Heat transfer processes in parallel-plate heat exchangers of thermoacoustic devices numerical and experimental approaches

Transcription

1 Heat transfer processes in parallel-plate heat echangers of thermoacoustic deices numerical and eperimental approaches Artur J Jaworski,* and Antonio Piccolo Department of Engineering, Uniersit of Leicester LE 7RH, United Kingdom Department of Ciil Engineering, Uniersit of Messina, Contrada di Dio 9866 S. Agata (Messina), Ital Abstract This paper addresses the issues of heat transfer in oscillator flow conditions, which are tpicall found in thermoacoustic deices. The analsis presented concerns processes taking place in the indiidual channels of the parallel-plate heat echangers (HX), and is a miture of eperimental and numerical approaches. In the eperimental part, the paper describes the design of eperimental apparatus to stud the thermal-fluid processes controlling heat transfer in thermoacoustic heat echangers on the micro-scale of the indiidual channels. Planar Laser Induced Fluorescence (PLIF) and Particle Image Velocimetr (PIV) techniques are applied to obtain spatiall and temporall resoled temperature and elocit fields within the HX channels. The temperature fields allow obtaining the local and global, phase-dependent heat transfer rates and Nusselt numbers, and their dependence on the Renolds number of the oscillating flow. The numerical part of the paper deals with the implementation of CFD modelling capabilities to capture the phsics of thermal-fluid processes in the micro-scale and to alidate the models against the eperimental data. A two-dimensional low Mach number computational model is implemented to analze the time-aeraged temperature field and heat transfer rates in a representatie domain of the HXs. These are deried b integrating the thermoacoustic equations of the standard linear theor into a numerical calculus scheme based on the energ balance. The comparisons between the eperimental and numerical results in terms of temperature and heat transfer distributions suggest that the optimal performance of heat echangers can be achieed when the gas displacement amplitude is close to the length of hot and cold heat echanger. Heat transfer coefficients from the gas-side can be predicted with a confidence of about 4% at moderate acoustic Renolds numbers. Kewords: Heat Transfer; Heat Echangers; Thermoacoustic Engines and Coolers; Oscillator Flow, Eperimental, Numerical * Corresponding author: Tel: 44()6 3 33; Fa: 44()6 5 55; a.jaworski@le.ac.uk

2 Nomenclature a speed of sound (m s - ) Greek smbols A surface area (m ) β thermal epansion coefficient (K - ) c P isobaric specific heat (J kg - K - ) γ ratio of isobaric to isochoric specific heat d dimension of the square side of the cross δ penetration depth, m section of the resonator Δz HX length along the z direction e& energ flu densit (Wm - ) η shear iscosit coefficient (N m - s - ) e& time aeraged energ flu densit (Wm - ) θ dimensionless temperature, (T-T w )/(T H -T C ) f spatiall aeraged h function, gas thermal diffusiit (m s - ) resonance frequenc (Hz) λ waelength of the sound wae (m) h thermoiscous function, kinematic iscosit (m s - ) specific enthalp (J m -3 ), ξ generic acoustic ariable heat transfer coefficient (Wm K - ) ξ C displacement amplitude at the joint j imaginar unit and centre of the channel (m) k wae number (m - ), ρ densit of the gas (kg m -3 ) thermoiscous function (m) σ component of the iscosit stress K gas thermal conductiit (Wm - K - ) tensor (N m - s - ) l half fin thickness (m) τ component of the iscosit stress L heat echanger length (m) tensor (N m - s - ) Nu Nusselt number Φ phase of the acoustic ccle (deg, rad) p acoustic pressure amplitude (Pa) angular frequenc (rad s - ) p C pressure amplitude at the joint and Ω blockage ratio centre of the channel (Pa) P A pressure amplitude at a pressure Subscripts antinode (Pa) mean, time aeraged Pr Prandtl number first order acoustic ariable q time-aeraged heat flu densit (W m - ) A acoustic amplitude at a pressure antinode Re Renolds number HX referring to the HX t time (s) P isobaric T instantaneous fluid temperature (K) Res resonator T amplitude of oscillating temperature (K) X longitudinal, -component T w wall (fin) temperature (K) Y transerse, -component T H temperature of the hot fin (K) Z along the z direction T C temperature of the cold fin (K) thermal T-T thermocouples iscous u specific internal energ (J m -3 ) amplitude of longitudinal elocit (m s - ) amplitude of transerse elocit (m s - ) C s elocit amplitude at the joint and centre of the channel (m s - ) aial coordinate/direction (m) HXs coordinate (m) transerse coordinate/direction (m) half fin spacing (m)

3 . Introduction Thermoacoustic technologies are concerned with deeloping new concepts of engines, coolers and heat pumps which operate on the basis of a range of thermoacoustic effects. These are broadl understood as energ transfer between a compressible fluid and solid boundar in the presence of an acoustic wae. The correct phasing between the acousticall-induced fluid displacement and its compression or epansion, coupled with heat transfer processes between the solid and fluid, will lead to the implementation of Stirling-like thermodnamic ccles which are of practical engineering importance. A number of practical thermoacoustic deices, both standing- and traelling-wae, hae alread been demonstrated [, ]. There are significant adantages of utilising thermoacoustic technologies, in certain areas of application. Firstl, as there are no moing components, the thermoacoustic deices can be around one order of magnitude cheaper than man conentional ccles (e.g. tpical Stirling machine) and can offer longeit and low maintenance costs. In addition, as the thermodnamic ccle uses enironmentall benign noble/inert gases as the working medium, there are significant enironmental benefits of the technolog. Finall, thermoacoustic sstems scale er well and are ideal for utilising the low grade heat input (waste heat from industrial processes and solar energ and other forms of energ haresting) which offers new opportunities for energ conseration. A tpical thermoacoustic deice consists of (i) an acoustic network, (ii) an electro-acoustic transducer, (iii) a porous solid medium (namel a regenerator in traelling-wae sstems [3] or a stack in standing-wae sstems [4]) and (i) at least a pair of heat echangers (HXs) [5]. The stack/regenerator is the component where the desired heat/sound energ conersion takes place. Hot and cold heat echangers (CHX and HHX, respectiel), placed in close proimit of both ends of the stack/regenerator, absorb (or suppl) energ from (or to) its ends thus enabling heat communication with eternal heat sources and sinks. It is worth noting at this point that compared to the well-known tpes of HXs for stead flows HXs in the thermoacoustic contet proide a range of new challenges. This is due to the fact that the flow is oscillator. While the eisting knowledge for stead flow arrangements is of little practical alue [6,7], the releant inestigations in the oscillator flow are er scarce [8-]. Therefore, this work is drien b the need for a better understanding and quantitatie description of heat transfer under oscillator flow conditions in thermoacoustic HXs with the aim of improing the oerall sstem performance. The research presented in this paper had two tpes of actiities carried out in tandem. Firstl, eperimental facilities had to be deeloped for careful inestigations of the thermal-fluid processes within the indiidual channels of parallel plate HXs. Secondl, CFD modelling capabilities had to be deeloped in order to capture the phsics of thermal-fluid processes and to alidate the models against the eperimental data. In order to simplif as much as possible the eperimental implementation and the numerical modelling the phsical arrangement of the HXs was simplified compared to real thermoacoustic deices. In particular, onl a pair of mock-up HXs is studied, with no stack/regenerator installed between them. These parallel-plate tpe HXs are arranged side b side with no gap between them. The eperimental actiities inoled two-dimensional temperature and elocit field measurements using Planar Laser Induced Fluorescence (PLIF) and Particle Image Velocimetr (PIV) techniques, respectiel, to obtain spatiall and temporall resoled temperature and elocit fields within the thermoacoustic HX samples. On the basis of recorded temperature fields, the eperimental data were processed to obtain the local and global, phase-dependent heat transfer 3

4 rates and Nusselt numbers, and their dependence on the Renolds number of the oscillating flow. The CFD modelling entailed the deelopment of a two-dimensional low Mach number computational model to analze the time-aeraged temperature field and heat transfer rates in a representatie domain of the HXs. These were generated b integrating the thermoacoustic equations of the standard linear theor into an energ balance-based numerical calculus scheme. The model accounts for hdrodnamic heat transport along the transerse direction normal to the fin surfaces, temperature dependent thermophsical gas/solid parameters and flow losses. The comparison between simulation results and eperiments is based on the phase-aeraged temperature fields and on the integrated data of heat flu and Nusselt number. This work is the first attempt of its kind to alidate the heat transfer prediction codes in the contet of a thermoacoustic deice b the detailed in-situ measurements of temperature fields obtained for an indiidual pore/channel of the heat echanger assembl. Therefore, it is hoped that the results of this work constitute an important scientific and technological foundation for future improements of the performance of thermoacoustic engines where this is dependent on heat communication.. Eperimental setup The general schematic of the eperimental apparatus and the PLIF instrumentation is shown in Fig.. The eperimental apparatus consists of a 7.4 m long resonator of a square internal cross section. It is joined with a cubical loudspeaker bo with a side length of.6 m through a.3 m long transition part, which increases the cross section from 34 mm 34 mm on its resonator end to 5 mm 5 mm on the bo end. The resonator is filled with nitrogen at atmospheric pressure and room temperature. A subwoofer tpe loudspeaker is used to generate a standing wae of one quarter waelength mode in the resonator, at a frequenc (f) of 3. Hz. Figure. Schematic drawing of the eperimental apparatus with a PLIF sstem A sinusoidal signal from a function generator (TTi TGA) is sent to a power amplifier, whose output is then connected to the loudspeaker. The acoustic oscillation amplitude is aried b controlling the amplitude of the sinusoidal signal from the function generator. The acoustic pressure is monitored b a microphone (B&K Model 436) flushmounted on the end plate of the resonator. The pressure signal from the microphone is also used as a phase reference for Planar Laser Induced Fluorescence (PLIF) imaging. 4

5 A pair of mock-up HXs was placed in the resonator, about 4.6 m (.7 of the waelength) from the closed end of the resonator. The configuration of the HX pair is schematicall shown in Fig. a, with a photograph of the actual implementation shown in Fig. b. Each HX consists of fie actie fins and eight dumm aluminium fins to fill up the whole cross-section of the resonator. Each hot HX fin is made out of a brass plate with a cable heater embedded in a prepared channel (Fig. c). High-temperature epo was applied to fill up the channel and form a flat surface. Each cold HX fin is also made out of a brass plate, with a wide meandering channel prepared for water circulation (Fig. c). Water inlet and outlet are connected to the channel ends ia steel tubing. The channel was coered with a thin brass sheet, which was soldered on top; the whole assembl was further machined to obtain a flat surface. All fins are 3. mm thick (l) and 35 mm long (L) in the direction of the oscillating flow, with a span-wise width (Δz) of 3 mm. Figure. Schematic of the configuration of the heat echangers (a); photograph of the phsical implementation (b); photograph of machined hot (left) and cold (right) heat echanger fins (c) Ceramic spacers (6 mm mm 5 mm) were used in the front, as well as in the back (not shown) to maintain the channel height ( ) at 6. mm, and also to preent the channel from seere deformation due to the thermal stresses in the fins. These ceramic blocks and HX fins were strengthened b supporting pillars. Onl one channel, in the centre of the resonator, was left unobstructed for PLIF imaging. The surface temperature of the HX fins that form the channel under inestigation were monitored b using twele K- tpe thermocouples (Fig. a). The are mounted in the centre of the fins in the span-wise direction. Four thermocouples (i.e. T, T4, T9 and T ) are 3 mm awa from the joint between hot and cold fins. Another four are in the middle of the cold ( T and T5 ) and hot ( T8 and T ) fins. The last four ( T3, T6, T7 and T ) are placed 3 mm awa from the far edge of the cold and hot fins. The heating on the hot fins was controlled b a PIDmode temperature controller, which maintained a preset temperature alue (e.g. C) at the monitored point ( T7 ) within the range of ± C. Due to the thermal contact on the joints between hot and cold fins in the present arrangement, unwanted heat conduction was introduced, which makes the surface temperatures on the hot and cold fins decrease from 5

6 left to right. The maimum temperature difference between T7 and T9 or between T and T can be up to C. It is known from tests that the surface temperature on the fins along the span-wise direction was nearl constant. Before the temperature field measurement, the flow elocit fields were obtained with Particle Image Velocimetr (PIV) techniques. The elocit amplitude ( C ), at the joint and centre of the channel, and the corresponding displacement amplitude of gas parcel (ξ C = C /, where is the angular frequenc) were aried from.3 m/s to 3.84 m/s and from 6 mm to 46 mm respectiel. The corresponding acoustic Renolds numbers based on the thickness (l) of the fin (Re = C l/) being the kinematic iscosit aried from 5 to 7. Temperature field measurements were then carried out at the same oscillation amplitude, b emploing a PLIF sstem (LaVision) as shown in Fig.. Acetone was used as the fluorescent tracer. UV light sheet at 66 nm is generated b a Nd:YAG laser. It enters the resonator perpendicularl to its ais through a quartz window of 3 mm thickness, is reflected b a UV mirror and becomes parallel to the resonator ais and normal to the surface of the heat echanger plates. The laser sheet plane was 33 mm awa from the front end of the heat echangers, rather than at the centre of the HXs (Fig. ), to aoid the effect of the surface mounted thermocouples on the flow and temperature fields. Two UV mirrors were used, to enable the inestigation of the temperature fields inside and outside of the channel. The laser could be moed along the resonator side wall b using an automated traerse sstem to use either of the two UV mirrors as necessar. The UV mirrors are placed 3 mm awa from the heat echanger not to disturb the flow and temperature fields around the heat echangers. Quartz windows are mounted on the resonator at the locations of the heat echangers and the mirrors for their higher transmission to UV and fluorescence emissions. Figure 3. Reference pressure, displacement and elocit ariation in an acoustic ccle and the inestigated phases. The gas displacement amplitude is 6mm. Acetone of optimal concentration was seeded into the flow b a purpose-designed seeding mechanism. PLIF images were captured b an intensified CCD camera. Images with effectie piels of 4 4 were obtained, giing a spatial resolution of 64μm per piel. Images were post-processed using LaVision DaVis 7. software package. Three iews were recorded with one net to another in the longitudinal direction to coer the area stretching beond the length of the channel in front of the hot/cold HX fins. Prior to the acquisition of eperimental measurement of temperature fields, additional calibration procedure was undertaken to establish the correlation between the fluorescence intensit 6

7 and the fluid temperature. Further information about the temperature measurement with PLIF techniques can be found in []. Measurements of temperature and elocit fields were carried out for phases within an acoustic ccle as illustrated in Fig. 3, which shows the measured reference pressure p C, the measured elocit C and the calculated displacement ξ C. The directions of the elocit and displacement are defined within the coordinate sstem gien in Fig.. For each phase, frames were taken to calculate the phase-aeraged temperature field. The increase of the number of frames beond proides negligible accurac gains in terms of the resulting temperature fields. 3. Computational model The numerical model used in this stud is the same as that applied in [], so a more snthetic description is gien here. For additional information the reader is directed to []. The periodicit of the HXs structure along the transerse direction allows calculations to be performed in a single channel of the HX assembl. The computational domain is further reduced b smmetr from half a gas duct to the solid surfaces of the coupled hot and cold fins which are treated as two isothermal surfaces respectiel at temperature T H and T C (see Fig. 4). Figure 4. Magnified iew of the computational domain (light gre area) The implementation of the numerical model relies on a number of assumptions: The problem is considered as twodimensional. The processes taking place are within a low Mach number regime, so that an acoustic ariable ξ can be epressed in the comple notation b the conentional first-order epansion { j ξ (, e t } ξ,, t) = ξ (, ) Re ), where t is the time, j the imaginar unit, the angular frequenc, ξ the ( (real) mean alue, ξ the (comple) amplitude of oscillation, and where Re{ } denotes the real part. The fluid is Newtonian and obes the ideal gas state equation. Temperature dependent thermo-phsical gas parameters are used in the model. HXs are significantl shorter than the acoustic waelength and acousticall non-intrusie so that the acoustic field can be approimated in the HX neighbourhood b a D lossless standing wae: p P = j () ρ A = PA cos( k s) = p sin ( k s) = j a 7

8 where p is the amplitude of the dnamic pressure, P A is the amplitude of the dnamic pressure at a pressure anti-node, is the amplitude of the longitudinal particle elocit, k is the wae number (k=π/λ), a is the sound elocit and s the distance of the HXs from the closed end of the resonator. Amplitudes of the oscillating temperature T, longitudinal elocit, transerse gradient of the longitudinal elocit /, transerse elocit and transerse gradient of the transerse elocit / inside a gas pore are described b the standard equations of the classical thermoacoustic theor. According to the assumption that the specific heats of the plate/fin materials are notabl greater than the isobaric specific heat of the gas c p (i.e. temperature oscillations in the solid are negligible), the following equations can be formulated [,6]: dp T T = ( h ) p [( h ) Pr ( )] h () ρ c ρ ( Pr) d P j dp = ( h ) (3) ρ d dp = ρδ d k [ ( γ ) f ]( k ) [ ( γ ) k ] ( f ) p = j ρa ( f ) (4) β f ( k ) f ( k ) ( k k ) T dp ρ ( Pr) ( f ) d j (5) = j ρ a [ ( γ ) f ]( h ) [ ( γ ) h ] ( f ) ( f ) p where h [( j) / δ ] [( j) / δ ] β f ( h ) f ( h ) ( h h ) T dp ρ ( Pr) ( f ) d j [( j) / δ ] [( j) / δ ] cosh cosh = h = (7) cosh cosh (6) k δ sinh [( j) / δ ] [( j) / δ ] conductiit coefficient). 8 δ [( j) / δ ] [( j) / δ ] = k = (8) j cosh j cosh f [( j) / δ ] [( j) / δ ] sinh [( j) / δ ] [( j) / δ ] tanh tanh = f = (9) and where Pr is the Prandtl number, β the thermal epansion coefficient, γ the ratio of isobaric to isochoric specific heat, ρ the mean densit, T the time-aeraged temperature, the shear iscosit coefficient) and δ = η / ρ the iscous penetration depth (η being δ = K / ρ c the thermal penetration depth (K being the gas thermal P

9 9 The longitudinal pressure gradient dp /d along the HX assembl is calculated b imposing continuit of olume flow rate at the entrance of the HXs ) ( f d dp Ω = ρ () where the blockage ratio Ω=A HX /A res =/(l/ ) describes the porosit of the HXs (A res being the cross sectional area of the resonator and A HX is the cross sectional area of the HXs open to gas flow). The gas temperature is goerned b the general equation of energ conseration [3]: = e& u t ρ ρ () where u is the specific internal energ and where e&, the energ flu densit, is defined as T K h = σ e ρ & () where h is the specific enthalp and σ the iscous stress tensor. Time aeraging Eq. () oer an acoustic ccle leads to = e& (3) where oer-bar means time-aeraged. Equation (3) is applied to each cell of the computational domain to impose the local energ balance. To accomplish this, a finite difference technique is emploed where the quantitatie results of standard linear theor are substituted for the and the components of the time-aeraged energ flu densit [3]. = π π π τ σ ρ π / / / ) ( dt T K dt dt h e & (4) = π π π σ τ ρ π / / / ) ( dt T K dt dt h e & (5) where = η η σ 3 3 (6) = = η η τ τ (7) = η η σ (8) the elocit gradients haing been neglected because of the order of δ /λ being smaller than the elocit gradients. Making use of the comple notation and retaining onl terms up to second order Eqs. (4) and (5) can be written as: { } T K T c q P = ~ Re ~ Re 3 ~ Re η η ρ (9) { } T K T c q P = ~ Re 3 ~ Re ~ Re η η ρ ()

10 where energ flu densities hae been conerted to heat flu densities ( e& q, e& q ) because in the short stack approimation the work flu contribution to the total energ flu is small []. The eplicit epressions of these equations are reported in []. The calculation of the stead-state two-dimensional time-aeraged temperature distribution is performed using a finite difference methodolog where temperature spatial gradients are discretized using first order nodal temperature differences. To this end, the computational domain is subdiided using a rectangular grid. In the direction the computation mesh size is tpicall.5 L while in the direction the computation mesh size is tpicall.. The imposed boundar conditions took into account: the smmetr on the nodal line at =; the continuit of temperature and transerse heat flues at the gas-solid interfaces (= ); and the anishing energ flu at the HX pore ends (=; =L) facing the duct (gas oscillations outside the HXs are adiabatic and thus the thermoacoustic effect anishes). Appling Eq. (3) in each cell of the computational grid, a sstem of quadratic algebraic equations with respect of the unknown ariable T is generated. The sstem is soled b a code deeloped b the authors in FORTRAN-9 language which eecutes the recursie Newton-Raphson method [4]. At each iteration the sstem of linear algebraic equations proiding the temperature corrections for the net step is soled using a LU decomposition with partial pioting and row interchanges matri factorization routine. The latter is taken from the LAPACK librar routines aailable online at [5], where details of the accurac, computational cost, etc. can be found. Once the time-aeraged temperature distribution is known, it can be substituted in Eqs. (9) and () to determine the energ flu distributions along the and directions in the gas. In the net section the code is alidated b comparing numerical predictions against eperimental measurements. 4. Results and discussion The temperature field measurements were performed in phases within an acoustic ccle. Here, the most characteristic temperature distributions at si phases are presented to discuss the heat transfer process in an acoustic ccle. Figure 5 illustrates the temperature distribution in these si phases, Φ ( ), Φ 6 (9 ), Φ 8 (6 ), Φ (98 ), Φ 6 (7 ) and Φ 8 (36 ), for a Renolds number equal to 5 (with the corresponding fluid displacement amplitude of 6 mm). At phase Φ, the fluid displacement reaches the maimum negatie alue, which indicates that the gas parcel moes the farthest to the left. The cold gas in the channel of the cold fin (referred as the cold channel thereafter) moes far into the channel of the hot fin (referred as the hot channel thereafter) and is heated b the plates. Some part of hot gas in the hot channel flows out of it and enters the open area behind the hot fins. It can be clearl seen that the gas in the top region behind the hot fins is heated up, due to the cumulatie effect of natural conection oer time. After this phase, the gas parcel starts to accelerate to moe to the right. At phase Φ 6, the cold gas heated b the hot fins partl flows out of the hot channel and enters into the cold channel. Hot gas entering from the open area behind the hot fins is dominant at the left end of hot channel. The gas parcel continues to moe to the right. At phase Φ 8, the hot gas has dominated the whole hot channel and partl penetrates into the cold channel. The cold gas in the cold channel partl moes into the open area behind the cold fins and mies with the colder gas in this region. It can be found that the natural conection effect also eists. At phase Φ, the gas parcel just starts to moe to the left. The cold gas in the cold channel graduall moes into the hot channel and, at the same time, the cold gas in the open area behind the cold fins partl flows into the cold channel. At phase Φ 6, the cold gas penetrates far into the hot channel and the hot gas in the hot channel also migrates

11 into the open area behind the hot fins. At phase Φ 8 the hot channel is occupied b the cold gas and the hot gas in the hot channel moes further to the left. After this phase, the gas parcel keeps moing to the left. Then another acoustic ccle repeats itself. Analogous processes can be identified at the higher Renolds number where the relatiel large elocit deelops a er thin thermal boundar laer. Figure 5. Temperature distributions for si selected phases and for Re=5. Red rectangles mark the heated plates and blue rectangles mark the cooled plates in the heat echanger assembl. The black dashed lines mark the position where the three iews (left, central and right) join to one another Figure 6. Time aeraged temperature distribution in half a gas channel generated b the computational model for Re=5.

12 For comparison, in Fig. 6 the time-aeraged temperature distribution generated b the computational model is shown for the same alue of the Renolds number. At the joint position and in the mid cannel regions (far from the fin surface) the temperature is nearl uniform oer a cross section (isotherms are ertical). In the direction awa from the joint towards both the cold and hot channel the isotherms become een more cured and practicall horizontal at the gassolid interface. Isotherms, howeer, become more densel packed when approaching the joint point and this trend is compatible with a growing transerse temperature gradient T / near this region. The joint inoles a great temperature gradient in both the longitudinal and the transerse direction and most of the heat transfer between the gas and the plate should be localized in this region. This is confirmed b the longitudinal distribution of the transerse heat flu densit at the gas-solid interface calculated b Eq. 9. Results are shown in Fig.7 at selected Renolds numbers. Figure 7. The time-aeraged transerse heat flu densit at the hot and cold fin surface (= ) as a function of the aial coordinate for selected Renolds numbers. Since the time-aeraged transerse energ flu at the plate surface is positie when entering the plate and negatie when leaing the plate, Fig. 7 implies that oer the period of an acoustic ccle, energ flows out of the hot fin on the left, then hdrodnamicall along the thermal boundar laer in the gas and into the cold fin on the right. Information on the time eolution of the heat flu echanged between the oscillating gas and the solid fins can be obtained from the cross sectional temperature profiles. In Fig. 8 the transerse temperature profiles measured mm awa from the centre of the HX assembl both in the cold and hot fin are illustrated for the preiousl selected si phases and for Re=5. The si phases denoted with red smbols correspond to the position in the hot channel mm awa from the joint, while the other si phases denoted with blue smbols correspond to the position in the cold channel mm awa from the joint. The distance from the fin surface is normalized b the channel width, = 6 mm while the gas temperatures are normalized b T T = T T w θ () H C

13 T, T w, T H and T C being the instantaneous gas temperature, wall (fin) temperature, reference hot temperature (ºC) and reference cold temperature (3ºC), respectiel. The temperature profiles clearl show the large temperature ariations in a er thin laer of gas near the wall. For all si phases in the hot channel the heat transfer direction is from the plate to the gas, while in the cold channel the heat is alwas transferred from the gas to the plate. It is also found that in the cold channel at phase Φ, the temperature of the gas beond the thin laer near the wall is lower than the wall temperature (a negatie θ). It is understood that at this phase the gas parcel is in the left-most position and the cold gas from the cold channel moes deepl into the hot channel. The colder gas from the right takes oer this position. This part of gas affects the processes for a while so that at phase Φ 6 the temperature in the centre of the cold channel is still lower than the wall temperature. Figure 8. Cross sectional temperature profiles measured mm awa from center of the HX assembl both in the cold and in the hot fin for the preious selected si phases and for Re=5. The temperature profiles enable calculation of local, phase-dependent, transerse heat flu densities on the HX surface from first principles: dt (,, Φ) q (, ) = K d = Φ () Integrating Eq. () in phase oer an acoustic ccle and/or in space oer the fin length it is possible to obtain timeaeraged and/or space aeraged heat flu densities. In particular, phase-and-area aeraged heat transfer coefficients are calculated as h = πl L d [ q (, Φ)/( Tw T )] dφ π. (3) T is the reference fluid temperature ealuated in the centre of the channel, at the same stream-wise location as the joint between hot and cold channels. The alues of h can then be conerted into non-dimensional Nusselt numbers based on the fin thickness l: h (l) Nu = (4) K 3

14 Note that when manipulating data generated b the computational model the phase integration is not required since the model directl proides time aeraged quantities. The comparison between the eperimental and computational data is presented in Fig. 9 in terms of the area-and-phase aeraged transerse heat flu densit at the fin surface. The graph clearl shows that the two fins ehibit a rather different behaiour. The heat flu echanged at the hot HX (open circles) appears to be an increasing function of the Renolds number. This is due to the high elocit and displacement amplitudes at high Renolds numbers which increase the temperature gradient in the hot and cold channel. The heat flu echanged b the cold HX (full circles) reaches a maimum for Re=535. This behaiour can be interpreted as an effect of the ariation of the effectie length of heat transfer with the gas displacement amplitude. At low Renolds number, in fact, the gas displacement amplitude is less than the length of hot and cold HX, so the gas portion heated b the hot echanger can onl partl transfer heat to the cold HX. In particular, the gas heated at the left-most end of the hot HX cannot echange heat with the cold HX. Figure 9. Area-and-phase aeraged transerse heat flu at the cold (full circles) and hot (open circles) heat echanger. Solid lines are predictions of the simulation model When the Renolds number is 535, the corresponding gas displacement amplitude is 36 mm, which is close to the length of hot and cold HX (35 mm). In this case the gas heated b the hot HX can full transfer heat to the cold HX. When the gas displacement eceeds the length of hot and cold heat HXs, the colder gas from outside of the hot HX moes into the cold HX and cannot effectiel echange heat with the cold HX. In the considered Re range the eperimental heat flues are matched b the computational ones with an error of about 4%. The model is able to take into account the saturation of the heat transfer at increasing Re numbers (the growth of the cures diminishes with Re) but not the maimum obsered in q at the cold HX. Furthermore, in contrast with eperiment, the heat echanged b the cold HX appears to be higher than the heat echanged b the hot HX. This result is not surprising since in the proposed model the channel ends are considered as thermall insulated, based on the hpothesis that oscillations in fluid outside the HXs are adiabatic, and so no energ can escape from the channel. Thus, the cold HX is loaded b the heat deliered b the hot HX plus eentuall the heat generated b iscous dissipation at the solid walls. 4

15 Howeer, Fig. 5 clearl shows that a large fraction of the heat transmitted to the gas b the hot HX is spread out in the resonator hot duct b free conection. This phenomenon is not negligible at the hot duct side since its strength is proportional to d 3, (d =34 mm being the side of the resonator s square cross section) and to the difference between the mean temperature of the gas leaing the hot HX pores and the mean temperature of the hot duct (approimatel the difference between 5 and C). At the cold duct side the free conection process is much less marked due to the lower temperature difference between the gas and cold duct (nearl zero). Inside the HX channels the phenomenon is clearl negligible since << d, as confirmed b inspection of Fig 5 that reeals that the gas-fin heat echange is smmetric along the transerse direction. This proides an a posteriori justification of the reduced computational domain selected on the basis of the (assumed) smmetr of the problem along the coordinate. Figure. Area-and-phase aeraged transerse heat flu at the cold (full circles) and hot (open circles) HX. Solid lines are predictions of the simulation model modified to account for heat leakage to the resonator duct The strong free conection processes taking place at the hot duct side facing the HXs assembl entail, howeer, that the hpothesis of considering the gas oscillations just outside the hot HX fins as adiabatic (applied in the implementation of the model) is probabl no longer alid. These circumstances are responsible for the higher measured heat transfer rates at the hot HX compared to those computed b the model, and for the breakdown of the smmetr at the inlet and outlet of the HX pores. Therefore, as a first step to take into account these er comple free conection processes in the framework of the simplified computational model used, the numerical code has been modified b introducing two fictitious thermal reseroirs on both the side of the cold duct and the side of the hot duct. These thermal reseroirs are modelled to thermall interact with the oscillating gas inside the HX channel through two heat transfer coefficients that, in addition to the thermal reseroir temperatures, are considered as fitting parameters. Results are shown in Figs. and in terms of the transerse heat flues and the Nusselt numbers. The corresponding heat leakages to the hot and cold side of the duct computed b the simulation model are reported in Fig.. 5

16 Figure. Area-and-phase aeraged Nusselt numbers at the cold (full circles) and hot (open circles) heat echanger. Solid lines are predictions of the simulation model modified to account for heat leakage to the resonator duct. Figure. Numericall computed heat leakages to the hot side and cold side of the resonator duct The eperimental data are now described b the simulation cures with an error of 3% and 35% at the hot and cold HX, respectiel. The measured Nusselt numbers ehibit a dependence on Re similar to that of q (Fig. ). A maimum in Nu ealuated at the cold HX for Re=535 is obsered to which a saturation in Nu ealuated at the hot HX corresponds. The eperimental Nu numbers are fitted b the numerical model with an error of 9% and 35% at the hot and cold HX, respectiel. 6

17 The preious analsis suggests that in the inestigated range of flow conditions, Re=535 is the optimal flow parameter for the thermal performance of the tested HXs. This is likel due to the circumstance that the gas displacement amplitude equals the length of the hot and cold echanger for this Renolds number. Heat transfer coefficients from the gas-side can be predicted with a confidence of about 4% at moderate acoustic Renolds numbers. The matching of the eperimental data improes if heat transport from the couple of HXs to the surrounding enironment (hot and cold ducts) is considered. These heat losses are most likel caused b the entrance/eit effects localized at the resonator-hx cross section interfaces which are responsible of comple non-linear temperature and flow patterns (strong orte flows and turbulence generation effects [6,7]). Although the simplified model proposed to account for this effect doesn t constitute at this stage a predictie algorithm for quantitatie estimations, it proides some eidence that this kind of phenomena could hae a non-negligible influence on the alue of the heat transfer coefficients at the gas side of the HXs (especiall at the hot HX). Further eperimental inestigations and theoretical modelling are required for more detailed insight into the phsics of thermo-fluid processes. 5. Conclusions The work described in this paper utilises the noel measurement techniques deeloped in order to obtain instantaneous temperature and elocit fields in the interstices of the internal structures of thermoacoustic deices. This is a challenging task, which has not been attempted before, and which enables a unique insight into the phsics of thermofluids processes within the oscillator compressible flows that goern the associated energ transfer. Acetonebased Planar Laser Induced Fluorescence (PLIF) and Particle Image Velocimetr (PIV) techniques hae been applied to obtain spatiall and temporall resoled temperature and elocit fields within parallel-plate thermoacoustic heat echangers. The local temperature distribution profiles within an acoustic ccle and space-aeraged phase-dependent heat flues reeal the heat transfer ariations phase b phase. The dependence of the space-ccle aeraged heat flues and the Nusselt numbers on the Renolds number shows that the optimal performance of heat echangers can be achieed when the gas displacement amplitude is close to the length of hot and cold heat echanger. The eperimental data obtained from the measurement of the temperature field proide an important reference for alidation of a CFD code implemented on the basis of the classical thermoacoustic linear theor. Comparisons between the detailed spatiall- and temporall-resoled measurements proide a unique opportunit to test the predictie capabilities of the CFD code deeloped in the course of this research. It is shown that the code is able to predict heat transfer coefficients from the gas-side with a confidence of about 4% at moderate acoustic Renolds numbers. Howeer, better estimates can be achieed when the entrance/eit effects localized at the resonator-hx cross section interfaces are taken into account. These effects are responsible for considerable heat losses from the couple of heat echangers into the surrounding enironment (the resonator etending beond the heat echangers). The results of this stud demonstrate the need for deeloping the new thinking for the design of CFD codes to encapsulate the nature of thermal-fluid interactions present in the thermoacoustic deices, and in the long term to encourage new paradigms and concepts for the future design of heat echangers in thermoacoustic sstems. Acknowledgments This research is financiall supported b the European Commission under the FP7 framework grant THATEA: THermoAcoustic Technolog for Energ Applications (Grant 645 FP7-ENERGY-8-FET). Dr Xiaoan Mao of Leicester Uniersit is acknowledged for collecting the PLIF data used in this paper. 7

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