Synthetical optimization of hydraulic radius and acoustic field for thermoacoustic cooler

Energy Conversion and Management 50 (2009) 2098–2105

Contents lists available at ScienceDirect

Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Synthetical optimization of hydraulic radius and acoustic field for thermoacoustic cooler Kang Huifang a,b, Li Qing a,*, Zhou Gang a a b

Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, P.O. Box 2711, Beijing 100190, China Graduate School of the Chinese Academy of Sciences, Beijing 100039, China

a r t i c l e

i n f o

Article history: Received 19 December 2007 Received in revised form 10 August 2008 Accepted 14 March 2009 Available online 1 May 2009 Keywords: Thermoacoustic Hydraulic radius Acoustic filed Cooling power Acoustic power dissipation

a b s t r a c t It is well known that the acoustic field and the hydraulic radius of the regenerator play key roles in thermoacoustic processes. The optimization of hydraulic radius strongly depends on the acoustic field in the regenerator. This paper investigates the synthetical optimization of hydraulic radius and acoustic field which is characterized by the ratio of the traveling wave component to the standing wave component. In this paper, we discussed the heat flux, cooling power, temperature gradient and coefficient of performance of thermoacoustic cooler with different combinations of hydraulic radiuses and acoustic fields. The calculation results show that, in the cooler’s regenerator, due to the acoustic wave, the heat is transferred towards the pressure antinodes in the pure standing wave, while the heat is transferred in the opposite direction of the wave propagation in the pure traveling wave. The better working condition for the regenerator appears in the traveling wave phase region of the like-standing wave, where the directions of the heat transfer by traveling wave component and standing wave component are the same. Otherwise, the small hydraulic radius is not a good choice for acoustic field with excessively high ratio of traveling wave, and the small hydraulic radius is only needed by the traveling wave phase region of likestanding wave. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction Thermoacoustic cooler pumps heat from the cold by thermoacoustic principles. Compared with traditional coolers, the thermoacoustic cooler has many advantages, such as, simple structure, low cost of manufacture, and reliability. By using inert gases as working fluid, this kind of cooler is environmentally friendly. The regenerator is the heart of the thermoacoustic cooler, and it performs the thermoacoustic conversion processes under a certain acoustic field. A temperature gradient with a certain coefficient of performance (COP) is set up along the regenerator, when the acoustic wave with the acoustic pressure p1 = |p1|ei(xt+/) and the velocity u1 = |u1|eixt passes through the cooler, where x is the angular frequency and / is the leading phase of p1 to u1. The acoustic wave drives gas parcels in the regenerator to experience a certain thermodynamic cycle consisting of compression, cooling, expansion, and heating. As a result, the heat pumping occurs without moving parts [1,2]. Generally, as described by Swift [3] and Ceperley [4], thermoacoustic devices can be classified into standing wave device and traveling wave device according to the leading phase /. In the standing wave device, gas parcels with |/| = 90° convert the energy between the thermal and acoustic energy * Corresponding author. Tel.: +86 10 82543660; fax: +86 10 62554670. E-mail address: [email protected] (Q. Li).

through irreversible thermal contacts between the working gas and the stuff in the regenerator. Such type of cooler has been used to liquefy natural gas [5]. However, the thermal efficiency of the standing wave engine is less than 20%, because the energy conversion is irreversible. A traveling wave thermoacoustic device was originally proposed by Ceperley [4] in 1979. In that device, gas parcels with / = 0° convert the energy between thermal and acoustic energy through reversible thermal contacts between the working gas and the stuff in the regenerator. In 1999, Backhaus and Swift [6] achieved the efficiency up to 30% by introducing a resonator to a looped tube thermoacoustic engine. Such high efficiency was achieved by two conditions of the acoustic field in the regenerator: the traveling wave phase and the high specific acoustic impedance Z. The high specific acoustic impedance Z implies low velocity of working gas which leads to low viscous loss. The specific acoustic impedance Z is defined as

Z¼

p1 u1

ð1Þ

Additionally, to achieve the high efficiency, it is necessary that the hydraulic radius is much smaller than the thermal penetration depth dk. The hydraulic radius is conventionally defined as [7]

rh ¼ A=P where A is the cross-sectional area and P is the perimeter.

0196-8904/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.03.021

ð2Þ

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

Though, the importance of rh and these two acoustic field conditions were emphasized to achieve the high efficiency, the optimal values of rh, /, and Z in their engine had not been discussed in detail. Several published literatures studied the acoustic field effect on the thermoacoustic conversion. Raspet [8] gave the coefficient of performance of the thermoacoustic cooler as a function of standing wave ratio, and fixed the hydraulic radius with rh/dk = 2.05 and rh/ dk = 0.025 to optimize the standing wave ratio. He pointed out that the highest efficiency for a thermoacoustic cooler was achieved when there was a significant traveling wave component. Poignand [9] designed a thermoacoustic device, and investigated the optimal acoustic field in the regenerator with rh/dk = 1.47. He concluded that the optimal acoustic field was different from standing wave and traveling wave. Biwa et al. [10] demonstrated thermoacoustic conversions in the regenerators with rh/dk = 0.36 and rh/dk = 1.87 by making full use of the acoustic field induced in the resonator. He obtained that, in order to achieve both high efficiency and gain, one had to choose an optimum combination of traveling wave component and standing wave component. However, those studies [8–10] optimized the parameters of acoustic fields using the fixed hydraulic radius, which was also one of the most important parameters which could influence the thermoacoustic conversion. And the optimal acoustic field is strongly dependent on the hydraulic radius. Several publications studied the hydraulic radius effect on the thermoacoustic conversion. Swift [3] studied the rh in the thermoacoustic system, and obtained the empirical values: rh/dk  1 for standing wave devices and rh/dk  1 for traveling wave devices. Tijani et al. [11] studied a standing wave thermoacoustic cooler and pointed out that rh = 1.25dk was optimum for the cooling power, rh = 2.0dk led to the lowest temperature, and the highest efficiency could be reached with rh = 1.5dk. Yu et al. [12] studied a traveling wave thermoacoustic engine and pointed out that a measured optimal value was rh  (0.3–0.2)dv for the excitation of the system, where dv is the viscous penetration depth. However, those publications [3,11,12] studied the optimization of the hydraulic radiuses on the assumption of standing wave mode or traveling wave mode, separately. According to those methods, it was not easy to deal with such complicated thermoacoustic process in detail. And the optimization of hydraulic radius strongly depends on the acoustic field in the regenerator. Those methods are unilateral to understand the optimal condition for the thermoacoustic conversion. As thermoacoustic devices developing from standing wave devices [3] to traveling wave devices [4,6,13] and then to hybrid thermoacoustic devices [14,15], it is significant to optimize the regenerator combining the hydraulic radius with the acoustic field. However, most researchers focus their particular attentions on the standing wave [16,17] or the traveling wave [18–21], and few published literatures studied on the traveling–standing wave mode due to its complexity. This paper combines hydraulic radiuses with acoustic fields together to analyze the heat flux, temperature gradient and COP, considering the acoustic power dissipation. Furthermore, the relationships among the parameters of acoustic fields are considered in calculations. Such parameters are the pressure p1, the velocity u1, the specific acoustic impedance Z and the leading phase / of p1 to u1.

order of acoustic approximation, the x component of the momentum equation is

qm

@u1 @p1 ¼ @t @x

ð3Þ

where qm is the mean density of the working gas. In the ideal acoustic channel, the pressure of the standing wave can be expressed as

ps ¼ pa cos kx cos xt ¼

1 p Re½eiðxtþkxÞ þ eiðxtkxÞ  2 a

ð4Þ

where Re[] denotes the real part, k is the wave vector, and pa is the pressure amplitude. The pressure of traveling wave is expressed as

pt ¼ pa cosðkx þ xtÞ ¼ pa Re½eiðxtkxÞ 

ð5Þ

Based on the superposition principle of the wave, the pressure of the traveling-standing wave can be written as

Reðp1 Þ ¼ ð1  sÞps þ spt   1 ¼ Re pa ½ð1  sÞeikx þ ð1 þ sÞeikx eixt 2

ð6Þ

where s is defined as the traveling wave ratio in this paper, and 0 6 s 6 1. s = 0 means a pure standing wave and s = 1 means a pure traveling wave. Substituting Eq. (6) into Eq. (3)

u1 ¼ 

1

qm

Z

dp1 p dt ¼  a ½ð1  sÞeikx  ð1 þ sÞeikx eixt dx 2qm c

ð7Þ

where c is the sound speed. Substituting Eqs. (6) and (7) into Eq. (1)

Z¼

p1 ð1  sÞeikx þ ð1 þ sÞeikx ¼ qm c  u1 ð1  sÞeikx  ð1 þ sÞeikx

ð8Þ

Normalized pressure, velocity, and impedance are given by

1 jð1  sÞeikx þ ð1 þ sÞeikx j 2 1 Gu ¼ ju1 jqm c=pa ¼ jð1  sÞeikx  ð1 þ sÞeikx j 2 ð1  sÞeikx þ ð1 þ sÞeikx Gz ¼ Z=qm c ¼  ð1  sÞeikx  ð1 þ sÞeikx Gp ¼ jp1 j=pa ¼

ð9Þ ð10Þ ð11Þ

As shown in Eqs. (9)–(11), Gp, Gu, and Gz are functions of the traveling wave ratio s and the normalized position kx. The phase of Gz represents the leading phase / of p1 to u1. So s and kx can characterize p1, u1, /, |Z|, as well as their relationships. At first, we discuss the typical case of s = 0.5, where the traveling wave component equals to the standing wave component. The results are plotted in Fig. 1, which shows Gp, Gu, |Gz| (the left

2. Traveling-standing wave Consider wave propagating in the x direction in an ideal gas within a resonator tube, dv  rh, dk  rh. So, in a channel with large cross-section, the viscosity due to the interaction between the sound wave and the solid channel wall is neglected. With the first

2099

Fig. 1. Gp, Gu, |Gz| and / versus kx, when s = 0.5.

2100

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

along x direction is entropy flow of the oscillation wave, carried by the oscillatory velocity u1. The time-averaged thermoacoustic heat flux per unit area along x is written as [7],

" !# 1 fk  ~f v ~ Q ¼ Re p1 u1  2 ð1 þ rÞð1  ~f v Þ þ

Fig. 2. / Versus kx with different s: 0.0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0.

dT m ju1 j2 qm cp Imðfk þ r~f v Þ dx 2Axð1  r2 Þj1  fv j2

ð12Þ

where Im[] denotes the imaginary part, the tilde denotes complex conjugation, cp is the isobaric specific heat per unit mass, Tm is the mean temperature of the fluid, r is the Prandtl number, and fv, fk are the Rott’s functions which depend on the specific channel geometry under consideration. The functions of fv and fk are known for many geometries. As shown in Fig. 4.14 of Ref. [7], the solutions for most of these geometries are actually very similar. Therefore, this paper only takes the parallel plate-type regenerator as a case to investigate, and the other types of regenerators can be analyzed in the similar way. y = 0 is defined at the centerline of the spacing between two adjacent parallel plates,

tanh½ð1 þ iÞy0 =dv  ð1 þ iÞy0 =dv tanh½ð1 þ iÞy0 =dk  fk ¼ ð1 þ iÞy0 =dk

fv ¼

ð13Þ ð14Þ

where 2y0 is the thickness of each spacing, and rh = y0. The relative hydraulic radius is defined as f = rh/dk, the ratio of the hydraulic radius rh to the thermal penetration depth dk. The functions of fv and fk can be written as follows:

pffiffiffiffiffiffiffiffiffi tanh½ð1 þ iÞf 1=r pffiffiffiffiffiffiffiffiffi ð1 þ iÞf 1=r tanh½ð1 þ iÞf fk ¼ ð1 þ iÞf

fv ¼ Fig. 3. |Gz| versus kx with different s: 0.0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0.

vertical coordinate), and / (the right vertical coordinate) versus kx. There are three points, kx = p/2, 0, and p/2, which have the property of / = 0°. However, only the point of kx = 0 is useful for the operation of regenerator in the thermoacoustic engine, because |Gz| reaches the maximum at this point. And the maximum |Gz| means the low velocity of the working gas which leads to low viscous loss. Whereas, at the other two points kx = ±p/2, |Gz| becomes the minimum. This results in the highest velocity of the working gas and consequently serious viscous loss. As mentioned above, the traveling wave component equals to the standing wave component in the case of s = 0.5. So the critical values of the traveling wave phase is defined as / = ±37° which is the peak points of / when s = 0.5. Accordingly, the region of 37° < / < 37° is the traveling wave phase region (TWPR). Furthermore, we analyze the acoustic field features with different s. Fig. 2 shows / versus kx for different s: 0.0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0. It is standing wave when s = 0.0. Accordingly, / = 90° when 0 < kx < p/2, and / = 90° when p/2 < kx < 0. This means that the length of the TWPR is 0. With the increase of s, |/| decreases. Accordingly, the length of the TWPR increases. The whole acoustic fields are in TWPR when s > 0.5, and when s = 1.0, it is pure traveling wave. As shown in Fig. 3, |Gz| is plotted as a function of kx for different s: 0.0, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0. |Gz| reaches the maximum at kx = 0 for each s. With the increase of s, |Gz| increases in the region of p/4 < kx < p/4. It is standing wave when s = 0.0 and |Gz| ? 1 at kx ? 0. When s = 1.0, it is traveling wave, and |Gz| = 1.0 anywhere. 3. Heat flux (Q) In the regenerator model, ordinary thermal conductivity in the x direction is neglectable. The only way that heat is transported

ð15Þ ð16Þ

For simplicity, the heat flux can be expressed as follows:

Q ¼ Q 0 þ Aq

dT m dx

ð17Þ

where the first term Q0 denotes the heat flux in the absence of a longitudinal temperature gradient, and the second term Aq dTdxm denotes the modification due to a temperature gradient. The expressions of Q0 and Aq have their own forms as follows:

Q0 ¼ Aq ¼

1 fk  ~f v ~ 1 ð Re½p1 u Þ 2 ð1 þ rÞð1  ~f v Þ ju1 j2 qm cp 2Axð1  r2 Þj1  fv j2

Imðfk þ r~f v Þ

ð18Þ ð19Þ

The heat flux of a single plate regenerator is calculated with different acoustic fields and hydraulic radiuses, using the parameters: helium as working gas, dTm/dx = 0.0, f = 200Hz, pa = 1.0 bar, pm = 35.0 bar, Ta = 300 K. The calculation results are plotted in Fig. 4. When s = 0.0, it is pure standing wave. As shown in Fig. 4a, when dTdxm ¼ 0, Q P 0.0 in the region of p/2 < kx < 0, and Q 6 0.0 in the region of 0 < kx < p/2. It is indicates that the heat flux is always towards the pressure antinode. For the adiabatic regenerator, as the heat flux departs from pressure node and cumulates in pressure antinode, the temperature gradient is set up along the regenerator. The end of the regenerator towards the pressure node is cold, while the other end towards the pressure antinode is hot. This is the reason why the temperature decreased in the regenerator’s end towards the pressure node when the standing wave was formed in the regenerator, as described in Refs. [16,17]. When s = 1.0, it is pure traveling wave. As shown in Fig. 4d, the direction of heat flux is always opposite to the propagation

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

2101

Fig. 4. Heat flux as a function of kx and f with different s: (a) s = 0.0, (b) s = 0.2, (c) s = 0.8, and (d) s = 1.0.

direction of acoustic wave in the cooler’s regenerator. The direction of the temperature gradient is also opposite to the propagation direction of acoustic wave, because the direction of temperature gradient is the same as the direction of the heat flux in the cooler’s regenerator. This property can be used to explain that the temperature gradient was set up along the regenerator when the traveling wave passed through the regenerator, as described in Refs. [18–21]. When 0.0 < s < 1.0, it is the mixture of traveling wave and standing wave. As shown in Fig. 4, when f ? 0.0, the direction of the heat flux and temperature gradient is opposite to the propagation direction of the traveling wave component. The reason is that the energy conversion with f ? 0.0 lies on reversible thermal contacts between the working gas and the stuff in the regenerator, which is beneficial to energy conversion by the traveling wave. When f  0.0, as shown in Fig. 4, the direction of the heat flux and temperature gradient is towards the pressure antinode for the regenerator of cooler. The reason is that the energy conversion with f  0.0 lies on irreversible thermal contacts, which is beneficial to energy conversion by the standing wave. In Fig. 4, the maximal heat flux always appears in 0.0 6 kx 6 p/ 2, where the directions of the heat fluxes generated by the traveling wave component and the standing wave component are the same. In the region of p/2 6 kx 6 0.0, the heat flux is smaller than that generated in the region of 0.0 6 kx 6 p/2, because the directions of the heat fluxes generated by the traveling wave component and the standing wave component are reverse.

the acoustic power dissipation in the thermoacoustic cooler. The time-averaged acoustic power dE produced in a length dx of regenerator per unit area is expressed as [7]

dE_ 2 rv 1 1 ~1 j2 þ Re½g p ~1 u1  jp ¼  ju1 j2  2r k 2 dx 2

ð20Þ

where rv is the viscous resistance per unit length, 1/rk is the thermal-relaxation conductance per unit length, and g is complex gain factor arising in continuity equation. The expressions of rv, 1/rk, and g have their own forms as follows:

rv ¼ xqm

Im½fv 

j1  fv j2 1 c  1 xIm½fk  ¼ rk pm c ðfk  fv Þ 1 dT m g¼ ð1  fv Þð1  rÞ T m dx

ð21Þ ð22Þ ð23Þ

where c is the ratio of isobaric to isochoric specific heats. Integrating Eq. (20) with respect to x over the length Dx = lreg of the regenerator we can obtain the work dissipation Eloss = DE = Ein  Eout. For a short enough regenerator lreg  k, we can replace spatially averaged values by the values at the center x of the regenerator

Eloss ¼ 

dE_ 2 dT m lreg ¼ Ev þ Ek þ Ae dx dx

4. Acoustic power dissipation (Eloss)

where Ev, Ek and Ae is expressed as follows:

To realize the thermoacoustic heat pumping process, a net work input (Eloss) is required. The following calculations aim at analyzing

Ev ¼

lreg xqm Im½fv  ju1 j2 2 j1  fv j2

ð24Þ

ð25Þ

2102

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

lreg c  1 xIm½fk  ~ 1 j2 jp pm 2 c   lreg 1 fk  fv ~ 1 u1 Ae ¼ Re p 2 Tm ð1  fv Þð1  rÞ

Ek ¼

ð26Þ

5. Cooling power (Qc) Although we can obtain the maximal cooling power Qc directly from Eq. (17), the regenerator of cooler cannot work successfully with the maximal cooling power Qc, if the other parts of regenerator could not transfer the heat Qx = Qc + Ex,loss > Qc. The maximal heat flux in the regenerator is Qh at the hot end. Thus, we should calculate Qh. Then the heat removing from the cold end is calculated by Qc = Qh  Eloss, which includes the acoustic power dissipation in the regenerator. The directions of the heat flux and the temperature gradient have been analyzed in Section 3. Assuming the traveling wave propagation direction is positive, the heat flux and the temperature gradient are classified into two groups as shown in Fig. 5. For the adiabatic regenerator, the energy flow in the x direction is uniform, i.e.

ð28Þ

where Hin and Hout are the energy flows in the inlet and outlet of regenerator, respectively. And

Hin ¼ Ein þ Q in ; Hout ¼ Eout þ Q out

ð29Þ

Substituting Eq. (29) into Eq. (28), we obtain

Ein þ Q in ¼ Eout þ Q out

ð31Þ

Substituting Eq. (31) into Eq. (30)

ð27Þ

The first two terms in Eq. (24) are independent of temperature gradient along x, and are always positive which describes the consumed acoustic power. The first term describes the viscous dissipation of wave. It is proportional to |u1|2 and independent of thermal conductivity. The second term describes the thermal-relaxation dissipation. It is proportional to |p1|2 and independent of viscosity. The third term is called the source/sink term, because it can either produce or consume acoustic power. It exists only if dTm/dx – 0.

Hin ¼ Hout

Q c ¼ Q out ; Q h ¼ Q in

ð30Þ

For the case of Fig. 5a, Qout 6 Qin 6 0, dTm/dx 6 0. The directions of the heat flux and the temperature gradient are negative, and

Fig. 5. Two cases of the heat flux and the temperature gradient in the cooler’s regenerator.

Q c ¼ Q h þ Eout  Ein ¼ Q h  Eloss

ð32Þ

For the case of Fig. 5b, Qout P Qin P 0, dTm/dx P 0. The directions of the heat flux and the temperature gradient are positive, and

Q c ¼ Q in ; Q h ¼ Q out

ð33Þ

Substituting Eq. (33) into Eq. (30)

Q c ¼ Q h þ Eout  Ein ¼ Q h  Eloss

ð34Þ

When Qh = |Q| 6 Eloss, the regenerator of the cooler cannot remove heat from the cold end to hot end. That device cannot work as a cooler successfully, and in this case

Qc ¼ 0

ð35Þ

According to Eqs. (32), (34), and (35), the cooling power (Qc) removed from the low temperature (Tc) end is expressed as follows:

Qc ¼



jQ j  Eloss

jQ j > Eloss

0

jQ j 6 Eloss

ð36Þ

6. Temperature gradient An estimated upper limit of the temperature gradient dTm/dx is obtained when assuming that no heat quantity removes from the cold end, i.e. Qc = 0. According to Eqs. (17), (24), and (36), the upper limit of the temperature gradient dTm/dx is expressed as follows:

8 Q E E v k 0 > >  Aq þAe dT m < Q þE þE ¼  0 v k Aq Ae > dx > : 0

Q 0 P Ev þ Ek Q 0 6 ðEv þ Ek Þ

ð37Þ

jQ 0 j < Ev þ Ek

In order to discuss the temperature gradient of a cooler’s regenerator with different acoustic fields and hydraulic radiuses, calculations have been performed for the regenerator of lreg = 0.05 m length. The other parameters are: helium as working gas, f = 200 Hz, pa = 1.0 bar, pm = 35.0 bar, Ta = 300 K. The calculation results are plotted in Figs. 6–8. Fig. 6 shows dTm/dx as function of kx and f with different s. Additionally, the maximum dTm/dx, the optimal f and the optimal kx with different s are summarized in Fig. 7. Furthermore, according to Eq. (11), |Gz| and / of the optimal acoustic fields versus s are shown in Fig. 8. From Fig. 6, it can be found that the maximum dTm/ dx ? 5484 occurs when f ? 0.35, kx ? 0.0 and s ? 0.0. As a result, this leads to |Gz| ? 1 and / ? 38.0° shown in Fig. 8. From Figs. 6 and 7, the maximal temperature gradient appears in the region of 0.0 6 kx 6 p/2, where the directions of the temperature gradient generated by the traveling wave component and the standing wave component are the same. As shown in Figs. 6 and 7, the higher temperature gradient region appears in the region of kx ? 0, and the optimal f is in the region of 0.0 < f 6 0.65, except for the case of s = 0.0. Those can be understood by the traveling-standing wave acoustic field. When s = 0.0, it is the pure standing wave. Although |Gz| ? 1 at kx = 0, the temperature gradient is zero because of the zero velocity. The optimal acoustic field is near the position of kx = 0. For the pure standing wave, / = 90°, which implies that the heat is pumped by the thermal-relaxation, and then the moderately large hydraulic radius is needed. When 0.0 < s < 1.0, the acoustic field is the mixture of the traveling wave and the standing wave, and the TWPR appears, as shown in Fig. 2. The optimal acoustic field appears in the TWPR, while the moderately small hydraulic radius is needed to realize the reversible energy conversion. This can explain the

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

2103

Fig. 6. The upper limit temperature gradient dTm/dx as a function of kx and f with different s: (a) s = 0.0, (b) s = 0.2, (c) s = 0.8, and (d) s = 1.0.

Fig. 7. The maximum dTm/dx, the optimal f and the optimal kx for the maximum |dTm/dx| versus s.

Fig. 8. |Gz| and / of the optimal acoustic field versus s, for the maximum |dTm/dx|.

experimental results of Refs. [3,11,12] that the optimal hydraulic radius for the standing wave device is much larger than that for the traveling wave device. As shown in Fig. 6a, when s = 0.0, the optimal f is larger than 5.0 because the thermal conductivity is neglected in this paper. With f increasing in the region of f > 1.8, although the heat pumped by the acoustic wave decreases as shown in Fig. 4a, the temperature gradient |dTm/dx| still increases because the acoustic power dissipation decreases faster. However, when the heat pumped by the acoustic wave is small enough, the thermal conductivity will influence the heat flux strongly. Thus, as discussed in Ref. [11], the optimal f appears in the region of f  2.0, where the heat pumped by the acoustic wave is much larger than that transferred by thermal conductivity. As shown in Fig. 8, when 0.0 < s 6 1.0, with s increasing, |Gz| of the optimal acoustic field decreases, while the optimal f increases. This is because both of the viscous and thermal-relaxation dissipations have effects on the thermoacoustic conversion, and must be balanced synthetically. When s ? 0.0, due to the high |Gz| of the optimal acoustic field, the velocity is low at the optimal position, which leads to low viscous dissipation. In this case, the thermalrelaxation dissipation is the main loss. The moderately small hydraulic radius is needed. If s = 1.0, |Gz| = 1.0, which means that the hydraulic radius should be moderately large to reduce the viscous dissipation. Therefore, in the acoustic field of more traveling wave component, the small hydraulic radius is not a good choice, and the moderately small hydraulic radius is needed just for the traveling wave phase region with more standing wave component, where |Gz| is high. Fig. 8 shows that the highest temperature gradient decreases with the increase of s. The reason is that, with f increasing, the thermal-relaxation dissipation increases; and with |Gz| decreasing, the viscous dissipation increases. Consequently, with s increasing, the optimized temperature gradient decreases.

2104

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

7. Coefficient of performance

8. Conclusions

A cooler pumps heat from cold end to hot end, and then releases to the surroundings. To realize this process a net work input (Eloss) is required. The coefficient of performance (COP) is defined as

In this paper, the optimal combinations of hydraulic radius and acoustic field of thermoacoustic cooler have been discussed including heat flux, cooling power, temperature gradient and COP, while the acoustic power dissipation is taken into account during the calculation of cooling power. The directions of heat flux and temperature gradient are consistent in the cooler’s regenerator. When s = 0.0, it is the pure standing wave, and the directions of heat flux and temperature gradient are towards the pressure antinodes in the regenerator of cooler. When s = 1.0, it is pure traveling wave, and the directions of heat flux and temperature gradient are reverse to the wave propagation. The better acoustic field always appears in the region where both traveling wave component and standing wave component pump heat from the cold end of regenerator. It is better for the thermoacoustic conversion. The better acoustic field in regenerator is neither pure standing wave because of the irreversible energy conversion, nor pure traveling wave because of the viscous dissipation. The better working condition for regenerator is in the traveling wave phase region with moderately low ratio of traveling wave component. Otherwise, the small hydraulic radius is not a good choice for acoustic field with excessively high ratio of traveling wave. The relatively small hydraulic radius is only needed in the traveling wave phase region of like-standing wave, where the specific acoustic impedance is high. Further studies will depend on the optimal match between the hydraulic radius and the acoustic field precisely. Basing on the synthetical optimization principle, we will design a new thermoacoustic device to prove the mentioned conclusions.

COP ¼

Qc Eloss

ð38Þ

For the regenerator of a general refrigerator, the temperature difference is near to 20 K, which leads to the temperature gradient near to 400 K/m in the previous regenerator. The directions of heat flux and temperature gradient have been analyzed in Section 3, and is shown in Fig. 5. Thus, we give the temperature gradient as

dT m ¼ dx



400

Q0 > 0

400 Q 0 < 0

ð39Þ

Combining Eqs. (24), (36), and (38), the COP of the previous regenerator with different acoustic fields and hydraulic radiuses have been calculated. The calculation results are plotted in Fig. 9. As shown in Fig. 9, the relationship between the COP and (kx, f) is similar to the relationship between the temperature gradient and (kx, f). For example, the high COP region appears in the positions of kx ? 0, and the optimal relative hydraulic radius is in the region of 0.0 < f 6 1.5, except for the case of s = 0.0. And the highest COP decreases with s increasing. This also can be understood by the similar analysis of the temperature gradient. From Fig. 9, the maximal COP also appears in the region of 0.0 6 kx 6 p/2, where the directions of the temperature gradient generated by traveling wave component and standing wave component are the same.

Fig. 9. COP a function of kx and f with different s: (a) s = 0.0, (b) s = 0.2, (c) s = 0.8, and (d) s = 1.0.

H. Kang et al. / Energy Conversion and Management 50 (2009) 2098–2105

Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 10804114). References [1] Wheatley JC, Swift GW, Hofler TJ. Heat-driven acoustic cooling engine having no moving parts. United States Patent 4858441; 1989. [2] Hofler TJ. High-efficiency heat-driven acoustic cooling engine with no moving parts. United States Patent 5901556; 1997. [3] Swift GW. Thermoacoustic engines. J Acoust Soc Am 1988;84(4):1145–80. [4] Ceperley PH. A pistonless stirling engine – the travelling wave heat engine. J Acoust Soc Am 1979;66:1508–13. [5] Radebaugh R, McDermott KM, Swift GW, Martin RA. Development of a thermoacoustically driven orifice pulse tube refrigerator. In: Proceedings of the interagency meeting on cryocoolers, October 1990, Plymouth (MA); 1990. [6] Backhaus S, Swift GW. A thermoacoustic-stirling heat engine: detailed study. J Acoust Soc Am 2000;107:3148–66. [7] Swift GW. Thermoacoustics: a unifying perspective for some engines and refrigerators. New York: Acoustical Society of America; 2002. [8] Raspet R, Brewster J, Bass HE. A new approximation method for thermoacoustic calculations. J Acoust Soc Am 1998;103(5):2395–402. [9] Poignand G, Lihoreau B, Lotton P, Gaviot E, Bruneau M, Gusev V. Optimal acoustic fields in compact thermoacoustic coolers. Appl Acoust 2007;68:642–59.

2105

[10] Biwa T, Tashiro Y, Mizutani U. Experimental demonstration of thermoacoustic energy conversion in a resonator. Phys Rev E 2004;69:066304(6). [11] Tijani MEH, Zeegers JCH, Waele TAM. The optimal stack spacing for thermoacoustic refrigeration. J Acoust Soc Am 2002;112(1):128–33. [12] Yu ZB, Li Q, Chen X, Guo FZ, Xie XJ. Experimental investigation on a thermoacoustic engine having a looped tube and resonator. Cryogenics 2005;45:566–71. [13] Zhou G, Li Q, Li ZY. A miniature thermoacoustic stirling engine. Energy Convers Manage 2008;49:1785–92. [14] Gardner DL, Swift GW. A cascade thermoacoustic engine. J Acoust Soc Am 2003;114(4):1905–19. [15] Hu ZJ, Li Q, Li ZY. Orthogonal experimental study on high frequency cascade thermoacoustic engine. Energy Convers Manage 2008;49:1211–7. [16] Tijani MEH, Zeegers JCH, Waele TAM. Design of thermoacoustic refrigerators. Cryogenics 2002;42:49–57. [17] Paek I, Braun JE, Mongeau L. Evaluation of standing-wave thermoacoustic cycles for cooling applications. Int J Refrig 2007;30:1059–71. [18] Yazaki T, Iwata A. Traveling wave thermoacoustic cooler in a looped tube. Phys Rev Lett 1998;15:3128–31. [19] Yazaki T, Biwa T, Tominaga A. A pistonless stirling cooler. Appl Phys Lett 2002;80:157–9. [20] Tijani MEH, Spoelstra S. Study of a coaxial thermoacoustic-stirling cooler. Cryogen 2008;48:77–82. [21] Dai W, Luo EC, Zhang Y, Ling H. Detailed study of a traveling wave thermoacoustic refrigerator driven by a traveling wave thermoacoustic engine. J Acoust Soc Am 2006;119:2686–92.