[American Institute of Aeronautics and Astronautics 9th Annual International Energy Conversion Engineering Conference - San Diego, California ()] 9th Annual International Energy Conversion

American Institute of Aeronautics and Astronautics

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Resonator Shape Effect on the Performance of a Standing-Wave

Thermoacoustic Heat Engine

Mazen A. Eldeeb1, Essam E. Khalil

2 and Mahmoud A. Fouad

3

1Teaching Assistant, Mechanical Power Engineering Department

2 Professor, Mechanical Power Engineering Department, AIAA Fellow, ASME Fellow

3 Professor, Mechanical Power Engineering Department

Faculty of Engineering, Cairo University, Cairo, Egypt

Abstract This thesis demonstrates an attempt to make a design of an about 1-meter-long thermoacoustic heat

engine that has an optimum efficiency. This will be done using DeltaEC, software which was developed

especially for low amplitude thermoacoustic devices modeling. The optimization process includes

geometrical parameters of the resonator tube and the stack, the working fluid, and the heat input to the

engine. The present optimization process has shown that slab stacks made of Celcor (a Ceramic material)

demonstrated much better performance than other stack shapes and materials. For a 1.1239-meter-long and

0.011 m2 square-shaped resonator tube, a 7.75 cm long slab stack made of Celcor having 0.304 mm-thick-

plates, spaced by 0.648 mm, giving a porosity ratio of 0.68067, will theoretically convert heat to acoustic

power at an efficiency of 30.611% which is equivalent to 47.97% of Carnot’s efficiency. The thesis ends

with a brief summary of conclusions.

1. Introduction

Thermoacoustics is a branch of science concerned mainly with the conversion of heat energy into

sound energy and vice versa. The device that converts heat energy in sound or acoustic work is called

thermoacoustic heat engine or prime mover and the device that transfers heat from a low temperature

reservoir to a high temperature reservoir by utilizing sound or acoustic work is called thermoacoustic

refrigerator. There are several advantages of heat engines based on thermoacoustic technology as compared

to the conventional ones. These devices have fewer components with at most one moving component with no

sliding seals and no harmful refrigerants or chemicals are required. Air or any inert gas can be used as

working fluids which are environmentally friendly. Furthermore, the simple design of the devices reduces

the fabrication and maintenance costs. However, significant efforts are needed to bring this technology to

maturity and develop competitive thermoacoustic devices. The thermoacoustic (TA) procedure uses a sound

wave to achieve local heat exchange between the gas in which it propagates and a solid medium. Heat

transfer occurs simultaneously along the length of the solid walls of the structure in which the gas is held. A

sound wave is the propagation of a disturbance, the passage of which induces a reversible variation in the

local physical properties (temperature, pressure) of the medium in which it propagates. It transports energy,

but not matter. The propagation medium undergoes macroscopic displacement in the same direction as the

propagating wave, and is therefore a longitudinal wave. The pressure wave causes the volumes of gas to

oscillate around a mean value. Thus, half-way through the cycle, the gas is on one side of this mean and is

compressed and hot, whereas at the end of the cycle, it is on the other side of the mean and is expanded and

cold. If a solid medium, such as a metal plate, is used, this solid medium is likely to accumulate heat or to

slow heat transfer. During the phases of compression and expansion, heat is exchanged with the wall,

generating a difference in temperature between the two ends. In this study, four different resonator shapes

are investigated and compared for a thermoacoustic heat engine of 1.12 m length to select the resonator

shape that gives the best efficiency of the device. The selected shape will then undergo some changes in the

geometrical parameters in order to obtain further performance enhancement.

2. Design Strategy

“Four different designs are investigated and compared with a previously studied design; see Figure 1

and Table 1. Design E, the previously studied design, had a 1.124 m long and 0.011 m2 square-shaped

resonator tube, a 7.75 cm long slab stack made of Celcor having 0.304 mm-thick-plates, spaced by 0.648

mm, which gave a heat to acoustic power conversion efficiency of 30.611% which is equivalent to 47.97%

9th Annual International Energy Conversion Engineering Conference31 July - 03 August 2011, San Diego, California

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of Carnot’s efficiency. The new designs are modifications of the previous design; the difference is the

resonator tube shapes. The previous design was a straight square duct, but the new designs do not have a

constant cross section area. The design A has a truncated pyramidal shape, with an increase of cross section

area (i.e. the smallest area is at the beginning and the biggest is at the end). The design B is the opposite of

the design A, as it has the same shape but the cross section area is decreasing. The design C is made of 4

truncated pyramids, with a 2 degrees increase of the taper angle in each part. The design D is similar to C but

the increase in the pyramid angle is 5 degrees instead of 2. The design E is the basic straight design. All

designs are subjected to the same operation conditions (Frequency: 169.24 Hz, Heat input: 670.6 W, Initial

gas temperature: 900 K). The designs are shown in figure 1.

A

B

C

D

E


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Figure 1: Schematic of the compared designs

Table 1: Summary of the compared designs description and dimensions

Design Name Description

A1 • Truncated pyramid (Square Cross Section)

• Initial square side length: 10.4 cm

• Final square side length: 30.1 cm

• Divergence angle: 5 degrees













B1 • Truncated pyramid (Square Cross Section)



• Convergence angle: 5 degrees

B2 • Truncated pyramid (Square Cross Section)



• Convergence angle: 4 degrees

C • Four truncated pyramid segments of equal length (28.1 cm)

• First segment: Initial 10.4 cm, Final 12.5 cm, angle 2 degrees

• Second segment: Final 15.4 cm, angle 3 degrees

• Third segment: Final 19.2 cm, angle 4 degrees

• Forth segment: Final 24.2 cm, angle 5 degrees

D • Four truncated pyramid segments of equal length (28.1 cm)

• First segment: Initial 10.4 cm, Final 12.5 cm, angle 2 degrees

• Second segment: Final 15.4 cm, angle 3 degrees

• Third segment: Final 19.2 cm, angle 4 degrees

• Forth segment: Final 24.2 cm, angle 5 degrees

E • 1.12 m long constant area square duct

• Square side length: 10.4 cm


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Figure 2: Comparison between the efficiencies of the investigated designs

By comparing the previous designs at the mentioned operating conditions, it is clear that the design

D is the most efficient design at the same resonator length and operating conditions. It has produced a

thermal to acoustic power conversion efficiency of 34.18%, equivalent to 48.73% of Carnot’s efficiency. The

acoustic power output is 229.23 W and the temperature difference is 631.36 K. Design B showed the worst.

3. Modification of the Selected Design

The next step will be performing a set of modifications on the selected design. These modification

will not approach the geometrical parameters of the resonator tube itself, it will be focused on the heat input,

and also the stack parameters (i.e. stack material, stack length and plate spacing). This process will be made

using DeltaEC (Design Environment for Low-Amplitude Thermoacoustic Energy Conversion) which is a

computer program that can calculate details of how thermoacoustic equipment performs, or can help the user

to design equipment to achieve desired performance. DeltaEC numerically integrates in one spatial

dimension using a low-amplitude, acoustic approximation and sinusoidal time dependence. It integrates the

wave equation and sometimes other equations such as the energy equation, in a gas (or a very compressible,

thermodynamically active liquid), in a geometry given by the user as a sequence of segments such as ducts,

compliances, transducers, and thermoacoustic stacks or regenerators [1]. The stack shapes to be used in all

designs is the slab stack as it has proven to be the most effective stack shape especially when compared to

other shapes like honeycomb and rectangular stacks [2].

The effect of heat input on the selected design’s performance

The heat input here represents the effect of the inlet volume flow rate as long as the inlet temperature is

fixed, knowing that:

The heat input will be changed over a range between 670 W to 5000 W with a 21.6 W step which makes 201

points. The heat input’s effect on the efficiency of the device is shown in figure 3.


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Figure 3: Heat input vs. efficiency

The previous figure shows that the efficiency increases with the heat input. The increment is

significant in the beginning, but when a heat input of 3 kW is reached, the efficiency keeps increasing but

with a less steep trend. So, the heat input increase is no longer very effective in obtaining a significantly

better efficiency, because the efficiency improvement is negligible in this case. The best practical efficiency

is attained at a heat input of 3000 Watts, where the efficiency is 38.4%, equivalent to 54.755% of Carnot’s

efficiency. This design will be called “D2”, and its efficiency is 12.28% higher that the selected design “D”.

The acoustic power output is 1.151 kW.

Stack material and plate spacing selection

The stack should be made of a low-thermal-conductivity-material, because if the thermal conductivity of

the stack is high, the heat flux in the x direction of the stack will increase by simple heat conduction, and this

reduces the engine’s efficiency [3]. Hence, it is preferred that the stack is manufactured from a non-metallic

material. The most common non-metallic materials used in stack manufacturing are Celcor, which is a

ceramic material made by Corning Incorporated, a company based in the United States, Mylar, which is a

plastic material made from the resin Polyethylene Terephthalate (PET), and it is made by DuPont Teijin

Films, a company based in the United States, and Kapton, which is produced from the condensation of

pyromellitic dianhydride and 4,4'-oxydiphenylamine, and also made by DuPont Teijin Films. Those

materials are inexpensive and commercially available, which will facilitate the manufacturing process. Now,

slab stacks made of Celcor, Mylar, and Kapton will be compared. At a fixed length of 7.75 cm, the plate

spacing will be swept over a range between 0.2 mm to 1 mm for all materials, to check which material

provides better performance.

The process shows that Mylar and Kapton demonstrate an almost identical behavior, both stacks

provide better performance than Celcor stacks. That makes sense as both Mylar and Kapton have much

lower thermal conductivity and higher heat capacity than Celcor, which makes them perfect for stacks. The

maximum efficiency of Mylar stacks was 40.008% at 0.284 mm half plate spacing, and for Kapton it was

40.021% at the same half plate spacing, compared with the maximum obtainable efficiency of Celcor stacks

which was 38.973% at the same half plate spacing (figure 4). However, the increase in efficiency

accompanied by changing the stack material from Celcor to Kapton or Mylar is negligible, as the efficiency

increases by only 2.68% and 2.65% respectively, an improvement which can be neglected. Moreover, there

is a major drawback in Mylar and Kapton as stack materials which makes it necessary to use Celcor in stack,


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which is the low melting point of them both. The highest known melting point of Mylar is between 218 - 232

°C [4], and for Kapton it is 400 °C [5], while the maximum working temperature in this model is 900 K (i.e.

627 °C). Celcor is a ceramic material which definitely has much higher working temperature (up to 1600

°C), that makes Celcor a perfect choice especially for heat engines as they typically operate in relatively high

temperatures [6]. So, the new modification “D3” will use a slab stack made of Celcor, but the new plate

spacing will be 0.284 mm instead of 0.324 mm, and the efficiency is 38.973%, equivalent to 55.418% of

Carnot’s efficiency. This modification has increased the efficiency by about 1.5%. The acoustic power

output became 1.17 kW.

Figure 4: Comparison between some stack materials

The effect of stack length on performance

The final step towards enhancing the efficiency of the device is checking the effect of the Celcor slab

stack on the engines efficiency. The length is investigated over a range between 7 and 10 cm, while fixing

the total resonator length to the original length of 1.12 m. The step will be 3 mm making 11 points. The

process is shown in figure 5.

The maximum efficiency could be obtained at a stack length of 8.5 cm, and it is 39.13% which is

equivalent to 55.5% of Carnot’s efficiency. That means that the new modification “D4” using a stack length

of 8.5 cm instead of 7.75 cm increases the efficiency by 0.4% as a result of a stack length increment of

9.67% compared with the design D3 which is a slight enhancement. The acoustic power output has jumped

to 1.174 kW.

http://www.matweb.com/tools/unitconverter.aspx?fromID=2&fromValue=218

http://www.matweb.com/tools/unitconverter.aspx?fromID=2&fromValue=232


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Figure 5: Stack length vs. efficiency

4. The final design

Table 2: Final design’s geometrical parameters

Parameter Description Value

A1 First segment initial Square cross sectional area 0.011 m2

S1 First segment initial square side length 10.488 cm

θ1 First segment taper angle 4°

A2 Second segment initial Square cross sectional area 0.0208 m2

S2 Second segment initial square side length 14.418 cm

θ2 Second segment taper angle 6°

A3 Third segment initial Square cross sectional area 0.0413m2

S3 Third segment initial square side length 20.323 cm

θ3 Third segment taper angle 8°

A4 Fourth segment initial Square cross sectional area 0.0796 m2

S4 Fourth segment initial square side length 28.221 cm

θ4 Fourth segment taper angle 10°

Af Final square cross sectional area 0.1454 m2

Sf Final square side length 38.13 cm

Ls Segment length 28.1 cm

LC,HEX Length of the cold (ambient) heat exchanger. 2.155 cm

2yo,cold hex Ambient HEX plate spacing 0.797 mm

LStack Stack length 8.5 cm

2yo The stack’s plate spacing 0.568 mm

2l The stack’s plate thickness 0.304 mm

PR Stack’s Porosity ratio 0.6513

Ltot Total engine’s length 1.1239 m


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Table 3: Final design operation conditions and performance

Working Fluid Air

Inlet Mean Pressure 1 bar

Inlet gas temperature 900 K (627 °C)

Operating frequency 169.24 Hz

Heat Input 3 kW

Acoustic Power Output 1.174 kW

TAHE efficiency 39.13 %

Relative normalized eff. (to Carnot) 55.5 %

Figure 6: Final design schematic

5. Conclusions

From the previous enhancement process, some conclusions can be made based on the results

obtained:

1. The utilization of the Celcor slab stack with enhanced length (8.5 cm) and plate spacing (0.569 mm)

which is used in the final design D4 has increased the efficiency and maximum power output by 1.9%

from the efficiency obtained before improving the stack for the design D2, which means the stack

enhancement, had a contribution in the efficiency enhancement.

2. The heat input modification (design D2) has increased the efficiency of the device by 12.32% of the

efficiency of the design D which was before changing the heat input. That means that the heat input is a

major parameter in the efficiency enhancement process.

3. The selection of the design D as the best design has given an efficiency which is 11.69% higher than

the original constant cross section area design (Design E), this is due to the divergence of the cross

4° 6°

8°

10°

1.124 m

0.281 m

0.085 m


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section area which allows the gas to flow with ease at high velocities without being subjected to the

danger of choking.

4. The total enhancement process has improved the efficiency from 30.611% to 39.128%, which means

a 27.82% increase in efficiency from the original design E to the final design D4.

5. The stack should be made of a material which has low thermal conductivity in order to prevent heat

conduction along the stack. Moreover, heat engines usually operate at high temperatures. Celcor is a

ceramic material which typically resists high temperature, as it has a melting point of about 1600 °C,

and it also has a low thermal conductivity (2.5 W/m.K) and high heat capacity (1121.2 J/kg.K), that

means it is perfect as a stack material as it satisfies the condition of the stack’s low thermal

conductivity.

6. Nomenclature

A ………Cross Sectional Area [m

2]

l …… Stack Half Thickness [m]

L …… Length [m]

Ls ……… Segment length [m]

LStack …… Stack length [m]

PR …… Porosity Ratio

…… Heat Energy [W]

S …… Square Side Length [m]

yo …… Stack half plate spacing [m]

…… Efficiency

θ …… Taper angle [degree]

7. References

1. Ward, B., Clark, J., & Swift, G. W. (2008). Design Environment for Low-amplitude Thermoacoustic

Energy Conversion (DeltaEC Version 6.2) Users Guide. Los Alamos: Los Alamos National Laboratory.

2. Eldeeb, M. A., Fouad, M. A., & Khalil, E. E. (2011). Efficiency Optimization of a Standing-Wave

Thermoacoustic Heat Engine. Proceedings of the 49th AIAA Aerospace Sciences Meeting including the

New Horizons Forum and Aerospace Exposition. Orlando.

3. Swift, G. W. (1988, October). Thermoacoustic Engines. Journal of the Acoustical Society of America ,

1145-1180.

4. DuPont™ Teijin Films Mylar® 100 CL Polyester Film, Cap Liner and Ovenable Lidding, Heat

Sealable, 100 Gauge. Retrieved from MatWeb: http://www.matweb.com/

5. DuPont™ Kapton® 50HN Polyimide Film, 13 Micron Thickness. Retrieved from MatWeb:

http://www.matweb.com/

6. Honeycomb Ceramic - China Industrial ceramic, honeycomb ceramic, ceramic packing in Chemical

Filling. Retrieved from Made in China: http://jxtianmei.en.made-in-china.com/

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[American Institute of Aeronautics and Astronautics 9th Annual International Energy Conversion Engineering Conference - San Diego, California ()] 9th Annual International Energy Conversion