. To assess the
energy efficiency, Rosenberg introduced the concept of
erosion-acoustic efficiency:

where *Е*_{m} – energy spent on mechanical erosion; *Е* – ultrasonic vibration energy.
The local rate of cavitation-induced destruction in an inhomogeneous ultrasonic field is determined by the
following formula:

where *w*_{m} – local rate of erosion in the cavitation field;
Δ*G* – mass of solid particles dispersed in the liquid during erosion that can be separated during analysis; *V* – the volume of the product (liquid); *t* – time.

A rather original mathematical apparatus was used to analyze the erosion efficiency of the cavitation field in [6]. Unfortunately, the lack of data makes it impossible to use it for these calculations.

##### ОБЪЕКТЫ И МЕТОДЫ ИССЛЕДОВАНИЯ

This research continues the studies of erosion patterns of cavitation in aquatic environment started at K.G. Razumovsky Moscow State University of Technology and Management [7].

Ultrasound is a promising method of mechanical processing of food masses to produce fine-dispersed systems. However, the product unavoidably comes in contact with the surface of the ultrasonic concentrator, which might result in the diffusion of the solid matter into the liquid medium [8, 9]. This phenomenon is multiplied by vibration, including ultrasound. Figure 1 illustrates erosion on the surface of concentrator (or any other ultrasonic device).

An analytical review of theoretical and experimental studies of cavitation and erosion of solid surfaces in ultrasonic field proved that this process still remains understudied, especially the effect of cavitation on ultrasonic emitters. The theoretical and experimental studies given below represent an initial stage in the research and practical application of data on the interaction between ultrasonic field and ultrasonic emitters, as well as the role of this process in the food safety.

Cavitation produces cavities. When they collapse, they produce an energetic effect on liquid and elements submerged in it. Ertugay and Sengul described an assessment method for the energy impact of cavitation. Evaluation is carried out on a screen of aluminum or tin foil [8]. When the cavity collapses and as it hits the surface of the foil, dents or tears appear on the screen. The erosive capacity of cavitation is defined by the relative size of the area of dents and tears [14].

The generation of harmonic acoustic vibrations by a solid ultrasonic emitter is a convenient and easily controllable method for producing cavities in liquid medium. Wave theory is the basis for the design of solid emitters. It determines the most active areas of emitters of various shapes. However, it cannot describe the cavitation interactions of individual cavities or their groups [15]. Therefore, the cavitation interactions in ultrasonic fields of various intensities, the propagation of ultrasonic waves, and their ability to create cavities are important research and practical tasks in ultrasonic processing of liquid food media for homogenization, suspensions, and emulsions.

The present research featured cavitation in ultrasonic
field created by the cylindrical surface of a solid emitter
immersed in distilled water to a depth *h*. An aluminum
foil screen was installed at distance *x*. The number of
cavities was calculated at distance *h*^{1} from the liquid
surface in the normal vector to this surface.

Laboratory device IL-10-6/2 was used to create ultrasonic waves as described in [16, 17]. Cavities created by the laboratory installation in the liquid deformed the aluminum foil screen, causing either indentations in the foil or tears in the screen. This kind of violation of the integrity of the screen will be called “holes”.

##### РЕЗУЛЬТАТЫ И ИХ ОБСУЖДЕНИЕ

The technological effect results from ultrasonic cavitation in liquid media as high-density energy localizes in microvolumes of the treated medium.

After a certain amount of pulsations, cavities collapse and trigger a shock wave, which destroys the nearby solid surfaces. Almost all related studies were focused on the process of cavitation-induced destruction of the medium components as a way to increase its homogeneity [18, 19]. The obtained data on kinetics and thermodynamics of cavity formation, time and forms of their existence, etc. make it possible to describe the kinetic and thermodynamic parameters of both cavities and the medium. The part of the ultrasonic device that comes in contact with the processed product is called the “oscillatory system”. The surfaces of oscillatory systems of ultrasonic devices are also subject to cavitationinduced erosion. As a result, a certain amount of the material can mix with the product [20–22].

The present research featured the types and nature of erosion of ultrasonic devices and the safety of the product.

A set of experiments provided foil screens deformed by ultrasonic field. The state of the screen revealed the erosion capabilities of the ultrasonic field generated by a cylindrical acoustic emitter in water. The surface of the screen was also given a cylindrical shape before it was installed in the test medium. The screen was placed in the test medium equidistantly to the surface of the emitter. The exposure of the screen and the curvature of its surface varied in different experiments. Figures 2 and 3 show the aluminum foil screens deformed by ultrasonic treatment.

The surface of the screens indicated a significant inhomogeneity of deformation. However, the agglomeration pattern of acoustic caverns was quite stable in certain areas of the screen surface. The experiments also determined the average value of the density of screen deformations depending on the time of the acoustic impact and the distance from the surface of the emitter (Table 1).

The amount of deformations on the screen decreased
as the distance from the surface of the emitter increased.
Figure 4 shows the number of deformations per
second (*N*, pcs) per 0.0004 m^{2} of the screen surface,
depending on the distance from the surface of the
emitter (*x*, m). The diameter of the emitter was 0.03 m,
the depth of immersion was 0.12 m, and the thickness of
water layer was 0.25 m.

Figure 5 illustrates that as the distance from the emitter increased, the amount of screen deformations decreased according to the hyperbolic function:

where a, b, and c are constant: *a* = 0.05, *b* = 2, and
*c* = 20.

Figure 6 clearly shows the distribution of cavitationinduced deformations after different acoustic exposure periods.

The difference in the distributions shown in Figures 6 and 7 can be explained by such factors as acoustic wind, Bjerknes force, and Stokes force. Figure 6 suggests that the holes on the radially arranged screen appear as a result of ultrasound waves in the vessel. Thus, the expression for the wave interference for the number of holes will be the following:

where *ψ* – constant equal to the maximum number of
cavities on 0.0004 m^{2}.

Curves 1 and 2 in Fig. 6 can also be simulated, like
curve 3. However, in their cases, the speed of sound is
somewhat less than it is universally accepted, i.e. instead
of *c* = 1500 m/s, it is *c* = 1140 m/s because the water flow
is intense and contains air bubbles, which obviously
reduces the speed of sound in the water.

The number of holes in the foil screen makes it possible to characterize the energy of cavities formed in this volume. In a first approximation, the work of cavities can be calculated by the following formula:

where *А*_{E} – work produced by the collapsing cavities, W·s; *R*_{max} – maximal size the cavity reaches as it
oscillates around the equilibrium state (max 100 μm); *P* – pressure in the cavity as it collapses (103 MPa); *N* – number of cavities per volume.

As the cavity collapses, it produces spherical shock waves (Fig. 8) at a pressure of thousands of MPa. This pressure is much higher than the tensile strength of aluminum, which begins to flow like an ideally plastic body according to Saint-Venant. Therefore, the theory of plasticity can be applied to the theoretical analysis of the effect of shock wave on aluminum foil.

Considering that the area of the foil around the hole formed by the cavity assumes hemispherical shape and that aluminum outside the strength limits behaves like an ideally plastic body, the model of the hole can be represented as the destruction of an ideally plastic sphere by a spherical shock wave (Fig. 9).

The following equation is valid for an ideally plastic body:

where *σ _{r}* and

*σ*– normal stresses (tangential and along coordinate

_{t}*r*), Pa.

Based on the deformation of the sphere:

where *σ _{f}* – yield stress, Pa.

Considering (27) and (28):

at *r* = *d*/2*σ _{f}* = –

*p*; at

_{d}*r*= (

*d*/2) +

*δσ*= –

_{r}*p*where

_{g}*p*is the hydrostatic pressure of the liquid, Pa.

_{g}By integrating (29) under the indicated boundary conditions, we obtain:

where *С* – arbitrary constant.

Expressing the total pressure of the internal and external hydrostatic (plastic) flow, we get the following equation:

At *р _{d}* >>

*р*:

_{g}

The resulting formula (33) makes it possible to calculate the pressure inside the sphere formed by the shock wave after the cavity collapses at a distance close enough to the foil for a hole to occur.

The formula was tested using the data on the
ultimate strength of aluminum. The ultimate
strength, or the yield condition, of aluminum is
*σ _{f}* = 60–100 MPa.

The size of the holes in the foil varied from 5×10^{–4}
to 10^{–6} m. Therefore, the range of pressures in the
corresponding cavities can be defined by formula (33). Figure 10 shows the dependences of the pressure in
spherical shock wave on the diameter of the collapsing
cavity at different strength limits of aluminum.

The pressure in the wave calculated by formula (33) was compared to that calculated by Kirkwood-Bethe formula (17). Figure 11 shows the pressure graph in the shock wave calculated by Kirkwood-Bethe formula for the distances from the center of the cavity r under the collapse of the cavities.

Figures 10 and 11 show that the calculations correlate almost entirely. Therefore, the approach to calculating the collapse pressure in collapsing cavities based on comparing this pressure with the strength of solid materials under erosion can be used to assess the intensity of erosion.

##### ВЫВОДЫ

The present research revealed the following erosive effect of cavitation on highly rigid materials of ultrasonic vibrating systems.

Figure 12 shows how the corresponding problem from the theory of elasticity can be applied to the task in hand. The spherical shock wave from the collapse of the ultrasonic cavity was considered here as an absolutely rigid ball penetrating into the elastic surface of the ultrasonic emitter.

The problem has the following solution:

where *а* – radius of the contact spot of the ball with radius *R* with elastic half-space; *q* and *q*_{0} – current and maximal (in the center of the spot) contact stresses, respectively; *β* = 1/2*R*; *k* = (1–*μ*^{2})/*πE*; *w*, *w*_{0} – current and maximal values of the deflection of the elastic space; *P* – force pushing the ball into the elastic half-space.

The previous studies defined the range of pressure values in cavities. Using these values, assuming that the maximal pressure in the cavity that erodes the emitter, is *p* = *q*_{0}, and expressing the size of the contact spot as а and the penetration depth as *w*_{0} via *q*_{0}, we get the following equation:

The obtained results proved that contact spot size

*a*and penetration depth

*w*

_{0}can serve as a criterion for the erosion of the surface of the ultrasonic emitter.

##### КОНФЛИКТ ИНТЕРЕСОВ

The authors declare that there are no conflicts of interests regarding the publication on this article.##### ФИНАНСИРОВАНИЕ

The research was carried out on the premises of the K.A. Timiryazev Russian State Agrarian University (RT SAU), with financial support from the Ministry of Science and Higher Education of the Russian Federation (Minobrnauka) as part of the program for the development of world-class research center “Agrotechnology of the Future” (The grant was delivered as subsidies from the federal budget for state support of world-class research centers, performing R&D in scientific and technological priority areas, No. 075-15-2020-905 (internal number 00600/2020/80682), November 16, 2020).##### СПИСОК ЛИТЕРАТУРЫ

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