Cryogenics 41 &2001) 513±520 www.elsevier.com/locate/cryogenics

Thermodynamic analysis of GM-type pulse tube coolers q Y.L. Ju *,1

Cryogenic Laboratory, Chinese Academy of Sciences, P.O. Box 2711, Beijing 100080, People's Republic of China Received 29 January 2001; accepted 28 June 2001

Abstract The thermodynamic loss of rotary valve and the coecient of performance &COP) of GM-type pulse tube coolers &PTCs) are discussed and explained by using the ®rst and second laws of thermodynamics in this paper. The COP of GM-type PTC, based on two types of pressure pro®les, the sinusoidal wave inside the pulse tube and the step wave at the compressor side, has been derived and compared with that of Stirling-type PTC. Result shows that additional compressor work is needed due to the irreversible entropy productions in the rotary valve thereby decreasing the COP of GM-type PTC. The e€ect of double-inlet mode on the COP of PTC has distinct improvement at lower temperature region &larger TH=TL). It is also shown that the COP of GM-type PTC is independent of the shape of the pressure pro®les in the ideal case of no ¯ow resistance in the regenerator. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Pulse tube cooler; GM-type; Thermodynamic analysis

1. Introduction achieved [13]. All these machines use a compressor and a valve system to produce pressure oscillation in the In general pulse tube cooler &PTC) requires a phase cooler system. They are called GM-type PTCs. shifter system, located at the hot end of the pulse tube, A GM-type PTC, shown in Fig. 1, only di€ers from to achieve an optimum phase angle h between the gas the Stirling-type at the compressor side of the cooler. ¯ow rate and the pressure oscillation inside the pulse Instead of a piston compressor the GM-type uses a ro- tube to increase the cooler performance. The so-called tary valve or several electromagnetic valves to generate ori®ce [1] and double-inlet [2] modes are two of the most the pressure oscillations in the cooler. In the ideal case well-known con®gurations. Other innovations are multi- the compressor is isothermal and reversible. The com- bypass [3], two-piston [4], four-valve [5], inertance tube pressor heat is removed in the aftercooler. The rotary [6], active-bu€er [7], inter-phasing [8], and double-ori®ce valve connects the PTC system alternatively to a con-

[9]. stant high pressure pH and a low pressure pL. In recent years multi-stage 4 K-PTCs [9±11] have been This paper is a continuation of our previous papers on reported with multi-layered hybrid magnetic materials the thermodynamic aspects of pulse tubes [14±17]. We in the coldest regenerator region. Two-stage PTCs can will discuss the thermodynamic losses of the rotary valve provide more than 0.5 W cooling power at 4.2K and and the coecient of performance &COP) of GM-type meet the cooling requirements of superconducting de- PTC in this paper. General expressions for the COP of vices operating at 4 K [12]. By using 3He as the working GM-type PTC, based on two types of pressure pro®les, ¯uid the lowest temperature below 1.8 K has been the sinusoidal wave inside the pulse tube and the step wave at the compressor side, have been derived and compared with those of Stirling-type PTCs.

q This work was partly carried out at Faculty of Applied Physics, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands. 2. Thermodynamic analysis * Tel.: +86-10-6262-7302; fax: +86-10-6256-4049. E-mail address: [email protected] &Y.L. Ju). Fig. 2shows the compressor and the rotary valve 1 On leave from Faculty of Applied Physics, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Nether- parts of a GM-type PTC with enthalpy ¯ows and en- lands. tropy ¯ows in it. Consider the control volume at left

0011-2275/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 227 5 & 0 1 ) 0 0 1 23 - 0 514 Y.L. Ju / Cryogenics 41 82001) 513±520

Nomenclature q gas density A area h phase angle C ¯ow conductance g viscosity

CP heat capacity at constant pressure fV dissipation rate, see Eq. &11) CV heat capacity at constant volume w ratio of ¯ow conductance, see Eq. &24) H enthalpy T notation for general parameter, see Eq. &56) l length U notation for general parameter, see Eq. &59) L length of regenerator W dissipation parameter, see Eq. &61) nà molar ¯ow rate s time p pressure x angular frequency Q heat Subscripts R ideal gas constant 0 average value S entropy 1 ori®ce t time 2double-inlet t period c A amplitude T temperature ac aftercooler u velocity c compressor U internal energy chx cold heat exchanger V volume à hhx hot heat exchanger V volume ¯ow rate H hot end, high-pressure side W work L cold end, low-pressure side Z ¯ow impedance m molar quantity Greeks r regenerator a ratio of molar ¯ow rate, see Eq. &25) t tube, total

aV volumetric thermal expansion coecient V rotary valve side, which contains the compressor and the aftercooler. The system is in steady state unchanged over one Applying the ®rst law of thermodynamics to this system, cycle, so the average internal energy U_ ˆ 0. Eq. &1) be- the rate of increase of the internal energy U_ is given by comes

à à _ à à _ _ W_ ˆ Q_ ‡ H À H 2† U ˆ H mL À H mH ‡ W À Qac; 1† ac mH mL _ If the gas is an ideal gas and if it enters and leaves the where Qac is the heat extracted in the aftercooler. The heat ¯ow and the molar ¯ow rates are positive when control system at room temperature TH, the enthalpy at they ¯ow into the control volume. We consistently de- note ¯ows of extensive quantities with an asterisk * on à top &like H) and the rate of change of extensive quan- tities of a certain system with a dot on top &like U_ ). For heat ¯ows and entropy production rates, which are pa- rameters rather than changes in functions of state, we will use the dot notation.

Fig. 1. Schematic diagram of a GM-type PTC: &1) compressor; &2) Fig. 2. Control volume of the compressor and the rotary valve parts aftercooler; &3) rotary valve; &4) regenerator; &5) cold end heat ex- with enthalpy ¯ows and entropy ¯ows in it. The control volume at the changer; &6) pulse tube; &7) hot end heat exchanger; &8) bu€er; &9) or- left side contains the compressor and the aftercooler; the right side i®ce; &10) double-inlet. contains the rotary valve and the connecting tubes. Y.L. Ju / Cryogenics 41 82001) 513±520 515 the low- and the high-pressure sides are equal. Eq. &2) _ THSV becomes fV ˆ : 11† W_ _ _ W ˆ Qac: 3† The COP of GM-type PTC is given by The second law gives the rate of increase of the en- ! Q_ T T S_ _ L L H t tropy S: COPGM ˆ ˆ 1 À : 12† W_ TH À TL W_ _ à à _ Qac _ S ˆÀ ‡ S mL À S mH ‡ Sac: 4† TH The system is in the steady state over one cycle, so the 3. COP of GM-type PTC !sinusoidal function) average entropy production rate S_ ˆ 0. Eq. &4) can be rewritten as Based on our previous paper [15] about Stirling-type PTC we will derive below the expressions for the COP of Q_ à à ac _ ˆ S mL À S mH ‡ Sac: 5† GM-type PTC. Assuming the pressure pro®les inside the TH tube are sinusoidal For an ideal gas, we have pt ˆ p0 ‡ p1 cos xt: 13† à à à pH The volume ¯ow rate through the ori®ce and double- S mH À S mL ˆÀnR ln : 6† pL inlet valves can be expressed as For the ideal case the compressor works reversibly, à _ V 1 ˆ C1 pt À p0†ˆC1p1 cos xt; 14† Sac ˆ 0. Combining with Eq. &3), the work needed to compress the gas from the low pressure p to the high à L V 2 ˆ C2 pc À pt†; 15† pressure pH sides of the compressor is given by where C1 and C2 are the ¯ow conductances of the ori®ce _ à pH W ˆ nTHR ln : 7† and double-inlet valves, respectively. The gas velocity at pL the hot end of the tube can be approximated as Consider the control volume at the right-hand side in 1 à à 1 u ˆ ‰V À V Šˆ ‰C p À p †ÀC p À p †Š: 16† Fig. 2, which contains the rotary valve and the con- H A 1 2 A 1 t 0 2 c t necting tubes. The system is also in steady state over one t t cycle. The enthalpy ¯ows entering and leaving the sys- The gas velocity at the cold end of the tube is given by tem at the compressor side are equal, so there is no net [15] enthalpy change due to this gas ¯ow. Also there are no CV Lt dpt enthalpy ¯ows entering and leaving this system at the uL ˆ uH ‡ : 17† CP p0 dt regenerator side. The rotary valve works isothermally, so in the case of an ideal gas no heat has to be extracted We assume here that the ®lling factor of the regenerator from it. is equal to unity so that the molar ¯ow rate is inde- The second law applied to this control system gives pendent of the length l of the regenerator, so the pres- sure drop in the regenerator with ¯ow impedance Zr and à à à _ length Lr can be expressed as SV ˆ S mL À S mH ‡ S mR; 8† à Z à Lr _ nrR Zr nrRTe here SV is the irreversible entropy production in the pc À pt ˆ T gdl ˆ 18† rotary valve. With Eq. &6), we have p0 Lr 0 p0CrH

à with C being the ¯ow conductance of the regenerator _ à pH rH SV ˆ nR ln ‡ S mR: 9† at room temperature pL 1 For the whole cooler system, the relationship between CrH ˆ 19† the cooling power and the work from compressor is ZrgH given by [14] and the e€ective temperature   Z L _ TH _ _ 1 r g W ˆ À 1 Q ‡ THSt: 10† T ˆ T dl: 20† T L e L Lr 0 gH _ In this expression St is the sum of the entropy pro- So the volume ¯ow rate through the regenerator can be duced by all irreversible processes in the cooler system simply expressed as during one cycle. à The modi®ed thermodynamic loss &dissipation rate) of à n rRTL VL ˆ ˆ Cr pc À pt† 21† the rotary valve is de®ned as p0 516 Y.L. Ju / Cryogenics 41 82001) 513±520 with Integrating and averaging Eq. &32) over a half cycle T gives L   Cr ˆ CrH : p T à w C C p p e r 2 0 1 2 n c ˆ ‡ 1 ‡ a : 33† The gas velocity at the cold end of the tube is p TL TH R 1 From Eq. &7) the work needed to compress the gas from uL ˆ Cr pc À pt†: 22† the low-pressure side to the high-pressure side is given At by Combining with Eqs. &16), &17) and &22), the pressure drop across the regenerator is given by _ à pH à 2pA W ˆ n cRTH ln  n cRTH : 34† pL p0 pc À pt ˆ wp1 cos xt À a sin xt† 23† Comparing with Eqs. &28) and &33), Eq. &34) becomes with   4 1 C C T C W_ ˆ C p2 r 2 ‡ H w ˆ 1 ; 24† p 2 1 1 C ‡ C C T C ‡ C 2 r r L 2 r qp here a is the ratio between the components of the molar  1 ‡ w†2 ‡ aw†2 1 ‡ a2: 35† ¯ow rate which are in-phase and 90° phase with the The cooling power can be expressed as [15] pressure, and it is de®ned as

Cr 1 2 CV Ltx Q_ ˆ C p : 36† a ˆ A : 25† L 1 1 t C2 ‡ Cr 2 CPC1 p0 Substituting Eq. &36) into Eq. &35) gives So the phase angle between the gas velocity and the  q pressure at the cold end of the tube is given by 4 C T p _ 2 H 2 2 2 _ W ˆ ‡ 1 ‡ w† ‡ aw† 1 ‡ a QL: 37† h ˆ arctan a: 26† p Cr TL Therefore the pressure oscillation at the compressor side Therefore the COP of GM-type PTC is is given by _ QL p ˆ p ‡ p cos xt ‡ wp cos xt À a sin xt†: 27† COPSin ˆ c 0 1 1 W_ It can be rewritten as p 1 q ˆ qp: 4 2 2 2 2 2 ‰C2=Cr ‡ TH=TLŠ 1 ‡ w† ‡ aw† 1 ‡ a pc À p0 ˆ p1 1 ‡ w† ‡ aw† cos xt ‡ u†: 28† 38† Therefore, the pressure amplitude q Since we discuss it in the case of the sinusoidal wave, we 2 2 pA ˆjpc À p0jˆp1 1 ‡ w† ‡ aw† : use subscript Sin. Substituting Eq. &24) into Eq. &38) gives Combining with Eqs. &21)±&23), the molar gas ¯ow rate through the regenerator is found COPSin

à p0 p 1‡ C2=Cr n r ˆ wCrp1 cos xt À a sin xt†: 29† ˆ qp: RTL 4 2 2 2 ‰C2=Cr ‡TH=TLŠ 1 ‡C2=Cr ‡ C1=Cr† ‡ aC1=Cr† 1 ‡a From Eq. &14), the molar gas ¯ow rate through the or- 39† i®ce valve is Two ideal cases help us to interpret Eq. &39). The ®rst à p0 n 1 ˆ C1p1 cos xt: 30† is the case when there is no double-inlet valve &single- RTH ori®ce PTC), this means C2 ˆ 0, then Eq. &39) becomes From Eq. &15), the molar gas ¯ow rate through the T p 1 double-inlet valve is COP ˆ L q : Sin T 4 p à p H 2 2 2 n ˆ 0 wC p cos xt À a sin xt†: 31† 1 ‡ C1=Cr† ‡ aC1=Cr† 1 ‡ a 2 RT 2 1 H 40† Combining with Eqs. &29) and &31), the molar ¯ow rate through the compressor side is The second case is that if the ¯ow resistance of the regenerator can be neglected, this means Cr !1 à à à n c ˆ n r ‡ n 2 Zr ! 0†. In this limit   C C p p p T p 1 r 2 0 1 2 L ˆ w ‡ 1 ‡ a cos xt ‡ u†: 32† COPSin ˆ p : 41† TL TH R TH 4 1 ‡ a2 Y.L. Ju / Cryogenics 41 82001) 513±520 517

For the Stirling-type PTC, we have &Eq. &79) in [15]) C2=Cr for the di€erent refrigeration temperatures TL at _ the ratio C1=Cr ˆ 0:468 [15], which is a alue in our ex- QL COPSt ˆ perimental set-up [17]. These two ®gures eliminate the W_ in¯uences of di€erent refrigerator temperatures. Com- 1 ‡ C =C paring Figs. 3&a) and &b), it is found that COP is al- ˆ 2 r : Sin 2 ‰C2=Cr ‡ TH=TLŠ‰1 ‡ C2=Cr ‡ C1=Cr 1 ‡ a †Š ways lower than that of COPSt for the same refrigeration 42† temperature TL. We also ®nd that the e€ect of double- inlet on the COP of PTC has distinct improvement at If there is no double-inlet valve, C ˆ 0, then Eq. &42) 2 the lower temperature region. becomes From these two ®gures, we ®nd that if TL is lower than T 1 COP ˆ L : 43† a certain critical value TLC, it is given by St T ‰1 ‡ C =C † 1 ‡ a2†Š H 1 r 1 ‡ a2†T T ˆ H : 50† If neglecting the ¯ow resistance of the regenerator, LC 2 1 ‡ a †‡Cr=C1 Cr !1, we obtain the well-known result for the COP of PTC In the case of a ˆ 1, C1=Cr ˆ 0:46, TLTLC  145 K, there is an optimum value of the ¯ow conductance C TL 2 COPSt ˆ : 44† of the double-inlet for achieving a maximum COP TH Comparing Eq. &44) with Eq. &41), we have COP p 1 Sin ˆ p < 1: 45† COPSt 4 1 ‡ a2 This is due to the dissipation of the rotary valve of GM-type PTC compared to that of Stirling-type PTC. Since the GM-type PTC only di€ers from the Stirling- type at the compressor side of the cooler, it uses a rotary valve, so that an additional compressor work is needed due to the irreversible entropy production in the rotary valve thereby decreasing the COP of GM-type PTC. From Eq. &25), we have a / 1=C1: 46† Substituting Eq. &46) into Eq. &42), we can ®nd the maximum COP for Stirling-type PTC with respect to a ˆ 1. From Eq. &26), this means that the phase angle h between the gas molar ¯ow rate &velocity) and the pressure at the cold end of the pulse tube is equal to 45°. This phase angle has been detected experimentally [18]. Substituting a ˆ 1 into Eq. &45) gives COP p 1 Sin ˆ p ˆ 0:55: 47† COPSi 4 2 This means that in the ideal case of no ¯ow resistance in the regenerator, the COP of ideal GM-type PTC is only 55% compared with that of ideal Stirling-type PTC. Substituting a ˆ 1 into Eq. &39) and Eq. &42) gives

COPSin 0:55 1 ‡ C =C † ˆ q2 r ; 2 2 ‰C2=Cr ‡ TH=TLŠ 1 ‡ C2=Cr ‡ C1=Cr† ‡ C1=Cr† 48†

1 ‡ C2=Cr† COPSt ˆ : 49† ‰C2=Cr ‡ TH=TLŠ‰1 ‡ C2=Cr ‡ 2C1=CrŠ Fig. 3. Variations of COP= T =T † as the function of the ratio of Figs. 3&a) and &b) show the variations of COP = L H Sin C2=Cr for di€erent refrigeration temperatures TL with TH ˆ 300 K, TL=TH† and COPSt= TL=TH† as a function of the ratio C1=Cr ˆ 0:468. &a) COPSin= TL=TH† and &b) COPSt= TL=TH†. 518 Y.L. Ju / Cryogenics 41 82001) 513±520 corresponding to TL, both for GM-type and Stirling- This equation is exactly the same as Kircho€'s law of a type PTC. Otherwise, if the refrigeration temperature TL resistance±inductance circuit, its solution is is above the critical value TLC, the double-inlet is p p ˆ p ‡ 1 ‡ CeÀt=T 55† harmful for COP of PTCs. t 0 1 ‡ w with 4. COP of GM-type PTC !step function) aw a C T ˆ ˆ 1 : 56† 1 ‡ w†x x C1 ‡ C2 ‡ Cr Next we will derive and discuss the COP of GM-type Constant C can be determined by the boundary condi- PTC based on the step function of the pressure pro®les tion. For reasons of symmetry and continuity, the am- generated by a GM compressor and a rotary valve. For plitude of the pressure oscillation inside the tube plus the this kind of pressure pro®le, an ideal time-dependent pressure di€erence between the pressures in the com- pressure pro®le before the regenerator is shown in Fig. pressor and the tube should be equal to the amplitude of 4, which is divided into four steps. The pressure in- the pressure oscillation at the compressor side, and this creases from low pressure p to high pressure p in step L H gives &a) and remains at high pressure during step &b), de- 2p 1 creases to low pressure pL in step &c) and remains con- C ˆÀ 1 : 57† stant at low pressure in step &d). It can be expressed as 1 ‡ w 1 ‡ eÀtc=2T 8 >p À p ‡ 2p =s†t 06t < s† 1a†; Therefore, the pressure oscillation inside the tube can be <> 0 1 1 p ‡ p s6t < 1t † 1b†; expressed as p t†ˆ 0 1 2 c c p ‡ p À 2p =s† t À 1t † 1t 6t < 1t ‡ s† 1c†; > 0 1 1 2 c 2 c 2 c p ˆ p ‡ Up 58† : 1 t 0 1 p0 À p1 tc ‡ s6t < tc† 1d†; 2 with parameter 51†   1 2eÀt=T where p0 and p1 are the mean pressure and the pressure U ˆ 1 À : 59† 1 ‡ w 1 ‡ eÀtc=2T amplitude, respectively. For convenience, we assume s  tc &tc ˆ 2p=x the period), and only consider the Fig. 5 shows a typical pressure pro®le inside the pulse period of the ®rst half-cycle where the pressure pro®le is tube given by Eq. &58). above the mean average pressure p0 &similar to that of Similar to the derivation of Eq. &33), a detailed anal- the part below p0 for reasons of symmetry and conti- ysis shows the average molar gas ¯ow through the nuity) compressor side     pc ˆ p0 ‡ p1: 52† à w Cr C2 p0p1 4T W n c ˆ ‡ 1 ‡ : 60† Similar to the derivation of Eq. &23), we can obtain the 2 1 ‡ w† TL TH R wtc pressure drop across the regenerator Here we introduced a dissipation parameter   a dpt 1 À eÀtc=2T pc À pt ˆ w pt À p0†‡ : 53† W ˆ : 61† x dt 1 ‡ eÀtc=2T It can be rewritten as aw dp t ‡ 1 ‡ w†p ˆ p ‡ wp : 54† x dt t c 0

Fig. 4. An ideal step pressure wave at the compressor side generated Fig. 5. A typical pressure pro®le inside the pulse tube &open box), the by means of a GM compressor and a rotary valve. solid line is the pressure pro®le at the compressor side. Y.L. Ju / Cryogenics 41 82001) 513±520 519

The maximum W ˆ 1atT ˆ a=x†fC1= C1 ‡ C2 ‡ resistance in the regenerator, the COP of GM-type PTC Crgˆ0 means that there is no ¯ow resistance &Cr !1) is independent of the shape of the pressure waves, which in the regenerator. According to Eq. &34), the work is the same as the result given in [16]. needed from the compressor is given by    C 1 T C 4T W _ r 2 H 2 W ˆ C1p1 ‡ 1 ‡ : 62† 5. Double-inlet pulse tube Cr ‡ C2 1 ‡ w† TL Cr wtc The general entropy production in the ori®ce can be For the double-inlet pulse tube the cooling power is expressed as [14] given by Z 1 tc=2 _ à Cr THSO1 ˆ V pt À p0†dt _ _ 1 QL ˆ THSO1 tc=2 0 C ‡ C Z 2 r   2 tc=2 C C p2 4T ˆ C p À p †2 dt: 63† ˆ r 1 1 1 À W : 71† 1 t 0 2 tc 0 C2 ‡ Cr 1 ‡ w† tc Substituting Eq. &58) into Eq. &63) yields Similarly to Eq. &63), the entropy production in the Z 2 tc=2 double-inlet &second ori®ce) is T S_ ˆ C p2U2 dt: 64† H O1 t 1 1 Z c 0 1 tc=2 à T S_ ˆ V p À p †dt Integrating Eq. &64) gives H O2 t =2 2 c t   c 0 2 Z C p 4T W tc=2 _ 1 1 2 2 THSO1 ˆ 1 À : 65† ˆ C p2 1 À U† dt: 72† 2 t 2 1 1 ‡ w† c tc 0 For simplicity, we ®rst consider here the single-ori®ce Integrating Eq. &72) gives pulse tube, which means that there is no double-inlet   2 C ˆ 0, T ˆ a=x†fC = C ‡ C †g with xt ˆ 2p. Eq. C1p C2 8w ‡ 4†T W 2 1 1 r c T S_ ˆ 1 w2 ‡ : 73† H O2 2 &62) becomes 1 ‡ w† C1 tc   T C C 2a _ H r 2 r The entropy production in the regenerator is W ˆ C1p1 1 ‡ W : 66† TL C1 ‡ Cr C1 ‡ Cr p Z tc=2 à _ TH 1 In this case, the cooling power is equal to the entropy THSr ˆ V r pc À pt†dt TL tc=2 0 production in the ori®ce &Eq. &44) in [15]). From Eq. &66) Z T 2 tc=2 2   H 2 2 C C 2a ˆ Crp 1 À U† dt: 74† _ r 2 1 T t 1 QL ˆ C1p1 1 À W : 67† L c 0 C1 ‡ Cr C1 ‡ Cr p Integrating Eq. &74) yields Therefore, the COP of GM-type PTC based on step   C p2 C T 8w ‡ 4†T W function at the compressor side is T S_ ˆ 1 1 r H w2 ‡ : 75† H r 2 C T t _ 1 ‡ w† 1 L c QL TL Cr COPStep ˆ ˆ Summing Eqs. &65), &73) and &75) gives the total entropy W_ TH C1 ‡ Cr    production C1 2a Cr 2a    1 À W 1 ‡ W : C p2 4T W C ‡ C T =T C1 ‡ Cr p C1 ‡ Cr p T S_ ˆ 1 1 1 À ‡ 2 r H L H t 2 1 ‡ w† tc C1 68†   2 8w ‡ 4†T W If there is no resistance in the regenerator, Cr !1,so  w ‡ : 76† W ˆ 1, Eq. &66) is deduced as tc

TL p Eqs. &62), &71) and &76) satisfy the general relationship COPStep ˆ : 69† Eq. &10). Therefore the COP of GM-type PTC based on TH p ‡ 2a the step function at the compressor side is found by Eqs. Comparing Eq. &69) to Eq. &41) based on the sinusoidal &62) and &71) function yields COP p ‡ 2a Cr C2 ‡ Cr† Sin ˆ p : 70† COPStep ˆ Ct C2 ‡ Cr TH=TL†† COPStep 4 1 ‡ a2    C 2a C ‡ C 2a For a ˆ 1, this is 0.908, which means that the sinu-  1 À 1 W 1 ‡ 2 r W : C p C p soidal function is slightly more inecient than the step t t function. In the ideal case of no double-inlet and no ¯ow 77† 520 Y.L. Ju / Cryogenics 41 82001) 513±520 pressure waves in the ideal case of no double-inlet and no ¯ow resistance in the regenerator.

Acknowledgements

The author would like to thank the members of Low Temperature Group, Eindhoven University of Tech- nology, The Netherlands, especially Professor A.T.A.M. de Waele, for many stimulating discussions.

References

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Fig. 6. Variations of COPStep= TL=TH† as a function of the ratio of [3] Zhou Y, Han Y. Pulse tube refrigerator research. In: Proc. 7th C2=Cr for di€erent refrigeration temperatures TL with TH ˆ 300 K, Internat. Cryog. Conf., USA, vol. 7, 1992, p. 147. C1=Cr ˆ 0:468. [4] Ishizaki Y, Ishizaki E. Experimental study and medullization of pulse tube refrigerator below 80 K down to 23 K. In: Proc. 7th Internat. Cryog. Conf., USA, vol. 7, 1992, p. 140. [5] Matsubara Y, et al. Four-valves pulse tube refrigerator, JSJS-4, With Beijing, vol. 4, 1994, p. 54. [6] Gardner DL, Jin C, et al. Characterization of 350 Hz thermoa- Ct ˆ C1 ‡ C2 ‡ Cr 78† coustic driven ori®ce pulse tube cryocooler with measurements of the phase of the mass ¯ow and pressure. Adv Cryog Eng is the total ¯ow conduction in the PTC. 1996;41:1411. An example is given in Fig. 6, which shows the varia- [7] Zhu SW, Kakimi Y, et al. Active-bu€er pulse tube refrigerator. In: Proc. CEC16/ICMC, Japan, vol. 16, 1996, p. 291. tions of COPStep= TL=TH† as a function of the ratio C2=Cr [8] Gao JL, Matsubara Y. An inter-phasing pulse tube refrigerator for the di€erent refrigeration temperatures TL at the ratio C =C ˆ 0:468. It can be found that the variations of for high refrigeration eciency. In: Proc. ICEC16/ICMC, Japan, 1 r vol. 16, 1996, p. 295. COPStep are similar to those of COPSin in Fig. 4&a). [9] Chen GB, Qiu LM, et al. Experimental study on a double-ori®ce two-stage pulse tube refrigerator. Cryogenics 1997;37:271. [10] Matsubara Y, Gao J. Novel con®guration of three-stage pulse tube refrigerator for temperatures below 4 K. Cryogenics 6. Conclusions 1994;34:259. [11] Wang C, Thummes G, Heiden C. A two-stage pulse tube cooler By using the ®rst and second laws of thermodynamics, operating below 4 K. Cryogenics 1997;37:159. [12] Wang C, Thummes G, et al. Use of a two-stage pulse tube the thermodynamic loss of the rotary valve and the COP refrigerator for cryogen free operation of a superconducting of GM-type PTC have been discussed and explained. Niobium±Tin magnet. In: Proc. ICEC17, vol. 17, 1998, p. 69. General expressions of the COP for GM-type PTCs, [13] Xu MY, de Waele ATAM, Ju YL. A pulse tube refrigerator below based on two-type pressure pro®les, the sinusoidal wave 2K. Cryogenics 1999;39:865. inside the pulse tube and the step wave at the com- [14] de Waele ATAM, Steijaert PP, Gijzen J. Thermodynamical pressor side have been derived and compared with those aspects of pulse tube. Cryogenics 1997;37:313. [15] de Waele ATAM, Steijaert PP, Koning JJ. Thermodynamical of Stirling-type PTC. Result shows that the additional aspects of pulse tube II. Cryogenics 1998;38:329. compressor work is needed due to the irreversible en- [16] de Waele ATAM. Optimization of pulse tubes. Cryogenics tropy productions in the rotary valve thereby decreasing 1999;39:13. the COP of GM-type PTC. The e€ect of double-inlet [17] Steijaert PP. Thermodynamical aspects of pulse-tube refrigerators. Doctoral Thesis, Eindhoven University of Technology, 1999, The mode on the COP of PTC has distinct improvement in Netherlands. lower temperature region. It is also shown that the COP [18] Ju YL, Wang C, Zhou Y. Dynamic experimental investigation of a of GM-type PTC is independent of the shape of the multi-bypass pulse tube refrigerator. Cryogenics 1997;37:357.