Control Laboratory Experiments in ThermoAcoustics using the Rijke Tube Jonathan P. Epperlein, Bassam Bamieh and Karl J. Astr˚ om¨ Abstract— We report on experiments that investigate the dynamics. The advantage of the Rijke tube experiment is dynamics, identification and control of thermoacoustic phe- the ability to produce thermoacoustic instabilities without a nomena in a Rijke tube apparatus. These experiments are combustion process. Many of the identification and feedback relatively simple to construct and conduct in a typical, well- equipped undergraduate controls laboratory, yet allow for control issues involved in combustion instabilities are present the exploration of rich and coupled acoustic and thermal in the Rijke tube experiment. Thus, this experiment provides dynamics, the associated thermoacoustic instabilities, and the an easily accessible platform within which one can explore use of acoustic feedback control for their stabilization. We the myriad issues relevant to thermoacoustic instabilities and describe the apparatus construction, investigation of thermoa- their control. coustic dynamics and instabilities in both open-loop and closed- loop configurations, closed-loop identification of the underlying The present paper aims at introducing the Rijke tube as an dynamics, as well as model validation. We also summarize experimental platform to explore thermoacoustic dynamics a transcendental transfer function analysis that explains the and their control. We present an empirical investigation — underlying phenomena. These experiments are notable for the which can be easily reproduced in a controls lab — of the fact that rich thermoacoustic phenomena can be analyzed using dynamics of the Rijke tube using closed-loop identification, introductory concepts such as the frequency response and root locus, and thus can be performed and understood by controls and standard linear techniques such as root locus and Nyquist students with relatively little background in acoustics or heat criterion. It is remarkable that one can obtain rather useful transfer. and predictive models of the system with this approach. A Rijke tube can be made out of a vertical, long, narrow I. INTRODUCTION and hollow tube, typically made out of glass (Pyrex) in our The Rijke tube is an experiment that is relatively simple case for ease of visualization. Figure 1(a) shows a basic and inexpensive to build in a typical university laboratory. diagram. A heating element (typically a resistive coil) is Despite its construction simplicity, it can serve to illustrate a placed towards the lower end of a vertical, open, hollow wide variety of mathematical modeling, empirical identifica- tube. If the coil is sufficiently hot, a steady upwards flow tion, verification and feedback control techniques. As such, it of air is achieved. An increase in the power to the coil is suitable for use in both advanced undergraduate and grad- z mic uate controls laboratory courses. The Rijke tube is perhaps the simplest illustration of the phenomena of thermoacoustic instabilities. These phenomena typically occur whenever heat Open glass is released into a gas in underdamped acoustic cavities. The tube heat release can be due to combustion or solid/gas conductive and convective heat transfer. Under the right conditions, the Controller coupling between the acoustic and heat release dynamics zo in the cavity becomes unstable. This instability manifests heating coil itself as a sustained limit cycle resulting in audible, powerful pressure oscillations. Thermoacoustic instability phenomena 0 are most often encountered in combustors [1], [2], [3], speaker w where the resulting powerful pressure waves are undesirable (a) (b) due to the danger of structural damage as well as perfor- Fig. 1. (a) The Rijke tube shown with a heating element placed towards mance degradations. In this context, they are often referred the bottom (suspension mechanism for coil not shown). Upward arrow to as combustion instabilities, and are notoriously difficult indicates steady air flow caused by the coil’s heat. (b) The Rijke tube to model due to the additional complexity of combustion with microphone, speaker and feedback controller. The external signal w is used for closed-loop identification. This research is partially supported by NSF grants ECCS-0937539 and CMMI-0626170. causes an increase in the air flow, and at some critical Jonathan Epperlein is with the Department of Electrical and Computer value, the tube begins to emit a loud steady hum like Engineering, University of California, Santa Barbara, Santa Barbara, Cali- fornia 93117, USA. [email protected] a pipe organ. Proportional acoustic feedback as shown in Bassam Bamieh is with the Department of Mechanical Engineering, Figure 1(b) can make this hum disappear with an appropriate University of California, Santa Barbara, Santa Barbara, California 93117, setting of the gain. It is important to note that this is USA. [email protected] Karl J. Astr˚ om¨ is with the Department of Automatic Control, Lund not a noise cancellation scenario, but rather a stabilization University, Lund, Sweden. [email protected] problem, in that the acoustic feedback actually stabilizes the underlying thermoacoustic instability that generates the collecting data, would do. A photograph and a diagram of sound. Distinguishing between these two settings is one part this particular arrangement is shown in Figure 2. The glass of the experimental investigation. This paper is organized as follows. We first describe the basic aspects of the apparatus construction, which though relatively simple, requires some careful attention to cer- Microphone tain parameters so as to obtain an easily humming tube. PC Section III contains the basic initial observations of the with thermoacoustic instabilities in open loop, and with stabilizing DAQ Board as well as further destabilizing feedback gains. Some of the elementary acoustic physics is diagrammatically illustrated. Section IV describes the procedure and typical results of frequency-domain closed-loop empirical identification. This Coil produces an open-loop plant transfer function that can be used for model validation. Though this transfer function has Audio Amplifier been arrived at without any underlying physical modeling, it is predictive in that it explains the initial thermoacoustic instability, the proportional feedback stabilization, as well as Power Supply Speaker the high gain instabilities of the controlled system. This is done in Section V using frequency responses and root loci of the identified open-loop model. The open-loop poles of the Fig. 2. Photograph and diagram of the Rijke tube experimental apparatus. system as well as its right half plane zeros play an important role in this analysis. Finally, we include a short analysis sec- tube is vertically mounted to a rigid frame, with the heater tion (Section VI) in which a transcendental transfer function coil mounted about 1/4 of the way up from the bottom of with an infinite number of poles (representing acoustics) is the tube. The power supply is used to heat the coil. The analyzed in feedback with heat release dynamics. A root microphone is mounted on top and in the center of the tube. locus analysis shows clearly how the coupling between The microphone signal (AC coupled) is fed via the DAQ acoustics and convective heat release is the underlying cause board to Simulink, where it is recorded and multiplied with of the thermoacoustic instability. Although the analytical the variable gain. The interrogation signal (see Figure 3) is derivations of these transfer functions are beyond the scope also added there. The generated signal is then routed from of the present paper, it is included to illustrate the tight the DAQ board to the audio amplifier and to the speaker. correspondence between models derived from the underlying Our working assumption is that the the power and pre- physics, and those obtained from empirical identification. interrogation power amp speaker mic pre-amp signal II. CONSTRUCTION OF THE RIJKE TUBE APPARATUS Rijke w We describe here the particular hardware configuration Tube used in the controls laboratory at the University of Cal- ifornia at Santa Barbara (UCSB). Earlier versions of this experiment have appeared elsewhere, namely in [4] where it Fig. 3. Equivalent block diagram of the closed-loop identification setup. was specifically used in a controls laboratory. Details of our amplifier, as well as the microphone and speaker can all basic set up can be easily modified according to the user’s be described by pure proportional gains. In reality, they particular laboratory facilities. Our basic Rijke tube apparatus each have their characteristic frequency response which may used for this experiment is composed of the following main not be flat. However, in the frequency range of interest in components: this experiment (typically 50-1000 Hz) where the Rijke tube Pyrex R Glass tube, length = 4ft, internal diameter • acoustic dynamics are dominant, we take these components 3in (A very high aspect ratio is necessary to achieve to be pure gains and regard the Rijke tube system with the thermoacoustic instability with only moderate heater this acoustic feedback as reasonably well modeled by the power.), conceptual diagram shown in Figure 3. Heater coil made from 24 gauge NiCr wire, • Simple clip-on microphone (with built-in preamplifier), III. EMPIRICAL INVESTIGATION OF THE RIJKE TUBE • Audio amplifier, • The experimental exploration of the Rijke tube begins