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 CO 2, N2, O2 and H2O, at a temperature of
approximately 150 ℃, a volume flow rate of 1,288 𝑚3 /ℎ𝑟, and the corresponding density of the exhaust gases is
0.797 kg/m3. Therefore, the waste heat is calculated as 𝑄̇ = 𝑚̇ (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], 𝑄̇ 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 𝐻̇ and the acoustic power 𝑊̇ , which is the useful mechanical work that is produced in the stack.
1
̃1 ] − (𝐴𝑘 + 𝐴𝑠𝑜𝑙𝑖𝑑 𝑘𝑠𝑜𝑙𝑖𝑑 ) 𝑑𝑇𝑚
𝐻̇ = 𝜌𝑚 𝑅𝑒 [ℎ1 𝑈
2
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 to 𝑑𝑥 𝑑𝑦 𝑑𝑧 + 𝑑𝑉, and so the work
is 𝑝𝑑𝑉.The time-averaged acoustic power is the product of 𝑝1 and 𝑈1 that is produced near the surface of the
stack plate,
𝑊̇ =
1
̃1 ]
𝑅𝑒[𝑝1 𝑈
2
(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 𝑝1 to the mean
pressure, yield greater power density. However, higher values for these two design parameters lead to an
increase in the viscous penetration depth 𝛿𝜈 , 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 𝛿𝜅 (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 𝑄̇𝑖𝑛
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 (𝛿𝜈 /𝛿𝜅 ) 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 𝑋𝐴 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 , 𝑘𝑠 ) 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 conductivity 𝑘𝑠 .
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 gradient Δ𝑇/Δ𝑥𝑠 , 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 𝑈1 at 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 𝛿𝜅 and 4𝛿𝜅 [19,21,22]. In this work the half plate
spacing is kept constant at 𝛿𝜅 . 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 𝛿𝜅 and 4𝛿𝜅 .
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 𝑑2 𝑝1
=0
𝜔 2 𝑑𝑥 2
(3)
This second-order equation can be reduced into a system of two first-order equations with respect to pressure 𝑝1
and volume flow rate 𝑈1 :
𝑑𝑝1
𝑖𝜔𝜌𝑚
= −
𝑈1
𝑑𝑥
𝐴
(4)
𝑑𝑈1
𝑖𝜔𝐴
= −
𝑝
𝑑𝑥
𝜌𝑚 𝑎2 1
(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,
𝑑𝑝1
𝑖𝜔𝜌𝑚
= −
𝑈
𝑑𝑥
𝐴(1 − 𝑓𝑣 ) 1
(6)
𝑑𝑈1
𝑖𝜔𝐴
𝛾−1
= −
(1 +
𝑓 )𝑝
𝑑𝑥
𝜌𝑚 𝑎2
1 + 𝜖𝑠 𝑘 1
(7)
where 𝑓𝑣 and 𝑓𝑘 are the spatially averaged thermal and viscous functions, and ∈𝑠 is the plate heat capacity
correction factor. If 𝑓𝑣 ≥ 0 then the pressure gradient of Eq.(6) is completely inertial, but if Im[ 𝑓𝑣 ] ≠ 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 𝑓𝑘 represents the thermal
contact between the working fluid and solid plate, if 𝑓𝑘 = 1 then thermal contact is perfect and if 𝑓𝑘 = 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 rate 𝑈1 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.
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[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
rate 𝑼𝟏 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, 𝑸̇𝒊𝒏 (kW)
Independent variable
1.00-19.00
2.
Mean pressure, 𝑷𝒎 (MPa)
Independent variable
1.00-3.00
3.
Resonant Frequency, 𝒇𝒓 (Hz)
Dependent Variable
43.38-137.07
4.
Temperature difference, 𝚫𝑻 = 𝑻𝑯 − 𝑻𝑨 (K)
Constant
144
5.
Mean temperature, 𝑻𝒎 = (𝑻𝑯 + 𝑻𝑨 )⁄𝟐 (K)
Independent Variable
351
Dependent Variable
calculated by simulator
[16,18]
6. Drive ratio, 𝑫𝑹
Material Parameters:
[19,20]
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]
Constant
0.80
[2,17]
Dependent variable
Output parameter
dependent on XA, DR, 𝑇𝑚 , y0, &𝑥𝑐
Independent Variable
0.12-0.98
[21,22]
Constant
𝛿𝜅
[2,17]
Independent Variable
0.10 – 0.20m
[19]
Geometric Parameters:
9.
Blockage ratio, 𝑩𝑹 = 𝑨𝒈𝒂𝒔 /𝑨𝒕𝒐𝒕
10. Stack length, 𝚫𝒙𝒔 (m)
11. Stack centre position, 𝒙𝒄 (m)
12. Half plate spacing, 𝒚𝟎 (m)
13. Diameter, 𝑫 ( m2)
𝑸̇𝒊𝒏 (kW)
𝑷𝒎
DR
𝒇𝒓 (Hz)
𝑿𝑨
∆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
Simulation
number
Optimal design parameters
𝑸̇𝒊𝒏 (kW)
𝑷𝒎 (MPa)
19.00
3.00
𝒇𝒓 (Hz)
DR
𝑿𝑨
∆𝒙𝒔 (m)
𝒙𝒄(m)
𝑫(m)
𝑾̇𝒐𝒖𝒕
137.06
0.60
1.00
0.08
0.28
0.20
1029.10