Transport Phenomena in an Orifice Pulse Tube Refrigerator/Cryocooler

Transcription

1 Termal Siene & Engineering Vol. 6 (,. 45-5, 998 Transort Penomena in an Oriie Pulse Tube Rerigerator/Cryoooler P. Ceng and T. S. Zao Abstrat Tis artile resents a review o transort enomena o a reiroating low in te eat eanger and te regenerator o an oriie ulse tube rerigerator/ryoooler. Some imortant indings in onnetion wit te onset o turbulene and ritional losses as well as te assoiated eat transer arateristis in a reiroating ie low wit aliation to te eat eanger will be disussed. Correlation equations o ritional losses and eat transer rate in te reiroating low in terms o te kineti Reynolds number Re and te dimensionless osillation amlitude o luid will be resented. Eerimental results on te ressure dros troug a tube aked wit stainless steel wire sreens subjeted to a reiroating low in a regenerator will be also addressed. KEYWORS: Reiroating low; osillatory eat transer; eat eanger; regenerator; rerigerator. Nomenlature A untion deined in Eq. (7 A m ross-setional area o te aked olumn dimensionless osillation amlitude o luid ( dimensionless osillation amlitude o luid on ydrauli diameter A amlitude o te imosed ressure gradient B untion deined in Eq. (8 rition oeiient yle-averaged rition oeiient deined in diameter o te ie ydrauli diameter ressure dros yle-averaged ressure dro te yle-averaged ressure dro ator F eression deined in Eq. (4 Nu sae-yle averaged Nusselt number Pr Prandtl number Re kineti Reynolds number T, θ dimensional and dimensionless temerature u,u dimensional and dimensionless aial veloity, X dimensional and dimensionless aial distane α termal diusivity o luid δ ase lag λ Stokes number ϕ orosity φ ase angle ρ density o luid τ w searing stress at te wall µ visosity o luid osillatory requeny Sub/Suersrits m ross-setional mean value ma maimum value w wall o te ie Introdution As sown in Fig., an oriie ulse tube rerigerator/ryoooler onsists o a reiroating omressor, a ulse tube wit two eat eangers at bot ends, a regenerator onneted to te omressor, and a reservoir onneted troug an oriie to te ot end o te ulse tube. uring te omression stroke, gas moves toward te reservoir end via te regenerator, te ulse tube and te oriie. Te gas is ooled in te regenerator wile in te ulse tube it is eated due to adiabati omression. uring tis ig ressure eriod, te gas rejets to te eat ot eat eanger and undergoes adiabati eansion wen lowing to te reservoir via te oriie. uring te eansion stroke, gas lows out te ulse tube and bak troug te regenerator. Gas in te ulse tube is ten ooled due to adiabati eansion and absorbs eat wile lowing troug te old end eat eanger. Gas is ten eated as it moves toward te omressor end via te regenerator Te yle results in an average entaly low rom te old end to te ot end wi establises a onstant temerature gradient in te regenerator and rovides a ontinuous rerigeration eet. Reently, oriie ulse tube ryooolers ave been etensively studied due to teir ig reliability and ig rerigeration aaity [-4]. Some interest as been given to use tis devie as a rerigerator beause it uses non-cfc as te rerigerant. For eartment o Meanial Engineering Te Hong Kong University o Siene and Tenology Clear Water Bay, Kowloon, Hong Kong

2 an otimum design o an oriie ulse tube rerigerator/ryoooler, Piston Regenerator Fig. Semati o an oriie ulse tube ryoooler it is imortant to redit aurately eat transer and ritional losses in te eat eangers and te regenerator were a reiroating low eists. U to now, te design o te reiroating low eat eangers and regenerator as been based on steady low orrelation equations wi are unrealisti. In te ast ew years, we ave undertaken a series o studies on te arateristis o luid low and assoiated eat transer in a reiroating eat eanger and a regenerator [5-9]. In tis aer, we sall reort our revious work and summarize some imortant indings related to tis ield. In wat ollows, we sall irst disuss te governing similarity arameters or a reiroating low in a ie and te assoiated eat transer, ollowed by resenting major indings on te transition to turbulene. Correlation equations or ressure dros in a laminar and turbulent reiroating inomressible low will also be disussed. Osillatory eat transer arateristis in a reiroating ie low will also be addressed. Finally, we sall resent our eerimental results on te ressure dros aross a tube aked wit stainless steel wire sreens subjeted to a reiroating low. Reiroating Flow Heat Eanger In tis setion, we sall disuss bot te luid meanis and eat transer asets o a reiroating inomressible low. Te results on onset o turbulene, ritional losses, eat transer rate in a reiroating ie low will be summarized.. Similarity Parameters We onsider irst a reiroating inomressible low in a irular ie (wit diameter in wi te reiroating motion o te luid is driven by a sinusoidal dislaer su tat te luid dislaement m varies aording to m Reservoir Oriie Heat Eanger Q Pulse tube Heat Eanger Q ma ( os t ( were is te osillation angular requeny and t is time, and te luid dislaement m is deined by assuming te luid moves as a lug low at a mean veloity u m wit te maimum luid dislaement m. ierentiating Eq. ( wit reset to time gives te ross-setional mean veloity u m u u m ma sin φ ( were te maimum ross-setional mean veloity u ma is related to te maimum luid dislaement ma by um ma (3 and te rank ase angle φ is related to te dimensionless time τ by τ t π( i + φ, wit i being te number o yles. It as been sown by Zao and Ceng [5] tat te governing similarity arameters or su a ie low and assoiated eat transer are te kineti Reynolds number Re wit ν ν being te kinemati visosity o te luid, te dimensionless osillation amlitude o luid Ao ma, and te Prandtl number Pr ν α. Onset o Turbulene wit α being te termal diusivity o luid. Some work as been erormed to study transition to turbulene in a reiroating inomressible low [6, -3]. Measurements sow tat te onset o turbulene in a reiroating low is dierent rom tat o a steady low. For a reiroating low, or eamle, a laminar-like low eists during te aeleration ase o te al yle wereas a turbulent-like low eists during te deeleration ase. It was ound wen te values o and Re were inreased to ertain values, eriodi turbulent bursts ourred near te wall during te deeleration ase o te yle [6]. It was also ound tat te veloity lutuations near te wall are mu stronger tan in te enterline o te ie. Tis inding imlies tat te instabilities were generated near te wall and te radial momentum transer aused a lower level o veloity lutuations as te enterline is aroaed. Tis is beause at ig kineti Reynolds numbers te aial veloity near te wall is iger tan te enterline veloity and tere eists an inleion oint in te veloity roile near te wall. Tus, i te kineti Reynolds number eeeds a ritial value at a given dimensionless luid dislaement, te luid low near te wall may irst beome unstable and eddies our near te wall. Tese eddies are transerred to te ore low wi ause small lutuations. Te ollowing orrelation equation as been obtained or te redition o te onset o turbulene in a eriodially reversing low [6]: ( ri 76 ( Re (4 were ( ri is te ritial dimensionless luid dislaement and ( Re ri is te ritial kineti Reynolds number. Equation ri

3 (4 was obtained based on eerimental data in te ranges o and 3 Re Laminar Flow Consider te osillating motion o te luid wi is driven by a sinusoidally varying ressure gradient given by ρ A os t (5 were A is te osillation amlitude o te eternally imosed ressure gradient. An eat solution or te aial veloity roile o a ully-develoed laminar reiroating low is given by Uida [4] as well as by Zao and Ceng [7] as: A u [ B os t + ( A sin t ] (6 4λ ν were A and B are given resetively by A ber λ ber λr+bei λ ber λr (7 ber λ + bei λ ( δ Re. 647[ 5. e(. 9 Re ] (5 were δ is te ase angle dierene between te rosssetional mean veloity u m and te wall searing stress. It is interesting to note tat altoug te veloity in a ullydeveloed osillatory low, given by Eqs.(6-(9, deends only on te kineti Reynolds number wile te rition oeiients o a ully-develoed osillatory low, given by Eqs.(-(5, deend not only on te kineti Reynolds number Re but also on te dimensionless osillation amlitude o te luid as well. Te instantaneous ressure oeiient an be eiter ositive or negative during a yle. Te ositive sign o te rition oeiient means tat te luid low moves in te ositive diretion wile te negative sign imlies tat it moves in te negative diretion. Te yle-averaged rition oeiient o a ully-develoed laminar reiroating low given by Eq. (3 is lotted as solid line or, Ao versus Re in Fig.. Te eerimental data obtained by Zao and Ceng [7] is also lotted in te same gra or omarison uroses. It is sown tat te analytial solution is in good agreement wit te eerimental data in te laminar low regime.. B ber λ ber λr -bei λ bei λr (8 ber λ + bei λ wit Rr/ being te dimensionless radial oordinate and λ being te Stokes number deined by λ ν Re (9 We now deine te instantaneous rition oeiient and te yle-averaged rition oeiient as, Turbulent low: Eq. (49 Laminar low: Eq. ( Re u τ µ ( w ρu ρu ma r r / ma ( Fig. Comarison o te orrelation equations o te yle averaged rition oeiient between te reiroating laminar low and turbulent low π π dτ ( were τ w is te wall searing stress and µ is te visosity o te luid. An eat solution or te instantaneous and te yleaveraged rition oeiients o a ully develoed laminar osillatory low (, and, were obtained by Zao and Ceng [7]: wit 3F, sin( φ + δ ( A o F, 64 πa ( Re o 66. (3 F ( (Re Turbulent Flow Te yle-averaged rition oeiient o a ylially turbulent reiroating low, t was also measured [6] and is resented in Fig.. It was ound tat te ollowing algebrai equation its well wit te measured data ( A Re, t. o (6 wi is valid in te range o 8 Re 54 and In omarison wit te orrelation equation o te yle-averaged rition oeiient or te laminar low given by Eq. (3, it is evident rom Fig. tat te value o given by Eq. (6 or a turbulent low is signiiantly iger tan tat o a laminar low at te same kineti Reynolds number.

4 .5 Osillatory Heat Transer In tis setion, te osillatory eat transer in a eated ie wit uniorm eat lu subjeted to a laminar eriodially inomressible reversing low will be disussed. Te test setu onsisted o a eated test setion and two ooled tubes at bot ends, as sown in Fig. 3. Figure 4 illustrates tyial temoral temerature variations measured at te let and rigt miing ambers during one yle or Re. and As sown in tis igure, wen te luid low moves rom te let to rigt, te luid temerature in te let miing amber ontinues to dro until te luid low reverses its diretion at te end o te irst al yle Nu. Re (7 were te sae-yle averaged Nusselt number Nu is deined as qi Nu k ( T T w m (8 wit T w being te dimensional sae-yle averaged temeratures measured at te surae o te eated tube and T m being te yle averaged dimensional temerature at te let miture amber. Equation (8 wit eerimental data is resented in Fig. 5. Te sae-yle averaged Nusselt number Nu generated rom te numerial simulation by Zao and Ceng [8] was also resented in Fig. 5. It is sown tat te numerial solution (reresented by te dased line is sligtly lower or iger tan Eq. (7 deending on weter Re <8 or Re >8. Note tat Eq.(7 indiates tat te eat transer rate inreases wit bot te dimensionless osillation amlitude o luid and te kineti Reynolds number Re ; te inrease in eat transer is more sensitive to tan to Re beause te eonent o is greater tan tat o Re..7 Numerial solution.55 Nu/.85.Re.583 Fig. 3 Semati o te aaratus Nu/ y let side rigt side Re. Fig. 5 A orrelation equation based on te eerimental data or te yle-averaged Nusselt number Fig. 4 θ m , Re φ Temoral luid temerature variations at te let and rigt miing ambers or 34.9 and Re., and L/44.8 Based on te eerimental data or 8.5 to 34.9 and Re 3 to 46, te ollowing orrelation equation o te sae-yle averaged Nusselt number Nu or an osillatory laminar low o air in a long tube wit onstant eat lu was obtained [8]: 3 Reiroating Flow Regenerator Previous setions are onerned wit te luid low and eat transer arateristis in a emty tube subjeted to a reiroating low wit aliation to eat eangers. Tis setion deals wit osillatory ressure dro arateristis in a regenerator (omosed o tree dierent sizes o woven sreens under a eriodially reiroating low o air. 3. Eerimental etails Figure 6 sows te test rig used to measure osillatory ressure dros troug a regenerator. Te test setion onsisted o a aked olumn (33.3 mm inside diameter and 4 mm in lengt wit ea o its end onneted to a oer tube o te same diameter (33.3 mm in diameter and 5 mm in lengt, and troug a onverging nozzle to anoter oer tube o

5 smaller diameter (3.5 mm in diameter. Tis test setion was onneted to te um troug a leible tube. Te tree aked olumns were made o staks o stainless steel lainlywoven wire sreens wit tree dierent mes sizes as listed in Table. Te sinusoidal luid dislaement in te aked olumn varies aording to Mes Size Table Proerties o Stainless Steel Wire Sreens used Num. Wire ia Pit Mes Porosity o w( mm l distane ϕ sreens (mm β (mm n Hydrauli ia (mm ( ma ( Ao (3 (Re ν (4 Eeriments were arried out or tree dierent mes numbers (, 5, o te woven sreens, tree values o te luid dislaement ma (8.98,., and 5.6 mm wi were varied by anging te stroke o osillations, and at various kineti Reynolds numbers (Re ranging rom. to.3. A total o nine test ases wit nine dierent values o ( are tabulated in Table. Table : Eerimental Conditions or te Tested Cases (.< (Re <.3 Case Mes Size ma (mm (A o Temoral Variations o Pressure ro Fig. 6 Semati o te aaratus ( ma ( os φ (9 Tus te maimum luid dislaement in te emty tube ma and in te aked olumn ( ma are related by We now resent te eerimental data or te ensembleaveraged ressure dro in te regenerator. Tyial variations o te ensemble-averaged ressure dros aross te tube aked wit mes sreens during a omlete yle at ( or (Re.5,.377 and.559 are illustrated in Fig. 7. At a ied value o te dimensionless luid dislaement, it is seen tat te ressure dro inreases wit ma ( ma ( ϕ were ϕ is te orosity o te sreens. ierentiating Eq. (9 wit reset to time, we obtain te ross-setional mean luid veloity ( u m (Pa 4 (Re mes: ( ( um ( u ma sin φ ( - were te maimum ross-setional mean veloity ( u ma is related to te maimum luid dislaement by ( u ma ( ma ( Te similarity arameters o an osillatory low in a aked olumn are te dimensionless luid dislaement and te kineti Reynolds number wi are deined resetively by [9] Fig φ Tyial variations o te ensemble-averaged ressure dro or ( at (Re.5,.377 and.559

6 te inrease o te kineti Reynolds number during most o te yle. It is noted tat or a small value o te kineti Reynolds number, (Re.5, or eamle, te ressure dro varies sinusoidally almost witout ase lags wit reset to te ross-setional mean luid veloity as given by Eq. (, wit te maimum value ourring at φ 9 o. But or iger values o te kineti Reynolds number, signiiant ase lags were observed. For eamle, te ase angle is delayed by 8 o at (Re.377 and is delayed by 4 o at (Re.559. Figure 8 sows tyial temoral variations o te ensembleaveraged ressure dro aross te tube aked wit mes sreens during a omlete yle at (Re.454 or ( 64.7, and Again, it is aarent tat te ressure dro inreases wit te inrease o te dimensionless luid were is te yle-averaged ressure dro wi is determined rom N N i j i j ~ (6 wit Ni being te total number o samling intervals in one yle and ~ j being te ensemble-averaged data at jt interval. Based on te eerimental data, te yle-averaged ressure dro ator was omuted aording to Eq. (5 and resented in Fig. 9. It is sown tat te eerimental data are well itted by te ollowing orrelation equation: 47. [ ] ( (Re (7 5 5 ( mes: (Re.454 Equation (7 as a maimum relative error o 4% in omarison wit eerimental data. 6 ( 47.3/Re d +3.6 (Pa 5-5 ( d φ 4 Fig. 8 Tyial variations o te ensemble-averaged ressure dro at (Re.454 or ( 64.7, and 43.3 dislaement at a ied value o te kineti Reynolds number. However, it aears tat te temoral variations o ressure dro are almost in ase wit one anoter or te tree dierent values o (. Tereore, we an onlude tat te dimensionless osillation amlitude o te luid as no eet on te ase lag o ressure dro variations. From Figs. 7 and 8, we an onlude tat te ressure dro in a aked olumn subjet to an osillatory low inrease wit bot te dimensionless osillation amlitude o te luid ( and te kineti Reynolds number (Re. Te ase lag o te ressure dro wit reset to te mean veloity inreases wit te inrease o (Re, but is relatively indeendent o (. 3.3 Cyle-Averaged Pressure ro Fator We now deine te yle-averaged ressure dro ator as ollows: ρ( u L ma (5 Fig (Re d Correlation equation o te yle-averaged ressure dro ator in terms o (Re and ( Equation (7 an be used to redit te yle-averaged ressure dro in te design o te regenerator o a Stirling engine or a ulse tube rerigerator/ryoooler. 5 Conluding Remarks Reent work on reiroating low eat eangers and regenerators in a ulse tube rerigerator/ryoooler ave been reviewed. Correlation equations or te onset o turbulene, ressure dro oeiients or bot laminar and turbulent ie lows in terms o aroriate arameters are resented. Heat transer asets o a reiroating laminar ie low eated at a uniorm eat lu ave also been disussed. Te eerimental results o te ressure dros in a regenerator subjeted to a reiroating low are also resented. Aknowledgment Tis work was suorted by Hong Kong RGC Earmarked Resear Grant No. HKUST 79/95E.

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