Experimental and numerical investigations of resonant acoustic waves in near-critical carbon dioxide

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11 Experimental and numerical investigations of resonant acoustic2 waves in near-critical carbon dioxide

32 NusairHasanandBakhtierFarouka)

4 Mechanical Engineering and Mechanics, Drexel University, Philadelphia, Pennsylvania 19014, USA

5 (Received 8 July 2014; revised 16 August 2015; accepted 30 August 2015; published online xx xx6 xxxx)

7 Acoustically augmented flows represent an efficient way of enhancing mixing and mass transport8 processes in supercritical fluids and has applications in the chemical separation and extraction9 industry. Flow and transport induced by resonant acoustic waves in a near-critical fluid filled

10 cylindrical enclosure is investigated both experimentally and numerically. Supercritical carbon11 dioxide (near the critical or the pseudo-critical states) in a confined resonator is subjected to12 acoustic field created by an electro-mechanical acoustic transducer and the induced pressure13 waves are measured by a fast response pressure field microphone. The frequency of the acoustic14 transducer is chosen such that the lowest acoustic mode propagates along the enclosure. For numer-15 ical simulations, a real-fluid CFD3 model representing the thermo-physical and transport properties16 of the supercritical fluid is considered. The simulated acoustic field in the resonator is compared17 with measurements. The formation of acoustic streaming structures in the highly compressible18 medium is revealed by time-averaging the numerical solutions over a given period. Due to19 diverging thermo-physical properties of supercritical fluid near the critical point, large scale20 oscillations are generated even for small sound field intensity. The strength of the acoustic wave21 field is found to be in direct relation with the thermodynamic state of the fluid. The effects of22 near-critical property variations and the operating pressure on the formation process of the23 streaming structures are also investigated. Irregular streaming patterns with significantly higher24 streaming velocities are observed for near-pseudo-critical states at operating pressures close to the25 critical pressure. However, these structures quickly re-orients to the typical Rayleigh streaming26 patterns with the increase operating pressure. VC 2015 Acoustical Society of America.

[http://dx.doi.org/10.1121/1.4930951]

[JDM] Pages: 114

27 I. INTRODUCTION

28 Acoustic wave induced convective transport inside an29 enclosure is a well-studied problem which hasbeen investi-30 gated by several researchers.18 Hamiltonet al.9 derived an31 analytic solution for the secondary flow field generated by32 standing waves confined by parallel plates. Acoustically33 augmented convection has the application in accelerating34 certain kinds of rate processes, especially the cooling process35 of electronic systems under micro-gravity conditions, where36 nature convections are greatly reduced or completely37 eliminated. Transport mechanisms such as this can also be38 employed to enhance mixing processes and to augment heat39 and mass transfer through resonator walls. In recent years,40 the problem has beenextensively studied both from experi-41 mental7 and numerical1,8 perspectives. From a mass trans-42 port point of view, Kawahara et al.10 studied the effect of43 standing wave fields on transport in an ideal gas. Convective44 heat transfer from an isolated sphere in a standing sound45 field was examined by Gopinath and Mills3 for large flow46 Reynolds numbers. Heat transfer enhancement by acoustic47 augmentation was also studied by Tajik et al.6 However,48 most of these studies are limited to ideal gases at near atmos-49 pheric pressures and at present, investigations of acoustically

50augmented transport phenomena in high pressure/supercriti-51cal fluids are absent in the literature.52Supercritical fluids are commonly used in industries

53nowadays as solvents, efficient thermal storage, and trans-54port media.11 Due to the increasing number of applications55of supercritical fluids in chemical and thermal process indus-

56tries, convective transport in near-critical and supercritical57fluids has drawn a lot attention to the researchers in the58recent decade.1215 While supercritical fluids, in general, ex-

59hibit interesting physical properties,1 specific interest in CO260is magnified by its perceived green propertiescarbon61dioxide is non-flammable, non-toxic, and relatively inert. In

62addition, unlike water, the supercritical regime of CO2 is

63readily accessible, given its critical temperature of only

64304 K (and critical pressure of 7.38 MPa). However, the65transport dynamics of CO2is relatively slow near the critical

66point and, therefore, improvements in convective transport

67(both thermal and mass) are required.16 Application of68mechanically driven acoustic waves represents a potential

69efficient way of enhancing transport processes in supercriti-

70cal fluid medium.17,18 The enhancement in transport proc-71esses is induced due to effects produced by compressions

72and decompressions of the acoustic waves as well as the for-

73mation of the secondary flow-field (acoustic streaming).

74Balachandranet al.18 studied the influence of ultrasound on75supercritical extraction based on extraction of gingerol from

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J_ID: JASMAN DOI: 10.1121/1.4930951 Date: 11-September-15 Stage: Page: 1 Total Pages: 15

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J. Acoust. Soc. Am.138 (3), September 2015 VC 2015 Acoustical Society of America 10001-4966/2015/138(3)/1/14/$30.00
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76 a freeze-dried ginger sample. In a relatively recent experi-77 mental study, Riera et al.19,20 proposed power ultrasound78 assisted supercritical fluid extraction to enhance the mass79 transport in almond oil extraction. Although results from80 both of these studies show that power ultrasound signifi-81 cantly accelerates the kinetics of the process and improves82 the final extraction yield, a clear physical description of the83 transport processes were not specified in either one of the84 papers. Hasan and Farouk21 developed a computational fluid85 dynamic model of acoustic field assisted supercritical fluid86 extraction of caffeine from coffee beans in a fixed bed87 extractor. However, the transport enhancement was observed88 mainly due to localized convection (primary oscillatory flow89 field) as no secondary flow structures were developed in the90 reactor geometry.91 In the present study, the generation and propagation of92 mechanically driven acoustic waves in sub- and supercritical93 carbon dioxide are experimentally studied and numerically94 simulated. A closed, cylindrical shaped resonator filled with95 carbon dioxide (at sub- and supercritical states) is consid-96 ered. The primary oscillatory flow field in the enclosure is

97 generated by the vibration of one of the end walls of the res-98 onator (using an electro-mechanical acoustic transducer).99 The driving frequencies of the vibrating wall are chosen

100 such that the lowest acoustic mode propagates along the res-101 onator and a standing wave field is produced. The amplitude102 of the generated acoustic waves is measured at the pressure103 anti-node (fixed end wall) by a fast response piezo-resistive104 pressure field microphone. The variations in acoustic and105 flow fields are also studied as a function of space and time.106 Interaction of acoustic waves in highly compressible fluids107 (e.g., supercritical fluids) and solid boundaries produces a108 second-order steady flow field known as acoustic stream-109 ing.1,22 It is often observed in compressible media exposed110 to an acoustic field. Formation of the standing wave and111 acoustic streaming are numerically simulated by directly112 solving the full compressible form of the Navier-Stokes113 equations. The NIST234 Standard Reference Database 23 is114 used to obtain the q-p-T relations for supercritical carbon115 dioxide as well as the different thermo-physical and trans-116 port properties of the fluid. With the developed model, phys-117 ical processes including the interaction of the wave field118 with viscous effects and formation of streaming structures119 are simulated. The effects of near-critical property variations120 and fluid pressure on the formation process of the streaming121 structures are also investigated in detail.

122 II. THERMO-PHYSICAL PROPERTIES AT123 SUPERCRITICAL STATE

124 The critical point of a fluid specifies the conditions125 (temperature and pressure) at which a phase boundary ceases126 to exist and above which distinct liquid and gas phases do127 not exist. For pure substances, there is an inflection point in128 the critical isotherm on a p-v diagram, leading to @p=@vT129 @2p=@v2T0. The critical pressure (pc) for carbon130 dioxide is 7.3773 MPa, the critical temperature (Tc) is131 304.1282 K and the critical density (qc) is 467.6 kg/m

3.23 If132 both pressure and temperature are beyond each critical

133value, the fluid is in a supercritical state. A supercritical fluid134can thus be categorized as a single phase fluid that is dense135as a liquid, yet compressible as a gas.136Near the critical point, the thermo-physical and trans-137port properties of fluids exhibit unusual behaviors; the spe-138cific heat (cp) and the isothermal compressibility (b) show139strong divergence, causing a vanishing thermal diffusivity140(a). The near-critical thermo-physical property variations are141also influenced by the corresponding pseudo-critical states.142A pseudo-critical state of a pure fluid can be defined as a143state in near-critical supercritical region at which the density144of the fluid is equal to its critical density (i.e., whereq qc)145and the thermodynamic and transport properties have their146maximum rate of change with temperature at a constant147supercritical pressure.24 Below the temperature correspond-148ing to the pseudo-critical state (where q=qc> 1), the fluid149has liquid-like properties while above (where q=qc < 1), it150more closely resembles a vapor (gas). Figure 1 shows the151density vs temperature diagram for supercritical carbon152dioxide with changing pressure. The horizontal line refers to153the critical density (qc 467.6 kg/m

3). At atmospheric pres-154

sure, carbon dioxide behaves like an ideal gas and the 155density is almost constant, while the density in the156supercritical-pressure condition widely varies across the phase157interface from the liquid or gas phase to the supercritical fluid158phase. The isothermal compressibility also shows a strong159divergence around the near-critical pseudo-critical states.160Hence, even around 10 K above the critical temperature161(304.1282 K), carbon dioxide is found to be highly compressi-162ble but the density remains high. Acoustic speed of the fluid163medium is an important factor for generation standing waves164in a resonator. The acoustic speed also exhibits divergence165around the near-critical pseudo-critical states. Due to this166wide variation of the thermo-physical and transport properties

167of near-critical fluids, it is expected that the behavior of168the generated acoustic field and the formation of acoustic

FIG. 1. Variation of density,q as functions of temperature and pressure in

near-critical carbon dioxide.



2 J. Acoust. Soc. Am.138 (3), September 2015 Nusair Hasan and Bakhtier Farouk


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169 streaming structures would be significantly different than that170 in ideal gases.

171 III. PROBLEM DESCRIPTION

172 A cylindrical resonator with a length (L) of 25.0 cm is173 considered. The aspect ratio (L/D) of the resonator is 12.5

174 (diameter of the resonator is 2.0 cm). The schematic of the

175 geometry considered is shown in Fig.2. One of the end walls

176 (see Fig. 2) of the resonator is vibrated at a specific fre-

177 quency. The frequency of vibration is chosen such that the178 lowest acoustic mode propagates through the fluid medium

179 (i.e.,f csi/2 L; wherecsiis the acoustic speed at the initial180 state of the fluid). The axial symmetry of the problem181 geometry is utilized and two dimensional (2D) cylindrical

182 coordinate (r- z) system is considered for the numerical cal-183 culations to simplify the computational efforts. Details

184 regarding the experimental setup used for the measurement185 of the acoustic waves in sub- and supercritical carbon diox-186 ide are discussed in Sec.IV.

187 IV. EXPERIMENTAL SETUP

188 Achieving supercritical state safely is a key point for the

189 success of generation and measurement of resonant acoustic

190 waves in supercritical carbon dioxide. The experimental191 setup is designed following our previous work25 with super-

192 critical carbon dioxide. Figure4 shows a schematic illustra-193 tion of the experimental setup.194 The supercritical chamber consists of gas inlet and

195 outlet ports fitted with high pressure (103.4 MPa) ball valves

196 and the chamber is connected to a carbon dioxide tank

197 (p 6.0 MPa). A hand pump (HiP, 34.5 MPa, 60 mL/stroke)198 is used to raise the pressure of the carbon dioxide in the199 supercritical chamber from the tank pressure to a pressure

200above the critical point of carbon dioxide. Heating tape201(Thermolyne, 0.5 in.4 ft) wrapped around the supercritical202chamber is used to raise the temperature of the carbon diox-203ide in the supercritical chamber from the room temperature204(usually295 K) to the operating temperature. The mechan-205ically driven acoustic waves in supercritical carbon dioxide206are studied in a PTFE (Polytetrafluoroethylene) tube (inside207diameter 2.0 cm, length 25.0 cm) snugly fitted inside the208supercritical chamber. A small hole in the PTFE tubing209aligned with the gas inlet port provides flow path for the gas210to and from the supercritical chamber. The resonator is211mounted horizontally with the acoustic driver on the right212end (see Fig. 3) and a plug made of polyoxymethylene213(Delrin) is used to close the opposite end. Delrin is a hard,214high stiffness thermoplastic that prevents any absorption/215attenuation of the acoustic waves during reflection from the216end wall.217The acoustic driver is an electro-mechanical driver type218loudspeaker (CUI CMS0401KL-3X) . The sinusoidal driving219signal of the acoustic driver is generated by a function gener-220ator (BK Precision 4011A) and amplified by a Crown221CE1000 type power amplifier. The signal generator is capa-222ble of providing 65.0 V sine waves up to a frequency of 5.0223MHz. A wattmeter (Powertek, ISW 8000) is connected

224between the amplifier and the loudspeaker to measure the225root mean square (RMS) values of the delivered input power,226the applied voltage, the applied current and the phase angle227between the voltage and the current. An Endevco 8507C-1228series piezo-resistive pressure transducer is used to detect229and quantify the acoustic field. The transducer is installed in230the Delrin plug located at one end of the resonator. The231cross-sectional area of the microphone is approximately one232percent of the resonator area, therefore the error introduced233by the presence of the probe in the sound field is assumed

FIG. 2. (Color online) Schematic dia-

gram of the problem geometry.

FIG. 3. (Color online) Schematic dia-gram of the experimental setup.



J. Acoust. Soc. Am.138 (3), September 2015 Nusair Hasan and Bakhtier Farouk 3


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234 negligible. Lead wires from the loudspeaker and the pressure

235 transducer are introduced inside the supercritical chamber

236 through compression fittings. An Endevco 4428A signal con-

237 ditioner is used to process the signal (pressure) from the238 transducer and provide excited voltage.239 Two type E thermocouples (Thermocouple 1 and 2,

240 Fig.3) are used to measure the steady state initial tempera-

241 ture of the supercritical fluid. A high pressure transducer

242 (Omega PX309) is used to measure and monitor the steady

243 state pressure inside the supercritical chamber. The analog

244 temperature and pressure measurements are recorded, digi-

245 tized and saved through a National Instrument SCB-68 ter-

246 minal block and a 6052E data acquisition (DAQ) board.

247 High sample rate (333 kHz) of the 6052E DAQ board guar-

248 antees that the signals are recorded with high fidelity. Details249 of the experimental setup are summarized in TableI.

250 V. MATHEMATICAL MODEL

251 The numerical model of the flow field developed in the

252 acoustic resonator is described below. Specific details about

253 the governing equations solved, the initial and boundary con-254 ditions and the numerical scheme used are also provided.

255 A. Governing equations

256 The transport processes in supercritical and near-critical

257 fluids can be modeled by the hydrodynamic description for

258 an isotropic, Newtonian, compressible,and dissipative (vis-

259 cous and heat-conducting) fluid.14,2628 The governing equa-

260 tions corresponding to mass, momentum, and energy261 balances are as follows:

@q

@t

1

r

@ qrur

@r

@ quz

@z 0; (1)

@ qur

@t

1

r

@ qru2r @r

@ quruz

@z

@p

@r

1

r

@ rsrr

@r

@srz@z

shh

r ; (2A)

@ quz

@t

1

r

@ qruruz

@r

@ qu2z @z

@p

@z

1

r

@ rsrz

@r

@szz@z

; (2B)

@ qh0

@t

1

r

@

@r qrurh0kf

@T

@r

@@z

quzh0kf@T@z

@p

@t U: (3)

262The viscous dissipation term, U is given as

U 1

r

@

@r r ursrruzsrz

@

@z ursrzuzszz : (4)

263The components of stress tensor are

srr l0 1

r

@ rur

@r

@ uz

@z

2l

@ur@r

; (5A)

szz l0 1

r

@ rur

@r

@ uz

@z

2l

@uz@z

; (5B)

shh l0 1

r

@ rur

@r

@ uz

@z

2l

ur

r

; (5C)

srz l @ur@z

@uz@r

: (5D)

264Here, p is pressure and T is temperature. h0 is the total en-

265thalpy of the fluid given by h0 h12

u2r u2z

. l is the

266dynamic viscosity and l0 is the second coefficient of viscos-

267ity [l

0

2

3 l; considering zero bulk viscosity]. ur and uz268are the velocities along the radial (r) and axial (z) coordi-

269nates, respectively,p is pressure, and Tis temperature of the

270fluid. The equation of state describing the q-p-T relation of

271supercritical fluids (including the near-critical states) is not

272well-represented by the van der Waals equation of state.29 In

273this study, we use the NIST23 Standard Reference

274Database 30 for the qf (p, T) relations and for evaluation275of other thermodynamic properties of supercritical and near-

276critical carbon dioxide. The NIST23 (Ref. 30) equation of

277state describing theq-p-Trelation of carbon dioxide is based

278on the equation of state proposed by Span and Wagner,31

279which is mainly empirical in nature and includes special

280non-analytic terms to predict the correct behavior of the fluid281to the immediate vicinity of the critical point. Two dimen-

282sional look-up tables are developed in the present study to

283represent the density,q f (p, T), thermal conductivity,kff284(p, T), dynamic viscosity,l f (p, T), specific enthalpy,h f285(p, T) and specific heat, cpf (p, T) data provided by the286NIST Standard Reference Database 23.

287B. Initial and boundary conditions

288Initially thermally quiescent and motion-free sub-/super-

289critical carbon dioxide inside the domain is considered for

290the calculations. The initial state (temperature and pressure)

291of the fluid is described as, Tiand pi. The wall boundaries at292the rigid end are maintained constant at the initial

TABLE I. Specifications of the experimental system.

No. Part Make and Model Specification

1 Hand pump HiP 87-6-5 34.5 MPa 60 mL/stroke

2 Glands Conax TG-24 T(E)-A2-T Pressure rating: 22 MPa

3 Thermocouples Omega EMTSS-125 (Probe Type) Type E3.175 mm diam.

4 Pressure transducers Endevco 8507C-1 Omega PX309-2 KG5V Sensitivity: 200 mV/psi Pressure rating: 1 psi pressure rating: 13.8MPa

5 Acoustic driver CUI CMS0401KL-3X Rated power: 10 W Impedance: 8X

6 Data acquisition board NI 6052E Sampling rate: 333 kHz





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293 temperature of the fluid, i.e., isothermal, Tr; z L Ti.294 The isothermal boundary condition is the closest approxima-

295 tion to an experiment and has been shown in paststudies to

296 have the best agreement with experimental results.32,33 The

297 vibrating wall and the wall along the length of the resonator298 are maintained adiabatic, i.e.,

@T

@z z0 0 @T

@r rR:299 For generating the acoustic waves, the vibrating wall is har-

300 monically oscillated to model the motion of the speaker. The

301 moving wall (atz 0) is vibrated according to the following302 equation:

zt A0sin2pft: (6A)

303 Here, A0is the amplitude of vibration and fis the frequency

304 of oscillation. The frequency of oscillation of the vibrating

305 wall is chosen such that the lowest acoustic mode propagates

306 through the fluid medium (i.e., f csi / 2 L, where csi is the307 acoustic speed at the initial state). For the numerical simula-

308 tions, the amplitude of vibration of the moving wall is calcu-309 lated from the power consumption of the electro-mechanical

310 speaker using the model developed by Beranek.34 Axial ve-311 locity of the fluid in the immediate vicinity of the moving312 wall is given by

uz t @z t

@t 2pfA0cos 2pft : (6B)

313 Radial velocity of the fluid at the same location is considered

314 to be zero. No-slip boundary conditions are considered at the

315 adiabatic side wall (at rR) and at the end wall (at z L).316 Symmetry boundary condition for both velocity and temper-

317 ature is imposed along the resonator axis.

318 C. Numerical scheme

319 The numerical scheme for solving the governing equa-

320 tions is based on the finite volume approach. The continuity,

321 momentum and energy equations are solved for the fluid

322 using the central difference scheme. The motion of the

323 vibrating wall is captured by a moving grid scheme near the

324 piston wall. The re-meshing scheme used in the simulations325 is the Transfinite Interpolation scheme.35 A second order

326 CrankNicholson scheme (with a blending factor of 0.7) is327 used for the time derivatives in the continuity, momentum

328 and energy equations. A convergence criterion (i.e., percent-329 age relative error) of 104 is used for all the variables in the330 iterative implicit numerical solver.331 As the cylindrical resonator geometry is axisymmetric,

332 2D cylindrical coordinate system (rvarying between 0 and333 r0) is used for simulation. The problem geometry is studied334 with non-uniform structured grid and fine grid is used in

335 vicinity of the wall boundaries to provide adequate spatial336 resolution and capture the oscillating boundary layer (dv). A

337 grid dependency study is performed considering four differ-

338 ent grid sizes ranging from coarse (100 30; minimum grid339 size in axial direction, Dzmin 1.0 mm) to fine (400 100;

340minimum grid size in axial direction, Dzmin 0.25 mm).341Supercritical carbon dioxide (pi 7.6 MPa; Ti 308K) in342the cylindrical resonator with a resonant driving frequency

343(f) of 391.0 Hz and vibration amplitude (A0) of 4.0lm is

344chosen as the test case. The results from the grid dependency

345study are shown in Fig. 4. The maximum amplitude of the

346generated acoustic wave (pmax) and the normalized axial

347streaming velocity (ust/uR) are selected as the characteristic

348variables indicating grid independency of the model. It is

349observed that for the 250 100 grid, the characteristic varia-350bles (pmax and ust/uR) are independent of the grid size. The

351time-step (Dt) for the simulations is dependent on the acous-

352tic speed in the fluid. For the thermodynamic states of carbon

353dioxide considered in the present study, the simulation time-

354step is around 0.51.0ls and obeys the CFL condition with

355a Courant number (C csiDt/Dx, wherecsiis the acoustic356speed for the given conditions) around 0.4.

357VI. RESULTS AND DISCUSSIONS

358The numerical model is validated first with results from

359past studies. The validation of the numerical model as well

360as the numerical and experimental results obtained from the361present study is discussed in Sec.VI A.

362A. Model validation

363The numerical prediction of mechanically driven acous-

364tic waves with the present model is compared with a previ-

365ous numerical study by Aktas and Farouk.1 Aktas and

366Farouk numerically investigated mechanically driven acous-

367tic waves in atmospheric pressure nitrogen at 300 K in a two

368dimensional rectangular enclosure of length, L 8.825 mm369and varying width. The frequency and amplitude of the

370vibrating wall is always kept at 20 kHz and 10 lm, respec-

371tively. The length of the enclosure is chosen such that372L k/2 (kis the wavelength calculated from csfk) and the373lowest acoustic mode propagates through the fluid medium.

FIG. 4. (Color online) Maximum amplitude of the generated acoustic wave

(pmax) and the normalized axial streaming velocity (ust/uR) as a function of

the grid size.





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374 For the case considered for validation of our present model,

375 the width of the rectangular enclosure is 20dv. Here,dvis the376 viscous penetration depth given by

d

ffiffiffiffiffiffiffi2l

qx

s ; x 2pf: (7)

377 The streaming velocities in the enclosure are calculated

378 using the following formula:

usthqui

hqi ; vst

hqvi

hqi : (8)

379 Here, ust and vst describes the x and y components of the

380 streaming velocities, respectively. Thehi sign indicates the381 time-averaged quantities. The time averaging is performed

382 during the 100th vibration cycle (between cycle #99 and

383 #100). The streaming velocities calculated based on the

384 time averaging during 80th and 100th cycles do not differ

385 significantly. Hence, the average mass transport velocities

386 are assumed to be cycle independent by this time (cycle

387 #100).388 Spatial variation of thex component of streaming veloc-

389 ity at x 3L/4 for the validation case is shown in Fig.5(a).390 In this figure, the vertical axis is the x component of the

391 dimensionless streaming velocity (ust/uR) anduR is given by

392 uR 3u02/16cs. Here, u0 is the maximum oscillatory

393 velocity. This reference velocity value represents the maxi-

394 mum streaming velocity in case of a perfect sinusoidal wave

395 form obtained by Rayleigh. Figure5(b)shows the variation

396 of theycomponent of streaming velocity along the enclosure

397 semi-height at x L/2. The vertical axis represents the y398 component of the non-dimensional streaming velocity (vstx0399 /uRy0). Here x0 is the length of the enclosure and y0 is the

400 semi-height of the enclosure. It is observed that the dimen-

401 sionless streaming velocities calculated from the present

402 model are in good agreement with that predicted by Aktas

403 and Farouk.1 Figure 5(a) indicates the existence of two

404 different vortical structures at x 3L/4; one formed at the405 vicinity of the side wall (inner streaming) while the other is

406 formed in the bulk fluid (outer streaming). The height of the407 circulatory flow structures (inner streaming) observed in the

408 vicinity of the horizontal walls is characterized by the thick-409 ness of the acoustic boundary layer. The streaming structures

410 seen in the middle section of the enclosure (outer streaming)

411 have larger sizes. These predicted streaming structures are

412 also in good agreement with the results reported by earlier413 studies.1,9 On an absolute scale, Aktas and Farouk1 reported

414 a maximum streaming velocity of approximately 0.06 m/s

415 with a maximum instantaneous velocity of 12 m/s in the pri-

416 mary oscillatory flow field in the enclosure, while the present

417 model slightly under predicts these velocities. The maximum

418 streaming velocity and the instantaneous velocity calculated

419 from the present model are 0.057 and 11.2 m/s, respectively.

420 This slight deviation is mainly due to the implicit nature of

421 the present numerical model (the model used by Aktas

422 and Farouk1 is explicit in nature and have less numerical423 diffusion).

424B. Acoustic streaming in sub- and supercritical fluids

425As described in Sec.I, acoustic streaming phenomena in

426ideal gases are well studied. However, the same in near-

427critical fluids is expected to be significantly different due to

428high density and compressibility and lower viscosity of the429fluid. To thoroughly investigate the formation of streaming

430structures in near-critical carbon dioxide and the transition

431from the ideal gas acoustic streaming phenomena, four cases

432are studied (cases 14) first, see Table II. These four cases

433correspond to carbon dioxide at states ranging from ideal

434gas-like to near-critical; excited at their resonant frequencies

435and at various amplitudes of vibration. At this stage the ini-

436tial temperature of the fluid is kept constant (Ti 308K) and437the undisturbed pressure is varied.438The fluid in case 4 is in the supercritical state and this439case is considered as the base case for the rest of the paper.

FIG. 5. (Color online) Variation of the (a) x component of the streaming ve-

locity at x 3L/4 and (b)y component of the streaming velocity at x L/2compared with Aktas and Farouk (Ref.1).





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440 In our experiments, acoustic waves are generated inside the441 cylindrical shaped resonator filled with supercritical carbon442 dioxide at this state (7.6 MPa and 308 K) with a speaker443 power of 10.0 W. The vibrational amplitude (A0) of the mov-444 ing wall corresponding to the acoustic impedance (qcs) of445 the fluid and the input power at case 4 is 4.0lm (Beranek34).446 For the acoustic wave measurements in cases 13, the input447 power is kept constant (10.0 W) and the vibrational ampli-448 tude is calculated using the same model.34 For the same449 electro-mechanical variables of the speaker and input power,450 these amplitudes can also be approximated using the ampli-451 tude for case 4 and the following equation:

A0;n A0;iqiqn

fi

fn

2: (9)

452 Equation(9) is based on a simple force balance on the453 vibrating wall where the subscript i refers to the values

454corresponding to the base case (case 4) and the subscript n455refers to the values corresponding to the nth case.456Transient variation of the pressure (gage) at the mid-457point of the end wall (pressure anti-node) of the cylindrical458resonator at a pseudo-steady state for cases 14 are shown in459Fig.6. The time-scale is normalized with the acoustic time460period (s 1=f). Both the computed and measured pressure461transients are shown in this figure. A monotonic increase of462the maximum pressure amplitude is observed with the463increase in operating pressure. The maximum pressure464amplitude reaches approximately 5.0 kPa at supercritical465state (case 4), while it is around 0.6 kPa at atmospheric pres-466sure (case 1). This monotonic increase is mainly due to the467increase in density of the fluid with pressure. Also at atmos-468pheric pressure (case 1), the pressure wave form is much469sharper and shock wave-type profile is observed [Fig. 6(a)],470while at higher sub-critical pressures (cases 2 and 3), the471pressure profile is near-sinusoidal [Figs. 6(b) and 6(c)]. At472the supercritical state (case 4), the pressure profile again473becomes sharper and a shock-wave type [Fig. 6(d)]. This474behavior can be explained by the isothermal compressibility,475

b 1=q@q=@pTof the fluid. At atmospheric pressure the 476isothermal compressibility is high. However the compressi-477bility quickly decreases with the increase of operating478pressure until the supercritical state is reached, where the479compressibility starts to increase again.480For all the cases discussed (cases 14), the computed481and the measured transient pressure distribution compares

TABLE II. List of cases simulated for investigation of acoustic streaming

formation in sub- and supercritical fluids.

Case no. Ti(K) pi(MPa) q/qc f(Hz) xmax(lm) dv(lm) Pr

1 308 0.1 0.0037 545.3 346.7 72.2 0.760

2 308 2.0 0.0812 517.6 17.5 15.9 0.853

3 308 4.0 0.1854 484.2 8.8 11.1 1.02

4 308 7.6 0.6224 391.0 4.0 8.0 3.53

FIG. 6. (Color online) Transient varia-

tion of computed and measured pres-

sure (gage) at the end wall of the

cylindrical resonator at a pseudo-

steady state for (a) p 0.1MPa, (b)p 2.0MPa, (c) p 4.0MPa, and (d)p 7.6 MPa and for four acoustic

cycles.





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482 well with the predictions. For the cases showing shock483 wave-type profiles (cases 1 and 4), existence of a slight484 variation between the measured and computed pressure485 amplitude is observed at the troughs and the trenches of the486 wave form. This is mainly due to the high experimental487 noise in the system when the temporal pressure gradient is488 high (i.e., at troughs and trenches of the wave form).489 For all these cases (cases 14), the streaming flow490 patterns are observed in the shape of regular structures.491 Four outer steaming structures are observed in all these492 cases. Variation of axial component of the non-dimensional493 streaming velocity at z 3L/4 for cases 14 is shown in494 Fig. 7(a). Similar to the maximum pressure amplitude, a495 monotonic increase in the streaming velocities with increas-496 ing pressure is observed in this figure. The difference in497 the streaming structures are also noticed from this figure.498 At atmospheric pressure (case 1), existence of the anti-499 clockwise rotating inner streaming structure gives rise to the500 negative velocity near the side wall (r/r0 1). However,501 with the increase of operating pressure, the inner streaming502 structures are dissolved while the outer streaming structures

503expand and cover the entire domain. At a supercritical state504(case 4), the streaming structures are characterized by rela-505tively high axial streaming velocity near the wall and a flat506plug-flow like profile in the rest of the domain as compared507to a parabolic velocity profile at the atmospheric pressure508(case 1). Variation of the radial component of the non-509dimensional streaming velocity at z L/2 for cases 14 is510shown in Fig. 7(b). A strong radial velocity from the bulk511fluid to the wall in the supercritical state (case 4) is observed512from this figure. This strong velocity corresponds to a jet513like flow at z L/2. This behavior is highly desired for514mixing applications, where the jet can efficiently carry515the bulk fluid to the wall boundary layer.

516C. Effect of thermo-physical property variations517across the pseudo-critical state (q=q

c51)

518The effect of the thermo-physical property variations519across the pseudo-critical state on the mechanically driven520acoustic waves and acoustic streaming formation is investi-521gated in this section. Three additional cases (cases 57)522along with case 4 are considered for this study. For these

523four cases, the operating pressure is supercritical and is kept524constant at p 7.6 MPa. The pseudo-critical state (i.e.,525where q=qc 1) corresponding to this pressure is at526305.43 K. Hence, the pseudo-critical state is approached by527reducing the operating temperature. The four cases corre-528spond to fluids ranging from gas-like (i.e., q=qc< 1:0) to529liquid-like (i.e.,q=qc > 1:0) properties. TableIII below lists530the details of the cases studied. Here, the fluid corresponding531to cases 4 and 5 are gas-like with relatively low density, ther-532mal conductivity, and acoustic speed, while that correspond-533ing to case 7 is liquid-like with higher density, thermal534conductivity and acoustic speed. The fluid corresponding to535case 6 is very close to the pseudo-critical state.536Temporal variation of the computed and measured537pressures (gage) at the end wall (pressure anti-node) of the538cylindrical resonator for cases 47 are shown in Fig. 8. Two539specific phenomena are observedfirst, the pseudo-critical540state has a strong effect on the maximum pressure amplitude.541It is observed that, as the pseudo-critical state (i.e.,

542T 305.43 K at p 7.6 MPa) is approached, the maximum543pressure amplitude increases. For the cases studied (cases54447), the maximum pressure amplitude is observed for case5456 (T 305.5 K) closest to the pseudo-critical state. This is546due to the high compressibility and density of the fluid as it547approaches the pseudo-critical state. The maximum pressure

548amplitude decreases as the thermodynamic state of the fluid549moves away from the pseudo-critical state (case 7,

550T 305.1 K). At this state, density of the fluid is very high,

FIG. 7. Spatial variation of the (a) axial component of the streaming veloc-

ity atz 3L/4 and (b) radial component of the streaming velocity at z L/2for different operating pressures (cases 14).

TABLE III. List of cases simulated for investigation of the effect of pseudo-

critical states on acoustic streaming.


4 308 7.6 0.6224 391.0 4.0 8.0 3.53

5 305.8 7.6 0.8 352.6 3.81 8.1 11.26

6 305.5 7.6 0.937 325.6 3.83 8.3 26.60

7 305.1 7.6 1.2 373.1 2.26 7.9 13.66





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551 but the compressibility is low. The second phenomenon552 observed from the transient pressure is related to the pressure553 wave form. The pressure wave forms corresponding to all554 these cases (cases 47) show a shock wave-type profile with555 a sharp rise and long tail in general. However, the pressure556 wave form corresponding to the case closest to the pseudo-557 critical state (case 6, T 305.5 K) exhibits a slightly differ-558 ent profile with a non-linear distortion and a shift in mean559 value. Thisdistortionof the pressure transient is mainly due560 to the strong variation of acoustic speed of the near-pseudo-561 critical fluid at this case (case 6). At near-pseudo-critical

562 states close to the critical point, the acoustic speed of the563 fluid exhibits relatively large variations with changing pres-564 sure and temperature. Hence, in an oscillating pressure (and565 hence temperature) field such as this, the acoustic speed of566 the fluid also oscillates to a great extent which in turn affects567 resonance and hence the standing wave field. The measured568 pressure (see Fig.8symbols) also shows similardistortion569 of the wave form discussed above. High frequency distur-570 bances are observed from the measurements in the

571supercritical state (cases 47). Similar disturbances were

572also reported in earlier experimental studies in supercritical

573fluids.25,36 These are caused mainly due to the high compres-

574sibility of the medium and non-uniform heating of the super-575critical chamber.576The quasi-steady (time-averaged) streamlines in the res-

577onator corresponding to cases 47 are shown in Fig. 9.

578Acoustic streaming structures are observed for all these

579cases (cases 47). For the cases relatively far from the

580pseudo-critical state (i.e., cases 4, 5, and 7), four outer

581steaming structures are observed. This behavior is in accord-

582ance with that observed in Sec. VI B . However, at the near-

583pseudo-critical state (case 6), the streaming structures do not

584exhibit the regular pattern. It is observed that the existence

585of the distorted pressure field gives rise to eight outer

586streaming cells in the system instead of four. Out of these

587eight, four of the cells are large and prominent in shape.

588While, the other four are relatively small in size and are

589derived from the variation of the thermo-physical properties

590(e.g., acoustic speed of the medium) due to the wave field591distortion.

592The variation of the axial component of the non-593dimensional streaming velocity at z 3L/4 for cases 47 is594shown in Fig. 10(a). For the cases relatively far from the

595pseudo-critical state (i.e., cases 4, 5, and 7), the streaming

596velocities also increase monotonically as the corresponding

597pseudo-critical state is approachedsimilar to the maximum

598pressure amplitude. The streaming velocity distribution at

599these states shows a flat plug-like profile in the bulk fluid

600with a sharp gradient near the wall. This sharp gradient indi-

601cates a very thin acoustic boundary layer (dv) in the fluid.

602This is also confirmed by the analytical calculation of the

603acoustic boundary layer (8.0lm) as presented in TableIII.604The plug-like profile (similar to a turbulent flow profile) is

605due to the small dynamic viscosity of the supercritical me-

606dium. However, the near-pseudo-critical state (case 6,

607T 305.5 K) exhibits none of these trends. The streaming608velocity gradient is relatively small near the wall and the

609streaming velocity distribution in the bulk fluid exhibits a

610parabolic profile (similar to the profile at atmospheric pres-

611sure, case 1). Although analytical calculation ofdv provides612a similarly thin boundary layer (8.3lm), the oscillating

FIG. 8. (Color online) Transient variation of computed and measured pressure

(gage) at the end wall of the cylindrical resonator at a pseudo-steady statefor different operating temperatures at p 7.6MPa (cases 47). Measuredpressures are shown with symbols: T 308.0K; T 305.8K; T 305.5K, and T 305.1 K.

FIG. 9. Cycle averaged streamlines

(acoustic streaming) in the cylindrical

resonator at a pseudo-steady state for

(a) T 308.0 K (case 4), (b)T 305.8K (case 5), (c) T 305.5K(case 6), and (d) T 305.1K (case 7)for an operating pressure of 7.6M Pa.





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613 thermo-physical properties of the near-pseudo-critical fluid

614 hinders resonance in a standing wave field and the formation615 of a pseudo-steady boundary layer.616 Variation of the radial component of the non-

617 dimensional streaming velocity at z L/2 for cases 47 is618 shown in Fig.10(b). Similar to case 4 (Sec. VI B), a strong

619 radial velocity from the bulk fluid to the wall in the super-

620 critical states (cases 47) is observed. The jetlike flow due

621 to the radial velocity strengthens as the pseudo-critical state622 is approached and it becomes maximum for the near-pseudo-623 critical state (case 6).

624 D. Effect of operating pressure

625 As observed in Figs.1and 2, the significant variation in

626 thermo-physical properties near the pseudo-critical states

627 become milder as the pressure is increased (from the critical

628 pressure). The effect of this transition in thermo-physical

629 properties on acoustic streaming phenomena in supercritical630 fluids is investigated in this section. Two different operating

631pressures (8.0 and 8.5 MPa) are considered for this study. To

632understand the effect of operating pressure on the near-

633pseudo-critical acoustic streaming formation, three cases

634(q=qc< 1:0, q=qc 1:0; and q=qc> 1:0) are considered635for each of these pressures. Table IVbelow lists the details636of the cases studied.637Temporal variation of the computed and measured pres-

638sures (gage) at the end wall (pressure anti-node) of the cylin-

639drical resonator for cases 810 (p 8.0 MPa) are shown in640Fig. 11(a). The same features of the pressure wave form

641(monotonic increase as pseudo-critical state is approached,

642sharp and shock wave-type profile, etc.) are also observed in

643this pressure. However, for the near-pseudo-critical state

644(case 9,T 308.0 K), the non-linear variation in the pressure645wave form is not as prominent as observed forp 7.6 MPa646(case 6). The experimental measurements for this case also

647confirm this observation. Temporal variation of the com-

648puted and measured pressures (gage) at the end wall

649(pressure anti-node) of the cylindrical resonator for cases

6501113 (p 8.5 MPa) are shown in Fig.11(b). It is observed651that, the near-pseudo-critical effect on the pressure wave

652form (i.e., the non-linear variation) is further diminished at653this pressure.654This behavior of the pressure (acoustic) wave can be

655explained by the near-critical thermo-physical property var-656iations. As the operating pressure is increased, the thermody-

657namic state of the fluid is moved far away from the critical

658point and the thermo-physical property variations (including659the acoustic speed) at these higher pressures (e.g., p 8.0660and 8.5 MPa) are not very strong. Hence, the acoustic speed661variation of the fluid (as discussed in Sec. II) in the resonator

662at the near-pseudo-critical state becomes negligible and as a663consequence of that, the pressure wave form becomes simi-

664lar to that observed at the cases far from the pseudo-critical665states.666The quasi-steady (time-averaged) streamlines in the

667resonator corresponding to the near-pseudo-critical cases at

668two different operating pressures (cases 9 and 12) are shown

669in Fig.12. It is observed that for the cases far from the corre-

670sponding pseudo-critical states (i.e., cases 8, 10, 11, and 13)

671the operating pressure has a negligible effect on the stream-

672ing structures. The regular outer streaming structures are

673observed for these cases (not shown in Fig. 12). However,

674for the near-pseudo-critical states (cases 9 and 12), the addi-

675tional streaming cells (corresponding to the distortion of

676the standing wave field) are not significant at this elevated

677pressures. At p 8.0 MPa, the additional streaming cell can

FIG. 10. (Color online) Spatial variation of the (a) axial component of thestreaming velocity atz 3L/4 and (b) radial component of the streaming ve-locity at z L/2 for different operating temperatures at p 7.6MPa (cases47).

TABLE IV. List of cases simulated for investigation of the effect of operat-

ing pressure on acoustic streaming.


8 307 8 1.2 401.1 1.97 7.6 9.58

9 308 8 0.933 360.9 3.13 7.9 11.99

10 308.75 8 0.8 373.0 3.42 7.8 6.95

11 309.3 8.5 1.2 422.8 1.78 7.4 7.13

12 311.1 8.5 0.936 384.1 2.76 7.7 7.57

13 312.5 8.5 0.8 392.5 3.10 7.7 4.95





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678 be noticed and is almost merged with the regular outer679 streaming structure [Fig. 12(a)]. At p 8.5 MPa, the addi-680 tional structure is not noticed at all.681 The variation of the axial component of the non-682 dimensional streaming velocity atz 3L/4 corresponding to683 the cases at two different operating pressures (cases 813) is684 shown in Fig.13. It is observed that the operating pressure685 has a deteriorating effect on the streaming velocity. The686 maximum amplitude of the non-dimensional streaming687 velocity decreases with increasing pressure. Although the688 additional streaming structures are not observed in the near-689 pseudo-critical states (cases 9 and 12), the distribution of the690 axial streaming velocity remains parabolic. A slight differ-691 ence between the axial streaming velocities in the bulk fluid692 (along the axis of the resonator) for the cases in the gas-like693 (q=qc < 1:0) and liquid-like (q=qc < 1:0) property regimes694 are observed at near-critical pressure [Fig.10(a)]. With the

695increase in operating pressure, this difference is diminished696(Fig.13).697The observations from Figs. 10(a) and 13are summar-698ized and shown in Fig. 14. Figure14 shows the variation of699the axial streaming velocity (non-dimensional) along the700axis of the resonator with reduced density for the three dif-701ferent operating pressures studied (p 7.6, 8.0, and7028.5 MPa). The reduced density axis represents the proximity703of the thermodynamic state to the pseudo-critical state

704(q=qc 1:0). The dotted lines represent the best fittedFIG. 11. (Color online) Transient variation of computed and measured pres-sure (gage) at the end wall of the cylindrical resonator at a pseudo-steady

state for different operating temperatures at (a) p 8.0MPa (cases 810).[Measured pressures are shown with symbols: T 308.75K; T 308.0K, and T 307.0 K] and (b) p 8.5MPa (cases 1113).[Measured pressures are shown with symbols: T 312. 5 K ; T 311.1 K, and T 309.3 K].

FIG. 12. Cycle averaged flow-field (acoustic streaming) in the cylindricalresonator at a pseudo-steady state for (a) p 8.0MPa, T 308.0 K (case 9)and (b)p 8.5MPa,T 311.1K (case 12).

FIG. 13. (Color online) Spatial variation of the axial component of the

streaming velocity at z 3L/4 for different operating temperatures atp 8.0MPa (cases 810) andp 8.5 MPa (cases 1113).





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705 curves for the data points. Parameters for the curves are pro-

706 vided in TableV. It is observed that, the streaming velocity

707 in the bulk fluid (i.e., along the axis of the resonator) is maxi-

708 mized at the pseudo-critical point. At pressures close to the

709 critical pressure (e.g., p 7.6 MPa), the axial streaming710 velocity is slightly higher in the gas-like property regime

711 (q=qc < 1:0) than that in the liquid-like property regime712 (q=qc > 1:0). This variation is mainly due to the reduced713 viscosity and higher compressibility of the gas-like fluid.

714 With the increase in pressure, the thermo-physical property715 variations become weak and this trend is barely noticed.

716 E. Thermal effects due to acoustic streaming

717 The oscillatory pressure and the streaming flow-field (as

718 discussed in the previous sections0 ) has a significant effect on

719 the temperature field in the resonator. At sub-critical pres-

720 sures (cases 13), a cold zone is observed in the temperature

721 contours roughly at the location of the pressure node (z/L

722 0.5), while two hot zones are observed at the pressure723 anti-nodes (z/L 0, 1). However, for the supercritical state,724 the cold zone almost disappears. The phenomena observed

725 in the sub-critical regime has been reported in earlier stud-

726 ies37 and is utilized in the development of thermoacoustic

727 refrigerators.38 The disappearance of the cold zone in the

728 supercritical state is due to the high thermal conductivity (kf)729 and Prandtl number (Pr) coupled with the high isothermal730 compressibility of the fluid at this state.

731The thermal effect due to acoustic streaming in near-

732critical fluids is summarized in Fig.15. The normalized max-

733imum and minimum temperatures along the cylindrical axis

734in a standing wave resonator as a function of the reduced735density of the near-critical carbon dioxide are shown in these

736figures. It is observed that the cases with gas-like properties737(i.e.,q=qc < 1:0) exhibits a strong temperature difference in738the resonatorwith hot zones near the pressure anti-nodes739and a cold zone near the pressure node. This temperature

740difference is rather weak for the near-pseudo-critical state

741(case 6) due to very high specific heat of the fluid. The

742temperature difference is slightly larger in the case with

743liquid-like properties (i.e., q=qc > 1:0). This is due to the744relatively high thermal conductivity, but lower specific heat

745as compared to the near-critical case. At elevated pressures,746the temperature field for the cases simulated (cases 813)

747also show similar trend. But this trend is relaxed as the oper-

748ating pressure is increased. At the highest operating pressure

749(p 8.5 MPa), the temperature difference is almost the same750for the three cases studied.

751VII. SUMMARY AND CONCLUSIONS

752Mechanically driven resonant acoustic waves in near-

753critical supercritical carbon dioxide are investigated in this

754chapter. The formation of acoustic (pressure) waves, acoustic-

755viscous boundary layer interactions, and associated flows in a

756cylindrical resonator are numerically studied by solving the

757unsteady, compressible NavierStokes equations in an axi-

758symmetricx-rcoordinate system. The acoustic field in the en-

759closure is created due to the harmonic vibration of the end

760wall. The effects of the pseudo-critical state (q=qc 1:0 A) and761operating pressure on the acoustic field and the formed flow

762structures are determined by utilizing a highly accurate nu-

763merical scheme. Acoustic waves generated by an electro-764mechanical driver in a cylindrical resonator filled with super-

765critical carbon dioxide are measured using a fast-response766pressure field microphone. The experimental measurements

FIG. 14. (Color online) Variation of non-dimensional axial streaming veloc-

ity in the bulk fluid (at z 3 L/4and r 0) with reduced densityq=qc atdifferent isobars.

TABLE V. Parameters for best fitted curve of the data presented in Fig.14.

just=uRj c0c1q=qc c2q=qc2 c3q=qc

3

c0 c1 c2 c3

p 7.6 MPa 59.03 225.80 281.10 111.10

p 8.0MPa 29. 80 64.02 32.19 -

p 8.5MPa 23.48 50.17 25.14 -FIG. 15. (Color online) Variation of non-dimensional maximum (solid lines)

and minimum (dashed lines) temperatures in the bulk fluid (along the reso-

nator axis) with reduced densityq=qcat different isobars.





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767 are compared with results from the numerical simulations768 with accurately developed boundary conditions.769 Several interesting phenomena are observed from the

770 numerical simulations and confirmed by the experimental

771 measurements. The observed primary oscillatory and

772 secondary steady flow fields in the supercritical fluid me-

773 dium demonstrate significant effects of the thermodynamic

774 state (sub-critical/supercritical/pseudo-critical). The strength

775 of the acoustic wave field is found to be in direct relation

776 with the thermodynamic state (near-pseudo-critical/far from

777 the pseudo-critical state). It is observed that due to the strong

778 thermo-physical property variations, maximum pressure am-

779 plitude in the standing wave field increases as the thermody-

780 namic state of the fluid approaches the corresponding781 pseudo-critical state. In the near-pseudo-critical state, the

782 pressure wave form exhibits a non-linear and distorted783 profile. In a standing wave field with near-critical fluid, the

784 acoustic speed of the fluid also oscillates to a great extent

785 which in turn affects resonance and hence the formation of

786 acoustic streaming structures. Far from the pseudo-critical

787 states, the streaming structure exhibits four counter rotating

788 cells with a jet like flow-field along the semi-length of the789 resonator (at the pressure node)a typical Rayleigh

790 streaming like behavior. While near the pseudo-critical state,

791 irregular streaming structures consisting of eight outer

792 streaming cells in the resonator are observed. The evolved

793 flow structures are also dependent on the operating pressure.

794 The irregular streaming patterns are observed mainly for

795 near-pseudo-critical states at operating pressures close to the

796 critical pressure (pc 7.377 MPa). However, these structures797 quickly re-orients to the regular streaming patterns (four

798 outer streaming cells) with the increase of operating

799 pressureas the singularity effect of the critical point dimin-

800 ishes. The acoustic streaming phenomena observed numeri-

801 cally and the near-critical standing wave field measurements

802 in this study can be utilized for mixing applications in super-

803 critical fluid medium, especially for enhancing the transport804 characteristics in supercritical fluid extraction processes.805

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Experimental and numerical investigations of resonant acoustic waves in near-critical carbon dioxide

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