Measurements of the flow resistance and inductance of inertance tubes at high acoustic amplitudes

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Daniella Booker
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1 Mesrements of the flow resistnce nd indctnce of inertnce tbes t high costic mplitdes J Y. L. Yn K. He G. Q. Ho Y. K. Ling J. T. nd Zho, Y. Technicl Institte of Physics nd Chemistry, Chinese cdemy of Sciences, Beijing 8, Chin BSTRCT We recently mesred the flow resistnce nd flow indctnce of inertnce tbes t high costic mplitdes for for different inner dimeters of.6,.,.5 nd.mm t vrios tbe length rnging from to 5mm, t freqencies of 3,, 5, 6 nd 7 Hz. The experimentl dt were compred with the explicit soltion to the liner momentm eqtion for smll costic mplitdes nd were fitted with the modifiction coefficients in terms of the operting freqency nd Reynolds nmbers chrcterized by the mplitde of gs velocities. INTRODUCTION Recent stdies [-] described simple wy of n inertnce tbe, to replce the orifice, t hot end of the plse tbe to generte proper phse shift to improve the plse tbe performnce. The inertnce tbe is long, thin tbe tht provides complex impednce t the wrm end of the plse tbe rther thn simple resistive impednce provided by the orifice. The inertnce tbe dds rective impednce, nlogos to indctnce in simple C electricl circit, tht llows the phse difference between the pressre nd mss flow rte in the plse tbe to be djsted to n extent s efficiently s Stirling coolers. In brief, the inertnce tbe is n costic term connoting both inerti resistnce nd indctnce of moving gs []. Stdies show tht se of the inertnce tbe is significntly beneficil for lrge-scle plse tbes or t higher operting freqencies [-3]. They indicte tht the improvement in the performnce by sing the inertnce tbe comes from: it cn prodce desirble phse shift between the pressre nd mss flow in the plse tbe, it cn blnce the intrinsic tendency for DC gs flow in the doble-inlet plse tbe cooler; nd 3 it cn increse work flow per mss flow within the plse tbe. However, little informtion is vilble on experimentl mesrements of the flow resistnce nd flow condctnce of the inertnce tbe, prticlrly, t high costic mplitdes. We recently mesred the flow resistnce nd flow indctnce of inertnce tbes t high costic mplitdes []. In this pper, we briefly smmrize or previos work nd present some sefl reslts. THEORETICL CONSIDERTION ND FORMULTION Considering n incompressible flow oscillting sinsoidlly in n inertnce tbe of inner rdis L r =, nd of length connecting two lrge reservoirs t room tempertres T = T. The oter sidewll of the tbe is dibtic. The tbe is filled with pressrized idel helim gs, which moves bck nd forth with the oscilltory

2 pressre t the nglr freqency. We ssme the mximm Reynolds nmber ssocited with the oscilltion is not too high lminr flow nd one cn neglect end effects. Gs velocity is in the x direction only nd vries only in the r-coordinte direction norml to the tbe wll. Within these limittions, nd considering the nonliner inerti term cn be neglected t low costic mplitdes, one cn esily derive tht the gs velocity in the tbe is in the form p = ρ ω x J J r / δ v / δ v Here nd re the mplitde of the gs velocity nd pressre oscilltion, respectively, J is the Bessel p fnction of the first kind of order zero, nd = µ / ρ ω is the viscos penetrtion depth denoting the relevnt δ ν bondry-lyer thickness. Integrting nd verging Eq. over the cross-sectionl re S = π one yields the cross-sectionl men velocity p J = = K dr ρ ω x KJ K Where K = /δ. The term p / x is the pressre drop per nit tbe length nd S p / is the force on the V x tbe per nit length, ths the force impendence per nit tbe length cn be expressed s p Z' M = S /. 3 x Sbstitting Eq. into Eq. 3, nd integrting it over the totl length L of the tbe nd ssming L is mch smller thn the locl sond wvelength, we find the force impendence long the tbe ρ ωl / S Z = J K / KJ K The flow impendence is complex in generl nd cn be simply pproximted s 8µ L ρωl Z = + K / 3 + j + 5 π π 3 + K / Its rel prt nd imginry prt indictes the flow resistnce nd flow indctnce, respectively, 8µ L R = + K / 3 6 π ρ 7 ωl X = + π 3 + K / The flow resistnce coefficient for the inertnce tbe nder oscilltory flow conditions cn be rewritten s 6 K f = + 8 Re 3 Compring Eq. 8 to the flow fctor of the tbe for the stedy-stte flow f st = 6 / Re one yields β = f / f = + K / 3 > 9 st Eq. 9 indictes tht t low costic mplitdes the flow resistnce coefficient of the inertnce tbe in the oscilltory flow is lrger by fctor of β thn tht of stedy-stte flow t the sme Reynolds nmbers. Fig.

3 shows the fctor of β of inertnce tbes with different. dimeters of.6,.,.5 nd. mm s fnction of freqency. It shows the fctor is in the rnge of.~. for.6mm tbe, nd of.~.8 for mm tbe, respectively. However, the correltion given bove bsed on the ssmption of the low costic mplitdes is slly not pplicble for plse tbe cooler operting t high costic mplitdes lrge pressre mplitdes, in which the nonliner inerti term cnnot be neglected. Therefore, we introdce here two modifiction coefficients C nd C for β d=.mm d=.5mm d=.mm d=.6mm Freqency Hz considering the nonliner effects t high costic mplitdes. Figre β s fnction of freqency R '= C R X '= C X Below we will describe the experimentl procedres for the two prmeters mesrements to determine the modifiction coefficients C nd C.to determine the modifiction coefficients of C nd C. EXPERIMENTS The detil experimentl set-p to determine the vle of C nd C hs been shown in Ref. []. The oscilltory flow is generted by mens of self-mde liner compressor with swept volme of cm 3, nd its working freqency cn be djstble from to 8Hz. The test section of different inertnce tbes is mde of thin wll stinless steel tbe. t one end of the test section is plced flow strightener. reservoir is directly connected to the other end of the test section. Two smll pressre trnsdcers, connected to chrge mplifiers re sed to mesre trnsient pressres t the inlet of the inertnce tbe, nd inside the reservoir. Two thermocoples re plced on the srfce of the flow strightener nd in the reservoir to mesre the tempertres. The cross-sectionl men velocity of the oscilltory flow is determined by mesring the instntneos pressre oscilltions in the reservoir. Since this pproch hs lredy been described in the litertres [5,6], we show the reslts withot describing the detils. The trnsient pressre in the reservoir hs slly smll pressre mplitde; it is resonble to ssme the pressre oscilltions re the stedy nd oscilltory prt of the pressre in the reservoir, respectively. The mplitde of the gs velocity t the inlet of the reservoir cn be expressed s π j ωt + pv resω = e ρc t Here t is the cross-sectionl re of the flow strightener connecting to the reservoir nd c = γrt is the locl sond speed. By mesring the trnsient pressre in the reservoir nd sing Eq. one cn determine the cross-sectionl men velocity of the oscilltory gs flow. We hve sed hot wire nemometer to mesre the velocity of the oscilltory flow [7]. The verge reltive derivtion between the velocities obtined by the hot wire nemometer nd evlted by Eq. is bot 3.35%, which is within experimentl error. Strictly speking, Eq. is vlid only t: the volme of inertnce tbe is mch smller thn the volme of the reservoir, nd the tbe length L is mch smller thn the locl sond wvelength of the working gs.

4 EXPERIMENTL RESULTS ND DISCUSSES Experiments were crried ot for for different inner dimeters.6,.,.5 nd.mm t vrios tbe length rnging from to 5mm, t freqencies of 3,, 5, 6 nd 7 Hz. The working gs ws helim nd the system men pressre ws.mp t room tempertre. In order to determine the pressre mplitdes nd their phse ngle, the rw experimentl dt mesred from the pressre trnsdcers need dt processing. The otpt signl of the pressre trnsdcer ws first mplified, pressre trnsformtion nd correction, nd then collected with the spectrm nlysis for high freqency hrmonic filtrtion nd Forier nlysis. The verge ncertinty in the pressre mplitde nd phse ngle ws evlted to be less thn %. fter obtined the trnsient pressres of p nd p t the inlet of the inertnce tbe nd in the reservoir nd t the mplitde of the gs velocity by sing Eq., one cn clclte the flow impendence by sing res Z = p p = R ' + jx ' 3 t res t Together with Eqs. nd we cn determine the coefficients of C nd C in terms of Re nd f. series of experiments were crried ot to determine the modifiction coefficients C nd C. Experimentl reslts demonstrted tht the flow resistnce nd indctnce of inertnce tbes t higher costic mplitdes strongly depend on the operting freqency nd Reynolds nmbers bsed on the mplitde of the cross-sectionl men velocity. The selected experimentl reslts re smmrized in Figs. nd 3. Fig. nd b show the experimentl dt of the correltion coefficient C of the flow resistnce of inertnce tbes for three different tbe dimeters of.6mm nd.mm, respectively, in terms of Re nd. The correltion coefficient C of the flow resistnce pre flow resistnce vries with different tbe dimeters, bt is nerly independent of the length of the inertnce tbe. It is clerly shown tht C increses monotony with incresing Reynolds nmbers nd with decresing operting freqencies. It mens tht the nonliner effect becomes lrger with the incresing of the mplitde of velocities. The correltion coefficient C grdlly tends to when the Reynolds nmbers rech to zero. Therefore, the explicit soltion of Eq. 6 for smll costic mplitdes is only pplicble to smll Reynolds nmbers. comprison of the correltion coefficient C for different tbe dimeters t the sme Reynolds nmbers shows tht the lrger the tbe dimeter, the smller the C. The reson is obvios: the velocity is smll for lrge inner dimeter tbe t the sme Reynolds nmbers thereby the nonliner effects become reltively wek. Fig. 3 nd b present the experimentl dt of C of the flow indctnce of inertnce tbes for two f different tbe dimeters of.6mm nd.mm, respectively, in terms of Re nd f. It is shown tht C re less thn, which mens tht the flow indctnce of the inertnce tbe t high costic mplitdes is smller thn tht t low costic mplitdes. The C decreses with incresing Reynolds nmbers, nd increses with incresing operting freqencies, which is contrry to the tendency of the correltion coefficient C of the flow resistnce. However, the inflence of the operting freqency on C grdlly becomes smller with the incresing of the tbe dimeter. The C for the inertnce tbe with inner dimeters of.6mm nd.mm re nerly independent of the length of the inertnce tbe, similr s tht of C. From these experimentl reslts we cn see evidently the difference of the flow resistnce nd flow indctnce of inertnce tbes t high nd low costic mplitdes. It is helpfl for nderstnding the physicl mechnism of the inertnce tbe sbjected to the oscillting gs flow in high freqency plse tbe opertions nd frther condct theoreticl nlysis.

5 . 6 C Hz Hz 6Hz 5Hz 7Hz C 5 3 7Hz 6Hz 5Hz Hz Re Re b Figre The coefficient C in terms of Reynolds nmbers nd freqency for.6mm nd b.mm C C Hz Hz 5Hz 6Hz 7Hz Re o Re o.. 3Hz Hz 5Hz 6Hz 7Hz Figre 3 The coefficient C in terms of Reynolds nmbers nd freqency for.6mm nd b.mm b CKNOWLEDGMENT The work is fnded by the Ntionl Ntrl Science Fondtion of Chin Grnt No REFERENCES. Godshlk K. M., Jin C., et l. Chrcteriztion of 35Hz thermocostic driven orifice plse tbe refrigertor with mesrements of the phse of the mss flow nd pressre, dv. Cry. Eng. 996,.. Grdner D. L. nd Swift G. W., Use of inertnce in orifice plse tbe refrigertors, Cryogenics ,7. 3. Kotsbo V., Hng P. nd Nst T.C., Observtion of DC flows in doble inlet plse tbe, Cryocoolers 999, 99.. J Y. L., He, G. Q. et l., Experimentl mesrements of the flow resistnce nd indctnce of inertnce tbes t high costic mplitdes, Cryogenics in press 5. Nishitni T., Nkno K. et l., Investigtion of costic streming nd stedy flow in the orifice of single stge plse tbe refrigertor, JSJS , L G. Q. nd Cheng P., Flow chrcteristics of metering vlve in plse tbe refrigertor, Cryogenics, J Y. L., Jing Y. nd Zho Y., Experimentl stdy of the oscillting flow chrcteristics for regenertor in plse tbe cryocooler, Cryogenics , 69.

R. I. Badran Solid State Physics

R. I. Badran Solid State Physics I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position