IJETED Desig of TAR-libre

International Journal of Emerging trends in Engineering and Development ISSN 2249-6149 Issue1, Vol. 2

RS Publication Page 47

Design and Optimization of a Thermo Acoustic Refrigerator

Bheemsha1, Ramesh Nayak. B 2, Pundarika.G3

1- Associate Professor, Dept of Mechanical Engineering, B M S College of Engineering, Bull Temple Road, Bangalore-560019, Karnataka, India

2- Associate Professor, Dept of Industrial Engineering and Management,B M S College of Engineering, Bull Temple Road,Bangalore-560019, Karnataka, India

3- Principal, Government Engineering College, Ramanagara-562159, Karnataka, India

Abstract The work reported here deals with the design and optimisation of a thermo acoustic

refrigerator (TAR) as an attempt to address the future generation environment friendly energy systems. The literature survey gives a complete picture on the history of thermo acoustics and the work carried out in the field of thermo acoustics till today. The motivation of the design of thermo acoustic refrigerator explains the reasons for carrying out the work illustrating its

benefits and how the performance of the TAR in future can be made efficient in comparison with the performance of a conventional refrigerator. The general linear theory of thermo

acoustics is used as a basis for the design taking simplified assumptions into consideration of the design. Optimization is carried out using MATLAB. The optimisation of different

components is carried out to improve the performance and minimise losses in the thermo acoustic refrigerator. For a cooling power of 10 watts, the stack COP was optimized and found to be 2.4948. The acoustic losses in the small diameter resonator tube were minimized by taking the optimized value of the diameter ratio (D2/D1) equal to 0.43.

Keywords: Thermoacoustic, Stack, Heat exchangers, Resonator tube, Buffer volume ___________________________________________________________________________

1 Introduction 1.1 A Standing-Wave Thermo Acoustic Refrigerator

The configuration of standing-wave thermo acoustic refrigerators is simple. A

standing-wave TAR comprises a driver, a resonator, and a stack. The practical device also



utilizes two heat exchangers; however, they are not necessary for creating a temperature

difference across the stack. The parts are assembled as shown in Figure 1.

Fig. 1 Configuration of a standing-wave thermoacoustic refrigerator

The driver, which is often a modified electro dynamic loudspeaker, is sealed to a resonator. Assuming the driver is supplied with the proper frequency input, the resonator will respond with a standing pressure wave, amplifying the input from the driver. The standing

wave drives a thermo acoustic process within the stack. The stack is so called because it was first conceived as a stack of parallel plates; however, the term stack now refers to the thermo acoustic core of a standing-wave TAR no matter the cores geometry. The stack is placed within the resonator such that it is between a pressure antinode and a velocity antinode in the

sound wave. Via the thermo acoustic process, heat is pumped toward the pressure antinode. The overall device is then a refrigerator or heat pump depending on the attachment of heat exchangers for practical application.

A temperature gradient can be created along the stack with or without heat

exchangers. The exchangers merely allow a useful flow of heat. If the hot end is thermally anchored to the environment and the cold end connected to a heat load, the device is then a refrigerator. If the cold side is anchored to the environment and the load applied at the hot end, the device operates as a heat pump. In any case, a few simple parts make up the thermo

acoustic device, and no sliding seals are necessary. Before introducing quantitative thermo acoustic theory, a simplified qualitative

Lagrangian explanation of the thermo acoustic refrigeration cycle is helpful. Consider a parcel of gas in a channel between two plates, as in Figure 2, where the gas is acted upon by an

acoustic standing wave. To keep things simple, the acoustic wave is considered a square wave and no losses are taken into account. There is a relatively small temperature gradient imposed on the walls of the channel such that the top is hot and the bottom cold. The thermo acoustic process can be conceptually simplified into four steps.



Fig. 2 Thermoacoustic cycle

First, the gas parcel undergoes adiabatic compression and travels up the channel due to

the acoustic wave. The pressure increases by twice the acoustic pressure amplitude, so the temperature of the parcel increases accordingly. At the same time, the parcel travels a distance that is twice the acoustic displacement amplitude. Then the second step takes place. When the parcel reaches maximum displacement, it is has a higher temperature than the adjacent walls, assuming the imposed temperature gradient is sufficiently small. Therefore, the parcel undergoes an isobaric process by which it rejects heat to the wall, resulting in a decrease in the size and temperature of the gas parcel. In the third step, the second half-cycle of the acoustic oscillation moves the parcel back down the temperature gradient. The parcel

adiabatically expands as the pressure becomes a minimum, reducing the temperature of the gas. The gas reaches its maximum excursion in the opposite direction with a larger volume and its lowest temperature. Finally, in step four, the parcels temperature has become lower than the local wall temperature (again assuming a small temperature gradient) so that heat flows from the wall to the gas parcel. The process then repeats so that small amounts of heat

can be transported up the temperature gradient along the wall. Although the actual thermo acoustic process is much more complicated than this idealized description, this view of thermo acoustics yields a few useful ideas. Each gas parcel can only move a small amount of heat over a small temperature difference in this manner, so to move the heat across a larger

temperature difference or move more heat (increase the power output), the situation must be modified. To move heat over a larger temperature difference, the length of the channel can be extended to allow more gas parcels to participate in moving the heat. Then, the temperature gradient is the same, but the total temperature difference increases. If the goal is to move



more heat, then adding more channels in parallel will effectively increase the heat capacity of the gas so that the cooling/heating power of the process is increased. Alternatively, the

working gas parameters can be modified so that its temperature fluctuates over a wider range, or the acoustic pressure can be increased to achieve the same effect.

If the temperature gradient perfectly matches the adiabatic temperature change in the gas, then there is no heat transfer in the second and fourth steps; the necessary temperature

distribution is called the critical temperature gradient. If the gradient is smaller than this value, then the cycle will perform a heat pumping action; however, if the gradient is larger than the critical value, then the cycle will produce work in the form of an acoustic oscillation. Therefore, both thermo acoustic engines (TAEs) and thermo acoustic refrigerator (TARs) utilize the same process, differing only in the temperature boundary condition. When losses are considered, the critical temperature gradient becomes a critical range rather than a single value so that no useful work is done in this range; acoustic power is absorbed and heat is moved down the temperature gradient. As stated above, this Lagrangian view of thermo

acoustics is extremely simplified.

2 Design of Thermo Acoustic Refrigerator A thermo acoustic refrigerator consists of an acoustic driver/speaker with a housing

attached to a gas filled resonator tube in which a stack and two heat exchangers are placed and a vacuum vessel in which the resonator is contained. The system is constructed in such a way that all components are independent so that specific parts can be exchanged when the design

changes or for other reasons. In the following section, the design and construction of the different parts will be described in detail. A schematic illustration of a thermo acoustic refrigerator is as shown in Figure 3.

Fig. 3 Thermoacoustic refrigerator experimental setup



2.1 Design Strategy

The stack is considered as the heart of any thermo acoustic system. It can be seen that these systems have very complicated expressions which cannot be solved, which necessitates the use of approximations. The coefficient of performance of the stack for example which can be defined as the ratio of heat pumped by the stack to the acoustic power dissipated in the stack.

A simplified expression is derived from the short stack and boundarylayer approximation [1]. However, even after the approximation, the expression looks complicated. They contain a large number of parameters such as working gas, material and geometrical parameters of the stack. It is difficult to deal with so many parameters in engineering. However, one can reduce

the number of parameters by choosing a group of dimensionless independent variables. Some dimensionless parameters can be deduced directly. Others can be defined from the boundary-layer and short stack assumptions [1, 2]. The parameters of paramount importance in thermo acoustics, which are contained in the work flow and heat flow expressions are given in Table

1 and 2.

Table 1 Operating and working gas variables

Operational variables Working gas properties

Operating frequency Average pressure PM Dynamic pressure amplitude PO Mean temperature TM Temperature gradient TM Mach number M Drive ratio D

Cooling power QC

Dynamic viscosity Thermal conductivity K Sound velocity a Ratio of isobaric to isochoric specific heat Specific heat CP Gas density m Prandtl number

The goal in the design of a thermo acoustic refrigerator is to meet the requirements of a given cooling power QC and a given low temperature TC. These requirements are added to the operation parameters as shown in Table 1. The low temperature TC is shown indirectly in the form of Temperature gradient TM.



Table 2: Normalised and non-dimensional design parameters

Non-Dimensional Parameters

Normalised thermal penetration Depth

Normalised Stack center position xsn Normalised Stack Length Lsn

Normalised Acoustic power loss in Small Diameter resonator tube

Normalised Cold heat exchanger position from the driver end xchn Normalised Hot heat exchanger position from the driver end xhhn Normalised Length of Cold heat exchanger Lchn Normalised Cold heat exchanger position from the driver end Lhhn Acoustic Mach Number M

The boundary layer and short stack approximations assume the following [1, 2]: i. The reduced acoustic wavelength is larger than the stack length /2 very much

greater than LS, so that the pressure and velocity can be considered as constant over the stack and that the acoustic field is not significantly disturbed by the presence of

the stack.

ii. The thermal and viscous penetration depth is smaller than the spacing in the stack x and V are very much less than y0 .This assumption leads to the simplification of Rotts functions, where the complex hyperbolic tangents can be set equal to one.

iii. The temperature difference is smaller than the average temperature TM very much greater than TM, so that the thermo physical properties of the gas can be considered as constant within the stack.

2.2 Design Operating Conditions For this investigation a design of refrigerator for a temperature difference of TM

=60K and a cooling power of 10 watts was chosen. In the following, the selection of some operating parameters, the gas and stack materials was discussed.

2.3. Design Assumptions The assumptions made in the design process are:

1. The thermal conduction (i.e. heat leak from cold side to the hot side) along both the



stack material and the gas in the stack is neglected. 2. The stack is short compared to the wavelength of the acoustic standing wave.

3. The temperature difference across the stack is a small fraction of the mean temperature of the stack and gas. 4. The heat and work flow are steady state. 5. The viscosity at the boundary layer is assumed to be zero.

2.4 Design Procedure A total of 5 basic components have to be designed, of which stack is the most important. Its not only the most critical component when it comes to the functioning of the thermo acoustic refrigerator, but also has a determining effect on the design and dimensioning of all remaining components. In order to begin with the design of the stack, first the values of all parameters are obtained and finalized. Sometimes direct values are not available. Values at particular temperatures are accurately available and using appropriate formulae, the values at

operating temperatures can be calculated. The temperature gradient TM is indicative of the range of temperatures within which the system is going to be operating. Given the lowest temperature TC and the highest TH one can obtain the operating temperature range which is

nothing but TM. (1)

Assuming that the maximum operating temperature is 45oC; and minimum -15oC, a temperature difference of 60oC was obtained. In an effort to simplify calculations a concentrated effort has been put into converting all parameters into dimensionless form so

that computations are simpler. This is achieved by normalising them. Hence TM is converted into TMN, the normalised mean temperature difference by dividing with mean temperature TM. Assuming mean temperature to be 300 K, a value of 0.2 is obtained.

(2)

2.5 Dynamic Pressure The dynamic pressure amplitude PO is limited by the following three factors:

1. The maximum force of the driver 2. Non-linearities

3. Drive ratio



The acoustic ach number can be defined as, (3)

The acoustic Mach number for noble gases has to be limited to 0.1 in order to avoid

any nonlinear effects [1]. Correspondingly, the drive ratio has to be less than 3%. The value of 2% was chosen.

2.6 Average Pressure and Drive Ratio After fixing the temperatures, the average pressure was calculated. Since the power

density in the thermo acoustic device is proportional to the average mean pressure PM and

drive ratio . Hence it is preferable to choose PM and drive ratio D as large as possible.

This is determined by the mechanical strength of the resonator. On the other hand K, the thermal penetration depth is inversely proportional to the square root of PM. So a high

pressure results in a small K and small stack plate spacing. This makes the construction difficult. Taking into account these effects and also making the preliminary choice for Helium

as the working gas, the maximal pressure is 12 bars and drive ratio should be D less than 3% [2]. An average mean pressure of 10 bars and drive ratio 2% was chosen. To minimize the heat conduction from the hot side of the stack to the cold side, a holder made of Ertacetal material with low thermal conductivity was used.

Drive ratio D = 2%

Hence

Hence the maximum dynamic pressure amplitude is limited to P0 = 0.2 bar

2.7 Sound Speed Speed of sound is given by the expression

(4)

When a sound wave travels through an ideal gas, the longitudinal wave is expected to be polytrophic or adiabatic and therefore the pressure and volume obey the relationship

(5)



= 1.67 for Helium The association with the sound wave happens so quickly that there is no opportunity

for heat to flow in or out of the volume of air. This is the adiabatic assumption. Density of gas

is

(6)

Now, (7)

Therefore speed of sound is

(8)

The ideal gas relationship:

(9)

(10)

This can also be written as [3] (12)

The conditions for these relationships are that the sound propagation process is adiabatic and the gas obeys the ideal gas laws.

R = 8.314 J/ mol-K and M = 0.004002602 kg/ mol

Hence, for a mean absolute temperature of 300 K the sound speed is found to be



2.8 Frequency As the power in the thermo acoustic device is a linear function of the acoustic

resonance frequency [2] an obvious choice is thus a high resonance frequency. On the other hand k is inversely proportional to the square root of the frequency which again implies a stack with very small plate spacing. Making a compromise between these two effects and the fact that the driver resonance has to be maintained to the resonator resonance for high efficiency of the driver, the frequency of 400Hz was chosen [2].

2.9 Working Gas After fixing the above values, its time to choose the working gas. It should not be

chemically reactive and should also have a high thermal conductivity. Owing to these reasons,

helium is chosen. It is a noble gas and hence not reactive and has a high thermal conductivity [3]. It also has the highest sound velocity of all inert gases. Furthermore helium is cheap in comparison with other noble gases. A gas with a high thermal conductivity is used since k is proportional to the square root of the thermal coefficient (k).

Having chosen Helium, its properties have to be noted down. Sometimes, accurate values at desired temperatures are not available necessitating the use of formulae to

extrapolate those values to arrive at the values at the required temperature. Hence the following formulae are used to arrive at accurate values. Thermal conductivity of helium at required temperature is given by [3]

(13)

The value of bk is 0.72 for Helium. At 300K the thermal conductivity is 0.1513 W/mK. The specific heat ratio for a mono-atomic gas like helium is 5/3 [3]. Specific heat is

found to be 5192.872 [3]. Gas density of helium is 1.626 kg/m3 calculated using the formula [3].

(14)

(15)

Prandtl number is a dimensionless parameter characterizing the ratio of kinematic viscosity to thermal diffusivity, which is a very important parameter to study the behaviour of gases in thermo acoustic devices. The Prandtl number is given by [3]



(16)

The prandtl number written in terms of thermal and viscous penetration depth is

(17)

Viscous friction has a negative effect on the performance of a thermo acoustic system.

Decreasing the prandtl number generally increases the performance of a thermo acoustic device. Kinetic gas theory has shown that the prandtl number for monatomic gases is about

0.2. Lower prandtl number can be realized using noble gases like Helium and coefficient of performance of refrigerator can be maximized. The value of can be obtained from either equation. They yield the same answer. Prandtl number of 0.6835 is obtained for pure Helium gas [4-7].

(18)

(19)

A value of V = 0.0987 mm and K = 0.11941 mm was got by calculations.

3 Optimisation of the Thermo Acoustic Refrigerator In mathematics and computer science, optimization or mathematical programming

refers to choosing the best element from some set of available alternatives.

3.1. Dimensionless Optimization The motive behind optimization is to improve the performance of the thermo acoustic

refrigerator and to minimize the losses occurring in various components of thermo acoustic refrigerator. Optimization is done to get the maximum co-efficient of performance taking practical aspects into consideration.

Dimensionless Optimization (DO) is used in design problems, and consists of the following steps: Rendering the dimensions of the design dimensionless Selecting a local region of the design space to perform analysis on.



Creating an I-optimal design within the local design space. Forming response surfaces based on the analysis. Optimizing the design based on the evaluation of the objective function, using the response surface models.

First, the standing wave thermo acoustic refrigerator was designed for a cooling power of 10 watts and the required dimensions of the components of thermo acoustic refrigerator were calculated. Secondly, a MATLAB program was devised for the design equations of thermo

acoustic refrigerator. This MATLAB program was used to optimise the design of thermo acoustic refrigerator to get the optimum co-efficient of performance by minimizing the total losses in the stack, cold and hot heat exchangers, small and large diameter resonator tube. Finally, the various values for stack length and stack centre position were got from this. From

the values obtained it was tabulated in Microsoft excel and using scatter option the graph of COPSTACK versus Lsn for various Xsn values were plotted to locate the maximum value of COPSTACK. From the maximum value of COPSTACK were noted the corresponding dimensions for stack and heat exchangers.

The acoustic power loss in small diameter tube (Wsrs) was minimised by plotting the graph of diameter ratio (D2/D1) with respect to Wsrs. The least value of acoustic power loss was found and the value of diameter ratio (D2/D1) corresponding to least value of acoustic power loss was found out.

3.2. Optimization of the stack geometry In the performance calculations, the data shown in table 3 are used.

Table 3 Parameters used in the performance calculations

Operation parameters Gas parameters Pm =10 bar Tm = 300K

Tmn = 0.2 D = 0.02 f = 400Hz, k = 2.46615m-1

a = 1019.1047 m/sec. = 0.6835 = (5/) B = 0.75 kn = (2/3)



Fig. 4 Performance calculations for the stack, as function of the normalized length and normalized stack centre position

Figure 4 shows the performance calculations as function of the normalized stack length Lsn, for different normalized stack positions xn. The normalized position xn = 0,

corresponds to the driver position (pressure antinode). In all cases the COP shows a maximum. For each stack length there is an optimal stack position. As the normalized length of the stack increases, the performance peak shifts to larger stack positions, while

performance decreases. This behaviour is to be understood in the following way: A decrease of the centre position of the stack means that the stack is placed close to the driver. This

position is a pressure antinode and a velocity node which will be discussed in the acoustic concepts [4-7].

From the fact that sin (kx) is zero where cos (kx) is maximum and vice versa, it follows that pressure antinodes are always velocity nodes and vice versa; the pressure and

velocity are spatially 90 degrees out-of-phase. The frequency of the acoustic standing wave is determined by the type of gas, the length L of the resonator and the boundary conditions. A



quarter-wavelength resonator is suitable. But a half-wavelength resonator can also be used. This depends on how the standing wave fits in the tube. A half-wavelength resonator has two

closed ends, so that the velocity is zero at the ends and the pressure is maximal (antinodes). The resonance modes are given by the condition that the longitudinal velocity vanishes at the ends of the resonator is used.

Hence,

(20)

Where,

(21)

In this case it is found that the first (fundamental) mode which is usually used in thermo acoustic devices, corresponds to L = /2, ergo the name half-wavelength resonator. For a quarter-wavelength resonator, one end is open and the other end is closed. This requires a pressure node at the open end, hence

cos kL = 0 (22) so that

(n = 1, 2, ,) (23)

The fundamental mode corresponds to L = /4, ergo the name quarter-wave resonator.

The refrigerator shown in Figure 5 is assumed to be a half-wavelength device. Thus, in the resonance tube, the pressure and velocity distributions will be obtained indicated in

Figure.5

Equation (24) shows that the viscous losses are proportional to the square of the acoustic velocity. Thus decreasing the velocity will result in a decrease of the losses and hence a higher COP. The effect of the position of the stack in the standing wave is discussed.



In this section, the effect of the position of the stack in the standing-wave thermo acoustic devices will be illustrated. In Figure 5 some important length scales are also shown: the longitudinal lengths, wavelength = 2L, gas excursion x1, stack length LS, and the mean stack position from the pressure antinode xS, and the transversal length: spacing in the stack 2k.For audio frequencies, and typically for thermo acoustic devices

(25)

Fig. 5 Model of a thermoacoustic refrigerator. a) An acoustically resonant tube containing a gas, a stack of parallel plates and two heat

exchangers. An acoustic driver is attached to one end of the tube and the other end is closed. Some length scales are also shown: the gas

excursion in the stack x1, the length of the stack LS, the position of the center of the stack from the closed end Xs and the spacing in the stack

2k. (b) Illustration of the standing wave pressure and velocity in the resonator.

Fig. 6 Amount of heat is shuttled along the stack plate



Due to the heat transfer of the parcel of gas to the next, as a result heat Q is transported from the left end of the plate to the right end, using work W. The heat increases in

the stack from Qc to Qh. Since the sinusoidal displacement x1 of gas parcels is smaller than the length of the

plate LS, there are many adjacent gas parcels, each confined in its cyclic motion to a short region of length 2x1, and each reaching the extreme position as that occupied by an adjacent parcel half an acoustic cycle earlier Figure 6. During the first half of the acoustic cycle, the individual parcels move a distance x1 toward the pressure antinode and deposit an amount of heat Q at that position on the plate. During the second half of the cycle, each parcel moves back to its initial position, and picks up the same amount of heat, that was deposited a half

cycle earlier by an adjacent parcel of gas. The net result is that an amount of heat is passed along the plate from one parcel of gas to the next in the direction of the pressure antinode as shown in Figure 6. Finally, it can be noted that, although the adiabatic temperature T1 of a given parcel may be small, the temperature difference Tm over the stack can be large, as the number of parcels LS/x1, can be large Figure 6. Due to the heat transfer of the parcel of gas to the next, as a result heat Q is transported from the left end of the plate to the right end, using work W. The heat increases in the stack from Qc to Qh.

From the above acoustic concept it is clear that the maximum cooling power is

roughly between the pressure antinode and pressure node(/8). From the Figure 4 it is clear that for a normalized stack length above 0.35 the COP is lower than one. Considering the above remarks and for practical reasons, a normalized stack center position of Xn =0.15 was chosen. To achieve optimum performance this requires a stack length of Lsn = 0.1. Expressed

in terms of the normal stack center position and length, Xs= 6.0823331970753 cm and Ls = 4.0548887980502 cm. This is equivalent to place the hot end of the stack at a distance of 4.0548887980502 cm from the driver. Under these conditions the dimensionless cooling power is Qcn =2.599747597328005e-006.

Since the required cooling power is 10 watt this leads to a cross sectional area A =37.25061041907 cm2. This means a radius of r = 3.4434339782798 cm for a cylindrical resonator. To pump 10 watt of heat the stack uses 4.008313650454014 watts of acoustic power (COP = 2.494814745564465).

4 Conclusions



The design and optimisation of thermo acoustic refrigerator for a cooling power of 10 watt was designed and analysed. From the available literatures initially it was assumed that the

optimised value of COP was found at normalised stack centre position, xsn=0.22 and normalised stack length, Lsn=0.23 for a cooling power of 4 watt. The optimisation graphs for 4 watt and 10 watt cooling power was compared. It was found that the optimised value of stack COP for 10 watt cooling power was found to be 2.4948 at xsn=0.15 and Lsn=0.1 which is greater than the stack COP of 1.3 [2] .

From the optimization graph shown in Figure 5, it was clear that the maximum cooling effect of the thermo acoustic refrigerator was found at the position of maximum pressure

amplitude (Pressure-antinode). Hence; the stack centre position should be placed at the maximum pressure amplitude.

The acoustic power loss in the resonator tube was minimised by optimizing the geometry of the resonator tube. It was found that the losses in small diameter tube were minimum at the diameter ratio of 0.43.

The design has a lot of scope for further improvement by experimenting and investigating the design and optimisation of the heat exchangers for various geometries for copper material as it has a high thermal conductivity. The design of the heat exchangers can be carried for different geometries and the particular geometry can be chosen which dissipates minimum acoustic power loss. The design of the resonator can be further

improvised by incorporating various different configurations of resonator tube (square tube, rectangular tube, hollow tube, etc) and the energy losses at the transition between the large diameter and small diameter tube can be found. A method can be devised to reduce the energy losses at the transition point (A point between large and small diameter tube).The main aim of improvising the design of resonator tube geometry is to ensure that the acoustic pressure waves do not create any turbulence problems. The variation of the COP with respect to the cooling power can be studied and analysed. The PLL controller (Phase Loop Locked) can be designed and its performance can be studied and suitable steps can be incorporating to improve its design. Suitable mechanism to detect the gas leakage can be found such as pressure relieving valve or safety valve as the leakage of the inert gases can lead to asphyxiation problems.



Finally, it can conclude that there is nothing like the universal best design since the thermo acoustics is an ongoing research area. In future an improvised design can be expected so that its working can be made feasible and applicable in various fields such as Automobiles, Aeronautics and electronic cooling equipments (Feasibility study of thermo acoustic heat engines on the performance of automobiles and aircrafts can be studied).The support of industries and governmental agencies is must for this technology to be made a viable option. Thermo acoustic Technology is still relatively young, but it holds much promise for a more sustainable future. The reason for making thermo acoustic a viable option is to solve the environmental problems such as global warming and green house

gas effects.

References

[1] Swift G W, Thermoacoustic engines, Journal of Acoustic Society of America, 84, 1988 pp. 1146-80 [2] Tijani M E H, Loud speaker driven Thermoacoustic Refrigerators, Ph. D Thesis [3] Swift G W. Thermoacoustic A unifying perspective for some engines and refrigerators, Acoust Society of America, American institute of physics, 2002 [4] Tijani MEH, Zeegers JCH, de Waele ATAM, A gas-spring system for optimizing loudspeakers in thermoacoustic refrigerators, Journal of Applied Physics

[5] Tijani MEH, Zeegers JCH, de Waele ATAM, Design of thermoacoustic refrigerators, Journal of Cryogenics, 42 (2002), pp 49-57 [6] Tijani MEH, Zeegers JCH, de Waele ATAM, Construction and performance of thermoacoustic refrigerators, Journal of Cryogenics, 42 (2002), pp 59-66 [7] Feng Wu, Lingen chen, anqing Shu, Xuxian Kan, Kun Wu, Zhichun Yang, Constructal design of stack filled with parallel plates in standing-wave thermo-acoustic cooler,Journal of Cryogenics, 49 (2009), pp 107-111



NOMENCLATURE Lower case

a Sound velocity [m/s] Cp Isobaric specific heat [J/KgK] Cv Isochoric specific heat [J/KgK] Cs Specific heat of the stack material [J/KgK] p pressure [Pa] r Resonator Radius [m] X Position along Sound Propagation [m] Xs Position of the Stack [m] y Position perpendicular to sound propagation [m] yo Plate half-gap [m] Upper case m Molecular weight [Kg/mol] M Mach number

T Temperature [K] V Volume [ ] W Acoustic power [watt] Tm Mean temperature [K]

Lower case Greek Thermal expansion co-efficient [K-1] Prandtl number Density [kg/m3] Dynamic viscosity [Pa/s] Angular frequency [rad/s] Penetration depth [m] s Stack heat capacity ratio Ratio of isobaric to isochoric specific heats Wave length [m] Upper case Greek ormalized temperature gradient Perimeter [m] Sub and Superscripts k Thermal

res Resonator

Viscous

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IJETED Desig of TAR-libre