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СКАЧАТЬ power density can be determined as the amount of actual electrical energy input per unit volume of the medium.

      The value of cavitational yield roughly indicates the capability of performing the desired chemical reaction at a given input of electrical energy per unit reaction mixture. On the other hand, calorimetric analysis is used to determine the power dissipated in the reaction mixture. In the calorimetric study, the variation in temperature (rise) of a known quantity of water is measured for a given period. Based on these data the actual power dissipated can be determined using the following expression [19, 87]:

      (2.2)

      where Cp – specific heat capacity of water (solvent) in J/kg K; m – mass of water (solvent) in kg; dT is the temperature difference (in K) over the time period dt (in s).

Based on the power dissipated in the system, the energy efficiency can be determined as the ratio of power dissipated to the electrical power supplied to the system. This energy efficiency represents the electrical power fraction used for cavity generations. Ultimately, energy efficiency and cavitational yield measures the amount of electrical power utilized in formation and generation of cavitational effects resulting in occurrence of desired chemical reaction. The approach is experimental and can be employed to determine the said parameters in acoustic as well as hydrodynamic cavitational reactors.

      The alternate procedure to evaluate the effectiveness or performance of cavitational reactors is mathematical modeling or simulations. The primary objective of mathematical modelling is to determine the pressure and temperature attained during the cavitational process, along with numbers and types of free radicals generated. There are several models available to estimate the cavitational intensity; most of these models are predictive only as assumptions defined in forming and solving mathematical equations makes them unrealistic in nature. For example, all the models assume that the shape of bubbles is spherical and remains the same throughout the process, which holds good but due to sudden change in pressure amplitude on either side of bubbles deformed the shape of bubbles. Secondly, these models are representative in nature as the number of formation of bubbles are not uniform throughout the course of operation but the model assumes it to be constant. Moreover, these models do not account for energy and heat losses as well as a molecule – molecule interaction during the process of cavitation. Most of these models ignore the change in viscosity, pressure, and temperature gradient as well as Bjerknes forces during the transient cavitation process. Still, the results (simulated profiles) of temperature, pressure, size of bubbles, etc. obtained from these models meaningfully predicts the cavitational intensity and corresponding effect on yield of the reaction precisely. The most commonly employed model for determination of cavitational effects is single–bubble dynamics model – based on considering the single bubble in isolation, as discussed below:

      The model consists of four ODE (ordinary differential equations) and can be solved by taking boundary conditions based on the physical properties of liquids. The model starts with the radial motion equation for a bubble derived from the work of Keller and Miksis [105]:

      (2.3)

Internal pressure inside the bubble as well as bulk pressure in the liquid medium can be determined by using following equations [18]:

      (2.4)

      (2.5)

      The ODE for determining the diffusive flux of solvent is as follows:

      (2.6)

      During the expansion of bubbles, the instantaneous diffusive penetration depth of solvent vapors into the cavitation bubbles can be predicted by:

      (2.7)

      As the compression and expansion of bubbles occurs, the heat conduction across the boundary walls of bubbles can be determined using following equations:

      (2.8)

      where, thermal diffusion length is expressed as:

      (2.9)

      Finally, the overall energy balance can be carried out using equations given below:

      Mixture heat capacity: ; where i = N₂/O₂/Solvent

      (2.10)

      (2.11)

      (2.12)

      (2.13)

      It was assumed that during the transient collapse of bubbles, the temperature and pressure reach the extreme values inside an exceptionally small volume of the bubble. The kinetics of dissociation reaction is much higher than bubble dynamics, and thus thermodynamic equilibrium is presumed inside the bubble [28, 29, 106].

      Thus, the mathematical models presented herein are the most relevant and developed, repeatedly used in the estimation of cavitational effects in the reaction mixture. Moreover, more complex equations are needed to fully explain the intricate working of the cavitation zone, which considers non-linear behaviour of bubbles or clusters, non-uniform size of the bubbles or clusters, and heat and energy loss from or clusters during the growth of bubbles. The next section deals with the scale-up process of cavitational reactors based on computational fluid dynamics (CFD) approach.

2.5 Scale-Up of Cavitational Reactors

The ultrasound has been useful in resolving the mixing issues, challenges associated with clogging of particles for small-scale flow reactors. Similarly, large-scale batch reactors were designed to intensify mixing, increase reaction rates, and mass transfer in many chemical and biological processes [11, 107]. However, scaling up of conventional flow reactor in combination with ultrasound is a challenging task, and therefore, the application of large-scale industrial ultrasonic reactors is seldom seen. The main challenges are improving the energy efficiency of the reactor, the efficient transfer of ultrasound energy, and enhancing ultrasonic effects. The presence of complex flow patterns in a conventional flow reactor and non-uniformity of acoustic fields, scaling-up, is difficult. Cavitational reactors operate at two frequency ranges: low frequency (20-100kHz) and high frequency (>1 MHz). The low-frequency ultrasound helps to break up agglomerates in solid particles by shear forces induced by the collapse of cavitation bubbles. In contrast, the high-frequency ultrasound helps to prevent clogging due to the centering of particles in flow channels. Since the high-frequency ultrasound operates at a power level below СКАЧАТЬ