Mathematical modeling of acoustic oscillations of chemically reacting two-phase mixture of monodisperse solid particles in a gaseous oxidant

Introduction. The phenomenon of self-excitation of acoustic oscillations due to heat transfer has been known for a long time, but its study has only recently begun due to the development of high- pressure combustion chambers. These oscillations can destabilize combustion, but they can also be beneficial by increasing the heat load and reducing the combustion time. The study of wave modes in combustion is important both theoretically and practically. Most studies have focused on homogeneous media, but real systems such as "liquid droplets-oxidizer" or "solid particles-oxidizer" exhibit unique wave dynamics. A deeper understanding is needed to comprehend these processes and manage fluctuations.

Goal. Based on the model of interacting continua, the problem of weakly nonlinear wave disturbances in a limited volume of a chemically reacting two-phase mixture of monodisperse solid particles in a gaseous oxidant is considered. The study takes into account that the dynamic and thermal interaction of the phases affects the dissipation and dispersion of the phase sound velocity. The method of slowly changing amplitudes allowed the system of mass, energy, and momentum conservation equations to be reduced to a nonlinear wave equation. Equations for the steady-state amplitudes of oscillations were obtained. The effect of dispersion caused by the difference in temperatures and phase velocities on the nonlinear interaction of standing waves is discussed. It is shown that the dependence of the speed of sound on frequency limits the transfer of energy up the spectrum, increasing the amplitudes of the first overtones.

Materials and methods. Using the method of slowly varying amplitudes, the system of mass, energy, and momentum conservation equations for both phases is reduced to a single nonlinear wave equation.

Results and discussion. Equations are obtained for determining the values of the established amplitudes of oscillations. The influence of dispersion caused by the non-coincidence of temperatures and velocities of gas-suspension phases on the nonlinear interaction of standing waves is investigated.

Conclusion. This article presents a study of the behavior of acoustic disturbances in a limited volume of burning gas suspension. The goal of the study was to obtain a single nonlinear wave equation that describes the evolution of pressure in the furnace. The analysis is based on the assumption that the effects of nonlinearity, dispersion, and non-conservation of oscillations on wave amplitudes are negligible. This assumption allows us to use the method of decomposition of the solution into eigenmodes of the linear conservative problem to solve the obtained equation. The decomposition procedure reduces the original wave equation to an infinite system of ordinary differential equations for complex amplitudes. Within this approach, the values of the steady-state amplitudes of the standing waves were found, which represents an important contribution to the understanding of the dynamic processes in combustion systems. Thus, the study demonstrates a high degree of analytical rigor and mathematical accuracy, as well as a deep understanding of the fundamental principles.

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