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14
Erosion potential of ultrasonic food processing
Sergei A. Bredihin1 , Vladimir N. Andreev1 , Alexander N. Martekha1* , Matthias G. Schenzle2 , Igor A. Korotkiy3
1 K.A. Timiryazev Russian State Agrarian University"0", Moscow, Russia 2 VA GmbH Society for Food Processing, Stuttgart, Germany 3 Kemerovo State University"0", Kemerovo, Russia * e-mail: [email protected] Received April 30, 2021; Accepted in revised form May 21, 2021; Published online October 15, 2021
Abstract:
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.
Keywords: Ultrasound, cavitation, aqueous medium, foil screen, erosion, oscillatory system
Funding: Hie research was carried out on the premises of the KA. Timiryazev Russian State Agrarian University (RT SAU)"011, with financial support from the Ministry of Science and Higher Education of the Russian Federation (Minobmauka)"0" 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).
Please cite this article in press as: Bredihin SA, Andreev VN, Martekha AN, Schenzle MG, Korotkiy IA. Erosion potential of ultrasonic food processing. Foods and Raw Materials. 2021;9(2):335-344. https://doi.org/10.21603/2308-4057-2021-2-335-344.
Foods and Raw Materials, 2021, vol. 9, no. 2
E-ISSN 2310-9599 ISSN 2308-4057
Research Article Open Access
Ê)
https://doi.org/10.21603/2308-4057-2021-2-335-344 Available online at http://jfrm.ru/en
INTRODUCTION
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
Copyright © 2021, Bredihin et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.
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:
1 dp
(1)
du du
--+ u —
dt dr
p dr
where t - time, s; r - current radius, m; p - liquid density, kg/m3; p - hydrostatic pressure, Pa.
The continuity equation has the foilowingform:
_5
dr
( r 2u ) = 0
(2)
The vortex-free motion of the surface can be considered as a potential with potential 9, i.e.:
u = -
dp dr
(3)
Provided the bubble surface is r = R, u = U, then:
u = UI R
(4)
Equations (3) and (4) produce the following result:
R2
p = U
r
(5)
(6)
Integration of (1) from r to r = œ results in:
2 p(r) j
u (dp
u--+ I — = 0
2 1 P
The condition of incompressibility of liquid means that p = p0 = const; therefore, (6) is:
t+-•[ P-~ p ( r )] = 0
2 p0
(7)
Inserting (5) in (7) at r = h, we get the following equation for an empty cavity:
11 R dU - + 2 RU 2|-1U2 R R
r \ 2 dR
-[P»-P(r)] = 0 (8)
Po
For cavity surface r = h, U = gh/gt; thus, equation (8) looks like this:
R
d2 R dt2
(f)' op; [P-- P (f)] <9)
where P(h) is cavity surface pressure.
The resulting equation describes the motion of the cavity surface depending on the regularity of the pressure change P(h).
Lamb and Rayleigh obtained a solution to equation (9) assuming the simplest boundary conditions: P(h) = 0, = P0, i.e. pressure at a sufficient distance equals hydrostatic pressure:
u 2 = 2 p
3 Po
( R
Rm - 1
R3
(10)
where hm is maximal cavity radius early during collapse. Provided U = gh/gt, we get Rayleigh's equation:
z= 0 915R„
P
\L 0 y
(11)
where t 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:
R
d R 3 -+—
2
dt2
dR dt
J_
P0
P0 - Ps - Pm sin (at) +
2ct i 2CT
+--1 P +
R
where P - steam
R
0 y
pressure
37
= 0 (12) in the cavity;
m = 2f f - oscillation frequency; c - Laplace surface tension; y - adiabatic exponent (4/3), P - density of liquid unaffected by fluctuations, am - maximal pressure in the cavity attained at its minimal radius.
The numerical solutionp x>roposed in [6, 2] ddscribe 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:
dp 1 d t 2 \ „
—( r pu ) = 0
dt r2 drv '
(13)
In the considered approximation u2/c2 << 1, whereas:
R
1-u
c0 J
d 2R 3 - + —
2
dt2
1-4U li ^ I2 [*-
- p ( R )]+-U
p0 c0
3 c0 1-u
. c0 J
dt J p0 ^dP ( R )
dR
= 0
(14)
where U - the speed of movement of the cavity surface during collapse and pulsation; c0 - 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:
R f 1 - 2 U J
cn dR2
3 c„
3 f 1 - 4 U Jf d_R J2+_L
dt
P0
P - p - P* » c >+¥+4U -f P.+1 J( R
-—Î1-U l dPiR), 0 (15)
P0 c0 I c0 J dR
Kirkwood and Bethe introduced the concept of specific enthalpy h and kinetic enthalpy:
u
Q = — = h + dt 2
(16)
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:
- U>| djR + 3 r 1 - 1 U
dt2 21 3 c
dR dt
-11 - UIH -
U r 1 - U l RdH=0,
dR
(17)
where c2 = c„21 P
p dp = P0 J p0 p n -1 c = [c^02+(n -1)H]; c02 = An / p
; h =f d-P-J p
( p\n-1
p0
-1
where H - free enthalpy on the sphere surface; A and n -constants for water (A = 3.001x108 Pa = 3.001x105 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 1within 7the 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:
U2
G ( R, t ) = R( H +— )
Afternecessary transformations, equation (18) looks like this:
(19)
[rr rRl2] L n+1 ^ rRl2]
rc U •l-I •n+— U •l-I
[ V r J _ L 4c [ V r J _
where thespeed of the cavity surface U is determined by the following transcendental equation:
U2 =-
2 P + P_
p0
R
3 (
1 —
U
3c,
-1
0 J
(20)
The pressure alongthe shock frontcan becalculated using the formula given in [16]:
p = A
n -1
n +1 n + 1
1+ G rc„
2n
n -1
-B (21)
where B - constant for water (3.0*105 MPa); constants A 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 3cavitation field occurs following the destruction of bonds in the crystal lattice of the object. Cavitation erosionis 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:
n = E • E
1er m
(22)
where Em - energy spent on mechanical erosion; E - ultrasonic vibration energy.
The local rate of cavitation-induced destruction in
Figure 1 Valve seat of milk homogenizer subjected to cavitation: 1 - erosion canals; 2 - areas of erosion damage to the valve working surface; 3 - guide ways
an inhomogeneous ultrasonic field is determined by the following formula:
d 2AG
W =ljg (23) where w - local rate of erosion in the cavitation field; AG - 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.
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 h1 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".
STUDY OBJECTS AND METHODS
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
RESULTS AND DISCUSSION
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 cavitation-induced 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
Figure 2 Foil screen deformed by ultrasonic treatment. Processing time - 20 s, distance from the surface of the emitter - 30 mm
Figure3 Foilscreen deformedby ultrasonictreatment. Processing time - 20 s, distance from the surface of the emitter - 150 mm
Table 1 Distribution of deformations on the screen depending on the distance from the surface of emitter x and the distance from the surface of liquid h1.
mm mm
Distribution of screen deformations at ultrasonic treatment time t and distance h
5 s 10 s 15 s 20 s
30 5 23 129 115 110
20 50 126 184 172
40 104 135 202 304
60 114 110 125 222
80 32 31 78 40
Z 323 Z 531 Z 704 Z 848
45 5 12 76 140 81
20 52 196 230 74
40 51 62 126 60
60 49 124 112 52
80 17 40 60 40
Z 181 Z 498 Z 668 Z 308
60 5 1 34 24 24
20 24 58 48 56
40 116 46 44 44
60 26 44 42 31
80 35 31 11 110
Z 202 Z 213 Z 169 Z 265
75 5 22 29 40 72
20 49 61 48 86
40 37 55 36 32
60 24 55 155 40
80 12 54 36 24
Z 144 Z 254 Z 315 Z 254
150 5 2 27 10 8
20 30 26 47 17
40 40 21 41 30
60 21 33 27 44
80 10 14 26 41
Z 103 Z 121 Z 151 Z 140
250 5 34 8 15 16
20 32 24 11 16
40 26 38 30 25
60 25 24 34 26
80 4 4 8 11
Z 121 Z 98 Z 98 Z 94
320 5 5 16 25 1
20 21 10 32 8
40 38 37 50 20
60 38 20 24 16
80 2 9 4 6
Z 104 Z 102 Z 135 Z51
a
Si
70
55
40
25
10
0.1
0.2
0.3
x, m
Figure 4 Effect of the distance from the surface of the cylindrical emitter on the amount of screen per 1 s of ultrasonic exposure
a cylindrical acoustic emitter in water. The surface of the sc0een was also given a cylindrical shape before it was installed in the test medium. The screen was placed in th60 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 snow the aluminum foil r2reenr defomied ty ultrasonic treatment.
The surface of the screens indicated a significant inhomogeneity of deformation. However, the agglomeration 9attern of acoustic caverns was quite stable in certain areas of the screen surface. The experiments also .0^^^]-mined the average value of the density of screen deformations depending on the time of the acous0io impact and the distance from the surface of the emitter(T8ble 1).
The amount of deformations on the screen decreased as ^di^nce from the surface of the emitter increased.
Figure 4 shows the number of deformations per secon0^, pcs) per 0.0004 m2 of the screen surface, dependi^ on the distance from the surface of the emitter (x, mo The diometeз 0O 0he 8mitter was0.03 m, the depth of immersion was 0.12 m, and the thickness of water layer was 0.25 m.
0
E milter ^
Figure 5 Distribution of cavitation-induced deformations and holes on the screen located equidistantly at distance x from the surface of the cylindrical emitter
Figure 4 illustrates that as the distance from the emitter increased, the amount of screen deformations decreased according to the hyperbolic function:
N = axAb+c (24)
where a, b, and c are constant: a = 0.05, b = 2, and
c = 20.
Figure 6 clearly shows the distribution of cavitation-induced deformations after different acoustic exposure periods.
The difference in the distributions shown in Figures 6 and 8 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:
N = y
2n
x --1
2n (x
-1
(25)
210 180 150
* 120
o &
90 60 30
0
0.00 0.02 0.04 0.07 0.09 0.12 0.13 0.16 0.18 0.19 0.21
Figure 6 Effect of the distance from the surface of the cylindric1.5E+09ter on the distribution of deformations of the
radially located screen during 10 s (1), 15 s (2), and 20 s (3)
where y - constant equal to the maximum number of cavities on 0.0004 m2.
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:
Af = -nR3 PN
E 3 max
(26)
where AE - work produced by the collapsing cavities, W s; R - maximal size the cavity reaches as it
7 max J
osciKates around the equilibrium state (max 100 ^m); P - pressure in the cavity as it collapses (103 MPa); N - number of cavities per volume. c sAs the cavity collapses, it produces spherical sho N, 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 plas10city can be applied to the theoretical analysis of the effect of shock wave on aluminum foil.
210 180 150
¡3 120 &
^ 90 60 30 0
0.00 0.05 0.07 0.10 0.12 0.14 0.17 0.21 0.23
x, m
Figure 7 Graphic approximation of curve 3 in Fig. 6
x, m
3
2
1
\
Figure 8 Expansion of hollow sphere made of ideally plastic material under internal pressure
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 Mtowing equation is vdid foa an sdeally plastic body:
+ = 0 (
where cr and ct - normal stresses (tangential and along coordmate r), Pa.
Based on the deformation of the sphere:
a, - a = a,
where cf - yield stress, Pa. Considering (27) and (28):
da = ±2a
dr
f~
(28)
(29)
at r = d/2of = -pd; atr = (d/2) +5Gr = - pg where pg is the hydrostatic pressure of the liquid, Pa.
By integrating (29) under the indicated boundary conditions, we obtain:
-pd =±2ofln(d /2) + C (30)
- pg = ±2of ln [( d/2 ) + ]] + C (31)
whereC - arbitrary constant.
Figure 9 Formation of a hole in the foil under the effect of spl1erical shock wave as the cavity collapses
Expressing the total pressure of the internal and external hydrostatic (plastic) flow, we get the following equation:
[(d /2) + £]
Pf = Pd - Ps =±2aflni
d/2
(32)
At Pd>>Ps-
pf = 2af ln
[( d/2) + ^^
d/2
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 0nough to the foil for a hole to occur.
The formula was tested using the data on the dtimate strength of aluminum. The ultimate 0trength, or the yield condition, of aluminum is 0f= 60-100 MPa.
The size of the holes in the foil varied from 5x10-4 to m. Thcrefoce, the range of pressures in the corresponding cavities can be defined by formula (33).
1.5E+09 1.2E+09 9.0E+08 6.0E+08
3.0E+08
0.0E+00
0.0E+00 1.0E-03 3.0E-03 5.0E-03 7.0E-03 9.0E-03
d, m
Figure 10 Effect of the diameter of the collapsing cavity on the pressure in spherical shock wave at the strength limits of aluminum = 60 MPa (curve 1) and 100 MPa (curve 2)
1.2E+09
1.0E+09
8.0E+08
Ph 6.0E+08
4.0E+08
2.0E+08
0.0E+00
2.5E-06 1.0E-05 3.0E-05 8.0E-05 1.3E-04 1.8E-04 r, m
Figure 11 Pressure in the shock wave as calculated by Kirkwood-Bethe formula for distances from the center of the cavity r: the radii of the collapsed cavities - 10-5 m (curve 1) and 10-6 m (curve 2)
Figure 12 Penetration of an absolutely rigid ball into elastic half-space
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.
CONCLUSION
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:
3n kP_
T D
=J24 (D p
(34)
(35)
ilf D 2 P 2
(36)
where a - radius of the contact spot of the ball with radius R with elastic half-space; q and q0 - current and maximal (in the center of the spot) contact stresses, respectively; ft = 1/2R; k = (1-^2)/nE; w, w0 - 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 = q0, and expressing the size of the contact spot as a and the penetration depth as w0 via q0, we get the following equation:
nR (1 -ju2)
a - q0-
- O. - R R
2E
q0n(1 )
2E
(37)
(38)
The obtained results proved that contact spot size a and penetration depth w0 can serve as a criterion for the erosion of the surface of the ultrasonic emitter.
CONTRIBUTION
The authors were equally involved in the research and are equally responsible for any potential plagiarism or other unethical issues.
CONFLICT OF INTEREST
The authors declare that there are no conflicts of interests regarding the publication on this article.
2
a = 3
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ORCID IDs
Sergei A. Bredihin https://orcid.org/0000-0002-6898-0389 Vladimir N. Andreev https://orcid.org/0000-0002-4890-379X Alexander N. Martekha https://orcid.org/0000-0002-7380-0477 Matthias G. Schenzle https://orcid.org/0000-0001-6334-0481 Igor A. Korotkiy https://orcid.org/0000-0003-4379-9652
