Influence of buffer on resonance frequency of thermoacoustic engine

Cryogenics 42 (2002) 223–227 www.elsevier.com/locate/cryogenics

Influence of buffer on resonance frequency of thermoacoustic engine G.B. Chen *, J.P. Jiang, J.L. Shi, T. Jin, K. Tang, Y.L. Jiang, N. Jiang, Y.H. Huang Cryogenics Laboratory, Zhejiang University, Hangzhou 310027, PR China Received 30 January 2001; accepted 30 January 2002

Abstract Frequency matching is of great importance to a thermoacoustically driven pulse tube refrigeration system. To compute the resonance frequency of thermoacoustic engines, the fluid impedance method is introduced. The calculations of the thermoacoustic engines with different arrangements of buffer have been carried out. The influence of the buffer arrangements and the volume on the resonance frequency as well as the acoustic power of thermoacoustic engines is also discussed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Thermoacoustic engine; Frequency matching; Pulse tube refrigeration

1. Introduction Thermoacoustic engine is a machine capable of converting thermal energy into acoustic energy. A pulse tube refrigerator driven by the thermoacoustic prime mover appears attractive because of its simplicity, high reliability, long lifetime and environmentally safety. Over the past two decades, substantial progress has been made in thermoacoustic engine and refrigerator development [1–3]. The efficiency of such engine is greatly influenced by the regenerator and phasing method, both of which are firmly related to one operating parameter – frequency. Frequency matching is important to a thermoacoustically driven pulse tube refrigeration system. It is true that the inefficiency of the generator is mainly caused by heat transfer due to finite temperature difference [4,5], but the working frequency mismatch is another cause. Furthermore, the optimal operating frequencies of the pulse tube refrigerator and the thermoacoustic prime mover are quite different. The former usually works at a fairly low frequency about several hertz, while the later works at a relatively high resonance frequency about several tens to hundreds of hertz [6]. Obviously, it is necessary to decrease the resonance frequency of a

*

Corresponding author. Tel.: +86-571-87951771; fax: +86-57187952464. E-mail address: [email protected] (G.B. Chen).

thermoacoustic prime mover, so as to match the lower frequency requirement of the pulse tube refrigerator. To a 1/2 wavelength standing thermoacoustic prime mover, the resonance frequency may be expressed by [1] pffiffiffiffiffiffiffiffiffi kRT a f ¼ ; ð1Þ ¼ 2L 2L where f is resonance frequency, a is acoustic speed, L is length of the resonator, k ¼ cp =cv is specific heat ratio of the working fluid, R is gas constant and T is the fluid temperature. Eq. (1) shows that the resonance frequency of the system depends on the resonator length and the acoustic speed. In order to obtain a lower resonance frequency, a longer resonator or a working fluid with lower acoustic speed must be adopted. However, the length of the resonator is limited by the size of the device and the selection of working fluids is limited by the refrigeration requirement. It seems that one of the effective ways of decreasing frequency is by means of remodeling buffers in the thermoacoustic system.

2. Approach to calculate resonance frequency In acoustics literatures, the articles on frequency calculation of a practical device are fairly little. Recently, Zhou and Matsubara [7] proposed a method to estimate the frequency of a standing wave thermoacoustic prime mover. When the thermoacoustic prime

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mover oscillates at a 1/2 wavelength mode, the total length L of the system equals k=2, that is f ¼ a=ð2LÞ. If diameters of the hot buffer, stack and resonator tube are close enough each other, we may use an equivalent length of the system, which is defined as the total volume of the system divided by the cross-sectional area of the resonator tube, to compute the resonance frequency of the system. Apparently, this is only an empirical formula, and can be adopted just in the double arrangement. Fluid impedance method [8] is a simple approach to deal with steady oscillating flow. From the impedance relation of a piping system based on continuity equation, the resonance frequency of a fluid piping system can be solved. Comparing with the experiment method and the above calculation method dealing with transient flow, the calculating result of the flow impedance method, which considers only a few factors, may be rough, but provides a simple and reasonable way to assess system frequency. In terms of the fluid network principle, to a pipe with length L and flow area A, when the gas viscosity is ignored, the relation between the input impedance and output impedance of the pipe, is given by [8] Z0 ¼

1 þ j ZZLc tg xL a 1 þ j ZZLc tg xL a

ZL ;

ð2Þ

where a is acoustic speed, Zc is characteristic impedance which is equal to qa=A to an idealized pipe, q is fluid density, x is angular frequency, Z0 is input impedance, ZL is output impedance, and tg is the angular function tangent. When Z0 and ZL are given to a special pipe, the resonance frequency can be solved by Eq. (2), which is called Frequency Equation. Buffers are commonly used in the thermoacoustic engine and the pulse tube refrigerator, and have a great effect on system frequency as well as other performance. We will discuss the influence of buffers with different arrangements on resonance frequency of thermoacoustic engines as follows: (1) A pipe with two ends closed, as shown in Fig. 1(a). The boundary conditions: Z0 ¼ 1; ZL ¼ 1 Combining above with Eq. (2), the frequency equation is tg

2pfL ¼ 0: a

ð3Þ

So the resonance frequency is: f ¼ n=2  a=L. (2) A pipe with a buffer of volume V at the right side, as shown in Fig. 1(b). The boundary conditions: Z0 ¼ 1; ZL ¼ a2 q=jxV . Combining the above with Eq. (2), the frequency equation is: 2pfV þ aAtg

2pfL ¼ 0: a

ð4Þ

Fig. 1. Different arrangements of the buffer: (a) a resonant pipe closed at both ends; (b) a resonant pipe with a buffer at one end (single arrangement); (c) a resonant pipe with one buffer at each end (double arrangement); (d) a resonant pipe with a buffer in the middle (middle arrangement); (e) a resonant pipe with a hybrid buffer.

(3) A pipe with the buffer of volume V at both the sides, as shown in Fig. 1(c). The boundary conditions: Z0 ¼ a2 q=jxV ; ZL ¼ 2 a q=jxV . Combining above values with Eq. (2), the frequency equation becomes: tg

2pfL 2 ¼ 2pV : Aa a f  2pfV Aa

ð5Þ

(4) A pipe with a buffer of volume V in the middle of the tube, as shown in Fig. 1(d). With the connection

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point of the pipe and the buffer as the interface, the piping system can be divided into three parts, symbolized as part I, part II and part III, respectively. The parameters of each part are expressed with subscripts 1, 2 and 3, respectively. The equations of continuity and mass conservation lead to: PL1 ¼ P02 ¼ P03 ;

3.1. Influence of buffer volume on frequency

where P is gas pressure and Q is volume flow rate. In terms of the definition of impedance, we get: 1 1 1 ¼ þ : ZL1 Z02 Z03

ð6Þ

Using Eq. (2) to part I and part II, we arrive: Zc aq ¼ ; jtg xLa 1 jAtg xLa 1

Z02 ¼ 

Z03 ¼

ð6:1Þ

Zc aq ¼ ; jAtg xLa 2 jtg xLa 2

a2 q : jxV

ð6:2Þ

ð6:3Þ

Combing Eqs. (6.1)–(6.3) with Eq. (6), the frequency equation can be solved in a simpler form: tg

ture of the angular function tangent, which is in coincidence with the existence of the harmonics in the real system. For simplicity, we show only the solutions corresponding to the fundamental harmonic component, which is the dominator. 3. Calculations

QL1 ¼ Q02 þ Q03 ;

ZL1 ¼

2pfL1 2pfL2 2pfV : þ tg ¼ Aa a a

ð7Þ

(5) Hybrid buffer. As shown in Fig. 1(e), it combines the middle and double arrangements. At the connection point of the pipe, the piping system can be divided into three parts, symbolized as part I, part II and part III, respectively. The parameters of each part are expressed with subscripts 1, 2 and 3, respectively. The equations of continuity and mass conservation lead to

The influence of buffer volume for various arrangements (see Fig. 1) on resonance frequency of the thermoacoustic engine is calculated with Eqs. (4), (5), (7) and (9), respectively. The specific buffer volume V =V0 is used in the calculation, where V is calculation buffer volume, and V0 is the initial volume. The calculation conditions are as follows: Pressure of working fluid P ¼ 2:0 MPa. Temperature of the resonator T ¼ 300 K. Length of the resonator corresponding to the experimental device is as follows: L ¼ 4:9 m for the single and double arrangements; L1 ¼ 0:45 m, L2 ¼ 4:45 m, L3 ¼ 0:070 m for the middle and hybrid arrangements. The computed result of the dependence of specific buffer volume on resonance frequency with nitrogen as the working fluid is shown in Fig. 2. From Fig. 2 we can see that the resonance frequency of the resonator can obviously be decreased by means of increasing the buffer volume in each arrangement, and the hybrid arrangement is the best for the frequency reduction.

P01 ¼ P02 ¼ P03 ; Q01 þ Q02 þ Q03 ¼ 0: In terms of the definition of impedance, we get: 1 1 1 þ þ ¼ 0: Z01 Z02 Z03

ð8Þ

Using Eq. (2), the frequency equation can be expressed as 1 þ j ZZL1 tg xLa 1 jxV1 1 þ j ZZL2 tg xLa 2 jxV2 c1 c2 þ 1 þ j ZZc1 tg xLa 1 qa2 1 þ j ZZc2 tg xLa 2 qa2 L1

225

L2

1 þ j ZZL3 tg xLa 3 jxV3 c3 þ ¼ 0: 1 þ j ZZc3 tg xLa 3 qa2

ð9Þ

L3

The above five Eqs. (3)–(5), Eqs. (7) and (9) are associated with the diverse arrangements, respectively, and have many solutions originating from the periodic na-

Fig. 2. Resonance frequency versus specific buffer volume.

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As to the middle arrangement, the resonance frequency depends on not only the volume of the buffer but also the location along the resonator. The calculation results under the condition of specific buffer volume V =V0 ¼ 1 are showed in Fig. 3. Apparently, when the specific resonator location L1 =L equals to zero (L1 ¼ 0 m, L2 ¼ L ¼ 4:9 m) or unit (L1 ¼ L ¼ 4:9 m, L2 ¼ 0 m), the middle arrangement is reduced to the single arrangement. The frequency increases gradually with the specific resonator location approaching to 0.5, then decreases gradually from nearby L1 =L ¼ 0:5 to unit as shown in Fig. 3. Worthy of note, there is a frequency jumping at the point of L1 =L ¼ 0:5, which is not showed in Fig. 3. Mathematically, this is a removable discontinuity. However, that is out of accord with the fact. Since the value of pressure at the midpoint (it is the pressure node of the resonator with two ends closed) is a constant theoretically, there is no working fluid coming in or out of the buffer. Thus in case of L1 =L ¼ 0:5, the middle arrangement is the same as the resonator closed at both ends in Fig. 1(a) and the frequency may be computed by Eq. (3). By this means, we find that in Fig. 3, the curve, which indicates the variety of frequency versus the specific resonator location for the middle arrangement, is continuous at L1 =L ¼ 0:5.

Table 1 Calculated wavelength number Specific volume

0 1 2 5 10

Wavelength number x ðfL=aÞ Single arrangement

Double arrangement

Middle arrangement

0.50 0.42 0.37 0.32 0.29

0.50 0.36 0.29 0.20 0.15

0.50 0.43 0.39 0.34 0.31

Let us define a wavelength number x as the resonator length L divided by wavelength k, that is x ¼ L=k ¼ fL=a. The system wavelength numbers of different arrangements of the buffers are computed as shown in Table 1. We can see obviously from Table 1 that the wavelengths of various arrangements in Fig. 1 are greatly deviated from k=2 and k=4. It means that a thermoacoustic engine cannot be simply treated as k=2 or k=4 system. The difference of the wavelength number will cause the different pressure and flow volume velocity distribution of the system. Therefore, we should consider the best location of the exit for the maximum sound output, where the pressure amplitude is largest while the velocity volume is at its node.

3.2. Influence of buffer volume on wavelength number

4. Experiments

It is important to know the exact value of wavelength for a practical device of thermoacoustic engine. In a general way, one may roughly define a thermoacoustic system as k=2 or k=4. However, this will not be helpful for designing a thermoacoustically driven pulse tube refrigerator system, since a practical system is neither k=2 nor k=4 exactly. Based on the above computed frequencies of various arrangements, an exact wavelength of a thermoacoustic system may be decided.

4.1. Experiment setups

Fig. 3. Resonance frequency versus specific resonator location.

The experimental setup of self-made thermoacoustically driven pulse tube refrigerator includes: thermoacoustic prime mover, pulse tube refrigerator and the measuring system as shown in Fig. 4. (1) The thermoacoustic prime mover is symmetrically structured system, i.e. a resonator accompanied with two same thermoacoustic generators at its both sides. Each thermoacoustic generator mainly consists of the heater, stack, water cooler and the buffer. The dimensions of the main parts are tabulated in Table 2. Since

Fig. 4. Experimental setup drawing of standing wave thermoacoustic engine.

G.B. Chen et al. / Cryogenics 42 (2002) 223–227 Table 2 Dimension of the thermoacoustic engine

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Table 3 Comparison between experimental and calculation results

Item

Resonator

Stack (including heat exchanger)

Buffer

Working gas

Arrangement

Frequency (Hz) Calculated

Experimental

Relative error (%)

Dimension (mm)

/36 4000

/56 450

1.0 (l)

Nitrogen

Single Double Hybrid

33 26 24

30a 24 21

10 7.7 12.5

Helium

Single Double Hybrid

92 72 69

87a 68 64

5.4 5.6 7.2

the volume of the buffer is not large, the length of the device is between k=4 and k=2. Additionally, the length of the stack is short compared with the resonator. And the diameter difference between the stack and the resonator tube can be neglected due to their same flow volume. Thus, the length of the double buffer device approximates to be L ¼ 4 þ 2 0:45 ¼ 4:90 m, where the length of each stack including heat exchanger equals 0.45 m. When the buffer at one side of the resonator to be moved away, the system becomes the single buffer arrangement, and L ¼ 4 þ 0:45 ¼ 4:45 m. (2) Pulse tube refrigerator. A coaxial single-stage double-inlet pulse tube refrigerator is connected to the prime mover. In order to test the hybrid buffer arrangement, an additional middle buffer of 1 l should be added between the prime mover and the pulse tube refrigerator. (3) Measuring system. Thermocouples are added at the cold and hot ends of the stack of each side of the resonant, so as to obtain more accurate information to assess the temperature gradient over the stacks. Temperatures at the hot buffer and the hot end of the stack are measured with NiCr–NiSi thermocouples, while those at the cold heat exchanger for the prime mover are measured with Cu–Constantan thermocouples. An Rh– Fe resistance thermometer (with 0.1 K accuracy) is applied to measure refrigeration temperature at the cold end of the pulse tube. The pressure measuring is accomplished by a PCbased digital acquisition system, which includes a piezoresistive silicon pressure sensor, data acquisition card (produced by NI) and a PC. The plots of the pressure are displayed on the screen, from which we can read the amplitude, mean pressure and frequency. Normally we use only one pressure sensor in front of the pulse tube. When it comes to the hybrid arrangement, we use two pressure sensors, which are placed in front of and behind the middle buffer, respectively. 4.2. Experiment The experiments of frequency measuring of the single arrangement, the double arrangement and the hybrid arrangement have been carried out. Because it is not convenient to change volume of the buffers, the experiments we have done are only under the condition of specific volume V =V0 ¼ 1. We observed the resonance frequencies of each arrangement mentioned above, and

a

The resonator length of the single arrangement experimental device is 4.45 m.

the results have been shown in Table 3. Table 3 also shows the calculated results, which we can partly get from Fig. 2 when the specific volume is equal to 1 with nitrogen as working fluid. Experimental results and calculated results agree well. From these results, we can see that as far as the resonance frequency considered, the hybrid arrangement is better than the double arrangement.

5. Conclusions The fluid impedance method can be used not only to calculate the resonance frequency of a simple piping system, but also to a complex piping one, such as hybrid arrangement. Acoustic structure such as the buffer has a vital influence on the resonance frequency and other performance of a thermoacoustic engine.

Acknowledgements The authors wish to express their appreciation for the financial support from the National Natural Sciences Foundation of China. References [1] Swift GW. Thermoacoustic engine. J Acoust Soc Am 1988;84: 1145–80. [2] Backhaus S, Swift GW. A thermoacoustic-Stirling engine: detailed study. J Acoust Soc Am 2000;107(6):3148–66. [3] Ceperley PH. A pistonless Stirling engine – the traveling wave heat engine. J Acoust Soc Am 1979;66(5):1508–13. [4] Tominaga A, Narahara Y, Yazaki T. Thermoacoustic effects of invisicid fluids. J Low Temp Phys 1984;54:36–45. [5] Tominaga A. Thermoacoustic theory and its application to refrigerator. Proc JSJS 1989;3:141–6. [6] Chen GB, Jin T, et al. Experimental study on a thermoacoustic engine with brass screen stack matrix. ACE 1998;43b:713–8. [7] Zhou S, Matsubara Y. Experimental research of thermoacoustic prime mover. Cryogenics 1998;38(8):813–22. [8] Kirshner JM, Katz S. Design theory of fluidic components. New York: Academic Press; 1975.