# Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications

## Abstract

**:**

## 1. Introduction

_{he}) near 18%. Standard electrodynamic loudspeakers (LSs) operated in reverse mode can be also used in lieu of LAs, but in low-power applications (a few hundred watts). This is due to the fact that an LS, generally designed for high-fidelity applications, is characterized by a weak and fragile cone, a short stroke, low power handling, and a low acoustic-to-electric efficiency (η

_{ae}). In any case, when a particular application requires low power levels, low cost for generated kW

_{e}, and not very high transduction efficiencies (~50%) the use of LSs could be justified. Relevant examples of this new class of electricity generators integrating LSs in TA engines are the prototypes developed by Yu et al. [6] and Kang et al. [7] working at thermal-to-electric efficiencies (η

_{he}) below 5%.

## 2. Theoretical Modeling of Linear Alternators

_{0}as

_{L}, the canonical equations assume the form of the following two linear equations [8]

_{1}and p

_{2}are the complex amplitudes of the acoustic pressures acting on the front and back sides, respectively, of the diaphragm of area A

_{d}, I is the complex amplitude of the current flowing in the voice coil, U is the complex amplitude of the volumetric velocity due to the diaphragm motion, Bl is the “force factor” (the product of the magnetic field, B, times the length of the voice coil wire, l), and Z

_{e}(the electrical impedance with blocked diaphragm) and Z

_{m}(the mechanical impedance with open electric circuit) are

_{e}and L

_{e}being the electrical resistance and inductance of the voice coil, R

_{m}the mechanical resistance of the system, m the mass of the diaphragm and voice coil, and K

_{m}the mechanical stiffness of the elastic suspensions.

_{pU}is the phase angle between the pressure difference (p

_{1}− p

_{2}), and U and the tilde symbol indicate complex conjugation. Substitution of Equation (6) into Equation (7) provides

_{m}, another fraction is consumed in the coil electrical resistance R

_{e}, and the remaining fraction is extracted by the load R

_{L}as electric power. The last term can be also written as

_{ae}, W

_{a}, and W

_{e}) grows with increasing force factor (Bl) and with decreasing mechanical (R

_{m}) and electrical (R

_{e}) resistances (in the ideal case R

_{m}= R

_{e}= 0 the device would have an acoustic-to-electric efficiency of 100%). The performance also increases when the electrical reactance (X

_{e}) is decreased. An additional condenser could be added to the coil circuit to use the alternator in favorable electrical resonant conditions (X

_{e}= 0) at the selected operation frequency [9] (even if the inductance of the voice coil, ωL

_{e}, is generally negligible at the typical working frequencies of TA devices for standard woofers and sub-woofers). Finally, the amount of acoustic power absorbed by the alternator also grows with decreasing mechanical reactance X

_{m}so the performance of the device is highest when it works in both mechanical and electrical resonant conditions (X

_{m}= 0, X

_{e}= 0) for which the alternator acoustical impedance becomes real (p

_{1}− p

_{2}in phase with U).

^{2}/R

_{e}R

_{m}.

_{max}| the maximum admissible diaphragm excursion (the corresponding maximum volume velocity being |U

_{max}| = ωA

_{d}|x

_{max}|), the maximum electrical output power is

_{0}is the mean density, c is the sound velocity, γ is the ratio of isobaric to isochoric specific heats, T

_{0}is the mean temperature, β is the thermal expansion coefficient, Pr is the Prandtl number, c

_{P}is the gas isobaric specific heat, K and K

_{s}are the thermal conductivity of the gas and solid, respectively, H is the enthalpy flux along the direction of acoustic vibration x, f

_{κ}and f

_{ν}are spatially averaged thermal and viscous functions depending on the geometry of the gas pores, A and A

_{s}are the area of the transverse section open to gas flow and obstructed by the solid, respectively, ε

_{s}is the ratio of specific heats per unit area of gas and solid, and q is the heat exchanged with external thermal reservoirs along the direction normal to x per unit length in the heat exchangers.

## 3. Acoustic Impedance Matching

_{1}− p

_{2}) across the LA and the local volume velocity (U) must always satisfy Equation (6). This requires high-impedance (or non-compliant) transducers, with high masses and suspension stiffness, to be preferably installed near high-impedance regions of the acoustic field and, conversely, low-impedance (or ultra-compliant) transducers, with low masses and suspension stiffness, to be preferably installed near low impedance regions of the acoustic field [1].

_{e}, and the maximum diaphragm excursion |x

_{max}|. Observing that at the operation angular frequency ω the following two equations have to be verified

_{L}, the operation frequency f compatible with the target output power and the maximum allowable stroke of the diaphragm. The f, and η

_{ae}values calculated through Equations (23) and (11) respectively, corresponding to the target output powers W

_{e}(=50 and 100 W), are reported in Figure 1 as a function of R

_{L}. The volume flow rates generating the target power outputs can then be calculated by substituting these frequencies in Equation (21).

_{1}and p

_{2}and U near zero, that enhances the acoustic power absorbed by the LA. For configurations involving LAs with enclosed housings, this condition could be met by adjusting the combined LA/housing impedance. The gas-spring effect caused by the back volume V

_{b}of the housing is equivalent, in fact, to an increment of the suspension stiffness of the LA given by

_{b}.

_{a}(=R

_{a}+ jX

_{a}) to match the above condition has however a strong impact on the engine efficiency in converting heat to acoustic power (η

_{ha}) since Z

_{a}acts as an acoustic load to the TA engine. Therefore, it contributes in determining the resonance frequency of the integral acousto-mechanical system and influences the structure of the acoustic field near the stack. So, an improper maximization of η

_{ae}and W

_{a}could produce the detrimental effect of degrading the η

_{ha}performance. Proper impedance matching requires instead that the working conditions of the coupled system be optimal both for the TA engine in converting heat to acoustic power and the LA in converting acoustic power to electrical energy. Therefore, once a range of optimal working conditions for the LA alternator are selected, the resulting ω and Z

_{a}values should be used as inputs in the optimization procedure of the TA engine for realizing impedance matching at the location where the LA is installed. In general, the above requirements cannot be met simultaneously and compromises should be made among them in the design phase when coupling the LA to the TA engine.

## 4. Results and Discussion

_{e}= 50 W for one-stage engines and W

_{e}= 100 W for two-stage engines. All the devices are standing wave-type engines (characterized by a 90° out-of-phase relationship between pressure and velocity oscillations) operated with air at atmospheric pressure. This choice meets the goal of the present study of developing low-cost TAEGs. The use of near atmospheric air, in fact, avoids the use of costly pressure vessels, eliminates the problems associated to the availability and cost of noble (or other exotic) gases, and reduces the engines’ size (and associated cost) due to the relatively low sound velocity.

_{0}, equal to 1 mm and porosity 59% to avoid much too specific manufacturing requisites and relative high costs. The HHX temperature (T

_{H}) is assumed at a maximum to be equal to 500 °C to not exceed the tolerances of typical materials used for its construction. The AHX temperature (T

_{A}) is assumed to be maintained at T

_{A}= $295\mathrm{K}$ and to be made of copper. A commercially available LS (the B&C 6PS38 woofer manufactured by B&C speakers) is selected as an LA due to its relatively low mechanical resistance R

_{m}, low electrical resistance R

_{e}, high force factor Bl, high η

_{max}parameter (=72.5%) and relatively small dimension (7-inch nominal diameter). The parameters of the LS are listed in Table 1.

#### 4.1. One Stage Engine

_{L}) to make the imaginary part of the input impedance of the coupled LA/housing negative.

_{L}and back volume V

_{b}) have been varied in order to find the optimal matching between the LA and the engine. In each simulation, the output electrical power and the peak-to-peak stroke of the diaphragm are fixed to 50 W and 12 mm, respectively, while the HHX and AHX lengths are modeled to match exactly the local peak-to-peak acoustic particle amplitude [1]. The geometrical dimensions of the engine deriving from the optimization procedure are given in Table 2. Results concerning the TAEG performance are shown in Figure 3 where the power values W

_{a}, W

_{e}, the efficiencies η

_{ha}, η

_{ae}, η

_{he}, and the impedance parameters |Z

_{a}|, ϕ

_{pU}are plotted as a function of f.

_{ha}of the engine exhibits a maximum at f ≈ 180 Hz (with R

_{L}= 42.2 Ω). The behavior of η

_{ha}vs. f can be interpreted observing that decreasing the frequency starting from high values has the effect of reducing the absolute value of the phase angle ϕ

_{pU}. This should have the positive effect of increasing the amount of acoustic power absorbed by the LA. This circumstance explains the observed increase of η

_{ha}. Further reduction of the frequency (below 180 Hz) has however the detrimental effect of decreasing η

_{ha}. This trend reflects the worsening of the stack performance in converting heat to sound. For a stack with square pores of hydraulic radius R

_{h}, in fact, the best performances should be reached when R

_{h}≈ δ

_{κ}[12]. For the working fluid modeled in the present study (air at atmospheric pressure) the above condition is matched at frequencies ≥200 Hz. So, when the frequency is lowered below 180 Hz, the reduction in the performance of the stack dominates over the increase in the ability of the alternator in absorbing acoustic power and a fall in η

_{ha}is observed.

_{L}) (f ≈ 165 Hz, R

_{L}= 33.4 Ω). Simulations reveal (results not shown) that that the optimal load resistances are related to the working frequency by the same functional dependence shown in Figure 1. This guarantees that in each configuration the LA is working at its highest acoustic-to-electric conversion efficiency that explains the behavior of η

_{ae}vs. f. Note that the phase angle ϕ

_{pU}between the driving pressure and volume velocity is quite high (82°). In this case, the regulation operated by the back volume is evidently ineffective to fulfill at the same time the requirement of a high impedance (dictated by the LA position) and of a low X

_{a}(dictated by acoustic power enhancement purposes). On the other hand, the last requirement (low X

_{a}) is not compatible with the dynamics of the gas oscillations (standing-wave oscillations).

_{L}= 39.2 Ω) where the global η

_{he}maximizes (η

_{he}≈ 4.6%) is clearly a compromise between the operative conditions maximizing η

_{ha}and the ones maximizing η

_{ae}. The high W

_{a}values found at high and low frequencies simply reflect the drop in performance of the engine in these frequency ranges.

#### 4.2. Two-Stage Engine with Alternator Coupled in Push-Pull Arrangement

_{max}(results not shown). A two-stage engine with a stack/HXs assembly placed on each side of a λ/2 standing wave resonator and the LA installed in the middle section (see Figure 4) is firstly considered in this study. Since the middle sections of the resonator are low-impedance regions of the standing wave, the acoustic coupling of the LA to the engine requires that it behave as a low-impedance transducer. This configuration where the acoustic pressures acting on both sides of the LA have the same amplitude |p| but are 180° out of phase is called “push–pull” coupling mode.

_{a}absorbed by the alternator compared to the one-stage case when considering that, with p

_{1}and p

_{2}being in phase opposition, |p

_{1}− p

_{2}| = 2|p| results, and observing that in the one-stage configuration the LA is driven by the pressure acting only on one side [13]. Compared to the one-stage case, however, this arrangement does not allow regulations of the alternator impedance through regulations of the acoustic compliance introduced by the back volume. In the simulations performed by DeltaEC for each working frequency f, the engine parameters (stack location, stack length, loudspeaker location x

_{L}) and the loudspeaker parameters (load resistance R

_{L}) have been varied in order to find the optimal matching between the LA and the engine. In each run, the peak-to-peak stroke of the diaphragm is fixed to 12 mm while the HHX and AHX lengths are varied, as in the one-stage case, to match exactly the local peak-to-peak acoustic particle amplitude. Furthermore, the distance of the alternator from the AHX (equal on both sides) is modeled to be not lower than 8 cm to allow the accommodation of the alternator in a real technical implementation.

_{ha}, η

_{ae}, the power values W

_{a}, W

_{e}, and the impedance parameters |Z

_{a}|, ϕ

_{pU}are plotted as a function of f. The graph shows how the TAEG is characterized by a very low efficiency in converting heat to electricity (η

_{he}≈ 2% at a maximum with η

_{ha}≈ 3% and η

_{ae}≈ 67.6%) and the maximum obtainable electric output power amounts to W

_{e}= 35.7 W (with W

_{a}= 54.1 W). Note that in this case the efficiencies η

_{ha}, η

_{ae}, η

_{he}are referred to maximum power output. These performance levels are considerably lower than the ones achieved by the one-stage engine previously considered. This result can be interpreted on the basis of the following observations:

- -
- The LA is installed in a low-impedance region of the standing wave so the acoustic coupling is realized for relatively low values of |Z
_{a}| (|Z_{a}| ≈ 10^{4}Pa s/m^{3}compared to |Z_{a}| ≈ 10^{5}Pa s/m^{3}of the one-stage case). This entails that at a given working frequency the stroke of the alternator reaches the maximum allowable value |x_{max}| for low values of the driving pressure |p_{1}− p_{2}| (|p_{1}− p_{2}| ≈ 10^{3}Pa compared to |p_{1}| ≈ 10^{4}Pa of the one-stage case). This precludes the stacks from generating high acoustic power levels which scale proportionally to |p|^{2}[1]. - -
- In both the one-stage and two-stage engine simulations the operation volume velocities are almost equal (|U| ≈ 9 × 10
^{−2}m^{3}/s). This circumstance derives from having fixed in simulations the peak-to-peak stroke to its maximum allowable value 2|x_{max}|. So, since the two engines work at comparable operation frequencies, the same volume velocities are involved (|U| = ω A_{d}x_{max}). - -
- The absolute value of the phase angle ϕ
_{pU}between the driving pressure and the volume velocity for which the impedance matching is realized does not change appreciably with the frequency and is approximately equal in the one-stage and two-stage cases, amounting to around 82°.

_{L}for which the impedance matching is realized are not related to the working frequency by the functional dependence shown in Figure 1, which contributes to the degradation of overall performance. The interpretation of the behavior of the efficiencies η

_{ha}, η

_{ae}with f parallels the one made in the previous subsection.

#### 4.3. Two-Stage Engine with Loudspeaker Coupled in Side Branch Arrangement

_{b}= R

_{b}+ jX

_{b}, is then used in the optimization process of the engine. Making the choice of operating the LA at an acoustic-to-electric conversion efficiency of at least 70%, the plots of Figure 1 allow us to determine the corresponding values of the operation frequency (f = 194.5 Hz) and load resistance (R

_{L}= 19 Ω) compatible with the target output power (100 W) and maximum allowable stroke (±6 mm). The driving pressure needed to match the target output power is |p

_{1}| ≈ 3 kPa, which corresponds to a volume flow rate amplitude of |U| = 0.0968 m/s).

- -
- The impedance at the branch input section should be sufficiently higher than the local trunk impedance to preserve the acoustic wave in the trunk from being distorted and the branch from absorbing too great amounts of power that could lead to unwanted high onset temperatures or, worse, to a not-starting engine. At the same time, the impedance should be not too large to allow the target acoustic power to enter the branch with (as far as possible) low driving pressures (which reduce termoviscous dissipation on the resonator walls proportional to |p|
^{2}). So, a trade-off between these two requirements has to be found. - -
- The impedance at the other side of the branch, where the LA is installed, should match the low impedance of the transducer. The latter can be adjusted, as in the one-stage case, by regulation of the back volume compliance with the aim of decreasing ϕ
_{pU}. The back volume compliance, however, also affects the input impedance of the branch and this leads again to the issues discussed at the previous point. Hence, also in this case a trade-off between different requirements has to be found.

^{5}Pa s/m

^{3}and 44.5°, respectively (R

_{b}= 1.84 × 10

^{5}Pa s/m

^{3}; X

_{b}= 1.81 × 10

^{5}Pa s/m

^{3}). The back volume, modeled by a compliance, is adjusted to bring the resonance frequency of the composed LA/housing near the target operation frequency (194.5 Hz). For the selected back volume (equal to 0.0015 m

^{3}after subtraction of the LA volume) the resonance frequency of the LA/housing system is about 187.6 Hz and the phase angle between p

_{1}and U is near 10°. For this configuration, the alternator extracts 100 W of electric power when 153.6 W of acoustic power flow through the branch input section, the peak-to-peak stroke of the loudspeaker being 12 mm.

_{b}and X

_{b}the engine is then optimized by varying the stack location, the stack length, and the branch location while adjusting the HXs length to match exactly the local peak-to-peak acoustic particle amplitude. The geometrical dimensions of the engine deriving from the optimization procedure are given in Table 2 (excluding the branch whose dimensions are reported in Figure 8).

_{a}, W

_{e}and the conversion efficiencies η

_{ha}, η

_{he}are reported as a function of the heating temperature T

_{H}which varies as a consequence of heat input change (from 100 to 2200 W by steps of 100 W). In these simulations, the HX lengths were held fixed to the values relative to W

_{e}= 100 W. From the graph it can be observed how all the investigated parameters grow with T

_{H}(reflecting the growth of the stacks conversion efficiency with T

_{H}) and that the onset temperature of the engine falls around 360 °C. When the temperature of the HHX reaches 719 K the LA extracts 50 W of electrical power with the design η

_{ae}efficiency (70%) and with an overall η

_{he}efficiency equal to 4.3%. At this operative point the amplitude of the driving pressure in front of the diaphragm and its peak-to-peak stroke amount to 2 × 10

^{3}Pa and 8.7 mm, respectively. When T

_{H}= 773 K, the engine produces the target (maximum) power (W

_{e}= 100 W) with an the overall η

_{he}efficiency equal to 4.6%. In this case the amplitude of the driving pressure and its peak-to-peak stroke amount to 3 × 10

^{3}Pa and 12 mm, respectively.

## 5. Conclusions

_{a}| ≈ 10

^{5}Pa s/m

^{3}). This entails relatively high driving pressures (|p

_{1}| ≈ 10

^{4}Pa) which, in turn, enable the stack for producing high acoustic power levels. In this case, the phase angle ϕ

_{pU}is quite high (ϕ

_{pU}≈ 82°) as is typical of standing waves.

_{a}| ≈ 10

^{4}Pa s/m

^{3}), and this entails low driving pressures (|p

_{1}| ≈ 10

^{3}Pa). In this case, however, the branch decouples the acoustic field driving the alternator from the one sustained by the trunk, allowing the stacks to work under high pressure levels, which enables the production of high acoustic powers. Furthermore, the impedance change operated by the branch and by the compliance associated to the back volume brings a favorable ϕ

_{pU}near low values (~10°).

_{a}| ≈ 10

^{4}Pa s/m

^{3}) and the driving pressures are low (|p

_{1}− p

_{2}| ≈ 10

^{3}Pa). As a consequence, the stacks generate a low acoustic power which, with the phase angle ϕ

_{pU}being quite high (ϕ

_{pU}≈ 82°), is poorly absorbed by the LA. In this case, poor impedance matching is realized.

## Conflicts of Interest

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**Figure 1.**The operation frequencies of the loudspeaker giving rise to an electrical output power values of 50 and 100 W, and the acoustic-to-electric conversion efficiency as a function of the load resistance.

**Figure 2.**Schematic of the one-stage standing-wave thermoacoustic electricity generator (TAEG) model. AHX: ambient heat exchanger; HHX: hot heat exchanger.

**Figure 3.**The power values W

_{a}, W

_{e}, efficiencies η

_{ha}, η

_{ae}, η

_{he}, and the impedance parameters |Z

_{a}|, ϕ

_{pU}as a function of f.

**Figure 5.**The power values W

_{a}, W

_{e}, the efficiencies η

_{ha}, η

_{ae}, η

_{he}, and the impedance parameters |Z

_{a}|, ϕ

_{pU}as a function of f.

**Figure 7.**The pressure and volumetric velocity distributions along the branch to the alternator for f = 194.5 Hz and R

_{L}= 19 Ω.

**Figure 8.**The power values W

_{a}, W

_{e}and the efficiencies η

_{ha}, η

_{he}as a function of the heating temperature.

**Table 1.**Specifications of the loudspeaker. The parameter η

_{max}is calculated through Equation (13).

Parameters | Symbol | Value |
---|---|---|

Mechanical resistance | R_{m} (kg/s) | 0.562 |

Moving mass | m (kg) | 0.014 |

Mechanical stiffness | K_{m} (N/m) | 3099.3 |

Coil electrical resistance | R_{e} (Ω) | 5.4 |

Force factor | Bl (Tm) | 10.8 |

Coil inductance | L_{e} (H) | 0.0006 |

Resonance frequency | f_{0} (Hz) | 75 |

Maximum stroke | x_{max} (mm) | ±6 |

Diaphragm area | A_{d} (m^{2}) | 0.0132 |

Maximum transduction efficiency | η_{max} (%) | 72.5 |

TAEGs | Working Frequency (Hz) | R_{L} (Ω) | ||||
---|---|---|---|---|---|---|

One-stage TAEG | 174 | 39.2 | ||||

Two-stage TAEG with LA in push-pull mode (A) | 200 | 81.6 | ||||

Two-stage TAEG with LA in side branch (B) | 194.4 | 19 | ||||

One stage engine | Length (m) | Diameter (m) | Porosity | R_{h} (m) | Taper angle (deg) | Volume (m^{3}) |

Hot duct | 0.044 | 0.13 | - | |||

stack | 0.0523 | 0.881 | 0.00024 | |||

HHX | 0.0197 | 0.592 | 0.0005 | |||

AHX | 0.0287 | 0.592 | 0.0005 | |||

Ambient duct | 0.795 | 0.13 | ||||

Connecting cone | 0.03 | 0.13–0.147 | 31.5 | |||

Connecting duct | 0.01 | 0.147 | ||||

Back volume | 0.0005823 | |||||

Two-stage engine (A) | Length (m) | Diameter (m) | Porosity | R_{h} (m) | Taper angle (deg) | Volume (m^{3}) |

Hot duct | 0.0862 | 0.13 | ||||

stack | 0.0168 | 0.881 | 0.00024 | |||

HHX | 0.0117 | 0.592 | 0.0005 | |||

AHX | 0.01 | 0.592 | 0.0005 | |||

Ambient duct | 0.04 | 0.13 | ||||

Connecting cone | 0.03 | 0.16–0.147 | 31.5 | |||

Connecting duct | 0.01 | 0.147 | ||||

Two-stage engine (B) | Length (m) | Diameter (m) | Porosity | R_{h} (m) | Taper angle (deg) | Volume (m^{3}) |

Hot duct | 0.0446 | 0.13 | - | |||

stack | 0.047 | 0.881 | 0.00024 | |||

HHX | 0.0179 | 0.592 | 0.0005 | |||

AHX | 0.0251 | 0.592 | 0.0005 | |||

Ambient duct | 0.018 | 0.13 | ||||

Connecting cone | 0.189 | 0.13–0.11 | 7 | |||

Connecting duct | 0.294 | 0.11 | ||||

Back volume | 0.00151 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Piccolo, A. Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications. *Appl. Sci.* **2018**, *8*, 287.
https://doi.org/10.3390/app8020287

**AMA Style**

Piccolo A. Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications. *Applied Sciences*. 2018; 8(2):287.
https://doi.org/10.3390/app8020287

**Chicago/Turabian Style**

Piccolo, Antonio. 2018. "Study of Standing-Wave Thermoacoustic Electricity Generators for Low-Power Applications" *Applied Sciences* 8, no. 2: 287.
https://doi.org/10.3390/app8020287