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Simple, numerical check of calibration of germanium resistance thermometers
A vacuum pumping line to the pressure vessel is also provided, and to protect the vacuum pumps an electrical interlock using a pressure switch, solenoid, solenoid valve, and microswitch is provided to prevent either the vacuum line valve being opened while the system is pressurized or the pressurizing valve being opened while the system is being evacuated. The instrumentation in the pressure vessel consists of a temperature measuring circuit and a depth gauge. Temperatures are measured by recording the resistances of four calibrated carbon resistors, which can be mounted anywhere inside the pressure vessel; each resistor has four wires brought to the cryostat top plate, so that any type of failure on one resistor can be completely isolated. Normally the resistors are all connected in series using two of the wires from each resistor, and a current of 10/aA passed through them. The remaining two wires are connected to the voltage recording device. The depth gauge also uses carbon resistors; in this case the change in resistance detected is caused by the change in cooling when the resistor passes from vapour into liquid, or vice versa. A similar depth gauge is installed in the cryostat to indicate the level of the liquid helium surrounding the pressure vessel. The pressure in the pressure vessel is controlled by the use of a pressure controller (Hale Hamilton type L15S) and relief controller (Hale Hamilton type L17) and measured by an accurate mechanical pressure gauge with an accuracy of -+0.02 bar. During preliminary runs with the rig, it was found necessary to line the pressure vessel with strips of copper of high
thermal conductivity to reduce the vertical temperature gradient in the vessel. By doing this the gradient has been reduced to 4 K m-l, nearly an order of magnitude below that existing before the lining was installed. In operation, the rig is cooled by transferring liquid helium into both the pressure vessel and the cryostat. The pressure vessel is then pressurized from a room temperature high pressure gas store to the required pressure, causing its temperature to rise, and some of the helium in the cryostat to boil off. More liquid helium is then transferred into the cryostat to reduce the temperature to the required value; since this causes more warm gas to be drawn into the pressure vessel, a time lapse must be allowed before equilibrium is reached, about 40 minutes being sufficient. The liquid helium transfers are aided by the installation of solenoid valves in the venting and pressurizing lines of the storage dewars, which enables the process to be started and stopped from the rig control panel, where the temperatures are monitored. Provision has been made for the cryostat to be pressurized up to 2.25 bar, to enable the pressure vessel to be surrounded by liquid helium at temperatures up to 5.2 K. This was done to allow better temperature control but in practice has been little used since the temperature gradient of 4 K m-1 and rate of rise, 1.6.10-4 K s-l, are adequate for the experiments performed. This apparatus was constructed at the Central Electricity Research Laboratories and this description is published by permission of the Central Electricity Generating Board.
Simple, numerical check of calibration of germanium resistance thermometers J. J. Grodski and A. E. Dixon Germanium-resistance thermometers (GRTs) are widely used as cryogenic temperature sensors. They are available from a number of commercial suppliers and are often supplied with calibration data for a certain range of temperatures. The calibration of a GRT is usually obtained by measuring the resistance (R) at particular levels of temperature (T). A secondary standard such as another GRT is commonly employed for checking the T levels, which are usually spaced at constant intervals over a certain range of T. The resistance of germanium cannot be expressed as a simple function 1-3 of temperature. However, numerous papers suggest interpolations 4 - 6 o r fits t o v a r i o u s functions 7 - 9 characterizing the temperature dependence of a GRT. Numerical tests were used to show that these representations are suitable, accurate, and consistent. None of these papers, however, point out the importance of the tests with respect to every GRT calibration. Such tests are particularly important when interpolation between calibration points is used since those techniques force a curve The authors are with the Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. Received 20 May 1973.
614
of some description to pass through all the data points and do not directly check the self-consistency of the data. It is the purpose of this note to stress the importance of checking GRT calibration data from commercial sources and to describe a simple numerical test allowing an easy and quick check of such calibration data for the possible existence of deviations larger than expected. The nature of the check is analogous to Ward's 6 approach in testing the self-consistency between several methods of interpolation. The basic procedure is to remove a set of points from the calibration data and to interpolate at these points using the remaining data. For example, using the set of the 'odd' points of the calibration data, an interpolation may be performed at the intermediate 'even' values and vice versa. The interpolated values are compared with the calibration data for self-consistency. The error (A) is defined as the difference between the measured and the interpolated values. Then, results of the two interpolation sequences lead to two complementary sets of values of A versus T, which can be displayed graphically as shown in the figures. Calibration is usually done at relatively small T intervals. Also, the dependence log R versus log T is almost linear,
CRYOGENICS
. OCTOBER
1973
same GRT. Fig.la shows results of a check which was used to prove to a commercial supplier that the GRT was not calibrated with the guaranteed accuracy of 5 inK. For example, interpolations between the first and third values and between the second and fourth values of the calibration data yield respectively As corresponding to points 2 and 3 (which are found in the two complementary sets in Fig. la) which differ by 55 mK and moreover have opposite signs. This is readily interpreted; a curve corresponding to such data would have to contain oscillations of about 30 mK amplitude, which is contrary to what is known about typical GRTs. Upon recalibration by the supplier, the new data were tested and results are shown in Fig. lb. The agreement between the two complementary sets is much better and the relative difference between them is just within 5 mK, the claimed accuracy of the calibration. The largest errors appear to be in the vicinity of the X-point and at about 4.2 K. This might be expected since differences between the thermal properties of the He 4 bath above and below the X-point are very large and they may be responsible for some variations. Also, geometry and design of the calibration system may allow small local temperature differences in the system affecting the calibration, particularly when the helium bath is subjected to
i
(a)
5oJ
40 30 20 ~O
-t0 /
/
/
/
/ #
F
#/
-20
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I
I
I (b)
IO
40 O
I
I
I
30 1.O
I 2.0
[ 3.0 Temperature, K
I 4.0
5.0 2C
Fig.1 Variations of the difference (A) between the measured and the linearly interpolated values versus temperature. Triangles and circles correspond respectively to points obtained using the 'odd' and 'even' values of the calibration data. a Large differences in A between the 'odd' and 'even' sets of values indicate that the calibration is erroneous b Results showing that the calibration is just w i t h i n the quoted 5 m K accuracy
10
O
-t0
g'
E
although the exact shape of the relation for a GRT is dependent on its particular doping. Therefore, the check uses linear interpolation upon converting R and T values to logarithms. The equidistant intervals on a linear T scale correspond to continuously increasing intervals on the log T scale as T decreases. If the dependence is not exactly finear, it is expected that linear interpolations at different Ts will yield different As, even for perfect calibration data. When linear interpolation is used, a is related to the derivative and curvature of the dependence of log R on log T and to the interval between logarithms of Ts used in a particular interpolation. However, this is not important in the check, while it would be important in an interpolation per se. GRTs are known to exhibit only slight variations in the curvature of the log R versus log T curve and except for a possible inflection 6,8 in the region above 10 K (depending on the doping), are known not to contain oscillations. It is therefore expected that if perfect data was involved, variations of A with T should be basically continuous. This is not the case when real data is used. Errors in the calibration are immediately reflected in a scatter of the As. Fig. 1 shows two complementary sets of As and their variation with T for two independent calibrations of the
CRYOGENICS. OCTOBER 1973
lI #
I
!
! ! # # # ! #
-20
-30
/ # # # # ! #
-40
d
i
t
I
5-
(b)
o
1.O
] 2.0
I 3.0 Temperature, K
I 4.O
5.0
Fig.2 Variations of the difference (A) between the measured and the spline-function interpolated values versus temperature. Triangles and circles correspond respectively to points obtained using the 'odd' and 'even' values of the calibration data. a Large differences in A between the 'odd' and 'even' sets of values indicate that the calibration is erroneous b Results showing that the calibration is just within the quoted 5 m K accuracy
615
pressure, to reach temperatures higher than 4.2 K. As was expected from Fig.la, the original data for the range from 2.25 to 5 K was found to be correct. However, data below the X-point (2.17 K) differed drastically in the two calibrations. The supplier could not explain this discrepancy, and the situation was further complicated by a change in the calibration current just above the X-point. More sophisticated interpolation methods may also be used to perform such a check. For example, a spline-function interpolation was used with the same sets of data and results of these checks are shown in Fig.2. The advantage of this method is that here A is dependent mainly on the error and therefore results are scattered around the T axis rather than along some other curve of undetermined shape, as it is when linear interpolation is used. However, this method has major disadvantages when compared with the linear interpolation: a. It is difficult to convey the impact of the results of this calculation to the suppliers. b. A computer is necessary to perform the calculation, whereas the linear interpolation may be performed on a calculator with five digits accuracy; or even manually. Note that data which is spaced at irregular intervals might not be easily interpretable, particularly when linear interpolation is used. Ward 6 showed that non-uniform spacing of calibration points may cause considerable deviations from the real relationship when the spline-function inter-
polation is used. Also, neither one of these checks would detect the existence of constant errors in a calibration or continuous changes varying the curvature of a calibration curve. However, any discontinuous variations are readily found by either one of the checks. The probability of detecting errors in the calibration current settings is increased when there is a change in the calibration current somewhere in the temperature range over which the device is calibrated; errors resulting from poor thermal contact between the GRT and the calibration standard thermometer are more likely to be detected when the device is calibrated through the X-point because of the increased thermal contact between the two thermometers and the superfluid helium bath. This work was supported by a grant from the National Research Council o f Canada.
References l Kunzler, J. E., Geballe, T. H., Hull, G. W. Rev Sci lnstr 28 (1957) 96 2 Biakemore, J. S. Rev Scilnstr 33 (1962) 106 3 Ahlers,G., Macre, J. F. Rev Sci lnstr 37 (1966) 962 4 Penar, J. D., Campi, M. RevScilnstr42 (197~) 528 5 Wepner, W. J P h y s E 4 (1971) 761 6 Ward, D. A. C~ogenics 12 (1972) 209 7 Sehriempf, J. T. Cryogenics6 (1966) 362 8 Osborne, D. W., Flotow, H. E., Schreiner, F. Rev Scilnstr 38 (1967) 159 9 Biakemore, J. S., Winstei, J., Edwards, R. V. Rev Sci Instr 41 (1970) 835
Critical two-phase flow of helium R. V. Smith Nomenclature
g
gas
a
sonic velocity
H
homogeneous
A
flow area
o
stagnation conditions
t
flow area, A t = total flow area
(CF) correction factor for separated-phase model (see equation 1) G
mass flow rate per unit area M / L 2
h
specific enthalpy L 2 / t 2
p
pressure M / L t 2
R
gas constant
T
temperature
x
quality
u
velocity L / t
~,
specific heat ratio =
specific heat constant pressure specific heat constant volume
Subscripts
c
critical flow
E
equilibrium
The author is with the Wichita State Received 10 May 1973.
616
University,Wichita, USA.
Critical, or choking flow may occur for any compressible fluid whether it is flowing by itself or in a multi-phase mixture. It will occur when there are sufficient pressure differences in a conduit of either constant or converging cross-sectional flow areas. Thus, in such a section, beginning with low flow rates, the velocity of the fluid at the exit will be increased as the downstream pressure is reduced. This velocity will continue to increase until the flow condition is reached where further reduction in the downstream pressure will no longer increase the exit flow velocity or the mass rate of flow. At this point and for all lower downstream pressures critical flow exists at the exit of the section. Critical two-phase flow is characterized by shock waves and relatively large pressure drops. These large pressure drops and the mass limiting characteristics of the flow make it very important that the designer know whether or not his system is likely to experience critical flow. Critical, two-phase (gas-liquid) flow has been studied and reviewed by a number of investigators but helium has not
C R Y O G E N I C S . OCTOBER 1973
thermal conductivity to reduce the vertical temperature gradient in the vessel. By doing this the gradient has been reduced to 4 K m-l, nearly an order of magnitude below that existing before the lining was installed. In operation, the rig is cooled by transferring liquid helium into both the pressure vessel and the cryostat. The pressure vessel is then pressurized from a room temperature high pressure gas store to the required pressure, causing its temperature to rise, and some of the helium in the cryostat to boil off. More liquid helium is then transferred into the cryostat to reduce the temperature to the required value; since this causes more warm gas to be drawn into the pressure vessel, a time lapse must be allowed before equilibrium is reached, about 40 minutes being sufficient. The liquid helium transfers are aided by the installation of solenoid valves in the venting and pressurizing lines of the storage dewars, which enables the process to be started and stopped from the rig control panel, where the temperatures are monitored. Provision has been made for the cryostat to be pressurized up to 2.25 bar, to enable the pressure vessel to be surrounded by liquid helium at temperatures up to 5.2 K. This was done to allow better temperature control but in practice has been little used since the temperature gradient of 4 K m-1 and rate of rise, 1.6.10-4 K s-l, are adequate for the experiments performed. This apparatus was constructed at the Central Electricity Research Laboratories and this description is published by permission of the Central Electricity Generating Board.
Simple, numerical check of calibration of germanium resistance thermometers J. J. Grodski and A. E. Dixon Germanium-resistance thermometers (GRTs) are widely used as cryogenic temperature sensors. They are available from a number of commercial suppliers and are often supplied with calibration data for a certain range of temperatures. The calibration of a GRT is usually obtained by measuring the resistance (R) at particular levels of temperature (T). A secondary standard such as another GRT is commonly employed for checking the T levels, which are usually spaced at constant intervals over a certain range of T. The resistance of germanium cannot be expressed as a simple function 1-3 of temperature. However, numerous papers suggest interpolations 4 - 6 o r fits t o v a r i o u s functions 7 - 9 characterizing the temperature dependence of a GRT. Numerical tests were used to show that these representations are suitable, accurate, and consistent. None of these papers, however, point out the importance of the tests with respect to every GRT calibration. Such tests are particularly important when interpolation between calibration points is used since those techniques force a curve The authors are with the Department of Physics, University of Waterloo, Waterloo, Ontario, Canada. Received 20 May 1973.
614
of some description to pass through all the data points and do not directly check the self-consistency of the data. It is the purpose of this note to stress the importance of checking GRT calibration data from commercial sources and to describe a simple numerical test allowing an easy and quick check of such calibration data for the possible existence of deviations larger than expected. The nature of the check is analogous to Ward's 6 approach in testing the self-consistency between several methods of interpolation. The basic procedure is to remove a set of points from the calibration data and to interpolate at these points using the remaining data. For example, using the set of the 'odd' points of the calibration data, an interpolation may be performed at the intermediate 'even' values and vice versa. The interpolated values are compared with the calibration data for self-consistency. The error (A) is defined as the difference between the measured and the interpolated values. Then, results of the two interpolation sequences lead to two complementary sets of values of A versus T, which can be displayed graphically as shown in the figures. Calibration is usually done at relatively small T intervals. Also, the dependence log R versus log T is almost linear,
CRYOGENICS
. OCTOBER
1973
same GRT. Fig.la shows results of a check which was used to prove to a commercial supplier that the GRT was not calibrated with the guaranteed accuracy of 5 inK. For example, interpolations between the first and third values and between the second and fourth values of the calibration data yield respectively As corresponding to points 2 and 3 (which are found in the two complementary sets in Fig. la) which differ by 55 mK and moreover have opposite signs. This is readily interpreted; a curve corresponding to such data would have to contain oscillations of about 30 mK amplitude, which is contrary to what is known about typical GRTs. Upon recalibration by the supplier, the new data were tested and results are shown in Fig. lb. The agreement between the two complementary sets is much better and the relative difference between them is just within 5 mK, the claimed accuracy of the calibration. The largest errors appear to be in the vicinity of the X-point and at about 4.2 K. This might be expected since differences between the thermal properties of the He 4 bath above and below the X-point are very large and they may be responsible for some variations. Also, geometry and design of the calibration system may allow small local temperature differences in the system affecting the calibration, particularly when the helium bath is subjected to
i
(a)
5oJ
40 30 20 ~O
-t0 /
/
/
/
/ #
F
#/
-20
!
I
I
I (b)
IO
40 O
I
I
I
30 1.O
I 2.0
[ 3.0 Temperature, K
I 4.0
5.0 2C
Fig.1 Variations of the difference (A) between the measured and the linearly interpolated values versus temperature. Triangles and circles correspond respectively to points obtained using the 'odd' and 'even' values of the calibration data. a Large differences in A between the 'odd' and 'even' sets of values indicate that the calibration is erroneous b Results showing that the calibration is just w i t h i n the quoted 5 m K accuracy
10
O
-t0
g'
E
although the exact shape of the relation for a GRT is dependent on its particular doping. Therefore, the check uses linear interpolation upon converting R and T values to logarithms. The equidistant intervals on a linear T scale correspond to continuously increasing intervals on the log T scale as T decreases. If the dependence is not exactly finear, it is expected that linear interpolations at different Ts will yield different As, even for perfect calibration data. When linear interpolation is used, a is related to the derivative and curvature of the dependence of log R on log T and to the interval between logarithms of Ts used in a particular interpolation. However, this is not important in the check, while it would be important in an interpolation per se. GRTs are known to exhibit only slight variations in the curvature of the log R versus log T curve and except for a possible inflection 6,8 in the region above 10 K (depending on the doping), are known not to contain oscillations. It is therefore expected that if perfect data was involved, variations of A with T should be basically continuous. This is not the case when real data is used. Errors in the calibration are immediately reflected in a scatter of the As. Fig. 1 shows two complementary sets of As and their variation with T for two independent calibrations of the
CRYOGENICS. OCTOBER 1973
lI #
I
!
! ! # # # ! #
-20
-30
/ # # # # ! #
-40
d
i
t
I
5-
(b)
o
1.O
] 2.0
I 3.0 Temperature, K
I 4.O
5.0
Fig.2 Variations of the difference (A) between the measured and the spline-function interpolated values versus temperature. Triangles and circles correspond respectively to points obtained using the 'odd' and 'even' values of the calibration data. a Large differences in A between the 'odd' and 'even' sets of values indicate that the calibration is erroneous b Results showing that the calibration is just within the quoted 5 m K accuracy
615
pressure, to reach temperatures higher than 4.2 K. As was expected from Fig.la, the original data for the range from 2.25 to 5 K was found to be correct. However, data below the X-point (2.17 K) differed drastically in the two calibrations. The supplier could not explain this discrepancy, and the situation was further complicated by a change in the calibration current just above the X-point. More sophisticated interpolation methods may also be used to perform such a check. For example, a spline-function interpolation was used with the same sets of data and results of these checks are shown in Fig.2. The advantage of this method is that here A is dependent mainly on the error and therefore results are scattered around the T axis rather than along some other curve of undetermined shape, as it is when linear interpolation is used. However, this method has major disadvantages when compared with the linear interpolation: a. It is difficult to convey the impact of the results of this calculation to the suppliers. b. A computer is necessary to perform the calculation, whereas the linear interpolation may be performed on a calculator with five digits accuracy; or even manually. Note that data which is spaced at irregular intervals might not be easily interpretable, particularly when linear interpolation is used. Ward 6 showed that non-uniform spacing of calibration points may cause considerable deviations from the real relationship when the spline-function inter-
polation is used. Also, neither one of these checks would detect the existence of constant errors in a calibration or continuous changes varying the curvature of a calibration curve. However, any discontinuous variations are readily found by either one of the checks. The probability of detecting errors in the calibration current settings is increased when there is a change in the calibration current somewhere in the temperature range over which the device is calibrated; errors resulting from poor thermal contact between the GRT and the calibration standard thermometer are more likely to be detected when the device is calibrated through the X-point because of the increased thermal contact between the two thermometers and the superfluid helium bath. This work was supported by a grant from the National Research Council o f Canada.
References l Kunzler, J. E., Geballe, T. H., Hull, G. W. Rev Sci lnstr 28 (1957) 96 2 Biakemore, J. S. Rev Scilnstr 33 (1962) 106 3 Ahlers,G., Macre, J. F. Rev Sci lnstr 37 (1966) 962 4 Penar, J. D., Campi, M. RevScilnstr42 (197~) 528 5 Wepner, W. J P h y s E 4 (1971) 761 6 Ward, D. A. C~ogenics 12 (1972) 209 7 Sehriempf, J. T. Cryogenics6 (1966) 362 8 Osborne, D. W., Flotow, H. E., Schreiner, F. Rev Scilnstr 38 (1967) 159 9 Biakemore, J. S., Winstei, J., Edwards, R. V. Rev Sci Instr 41 (1970) 835
Critical two-phase flow of helium R. V. Smith Nomenclature
g
gas
a
sonic velocity
H
homogeneous
A
flow area
o
stagnation conditions
t
flow area, A t = total flow area
(CF) correction factor for separated-phase model (see equation 1) G
mass flow rate per unit area M / L 2
h
specific enthalpy L 2 / t 2
p
pressure M / L t 2
R
gas constant
T
temperature
x
quality
u
velocity L / t
~,
specific heat ratio =
specific heat constant pressure specific heat constant volume
Subscripts
c
critical flow
E
equilibrium
The author is with the Wichita State Received 10 May 1973.
616
University,Wichita, USA.
Critical, or choking flow may occur for any compressible fluid whether it is flowing by itself or in a multi-phase mixture. It will occur when there are sufficient pressure differences in a conduit of either constant or converging cross-sectional flow areas. Thus, in such a section, beginning with low flow rates, the velocity of the fluid at the exit will be increased as the downstream pressure is reduced. This velocity will continue to increase until the flow condition is reached where further reduction in the downstream pressure will no longer increase the exit flow velocity or the mass rate of flow. At this point and for all lower downstream pressures critical flow exists at the exit of the section. Critical two-phase flow is characterized by shock waves and relatively large pressure drops. These large pressure drops and the mass limiting characteristics of the flow make it very important that the designer know whether or not his system is likely to experience critical flow. Critical, two-phase (gas-liquid) flow has been studied and reviewed by a number of investigators but helium has not
C R Y O G E N I C S . OCTOBER 1973