Applied Mechanics and Materials Vol. 331 (2013) pp 166-169
© (2013) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.331.166
Effects of heat leakage and internal irreversibility on quality factor of
thermoacoustic system
Jinhua Fei1,a, Feng Wu2,b, and Tuo Wang1,c
1
School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan , China
2
School of Science, Wuhan Institute of Technology, Wuhan , China
a
[email protected], [email protected], [email protected]
Keywords: quality factor, thermoacoustic, heat leakage, irreversibility factor, resonant frequency.
Abstract: The quality factor of the thermoacoustic system is calculated, considering the
dissipations caused by viscous and thermal effects at the resonator walls, heat leakage and internal
irreversibility. The quality factor was analyzed in several situations. And it was analyzed
numerically with the stainless steel wire mesh as an example. It is found that the quality factor is
increased monotonically with the resonant frequency, and decreased monotonically with heat
leakage conductivity and irreversibility factor. The relationship between quality factor and the
regenerator packing material was analyzed, it is found that quality factor is increased monotonically
with porous materials characteristic factor. It is a certain theoretical significance for future research.
Introduction
Thermoacoustic engine is a new type engine, which achieve energy conversion depended on the
thermoacoustic effect [1]. In the process, the regenerator, as a resonant transducer, plays an
important role in driving and maintaining the self-excited oscillation. The regenerator periodically
changes the parameters of the resonance system with a certain frequency phase to transform its
energy storage state. It is called the frequency characteristic of resonance system [2]. And quality
factor, as a target to measure the energy storage capability of the oscillation system [3], combines
the energy stored in the system with the energy dissipated per cycle. It can be used to measure the
sharp level of the system resonance curve.
In the article, when calculating thermoacoustic system’s energy dissipation, it was calculated by
considering the viscous of the near-wall boundary layer in resonance tube and the thermal effects.
Moreover, the dissipations caused by heat leakage and system’s internal irreversibility were
considered. This article has been calculated the quality factor and study the influence of resonant
frequency, heat leakage, system’s internal irreversibility and regenerator packing material on quality
factor, and used the target to measure the performance of resonance tube.
Theoretical calculation for quality factor
A coordinate system can be established, making the center of the resonance tube as axes. According
to the oscillation characteristics of thermoacoustic system, the distribution of pressure and speed in
resonance tube could be obtained [4], as shown in Fig.1. For standing wave field, there are
（1）
p1 = P0 sin(2π x / λ ) ≡ p1s ( x )
u1 = i( P0 / ρ0 a) cos(2π x / λ ) ≡ iu1s ( x)
The quality factor is defined as [5]:
E
Q = 2π f 0 st
E
（2）
（3 ）
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,
www.ttp.net. (ID: 210.42.24.129, Wuhan Institute of Technology, Wuhan, China-16/04/13,02:27:48)
Applied Mechanics and Materials Vol. 331
167
which, f 0 is resonant frequency of the system, Est is the energy stored in the system, E is
dissipation energy per cycle. From the above equation, it could be known that Est and E should be
obtained firstly if we want to get the quality factor.
The time-average acoustic energy density of thermoacoustic system and resonant dissipation
energy per unit area are[4]
est =
1 ( p1s ) 2 1
+ ρ 0 (u1s ) 2
4 ρ0 a 2 4
（4）
er =
1 ( p1s )2
γ −1
1
δk
ω + ρ0 (u1s ) 2 δ vω
2
4 ρ0 a
1+ εs
4
（5）
Eq. (4) and Eq. (5) are integrated with volume, then the energy stored in the system and the
energy dissipated per cycle can be obtained:
E st =
Er =
1 p02
πR 2 L
4 ρ0a 2
 γ −1  2R 

1 p02 2
π f0 RL δk
1 +  + δ v 
2
L 
2 ρ0a
 1+ ε s 

（6）
（7）
which, p 0 is static pressure, ρ 0 is average density of the working gas, a is sound speed,
R and L are radius and length of resonance tube, δ k is thermal penetration depth, δ k = K / π f 0 ρ0 c p ,
K is the thermal conductivity of the gas, δ v is viscous penetration depth, δ v = µ / π f 0 ρ0 , µ
and γ are dynamic viscous coefficient and specific heat ratio of the gas, ε s is regenerator porous
material characteristic factor, ε s = ρ0c pδ k ρ s csδ s , δ s is thermal penetration depth of regenerator
filler, δ s = K s / π f 0 ρ s cs , cs and ρ s are specific heat capacity, density of filler
For Eq. (7), the system dissipation Er is only considered with viscous of the near-wall boundary
layer and thermal effects. For thermoacoustic system, there is direct heat leakage in energy
exchange process, it also has an impact on energy conversion of thermoacoustic engine. The
thermoacoustic engine energy flow is shown in Fig. 2.
Based on the Bejan linear heat leakage model [6], the heat leakage amount per unit time (heat
leakage flow rate) can be expressed as
q = α (T H − T L )
（8）
which, α is a longitudinal heat leakage thermal conductivity, TH and TL are high side and cold side
temperature of heat exchangers. The heat leakage amount per cycle can be obtained with Eq.(8).
El = qτ = α (TH − TL ) f 0
（9）
In thermoacoustic system, in addition to energy loss described above, due to the friction and
other non-balance movement, there are other irreversibility losses in system. When the time-average
acoustic energy density is constant, the actual resonance dissipation energy is larger than the
theoretical resonant dissipation value.
To express thermoacoustic system internal irreversibility, the irreversible factor is defined as:
ϕ =
E r′
≥1
Er
（10）
So the energy dissipation in thermoacoustic system per cycle is
E = ϕ Er + El
（11）
With Eq.(3), Eq.(6), Eq.(7), Eq.(9), Eq.(11), the equation of quality factor for thermoacoustic
system can be obtained:
Q = {ϕ [
δv
R
+
2 ρ 0α a 2 (T H − T L ) − 1
δ k ( γ − 1 ) 2δ k ( γ − 1 )
+
]+
}
π 2 f 0 2 P0 2 R 2 L
R 1+ εs
L (1 + ε s )
When quality factor is considered without heat leakage and irreversible factor, α = 0 , ϕ = 1
（12）
168
Q =[
2013 International Conference on Process Equipment, Mechatronics
Engineering and Material Science
δv
R
+
δ k ( γ − 1) 2δ k ( γ − 1) −1
+
]
R 1+ εs
L (1 + ε s )
（13）
When quality factor is considered with irreversible factor, but without heat leakage, α = 0 , ϕ > 1
Q = {ϕ[
δv
+
δ k (γ − 1) 2δ k (γ − 1) −1
+
]}
R 1+ εs
L (1 + ε s )
（14）
When quality factor is considered with heat leakage, but without irreversible factor, α ≠ 0 , ϕ = 1
Q = {[
R
δv
R
+
δ k ( γ − 1) 2δ k (γ − 1) 2ρ0α a 2 (TH − TL ) −1
+
]+
}
π 2 f 02 P02 R 2 L
R 1+ εs
L (1 + ε s )
（15）
Numeric Calculation
In the numerical simulation analysis, we choice half wavelength of stainless steel equal diameter
circular tube for resonator tube. The inner diameter is 76mm , the wall thickness is 3mm , the length
is 0.45 ~ 1.00m . The resonant frequency could be defined by L = λ / 2 and f = a / λ . Putting a
stainless steel wire mesh regenerator in the tube, ρs = 7.93×103 kg / m3 , cs = 508 J / (kg ⋅ K ) ,
Ks = 16.2W / (m ⋅ K ) . Choosing nitrogen as working gas, the pressure is 0.8MPa , TH = 450K , TL = 350K ,
γ =1.4 , a = 407.676m/ s ,
ρ0 = 6.7235kg / m3 , K = 32.51×10−3W / (m⋅ K) , cp =1.0510kJ / (kg ⋅ K) ,
µ = 21.67 ×10−6 Pa ⋅ s , take α and ϕ are: 0kW / K , 0.2kW / K , 0.5kW / K and ϕ = 1.0 , ϕ = 1.5 , ϕ = 2.0 .
The relationship between Q and f 0 is shown in Fig.3. The quality factor increases monotonically
with resonant frequency, the higher the resonance frequency, the larger the quality factor, so the
energy conversion capability of thermoacoustic system is bigger.
With Fig.4 and Fig.5, it is known that the quality factor decreases monotonically with the heat
leakage conductivity α and irreversible factor ϕ respectively. The increasing of α and ϕ make the
dissipation increase, and reduce the energy storage capacity of the thermoacoustic system.
It also can be known that the quality factor, considering with the heat leakage and irreversible
factor, is smaller than the quality factor considering without these factors. The former is closer to
the actual, and it is more accurately to reflect the ability of the system to store energy.
The length of the resonant tube take 0.7m, 0.8m, 0.9m, and select different material fillers for
regenerator to analysis, The relationship between the Q and ε s is shown in Fig. 6. When other
things are equal, the quality factor increases monotonically with the porous material characteristic
factor. Therefore, in order to obtain a high quality factor, choice of materials should be considered.
Conclusion
The quality factor is an important target to measure thermoacoustic system’s energy storage
capacity. In order to improve the energy conversion efficiency, it should have a high quality factor
for thermoacoustic system. In this article, considering with the heat leakage and system’s internal
irreversibility, the quality factor is analysed by calculating and numerical analysis. It can be seen
that: the greater the resonant frequency, the higher the quality factor; increasing of heat leakage
and irreversible factor make the quality factor decreases; and the larger the porous material
characteristic factor, the higher the quality factor, it should be considered when choice material for
filler. It provides guidance for future research.
Acknowledgement
This paper is supported by the National Natural Science Fund, People’ Republic of China (Project
No. 51176143)
Applied Mechanics and Materials Vol. 331
169
Fig.1 Pressure and velocity in resonance tube
Fig.2 Energy flow in thermoacoustic engine
Fig.3 The relationship between quality factor
and resonance frequency
Fig.4 The relationship between quality factor
and heat leakage conductivity rate（ ϕ = 1.5 ）
Fig.5 The relationship between quality factor
and irreversible factor（ α = 0.2kW / K ）
Fig.6 The relationship between quality factor
and porous material characteristic factor
References
[1] Feng Wu, Q Li., F Z Guo: Advance in thermoacoustic theory, Journal of Wuhan Institute of
Technology, 2012,34(1):1-6.
[2] Zhang Chunping, Experimental Study of Characteristic Parameters of Thermoacoustic Core and
Development of High Frequency Miniature Thermoacoustic Experimental Devices[D].Wuhan:
Huazhong University of Science and Technology,2011.
[3] F Z Guo,Q Li:Thermodynamics. Huazhong University of Science and Technology.Wuhan, 2007.
[4] G.W.swift, Thermacoustic engines, J. Acoust. Soc. Am, Vol. 84(4), 1988, pp.1145-1180.
[5] Charles K. Alexander & Matthew Sadiku, Fundamentals of Electric Circuits. 2001. pp600-605.
[6] Bejan A. Theory of heat transfer irreversible power plants[J]．Int.J.Heat Mass transfer, 1988,
31(6):1211-1219