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Cryocooler modelling methodology
Cryogenics 46 (2006) 183–190 www.elsevier.com/locate/cryogenics
Cryocooler modelling methodology Jayne Fereday a,*, Tom Bradshaw a, Martin Crook a, Anna Orlowska a, Ravinder Bhatia b, Martin Linder b, Olivier Pin b, Arnaud Scommegna b, Sylvain Vey b
b
a Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK European Space Agency, ESTEC, Keplerlaan 1, AG2200 Noordwijk, The Netherlands
Received 17 October 2005; accepted 18 October 2005
Abstract Future spacecraft will increasingly use cryocoolers to improve the performance of their payloads. For system level studies, it is necessary to quickly and easily assess the impact of these coolers on the overall thermal budget of a spacecraft. This requires the capability of simulating, in a simplified yet representative manner, the interaction of a cooler mathematical model with the overall thermal mathematical model of the spacecraft and instruments. This paper discusses potential methodologies for creating basic models for system level studies. It covers the prototype cooler models created to assess the methods and algorithms for Stirling, pulse tube, Joule–Thomson and Peltier coolers. 2005 Elsevier Ltd. All rights reserved. Keywords: Joule–Thomson coolers (E); Pulse tube (E); Stirling (E); Space cryogenics (F)
1. Introduction Future spacecraft will increasingly use cryocoolers to improve the performance of their payloads, whether it is for space science, Earth observation or telecommunications. For system level studies, it is necessary to quickly and easily assess the impact of these coolers on the overall thermal budget of a spacecraft. This requires the new capability of simulating, in a simplified yet representative manner, the interaction of a cooler model with the overall thermal mathematical model of the spacecraft and instruments. No such models or established methodology currently exist in an accessible form for mission level designers. For each mission it has been necessary to create these models to perform even basic analyses. Whilst it is clear that detailed cooler models are needed for final optimisation of thermal design, it was the purpose and innova-
*
Corresponding author. Tel.: +44 1235 445031; fax: +44 1235 445848. E-mail address: [email protected] (J. Fereday).
0011-2275/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2005.10.003
tion of this programme of work to explore how to create basic models for system level studies. In 2004 the Rutherford Appleton Laboratory (RAL) carried out a programme of work for the European Space Agency (ESA) aimed at understanding how to develop components to carry out system level thermal analyses of spacecraft that use cryocoolers, and to identify the most promising modelling methodologies. Prototype cooler models were created to assess the methods and algorithms for Stirling, pulse tube, Joule–Thomson (JT) and Peltier coolers. This variety of cooling techniques is commonly of interest to spacecraft designers. ESA has subsequently continued the study for single-stage Stirling and pulse tube coolers to cover the complete operating range of the coolers in a simple manner. The modelling of multi-stage Stirling coolers has also been investigated. The study began with a review of existing cooler models and system modelling methods. Many papers are available on the stand-alone modelling of specific cooler types [1,2], but few describe system level modelling of cryocoolers in space. While this modelling will undoubtedly have been
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carried out, it is not widely reported in the available literature. The Cryogenics Systems Integration Model (CSIM) developed by The Aerospace Corporation [3] is an interactive PC Windows based tool for the simulation and analysis of spacecraft cryogenic systems. The choice of coolers covers Peltier, pulse tube and Stirling type, where input data, performance curves must be entered into a database. CSIM employs a variety of modelling methods including design algorithms, subroutines and databases. NASA AMES has developed the online ARCOPTR (Ames Research Centre Orifice Pulse Tube Refrigerator) pulse tube model [4] to model the behaviour of orifice and inertance coolers. Detailed design data is required as an input. NASA AMES has also set up an online database giving comparative tables and bibliographies of pulse tubes [4]. For Peltier coolers, Melcor and Harvard Thermal have produced the ÔTAS-PCBÕ Printed Circuit Board Thermal Analysis software [5] allowing users to select a Melcor product from a component library, which is then added to a system level model. 2. Investigation of methodologies 2.1. General modelling methods Analyses of coolers can vary in their degree of complexity from simple calculations to nodal network analyses and detailed thermodynamic calculations. A zero order analysis for a cooler using a regenerative heat exchanger would typically be of the form below: Cooling power ¼ functionðfrequency; temperature; pressure; volumeÞ
ð1Þ
This type of relationship can be found empirically from measured data, and reasonably realistic predictions can be produced if equations are well correlated. A first order Schmidt analysis, relevant for reciprocating regenerative coolers, assumes ideal conditions, while a second order analysis takes the ideal Schmidt analysis and includes losses from the system, thus giving more realistic results. A third order analysis involves creating a network of thermal nodes and solving equations for conservation of mass, momentum and energy at each node [6]. If sufficiently discretised, this method potentially provides the most accurate results, but at the expense of computer processor time. A combination of methods can be used, with some para-
meters represented on a basic level while more critical elements are dealt with in greater detail. Coolers can also be represented using databases rather than modelling the underlying processes. This provides the user with a large amount of data but also relies on correct interpretation and integration of that information within a system level thermal model. Underlying databases would nonetheless be very useful for users wishing to extend the models beyond a top-level analysis and also for developers creating and verifying model libraries. 2.2. Requirements and constraints In order to ensure that an appropriate methodology is adopted, system and thermal engineers were consulted to obtain an understanding of the tools typically needed for system level analysis involving cryocoolers and of the level of detailed interaction necessary. Five types of cooler were assessed and typical specifications based on current available technology, shown in Table 1, were used as a starting point to frame the approach adopted. There are a number of important requirements for and constraints on the modelling methodology chosen. Models should be widely available to the thermal systems engineering community. A significant constraint on system model development is a lack of cooler performance data. Commercial manufacturers supply data sheets but the data presented is often difficult to apply to the thermal system being analysed. Also, many of these coolers are based on relatively new technology and have not been widely used for space applications. When modelling cooler interface temperatures, it should be clear which side of the interface is relevant, as there may be a significant temperature difference between the two sides. Computationally, it is essential that the speed of execution and memory requirements are such that implementation of a model does not lead to unacceptable CPU times. Any complex thermodynamic computation is likely to slow down the run-time. It is essential the system is robust and handles errors in a well-defined manner. Routines must be in place to ensure that input parameters and calculations are valid, and these routines must show the nature of the error in a helpful way. Models must be fully tested and the test programme documented to demonstrate that the models accurately predict performance for a number of scenarios. Finally, the methodology chosen must allow the cooler library to be extended to include different types of coolers.
Table 1 Typical cryocooler performance parameters
Base temperature, K Cold tip temperature, K Precooling stage temperature, K Heatlift, W Input power, W
Single-stage Stirling
Two-stage Stirling
Single-stage pulse tube
Helium JT
Single-stage Peltier
50 80 300 1.0 44
14 20 150 0.2 100
55 80 300 1.2 40
4 4 20 0.014 70
240 250 290 1 2
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Table 2 Suggested uncertainty levels for system model parameters
Cold head temperature, K Heat lift Input power, W
Single-stage Stirling
Two-stage Stirling
Single-stage Pulse Tube
Helium JT
Single-stage Peltier
±2 ±0.1 W ±3
±1 ±0.05 W ±3
±2 ±0.1 W ±3
±0.1 ±1 mW ±3
±1 ±0.1 W ±0.5
The level of accuracy of the model output must be considered in the context of functional and non-functional requirements. It must not be so high as to impose undue constraints on the required detail in a system model. Sufficient representation of boundary stages is required but the level of detail there may be minimal. Typical uncertainty levels might be as shown in Table 2. 2.3. Required parameters The following items have been identified as the most important to understand the capabilities of the cooler and its impact on other spacecraft budgets: • temperature of the cryocooler cold tip, • cryocooler heatlift over the required operating range, • cryocooler dissipated power at precooler stages and warm end components, • input power needed by the cooler system, • non-operational heat loads and parasitics. Some of the above parameters are already defined by the overall system, whereas the cooler model must provide as an output the remaining parameters. For most applications, one can distinguish between three cases, which need to be covered by a model of any cryocooler: (a) Design case—a cryocooler is required to cool a certain element (e.g., detector) to a required temperature (Tcold). The heatload (Qnet) at this temperature is derived from the Thermal Mathematical Model (TMM) of the system to be cooled. The cooler model has to provide the required input power and the dissipated power at pre-cooler stages and warm end components. In this case, the cold tip temperature is fixed to a boundary temperature as required. (b) Verification case—for transient runs, verification of cooler drive electronics (CDE) control algorithms and correlation with measured data. The model must be able to provide the cooling power as a function of input power and the cold tip temperature. During this phase the input power is defined by the model user whereas the cold tip temperature is an output of the TMM thermal solver. For coolers which use mechanical compressors, the compressor stroke may be an alternative parameter to the input power. (c) Off case—to simulate a non-operating cooler at system level. The model must provide as an output the
parasitic heat losses as a function of cold tip and warm end temperature.
2.4. Possible implementations The requirements and constraints identified have been used to identify feasible options for a software package or language with which to develop cooler models. The relative merits of the various options identified as suitable for more detailed analysis are summarised in Table 3. 2.5. Selected methodology The emphasis of this study is system modelling rather than cooler selection and as such these models should enable the user to analyse whether a cooler works or does not work for a particular application. There already exist software packages that aid users when selecting coolers. Packages such as MelcorÕs Aztec [7] aid the user in choosing a Peltier cooler to suit their requirements. This type of package is useful but does not provide a unified approach to cooler selection and system level thermal modelling. The user is still required to enter the extracted data into their own thermal model to assess the impact of the cooler on their system. Packages such as ESATAN [8] or I-DEAS TMG [9] offer an advantage in that the cooler models and system level thermal model are integrated. Additionally, a ÔpureÕ thermal formulation will have advantages over thermo-hydraulic solvers. Thermal solvers are already well understood by thermal engineers and are likely to be faster and more reliable. ESATAN is the most common tool used throughout Europe and it is a tool that is fully supported by ESA. For this reason, ESATAN has been selected as the focus for developing prototype models. The methods and approaches used are applicable to other thermal analysis packages. Additionally, conversion utilities are available for translating ESATAN models into other thermal software packages, although these tools have not always been found to be robust. The implementation method selected by RAL for the modelling presented here makes use of the $ELEMENTS functionality. This is essentially an environment for creating library sub-models that can then be called from a system level thermal model. A generic model can be written for each cooler type, with parameters such as input power made available to the user for substitution from their default value.
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Table 3 Summary of implementation advantages and disadvantages Method
Advantages
Disadvantages
ESATAN
Standard ESA thermal engineering tool Familiar to European thermal engineers Provides submodel functionality Flexible format Can handle large models Suitable for whole project life cycle
No diagrammatic representation European focus
SINDA, IDEAS/TMG, etc.
In built thermal solvers Flexible format
Not standard ESA tools
Thermo-hydraulic solver
Additional functionality over pure thermal solvers Not widely used by thermal engineers Longer processing time
Visual Basic within Excel/Access
Widely available Diagrammatic representation possible Underlying databases available
User must include data in system model Compatibility with Unix/Linux
Visual Basic + ThermXL (an Excel based thermal solver)
Easy interface In built thermal solvers
Not widely used
GUI with C++ code
Easy user interface if well designed Diagrammatic representation possible
Inflexible format User must include data in system model
Web browser interface with C++ code
Easy user interface if well designed Widely available Easy to update Diagrammatic representation possible
User must include data in system model Maintenance of web server Reliant on network
3. Cooler models
3.1. Joule–Thomson cooler
System-level prototype models have been produced with the aim of establishing a practical methodology for further development. Preliminary models have been created of Joule–Thomson and Peltier coolers. More detailed modelling has been undertaken of Stirling and pulse tube coolers. Note that these detailed models must take into account the thermal environment not just at the cold but also at the warm stage. For example, requirements on electronic components mean the cooler electronics unit must typically lie within a temperature range of 0–40 C. Electronics dissipation is included in all the models. Also, as the heat rejection interface temperature is increased, for a given input power the cooling power decreases, or correspondingly for a given heat load, the cold end temperature increases.
Expansion of a non-ideal working fluid from below its inversion temperature results in a cooling effect. This effect is made use of in the Joule–Thomson cycle, which is commonly used because of its simplicity, reliability and efficiency. A RAL helium-based cooler (see Fig. 1) with mechanical compressors and a base temperature of 4 K has been selected for the Planck mission and this type is discussed here. The system uses two compressors to provide high and low pressure gas across a 4 K Joule–Thomson expansion orifice. Cold end performance depends on compressor pressure, mass flow rate, heat exchanger efficiency, mechanical configuration and precooling temperature. Precooling to a final temperature below the inversion temperature of the working fluid is required,
Fig. 1. Schematic of RAL 4 K Joule–Thomson cooler.
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and this must be represented in the system level model. For the Planck configuration, two precooling stages are available, at 50 K and 18 K. Thermodynamic modelling is computationally intense and thus the adopted approach takes available performance data and derives parametric equations to simulate that performance. Precooling loads from the heat exchangers are represented as a function of precooling stage and compressor temperatures, derived from theoretically modelled values and measured data. For a fixed compressor pressure and coldhead temperature, the cooling power can be modelled as a function of the temperature of the 18 K precooling stage. Coldhead temperature is a function of the pressure supplied by the compressors. For non-operational cases, parasitic loads due to conductance from the pipes and cryoharness are incorporated into the model. Table 4 briefly summarises the modelled variables and the implementation approach used. Fig. 2. Schematic of Peltier cooler.
3.2. Peltier cooler Peltier cooling results when an electric current flows through a junction of two dissimilar metals. Practical Peltier modules (see Fig. 2) consist of semi-conductors connected by a metallic conductor and sandwiched between two ceramic plates. When a voltage is applied, heat from the cold side is absorbed and rejected at the warm side. Peltier modules are typically used for local cooling applications when only small changes in temperature are required; multi-stage modules are available but efficiency is reduced when large temperature gradients are required. The cooling power at a junction is directly proportional to the current, I, flowing across it and the time for which it flows. A moduleÕs Peltier coefficient depends upon the temperature and material of the cold junction but is independent of the temperature of the hot junction. Qdot ¼ ðpA pBÞ I
ð2Þ
with Qdot
power at the junction of an ideal Peltier, where negative denotes cooling; I current of magnitude I across the junction of two different conductors, A and B; pA, pB Peltier coefficients of materials A and B. The Seebeck coefficient is the Peltier coefficient divided by the absolute temperature, and is the more usual means of definition for commercially available Peltier modules. The heat removed from the cold side is the difference between the Peltier cooling effect and the Joule heating effect (I2R heating), plus heat conducted from the hot to the cold side. The material thermal conductivity of a module can be approximated by a constant or modelled as temperature dependent. Manufacturer data sheets typically give temperature dependent equations for these properties
Table 4 Summary of Joule–Thomson cooler thermal model variables and implementation approach
Required parameters Final operating temperature Heatlift Dissipated power at precooling stages and warm end components Required input power Non-operational heat loads Desirable parameters Thermal link conductance between cooler and spacecraft Heat lift at non-nominal temperatures Dependence of heatlift on the heat rejection temperature and the stability of this level
Dependent variables
Implementation approach
Compressor low pressure Cold end temperature and 20 K stage temperature Precooling stage and compressor temperatures Compressor pressure
Calculated using fit functions based on theoretical modelled values and correlated using available performance data
Cold end, precooling stage and compressor temperatures Cold end, precooling stage and compressor temperatures Precooling stage temperatures 20 K stage temperature
Calculated based on performance data and user supplied efficiency value Linear conductances based on typical cooler geometry To be added by user—typical values suggested Not yet implemented Stability not yet implemented
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based on a polynomial with specified coefficients for particular modules. Alternatively, the data are supplied in tabular form. Equations modelling the performance of Peltier coolers based on their material properties are widely available in the literature (e.g., [10]). Unlike the other types of cryocooler being investigated, the equations are relatively simple and are not computationally intense. They have therefore been used in full to create a prototype Peltier model. Typical values for the Seebeck coefficient, electrical resistance and thermal conductance are used, with a database of co-efficients for library modules envisioned for future developments. Since this preliminary study was carried out, a similar Peltier model has been developed by Alstom as part of the ESA supported ESATAN software package and thus this part of the work has not been taken further. 3.3. Single-stage Stirling/pulse tube cooler
different fit functions) has been used to predict the performance of pulse tube and of Stirling coolers. For the purposes of modelling the cooler the real performance can be derived from measured data, since thermodynamic calculations are computationally intense and would require design details which are not available at system level. In order to cover a large operating range of the cooler with a limited number of measured data, fit functions based on the second order Schmidt analyses have been obtained. This has the advantage that the temperature dependence of the cryocoolers is already covered by a physical description of the equations and does not require any additional measurements. Based on the measured performance of an Astrium 50– 80 K Stirling cooler, it has been found that the cooler can be reasonably well described by the expression: Qgross Qparasitic Qloss Qnet ¼ 0
ð3Þ
with A typical single stage system comprises a cooler electronics box (for motor drive and control), a compressor, displacer/pulse tube body, the cold end and the cold end interface. The electronics box can be considered as a separate coupled object, whereas the other items are thermodynamically linked. The compressor produces a pressure wave, which is transmitted to the displacer/pulse tube body, where it is used to lift the heat from the cold end. The displacer/pulse tube body is connected to the cold end via the coldfinger. The cold end is in turn connected to the detector by a thermal link. The parasitic heatload from these connections must be taken into account, particularly for non-operating conditions (Fig. 3). Future European pulse tube coolers will use a coaxial architecture, which has comparable thermodynamic efficiency and significantly easier integration compared to the inline version. Therefore, the same thermal model (with
Qgross ¼ e Qinput T cold =ðT hot T cold Þ cooling power of an ideal Stirling cooler Qparasitic ¼ c ðT hot T cold Þ þ d
ðT 4hot
ð4Þ
T 4cold Þ
obtained from parasitic loss measurements of a non-operating coolerðconductive and radiativeÞ ð5Þ Qloss ¼ Regenerator loss þ shuttle loss ¼ e Qinput =T hot ðT hot T cold Þ a expðb T hot =T cold Þ it was not possible to separate the shuttle loss from the regenerator loss. a and b both depend on the Qnet
displacer stroke
ð6Þ
available cooling power
ð7Þ
Fig. 3. Schematic of single-stage Stirling or pulse tube cooler model.
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A similar approach can be applied for pulse tube coolers, by adapting the terms as Qgross ¼ e Qinput T cold =T hot cooling power of an ideal pulse tube cooler Qparasitic ¼ cond ðT hot T cold Þ þ rad
ðT 4hot
ð8Þ
T 4cold Þ
obtained from parasitic loss measurements of a non-operating coolerðconductive and radiativeÞ ð9Þ Qloss ¼ regenerator loss þ pressure loss ¼ e Qinput ð1=T hot ðT hot T cold Þ a expðb T hot =T cold Þ þ ðc þ d Qinput Þ T cold =T hot Þ the dominating losses are regenerator and pressure losses Qnet
available cooling power
Fig. 4. Percentage error in input power as a function of measured cold tip temperature, for various warm end temperatures.
ð10Þ ð11Þ
This has then been implemented into a thermal model of a Stirling/pulse tube cooler as shown below, implementing Qparasitics with the corresponding conductive and radiative links. Qcompressor and Qhot gas have been obtained by interpolating measurements and depend on the input power to the cooler. For consistency, it is required that the following condition is fulfilled: Qcompressor þ Qhotgas ¼ Qinput þ Qnet
ð12Þ
Depending on the case to be considered, the following restrictions apply: (a) Nominal design case—the coldfinger temperature is set to the required boundary value and Qinput is calculated by solving the cooler function, Eq. (3), with Qnet and Qparasitc provided by the thermal solver (e.g., as heat imbalance on the coldfinger node). (b) Verification case—all power dissipations are calculated as a function of Qinput. The coldfinger temperature is provided by the thermal solver. (c) Off case—as verification case, but Qinput = 0. The above approach has been verified using performance data of an Astrium 50–80 K Stirling cooler [11] and an air liquid miniature pulse tube cooler. It can be seen from Figs. 4 and 5 that the input power of these coolers can be predicted to within an uncertainty of less than 6% and the temperature can be predicted to within 2 K, covering an ambient operating temperature range between 260 K and 313 K. 3.4. Two-stage Stirling cooler Two-stage Stirling coolers for space applications have a very similar design to the single-stage coolers. They are
Fig. 5. Difference between measured and predicted cold tip temperatures as a function of measured cold tip temperature, for various warm end temperatures.
composed of a compressor unit, cooler electronics, and two stages of Stirling coolers, i.e., two sets of regenerator and displacer body. The moving parts of the stages are on the same shaft such that the two regenerators and displacer bodies thus follow the same cycle. A direct consequence is that it is not possible to optimise each stage independently and the cooler has to be treated indeed as a single system. The two stages are simply attached together, the cold end of the first stage becoming the hot end of the second one. There are other alternative architectures but these tend to increase the number of moving parts and mechanisms, resulting in a decrease of the overall reliability of the system which is so important for space applications. The modelling of the two stage Stirling cooler is based on the data provided by an Astrium (formerly BAe) 20– 50 K cooler (Serial No. EM001). The coldest stage of this cooler is linked to a double compressor via its first stage. It is necessary to determine how the pneumatic work is split between the two stages. The system level approach taken has been to define a coefficient expressing the percentage
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of work done at each stage. This method has been chosen to avoid the computationally intensive modelling of detailed gas dynamics, which would be too detailed for the fit function modelling approach adopted here. After several investigations, the most efficient ratio was found to be of the order of 85% for the first (intermediate temperature) stage and 15% for the second (coldest) stage. Each stage of a multi-stage cooler can be considered as a single-stage cooler for developing the different fit functions which describe the gross cooling power and the relevant losses mechanisms. Thus, the same equations as those developed for single-stage coolers have been used to determine these parameters. The regenerator losses are described by an exponential function and the shuttle heat transfer losses by a third order polynomial fit function. The parasitic losses between the hot end and the cold tip of each stage have been modelled based on experimental measurements. The interpolation functions developed fit the data well; the remaining losses represent at maximum around 1% of the gross cooling power for each stage, and thus can be neglected. To validate the fit functions, a simple 5-node model has been implemented using a standard thermal solver. The hot stage, the middle stage, the cold stage and the two heat loads applied on the cooler (one at the middle stage and the last one at the cold stage) are modelled. Setting the hot stage as a boundary and including the equations found earlier, the model is processed using the iterative solver. The results of these simulations are not convincing. Few of the experimental data points have been re-found via the solver and most of them diverged. A change in the pneumatic work coefficient has been tried, relating it to the temperature ratio, and thus having a variable coefficient throughout the simulation. The results did not improve drastically. Such a simple approach for a two-stage Stirling cooler does not take into account all the relevant phenomena involved between the two stages, and thus a simple coefficient for the pneumatic work is not sufficient. Nevertheless, by looking more deeply into the gas dynamics taking place in a cycle of the cryocooler, or by taking another approach for the simulation of the cooler in the solver (for example two separate Stirling coolers connected via a thermal link, or the cooler being treated as a large single-stage cooler)
might enhance the results. But for now, the method using fit functions describing gross cooling and losses combining boundary nodes method does not work for two-stage Stirling cryocoolers. 4. Summary and future developments An investigation has been performed into methodologies suitable for the development of library cryocooler models for performing system level studies of spacecraft. Recommendations for a suitable approach are given and prototype models of a Joule–Thomson and a Peltier cooler are briefly described. Models of single-stage Stirling and pulse tube coolers using fit functions based on available performance data have been further developed. Verification of these fit functions using simple thermal models gives good correlation. Investigations into the modelling of twostage coolers have begin but initial results at correlation have not been successful. Further development is planned for the future with the aim of developing a library of simple yet representative cryocooler models. References [1] Yuan SWK, Spradley IE. A third order computer model for Stirling refrigerators. Adv Cryogen Eng 1992;37(B):1055–62. [2] Yuan SWK et al. Computer simulation model for Lucas Stirling refrigerators. Cryogenics 1992;32(2):143–8. [3] Miller SD et al. Cryogenic Systems Integration Model (CSIM). Cryocoolers 10. New York: Kluwer Academic/Plenum Publishers; 1999. [4] Available from: http://irtek.arc.nasa.gov/CryoDev.html, the NASA AMES Cryogenic Group homepage. [5] Details available from: http://www.harvardthermal.com, homepage of Harvard Thermal, USA. [6] Walker G. Cryocoolers, Part 1: Fundamentals. New York: Plenum Press; 1983. [7] Details available from: http://www.melcor.com, homepage of Melcor, USA. [8] Details available from: http://www.aerospace.power.alstom.com, homepage of Alstom. [9] Details available from http://www.mayahtt.com, homepage of Maya. [10] Metzger T, Huebener RP. Modelling and cooling behaviour of Peltier cascades. Cryogenics 1999;39(3):235–9. [11] Development of an engineering model 50–20 K two stage Stirling cycle cooler, Final report, British Aerospace Space Systems Limited, ESA contract 9004/90/NL/PP, May 1992.
Cryocooler modelling methodology Jayne Fereday a,*, Tom Bradshaw a, Martin Crook a, Anna Orlowska a, Ravinder Bhatia b, Martin Linder b, Olivier Pin b, Arnaud Scommegna b, Sylvain Vey b
b
a Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX, UK European Space Agency, ESTEC, Keplerlaan 1, AG2200 Noordwijk, The Netherlands
Received 17 October 2005; accepted 18 October 2005
Abstract Future spacecraft will increasingly use cryocoolers to improve the performance of their payloads. For system level studies, it is necessary to quickly and easily assess the impact of these coolers on the overall thermal budget of a spacecraft. This requires the capability of simulating, in a simplified yet representative manner, the interaction of a cooler mathematical model with the overall thermal mathematical model of the spacecraft and instruments. This paper discusses potential methodologies for creating basic models for system level studies. It covers the prototype cooler models created to assess the methods and algorithms for Stirling, pulse tube, Joule–Thomson and Peltier coolers. 2005 Elsevier Ltd. All rights reserved. Keywords: Joule–Thomson coolers (E); Pulse tube (E); Stirling (E); Space cryogenics (F)
1. Introduction Future spacecraft will increasingly use cryocoolers to improve the performance of their payloads, whether it is for space science, Earth observation or telecommunications. For system level studies, it is necessary to quickly and easily assess the impact of these coolers on the overall thermal budget of a spacecraft. This requires the new capability of simulating, in a simplified yet representative manner, the interaction of a cooler model with the overall thermal mathematical model of the spacecraft and instruments. No such models or established methodology currently exist in an accessible form for mission level designers. For each mission it has been necessary to create these models to perform even basic analyses. Whilst it is clear that detailed cooler models are needed for final optimisation of thermal design, it was the purpose and innova-
*
Corresponding author. Tel.: +44 1235 445031; fax: +44 1235 445848. E-mail address: [email protected] (J. Fereday).
0011-2275/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2005.10.003
tion of this programme of work to explore how to create basic models for system level studies. In 2004 the Rutherford Appleton Laboratory (RAL) carried out a programme of work for the European Space Agency (ESA) aimed at understanding how to develop components to carry out system level thermal analyses of spacecraft that use cryocoolers, and to identify the most promising modelling methodologies. Prototype cooler models were created to assess the methods and algorithms for Stirling, pulse tube, Joule–Thomson (JT) and Peltier coolers. This variety of cooling techniques is commonly of interest to spacecraft designers. ESA has subsequently continued the study for single-stage Stirling and pulse tube coolers to cover the complete operating range of the coolers in a simple manner. The modelling of multi-stage Stirling coolers has also been investigated. The study began with a review of existing cooler models and system modelling methods. Many papers are available on the stand-alone modelling of specific cooler types [1,2], but few describe system level modelling of cryocoolers in space. While this modelling will undoubtedly have been
184
J. Fereday et al. / Cryogenics 46 (2006) 183–190
carried out, it is not widely reported in the available literature. The Cryogenics Systems Integration Model (CSIM) developed by The Aerospace Corporation [3] is an interactive PC Windows based tool for the simulation and analysis of spacecraft cryogenic systems. The choice of coolers covers Peltier, pulse tube and Stirling type, where input data, performance curves must be entered into a database. CSIM employs a variety of modelling methods including design algorithms, subroutines and databases. NASA AMES has developed the online ARCOPTR (Ames Research Centre Orifice Pulse Tube Refrigerator) pulse tube model [4] to model the behaviour of orifice and inertance coolers. Detailed design data is required as an input. NASA AMES has also set up an online database giving comparative tables and bibliographies of pulse tubes [4]. For Peltier coolers, Melcor and Harvard Thermal have produced the ÔTAS-PCBÕ Printed Circuit Board Thermal Analysis software [5] allowing users to select a Melcor product from a component library, which is then added to a system level model. 2. Investigation of methodologies 2.1. General modelling methods Analyses of coolers can vary in their degree of complexity from simple calculations to nodal network analyses and detailed thermodynamic calculations. A zero order analysis for a cooler using a regenerative heat exchanger would typically be of the form below: Cooling power ¼ functionðfrequency; temperature; pressure; volumeÞ
ð1Þ
This type of relationship can be found empirically from measured data, and reasonably realistic predictions can be produced if equations are well correlated. A first order Schmidt analysis, relevant for reciprocating regenerative coolers, assumes ideal conditions, while a second order analysis takes the ideal Schmidt analysis and includes losses from the system, thus giving more realistic results. A third order analysis involves creating a network of thermal nodes and solving equations for conservation of mass, momentum and energy at each node [6]. If sufficiently discretised, this method potentially provides the most accurate results, but at the expense of computer processor time. A combination of methods can be used, with some para-
meters represented on a basic level while more critical elements are dealt with in greater detail. Coolers can also be represented using databases rather than modelling the underlying processes. This provides the user with a large amount of data but also relies on correct interpretation and integration of that information within a system level thermal model. Underlying databases would nonetheless be very useful for users wishing to extend the models beyond a top-level analysis and also for developers creating and verifying model libraries. 2.2. Requirements and constraints In order to ensure that an appropriate methodology is adopted, system and thermal engineers were consulted to obtain an understanding of the tools typically needed for system level analysis involving cryocoolers and of the level of detailed interaction necessary. Five types of cooler were assessed and typical specifications based on current available technology, shown in Table 1, were used as a starting point to frame the approach adopted. There are a number of important requirements for and constraints on the modelling methodology chosen. Models should be widely available to the thermal systems engineering community. A significant constraint on system model development is a lack of cooler performance data. Commercial manufacturers supply data sheets but the data presented is often difficult to apply to the thermal system being analysed. Also, many of these coolers are based on relatively new technology and have not been widely used for space applications. When modelling cooler interface temperatures, it should be clear which side of the interface is relevant, as there may be a significant temperature difference between the two sides. Computationally, it is essential that the speed of execution and memory requirements are such that implementation of a model does not lead to unacceptable CPU times. Any complex thermodynamic computation is likely to slow down the run-time. It is essential the system is robust and handles errors in a well-defined manner. Routines must be in place to ensure that input parameters and calculations are valid, and these routines must show the nature of the error in a helpful way. Models must be fully tested and the test programme documented to demonstrate that the models accurately predict performance for a number of scenarios. Finally, the methodology chosen must allow the cooler library to be extended to include different types of coolers.
Table 1 Typical cryocooler performance parameters
Base temperature, K Cold tip temperature, K Precooling stage temperature, K Heatlift, W Input power, W
Single-stage Stirling
Two-stage Stirling
Single-stage pulse tube
Helium JT
Single-stage Peltier
50 80 300 1.0 44
14 20 150 0.2 100
55 80 300 1.2 40
4 4 20 0.014 70
240 250 290 1 2
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Table 2 Suggested uncertainty levels for system model parameters
Cold head temperature, K Heat lift Input power, W
Single-stage Stirling
Two-stage Stirling
Single-stage Pulse Tube
Helium JT
Single-stage Peltier
±2 ±0.1 W ±3
±1 ±0.05 W ±3
±2 ±0.1 W ±3
±0.1 ±1 mW ±3
±1 ±0.1 W ±0.5
The level of accuracy of the model output must be considered in the context of functional and non-functional requirements. It must not be so high as to impose undue constraints on the required detail in a system model. Sufficient representation of boundary stages is required but the level of detail there may be minimal. Typical uncertainty levels might be as shown in Table 2. 2.3. Required parameters The following items have been identified as the most important to understand the capabilities of the cooler and its impact on other spacecraft budgets: • temperature of the cryocooler cold tip, • cryocooler heatlift over the required operating range, • cryocooler dissipated power at precooler stages and warm end components, • input power needed by the cooler system, • non-operational heat loads and parasitics. Some of the above parameters are already defined by the overall system, whereas the cooler model must provide as an output the remaining parameters. For most applications, one can distinguish between three cases, which need to be covered by a model of any cryocooler: (a) Design case—a cryocooler is required to cool a certain element (e.g., detector) to a required temperature (Tcold). The heatload (Qnet) at this temperature is derived from the Thermal Mathematical Model (TMM) of the system to be cooled. The cooler model has to provide the required input power and the dissipated power at pre-cooler stages and warm end components. In this case, the cold tip temperature is fixed to a boundary temperature as required. (b) Verification case—for transient runs, verification of cooler drive electronics (CDE) control algorithms and correlation with measured data. The model must be able to provide the cooling power as a function of input power and the cold tip temperature. During this phase the input power is defined by the model user whereas the cold tip temperature is an output of the TMM thermal solver. For coolers which use mechanical compressors, the compressor stroke may be an alternative parameter to the input power. (c) Off case—to simulate a non-operating cooler at system level. The model must provide as an output the
parasitic heat losses as a function of cold tip and warm end temperature.
2.4. Possible implementations The requirements and constraints identified have been used to identify feasible options for a software package or language with which to develop cooler models. The relative merits of the various options identified as suitable for more detailed analysis are summarised in Table 3. 2.5. Selected methodology The emphasis of this study is system modelling rather than cooler selection and as such these models should enable the user to analyse whether a cooler works or does not work for a particular application. There already exist software packages that aid users when selecting coolers. Packages such as MelcorÕs Aztec [7] aid the user in choosing a Peltier cooler to suit their requirements. This type of package is useful but does not provide a unified approach to cooler selection and system level thermal modelling. The user is still required to enter the extracted data into their own thermal model to assess the impact of the cooler on their system. Packages such as ESATAN [8] or I-DEAS TMG [9] offer an advantage in that the cooler models and system level thermal model are integrated. Additionally, a ÔpureÕ thermal formulation will have advantages over thermo-hydraulic solvers. Thermal solvers are already well understood by thermal engineers and are likely to be faster and more reliable. ESATAN is the most common tool used throughout Europe and it is a tool that is fully supported by ESA. For this reason, ESATAN has been selected as the focus for developing prototype models. The methods and approaches used are applicable to other thermal analysis packages. Additionally, conversion utilities are available for translating ESATAN models into other thermal software packages, although these tools have not always been found to be robust. The implementation method selected by RAL for the modelling presented here makes use of the $ELEMENTS functionality. This is essentially an environment for creating library sub-models that can then be called from a system level thermal model. A generic model can be written for each cooler type, with parameters such as input power made available to the user for substitution from their default value.
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Table 3 Summary of implementation advantages and disadvantages Method
Advantages
Disadvantages
ESATAN
Standard ESA thermal engineering tool Familiar to European thermal engineers Provides submodel functionality Flexible format Can handle large models Suitable for whole project life cycle
No diagrammatic representation European focus
SINDA, IDEAS/TMG, etc.
In built thermal solvers Flexible format
Not standard ESA tools
Thermo-hydraulic solver
Additional functionality over pure thermal solvers Not widely used by thermal engineers Longer processing time
Visual Basic within Excel/Access
Widely available Diagrammatic representation possible Underlying databases available
User must include data in system model Compatibility with Unix/Linux
Visual Basic + ThermXL (an Excel based thermal solver)
Easy interface In built thermal solvers
Not widely used
GUI with C++ code
Easy user interface if well designed Diagrammatic representation possible
Inflexible format User must include data in system model
Web browser interface with C++ code
Easy user interface if well designed Widely available Easy to update Diagrammatic representation possible
User must include data in system model Maintenance of web server Reliant on network
3. Cooler models
3.1. Joule–Thomson cooler
System-level prototype models have been produced with the aim of establishing a practical methodology for further development. Preliminary models have been created of Joule–Thomson and Peltier coolers. More detailed modelling has been undertaken of Stirling and pulse tube coolers. Note that these detailed models must take into account the thermal environment not just at the cold but also at the warm stage. For example, requirements on electronic components mean the cooler electronics unit must typically lie within a temperature range of 0–40 C. Electronics dissipation is included in all the models. Also, as the heat rejection interface temperature is increased, for a given input power the cooling power decreases, or correspondingly for a given heat load, the cold end temperature increases.
Expansion of a non-ideal working fluid from below its inversion temperature results in a cooling effect. This effect is made use of in the Joule–Thomson cycle, which is commonly used because of its simplicity, reliability and efficiency. A RAL helium-based cooler (see Fig. 1) with mechanical compressors and a base temperature of 4 K has been selected for the Planck mission and this type is discussed here. The system uses two compressors to provide high and low pressure gas across a 4 K Joule–Thomson expansion orifice. Cold end performance depends on compressor pressure, mass flow rate, heat exchanger efficiency, mechanical configuration and precooling temperature. Precooling to a final temperature below the inversion temperature of the working fluid is required,
Fig. 1. Schematic of RAL 4 K Joule–Thomson cooler.
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and this must be represented in the system level model. For the Planck configuration, two precooling stages are available, at 50 K and 18 K. Thermodynamic modelling is computationally intense and thus the adopted approach takes available performance data and derives parametric equations to simulate that performance. Precooling loads from the heat exchangers are represented as a function of precooling stage and compressor temperatures, derived from theoretically modelled values and measured data. For a fixed compressor pressure and coldhead temperature, the cooling power can be modelled as a function of the temperature of the 18 K precooling stage. Coldhead temperature is a function of the pressure supplied by the compressors. For non-operational cases, parasitic loads due to conductance from the pipes and cryoharness are incorporated into the model. Table 4 briefly summarises the modelled variables and the implementation approach used. Fig. 2. Schematic of Peltier cooler.
3.2. Peltier cooler Peltier cooling results when an electric current flows through a junction of two dissimilar metals. Practical Peltier modules (see Fig. 2) consist of semi-conductors connected by a metallic conductor and sandwiched between two ceramic plates. When a voltage is applied, heat from the cold side is absorbed and rejected at the warm side. Peltier modules are typically used for local cooling applications when only small changes in temperature are required; multi-stage modules are available but efficiency is reduced when large temperature gradients are required. The cooling power at a junction is directly proportional to the current, I, flowing across it and the time for which it flows. A moduleÕs Peltier coefficient depends upon the temperature and material of the cold junction but is independent of the temperature of the hot junction. Qdot ¼ ðpA pBÞ I
ð2Þ
with Qdot
power at the junction of an ideal Peltier, where negative denotes cooling; I current of magnitude I across the junction of two different conductors, A and B; pA, pB Peltier coefficients of materials A and B. The Seebeck coefficient is the Peltier coefficient divided by the absolute temperature, and is the more usual means of definition for commercially available Peltier modules. The heat removed from the cold side is the difference between the Peltier cooling effect and the Joule heating effect (I2R heating), plus heat conducted from the hot to the cold side. The material thermal conductivity of a module can be approximated by a constant or modelled as temperature dependent. Manufacturer data sheets typically give temperature dependent equations for these properties
Table 4 Summary of Joule–Thomson cooler thermal model variables and implementation approach
Required parameters Final operating temperature Heatlift Dissipated power at precooling stages and warm end components Required input power Non-operational heat loads Desirable parameters Thermal link conductance between cooler and spacecraft Heat lift at non-nominal temperatures Dependence of heatlift on the heat rejection temperature and the stability of this level
Dependent variables
Implementation approach
Compressor low pressure Cold end temperature and 20 K stage temperature Precooling stage and compressor temperatures Compressor pressure
Calculated using fit functions based on theoretical modelled values and correlated using available performance data
Cold end, precooling stage and compressor temperatures Cold end, precooling stage and compressor temperatures Precooling stage temperatures 20 K stage temperature
Calculated based on performance data and user supplied efficiency value Linear conductances based on typical cooler geometry To be added by user—typical values suggested Not yet implemented Stability not yet implemented
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based on a polynomial with specified coefficients for particular modules. Alternatively, the data are supplied in tabular form. Equations modelling the performance of Peltier coolers based on their material properties are widely available in the literature (e.g., [10]). Unlike the other types of cryocooler being investigated, the equations are relatively simple and are not computationally intense. They have therefore been used in full to create a prototype Peltier model. Typical values for the Seebeck coefficient, electrical resistance and thermal conductance are used, with a database of co-efficients for library modules envisioned for future developments. Since this preliminary study was carried out, a similar Peltier model has been developed by Alstom as part of the ESA supported ESATAN software package and thus this part of the work has not been taken further. 3.3. Single-stage Stirling/pulse tube cooler
different fit functions) has been used to predict the performance of pulse tube and of Stirling coolers. For the purposes of modelling the cooler the real performance can be derived from measured data, since thermodynamic calculations are computationally intense and would require design details which are not available at system level. In order to cover a large operating range of the cooler with a limited number of measured data, fit functions based on the second order Schmidt analyses have been obtained. This has the advantage that the temperature dependence of the cryocoolers is already covered by a physical description of the equations and does not require any additional measurements. Based on the measured performance of an Astrium 50– 80 K Stirling cooler, it has been found that the cooler can be reasonably well described by the expression: Qgross Qparasitic Qloss Qnet ¼ 0
ð3Þ
with A typical single stage system comprises a cooler electronics box (for motor drive and control), a compressor, displacer/pulse tube body, the cold end and the cold end interface. The electronics box can be considered as a separate coupled object, whereas the other items are thermodynamically linked. The compressor produces a pressure wave, which is transmitted to the displacer/pulse tube body, where it is used to lift the heat from the cold end. The displacer/pulse tube body is connected to the cold end via the coldfinger. The cold end is in turn connected to the detector by a thermal link. The parasitic heatload from these connections must be taken into account, particularly for non-operating conditions (Fig. 3). Future European pulse tube coolers will use a coaxial architecture, which has comparable thermodynamic efficiency and significantly easier integration compared to the inline version. Therefore, the same thermal model (with
Qgross ¼ e Qinput T cold =ðT hot T cold Þ cooling power of an ideal Stirling cooler Qparasitic ¼ c ðT hot T cold Þ þ d
ðT 4hot
ð4Þ
T 4cold Þ
obtained from parasitic loss measurements of a non-operating coolerðconductive and radiativeÞ ð5Þ Qloss ¼ Regenerator loss þ shuttle loss ¼ e Qinput =T hot ðT hot T cold Þ a expðb T hot =T cold Þ it was not possible to separate the shuttle loss from the regenerator loss. a and b both depend on the Qnet
displacer stroke
ð6Þ
available cooling power
ð7Þ
Fig. 3. Schematic of single-stage Stirling or pulse tube cooler model.
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A similar approach can be applied for pulse tube coolers, by adapting the terms as Qgross ¼ e Qinput T cold =T hot cooling power of an ideal pulse tube cooler Qparasitic ¼ cond ðT hot T cold Þ þ rad
ðT 4hot
ð8Þ
T 4cold Þ
obtained from parasitic loss measurements of a non-operating coolerðconductive and radiativeÞ ð9Þ Qloss ¼ regenerator loss þ pressure loss ¼ e Qinput ð1=T hot ðT hot T cold Þ a expðb T hot =T cold Þ þ ðc þ d Qinput Þ T cold =T hot Þ the dominating losses are regenerator and pressure losses Qnet
available cooling power
Fig. 4. Percentage error in input power as a function of measured cold tip temperature, for various warm end temperatures.
ð10Þ ð11Þ
This has then been implemented into a thermal model of a Stirling/pulse tube cooler as shown below, implementing Qparasitics with the corresponding conductive and radiative links. Qcompressor and Qhot gas have been obtained by interpolating measurements and depend on the input power to the cooler. For consistency, it is required that the following condition is fulfilled: Qcompressor þ Qhotgas ¼ Qinput þ Qnet
ð12Þ
Depending on the case to be considered, the following restrictions apply: (a) Nominal design case—the coldfinger temperature is set to the required boundary value and Qinput is calculated by solving the cooler function, Eq. (3), with Qnet and Qparasitc provided by the thermal solver (e.g., as heat imbalance on the coldfinger node). (b) Verification case—all power dissipations are calculated as a function of Qinput. The coldfinger temperature is provided by the thermal solver. (c) Off case—as verification case, but Qinput = 0. The above approach has been verified using performance data of an Astrium 50–80 K Stirling cooler [11] and an air liquid miniature pulse tube cooler. It can be seen from Figs. 4 and 5 that the input power of these coolers can be predicted to within an uncertainty of less than 6% and the temperature can be predicted to within 2 K, covering an ambient operating temperature range between 260 K and 313 K. 3.4. Two-stage Stirling cooler Two-stage Stirling coolers for space applications have a very similar design to the single-stage coolers. They are
Fig. 5. Difference between measured and predicted cold tip temperatures as a function of measured cold tip temperature, for various warm end temperatures.
composed of a compressor unit, cooler electronics, and two stages of Stirling coolers, i.e., two sets of regenerator and displacer body. The moving parts of the stages are on the same shaft such that the two regenerators and displacer bodies thus follow the same cycle. A direct consequence is that it is not possible to optimise each stage independently and the cooler has to be treated indeed as a single system. The two stages are simply attached together, the cold end of the first stage becoming the hot end of the second one. There are other alternative architectures but these tend to increase the number of moving parts and mechanisms, resulting in a decrease of the overall reliability of the system which is so important for space applications. The modelling of the two stage Stirling cooler is based on the data provided by an Astrium (formerly BAe) 20– 50 K cooler (Serial No. EM001). The coldest stage of this cooler is linked to a double compressor via its first stage. It is necessary to determine how the pneumatic work is split between the two stages. The system level approach taken has been to define a coefficient expressing the percentage
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of work done at each stage. This method has been chosen to avoid the computationally intensive modelling of detailed gas dynamics, which would be too detailed for the fit function modelling approach adopted here. After several investigations, the most efficient ratio was found to be of the order of 85% for the first (intermediate temperature) stage and 15% for the second (coldest) stage. Each stage of a multi-stage cooler can be considered as a single-stage cooler for developing the different fit functions which describe the gross cooling power and the relevant losses mechanisms. Thus, the same equations as those developed for single-stage coolers have been used to determine these parameters. The regenerator losses are described by an exponential function and the shuttle heat transfer losses by a third order polynomial fit function. The parasitic losses between the hot end and the cold tip of each stage have been modelled based on experimental measurements. The interpolation functions developed fit the data well; the remaining losses represent at maximum around 1% of the gross cooling power for each stage, and thus can be neglected. To validate the fit functions, a simple 5-node model has been implemented using a standard thermal solver. The hot stage, the middle stage, the cold stage and the two heat loads applied on the cooler (one at the middle stage and the last one at the cold stage) are modelled. Setting the hot stage as a boundary and including the equations found earlier, the model is processed using the iterative solver. The results of these simulations are not convincing. Few of the experimental data points have been re-found via the solver and most of them diverged. A change in the pneumatic work coefficient has been tried, relating it to the temperature ratio, and thus having a variable coefficient throughout the simulation. The results did not improve drastically. Such a simple approach for a two-stage Stirling cooler does not take into account all the relevant phenomena involved between the two stages, and thus a simple coefficient for the pneumatic work is not sufficient. Nevertheless, by looking more deeply into the gas dynamics taking place in a cycle of the cryocooler, or by taking another approach for the simulation of the cooler in the solver (for example two separate Stirling coolers connected via a thermal link, or the cooler being treated as a large single-stage cooler)
might enhance the results. But for now, the method using fit functions describing gross cooling and losses combining boundary nodes method does not work for two-stage Stirling cryocoolers. 4. Summary and future developments An investigation has been performed into methodologies suitable for the development of library cryocooler models for performing system level studies of spacecraft. Recommendations for a suitable approach are given and prototype models of a Joule–Thomson and a Peltier cooler are briefly described. Models of single-stage Stirling and pulse tube coolers using fit functions based on available performance data have been further developed. Verification of these fit functions using simple thermal models gives good correlation. Investigations into the modelling of twostage coolers have begin but initial results at correlation have not been successful. Further development is planned for the future with the aim of developing a library of simple yet representative cryocooler models. References [1] Yuan SWK, Spradley IE. A third order computer model for Stirling refrigerators. Adv Cryogen Eng 1992;37(B):1055–62. [2] Yuan SWK et al. Computer simulation model for Lucas Stirling refrigerators. Cryogenics 1992;32(2):143–8. [3] Miller SD et al. Cryogenic Systems Integration Model (CSIM). Cryocoolers 10. New York: Kluwer Academic/Plenum Publishers; 1999. [4] Available from: http://irtek.arc.nasa.gov/CryoDev.html, the NASA AMES Cryogenic Group homepage. [5] Details available from: http://www.harvardthermal.com, homepage of Harvard Thermal, USA. [6] Walker G. Cryocoolers, Part 1: Fundamentals. New York: Plenum Press; 1983. [7] Details available from: http://www.melcor.com, homepage of Melcor, USA. [8] Details available from: http://www.aerospace.power.alstom.com, homepage of Alstom. [9] Details available from http://www.mayahtt.com, homepage of Maya. [10] Metzger T, Huebener RP. Modelling and cooling behaviour of Peltier cascades. Cryogenics 1999;39(3):235–9. [11] Development of an engineering model 50–20 K two stage Stirling cycle cooler, Final report, British Aerospace Space Systems Limited, ESA contract 9004/90/NL/PP, May 1992.