PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Study of the relation between entropy flux density production andthermal efficiency of a thermoacoustic engine
Mariko SENGA1; Shinya HASEGAWA21, 2

Tokai University, Japan

ABSTRACTA traveling wave thermoacoustic device, which traveling wave performs energy conversion with theregenerator (a bundle of narrow channels) having a temperature gradient instead of a piston. This device
can realize Carnot cycle, which is an isothermal reversible cycle, ideally. However, irreversibility exists inthe actual traveling wave thermoacoustic device, and the isothermal reversibility of energy conversion canbe considered by the ratio of the thermal penetration depth δα and the channel radius r in the regenerator. Inorder to make the actual traveling wave thermoacoustic device approach the Carnot cycle, it is necessary toset r/δα in the regenerator to realize the small irreversibility due to viscous dissipation and the isothermal
thermodynamic cycle. In this study, entropy production is focused as an indicator of reversibility /irreversibility in the regenerator. The thermal efficiency and the entropy production in the regenerator wereanalyzed by changing r/δα, and the relationship between irreversibility and thermal efficiency wasinvestigated. In analysis results, it was found that r/δα of the maximum point of the thermal efficiency andthe minimum point of the entropy production almost agrees, and the thermal efficiency can be evaluated byusing the minimum point of the entropy production.
Keywords: Thermoacoustic engine, Thermal efficiency, Entropy production

1. INTRODUCTION
A traveling wave thermoacoustic device that performs energy conversion with a regenera tor (abundle of narrow channels) having a temperature gradient using traveling wave sound waves insteadof pistons can realize Carnot cycle, which is ideally a isothermal reversible cycle (1). This device
has the advantage of high efficiency (2) and simple structure. Recently, many research is promoted
as a waste heat recovery device actively (3, 4). The reversibility of the energy conversion of thetraveling wave thermoacoustic device can be considered by the thermal penetration depth δ α and thechannel radius r of the regenerator (5). In the case of r/δ α >> 1, the heat from the channel wall is nottransferred to the gas in the channel, resulting in an adiabatic reversible cycle. Because of this reason,
the energy conversion cannot be performed. On the other hand, in r/δ α << 1, the thermodynamic
cycle is isothermal because the heat of the channel wall is easily transferred to the gas in the channel.
However, if r/δ α is too small, viscous dissipation becomes dominant and the irreversibility due todissipation becomes large (5, 6). Therefore, it is necessary to set r/δ α appropriately in order to bringthe actual traveling wave thermoacoustic device closer to the Carnot cycle.
In general heat engines, the entropy production is used as an indicator of thereversibility/irreversibility. If the entropy production is large, the irreversibility becomes large . Since
the thermoacoustic device is also heat engine, it is possible to discuss reversibility / irreversibility byusing the entropy production.
In this study, the thermal efficiency and the entropy production in the regenerator were analyzed
by changing r/δ α, and the relationship between irreversibility and thermal efficiency wasinvestigated.

2. ANALYSIS METHOD
In this study, analysis of the entropy production and the thermal efficiency under the constraint
1
2

senga.mariko@gmail.com
s.hasegawa@tokai-u.jp5615

condition is based on the linear thermoacoustic theory (5, 6, 7). In analysis, only one flow channel ofthe regenerator is considered. By using the linear thermoacoustic theory, dP/dx (gradient of complex
pressure amplitude P), dU/dx (gradient of cross-sectional mean complex velocity amplitude U) andcross-sectional mean complex temperature amplitude T are expressed as follows. x is defined as thepropagation direction of the acoustic wave as positive along the axial direction of the flow channel.

dP െjωρmൌU,dx
1 െ χν(1)χα െ χνdUjω
1 dTmൌെൣ1 െ ሺγ െ 1ሻχα ൧P ൅UǡdxPm
൫1 െ χν ൯ሺ1 െ σሻ Tm dx(2)൫1 െ χα ൯ െ ߪ൫1 െ χν ൯ 1 dTm
1
൫1 െ χα ൯P െ ቆቇPǤCP ρmjω dx
൫1 െ χν ൯ሺͳ െ ߪሻ(3)T ൌHere, j: imaginary unit, ω㸸angular frequency, ρ m㸸mean density, γ㸸specific heat ratio, P m㸸meanpressure, T m 㸸 cross-sectional mean temperature, Cp 㸸 isobaric specific heat, σ㸸Prandtl number.
Furthermore, the parameters χ α, χν are thermoacoustic functions related to thermal diffusion andviscosity depending on r / δ α, ν (5, 6). The energy conversion between heat flux density Q and work
flux density I in the regenerator of the thermoacoustic engine can be written as Equation (4)corresponding to the first law of thermodynamics.
dሺQ ൅ I ሻ ൌ 0 ,dx
1෩U൧ ǡI ൌ ReൣP
21෩ ൧ െ I ǤQ ൌ CP ρm ReൣTU
2(4)

(5)

(6)

Here, if the entropy flux density S is Q/T m, the entropy flux density production per unit length dS/dx is
expressed as follows (8);
dSd Q
1 dQ
Q dTm
1 dI
Q dTm
ൌ ൬ ൰ൌ
െ 2ൌെെ 2.
dx dx Tm
Tm dx
Tdx
dx
Tm
Tm dxm(7)

Therefore, thermal efficiency η is given by the ratio of increment of I and amount of heat input Q inthe regenerator.
dIηൌቌቍ.
dI
െQ ൅ 2(8)dS/dx and η can be analyzed by substituting Equations (1)~(6) to Equations (7) and (8). In the analysisof dS/dx and η, it is assumed that the phase difference between P and U is zero in order to consider theenergy conversion by traveling wave. In this study, η and dS/dx are analyzed by changing r/δ α , theirdependence on r/δ α and the relationship between them was investigated. The analysis conditions areshown in Table1.

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Table 1 – Parameter in analysis
Tm

450 K

ρm

1.08 kg/m3dTm

300 K

Pm

1.01 MPa

dx

50 mm

P5000 Pa

γ

1.65

U

0.40 m/s

λ4.95r

5.23×10-3~5.23 mm(r / δα = 0.01 ~10)Cp

ω5232 J/(kg࣭K)

251.3 rad/s (= 40 Hz)

3. ANALYSIS RESULTS

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00.01

1600
1400
dS / dx [W/(m3 ʀK)]

ηFigure 1 shows the results of η and Fig. 2 shows dS/dx, respectively. The horizontal axis in bothfigures is r/δ α. The maximum point of η is shown by black circle in Fig. 1. From this figure, η ismaximized at r/δ α㸻0.19. This reason is considered from dS/dx. In Fig. 2, the minimum point dS/dx is
expressed by black triangle. The range in which dS/dx decreased is 0.05 < r/δ α < 1.7, and dS/dx
becomes minimum value at r/δ α㸻0.17. As described above, it is considered that dS/dx becomes largein r/δ α ≤ 0.05 because the viscous dissipation is dominant. On the other hand, dS/dx is small in 1.7 ≤r/δ α. However, in this region r / δ α, the energy conversion between I and Q is not performed since thethermodynamic cycle is adiabatic cycle. For this reason, η is decreased in this region. Furthermore, bycomparing the value r/δ α in Fig. 1 and Fig. 2, the r/δ α minimum point of dS/dx and maximum point ofη are almost same. From these results, η can be evaluated by using the minimum point of dS/dx.

12001000
8006004002000.11r / δαFigure 1 – η vs. r / δα1000.010.1110r / δαFigure 2 – dS/dx vs. r / δα

4. CONCLUSIONSIn this study, for the traveling wave thermoacoustic engine, the thermal efficiency and the entropy
production under a constraint condition in the regenerator were analyzed and the relation between
them was investigated. As results, the thermal efficiency is maximum at r/δ α㸻0.19, and the entropy
production became minimum at r/δ α㸻0.17 in the analysis range of this study. Therefore, it was foundthat r/δ α of the maximum point of the thermal efficiency and the minimum point of the entropy
production almost agrees, and the thermal efficiency can be evaluated by using the minimum point ofthe entropy production.

ACKNOWLEDGEMENTS
This research was partially supported by JSPS KAKENHI Grant Number JP16H06070.

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