Plastic low power cryocoolers

(1)

PLASTIC L

POWER CRYOCOOLERS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT,

OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. J.M. DIRKEN, IN HET OPENBAAR TE VERDEDIGEN TEN OVERSTAAN VAN HET COLLEGE VAN DEKANEN

OP DONDERDAG 29 MEI 1986 TE 16.00 UUR DOOR

NICOLAAS LAMBERT

NATUURKUNDIG INGENIEUR GEBOREN TE ROTTERDAM

TR diss

1489

(2)

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN: PROF.DR. G.J.C. BOTS EN PROF.DR.IR. J.E. MOOU

(3)

CONTENTS 1 1. 2 4 5 2. 7 10 12 17 18 3. 22 25 32 41 4. 42 43 44 48 55 58 59 5. 61 62 63 68 CONTENTS PREFACE INTRODUCTION Objectives Overview SQUID requirements REFRIGERATION SYSTEMS Idealized cycles I r r e v e r s i b l e gas expansion Reversible gas expansion Other r e f r i g e r a t i o n systems Current approaches MODELLING Introduction Basic equations Refinements, discussion Conclusions DESIGN Introduction

Sealing and contamination Pressure wave generator Displacer and sleeve Displacer drive Joule-Thomson stage Miscellaneous CONSTRUCTION Introduction 5-Stage displacer Conical displacer Joule-Thomson stage

6. OPERATION AND PERFORMANCE 70 Introduction 71 Sealing and contamination 72 Cooldown 74 Lowest t e m p e r a t u r e 76 J o u l e - T h o m s o n s t a g e 7. DISCUSSION 78 Loss analysis 80 Cycle optimization 87 Improvements 91 Conclusions APPENDIX A

93 Computer program listing APPENDIX B

105 Manufacturing scheme of conical sleeve 109 Manufacturing scheme of conical displacer

111 REFERENCES

119 SUMMARY 122 SAMENVATTING

(4)

PREFACE 1

PREFACE

The p r o j e c t described i n t h i s t h e s i s concerned t h e d e s i g n , c o n s t r u c t i o n and development of closed c y c l e p l a s t i c c r y o c o o l e r s f o r SQUID magneto meters, and s t a r t e d i n november 1981 . I t was sponsored by t h e D e l f t s Hogeschoolfonds and supported by two groups at t h e A p p l i e d Physics department: The Low Temperature group of p r o f e s s o r H. Postma and o f p r o f e s s o r G.J.C. Bots was i n t e r e s t e d in t h e a p p l i c a t i o n of small c r y o c o o l e r s and t h e use of p l a s t i c s at low t e m p e r a t u r e s ; i n t h e Solid State-Superconductivity group of professor J.E. Mooij the i n t e r e s t arose from the i n t e r n a l development of a high-performance DC-SQUID magnetometer. Part of the work was done in f r u i t f u l col laboration with doctor J.E. Zimmerman and doctor S. Barbanera during a stay at the National Bureau of Standards in Boulder, U.S.A. The epoxy techniques were developed in close cooperation with mr. J.B. Nieuwland, mr. A. Dijkshoorn and mr. H.J. Lukas at the glass shop of our department.

I sincerely hope t h a t t h i s thesis w i l l f i n d i t s way t o many researchers and t h a t the project as a whole w i l l c o n t r i b u t e t o the a v a i l a b i l i t y of simple, low cost c r y c o o l e r s , and thereby w i l l widen the range of applications f o r low temperature devices.

(5)

1. INTRODUCTION 2 Objectives

1. INTRODUCTION

Objectives

The operating temperature of present SQUID magnetometers and several other small devices is about 4.2 K. In a laboratory, the standard technique of cooling such a system to liquid helium temperatures is to immerse it in liquid helium. The helium is contained inside a cryostat, cooling is obtained by evaporation of the 1 ïquid. Liquid helïum is usually intermittent]y supplïed in dewars from a 1ïquefier. Advantages of this method are:

- Good thermal stability with flexible cooling capacity, - Good mechanical stability,

- Hagnetical quietness, - High reliability,

- Moderate design flexibility.

However for non-laboratory applications this method looks less feasible. Liquefiers may not be available or far away, transport of liquid helium is costly, logistics of supply can be troublesome, personnel should be trained and competent to avoid spillage and to garantee safety. In addition, many applications have special requirements that are difficult to achieve in a dewar-based system, such as independence from supply, low weight, low volume, use in remote areas, unattended service for several years (e.g. in space), freedom of orientation etc. Several commercial solutions to these problems are being developed. However, most of these systems are grossly mismatched to several applications in cost, dimensions, cooling capacity or complexity of operation.

1. INTRODUCTION 3 Objectives

The above considerations lead us to conclude that widespread use of low temperature devices depends heavily on the availability of easily operated, smal 1, low power, low cost and reliable cryocoolers. The objectives of the project described in this thesis were:

- To develop such refrigeration systems or components, suitable for smal 1 scale production. It is expected that the availability of a simple, low cost cooler will stimulate applications which at present seem impracticable or dre unknown.

- More specifically, to meet the special requirements for cooling a SQUiü. This objective arose from the development of a high performance SQUIO in our institute. The operating conditions of SQUIDs are rather extreme in several respects. Many other applica-tions could use the same type of cooler with less demanding requirements.

(6)

1. INTRODUCTION 4 Overview

Overview

Chapter 2 contains an introduction to refrigeration principles and an overview of recent activities in the development of small cryocoolers. Chapter 3 outlines a mathematical model for the plastic cryocoolers considered. The results of this analysis have been a guide throughout the project. The remaining chapters deal with the experimental work. Several cryocoolers have been built in the course of the project, and the considerations and results in this thesis are typical for a number of these coolers. Therefore, a division by design (chapter 4 ) , construction (chapter 5 ) , operating data (chapter 6) and a discussion of performance (chapter 7) seemed more appropriate than a full description of each type of cryocooler system separately. This has the added benefit that this thesis can be used as a general guide for developing new cryocoolers of this type. A computer program based on the theoretical model, and a detailed manufacturing scheme for the conical displacer with sleeve are listed in appendices. The complete list of references and a summary with pertinent conclusions are given at the end of this thesis.

1. INTRODUCTION 5 5QUI0 requirements

SQUID requirements

A SQUID is a ^ery s e n s i t i v e magnetometer, based on superconductive phenomena [ 1 ] . I t e s s e n t i a l l y measures magnetic f l u x through a

super--14

conducting r i n g , r e s u l t i n g in a s e n s i t i v i t y below 10 T f o r t y p i c a l devices. Actually, in many measurements s e n s i t i v i t y is 1imited by external interference rather than by the device i t s e l f . Thus, much a t t e n t i o n must be paid to s u i t a b l e electromagnetic s h i e l d i n g , f i l t e r i n g and compensation techniques. In the f o l l o w i n g , several aspects of cooling a SQUID are described.

- Ohmic d i s s i p a t i o n in a SQUID is n e g l i g i b l e compared with p r a c t i c a l losses in a cooling system, such as conducted heat and radiated heat. So, in p r i n c i p l e cold-end refrigeration capacity is not important. Design optimization concentrates on the cool ing d i s t r i b u t i o n between the cold end and ambient temperature.

- Superconductivity is obtained only at low temperatures. SQUIDs are usually designed f o r 4.2 K operating temperature. Maximum operating temperature depends on many parameters, the upper l i m i t being the c r i t i c a l temperature of the applied superconductor. This means about 8.5 K f o r present niobium SQUIDs, but in general, lower temperatures are highly d e s i r a b l e . Progress in high-T devices, as described by D i l o r i o and Beasley [ 2 ] , Konishi and Kuriki [ 3 ] , Zarembihski and Kachniarz [ 4 ] and Rogalla et a l . [ 5 ] , would relax these requirements considerably by extending the operating temperature range up t o 19 K.

- Because the SQUID'S operating parameters are temperature dependent, fluctuations in cold-end temperature i n t e r f e r e with measurements. Long-term v a r i a t i o n s ( d r i f t ) can be reduced by a c t i v e temperature c o n t r o l , the r e l a t i v e l y f a s t c y c l i c f l u c t u a t i o n s in r e c i p r o c a t i n g r e f r i g e r a t o r s can be damped by introducing more heat capacity at the cold end. An extension with additional thermal impedance and heat capacity plus an a c t i v e l y c o n t r o l l e d heater may be used t o f u r t h e r s t a b i l i z e the device temperature. Depending on the a p p l i c a t i o n , s u i t a b l e external f i l t e r i n g can be s u f f i c i e n t . In general SQUIDs are

(7)

1. INTRODUCTION 6 SQUID requirements

less temperature dependent at lower temperatures. In addition, an increment in cooling capacity enlarges the possibilities of special temperature stabilization.

- The extreme sensitivity of a SQUID requires an extremely low electro magnetic interference level of the cooler. Remanent magnetism should be avoided at all times. Low-temperature noise of materials with non zero susceptibility can be unacceptable. Influence of Johnson-noise, eddy currents etc. in conducting materïals should be kept to a minimum. Of course moving parts deserve special attention. Mechanical vibration is a serious problem; Tilting a simple SQUID in the earth's magnetic field over 10"J radians corresponds to a signal of about 10" tesla, which is too high for typical sensing levels.

When the interfering signal lies within the frequency band of interest the following standard techniques are available:

■ Shielding DC and low frequency magnet ie fields with a rnumetal enclosure,

■ Shlel ding low and high frequencies with a high-conductivity enclosure,

■ Active compensation with externally generated compensating fields, ■ Specïal input coil arrangements with selective sensitivity to the

fields of interest, e.g. gradiometers,

■ Special filtering techniques, e.g. averaging the signal of interest or electronic substraction of the interfering signal.

Zimmerman [6] describes application of some of these techniques.

- Depending on the type of measurement, several other parameters must be considered, such as cost, dimensions, simplicity of operation ("throwing a switch"?), reliability, service interval, cool down time etc. Even within one field of application like biomagnetism [7 ] or

geomagnetism [8], it is difficult to derive solid conclusions, although Simmonds [9] and Nisenoff [10] give some estimates. Sometimes, widely different approaches are possible for certain applications. For instance, a very short cooldown time makes entirely different operating schemes possible, and very low cost may sometimes

relax life time and mean time between failure demands.

2. REFRIGERATION SYSTEMS Idealized cycles

Ideal ized cycles

2. REFRIGERATION SYSTEMS

Figure 2 . 1 . Temperature-entropy diagram of 3 idealized cycles. (a) Carnot: isothermal and ïsentropic l i n e s .

(b) S t i r l i n g : isothermal and isochoric l i n e s . (c) Ericsson: isothermal and isobaric l i n e s .

Refrigeration cycles can be characterised by t h e i r temperature-entropy diagram. Figure 2.1 gives three examples of idealized cycles, known as respectively the Carnot, the S t i r l i n g and the Ericsson c y c l e . The absorbed heat related to entropy change is

dQ < TdS . (2.1)

If the process is reversible, refrigeration is obtained by absorbing heat at low temperature:

T , A S , (2.2)

represented by the area under the lower line, and by releasing heat at ambient temperature:

T A S (2.3)

represented by the area under the upper line, and by supplying a total amount of work

(8)

2. REFRIGERATION SYSTEMS 8 idealized cycles

represented by the enclosed area, to complete the cycle. Note that nothing has been assumed about how the work is accomplished. Therefore, the concepts generalized force and generalized displacement are introduced, corresponding to pressure and volume in a gas-liquid system, tension and length in e rubber band, magnetic field and magnetic moment in a magnetic system, electric field and dipole moment in a dielectric system, etc. In the Carnot cycle the two isotherms are connected by adiabatic lines, which in practice makes it difficult to reach the upper left corner in the TS-diagram of this cycle. This requires the generation of a very large generalized force. The Stirling cycle uses lines of constant generalized displacement, and the Ericsson cycle uses lines of constant generalized force. In these cases the working fluid rejects heat in going to low temperatures, and absorbs heat in going up as represented by the areas under these lines. This leads to the use of a counter flow heat exchanger in the case of synchronous flow, or a regenerator in the case of asynchronous flow. In the former, heat is exchanged directly between the two flows, and in the latter, heat is temporarily stored in the regenerator matrix material. In fact, a wide range of other idealized cycles is possible.

When they &re reversible, all three processes have the same efficiency,

equal to the maximum possible or Carnot efficiency:

W ( V T

l ) a S Ta - T,

Any real cycle will not be fully reversible, leading to a (much) lower efficiency.

The preceding is concerned with cycles ooerating between two fixed temperatures. When the heat leak from the surroundings dominates the heat generated in the device - the situation of our interest - it becomes advantageous to absorb heat over a range of temperatures. As can be seen from (2.5) it is more efficient to intercept the heat leaks at the higher temperatures, which reduces the more costly effort at the lowest temperature. In a conventional liquid helium bath this is more or

2. REFRIGERATION SYSTEMS 9 Idealized cycles

less automatically accomplished by using the heat capacity of the b o i l e d - o f f gas (70 times the heat of evaporation !) to gradually absorb the conducted heat and radiated heat from the surroundings. In the same way, special cryostats with r a d i a t i o n shields a c t i v e l y cooled by a small r e f r i g e r a t o r are f e a s i b l e only when the d e v i c e - d i s s i p a t i o n is very low. A closed cycle r e f r i g e r a t o r in contrast t o a l i q u i d helium bath -gives the opportunity t o choose an optimum r e f r i g e r a t i o n d i s t r i b u t i o n between ambient and low temperature.

The essence of any r e f r i g e r a t i o n system is the p o s s i b i l i t y to control the t r a n s i t i o n between ordered (low entropy) and disordered (high entropy) states using a generalized f o r c e . I t is useful to d i s t i n g u i s h cooling by doing r e v e r s i b l e work on the surroundings and cooling without external work. The former usually works in a wide range of temperatures, but some means must be provided to do work on the surroundings at low temperature. The l a t t e r is based on a change in i n t e r n a l energy or enthalpy and is usually l i m i t e d to a c e r t a i n temperature range. Another d i s t i n c t i o n is whether the system goes around the cycle in the time domain r e s u l t i n g in i n t e r m i t t e n t cooling , or in the s p a t i a l domain -using t r a n s p o r t between d i f f e r e n t locations - , or both. Radebaugh [11] discusses many possible systems, estimating r e f r i g e r a t i o n p o t e n t i a l by the maximum entropy change.

(9)

2. REFRIGERATION SYSTEMS 10 I r r e v e r s i b l e gas expansion

I r r e v e r s i b l e gas expansion

I r r e v e r s i b l e expansion of a gas by means of a c a p i l l a r y or a porous plug is known as Joule-Thomson expansion. Enthalpy is preserved in the process, no heat from the surroundings is absorbed. For a perfect gas the cooling e f f e c t is zero. In the Van der Waals model the a t t r a c t i v e force between molecules gives r i s e to cooling, while the f i n i t e dimensions of the molecules cause heating.

The Joule-Thomson valve consists of a c o n s t r i c t i o n , usually an o r i f i c e or a long narrow c a p i l l a r y . The l a t t e r can be flow-adjusted by inserting a w i r e . Small p a r t i c l e s and frozen alien gases clogging the c o n s t r i c t i o n are a serious problem, especially in miniature devices. A contamination-free compressor, f i l t e r s , adsorbers and procedures for regular purging are necessary.

heat exchanger

orifice

reservoir

Figure 2.2. Joule-Thomson process.

The cooling effect at s t a r t i n g temperature and pressures is s h i f t e d to lower temperatures by using counterflow heat exchangers, see f i g u r e 2 . 2 . L i t t l e [12a,b] discusses scaling laws for microminiature Joule-Thomson systems and observes that laminar flow can be acceptable in such systems. Seal ing down not only results in smal 1 dimensions and low cooling capacity, but also in low gas consumption, allowing the use of bottled high-pressure gas, and in a low cooldown time, reducible to seconds. A new etching process, suitable f o r mass production, was

2. REFRIGERATION SY5TEM5 11 I r r e v e r s i b l e gas expansion

developed to manufacture such small scale devices, even down to the dimensions of a standard DIL-16 chip package, in a glass substrate [12c]. Single stage devices for cooling to 80 K are now produced on a commercial basis [12d].

Unfortunately the temperature below which a net cooling e f f e c t can be obtained, the so called maximum inversion temperature, is about 40 K f o r hel lurn and 202 K f o r hydrogen, so precool ing is necessary when using these gasses. Precooling of the helium is obtained by adding separate stages using other gases ( e . g . hydrogen and n i t r o g e n J , or by using another cooling system, or by integrating Joule-Thomson valves for low temperatures and one or more expansion engines for higher temperatures in a single gas system. The last p o s s i b i l i t y is a t t r a c t i v e because i t reduces the number of components considerably as compared with a t y p i c a l l y large and complex multiple-gas system. Some experiments with the integrated type w i l l be described in a l a t e r chapter.

The advantages of Joule-Thomson coolers are the absence of moving parts at low temperatures, t h e i r simple construction and t h e i r f a s t cooldown. The disadvantages are the high pressures involved, the need for a high p u r i t y gas system, and the l i m i t e d temperature range which leads to a complex m u l t i - s t a g e system. Although substantial progress has been made to meet these challenges [12-16], an o f f - t h e - s h e l f system s u i t a b l e f o r our purposes is not yet a v a i l a b l e .

(10)

2. REFRIGERATION SYSTEMS 12 Reversible gas expansion

Reversible gas expansion

A reversible expansion always produces cooling and a wide range of cycles employs this principle. The idealized Stirling end Ericsson cycles, as shown in figures 2.1b and 2.1c, are two well known examples, but numerous variants (e.g. in which parts of the cycle are replaced by adiabatic lines) and arrangements in which these cycles are realized exist, along with a. profusion of confusing names. The general idea is to compress the gas and remove the heat of compression at ambient temperature, and to transport the gas to the cold section. Then, cooling is obtained by reversible expansion, and finally the gas goes up in temperature while absorbing the heat given off in going down. Important deviations from this scheme exist, however. Several possible arrange ments are summarized in the following paragraphs:

Low temperature arrangements. Several possibilities are known for the acLual cooling stage:

- Expansion turbine with counterf low heat exchanger. Turbines are difficult to lubricate and already of small size for large refrigera tors. Probably unsuitable for scaling down to milliwatt cool ing. - Expansion piston with regenerator. Heat leak and low temperature

sealing problems are difficult to solve, especially in small systems. - Displacer with regenerator. Expansion is control led Dy the

room-temperature part, the displacer takes care of gas transport only. - Pulse tube, acoustic systems. The pulse tube was proposed by Gifford and Longsworth [17], Recently, Wheatiey and Cox [18] reviewed the unusual principle of operation of such systems in which the natural gas transport in a pressure wave is used instead of a displacer. Several configurations are possible. The main advantage is the absence of moving parts at low temperature, it appears to be difficult to reach temperatures below 100 kelvin, although recent experiments indicate that the performance of a staged concept may be similar to other regenerative coolers [19],

- Combinations. E.g. Roubeau's "crossed cycles refrigerator" [20] can be seen as a combined compressor and displacer piston, without the regenerator. The cycle resembles that of the pulse tube.

2. REFRIGERATION SYSTEMS Reversible gas expansion

Currently, the displacer with regenerator is the proven concept for smal 1 cryocoolers. There is no significant pressure drop over the displacer, so driving mechanism and seal can be simplified or even omitted. In omitting the driving mechanism, the movement of the so called free displacer is controlled by the small pressure drop over the displacer. Resonance, usually obtained by using gas-springs, is needed for acceptable operation, emitting the seal results in using the annular gap as regenerator. While being mechanically trivial it results in a low pressure drop and it is well suited to low power systems in which the heat flow in the regenerator is relatively small. To obtain distributed cooling the displacer is usually staged to obtain expansion spaces at several temperature levels. Figure 2.3 shows some of these concepts.

,

H-^

V/

///

n_

I i

l

i

VA

ir

i

Figure 2.3. Displacer/regenerator configurations. (a) External -, (b) internal -, (c) gap regenerator. (d) Stepped displacer.

Counterflow versus regenerative heat exchanger. In counterflow heat exchangers the important parameters are high transversal heat conducti vity, low longitudinal heat conductivity and low flow losses. Regenera tive heat exchangers require in addition high heat capacity and low void volume. In spite of this, regenerative heat exchangers usually have very high efficiencies with low flow losses and smal 1 dimensions. The conventional matrix materials are fine copper wire or copper mesh and lead shot. Numerous research projects are directed towards finding the optimum regenerator matrix. The most serious problem in using regenera tors below about 20 K is that the specific heat of solids drops steeply

(11)

at lower temperatures. At about 10 K, the heat capacity per volume of the commonly used lead matrices drops below that of helium of 5 bar. It is not easy to improve on the heat capacity of a lead matrix, although some possibilities are known. Another approach is to lower the pressure of the working gas. In this way Zimmerman and Sullivan [2lf] were able to achieve cooling of several mW at 3 to 4 K, rejecting 50 to 100 mW at 8 to 14 K in a single-stage low-pressure Stirling cryocooler. Throughout the cycle liquid helium was present in the cold space. Counterflow heat exchangers are more sensitive to plugging than regenerators, but do not have the heat capacity problem. Several configurations have been proposed which combine the advantages of both in a hybrid system, e.g. by Daney [22].

Ambient temperature arrangements. The compressor, c.q. expander, can be located directly on top of the cooling unit, or connected to it with a gasline. The separation in the latter, so called split configuration is convenient in construction and avoids interference from the compressor to a sensor mounted in the cooler. In all cases the heat of compression is removed at ambient temperature. Possible arrangements are:

- Continuous high speed compressor. E.g. the Gifford-McMahon cooler uses a compressor connected with valves to a displacer with a regenerator. The irreversibility in the valves reduces efficiency (this can be avoided in more complicated configurations), but the reliability is usually very good.

- Discontinuous piston-compressor-expander. The Stirling cooler uses a piston compressor-expander connected without valves to a displacer and regenerator. A wide variety of arrangements of pistons, displacers, regenerators and driving mechanisms is possible.

- Thermo compressor-expander. The Vuilleumier cooler uses an extra, heated displacer with regenerator to generate a pressure wave for the low temperature stage. Mechanical work is replaced by heating and the driving forces on the two displacers are small, leading to low wear, high reliability and compact design. The pressure ratio is limited by maximum allowable temperature and void volumes. In some systems the displacers are self-driven or replaced by turbines and the entire

cooler can be hermetically sealed, heat being the only input.

Elimination of wear and contamination. The essential features of the compressor are r e l i a b i l i t y and very low contamination of the helium. Although impressive r e l i a b i l i t y has been obtained in conventional, o i l -lubricated compressors, the associated complexity of f i l t e r s , absorbers and gas conditioning is u n a t t r a c t i v e f o r small, low-power coolers. The important source of wear is rubbing of moving p a r t s , especially piston against cylinder w a l l . Zimmerman [21a-21c,21h-21k] achieved continuous t r o u b l e - f r e e operation over several months in a crank-driven piston compressor-expander with c a r e f u l l y made, s l i d i n g rubber 0-ring seals lubricated with grease. Reduction of wear can be expected in using dry-l u b r i c a t i n g seadry-ls or r o dry-l dry-l i n g membranes, using hardened and podry-lished surfaces and achieving lowest possible mechanical loads. Zimmerman [21m] and Heiden [23c] achieved excellent performance in a f i n e - g r a i n e d aluminum oxide ceramic piston and c y l i n d e r with r a d i a l clearance of a few pm. Such a close tolerance seal is not hermetically t i g h t , and has to be backed by another seal on the crankshaft, e . g . a small s l i d i n g 0-ring seal or metal bellow. Elimination of wear is obtained by using close tolerance seals combined with proper bearings such as e l a s t i c supports - r a d i a l s t r i n g s , metal s t r i p s , membranes - , or gas bearings. An approach of current i n t e r e s t are magnetic bearings, using accurate ('x. 3 \M) p o s i t i o n sensors. Combined with l i n e a r - d r i v e motors to eliminate crankshafts, hermetically sealed systems with very high l i f e -expectancy are p o s s i b l e . The f r e e - d i s p l a c e r concept i s e a s i l y incorpora t e d . Another p o s s i b i l i t y is the use of bellows instead of p i s t o n s . Roubeau [ 2 0 ] , Daunt and Heiden [23a], and Smith [24] describe t h e i r experiences. In order t o avoid f a t i g u e in metal, a l l deformation should be kept several orders of magnitude below the e l a s t i c l i m i t . This results in designs with a large area or with a high f o r c e , e . g . a wide diaphragm or a t i l t i n g s e a l . Very good r e s u l t s have been obtained with t e f l o n bellows [23b], but a f a i r amount of helium permeation must be allowed f o r in the design. In general, bellows and diaphragms have good prospects f o r sealing and s i m p l i c i t y . Elimination of a l l moving parts of a compressor can be obtained by using heating and c o o l i n g . Roubeau [20] suggests such an extremely simple system, c o n s i s t i n g of i n l e t and o u t l e t valve, a heating wire and cooled w a l l s . Considerable research by Hartwig and others [16] showed the f e a s i b i l i t y of using gas adsorption and

(12)

desorption in zeolites. Impressive quantities of gas can be stored in such an adsorber, to be released at high pressure by simply heating. The advantages are many: Hermetical seal, no contamination, low cost, very

high reliability and life expectancy, no vibration and the ability to use waste heat. A disadvantage is the low efficiency. At room tempera ture, most gases can be stored in standard absorbers. Hydrogen can be stored very effectively by chemisorption in La Ni^-powder, but for helium no room temperature solution is known.

2. REFRIGERATION SYSTEMS 17 Other refrigeration systems

Other refrigeration systems

Radebaugh [11] discusses many systems which exhibit an interesting entropy change. The first problem is to find a practical "handle", i.e. a. generalized force. This makes systems with for example rotational ordering rather unattractive. Still there are many other possibilities than the conventional gas expansion and gas-liquid systems. McCormick and Brauer [25] estimate that down to 70 kelvin a combination of thermoelectric (Peltier effect) and thermomagnetic (Ettinghausen effect) cooling is feasible. Magnetocaloric cooling (e.g. adiabatic demagnetiza tion) in several stages down from room temperature appears to be highly efficient in both power consumption and volume, but the presence of strong magnetic fields precludes the use for cooling SQUIDs. Electro-caloric cooling avoids this problem, but suitable semiconductor materials are not easily prepared due to clustering at high doping levels [26], Among the serious candidates is cool ing by desorption of

gases. The heat of desorption is much larger than the heat of evaporation and transportable from much higher temperatures. Contamina tion by alien gases will generally pose no problem, and the distributed cooling is particularly attractive in low power systems. Little research has been done along this line.

(13)

2. REFRIGERATION SYSTEMS 18 Current approaches

Current approaches

Although closed-cycle r e f r i g e r a t o r systems have d e f i n i t e advantages, the old-fashioned l i q u i d helium dewar continues t o be improved f o r cooling small sensors- E.g. Byrd and Hansen [27] describe a p p l i c a t i o n of a commercially available f l e x i b l e l i q u i d helium t r a n s f e r system between storage dewar and a SQUID, f o r freedom of o r i e n t a t i o n , reduced helium consumption and enhanced temperature s t a b i l i z a t i o n over a wide tempera ture range. Vincent [28] describes a small cryocooler l i q u e f a c t i o n system, in which l i q u i d helium is transferred by g r a v i t y - f l o w to a separate dewar with a SQUID susceptometer. The separation permits v i b r a t i o n damping and electromagnetic shielding between both u n i t s .

Along with r i s i n g interest f o r smal 1 cryocoolers, a series of four s p e c i a l i s t conferences were held during the l a s t 8 years:

"Applications of Closed-Cycle Cryocoolers t o Small Superconducting Devices", NBS Boulder, Colorado, Oct 3-4, 1977.

"Refrigeration f o r Cryogenic Sensors and Electronic Systems", NBS Boulder, Colorado, Oct 6-7, 1980.

"Refrigeration f o r Cryogenic Sensors and Electronic Systems",

NASA Goddard Space F l i g h t Center, Greenbelt, Maryland, Dec 7-8, 1982. "Third Cryocooler Conference",

NBS Boulder, Colorado, Sep 17-18, 1984.

The main stimulants f o r research a c t i v i t i e s in t h i s f i e l d are the commercial success of cryopumps, now being massproduced, growing i n t e r e s t in space applications and continuing i n t e r e s t in cooled infrared sensors for defense a p p l i c a t i o n s . The overriding requirement is r e l i a b i l i t y , and f o r airborne applications weigth, s i z e , energy con sumption and robustness. Space applications are usually intended f o r 5 or 10 years maintenance-free operation. Another push f o r new cryocooler concepts originated from the development of superconducting Josephson computers. In a l l these a p p l i c i o n s , high cost is not objectionable. A c t u a l l y , low costs tend to be viewed with suspicion: The manufacturer expects lower p r o f i t , and the user expects lower q u a l i t y . The r e l i a b i

-2. REFRIGERATION SYSTEMS 19 Current approaches

l i t y and l i f e expectance of conventional cryocooler systems f o r a few Watt cooling power, e . g . at 80, 20 or 4.2 k e l v i n , can be extremely good, even at 1 t o 3 years maintenance i n t e r v a l s [ 2 9 - 3 2 ] . A large e f f o r t towards smal Ier and wear-free cryocoolers consists of the previously mentioned concept of close tolerance seals with magnetic bearings and linear d r i v e [ 3 1 - 3 4 ] . Recently, new designs f o r adiabatic demagnetiza t i o n in the 20 K to 4 K range were developed [ 3 5 - 3 8 ] .

The previous concepts were mainly concerned with very high r e l i a b i l i t y and do not seem part i c u l a r l y suited to cool SQUIDs because of insuf f i c i e n t l y low temperature or magnetic interference by moving metal parts etc. Cox and Wolf [39] measured the magnetic and v i b r a t i o n a l signatures of a standard, commercially available Gifford-McMahon r e f r i g e r a t o r with Joule-Thomson stage. Their data suggest t h a t an o f f - t h e - s h e l f system of t h i s type is not compatible with the use of SQUIDs, the noise produced at the cold stations being greater than the operating range of the least sensitive SQUID. Heat t r a n s f e r techniques extend the p o s s i b i l i t i e s of dewars in freedom of o r i e n t a t i o n and dimensions and allow s u f f i c i e n t shielding between SQUID and some standard closed-cycle r e f r i g e r a t o r . These techniques are state of the a r t solutions t o t h e problem of cooling a SQUID. However, such systems are grossly mismatched t o the device in cooling capacity, cost and s i z e . L i t t l e has been done t o d i r e c t l y meet the special requirements of SQUIDs. Almost any optimiza t i o n is directed towards high e f f i c i e n c y , but trie low d i s s i p a t i o n in a SQUID makes the concept of e f f i c i e n c y almost meaningless. Also, the rigorous use of non-magnetic materials is scarcely seen. The f o l l o w i n g approaches are in the opposite l i n e .

Zimmerman and coworkers [21] pioneered the development of low power, p l a s t i c , s p l i t S t i r l i n g r e f r i g e r a t o r s with gap regenerator. A 3-stage u n i t achieving 13 K operated f o r more than 6C00 hours with uninterrupted periods of up t o 5 weeks. A 4-stage u n i t reached 8.5 K, allowing operation with a niobium point-contact SQUID. A 5-stage u n i t reaching

7 K was operated with a SQUID-gradiometer. The compressor used was an

0-ring sealed metal piston or a close tolerance ceramic p i s t o n . Recently, the compressor was replaced by a pneumatic a i r driven double

(14)

diaphragm, and the S t i r l i n g cooler was extended with a Joule-Thomson stage, reaching l i q u i d helium temperature [ 4 0 ] . These developments are described in a l a t e r chapter of t h i s t h e s i s . Daunt and Heiden [23a J b u i l t a s i m i l a r 3-stage p l a s t i c displacer u n i t , reaching 18 K. They report about the use of a contamination-free bellows compressor-expander driven by the gas pressure of a standard piston compressor-expander. In stainless s t e e l , l i f e t i m e was l i m i t e d t o only 10 strokes (fatigue f r a c t u r e ) . Teflon bellows were found to be excellent [ 2 3 b ] . Myrtle et a l . [41] successfully b u i l t a S t i r l i n g cooler with conical p l a s t i c displacer, reaching 9 K. Their current i n t e r e s t i s a variant of the Simon expansion cool er, mounted on a conventional Gifford-McMahon r e f r i g e r a t o r [ 4 2 a ] . This system is now commercially a v a i l a b l e [42b] and delivers 3 mW of continuous cooling power at 4.2 K, with a temperature s t a b i l i t y of 0.03 degrees. The main advantages are the inherent temperature s t a b i 1 i t y and the r e l i a b i 1 i t y compared with a plugging-sensitive Joule-Thomson stage.

Stephens [43] describes the assortment of Joule-Thomson coolers commer c i a l l y available from Hymatic. These very smal 1 devices operate on bottled high-pressure gas and are intended f o r infrared a p p l i c a t i o n s . Typical cooling power i s of the order of a few Watts at 80 K, 20 K or 4 K. Special geometry and reduction of thermal mass can r e s u l t in cool down times below 2 seconds. Some s e l f - r e g u l a t i n g variants combine rapid cooldown with low steady state gas consumption [ 4 4 ] . As was mentioned above, a set of scaling 1 aws f o r Joule-Thomson coolers was derived by L i t t l e [12] in 1977. Microminiature Joule-Thomson systems f o r nitrogen (80 K) are now mass produced in small glass plates by a special etching technique. The main problem appears to be the s e n s i t i v i t y to clogging by impurities (operated from 99.996% pure h-, high pressure b o t t l e s ) . A suitable compressor f o r closed cycle operation and a hydrogen l i q u e f i e r with integrated nitrogen precooler are being developed. If these systems can be successfully extended t o superconduc t i v e temperatures, they could be a t t r a c t i v e t o cool very small devices with minimum interference and dimensions. Tward [13] constructed a 4-stage Joule-Thomson cooler f o r cooling small sensors to l i q u i d helium temperature. A contamination f r e e , elastomere diaphragm compressor f o r

these four gases is being developed w i t h i n the same p r o j e c t . The work on adsorption-desorption compressors [16] could lead to highly r e l i a b l e , low cost and low interference systems. Their main disadvantage w i l l probably be the very low e f f i c i e n c y : about 10 kW input power was calculated f o r 10 mW of cooling at 4.2 K with present m a t e r i a l s . Jones [16g] indicated t h a t the use of hydrides makes operation down t o 10 K by hydogen sublimation possible at an e f f i c i e n c y better than most mechanical systems.

This Thesis. Extensive research in complex systems or new materials was beyond the scope of the small project described in t h i s t h e s i s . The system chosen was a p l a s t i c , S t i r l i n g - t y p e r e f r i g e r a t o r with gap regenerator, because at the s t a r t of the p r o j e c t , the only approach t h a t had been successful w i t h i n our objectives were the experiments of Zimmerman. This s i t u a t i o n has not changed during these years, although several other approaches, e . g . the e a r l i e r mentioned Joule-Thomson cooler of Tward, are on the verge of t e s t s in a p r a c t i c a l a p p l i c a t i o n now. The concept of Zimmerman i s p a r t i c u l a r l y a t t r a c t i v e because of the low cost and simple f a b r i c a t i o n techniques involved. In f a c t , t h i s aspect makes commercial research u n a t t r a c t i v e , which can be understood as another motivation f o r research along these 1ines in an u n i v e r s i t y environment. Apart from Zimmerman and coworkers at NBS Boulder, USA, and our project in D e l f t , Holland, work on the same type of coolers is being done in Giessen, Germany, by Heiden and in Rio de Janeiro, B r a z i l . To my knowledge other attempts, such as the e a r l i e r mentioned e f f o r t in Canada, by Myrtle et a l . and some commercial p r o j e c t s , have not been pursued.

(15)

3. MODELLING 22 Introduction

3. MODELLING

Introduction

Existing models. Many t h e o r e t i c a l models have been developed f o r

describing several types of S t i r l i n g coolers employing several types of operating cycles. A detailed analysis of each component ( e . g . regenerator, compressor, heat exchanger) can be made separately, and usually a set of semi-empirical parameters is derived which describes a s p e c i f i c class of component, over a given operating range ( e . g . a copper mesh regenerator at low temperatures in a Schmidt c y c l e ) . The general approach is then t o use these r e s u l t s in an overall system a n a l y s i s . The emphasis is on the p r e d i c t i o n or optimization of the system's e f f i c i e n c y or cooling capacity at c e r t a i n temperatures. Walker [45] presents an extensive review of such approaches. For large engines, and l i f e t i m e considerations a detailed mechanical analysis is also important. More r e c e n t l y , i n t e r e s t in f r e e piston and resonant systems stimulated research on the d e t a i l s of time-dependent i n t e r a c t i o n s in such coolers [ 4 6 ] .

Although the p l a s t i c coolers considered in t h i s thesis operate along the same basic p r i n c i p l e s , the construction and operating range deviate too much from the standard type of coolers, and the existent models can not be used. Radebaugh and Zimmerman [21d,e] pointed out that the s h u t t l e heat losses and the e f f e c t of thermal penetration in the p l a s t i c are important. A computer model was developed along these l i n e s t o optimize the geometry of a tapered displacer with gap regenerator f o r the lowest input power [21m,n].

Present model. The analysis presented here uses a s i m i l a r model, but

an attempt was made to use a f u l l y a n a l y t i c a l d e s c r i p t i o n wherever possible, and to include a detailed description of important deviations from the idealized model. I t applies to p l a s t i c , low-power cryocoolers with gap regenerator and a tapered d i s p l a c e r . The model includes regenerator losses, s h u t t l e heat t r a n s f e r , radiated heat and thermal conduction. The effects of low temperatures, void volumes and

e x c e n t r i c i t y of displacer in c y l i n d e r are discussed. No p a r t i c u l a r room temperature c o n f i g u r a t i o n is assumed ( i . e . compressor, valves, displacer d r i v e , heat exchanger), and a d e s c r i p t i o n in terms of d i f f e r e n t cycle parameters is used instead. The analysis leads to a simple equation f o r the temperature gradient, which can be solved numerically in a single i t e r a t i o n loop. The attractiveness of t h i s simple one-dimensional approach is t h a t i t allows f o r f a s t c a l c u l a t i o n s . Thus, i n s i g h t in a p a r t i c u l a r design can be gained by interactive computation. (A simple Pascal implementation is l i s t e d in appendix A of t h i s t h e s i s . On an IBM PC-type microcomputer with 8087 numerical coprocessor, i t calculates each i t e r a t i o n in less than a second. When a proper i t e r a t i o n algorithm is used, the program converges w i t h i n 10 t o 20 i t e r a t i o n s . ) This is very important f o r several reasons. F i r s t of a l l , the regenerator losses cause strong i n t e r a c t i o n s between d i f f e r e n t sections of the d i s p l a c e r . Secondly, in the low temperature range, several parameters have a strong temperature dependence. For these reasons, i t is generally d i f f i c u l t to predict the reaction of the e n t i r e system t o a small change. Also, a straightforward optimization of a design f o r a given condition may lead t o a very bad s o l u t i o n , because i t i s very s e n s i t i v e t o a small deviation from the ( i d e a l i z e d ) model. Therefore, i t is much better t o t e s t several v a r i a t i o n s of a p a r t i c u l a r geometry or operating condition manually, and to i n t e r p r e t the r e s u l t s w i t h c a u t i o n .

Thermal penetration. Cyclic heat flow through the surface of the

displacer and c y l i n d e r is generally l i m i t e d by thermal penetration in the p l a s t i c . When the (uniform) surface temperature T ( t ) is a harmonic function of time t w i t h frequency v

8 Ts( t ) = Ts( t ) - T_ - e '2" ^ , (3.1)

the temperature d i s t r i b u t i o n in a s e m i - i n f i n i t e s o l i d (see f i g u r e 3.1)

i s e a s i l y worked out ( e . g . by Carslaw and Jeager [ 4 7 ] ) t o be

- X / AT+ i ( 2 ï ï v t - x / A )

6 T ( x , t ) % e ' . (3.2)

The thermal penetration depth AT depends on the frequency v, and the s o l i d ' s thermal conductivity k$ and s p e c i f i c heat (per volume) c :

(16)

(3.3)

The temperature waves are strongly attenuated, so the solution f o r the s e m i - i n f i n i t e solid can be used in cases where the thickness exceeds a

The heat flow per unit area i n t o the surface i s

- f c J i r L o = ( 1 + i > — « Ts( t ) = el ï ï / 4(2 l tv kscs) V5{ t ) . (3.4) AT

The temperature's phase lags n/4 behind the heat f l o w ' s phase. (A simple heat capacity gives it/2 phase lag, while conductivity alone r e s u l t s in zero phase l a g . )

6T

Figure 3 . 1 . Thermal penetration in s e m i - i n f i n i t e s o l i d . Temperature 5 T ( x , t ) f o r t = n/12v , n = 0 , 1 , . . , 6 . 3. MODELLING Symbol reference. Basic equations v cycle frequency p helium pressure ( f i g u r e 3.3) s displacer stroke ( f i g u r e 3.2 and 3.3) N molar amount of gas, cycled each stroke n molar gas flow

\l volume

D displacer diameter (D., D?; radiation shield diameter) w wall thickness of sleeve (^cylinder)

g gap (3.5)

v void space (minimum gap, f i g u r e 3.2) Y e x c e n t r i c i t y of displacer in sleeve (3.22) L r a d i a t i o n shield length

a p o s i t i o n along displacer axis ( f i g u r e 3.2)

T temperature at a c e r t a i n level e along displacer

oSB Stefan-Boltzmann constant

e e f f e c t i v e e m i s s i v i t y of r a d i a t i o n shields R gas constant

k thermal c o n d u c t i v i t y of helium gas ks thermal c o n d u c t i v i t y of s o l i d ( p l a s t i c ) cs s p e c i f i c heat by volume of s o l i d ( p l a s t i c )

c molar s p e c i f i c heat of helium at constant pressure (3.7) AT thermal penetration depth (3.3)

r thermal impedance per u n i t area in s h u t t l e losses (3.13) z thermal impedance per u n i t area in regenerator losses (3.18) Nu Nusselt number

^cond conducted heat loss (3.10) rad radiated heat loss (3.11) 0 regenerator heat loss (3.19) Ösh t shuttle heat transfer loss (3.14)

(17)

3. MODELLING 26 asic equations

Basic equations

displacer

Figure 3.2. Geometrical variables, displacer at highest position. Definitions. Figure 3.2 schematically illustrates the geometrical variables of the displacer unit. Note that the gap g may consist of both a static part, the void gap v, and a dynamic part, when the displacer is tapered. Here we take g to be the value at maximum stroke:

1 dD

s ~ ₂_ds. (3.5)

The swept volume is the a c t i v e l y displaced volume in a stroke s:

I L -- l sü2 . (3.6)

3. MODELLING 27 Basic equations

&P

D

displacement

Figure 3 . 3 . Cycle parameters. Top: Pressure wave. Center: Displacer movement.

Bottom: Pressure-displacement diagram.

Figure 3.3 shows the basic cycle. For the time being, the pressure-displacement diagram is assumed t o be rectangular, but the pressure and displacement are approximated by harmonic functions of time with unaltered peak t o peak amplitude. Deviations of a p r a c t i c a l cycle from these ( i n c o n s i s t e n t ) assumptions are treated l a t e r .

The working gas, helium, is assumed t o be a monoatomic perfect gas:

pV = NRT

Cp - 2 R (3.7)

N is the amount of gas in moles, and R the gas constant. The heat absorbed by the gas in r e v e r s i b l e processes now s i m p l i f i e s t o

dQ = NcpdT - T §^|ndp = | 4 d T

9T|pU|J ~ 2 ~T u' " 'U | J * ^3'8^

(18)

f i g u r e 3.3 results in a simple description in which transport of the gas in the regenerator is described by the f i r s t term, while (isothermal) expansion is described by the last term.

Gross cold production. For a rectangular pressure-displacement cycle, the gross cold production below a certain level due to expansion of a perfect gas is the product of cycle frequency y, pressure v a r i a t i o n AP and swept vol urne (3.6 ) :

0 = -v A | ï | 4?dt =4 VSD2AP . (3.9)

gross i aT|p dt 4 F

Thermal conduction. Axial thermal conduction through the p l a s t i c of the displacer and sleeve i s :

Radiated heat. Radiated heat is intercepted by the r a d i a t i o n shields in the vacuum can. They are surrounded by a few loosely wrapped layers of alumimzed mylar ( s u p e r i n s u l a t i o n ) . This can be described by a Stefan-Boltzmann law in which the emissivity e of the surfaces can be adjusted to account f o r the superinsulation. When many layers of superinsulation are used, the heat flow is more or less linearized and a description in terms of an e f f e c t i v e thermal conductivity becomes more appropriate. Since t y p i c a l l y only a few layers of superinsulation are used, we choose a Stefan-Goltzman description for the heat flow from shield 2 to shield 1 :

d

rad ^ S B ' W l V ^

4

-

T

l

4

> • <

3

"

11

>

-8 2 4

ÖCB = 5.6703-10 W/m K is the Stefan-Boltzmann constant. The indices refer to the shields. The term T D . represents the bottom of the s h i e l d . L. is taken to be the e f f e c t i v e length of the s h i e l d , i . e . including the effect of the exposed sleeve area between shields 1 and 2. The radiated heat turns out t o be a minor loss in wel 1-designed coolers, so there is no need f o r high accuracy here. The data from the

manufacturer of our superinsulation correspond, for a few layers, to an e f f e c t i v e emissivity c below 0 . 0 1 .

Shuttle heat t r a n s f e r . The v e r t i c a l movement of the displacer in the thermal gradient gives r i s e t o shuttle heat losses. Heat flows from the sleeve to the displacer, when the displacer is up, and i t flows back at a lower level when the displacer is down. A sinusoidal movement gives a temperature difference

6T(t) =\ s ~j s i n ( 2 * v t ) (3.12) between displacer and sleeve. The unit area thermal impedance r- of the

gap between displacer and sleeve consists of the thermal impedance of the sleeve w a l l , of the gas gap and of the displacer w a l l , a l l in series. The gas flow in these narrow gap, low-speed machines is laminar (Re<500), so the thermal impedance of the gas gap is simply g/k . The thermal impedances of the two p l a s t i c walls, however, are dominated by

Introducing the a u x i l i a r y variables r , r ,, and r ... , we a r r i v e at the t o t a l thermal impedance rT by vector a d d i t i o n :

r

„A. - ! - „Jj£a

r9 *g ' "sd - V ( „ kscs) ' rs d / / g ' rs d +rg ■

| rT| ■ ^5 d +< V2 + rs d2) • c o s<a r9 rT> ' ' ^ ' ( 3- '3 )

The s h u t t l e heat loss is found by averaging the product of the horizontal heat flow per unit area 5 T ( t ) / rT, the circumference *D and

the position during one c y c l e . The harmonic movement and phase s h i f t r e s u l t in a f a c t o r i cos(arg r - J :

j, cos(arg rT) AT o * J T

^shut = f Ds 2|r-| il -i Ds <2 rs d+Vrs d / / g > B7 • (3.14) The estimations of Radebough and Zimmerman [21(1] deviate s l i g h t l y from t h i s r e s u l t . Their numerical c o e f f i c i e n t f o r the gas gap is a factor H/n and f o r the p l a s t i c a f a c t o r ( v 2 ) 4 / i higher than derived here.

(19)

3- MODELLING 30 Basic equations

Regenerator losses. The regenerator acts as a heat sponge. Heat is absorbed from the helium when it flows towards the cold tip, and returned when the helium flows back. The heat exchange is not perfect; the temperature of the gas is slightly higher in the downward flow than in the upward flow. The driving factor is the gasflow n(t) in mole/s. The total amount of gas N that passes a certain level in each half cycle consists of the gas in the swept volume, and the gas in the void volume:

N = ƒ § f s ^ pc + V A P ) d* . (3.15)

We assume a sinusoidal gas flow, transporting N mole of gas in half a cycle:

nft) = jtvN sin(2nvt) . (3.16)

The horizontal heat flow is not only caused by the gas flow in the thermal gradient, but al so by the compress ion/expansion of the gas in the void volumes. When the unit area thermal impedance between gas and regenerator is zT, the temperature difference between gas and regenerator is

öT(t) = Zy ( ^ - c - j j j + *vvAp) sln(2itvt) . (3.17)

The phase difference between the term from the swept volume and the terms from the void volume has been ignored. This is certainly not correct, but the error is small when the void volume is reasonably small. In the gap regenerator-type cryocoolers considered here, the second term in (3.17) is generally small, and the phase difference between the terms in (3.15) can usually be ignored. In fact, the sinusoidal flow is the crudest assumption.

The thermal impedance between regenerator surfaces and the gas is g/k Nu. For laminar flow in a narrow flat box with a stabilized temperature profile, the Nusselt number for a constant, homogeneous heat

load is easily worked out to be Nu = 8.23 . The total thermal impedance zT is the impedance of the sleeve wall and displacer wall in parallel with the impedance of the gas in series (z and z . are auxiliary variables):

g k Nu '

y g

|zT| = V ( ( zs d +zg)2 +zs/ ) , cosfarg Zj) - ^ ^ ~ ■ (3-18)

The regenerator heat loss is found by averaging the heat content of the gasflow c n ( t ) ó T ( t ) over one c y c l e :

«reg = \ "\$ cp £ + »*P> <zsd+zg » ■ <3'1 9>

Stationary c o n d i t i o n . For a stationary temperature p r o f i l e we have at each level the simple condition

«gross ' Qcond + «rad + Qshut + % ' <3-2 0»

which can be w r i t t e n as:

d T jvsD^Ap - j v2NcDVAP(zsd+2q) - c O s ^ q ^ ^ M T . ,4- ! ,4)

61 Z < " VV 1<zs d + V +? ( ^ w ) \ 4 D s2( 2 rs d +rg- rs d / / g) -1

(3.21) For a given set of parameters, the temperature profile can be calculated numerically from this equation by estimating shield temperatures - T. etc. - and cold tip temperature, and then integrating the temperature gradient and the cycled amount of gas N from the tip upwards. Tne estimated temperatures are adjusted until the solution is consistent with the assumptions - shield temperatures consistent with temperature profile, top is at room temperature, etc -. When a proper estimating algorithm is used, this scheme gives reasonable convergence within 10 or 20 iterations.

(20)

3. MODELLING 32 Refinements, discussion

Refinements and discussion

Anisotropy. When us ing laminated plastic, thermal conductivity tends to be anisotropic. This can be taken into account by using the axial thermal conductivity in (3.10), and the radial thermal conductlvity in

(3.13) and (3.18). In fact, the sleeve and displacer materials are generally different. Although it is a simple matter to include this, the symmetric cases are shown here to stress the difference between gas gap and plastic contributions. It is interesting to note that the materials requirements for low shuttle heat loss are directly opposite to those for low regenerator losses. In principle, when one side of the gap (the sleeve) is made of a poor heat absorber, and the other side (the displacer) of an extremely good one, both losses can be made low simultaneously. Unfortunately, in practice it is quite difficult to improve the worst of the two anyway.

Gap width. In the case of a conical displacer section, the gap varies during the stroke. The main effect is that the effective gap g is slightly narrower than the widest gap - at full stroke -. The exact effect depends on the details of cycle employed. A more serious effect can be expected when the displacer is not exactly centered in the sleeve. Excentricity of the gap g is described by the dimension less parameter 7, see figure 3.4:

Figure 3.4. Excentricity of displacer in sleeve.

g(y,a) = (1 - ïcosa) g , 0 <y < 1 , 0 £ a < 2n . (3.22)

The flow resistance at a certain angle varies with the cube of the gap. Taking into account that the total mass flow is fixed, we find a correct'on factor for the thermal impedance of the gas gap:

r / JA , r m >3 . ,2 , 21 2 105 4 35 6

Z (0) /g(0f«)7da ( / g ( 7 , a ) V )2

For the thermal impedance of the plastic:

_zsd (3.23) ('O / g ( T ^ )5d a ( / g ( 0 , M3d a )2 J + ! | y1 + « / + , 5 ^6 (0) / g ( 0 , a )6 da ( J g f ï . e o V )2 1 3 2 , 2 1 + j y ) prefactor (3.24) 0 0.5 ^ ^ ^ excentricity

Figure 3.5. Prefactors in regenerator losses due to e x c e n t r i c i t y .

As shown in f i g u r e 3.5, t h i s rather d r a s t i c a l l y d e t e r i o r a t e s the regenerator losses. For completeness, a s i m i l a r f a c t o r is given f o r the compression and expansion heat in the void volume - the second term in (3.17) - although i t should be noted that in t h i s case the e f f e c t i v e e x c e n t r i c i t y 1 may be q u i t e d i f f e r e n t f o r the void v part of i t :

(21)

/g(Y,a)5d« fg(0,o)3da 1 + 5 -,2 + -g -,

5 r~" n —

(3

-

25) /g(0,a)Dd« /g(-,,«) da 1 * f i

f o r the thermal impedance of the gas gap, and

rg(-,,a)4d« fg(0,a)3d„ 1 + 3 ■? + f /

4 3~ = 3 ~ 2 ( 3-2 6 )

Jg(0,o) da f g (T, a ) da 1 + f y

for the thermal Impedance of the plastic. In a well designed machine, the compression-expansion term is rather small and we did not analyse the effect further. In the case of the shuttle heat transfer, the effect on the gas flow is absent, but the calculation is more complicated because the gap and plastic interact. When the thermal impedance of the plastic has a significant contribution, as is usually the case, the effect will be small.

Thermal lengths. Figure 3.6 shows the gap width where the thermal impedances of gas and plastic (3.13) are equal for a typical epoxy laminate at 1 cycle/s, as a function of temperature. Generally, the actual gap is of the same level, and both gas and plastic contribute In the calculations. Also shown are the thermal penetration depths In the plastic and helium at 1 cycle/s and atmospheric pressure. At low temperatures, the thermal penetration depth in the plastic may rise to the same order of magnitude as tip diameter, and corrections on the semi-infinite solid model are necessary. Similarly, thermal penetration in the gas may drop below the diameter of the expansion space at low temperatures. This causes non-isothermal expansion of the gas. When the gas is forced to be in good thermal contact on leaving the expansion space, as is easily accomplished by using a metal wall in this first part of gap, the main effect at the tip is a phase shift in the gas and heat flow, and we expect no serious loss in performance.

l

\

_\

\

_\

\

_\

\

_\

\

Gas gap Penetration in helium (0,1 MPa)

/

• ^ Penetration in

***•"•*, glass fibre epoxy

that matches plastic loss

10 100 temperature [ K ]

Figure 3.6 Thermal lengths f o r a t y p i c a l epoxy and helium at 1 cycle per second.

Real gas. Throughout, helium was treated as a perfect gas. At low temperatures t h i s assumption is no longer v a l i d . The parameters of

interest are the volumetric expansion c o e f f i c i e n t in (3.8) and ( 3 . 9 ) :

I Ml ? .

V a T j p ~ ' '

and correspondingly, the density in (3.15) and the specific heat c in (3.17). Down to about 6 kelvin, at the low pressures (6 atmospheres at most) involved, we expect more gross cooling, larger regenerator losses - due to larger gas flow and larger specific heat - and an extra loss from the enthalpy deficit between the low and high pressure gas flow (regenerator imbalance). The net result may be positive or negative, depending on the particular system. At even lower temperatures and/or higher pressures, the helium properties change dramatically, and our simple analysis should be replaced by a full treatment.

(22)

displacement

Figure 3.7. Truncated rectangular pressure-displacement diagram. Regenerator losses (mass flow squared) are reduced considerably. gross cooling (area) is almost the same. Dotted hyperboles are lines of constant pressure-displacement product (amount of gas).

Cycles. To achieve low temperatures, one has to concentrate on the r a t i o of gross cooling to regenerator losses. From equation (3.9) versus (3.15) in (3.19) i t is clear that a high compression r a t i o and a high active to void volume r a t i o are c r u c i a l in low temperature performance. In p r a c t i c e , r a t i o s of 2.5 are marginal and r a t i o s above 3 are preferable. For s i m p l i c i t y , a rectangular pressure-displacement diagram was assumed in the cryocooler. The actual shape depends on the room temperature arrangements. In the S t i r l i n g system, a piston compressor-expander is used, and both piston and displacer are driven more or less harmonically. This w i l l r e s u l t in a warped egg-shaped pressure-displacement diagram. In a Gifford-fitf4ahon system however, a continuous high speed compressor is used, and valves are used to achieve the pressure wave. For some valve timings, t h i s may actually match our

rectangular assumption q u i t e c l o s e l y . The differences in pressure-displacement diagrams can be taken into account by two correction f a c t o r s : One in the gross cooling in ( 3 . 9 ) , to compensate for the smaller area in the diagram compared with the maximum rectangular case, and the second in (3.15) to compensate f o r smaller cycled amount of gas compared with the maximum rectangular case (diagonal). In f a c t , i t is very important to use such cycles at low temperatures. By the use of proper t i m i n g , i t is possible to truncate the rectangular diagram as shown in f i g u r e 3.7, and i t is usually possible to reduce regenerator losses by a f a c t o r of two, while the gross cooling drops only to 90% of the rectangular value.

Apart from these gross numbers, the time dependence of the pressure-displacement wave during a cycle is important as w e l l . In the calculation of the s h u t t l e and regenerator losses a sinusoidal wave was assumed for both the displacer movement and the gasflow. E.g. in the case of a purely r e s i s t i v e thermal impedance zT (gas gap dominates), a square wave-shaped mass flow is optimal. The sine wave considered above is only a factor

(1-1) 5(2) » | * 1.23

worse, but a peaked wave is better avoided (z is the Riemann zetha

function). When the plastic impedance dominates, zT contains 45 degrees phase shift and corresponding (frequency)~T dependence in all Fourier components. Optimisation of all Fourier components now leads to an optimum wave form with sharp (peaked), symmetric transitions between upward and downward flow, - §{{1+2i)v)'*siri{2ir(1+2i)vt). A sine wave is

(1-gJg) C(l-5) a 1.68 worse, and a square wave is

( 1 - 2 ^ ) 5(1.5) (1-jlj) c(2.5)

i ^ 5 ~ 1 .22

(d-j) c(2)r

(23)

P^B

the displacer, and that the exact phase relations between gas transport and gas expansion/compression were not taken into account.

A similar optimization for the shuttle losses leads to unrealistic wave forms. When the gas gap dominates, a square-wave-like rocking movement is a factor 2 worse than the sine wave, but a triangle wave is better than the sine wave by a factor 2/3. When the plastic dominates, these factors are 2.73 and 0.675, respectively.

Edges. Al 1 parameters are considered smooth functions of position (and

temperature). A notable deviation occurs at the edges of the displacer. Our shuttle loss model does not account for the edge moving up and down over one stroke length, or, in other words, for the details of the boundary conditions. When the stroke is smal 1 compared with the displacer length - as is the case in existing plastic coolers -, the omission is not serious. Harness and Newmann [48] performed numerical calculations on shuttle losses for certain situations which included the effect.

Similarly, regenerator losses are affected by the edges. Not only because the edges move, but also because the gas flow will be mixed in the expansion space, and because the thermal contact with the walls will be different. The order of magnitude of the effect can be estimated by dividing the gas temperature fluctuation in the regenerator by the vertical thermal gradient. This length is usually small (1 or 2 cm) in the coolers considered here, and the effect is ignored in this model.

Dissipation. Friction between displacer and sleeve, and dissipation in the gas flow were ignored. Other effects will render this calculation useless long before friction becomes noticeable (e.g. due to frozen particles). Dissipation in the device (sensor) or in the device's electric wiring is easily taken into account by substracting all dissipation at lower positions from the gross cooling. Thermal conduction in the electric wiring along the cone can be included in the conduction loss term.

Final remarks. Thermal penetration and conduction are essential

ingredients in our model. Regenerators in large cryocoolers. however, are usually restricted by heat capacity alone. Although it may not be immediately evident from our equations, it is good to realize that our model would lead to no regenerator losses in this case, the phase shift between gas flow and temperature fluctuation being 90 degrees. The point is that phase shifts due to the large void volume, several end effects and non-isothermal expansion are dominating the regenerator losses of such coolers. In addition, the parameter of interest in such coolers is the overall efficiency, whereas in our model the actual input power was not considered.

Figure 3.8 shows the results of a complete calculation in our model, based on the computer program listed in appendix A. Further details are provided in the caption.

(24)

JlÊBÊBÊkmm

3 . MODELLING 40 R e f i n e m e n t s , d i s c u s s i o n 4 0 d i a m e t e r r n 3 0 [ m m J 2 0 10 0 l o s s e s

[W]

3 0 0 t e m p e r a t u r e

[K]

200 1 0 0 0 0 100 200 300 400 p o s i t i o n [ m m j F i g u r e 3 . 8 . R e s u l t o f c a l c u l a t i o n f o r c o n i c a l p l a s t i c d i s p l a c e r . T o p : D i s p l a c e r p r o f i l e , and r a d i a t i o n s h i e l d s u p p o r t p o s i t i o n s . Other i n p u t p a r a m e t e r s used i n c a l c u l a t i o n can be f o u n d n e a r t h e end o f t h e c o m p u t e r p r o g r a m l i s t i n g i n a p p e n d i x A. C e n t e r : C a l c u l a t e d g r o s s c o o l i n g and l o s s e s . D i s c o n t i n u i t i e s a r e caused by r a d i a t i o n s h i e l d s u p p o r t s o r changes i n g a p / e x c e n t r i c i t y . B o t t o m : T e m p e r a t u r e p r o f i l e . A l t h o u g h t h e c a l c u l a t i o n ( s o l i d l i n e ) matches t h e e x p e r i m e n t ( s q u a r e s ) s a t i s f a c t o r y h e r e , s e v e r a l i n p u t p a r a m e t e r s a r e n o t a c c u r a t e l y known f r o m t h e e x p e r i m e n t , w h i c h makes a r e a l i s t i c c o m p a r i s o n d i f f i c u l t . 3 . MODELLING 41 C o n c l u s i o n s C o n c l u s i o n s

Our model a p p l i e s t o p l a s t i c , low p o w e r , S t i r l i n g - t y p e c r y o c o o l e r s w i t h g a p - r e g e n e r a t o r and a t a p e r e d ( o r s t a g e d ) d i s p l a c e r .

The model i s a p o w e r f u l t o o l i n w e i g h i n g t h e d e s i g n t r a d e - o f f s and i n e v a l u a t i n g t h e p e r f o r m a n c e o f an o p e r a t i n g c o o l e r . Because i n p r a c t i c e m a t e r i a l s d a t a may be i n a c c u r a t e o r unknown and a number o f o t h e r i n p u t p a r a m e t e r s t e n d t o be u n c e r t a i n , an e x a c t p r e d i c t i o n i s n o t a l w a y s p o s s i b l e . S t i l l , t h e a n a l y s i s can be used t o e v a l u a t e t h e r a n g e o f a c c e p t a b l e v a l u e s i n a d e s i g n and s t r o n g l y s u g g e s t s w h i c h p a r a m e t e r s a r e d o m i n a t i n g t h e p e r f o r m a n c e .

The n u m e r i c a l i m p l e m e n t a t i o n o f t h e model i s f a s t , even on a m i c r o c o m p u t e r . T h i s a l l o w s f o r i n t e r a c t i v e c a l c u l a t i o n s .

A ' s i n g l e s e c t i o n ' a n a l y s i s i s d a n g e r o u s when c o m p a r i n g w i t h e x p e r i m e n t a l p e r f o r m a n c e . Small changes i n one s t a g e may have a p r o f o u n d e f f e c t i n o t h e r s t a g e s . The s t r o n g t e m p e r a t u r e dependence i n t h e m a t e r i a l o f d i s p l a c e r and s l e e v e must be t a k e n f u l l y i n t o a c c o u n t . E f f e c t s o f v o i d v o l u m e , e x c e n t r i c i t y and c y c l e a r e shown t o be e x t r e m e l y i m p o r t a n t . T h i s s u g g e s t s t h a t t h e y s h o u l d be c o n s i d e r e d i n any q u a l i t a t i v e o r q u a n t a t i v e e v a l u a t i o n o f a c o o l e r . I t a l s o s u g g e s t s changes t o p r e v i o u s d e s i g n s t h a t i g n o r e d t h e s e e f f e c t s .