Thermoacoustic System Research Articles

Design optimization and CFD analysis of the dynamic behavior of a standing wave thermoacoustic engine with various geometry parameters and boundary conditions

Thermoacoustic devices are converters of thermal energy into acoustic energy and vice versa. Although these machines contain simple components, the design of these machines is very challenging. In order to predict the behavior and optimize the efficiency of a standing-wave thermoacoustic engine designed to drive a thermally driven thermoacoustic refrigerator, considering changes in geometrical parameters and operating conditions, two analogies have been presented in this paper. The first analogy is based on a CFD simulation carried out to investigate the influence of stack parameters, working gas and boundary conditions on the thermoacoustic process. The second analogy is performed by the use of an optimization algorithm based on the simplified linear thermoacoustic theory to design and optimize the parameters investigated by the CFD study. Stack of parallel plates of normalized stack center positions of 0.007 to 0.26, normalized stack lengths of 0.018 to 0.11, and several gaps and thicknesses of plates and working gases are used. The results from the algorithm give the ability to design any thermoacoustic engine with high efficiency by picking the appropriate parameters. Simulation results show that decreasing thickness and position of the plates gives a significant efficiency. However, there are optimum values for length of the stack and the gap between two plates. The material chosen for the construction of plates should have a low thermal conductivity and gases with higher ratios of specific heats and lower Prandtl numbers are well suitable for thermoacoustic systems.

Experimental results of a waste-heat powered thermoacoustic refrigeration system for ships

Greenhouse gas emissions are a major concern for maritime transport. TheInternational Maritime Organization (IMO)presents Waste Heat Recovery Systems (WHRS) from engine exhaust gas as a viable solution to improve energy utilization and reduce greenhouse gas emissions for marine power plants. In this paper, we present an original thermoacoustic WHRS and its associated experimental setup for validation. This WHRS aims to transport heat from a lower to a higher temperature reservoir (heat pump) and maintain a fridge at 0 °C. For the experimental setup, exhaust gases from a 4-stroke marine diesel engine are used to generate pressure waves (work) in thermoacoustic engines. The main result is the Coefficient of Performance (COP) of the system, around 0.3, assessed for various cold loads. The first results estimate that the system could substitute part of the chiller cold production onboard. Also, a preliminary economic analysis suggests that this system could perform 2% fuel savings that allow for a payback time below one year.

Research on thermoacoustic refrigeration system driven by waste heat of industrial buildings

Industrial production process continuously produces a large amount of stable waste heat, which has great potential for recovery and utilization. As a new energy conversion and utilization technology, thermoacoustic technology has a good application prospect in low-grade heat sources utilization, with advantages of simple structure, long service life and low cost, and can be applied in waste heat recovery field. The current thermoacoustic studies pay more attention to the improvement of thermoacoustic conversion efficiency. Power output should be considered while paying attention to efficiency in the field of engineering application. The thermoacoustic refrigeration system proposed in this paper uses waste heat generated in industrial buildings to regulate indoor temperature. Based on the theory of traveling-standing wave acoustic field, a design concept of thermoacoustic system was proposed, that is, two thermoacoustic cores are placed on both sides of the high impedance zero phase position to limit the high impedance zero phase phenomenon at the independent thermoacoustic core position. On the basis of this study, the topology of thermoacoustic refrigeration system operating in traveling-standing wave phase was proposed. A two-stage thermoacoustic engine/refrigerator coupled loop structure was adopted to realize the traveling-standing wave acoustic field through the non-uniform alternate distribution of thermoacoustic units. Based on the topology, numerical simulation and system optimization were carried out, and an experimental prototype was set up for experimental studies. The experimental results show good consistency with the simulation results. Under the condition of no load, the lowest cooling temperature reached by thermoacoustic refrigerator is −12℃, and the maximum cooling capacity is 0.95 kW. This study shows that the thermoacoustic refrigeration system driven by waste heat can significantly reduce the energy consumption of industrial buildings in terms of indoor temperature regulation, and improve the energy utilization ratio of industrial buildings.

Experimental Study of Suppressing the Thermoacoustic Instabilities in a Rijke Tube Using Microsecond Discharge Plasma

Thermoacoustic instabilities occur when heat release is coupled with pressure fluctuation, which may cause performance degradation of the combustor and serious structural damage. This study focued on an active control method using discharge plasma and showed experimentally that discharge plasma can make a difference in controlling the thermoacoustic instabilities in a Rijke tube. A vertically placed Rijke tube thermoacoustic system using induction heating tungsten mesh as a heat source was built. The results show that the high repetition rate discharge can effectively suppress the thermoacoustic oscillations in the Rijke tube and that they will not re-occur for some time. Additionally, their effectiveness depended more on average power than energy per pulse. Combining the collected pressure, schlieren data, and theoretical analysis, it can be suggested that the plasma discharge could heat the inlet airflow, which could influence the heat exchange and then could break thermo-acoustic coupling, and its high-frequency pressure perturbation might increase the dissipation of the energy of sound.

The Nonlinear Thermoacoustic Eigenvalue Problem and Its Rational Approximations: Assessment of Solution Strategies

Abstract Nonlinear eigenvalue problems (NLEVPs) arise in thermoacoustics by considering the temporal evolution of small perturbations in the relevant governing equations. In this work, two solution strategies are compared: (i) a contour-integration-based method that guarantees to provide all eigenvalues in a given domain and (ii) a method that approximates the NLEVP by a rational eigenvalue problem (REVP), which is generally easier to solve. The focus lies on numerical speed, the completeness of the computed spectrum, and the appearance of spurious modes, i.e., modes that are not part of the original spectrum but appear as a result of the approximation. To this end, two prototypical thermoacoustic systems are considered: a single-flame Rijke tube and an annular model combustor. The comparison of both methods is preceded by a detailed analysis of the user-defined input parameters in the contour-integration-based method. Our results show that both methods can resolve all types of considered eigenvalues with sufficient accuracy for applications. However, the recast linear problem is overall faster to solve and allows a priori precision estimates—unlike the contour-integration-based method. Spurious modes as a by-product of the NLEVP approximation are found to play a minor role, and recommendations are given on how to eliminate them.

A unified approach for dynamic characteristic analysis of a standing-wave thermoacoustic engine with general impedance ends

As promising devices for energy conversion, thermoacoustic engines (TAEs) are attracting more and more research attention with regard to renewable thermal energy application. For a TAE, the oscillation frequency is a critical parameter in its design and operation, which can only be estimated currently through an often tedious process using a shooting method or a numerical iteration method, and an analysis model with classical boundary conditions. In this context, this study seeks to establish a unified model for dynamic analysis of a one-dimensional (1D) standing-wave TAE with general impedance ends. First, the differential equations governing the thermoacoustic behavior of the TAE and its impedance boundary conditions are simultaneously solved using the Galerkin method and an improved Fourier series expansion method. Then, the characteristic distributions of oscillation frequency, sound pressure, and volume flow rate are obtained by solving a standard eigenvalue problem. Finally, the effects of engine length, stack length and position, and impedance ends on the dynamic behavior of the TAE’s thermoacoustic system are investigated and analyzed. The results show that engine length has a significantly negative correlation with the modal frequencies in different working gases, and stack position and stack length have a slight effect on the oscillation frequency of the TAE. The arbitrary variations in impedance boundary conditions can be simulated from the open end to the rigid wall and all intermediate cases. The proposed model can efficiently predict changes in dynamic parameters of 1-D TAEs with impedance ends in a unified pattern and can serve as a reliable analysis tool for further research on TAEs coupled with various energy harvesters.

Abrupt transitions in turbulent thermoacoustic systems

Abrupt transitions to the state of thermoacoustic instability (TAI) in gas turbine combustors are a significant challenge plaguing the development of next-generation low-emission aircraft and power generation engines. In this paper, we present the observation of abrupt transition in three disparate turbulent thermoacoustic systems: an annular combustor, a swirl-stabilized combustor, and a preheated bluff-body stabilized combustor. Using a low-order stochastic thermoacoustic model, we show that the reported abrupt transitions occur when an initially stable, supercritical limit cycle becomes unstable, leading to a secondary bifurcation to a large amplitude limit cycle solution. The states of combustion noise and intermittency observed in these turbulent combustors are well captured by the additive stochastic noise in the model. Through amplitude reduction, we analyze the underlying potential functions affecting the stability of the observed dynamical states. Finally, we make use of the Fokker–Planck equation, educing the effect of stochastic fluctuations on subcritical and secondary bifurcation. We conclude that a high enough intensity of stochastic fluctuations which transforms a subcritical bifurcation into an intermittency-facilitated continuous transition may have little effect on the abrupt nature of secondary bifurcation. Our findings imply the high likelihood of abrupt transitions in turbulent combustors possessing higher-order nonlinearities where turbulence intensities are disproportionate to the large amplitude limit cycle solution. Consequently, secondary bifurcations would be much more resilient to control, requiring improved control strategies.

A displacer coupled thermoacoustic cooler driven by heat and electricity

A small-scale, heat-driven cooling system is required for the on-site liquefaction of unconventional natural gas in a distributed station. The thermoacoustic cooler has been considered a promising candidate for meeting such demands due to its high reliability and environmental friendliness. However, conventional thermoacoustic systems use a resonance tube to couple the thermoacoustic engine and cooler, which degrades cooling performancedue to poor phase-shifting capability and significant power dissipation of the resonance tube. To overcome the limitation, this paper introduces a novel driven by heat and electricity thermoacoustic cooler using displacers to replace the resonance tube, which enables better phase-shifting capability and less power dissipation, thus allowing better cooling performance. Numerical investigation and experimental study are performed on the displacer-coupled thermoacoustic cooler driven by heat and electricity. Firstly, the design method of the displacers is presented based on the acoustic impedance matching principle. Numerical investigations are then conducted to understand the operational characteristics better to explore the axial distribution of critical parameters. The effects of the displacer damping coefficient, operating frequency, and mean pressure are further studied. Experimental investigations are then carried out to understand the impact of operating temperatures on system performance. Experimental results show that the system achieves a record thermal-to-cooling exergy efficiency of 13.4 % and cooling power of 117 W at heating and cooling temperatures of 773 K and 110 K. This represents an over 34 % improvement in efficiency when compared to the previous record-holder thermoacoustic cooler.

Time-domain acoustic-electrical analogy investigation on a high-power traveling-wave thermoacoustic electric generator

Thermoacoustic electric generator is a novel heat-to-power conversion technology with potential high efficiency and reliability, showing great potential in distributed power generation. In this paper, we conduct time-domain acoustic-electrical analogy investigations on a high-power 4-stage looped traveling-wave thermoacoustic electric generator to shed lights on the system performance. The components of a thermoacoustic system are simplified as an acoustic–electrical analogy model, thus saving computational time. Additionally, the nonlinear effects in the regenerator, heat exchanger and resonance tube are considered in establishing the time-domain equations. Method validation shows a reasonable agreement between experimental and simulated results. Transient evolutions of the oscillating pressure and volume flow rate are firstly given. Investigation is then conducted on onset temperature using different working gases of helium, argon, nitrogen, and neon. The results indicate that when the external resistance reaches 70 Ω, the minimum onset temperature difference of 112 K can be achieved with argon as the working gas. For steady-state characteristics, the influences of electric resistance and heating temperature are explored. It is found that higher heating temperature contributes to larger electric power. A large electric resistance results in a large electric resistance at high heating temperature, while the highest efficiency is achieved with a medium electric resistance. Furthermore, higher heating temperature can lead to more irreversible losses. With helium employed, a maximum electric power of 13.3 kW with thermal-to-electric efficiency of 27.4% is achieved with an electric resistance of 150 Ω and a heating temperature of 950 K, and a highest thermal-to-electric efficiency of 33.5% with electric power of 1.38 kW is obtained when the electric resistance and heating temperature are 90 Ω and 750 K, respectively. This study shows that the proposed method provides a new perspective and an effective method for understanding thermoacoustic electric generator.

On the nonlinear behaviour of oscillatory flow in a high pressure amplitude standing-wave thermoacoustic heat engine

This article deals with the numerical study of the nonlinear features of oscillatory flow in a high pressure amplitude standing wave thermoacoustic heat engine. The overall performance of thermoacoustic devices can deteriorate due to nonlinearities caused by their operation at high pressure amplitudes. Acoustic mass streaming is one of these nonlinearities that can influence the performance of thermoacoustic systems by an undesirable convective heat transfer. Therefore, understanding the nonlinear phenomena within thermoacoustic devices is of fundamental and practical importance. The Navier‐Stokes and energy equations are solved on the structured meshes by resolving thermal and viscous boundary layers utilizing the open-source modelling software OpenFOAM. In particular, the paper looks in more detail at the causes of a non-uniform distribution of mean pressure along the resonator reported previously by both experimental and numerical studies. This is explained by an analogy to a “distributed fluidic pump” manifesting itself through the pumping effects of streaming patterns. Also, the non-linear regime of the mean dynamic pressure shows interesting start-up instabilities which can be influential on achievable efficiencies.

Turbulence and heat release rate network structure in hydrogen-enriched combustion

Complex network theory is used to analyze the spatiotemporal dynamics of the PRECCINSTA swirl burner, operating on hydrogen-methane fuel blends. At a power setting of 15 kW with equivalence ratio of 0.8 and hydrogen fuel fraction (HFF) ranging from 0% to 80%, period-1 and period-2 limit cycle oscillations as well as chaotic oscillations were observed. A turbulence network was constructed from the vorticity data obtained with particle image velocimetry. In addition, a heat release rate (HRR) correlation network was constructed from chemiluminescence images. Although the two networks concern a common thermoacoustic system, both exhibit significant differences: the turbulence network displays power law degree distribution, maintains small world property for all HFFs and is scale-free only in the absence of hydrogen enrichment. The HRR correlation network does not feature these properties, but hints at an asymmetric coupling between the heat release rate and the acoustic pressure for all HFFs. Furthermore, the HRR network is responsive to changes in HFF and the corresponding shifts in the dynamical state as well as in the root-mean-square of acoustic pressure. On the contrary, the turbulence network displays no such sensitivity and its properties are almost constant in the upper HFF range. It exhibits stationary hub structures for all the fuel blends tested, whereas the HRR correlation network is hub-free.

Transitions of thermoacoustic modes and flame dynamics in a centrally-staged swirl combustor

In the start-up process of practical gas turbine applications, the modulation of operating condition, especially the variation of global equivalence ratios (ϕglo), could cause the variations of stability regimes. The above process is accompanied by the increase of total air flow rates (ṁair) when approaching the stable operation condition, and thus the effect of this parameter should also be considered in the operating condition modulation process. This work presents an experimental investigation into the transitions of thermoacoustic modes and flame dynamics in a centrally-staged swirl combustor over a range of ϕglo and ṁair. The present thermoacoustic system exhibits two transitions of thermoacoustic modes with ϕglo variation, showing distinct characteristics of pressure oscillation (p′) and flame structures. To further explore the detailed variation of p′ characteristics and its link to flame dynamics during two transitions, a nonlinear dynamic analysis of p′ including its phase and recurrence plots as well as a proper orthogonal decomposition (POD) study have been conducted. The thermoacoustic network analysis and local Rayleigh index analysis have clarified the mode transition mechanism with varying ϕglo and ṁair. The present results can provide a guideline for quantifying the stability regimes of fuel-flexible combustors during the operating condition modulation processes.

Performance analysis of thermoacoustic plasma MHD generation

We develop a quasi-one-dimensional theoretical model describing the oscillatory plasma MHD conversion in a channel with parallel electrodes. Based on the model, a thermoacoustic plasma MHD generation system with a looped configuration is proposed and the characteristics of the operation of the system are investigated. Our results show that, for efficient oscillatory MHD generation, the channel dimension of the MHD duct should be much larger than the penetration depth (a proper Womersley number is above 100), otherwise the thermoacoustic effect will play a major role and consume considerable energy. Further, the electromagnetic induction parameter should be larger than 100S⋅T2/m, so that a majority of the kinetic energy of the oscillatory plasma can be harnessed for energy conversion. Moreover, a load ratio between 0.6 to 0.8 is suggested for the presented system, with which an efficiency above 20% can be achieved when the electromagnetic induction parameter is 500S⋅T2/m. In the optimal case, an efficiency of 24% can be obtained, with the corresponding output electricity of 1644 W. These results shed light on the feasibility of the closed-cycle MHD generation without any moving part, which endows the system with advantages of high reliability and long lifetime.

Occasional coupling enhances amplitude death in delay-coupled oscillators.

This paper aims to study amplitude death in time delay coupled oscillators using the occasional coupling scheme that implies intermittent interaction among the oscillators. An enhancement of amplitude death regions (i.e., an increment of the width of the amplitude death regions along the control parameter axis) can be possible using the occasional coupling in a pair of delay-coupled oscillators. Our study starts with coupled limit cycle oscillators (Stuart-Landau) and coupled chaotic oscillators (Rössler). We further examine coupled horizontal Rijke tubes, a prototypical model of thermoacoustic systems. Oscillatory states are highly detrimental to thermoacoustic systems such as combustors. Consequently, a state of amplitude death is always preferred. We employ the on-off coupling (i.e., a square wave function), as an occasional coupling scheme, to these coupled oscillators. On monotonically varying the coupling strength (as a control parameter), we observe an enhancement of amplitude death regions using the occasional coupling scheme compared to the continuous coupling scheme. In order to study the contribution of the occasional coupling scheme, we perform a detailed linear stability analysis and analytically explain this enhancement of the amplitude death region for coupled limit cycle oscillators. We also adopt the frequency ratio of the oscillators and the time delay between the oscillators as the control parameters. Intriguingly, we obtain a similar enhancement of the amplitude death regions using the frequency ratio and time delay as the control parameters in the presence of the occasional coupling. Finally, we use a half-wave rectified sinusoidal wave function (motivated by practical reality) to introduce the occasional coupling in time delay coupled oscillators and get similar results.

Thermoacoustic oscillations in a can-annular model combustor with asymmetries in the can-to-can coupling

Can-annular combustors are equipped with a set of nominally identical cans, circumferentially arranged around the shaft. Adjacent cans are coupled acoustically via a small gap at the downstream end, where the circular cross-section transitions into the annular turbine inlet. Recent experimental and theoretical work has shown that the coupling strongly affects thermoacoustic system stability and, hence, may give rise to damaging pressure oscillations that originate from a constructive interference of pressure waves and heat release rate fluctuations. A laboratory-scale can-annular combustor has been operated with premixed CH4–H2–air mixtures to study these instabilities. The combustor consists of eight identical cans, connected acoustically to their respective neighbours via size-adjustable side branches; the latter allow for a variation of the coupling strength. New experimental results are presented in which five asymmetric sets of coupling strengths are investigated and compared to a previously studied symmetric configuration. The sets are chosen such that the symmetry is gradually reduced until a fully asymmetric configuration is realized. The perturbations affect the observed mode type, frequency of oscillation and the pressure amplitude distribution over the eight cans. The perturbations via the can-to-can couplings do not trigger mode localization. The symmetric configuration considered previously, shows activity in an azimuthal mode of third and fourth order, with unsteady switching between the two. The mode shape is characterised through novel phase-averaged OH* chemiluminescence images capturing all eight cans.

Експериментальні дослідження термоакустичних двигунів з двофазним робочим тілом

A distinguishing feature of these days is the general tendency to decrease the temperature level of the waste heat of engines for power plants in industry, transport, and energy. These circumstances complicate the usage of traditional energy-saving technologies designed to transform this heat into mechanical work. Given the lack of effective technologies, large volumes of such heat emissions are lost. As an example, we can consider ship power engineering. On ships that are equipped with dual-fuel medium and low-speed engines, the thermal emissions of the cooling systems have a temperature of 355…365 K. Given the small exegetical potential, the use of such low-temperature waste energy sources by any heat engines is a difficult problem. Therefore, the task of improving existing energy-saving technologies or developing new ones remains relevant. Thermoacoustic technologies can be useful in solving this problem. A significant advantage of thermoacoustic heat machines is the ability to use any external heat source, in this case low-temperature sources, and produce mechanical work. There are known cases of thermoacoustic oscillations at small temperature differences between heat sources under conditions of high humidity of the working environment. This phenomenon can be used to create low-temperature energy-saving systems based on thermoacoustic engines (TAE) with a wet two-phase working environment. The practical use of thermoacoustic systems as part of ship power plants requires additional research to solve low-level issues, in particular, increasing the specific power of the TAE. This work provides a description of the experimental equipment, design of experimental TAE with a wet working body and research methods. The results of the experiments showed that in experimental TAEs with a two-component (moist) working environment, the temperature of spontaneous thermoacoustic oscillations was 355…359 K, while the longitudinal temperature gradient in the matrix was 1.19…1.30 K/m. The specific power of TAE with a moist environment increased by 1.7…7 times, and the acoustic pressure increased by 2…4.7 times compared to the operation in dry air. It is shown that in the existing TAEs, the condensation of water vapor in the ceramic matrix and on the surfaces of the heat exchangers can lead to a loss of power, roughly up to 25 %, while maintaining the operational capacity.

Self-coupling: an effective method to mitigate thermoacoustic instability

The presence of undesirable large-amplitude self-sustained oscillations in combustors resulting from thermoacoustic instability can lead to performance loss and structural damage to components of gas turbine and rocket engines. Traditional feedback controls to mitigate thermoacoustic instability possess electromechanical components, which are expensive to maintain regularly and unreliable in the harsh environments of combustors. In this study, we demonstrate the quenching of thermoacoustic instability through self-coupling—a method wherein a hollow tube is used to provide acoustic self-feedback to a thermoacoustic system. Through experiments and modeling, we identify the optimal coupling conditions for attaining amplitude death, i.e., complete suppression of thermoacoustic instabilities, in a horizontal Rijke tube. We examine the effect of both system and coupling parameters on the occurrence of amplitude death. We thereby show that the parametric regions of amplitude death occur when the coupling tube length is close to an odd multiple of the length of the Rijke tube. The optimal location to place the coupling tube for achieving amplitude death is near the antinode of the acoustic pressure standing wave in the Rijke tube. Furthermore, we find that self-coupling mitigates thermoacoustic instability in a Rijke tube more effectively than mutual coupling of two identical Rijke tubes. Thus, we believe that self-coupling can prove to be a simple, cost-effective solution for mitigating thermoacoustic instability in gas turbine and rocket combustors.

Dynamical states of thermo-acoustic system with respect to frequency–phase relationship based on probabilistic oscillator model

Dynamical states of thermo-acoustic system with respect to frequency–phase relationship based on probabilistic oscillator model

Numerical study of a heat-driven thermoacoustic refrigerator based on a time-domain lumped acoustic–electrical analogy model

Thermoacoustic technology is a promising clean energy solution for the recovery of low-grade heat from renewable energy sources and waste heat. Existing methods for simulating thermoacoustic systems are inconsistent or time-consuming. In this study, a time-saving and reliable time-domain lumped acoustic–electrical analogy model is proposed to investigate the performance of thermoacoustic systems. In the proposed model, each component of a thermoacoustic system is simplified as a lumped acoustic–electrical analogy model. The nonlinear effects of both regenerator and liquid acoustic resistances in gas–liquid resonators are considered to obtain nonlinear dynamic evolution equations. Case studies were performed on a looped heat-driven thermoacoustic refrigerator for low-grade heat recovery to investigate its onset and steady characteristics. The evolutions of the oscillating pressure and volume flow rate were explored, which initially increase rapidly until reaching the saturation state. The minimum onset temperature was 32.5 K at a pressure of 2.5 MPa with hydrogen as the working gas. In addition, the influences of the mean pressure, heating temperature, and cooling temperature on the steady-state system performance with various working fluids were investigated. The results indicated that the cooling power increased significantly with increasing mean pressure, heating temperature, and cooling temperature. A higher working liquid density resulted in a lower onset temperature, lower working frequency, and larger pressure ratio. With the use of helium or hydrogen, the refrigerator performed better in terms of a lower onset temperature and a larger pressure ratio, cooling power, and coefficient of performance. The proposed model provides a new perspective and an effective approach to characterise the onset and steady characteristics of thermoacoustic systems.

Theoretical analysis of sound propagation and entropy generation across a distributed steady heat source

Acoustic and entropy waves interacting in a duct with a steady heat source and mean flow are analysed using an asymptotic expansion (AE) for low frequencies. The analytical AE solutions are obtained by taking advantage of flow invariants and applying a multi-step strategy. The proposed solutions provide first-order corrections to the compact model in the form of integrals of mean flow variables. An eigenvalue system is then built to predict the thermoacoustic modes of a duct containing a distributed heat source or sink. Predictions from the AE solutions agree well with the numerical results of the linearised Euler equations for both frequencies and growth rates, as long as the low-frequency condition is satisfied. The AE solutions are able to accurately reconstruct the acoustic and entropy waves and correct the significant errors in the predicted entropy wave associated with the compact model. The analysis illustrates that the thermoacoustic system needs to account for the entropy wave generated by the interaction of acoustic wave and the distributed steady heat source, especially when density- or entropy-dependent boundary conditions are prescribed at the duct ends. Furthermore, a combination of the AE method and the modified WKB approximation method is discussed for a cooling case. The AE solutions remedy the disadvantage of the WKB solution in the low and very low-frequency domain and facilitate full-frequency theoretical analyses of sound propagation and entropy generation in inhomogeneous duct flow fields.