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As discussed in the previous sections, the chemical and physical effects of a cavitational phenomenon will be able to initiate as well as promote various chemical reactions. Thus, to design an efficient cavitational or sonochemical (either acoustic or hydrodynamic cavitation) reactor, one must be able to produce a cavitation zone, i.e., the area in the reactor in which formation, growth and the transient collapse of tiny bubbles should occur. Hence, the measurement of cavitation zone as well as cavitational intensity will ultimately impart the effectiveness of a selected cavitational reactor design. Along with cavitational intensity, analysis of pressure difference is also done, which leads to the development of a high number of cavitational bubbles and increases the impacts of physical effects of cavitational phenomena (especially in the case of a hetero-phasic reaction system). Moreover, application of either acoustic (either probe or ultrasound bath) or hydrodynamic (either orifice or venturi based) cavitational reactors is typically based on the size of reactants processed. In the following paragraphs, the basic methodology to determination of cavitational intensity in both cases is discussed elaborately.

As a standard procedure in chemical engineering practice, the lab-scale kinetics data is used to design a specific reactor followed by dimensional analysis; the design approach for a cavitational reactor will need an estimation of cavitational effects on the reaction mixture and variation of pressure amplitude throughout the reactor geometry. Gogate et al. [89] reported the methods commonly used to measure the variation in pressure amplitude and cavitational intensity. In this study, Gogate et al. [89] also reviewed the literature published in analyzing the cavitation intensity using an experimental- and mathematical-based approach. The authors reported the use of hydrophones to measure the variation of pressure amplitude experimentally. The obtained readings can be compared to the simulated (i.e., mathematically model-based) results to determine the efficiency and energy dispersion into the liquid under operating conditions. To estimate the chemical effects of cavitation, i.e., the formation of free radicals, authors suggested using degradation of potassium iodide as a model reaction. The quantity of iodine gas liberated can be measured using the UV/VIS spectrophotometer (measuring absorbance at a wavelength of 355 nm) [87, 89]. The cavitational yield for a given design and particular model reaction could be determined using the following equation [88]:

(2.1)

where 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.