The 21st International Congress on Sound and V
Vibration
13-17 July, 2014, Beijing/China
CONTROL STANDING WAV
WAVE
E RATIO IN THE LOOPLOOP
TUBE-TYPE
TYPE THERMOACOUSTIC SYSTEM
Huifang Kang, Peng Cheng,, Hongfei Zheng
Beijing Institute of Technology, Beijing, China, 100081
e-mail: kanghf@bit.edu.cn
In the loop-tube-type
type thermoacoustic system
system, the repetitive cycle of traveling wave makes the
traveling wave can also store the acoustic energy. Thus, in the loop, the acoustic energy can
be deposited with both state of traveling wave and standing wave, which make the acoustic
field much complicate and sensitivity in the loop. In this paper, an equivalent acoustic circuit
of a loop-tube-type
type thermoacoustic system is built and analyzed the standing wave ratio using
electro–mechano-acoustical
acoustical analogy. It is found that the resistance, the inertance and comco
pliance of thee components in the system can influence strongly on the standing wave ratio.
Furthermore,
more, the detail study on the standing wave ratio using the linear thermoacoustic
theory is described.
scribed. The effect on the standing wave of the key parameters will be discus
discussed,
and criteria will be given to obtain an optimal standing wave ratio in the system.
1.
Introduction
Thermoacoustic energy conversion technologies have undergone a substantial development
over the recent four decades thanks to the pioneering work of Rott1-4, Ceperley5, 6 and subsequent
subs
7-11
demonstration projects undertaken at Los Alamos National Laboratory . The acoustic wave is
excited, when a steep temperature gradient is set up along the regenerator of the thermoacoustic
the
engine (TAE). When the acoustic
coustic wave passes through the regenerator of the thermoacoustic
moacoustic cooler
(TAC), a temperature gradient is established along its length corresponding to a certain coefficient
of performance. The acoustic wave drives gas parcels in the thermoacoustic rregenerators
ators to experiexper
ence a certain thermodynamic cycle. Then, the conversion from thermal to acoustic energy and the
heat pumping occurs without any moving parts in the system. Compared with traditional thermodynamic systems, the cooler driven by thermoacoustic engine a has three main advantages: (1) It has a
simple structure, no moving parts, low cost of manufacture, and high reliability; (2) By using inert
gases as working fluids,, this kind of machines is environmentally friendly; (3) The thermoacoustic
devicess can be driven by low quality energy source such as the exhausting thermal energy and the
solar energy, so it is significant for remote rural areas where there may be no access to electricity
grid.
The early work concerned mainly a variety of standing wave devices11-13 which were built
based on the thermoacoustic theory. In the standing wave device
devices,, gas parcels realize the conversion
co
between thermal and acoustic energy using a thermodynamically irreversible process defined by an
imperfect thermal contact
ct between the working gas and the porous solid material, traditionally rer
ferred to as a stack. Although the standing
standing-wave
wave devices are relatively simple to build, their effieff
ciency is limited due to the irreversible thermodynamics on which their thermoacous
acoustic conversion
processes rely.
ICSV21, Beijing, China, 13-17 July 201
2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
With the development of thermoacoustic study, the traveling wave thermoacoustic device was
proposed by Ceperley6, 7 in 1979. In such engine, the gas in a traveling acoustic wave passing
through a differentially heated regenerator underwent a thermodynamic cycle similar to the Stirling
cycle, and thus correspondingly proposed a pistonless Stirling engine where traveling acoustic
waves acted as the piston of the conventional Stirling engine. In 1998, Yazaki et al.14 first built a
pistonless engine by using a thermoacoustic technique. However, both of the works did not obtain
high efficiency because the viscous losses due to high volumetric velocity in the regenerator were
not successfully suppressed. In 1999, Backhaus and Swift9, 10 solved this problem successfully by
introducing a resonator to a looped tube engine. This new thermoacoustic Stirling heat engine had a
high efficiency, reaching 40% Carnot efficiency, or 30% thermal efficiency. This efficiency reached
a level comparable to that of a car engine. With the high thermal efficiency and pressure amplitude,
these engines could be applied to liquefying natural gas, powering refrigerators, and so on. The
thermoacoustic travelling wave engines having looped tube are considered relatively efficient due to
the travelling wave having a zero phase difference between the oscillating pressure and velocity
controlling the acoustic field in the engine.
In practical thermoacoustic systems, there is neither pure travelling-wave mode nor pure
standing-wave mode. The thermoacoustic effect is the result of the combined interaction of the
travelling-wave component (TWC) and the standing-wave component (SWC) in the thermoacoustic
system. In the TAE, the generated acoustic energy can be transferred by the TWC or stored as the
SWC. In the TAC, the consumed acoustic energy can be supplied by the TWC or compensated from
the stored acoustic energy in the SWC. In the linear-type thermoacoustic system, the acoustic energy is stored as standing wave, and transferred by the traveling wave. The causal relationship between the acoustic field distribution and the acoustic field transmission make acoustic field in the
system is stabilization. Furthermore, there is a clear terminal boundary condition. So it is easy to
determine the acoustic field distribution in the linear-type thermoacoustic system. In the loop-tubetype thermoacoustic system, the repetitive cycle of traveling wave makes the traveling wave can
also store the acoustic energy. Thus, in the loop, the acoustic energy can be deposited with both
state of traveling wave and standing wave, which make the acoustic field much complicate and sensitivity in the loop.
The standing wave ratio can determine the acoustic field distribution15-17. Thus, this paper will
study on how to control the standing wave ratio in the loop-tube-type thermoacoustic system.
2.
Lumped-element model of the thermoacoustic-Stirling engine
Compliance cavity
Main Ambient heat exchanger
Recycling
inertance
Regenerator
Hot heat exchanger
Thermal buffer tube
Second ambient heat exchanger
Junction
Figure 1. Schematic of loop-tube-type heat engine, and lumped-element model of the engine9.
Figure 1 shows the simplest lumped-element model of the loop-tube-type engine that captures
all the most important features9. Each component of the engine is replaced with its idealized
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
lumped-element equivalent. The regenerator and adjacent heat exchangers is modeled as the resistance and volumetric velocity source inside the dashed box. The resistance R is due to viscous flow
in the tightly-spaced screen bed that forms the regenerator. The volumetric-velocity source is due to
the temperature profile. The feedback inertance is modeled as inertance L, and the compliance
cavity is modeled as the compliance C.
The volumetric velocity U1,c at the ambient end of the regenerator can be expressed as9
p1,c
w 2 LC
(1)
U1,c =
R 1 + iw L / R
where p1,c is acoustic pressure at the ambient end of the regenerator, w is the angular frequency, R
is the resistance due to viscous flow in the tightly-spaced screen bed that forms the regenerator. L is
the inertance of the feedback inertance, and C is compliance of the compliance cavity.
Using Eq. (1), the time-averaged power flow into the regenerator is given by
2
p1,c w 2 LC
1
&
(2)
%
Wc = Re [ p1U1 ] =
2
2
R
For the high efficiency, the regenerator should be located at the pressure antinode. Thus it is
can be assumed that
p max = p1,c
(3)
The acoustic power is transfer as traveling wave, and the minimum pressure amplitude is
equal to the amplitude of the traveling wave. So, the acoustic power can also be expressed as,
2
1 pmin
&
Wc =
(4)
2 rmc
Thus, the standing wave ratio (SWR) can be expressed as,
SWR =
2
pmax
2
2
p1,c
R
=
=
2W&c r c r cw 2 LC
(5)
2
pmin
As shown in Eq. (5), the SWR is proportional to the regenerator resistance R, and is inversely
proportional to the compliance C and the inertance L. The acoustic field of the loop-tube-type system can be control by those parameters.
3.
Simulation
Figure 2. Schematic drawing of the loop-tube-type thermoacoustic engine: (TAE- Thermoacoustic
engine, MAHE -Main ambient heat exchanger, Reg-Regenerater, HHE -Hot heat exchanger, TBT -Thermal
buffer tube, SAHE -Secondary ambient heat exchanger).
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
The loop-tube-type thermoacoustic engine is shown schematically in Figure 2. Along the engine loop there are: main ambient heat exchanger (MAHX), stacked screen regenerator (Reg), hot
heat exchanger (HHX), thermal buffer tube (TBT), secondary ambient heat exchanger (SAHX),
feedback pipe, and compliance.
In order to perform a more detailed analysis, a specialized design tool referred to as DeltaEC
(Design Environment for Low-amplitude ThermoAcoustic Energy Conversion) and developed by
Los Alamos National Laboratory is employed8. Its calculation capabilities and precision in modeling thermoacoustic devices have been validated by many researchers9, 15, 16. DeltaEC solves the onedimensional wave equation based on the usual low-amplitude acoustic approximation. It is suitable
for the standing wave, the travelling wave and the complex acoustic field of standing-traveling
wave. It can be used in the design process in order to optimize a thermoacoustic system, or to predict the performance of an existing build of a thermoacoustic device. DeltaEC integrates numerically the acoustic wave equation and energy equation segment by segment throughout the whole device based on the low amplitude acoustic approximation and the sinusoidal time dependence of the
variables8. In the current work, DeltaEC is used to simulate the acoustic field and the acoustic power flow in the thermoacoustic device under study.
Table 1. The main parameter of the model.
Element
Length (mm)
cross-sectional area of
compliance (cm2)
Radius(mm)
Porosity
MAHE
90
2.5
0.6
Reg
21
0.015
0.733
HHE
160
4.0
0.196
TBT
178
MAHE
30
2
0.6
21.4
Material
Copper
Stainless-steel
Copper
This thermoacoustic generator uses pressurized helium at 2.0 MPa as working gas, and operates at working frequency of around 276 Hz. The temperature of the hot heat exchangers is set at
850K, and the temperature of the ambient heat exchangers is 300K. The total length of the loop is
around 3.58 meters, and the other most important quantitative details of the construction of the three
stages are presented in Table 1.
According to the Eq. (5), the SWR will be influence by a lot of parameters. This paper take
the compliance as an example to discuss the influence of compliance on the SWR. In the calculation,
the cross-sectional area of compliance Ac was changed to demonstrate the influence of the compliance on the acoustic field. The calculation results of the distributions of the acoustic power flow and
the pressure amplitude in the system with different Ac are shown in Figure 3.
As shown in Figure 3a, the acoustic power in the system change strongly with different Ac .
When Ac increase from 26cm2 to 60cm2, the acoustic power inputted into the regenerator increases
from 37W to 798W. It is indicated that the acoustic power cycling in the system is sensitive to Ac ,
and the traveling wave component is also sensitive to Ac . As shown in Figure 3b, the difference
between the maximum and minimum of the pressure amplitude change strongly with different Ac .
It is indicated that the standing wave components in the system is also sensitive to Ac .
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 3. the distributions of the acoustic power flow and the pressure amplitude in the system with
different cross-sectional area Ac of compliance
The impacts of Ac on the thermoacoustic efficiency η, the standing wave ratio (SWR), the
phase difference j z between the pressure wave and the velocity wave, and the normalized impedance Z n are shown in Figure 4. As shown in Figure 4a, with the increase of Ac , the SWR decreases. The reason is that the SWR is inversely proportional to the compliance C as shown in Eq. (5),
and the compliance increases with the increase of Ac . With the decrease of the SWR, j z in the
regenerator closes to the 0, but Z n decrease. The reason is that the traveling wave ratio in the system increases with the decrease of the SWR, and j z and Z n decrease with the increase of traveling wave ratio15. The small j z is better for the efficiency, but the small Z n is worse for the efficiency. Thus, the efficiencyη is not the , and reach the maximum h = 0.401 when Ac = 30cm 2 .
Compared with engine designed by Yazaki et al.5, the thermoacoustic efficiency is much improved by adjusting the SWR in this paper. The thermoacoustic efficiency in the loop-tube-type
without branch tube can be successfully improved to similar efficiency as the engine designed by
Backhaus and Swift6.
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 4. The impacts of Ac on the thermoacoustic efficiency η, the standing wave ratio (SWR), the phase
difference j z between the pressure wave and the velocity wave, and the normalized impedance Z n .
4. Conclusions
In this paper, an equivalent acoustic circuit of a loop-tube-type thermoacoustic system is built
and analyzed the standing wave ratio using electro–mechano-acoustical analogy. It is found that the
inertance and compliance of the components in the system can influence strongly on the standing
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
wave ratio. Furthermore, the detail study on the standing wave ratio using the linear thermoacoustic
theory is described. The thermoacoustic efficiency is much improved to h = 0.401 by adjusting the
standing wave ratio in this paper. The thermoacoustic efficiency in the loop-tube-type system without branch tube can be successfully improved to similar efficiency as the engine designed by Backhaus and Swift6.
ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China (Grant no.
51006009), and Beijing Higher Education Young Elite Teacher Project (Grant No. YETP1172).
REFERENCES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Rott, N. Damped and thermally driven acoustic oscillations in wide and narrow tubes, Z.
Angew. Math. Phys., 20, 230-243, (1969).
Rott, N. Thermally driven acoustic oscillations, Part II: Stability limit for helium, Z.
Angew. Math. Phys., 24, 54-59, (1973).
Rott, N. Thermally driven acoustic oscillations, Part III: Second-order heat flux, Z.
Angew. Math. Phys., 26, 43-49, (1975).
Rott, N. and Zouzoulas, G. Thermally driven acoustic oscillations, Part IV: Tubes with
variable cross section, Z. Angew. Math. Phys., 27, 197-224, (1976).
Ceperley, P. H. A pistonless stirling engine-the travelling wave heat engine, J. Acoust.
Soc. Am., 66, 1508-13, (1979).
Ceperley, P. H. Gain and efficiency of a short traveling wave heat engine, J. Acoust. Soc.
Am., 77, 1239-1244, (1985).
Swift, G.W. Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators, Acoustical Society of America, New York , ISBN 0-7354-0065-2, (2002).
Ward, W. C., John, C., Swift, G. W. Design Environment for Low-amplitude Thermoacoustic Energy Conversion (DeltaEC). Software and users guide are available online
from http://www.lanl.gov/thermoacoustics/, (2008).
Backhaus, S., Swift, G. W. A thermoacoustic-Stirling heat engine: Detailed study. J.
Acoust. Soc. Am., 107, 3148-3166, (2000).
Backhaus, S., Swift, G. W. A thermoacoustic-Stirling heat engine. Nature, 399, 335-338,
(1999).
Swift, G. W. Thermoacoustic engines. J Acoust Soc Am, 84(4), 1145-80, (1988).
Hofler, T. J. High-efficiency heat-driven acoustic cooling engine with no moving parts.
U.S. Patent No.5901556, (1999).
Zhou, S. L., Matsubara, Y. Experimental research of thermoacoustic prime mover. Cryogenics, 38(8), 813-822, (1998).
Yazaki, T., Maekawa, I. T., Tominaga, A. Traveling wave thermoacoustic engine in a
looped tube. Phys. review lett., 81(15), 3128-3131, (1998).
Kang, H. F., Li, Q., Zhou, G. Optimizing hydraulic radius and acoustic field of the thermoacoustic engine, Cryogenics, 50, 450-458, (2010).
Kang, H. F., Li, Q., Zhou, G. Heat driven thermoacoustic refrigerator based on travellingstanding wave, Energy Conversion and Management, 51, 2103-2108, (2010).
Kang, H. F., Jiang, F., Zheng, H. F. Thermoacoustic travelling-wave cooler driven by a
cascade thermoacoustic engine, Applied Thermal Engineering, 59, 223-231, (2013).
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