Geometrical Optimization of Thermoacoustic Heat Engines A. H. Ibrahim, Nadim M. Arafa and Essam E. Khalil

Mechanical Power Engineering Department, Faculty of Engineering, Cairo University, Cairo-Egypt

ABSTRACT Thermoacoustics is concerned with the interaction of thermodynamics and acoustics. This interaction can

bring numerous innovative applications in power generation and refrigeration. Thermoacoustic phenomenon can be classified into two main categories: the first one where the thermal energy (heat) is transformed into acoustic energy (sound) is called thermoacoustic engine; the second type, where the acoustic energy is transformed into thermal energy, is called thermoacoustic refrigerator or heat pump [1]. The main objective of this work is to inspect the effect of the main geometrical parameters of the thermoacoustic heat engine on the performance, in order to increase the thermal efficiency. In this work, the thermoacoustic heat engine is modeled using the DeltaEC software [2]. The results show that the efficiency of the thermoacoustic engine can be increased from 9.6 % to 16 % by changing the mean pressure, stack spacing, stack mid-position and stack length. The physics behind these changes and the suggested design guidelines are presented.

I. Introduction

Thermoacoustics is the science concerned with the interaction of thermodynamics and acoustics. The compressions and expansions of a gas parcels, due to the existence of a standing acoustic wave, results in a consequence fluctuations in the gas parcels’ local temperature. This kind of interaction can bring many innovative applications in power generation and refrigeration. Thermoacoustic devices work with environmentally-friendly working fluids, have simple structure, can readily be operated with renewable energies and operate with no moving parts except for the oscillating diaphragm. However, these devices have very low power density and relatively low first-law thermal efficiencies. This leads to the emerging need for optimization of these devices. The main objective of this work is to investigate the effect of the main operating parameters (frequency, hot side temperature and mean pressure) and geometrical parameters (stack spacing, stack mid-position, and stack length) on the performance of a thermoacoustic heat engine in order to maximize its thermal efficiency.

Figure 1: Basic structure of thermoacoustic engine (Not to scale)

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II. PROCEDURE

This work starts by considering the well-documented thermoacoustic engine presented by Swift [3] as a

starting design for the optimization. The reason for this choice is the simplicity of its design and availability of both the numerical model and the experimental measurements. This engine has a resonator length of 4.32 m and diameter of 13 cm. It has a 27.9 cm stainless steel stack of parallel plates sandwiched between a couple of heat exchanger and placed at 54.2 cm from the left closed end (Xs =54.2 cm as shown in Figure1). This design produces a 217.6 W acoustic power by consuming 2210 W of thermal energy (supplied at a hot side temperature of 583K) with a first-law efficiency of 9.8 %. Then, the DeltaEC software is used to model this design and to predict its performance. Each of the governing parameters is swept over a wide range and the most optimum value for each parameter is selected. Then, the next parameter is optimized using the best values for the previous parameters. The results of each step are documented and the most efficient design is presented and explained at the end of this work.

III. RESULTS III.1. Effects of the Operating Frequency

The frequency of the standing wave plays a major rule in the power conversion process. The results show that a resonance frequency must be used. The resonance frequency depends on the general geometry and the gas used. This study investigates which frequency harmonics provides the best performance. The results show that the first resonance frequency results in higher thermal efficiency than the higher harmonics because the oscillating pressure and velocity are both higher in the first harmonics than in the higher subsequent harmonics. The amplitude of the generated acoustic pressure is drawn versus the operating frequency as shown in Figure (2) and the peak of the acoustic pressure amplitude decreases gradually for each of the higher harmonics. The acoustic (oscillating) pressure and oscillating velocity are drawn along the resonator for each of the first three harmonic as shown in Figure (3). Since the generated acoustic power is the result of the dot product of the pressure and velocity, the highest generated acoustic power (and so the thermal efficiency since the input heat is held constant) appears at the first resonance frequency as shown in Figure (4). Therefore, it is suggested to always use the first frequency harmonics.

Figure 2: The pressure amplitude for different harmonics for a fixed resonator length of 4.32 m

Figure 3: The oscillating pressure and velocity versus the resonator length for a standing wave in 4.32 m resonator for the different harmonics

Figure 4: Generated acoustic power along the resonator length for the different harmonics

III. 2. Effects of the Hot-Side Temperature The effect of the hot side temperature is shown in figure (5). The thermal efficiency increases as the hot side

temperature increases, although it does not affect the thermal or viscous boundary layer thicknesses but it increases the availability of the input heat. As per the second law of thermodynamics, it is recommended to increase the hot side temperature as much as the material can withstand and/or the available source allows. The efficiency of the thermoacoustic engine increased from 9.85 to 11.2 % when the hot side temperature increased from 600 K to 800 K.

Figure 5: Effect of the hot side temperature on the efficiency of the thermoacoustic engine

III. 3. Effects of the Stack Spacing The results show that the stack spacing between the two plates of the stack plays a critical role. Decreasing

the stack spacing ensures that all the gas parcels are within the thermal boundary layer thickness ( k) and hence interact thermally with the walls of the stack, which enhances the performance. However, a stack spacing that is too close causes the viscous boundary layers ( v) on both walls to intersect causing viscous losses, which decreases the performance. This leads to an optimum stack spacing that must be considered for each gas type. The results show that the optimum stack spacing is at approximately 2 v which corresponds to ~ 2.53 k for a Prandtl number of 0.667 for the Helium gas used.

Figure 6: The efficiency versus the stack spacing normalized by the thermal penetration depth , yo / k

Figure 6 shows that increasing the stack spacing from 0.8 k to 2.5 k increased the efficiency by 11% (from 11.2% to 12.5%). The effect of the mean operating pressure should be carefully examined here. It should be noted that the thickness of the thermal boundary layer depends on the pressure. Hence, for a given stack spacing there is a single optimum pressure, which is the pressure at which the stack spacing equals 2.5 k. However, if the stack spacing is not set yet, one should use the highest possible pressure (as the mechanical design allows), since this increases the efficiency [4] and then select the stack spacing accordingly. III. 4. Effects of the Stack Mid-Position

Then, using the first frequency harmonics and 511 K temperature difference across the stack (823 K-312 K) and a

stack spacing of 2.5 k (0.687 mm), the effect of the stack mid-position was investigated. The results indicate that each of the oscillating pressure and velocity has a profile along the resonator length.

Because the output acoustic power is proportional to the dot product of the two in the region where the stack is located, moving the stack along the resonator changes the output acoustic power and thus it is expected that the performance depends on the stack mid-position.

In this work, the results show that decreasing the stack mid-position from 12.5 % of the resonator length (stack

starts at 0.54 m) to 8.5% of the resonator length (stack starts at 0.17 m) causes an increase in the thermal efficiency by 27% (from 12.47 % to 15.85 %), as shown in Figure 7.

III. 5. Effects of the Stack Length

Figure 7: the efficiency versus the stack position inside the resonator. (The zero indicates the left closed end at

the hot side)

The effect of the stack length as shown in figure 8 indicated that as the stack length increases, the efficiency increases; because the stack is allowed to be receive more acoustic power along the resonator length. However, as the stack length increases, the temperature of the ambient heat exchanger decreases, which sets a limit on the maximum, stack length for a given ambient temperature, above which the operation of the heat exchanger cannot be sustained. The same effect was observed with high mid-stack positions. This resembles the condenser pressure in conventional power plants. In the case considered, using the optimized parameters above, the results show that increasing the stack length from 0.27 m (0.06 Lres) to.3 m (0.12 Lres) corresponds to a decrease in the ambient heat exchanger temperature from 310 K to 303 K and a corresponding increase in the thermal efficiency from 15.85 % to 16 %.

Figure 8: The efficiency and generated cold side temperature versus the stack length for a fixed resonator

length of 4.32 m

IV. SUMMARY OF CONCLUSIONS The efficiency of the thermoacoustic engine can be increased by understanding the underlying physics and the relationships between thermodynamics and acoustics. The following design guidelines are suggested:

1- Based on the resonator general dimensions and gas type, select the first resonance frequency (as opposed to the second or third). After the final design is set, recalculate the first resonance frequency to account for any changes in it. 2- Then, use the highest hot-side temperature possible, since this increases the availability of the incoming heat and does not affect the thermal or viscous boundary layer thicknesses. 3- Then, use the highest possible mean pressure (as the mechanical design allows) and select a stack spacing that is ~ 2 v at this pressure. 4- Then, select a stack mid-position that corresponds to a high dot product of the pressure and velocity waves in the region of the stack.

5- Then, select the larger stack length available, but consider the resulting temperature of the ambient heat exchanger and keep it hotter than the ambient by a suitable temperature difference. Table 1 bellow summarizes the numeric increases in the thermal efficiency for the heat engine presented in [3]: Table 1: steps in increasing the efficiency of the thermoacoustic engine reported for each parameter

Effect of Change in efficiency Percent Increase in efficiency

Pressure From 9.85 % to 11.2 % 13.7 %

Stack spacing From 11.2 % to 12.47 % 11.3 % Stack Position From 12.47 % to 15.85 % 27.1 %

Stack Length From 15.85 % to 16.03 % 1.1 %

REFERENCES

[1]. Swift G.W. Thermoacoustics: a unifying perspective for some engines and refrigerators. The Acoustical Society of America, NY: Melville; 2002

[2]. Ward W, Swift G.W. Design environment for low amplitude thermoacoustic engines. Journal of Acoustics. Acoustical Society of America .1994; 95:3671– 4.

[3]. Swift G.W. Thermoacoustic engines. Journal of Acoustics. Acoustical Society of America 1988;84:1146–80. [4]. Tijani MEH, Zeegers JCH, De Waele ATAM. Design of thermoacoustic refrigerators. Cryogenics, 2002;42:49–57.