This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 152.3.102.242

This content was downloaded on 11/11/2015 at 03:00

Please note that terms and conditions apply.

Rijke-type thermoacoustic oscillations

View the table of contents for this issue, or go to the journal homepage for more

2011 Eur. J. Phys. 32 305

(http://iopscience.iop.org/0143-0807/32/2/005)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 32 (2011) 305–327 doi:10.1088/0143-0807/32/2/005

Rijke-type thermoacoustic oscillations

Tamas Beke

Our Lady Catholic Institute, Kalocsa, Hungary

E-mail: tamasbe@gmail.com

Received 3 November 2010Published 5 January 2011Online at stacks.iop.org/EJP/32/305

AbstractThermoacoustic instability can appear in any thermal device when the unsteadyheat transfer is favourably coupled with the fluctuations of acoustic pressure.In this paper, we present a project type of physical measuring and modellingtask; the aim of our project is to help our students increase their knowledgeof thermoacoustics. Our paper proposes several experiments and describessome tools’ setups that are easy to obtain and work with. Free software isoffered to analyse the signals with a personal computer. In this paper, the basisof standing wave theory and the tie between thermodynamics and acousticaloscillations are also discussed; some devices and technical applications ofthermoacoustic oscillations are presented. The objective of this paper is topresent the theory of frequency shifting of thermoacoustic oscillations as well.The frequencies of the acoustic modes in the excited state are of interest forpractical purposes; the differences between the calculated and the measuredvalues of these frequencies are shown. The behaviour of the properties of theexited modes shows the complexity of the real thermoacoustic systems; themathematical modelling intended to simulate the effect of frequency shiftingis observed in tests. We think that these experiments can be implemented inphysics courses on thermodynamics for graduates or specialized courses forundergraduates.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Our school launched a project in 2007 with the aim of observing and analysing thermoacousticphenomena in the Rijke tube. A series of experiments were carried out first with gas-heatedRijke tubes and then with electrically heated pipes to determine the behaviour of the transitionto instability and to model the excited regimes of operation. Here, I would like to presentsome of our thermoacoustic project experiences. Our students were invited to volunteer in thisproject. We carried out the measurements in the afternoons in extracurricular physics classes.We executed the experiments in teams and each team consisted of three to six students.

0143-0807/11/020305+23$33.00 c© 2011 IOP Publishing Ltd Printed in the UK & the USA 305

306 T Beke

2. Thermoacoustic instability

Thermoacoustic instability can appear in any thermal device when the unsteady heat releaseis coupled with pressure perturbations. These thermoacoustic instabilities are important invarious technical applications: jet or rocket motors, thermoacoustic engines, pulse combustors,industrial burners, power generating gas turbines, etc.

Thermoacoustics study deals with the conditions under which thermal interaction canresult in the excitation of the acoustic modes of the system when unsteady heat release isconjoined with pressure perturbations. When a gas flame or another heat source is enclosedin a cavity, the potential for heat-driven oscillations exists (Beke 2010a). Every chamberof a combustor possesses certain acoustic properties: if the heat released in the combustordepends on the velocity and pressure fluctuations, then a feedback loop is established whichcan destabilize the system. When the heat release has a component in phase with the pressurefluctuation, then according to Rayleigh’s criterion, acoustic oscillations can be enhanced; thiseffect results in the excitation of eigenacoustic modes of the system (Rayleigh 1896).

The Rijke tube is the simplest device for studying thermoacoustic instability. It is aresonator cavity with natural or forced mean flow and a concentrated heat source (heatedwire gauze). It is a convenient thermodynamic system for studying the fundamental physicsof thermoacoustic phenomena and thermoacoustic instabilities, since at certain values of themain system parameters a loud sound can be generated, similar to that in real-world physical(industrial) devices prone to thermoacoustic instability.

The mathematical model, presented in this paper for the thermoacoustic oscillations, forpreliminary design and analysis of real-world thermal devices is recommended; so it canbe useful for undergraduate and graduate students, who are dealing with the research theme,where thermoacoustic instability is a concern. Our main goals are to obtain accurate frequencydata in the excited regimes and the scope of the unstable state, and to develop a theory thatexplains and predicts the effects of frequency shifting observed during the tests; thus, wehope that the demonstrated method and this essay can be profitable for students of physics atuniversity level.

3. The Rijke tube

The study of the thermoacoustic effect has a long and interesting history. One of the mostsimple thermoacoustic devices is the Rijke tube, named after Petrus Leonardus Rijke. Thistube is a simple pipe, with both ends open and a heat source placed inside; the heat sourcemay be a gas flame or an electrical resistance heating (see figure 1).

If the tube is positioned vertically and the heat source is introduced into the tube frombelow, depending on the position of the heat source within the pipe, the tube can emit sound.Rijke heated the gauze by gas flame in the lower half of the tube until it became red hot,then the flame was withdrawn and the pipe emitted an audible sound for a few seconds (Rijke1859). He also observed that a sound could be sustained for an indefinite period in the tubeusing an electrically heated mesh. Rijke found that instead of driving, damping occurred if thegauze was positioned in the upper half of the tube. When the pipe was turned horizontally, theoscillation ceased. The Rijke tube is one of the thermoacoustic devices which have the abilityto generate temperature differences using acoustic oscillations; it converts heat into sound bycreating a self-amplifying standing wave. If the tube emits sound, we can say that the systemis in an unstable (excited) state. If there is no sound emission, then the system is in a stablestate.

Rijke-type thermoacoustic oscillations 307

Figure 1. Vertical Rijke tubes with different heating.

The earliest explanation of the sounding mechanism was provided by Rijke himself. Hesuggested that the hot gauze transferred heat to the adjacent volume of air in the tube, whichthen expanded, became less dense and started rising up in the tube, thus setting up a meanupward flow of air in the tube (Sarpotdar et al 2003). The rising volume of air on comingin contact with the cooler walls of the upper half of the tube subsequently contracted andbecame more dense, thereby setting up a variation in density along the length of the pipe.According to Rijke, the resulting variation in pressure was such that fluid elements in thelower half of the tube always experienced expansion, while those in the upper part of thepipe always underwent compression. The acceleration and deceleration of the gas moleculeswere sinusoidal; the result was a self-sustained series of longitudinal sinusoidal air pressureoscillations (Fahey and Timbie 2006).

The flow of air past the grid is a combination of two motions: there is a uniform upwardmotion of air due to a convection current resulting from the gauze heating up the air; andsuperimposed on this is the motion due to the sound wave. For one half of the vibration cycle,the air flows into the tube from both ends until the pressure reaches a maximum. During the

308 T Beke

other half cycle, the flow of air is outwards until the minimum pressure is reached. All airflowing past the mesh is heated to the temperature of the gauze and any transfer of heat tothe air will increase its pressure according to the gas law, so reinforcing the vibration; so thiseffect is called a ‘sonically induced heat gradient’. During the other half cycle, when thepressure is decreasing, the air above the gauze is forced downwards past the gauze again; forit is already hot, no pressure change due to the reseau takes place, since there is no transfer ofheat. The sound wave is therefore reinforced once every vibration cycle, and it quickly buildsup to very large amplitude.

Stationary acoustic waves in tubes can easily be set up by any source of energy, but oncethe source of energy is discontinued, the acoustic waves usually damp out due to friction in thetube and energy lost at the open ends of the pipe. The role of the energy source in a soundingRijke tube is not merely to excite acoustic waves in the tube but also to build up and sustainthe already excited acoustic waves (Sarpotdar et al 2003).

4. Practical applications of the thermoacoustic phenomena

Petrus Leonardus Rijke was a professor of physics at Leiden University in the Netherlands.In 1859, he discovered a way of using heat to sustain a sound in a tube open at both ends.Inside the pipe, a quarter from the bottom, he placed a wire grid and heated the mesh byflame until it was glowing red hot. After removing the flame, he obtained a sound from thepipe which lasted until the grid cooled down. Rijke received complaints from his universitycolleagues because the sound was loud and could easily be heard three rooms away from hislaboratory. Rijke’s original interest in the phenomenon appears to have been from the pointof view of musical acoustics. Instead of heating the reseau with a flame, Rijke also triedelectrical heating. Making the grid with electrical resistance wire caused it to glow red whena sufficiently large current was passed. With the heat being continuously supplied, the soundwas also continuous and rather loud. However, the response of the tube did not satisfy Rijke’srequirement for musical acoustics.

The thermo-acoustically unstable state plays an important role in various technicalapplications. Important examples of thermoacoustic phenomena are the acoustically unstablecombustors: gas turbine engines and jet engines are susceptible to combustion instability,which is a special case of thermoacoustic instability. Pressure and flow oscillations inside theengine can lead to unacceptable levels of vibration and enhanced heat transfer that degradepropulsive efficiency, or even destroy the system. Both solid and liquid fuelled propulsionsystems can be prone to instability. These acoustic waves result in a loud, annoying sound(called a screech or buzz) and can also cause structural damage to the combustion chamber.(The combustion process in jet and rocket engines involves very high power densities of theorder of GW m−3, and a very small fraction of this energy is more than adequate to excite andsustain acoustic waves inside the combustion chamber.) The need to control thermoacousticphenomena in jet and rocket combustion chambers led to renewed interest in Rijke pipethermoacoustics (Sarpotdar et al 2003).

One of the most objectionable constituents of jet engine emissions is NOx. Nitrous oxideemission in combustion processes is proportional to the temperature. It is found that a high ratioof air to fuel in the form of lean, premixed and prevapourized (LPP) flame keeps the temperatureof the combustor within acceptable limits. However, LPP combustors, beyond a critical fuel-air ratio, tend to show low-frequency (50–150 Hz) longitudinal acoustic instabilities, knownas ‘buzz’, which can cause serious structural damage. The Rijke tube, which also showsheat-induced longitudinal acoustic instability, provides a convenient prototypical system forstudying this ‘buzz’ phenomenon in the physics laboratory (Sarpotdar et al 2003).

Rijke-type thermoacoustic oscillations 309

Combustion instabilities are not always negative phenomena: an active research erawitnessed the development of pulse combustors and thermoacoustic engines that can be usedas prime movers, refrigerators, etc. In conventional steady burners, several problems usuallyarise with heavy oil fuels: incomplete combustion, soot emission, long flames, and walldeposits; so specially designed Rijke-type pulse combustors use the thermoacoustic effect toincrease the burning rate of heavy fuel oils. Another industrial application where the Rijke-type oscillation is used is in pulse combustors and coal bed combustors. In a pulse combustor,the acoustic oscillations help to mix the fuel and air better, thus improving the efficiency ofcombustion. In coal bed combustors, almost 70% of the fine ash particles can escape throughthe filtering process, but the intense acoustic energy increases the collision rate between theseparticles, and these particles can be coalesced and the greater particles can then be effectivelyremoved by conventional ash removing methods. In these cases, the acoustic oscillationsare actually desirable and experiments with a Rijke pipe help to determine configurations forwhich the thermoacoustic effects in these combustors can be maximally exploited (Sarpotdaret al 2003).

Sound waves are generated when a sufficient temperature gradient is created inside thetube and the resulting acoustic energy can be converted to electricity or applied for otherpurposes. The thermoacoustic processes are a widely studied research area today, and theyhold many potential applications in the power, appliance, and space industries. Presumablythermoacoustic engines will be preferred in future because they are mechanically simple andenvironmentally friendly devices, and they will be used in hybrid electric vehicles, solartechnology, air conditioning, liquefaction of industrial gases, and specialized applications inspace technology.

5. The thermoacoustic school project

The simplest Rijke tube set up consists of a pipe and a gas-heated wire grid. In our experiments,this type of Rijke tube was employed first. In the course of the measurements, we were workingwith seven different tubes. We need the following apparatus in experiments:

• Rijke tubes with a support (we used aluminium, copper and glass tubes as well, becauseit is easy to demonstrate the position of wire mesh with these).

• Wire gauzes which match the tubes with given diameters, and with different grating—sothat this could be analysed as well.

• A gas burner (with controllable efficiency).• An electrical power source.• A thermometer, with a range of 200–1000 ◦C.• An air blower (e.g. vacuum cleaner).• For the cooling experiments, liquid nitrogen.• A tool for recording and measuring sound, e.g. a microphone and a computer with a sound

card.• A sound level meter, which we needed only for the quantitative results, otherwise the

experiment could be done without it too. We used a noise level meter 322 Datalog type.(Fortunately we do not necessarily need this tool, eventually the microphone and thecomputer will do so that we can ascertain the relative loudnesses—of course, in this casewe cannot count directly with numerical values.) The Sound Level Meter 322 measuresthe sound level impacting the built-in electret condenser; it is suitable as a test instrumentfor laboratory use or for scientific purposes to measure sound levels in a general fieldtesting up to an intensity of 130 dB. The unit has a built-in data logger function, i.e. it

310 T Beke

Rijketube

gas heater

infrared

thermometer

microphone

sound level meter

Figure 2. Vertical aluminium Rijke pipe with measuring arrangement.

can automatically record the measured values at specific recording intervals. The storedvalues can be read via software immediately or later. The unit has two weighting filters:

(1) Weighting per A curve enables the unit to evaluate frequencies as the human ear does,which perceives loudness differently in different frequency ranges. (The A weightingis selected for general measurements, for example for measurements of the workingenvironment.)

(2) Weighting per C curve enables a measurement without corresponding lift orattenuation in specific frequency ranges. The C weighting is used for measurementsfor determining the loudness of machines and engines, for example; so we chose theC curve for our experiments.

During the measurement, the meter was fixed to avoid vibration and movement, andthe unit was placed 10 cm sideways from the Rijke tube, in order to avoid operationunder unfavourable ambient conditions, since the excessive ambient temperatures wouldlead to damage to the sensitive electronics within the sensor. While we were carrying outmeasurements, we always aligned the microphone precisely towards the sound source to bemeasured, and we took care that no objects came between the microphone and the soundsource. In figure 2, the given measuring arrangements can be seen.

This apparatus can be found in every physics laboratory in schools or obtained at low cost;hence, this project does not require extra expenditure from the school, which corresponds withthe statements of the Rocard report (2007). Each of our student teams examined the behaviourof different tubes and shared some of the equipment (e.g. infrared thermometer). In addition,we had to be careful not to disturb the other team’s work with the Rijke pipe. We repeated

Rijke-type thermoacoustic oscillations 311

every measurement five times and calculated the average values. We recorded the results oncomputers (Beke 2009a, 2009b, 2009c).

In the course of measurements, we were looking for the answers to the following questions:

• How do the length of the tube and the position of the heat source influence the formationof sound and the frequency issued?

• How does altering the heat input influence the sound?• How much does the sound depend on the temperature that can be measured inside the

tube and the temperature of the reseau?• Can sound arise in the course of cooling?• How does the position of the tube influence the formation of sound?• If there is a separate air current in the tube as well, how does that influence sound emission?

5.1. The course of measurement

5.1.1. Measurement of temperature. We used two infrared thermometers. The IR-380 andIR-1000L infrared thermometers are precise non-contact measurement units with a built-inlaser pointer; they measure the surface temperature of an object. The unit’s optics senseemitted, reflected, and transmitted energy, which is collected and focused onto a detector.

5.1.2. The definition of the intensity of perfused air. In some of the experiments, we ensureda separate air current with the help of a vacuum cleaner. To characterize the device; we blewup a big (approx. 50 l) plastic bag then exhausted a small amount of air with the vacuumcleaner. We got the sucking power (�m/�t) (i.e. the mass flow rate or mass flow intensity)from the gas law,

�m = p · M · �V

R · T, (1)

where T is the temperature in Kelvin, and p is the pressure of air (it was about 105 Pa). Wesupposed that the pressure was almost constant, since we evacuated only a little air; that is, itwas enough to measure only the volume change of the bag (Beke 2009a).

5.1.3. Transmissivity of the grid. We did not have standard resistance heating wire mesh(e.g. nichrome), but we found that stainless steel mesh was suitable in terms of longevity andoxidation resistance. The next step was the definition of the transmissivity of the wire meshput into the Rijke tube. Our first idea was to attach a vacuum cleaner to the tube and measurethe volume of air flowing during a given time—first while the wire netting was in the tube,and then while there was no wire mesh in it. The ratio of the two measurements gives thetransmissivity of the wire grid. Unfortunately, this method did not work properly in practice ifthe transmissivity was high, since there was no measurable difference between the two cases;therefore, we chose another method. We made a digital photo of the mesh in super macromode, then measured the size of one empty square and counted the number of such squares inthe tube (taking the deformed squares on the brink as half) to find the empty area. The ratio ofthe total empty territory and the cross-section of the tube gave the transmissivity of the wiregrid (figure 3). The transmissivity of the ten different meshes were about 10%, 20%, 35%,50%, 60%, 70%, 80%, 85%, 90%, and 95%, respectively. (The three densest gauzes weremade from the metal disk drilled through round in some places.)

Measurements with all of the ten wire meshes showed significant differences. The densityof the grid is an important factor because the effective heat transfer from the gauze to the upwardair convection is proportional to the surface area of the grid exposed. If the mesh is more

312 T Beke

Figure 3. Grids with different transmissivity.

dense, it will provide more surface area for heat transfer, but it impedes the air flow throughthe grid too. In the case of 10% and 20% transmissivity, the meshes were not suitable togenerate sound while the air flow was too poor. A rare mesh provides less heat transfer area,but it is less restrictive to air convection. In the case of 95% transmissivity, the mesh was notsatisfactory to generate sound while the surface area of the gauze was too little.

For the reseau material, a variety of materials (copper, steel and stainless steel) were tried.We found that the material of the wire grid counts too; the stainless steel reseau with 80%density seemed to work best. (We had to change the other meshes, which were made of hardersteel, a lot more often.) Upon repeated heating, the wire mesh becomes excessively brittleand delicate to handle. If the power of heating was too high, then of course any mesh wouldmelt and this approach became unusable for the experiments. Consequently, it is practical touse some sort of flexible wire gauze with a higher melting point, and the density of the meshis therefore proposed between 50% and 90%.

The Rijke tube may be regarded as a ‘thermoacoustic pump’ in which the pumping-liketemperature oscillations raise the acoustic energy to audible levels. We found that oscillationsreach maximum amplification when the grid is placed in the bottom half about a quarter ofthe way up the pipe; in this case the pumping-like motions of acoustic pressure and velocityoccur in the same direction. If the mesh is positioned in the upper half of the pipe, theacoustic velocity and pressure exhibit unfavourable phases. We examined the sound emissionof the tube in the case of different burning powers as well. In this case, the position andtransmissivity of the mesh did not change. A minimum threshold value is found for the heatpower supplied, below which no self-sustained acoustic oscillations may be possible. Thereis an optimal point (an interval) in the power and duration of warming as well, at whichthe sound issued by the tube can be heard for the longest time; there is an ideal position for themesh as well. It is found that, as mentioned in many research papers, this optimum positionof the mesh is approximately 1/4 of the total length of the tube measured from the bottom.This position produces the strongest resonance for the least amount of energy input.

5.2. Cooling experiment

We hoped that we could also observe the Rijke sound phenomenon by putting a cooling systemin the upper half of the tube, as if it were the ‘negative pair’ of the heat source in the lowerpart; ‘it is not difficult to believe that the Rijke phenomenon can be equally well observed

Rijke-type thermoacoustic oscillations 313

by placing a cooling device in the upper half of the tube, and this has been demonstrated bysome researchers’ (Sarpotdar et al 2003). This ‘reverse’ Rijke effect—where a Rijke pipewill also produce audio oscillations if hot air flows through a cold mesh—was first observedby Rijke’s assistant Johannes Bosscha in 1859 too. A similar phenomenon is the Taconisoscillation; when the open end of a gas-filled tube was immersed in liquid nitrogen and cooledto a cryogenic temperature, and later the tube was removed from the coolant, it began to vibrateand loud sound appeared. We were trying to cool down the gauze in the upper half of thetube in liquid nitrogen, and then set the tube back into the vertical position, but unfortunatelywe could not hear any sound (Beke 2009a). We performed the cooling attempt in anotherway, too: we also created a separate air current in the tube (we sucked the air from thetube with a vacuum cleaner), but we did not experience any sound effect. The explanationfor this may be that between the temperature of the wire mesh and the tube (the end of thepipe) there should be a big enough difference so that the tube issues a sound. As a matterof fact, we did not manage to reach the lower power threshold considering the fluctuation ofheat transfer, since the necessary temperature gradient did not exist between the reseau andthe nose pipe. So we decided that we would warm the grid with a gas flame parallel to thecooling tube. In this case, the system became unstable and we got sound. The minimumthreshold heating power was lower in this case as if there was no cooling, and the temperaturegradient was already sufficient between the mesh and the end of the tube to generatesound.

5.3. Horizontal and slanting tubes

If we turn the tube to a horizontal position, and then we try to heat the wire mesh, normally(without a separate air current) there will be no sound issued, since the air does not have anatural convectional flow. We can say that the ‘chimney affection’ does not work, and becauseof this, pressure fluctuation will not evolve either. At this time, neither the mesh position northe burning power input matters—the tube will not issue any sound. The situation is differentwhen we ensure another air current in the pipe with the help of a vacuum cleaner. In thiscase, there will be a sound; moreover, it will be similar to that issued in a tube which is setvertically. The mesh’s optimal position is about xg = L/4 too. We experienced that in thecase of a horizontal tube as well, even with an optimal wire mesh position, the upper thresholdeffect is observable. When we increased the power input while the air current intensity wasat a maximal level, above a certain value the Rijke tube did not issue any sound. This upperthreshold power depended on which tube we used (Beke 2009a).

If the tube is placed at a different angle (α) with the horizontal, the system may be stableor unstable depending on the extra mass air flow rate and the heating power. If the angleis greater than 60◦, then the tube can sound without a separate air flow, when the heatingpower is sufficient. We tried to regulate the sucking power of the vacuum cleaner to determinethe minimal air current intensity from which sound arises in the Rijke tube and the highestpossible air current intensity that will still result in a sound. Unfortunately, we could notdefine the exact power values as the gas burner’s power significantly changed because of thejet’s slanting position and because we were sucking the air out of the pipe with the help of avacuum cleaner. The graph in figure 4 shows the minimum flow of extra air required to issuesound depending on the angle.

To sum up, we can tell that even in the case of horizontal and slating pipes, both the lowerand the upper threshold are apparent in the sound emission. This is dependent on the power ofthe gas burner, the angle with the horizontal, the intensity of the air flowing through the tube,and the position of the wire mesh.

314 T Beke

Mass flow rate (g/s)

00,10,20,30,40,50,60,70,8

0 10 20 30 40 50 60 70 80 90Angle (degree)

Mass flow rate(g/s)

Figure 4. Minimum extra mass air flow rate demand to sound emission, depending on the angle(α) with the horizontal. (The 470 mm long copper Rijke tube was placed at a slant, and the heatpower was about 320 W.)

6. Thermoacoustic oscillation in the Rijke tube

A standing (stationary) wave is a wave that remains in a constant position; this phenomenoncan arise in a stationary medium as a result of interference between two waves travelling inopposite directions. The length of the Rijke pipe was about 10–15 times its diameter in ourexperiments, so the acoustic waves can be regarded as one dimensional. Standing waves areproduced as a result of the repeated interference of two waves of identical frequency whilemoving in opposite directions with equal amplitudes along the air column inside the Rijke tube.This results in nodes of zero displacement spaced every one-half wavelength, and antinodesof maximum displacement located halfway between the nodes. The number of nodes withinthe tube depends on the harmonic of the standing wave and the boundary conditions. In a tubewith two open ends (Rijke tube), the boundary conditions dictate maximum displacements atthe open ends (Fahey and Timbie 2006). The frequency is dependent on the harmonic number,the velocity of sound in the medium, and the pipe length,

f = vs

λ= vs

2 · L· n, (2)

and n = 1, 2, 3, . . . where L is the length of the tube, and n is the harmonic number; the firstharmonic (n = 1) is defined as the fundamental harmonic. As one would expect, increasingthe length of the tube increases the wavelength at the fundamental harmonic, and thus equatesto a lower frequency (for constant velocity of sound in the gas). This phenomenon can easilybe demonstrated with the help of three glass tubes (see figure 5).

As a matter of fact, the sound speed (vs) is not constant, since the temperature (T) and thepressure are changing all the time as well,

vs(T ) =√(

cp

cv

− 1

)· cp · T , (3)

where cp and cV are the specific heats by the constant pressure and constant volume case,respectively. The frequency spectrum of the issued sound was examined with the help of theAudacity 1.3 Beta (freeware) program, while the temperature was measured inside the tube.The program visualized the relative spectrum of the recorded sound graphically, which can beseen in figure 6.

Analysing the graphs, it is found that the most powerful sounding frequency of eachtube is the fundamental harmonic. In the case of the shortest glass tube (0.2 m long),

Rijke-type thermoacoustic oscillations 315

Figure 5. (a) Gas-heated wire mesh in a glass Rijke tube; (b) experiments with three glass tubeswere carried out to demonstrate that increasing the length of the tube increases the wavelength atthe fundamental harmonic.

(a) (b) (c)

Figure 6. Relative spectrum of the recorded sound of the Rijke tubes: (a) 0.2 m long tube;(b) 0.4 m long tube; (c) 0.8 m long tube.

the measured frequency was 902 Hz, while the calculated value was 998 Hz—if we usevs(Tavg = 400 K) = 399 m s−1. In the case of the 0.4 m long tube, it was 513 Hz, which is notfar from the theoretical fundamental harmonic (487 Hz) frequency, if we use vs(Tavg = 380 K) =389 m s−1, according to equation (3). This fundamental harmonic measured value was 233 Hz

316 T Beke

in the case of a 0.8 m long tube, while its calculated value was 237 Hz—if we used vs(Tavg =360 K) = 379 m s−1. It can be seen that the measured and calculated fundamental frequencyvalues are not far from each other. The reason for the excursion in the case of the smallesttube may be that this pipe become hot sooner, and that significantly changed the sound speed.While analysing the illustrations, it is also found that besides the fundamental harmonic, thefirst few harmonics sound with the highest intensity, and the intensity of the higher harmonics(n > 4) is less significant (Beke 2009c).

We ascertained that the simplified frequency model generally overestimated the testresults. In the case of longer tubes at intermediate air flow rate range, the error is about3%, and in the case of shorter tubes and high air flow rate range, the error is about 5–10%.We know that the large quantitative errors are not acceptable for the real-world practical(industrial) application of this simplified model. The sources of errors must be, for example,due to the ignorance of the unsteady temperature inside the system. Our next goal was todevelop an advanced theoretical model based on the simple model that would describe thefrequency of the thermoacoustic unstable state more accurately than the previous simplifiedmodel.

7. Electrically heated Rijke tube apparatus

We did not find the results of the project work with gas-heated Rijke tubes satisfactory, so wedecided to build an electrically heated Rijke apparatus to get precise data. We carried out the‘electrically heated Rijke tube’ project in extracurricular physics classes with science-mindedvolunteer students. We have only one Rijke tube, and it was observed by three teams; eachteam consisted of four or five students and examined the tube alternately.

Our Rijke tube setup consists of a horizontal aluminium tube (the length is 1200 mm,the internal diameter is 72 mm and the external diameter is 78 mm) and electrically heatedwire grid inside the pipe, and we need to provide extra air flow through the tube, too. Thehorizontal orientation of the Rijke tube was implemented in order to exclude the influence ofnatural convection on the mean air flow rate. If the tube was kept vertically or at a slant, itwould be necessary to take into account a mean flow component caused by natural convection,and it would be difficult to provide low rates of mean flow at high heating powers. Themean air flow is provided by an industrial vacuum cleaner at the ‘upper’ tube end, whichsucks air through the pipe. This allows us to control the mean air flow rate, a major systemparameter, precisely and independently of the thermal power release, which is also regulated.A damping chamber, inserted between the Rijke pipe and the vacuum cleaner, is intended toprevent any interaction between the vacuum cleaner and the tube acoustics. The chamber is astrong cardboard box with dimensions of 450 mm × 450 mm × 500 mm (Beke 2010b). Thestructure of the experimental apparatus employed in our project is shown in figure 7.

The mean air flow through the Rijke tube is provided by ETA 3404 industrial vacuumcleaner, with a maximum throughput of 0.0026 m3 s−1 (i.e. 3 g s−1) at standard atmosphericconditions. The power of the vacuum cleaner is electrically regulated and a choker is insertedso that we can control the amount of air drawn through the damping chamber and the Rijke tube.The temperature is measured with the help of IR-380 and IR-1000L infrared thermometers.

The wire grid is suspended on a ceramic support pipe in order to eliminate electric andreduce thermal contact with Rijke tube walls. The length of the ceramic tube is 65 mm, theinternal diameter is 52 mm, the external diameter is 71.5 mm and it is drilled through in someplaces. To connect the power source to the heater element, we applied two 1000 mm longcopper rods with diameters of 4.5 mm. The rods are threaded into the ceramic tube and are

Rijke-type thermoacoustic oscillations 317

air sucking

damping chamber Rijke tubepower supply

Figure 7. Our students performed the experiments with the electrically heated horizontally orientedRijke tube in teams.

Figure 8. Electrically heated wire mesh.

directly screwed to the wire mesh. The location of the heater can be easily changed within thetube (Beke 2010a, 2010b). The scheme of the heater assembly is shown in figure 8.

The density (porosity) of the meshes is proposed between 60% and 90%. The gauze with80% transmissivity is a flexible wire mesh with the highest melting point; it was the finest, asit could be heated generally about 10–20 times without impairment, so we decided to employthe wire mesh with 80% transmissivity for each measurement.

We have to obtain a suitable means for electrically heating the gauze. Resistance heatingis the simplest and the most cost effective method, so the power source is a Trakis Hetra101 SM welding machine, capable of producing over 100 amp of current; its nominal maximalpower is 4 kW.

The position of the heated grid (xg), mass air flow rate and heating power are the controlledparameters. The temperature of the mesh and the temperature profile along the tube weremeasured with infrared thermometers. The heating voltage and current were also metered, so

318 T Beke

the heating power was available for us. We decided that the horizontal position of the tube andthe transmissivity of the mesh would be constant throughout the experiments. There remainedfour basic parameters that could be controlled by the students: the heated grid position, massair flow rate, power supplied to the heater and time of heating.

In the experiments in each individual run, the mesh’s position and the mass air flowrate were fixed, and the heating power was varied slowly in both directions, increasing anddecreasing, in order to capture history-dependent properties of the thermoacoustic system andreach the critical power when the transition to instability occured. Following a change ofpower, the temperature field in the tube slowly responded. It was observed in the experimentsthat the critical power obtained at large steps of power increments may significantly differ (tolower values) from the critical power obtained at small steps. To avoid this early initiationof instability, known as ‘nonlinear triggering’, we need to obtain a critical power accurately;the power increments should be made as ‘small’ as possible; the step in power variation issuccessively reduced until a converged value of the critical power is obtained (Matveev 2003).Before commencing an experimental run, the tube was subjected to a ‘warm-up’ procedure,to minimize temperature variations as the power input was increased. This ‘warm-up’ processdepends on the actual conditions under which the experimental run takes place; the duration of‘warming-up’ is typically 1–6 min. When the transition to instability was studied, the powerwas driven slowly in order to obtain quasi-steady conditions. We can say that the duration of thewarming was not so important if we allowed sufficient time to ‘warm-up’ through quasi-steadystates. So we found that there are three important parameters. In steady state, correspondingto a given triplet of the system parameters (heater location, mass air flow rate, and heatingpower), it was determined whether the system was in the stable or excited state (Beke 2010a).

Our goal in the project was the investigation of the most important factors; thus, amathematical theory involving heat transfer, acoustics and thermoacoustic interactions isdiscussed to get the transition to the excited state. The general behaviour of the stability–instability boundary has been captured from this model. These results were presented inprevious articles (Beke 2010a, 2010b).

When the heated mesh is positioned in the ‘downstream’ half of the pipe (i.e. xg < L/2),the first acoustic mode is responsible for the unstable state of the system; the issued frequencyis about 154 Hz. It is found that there is an optimal point (interval) in the intermediate rangeof the mass air flow intensity where the power needed to excite a system is minimal. Whenthe heated grid is placed in the ‘upstream’ half of the pipe (i.e. xg > L/2), the system could beunstable at this condition too. However, in a different way than in the former case, when theheater is located in the ‘downstream’ part of the tube, the first acoustic mode is impossible atthe ‘upper’ gauze position. We were able to get the system excited with a frequency of about310 Hz; that is approximately twice the natural frequency perceived in the first acoustic mode.The most favourable grid position is found at about xg = 5L/8 experimentally, so the locationof the gauze was chosen to be 5/8 of the tube length in further tests. The method to obtainthe stability–instability boundary was similar to the measurements at the ‘downstream’ meshposition: for the fixed mass air flow rate, the power was increased until the excited regimewas attained.

8. Modelling of the frequency shift

Some shifts in frequencies were observed in the excited state during experiments. Our nextgoal in modelling was the investigation of the main system factors of this phenomenon.The finite-dimensional or Galerkin models have been widely used to study thermoacousticinteractions (Meirovitch 1967, Dowling 1995). For simplicity, we assume that these acoustic

Rijke-type thermoacoustic oscillations 319

Galerkin modes are linearly uncoupled, as shown by Culick (1976, 1997) and Culick et al 1995.The purpose of this assumption is to obtain closed form solutions for changes in frequency.Culick claimed that the linear coupling terms can be ignored for analytical and numericalconvenience. In this paper, using a simple model for the Rijke pipe, we will demonstratethat in the absence of ‘linear coupling’ in the Galerkin modes, there is a small shift in theeigenacoustic frequencies, similar to the former analysis of Kedia (2008).

We are going to use a coupled ordinary differential equation system for describing thethermoacoustic interactions in the Rijke tube. In this model, a perfect gas is assumed, the heatconduction is neglected, and the nonlinear gas dynamics terms are ignored, too. The one-dimensional linear equation systems for the pressure fluctuation (p′) and velocity fluctuation(u′) are, if the heat addition is not ignored,

ρ0∂u′

∂t+

∂p′

∂x= 0, (4)

and∂p′

∂t+ γp0

∂u′

∂x= (γ − 1)q ′, (5)

where γ is the gas constant, q′ is the fluctuation of heat addition rate per unit of volumeand ρ0 is the averaged density (Matveev 2003). Pressure and density are functions of thehorizontal coordinate x directed along the Rijke tube. The heat transferred from the gauzeto the surrounding air can be determined to be a function of mesh temperature, ambient airtemperature, source area, and average convection heat transfer coefficient. Since all propertiesare intimately related, the slightest disturbance in a given quantity is echoed in the signalsobtained from the remaining variables; the acoustically driven motion is established oncetemperature oscillations become self-sustained (Entezam et al 2002).

9. Results

Here we apply the results, and the details of the calculation are given in the appendix. Thelength of the electrically heated Rijke pipe is 1200 mm (L); the internal and external diametersof the tube are 72 and 78 mm, respectively. The issued frequencies in the experiments andthe calculated values at the heater locations xg = L/4 and xg = 5L/8 are shown in figure 9 forincreasing and decreasing heating power. If the heated mesh is located in the ‘upstream’ halfof the tube, the excitable mode is the second eigenacoustic mode of the Rijke pipe, and thefrequencies are about twice as high as that of the first mode. The frequencies increase withthe growth of the heating power, since the temperature and consequently the speed of soundincrease; on the reverse path, the frequencies decrease for the same reason. The frequencyvalues are higher during reverse power variation due to the higher temperatures caused bythe system thermal inertia (Matveev 2003). In the range of intermediate air flow rates, theresults are very close to experimental data; the tendency in the frequency behaviour in both the‘downstream’ and the ‘upstream’ part of the tube is modelled well, but the numerical valuesdiffer by about 1% from the measured ones (see figure 9).

This theory can be considered satisfactory for estimations of frequency components in thesignal after transition to instability (see figure 10): in case (a) for a heater location around xg

= L/4, the dominant frequencies are about 154 Hz, corresponding to the first acoustic modeof the Rijke pipe; in case (b) the heated grid is positioned around xg = 5L/8, the second modeis dominant, and the measured frequencies are within the 306–312 Hz interval.

We can see a great improvement achieved in the accuracy of the results by the advancedtheoretical frequency model compared to numerical results obtained by a simplified model.

320 T Beke

(a)

145

150

155

160

200 240 280 320 360 400

Power [W]

Fre

qu

ency

[H

z]Calculated frequency [Hz]Measured frequency [Hz]

(b)

145

150

155

160

200 240 280 320 360 400

Power [W]

Fre

qu

ency

[H

z]

Calculated frequency [Hz]Measured frequency [Hz]

(c)

305

310

315

320

325

600 620 640 660 680 700Power [W]

Fre

qu

ency

[H

z]

Calculated frequency [Hz]Measured frequency [Hz]

(d)

305

310

315

320

325

600 620 640 660 680 700Power [W]

Fre

qu

ency

[H

z]

Calculated frequency [Hz]Measured frequency [Hz]

Figure 9. Shift in the eigenacoustic frequencies of the Rijke tube: the dominant frequencies in thesignal, (a) xg = L/4, power is increasing; (b) xg = L/4, power is decreasing; (c) xg = 5L/8, poweris increasing; (d) xg = 5L/8, power is decreasing.

152

152,5

153

153,5

154

154,5

155

0,2 0,22 0,24 0,26 0,28 0,3Relatíve grid position

Fre

qu

ency

[H

z]

Calculated frequency [Hz]

Measured frequency [Hz]

306

307308

309

310

311312

313

0,6 0,62 0,64 0,66 0,68 0,7Relatíve grid position

Fre

qu

ency

[H

z]

Calculated frequency [Hz]

Measured frequency [Hz]

Figure 10. The calculated and the measured dominant frequencies in the signal after the transitioninto the unstable state. The relative heated grid position: (a) around xg/L = 1/4; (b) aroundxg/L = 5/8.

In the simplified model, the computed frequency differs 3–10% from that obtained duringthe experimental tests generally. Applying the advanced frequency model, the agreement ismuch better: in the intermediate mass air flow rate and heating power, the numerical results fitwith experimental data very well. In both low and high ranges of system parameters, where

Rijke-type thermoacoustic oscillations 321

assumptions for some parts of the modelling are not perfect, errors are larger, but we can statethat the advanced model predicts the results better than the simplified model.

10. Conclusions

In our project work, a series of experiments are carried out to determine the behaviour ofthe transition to instability and the excited regimes for Rijke tubes. This device consists ofa duct with a heated wire mesh inside. In the case of horizontal orientation, the air flowthrough the tube is provided by a vacuum cleaner. For some combinations of the main systemparameters (mean air flow rate, gauze location, and heating power), a high-intensity sound canbe generated in the pipe. Basically, we used relatively simple and cheap apparatus that can befound in every school laboratory, so the expense of the experiment could be minimized.

In this paper, some tools and experiments involving the basic principles of generation anddetection of Rijke-type thermoacoustic phenomena have been described. The same principlesare used in contemporary rocket and gas jet motors that convert heat to mechanical energy. Wecan say that such experiments are rarely demonstrated in high schools and university courses;but we hope that the use of the described simple and inexpensive tools and setups that areavailable at schools may change the situation.

We presented a mathematical model capable of predicting the shifting of frequenciesby transition to instability in the Rijke tube for specified boundary conditions. The modeldemonstrates good agreement with test results for the ranges of system parameters wherethe model assumptions are fulfilled. The behaviour of the dominant frequencies of the Rijkepipe was observed in the tests and explained in our model. The shift in the eigenacousticfrequencies demonstrates satisfactory agreement with experimental results for two positionsof the heated wire gauze (xg = L/4 and xg = 5L/8), corresponding to the different excitedmodes of the Rijke tube. This model is recommended for approximate analysis of the thermaldevices where thermoacoustic instability is present. For more accurate knowledge of theprocess in a real-world industrial system, a sophisticated CFD analysis must be carried out.

We hope that this paper meets the intellectual interests of college and university physicsteachers and students, and provides a deeper understanding of thermoacoustic phenomenataught at undergraduate and graduate levels. I would really suggest my colleagues to try these,or similar thermoacoustic experiments, because they are easily worked out but can producevery spectacular results.

Acknowledgments

This paper has been written as part of ‘Researches for the Development of the Education ofthe Secondary and Advanced Physics’ at the University of Szeged (SZTE), Faculty of Scienceand Informatics, Institute of Physics, Department of Experimental Physics. I would like tothank Dr Katalin Papp (the Director of studies of the research in SZTE) who helped me inwriting this paper by giving useful information and I would also like to specially thank theeditors and reviewers.

Appendix. Details of the calculation of the perturbed frequencies

Here we supply some details on the calculation with intermediate expressions that canbe useful to better understand the method and check the results. We are going to write

322 T Beke

equations (4) and (5) in a non-dimensional form, so we need the following non-dimensionalvariables.Mach number:

Ma = u0

vs

, (A.1)

gas constant:

γ = ρ0v2s

p0, (A.2)

non-dimensional horizontal coordinate:

x = 1

Lx, (A.3)

non-dimensional time:

t = vs

Lt, (A.4)

non-dimensional wave number:

k = Lk, (A.5)

non-dimensional velocity fluctuation:

u′ = 1

u0u′, (A.6)

non-dimensional pressure fluctuation:

p′ = 1

p0p′, (A.7)

non-dimensional fluctuation of heat addition rate per unit of volume:

q ′ = L

ρ0v3s

q ′, (A.8)

where L is the length of the Rijke tube. The subscript ‘0’ means the average value of the givenquantity. In our thermoacoustic system, the mean air flow is assumed to be steady, and theMach number is very small (Ma < 0.0018).

Equation (A.9) follows from substituting equations (A.3), (A.4), (A.6) and (A.7) intoequation (4),

ρ0u0∂u′Lvs

∂ t+

p0∂p′

L∂x= ρ0v

2s

p0

u0

vs

∂u′

∂t+

∂p′

∂x= 0. (A.9)

Equation (A.10) follows from substituting equations (A.1) and (A.2) into equation (A.9),

γ · Ma

∂u′

∂t+

∂p′

∂x= 0. (A.10)

Equation (A.11) follows from substituting equations (A.3), (A.4), (A.5), (A.6) and (A.7) intoequation (5),

p0∂p′Lvs

∂ t+ γp0

u0∂u′

L∂x= (γ − 1)

ρ0v3s

Lq ′. (A.11)

Equation (A.12) follows from substituting equations (A.1) and (A.2) into equation (A.11),

∂p′

∂t+ γ · Ma

∂u′

∂x= (γ − 1)γ · q ′. (A.12)

Rijke-type thermoacoustic oscillations 323

The partial differential equations (A.10) and (A.12) can be reduced to a set of ordinarydifferential equations using the Galerkin technique. The non-dimensional acoustic velocityand the non-dimensional acoustic pressure can be written in terms of the first N natural acousticmodes of the pipe, as shown by Dowling and Stow (2003),

u′(x, t ) = ε′(

N∑n=1

ˆun(t) · cos(knx)

)(A.13)

and

p′(x, t ) = −ε′(

N∑n=1

γ · Ma

kn

˙un(t) · sin(knx)

), (A.14)

where

kn = nπ (A.15)

is the non-dimensional wave number, ε′ is the order of the velocity perturbation over the meanflow and ˆun is the amplitude of the non-dimensional velocity in the nth mode.

We obtain equation (A.16) by substituting equations (A.13) and (A.14) intoequation (A.12),

−ε′ · γ · Ma ·(

N∑n=1

1

kn

· sin(knx)∂2 ˆun

∂t2

)− γ · Ma · ε′

(N∑

n=1

kn · sin(knx) · ˆun

)= γ (γ − 1)q ′.

(A.16)

Heckl (1990) proposed the following expression for the heat release rate,

q ′ = μ

{2Lw(Tg − T0)

S√

3

√πλacV ρ0u0rw

[√∣∣∣∣13 + u′(t − τ )

∣∣∣∣−√

1

3

]L−1δ(x − xg)

},

(A.17)

where Tg is the temperature of the heated grid, T0 is the mean temperature, S is the cross-sectional area of the tube, λa is the heat conductivity of air, Lw is the length of the heated wire(mesh), rw is the radius of the wire, cV is the specific heat capacity of air by constant volume,xg is the non-dimensional coordinate of gauze, τ is the non-dimensional time delay and μ isthe order of mean flow. The interaction between the heat transfer process and the mean flow isof the order of the mean flow μ; thus, the order of the oscillatory heat release is μ times overthe order of the perturbation ε′. Equation (A.18) follows from substituting equation (A.17)into equation (A.16), as shown by Kedia (2008),

d2 ˆun

dt2+ k2

nˆun = −μ

ε′

{2bkn

γMa

[√∣∣∣∣13 + u′(t − τ )

∣∣∣∣−√

1

3

]sin(knxg)

}, (A.18)

where the constant b is

b = γ (γ − 1)2Lw(Tg − T0)

ρ0v3s S

√3

√πλacV ρ0u0rw. (A.19)

The non-dimensional velocity fluctuation (u′) is a small value, so we can write√∣∣∣∣13 + u′(t − τ )

∣∣∣∣ ≈√

1

3+

√3

2u′(t − τ ). (A.20)

324 T Beke

We obtain equation (A.21) by substituting equation (A.20) into equation (A.18),

d2 ˆun

dt2+ k2

nˆun = − u

ε′

{2bkn

γMa

√3

2u′(t − τ ) sin(knxg)

}

= −μ

ε′ {cknu′g(t − τ ) sin(knxg)}, (A.21)

where the constant c is defined as

c = b√

3

γMa

. (A.22)

Equation (A.23) follows from substituting equation (A.13) into equation (A.21), and ignoringthe small non-dimensional time lag, as shown by Culick (1997) and Kedia (2008),

d2 ˆun

dt2+ k2

nˆun = −μ

⎧⎨⎩ckn sin(knxg)

N∑j=1

ˆuj cos(kj xg)

⎫⎬⎭ . (A.23)

The evolution of the individual Galerkin modes in equation (A.27) can be written (as shownby Culick (1997)), as⎛⎜⎜⎜⎜⎜⎜⎜⎝

d2

dt2 +(k2

1 + μA11)

μA12 μA13 · · · μA1N

μA21d2

dt2 +(k2

2 + μA22)

μA23 · · · μA2N

μA31 μA32d2

dt2 +(k2

3 + μA33) · · · μA3N

· · · · · · · · · · · · · · ·μAN1 μAN2 μAN3 · · · d2

dt2 +(k2N + μANN

)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

·

⎛⎜⎜⎜⎜⎝

ˆu1

ˆu2

ˆu3

· · ·ˆuN

⎞⎟⎟⎟⎟⎠ = 0, (A.24)

where

Aij = cki sin(ki xg) cos(kj xg), (A.25)

and the indices are i = 1, 2, . . . , N and j = 1, 2, . . . , N.We find out the eigenacoustic frequencies and the eigenacoustic modes by assuming that

the amplitudes have the same time dependence (Culick 1997, Kedia 2008),

ˆun = ˆun eλt , (A.26)

where ˆun is the initial value of the amplitude of the non-dimensional velocity in the nth modeand n = 1, 2, . . . , N; thus, it can be written as

d2 ˆun

dt2= λ2 ˆun eλt = λ2 ˆun. (A.27)

For simplicity, we examine only two mode expansion (N = 2). Equation (A.28) follows fromsubstituting equation (A.27) into equation (A.24),(

λ2 +(k2

1 + μA11)

μA12

μA21 λ2 +(k2

2 + μA22)) ·(

ˆu1

ˆu2

)= 0, (A.28)

i.e.

λ4 +[(

k21 + μA11

)+(k2

2 + μA22)]

λ2 +(k2

1 + μA11) · (k2

2 + μA22)− μ2A12A21 = 0. (A.29)

Rijke-type thermoacoustic oscillations 325

Equation (A.29) is a quadratic equation to λ2, and the two roots of this equation are

λ21,2 =

(k2

1 + μA11)

+(k2

2 + μA22)

−2

± 1

2

√((k2

1 + μA11)

+(k2

2 + μA22))2 − 4

((k2

1 + μA11)(

k22 + μA22

)− μ2A12A21)

=(k2

1 + μA11)

+(k2

2 + μA22)

−2±(k2

1 + μA11)− (k2

2 + μA22)

2

×√√√√1 +

4μ2A12A21((k2

1 + μA11)− (k2

2 + μA22))2 , (A.30)

and we can write√√√√1 +4μ2A12A21((

k21 + μA11

)− (k22 + μA22

))2∼= 1 +

2μ2A12A21((k2

1 + μA11)− (k2

2 + μA22))2 − 2

(μ2A12A21

)2((k2

1 + μA11)− (k2

2 + μA22))4 . (A.31)

Equation (A.32) follows from substituting equation (A.31) into equation (A.30) and ignoringthe very small values and regarding the results as small perturbations,

λ21p

∼= −(k21 + μA11

)− μ2A12A21(k2

1 + μA11)− (k2

2 + μA22) +

(μ2A12A21)2((

k21 + μA11

)− (k22 + μA22

))3= −k2

1

⎡⎣1 +

μA11

k21

+μ2A12A21

k41

(1 + μA11

k21

)− k22 k

21

(1 + μA22

k22

)⎤⎦

≈ −k21

[1 +

μA11

k21

+μ2A12A21

k21

(k2

1 − k22

)]

, (A.32)

and

λ22p

∼= −(k22 + μA22

)− μ2A12A21(k2

2 + μA22)− (k2

1 + μA11) +

(μ2A12A21)2((

k22 + μA22

)− (k21 + μA11

))3= −k2

2

⎡⎣1 +

μA22

k22

+μ2A12A21

k42

(1 + μA22

k22

)− k22 k

21

(1 + μA11

k21

)⎤⎦

≈ −k22

[1 +

μA22

k22

+μ2A12A21

k22

(k2

2 − k21

)]

. (A.33)

The perturbed non-dimensional wave numbers are

k1p = k1

√1 +

μA11

k21

+μ2A12A21

k21

(k2

1 − k22

)∼= k1

⎡⎣1 +

μA11

2k21

+μ2A12A21

2k21

(k2

1 − k22

) − 1

8

(μA11

k21

+μ2A12A21

k21

(k2

1 − k22

))2⎤⎦

≈ k1

[1 +

μA11

2k21

+μ2A12A21

2k21

(k2

1 − k22

) −(μA11

)28k4

1

], (A.34)

326 T Beke

and

k2p = k2

√1 +

μA22

k22

+μ2A12A21

k22

(k2

2 − k21

)∼= k2

⎡⎣1 +

μA22

2k22

+μ2A12A21

2k22

(k2

2 − k21

) − 1

8

(μA22

k22

+μ2A12A21

k22

(k2

2 − k21

))2⎤⎦

≈ k2

[1 +

μA22

2k22

+μ2A12A21

2k22

(k2

2 − k21

) − (μA22)2

8k42

]. (A.35)

The perturbed frequency is

fp = vs(T )

2πkp, (A.36)

where vs(T) means the velocity of sound that depends on temperature.Equation (A.37) follows from substituting equations (A.5) and (A.34) into

equation (A.36),

f1p = vs(T )

2πk1p

∼= vs(T )

2π

k1

L

[1 +

μA11

2k21

+μ2A12A21

2k21

(k2

1 − k22

) − (μA11)2

8k41

]. (A.37)

Equation (A.38) follows from substituting equations (A.5) and (A.35) into equation (A.36),

f2p = vs(T )

2πk2p

∼= vs(T )

2π

k2

L

[1 +

μA22

2k22

+μ2A12A21

2k22

(k2

2 − k21

) − (μA22)2

8k42

]. (A.38)

With the help of this method, we can obtain the first two perturbed eigenacoustic frequenciesof the system.

References

Beke T 2009a Observation of thermoacoustic phenomena in school project Phys. Educ. 44 536–48Beke T 2009b Termoakusztikus jelensegek vizsgalata iskolai projektfeladatban Fiz. Tanıtasa 17 7–14Beke T 2009c Termoakusztikus projektfeladat Rijke-cso vizsgalatara Fiz. Sz. 59 253–7Beke T 2010a Thermoacoustic school project work with an electrically heated Rijke tube Phys. Educ.

45 516–28Beke T 2010b Modelling of thermoacoustic phenomena in an electrically heated Rijke tube Eur. J. Phys.

31 1331–44Culick F E C 1976 Nonlinear behavior of acoustic waves in combustion chambers: parts I and II Acta Astronaut. 3

714–57Culick F E C 1997 A note on ordering perturbations and insignificance of linear coupling in combustion instabilities

Combust. Sci. Technol. 126 359–79Culick F E C, Burnley V and Swenson G 1995 Pulsed instabilities in solid propellant rockets J. Propul.

Power 11 657–65Dowling A P 1995 The calculation of thermoacoustic oscillations J. Sound Vib. 180 557–81Dowling A P and Stow S R 2003 Acoustic analysis of gas turbine combustors J. Propul. Power 19 751–64Entezam B W, Van Moorhem W K and Majdalani J 2002 Two-dimensional numerical verification of the unsteady

thermoacoustic field inside a Rijke-type pulse combustor Numer. Heat Transfer A 41 245–62Fahey D and Timbie P 2006 Thermoacoustic oscillations Wave Motions and Optics (Berlin: Springer)Heckl M A 1990 Nonlinear acoustic effects in the Rijke tube Acoustica 72 63–71Kedia K S 2008 Thermoacoustic instability in a Rijke tube: impact of linear coupling and stochastic sources Project

Report, Department of Aerospace Engineering, Indian Institute of Technology, MadrasMatveev K I 2003 Thermoacoustic instabilities in the Rijke tube: experiments and modeling PhD Thesis, California

Institute of Technology, Pasadena, CAMeirovitch L 1967 Analytical Methods in Vibrations (New York: Macmillan)Rayleigh L and Strutt J W 1896 The Theory of Sound vol 2 2nd edn (London: Macmillan)

Rijke-type thermoacoustic oscillations 327

Rijke P L 1859 Notiz uber eine neie art, die luft in einer an beiden enden offenen Rohre in schwingungen zu versetzenAnn. Phys., Lpz. 107 339–43

Rocard-report 2007 Science education NOW: a renewed pedagogy for the future of Europe 7–14 Brussels Office forOfficial Publications of the European Communities

Sarpotdar S M, Ananthkrishnan N and Sharma S D 2003 The Rijke tube—a thermo-acoustic device Resonance8 59–71