Mathematical Modeling of Self-Oscillations in a Rijke’s Tube with Variable Heat Flow Power

Boris Basok

doi:10.11648/j.ajmie.20170201.18

Abstract

In this paper the mathematical model of self - oscillation in Rijke's tube is found. We introduce the characteristic of the pressure of the heat supply. Using the energy equation in the form of the first law of thermodynamics to flow defined mechanisms of thermoacoustic instability in this problem. Using the pressure characteristic of the supply of heat and the classical Lyapunov’s theory of stability defines the conditions for self-excitation of oscillation. It was found that when the increasing combustion delay the harmonic self-oscillations of the "singing" flame are converted to the relaxation oscillations.

Highlights

Small-amplitude thermoacoustic self-oscillations of oscillating combustion improve the economic and performance indices of combustion chambers of industrial units
Its phenomenon lies in excitation of self-oscillations by heat supply from an internal heat source located in the lower part of a vertical tube that is open at both ends on formation of direct motion in the tube
Apart from the phenomenological delay of the process of combustion, negative thermal resistance contributes to the formation of the "singing" flame mechanisms, which, under the corresponding conditions [4], can generate self-oscillations in the Rijke’s tube when a burner is used for supply of heat instead of the electric coil

Summary

Introduction

Small-amplitude thermoacoustic self-oscillations of oscillating (pulsating) combustion improve the economic and performance indices of combustion chambers of industrial units. W of the heat flux is constant, which decreases its viscosity This is responsible for the reduction in the hydraulic loss along the length with growth in the flow rate Q in the laminar regime and for the formation of the descending branch hfr (Q) of the negative hydraulic resistance. Negative thermal resistance is responsible for the excitation of self-oscillations for a variable (flow-ratedependent) power W of the heat flux. Apart from the phenomenological delay of the process of combustion, negative thermal resistance contributes to the formation of the "singing" flame mechanisms, which, under the corresponding conditions [4], can generate self-oscillations in the Rijke’s tube when a burner is used for supply of heat instead of the electric coil. Rijke’s phenomenon, they are reliable and can be modeled mathematically but are very limited [5]

Formulation of the Problem

Mechanisms of Thermoacoustic

Conditions of Excitation of Vibrational Combustion

The Results of Mathematical Modeling

Conclusions