babaei design and optimization of thermoacoustic devices
TRANSCRIPT
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Design and optimization of thermoacoustic devices
Hadi Babaei, Kamran Siddiqui *
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada
a r t i c l e i n f o
Article history:
Received 6 December 2007
Accepted 21 July 2008
Available online 11 September 2008
Keywords:
Thermoacoustics
Sustainable refrigerator
Design procedure
a b s t r a c t
Thermoacoustics deals with the conversion of heat energy into sound energy and vice versa. It is a new
and emerging technology which has a strong potential towards the development of sustainable and
renewable energy systems by utilizing waste heat or solar energy. Although simple to fabricate, the
designing of thermoacoustic devices is very challenging. In the present study, a comprehensive design
and optimization algorithm is developed for designing thermoacoustic devices. The unique feature of
the present algorithm is its ability to design thermoacoustically-driven thermoacoustic refrigerators that
can serve as sustainable refrigeration systems. In addition, new features based on the energy balance are
also included to design individual thermoacoustic engines and acoustically-driven thermoacoustic refrig-
erators. As a case study, a thermoacoustically-driven thermoacoustic refrigerator has been designed and
optimized based on the developed algorithm. The results from the algorithm are in good agreement with
that obtained from the computer code DeltaE.
2008 Elsevier Ltd. All rights reserved.
1. Introduction
Thermoacoustic is a branch of science dealing with the conver-
sion of heat energy into sound energy and vice versa. Device that
converts heat energy in sound or acoustic work is called thermoa-
coustic heat engine or prime mover and the device that transfers
heat from a low temperature reservoir to a high temperature res-
ervoir by utilizing sound or acoustic work is called thermoacoustic
refrigerator. Although the thermoacoustic phenomenon was dis-
covered more than a century ago, the rapid advancement in this
field occurred during the past three decades when the theoretical
understanding of the phenomenon was developed along with the
prototype devices based on this technology [1,2]. The thermoacou-
stic technology has not reached the technical maturity yet, as a re-
sult, the performance of thermoacoustic devices is still lower than
their convectional counterparts. Thus, significant efforts are
needed to bring this technology to maturity and develop compet-
itive thermoacoustic devices. There are several advantages of heat
engines and refrigerators based on thermoacoustic technology as
compared to the conventional ones. These devices have fewer com-
ponents with at most one moving component with no sliding seals
and no harmful refrigerants or chemicals are required. Air or any
inert gas can be used as working fluids which are environmentally
friendly. Furthermore, the fabrication and maintenance costs are
low due to inherent simplicity of the thermoacoustic devices.
The main components of a typical thermoacoustic engine or
refrigerator are a resonator, a stack of parallel plates and two heat
exchangers. A half wavelength (or a quarter wavelength) acoustic
standing wave is generated in the resonator. The thermoacoustic
phenomenon takes place in the stack when a nonzero temperature
gradient imposed along the stack plates (i.e. parallel to the direc-
tion of the sound wave propagation) interacts with the sound wave
oscillations. The heat exchangers are responsible of transferring
heat in and out of a thermoacoustic device at their desired temper-
atures, thus maintaining a given temperature gradient along the
stack.
Thermoacoustic refrigerators can be classified based on the
source of the acoustic energy input. If the acoustic energy is pro-
vided by a thermoacoustic engine, the refrigerator is called ther-
moacoustically-driven thermoacoustic refrigerator (TADTAR).
Whereas, if the acoustic energy is provided by an acoustic driver
e.g. a loudspeaker, it is termed as acoustically-driven thermoacou-
stic refrigerator. During the past decades, several acoustically-dri-
ven thermoacoustic refrigerators have been developed [3–5].
Although the form of energy consumed in these refrigerators is
acoustic, the energy source for the acoustic driver is typically elec-
trical from conventional energy resources. During recent years,
there is an increased interest in the development of thermoacous-
tically-driven thermoacoustic refrigerators. These devices are built
by coupling a thermoacoustic refrigerator to a thermoacoustic en-
gine. Thermoacoustic engines are capable of producing acoustic
energy from any source of heat energy. Thus, the primary energy
source to drive the refrigerator could be conventional or unconven-
tional that includes industrial waste heat, solar energy and fossil
fuels. If the heat source for the thermoacoustic engine is the indus-
trial waste heat or solar energy then this device has two major
advantages. Firstly, it does not require any addition conventional
0196-8904/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.enconman.2008.07.002
* Corresponding author. Tel.: +1 514 848 2424x7940; fax: +1 514 848 3175.
E-mail address: [email protected] (K. Siddiqui).
Energy Conversion and Management 49 (2008) 3585–3598
Contents lists available at ScienceDirect
Energy Conversion and Management
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n c o n m a n
mailto:[email protected]://www.sciencedirect.com/science/journal/01968904http://www.elsevier.com/locate/enconmanhttp://www.elsevier.com/locate/enconmanhttp://www.sciencedirect.com/science/journal/01968904mailto:[email protected]
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energy resource and secondly, by utilizing the waste heat, theamount of total waste heat rejected to the thermal energy sink will
be reduced which will increase the overall performance of the en-
tire system. Thus, a complete thermoacoustic refrigeration system
in which the heat engine (which operates on waste heat) drives a
refrigerator and the entire system has no harmful affects on the
environment can be termed as a ‘‘sustainable refrigeration sys-
tem”. In contrast to the acoustically-driven thermoacoustic refrig-
erator which has one moving component i.e. the acoustic driver,
thermoacoustically-driven thermoacoustic refrigerator has no
moving parts thus; chances of mechanical failure are extremely
low.
Recently, some efforts have been made to develop heat engines
that operate on waste heat. Symko et al. [6] designed and devel-
oped a thermoacoustic heat engine that utilizes heat from a micro-circuit to produce sound. Hatazawa et al. [7] proposed a heat
engine that utilizes waste heat from a four-stroke automobile gas-
oline engine. Adeff and Hofler [8] developed a prototype thermoa-
coustic refrigeration system that operates on the solar energy.
Babaei et al. [9] have proposed a thermoacoustic refrigeration sys-
tem for a gas turbine trigeneration system that operates on the
waste heat from the gas turbine. It has been demonstrated that
the thermoacoustic refrigeration system has the ability to enhance
the overall efficiency of a trigeneration system by 5%.
Some recent theoretical studies have demonstrated the strong
potential of thermoacoustic devices in energy conservation and
reduction of harmful emissions. A study shows that if all the indus-
trial waste heat above 140 C in Netherlands can be used in ther-
moacoustic devices, this would save 16 PJ per year whichcorresponds to the saving of more than 5 billion m3 of natural
gas [10]. It is estimated that over 32 billion liters of fuel is con-sumed annually for the operation of vehicle air-conditioners in
the US alone. Modern vehicle refrigeration systems use R-134a,
with a global warming potential still 1300 times that of carbon
dioxide [11]. Zoontjens et al. [12] theoretically investigated the
feasibility of using thermoacoustic devices as the air conditioning
system of an automotive by utilizing the automotive waste heat.
They concluded that the thermoacoustic refrigerator has a strong
potential to replace the existing automotive air conditioning
systems.
Although thermoacoustic devices are easy to build and main-
tain, designing of these devices involves significant technical chal-
lenges. These challenges become more substantial when designing
a thermoacoustically-driven thermoacoustic refrigerator. This is
attributed to the complicated thermoacoustic theory which is notdirectly applicable for design purposes. Thus, a systematic ap-
proach is necessary to design and optimize thermoacoustic
devices.
Wetzel and Herman [13] developed a design algorithm for
acoustically-driven thermoacoustic refrigerators. They developed
the design algorithm by using the simplified linear thermoacoustic
model, and normalizing the position and length of the refrigerator
stack and the equations of the total power flow and consumed
acoustic power in the stack. By applying the algorithm, the de-
signer can decide the stack position and length at the given tem-
peratures of heat exchangers to have the maximum performance
of the stack. The geometrical parameters such as stack plate thick-
ness and spacing as well as the cross-sectional area of the resona-
tor can also be estimated. In this study, however, it is not describedhow the desired cooling power of the refrigerator, the stack
Nomenclature
A cross-sectional area (m2)a speed of sound (m/s)BR blockage ratioCOP coefficient of performanceCOPR coefficient of performance relative to Carnot
c p isobaric heat capacity of the working gas (J kg1 K1)c solid heat capacity of the stack plates (J kg
1 K1)DR drive ratio_E 2 work flux (W)D _E 2 produced or consumed work flux (W) f resonant frequency (Hz)_H 2 total energy flux (W)
HX heat exchangerK thermal conductivity of the working gas (W m1 K1)k wave number (m1)L length (m)l half of the plate thickness (m)P m mean pressure (Pa)P A antinode pressure amplitude (Pa) p1 pressure amplitude (Pa)
Pr Prandtl numberQ heat flux (W)r h hydraulic radius (m)S surface area (m2)_S entropy flux (W/K)T temperature (K)DT temperature difference (K)rT temperature gradient (K/m)U 1 volume flow rate (m
3/s)u1 velocity amplitude (m/s) x1 gas displacement amplitude (m) xc stack center position (m)
y0 half of the plate spacing (m)a thermal diffusivity (m2 s1)b thermal expansion coefficient (K1)dk thermal penetration depth (m)dv viscous penetration depth (m)
es plate heat capacity ratioc ratio, isobaric to isochoric specific heatsC normalized temperature gradientgth thermal efficiencyk wavelength (m)l dynamic viscosity (kg m1 s1)P perimeter (m)q density (kg m3)x angular frequency (rad s1)
Subscripts, superscriptsa ambientc coldcon consumedcrit criticald ducteng enginegen generatedh hotm meann normalizedpro producedr resonatorref refrigerators stackt total
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consumed acoustic power and the total power flow in the stack are
correlated, and under which configuration of the refrigerator stack
this correlation is valid.
Tijani et al. [14] also described the design algorithm for acous-
tically-driven thermoacoustic refrigerators by considering a corre-
lation between the desired cooling power of the refrigerator, the
stack consumed acoustic power and total power in the stack,
which is different from that of Wetzel and Herman [13]. It is how-ever, not well described how this correlation is derived and at
which refrigerator configuration it may be applied.
The above design algorithms are applicable only to design
acoustically-driven thermoacoustic refrigerators. These algorithms
cannot be used in designing a thermoacoustically-driven thermoa-
coustic refrigerator (TADTAR), as designing of TADTAR involves
more parameters and it is more challenging than the acousti-
cally-driven thermoacoustic refrigerators. Therefore, to design
and develop efficient sustainable thermoacoustic refrigeration sys-
tems, a detailed design and optimization procedure is necessary.
To the best of authors’ knowledge no such design and optimization
procedure or algorithm is available.
In this paper, a comprehensive systematic procedure has been
developed for the design and optimization of thermoacoustic de-
vices by applying the simplified linear thermoacoustic model. This
procedure which is mainly intended to design and optimize a ther-
moacoustically-driven thermoacoustic refrigerator (TADTAR) can
also be used to design and optimize individual thermoacoustic en-
gines and acoustically-driven thermoacoustic refrigerators. It
should be noted that the procedure presented in this study pro-
vides a more comprehensive discussion on the design and optimi-
zation of acoustically-driven thermoacoustic refrigerators than
previous studies. The design procedure which is based on the en-
ergy and entropy balances applied on different components of
the device is a simple and effective tool to design and optimize a
thermoacoustic device to meet its requirements. The goal of
designing a thermoacoustically-driven thermoacoustic refrigerator
is to meet the required cooling power at the desired cooling tem-
perature and at the given heat input temperature while rejectingsome heat to the environment.
The developed algorithm not only provides a step by step pro-
cedure to design and optimize a thermoacoustic device but also en-
ables to evaluate the influence of different parameters on the
behavior and performance of the device.
Finally, a thermoacoustically-driven thermoacoustic refrigera-
tor is designed and optimized based on the developed procedure
and simulated by the computer code DeltaE to compare and verify
the design parameters.
It is worth mentioning that using DeltaE to design thermoacou-
stic devices from scratch needs tremendous amount of effort espe-
cially in the case of thermoacoustically-driven thermoacoustic
refrigerator. The presented procedure significantly reduces the
technical challenges associated with the designing of thermoacou-stic devices.
2. Thermoacoustic principle
Phasing plays an important role in the operation of thermoa-
coustic devices. To attain a proper phasing in a thermoacoustic de-
vice, a rather poor thermal contact is essential between the gas
parcel and its adjacent solid plate. This imperfect thermal contact
causes the heat flow between the gas and the plate, not to produce
instantaneous changes in the gas temperature. Instead, the heat
flow creates a time phasing between temperature, pressure and
displacement needed to drive the gas parcels through a thermody-
namic cycle [2].
Consider a solid plate aligned in the direction of the acousticwave propagation with an imposed temperature gradientrT along
the plate. The length of the plate is assumed equal to the peak to
peak displacement of gas parcels (2 * j x1j) oscillating along the
plate (see Fig. 1a). The figure shows a magnified view of a single
plate stack and a gas parcel oscillating next to it in a half wave-
length thermoacoustic refrigerator. The gas parcel oscillates under
the influence of standing wave generated by the acoustic power in-
put. The heat energy is transferred in and out of the device by the
cold and ambient heat exchangers located at the edges of the stackplate, respectively. The temperatures of the heat exchangers im-
pose a temperature gradient along the stack plate (rT ). The varia-
tion of the pressure and velocity magnitudes of the acoustic wave
along the resonator is shown in Fig. 1b.
In a real thermoacoustic device, the oscillations are sinusoidal;
but for simplicity, the square wave motion is considered to explain
the basic thermodynamic cycle that a gas parcel undergoes. The
gas parcel experiences two adiabatic processes while moving along
the solid plate and two irreversible constant pressure processes
while exchanging heat with the solid plate [2]. Two temperatures
are important to the parcel. The temperature of the gas parcel after
adiabatic compression and expansion (imposed by the sound wave
and related to the sound wave pressure oscillation) and the local
temperature of the solid plate (imposed by the heat exchangers)
adjacent to the gas parcel after adiabatic compression and expan-
sion and displacement of the gas parcel. Note that the acoustic
wave is responsible for both adiabatic compression and expansion,
and the displacement of the gas parcel along the solid plate. If the
temperature of the gas is higher than that of the plate, heat flows
from the gas to the plate. If the temperature of the gas is lower
than that of the plate, heat flows from the plate to the gas. Thus,
it is the imposed temperature gradient rT along the plate that
makes a thermoacoustic device to operate as an engine or a refrig-
erator. A zero or low temperature gradient is the condition for a
refrigerator and a high temperature gradient is the condition for
an engine. If rT along the plate be selected in such a way that
the temperature change along the plate (2rT j x1j) as seen by the
parcel just matches the parcel’s temperature change due to adia-
batic compression and expansion 2 T mb p1qmc p
, no heat would flow
between the parcel and the solid plate. This temperature gradient
is called the critical temperature gradient and is defined as [2],
Fig. 1. (a) Schematic of a thermoacoustic refrigerator with a single plate stack, (b)
variation of pressure and velocity amplitudes along the resonator tube, solid line:pressure, dashed line: velocity.
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rT crit ¼ T mbx p1qmc pu1ð1Þ
Usually the length of the plate is larger than the displacement of a
given gas parcel. Thus, the heat transfer from one end of the plate to
the other end is occurred by a series of gas parcels along the plate as
depicted in Fig. 2. In Fig. 2, gas parcel A absorbs heat from the plate
at location a and transfers it to the plate at location b. Half a cycle
later, the adjacent gas parcel B picks this heat from the location b
(at this instant, parcel A is at the location a). This heat is transferred
to location c by the parcel B from where the gas parcel C picks it and
delivers to the location d. Thus, the heat is transferred from one end
to the other by gas parcels as bucket brigade. It should be noted that
the plate is used only for the temporary storage of heat [2].
2.1. Simplified linear model of thermoacoustic devices
Consider a stack of parallel plates with x axis along the direction
of the acoustic wave propagation and y axis perpendicular to the
plane of the stack. The plate thickness is equal to 2l and the plate
spacing is equal to 2 y0. The simplified thermoacoustic model is
developed by linearizing momentum, continuity and heat flow
equations, and considering the following three assumptions [2].First, it is assumed that y0dj, y0 dv where dk is the thermal
penetration depth defined as the thickness of the layer around
the stack plate through which the heat can diffuse in the fluid,
whereas, dm is the thickness of the layer around the stack plate
where the viscous effects are significant. This assumption is called
the boundary layer approximation and typically in thermoacoustic
devices, dj 6 y0 6 2dj [2]. Second, the length of the stack is consid-
ered to be significantly less than the wavelength of the standing
acoustic wave (i.e. Ls k) such that it does not perturb the acoustic
standing wave (short stack approximation). With this approxima-
tion and assuming standing wave phasing between pressure and
velocity, the velocity and pressure can be expressed as [2],
p1 ¼ P A cosðkxÞ; u1 ¼ 1 þ l y0
P Aqma
sinðkxÞ ð2Þ
Finally, it is assumed that the stack is short enough that p1 and u1could be regarded as independent of x within the stack, and the
temperature difference along the stack is less than the stack mean
temperature (i.e. DT T m). So the thermophysical properties of the
gas are assumed to be independent of x within the stack. Thus, p1,
u1 and thermophysical properties are evaluated at the stack mid
point, i.e. the stack mid temperature [2].
The simple linear expressions for total power flow _H 2 (i.e. total
energy flux) through the stack, and the acoustic power D _E 2 (work
flux) produced in (or consumed by) the stack are expressed by
the following equations [2,15]:
_H 2 ¼ A4
dk
r h
T mb j p1kU 1 j Að1þ esÞð1þPr ÞK C
1þ ffiffiffiffiffiPr
p þPr þPr es
1þ ffiffiffiffiffiPr
p 1þ ffiffiffiffiffiPr
p dv
y0
" #
ð AK þ AsolidK solidÞrT ð3ÞD _E 2 ¼ A
4
Lsr h
ðc1Þ j p1j2dkxc pmð1þesÞ
C
1þ ffiffiffiffiffiPr
p K
10@
1Aqm jU 1j2dvx
A2K
24
35
ð4Þwhere C is the normalized temperature gradient defined as rT
rT crit,
K ¼ 1 dvr hþ
d2v2r 2
h
, A and Asolid are the fluid and solid cross-sectional
areas in the stack, respectively, andes is the plate heat capacity ratio
defined as, es ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qmc p K qsolidc solidK solid
q .
By assuming that all dimensions of the resonator are much lar-
ger than the penetration depths and the temperature gradient
along the axis of the resonator is zero, the acoustic power loss
per unit surface area of the resonator can be estimated as [15],
d _E 2dS
¼ 14qm
U 1 A
2
dvx 14
j p1j2c pm
ðc 1Þdjx ð5Þ
The first term on the right hand side of Eq. (5) represents the energy
dissipated due to the viscous shear and the second term on the right
hand side represents the energy dissipated due to the thermal
relaxation.
It should be noted that the stack plates are assumed ideal so
that the plate heat capacity ratio es is zero and the last term inthe right hand side of Eq. (3) which represents the axial conduction
in the stack plates and the working gas is neglected. These two
terms have a negligible effect on the calculations [13]. By consider-
ing only ideal gases close to their critical point as the working gas,
the parameter T mb in Eq. (3) can be set equal to unity [2].
3. Design and optimization procedure
Besides the available features from previous studies, following
new features are applied in the present study to develop the com-
prehensive design and optimization procedure for thermoacoustic
devices.
The simplified linear thermoacoustic model is used to evaluate
the engine part of the device. All the dimensions in the direction of
the acoustic wave propagation including the length and position of
the stacks and heat exchangers are normalized. The normalized
acoustic power equation is applied to estimate the dissipated
acoustic power in the heat exchangers. The equation estimatingthe acoustic power losses in the resonator’s wall surface area is
normalized (see Eq. (9)). A comprehensive discussion is presented
to correlate the desired cooling power and the required heat input
to the total power flow in the stack and the acoustic power flow, by
applying the energy balance on the cold and hot heat exchangers
(see Eqs. (11), (15), (16), (18), and (20)). The normalized engine
stack position and length are selected by applying the energy bal-
ance on the whole device and these selections are then modified to
have the engine stack performs at the maximum efficiency at the
given temperatures of the heat exchangers. This behavior is also
shown by applying the entropy balance and energy balance on a
device, simultaneously. It is shown that the engine stack position
and length could be selected to have the minimum entropy gener-
ation within the system while producing the required acousticpower to run the system at the desired temperatures of the heatFig. 2. Mechanism of heat transfer by the gas parcels along the stack plate of athermoacoustic device.
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exchangers (Eq. (33)). The designer could also estimate the re-
quired heat input at the desired temperature to run the engine sec-
tion of the device (Eq. (35)).
Heat exchangers are the least understood components of ther-
moacoustic devices and their proper designing is a critical task,
as the literature contains very little experimental or analytical
guidance. Swift [2] originally argued that the optimum length of
a heat exchanger should be equal to the peak to peak gas displace-ment amplitude at the heat exchanger location. The ambient heat
exchanger is always closer to the velocity node, so the peak to peak
gas displacement at its location is smaller than that of the cold heat
exchanger in a thermoacoustic refrigerator. On the other hand, the
ambient heat exchanger transfers more heat compared to the cold
heat exchanger in a thermoacoustic refrigerator as it must transfer
both the transferred heat by the cold heat exchanger and portions
of dissipated acoustic power in the device to the outside environ-
ment. So it is safe to say that by assuming the same heat transfer
coefficient and temperature difference between the solid plate
and the working gas, the ambient heat exchanger requires more
heat transfer area compared to the cold heat exchanger. The same
argument can be raised for thermoacoustic engines. The heat
transferred by the ambient heat exchanger in a thermoacoustic en-
gine is smaller than that transferred by the hot heat exchanger
while the hot heat exchanger is always closer to the velocity node.
For the design procedure, following assumptions are made for
heat exchangers. All heat exchangers are assumed parallel plate.
The length of the cold heat exchanger and ambient heat exchanger
in a thermoacoustic refrigerator are assumed equal to the peak-to-
peak gas displacement amplitude at the cold heat exchanger loca-
tion. The length of the hot heat exchanger and ambient heat
exchanger in a thermoacoustic engine are assumed equal to the
peak-to-peak gas displacement amplitude at the ambient heat ex-
changer location. The blockage ratio of heat exchangers is assumed
equal to that of their respective stack.
In the following subsections, normalization of thermoacoustic
parameters and equations are described first followed by the
description of the energy balance (first law of thermodynamics)and entropy balance (second law of thermodynamics) on the se-
lected control volumes of the device.
3.1. Normalization
The energy flux equations (Eqs. (3)–(5)) indicate that there are
different sets of independent parameters that play important roles
in evaluating the performance of a thermoacoustic system. These
parameters can be categorized in three main categories. Geometri-
cal variables, which are stack plate thickness and spacing, position
and length of the stack, and stack cross-sectional area. Material re-
lated variables that include thermophysical properties of the work-
ing gas and the stack. Design related variables which are resonance
frequency, mean pressure and pressure amplitude of the workinggas, mean temperature and temperature difference along the stack
and the desired cooling power of the system [13]. Due to a large
number of design parameters, the number of independent param-
eters can be reduced through normalization. Table 1 shows the
independent variables, normalizing parameters and the normal-
ized form [16]. In the present study, the maximum value for the
normalized stack position (measured from pressure antinode)
and the normalized stack length are assumed equal to 0.5 to avoid
large viscous dissipation which decreases the overall performance
of the device.
As recommended in the literature, the normalized plate spacing(i.e. blockage ratio) was set equal to 0.8 [2,4,17].
The normalized thermal penetration depth and the normalized
viscous penetration depth can be assumed in the range 0.5–1 and
0.5 Pr 2 to Pr 2, respectively [2].
As the thermoacoustic model is based on the linear wave the-
ory, to avoid nonlinearities, it is recommended that the drive ratio
(DR = p1/ pm) should be smaller than 3% so that the acoustic Mach
number and acoustic Reynolds number would be smaller than
0.1 and 500, respectively [18,19].
The normalized temperature difference along the refrigerator
stack and engine stack are assumed in the range 0–0.17 and
0.35–0.95, respectively. The mean temperature along the refriger-
ator stack and engine stack are assumed in the range 288–303 K,
and 390–600 K, respectively. It should be noted that to satisfy
the third assumption described in the previous section, thermo-
physical properties of the gas inside the refrigerator stack (and res-
onator) are calculated based on the refrigerator stack mean
temperature and the thermophysical properties of the gas inside
the engine stack are calculated based on the engine stack mean
temperature.
The normalized temperature gradient, C ¼ DT =LsrT crit
¼ rT rT crit
can be
expressed as the function of other normalized parameters as [2],
C ¼ DT nBR ðc 1ÞLsn cotð xcnÞ ð6Þ
The above equation shows that for a stack with specified length and
position, there is a range of normalized temperature differences at
which the stack operates as a refrigerator (C
< 1) and there is arange of normalized temperature differences at which it operates
as an engine (C > 1).
By dividing the total power and acoustic power equations (Eqs.
(3) and (4)) by the product AP ma, and assuming a parallel plate
stack (r h = y0), the following normalized equations for the total
power flow through the stack ð _H 2nÞ and the acoustic power pro-
duced in (or consumed by) the stack ðD _E 2nÞ are expressed as [16],
_H 2n 18c
dknDR 2 sinð2 xcnÞð1 þ Pr ÞK C
1 þ ffiffiffiffiffiPr
p þ Pr
1 þ ffiffiffiffiffiPr
p 1 þ ffiffiffiffiffiPr
p dvn
" #
ð7Þ
D _E 2n 14c
dknDR 2Lsn BR ðc 1ÞCos2ð xcnÞ C
1 þ ffiffiffiffiffiPr p K 1
0@
1A
24
Sin2ð xcnÞ
ffiffiffiffiffiPr
p
BR K
# ð8Þ
Table 1
Normalized parameters
Independent parameters Normalizing parameters Normalized parameters
Length and position k2p Normalized length and position
Plate spacing The sum of plate spacing and thickness Blockage ratio
Penetration depths Half of the stack plate spacing Normalized penetration depths
Pressure amplitude Mean pressure Drive ratio
Temperature difference along the stack Mean temperature of the stack Normalized temperature difference
Temperature gradient along the stack Critical temperature gradient Normalized temperature gradient
Power The product of mean pr es sure, s ound velocit y and gas cross -sectional area in the s tack Normalized power
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In this study, by using the same normalizing parameters as for Eqs.
(7) and (8), the normalized acoustic power dissipated in a half
wavelength resonator is estimated as,
D _E 2n;r p8c
ffiffiffiffiffiPr
p DR 2
dk
r hr
ðc 1Þp
8c DR 2
dk
r hr
ð9Þ
The energy dissipated in the resonator is proportional to the wall
surface area of the resonator. So to determine the normalizedacoustic power dissipated in a quarter wavelength resonator, the
above equation should be divided by two.
3.2. Energy balance
One purpose of the analyses presented in this section is to cor-
relate some important thermoacoustic parameters. The required
heat input to the device is correlated with the total energy flux
through the engine stack and acoustic energy flux, by applying
the energy balance on the hot heat exchanger. The desired cooling
power of the device is correlated with the total energy flux through
the refrigerator stack and the acoustic energy flux by applying en-
ergy balance on the cold heat exchanger. The analyses cover all
possible characteristics that could be assumed for thermoacoustic
engines and refrigerators. These relationships would also provide a
better estimation of the engine and refrigerator performance.
Consider a thermoacoustic engine with its stack located near
the left pressure antinode of the resonator as illustrated in Fig. 3.
The energy balance is applied on the control volume outlined with
the dashed line which encloses the hot heat exchanger. Thus,
Q h ¼ _H 2;s þ _H 2;hd ð10Þwhere _H 2;hd is the acoustic power leaving the control volume into
the hot duct, which is dissipated in the hot duct. If it is assumed that
the heat generated by the dissipation of the acoustic power in the
hot duct ðD _E 2;hdÞ is rejected to the environment through the hot
duct walls then, ð _H 2;hd ¼ D _E 2;hdÞ and we have Q h ¼ _H 2;s þD _E 2;hd. As
the hot duct is always a small portion of the resonator, the magni-
tude of D _E 2;hd is very small and can be neglected, thus,
Q h ¼ _H 2;s ð11Þ
This implies that the total energy flux into the engine stack is
approximately equal to the heat input by the hot heat exchanger.
It should be noted that Eq. (11) will also be applicable when the en-
gine stack is placed near the right-end pressure antinode or when
the resonator walls are insulated.
The ratio of the acoustic work produced by the engine stack to
the energy flux delivered to the system by HXh is defined as the
thermal efficiency of the engine stack, gth,s
, expressed as,
gth;s ¼D _E 2;s;eng
Q h¼ D
_E 2n;s;engQ hn
ð12Þ
where Q h(Q hn) is defined in Eq. (11).
Fig. 4a and b shows two possible configurations of a thermoa-
coustic refrigerator, i.e. the refrigerator stack located near the left
pressure antinode or right pressure antinode of the resonator,
respectively. Note that the acoustic power to the refrigerators
(either by an engine or a loud speaker) is provided from the left
side of the resonator in both cases. In other word, the engine stack
(or loud speaker) is located on the left side of the refrigerator stack.
A thermoacoustic refrigerator with its stack located near the left
pressure antinode is illustrated in Fig. 4a. The energy balance is ap-
plied on the control volume outlined with the dashed line in Fig. 4a
which encloses the cold heat exchanger. Thus,
Q c ¼ _H 2;s þ _H 2;cd ð13Þwhere Q c is the desired cooling power, _H 2;s is the total energy flow
towards the stack which is equal to the sum of heat extracted from
the cold heat exchanger and the heat produced by the acoustic
power dissipation in the cold heat exchanger, minus the acoustic
power enter the control volume. _H 2;cd is the acoustic power leaving
the control volume into the cold duct. This acoustic power is dissi-
pated in the cold duct. If it is assumed that the heat generated by
the dissipation of the acoustic power in the cold duct ðD _E 2;cdÞ is re-
jected to the environment through the cold duct walls then,
_H 2;cd
¼D _E 2;cd
ð14
Þand,Q c D _E 2;cd ¼ _H 2;s ð15ÞSince the actual net cooling power is the amount of energy flux re-
moved from the cold heat exchanger and pumped uphill by the
stack, the actual cooling power of the device is Q c D_E 2;cd in this
case.
If the cold duct is insulated then it could be assumed that D _E 2;cdis not rejected to the environment and it appears as a load on the
cold heat exchanger. This heat leaves the cold duct and enters the
control volume and flows into the stack.
Thus, _H 2;cd ¼ 0 and,
Q c
¼ _H 2;s
ð16
ÞIn other configuration of the thermoacoustic refrigerator, the stackcan be located near the right pressure antinode of the device as
Fig. 3. Schematic of a thermoacoustic engine. The control volume is outlined with
dashed lines which encloses the hot heat exchanger (HXh).
Fig. 4. Schematic of a thermoacoustic refrigerator with two possible configurations: (a) Refrigerator stack located near the left pressure antinode, (b) refrigerator stacklocated near the right pressure anitnode. The control volume is outlined with dashed lines which encloses the cold heat exchanger (HX c).
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shown in Fig. 4b. Consider the control volume outlined with the
dashed line which encloses the cold heat exchanger. If it is assumed
that the heat generated by the dissipation of the acoustic power in
the cold portion of the resonator is rejected to the environment,
then from the energy balance, we have,
Q c ¼ _H 2;s _H 2;sum ð17Þ
where _
H 2;sum is the acoustic power entering the control volumewhich is the sum of the acoustic power to be dissipated by the cold
heat exchanger, stack, ambient heat exchanger and ambient duct.
Neglecting the acoustic power dissipated in the ambient duct, and
defining D _E 2;t as the acoustic power dissipated/consumed in the
cold heat exchanger, stack, ambient heat exchanger, we have,
Q c ¼ _H 2;s D _E 2;t ð18ÞThis equation implies that the total energy flux of the stack is the
cooling power of the system and the acoustic power consumed by
the stack and the heat exchangers. If the cold duct is insulated,
D _E 2;cd is not rejected to the environment through the cold duct
walls, and it appears as a load on the cold heat exchanger. Thus,
_H 2;sum
¼D _E 2;t
þD _E 2;cd
ð19
Þand,
Q c ¼ _H 2;s ðD _E 2;t þ D _E 2;cdÞ ð20ÞThe ratio of the cooling power of a thermoacoustic refrigerator to
the consumed acoustic power by the stack is defined as the coeffi-
cient of performance of the refrigerator stack.
COPs ¼ Cooling powerD _E 2;s;ref
¼ Normalized cooling powerD _E 2n;s;ref
ð21Þ
where cooling power is defined in Eqs. (15), (16), (18) and (20).
Considering a thermoacoustically-driven thermoacoustic refrigera-
tor, the acoustic power produced by the thermoacoustic engine
must be consumed by the thermoacoustic refrigerator and the res-
onator. That is,
D _E 2;s;eng ðD _E 2;HXh þ D _E 2;HXa;eng Þ¼ D _E 2;s;ref þ D _E 2;HXc þ D _E 2;HXa;ref þ D _E 2;r ð22Þ
The left hand side is the net acoustic power output of the thermoa-
coustic engine. That is, the total acoustic power available for the
refrigeration purpose. The first three terms on the right hand side
are the total acoustic power consumed by the refrigerator stack
and its exchangers and the last term is the dissipated acoustic
power in the resonator. The above equation could be summarized
as,
D _E 2;t;eng ¼ D _E 2;t;ref þ D _E 2;r ð23Þ
Thus,D _E 2;pro ¼ D _E 2;con ð24Þ
where D _E 2;pro is the acoustic power produced in the engine and
D _E 2;con is the acoustic power consumed in the refrigerator and res-
onator. In the normalized form, the above equation can be ex-
pressed as,
D _E 2n;pro ð AP maÞeng ¼ D _E 2n;con ð AP maÞref ð25ÞParameters A and P m are the same for both engine and refrigerator
whereas, the speed of sound is different in both stacks due to the
difference in the mean temperatures of the stacks. Eq. (25) can fur-
ther be simplified as,
D _E 2n;pro ¼ aref aeng
D _E 2n;con ð26Þ
3.3. Entropy balance
To determine the entropy generation within a thermoacoustic
device, the entropy balance and energy balance are applied on
two control volumes. The first control volume (system I) is the
acoustic power producing system which consists of the hot heat
exchanger, engine stack and the engine ambient heat exchanger,
as outlined in Fig. 5a with the dashed line. The second control vol-
ume (system II) is the acoustic power consuming system whichconsists of the resonator, cold heat exchanger, refrigerator stack
and ambient heat exchanger, as outlined in Fig. 5b by the dashed
line. Since the entropy change of a steady state control volume is
zero, the second law of thermodynamics indicates that the entropy
leaving the control volume must equal the sum of the entropy
entering the control volume and the entropy generation within
the control volume. It is useful to mention that there is no entropy
associated with energy transfer as work [20].
Following equations show the energy balance on systems I and
II, respectively.
Q h ¼Q a;eng þ D _E 2;pro ð27ÞQ c
þD _E 2;con
¼Q a;ref
þQ r
ð28
Þwhere Q r represents the amount of dissipated acoustic power in theresonator leaving through the resonator’s wall in the form of heat
energy at the ambient temperature.
Following equations show the entropy balance on systems I and
II, respectively.
Q a;engT a
¼Q hT h
þ _S gen;eng ð29ÞQ a;ref T a
þQ rT a
¼ Q cT c
þ _S gen;r;ref ð30Þ
By substituting the values of Q a,eng and Q a,ref from Eqs. (27) and (28)
into Eqs. (29) and (30), respectively, we get,
T a
_S gen;eng ¼
Q h 1
T a
T h D
_E 2;pro ð
31
ÞT a _S gen;r;ref ¼ Q c 1
T aT c
þ D _E 2;con ð32Þ
Fig. 5. Two thermodynamic systems outlined by dashed lines, (a) system I, acoustic power producing system (engine), (b) system II, acoustic power consuming system(resonator and refrigerator).
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Considering a thermoacoustically-driven thermoacoustic refrigera-
tor (TADTAR), the total entropy generation within the TADTAR
could be estimated by adding the entropy generation in the two
systems (since the two systems form a TADTAR). Thus,
T a _S gen;t ¼ T a _S gen;eng þ T a _S gen;r;ref ¼ Q h 1
T aT h þ Q c 1
T aT c
þ ½D _E 2;con D _E 2;pro ð33ÞTo satisfy Eq. (24) which implies that the produced acoustic power
by the engine stack must be equal to the consumed acoustic power
by the other components of the device, the engine characteristics
must be selected to make the last term on the right hand side of
Eq. (33) equal to zero. Thus, the first term on the right hand side
of Eq. (33) shows the total entropy generation within the TADTAR.
To have the maximum efficiency, the engine characteristics must be
selected to make the entropy generation minimum. For an acousti-
cally-driven thermoacoustic refrigerator (ADTAR), the total entropy
generation could be determined by applying Eq. (32).
In the following subsections, the developed design and optimi-
zation algorithm is explained in detail and, an example is pre-
sented as a case study to demonstrate the functioning of thealgorithm. Some important issues related to the designing are also
discussed.
3.4. Design and optimization algorithm
The algorithm proposed by Wetzel and Herman [13] to design
acoustically-driven thermoacoustic refrigerators was used as the
base model to develop a new systematic comprehensive algorithm
in this study. The developed algorithm can be used to design and
optimize not only thermoacoustically-driven thermoacoustic
refrigerator but also individual thermoacoustic engines and
acoustically-driven thermoacoustic refrigerators. Furthermore,
the present algorithm includes new features for designing an
acoustically-driven thermoacoustic refrigerator by incorporatingcorrelations between different design parameters based on the
energy balance, that were not available in the previous studies.
The complete design algorithm is shown in Fig. 6. As mentioned
earlier, meeting the required cooling power at the desired cooling
temperature and at the given hot heat temperature while rejecting
someheat to the environment can bedefined as the goals of design-
ing a thermoacoustically-driven thermoacoustic refrigerator.
The design procedure starts with the refrigerator section of the
device followed by the resonator and the heat engine. As a first
step, the designer must pick the working gas, blockage ratio, ther-
mal penetration depth and drive ratio of the device (similar to
previous studies). Using the values of the given cooling tempera-
ture (i.e. the temperature of HXc), heat input temperature (i.e.
the temperature of HXh) and, the surrounding ambient tempera-ture (i.e. the temperature of HXa), the mean temperatures of the
refrigerator and engine stacks can be calculated. The thermophys-
ical properties of the working gas are then computed in the refrig-
erator section of the device (and the resonator) and engine section
of the device based on the refrigerator and engine stack mean tem-
peratures, respectively.
As mentioned above, the designing process starts with the
refrigerator. In the energy balance section, two configurations are
presented for a thermoacoustic refrigerator (Fig. 4a and b). For each
configuration, two conditions are presented, i.e. cold duct insulated
and uninsulated. The relationship between cooling power, total en-
ergy flux to the stack and the acoustic power for all cases are also
presented (see Eqs. (15), (16), (18) and (20)). The designer must
select the desired configuration and apply the appropriate energy
balance on HXc.
By plotting COPs and COPR s as functions of xcn,ref , Lsn,ref , DT n,ref ,
the normalized refrigerator stack length and position at the desired
normalized temperature difference is selected. The resonator
cross-sectional area (and then the resonator hydraulic radius)
can be determined by,
Ar ¼ Q cQ cn
P m
a
BR
ð34Þ
In the next step, the normalized acoustic power dissipated in the
refrigerator stack ðD _E 2n;s;ref Þ is determined by applying Eq. (8). This
equation is also used to determine the normalized acoustic power
dissipated in heat exchangers ðD _E 2n;HXc þ D_E 2n;HXa;ref Þ by substituting
the normalized heat exchanger length and position in Eq. (8) and
assuming no temperature gradient along the exchangers (i.e. along
the direction of acoustic wave propagation). The value D _E 2n;t;ref which is the sum of the above mentioned values can then be
determined.
To design the resonator, select the resonance frequency of the
thermoacoustic device. Compute the thermophysical properties
of the working gas in the resonator based on stack mean tempera-
ture of the refrigerator. Calculate the thermal penetration depth at
the resonator’s wall. The length of the resonator can be set equal tohalf wavelength or quarter wavelength of the resonant standing
wave in the device. Determine the normalized acoustic power dis-
sipated in the resonator ðD _E 2n;rÞ by using Eq. (9).
The resonance frequency of the acoustic standing wave is an
important design parameter. Although it has been selected at this
stage, it could be modified afterwards if necessary. Higher reso-
nance frequency results in lower penetration depth i.e. small plate
spacing in the stack which increases the manufacturing challenge,
however, higher resonance frequency increases the power density
and reduces the length of the resonator. In the next step, the total
consumed acoustic power ðD _E 2n;conÞ can be computed by summing
the dissipated acoustic power in the refrigerator stack, its heat
exchangers ðD _E 2n;t;ref Þ and the resonator ðD _E 2n;rÞ.
The purpose of the heat engine is to produce the acoustic powerthat is consumed by the device. Once the total consumed acoustic
power is computed ðD _E 2n;conÞ, the acoustic power to be produced by
the engine ðD _E 2n;proÞ can be estimated by using Eq. (26). This
parameter serves as a basis to design the heat engine which can
meet the given requirements.
To design the engine stack of the device, estimate the stack effi-
ciency based on the energy balance applied on the hot heat ex-
changer using Eq. (12). The results from this energy balance
would also be used in estimating the heat input to the engine. In
the next step, the length and position of the engine stack must
be selected that could produce the estimated acoustic power at
the desired heat exchangers’ temperature while having the maxi-
mum possible efficiency. The appropriate length and position for
the engine stack can be selected by plotting the engine thermalefficiency (gth), the engine normalized thermal efficiency (gthn)and D _E 2n;pro as functions of xcn,eng, Lsn,eng, D T n,eng. At this stage,
the entropy balance described in the previous section is applied
on the device to evaluate the overall entropy generation within
the thermoacoustic device (Eq. (33)). This analysis is useful to
examine the variation of the generated entropy as a function of
xcn,eng, Lsn,eng, so the engine stack position and length are selected
to minimize the entropy generation while providing the required
acoustic power at the given temperatures of the heat exchangers.
The point of the minimum entropy generation is the same as the
point of the maximum efficiency of the device.
After selecting the optimized engine stack length and position,
estimate the amount of heat input to the hot heat exchanger to
produce the estimated acoustic power at the desired temperatures.
The heat input can be computed by,
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Q h ¼ Q hn ð Ar BR P m aengÞ ð35ÞIn the last step, the plate thickness and plate spacing of engine and
refrigerator stacks are estimated.
As mentioned earlier, the present algorithm can also be used to
design an acoustically-driven thermoacoustic refrigerator. In this
case, the same procedure and steps are used to calculate the total
acoustic power consumed by the refrigerator stack, heat exchang-
ers and the resonator ðD _E 2n;conÞ. A loud speaker is then selected
based on this total acoustic power to run the apparatus.
It is useful to mention that this study is more comprehensive
compared to previous studies for designing and optimizing ther-moacoustic refrigerators;since, it usesthe cooling power andenergy
flux relations for different configurations of the thermoacoustic
refrigerator.
3.5. Case study
A case study is presented to demonstrate the working of the
developed procedure. The case study comprised of designing a
TADTAR with 30 W of cooling power at the desired cooling temper-
ature of 277 K and the desired hot heat exchanger temperature of
623 K. The ambient heat exchangers are assumed to operate at
300 K. Helium at a mean pressure of 700 kPa is selected as theworking gas, which is one of the recommended gases for thermoa-
Fig. 6. Schematic of developed design and optimization algorithm for thermoacoustically-driven thermoacoustic refrigerators.
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coustic devices [21]. The other parameters required at the begin-
ning of the designing procedure are set as follows; BR = 0.8,
dkn = 0.66 and DR = 0.03, which are consistent with the previous
studies [17,22]. Based on the given temperatures for hot, cold
and ambient heat exchangers, the mean temperature of the refrig-
erator stack is 288.5 K (DT n,ref = 0.08) and the mean temperature of
the engine stack is 461.5 K (DT n,eng = 0.7).
As mentioned in the previous section, the design procedurestarts with the designing of the refrigerator section. The second
configuration of the refrigerator is selected with no insulation on
the cold duct. That is, the refrigerator stack is located near the right
pressure antinode of the resonator (see Fig. 4b), and the dissipated
acoustic power in the resonator is rejected to the environment
through the resonator’s wall.
The next step is the selection of the normalized refrigerator
stack length and position by plotting COPs and COPR s as functions
of xcn,ref , Lsn,ref at the desired normalized temperature difference.
Fig. 7a and b show the variation of COPs and COPR s as the function
of Lsn,ref at different xcn,ref and DT n,ref = 0.08. The plots show that by
shifting the stack center away from the pressure antinode (i.e.
increasing x cn,ref ), the stack length must be increased to have the
performance peak. However, the peak magnitude decreases with
increasing xcn,ref . By selecting the length of the stack to have the
performance peak, two problems arise. First, as the figures show,
the COPs values are very sensitive to the stack length near the peak.
A slightly smaller stack length causes a sharp decrease in the per-
formance of the refrigerator. Second, the apparatus does not per-
form as a refrigerator at higher values of the normalized
temperature differences (discussed in a later section). Taking into
considerations these issues, it is decided to assume xcn,ref = 0.11
and Lsn,ref = 0.035 (COPs = 4.47, COPR s = 0.37). The cross-sectional
area of the resonator is computed by using Eq. (34), which for
the present case is equal to 0.0123 m2.
The variation of the normalized consumed acoustic power
ðD _E 2n;conÞ is plotted versus the resonance frequency at the selected
specifications of the refrigerator stack in Fig. 8. The values of
D _E 2n;con are computed using Eqs. (8) and (9), as described in theprevious section. The plot shows that at a given refrigerator stack
temperature difference, the consumed acoustic power decreases
by increasing the resonance frequency. As the resonance frequency
increases, the viscous and thermal penetration depths decreases
causing the acoustic power dissipated in the resonator to decrease.
Thus, it is desirable to have a thermoacoustic device operating at
higher resonance frequency. However, higher frequency results
in lower peak-to-peak displacement of the gas particles which
could affect the performance of the heat exchangers [2].
In the present study, the resonance frequency of 400 Hz is se-
lected. For this frequency, the length of the resonator based on
the half wavelength of the acoustic standing wave is estimated
to be 1.25 m. The normalized consumed acoustic power and nor-
malized acoustic power to be produced by the engine are esti-
mated to be D _E 2n;con ¼ 3:2 106 and D _E 2n;pro ¼ 2:5510
6,
respectively.
The engine stack is considered to be located near the left pres-
sure antinode. In the next step, the length and position of the en-
gine stack is selected for the given normalized temperature
difference. The selection is done from the graphs of gth,s, gthn,sand D _E 2n;pro versus the normalized engine stack length (Lsn,eng) at
different values of xcn,eng for the given DT n,eng, that are obtained
by applying the energy balance on the hot heat exchanger. Fig. 9
shows the variations of gth,s, gthn,s and D _E 2n;pro versus Lsn,eng at
Fig. 7. Normalized length of the refrigerator stack (Lsn,ref ) versus (a) coefficient of performance of refrigerator stack (COPs), (b) coefficient of performance relative to Carnotcycle (COPR s), at different normalized stack center positions ( xcn,ref ) at DT n,ref = 0.08.
Fig. 8. Normalized consumed acoustic power ðD _E 2n;conÞ versus the resonance
frequency ( f ) at xcn,ref = 0.11, Lsn,ref = 0.035 and DT n,ref = 0.08.
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different xcn,eng and at DT n,eng = 0.7. Fig. 9a shows that at a given
stack position, the acoustic power produced by the engine de-
creases with an increase in the stack length, whereas, as a given
stack length, the acoustic power produced by the engine increases
as the stack moves away from the pressure antinode. For the esti-
mated value D_
E 2n;pro ¼ 2:55106
in the present study, severalcombinations of the stack length and position are available from
short stack close to the pressure antinode to long stack away from
the pressure antinode. Based on the combination of Lsn,eng and
xcn,eng that can produce the required acoustic power, the corre-
sponding values of gth,s and gthn,s can be estimated from Fig. 9band c, respectively, to evaluate the stack performance. Fig. 9 also
shows that for a specified stack center position there is not more
than one stack length that could produce the required acoustic
power at the desired normalized temperature difference. The re-sults show that at the combination xcn,eng = 0.14 and Lsn,eng = 0.09,
the stack performance is the best i.e., gth,s = 18% and gthn,s = 0.347.Fig. 9a shows that certain combinations of Lsn,eng and xcn,eng cannot
produce the required acoustic power. It also shows that below a
certain value of xcn,eng, there is no length of stack that could pro-
duce the required acoustic power, which in the present case is
xcn,eng 6 0.06. Thus, for a specified thermoacoustic refrigerator,
there could be an engine stack with specified position and length
that produces the required acoustic power with the maximum pos-
sible efficiency. Thus, the device must generate the least entropy at
these specifications.
To check if the total entropy generation ð _S gen;tÞ in TADTAR is
minimum, _S gen;t is computed using Eq. (33) at the selected combi-
nations of xcn,eng and Lsn,eng at which the required D _E 2n;pro is ob-
tained. The _S gen;t is plotted as a function of xcn,eng in Fig. 10. For
xcn,eng 6 0.06, there is no engine stack length that could produce
the required acoustic power of the system. Fig. 10 shows that the
device would generate minimum entropy by placing the engine
stack at xcn,eng = 0.14 with the corresponding value of Lsn,eng = 0.09,
which confirms the above mentioned discussion. Thus, it can be
concluded that while selecting the position and length for the en-
gine stack that produces the required acoustic power, the designer
should confirm that the entropy generation of the device is mini-
mum at the selected specifications.
In the final step, the amount of heat input to the hot heat ex-
changer (Q h) to produce the required acoustic power at the desired
temperatures is estimated using Eq. (35). For the present TADTAR,
Q h = 162.2 W. Thus, for the thermoacoustically-driven thermoa-
coustic refrigerator to produce the cooling power of 30 W,162.2 W of heat input is required.
The plate thickness and plate spacing of the refrigerator stack
are estimated to be approximately 0.10 mm and 0.42 mm, respec-
tively. The plate thickness and plate spacing of the engine stack are
estimated to be about 0.16 and 0.63 mm, respectively.
Fig. 10. Total entropy generation in the device ð_S gen;tÞ versus normalized stack
center positions ( xcn,eng).
Fig. 9. Normalized length of the engine stack (Lsn,eng) versus (a) normalized acoustic
power produced ðD _E 2n;proÞ, (b) thermal efficiency of engine stack (gth,s), (c)normalized thermal efficiency of engine stack (gthn,s), at different normalized stackcenter positions ( xcn,eng) at DT n,eng = 0.7.
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3.6. Effects of stack temperature difference (DT n) on the design of a
thermoacoustic device
As described in the design algorithm, the designer must select
the temperatures of all heat exchangers (i.e. stack temperature dif-
ference) at the beginning of the design procedure. The equations
show that the stack temperature difference (DT n) is an important
design parameter and has a significant influence on the perfor-mance of the respective engine or refrigerator. Therefore, it is
important for a designer to have a good understanding of the influ-
ence of DT n on the performance of the device. Since the design pro-
cedure is based on selected values of DT n for engine and
refrigerator, the effect of DT n on their performance cannot be eval-
uated in the previous sections. In this section, the effects of DT n on
the overall design of the device are discussed in detail.
The impact of the refrigerator stack temperature difference
(DT n,ref ) on the performance of the refrigerator stack is illustrated
in Fig. 11a and b, where the variations of COPs and COPR s are plot-
ted as a function of Lsn,ref at different DT n,ref and at a given value of
xcn,ref . The figure shows that the peak performance of the refriger-
ator stack reduces by increasing the stack temperature difference.
The trends in the given figure also indicate that for a given stack
temperature difference, the COP drops to zero if the length of the
stack is lower than a certain value. That is, for a stack to operate
as a refrigerator, the length of the stack should be higher than a
cutoff value (also see Eq. (6)). The figure shows that the cutoff va-
lue of the stack length increases with decreasing the stack temper-
ature difference. In the case study, at DT n,ref = 0.08, the best
performance of the refrigerator stack is at Lsn,ref = 0.025. However,
if the temperature difference is increased, the stack may not per-
form as a refrigerator. This could happen when developing the ac-
tual device as the actual stack temperature difference may vary
from its designed value. Therefore, it is safer to select the stack
length slightly larger than that correspond to the peak perfor-
mance. Therefore, in the case study the length of the refrigerator
stack was selected as Lsn,ref = 0.035.
The influence of stack temperature difference on the resonancefrequency is shown in Fig. 12, where the variation of the normal-
ized consumed acoustic power ðD _E 2n;conÞ is plotted versus the nor-
malized refrigerator stack temperature difference (DT n,ref ) at
different resonance frequencies at xcn,ref = 0.11 and Lsn,ref = 0.035.
At a given resonance frequency, the consumed acoustic power de-
creases with an increase in the stack temperature difference. As the
temperature difference along the refrigerator stack increases, the
thermal penetration depth decreases, causing a reduction in the
acoustic power consumed in the stack.
The influence of the engine stack temperature difference on the
performance of the engine stack is shown in Fig. 13. In this figure,
the variations of gth,s, gthn,s and D _E 2n;pro are plotted versus the nor-malized engine stack length (Lsn,eng) at different values of D T n,eng.
This figure shows that the peak efficiency of the engine stack in-
creases by increasing the normalized temperature difference along
the engine stack. The trends in Fig. 13 also indicate that for a given
stack temperature difference, the stack efficiency drops to zero if
the length of the stack is greater than a certain value. That is, for
a stack to operate as an engine, the length of the stack should be
lower than a cutoff value (also see Eq. (6)). The figure shows that
the cutoff value of the stack length decreases with decreasing the
stack temperature difference.
The influence of stack temperature difference for engine and
refrigerator on the heat input to the device is shown in Fig. 14.
The heat input is plotted as a function of normalized engine stack
Fig. 11. Normalized length of the refrigerator stack (Lsn,ref ) versus (a) coefficient of performance of refrigerator stack (COPs), (b) coefficient of performance relative to Carnotcycle (COPR s), at different values of normalized refrigerator stack temperature difference (DT n,ref ) at xcn,ref = 0.11.
Fig. 12. Normalized consumed acoustic power ðD _E 2n;conÞ versus normalized refrig-
erator stack temperature difference (DT n,ref ) at different resonance frequencies, at
xcn,ref = 0.11, Lsn,ref = 0.035.
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temperature difference at different normalized refrigerator stack
temperature difference at the specified engine and refrigerator
stack positions and lengths. The figure shows that by increasing
D
T n,eng or D
T n,ref or both, the required heat input to the deviceincreases.
4. DeltaE
The computer code DeltaE can be used to simulate the devices
designed and optimized by the procedure presented in this study.
DeltaE solves the one-dimensional wave equation in gas or liquid,
based on the low amplitude acoustic approximation in user de-
fined geometries [23]. The desired parameters initially selected
and the parameters computed by using the design and optimiza-
tion algorithm developed in this study are summarized in Table
2. Also presented in the table are the values obtained from DeltaE
for comparison. A good agreement is observed between the devel-
oped procedure and the computer code DeltaE. Although the
parameters calculated by the computer code DeltaE are slightlydifferent from those of estimated by the developed procedure, it
is reasonable to say that the developed procedure can serve as a
great tool to design and optimize thermoacoustic devices since
designing a TADTAR by using the computer code DeltaE to meet
the designer’s requirements requires tremendous numbers of trials
and errors making this job tedious. The small differences between
the two approaches are mainly due to the assumptions that were
made to linearize and simplify the governing equations to develop
the design and optimization procedure. The inaccurate expression
used to estimate the temperature difference between the metal
and working gas in the heat exchangers in the computer code
DeltaE could be another reason for the deviations [23]. One or
Fig. 13. Normalized length of the engine stack (Lsn,eng) versus (a) thermal efficiency
of engine stack (gth,s), (b) normalized thermal efficiency of engine stack (gthn,s), (c)normalized acoustic power produced ðD _E 2n;proÞ, at different normalized engine stack
temperature difference (DT n,eng) at xcn,eng = 0.09.
Fig. 14. Heat input (Q h) to the device versus normalized engine stack temperature
difference (DT n,eng) for different values of normalized refrigerator stack temperature
difference (DT n,ref ).
Table 2
Comparison between results from present algorithm and DeltaE simulations
Present algorithm DetlaE
f 400 402.7
T HXh 623 630
T HXa;eng 3 00 300.5
DT n,eng 0.7 0.708
T HXc 277 277
T HXa;ref 300 303.3
DT n,ref 0.08 0.09
Q h 164.7 166.1
Q c 30 30
D _E 2;s;eng 29.7 32.9
D _E 2;s;ref 6.72 6.9
gth,s (%) 18 19.8COPs 4.47 4.3
Overall efficiency (%) 18.5 18.1
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more parameters such as stack center position, stack length or
resonator cross-sectional area can be adjusted to meet the desired
values in DeltaE.
5. Conclusion
Thermoacoustic devices operate by the energy conversion
between heat and sound, and have no harmful effects on the envi-
ronment. The designing of thermoacoustic devices involves signif-
icant technical challenges. In the present study, a comprehensive
design and optimization algorithm is developed for designing ther-
moacoustic devices. The unique feature of the present algorithm is
its ability to design thermoacoustically-driven thermoacoustic
refrigerators that can serve as sustainable refrigeration systems.
In addition, new features based on the energy balance are also in-
cluded to design individual thermoacoustic engines and acousti-
cally-driven thermoacoustic refrigerators. The algorithm is based
on the simplified linear thermoacoustic model. It includes different
correlations based on the energy balance for different device con-
figurations. Another important feature of the algorithm is the
implementation of the entropy balance on the device to refine
the optimization process. A step-by-step design and optimization
procedure is described which is followed by a case study inwhich a thermoacoustically-driven thermoacoustic refrigerator is
designed and optimized to demonstrate the working of the algo-
rithm. The results from the algorithm are in good agreement with
that obtained from the computer code DeltaE.
Acknowledgement
This research is funded by a grant from the Concordia Univer-
sity to Kamran Siddiqui.
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