babaei design and optimization of thermoacoustic devices

8/19/2019 Babaei Design and Optimization of Thermoacoustic Devices

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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

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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

3586 H. Babaei, K. Siddiqui/ Energy Conversion and Management 49 (2008) 3585–3598


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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.

H. Babaei, K. Siddiqui / Energy Conversion and Management 49 (2008) 3585–3598 3587


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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


p 1þ ffiffiffiffiffiPr

p dv

y0

" #

ð AK þ AsolidK solidÞrT ð3ÞD _E 2 ¼ A

4

Lsr h

ðc1Þ j p1j2dkxc pmð1þesÞ

C


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



6/14

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).



7/14

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).



8/14

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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9/14

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.



10/14

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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11/14

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.



12/14

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.



13/14

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



14/14

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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http://www.ecn.nl/fileadmin/ecn/units/eei/Onderzoeksclusters/Restwarmtebenutting/b-07-007.pdfhttp://www.ecn.nl/fileadmin/ecn/units/eei/Onderzoeksclusters/Restwarmtebenutting/b-07-007.pdfhttp://www.ecn.nl/fileadmin/ecn/units/eei/Onderzoeksclusters/Restwarmtebenutting/b-07-007.pdfhttp://www.ecn.nl/fileadmin/ecn/units/eei/Onderzoeksclusters/Restwarmtebenutting/b-07-007.pdf

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