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Evanescent modes and anomalous streaming in a thermoacoustic device
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Applied Acoustics 69 (2008) 23–30 www.elsevier.com/locate/apacoust
Evanescent modes and anomalous streaming in a thermoacoustic device Gordon P. Smith a
a,*
, Richard Raspet b, Robert Hiller c, Joseph McDaniel
d
Department of Physics and Astronomy, Western Kentucky University, Bowling Green, KY 42101, USA b National Center for Physical Acoustics, University of Mississippi, MS 38677, USA c 1231 Seminole Avenue, W. St. Paul, MN 55118, USA d Department of Physics and Astronomy, Florida State University, Tallahassee, FL 32306, USA Received 14 July 2005; received in revised form 30 July 2006; accepted 1 August 2006 Available online 27 October 2006
Abstract An oil-heated thermoacoustic refrigerator was constructed in order to investigate the use of waste-heat sources to operate a refrigerator. Fluid flows within the resonator in the vicinity of the stack/heat exchanger assemblies were measured through optical means. During the course of the experiment, anomalous centerline steady flows were observed at magnitudes of up to three times the acoustic amplitudes within the resonator of the thermoacoustic device. An evanescent component of the acoustic field was also measured at the same location. An order of magnitude calculation indicates that the body force induced by the evanescent mode is of sufficient magnitude and structure to be the source of the streaming. 2006 Elsevier Ltd. All rights reserved. PACS: 43.35.Ud; 43.25.Nm Keywords: Thermoacoustic; Streaming; Evanescent
1. Introduction An interesting application of thermoacoustics is the socalled ‘‘beer cooler’’. Waste-heat is utilized to drive an acoustic wave in a resonator, and the acoustic wave in turn drives a refrigerator stack [1,2]. To study the conversion of relatively low quality waste-heat for refrigeration, a thermoacoustically driven, standing wave device designed to operate between 305 K and 600 K in the driver stack and 305K and 295 K in the refrigeration stack with a working fluid mixture of 60/40 He/Ar at 100 psi was constructed [3]. As part of the design, two spacing duct sections were constructed, one of which contained an optical port for measurements of the fluid motion *
Corresponding author. Tel.: +1 270 745 5003; fax: +1 270 745 2014. E-mail addresses: [email protected] (G.P. Smith), raspet@ olemiss.edu (R. Raspet), [email protected] (R. Hiller), jcm04j@ garnet.acns.fsu.edu (J. McDaniel). 0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.08.005
via a laser doppler anemometer (LDA) [4]. By interchanging the location of the port and spacer, measurements were obtained beneath the cold side of the prime mover stack and heat exchanger assembly, and beneath the location of a refrigerator stack. In operation, the device suffered significant unaccounted losses, producing acoustic amplitudes of approximately 60% predicted by DeltaE [5]. This, in itself, is not unusual, as errors of 50% [6] or more have been reported between initial results and the design goals. However, in this device, the measurement of the fluid flow revealed both significant streaming flow and a secondary evanescent acoustic mode in the resonator below the stack. Subsequent temperature measurements revealed temperature profiles consistent with streaming flows extending within the stack. The observations within a thermoacoustic prime mover revealed a candidate for additional dissipation heretofore unconsidered in thermoacoustic modeling – streaming caused by an evanescent acoustic mode.
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2. Experimental setup The details of the experimental device are shown to scale in Fig. 1. Hot and cold heat exchangers bracket a stack. The resonator was composed of 800 standard T304 steel flanged components, with a 400 reduction in the center. This reduction served to reduce the physical length of the device without adjusting the operating wavelength of the device. The system was pressurized with a 60/40 He/Ar mixture, with the inlet located at the velocity antinode (in the narrow section of the resonator) to minimize the system disturbance. The inlet was also used to introduce cotton smoke into the system to provide a scattering agent for velocity measurements as described below. An optical window allowing for probing with an LDA beam was located beneath the cold heat exchanger. Microphones were located at the rigid endcap opposing the prime mover, and at the location of the LDA. The stacks were constructed of stainless steel, utilizing a hexagonal geometry for the pores. Each stack consisted of a series of 1 inch thick layers of hexagonal core material formed from 0.004 inch thick strips, with hexagonal pore diameters from flat face to flat face measuring 0.060 inches. These layers were placed on top of each other, with random orientation from one layer to another, and brazed into a stainless steel flange to form stacks 5 inches long. This construction was derived from the work of Bo¨sel, et al. [7] reasoning that the layered construction would help reduce thermal conductivity down the stack, as well as increase the opportunity for fluid streamlines to intermix. The prime mover stack con-
Fig. 1. Representative experimental arrangement. Shown is the prime mover arrangement (drawn to scale) used in collecting the presented data. The entirety of the resonator is empty, save for the two heat exchangers and stack structure is in between. Endevco 8510b piezoelectric transducers are located at the points marked with an ‘E’.
sisted of five layers of stack cores. We estimated the total porosity to be approximately 80%, believing the cumulative contributions of the random layers to be minimal as the displacement amplitudes of the gas significantly smaller than the thickness of a given stack layer. The heat exchangers were constructed of copper, consisting of a 1 inch thick solid ring enclosing a porous central structure as shown in Fig. 2. Two strips of 0.004 inch copper foil, one flat and one corrugated, were brazed together to form a strip of half-hexagons with a side dimension of 0.040 inches. This strip was then wound around a mandrel to form a porous disk, 800 in diameter, consisting of half-hexagonal cells. Finally, the disc was brazed to the interior of the copper ring. Heat transport to or from the exchanger was accomplished by milling a channel into the face of the exchanger, and imbedding a sinuous copper tube, whose path followed a near-radial symmetry. As a manufacturing necessity, the hydraulic radius of the heat exchanger pores was smaller than the pores of the stack, resulting in a reduced porosity to approximately 90% that of the stacks. With the blockages introduced by the heat transport tubes, the overall porosity of the heat exchangers was estimated at 65–70%. The system typically operated using hot temperatures from approximately 450 K–510 K, an ambient temperature of approximately 305 K, and a cold temperature of approximately 295 K. Typical acoustic parameters were a 42 Hz operating frequency, at sound pressure levels of less than 20 kPa (3% P0). Prime mover arrangements operated using the hot and ambient temperatures, and refrigerator arrangements utilized the ambient and cold temperatures, respectively. In operation, flow data for the experiment was taken via a LDA, which utilizes backscattered radiation from the intersection of two interfering beams. This intersection point defines the effective probe volume for the device, forming an ellipsoidal volume with dimensions 0.22 mm · 0.036 mm · 0.036 mm. The process converts burst radiation arising from a passing scattering particle to a velocity measurement in the direction perpendicular to the angular bisector of the intersecting beams. This arrangement allows for a precise velocity measurement at a highly localized point. The scattering particles for the experiment were provided via charred cotton smoke. A ‘‘smoke detonator’’ was constructed, consisting of an electric heater contained within a cylinder. The intervening space was filled with cotton yarn. In use, the device was inserted into the gas inlet line. The operating fluid of helium and argon was pumped through the device. Due to the lack of oxygen, the yarn surrounding the heater would only char, providing copious amounts of scattering smoke particles. The inlet was then closed, and a few minutes were allowed to pass before collecting data, so that any transitional effects would subside. Heuter and Bolt [8] deter-
G.P. Smith et al. / Applied Acoustics 69 (2008) 23–30
25
Fig. 2. Heat exchanger details. Shown is the construction of the heat exchangers, including the path of thermal transport tubes. Note the constriction at the center formed by the close proximity of the four thermal transport tubes.
mine the relationship between fluid motion and particle motion as: nP 1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ n 2 ; F 2qxr2 1 þ 9g where fP and fF are the particle and fluid displacements, q is the particle density, x is the frequency of interest, and g is the viscosity of the fluid. Utilizing tabulated values of r = 0.31 microns and q = 0.70 kg/cm3 for cotton wick smoke particles [9], and gas properties as determined through DeltaE, the particles were determined to travel nearly perfectly with the fluid (jfP/fFj = 0.9999). The optical probe traversed across the face of the heat exchanger to take a series of data points, yielding an excellent degree of resolution for the net velocity of the fluid within the resonator. To facilitate closer face measurements, the apparatus was inclined 5.4 to allow one beam to enter parallel to the face of the heat exchanger. This adjustment required a minimal cosine correction to be applied to the axial velocity measurements. This velocity was then decomposed into an acoustic component as well as a DC streaming component within the experimental apparatus. The capabilities of the LDA yielded a velocity resolution on the order of 104 m/s. The experimental arrangement for data collection also recorded relative phases between the acoustic component of the velocity and the endcap pressure oscillations as determined by an Endevco piezoresistive microphone. Velocity measurements at specific points were repeatedly made over a 30 sec time span, resulting in an ensemble ranging from 30 to 300 measurements from which averages and deviations could be determined.
3. Experimental data 3.1. Velocity measurements In the measurement region, the total gas motion is a mixture of acoustic motion and the steady motion of a significant steady flow in the vicinity of a geometry transition between the pores and blockages of the heat exchanger, and the open empty volume of the resonator. The total velocity of the gas was measured via the LDA, which was then separated into acoustic and steady-state data. Typical acoustic velocities measured just outside of the cold side of the prime mover assembly are shown in Fig. 3. Traditionally, it has been assumed that planar acoustics are the dominant mode (and hence, the only mode of interest) within thermoacoustic devices. However, the recorded data reveals the presence of an evanescent mode superimposed upon the planar mode. This evanescent mode was empirically determined to be in phase with the planar acoustic mode, and decayed in accordance with theoretical models as described in Section 4. The z-direction was measured from the outside face of the cold heat exchanger, with the positive direction associated into the open resonator. Significant deviations about the mean values were observed, interpreted as an indication of a turbulent flow. The streaming profile (also shown in Fig. 3) revealed unprecedented amplitudes of up to three times the magnitude of the acoustic velocity. There was also considerable structure within the profile. Significant deviations about the mean flow values were also recorded, reinforcing an interpretation of turbulent flow. At a physically identical location from the endcap on the opposing side of the resonator, additional data was taken
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Fig. 3. Experimental data. Acoustic and steady-state data from beneath the prime mover arrangement is shown. The solid and open dots are ensembles averaging from two distinct data runs under identical pressure and temperature parameters. The dashed line in the acoustic data represents a theoretical fit to a combination of a planar acoustic mode superimposed with an evanescent mode. The dashed line in the steady-state data represents a theoretical fit to that process. Negative velocities are indicative of flow directed toward the thermoacoustic components. Data is presented for (a) 1 cm from the face of the cold heat exchanger; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm.
to determine the nature of the fluid motion at a relatively similar location within the resonator as shown in Fig. 4. The data was collected, while the device operated under the same conditions as when the prime mover data was collected. The optical port position was not located at the precise location relative to the endcap as the prime mover data location, however, due to the missing impedance associated with the thermoacoustic components. However, we believe that our arguments regarding the origins of the evanescent
mode do not significantly depend on the location of the apparatus. With no structures in the resonator as shown in Fig. 4a, the acoustic motion was determined to follow simple planewave behavior, with some deviations near the wall due to the presence of an optical window, which could not be manufactured flush to the wall surface. The steady flow appeared consistent with Raleigh streaming. With the placement of a stack into the resonator (operating as a par-
G.P. Smith et al. / Applied Acoustics 69 (2008) 23–30
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Fig. 4. Comparative steady-state data. Comparative data at a similar position from the rigid endcap as in the prime mover data. (a) Empty resonator; (b) resonator with only a thermoacoustic stack; and (c) resonator with stack and heat exchangers. Note the significant departure from Rayleigh streaming behavior with the addition of the heat exchangers.
asitic refrigerator, as shown in Fig. 4b), no significant evanescent mode or steady flows were observed. With the addition of heat exchangers around the stack to construct a genuine refrigeration device as shown in Fig. 4c, the evanescent mode and significant steady flows were again observed. We believe that the evanescent mode and the subsequent streaming arise due to the considerable central constriction in the heat exchangers resulting from the installation of the heat transport tubes. Several of the zero-flow structural features of the steady flow correspond to the location of blockages introduced by thermal transport tubes. Additional possibilities include the sudden change in porosity from the heat exchanger to the open resonator, gravitational streaming flows, or the presence of a non-uniform temperature gradient within the stack structure. 3.2. Temperature measurements To investigate the extent of the streaming flows inside the thermoacoustic prime mover stack itself, additional
thermocouples were inserted into the stack directly above the optical port at 1 inch intervals, along radial coordinates of r/r0 = 0, r/r0 = 0.25, and r/r0 = 0.95. This arrangement formed a two-dimensional plane for temperature measurements within the stack, directly aligned with the plane of the observed velocity data. The prime mover was operated under the same conditions as the original flow data; the resulting flow behavior as measured beneath the prime mover assembly were consistent with the earlier streaming and acoustic profiles. Temperature data within the stack was collected for preonset values, and post-onset steady values as shown in Fig. 5. While some temperature depression is expected under normal operation of a prime mover, the temperature measurements before and after onset implied that considerable flows existed within the stack. Along the centerline, the flows appear to be consistent with the direction of the flow outside the stack, implying that streaming penetrates the stack through the cold heat exchanger. Near the resonator wall, the inferred stack flows appear to be opposite in
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Fig. 5. Evidence for streaming within the stack. Steady-state temperature measurements within the stack structure before (open symbols) and after (filled symbols) onset are shown. Resonator geometry and steady-state streaming data 1 cm from the cold heat exchanger are overlaid for context. Heavy-dashed lines indicate planes of measurement: (1) 1 inch from hot heat exchanger (20% of the overall stack length); (2) 2 inches from hot heat exchanger (40%); (3) 3 inches from hot heat exchanger (60%); and (4) 4 inches from hot heat exchanger (80%). Certain data points from the post-onset data have been removed for clarity due to thermocouple failure. We interpret, increases in temperature as indicative of streaming flows from the hot heat exchanger toward the cold heat exchanger. The overlaid arrows in the stack are for directional purposes only, not indicative of flow magnitude within the stack.
direction compared to those measured outside the stack. Further, on several occasions, the inferred flows from the stacks appeared absent, or even reversed themselves, indicating that the stack behavior is not necessarily consistent. These impressions are presented purely as qualitative observation, and no attempt is made herein to explain the behavior, lacking sufficient experimental flexibility in the apparatus to obtain sufficient data.
4. Modeling of the acoustic modes
the acoustic and evanescent mode. The radial and evanescent wave numbers are given by: a0 ð4Þ kr ¼ ; r0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ev ¼ k 2r k 2ac ; ð5Þ where a0 is the lowest non zero root of J 00 ðgÞ ¼ 0, equal to approximately 3.832. The velocities are readily available from the pressures and the equation of state vac ðz; tÞ ¼ Aac sin½k ac ðzs zÞeixt ; kr ðrÞ ðr; z; tÞ ¼ Aev J 1 ðk r rÞeikev z ei/ eixt ; vev k ac k ev J 0 ðk r rÞeikev z ei/ eixt : vðzÞ ev ðr; z; tÞ ¼ iAev k ac
ð6Þ
We consider a radially symmetric system, wherein an evanescent acoustic mode co-exists with an acoustic standing wave. The face of the cold heat exchanger is the origin of the system for overall mathematical clarity, with the positive direction directed away from heat exchanger/stack assembly. A distance zs separates the rigid termination and the cold heat exchanger. The acoustic pressure may be expressed as:
The total axial velocity and pressure relations may simply be determined by adding the components of the two modes
pac ðz; tÞ ¼ iq0 cAac cos½k ac ðzs zÞeixt ;
u1r ðr; z; tÞ ¼ UðzÞvðrÞ ev ðr; z; tÞ;
ð2Þ
where Aac is the acoustic amplitude, q0 is the mean density of the gas, kac is the acoustic wave number, and c is the speed of sound. This notation allows the focus to be placed at the origin of the evanescent mode (the interface between the empty resonator and the thermoacoustic components), with positive z-coordinates to be interpreted as extending into the empty resonator. The evanescent pressure may be expressed [10] as: pev ðr; z; tÞ ¼ iq0 cAev ; J 0 ðk r rÞeikev z ei/ eixt ;
ð3Þ
where Aev is the evanescent amplitude, J0 is a Bessel function of the zeroth-order, and / is the phase between
ð7Þ ð8Þ
ð9Þ
UðzÞvðzÞ ev ðr; z; tÞ;
ð10Þ
p1 ðr; z; tÞ ¼ pac ðz; tÞ þ UðzÞpev ðr; z; tÞ;
ð11Þ
u1z ðr; z; tÞ ¼ vac ðz; tÞ þ
where U(z) is the Heaviside step function 1 for z P 0 UðzÞ ¼ 0 for z < 0;
ð12Þ
which serves to restrict the existence of the evanescent mode to the region external to the stack in the vicinity of the measurements. The comparison of the acoustic theory to the collected data is shown in Fig. 3 as the dashed line to the acoustic data.
G.P. Smith et al. / Applied Acoustics 69 (2008) 23–30
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5. Modeling of the steady-state flow
Dividing both sides by the volume, and recognizing the relation between acceleration and velocity, one finds:
Streaming theories have been developed to describe streaming in various acoustic field geometries [11–13]. These theories are commonly based on the analysis of traveling or standing waves formed from propagating wave modes. This assumption usually yields streaming magnitudes that are smaller than the acoustic particle velocity. With the evidence of an evanescent mode and a high-amplitude centerline streaming profile, it is of value to investigate the effect that such a mode would have in basic streaming theories. The gross features of the observed streaming behavior may be described by the following Bradley [11], who develops a body force density relation from the second-order equations of mass and momentum conservation as: 1 o ~ ~ u1 i; f b ¼ 2 p1 ~ u1 rÞ~ ð13Þ u1 q0 hð~ c ot
F dv ¼q : V dt
where the brackets indicate time-averaging. The axial component of this force density may be expressed as: 1 o ~ 1z i: fbz ¼ 2 p1 ~ u1z q0 hð~ u1 rÞu ð14Þ c ot
fbz dz ¼ qvz dvz ; Z 2i=kev 2 v2z ¼ fbz dz: q0 zs
This results in a dominantly central force as shown in Fig. 6, considerably different from typical streaming calculations, whereby the streaming flows are driven by effects at the boundary layer at the edge of the resonator. A fully determined solution for the steady motion would require a detailed calculation using computational fluid dynamics, which is beyond the scope of this paper. Instead, we will show that the body force density (14) is sufficient to accelerate initially stationary particles to velocities comparable to those measured. We use a zeroth-order approach, ignoring the higher-order effects of friction and mass conservation. We begin with Newton’s second law: F ¼ ma:
ð15Þ
ð16Þ
The ratio of force per unit volume is the body force density fb. Restricting our approach to the axial direction, we may express the axial body force density as: fbz ¼ q
dvz : dt
ð17Þ
Expanding the derivative along the axial coordinate, one finds: fbz ¼ q
dvz dz : dz dt
ð18Þ
Rearranging, and integrating from the rigid termination to the point of interest yields the steady velocity at the point of interest. ð19Þ ð20Þ
The choice of integration limits determines the predicted velocity after passing through two decay lengths, and serves to estimate the magnitude of expected streaming. This prediction (shown as the dashed line in the Fig. 3a DC velocity data) agrees fairly well with the character of the data points, as well as with our expectation that a second-order effect should be of reduced magnitude than the first-order source. However, there remain inconsistencies. Primarily, there is a significant discrepancy with the streaming flow in the vicinity of r/r0 = 0. We suspect that this center flow would be predicted as a return flow mechanism, satisfying fluid dynamic calculations of the flow beyond the scope of this treatment. This flow may be further amplified due to manufacturing necessities involved in installing the thermal transport tubes in the heat exchangers resulted in a ring-like blockage. 6. Conclusions
Fig. 6. Body force distribution resulting from an evanescent source mode. Shown is the resulting body force density arising from an evanescent mode in the experimental apparatus, presumed to drive the anomalous observed streaming behavior. Instead of forces from propagating-mode acoustics that exist at the resonator walls (driving a central flow merely from continuity requirements), the body force from the evanescent mode contributes to the flow directly in the central region of the resonator.
This experiment revealed two important processes within a thermoacoustic device that have not yet been incorporated into standard thermoacoustic theory. First, a non-propagating evanescent acoustic mode was measured in the vicinity of the face of the stack/heat exchanger assembly. Second, we detected structured, large-amplitude streaming flows that, we believe result from the evanescent acoustic mode. It is believed that anisotropies in the construction of the heat exchangers produces the evanescent acoustic mode, which serves as a source for the streaming. These streaming flows provide a plausible explanation for the observed deficient performances within certain thermoacoustic devices. Such flows can carry heat directly through the stack to the heat exchangers, bypassing the thermoacoustic
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conversion process, and reducing the overall efficiency. Furthermore, the streaming flows can result in dissipative turbulence, increasing the entropy of the system, resulting in further reduced efficiencies. The streaming at the surface of the heat exchanger appears to induce flow in the stack itself. Measurements of the temperature gradient within the stack indicate the existence of flows in the same direction as measured at the surface of the heat exchanger. This results in a radially-dependent temperature gradient, which significantly increases the complexity of the thermal model of the system. In addition to the thermal transport in the axial direction supplied by the thermoacoustic process, an additional radial conductive process is anticipated, which should further impact the efficiency of the thermoacoustic device. The effects of a non-uniform temperature gradient within a stack have not been sufficiently studied to adequately predict the effects, however, this research demonstrates that anisotropies in heat exchanger design can lead to additional energy losses in thermoacoustic devices. Acknowledgements The authors thank Dr. H.E. Bass for his helpful discussions and guidance regarding this work. Support by the Office of Naval Research is also gratefully acknowledged, as the support from the Office of Sponsored Programs at Western Kentucky University.
References [1] Wheatley John C, Swift Gregory W, Hofler Thomas J. Heat-driven acoustic cooling engine having no moving parts. J Acoust Soc Am 1990;88(4):2047. [2] Hofler TJ. Thermoacoustic refrigerator design and performance. Ph.D. dissertation, Physics Department, University of California at San Diego, 1986. [3] Smith GP. ‘‘Examination of streaming regions within a waste-heat driven thermoacoustic refrigerator’’. Doctoral dissertation from the University of Mississippi, August 2000. [4] Dantec Dynamics, Inc., 200 Williams Dr, Ramsey, NJ 07446. Available from:. [5] Ward WC, Swift GW. Design environment for low-amplitude thermoacoustic engines. J Acoust Soc Am 1994;95(6):3671–2. [6] Swift GW. Thermoacoustics: a unifying perspective for some engines and refrigerators. Acoust Soc Am; 2002, ISBN 0-7354-0065-2. [7] Bo¨sel J, Trepp Ch, Fourie JG. An alternative stack arrangement for thermoacoustic heat pumps and refrigerators. J Acoust Soc Am 1999;106(2):707–15. [8] Hueter TF, Bolt RH. Sonics: techniques for the use of sound and ultrasound in engineering and science. John Wiley & Sons, Inc; 1955 [Chapter 6]. [9] Weinert DW, Cleary TG, Mulholland GW, Beever PF. Light scattering characteristics and size distribution of smoke and nuisance aerosols. In: Fire safety science – Proceedings of the seventh international symposium. p. 209–20. [10] Pierce AD. Acoustics: an introduction to its physical principles and applications. Acous Soc Am; 1989, ISBN 0-8831-8612-8. [11] Bradley CE. Acoustic streaming field structure: the influence of the radiator. J Acoust Soc Am 1996;100(3):1399–408. [12] Nyborg WL. Acoustic streaming. In: Hamilton MF, Blackstock DT, editors. Nonlinear acoustics. San Diego: Academic; 1998. Chap. 7. [13] Westervelt PJ. The theory of steady forces caused by sound waves. J Acoust Soc Am 1951;23(4):312–5.
Applied Acoustics 69 (2008) 23–30 www.elsevier.com/locate/apacoust
Evanescent modes and anomalous streaming in a thermoacoustic device Gordon P. Smith a
a,*
, Richard Raspet b, Robert Hiller c, Joseph McDaniel
d
Department of Physics and Astronomy, Western Kentucky University, Bowling Green, KY 42101, USA b National Center for Physical Acoustics, University of Mississippi, MS 38677, USA c 1231 Seminole Avenue, W. St. Paul, MN 55118, USA d Department of Physics and Astronomy, Florida State University, Tallahassee, FL 32306, USA Received 14 July 2005; received in revised form 30 July 2006; accepted 1 August 2006 Available online 27 October 2006
Abstract An oil-heated thermoacoustic refrigerator was constructed in order to investigate the use of waste-heat sources to operate a refrigerator. Fluid flows within the resonator in the vicinity of the stack/heat exchanger assemblies were measured through optical means. During the course of the experiment, anomalous centerline steady flows were observed at magnitudes of up to three times the acoustic amplitudes within the resonator of the thermoacoustic device. An evanescent component of the acoustic field was also measured at the same location. An order of magnitude calculation indicates that the body force induced by the evanescent mode is of sufficient magnitude and structure to be the source of the streaming. 2006 Elsevier Ltd. All rights reserved. PACS: 43.35.Ud; 43.25.Nm Keywords: Thermoacoustic; Streaming; Evanescent
1. Introduction An interesting application of thermoacoustics is the socalled ‘‘beer cooler’’. Waste-heat is utilized to drive an acoustic wave in a resonator, and the acoustic wave in turn drives a refrigerator stack [1,2]. To study the conversion of relatively low quality waste-heat for refrigeration, a thermoacoustically driven, standing wave device designed to operate between 305 K and 600 K in the driver stack and 305K and 295 K in the refrigeration stack with a working fluid mixture of 60/40 He/Ar at 100 psi was constructed [3]. As part of the design, two spacing duct sections were constructed, one of which contained an optical port for measurements of the fluid motion *
Corresponding author. Tel.: +1 270 745 5003; fax: +1 270 745 2014. E-mail addresses: [email protected] (G.P. Smith), raspet@ olemiss.edu (R. Raspet), [email protected] (R. Hiller), jcm04j@ garnet.acns.fsu.edu (J. McDaniel). 0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.08.005
via a laser doppler anemometer (LDA) [4]. By interchanging the location of the port and spacer, measurements were obtained beneath the cold side of the prime mover stack and heat exchanger assembly, and beneath the location of a refrigerator stack. In operation, the device suffered significant unaccounted losses, producing acoustic amplitudes of approximately 60% predicted by DeltaE [5]. This, in itself, is not unusual, as errors of 50% [6] or more have been reported between initial results and the design goals. However, in this device, the measurement of the fluid flow revealed both significant streaming flow and a secondary evanescent acoustic mode in the resonator below the stack. Subsequent temperature measurements revealed temperature profiles consistent with streaming flows extending within the stack. The observations within a thermoacoustic prime mover revealed a candidate for additional dissipation heretofore unconsidered in thermoacoustic modeling – streaming caused by an evanescent acoustic mode.
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G.P. Smith et al. / Applied Acoustics 69 (2008) 23–30
2. Experimental setup The details of the experimental device are shown to scale in Fig. 1. Hot and cold heat exchangers bracket a stack. The resonator was composed of 800 standard T304 steel flanged components, with a 400 reduction in the center. This reduction served to reduce the physical length of the device without adjusting the operating wavelength of the device. The system was pressurized with a 60/40 He/Ar mixture, with the inlet located at the velocity antinode (in the narrow section of the resonator) to minimize the system disturbance. The inlet was also used to introduce cotton smoke into the system to provide a scattering agent for velocity measurements as described below. An optical window allowing for probing with an LDA beam was located beneath the cold heat exchanger. Microphones were located at the rigid endcap opposing the prime mover, and at the location of the LDA. The stacks were constructed of stainless steel, utilizing a hexagonal geometry for the pores. Each stack consisted of a series of 1 inch thick layers of hexagonal core material formed from 0.004 inch thick strips, with hexagonal pore diameters from flat face to flat face measuring 0.060 inches. These layers were placed on top of each other, with random orientation from one layer to another, and brazed into a stainless steel flange to form stacks 5 inches long. This construction was derived from the work of Bo¨sel, et al. [7] reasoning that the layered construction would help reduce thermal conductivity down the stack, as well as increase the opportunity for fluid streamlines to intermix. The prime mover stack con-
Fig. 1. Representative experimental arrangement. Shown is the prime mover arrangement (drawn to scale) used in collecting the presented data. The entirety of the resonator is empty, save for the two heat exchangers and stack structure is in between. Endevco 8510b piezoelectric transducers are located at the points marked with an ‘E’.
sisted of five layers of stack cores. We estimated the total porosity to be approximately 80%, believing the cumulative contributions of the random layers to be minimal as the displacement amplitudes of the gas significantly smaller than the thickness of a given stack layer. The heat exchangers were constructed of copper, consisting of a 1 inch thick solid ring enclosing a porous central structure as shown in Fig. 2. Two strips of 0.004 inch copper foil, one flat and one corrugated, were brazed together to form a strip of half-hexagons with a side dimension of 0.040 inches. This strip was then wound around a mandrel to form a porous disk, 800 in diameter, consisting of half-hexagonal cells. Finally, the disc was brazed to the interior of the copper ring. Heat transport to or from the exchanger was accomplished by milling a channel into the face of the exchanger, and imbedding a sinuous copper tube, whose path followed a near-radial symmetry. As a manufacturing necessity, the hydraulic radius of the heat exchanger pores was smaller than the pores of the stack, resulting in a reduced porosity to approximately 90% that of the stacks. With the blockages introduced by the heat transport tubes, the overall porosity of the heat exchangers was estimated at 65–70%. The system typically operated using hot temperatures from approximately 450 K–510 K, an ambient temperature of approximately 305 K, and a cold temperature of approximately 295 K. Typical acoustic parameters were a 42 Hz operating frequency, at sound pressure levels of less than 20 kPa (3% P0). Prime mover arrangements operated using the hot and ambient temperatures, and refrigerator arrangements utilized the ambient and cold temperatures, respectively. In operation, flow data for the experiment was taken via a LDA, which utilizes backscattered radiation from the intersection of two interfering beams. This intersection point defines the effective probe volume for the device, forming an ellipsoidal volume with dimensions 0.22 mm · 0.036 mm · 0.036 mm. The process converts burst radiation arising from a passing scattering particle to a velocity measurement in the direction perpendicular to the angular bisector of the intersecting beams. This arrangement allows for a precise velocity measurement at a highly localized point. The scattering particles for the experiment were provided via charred cotton smoke. A ‘‘smoke detonator’’ was constructed, consisting of an electric heater contained within a cylinder. The intervening space was filled with cotton yarn. In use, the device was inserted into the gas inlet line. The operating fluid of helium and argon was pumped through the device. Due to the lack of oxygen, the yarn surrounding the heater would only char, providing copious amounts of scattering smoke particles. The inlet was then closed, and a few minutes were allowed to pass before collecting data, so that any transitional effects would subside. Heuter and Bolt [8] deter-
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Fig. 2. Heat exchanger details. Shown is the construction of the heat exchangers, including the path of thermal transport tubes. Note the constriction at the center formed by the close proximity of the four thermal transport tubes.
mine the relationship between fluid motion and particle motion as: nP 1 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ n 2 ; F 2qxr2 1 þ 9g where fP and fF are the particle and fluid displacements, q is the particle density, x is the frequency of interest, and g is the viscosity of the fluid. Utilizing tabulated values of r = 0.31 microns and q = 0.70 kg/cm3 for cotton wick smoke particles [9], and gas properties as determined through DeltaE, the particles were determined to travel nearly perfectly with the fluid (jfP/fFj = 0.9999). The optical probe traversed across the face of the heat exchanger to take a series of data points, yielding an excellent degree of resolution for the net velocity of the fluid within the resonator. To facilitate closer face measurements, the apparatus was inclined 5.4 to allow one beam to enter parallel to the face of the heat exchanger. This adjustment required a minimal cosine correction to be applied to the axial velocity measurements. This velocity was then decomposed into an acoustic component as well as a DC streaming component within the experimental apparatus. The capabilities of the LDA yielded a velocity resolution on the order of 104 m/s. The experimental arrangement for data collection also recorded relative phases between the acoustic component of the velocity and the endcap pressure oscillations as determined by an Endevco piezoresistive microphone. Velocity measurements at specific points were repeatedly made over a 30 sec time span, resulting in an ensemble ranging from 30 to 300 measurements from which averages and deviations could be determined.
3. Experimental data 3.1. Velocity measurements In the measurement region, the total gas motion is a mixture of acoustic motion and the steady motion of a significant steady flow in the vicinity of a geometry transition between the pores and blockages of the heat exchanger, and the open empty volume of the resonator. The total velocity of the gas was measured via the LDA, which was then separated into acoustic and steady-state data. Typical acoustic velocities measured just outside of the cold side of the prime mover assembly are shown in Fig. 3. Traditionally, it has been assumed that planar acoustics are the dominant mode (and hence, the only mode of interest) within thermoacoustic devices. However, the recorded data reveals the presence of an evanescent mode superimposed upon the planar mode. This evanescent mode was empirically determined to be in phase with the planar acoustic mode, and decayed in accordance with theoretical models as described in Section 4. The z-direction was measured from the outside face of the cold heat exchanger, with the positive direction associated into the open resonator. Significant deviations about the mean values were observed, interpreted as an indication of a turbulent flow. The streaming profile (also shown in Fig. 3) revealed unprecedented amplitudes of up to three times the magnitude of the acoustic velocity. There was also considerable structure within the profile. Significant deviations about the mean flow values were also recorded, reinforcing an interpretation of turbulent flow. At a physically identical location from the endcap on the opposing side of the resonator, additional data was taken
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Fig. 3. Experimental data. Acoustic and steady-state data from beneath the prime mover arrangement is shown. The solid and open dots are ensembles averaging from two distinct data runs under identical pressure and temperature parameters. The dashed line in the acoustic data represents a theoretical fit to a combination of a planar acoustic mode superimposed with an evanescent mode. The dashed line in the steady-state data represents a theoretical fit to that process. Negative velocities are indicative of flow directed toward the thermoacoustic components. Data is presented for (a) 1 cm from the face of the cold heat exchanger; (b) 2 cm; (c) 3 cm; (d) 4 cm; and (e) 5 cm.
to determine the nature of the fluid motion at a relatively similar location within the resonator as shown in Fig. 4. The data was collected, while the device operated under the same conditions as when the prime mover data was collected. The optical port position was not located at the precise location relative to the endcap as the prime mover data location, however, due to the missing impedance associated with the thermoacoustic components. However, we believe that our arguments regarding the origins of the evanescent
mode do not significantly depend on the location of the apparatus. With no structures in the resonator as shown in Fig. 4a, the acoustic motion was determined to follow simple planewave behavior, with some deviations near the wall due to the presence of an optical window, which could not be manufactured flush to the wall surface. The steady flow appeared consistent with Raleigh streaming. With the placement of a stack into the resonator (operating as a par-
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Fig. 4. Comparative steady-state data. Comparative data at a similar position from the rigid endcap as in the prime mover data. (a) Empty resonator; (b) resonator with only a thermoacoustic stack; and (c) resonator with stack and heat exchangers. Note the significant departure from Rayleigh streaming behavior with the addition of the heat exchangers.
asitic refrigerator, as shown in Fig. 4b), no significant evanescent mode or steady flows were observed. With the addition of heat exchangers around the stack to construct a genuine refrigeration device as shown in Fig. 4c, the evanescent mode and significant steady flows were again observed. We believe that the evanescent mode and the subsequent streaming arise due to the considerable central constriction in the heat exchangers resulting from the installation of the heat transport tubes. Several of the zero-flow structural features of the steady flow correspond to the location of blockages introduced by thermal transport tubes. Additional possibilities include the sudden change in porosity from the heat exchanger to the open resonator, gravitational streaming flows, or the presence of a non-uniform temperature gradient within the stack structure. 3.2. Temperature measurements To investigate the extent of the streaming flows inside the thermoacoustic prime mover stack itself, additional
thermocouples were inserted into the stack directly above the optical port at 1 inch intervals, along radial coordinates of r/r0 = 0, r/r0 = 0.25, and r/r0 = 0.95. This arrangement formed a two-dimensional plane for temperature measurements within the stack, directly aligned with the plane of the observed velocity data. The prime mover was operated under the same conditions as the original flow data; the resulting flow behavior as measured beneath the prime mover assembly were consistent with the earlier streaming and acoustic profiles. Temperature data within the stack was collected for preonset values, and post-onset steady values as shown in Fig. 5. While some temperature depression is expected under normal operation of a prime mover, the temperature measurements before and after onset implied that considerable flows existed within the stack. Along the centerline, the flows appear to be consistent with the direction of the flow outside the stack, implying that streaming penetrates the stack through the cold heat exchanger. Near the resonator wall, the inferred stack flows appear to be opposite in
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Fig. 5. Evidence for streaming within the stack. Steady-state temperature measurements within the stack structure before (open symbols) and after (filled symbols) onset are shown. Resonator geometry and steady-state streaming data 1 cm from the cold heat exchanger are overlaid for context. Heavy-dashed lines indicate planes of measurement: (1) 1 inch from hot heat exchanger (20% of the overall stack length); (2) 2 inches from hot heat exchanger (40%); (3) 3 inches from hot heat exchanger (60%); and (4) 4 inches from hot heat exchanger (80%). Certain data points from the post-onset data have been removed for clarity due to thermocouple failure. We interpret, increases in temperature as indicative of streaming flows from the hot heat exchanger toward the cold heat exchanger. The overlaid arrows in the stack are for directional purposes only, not indicative of flow magnitude within the stack.
direction compared to those measured outside the stack. Further, on several occasions, the inferred flows from the stacks appeared absent, or even reversed themselves, indicating that the stack behavior is not necessarily consistent. These impressions are presented purely as qualitative observation, and no attempt is made herein to explain the behavior, lacking sufficient experimental flexibility in the apparatus to obtain sufficient data.
4. Modeling of the acoustic modes
the acoustic and evanescent mode. The radial and evanescent wave numbers are given by: a0 ð4Þ kr ¼ ; r0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ev ¼ k 2r k 2ac ; ð5Þ where a0 is the lowest non zero root of J 00 ðgÞ ¼ 0, equal to approximately 3.832. The velocities are readily available from the pressures and the equation of state vac ðz; tÞ ¼ Aac sin½k ac ðzs zÞeixt ; kr ðrÞ ðr; z; tÞ ¼ Aev J 1 ðk r rÞeikev z ei/ eixt ; vev k ac k ev J 0 ðk r rÞeikev z ei/ eixt : vðzÞ ev ðr; z; tÞ ¼ iAev k ac
ð6Þ
We consider a radially symmetric system, wherein an evanescent acoustic mode co-exists with an acoustic standing wave. The face of the cold heat exchanger is the origin of the system for overall mathematical clarity, with the positive direction directed away from heat exchanger/stack assembly. A distance zs separates the rigid termination and the cold heat exchanger. The acoustic pressure may be expressed as:
The total axial velocity and pressure relations may simply be determined by adding the components of the two modes
pac ðz; tÞ ¼ iq0 cAac cos½k ac ðzs zÞeixt ;
u1r ðr; z; tÞ ¼ UðzÞvðrÞ ev ðr; z; tÞ;
ð2Þ
where Aac is the acoustic amplitude, q0 is the mean density of the gas, kac is the acoustic wave number, and c is the speed of sound. This notation allows the focus to be placed at the origin of the evanescent mode (the interface between the empty resonator and the thermoacoustic components), with positive z-coordinates to be interpreted as extending into the empty resonator. The evanescent pressure may be expressed [10] as: pev ðr; z; tÞ ¼ iq0 cAev ; J 0 ðk r rÞeikev z ei/ eixt ;
ð3Þ
where Aev is the evanescent amplitude, J0 is a Bessel function of the zeroth-order, and / is the phase between
ð7Þ ð8Þ
ð9Þ
UðzÞvðzÞ ev ðr; z; tÞ;
ð10Þ
p1 ðr; z; tÞ ¼ pac ðz; tÞ þ UðzÞpev ðr; z; tÞ;
ð11Þ
u1z ðr; z; tÞ ¼ vac ðz; tÞ þ
where U(z) is the Heaviside step function 1 for z P 0 UðzÞ ¼ 0 for z < 0;
ð12Þ
which serves to restrict the existence of the evanescent mode to the region external to the stack in the vicinity of the measurements. The comparison of the acoustic theory to the collected data is shown in Fig. 3 as the dashed line to the acoustic data.
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5. Modeling of the steady-state flow
Dividing both sides by the volume, and recognizing the relation between acceleration and velocity, one finds:
Streaming theories have been developed to describe streaming in various acoustic field geometries [11–13]. These theories are commonly based on the analysis of traveling or standing waves formed from propagating wave modes. This assumption usually yields streaming magnitudes that are smaller than the acoustic particle velocity. With the evidence of an evanescent mode and a high-amplitude centerline streaming profile, it is of value to investigate the effect that such a mode would have in basic streaming theories. The gross features of the observed streaming behavior may be described by the following Bradley [11], who develops a body force density relation from the second-order equations of mass and momentum conservation as: 1 o ~ ~ u1 i; f b ¼ 2 p1 ~ u1 rÞ~ ð13Þ u1 q0 hð~ c ot
F dv ¼q : V dt
where the brackets indicate time-averaging. The axial component of this force density may be expressed as: 1 o ~ 1z i: fbz ¼ 2 p1 ~ u1z q0 hð~ u1 rÞu ð14Þ c ot
fbz dz ¼ qvz dvz ; Z 2i=kev 2 v2z ¼ fbz dz: q0 zs
This results in a dominantly central force as shown in Fig. 6, considerably different from typical streaming calculations, whereby the streaming flows are driven by effects at the boundary layer at the edge of the resonator. A fully determined solution for the steady motion would require a detailed calculation using computational fluid dynamics, which is beyond the scope of this paper. Instead, we will show that the body force density (14) is sufficient to accelerate initially stationary particles to velocities comparable to those measured. We use a zeroth-order approach, ignoring the higher-order effects of friction and mass conservation. We begin with Newton’s second law: F ¼ ma:
ð15Þ
ð16Þ
The ratio of force per unit volume is the body force density fb. Restricting our approach to the axial direction, we may express the axial body force density as: fbz ¼ q
dvz : dt
ð17Þ
Expanding the derivative along the axial coordinate, one finds: fbz ¼ q
dvz dz : dz dt
ð18Þ
Rearranging, and integrating from the rigid termination to the point of interest yields the steady velocity at the point of interest. ð19Þ ð20Þ
The choice of integration limits determines the predicted velocity after passing through two decay lengths, and serves to estimate the magnitude of expected streaming. This prediction (shown as the dashed line in the Fig. 3a DC velocity data) agrees fairly well with the character of the data points, as well as with our expectation that a second-order effect should be of reduced magnitude than the first-order source. However, there remain inconsistencies. Primarily, there is a significant discrepancy with the streaming flow in the vicinity of r/r0 = 0. We suspect that this center flow would be predicted as a return flow mechanism, satisfying fluid dynamic calculations of the flow beyond the scope of this treatment. This flow may be further amplified due to manufacturing necessities involved in installing the thermal transport tubes in the heat exchangers resulted in a ring-like blockage. 6. Conclusions
Fig. 6. Body force distribution resulting from an evanescent source mode. Shown is the resulting body force density arising from an evanescent mode in the experimental apparatus, presumed to drive the anomalous observed streaming behavior. Instead of forces from propagating-mode acoustics that exist at the resonator walls (driving a central flow merely from continuity requirements), the body force from the evanescent mode contributes to the flow directly in the central region of the resonator.
This experiment revealed two important processes within a thermoacoustic device that have not yet been incorporated into standard thermoacoustic theory. First, a non-propagating evanescent acoustic mode was measured in the vicinity of the face of the stack/heat exchanger assembly. Second, we detected structured, large-amplitude streaming flows that, we believe result from the evanescent acoustic mode. It is believed that anisotropies in the construction of the heat exchangers produces the evanescent acoustic mode, which serves as a source for the streaming. These streaming flows provide a plausible explanation for the observed deficient performances within certain thermoacoustic devices. Such flows can carry heat directly through the stack to the heat exchangers, bypassing the thermoacoustic
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conversion process, and reducing the overall efficiency. Furthermore, the streaming flows can result in dissipative turbulence, increasing the entropy of the system, resulting in further reduced efficiencies. The streaming at the surface of the heat exchanger appears to induce flow in the stack itself. Measurements of the temperature gradient within the stack indicate the existence of flows in the same direction as measured at the surface of the heat exchanger. This results in a radially-dependent temperature gradient, which significantly increases the complexity of the thermal model of the system. In addition to the thermal transport in the axial direction supplied by the thermoacoustic process, an additional radial conductive process is anticipated, which should further impact the efficiency of the thermoacoustic device. The effects of a non-uniform temperature gradient within a stack have not been sufficiently studied to adequately predict the effects, however, this research demonstrates that anisotropies in heat exchanger design can lead to additional energy losses in thermoacoustic devices. Acknowledgements The authors thank Dr. H.E. Bass for his helpful discussions and guidance regarding this work. Support by the Office of Naval Research is also gratefully acknowledged, as the support from the Office of Sponsored Programs at Western Kentucky University.
References [1] Wheatley John C, Swift Gregory W, Hofler Thomas J. Heat-driven acoustic cooling engine having no moving parts. J Acoust Soc Am 1990;88(4):2047. [2] Hofler TJ. Thermoacoustic refrigerator design and performance. Ph.D. dissertation, Physics Department, University of California at San Diego, 1986. [3] Smith GP. ‘‘Examination of streaming regions within a waste-heat driven thermoacoustic refrigerator’’. Doctoral dissertation from the University of Mississippi, August 2000. [4] Dantec Dynamics, Inc., 200 Williams Dr, Ramsey, NJ 07446. Available from: