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Efficiency Optimization of a Standing-Wave Thermoacoustic Heat Engine

Mazen A. Eldeeb1, Mahmoud A. Fouad2, and Essam E. Khalil2

1Teaching Assistant, Mechanical Power Engineering Department 2 Professor, Mechanical Power Engineering Department Faculty of Engineering, Cairo University, Cairo, Egypt

ABSTRACT Thermoacoustic heat engines (TAHE) are capable of producing acoustic energy from any source of heat energy. Thus, the primary energy source to drive the engine could be conventional or unconventional that includes industrial waste heat, solar energy and fossil fuels. It has no moving parts thus; chances of mechanical failure are extremely low. The present work main goal is to demonstrate an optimization process that would yield a better efficiency of thermoacoustic engine model. Computational investigations were carried out to improve the efficiency of a 1.05 meter thermoacoustic heat engine using air at atmospheric pressure and 900 K temperature as the working fluid. The efficiency optimization process was implemented by performing an optimization process of stack parameters, like stack shape (i.e. Rectangular, Honeycomb, Slab, and etc…), stack plates spacing, stack length and stack material. The present optimization process has shown that slab stacks made of Celcor (a Celcor material) demonstrated much better performance than other stack shapes and materials which resist such high temperatures. For a 1.124-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 paper ends with a brief summary of conclusions.

I. 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. Although the thermoacoustic phenomenon was discovered more than a century ago, the rapid advancement in this field occurred during the past three decades when the theoretical understanding of the phenomenon was developed along with the prototype devices based on this technology. The thermoacoustic technology has not reached the technical maturity yet, as a result, the performance of thermoacoustic devices is still lower than their convectional counterparts. Thus, significant efforts are needed to bring this technology to maturity and develop competitive thermoacoustic devices.

There are several advantages of heat engines and refrigerators based on thermoacoustic technology as compared to the conventional ones. These devices have fewer 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 fabrication and maintenance costs are low due to inherent simplicity of the thermoacoustic devices.

On the other hand, the thermoacoustic devices didn’t reach an energy conversion efficiency which is equivalent to their conventional counterparts; but that may be because the research in the field in thermoacoustics is still in its beginning, it’s no secret that the conventional engines and refrigerators have reached high efficiencies after decades of research and industrial progress, that’s why there is a big hope that the thermoacoustic devices will take over from the conventional ones after a sufficient amount of research in this field regarding their advantages compared with the conventional devices.

The main components of a typical thermoacoustic engine or refrigerator are a resonator, a stack of parallel plates and two heat exchangers. A half wavelength (or a quarter wavelength) acoustic standing wave is generated in the resonator. The thermoacoustic phenomenon takes place in the stack when a nonzero

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temperature gradient imposed along the stack plates (i.e. parallel to the direction of the sound wave propagation) interacts with the sound wave oscillations. The heat exchangers are responsible of transferring heat in and out of a thermoacoustic device at their desired temperatures, thus maintaining a given temperature gradient along the stack.

Thermoacoustic engines are capable of producing acoustic energy from any source of heat energy. Thus, the primary energy source to drive the refrigerator could be conventional or unconventional that includes industrial waste heat, solar energy and fossil fuels. If the heat source for the thermoacoustic engine is the industrial waste heat or solar energy then this device has two major advantages. Firstly, it does not require any additional conventional energy resource and secondly, by utilizing the waste heat, the amount of total waste heat rejected to the thermal energy sink will be reduced which will increase the overall performance of the entire system. Thus, a complete thermoacoustic refrigeration system in which the heat engine (which operates on waste heat) drives a refrigerator and the entire system has no harmful effects on the environment can be termed as a ‘‘sustainable refrigeration system”. In contrast to the acoustically-driven thermoacoustic refrigerator which has one moving component i.e. the acoustic driver, thermoacoustically-driven thermoacoustic refrigerator has no moving parts thus; chances of mechanical failure are extremely low.

Thermoacoustic refrigerators can be classified based on the source of the acoustic energy input. If the acoustic energy is provided by a thermoacoustic engine, the refrigerator is called thermoacoustically-driven thermoacoustic refrigerator (TADTAR). Whereas, if the acoustic energy is provided by an acoustic driver (e.g. a loudspeaker), it is termed as acoustically driven thermoacoustic refrigerator. During the past decades, several acoustically-driven thermoacoustic refrigerators have been developed. Although the form of energy consumed in these refrigerators is acoustic, the energy source for the acoustic driver is typically electrical from conventional energy resources. During recent years, there is an increased interest in the development of thermoacoustically-driven thermoacoustic refrigerators. These devices are built by coupling a thermoacoustic refrigerator to a thermoacoustic engine.

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 (Figure 1). During the phases of compression and expansion, heat is exchanged with the wall, generating a difference in temperature between the two ends.

The thermoacoustic devices (i.e. heat engines and refrigerators) have a very simple structure, as it consists of a long tube working as a resonator, a stack of parallel plates where the thermoacoustic effect occurs, two heat exchangers on both sides of the stack. In thermoacoustic engines, the hot heat exchanger is subjected to high temperature and given an amount of heat, while the other heat exchanger kept at ambient temperature. As explained previously, acoustic power will be a result of the process and it will be transmitted through the cold duct and can be used directly or transformed into electricity by linear alternator or a piezoelectric material. The remaining heat energy will be transmitted to the outside environment.

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Figure 1:

Figure 2: Heat Engine Schematic

II. Model Consider a solid plate of length x, width /2, and negligible thickness. The length x is aligned along

the x axis, and there is an ordinary acoustic standing wave directed along x in the fluid around the plate, so that the acoustic pressure is:

1) and the acoustic velocity is:

The plate will be introduced now into the standing wave and begin our thermoacoustic calculations by

first finding the temperature of the fluid near the plate. It will be clear that the plate modifies the original, unperturbed temperature oscillations in both magnitude and phase, for fluid about a

thermal penetration depth away from the plate, where is the fluid's thermal diffusivity, and K is its thermal conductivity. To make rapid progress, several assumptions are needed: [1] Assume that a steady state exists.

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Assume that the plate is short enough and far enough from both velocity and pressure nodes that p1 and u1 can be considered uniform over the entire plate. Assume the fluid has zero viscosity, so that u1 does not depend on y. Assume that the plate has a large enough heat capacity per unit area that its temperature does not change

appreciably at the acoustic frequency. Temperature dependence of the thermo physical properties of the fluid and plate will be neglected. Assume that the plate has a given mean-temperature gradient in the x direction and neglect the

plate's thermal conductivity in the x direction. Neglect the fluid's thermal conductivity along x.

Consistent with these assumptions, the mean fluid temperature Tm (x) is also taken to be independent of y

and to be the same as that of the plate. To calculate the oscillating fluid temperature T1, begin with a small mass of fluid, knowing that the second law of thermodynamics (in case of injection of small mass dM in an open system like the fluid element): [2]

Using the product rule to expand the derivatives, subtracting s times the continuity equation from the result, and multiplying by T yields

The final pair of equations can be taken as the expression of the Second law for a fluid (where v is the

velocity), which shows that the entropy at a point changes in time due to convective flow of entropy, conduction of heat, and generation of entropy by quadratic terms (e.g., viscosity). Keeping only first-order terms, and neglecting thermal conduction along x, Eq. (5) becomes

Applying entropy Maxwell relations,

The PDE is to be solved subject to the boundary condition T1 (0) = 0 imposed by the plate and the

boundary condition that T1 ( ) be finite. Knowing that The solution is:

The first term here is simply due to the adiabatic compressions and expansions of the fluid. The second

term comes from the mean-temperature gradient in the fluid; as the fluid oscillates along x with displacement

amplitude , the temperature at a given point in space oscillates by an amount even if the temperature of a given piece of fluid remains constant. The actual temperature oscillations are just a linear superposition of these two effects. Setting Eq. (9) equal to zero, it is clear that there is a critical mean-temperature gradient:

The critical temperature gradient is important because, as will be shown later, it is the boundary between

the heat pump and prime mover functions of thermoacoustic engines, and for efficient engine performance.

Since ordinary thermal conductivity in the x direction is being neglected, the only way heat can be transported along x is by the hydrodynamic transport of entropy, carried by the oscillatory velocity :

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The second term is zero because p1 and u1 are /2 out of time phase for a standing wave, i.e., is purely imaginary. Only Im[T1] contributes to the first term because is

purely imaginary, so that

The total heat flux can be obtained by integration,

where is the ratio of actual temperature gradient to the critical gradient defined in Eq. (10).

Now, the work flux, i.e., the acoustic power, will be discussed. It is known from thermodynamics that the work dw done by a differential volume of fluid dx dy dz as it expands from dx dy dz to dx dy dz + dV is

, and so the power per unit volume is:[1]

The work flux per unit volume is,

The total acoustic power produced is found by integrating over all space:

The acoustic power is proportional to the volume of fluid that is about a thermal penetration

depth from the plate. It is proportional to , and so is quadratic in the acoustic amplitude (as was the

heat flux) and vanishes at pressure nodes. Finally, is proportional to , the same temperature

gradient factor as appeared in When , , and there are no temperature oscillations in the fluid, and no acoustic power. For , and acoustic power is produced near the plate.

Whether this power increases the amplitude of the standing wave, is radiated away to infinity, is simply absorbed, or flows through an acoustic-to-electric transducer to generate electric power depends on details of the resonator, not on the plate itself or on the standing wave near the plate. For , and acoustic power is absorbed near the plate.

III. Procedure and Results The optimization process is performed by altering the stack geometrical parameters like length, shape,

spacing, and also the stack material itself, to obtain the maximum efficiency of a square cross section thermoacoustic heat engine made of stainless steel, of about 1.1 m length and 0.011 m2 cross sectional area, utilizing atmospheric air at 627 °C temperature as the working fluid, giving a heat-to-acoustic-power conversion efficiency of 19.613%. This process will be made using DeltaEC (Design Environment for Low-Amplitude Thermoacoustic Energy Conversion) 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. [3] 1. Stack Material Choice

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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. [1] 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, and 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. Both materials are inexpensive and commercially available, which will facilitate the manufacturing process. Now, honeycomb stacks made of Celcor and Mylar will be compared. At a fixed length of 5 cm, the pores radius will be swept over a range between 0.3 mm to 1 mm for both materials, to check which material provides better performance.

The process shows that Mylar stacks provide better performance than Celcor stacks. That makes sense as Mylar has much lower thermal conductivity and higher heat capacity than Celcor, which makes it perfect for stacks. The maximum efficiency of Mylar stacks was 24.364% at 0.49 mm pore radius, compared with the maximum obtainable efficiency of Celcor stacks which was 19.66% at 0.52 mm pore radius, which means that using Mylar instead of Celcor increased the efficiency by 23.9%, which is a really significant performance improvement. However, there is a major drawback in Mylar as a stack material which makes it necessary to use Celcor in stack, which is the low melting point of the Mylar. The highest known melting point of Mylar is between 218 - 232 °C [4], 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. [5]

Figure 3:

Figure 4:

2. Stack Shape and Plate Spacing Choice

The most commonly used stack shapes are slab stacks which consist of parallel slabs or plates, rectangular stacks which consist of a block having rectangular pores, and honeycomb stacks which differ

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from the rectangular stacks in the pores’ shape, in this case they are hexagonally shaped. The procedure of this optimization stage is simple. Honeycomb, rectangular, and slab 5 cm-long-stacks made of Celcor will be compared, by changing the pore radius and plate spacing over the range between 0.3 mm and 1 mm, and studying the behavior of efficiency and power output for each stack shape.

Figure 5:

Figure 6:

The previous figures show clearly that slab stacks give much better performance than any other stack

shape. A maximum efficiency of 19.66% and a maximum power of 131.75 W were previously obtained using Celcor honeycomb stack. When the stack shape is changed to rectangular stack, a maximum efficiency of 18.056%. But using slab stacks, the maximum efficiency jumped to 26.397% and the maximum power output increased to 176.99 W at half plate spacing of 0.324 mm. That means that the maximum efficiency has been improved by 34.26% of the previously obtained honeycomb stack maximum efficiency, along with another improvement in acoustic power output by 34.33%. This improvement is very huge, that means that using Celcor slab stack is the right decision for this design.

3. Stack Length

Further improvement of the thermoacoustic heat engine’s efficiency can be performed by optimizing the stack length. In this optimization stage, the stack length will be swept over a range between 5 cm and 10 cm, at a step of 0.25 mm, which makes 201 points.

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Figure 7:

Figure 8:

The previous figures show that the maximum efficiency is 30.611% and it can be obtained using a 7.75-

cm-long stack, and the maximum power is 205.24 W and it can be obtained at the same length. That means that using a 7.75 cm long stack instead of a 5 cm one at a half plate spacing of 0.324 mm has improved the efficiency and the maximum power output by 15.96%. 4. Final Design

Table 1: Parameter Description Value

A Square cross sectional area 0.011 m2 S Square side length 10.488 cm

Lhot,duct Length of the hot duct. 2.315 cm

Lcoldt,duct Length of the cold

(ambient) duct. 98.25 cm

LH,HEX Length of the hot heat

exchanger. 1.92 cm

2yo,hot hex Hot HEX plate spacing 0.2268

mm

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 7.75 cm 2yo The stack’s plate spacing 0.648 mm 2l The stack’s plate thickness 0.304 mm

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PR Stack’s Porosity ratio 0.68067 Ltot Total engine’s length 1.1239 m

Table 2:

Working Fluid Air Inlet Mean Pressure 1 bar Inlet gas temperature 900 K (627 °C) Operating frequency 169.24 Hz Heat Input 670.5 W Acoustic Power Output 205.24 W TAHE efficiency 30.611% Relative normalized eff. (to Carnot) 47.97%

Figure 9:

IV. Conclusions From the previous optimization process, some conclusions can be made based on the results

obtained:

1. Stack optimization is the most important process towards obtaining the highest possible efficiency. The utilization of the Celcor slab stack with optimized length and plate spacing have increased the efficiency and maximum power output by 40.5% from the efficiency obtained before optimizing the stack that means the stack optimization had the major contribution in the optimum efficiency. 2. The selection of stack material is one crucial factor of the performance improvement process. First of all, it should be a material which has low thermal conductivity in order to prevent heat conduction along the stack, so that a large temperature gradient between the two ends could be obtained, and the thermoacoustic effect could be sustained. That means metals could be eliminated from the selection. Moreover, heat engines usually operate at high temperatures. In the case shown in this study, the maximum gas temperature is 900 K (627 °C). The efficiency analysis has shown that using a polymer material called Mylar gives the best performance of the engine, but the problem is that it has a melting point of about 250 °C, which means it will not be suitable for operation in high temperatures. That’s why Celcor was chosen as the stack material. 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. 3. Stack’s shape is also a very important factor in the efficiency improvement process. The study has shown that slab stacks give much better performance than other stack shapes like honeycomb and rectangular stacks. Using slab stacks improved the efficiency and maximum power by 34.26% higher than honeycomb stacks, and 46.2% higher than rectangular stacks of the same length and material (Celcor). Also, this maximum efficiency has been obtained at a porosity ratio 13.68% higher than the porosity ratio at which the honeycomb stack gave its best performance, and 16.85% higher than the most efficient porosity ratio of the rectangular stack. This higher porosity ratio means that the open frontal area of the stack is increased, that means less solid material needed to manufacture the stack, consequently lower stack manufacturing cost, and most importantly, higher porosity ratio means less viscous dissipation losses in the stack. 4. Small cross sectional area of the resonator tube helps improve the performance of the thermoacoustic heat engine. According to G. W. Swift, the easiest way to decrease the acoustic power losses is to decrease the surface area of the resonance tube’s walls. That could be achieved for a fixed length tube by decreasing the cross sectional area. However, big decrease in cross sectional area may lead to choking and power losses inside the tube. Careful optimizing process of the cross sectional area is necessary. [6]

Nomenclature A ………Cross Sectional Area [m2]

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a ………Speed of Sound [m/s]

…… Isobaric Specific Heat Capacity [J/kg.K]

………Total Acoustic Power [W] F …… Heat Transfer Surface Area [m2] f ……… Frequency [Hz] h …… Enthalpy [J] K …… Thermal Conductivity [W/m.K]

…… Thermal Diffusivity [m2/s] l …… Stack Half Thickness [m] L …… Length [m] M …… Mass [kg] n ……Complex Heat Transfer Component

…… Peak standing wave pressure amplitude [Pa]

…… Complex Power Output [W] …… Acoustic Pressure [Pa] ……Mean Pressure [Pa] ……Dynamic pressure amplitude at a location x [Pa]

Q …… Quality Factor

…… Heat Energy [W]

……Total Heat Flux [W] …… Heat Flux per Unit Area [W/m2]

R …… Specific Gas Constant [J/kg.K] S …… Entropy [J/K] s …… Specific Entropy [J/kg.K]

…… Oscillatory Temperature [K] ……Mean Temperature [K] ……Volume Velocity [m3/s] ……Acoustic Particle Velocity [m/s]

v …… Velocity [m/s] …… Power per Unit Volume [W/m3] ……Stack Half Plate Spacing [m]

…… Overall Heat Transfer Coefficient [W/m2.K]

…… Thermal Expansion Coefficient [1/K] …… Ratio of actual to critical temperature gradients

…… Ratio of Isobaric to Isochoric Specific Heats ……Thermal Penetration Depth [m] ……Viscous Penetration Depth [m]

…… Efficiency

…… Complex Efficiency ……Carnot’s Efficiency

…… Wavelength [m]

…… Radian Wavelength ……Fluid’s Mean Density [kg/m3]

……Angular Frequency [rad/s]

References 1. Swift, G. W. (1988, October). Thermoacoustic Engines. Journal of the Acoustical Society of America , 1145-

1180. 2. Landau, L. D., & Lifshitz, E. M. (1987). Fluid Mechanics (2nd ed., Vol. 6). (J. B. Sykes, & W. H. Reid,

Trans.) Oxford: Pergamon Press. 3. 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.

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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. Honeycomb Ceramic - China Industrial ceramic, honeycomb ceramic, ceramic packing in Chemical Filling. Retrieved from Made in China: http://jxtianmei.en.made-in-china.com/

6. Swift, G. W. (2002). Thermoacoustics: A unifying perspective for some engines and refrigerators. Acoustical Society of America.