On the relationship between acoustic absorption and temperature gradient in a thermoacoustic liner

Abstract

The thermoacoustic effect provides a means to convert acoustic energy to heat and vice versa without the need for moving parts. This makes thermoacoustic devices of interest for multifunctional applications in harsh or remote service environments. Embedding thermoacoustic stacks within the framework of acoustic liners could provide opportunities to simultaneously optimize both acoustic absorption and thermal output as well as to integrate energy harvesting and sensing and monitoring functionalities. In this study, the influence of sound pressure level and frequency as well as stack length and position on the performance of a thermoacoustic liner is investigated using normal incidence impedance tube tests and DeltaEC simulations. Using Rott’s approximation, the stack’s pore width and wall thickness are optimized for additively manufactured thermoacoustic liner test articles. A peak steady-state temperature gradient of 9.5°C is obtained at 790 Hz for 140 dB tonal dwells. It is found that the temperature gradient correlates directly with the acoustic power available at the leading stack-face, which is highest when the stack is near the center of the resonator. In contrast, acoustic absorption diminishes when the stack is nearer to the closed-end of the resonator. At higher excitation pressure levels, the contribution of viscous heating to the deviation in the thermoacoustic gradient becomes significant. Further, the relationship between the acoustic absorption and the thermoacoustic temperature gradient was examined. In application scenarios requiring multifunctional optimization, a tradeoff may be required in terms of the absorption in order to maximize the temperature gradient. With current additive and hybrid fabrication processes and materials reaching commercial maturity, opportunities exist to realize thermoacoustic liners with enriched functionalities.

Keywords

Thermoacoustics acoustic liner

Introduction

Acoustic liners are widely used for noise control in various applications, such as in aircraft jet engines, civil or transportation infrastructures, and turbomachinery.^1–5 These liners are usually constructed from lightweight and porous materials such as perforate over honeycomb or foam structures, which reduce sound energy through various absorptive mechanisms.^6,7 Despite their widespread use, conventional acoustic liners have their limitations, including narrow and higher frequency bandwidths of effectiveness, lack of adaptivity to varying source characteristics,⁸ and unexploited potential for integration in multifunctional applications such as energy harvesting and sensing and monitoring. With new materials and manufacturing processes being adopted for the next generation of engineered structures, noise source characteristics evolving due to technological developments and regulations becoming more stringent, it is an opportune time to explore new liner concepts^9–12 that could deliver improved solutions. The incorporation of thermoacoustic phenomena opens new avenues for acoustic liner advancements. In this study, a thermoacoustic liner is studied, examining the relationship between acoustic absorption and thermoacoustic performance, exploring its potential for multifunctional applications beyond the scope of conventional liner.

Recent innovations in acoustic liners include a variety of novel designs and materials. Liu et al.¹³ devised a piezoelectric panel, which could be used to control the size and shape of the resonant chambers in a liner so that the absorption bandwidth could be actively controlled. Rego et al.¹⁴ investigated the use of curved cavity Helmholtz resonators for noise reduction for a flat plate and a subsonic jet arrangement. Simulations showed a noise reduction of up to 7 dB in a broader frequency range in the vicinity of resonance. Guo et al.¹⁵ investigated the noise attenuation properties of an acoustic liner consisting of Helmholtz resonators with extended necks and found them to be a promising solution for wide-band noise attenuation in both static and flow conditions while retaining a compact form-factor. Lam et al.¹⁶ demonstrated numerically that broadband noise reduction can be achieved using a liner with an elastic panel with a cavity beneath under low Mach number grazing flow.

Fan et al.¹⁷ examined the structural-acoustic interaction of an elastic panel liner backed by a cavity under subsonic ducted boundary layer flow and reported to be more effective when the noise source radiates against the flow. Xu et al.¹⁸ examined the importance of bias flow and its temperature in improving absorption and considered the modal effects on thermoacoustic instabilities in the design of acoustic liners. Neubauer et al.¹⁹ recently explored a novel liner concept utilizing flexible Helmholtz and plate resonators to demonstrate enhanced noise mitigation. Simultaneously, the need to reduce broadband fan noise in high bypass ratio turbofan engines has attracted interest with advancement in fan design and manufacturing techniques.^20,21 As mission requirements become more multifarious and stringent, acoustic liners of the future could be expected to be more actively or passively adaptive, multifunctional, and certainly better performing.

In this context, it is of interest to explore the incorporation of multi-physical sub-structures within the framework of the acoustic liner. Inspired by metamaterials,²² thermoacoustic liners that have a porous stack embedded in the liner cavity are a promising configuration. In a true multifunctional environment as in the case of core liners in turbofan engines, a confluence of innovative design approaches, interactive multiphysical mechanisms, as well as novel implementations of structural materials can lead to a synergistic enrichment in performance. Given the availability of a steep thermal gradient across the core walls of the engine, embedding a thermoacoustic functionality into the liner’s resonators provides a natural means to engineer additional tunability for performance enhancement in core liners. The thermoacoustic phenomenon is manifested due to an instability caused by pressure waves propagating in an acoustic waveguide inducing oscillations in a porous stack, the ends of which manifest a thermal gradient, thereby creating fluctuations in the rate of heat transfer. This in turn generates further pressure oscillations, leading to an instability that can be harnessed without the need for any moving parts. Thus, in comparison to conventional perforate-over-honeycomb acoustic liners, such thermoacoustic liners that combine multiple functionalities can offer many benefits such as adaptive control, transduction and energy harvesting in many applications especially in modern turbofan jet engines. Figure 1 illustrates a comparison between a conventional acoustic liner and a thermoacoustic liner for a potential application scenario in jet engines.

Figure 1.

Conventional perforate-over-honeycomb, double-degree-of-freedom liner⁴⁴ and the thermoacoustic liner for a potential application scenario in jet engines.

In the past, the development of thermoacoustic devices has been facilitated by advancements in materials, modeling and manufacturing techniques.^23–26 Thermoacoustic devices have potential applications in various fields,^27–31 including refrigeration,³² power generation,³³ and noise control³⁴ and mitigation.²² Recent research has enhanced their performance and efficiency. Watanabe et al.³⁵ explored the rotational dynamics of a looped-tube thermoacoustic engine with a flywheel, improving engine stability and efficiency. Huang et al.³⁶ reviewed thermoacoustic engines with unconventional working substances, highlighting the importance of selecting the right working fluids to maximize efficiency. Chen et al.³⁷ conducted numerical simulations on standing-wave thermoacoustic engines, showing that pin-array stacks outperform circular-pore stacks in power generation and thermal efficiency. Recent advancements in thermoacoustic systems have shown promise in utilizing waste heat for refrigeration and cooling. Ding et al.³⁸ explored a thermoacoustic refrigeration system that uses waste heat from industrial buildings to regulate indoor temperatures. The system employs a traveling-standing wave acoustic field and demonstrated a cooling temperature as low as −12°C with a maximum cooling capacity of 0.95 kW, highlighting its potential to enhance energy utilization in industrial settings. Xiao et al.³⁹ reported on a highly efficient heat-driven thermoacoustic refrigerator (HDTR) using a bypass configuration that synergizes acoustic and energy-flow fields.

In this study, a prototypical thermoacoustic liner test article was designed and fabricated using additive manufacturing techniques. Experiments conducted in the normal incidence impedance tube at NASA Langley and correlated with DeltaEC simulations were used to characterize the liner’s acoustic and thermoacoustic performance. A key aspect studied was the relationship between acoustic absorption and the thermoacoustic temperature gradient. Understanding this relationship is essential for optimizing thermoacoustic liner designs and facilitates the development of a new class of multifunctional acoustic liners.

Thermoacoustic liner

Test articles

The test articles used in this study were designed to be compatible with the normal incidence impedance tube at NASA Langley.⁴⁰ They consist of two subcomponents: the modular resonator sections and the stack. The CAD model and photographs of the test article are shown in Figure 2. Modular resonator sections were 3D-printed using polylactic acid (PLA) using the fused deposition modeling (FDM) method. On the other hand, stack blocks were 3D-printed using Formlabs^® standard resin and the stereolithography (SLA) method. Both the stack and resonator sections have a nominal cross-sectional dimension of 0.06 m × 0.06 m. The resonator sections have different lengths to build up the total length of the resonator. The stack pore geometry is an important consideration for optimal thermoacoustic performance. A square pore geometry is chosen for simplicity and ease of manufacturing, although there could be other geometries that may improve performance. Within the realm of thermoacoustic linearity, the recommended pore width is 2–4 times the thermal (or viscous) penetration depth for the choice of working fluid and ambient conditions.⁴¹ Using this guideline, a pore width of 1.0 × 10⁻³ m is determined for air as the working fluid under room temperature and pressure in the frequency range of interest (<∼1000 Hz). It is also necessary to consider how the pore wall thickness affects areal porosity, which is the ratio of the total pore area to the overall area of the stack. As pore wall thickness increases, areal porosity decreases, given a constant pore size.

Figure 2.

(a) CAD drawing and (b) photographs of the test article.

A higher porosity allows for a greater volume of working fluid (in this case, air) to interact with the stack, potentially improving thermoacoustic efficiency. We selected a pore wall thickness of 0.50 × 10⁻³ m. This choice was driven by the desire to maintain structural robustness under operation, in light of the limitations posed by the 3D printing technology at our disposal. This selection resulted in an areal porosity of 0.39 (measured), which falls within the ideal range for satisfactory thermoacoustic performance. These parameters are within the range of what is feasible using commercial FDM printing. The optimal lengths for the stack and resonator were determined through simulations such that the thermoacoustic output is maximized in the 500–1000 Hz range. It is expected that the actual (measured) dimensions for the 3D-printed components would have some variation from the designed (or nominal) dimensions. Microscopy-based digital image analysis was employed to estimate the actual dimensions of the pore width and wall thickness for both stack faces. The comparison of the nominal and measured values for the stack is presented in Table 1. The measured values including error bounds are well within about 20% of the nominal values for all cases. The measured values are used in the simulations for the various test cases in this study. The manufacturer-specified thermophysical properties for the stack material (standard resin) are listed in Table 2.

Table 1.

Geometric parameters of the stack.

Property	Nominal value	Measured value
Length, $L_{s}$ , (m)	0.028	0.025 ± 0.002
Areal porosity, $Φ$	0.44	0.39 ± 0.0068
Pore wall thickness, $2 l$ , ( $10^{- 3}$ m)	0.5	0.58 ± 0.021
Pore width, $2 y_{0}$ , ( $10^{- 3}$ m)	1.0	0.91 ± 0.021

Table 2.

Thermophysical properties of the stack.

Property	Value
Thermal conductivity, $k$ , (W/(m·K))	0.31
Specific heat capacity, $c$ , (J/(kg·K))	1466
Density, $ρ$ , (kg/m³)	1081

Experiments

All experiments were conducted in the NASA Langley normal incidence tube (NIT) shown in Figure 3, which has a 0.051 m × 0.051 m cross-section waveguide. The tube is equipped with six 120 W compression drivers that generate a plane-wave sound field for frequencies up to about 3300 Hz. This sound field impinges on the surface of the liner and combines with reflections from the liner to create a standing wave pattern.⁴⁰ Acoustic pressures at two prescribed locations 0.064 m and 0.095 m from the liner-waveguide interface were measured using two microphones mounted in a rotating plug. The two-microphone method (ASTM E1050)⁴² is used to determine the frequency dependence of the no-flow acoustic impedance of the liner. The liner test article is mounted on a movable platen at the open-end of the waveguide. Two K-type thermocouples are provided, one each on the leading and trailing stack faces within the resonator. Further, a third thermocouple is placed on the frame of the rig outside the test article to measure the ambient temperature. The temperature and acoustic data were collected using a time synchronized NI DAQ and LabView interface.

Figure 3.

Normal Incidence Tube (NIT) at NASA Langley with schematic details of the test article.

The frequency range of interest spans from 100 to 1400 Hz. Unless otherwise specified, the excitation sound pressure level was set at 140 dB OASPL. Tonal dwells were also conducted at SPLs between 130 and 150 dB. However, variation of stack position, stack length, and resonator length was considered. For the purposes of extracting the thermoacoustic temperature gradient across the stack, the steady-state criterion was defined as less than 0.1°C change in temperature gradient over 1 minute. The repeatability of the tests was ascertained using multiple trials for key cases demonstrating error bounds less than +/−5%. The baseline parameters for the experiments, including stack position at 0.04 m from the resonator’s top surface, stack length of 0.025 m, resonator length of 0.097 m, resonance frequency of 790 Hz, and driving pressure of 140 dB, were consistently applied across various test setups.

Modeling

The thermoacoustic liner consists of two types of sections: the duct and the stack. The modeling approach for each of these sections is different based on the thermal characteristics they exhibit. While the duct section does not exhibit a temperature gradient, the stack section does, and this distinction dictates the use of different mathematical models for accurate representation of their behaviors. For the duct section where there is no temperature gradient, the wave equation, which is expressed as the second-order Helmholtz differential equation for the complex pressure amplitude $p_{1} (x)$ is used. In the absence of any viscous or thermal-hysteresis losses and for an axially-invariant cross-sectional area A, the wave equation becomes:

p_{1} + \frac{a^{2}}{ω^{2}} \frac{d^{2} p_{1}}{d x^{2}} = 0

(1)

This second-order equation can be reduced to two coupled first-order equations in terms of the pressure, $p_{1}$ and the volume flow rate, $U_{1}$ using momentum and continuity considerations to yield:

\frac{d p_{1}}{d x} = - \frac{i ω ρ_{m}}{A} U_{1}

(2)

\frac{d U_{1}}{d x} = - \frac{i ω A}{ρ_{m} a^{2}} p_{1}

(3)

In this form, the equations can be numerically integrated simultaneously in the axial direction, $x$ to generate solutions for $p_{1} (x)$ and $U_{1} (x)$ in the duct section.

In the stack section, the presence of a temperature gradient due to thermoacoustic effects is modeled using Rott’s linear approximation, which utilizes boundary layer approximations, neglects any transverse gradients for acoustic pressure, average temperature and viscosity, as well as heat conduction in the axial direction in the acoustic medium and frictional effects due to axial gradients. With these assumptions, continuity, energy, and momentum considerations are used to derive the axial temperature gradient.

The steady-state sinusoidal oscillations of pressure, temperature, and density can be expressed using the following complex notation⁴³:

p = p_{m} + Re [p_{1} (x, y, z) e^{i ω t}]

(4)

u = Re [u_{1} (x, y, z) e^{i ω t}]

(5)

T = T_{m} (x) + Re [T_{1} (x, y, z) e^{i ω t}]

(6)

ρ = ρ_{m} (x) + Re [ρ_{1} (x, y, z) e^{i ω t}]

(7)

Here,

ρ_{m}

T_{m}

, and

p_{m}

represent the mean values of density, temperature, and pressure, respectively. The assertion that the mean values of these variables depend solely on

x

is simply a reflection of the assumption that nothing except for the oscillations can lead to gradients in the transverse direction. The time-dependent components are assumed to be small compared to the mean values.

The continuity equation for a microscopic control volume in the stack can be expressed as

i ω ρ_{1} + \frac{d}{d x} (ρ_{m} u_{1}) = 0

(8)

Noting that the existence of a non-zero axial spatial gradient for the mean temperature, ( $d T_{m} / d x$ ) does not influence the momentum equation in Rott’s approximation, the $x$ component of the momentum equation can be simplified as follows:

i ω ρ_{m} u_{1} + \frac{d p_{1}}{d x} = μ (\frac{\partial^{2} u_{1}}{\partial y^{2}} + \frac{\partial^{2} u_{1}}{\partial z^{2}})

(9)

The solution to the momentum equation can be written as:

u_{1} = \frac{i}{ω ρ_{m}} (1 - h_{v}) \frac{d p_{1}}{d x}

(10)

Here

h_{v}

is a complex function which depends on the stack geometry. Now, integrating both sides of Equation (10), over the cross-sectional area of the channel,

A

with the boundary condition,

u_{1} = 0

at the solid surface and subsequently solving for

d p_{1}

we get:

d p_{1} = \frac{i ω ρ_{m} d x}{A (1 - f_{ν})} U_{1}

(11)

where

f_{ν}

is the Rott’s viscous function and

U_{1}

is the volumetric flow rate. Further, a linearized form of the energy equation can be expressed as:

ρ_{m} c_{p} (i ω T_{1} + u_{1} \frac{d T_{m}}{d x}) - i ω p_{1} = k (\frac{{\partial^{2} T}_{1}}{\partial y^{2}} + \frac{\partial^{2} T_{1}}{\partial z^{2}})

(12)

Now, using a similar process as before, the solution of the energy equation can be written as:

T_{1} = \frac{1}{ρ_{m} c_{p}} (1 - f_{k}) p_{1} - \frac{1}{i ω A} \frac{d T_{m}}{d x} \frac{(1 - f_{κ}) - σ (1 - f_{ν})}{(1 - f_{ν}) (1 - σ)} U_{1}

(13)

where

f_{κ}

is the Rott’s thermal function. Next, the oscillatory component of the density is expressed as follows:

ρ_{1} = - \frac{ρ_{m}}{T_{m}} T_{1} + \frac{ρ_{m}}{p_{m}} p_{1}

(14)

Substituting equations (13) and (14) in (8) yields:

d U_{1} = - \frac{i ω A d x}{γ p_{m}} [1 + (γ - 1) f_{κ}] p_{1} + \frac{(f_{κ} - f_{v})}{(1 - f_{ν}) (1 - σ)} \frac{d T_{m}}{T_{m}} U_{1}

(15)

Combining equations (11) and (15), Rott’s thermoacoustic equation can be derived as:

[1 + (γ - 1) f_{k}] p_{1} + \frac{γ p_{m}}{ω^{2}} \frac{d}{d x} (\frac{1 - f_{v}}{ρ_{m}} \frac{d p_{1}}{d x}) - \frac{a^{2}}{ω^{2}} \frac{f_{k} - f_{v}}{1 - σ} \frac{1}{T_{m}} \frac{d T_{m}}{d x} \frac{d p_{1}}{d x} = 0

(16a)

Finally, introducing the correction factor for finite solid capacity $(ϵ_{s})$ and the thermal expansion coefficient $(β)$ yields:

(1 + \frac{(γ - 1) f_{κ}}{1 + ϵ_{s}}) p_{1} + \frac{ρ_{m} a^{2}}{ω^{2}} \frac{d}{d x} (\frac{1 - f_{ν}}{ρ_{m}} \frac{d p_{1}}{d x}) - β \frac{a^{2}}{ω^{2}} \frac{f_{κ} - f_{ν}}{(1 - σ) (1 + ϵ_{s})} \frac{d T_{m}}{d x} \frac{d p_{1}}{d x} = 0

(16b)

where

f_{k}

and

f_{v}

are the Rott’s thermal and viscous functions respectively and are given by

f_{κ} = \frac{\tanh [(1 + i) \frac{h_{0}}{δ_{κ}}]}{(1 + i) \frac{y_{0}}{δ_{κ}}};

(17a)

f_{ν} = \frac{\tanh [(1 + i) \frac{h_{0}}{δ_{ν}}]}{(1 + i) \frac{y_{0}}{δ_{ν}}}

(17b)

where,

δ_{κ}

is the thermal penetration depth,

δ_{ν}

is the viscous penetration depth,

y_{0}

is the half pore width and

h_{0}

is the hydraulic radius of the stack pore. Rott’s equation (equation (16a), (16b)) is restricted to low amplitude acoustic waves and linear temperature gradients across the stack. The gas displacement amplitude,

| U_{1} | / (ω A)

is assumed to be much smaller than any axial dimension in the device. It is also ensured that the stack length is much smaller than the acoustic wavelength so that the pressure and velocity changes across the stack due to acoustic standing wave formation can be assumed to be very small.⁴⁴

In addition to the aforementioned Rott’s thermoacoustic equation (equation (16a), (16b)), it is of interest to predict the acoustic and total powers in the thermoacoustic system. In the absence of any thermal inputs to the system, the power carried in the axial direction by second-order thermoacoustic effects can be represented by the acoustic power, $\dot{E_{2}}$ . This may be obtained via Rott’s energy equation (equation (12), which may be written as

\dot{E_{2}} = \frac{1}{2} Re [p_{1} {\tilde{U}}_{1} (1 - \frac{f_{κ} - \tilde{f_{ν}}}{(1 + ϵ_{s}) (1 + σ) (1 - {\tilde{f}}_{ν})})] + \frac{ρ_{m} c_{p} {| U_{1} |}^{2}}{2 A_{g a s} ω (1 - σ) {| 1 - f_{ν} |}^{2}} \frac{d T_{m}}{d x} Im [\tilde{f_{ν}} + \frac{(f_{κ} - \tilde{f_{ν}}) (1 + ϵ_{s} \tilde{f_{ν}}) (1 + ϵ_{s} f_{ν} / f_{κ})}{(1 + ϵ_{s}) (1 + σ)}] - (A_{g a s} k + A_{g a s} k_{s}) \frac{d T_{m}}{d x}

(18)

where

\dot{E_{2}}

is representative of the acoustic power input to the stack and the tildes over the symbols indicate complex conjugates.

The thermoacoustic liner configurations are simulated using DeltaEC (Design Environment for Low-amplitude Thermoacoustic Energy Conversion).⁴⁵ This tool uses continuity of oscillating pressure, oscillating volumetric velocity, and mean temperature to match the solutions of adjacent geometric segments. Within each segment, it integrates the one-dimensional wave equation appropriate to the geometry of that segment. For instance, it uses the lossy Webster horn equation⁴⁶ in conical segments, and Rott’s⁴⁷ wave equation (as extended by Swift⁴⁴) in thermoacoustic stacks, to determine pressure and volumetric velocity. In isothermal segments such as ducts, the wave equation is sufficient to determine the solution, however in thermoacoustic stacks, Rott’s enthalpy equation^45,47 must also be used to compute the mean temperature. To solve the governing equations, at least four boundary conditions are required, such as the magnitude and phase of acoustic pressure ( $p_{1}$ ) and the magnitude and phase of the volumetric flow rate ( $U_{1}$ ). One or more boundary conditions are usually given at the end segment, but if there is a missing boundary condition, DeltaEC uses the shooting method to complete the integration.

Further, extracting the acoustic parameters from DeltaEC, one can compute the absorption coefficient using the following relationships. The complex pressure and volume flow rate in a duct can be represented in terms of the real part ( $Re$ ) and the imaginary part ( $Im$ ) as follows:

p_{1} = Re [p_{1}] + i Im [p_{1}] = {| p_{1} | e}^{i ϕ_{p_{1}}}

(19)

U_{1} = Re [U_{1}] + i Im [U_{1}] = {| U_{1} | e}^{i ϕ_{U_{1}}}

(20)

The corresponding acoustic power can be expressed as the product of the magnitudes of the pressure, volume flow rate, and the cosine of the phase angle between them:

\dot{E_{2}} = \frac{1}{2} | p_{1} | | U_{1} | \cos ϕ

(21)

where

ϕ = ϕ_{p_{1 U_{1}}} = ϕ_{p_{1}} - ϕ_{U_{1}}

. The acoustic impedance is defined as the ratio of the pressure and volume flow rate (Z = p₁/U₁). Knowing the cross-section area (A) of the duct, the specific acoustic impedance can be expressed as

z = \frac{A p_{1}}{U_{1}}

. Further, the normalized specific acoustic impedance is obtained by dividing the specific acoustic impedance by the characteristic impedance of air:

z_{n} = \frac{Z A}{ρ a}

(22)

where

ρ

is the density of air and

a

is the speed of sound in air. For a plane wave in a duct reflected by a test liner under normal incidence, the sound absorption coefficient (α_n) is given by⁴⁸:

α_{n} = \frac{4 Re (z_{n})}{{[Re (z_{n}) + 1]}^{2} + Im {(z_{n})}^{2}}

(23)

Substituting the expression for $Z_{n}$ , the absorption coefficient can be expressed as:

a_{n} = \frac{4 Re (\frac{Z A}{ρ a})}{{[Re (\frac{Z A}{ρ a}) + 1]}^{2} + Im {(\frac{Z A}{ρ a})}^{2}}

(24)

Further, expressed in terms of the pressure, volume flow rate, and acoustic power, the absorption coefficient becomes:

α_{n} = \frac{8 \dot{E_{2}}}{{\frac{A}{ρ a} | p_{1} |}^{2} + \frac{ρ a}{A} {| U_{1} |}^{2} + 4 \dot{E_{2}}}

(25)

In this form the absorption coefficient is seen to be dependent on not just the magnitude of the pressure and volume flow rate, but the acoustic power as well, which in turn is also dependent on the phase difference between them. Since the acoustic power is known to correlate with the thermoacoustic temperature gradient (equation (18)), this demonstrates the inter-dependence of acoustic absorption and the thermoacoustic gradient. This relationship is further explored using comparison of simulations to the experiments.

Discussion of results

Frequency response

The response of the thermoacoustic liner in the frequency range of interest is identified from sine sweep tests as shown in Figure 4. Each sweep covered a 200 Hz window with an overlap of 25 Hz with the previous sweep window and a sweep time of 400 seconds. The intention of these measurements using sine sweeps was to conduct a diagnostic check to identify frequency ranges where the slope of the temperature gradient is high and has a reversal, indicating thermoacoustic resonance. At low frequencies, the absorption increases, tending towards the overall tube resonance of the apparatus which is below a 100 Hz. The absorption coefficient ranges from 0.51 to 0.66 in the frequency range of 400 – 1000 Hz. The educed normalized resistance and reactance, and the absorption coefficient are plotted. The absolute values of the normalized resistance and reactance values decrease with excitation frequency in this range. Inspecting the temperature measurements that accompany the absorption measurements under sine sweep, the slope of the temperature gradient with respect to the frequency is found to peak significantly in the vicinity of 790 Hz, indicating thermoacoustic resonance corresponding to the quarter wavelength response for the built height of the test article. Consequently, the focus of further interrogation using tonal dwells to steady state was restricted to the frequency range that encompasses this resonance frequency. It must be noted that under sine sweep excitation, absorption reaches steady-state very quickly, although the temperature difference or gradient takes much longer to get established. These results suggest that notwithstanding the narrowband spike in thermoacoustic performance around 800 Hz, sustained acoustic absorption is indicated under sine sweep excitation in a broadband frequency range that encompasses the frequency of peak thermoacoustic performance.

Figure 4.

Frequency response of the thermoacoustic liner from sine sweep excitation: (a) normalized resistance and reactance, (b) absorption coefficient, (c) temperature gradient and (d) slope of the temperature gradient curve.

To gain further insights, tonal dwells (held frequency of sound source constant) to steady-state were conducted in the 600 – 900 Hz range using the same liner configuration. As shown in Figure 5, a pronounced peak in the temperature gradient is observed at 790 Hz. This peak suggests a thermoacoustic resonance in our system. The time to steady-state (amount of time for which the temperature continues to change) was about 15 minutes for most excitation frequencies, and the cold-side occurred at the top end of the stack while the hot-side occurred at the bottom end.

Figure 5.

Frequency response of thermoacoustic parameters for tonal dwells: (a) top and bottom thermocouple outputs and (b) temperature gradient.

The acoustic performance under tonal dwells to steady-state is depicted in Figure 6. Unlike thermoacoustic transients, which require more time to subside, the acoustic transients are quite compact in time. Therefore, the acoustic responses from tonal dwells correlate well with those from the sine sweeps. This also indicates a relatively weak transient coupling between the thermoacoustic and acoustic absorption responses.

Figure 6.

Frequency response of acoustic parameters for tonal dwells: (a) normalized resistance and reactance, and (b) absorption coefficient.

Stack configuration

Given the predetermined optimum stack geometry tailored to the specific frequency range of interest and the constraints of physical durability and our 3D printing capabilities, this section focuses on the significant roles of stack location and length within the thermoacoustic liner. The location of the stack within the resonator controls the pressure differential across the stack which impacts how well heat is transferred between the working fluid to the stack ends. The length of the stack plays a role in determining the magnitude of the pressure oscillations and thermal transients in the working fluid, which can influence the stability and efficiency of the liner. In general, optimizing the position and length of the stack can lead to improved acoustic and thermoacoustic performance of the thermoacoustic liner.

Figure 7 depicts the thermoacoustic and acoustic performance for different stack positions. The stack position is the position of the leading stack-face as measured from the top surface of the test liner. In these cases, the overall resonator length was kept constant at 0.097 m (3.9”). The findings indicate that the system operates at its best when the leading stack-face is placed just short of the midpoint of the resonator. The highest steady-state temperature gradient at resonance (790 Hz) of nearly 9.5°C was measured when the stack was positioned at this location. It is noted that as the stack is moved away from this mid-position either towards the top surface of the liner or the rigid termination at the bottom, the thermoacoustic performance diminishes, while the acoustic performance diminishes as the stack is moved closer to the rigid termination. It is therefore of interest to investigate the underlying mechanisms that relate the thermoacoustic and acoustic performances using modeling.

Figure 7.

Variation of stack position: (a) temperature histories from top ( $T C_{1}$ ) and bottom ( $T C_{2}$ ) thermocouples, (b) temperature gradient, and (c) absorption coefficient, as a function of stack position at an excitation frequency of 790 Hz.

Similarly, the stack length was varied under the same testing conditions. The thermoacoustic and acoustic performance of the liner, varying by stack length, is depicted in Figure 8. The shorter stack, with a length of 0.025 m, displays a distinct temperature profile compared to the longer stack of 0.076 m. The absorption coefficient increases sharply with a slight increase in stack length and then gradually levels off, suggesting that longer stacks do not significantly increase absorption beyond a certain point.

Figure 8.

Variation of stack length: (a) temperature histories from top ( $T C_{1}$ ) and bottom ( $T C_{2}$ ) thermocouples, (b) temperature gradient, and (c) absorption coefficient, as a function of stack length at an excitation frequency of 790 Hz.

Sound pressure level

The linearity of the thermoacoustic liner’s response is worth inspecting by varying the excitation sound pressure level (SPL) using tonal dwells to steady-state at resonance (790 Hz). The hot- and cold-side temperature time histories, temperature gradient and absorption coefficient as a function of the excitation SPL are shown in Figure 9. Several interesting features are evident. It is true that at higher SPLs the steady-state temperature gradient is significantly enhanced while the absorption only varies slightly (<+/−10%). However, the time histories reveal a transition in the dominant heating mechanism involved. At lower excitation SPL (130 and 140 dB) the hot- and cold-side temperatures display the typical evolution seen from the thermoacoustic effect, wherein the hot- and cold-side temperatures straddle the ambient temperature (24.3°C) as they asymptote to their steady-state values. As the excitation SPL is raised to 145 and 150 dB, this trend is seen to be overridden by a monotonically increasing superposed heating contribution that increases the temperatures on both sides above the ambient temperature. This effect is quite drastic for the 150 dB case for which the hot-side temperature was about 54°C and the temperature gradient was nearly 27°C. It is a strong possibility that as the excitation SPL and consequently the acoustic energy available is increased, frictional (or viscous) heating within the stack pores become more dominant and therefore the thermoacoustic trends are overwhelmed. These mechanisms and their calibration could have implications for transduction of SPLs using ruggedized thermoacoustic devices.

Figure 9.

Response at different driving pressures for tonal dwell at the resonance (790 Hz): (a) temperature histories from top ( $T C_{1}$ ) and bottom ( $T C_{2}$ ) thermocouples, (b) temperature gradient, and (c) absorption coefficient.

Comparison with simulations

The experimental datasets were compared to numerical simulations conducted using DeltaEC. Stack configuration parameters such as length and position are examined through simulations. The predicted thermoacoustic parameters are presented as a function of the distance along the resonator for different stack positions in Figure 10.

Figure 10.

Simulated (a) pressure amplitude, (b) volumetric flow rate, (c) phase angle, and (d) acoustic power as a function of resonator length for various stack positions at an excitation frequency of 790 Hz. The shaded boxes indicate the stack position where the width of the box is the stack length (0.0254 m) and the height is proportional.

Note that the pressure amplitude envelopes in Figure 10(a) result from standing wave formation from multiple reflections at steady state due to the impedance mismatch that occurs at the stack-duct interfaces within the resonator. For $x_{s}$ = 0.015 m, the stack is placed near a node, causing the initial drop in pressure. As the distance from this node increases, the pressure amplitude rises towards the antinode. As can be noted, the velocity node and pressure antinode are located at the closed (far) end, while the velocity antinode is located at the driver end. As the acoustic energy flux transits through the stack, a small portion of it is transformed into thermal power due to the thermoacoustic effect. The position of the stack in the resonator is therefore optimal from a thermoacoustic perspective when the acoustic power at the leading stack face is maximized. The simulations indicate that the acoustic power at the leading stack-face is largest when the stack is positioned around 0.04 m below the top surface of the resonator (liner), which agrees well with experiments. The influence of stack position can also be explored through additional details. Figure 11 shows the thermoacoustic parameters just before and after the stack for a variation in the stack position, which is defined as the distance of the leading stack-face from the top surface of the liner.

Figure 11.

Simulated (a) pressure amplitude and volume flow rate, (b) phase angle, (c) acoustic power before and after the stack, and (d) comparison of simulated and experimental temperature gradient as a function of stack position.

From Figure 11, it is evident that at a stack position slightly above 0.04 m, there is a pronounced peak in the pressure amplitude, volume flow rate, and acoustic power. This peak signifies that the acoustic power available at the stack face closely aligns with the achieved thermoacoustic gradient. Comparison of the predicted and measured temperature gradients as a function of stack position shows very good agreement. The stack position (0.04 m) that corresponds to the maximum temperature gradient also agrees well between simulations and experiments. These results could be used to optimize the design and performance of thermoacoustic liners.

The results for the influence of stack length are presented in Figure 12. Note that for all other test settings, the stack is placed at 0.04 m, but for this test, the stack is placed at 0.015 m to accommodate the additional stack length while keeping other settings the same (as detailed in experiments section). In general, a longer stack length provides more pore surface area, which can lead to more significant temperature drops. However, as the stack length increases, the dissipative losses within the pores tend to mount up as well, degrading performance. For this specific configuration, the phase angle reverses as the stack length is increased beyond ∼0.04 m. This means that the location of the hot and cold sides of the stack switch between top or bottom, significantly impacting thermoacoustic output. Changing the stack length also shifts the locations of impedance mismatch due to the duct-stack-duct interfaces within the resonator leading to differences in the formation of nodes and antinodes of the standing wave patterns, which affects the pressure amplitude at the leading stack face. The sudden change in pressure amplitude at certain stack lengths (∼0.04 m) can be attributed to the interaction between the stack length and the standing wave pattern within the resonator. Therefore, optimal stack length is important to ensure that the acoustic power available at the leading stack face is maximized.

Figure 12.

Simulated (a) pressure amplitude and volume flow rate, (b) phase angle, (c) acoustic power, and (d) comparison of simulated and experimental temperature gradient, as a function of stack length.

The comparison with simulations for the influence of excitation or driving pressure on the thermoacoustic performance is presented in Figure 13. A nearly linear increase in both pressure amplitude and volume flow rate is seen with increasing driving pressure. At the same time, the phase angle is nearly constant, and therefore the acoustic power sees a nonlinear rise with driving pressure, which contributes towards achieving a higher temperature gradient at high driving pressure.

Figure 13.

Simulated (a) pressure amplitude and volume flow rate, (b) phase angle, and (c) acoustic power at the leading stack face and (d) comparison of simulated and experimental temperature gradients as a function of driving pressure.

However, the experimental temperature gradient exceeds the predictions at higher driving pressures (145 and 150 dB). This deviation reinforces the possibility that viscous heating and nonlinear effects, which are not captured in the simulations, contribute significantly to the temperature gradient at higher driving pressures. This provides an opportunity to add a simple viscous heating model in conjunction with DeltaEC to capture the temperature gradient predictions at higher driving pressures more accurately.

It is also to be noted that for Rott’s linear approximation, thermoacoustic parameters are accurately predicted for acoustic Mach numbers (particle velocities) up to 0.1. Figure 14 displays the predicted thermoacoustic parameters as a function of the drive ratio, which is the ratio of the peak pressure amplitude at the leading stack-face to the mean pressure. To fit with the assumption of linearity in DeltaEC, it is recommended that the drive ratio be less than 3%, which can be achieved by having an acoustic Mach number (M) less than 0.1 and a Reynolds number (Ry) less than 500.⁴⁹ As shown in Figure 14, the simulation correlates well with experiment up to 3% drive ratio but deviates above this value indicating the influence of nonlinear effects.

Figure 14.

(a) Simulated acoustic power at leading stack face and (b) comparison of simulated and experimental temperature gradient, as a function of drive ratio.

Absorption and thermal gradient

Figure 15 shows the comparison of the simulated and experimental temperature gradient and acoustic parameters as a function of frequency for the thermoacoustic performance liner. The trend for the temperature gradient aligns well with the simulations, although they tend to overpredict the gradient, likely due to unaccounted experimental losses. In the same frequency range, the resistance shows a slight drop and the reactance increases, while the absorption coefficient has an increasing trend. Simulations capture the trends favorably. Clearly, multiple reflections for the stack faces within the resonator could contribute to the primary absorption peak being associated with a shorter liner thickness and therefore resulting in the peak occurring at a higher frequency than this range.

Figure 15.

Comparison of simulated and experimental (a) temperature gradient, (b) normalized resistance and reactance and (c) absorption coefficient, as a function of excitation frequency.

As a measure of the influence of stack position within the resonator on the absorption, comparison of simulated and experimental acoustic parameters for varying stack positions is shown in Figure 16. They generally agree well but the experimental absorption is markedly higher than the predictions, likely due to losses that contribute to the absorption. From an acoustic perspective, it seems preferable to position the additional impedance due to the stack to be closer to the driver end in order to obtain slightly better absorption. However, a balance would need to be struck with the ideal position of the stack for thermoacoustic performance (∼0.04 m) in order to extract the best combination for a specific device implementation.

Figure 16.

Relationship between acoustic absorption and stack position: Comparison of simulated and experimental results for (a) normalized resistance and reactance, and (b) absorption coefficient for different stack positions.

Finally, Figure 17(a) and (b) illustrate the relationship between the absorption coefficient (α) and the acoustic power ( ${\dot{E}}_{2}$ ) available at the leading stack face, and the temperature gradient (ΔT) across the stack as a function of varying the excitation frequency ( $f$ ). At lower frequencies, the increase in acoustic power is more gradual as absorption increases. However, at higher excitation frequencies, the rate at which acoustic power grows becomes higher. At higher frequencies more acoustic power is available at the leading stack face within this frequency regime for the test article’s configuration, accompanied by an increase in absorption. Figure 17(b) shows the variation of the temperature gradient and absorption coefficient with excitation frequency. Notably, the temperature gradient peaks around 790 Hz corresponding to the optimal frequency where thermoacoustic resonance maximizes the temperature gradient. Beyond this peak value, the temperature gradient decreases, likely also due to increased energy loss in the conversion process due to viscous effects at higher frequencies.

Figure 17.

Relationship between acoustic absorption and (a) acoustic power ( ${\dot{E}}_{2}$ ) at the leading stack face and (b) temperature gradient (ΔT) as a function of the excitation frequency ( $f$ ).

Conclusion

This study presents the characterization of a thermoacoustic liner, exploring both its acoustic and thermoacoustic response through experiments and simulations. A compact, additively-manufactured thermoacoustic liner test article was designed and fabricated by embedding a porous stack within modular waveguide sections that make up the liner. The influence of factors such as frequency, sound pressure level as well as stack length and position on the acoustic and thermoacoustic performance was considered. Experimental evaluation of the test article was conducted using the normal incidence impedance tube as per the two-microphone method. Tonal dwells to steady state confirm that the peak temperature gradient (∼9.5°C) coincided with thermoacoustic resonance (790 Hz) for the liner. For the liner configuration studied, a stack position close to the middle of the resonator was found to have the best efficiency. Simulations using DeltaEC were found to correlate well with the experimental trends, although unmodeled losses contributed to deviations. At higher excitation pressure levels, the contribution of viscous heating to the deviation in the thermoacoustic gradient was observed to be significant and needs to be modeled in conjunction with Rott’s linear approximation in order to be captured. In all cases, the acoustic power available at the leading stack-face was found to correlate positively with the thermoacoustic temperature gradient achieved. Further, the relationship between the acoustic absorption and the thermoacoustic temperature gradient was examined. It was found that in application scenarios requiring multifunctional optimization, a tradeoff may be required in terms of the absorption in order to maximize the temperature gradient.

Footnotes

Acknowledgments

Thanks are also due to the members of the staff at the Liner Technology Facility at NASA Langley for their assistance in conducting the experiments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation (2033399).

ORCID iD

James M Manimala

Appendix

References

Huff

Sutliff

Jones

, et al. Acoustic liner for turbomachinery applications. NASA Technical Brief LEW-18438-1. Cleveland, OH: John H. Glenn Research Center, 2010.

Sadeghian

. Technologies for aircraft noise reduction. A Review. Journal of Aeronautics and Aerospace Engineering 2020; 9: 01–10.

Huff

DL.

Noise reduction technologies for turbofan engines. In 35th International Congress and Exposition on Noise Control Engineering (INTER-NOISE 2006) 2007; (No. E-15787).

Özkaya

Gauger

Hay

, et al. Efficient design optimization of acoustic liners for engine noise reduction. American Institute of Aeronautics and Astronautics 2020; 58(3): 1140–1156.

Greaves

Petersson

. Performance of acoustic liners for jet engine intake ducts. J Acoust Soc Am 1999; 105(2): 1141.

. Development of acoustic liner in aero engine: a review. Sci China Technol Sci 2020; 63: 2491–2504.

Jones

Watson

Nark

, et al. A review of acoustic liner experimental characterization at NASA Langley. NASA, 2020. Technical Publication NASA/TP–2020–220583.

Guo

Zhang

Fang

, et al. Reflected wave manipulation by inhomogeneous impedance via varying-depth acoustic liners. J Appl Phys 2018; 123(17): 174902.

Bravo

Maury

Pinhède

, et al. Micro-perforated materials for the reduction of flow-induced noise. Proceedings of meetings on acoustics 22ICA. Acoustical Society of America, 2016.

10.

Sutliff

Nark

Jones

, et al. Design and acoustic efficacy of a broadband liner for the inlet of the DGEN aero-propulsion research turbofan. In: 25th AIAA/CEAS Aeroacoustics Conference, Rome, Italy, 2019.

11.

Nark

Jones

Sutliff

. Broadband inlet liner design for the DGEN aero-propulsion research turbofan. In: AIAA/CEAS Aeroacoustics Conference. Atlanta, Georgia, 2018, p. 3608.

12.

Bozak

Podboy

. Evaluating the aerodynamic impact of circumferentially grooved fan casing treatments with integrated acoustic liners on a turbofan rotor. In: Turbo Expo: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2019.

13.

Liu

Yan

, et al. Design of tunable acoustic liner and adaptive control system. Acoust Aust 2021; 49: 83–91.

14.

Rego

Avallone

Ragni

, et al. Acoustic liners for jet-installation noise reduction. J Sound Vib 2022; 537: 117189.

15.

Guo

Fang

Jiang

, et al. An investigation on noise attenuation by acoustic liner constructed by Helmholtz resonators with extended necks. J Acoust Soc Am 2021; 149(1): 70–81.

16.

Lam

Leung

Fan

, et al. Effect of back cavity configuration on performance of elastic panel acoustic liner with grazing flow. J Sound Vib 2021; 492: 115847.

17.

Fan

Lam

Leung

. Spatio-temporal aeroacoustic-structural responses of cavity-backed elastic panel liner exposed to grazing duct flow. J Fluid Struct 2021; 102: 103228.

18.

Zhang

Wang

, et al. Effects of acoustic liner on thermoacoustic instabilities in a premixed swirl combustor. Aero Sci Technol 2021; 118: 107070.

19.

Neubauer

Genßler

Radmann

, et al. Experimental and numerical investigation of novel acoustic liners and their design for aero-engine applications. Aerospace 2022; 10(1): 5.

20.

Nark

Jones

. An acoustic liner design methodology based on a statistical source model. Int J Aeroacoustics 2021; 20(5–7): 441–457.

21.

Simon

Ghouali

Fascio

, et al. Development and assessment of a low frequency acoustic liner design for landing systems noise minimization. J Acoust Soc Am 2020; 148(4): 2539.

22.

Biswas

Manimala

. Quiet power: exploring the feasibility of a noise-mitigating, thermoacoustic energy harvester. J Acoust Soc Am 2022; 151(4): A179.

23.

Kajurek

Rusowicz

. Experimental investigation on the thermoacoustic effect in easily accessible porous materials. Energies 2020; 14(1): 83.

24.

Alamir

. Thermoacoustic energy conversion devices: novel insights. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 2021; 77(2): 130–144.

25.

Ramadan

Bailliet

Poignand

, et al. Design, manufacturing and testing of a compact thermoacoustic refrigerator. Appl Therm Eng 2021; 189: 116705.

26.

Sbrockey

Salagaj

Tompa

, et al. Synthesis and characterization of graphene-based thermoacoustic devices. J Cryst Growth 2018; 493: 41–44.

27.

Bansod

Raut

. Review on thermoacoustic refrigeration. International Journal of Innovations in Engineering and Science 2017; 2(3): 18–24.

28.

Karlsson

Åbom

Lalit

, et al. A note on the applicability of thermo-acoustic engines for automotive waste heat recovery. SAE International Journal of Materials and Manufacturing 2016; 9(2): 286–293.

29.

Babaei

Siddiqui

. Design optimization of an automotive thermoacoustic air conditioning system. Can Acoust 2007; 35(3): 166–167.

30.

Zhang

, et al. A 1 kW-class multi-stage heat-driven thermoacoustic cryocooler system operating at liquefied natural gas temperature range. Appl Phys Lett 2015; 107(3): 033905.

31.

Chen

Tang

Mace

, et al. Multi-physics coupling in thermoacoustic devices: a review. Renew Sustain Energy Rev 2021; 146: 111170.

32.

Alamir

Sidik

NAC

. Thermoacoustic refrigerators and heat pumps: new insights for a high performance. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 2021; 78(1): 146–156.

33.

Kisha

Hann

. Thermoacoustic electricity generator for rural dwellings in the developing world. In: Advances in Heat Transfer and Thermal Engineering. Singapore: Springer, 2021, pp. 893–897.

34.

Biswas

Krawczyk

Manimala

. Evaluation of additively manufactured stacks for thermo-acoustic devices. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, Institute of Noise Control Engineering, Washington, D.C, 2021, pp. 4376–4387.

35.

Watanabe

Shoji

Biwa

, et al. Rotational dynamics of a looped-tube thermoacoustic engine with a flywheel. Phys Rev Appl 2022; 18(4): 044079.

36.

Huang

Yang

, et al. Generalized thermoacoustic heat engines with unconventional working substances: a review. Appl Energy 2023; 347: 121447.

37.

Chen

Tao

Wang

, et al. Full-scale numerical simulations of standing-wave thermoacoustic engines with circular-pore and pin-array stacks. Int J Heat Mass Tran 2024; 228: 125605.

38.

Ding

Chen

Kang

, et al. Research on thermoacoustic refrigeration system driven by waste heat of industrial buildings. Sustain Energy Technol Assessments 2023; 55: 102971.

39.

Xiao

Luo

, et al. A highly efficient heat-driven thermoacoustic cooling system. Cell Reports Physical Science 2024; 5(2): 3.

40.

Jones

Brown

Howerton

, et al. Over-the-rotor liner investigation via the NASA Langley normal incidence tube. NASA Technical Memorandum NASA/TM–2019-220430. Langley Research Center, Hampton, VA; 2019.

41.

Wheatley

Hofler

Swift

, et al. Understanding some simple phenomena in thermoacoustics with applications to acoustical heat engines. Am J Phys 1985; 53(2): 147–162.

42.

Jones

Parrott

. Evaluation of a multi-point method for determining acoustic impedance. Mech Syst Signal Process 1989; 3(1): 15–35.

43.

Swift

. Thermoacoustics: a unifying perspective for some engines and refrigerators. New York: American Institute of Physics Press, 2003.

44.

Swift

. Thermoacoustic engines. J Acoust Soc Am 1988; 84(4): 1145–1180.

45.

Clark

Ward

Swift

. Design environment for low-amplitude thermoacoustic energy conversion (DeltaEC). J Acoust Soc Am 2007; 122(5): 3014.

46.

Pierce

Beyer

. Acoustics: an introduction to its physical principles and applications. Springer, 1990.

47.

Rott

. Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z Angew Math Phys 1969; 20(2): 230–243.

48.

Tijani

MEH

Zeegers

JCH

De Waele

ATAM

. Design of thermoacoustic refrigerators. Cryogenics 2002; 42(1): 49–57.

49.

Chambers

Manimala

Jones

. Design and optimization of 3D folded-core acoustic liners for enhanced low-frequency performance. American Institute of Aeronautics and Astronautics 2020; 58(1): 206–218.