Physical Mechanisms and Theoretical Computation of Efficiency of Submicron Particles Agglomeration by Nonlinear Acoustic Influence

This study models the agglomeration of submicron particles when they are exposed to different types of ultrasonic waves, viz., sinusoidal waves and shock waves (pulses). The nonlinear effects of the shock waves (the transfer of heat, drop in pressure, change in the particles’ collisional cross-section due to Brownian motion, and difference in particle concentration), which influence the particle coagulation rate, are simulated for the first time and evaluated. The results reveal the optimum duration for compression and depression shock-wave pulses. Furthermore, given the same total amount of ultrasonic energy, the submicron particles coagulate 20 times more quickly with shock waves than sinusoidal waves.


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Aerosols being formed in the atmosphere due to the human-induced impact, technogenic 28 accidents, terrorist acts, and natural processes, constitute a global challenge. Mankind has been 29 forced to continuously combat fogs, smog, dust, aerosols of harmful, toxic and radioactive 30 substances (Hansen et al., 2006;Khmelev et  Aerosols of finely dispersed particles (smaller than 1…2.5 µm in size) presents the greatest 33 danger. Those particles have a high total surface area (more than 55% of the total surface area of 34 particles emitted into the atmosphere) and number concentration (more than 95% of the total 35 number concentration) even at a small mass fraction (less than 1% of the total portion of aerosols 36 contained in the atmosphere) ( To date, the efficiency of the US coagulation has been proved many times for particles larger 48 than 2.5 µm. Many authors describe coagulation modes (Khmelev et al., 2016;Khmelev et al., 49 2017;Rozenberg, 1970;Timoshenko and Chernov, 2004), at which the highest degree of 50 agglomeration is achieved; and propose special-purpose equipment (Crook, 1995 The existing equipment, however, turns out not to be sufficiently efficient in coagulating 56 particles smaller than 2.5 µm (Langner et al., 2020), and especially smaller than 1 µm. This is 57 due to the peculiarity of sinusoidal-wave US exposure, making submicron particles be involved 58 in the vibratory motion within the boundaries of areas between the zones of the minimum 59 vibration speed (Hansen et al., 2006;Jajarmi, 2016;Khmelev et al., 2017;Riera et al., 2006; 60

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5 Since it is impossible to generate shock waves of compression or depression in practice in the 82 absence of co-occurring phases of depression or compression, we should consider cases of 83 exposure at various relations between durations of the depression phase ( d  ) and the 84 , including limit relations -0 (the compression wave 85 amplitude is zero) or + (the depression wave amplitude is zero). 86 The ultimate objective of research consists in determining the efficient conditions of exposure, 87 which provide the coagulation of different particles, and revealing variations in the coagulation 88 rate with respect to the influencing factors to be determined according to the following 89 formula (2), which follows from mass conservation law const 1 (1) 91 (2) 92 where D30 is mean volumetric diameter of particles, m; ni is number concentration of particles of 93 size Di, m -3 ; symbol < > means averaging of the particle concentration based on the ultrasonic 94 disturbance length. 95

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6 To research the coagulation process of submicron particles it is appropriate to apply the 96 probabilistic approach proposed by Smoluchowski, 1916, being the most developed for today by 97 Sheng andShen (2006, 2007), experimentally proved (Riera et al., 2006;Sheng and Shen, 2007) 98 and enabling the investigation of the evolution of particles given the collision probability 99 described by the formula (3): 100 (3) 101 where nk is concentration of particles with the nominal diameter of 3 0 k d , m -3 ; uk is average 102 motion velocity of particles with a diameter of 3 0 k d ; (the root-mean-square value of ultrasonic pressure does not exceed 500 Pa, the maximum 115 momentary disturbance is not more than 5000 Pa). 116 When the compression shock-wave duration is more than 2 μs, the force that must be applied 117 to accelerate the gas flow is (pA amplitude of pressure of shock-wave, 118 Pa; pRMS is root mean square of pressure of shock-wave; T is period between shock-waves, s; τ is 119 shock-wave duration, s; ρ is density of gas, kg/m 3 ; c is sound speed in gas, m/s), that is less than 3 120 kN·dm -3 . 121 This research of ultrasonic coagulation considers monodisperse and polydisperse aerosols with 122 a predefined initial number concentration, the size of particles within 0.1…0.5 µm, which is 123 usually not captured by known devices. 124 It is obvious that during the ultrasonic coagulation of such particles, one should take into 125 account changes in the collision cross-sectional area due to the motion of gas-phase molecules, as 126 well as changes in the particle concentration with great changes in gas phase pressure due to 127 inertia of particles and inter diffusion of gas molecules along the free path.
(4) 139 where < > is the symbol of averaging in a period of changes in pressure; V is selected volume of a 140 cloud which size is small as contrasted to the exposure duration, but large as contrasted to the 141 distance between particles, m 3 ; N is number of particles in the selected volume V; n is initial 142 number concentration of particles, m -3 ; 1 , 1  is probability of particle collision, m 3 ·s.

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The coagulation rate characterizes the increase in the mean volumetric diameter of particles per 144 unit time. 145 The collision probability of particles is determined using the following formula (5) (  , i.e. the 162 hydrodynamic interaction is determinative. The orthokinetic interaction is insignificant, since 163 each particle, regardless of its size, is equally involved in the vibratory motion. 164

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10 The probability of hydrodynamic interaction is proportional to the collision cross-sectional 165 area. According to the available data (Michaelides, 2015;Yang et al., 2015), the cross-sectional 166 area of submicron particle collision changes due to the influence of the Brownian motion. Taking 167 into consideration the Brownian motion, the collision cross-sectional area is increased by 168 dispersion r  of the spatial position of a particle according to expression (7).

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It is an obvious point that collisions between moving gas-phase molecules and particles ( Fig. 1) 170 cause changes in the collision cross-sectional area according to the formula (7) and a diagram 171 shown in Figure 2. 172 Dispersion of the particle position is determined by the Monte-Carlo method subject to the 173 random generation of times of collision between a molecule and a particle, the velocities of 174 molecules at the moment of collision and positions of a particle surface point at which the 175 molecule collides with it. 176 Consideration should be given to the following assumptions when generating the above 177 parameters: 178 the particle velocity after collision of a molecule with it is determined by the law of 179 conservation of momentum; 180 collisions are absolutely elastic; 181

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11 collisions are equally probable in all feasible directions, since the most probable 182 velocity of an arbitrarily chosen gas molecule is many times higher than the velocity of a particle, 183 because the mass of an individual particle is much greater than the mass of an individual gas 184 molecule. 185 The spatial position of a particle as a result of a sequence of collision elementary events 186 "particle-molecule of the gas phase" is determined according to the following formula: 187 where ti is occurrence time point of the i-th elementary event of collision between a molecule and 189 a particle, s; ui is particle velocity as a result of the i-th elementary event of collision with a 190 molecule, m/s; ri is coordinate vector of particle at the occurrence time point of the i-th 191 elementary event of collision, m. 192 The particle velocity as a result of the i-th elementary event of collision is determined 193 according to the law of conservation of momentum: 194 change in the velocity of a collided molecule in the center-of-momentum 195 frame of a particle; 196 change in the velocity of a collided molecule in the laboratory frame of 197 reference; 198

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-law of conservation of momentum for the system "particle-199 collided molecule"; 200 ;common expression for the changed velocity of a particle as a result of the 201 i-th elementary event of collision.

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An unknown variable w is determined on the basis of the energy conservation law. 203 (10) 204 where M is particle mass, kg; m is gas-phase molecule mass, kg.

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13 Preliminary calculations showed that the time sufficient to meet the condition (12) is much less 214 than the characteristic time of pressure changing. This gives us grounds for considering pressure 215 to be constant in determining an instantaneous effective collision cross-section of particles. 216 An approach set forth herein made it possible to establish relations between the effective 217 collision cross-section and the vibration amplitude for various ultrasonic effects: 218 where T is acoustic disturbance time period, s; pA is amplitude of US disturbances, Pa;  is 228 characteristic duration of a pulse, s; depr is characteristic duration of depression pulse, s; compr is 229 M A N U S C R I P T 14 characteristic duration of compression pulse, s; pA compr is amplitude of compression pulse, Pa; 230 pA depr is amplitude of depression pulse, Pa. 231 The forms of each type of shock-wave are shown in Fig. 3. 232 In particular, the effective collision cross section for continuous sinusoidal-wave 233 exposure does not change very much (less than 10 percent increase). For 234 , while forming a predominant compression phase 235 The resulting data on the collision area value allowed proceeding with further research of the 242 hydrodynamic interaction process which, according to (Rozenberg, 1970), is determined by 243 radiation pressure of the US effect reflected from a neighbouring particle. 244 (16) 245

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15 where f21 is interparticle force, N; p is pressure disturbance of gas near a particle created by a 246 neighbouring particle, Pa; u is gas velocity perturbation near a particle created by a neighbouring 247 particle, m; n is normal vector to the particle surface. 248 Since the particles in question are submicronic, and the distance between them is consistent 249 with the free length of gas-phase molecules, the following quasi-gasdynamic equations, 250 considering the inter diffusion of the gas phase and the nonequilibrium state of the operation of 251 pressure change process in the gas phase, were used to calculate the field of reflected ultrasonic 252 effects (Elizarova, 2005): 253 where jm is gas phase mass flux, kg·m -2 •s -1 ; u is gas-phase molecule motion velocity, m·s -1 ; ρ is 257 gas phase density, kg•m -3 ; τ is relaxation time required for gas phase to move to the equilibrium 258 state, s; Π is stress tensor in gas phase, Pa. 259 Solving the presented set of equations made it possible to determine the fields of pressure and 260 gas velocity perturbations around an individual particle for a given ultrasonic exposure 261 (sinusoidal one or one with predominant compression or depression phases) and calculate the 262 radiation pressure force on a neighbouring particle from the part of the found disturbance fields. 263

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16 Since the propagation process of ultrasonic exposure is adiabatic, the set of equations (17-20) 264 is rearranged: 265 (24) 269 where φ is velocity potential, m 2 /s. 270 The system is further supplemented with the boundary condition on the particle surface (26), 271 from which the reflection occurs, and with conditions (26, 27) at a distance exceeding the 272 distance between neighbouring particles: 273 where rel p is static pressure in gas with no ultrasonic disturbances, Pa; PA is pressure of 275 ultrasonic exposure, Pa; k is wave vector of field, m -1 . 276 The distribution of pressure disturbances near a ball-shaped particle will consists of three 277 where refl p is pressure in ultrasonic exposures reflected from a particle, Pa. 280

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17 Since particle sizes are massively smaller than the ultrasonic exposure area length, radiation 281 pressure on the surface of a neighbouring particle will be contributed by pressure prefl because the 282 for shock-wave influence with pulse amplitude more than 5000 Pa and particles 285 diameter less than 1 μm; where r1 and r2 diametrally opposite points on particle; PAmax is pressure 286 amplitude of shock-wave, Pa; d is particle diameter, m; k is wave number of shock-wave; τ is 287 pulse duration of shock wave, s; R is particle radius; Umax is maximum gaseous speed in shock-288 wave, m/s; PRMS is root mean square pressure of shock-wave, Pa), and the integral of components 289 A P and rel p is equal to zero. 290 In real gas-dispersion systems the interparticle distance is more than the free length of gas 291 molecules (70 nm), and determination of fields of velocity and pressure disturbances is based on (33) 301 (41) 307 and so on. 308

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19 The equations (35-41) were solved analytically. General scheme of analytical solution is 309 following: 310 (43) 330 where  is viscosity of gaseous phase, m; d is diameter of separate particle, m; n is particle 331 concentration, m -3 ; f21 is particles hydrodynamic interaction force, N; S is square of equivalent 332 collision cross-section, m 2 ;  is solid angle of particles centers line, sr; n is unit normal vector of 333 centers line orientation; N is normal vector to particle surface. 334 The physical meaning of last expression is sum by neighbour particles interacting with 335 observed particle from different angles between centers line and wave vector of acoustic field. 336 The hydrodynamic force depends on both acoustic pressure and angle. However collision cross-337 section depends on acoustic pressure only. 338 The resulting dependences of the coagulation rate are set forth in the next section. shock-wave has positive sign of pressure. Less pressure, larger collision cross-section (due to 367 Brownian motion). Larger collision cross-section follows more coagulation efficiency. 368 All of that confirms the highest efficiency of shock-wave action for small-size particles. 369 In view of the fact that it is impossible to carry out any pulse exposure with the compression 370 (or depression) phase only, it is to be supposed that the maximum effect of coagulation 371 acceleration (10 times and more) will be provided in practice on successive exposures to 372 vibrations in which the duration of the depression phase is less than the compression phase 373 duration. 374 Relationships between the agglomeration rate and the ratio of the depression phase duration 375

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23 As it follows from the presented dependences, the maximum coagulation rate is achieved when 377 the compression phase duration in ultrasonic exposure dominates over the depression phase 378 duration (the amplitude of the depression phase is higher than that of the compression phase). 379 Then the influence of acoustic disturbance on the particle size distribution in polydisperse 380 aerosol was established in terms of the determined value of the coagulation rate. The 381 Smoluchowski equation, upon substitution of the expressions obtained for the collision 382 probability allowed plotting histograms of the initial aerosol and aerosols after various ultrasonic 383 exposures (Fig. 6). However the methods have high computational complexity. Moreover, the methods don't take 391 into account difference between velocities of same sizes particles. Herewith, the main goal of the 392 paper is research of shock-wave influence on coagulation in spatial homogeneous case 393 (convective member is zero). Thus the using of CFD-PBM algorithms is impractical. 394

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(44) 398 five times increased (for the modal size of initial particles -100 nm, RMS -500 Pa) in the 413 quantitative portion of coarse fraction (particles whose diameter exceeds the modal diameter by 414 more than 30%) is found while the coarse fraction portion shows no more than three times 415 increase under the sinusoidal-wave ultrasonic exposure. 416 Time dependences of the mean volumetric surface diameter of aerosol particles at various root-417 mean-square values of sound pressure and various conditions of ultrasonic exposure were 418 constructed based on the resulted histograms (Fig. 7). 419 The presented dependences attest a reduction of time required to achieve the maximum size of 420 particles when applying shock-wave exposure. For example, for particles of 100 nm, at 421 sinusoidal-wave exposure, the maximum diameter is not achieved even in 150 minutes. While in 422 shock-wave exposure a diameter being 95% of the maximum diameter is achieved in 60 minutes.  is random velocity of the gas-phase molecular motion; u is random component of the particle 563 motion velocity determined by heat-induced collision between gas phase molecules and a particle; 564 U is deterministic component of the motion velocity caused by involvement of a particle in the 565 vibratory motion of gas flow) 566