Design of a thermoacoustic heat engine for low temperature waste heat recovery in food manufacturing

(A thermoacoustic device for heat recovery)

J-A. Mumith, C. Makatsoris*, T.G.Karayiannis

School of Engineering and Design, Brunel University, London, U.K.

* Corresponding Author. Tel.: +44 (0)1895 265063

E-mail address: Harris.Makatsoris@brunel.ac.uk, Jurriath-Azmathi.Mumith@brunel.ac.uk, Tassos.Karayiannis@brunel.ac.uk

ABSTRACT

There is currently an urgent demand to reuse waste heat from industrial processes with approaches that require

minimal investment and low cost of ownership. Thermoacoustic heat engines (TAHE) are a kind of prime

mover that convert thermal energy to acoustic energy, consisting of two heat exchangers and a stack of parallel

plates, all enclosed in a cylindrical casing. This simple design and the absence of any moving mechanical parts

make such devices suitable for a variety of heat recovery applications in industry. In this present work the

application of a standing-wave TAHE to utilise waste heat from baking ovens in biscuit manufacturing is

investigated. An iterative design methodology is employed to determine the design parameter values of the

device that not only maximise acoustic power output and ultimately overall efficiency, but also utilise as much

of the high volume waste heat as possible. At the core of the methodology employed is DeltaEC, a simulation

software which calculates performance of thermoacoustic equipment. Our investigation has shown that even at

such a comparatively low temperature of 150℃ it is possible to recover waste heat to deliver an output of

1,029.10 W of acoustic power with a thermal engine efficiency of 5.42%.

Keywords: Thermoacoustic heat engines; Heat recovery technology; Food Manufacturing; Simulation

1. Introduction

In recent years there has been a renewed interest in heat recovery technologies for the utilisation of waste heat

from industrial processes. This is due to new legislation, the urgent need to reduce dependency on fossil fuels

and concerns of the negative impact of many dated and unsustainable industrial processes on the environment.

In particular, in biscuit manufacturing the biscuit dough is heated during baking at elevated temperatures in gas

fired ovens. As a result of this baking process, exhaust gas is expelled from the baking oven and released into

the atmosphere via an exhaust gas flue.

In this paper, for the first time an investigation and assessment of the potential of thermoacoustic heat

engines to a real-life industrial process is presented. The aim of the investigation is the utilisation of high

volume flow rate but low temperature waste heat discharged from the baking process in high volume biscuit

manufacturing. Typically, applications of thermoacoustic heat engines are for low power levels, so far limited to

a maximum of 6 kW thermal power input [1]. The application of this technology in food manufacturing is very

attractive due to the low investment requirements and low cost of ownership. This is because overall designs are

very simple as they require no moving parts, exotic materials, close tolerances or critical dimensions [2].

Furthermore, they are small geometrically, therefore unlike other heat recovery technologies based for example

on the Organic Rankine or the Kalina cycle [3], TAHEs do not have significant ‘footprint’ [2]. Also, compared

to conventional heat recovery approaches, this technology can be used in a variety of applications because

energy can be converted into a more useful form.

1.1. Overview of thermoacoustic heat engine technology

Thermoacoustic heat engines have the following four essential elements, fig. 1:

1. high temperature heat exchanger (High T HX),

2. stack,

3. ambient temperature heat exchanger (Ambient T HX),

4. resonator.

The thermoacoustic heat engine in Fig.1 consists of two heat exchangers. Those are the engine’s heat source and

heat sink, a stack where input thermal power is converted to acoustic power (a form of mechanical power), and a

resonator which is a cylindrical tube encompassing all components and is the solid container for the acoustic

wave generated.

The key mechanism for energy conversion from thermal to acoustic is the thermoacoustic effect,

occurring in the TAHE when certain conditions are satisfied. A compressible fluid is used as the working fluid

within the engine, which in most cases is an inert gas such as helium. Acoustic waves occur naturally as a result

of a temperature gradient across the stack as heat transfer occurs between the compressible fluid and a solid

boundary (stack). The transfer of thermal energy to and from the compressible fluid and the stack creates local

changes of pressure and velocity in the working fluid. When there is the correct pressure-velocity phasing,

acoustic oscillations appear spontaneously creating an acoustic wave. Depending on the pressure-velocity

phasing either a standing-wave or a traveling-wave is created. A standing-wave pressure-velocity phasing is

shown Fig.2. The pressure the acoustic wave generates creates mechanical work, which can then be easily

recovered to generate for example electric power. In this work a standing-wave thermoacoustic heat engine is

evaluated due to its simple design, as can be seen in Fig.1.

The field of thermoacoustics is an emerging one, with its primary focus on a deeper understanding in an

effort to increase the performance of the engine [2,4-7]. A recent development in the field of thermoacoustics is

research that demonstrates the various ways in which TAHEs can be used in order to realise practical

applications including refrigeration, lifting temperature of a heat source and generating electricity [8-11]. One

example of such work is the utilisation of waste heat from an internal combustion engine to drive a

thermoacoustic refrigeration system for automotive applications. In this particular instance the TAHE only

harvests 6 kW from a total of 145 kW of thermal power that is rejected from the internal combustion engine [1].

Another example is the work carried out regarding a thermoacoustic heat pump for upgrading industrial waste

heat to a higher temperature by the Energy Research Centre in Netherlands, again only limited to a maximum

thermal power input of 5 kW [9], whereas this work looks at the effects of relatively high power levels on the

design of a TAHE in order to utilise high volume, low temperature waste heat. Also, attempts have been made

in recent years to design efficient thermoacoustic electricity generators, where the acoustic power produced in

the thermoacoustic engine is converted to electric power by coupling the engine with a type of transducer; these

have achieved acoustic-to-electric conversion efficiencies of up to 77% [12-14].

The aim of the paper is to investigate and assess the application of this technology to the biscuit baking

process in a large biscuit manufacturer. As the food manufacturing process results in waste heat that is

comparatively low temperature and high volume, the thermoacoustic heat engine must be carefully designed to

maximise performance of the device as well as maximising the utilisation of the waste heat. Various parameters

ranging from materials to the geometry of the engine are considered during the design process in order to meet

the criterion mentioned.

In the following section the design parameters, iterative methodology and simulation model are outlined

and discussed. Then in section 3, the results are presented and analysed. Finally the main conclusions as well as

limitations and further work are discussed.

2. Thermoacoustic heat engine for biscuit baking

The rejected gas mixture from the baking oven comprises CO2, N2, O2 and H2O, at a temperature of

approximately 150℃, a volume flow rate of 1,288m3/hr , and the corresponding density of the exhaust gases

is 0.797 kg/m3. Therefore, the waste heat is calculated asQ=m (h@ 150℃−h@ A ) depending on the outdoor air

temperature which varies over the course of the year. For the mean annual temperature in England of 9.6℃

[15],Q is 40.35 kW from a single exhaust gas flue.

The thermoacoustic heat engine is to be installed in the exhaust gas flue, perpendicular to the flow of the

exhaust gas so that heat can be transferred to the working fluid through the high temperature heat exchanger

(High T HX section in Fig.1). The material and geometric properties of the TAHE will be varied in order to

maximise the performance of the engine and utilise as much of the waste heat as possible.

3. Thermoacoustic heat engine design

3.1 Design parameters

There are two important parameters in thermoacoustics; the total thermal power that is available for conversion

in the TAHE H and the acoustic powerW , which is the useful mechanical work that is produced in the stack.

H=12

ρm ℜ [h1~U 1 ]− ( Ak+ A solid ksolid )

d Tm

dx(1)

where Re[] denotes the real part of the terms inside the bracket, and the tilde denotes the complex conjugate.

The energy available for conversion is related to the energy of a flowing fluid, and hence enthalpy is given by

Eq.(1). The second term in Eq.(1), is the thermal conduction that occurs in the working fluid and solid plate of

the stack, causing losses of energy as this energy does not contribute to the thermoacoustic effect and hence it is

taken away from the energy available for conversion.

Acoustic power generated in a thermoacoustic heat engine is related to the work done by a differential

volume of fluid dx dy dz in the stack section, as it expands from dx dy dz todx dy dz+dV , and so the work is

pdV .The time-averaged acoustic power is the product of p1andU 1 that is produced near the surface of the

stack plate,

W =12

ℜ [ p1~U 1 ] (2)

From these two equations that can be found in [2] and previous experimental work [2,16-18], 12 of the

most significant design parameters are identified and considered in this work for the design of the TAHE for

low temperature waste heat recovery, see Table 1. Also the thermal power input is another design parameter of

the engine, in order to observe the behaviour of the TAHE and how performance is affected with relatively large

thermal power levels.

Higher mean pressure Pm and drive ratio DR, which is the ratio of peak pressure amplitude p1to the

mean pressure, yield greater power density. However, higher values for these two design parameters lead to an

increase in the viscous penetration depth δ ν, the region in which power is dissipated due to viscosity and there is

a greater risk of nonlinear effects (i.e. turbulence) occurring which diminish the performance of the engine. This

suggests that there is an optimum mean pressure and drive ratio that provide good power density and do not

significantly affect the thermal penetration depth δ κ (the region in which heat from the solid plate diffuses into

the fluid and acoustic power is produced). It is also necessary to consider the maximum mean pressure that

reasonable fabrication methods can accommodate with regard to the resonator see Fig.1. Therefore, the range of

mean pressure value that is employed in the iterative process is 1MPa-3MPa [19,20], based on values used in

previous experimental work. The drive ratio is calculated by the simulator as a result of the thermal power input

specified.

Normal heat exchanger calculation methods cannot be used to determine the thermal power extracted

from the exhaust gas, as the volume flow rate in the TAHE is oscillatory and hence the time-averaged flow is

zero. Furthermore, the operating conditions which affect the thermal power input do not remain the same and so

the output temperature of the high temperature heat exchanger cannot be determined. Thus a wide range of Q¿

was adopted, to observe how the TAHE behaves at these larger power levels and how various thermal power

inputs affect the material and geometric parameter values that produce the greatest performance in the TAHE. A

single TAHE will not be able to handle a large thermal power input as the very large pressure amplitudes

created would cause nonlinear affects accounting for significant losses and degradation of performance [11].

Therefore a thermal power input of up to 19 kW is considered in this work. But this limitation of the engine can

be mitigated by using multiple TAHEs together.

The working fluid’s Prandtl number Pr (δ ν /δ κ) determines the fraction of the energy passing through

the stack that will dissipate due to viscosity. Also, higher speed of sound yields greater acoustic power, as the

time taken to create a standing-wave is reduced. In some cases a small fraction of lighter gases are mixed with a

heavier gas in an effort to reduce the Prandtl number, but sacrificing power density, as the added mass reduces

the speed of sound [24]. A mixture of Helium and a lighter gas Argon [25] was used for this work, varying the

mole fraction of Argon X A from 0-100% to determine numerically from the iterative process how this directly

affects the acoustic power output and acoustic losses of the system.

There are two aspects of the stack that have a direct effect on the acoustic power produced, the material

and geometric properties. The thermophysical properties (cs,k s) of the stack should be such that heat capacity

cs is as high as possible to enable heat to move along the stack in the x direction by the constant heat transfer

between the stack and the working fluid, but have low thermal conductivity to minimise ordinary conduction of

heat along the stack which causes dissipation of power [24]. Hence, stainless steel was used as it is readily

available and its heat capacity cs is approximately 30 times more than its thermal conductivityk s.

The stack length plays a direct role in the desired performance of the engine as it is in the stack region

that acoustic power is produced. It can be determined numerically from the temperature gradientΔT / Δ xs, as it

is above a critical temperature gradient that acoustic power is produced. Also care should be taken that the stack

is located where U1 is small to reduce viscous dissipation caused in the region of the viscous penetration depth,

which prohibits the transfer of heat from the stack to the working fluid. This is a fundamental aspect of the

thermoacoustic effect generating acoustic power. For this to take place, the stack position should be close to the

pressure antinode of a standing-wave, see Fig. 2. But standing-wave systems produce acoustic power

proportional to p1 and U 1at a particular location. Therefore, there is an optimum position in the engine where

maximum acoustic power is produced and minimum losses occur, which is typically between x=0 m and

x=λ /8 m [21,22], therefore this range of values is used in our design methodology. The Blockage Ratio BR is

defined as the ratio of the cross-sectional area occupied by the gas to the total cross sectional area Agas/ A, and

represents the extent to which the plates are tightly packed in the stack section [25]. It is a design parameter that

is intended to take into account the effect of the stack on the acoustic field. Previous experimental work has

shown that a value of 0.8 yields good results, and is therefore used in this work [2,6]. The plate spacing is a

crucial parameter as it determines the strength of the interaction between the working fluid and the stack in

terms of heat transfer and hence temperature of the working fluid. Typical values for the half plate spacing y0

for a standing-wave thermoacoustic heat engine is between δ κ and 4δ κ [19,21,22]. In this work the half plate

spacing is kept constant atδ κ. A gap which is any larger will weaken the interaction between the working fluid

and stack and ultimately negatively affect the performance of the engine. This has been found to be the case

when running simulations varying the half plate spacing between δ κ and 4δ κ.

3.2 Iterative Design methodology

An iterative design process has been developed with which an incremental change in a design parameter value is

made within the range of values shown in Table 1. Each time a new value of a design parameter is set as the

input, the simulation is run and the output values (thermal stack efficiency, thermal engine efficiency) are

assessed according to the following criteria: maximum power in stack and; minimum acoustic losses in engine

as shown in Fig.3. Also various design parameters are varied simultaneously during a simulation to observe the

relationship to each other and ultimately how they affect the performance of the engine. At the core of this

approach is DeltaEC, a simulation code developed by Ward et.al. [26] specifically for the design and assessment

of thermoacoustic systems.

3.3 Standing-wave model simulation

DeltaEC has been employed for analysis and simulation [26]. This is a simulation code that provides

information regarding the performance of thermoacoustic equipment. It also aids the user to design equipment to

achieve some desired performance. In this simulator, a thermoacoustic system is represented by a series of

segments, such as duct, cone, stack, heat exchanger, and is modelled in steady state conditions. The wave

equation employed in the simulator without viscous or thermal-hysteresis losses is,

p1+a2

ω2

d2 p1

dx2 =0 (3)

This second-order equation can be reduced into a system of two first-order equations with respect to pressure p1

and volume flow rateU1:

d p1

dx=

−iω ρm

AU1 (4)

d U 1

dx=−iωA

ρm a2 p1 (5)

These are the most fundamental equations required to find the pressure and volume flow rate of the system as a

function of x. Additionally the simulator takes into account the dissipation of acoustic power along the inner

walls of the cylinder due to viscous effects. Different segments use different equations to account for local

conditions. For example, the governing equations in large-diameter ducts and shallow cones are,

d p1

dx=

− iω ρm

A (1− f v)U 1 (6)

dU 1

dx=−iωA

ρm a2 (1+ γ−11+ϵ s

f k )p1 (7)

where f v and f k are the spatially averaged thermal and viscous functions, and ∈s is the plate heat capacity

correction factor. Iff v ≥ 0 then the pressure gradient of Eq.(6) is completely inertial, but if ℑ [ f v ] ≠ 0 then the

existence of viscosity and stationary boundaries adds a resistive component to the pressure gradient and also

effects the size of the inertial influence. The spatially averaged thermal function f k represents the thermal

contact between the working fluid and solid plate, if f k=1 then thermal contact is perfect and if f k=0 there is

no thermal contact between working fluid and solid plate [23]. It is these equations that are used for the hot and

ambient duct that is modelled in the simulator.

The configuration of the standing-wave thermoacoustic heat engine model created in DeltaEC is as

follows:

1. Drive ratio (guess) is calculated by DeltaEC as a result of the thermal power input (target) that is defined

by the user, using the guess-target feature of the simulation.

2. The overall length was kept constant at 4 m and the hot and ambient heat exchanger plate length was set at

a constant value of 4 cm and 3.6 cm respectively, similar to Swift’s standing-wave TAHE [19].

3. The thermal power (guess) taken away by the ambient heat exchanger was predicted by DeltaEC,

according to the target of constant temperature of 278 K using the simulator’s guess-target feature.

4. A thermoacoustic heat engine typically has a λ/2 or λ/4 wavelength resonator. Although a λ/4 wavelength

resonator provides greater power per unit volume, the engine will require a higher resonant frequency,

therefore in this case the engine is a λ/2 wavelength resonator.

5. The Blockage Ratio (BR) value was set similar to Swift’s standing-wave engine for both heat exchangers

[19] of 0.4, as it is likely that this would provide enough space for the working fluid to move through the

heat exchanger regions, and also provides sufficient heat transfer surface area.

A schematic of the standing-wave thermoacoustic heat engine created by DeltaEC is shown in Fig.4, which

forms the simulated model of the diagram in Fig.1. Each segment in Fig.4 represents a section of the TAHE in

Fig.1, such as segment 5HX and 20HX represents the high temperature and ambient temperature heat exchanger

respectively. Also segment 10 is the parallel plate stack and segment 4 and 22 is the high and ambient

temperature duct respectively. The join segments between the stack (segment 10) takes into account

discontinuities between stack and the heat exchangers with regard to temperature and volume flow rate.

By varying the design parameters identified in Table 1 systematically and solving the equations for

pressure p1 and volume flow rateU1 for each segment numerically using DeltaEC, it has been possible to

determine those values that maximise performance of the engine configuration shown in Fig.1.

4. Results

The results of the iterative design process shown in Fig. 3 are tabulated in Table 2, showing the values/range of

values used for the design parameters for each simulation, providing information that ultimately yields

maximum performance of the thermoacoustic heat engine, and utilises as much of the waste heat as possible.

The main conclusions made from the simulations are described below.

Drive ratio is directly proportional to the thermal power input as shown in Fig.5, therefore the greater the

thermal power input the greater the drive ratio in the engine. As shown in Fig.6 when drive ratio is increased,

beyond a certain point there are no gains in thermal efficiency as viscous losses increase in the stack. This is the

obstacle faced when attempting to use high level thermal power inputs. You can decrease drive ratio by

increasing mean pressure, also shown in Fig.5, but high mean pressure values also cause viscous losses and

nonlinear behaviour that diminishes the performance of the engine. All of this suggests that there is a mean

pressure and drive ratio value that yield optimal thermal efficiency in the engine.

The relationship to the thermal efficiency of the engine when varying the global parameter drive ratio

DR, with a constant mean pressure is shown in Fig.6. As the Drive Ratio increases, the thermal efficiency

increases as the increased pressure amplitudes in the stack yield greater acoustic power. However, as the drive

ratio continues to increase it does not have the same gains in thermal efficiency, until the engine reaches peak

performance at a drive ratio of 0.06, thermal power input of 19kW and thermal efficiency of 5.42%. This is

because local conditions in the stack of increased volumetric flow rate results in greater viscous losses in the

stack, limiting the engine’s efficiency and therefore, any increase in the DR leads to diminishing returns as there

is no additional increase in thermal efficiency.

Larger diameter resonators are able to manage larger thermal power inputs as they are able to deal with

the large pressure amplitudes generated in the engine as shown in Fig.7. A resonator diameter of 0.20 m is able

to manage a thermal power input of 19 kW at a thermal engine efficiency of 5.38%. But diameter cannot be

increased limitlessly, because if D /∆ xe is too large then this would create noise in the acoustic wave and

therefore lead to a loss in performance of the TAHE.

When varying the stack centre position from near the pressure antinode, moving it towards the pressure

node there is a significant difference in thermal engine efficiency as local conditions at the x position contribute

directly to the creation of acoustic power and viscous dissipation. Figure 8 clearly shows that there is one

optimal stack centre position. This position at x = 0.28 m is far closer to the pressure antinode than the pressure

antinode (see Fig.9). This is because as the stack moves closer to the velocity antinode, the local volumetric

flow rate is greater which means that acoustic power dissipation due to viscosity is greater.

Figures 10 and 11 shows the variation of Prandtl number and speed of sound as the mole fraction is

varied from 0 (where working fluid is 100% Argon) to 1 (where working fluid is 100% Helium). Ideally, the

Prandtl number should be as low as possible and the speed of sound should be as large as possible as the time

taken for the wave to complete an acoustic cycle is reduced. In Fig. 12 the mole fraction is varied at a mean

pressure of 2MPa and 3MPa, while keeping the Drive Ratio constant at 0.04. The optimal value for the mole

fraction, XA is 1.0, which provides a far greater speed of sound value of 1,096.50m/s even though the Prandtl

number is as high as 0.681.

An interesting behaviour is observed when both the mean pressure and the drive ratio are varied, while

thermal power input is kept constant at 6kW. For the range of values used in this work, optimal mean pressure is

not the same for varying values of drive ratio as shown in Fig.13. For a drive ratio of 0.04 optimal mean

pressure that yields maximum thermal efficiency is 2.2MPa, for a drive ratio of 0.05 optimal mean pressure is

1.4MPa and for a drive ratio of 0.06 the optimal mean pressure is 1.0MPa. Therefore it is not necessarily the

case that greater mean pressure values produce greater performance in the TAHE.

As shown in table 3 even at such a comparatively low temperature of 150℃, we have shown as a result

of the iterative design methodology employed with this study that it is possible to recover waste heat and obtain

at the output of the TAHE 1,029.1 W of acoustic power at a thermal engine efficiency of 5.42%. Additionally,

the TAHE has been able to recover a maximum of 19kW in this study. The thermal efficiency of the engine can

be further increased by considering other aspects of the TAHE design such as the geometry of the resonator and

the design of the heat exchangers to ensure that as much of the thermal power is utilised while minimising

pressure drops across these sections of the engine, which can cause significant losses.

5. Conclusions

By varying key design parameters of the TAHE, our investigation has shown that for even such a comparatively

low temperature industrial process it has been possible to recover waste heat to deliver at the output of our

engine 1,022.2W of acoustic power with a thermal engine efficiency of 5.38% using a very low cost, zero

maintenance device. The study thus demonstrates the potential of the technology as a low cost alternative for the

recovery and potential reuse of waste heat. Thermoacoustic behaviour with current heat exchanger design limits

the level of thermal power utilised, however using multiple TAHEs together can overcome this limitation.

Experimental validation of the results presented in this paper is necessary at a later stage of the

investigation for low temperature heat utilisation, focusing on how large thermal power input, mean pressure

and drive ratio causes dissipative nonlinear effects which have an impact on the acoustic power generated by the

engine. Also, it is clear from this work that the heat exchanger design is crucial for effective waste heat

utilisation of high power levels. Therefore, optimisation of the heat exchanger will add significant value to this

investigation.

References[1] D.L.Gardner, C.Q.Howard, Waste-heat-driven thermoacoustic engine and refrigerator. In Proceedings of ACOUSTICS (2009) 1-4.[2] G. W. Swift, Thermoacoustic engines, J. Acoust. Soc. Am. 84 (1988) 1145-1180.[3] Y. Ammar, S. Joyce, R. Norman, Y.Wang, and A.P. Roskilly, Low grade thermal energy sources and uses from the process industry in the UK, Appl. Energ. 89 (2012) 3-20.[4] G.Cannistraro, C. Giaconia, A. Piccolo, Influence of thermal irreversibilities on thermoacoustic engines. In AIP Conference Proceedings. 513(2000) 67-70.[5] S.Zhou, Y.Matsubara, Experimental research of thermoacoustic prime mover. Cryogenics, 38(1998) 813-822.[6] P.Merkli, H.Thomann, Transition to turbulence in oscillating pipe flow, J. Fluid Mech. 68(1975) 567-575.[7] A.Migliori, G. W. Swift, Liquid‐sodium thermoacoustic engine, Appl. Phys. Lett. 53(1988) 355-357.[8] M.Hatazawa, H.Sugita, T.Ogawa, Y.Seo, Performance of a thermoacoustic sound wave generator driven with waste heat of automobile gasoline engine, Trans.Jpn Soc. Mech. Eng. 70 (2004) 292–9.[9] http://www.ecn.nl/fileadmin/ecn/units/eei/Onderzoeksclusters/Restwarmtebenutting/b-07-007.pdf. [Accessed 26 July 2013][10] O. G. Symko, E.Abdel-Rahman,Y.S. Kwon, M.Emmi, R.Behunin, Design and development of high-frequency thermoacoustic engines for thermal management in microelectronics, Microelectr. J. 35(2004) 185-191.[11] L.Zoontjens, C.Howard, A.Zander, B.Cazzolato, Feasibility study of an automotive thermoacoustic refrigerator. In Proceedings of ACOUSTICS (2005) 9-11.

[12] C. Jenson, R. Raspet, Thermoacoustic power conversion using a piezoelectric transducer, J. Acoust. Soc. Am. 128 (2010) 98-103.[13] S. Backhaus, Traveling-wave thermoacoustic electric generator, Appl. Phys. Lett. 85(2004) 1085-1087.[14] E. Luo, Z. Wu, W. Dai, S. Li, Y. Zhou, A 100 W-class traveling-wave thermoacoustic electricity generator, Chinese Sci. Bull. 53 (2008) 1453-1456. [15] http://www.metoffice.gov.uk/climate/uk/summaries/2012/annual/regional-values[Accessed 3 August 2013][16] A. A. Atchley, T.J. Hofler, M.L. Muzzerall, D. Kite, C. Ao, Acoustically generated temperature gradients in short plates. J. Acoust. Soc. Am. 88 (1990) 251-263.[17] J.C. Weatley, Intrinsically irreversible or natural engines. Los Alamos National Laboratory, MSK764 (1986).[18] G.W.Swift, Thermoacoustic engines and refrigerators. Physics Today 48 (1995) 22-28.[19] G.W. Swift, Analysis and performance of large thermoacoustic engine, 92(1992) 1551-1563.Available at: http://www.lanl.gov/thermoacoustics/DeltaEC.html [Accessed at 5 July 2010].[20] K.Tang, Z.J.Huang, T.Jin, G.B.Chen, Influence of acoustic pressure amplifier dimensions on the performance of a standing-wave thermoacoustic system. Appl. Therm.Eng. 29 (2009) 950-956.[21] M. Wetzel, C. Herman, Design optimization of thermoacoustic refrigerators, Int.J.Refrig. 20 (1997) 3-21. [22] H. Babei, K. Siddqui, Design and optimization of thermoacoustic devices, Energ.Convers.Manage. 49(2008) 3585-3598.[23] G.W. Swift, Thermoaccoustic: A unifying perspective for some engines and refrigerator, Condensed matter and thermal physics group, Los Alamos National Laboratory, 2002.[24] X. H. Hao, Y. L. Ju, U. Behera, S. Kasthurirengan, Influence of working fluid on the performance of a standing-wave thermoacoustic prime mover. Cryogenics, 51(2011) 559-561 [25] E. Besnoin, Numerical Study of Thermoacoustic Heat Exchangers, Doctoral dissertation, Department of Mechanical Engineering, The Johns Hopkins University, 2001.[26] B. Ward, J. Clark, G.W. Swift, Design Environment for Low-amplitude Thermoacoustic Energy Conversion: DeltaEC Version 6.2 User Guide. [e-book].

Figure 1. Design of a thermoacoustic heat engine

Figure 2. (a) resonator of length λ /2 with diameter D, (b) pressure amplitude p1 and volume flow rateU1

distribution of a standing wave in resonator as a function of position x.

Figure 3. Flowchart of iteration process

Figure 4. Schematic of the standing-wave thermoacoustic heat engine in DeltaEC

Figure 5. Drive ratio against thermal power input, with mean pressure Pm values of 1, 2 and 3 MPa

Figure 6. Thermal efficiency of the engine against drive ratio, at a mean pressure value of 3 MPa

Figure 7. Thermal efficiency of engine against thermal power input at various resonator diameters

Figure 8. Thermal engine efficiency against stack centre position, at a mean pressure value of 3MPa

Figure 9. Pressure of working fluid against position in engine along x-direction (created in DeltaEC)

Figure 10. Prandtl number against mole fraction

Figure 11. Speed of sound against mole fraction

Figure 12. Mole fraction against thermal engine efficiency with mean pressure 2MPa and 3MPa

Figure 13. Thermal engine efficiency against mean pressure with DR of 0.04, 0.05 & 0.06

Table 1. Design parameters and the range of values employed during the iterative process

Table 2. Values of the design parameters considered in the design iterations

Table 3. Optimal design parameter values as a result of iterative design methodology

Design Parameters Type of Parameter Value/Range Ref.

Global Design Parameters:

1. Thermal power input, Q¿ (kW) Independent variable 1.00-19.00

2. Mean pressure, Pm (MPa) Independent variable 1.00-3.00 [19,20]

3. Resonant Frequency, f r (Hz) Dependent Variable 43.38-137.07

4. Temperature difference,

ΔT=T H−T A (K)

Constant 144

5. Mean temperature,

T m=(T H+T A) /2 (K)

Independent Variable 351

6. Drive ratio, DR Dependent Variable calculated by simulator [16,18]

Material Parameters:

7. Speed of Sound, a (m/s) Dependent Variable 347.07-1096.50 [21-23]

8. Prandtl number, Pr Dependent Variable 0.39-0.68 [21-23]

Geometric Parameters:

9. Blockage ratio, BR=Agas / A totConstant 0.80 [2,17]

10. Stack length, Δ xs (m) Dependent variable Output parameter

dependent on XA, DR, T m, y0, &xc

11. Stack centre position, xc (m) Independent Variable 0.12-0.98 [21,22]

12. Half plate spacing, y0 (m) Constant δ κ[2,17]

13. Diameter, D ( m2) Independent Variable 0.10 – 0.20m [19]

Simulation

numberQ¿ (kW) Pm

DR f r(Hz) X A ∆ xs(m) xc(m) D(m)

1 1.00-3.00 2.00 0.020-0.060 137.06 1.00 0.032-0.082 0.076-0.101 0.10

2 1.00-4.00 3.00 0.013-0.053 137.06 1.00 0.023-0.073 0.071-0.097 0.10

3 1.00-7.00 1.00-3.00 0.0123-

0.059

137.06 1.00 0.018-0.081 0.070-0.100 0.125

4 1.00-7.00 2.00 0.009-0.062 137.06 1.00 0.021-0.084 0.070-0.102 0.15

5 1-14.00 1.00-3.00 0.004-0.060 137.06 1.00 0.017-0.082 0.069-0.10 0.175

6 1.00-12.00 2.00 0.005-0.060 137.06 1.00 0.022-0.082 0.071-0.101 0.20

7 1.00-19.00 3.00 0.003-0.626 137.06 1.00 0.019-0.085 0.007-

0.0103

0.20

8 8.00 2.00 0.040 43.38-

137.06

0.00-1.00 0.013-0.039 0.067-0.080 0.15

9 9.00 2.00 0.040 43.38-

137.06

0.00-1.00 0.014-0.041 0.067-0.080 0.175

10 12.00 2.00 0.040 43.38-

137.06

0.00-1.00 0.014-0.040 0.067-0.080 0.20

11 8.00 3.00 0.040 43.38-

137.06

0.00-1.00 0.019-0.056 0.069-0.088 0.15

12 9.00 3.00 0.040 43.38-

137.06

0.00-1.00 0.020-0.059 0.070-0.090 0.175

13 12.00 3.00 0.040 43.38-

137.06

0.00-1.00 0.019-0.056 0.069-0.088 0.20

14 19.00 3.00 0.060 43.38-

137.06

0.00-1.00 0.027-0.079 0.073-0.100 0.20

15 19.00 3.00 0.01-0.08 137.06 1.00 0.041-0.146 0.062-0.133 0.20

Optimal design parameters

Q¿ (kW) Pm (MPa) f r(Hz) DR X A ∆ xs(m) xc(m) D(m) W out

19.00 3.00 137.06 0.60 1.00 0.08 0.28 0.20 1029.10