Modeling of three-dimensional linear pressure fields in sonochemical reactors with homogeneous and inhomogeneous density distributions of cavitation bubbles

A new model is presented for the numerical calculation of pressure fields in liquids with an inhomogeneous distribution of cavitation bubbles. To calculate the pressure field in a homogeneous single-phase fluid, the Helmholtz integral and the Kirchhoff integral are solved numerically. The Helmholtz integral equation and the Kirchhoff integral are used for the calculation of the acoustic field in a homogeneous fluid for all kinds of transducers of various shapes. The first term of the integral equation embodies a simple superposition of the pressure fields of several point sources, which serves to simulate a harmonic vibrating surface, while the Kirchhoff integral calculates the pressure field which emerges from the boundaries. With a new technique the three-dimensional time-independent pressure field is calculated gradually in the beam direction. With this procedure one is able to combine the Helmholtz integral with a wave propagation in liquids with inhomogeneous distributions of cavitation bubbles. Compared to a single-phase fluid, gas bubbles in a liquid lead to a heavy change of phase velocity and sound attenuation. These changes are determined and considered for every step in the beam direction. With this technique, one should be able to calculate the pressure field in a sonochemical reactor with a sufficient approximation which serves to predict the spatial distribution of cavitation events. These events are related to the yield of chemical reactions.