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Thermal model for a hybrid cryostat
Thermal model for a hybrid cryostat J.L Casse and E.E.M. Woestenburg Netherlands Foundation for Radio Astronomy, Dwingeloo Observatory, PO Box 2, 7 9 9 0 AA Dwingeloo, The Netherlands Received I October 1985; revised 13 November 1985 To maximize the hold time of a 2 dm 3 4 K helium cryostat, a system has been designed and built which makes use of a closed cycle refrigerator for cooling the radiation shields. The optimization of this so-called hybrid cryostat has been simplified by the use of a thermal model. The properties of a number of models have been obtained from simulation runs. Comparison between simulation results and measurements on the real system has shown good agreement The real system optimized from model results yields a hold time of 6 days.
Keywords: cryostat; refrigerator, helium; thermal modelling
Some of the detectors used in astronomy at optical, infrared or radio wavelengths have to be cooled to temperatures as low as or below 4 K. This is usually done using cryostats with liquid helium. The size and the insulation of the helium cryostats are subject to practical limitations, while these same parameters determine, in practice, the "hold time" of the cryostat, i.e. the time between helium refill. This refill operation is tedious and time consuming. One way to increase the hold time is to make use of a hybrid cryostat in which the radiation shields of the helium dewar are cooled by a closed cycle refrigerator. Whereas a 2 dm 3 free standing helium cryostat of 4 K has a hold time of ~1-1.5 days, a comparable hybrid system can last up to 6 days. To optimize the cryostat the major heat leakages need to be identified. This can be done by using a thermal model. Using a simulation software program it is possible to evaluate the model. By modifying the system parameters the sensitivity of the system to changes can be investigated and potential areas for improvement identified. The hybrid cryostat analysed in this Paper is based on a 2 dm 3 helium cryostat being built for the United Kingdom lntYa Red Telescope (UKIRT) on Mauna Kea, Hawaii. USA. A Cryogenic Technology Inc. (CTI) model 1020 closed cycle refrigerator is used to cool the two radiation shields of the dewar. Measurements on the real system have been of great assistance in the construction of the theoretical model. A standard simulation run yields the temperature at selected nodes in the model as well as the heat flow between the nodes. This latter information in particular has acted as a guide-line for identifying the main heat transfer mechanisms. Complete agreement between model and cryostat is very difficult to achieve, as a result of inaccuracies in system parameters and, in particular, in the infrared emissivities of the radiating surfaces. The model simulation yields a hold time of 9 days for a practical system with a 5 mW signal window. For comparison, the original free standing dewar has a hold time of 1.2 days. A real system optimized with the help of the model has reached a hold time of 6 days.
0011-2275/86/030165-06 $03.00 © Butterworth ~ Co (Publishers) Ltd
As the closed cycle refrigerator forms an essential part of the hybrid cryostat a simple phenomenological model of this refrigerator has been constructed. This model is described in the following section and is relevant for CTI refrigerator models 1020 and 21. The thermal model is then integrated into the cryostat thermal model in the section entitled Thermal model of the hybrid cryostat while in the section following this, a number of simulation runs are presented and discussed. Finally, conclusions are drawn on the usability of the model.
Thermal model refrigerator
of
the
closed
cycle
The thermal model that has been constructed is the simplest possible model satisfying the known boundary conditions. These relate the temperature at the two stations to the thermal loads on the system. These conditions, known from laboratory measurements, are not very accurate. The refrigerator type considered in this Paper is based on the Gifford-MacMahon cycle '. In Giflbrd and MacMahon's paper, the refrigerator cycle is analysed for a one stage machine. The authors derive the refrigeration power, Q j, at the cold station, j, per cycle, and the regenerator losses, Q ij. These parameters have the following form: Qj = Kj zSat' V 1 Qij - Ki j
(1)
T i - Tj Tj
(2)
where AP is the helium pressure difference between the compression and expansion phases and V is the expanding volume. T i and Tj are the ambient and station temperatures, respectively. Kj and K ij are proportionality factors. Equations (1) and (2) are approximations acceptable only for heat loads which are not too large. The
Cryogenics 1 9 8 6 Vol 26 March
165
Hybrid cryostat: J.L. Casse and E. E.M. Woestenburg
Ti'¢
I
Rij
mj
T 100
I
It m=
Figure 1
CTI 1020
(K)
O,j
so
I
Thermal model for a one stage refrigerator
0 0
I 5
I 10
I 15
I 20
020 K (Watts)
100 E
[ ~R12 ,,
1
i
_CTI _ _ 21
11
I."
"
T
R2 3
,
3
I
(K)
/- /
Q70= 0
5O
(13
a2
Figure 2 Thermal model of the two stage refrigerator showing the thermal losses, O2x and O3~ at the two stations 0
0 thermal model is based on these two Equations and neglects all other heat loss mechanisms The equivalent network for the model is shown on Figure 1, where Qj is given by Equation (1) and R ij by
Rij = TjKij
Model
I
I
t
2
3
4
Q20K (Watts) Figure 3 Plot of the temperatures at the 70 K (- - -) and 20 K ( ) stations for a CTI 1020 (top) and a CTI model 21 (bottom) refrigerator against thermal loading
(3)
The model for a two stage refrigerator is shown on Figure2, where two one stage networks have been connected in series. Nodes 1, 2 and 3 in the model represent the ambient, the 70 K and the 20 K stations, respectively. The thermal loads at the two stations are Q ~x and Q 3x. Determination of the model parameters for a given refrigerator is performed by relating the given model to the performance measurements. Solving the network for the case of CTI model 1020 and 21 refrigerators yields the parameters shown in Table 1. From the thermal model the refrigerator performance curves can be derived. The temperatures at the two
Table 1
i
1
parameters for the
CTI 1020
and 21
stations are calculated from the model shown on
Figure 2. T2 =
T3 =
T1 1 +Kl2 [Q2 + Q a - Q 2 x - Q a x ]
T2
(4)
(5)
1 +K2a [ Q a - Q a x ]
The calculated performance curves for the CTI 1020 and 21 refrigerators are shown on Figure 3. These model performance curves do fit the actual refrigerator data accurately enough for present purposes.
refrigerators 02
Model 1020 Model 21
166
(w)
(w)
Q3
K12
(w -1)
(w-')
K23
Thermal model of the hybrid cryostat
53.2 10
21.3 4.1
0.085 0.45
0.13 0.7
The modelled cryostat is shown inFigure4. It consists of a 2 dm 3 helium dewar surrounded by two radiation shields attached to the 70 K and the 20 K stations of a CTI 1020
Cryogenics 1986 Vol 26 March
Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg NECKOFL,KOEWAR; He FILLCONNECTION
2H
GAS FLOW
3H
CONNECTIONFOR VACUUMPUMP IN
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.
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Figure 5 Simplified thermal model of the hybrid cryostat The refrigerator is represented by nodes 1 , 2 and 3, while the helium vessel is at node 4. Nodes 2S and 3S are at the radiation shields, while 2N and 3N are the surface nodes in the neck of the dewar. Nodes 2H and 3H model the gas flowing in the neck of the dewar Figure 4 Prototype of the hybrid cryostat for the UKIRT common user receiver. In the centre is the helium dewar surrounded by two radiation shields connected to the 20 K and the 70 K stations of a CTI 1020 closed cycle refrigerator. The 690 GHz bolometer is attached to the bottom plate of the dewar and the 3 5 0 GHz mixer to the 20 K station. For both units backshort facilities have been implemented
closed cycle refrigerator. The Figure shows two signal windows (on the left) giving access to a bolometer for 690 G H z (top) and a Schottky mixer for 350 GHz (bottom). It also shows, on the right, the access openings for the adjustment of the backshorts of both devices. The radiation shields are attached to the refrigerator through flexible copper links. Their infrared emissivity in the reference model is 3%. The input signal to the bolometer passes through infrared filters located in both shields. The shields are made of aluminium and are gold-plated. The helium vessel is made of stainless steel. The thermal model of the hybrid cryostat is represented by a number of nodes linked together by thermal resistances and radiation links. The simulations performed on the model assume that the system has reached the steady state and hence all heat capacities are ignored. The simplest model of the hybrid cryostat is shown on Figure 5, where nodes 1, 2 and 3 represent the refrigerator. The neck of the dewar is modelled by the links I, 2N, 3N and 4, where node 4 represents the 4 K station. Nodes 2N and 3 N represent the nodes in the neck of the dewar where the radiation shields at 70 K and 20 K are attached, respectively. Nodes 2S and 3S are located at the centre of the shields. The model is determined by the heat transfer mechanisms proper to the system. The main mechanisms are considered below.
Convection between escaping gas and the dewar Heat transfer between the evaporating helium gas and the walls of the neck of the dewar plays a very important role. This effect has been modelled as shown in Figure5 where the gas flows from node4 to node 1, as indicated, while the cooling of the walls takes place via the convection resistances 2N-2H and 3N-3H. The process is similar to that of a heat exchanger. Assuming that the flow in the neck is laminar(small flows), the convection coefficient, h, can be then calculated from the Nusselt formula 2
k (Wm_2 K_I) h = 3.66 ~--
(6)
where k is the conductivity of the gas, which is temperature dependent and D is the diameter of the pipe. The gas conduction coefficient has been approximated by a polynomial k(T) = 6.27 x 10.3 + 0.75 x 10-3 T - 0.914 x 10-6 T 2 (7) where k is in W m -1 k -1 and T in K. The model of the dewar shown on Figure 5 is, in practice, too crude to accurately represent the heat exchange between the walls of the dewar and the helium gas. By using only four nodes to model the neck of the dewar, the cooling effect of the gas is underestimated. Table 2 gives the hold time in days for the model as a function of the number of nodes. It shows that a model Table 2 Hold time and gas temperature as a function of the number of nodes
Radiation transfer through the shields Radiation transfer is modelled through radiation links as shown on Figure5. These links connect the shields to the outer surface at ambient temperature (node I) and the helium cryostat (node 4). Nodes 1, 2S, 3S and 4 are the surface nodes; they are linked together via the so-called floating nodes z.
Number of nodes
Hold time (days)
Gas temperature at outlet (°C)
4 8 15 30
.5 .73 1.09 1.23
-75 -30 -13
Cryogenics 1986 Vo126 March
167
Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg with 30 nodes is accurate enough for present purposes. The temperature of the gas at the exit is also listed. The gas temperature at outlet quoted for the 30 node model (-13°C) agrees with the measured temperature
(-11)°C). Thermal conduction along the neck of the dewar The thermal conduction parameters lbr the neck of the dewar must also account for the fact that the conductivity of the material (stainless steel) depends strongly on temperature. The conductivity of the material k(T), has been approximated by a function
[ -l 'Til logk(T) = 4.123 1 - e
~l.6s r j _ l . 9 5 9
,mw
NODE6A
(8)
x
1 4rn~,
l
NODE 2N
16.Stow
08row
ImW
OSmW
NODE 3N
where k is in W m -~ k -t and T in K.
[]
Shield thermal resistance As can be seen on Figure5 the resistance of the shields has been taken into account These resistances should be small (1.7 K w-a): however, the link between shield and refrigerator needs to be flexible and an additional resistance of ~15 K W -~ has to be considered in the model for the 70 K link. This leak resistance is important since most of the heat absorbed in the 70 K shield ( ~ 3 W) flows through the link and hence determines the temperature at the connecting node in the neck of the dewar.
Radiation transfer through the signal window The signal window at 4 K is critical for the heat loss balance, as soon as hold times of several days are achieved. At the level of 5 days. for instance, the heat loss amounts to 12 mW. while the loss via the window can beas large as 5 mW lbr a 3% intYared transmissivity. The effect of the window has been modelled as a direct heat source into node 4, the 4 K station.
NODEb, {HELIUMI
Figure 6 Characteristic heat flow distribution in the neck of the dewar as a result of radiation. The neck is modelled by three surface nodes only (6A, 2N and 3N). The floating nodes, 6AX, 2NX and 3NY~ account for the emissivity of the surface
1
Dewar anchoring lines; they are made of stainless steel0.4 mm in diameter. The heat loss is ~0.2 mW for 4 wires, 3 cm long,
2
Bias lines: the thermal resistance from the 4 K and 20 K stations has been estimated to be 0.7 × I ( P K W -t.
3
Conduction through rarefied gas; because of cryopumping only a thermal loss between the 300 K node and the70 K shield has been considered. The estimate for the loss3 gives 40 mW indicating a resistance, R ,.2s of 5300 K W-L
4
The loss through gas conduction in the neck of the dewar is more difficult to estimate. The gas flows upwards while the heat diffuses downwards. Depending on the flow, the heat wilt reach further down the neck The heat diffusion velocity can be estimated from the gas thermal conductivity and heat capacity. It can be shown that with a flow ofl00 cm 3 min-' ( 10 days hold time) the heat will not reach beyond the 70 K shield connection. The model accounts for this loss by having a thermal resistance between nodes 1 (ambient) and2 NH (gas at level 2N) equal to 2680 K
Radiation through the neck of the dewar The analysis showed that the radiation heat loss through the neck of the dewar is quite important The hot top section of the dewar radiates, like the300 K signal window+ toward the 4 K station, the neck of the dewar acting like a waveguide tot the infrared radiation. The amount of heat reaching the liquid helium depends on the infrared emissivity of the walls of the neck and on the geometry. For simplicity a model with only five radiation nodes was used. as shown on Figure6, These nodes are 1 (ambient), 6A, 2N, 3N and 4 (4 K station). The floating nodes IX, 6AX, 2NX and 3NX account for the infrared emissivity of the wall surfaces. The radiation coupling between nodes depends on the geometry from which the form or viewing factors can be derived 2. Figure 6 shows the heat flows between the various nodes in the model for an infrared emissivity of 5% of the wall of the neck and 1/)% for the top node (node 1): some 7 mW is absorbed by the 4 K station.
Other heat losses The remaining heat transfer mechanisms are less critical. The following heat loss mechanisms have, nevertheless, been taken into account
168
Cryogenics 1 9 8 6 Vol 2 6 M a r c h
W
-l
"
Thermal simulation results The simulation program ESACAP 4 translates the thermal model into a set of differential equations which are then solved in the time domain or in the steady state. It yields the node temperatures of the system as well as the heat flows between nodes. The major heat flows lbr the reference model are shown on Figure 7, where the main nodes are indicated. The Figure shows that in this case. where there is no signal window, the main source of heat is
Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg
/
/
-93W1
.OSW 1.SW
.?lW
,2~.22w
.38W
q
.22W
. • 5mW
I
nWl
2L . t,81
~'~my/~j.2//i2////////A
17rnW
3
2
y/////////////////////////////////-//////////////,2////A
7 Main heat flow distribution for the reference hybrid cryostat in the absence of a signal window Figure
Figure8
radiation through the neck of the dewar. shows the heat flows for a free standing cryostat. The contribution of the refrigerator is clear after comparing the two Figures: it absorbs the radiation falling onto the shields. An obvious improvement to the system is to bring a shield into the neck of the dewar, but to allow the gas to escape (baffle). The improvement is significant as shown on which gives the hold time in days for a number of models with a 5 mW signal window, as well as the physical temperatures at the nodes 2N and 3 N in the neck of the dewar, and the main heat flows into the4 K node in mW. The various headings indicate the deviations from the reference case (model 1). The reference model yields a hold time of 5.1 days, while the improved system with a baffle in the neck of the dewar(model 2) reaches 8.9 days. When compared with model 3, where the radiation loss has been ignored, it can be seen that the radiation shield is quite effective. However, care must be taken when the model reaches hold times > ~ 10 days. It must be ensured that the model parameters are accurate enough and that the second order heat transfer mechanisms have been well
Table3
Figure 8 Main heat flow distribution for the free standing cryostat in the absence of a signal window
accounted for. It is doubtful that the accuracy of the present model is adequate to separate the second order heat contributions. Model 4 in is based on a smaller refrigerator of the type CT1 21. The analysis has shown that with the present geometry a small refrigerator is quite satisfactory. In models 5, 6 and 7 changes have been made to the reference model to study the sensitivity of the system to a number of factors such as infrared emissivity of the 70 K shield and the shield serial resistances. Model 7 in particular, where radiation in the neck of the dewar as well as the shield resistances are reduced, yields a substantial increase in the hold time. To quantify the cooling by the escaping helium gas, model 8 has been constructed, where it is assumed that convection is not active in the neck of the dewar. The hold time then reduces significantly, from 5.1 days to 3.2 days. This heat transfer mechanism is thus quite important for the accuracy of the model. For comparison purposes the case of the free standing cryostat is given (model 9). This model uses the same parameters as for the hybrid system, except that the
Table3
Cryogenics 1 9 8 6 Vol 2 6 March
169
Hybrid cryostat: J.L. Casse and E. E.M. Woestenburg. Table 3
Performance of the hybrid cryostat with a 5 mW signal window loss Heat input into 4 K node (mW)
Model Characteristics 1 2 3 4 5 6 7 8 9 10 11
Table 4
Reference Radiation shield in neck No radiation loss in neck dewar Refrigerator model 21 Improved flexible link (lOx) shield-refrigerator Emissivity 70 K outer shield of 6% Improved flexible link and radiation shield in neck No convection in neck dewar Free standing cryostat 10 mW signal window No signal window
Hold time (days)
T3N (K)
T2N (K)
Conduction via neck
5.1 8.9 9.7 4.9 5.6 4.7 11.5 2.5 1.2 3.8 7.8
16 18 19 20 11 11 11 22 83 15 18
93 97 97 1O0 111 118 112 102 195 89 96
0.23 1.0 1.1 0.5 0.06 0.5 0.2 11.5 19.0 O. 1 0.7
Radiation from shields
Radiation in neck 6.6 0.7
0.006 0.008
0
0.008
6.8 6.0 7.5
0.013 0.003 0.01
0.05
0.003
7.9
0.01
12.0
13.0
6.3 7.0
0.005 0.007
Measurements and simulation results for the hybrid cryostat without signal window Hold time (days)
Features of model
Simulation
Temperatures of 70 K shields (I0
Measured
Simulation
Measured
1 Model with 6% infrared emissivity of the 70 K shield outer surface and a central thin tube in the neck of the dewar
5.3
4.1
119
92
2 Model with infrared emissivity of the 70 K shield outer surface better than 3%
7.8
7.3
88
79
120
112
3 Model with 6% infrared emissivity of the 70 K shield outer surface and a baffle in the neck of the dewar
10
refrigerator is removed and the shields somewhat shortened. The importance of the signal window can be quantified from model 10, where l0 mW is leaking through and model 11 without a window. The simulation results have been compared with measurements on the cryostat The results are shown in Table 4 for three different variations from the reference model each without signal window. As can be seen, the match between simulation and measurement is not perfect This is because of the limited accuracy to which some of the physical parameters of the cryostat are known. The accuracy of the temperature and gas flow measurements also plays a role, while some of the heat transfer mechanisms could probably be improved upon. The third model in particular does not compare too well. This is probably due to the construction of the baffle which does not properly seal the neck of the dewar. The accuracy achieved with the m o d e l however, has proved quite useful for optimizing the design of the cryostat The thermal model is particularly useful in identifying the main heat losses in a particular system. It shows where improvements can be made to increase the cryostat hold time. The final design of the hybrid cryostat described in this Paper includes the following improvements on the reference system: lower emissivity of the 70 K shield and a radiation shield (baffle) in the neck of the dewar. The hold time of the 2 dm 3 system including a signal window is 6 days, while the thermal model simulation predicts a hold time of 8.9 days (model 2),
170
Cryogenics 1 9 8 6 Vol 26 March
8
Conclusions A thermal model for an existing hybrid cryostat has been constructed which takes into account all active heat transfer mechanisms The accuracy of the model has been tested by comparing the simulation results with real measurements. The comparison is quite satisfactory, although it is limited by the accuracy of the physical parameters of the device. The model has proved to be accurate enough to make identification of the important heat transfer mechanisms possible; it has shown the way to substantial improvements in cryostat hold time.
Acknowledgements The authors wish to thank ESA for making the program ESACAP available to them. The Dwingeloo Radio Observatory is operated by the Netherlands Foundation for Radio Astronomy with the financial support of the Netherlands Organization for the Advancement of Pure Research (ZWO).
References 1 2 3 4
MacMahon, H.O. and Gifford, W.F_, ddv Cryog Eng (1959) 5 354 Chapman, A.J. Heat Transfer MacMillan Publishing Co, New York. USA (1974) Bailey, C.A. Advanced Cryogenics Plenum Press, London. UK (1971) Stangerul~ P. European SpaceAgency J (1980) 6 301
Keywords: cryostat; refrigerator, helium; thermal modelling
Some of the detectors used in astronomy at optical, infrared or radio wavelengths have to be cooled to temperatures as low as or below 4 K. This is usually done using cryostats with liquid helium. The size and the insulation of the helium cryostats are subject to practical limitations, while these same parameters determine, in practice, the "hold time" of the cryostat, i.e. the time between helium refill. This refill operation is tedious and time consuming. One way to increase the hold time is to make use of a hybrid cryostat in which the radiation shields of the helium dewar are cooled by a closed cycle refrigerator. Whereas a 2 dm 3 free standing helium cryostat of 4 K has a hold time of ~1-1.5 days, a comparable hybrid system can last up to 6 days. To optimize the cryostat the major heat leakages need to be identified. This can be done by using a thermal model. Using a simulation software program it is possible to evaluate the model. By modifying the system parameters the sensitivity of the system to changes can be investigated and potential areas for improvement identified. The hybrid cryostat analysed in this Paper is based on a 2 dm 3 helium cryostat being built for the United Kingdom lntYa Red Telescope (UKIRT) on Mauna Kea, Hawaii. USA. A Cryogenic Technology Inc. (CTI) model 1020 closed cycle refrigerator is used to cool the two radiation shields of the dewar. Measurements on the real system have been of great assistance in the construction of the theoretical model. A standard simulation run yields the temperature at selected nodes in the model as well as the heat flow between the nodes. This latter information in particular has acted as a guide-line for identifying the main heat transfer mechanisms. Complete agreement between model and cryostat is very difficult to achieve, as a result of inaccuracies in system parameters and, in particular, in the infrared emissivities of the radiating surfaces. The model simulation yields a hold time of 9 days for a practical system with a 5 mW signal window. For comparison, the original free standing dewar has a hold time of 1.2 days. A real system optimized with the help of the model has reached a hold time of 6 days.
0011-2275/86/030165-06 $03.00 © Butterworth ~ Co (Publishers) Ltd
As the closed cycle refrigerator forms an essential part of the hybrid cryostat a simple phenomenological model of this refrigerator has been constructed. This model is described in the following section and is relevant for CTI refrigerator models 1020 and 21. The thermal model is then integrated into the cryostat thermal model in the section entitled Thermal model of the hybrid cryostat while in the section following this, a number of simulation runs are presented and discussed. Finally, conclusions are drawn on the usability of the model.
Thermal model refrigerator
of
the
closed
cycle
The thermal model that has been constructed is the simplest possible model satisfying the known boundary conditions. These relate the temperature at the two stations to the thermal loads on the system. These conditions, known from laboratory measurements, are not very accurate. The refrigerator type considered in this Paper is based on the Gifford-MacMahon cycle '. In Giflbrd and MacMahon's paper, the refrigerator cycle is analysed for a one stage machine. The authors derive the refrigeration power, Q j, at the cold station, j, per cycle, and the regenerator losses, Q ij. These parameters have the following form: Qj = Kj zSat' V 1 Qij - Ki j
(1)
T i - Tj Tj
(2)
where AP is the helium pressure difference between the compression and expansion phases and V is the expanding volume. T i and Tj are the ambient and station temperatures, respectively. Kj and K ij are proportionality factors. Equations (1) and (2) are approximations acceptable only for heat loads which are not too large. The
Cryogenics 1 9 8 6 Vol 26 March
165
Hybrid cryostat: J.L. Casse and E. E.M. Woestenburg
Ti'¢
I
Rij
mj
T 100
I
It m=
Figure 1
CTI 1020
(K)
O,j
so
I
Thermal model for a one stage refrigerator
0 0
I 5
I 10
I 15
I 20
020 K (Watts)
100 E
[ ~R12 ,,
1
i
_CTI _ _ 21
11
I."
"
T
R2 3
,
3
I
(K)
/- /
Q70= 0
5O
(13
a2
Figure 2 Thermal model of the two stage refrigerator showing the thermal losses, O2x and O3~ at the two stations 0
0 thermal model is based on these two Equations and neglects all other heat loss mechanisms The equivalent network for the model is shown on Figure 1, where Qj is given by Equation (1) and R ij by
Rij = TjKij
Model
I
I
t
2
3
4
Q20K (Watts) Figure 3 Plot of the temperatures at the 70 K (- - -) and 20 K ( ) stations for a CTI 1020 (top) and a CTI model 21 (bottom) refrigerator against thermal loading
(3)
The model for a two stage refrigerator is shown on Figure2, where two one stage networks have been connected in series. Nodes 1, 2 and 3 in the model represent the ambient, the 70 K and the 20 K stations, respectively. The thermal loads at the two stations are Q ~x and Q 3x. Determination of the model parameters for a given refrigerator is performed by relating the given model to the performance measurements. Solving the network for the case of CTI model 1020 and 21 refrigerators yields the parameters shown in Table 1. From the thermal model the refrigerator performance curves can be derived. The temperatures at the two
Table 1
i
1
parameters for the
CTI 1020
and 21
stations are calculated from the model shown on
Figure 2. T2 =
T3 =
T1 1 +Kl2 [Q2 + Q a - Q 2 x - Q a x ]
T2
(4)
(5)
1 +K2a [ Q a - Q a x ]
The calculated performance curves for the CTI 1020 and 21 refrigerators are shown on Figure 3. These model performance curves do fit the actual refrigerator data accurately enough for present purposes.
refrigerators 02
Model 1020 Model 21
166
(w)
(w)
Q3
K12
(w -1)
(w-')
K23
Thermal model of the hybrid cryostat
53.2 10
21.3 4.1
0.085 0.45
0.13 0.7
The modelled cryostat is shown inFigure4. It consists of a 2 dm 3 helium dewar surrounded by two radiation shields attached to the 70 K and the 20 K stations of a CTI 1020
Cryogenics 1986 Vol 26 March
Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg NECKOFL,KOEWAR; He FILLCONNECTION
2H
GAS FLOW
3H
CONNECTIONFOR VACUUMPUMP IN
/7 RADIATK)NSHIELO 20K RADIATIONSHIELD He VESSEL 4K STATION
70K
tNFRA REDF I L T E R S ~ lnSb BOLOMETER
1~
'A"" I
/
~
ZS
,. . . .
SHIELD NODES
",,.
/
',
i
/* INELIUMI
./
FLOATINGNODE
RADIATION LINK .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
BACKSItORI ORIVER
.
.
3 (20KSTATIONI
"20K STATION I S GHz FET AMPLIFIER
REFRIGERATOR
Figure 5 Simplified thermal model of the hybrid cryostat The refrigerator is represented by nodes 1 , 2 and 3, while the helium vessel is at node 4. Nodes 2S and 3S are at the radiation shields, while 2N and 3N are the surface nodes in the neck of the dewar. Nodes 2H and 3H model the gas flowing in the neck of the dewar Figure 4 Prototype of the hybrid cryostat for the UKIRT common user receiver. In the centre is the helium dewar surrounded by two radiation shields connected to the 20 K and the 70 K stations of a CTI 1020 closed cycle refrigerator. The 690 GHz bolometer is attached to the bottom plate of the dewar and the 3 5 0 GHz mixer to the 20 K station. For both units backshort facilities have been implemented
closed cycle refrigerator. The Figure shows two signal windows (on the left) giving access to a bolometer for 690 G H z (top) and a Schottky mixer for 350 GHz (bottom). It also shows, on the right, the access openings for the adjustment of the backshorts of both devices. The radiation shields are attached to the refrigerator through flexible copper links. Their infrared emissivity in the reference model is 3%. The input signal to the bolometer passes through infrared filters located in both shields. The shields are made of aluminium and are gold-plated. The helium vessel is made of stainless steel. The thermal model of the hybrid cryostat is represented by a number of nodes linked together by thermal resistances and radiation links. The simulations performed on the model assume that the system has reached the steady state and hence all heat capacities are ignored. The simplest model of the hybrid cryostat is shown on Figure 5, where nodes 1, 2 and 3 represent the refrigerator. The neck of the dewar is modelled by the links I, 2N, 3N and 4, where node 4 represents the 4 K station. Nodes 2N and 3 N represent the nodes in the neck of the dewar where the radiation shields at 70 K and 20 K are attached, respectively. Nodes 2S and 3S are located at the centre of the shields. The model is determined by the heat transfer mechanisms proper to the system. The main mechanisms are considered below.
Convection between escaping gas and the dewar Heat transfer between the evaporating helium gas and the walls of the neck of the dewar plays a very important role. This effect has been modelled as shown in Figure5 where the gas flows from node4 to node 1, as indicated, while the cooling of the walls takes place via the convection resistances 2N-2H and 3N-3H. The process is similar to that of a heat exchanger. Assuming that the flow in the neck is laminar(small flows), the convection coefficient, h, can be then calculated from the Nusselt formula 2
k (Wm_2 K_I) h = 3.66 ~--
(6)
where k is the conductivity of the gas, which is temperature dependent and D is the diameter of the pipe. The gas conduction coefficient has been approximated by a polynomial k(T) = 6.27 x 10.3 + 0.75 x 10-3 T - 0.914 x 10-6 T 2 (7) where k is in W m -1 k -1 and T in K. The model of the dewar shown on Figure 5 is, in practice, too crude to accurately represent the heat exchange between the walls of the dewar and the helium gas. By using only four nodes to model the neck of the dewar, the cooling effect of the gas is underestimated. Table 2 gives the hold time in days for the model as a function of the number of nodes. It shows that a model Table 2 Hold time and gas temperature as a function of the number of nodes
Radiation transfer through the shields Radiation transfer is modelled through radiation links as shown on Figure5. These links connect the shields to the outer surface at ambient temperature (node I) and the helium cryostat (node 4). Nodes 1, 2S, 3S and 4 are the surface nodes; they are linked together via the so-called floating nodes z.
Number of nodes
Hold time (days)
Gas temperature at outlet (°C)
4 8 15 30
.5 .73 1.09 1.23
-75 -30 -13
Cryogenics 1986 Vo126 March
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Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg with 30 nodes is accurate enough for present purposes. The temperature of the gas at the exit is also listed. The gas temperature at outlet quoted for the 30 node model (-13°C) agrees with the measured temperature
(-11)°C). Thermal conduction along the neck of the dewar The thermal conduction parameters lbr the neck of the dewar must also account for the fact that the conductivity of the material (stainless steel) depends strongly on temperature. The conductivity of the material k(T), has been approximated by a function
[ -l 'Til logk(T) = 4.123 1 - e
~l.6s r j _ l . 9 5 9
,mw
NODE6A
(8)
x
1 4rn~,
l
NODE 2N
16.Stow
08row
ImW
OSmW
NODE 3N
where k is in W m -~ k -t and T in K.
[]
Shield thermal resistance As can be seen on Figure5 the resistance of the shields has been taken into account These resistances should be small (1.7 K w-a): however, the link between shield and refrigerator needs to be flexible and an additional resistance of ~15 K W -~ has to be considered in the model for the 70 K link. This leak resistance is important since most of the heat absorbed in the 70 K shield ( ~ 3 W) flows through the link and hence determines the temperature at the connecting node in the neck of the dewar.
Radiation transfer through the signal window The signal window at 4 K is critical for the heat loss balance, as soon as hold times of several days are achieved. At the level of 5 days. for instance, the heat loss amounts to 12 mW. while the loss via the window can beas large as 5 mW lbr a 3% intYared transmissivity. The effect of the window has been modelled as a direct heat source into node 4, the 4 K station.
NODEb, {HELIUMI
Figure 6 Characteristic heat flow distribution in the neck of the dewar as a result of radiation. The neck is modelled by three surface nodes only (6A, 2N and 3N). The floating nodes, 6AX, 2NX and 3NY~ account for the emissivity of the surface
1
Dewar anchoring lines; they are made of stainless steel0.4 mm in diameter. The heat loss is ~0.2 mW for 4 wires, 3 cm long,
2
Bias lines: the thermal resistance from the 4 K and 20 K stations has been estimated to be 0.7 × I ( P K W -t.
3
Conduction through rarefied gas; because of cryopumping only a thermal loss between the 300 K node and the70 K shield has been considered. The estimate for the loss3 gives 40 mW indicating a resistance, R ,.2s of 5300 K W-L
4
The loss through gas conduction in the neck of the dewar is more difficult to estimate. The gas flows upwards while the heat diffuses downwards. Depending on the flow, the heat wilt reach further down the neck The heat diffusion velocity can be estimated from the gas thermal conductivity and heat capacity. It can be shown that with a flow ofl00 cm 3 min-' ( 10 days hold time) the heat will not reach beyond the 70 K shield connection. The model accounts for this loss by having a thermal resistance between nodes 1 (ambient) and2 NH (gas at level 2N) equal to 2680 K
Radiation through the neck of the dewar The analysis showed that the radiation heat loss through the neck of the dewar is quite important The hot top section of the dewar radiates, like the300 K signal window+ toward the 4 K station, the neck of the dewar acting like a waveguide tot the infrared radiation. The amount of heat reaching the liquid helium depends on the infrared emissivity of the walls of the neck and on the geometry. For simplicity a model with only five radiation nodes was used. as shown on Figure6, These nodes are 1 (ambient), 6A, 2N, 3N and 4 (4 K station). The floating nodes IX, 6AX, 2NX and 3NX account for the infrared emissivity of the wall surfaces. The radiation coupling between nodes depends on the geometry from which the form or viewing factors can be derived 2. Figure 6 shows the heat flows between the various nodes in the model for an infrared emissivity of 5% of the wall of the neck and 1/)% for the top node (node 1): some 7 mW is absorbed by the 4 K station.
Other heat losses The remaining heat transfer mechanisms are less critical. The following heat loss mechanisms have, nevertheless, been taken into account
168
Cryogenics 1 9 8 6 Vol 2 6 M a r c h
W
-l
"
Thermal simulation results The simulation program ESACAP 4 translates the thermal model into a set of differential equations which are then solved in the time domain or in the steady state. It yields the node temperatures of the system as well as the heat flows between nodes. The major heat flows lbr the reference model are shown on Figure 7, where the main nodes are indicated. The Figure shows that in this case. where there is no signal window, the main source of heat is
Hybrid cryostat: J.L. Casse and E.E.M. Woestenburg
/
/
-93W1
.OSW 1.SW
.?lW
,2~.22w
.38W
q
.22W
. • 5mW
I
nWl
2L . t,81
~'~my/~j.2//i2////////A
17rnW
3
2
y/////////////////////////////////-//////////////,2////A
7 Main heat flow distribution for the reference hybrid cryostat in the absence of a signal window Figure
Figure8
radiation through the neck of the dewar. shows the heat flows for a free standing cryostat. The contribution of the refrigerator is clear after comparing the two Figures: it absorbs the radiation falling onto the shields. An obvious improvement to the system is to bring a shield into the neck of the dewar, but to allow the gas to escape (baffle). The improvement is significant as shown on which gives the hold time in days for a number of models with a 5 mW signal window, as well as the physical temperatures at the nodes 2N and 3 N in the neck of the dewar, and the main heat flows into the4 K node in mW. The various headings indicate the deviations from the reference case (model 1). The reference model yields a hold time of 5.1 days, while the improved system with a baffle in the neck of the dewar(model 2) reaches 8.9 days. When compared with model 3, where the radiation loss has been ignored, it can be seen that the radiation shield is quite effective. However, care must be taken when the model reaches hold times > ~ 10 days. It must be ensured that the model parameters are accurate enough and that the second order heat transfer mechanisms have been well
Table3
Figure 8 Main heat flow distribution for the free standing cryostat in the absence of a signal window
accounted for. It is doubtful that the accuracy of the present model is adequate to separate the second order heat contributions. Model 4 in is based on a smaller refrigerator of the type CT1 21. The analysis has shown that with the present geometry a small refrigerator is quite satisfactory. In models 5, 6 and 7 changes have been made to the reference model to study the sensitivity of the system to a number of factors such as infrared emissivity of the 70 K shield and the shield serial resistances. Model 7 in particular, where radiation in the neck of the dewar as well as the shield resistances are reduced, yields a substantial increase in the hold time. To quantify the cooling by the escaping helium gas, model 8 has been constructed, where it is assumed that convection is not active in the neck of the dewar. The hold time then reduces significantly, from 5.1 days to 3.2 days. This heat transfer mechanism is thus quite important for the accuracy of the model. For comparison purposes the case of the free standing cryostat is given (model 9). This model uses the same parameters as for the hybrid system, except that the
Table3
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Hybrid cryostat: J.L. Casse and E. E.M. Woestenburg. Table 3
Performance of the hybrid cryostat with a 5 mW signal window loss Heat input into 4 K node (mW)
Model Characteristics 1 2 3 4 5 6 7 8 9 10 11
Table 4
Reference Radiation shield in neck No radiation loss in neck dewar Refrigerator model 21 Improved flexible link (lOx) shield-refrigerator Emissivity 70 K outer shield of 6% Improved flexible link and radiation shield in neck No convection in neck dewar Free standing cryostat 10 mW signal window No signal window
Hold time (days)
T3N (K)
T2N (K)
Conduction via neck
5.1 8.9 9.7 4.9 5.6 4.7 11.5 2.5 1.2 3.8 7.8
16 18 19 20 11 11 11 22 83 15 18
93 97 97 1O0 111 118 112 102 195 89 96
0.23 1.0 1.1 0.5 0.06 0.5 0.2 11.5 19.0 O. 1 0.7
Radiation from shields
Radiation in neck 6.6 0.7
0.006 0.008
0
0.008
6.8 6.0 7.5
0.013 0.003 0.01
0.05
0.003
7.9
0.01
12.0
13.0
6.3 7.0
0.005 0.007
Measurements and simulation results for the hybrid cryostat without signal window Hold time (days)
Features of model
Simulation
Temperatures of 70 K shields (I0
Measured
Simulation
Measured
1 Model with 6% infrared emissivity of the 70 K shield outer surface and a central thin tube in the neck of the dewar
5.3
4.1
119
92
2 Model with infrared emissivity of the 70 K shield outer surface better than 3%
7.8
7.3
88
79
120
112
3 Model with 6% infrared emissivity of the 70 K shield outer surface and a baffle in the neck of the dewar
10
refrigerator is removed and the shields somewhat shortened. The importance of the signal window can be quantified from model 10, where l0 mW is leaking through and model 11 without a window. The simulation results have been compared with measurements on the cryostat The results are shown in Table 4 for three different variations from the reference model each without signal window. As can be seen, the match between simulation and measurement is not perfect This is because of the limited accuracy to which some of the physical parameters of the cryostat are known. The accuracy of the temperature and gas flow measurements also plays a role, while some of the heat transfer mechanisms could probably be improved upon. The third model in particular does not compare too well. This is probably due to the construction of the baffle which does not properly seal the neck of the dewar. The accuracy achieved with the m o d e l however, has proved quite useful for optimizing the design of the cryostat The thermal model is particularly useful in identifying the main heat losses in a particular system. It shows where improvements can be made to increase the cryostat hold time. The final design of the hybrid cryostat described in this Paper includes the following improvements on the reference system: lower emissivity of the 70 K shield and a radiation shield (baffle) in the neck of the dewar. The hold time of the 2 dm 3 system including a signal window is 6 days, while the thermal model simulation predicts a hold time of 8.9 days (model 2),
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Cryogenics 1 9 8 6 Vol 26 March
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Conclusions A thermal model for an existing hybrid cryostat has been constructed which takes into account all active heat transfer mechanisms The accuracy of the model has been tested by comparing the simulation results with real measurements. The comparison is quite satisfactory, although it is limited by the accuracy of the physical parameters of the device. The model has proved to be accurate enough to make identification of the important heat transfer mechanisms possible; it has shown the way to substantial improvements in cryostat hold time.
Acknowledgements The authors wish to thank ESA for making the program ESACAP available to them. The Dwingeloo Radio Observatory is operated by the Netherlands Foundation for Radio Astronomy with the financial support of the Netherlands Organization for the Advancement of Pure Research (ZWO).
References 1 2 3 4
MacMahon, H.O. and Gifford, W.F_, ddv Cryog Eng (1959) 5 354 Chapman, A.J. Heat Transfer MacMillan Publishing Co, New York. USA (1974) Bailey, C.A. Advanced Cryogenics Plenum Press, London. UK (1971) Stangerul~ P. European SpaceAgency J (1980) 6 301