Theoretical and experimental studies of a double-inlet reversible pulse tube refrigerator

Thermal isolation of large loads at low temperature using Kevlar rope L. D u b a n d * , L. Hui and A. Lange Department of Physics, University of California, Berkeley, CA 94720, USA

Received 17 July 1992 A mechanical support system is described for a space-borne 3He refrigerator that relies on Kevlar 29 braided cords to achieve thermal isolation at temperatures ranging from 0.3 to 40 K. Measurements of the Young's modulus, tensile strength, creep and thermal conductivity of Kevlar cords at room and liquid nitrogen temperatures are discussed and used to predict the performance of the system. Results of vibration tests of the mechanical support are compared with these predictions. The system safely supports a 350 g mass against 30 G sinusoidal accelerations at frequencies of 20 Hz to 2 kHz. The calculated heat load between 2 and 0.3 K is 0.7/~W.

Keywords: space cryogenics; support systems; Kevlar

Cryogenic systems for use in space require strong mechanical support to survive the large accelerations and severe vibrations of launch. The support system must also provide excellent thermal isolation to maximize the performance of the cryogenic system. The required mechanical strength and thermal isolation can be difficult to achieve in a compact structure. We have developed a 3He refrigerator that can be recycled in zero-g for the Infrared Telescope in Space (IRTS), an orbital telescope to be launched in 1994 to study the diffuse galactic and extragalactic backgrounds at infrared and submillimetre wavelengths. The refrigerator will be used to cool the bolometric detectors of the Far Infrared Photometer (FIRP) from the 2 K provided by the IRTS 4He cold plate to 0.3 K. Our work on the design and tests of the 3He refrigerator has been presented before ~. The thermal isolation of the evaporator from the 2 K cold plate determines both the operating temperature and the duty cycle efficiency of the refrigerator. The mechanical support of the evaporator is thus a critical component in the design of the refrigerator. This paper reports in detail the design of the mechanical support and the thermal and mechanical properties of Kevlar 29 cords, at room and cryogenic temperatures. Good thermal isolation requires the use of materials with a high length to cross-section ratio. For most materials, being stronger under tension than in compression, the optimum combination of mechanical strength and thermal isolation is usually obtained from a support system consisting of cords or fibres under tension. An * Present address: Commissariat ~ I'Energie Atomique, Service des Basses Temperatures, CENG/SBT, BP 85 X, Grenoble 38041 Cedex, France

object suspended in three-dimensional space has six degrees of freedom (three translational and three rotational), requiring a minimum of six constraints to fix the object rigidly. In general, the minimum thermal conductivity is thus achieved by attaching the cords to six appropriately chosen points on the suspended object, as described by Timbie et al. 2. The system described here, shown in Figure 1, uses eight attachment points in order to take advantage of the rectangular symmetry of the available space. The small increase in thermal conductivity due to the addition of two supports is more than offset by the additional length of each of the cords in this symmetry. The material should have low thermal conductivity, high breaking strength and high Young's modulus. A high modulus ensures that the resonant frequencies are higher than the driving frequencies, which are typically below 100 Hz for most launch systems. We chose braided cords made out of Kevlar 29 as the candidate that fulfils all these requirements. Kevlar 29 is one of the high tensile strength and modulus aramid fibres produced by Du Pont. Table 1 is a comparison of the properties of Kevlar 29 (both fibre and braided cord) and stainless steel 3. The braided cords, with a specified breaking strength of 440 N (100 lb), are manufactured by Ashaway Line and Twine Company 4.

Design of support system Figure 1 shows a schematic of the support system. The refrigerator ~ consists of two elements, an evaporator and a sorption pump, that are supported independently in a similar manner. The eight 'attachment points' are actually pulleys: the support system for each element of the refrigerator consists of two separate Kevlar cords,

0011-2275/93/060643q35 © 1993 Butterworth-Heinemann Ltd

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Thermal isolation of large loads: L. Duband et al. equations we obtain a2z a2 t

AY

a2

16X--lm x ~ x z

where m is the mass of the suspended object and l is the length of cord between two pulleys. The translational frequency in the z direction is

N.

1 ./16AY

fz= x Hm x Figure 1 Top and front view of Kevlar support system. Heavy black dots indicate attachment pulleys

a 2

The translational frequencies in the x and y directions can be calculated in a similar manner and are equal to

1 ]8AY b 2 + r2 fx'Y= ~ " ~ l ~m X one on each end of the suspended object, each wound around free pulleys (which can be turned to adjust tension) and ending its course by being wound around a capstan. Such a design allows for easy adjustment and balancing of the tension in different parts of the cords (as compared to a support system using eight separate cords). The capstan serves as an excellent device to attach the end of the cord and maintain tension. It is important that the support system is not weakened by bad attachment of the end of the cord. The quantities in Figure 1 specify the configuration of the Kevlar cords. Ideally, the configuration should be chosen to equalize translational resonant frequencies in the x, y and z directions and so maximize the lowest resonant frequency of the three modes. (Raising the resonant frequency in one direction necessarily lowers the resonant frequency in other directions.) In our system, however, the frequencies in the three directions are not exactly equalized, due to other design constraints. Treating the pulleys as points, we can easily calculate the resonant frequencies for all the modes, as detailed below. There are six modes of vibrations: three translational modes along the x, y and z axes and three rotational modes around the x, y and z axes. Assuming equal tension in each portion of the cord, we first calculate the frequency for the translational mode in the z direction. The potential energy stored in the upper cord and in the lower cord for a displacement z are, respectively

1

8x-×--×

2 8 X

1

2

X

AY l0 AY

[x/(a+z) 2 + b 2+r 2-1012

X [ x / ( a - z) 2 + b 2

+

r 2 --

lo ] 2

lo

where A is the cross-section of the cord, Y the Young's modulus of the cord, 10 the natural or unstretched length of cord between two pulleys and a, b and r are as defined on Figure 1. Because the Young's modulus of Kevlar is very high, we can assume I = 10. For the same reason the displacement z is small and we can limit the calculation to the first order. Using the E u l e r - L a g r a n g e differential

644

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Similarly, the frequency of the rotational mode around the z axis is

where I is the moment of inertia of the suspended object. The frequencies of rotation around the x and y axes are higher because of higher torques as a result of the suspended object being longer in the z direction than in the other two. In fact, all of the rotational frequencies are higher than translational ones. The support system is vulnerable to three potential difficulties. The first is the weakening of Kevlar cord when it goes around a pulley. Kevlar, because of its anisotropic molecular chain structure 3, is strong in the longitudinal but weak in the transverse directions. A minimum pulley diameter is necessary in order not to weaken the cords. To establish the minimum pulley diameter, we hung a piece of Kevlar cord around a pulley and loaded it to failure. As expected, the cord breaks at the pulley. For the 440 N (100 lb) test cord, we found that a pulley diameter of 2.5 mm is adequate to realize the full rated strength of the cord. The second difficulty is the negative longitudinal thermal expansion of Kevlar (it expands longitudinally when it cools down). Such expansion lowers the tension of the cord and might loosen it up all together. The only data we are aware of on the thermal expansion of Kevlar at cryogenic temperature is that of a Kevlar 49 epoxy composite (60% Kevlar 49 by volume) 5. Its fractional integral expansion in the longitudinal direction from 293 to 0 K is 9 × 10 -4. We assume that this gives a reasonable estimate of the thermal expansion of our Kevlar 29 cord. (Kevlar 29 and 49 have the same coefficients of thermal expansion at room temperature.) Note that the estimate of thermal expansion of Kevlar 29 cord is further complicated by its weave structure. Our system has been designed to compensate for the thermal expansion of the Kevlar by the contraction of other parts of the support system. The pulleys are attached to materials with low coefficients o f thermal expansion, and the two brackets holding the outer set of pulleys are

Thermal isolation of large loads. L. Duband et al. connected to each other by aluminium, which contracts by an amount large enough to maintain tension of the Kevlar cord. Moreover, the Kevlar braided cord that we use has a lower Young's modulus than Kevlar fibre itself, which reduces the change in tension because of thermal expansion. The last difficulty is the creep of Kevlar cord which, again, has the danger of significantly reducing tension in the cord. Our measurements, discussed below, indicate that the creep is logarithmic in time, and that the effect is insignificant if the cord is retensioned at least one day after the initial tensioning.

Measurements on Kevlar 29 braided cords The performance of our support system depends critically on the properties of Kevlar 29 cord, at room and cryogenic temperatures. Since the properties of Kevlar at low temperature are not well known, we have performed some measurements ourselves, which are presented below. They are important for the prediction of the performance of any support system that makes use of Kevlar 29 cords.

Thermal conductivity Little information is available on the thermal properties of Kevlar. The thermal conductivity of a Kevlar 49 composite was measured at 6, 81, 196 and 277 K by Hust 6. In the range 1 . 8 - 4 . 2 K, the thermal conductivity of Kevlar 49 was also measured 7 and was described by k = 18.9 × 10 -6 T 2 W cm -I K -I. We have performed measurements on a Kevlar 29 braided cord in the range 5 - 4 0 K and found relatively good agreement with the above data. Our results can be represented by k = 2 1 . 5 × 10 -6 T l 5 8 W cm -~ K 1. The data are summarized in Figure 2. It is believed that Kevlar 29 and 49 have very similar thermal properties. Based on the values given by Zhang 7, the parasitic heat load through the Kevlar cords in our support system from a 2 K cold plate to a 0.3 K evaporator is estimated to be 0.67 #W, which contributes negligibly compared to other sources of heat load.

Thermal conductivity(W/cm.K) 10-~ . . . . . . . . . . .

Modulus and strength tests The modulus and strength were measured on a digital testing machine. The strain rate was kept constant at 2 5 . 4 # m s -~ (0.001in s ~). The Kevlar cord was wound around an upper pulley a few times, passed around a lower pulley and ended its course by being wound again on the upper pulley. The pulleys are free to turn so as to equalize tension on the two sides of the cord. The pulleys are pulled apart while load and stroke are recorded; the tests are essentially tensile tests in the longitudinal direction. This testing set-up is similar to commercially available capstan grips, which are commonly used for testing of yarns and cords to avoid breaking of the specimen at grips. In our tests, the specimen did indeed break between the pulleys. The gauge length is measured by the distance between the grips instead of being directly measured by a strain gauge because the attachment of a strain gauge to the specimen is difficult. The load versus gauge length graphs of Kevlar 29 braided cord at 290 and at 77 K are given in Figure 3. Note the jumps on the graph. There are especially big jumps towards bigger loads. These are due to slippage of the Kevlar cord at the upper pulley, which increases the gauge length for a given load. But between two big jumps, the data points are well fitted by straight lines with similar slopes. We estimate the Young's modulus using the slope o f the straight line on to which the largest number of data points fall. (The squares of the correlation coefficients for the straight lines shown in Figure 3 for room temperature and 77 K are 0.998 and 0.997, respectively). It is possible that there are also slippages in this portion of the data points, which would make our estimate of Young's modulus a lower limit to the actual value. In the support system using Kevlar cord, it is the lower limits of the resonant frequencies and, therefore, the lower limit of the Young's modulus that are important. Note that, initially, the load does not increase even while the gauge length is increasing, again due to slipping of cord over the upper pulley. We thus have to estimate the natural length of the specimen (length at no load) by extending the straight line on the graph to the x axis and taking the x intercept as the natural length.

Load (Newton) 1200 rupture of ~--Kevlar cord

Kevlar 29, This work -----

K e ~ l a r 4 9 , J. H . Kevlar

Zhang,

ref. 7

1000

4 9 , J. G . H u s t , r e f . 6

1 0 .2

800 ~ ~ m 103

400

/

10 4

temperature

600

•

/" J

20O 10s

,

Figure 2 Thermal temperature

,

. . . .

, ,I

,

10 Temperature (K) conductivity

,

,

,

.

0 7,4

. .

10O

of Kevlar as a function

of

i

7,6

7,8 Gage length(cm)

v

8,0

8,2

Figure 3 Load v e r s u s gauge length of Kevlar 29 braided cords at room and liquid nitrogen temperatures

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Thermal isolation of large loads: L. Duband et al.

The cross-sectional area of the cord is found by weighing a piece of known length and taking the density of Kevlar to be 1.44 g cm -3. The Young’s modulus is equal to (load/area) multiplied by (natural lengthiextension). Load/extension is given by half the slope of the best fit straight line (half because in our set-up we are essentially measuring the load/extension of two cords together, one on each side of the pulleys). Area and natural length are obtained as indicated above. The tensile strength equals the breaking load divided by the cross-sectional area. Like load/extension, the breaking load measured has to be divided by two to obtain the actual strength of a single braided cord. Note that since we do not know how much the cross-sectional area changes with temperature, we just use the cross-section at room temperature to evaluate Young’s modulus and tensile strength, even at 77 K. It is believed that the cross-sectional area does not change appreciably. The results are listed in Table 2. Compared with the properties of Kevlar 29 fibre, as listed in Table 1, Kevlar braided cord has lower tensile strength and modulus than Kevlar fibre but its properties improve quite significantly at cryogenic temperatures. The major sources of errors in the above measurements are: 1 Slippage of the cord wound around the pulley. It is hard to quantitatively estimate this error. However, since this has the effect of lowering the apparent Young’s modulus, we believe our number for Young’s modulus is a good lower limit. 2 Unequal tension in the two portions of the cord on the two sides of the pulley. This brings in errors in the estimates of both the load and the natural length. While both affect the modulus estimate, only the first affects the breaking strength estimate (it actually makes the estimate lower than the actual breaking strength). We measured the friction that prevents the pulleys from rotating to equalize tension and, therefore, the maximum difference in load in the two sides of the cord (this friction is smaller than that between the pulley and the braided cord). From this, we obtain 9% for the maximum fractional error in the estimate of the load on one side of the cord, while the Table 1 Comparison of properties of Kevlar 29 fibre, braided cord and stainless steel at room temperature

Kevlar fibre

Tensile

strength

(MPa) Young’s

modulus

(MPa) Density (g cmm3)

29

Kevlar 29 braided cord

Table 2 Properties of temperature and 77 K

Kevlar

29

We performed a creep experiment on Kevlar 29 braided cord under a load of 110 N (25 lb), which is 25 % of its breaking strength. The result is shown in Figure 4. The best fit straight line through the data points in the plot of strain versus logarithm of time has a correlation coefficient squared of 0.996. The creep is logarithmic: an

strain(%)

29

200000

1.44

1.44

Creep

580

29000

83000

Based on the above, the maximum error in the Young’s modulus is 15% and the maximum error in the tensile strength is 12%. Using these measurements, we can calculate by how much the tension of the Kevlar cord will change in our support system as it is cooled from room temperature to 77 K, taking into account the change in Young’s modulus and the thermal expansion of the Kevlar. For an initial tension of 133 N (30 lb) at room temperature, the tension at 77 K would be 253 N (57 lb). For the estimation we use Hartwig’s number for the thermal expansion of a Kevlar composite5. We assume in the calculation that the two ends of each portion of the Kevlar cord between two pulleys are fixed, which would not be true if the cord slips over the pulley. We ignore the change in distance between the two pulleys because that depends on the materials used and because our primary interest is in how the thermal expansion affects the tension. Note also that since most of the thermal expansion and change in Young’s modulus should occur between room temperature and 77 K, our calculation for 77 K should give a good indication of the behaviour at 4 K or even lower. In any case, we can see that the combined effects of the thermal expansion and the increase in Young’s modulus increase the tension in the cord by approximately a factor of two, assuming that the endpoints are kept fixed.

Stainless steel (type 304)

1500

3600

Kevlar

maximum fractional error for the natural length is 3%. 3 Fractional error in the estimate of the cross-sectional area of the Kevlar cord (0.4%). 4 Errors in measurements of the load (10 N). 5 Error in the measurement of the gauge length [2.5 X 1O-5 m (O.OOlin)] .

7.8

braided

cord

at

room

I

I Room temperature

77 K

1500 29000

1800 58000

10 Tensile strength (MPa) Young’s modulus (MPa)

I

I

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100

loo0

time (hours) Figure

4

Measured

creep

of Kevlar

29 braided

cords

loo00

Thermal isolation

increase in strain of 0.15 % per decade. This is about three times bigger than the value for Kevlar 29 fibre3, but is still small. In fact, it takes more than three years to halve the tension. Note that the calculation is based on the creep at 110 N (breaking strength at 440 N) but in our case the tension actually decreases as the cord creeps. Given the fact that lower load means smaller creep, the tension calculated from Figure 4 will underestimtae the real tension at various times. To be most careful, one can tighten the cord after it has been allowed to creep for some time. Since the creep is logarithmic, further increase in strain after the initial creep will be very small. Results

of vibration

tests

A mechanical prototype of the 3He refrigerator was built, with the Kevlar cord suspending the pump and evaporator, which have masses of 350 and 140 g, respectively. The system was vibration tested by being shaken in the y* and z directions, where the z direction is as defined before and y* lies in the x-y plane and lies at 45” to both the x and y axes. The resonant frequency of translation in the y* direction can easily be shown to be the same as that in the x and y directions. Tests were first performed at a vibration level of l/2 G from 5 to 2000 Hz to look at the resonance behaviour. The calculated and observed frequencies for the pump along the y* and z directions are compared in Table 3. Note that with the geometry we used, the resonant frequencies in the x, y and z directions are not exactly equal, due to other design constraints. The fractional errors in the frequencies are 18.8 and 7.1% for the y* and z directions, respectively. The refrigerator was shaken at 7.5 G peak acceleration from 30 to 400 Hz and then at 15 G from 400 to 2000 Hz, again in both directions. As expected, the resonances become less sharp at higher amplitude; in other words, the quality factor Q is decreased. Subsequent tests involved increasing the vibration levels until the Kevlar cord broke. Failure occurred during a 40 G acceleration at one of the pulleys because of intense rubbing of the cord against the pulley. The vibration tests

Table 3

Resonant

frequencies

of pump

y’

Calculated

resonant

0-k)

resonant

z direction

frequencies

(Hz) Measured

direction

260

300

320

280

frequencies

of large loads: L. Duband

et al.

show that the support system can safely withstand a vibrational level of 30 G and that all the resonant frequencies are well above 200 Hz. The vibration tests were performed at room temperature only. We expect the performance of the support system to improve at lower temperature due to increase of the Young’s modulus and of the breaking strength. Timbie8 has performed vibration tests on a similar system at = 100 K and the results indicate that the performance of the system is at least as good as at room temperature. Conclusions

A support system utilizing Kevlar braided cord provides high strength and excellent thermal isolation in a compact geometry. Calculations based on our measured value of Young’s modulus agree well with measured values of the resonant frequencies of our support system. The strength of the support under prolonged vibration seems to be limited by abrasion of the Kevlar cord against the pulley. Nevertheless, the results of our vibration tests indicate that our design is adequate for supporting large loads during launch. The support is expected to become stronger with higher resonant frequencies at lower temperatures. Acknowledgements

We wish to thank F. Lopez, M. Ambrosini and A. Baeza of the UC Berkeley machine shop for their expert mechanical work in building the mechanical prototype. J. Glazer and D. Chu were extremely helpful in the running of the tensile tests. Thanks are also given to G. Bernstein, P. Richards, P. Timbie and J.H. Zhang for many useful discussions. This work is supported by NASA-Ames University Interchange no. NCA2-240 and NASA grant no. NAGW-1597. References Duband, L., Hui, L. and Lange, A. A space-borne ‘He refrigerator Cryogenics (1990) 30 263-270 Timbie, P., Bernstein, G. and Richards, P. Development of an adiabatic demagnetization refrigerator for SIRTF Cryogenics (1990) 30 271-275 Pigliacampi, J.J. Organic fibers, in: Engineering Materials Handbook Vol 1, ASM International, Ohio, USA (1991) 54-57 Kevlar 29, 100 Ibs test, company report, Ashaway Line and Twine Company, Ashaway, Rhode Island, USA Hartwig, G. and Knaak, S. Fibre-epoxy composites at low temperatures Cryogenics (1984) 24 639-647 Hust, J.G. Low-temperature thermal conductivity of two fibreepoxy composites Cryogenics (1975) 15 126-128 Zhang, J.H. Kevlar thermal conductivity, personal communication, University of Wisconsin, Madison, USA (1990) Timbie, P. Vibration tests, personal communication, University of California, Berkeley, USA (1990)

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