Theoretical study of the influence of gas phase turbulent disturbances on the kinetics of ultrasonic coagulation of PM2.5

A numerical model of ultrasonic coagulation of PM2.5 aerosol particles in three-dimensional vortex and turbulent acoustic flows is proposed in the article. The model is intended to identify the possibility of increasing the efficiency of the process. A numerical analysis of the model using the example of PM2.5 aerosol made it possible to establish that the presence of three-dimensional turbulent disturbances leads to the fact that the coagulation efficiency of PM2.5 reaches almost 100% at a sound pressure level of no more than 165 dB.


Introduction
The effectiveness of existing dust collectors for PM2.5 particles, which pose a significant risk to human health and are found in areas previously considered free of such particles [1][2][3][4], is reduced to zero [5,6].
Therefore, it is necessary to search for new nonlinear physical effects in ultrasonic fields, different from the classical mechanisms of ultrasonic coagulation (orthokinetic and hydrodynamic interaction), and to create conditions for the emergence of new effects.
In earlier publications made by the authors of this article [12,13], an approach to increasing the efficiency of ultrasonic coagulation was proposed, based on the use of stationary vortex acoustic flows associated with: • absorption of vibrations in a gas, in which part of the absorbed energy is transferred into the movement of the gas with work performed against the forces of viscous friction; • the limitation of the voiced volume, which excludes the potentiality of this flow arising due to the volumetric source of energy.Stationary vortex flows lead to a local increase in the concentration of particles due to their inertial transfer to the periphery of the vortex by analogy with the physical effect of compaction of an aerosol cloud during the forced formation of swirling gas flows by feeding a gas suspension into a specially shaped chamber ("cyclone") [12,13].
To describe this effect, the authors previously created a theoretical model with the development of numerical calculation methods, which makes it possible to predict the efficiency of coagulation in these flows [12,13].
However, the proposed theoretical model considers only the 2D case, namely its variation 2Daxisymmetric.This does not consider chaotic turbulent disturbances of the gas phase along all three axes, which lead to an additional increase in the efficiency of particle collisions.In this case, turbulent disturbances also have a vortex character.
Therefore, the relevance of developing a numerical model of the influence of three-dimensional turbulent disturbances on the probability of coagulation is beyond doubt.

Mathematical formulation of the three-dimensional problem of calculating ultrasonic coagulation of aerosols in vortex and turbulent disturbances
The ultimate goal of the model study is to study the influence of three-dimensional vortex and turbulent disturbances on the probability of particle coagulation.When modeling ultrasonic coagulation of aerosols in vortex acoustic flows, the following assumptions are made: 1. Vortex acoustic flows contain two components: The laminar component is due to the presence of gas phase drift due to the absorption of ultrasonic vibrations in the volume.The laminar component of acoustic flows is determined by the model described in the monograph [20].
The turbulent component is initiated by the main oscillatory speed, which is many times higher than the speed of the laminar component and is equal to several tens of meters per second at a sound pressure level of 165 dB and above.
2. Laminar vortex acoustic flows are stationary.This assumption is determined by the fact that an increase in the speed of acoustic flows arising due to the absorption of the energy of ultrasonic vibrations leads to a proportional increase in the force of viscous friction, which impedes the flow.
In this case, the absorbed energy of ultrasonic vibrations includes a kinetic component -energy that turns into kinetic energy of the liquid, and a thermal component -energy that turns into heat due to fluctuations in the force of viscous friction.
As the speed of acoustic flows increases, the kinetic component of the absorbed energy of ultrasonic vibrations is balanced by the work against the forces of viscous friction in the flows.
3. In a stationary flow of the gas phase, particles drift at a constant speed ; where τp is the particle relaxation time, s; u -particle speed, m/s; rp(t) -vector of coordinates of the current position of the particle, m.This assumption is justified by decomposing the velocity of a particle in terms of a small parameter τp and substituting it into the equation of particle motion based on I. Newton's 2nd law.
4. The characteristic time of particle coagulation exceeds the time of complete rotation of the particle around the streamline.This assumption is due to the low concentration of particles when it comes to fine gas purification.
5. The characteristic time of transition of particles between streamlines of a laminar vortex flow is much greater than the time of a complete revolution around the streamline.
6.Because of assumptions 4, 5, the concentration of particles within one streamline can be considered the same and the contributions of inertial transfer of particles and coagulation among themselves are additive.
7. Turbulent gas movements can be considered in the probability of a collision, and their contribution to the probability can be determined according to expression (2).This is due to the high frequency and chaotic nature of the pulsations, as well as their direct dependence on the gas velocity at the current time.
The final collision probability is determined as follows: The following describes the results of numerical calculations of ultrasonic coagulation based on concentration evolution models [12,13] taking into account the modified collision probability.

Results of numerical simulation of ultrasonic coagulation
Initially, an analysis was carried out of the influence of the size of the air gap on the speed of the formed vortex flows.The optimal size of the air gap at which the speed of vortex acoustic currents is maximum has been established.It corresponds to half the wavelength of ultrasonic vibrations.
Next, a numerical analysis of the contribution of each of the studied mechanisms for increasing the efficiency of ultrasonic coagulation was carried out: Laminar flow, ensuring the movement of particles in a given direction and their compaction due to inertial transfer.
Turbulent disturbances that provide differences in the velocities of particles of different sizes.
The following are the dependences (Figures 1, 2) of coagulation efficiency on the sound pressure level (the maximum level created in the air gap) for different particle sizes.For almost all particles in the size range 0.6...3 μm, there is a critical sound pressure level, starting from which three-dimensional laminar flows contribute to the coagulation efficiency.
This sound pressure level ranges from 160 to 165 dB.The additional contribution of turbulent disturbances reduces the required sound pressure level for coagulation of fine particles to 163 dB.At the same time, the coagulation efficiency of even the smallest particles (less than 1 micron) reaches almost 100% already at a sound pressure level of 165 dB.This confirms the need to take this factor into account to increase the efficiency of coagulation of the smallest particles of 1 micron or less.
At the same time, for large particles from 2.5 microns, turbulent disturbances do not contribute to the critical level of sound pressure, since they are more efficiently compacted due to inertial transfer.

Conclusion
As a result of the research performed, a method was proposed for determining the modified probability of ultrasonic coagulation taking into account the contribution of turbulent acoustic flows.
Numerical analysis of the effectiveness of ultrasonic coagulation using the found modified probability allowed us to establish: • coagulation is most effectively realized in resonant acoustic fields, in which vortex flows and turbulent disturbances have maximum speed; • boundary levels of sound pressure at which various types of flows in three-dimensional space influence.Laminar vortex flows begin to influence (due to local compaction of the aerosol cloud associated with inertial forces) from a sound pressure level of 160...165 dB, and turbulent disturbances make an additional contribution in the range of sound pressure levels from 140 to 170 dB; • the presence of three-dimensional turbulent disturbances leads to the fact that the coagulation efficiency reaches almost 100% at a sound pressure level 5 dB lower than with only laminar flows (or at a sound pressure amplitude 3 times lower than that required for 100% -no efficiency of coagulation in laminar flows).The research conducted and the results obtained serve as a scientific basis for increasing the efficiency of ultrasonic coagulation by creating conditions for the emergence of three-dimensional laminar and turbulent vortex acoustic flows.

Figure 1 .Figure 2 .
Figure 1.Dependence of coagulation efficiency on sound pressure level for particles smaller than 2 microns.