J. Non-Equilib. Thermodyn. 2004  Vol. 29  pp. 209–220 J. Non-Equilib. Thermodyn.  2004  Vol. 29  No. 3 6 Copyright 2004 Walter de Gruyter  Berlin  New York. DOI 10.1515/JNETDY.2004.051 Non-linear phenomena in thermoacoustic engines Alejandro Rivera-Alvarez1,* and Farid Chejne2 1 Instituto de Energı ´a y Termodina ´ mica, Universidad Pontificia Bolivariana, Medellı ´n, AA 56006, Colombia and Ingenierı ´a Te ´ rmica Ltda., AA 80054, Medellı ´n, Colombia 2 Facultad de Minas, Universidad Nacional de Colombia, sede Medellı ´n, Colombia *Corresponding author (ajrivera@ingenieriatermica.com) Communicated by A. Bejan, Durham, USA Abstract A non-linear simplified model for a thermoacoustic engine is presented in this work, which assumes the form for the spatial dependence of the state variables and con- siders only e¤ects inside the stack. For such a model the limit cycle solution is found numerically using a shooting technique in the ‘T > ‘T crit region, where the station- ary, fixed-point, trivial solution is unstable. With this solution a bifurcation diagram for thermoacoustic phenomena is built. Through such diagram an analysis of the non-linear nature of the generated oscillations is conducted, focusing on the presence of harmonics and the higher-order time-average pressure. 1. Introduction The thermoacoustic phenomenon is an example of an oscillatory structure in a fluid system (thermoacoustic oscillations), which is being used currently by engineers to build useful devices that work as heat engines and refrigerators [1–3]. Thermoacous- tic engines are the pioneers of a new generation of engines named natural engines [4, 5], which use ‘natural’ structures in a complex system as the heart of its operation. In this paper we follow the so-called dynamical path to study such structures [6]. Most models developed for thermoacoustic engines have used the acoustic approxi- mation, which considers that the amplitude of the oscillations is small enough to ne- glect the second-order and higher-order terms in the corresponding equations. The use of this first-order model (linear model) does not take into account very important phenomena such as: harmonics generation, high amplitude behavior and transient stage. Some works dealing with non-linear models for thermoacoustic engines have been developed [7, 8], but much is still to be done in this direction. In the first phase of this work a stability analysis of a thermoacoustic engine was used to interpret the thermoacoustic phenomena [9, 10]. Furthermore, the path Brought to you by | Brown University Rockefeller Library Authenticated | 128.148.252.35 Download Date | 12/12/12 9:41 PM