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An accurate differential pressure gauge for use in liquid and gaseous helium
PII: S0011-2275(98)00028-9
Cryogenics 38 (1998) 673–677 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
An accurate differential pressure gauge for use in liquid and gaseous helium Chris J. Swanson*, Kris Johnson and Russell J. Donnelly Department of Physics, University of Oregon, Eugene, OR 97403, U.S.A.
Accepted 21 January 1998 We have developed and calibrated an accurate pressure gauge studying the flow of liquid or gaseous helium. The gauge is designed to make measurements of pressure drops in liquid helium flow experiments but could be used for a broad range of other purposes. The gauge is calibrated over a range of pressures from 0 to 200 Pa and can resolve pressures as small as 0.01 Pa. It is accurate to better than 0.5% of the reading or ± 0.01 Pa, whichever is greater, and maintains this accuracy even after warm up and cool down. 1998 Published by Elsevier Science Ltd. All rights reserved Keywords: B. helium gas; B. helium liquid; D. pressure sensor
Introduction Liquid helium and critical helium gas are becoming widely used in the study of high Reynolds number and high Rayleigh number turbulence due to their low kinematic viscosity1–5. However, the cryogenic environment renders most room temperature measuring devices inoperable. To study turbulence at helium temperatures, the standard devices for measuring quantities of interest in turbulence must be replaced by cryogenic alternatives. Many instruments such as thermometers and hot film velocity probes6 have already been developed for cryogenic application. Further, there is an excellent review of cryogenic pressure transducers for a variety of applications7,1. We have developed a gauge which complements previous gauges and meets the following requirements: a reading accuracy of 0.5% of the pressure reading or ± 0.01 Pa, whichever is greater, capability of spanning a range of pressures from 0 to 200 Pa, and most importantly, repeatability through many thermal cycles to room temperature without recalibration. Our gauge follows in large part the design of a high pressure metal diaphragm gauge8 except that the differential pressure range has been significantly reduced and the calibration remains unchanged upon repeated thermal cycling.
Design Low temperature capacitance pressure gauges generally fall into two categories, those using metal diaphragms and those *To whom all correspondence should be addressed 1 Besides capacitive pressure transducers, Validyne (Northridge, CA 91324) manufactures a variable reluctance pressure transducer which has been used in cryogenic applications.
using plastic diaphragms9,10. Initially we designed a gauge using a 25 m thick plastic diaphragm since the plastic style seemed to offer sensitivity to smaller pressure differences. Of the two plastics widely used, Mylar and Kapton11 we selected Kapton due to an observed anisotropic shrinkage of Mylar at 77 K2. However extensive testing found the Kapton calibration curve to vary by as much as 10% after a single thermal cycle. To overcome this problem we attempted to calibrate the gauge using an electrostatic technique10 which could be quickly performed prior to each usage. Comparison of the electrostatic method with a direct pressure calibration also yielded unsatisfactory results, leading us to a metal diaphragm gauge. A drawing of the pressure transducer is shown in Figure 1. The capacitor is made of two parallel plates, which are approximately 1 cm2 in area separated by roughly 40 m. One of the plates is connected by a thin post to a metal diaphragm 150 m thick, across which the differential pressure is applied. Changes in differential pressure move the diaphragm and hence change the capacitance between the plates, which is measured using a high accuracy AC bridge12. The capacitor has been calibrated using helium gas at 2.1 K as the working fluid. The fluid is admitted to the capacitor through top and bottom ports by means of short, 1/8th inch copper tubes. The short copper tubes each attach to 20–25 m filter paper filters. The filters are included to prevent ice or other particles from getting between the electrodes of the capacitor and causing changes in the dielectric constant. The filter 2 We observed that upon submerging a uniformly stretched circular film of Mylar in liquid nitrogen, a series of wrinkles developed. The wrinkles were in a pattern of regular straight lines, such as would be expected if the membrane were stretched in the direction of the lines.
Cryogenics 1998 Volume 38, Number 6 673
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al.
Figure 1 Drawing of the pressure gauge
elements are installed in custom housings built to allow for indium seals. The electrical connections are made through the sides of the main body, as shown in Figure 1. In order to insure that the wires were completely shielded, miniature connectors were screwed into tapped holes in the sides. A thin metal post, soldered to the backside of the connector, passes through the wall to the inside cavity of the capacitor. Embedding the post in Stycast 2850 GT epoxy3 creates a leak tight seal and provides electrical insulation. The most critical feature of the capacitor design involved the placement of the capacitor electrodes. Both centering and spacing were important factors to consider. The copper electrode was centered by gluing it into a short plastic sleeve that fit snugly into the brass body. Subsequently the remaining gap between the electrode and the brass was filled with epoxy. After curing, the electrode was polished flush with the surrounding brass face. In order to minimize any stray capacitance due to charges bound to the epoxy, we trimmed the epoxy back from the surface of the electrode by approx. 0.5 mm. The other side of the device contained the pedestal shaped electrode. The pedestal electrode and membrane are machined out of a single piece of brass. We chose brass in order to match the thermal expansion coefficient of the body. We found that using metals with mismatched expansion coefficients caused the diaphragm to bow slightly in the center, with the maximum deflection being 5–10 times greater than the differential shrinking in the radius. This epoxy is filled with 苲 70% aluminum oxide grit to better match the linear expansion coefficient of the brass. Further, we found it to adhere much better to the metallic surfaces. It is available from Emerson and Cumming, Woburn, MA 01888.
3
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Cryogenics 1998 Volume 38, Number 6
The machining requires special procedures to obtain a flat thin diaphragm of the desired thickness. First, a brass rod is annealed to relieve any internal stresses in the material. It is then turned down to the desired radius, and the indentation at the top of the piece is cut. The brass is then annealed again on a hot plate at 苲 380°C. Failure to pre-relieve these stresses in the metal creates curvature in the diaphragm. A brass rod is then epoxied into the indentation. Holding the work with the brass rod, the rest of the piece can be machined. The solid support of the epoxied brass allows for the metal to be cut down to a thickness of 0.015 cm without tearing or deforming the diaphragm. The unit is then heated to release the epoxy joint, after which it is cleaned with a solvent. We expect that this technique could be used to fabricate a diaphragm down to about 0.0025 cm. Next the pedestal electrode must be installed into the brass body. The body and the pedestal are placed face down on a flat glass plate. The pedestal is then centered by eye and epoxied to the body. While this technique attempts to align the capacitor plate flush with the outside surface, some level of height differential is inevitable. Additionally, the critical distance is not the distance between one electrode and its surrounding surface but the total distance between the electrodes. The simplest and most accurate way to measure the distance between the electrodes is to measure the capacitance. Our capacitance at room temperature is roughly 25 pF, corresponding to a separation of 40 m, and is slightly larger than expected. The source of the larger than expected separation could be from either of two sources. First, the pedestal shaped electrode could be slightly recessed from the surrounding surface, despite our efforts to keep surfaces flush. Second, when the two halves of the capacitor are assembled, indium is squeezed into an O-ring groove. A very thin layer of indium typically spills over the edge of the groove creating a small gap between the halves. In any case, the pressure gauge has sufficient sensitivity for a wide range of plate separations so that exact control over the plate spacing is unnecessary. After assembly, the capacitor was thermally cycled between 77 K and room temperature a few times in order to relax any residual thermal stresses. We also thoroughly leak checked the capacitor at room temperature and 77 K.
Calibration The sensitivity of the device can be predicted from known properties of the material. Assuming a parallel plate configuration, the capacitance is to first order C=
冉 冊
⑀0A y 1+ , d0 d0
(1)
where is the dielectric constant of the fluid, ⑀0 is the permittivity of free space, A is the area of the plates, d0 is the gap between the plates with no pressure applied and y is deflection of one of the plates. Note that a positive value of y decreases the gap. The pressure dependence of the deflection y is given by y=
a4 P, Et3
(2)
where a is the radius of the flexible diaphragm, E is the
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. elastic modulus of the metal, t is the thickness of the diaphragm and  is a coefficient which depends on the boundary condition of the membrane at the wall13. Values of  have been calculated for various boundary conditions and are tabulated in a book by Roark and Young413. Our situation is closest to the case where the membrane is fixed at the wall and guided in the center with pressure being applied to the membrane. In this case  is found to be 0.090. Our geometry differs slightly from the geometry in the book in that a force is applied not only to the membrane but also to the guiding post. We expect this force to increase . To determine an upper limit to how much this could affect , we examine the case where there is no guiding post but a simple membrane with a fixed outer edge. In this case  is 0.188. Thus, we conclude that  is between 0.090 and 0.188. Substituting Equation (2) into Equation (1) we can write the capacitance as C = C0 +
␣C20 P,
(3)
where C0 is the capacitance with no pressure applied, and ␣ ⬅ a4/⑀0AEt3. All of the factors which contribute to ␣ should remain nearly independent of temperature below 5 K and independent of the thermal cycling history of the device. Thus it is expected that ␣, once determined, should remain constant. The explicit dependence on C0 and are included in Equation (3) so that the calibration can be adjusted for differing initial gaps and differing dielectric constants. This explicit dependence on C0 is critical since it renders the calibration effectively insensitive to a host of effects that change the plate separation with no differential pressure applied5. For our capacitor we have a = 0.632 cm, A = 1.105 cm2, E = 1.1 × 1011 N/m2, and t = 0.0151 cm. Using these values the expected value of ␣ ranges from 4.3 to 8.1 × 10−7 (pF Pa)−1 depending on which value of  is most appropriate. It is worth noting also that although we have calibrated this gauge only up to 200 Pa, the maximum pressure that the gauge can withstand without deformation is much higher. It is determined by the yield stress of brass at 4 K, which is 7.6 × 108 Pa. The relation between radial stress, s, and pressure is13 s = 3a2p/4t2.
ferential across the gauge and measuring the resulting capacitance. To obtain the highest accuracy calibration we have used a system represented schematically in Figure 2. The key features of the system are as follows. The gauge is submerged in a liquid helium bath maintained at T = 2.1 K. The gauge is filled with helium gas which is cold trapped at 77 K and filtered at 2.1 K. The gas is maintained at approximately 1300 Pa (i.e. 10 Torr). The differential pressure is measured with a high accuracy MKS Baratron capacitance manometer14, which is connected to the low temperature pressure gauge by means of two 1/4 inch stainless steel tubes. The capacitance is measured with a high accuracy capacitance bridge. Lastly, the two volumes on either side of the capacitor diaphragm are connected to large thermally insulated reservoirs to minimize P-T drifts. Of the two measurements, differential pressure and capacitance, the differential pressure measurement was the most uncertain due to a wide variety of sources. These sources of uncertainty are the Baratron standard, the dielectric constant, drifts in room temperature, hydrostatic pressure differences, and thermomolecular pressure differences. The MKS Baratron differential gauge measures up to a maximum differential pressure of 1330 Pa, with a resolution of 0.002–0.02 Pa depending on the pressure range. At the lowest differential pressures measured, from 0 to 20 Pa, the accuracy of the Baratron is rated at ± 0.006 Pa, and at the higher differential pressures the accuracy is 0.1%. Another source of uncertainty, changes in the dielectric constant of the gas in the low temperature gauge, turns out to be negligible. The dielectric constant is affected by impurities in the helium gas and by variations in temperature and pressure. The uncertainty due to temperature and pressure dependence is negligible since for helium gas is so close to 1. Even the larger of the two effects, uncertainty in the absolute pressure of the gas between the plates, causes variations in of less than 0.01%. The dielectric
(4)
If we set the maximum stress we can apply to be 5-fold lower than the yield stress of brass, we obtain pmax 苲 1 bar. In fact, the electrodes touch each other at a differential pressure of 0.5 bar, further protecting the gauge from inadvertent over-pressure. The calibration is performed by applying a pressure dif-
4
See also 9 for further details on design criteria for determining an optimum membrane thickness. 5 Prominent among these effects is the variation in the internal stresses in the capacitor. We found the capacitor to be extremely sensitive to how the body was mounted. Mounting the capacitor from two different supports with different expansions, for example, caused stresses that required hours to relax. Other variations in C0 were due to differing protocols for cooling the capacitor and variations in gravitational loading due to orientation.
Figure 2 Schematic diagram of the calibration apparatus
Cryogenics 1998 Volume 38, Number 6 675
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. constant can also be affected if ice or other contaminants become lodged between the plates. To eliminate these problems we have cold trapped the helium gas and filtered it with 20–25m filters at 2.1 K. Consider next the uncertainty due to room temperature variations. Small thermal drifts of the tubes connected to the gauge cause similar drifts in the pressure. We found that simply touching one of the copper tubes for a period of 2 s caused a significant change in the differential pressure between the sides. Since we simultaneously measure capacitance and differential pressure, variations in differential pressure do not necessarily contribute to any error. However, the lack of signal stability makes it impossible to signal average to improve precision and raises the concern that the cold gauge is not in thermodynamic equilibrium with the room temperature equipment. To minimize these temperature drift effects we increased the volume of the system and insulated it. This was accomplished by attaching a thermally insulated, 30-l, helium gas bottle to each side of the pressure manifold. This reduced the pressure drifts by more than an order of magnitude. Although helium gas is light at room temperature, the density increases significantly at 2.1 K, giving rise to a small hydrostatic head. Thus, the absolute pressure in the cold gauge at the bottom of the dewar differs from the absolute pressure at the top. This difference is approx. 1– 2 Pa depending on the absolute pressure in the lines and the level of the bath. If the absolute pressure in both tubes is the same then the hydrostatic head is the same and the pressure differential is unaffected. However, on one of the sides the pressure is necessarily elevated above the other in order to obtain the measured pressure differential. This causes our measurement of the pressure differential to be systematically too low by about 0.1%. Finally, we can calculate uncertainties from thermomolecular pressure. If the tubes to the gauge were identical then there would be no thermomolecular effect. However, the junction from 1/8th inch tubing to 1/4th inch tubing occurs at different heights so some thermomolecular pressure difference could occur. The correction for the thermomolecular pressure varies inversely with the absolute pressure in the tube. Thus, we want to keep the pressure as high as possible. On the other hand, we want to keep the pressure well below the saturated vapor pressure of 4176 Pa to prevent condensation. We chose an intermediate value for the absolute pressure of approximately 1300 Pa. At this absolute pressure, the thermomolecular pressure correction is approx. 0.06 Pa15. Thus, we expect differences in thermomolecular pressure between the two sides to be insignificant, even at the low pressures. The capacitance measurement is made using a high precision capacitance bridge from Andeen-Hagerling12. The bridge is an auto balancing bridge simultaneously measuring both capacitance and loss. The capacitance standards are highly stable fused silica capacitors, giving the bridge a stated accuracy of 0.1 × 10−6 pF. The largest uncertainty involved in the measurement of capacitance came from random vibrations of the gauge. We noticed that fluctuations in capacitance decreased by roughly a factor of ten upon cooling the surrounding bath through the lambda transition. We believe that the boiling of the helium in the surrounding bath during cooldown creates vibrations of the capacitor and consequently capacitance fluctuations. At 2.1 K we find that the RMS capacitance fluctuations are approx. 2 × 10−6 pF.
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Cryogenics 1998 Volume 38, Number 6
In conclusion, the two most significant errors were probably due to variations in the gas pressure head and room temperature fluctuations. Since we did not independently monitor the liquid level in the bath we cannot correct for pressure head differences from day to day. Similarly, we are unable to correct for variations in the absolute pressures due to room temperature fluctuations. To better estimate the size of these uncertainties we examine below the variation in calibration on three consecutive days.
Response time The pressure gauge is designed primarily for measurement of mean differential pressure. However, it can be used to measure pressure fluctuations in the fluid. For such measurements the temporal response is of interest. There are at least three factors that will affect the response time of the gauge; the response time of the bridge electronics, the natural frequency of the membrane, and the delay time for a pressure fluctuation to be transmitted through the tubes to the capacitor. The response time of the electronics is set by the manufacturer of our bridge. The bridge excitation frequency is 1 kHz, and the minimum output time constant is 0.01 sec. The natural frequency of the system can be estimated by calculating the mass of the pedestal and the spring constant of the membrane6. The calculation gives f = 630 Hz, well above the 100 Hz limit of the bridge. Finally, one can determine the delay time in the long thin tubes16. In liquid helium for nearly all tubes commonly used this time constant is the time for sound to propagate in the tube. Hence, the greatest limitation is that of the electronics. It should be noted, however, that because the averaging time is reduced, the minimum resolution of the gauge is correspondingly reduced.
Figure 3 Calibration curve. This particular curve corresponds to day 3-run A
6
The response of the membrane to a central force is given in in table 24, case 1f.
16
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. Table 1 Regression coefficients from calibration curves Slope (pF/Pa) Day Day Day Day Day Day
1-run 1-run 2-run 2-run 3-run 3-run
A B A B A B
4.1045 4.1023 4.0615 4.0623 4.0651 4.0649
× × × × × ×
10−4 10−4 10−4 10−4 10−4 10−4
Uncertainty (pF/Pa) 0.0010 0.0009 0.0012 0.0010 0.0010 0.0011
× × × × × ×
10−4 10−4 10−4 10−4 10−4 10−4
Results and conclusions A calibration was performed on three consecutive days, warming the capacitor up to room temperature overnight between each day. On each day two different calibration curves were measured. A typical calibration curve is shown in Figure 3. The slopes and intercepts along with their associated uncertainties are determined with a linear regression analysis and are displayed in Table 1. One can see that on any particular day the variation of the slope was in the order of 0.02%, while variations from day to day were much larger. The primary difference between the slopes from day to day is due to variations in C0, in accordance with Equation (3). We suspect that the reason for the variation of C0 is due to differences in rates of cool down. The differences in C0 seen in Table 1 can be accounted for by changes in the capacitor plate separation of only 0.1 m. It should be emphasized that variations from day to day of C0 are in no way problematic, since the pressure depends on C − C0. To properly compare the different calibrations, we use the coefficient ␣. The average value of ␣ is 8.079 × 10−7 with a standard deviation of 0.022 × 10−7 or ± 0.26%. This number agrees reasonably well with the calculated values from 3.8 × 10−7 to 8.1 × 10−7 (pF Pa)−1. To be conservative we estimate that the uncertainty in our value of ␣ is approx. 0.5% and is primarily due to the abovementioned pressure head and room temperature variations. We conclude then that the calibration coefficient, ␣, of the gauge remains constant over a range of differential pressures from 0 to 200 Pa to an accuracy of 0.5%, even after thermal cycling. It is thus possible that in some cases the accuracy of the instrument in liquid helium and critical helium gas applications will be limited by uncertainty in our knowledge of the dielectric constant , not uncertainty in the gauge itself. In an actual flow application the gauge and measurement ports will all typically be submerged in the helium. They will not be connected to long stainless steel tubes spanning a temperature difference of 300 K. We
Intercept (pF)
Uncertainty (pF)
␣ (pF Pa)−1
22.51218 22.51239 22.44146 22.44166 22.43768 22.43790
0.000014 0.000011 0.000009 0.000015 0.000013 0.000014
8.099 8.094 8.065 8.066 8.074 8.074
× × × × × ×
10−7 10−7 10−7 10−7 10−7 10−7
expect that in this case many of the uncertainties involved in the calibration will disappear. Thus, the gauge should be accurate down to its lowest resolution of 0.01 Pa.
Acknowledgements We gratefully acknowledge support from the National Science Foundation under grant DMR 96-14058.
References 1. Donnelly, R.J., High Reynolds Number Flows Using Liquid and Gaseous Helium. Springer, New York, 1991. 2. Barenghi, C.F., Swanson, C.J. and Donelly, R.J., Emerging issues in helium turbulence. J. Low Temp. Phys., 1995, 100, 385–413. 3. Maurer, J., Tabeling, P. and Zocchi, G., Statistics of turbulence between two counter-rotating disks in low temperature helium gas. Europhys. Lett., 1994, 26, 31–36. 4. Wu, X.Z. and Libchaber, A., Non-Boussinesq effects in free thermal convection. Phys. Rev. A., 1991, 43, 2833. 5. Smith, M.R., Donnelly, R.J., Goldenfeld, N. and Vinen, W.F., Decay of vorticity in homogenous turbulence. Phys. Rev. Lett., 1993, 71, 2583–2586. 6. Castaing, B., Chabaud, B. and Hebral, B., Hot wire anemometer operating at cryogenic temperatures. Rev. Sci. Instr., 1992, 63, 4168. 7. Adams, E.D., High-resolution capacitive pressure gauges. Rev. Sci. Instr., 1993, 64, 601–611. 8. Straty, G.C. and Adams, E.D., Highly Sensitive Capacitive Pressure Gauge. Rev. of Sci. Instr., 1969, 40, 1393–1397. 9. Lorenson, C.P., Dynamical properties of superfluid turbulence. Ph.D. thesis, Ohio State University, 1985. 10. Steinhauer, J., Schwab, K., Mukharsky, Y., Davis, J.C. and Packard, R.E., Determination of the energy barrier for phase slips in superfluid 4He. J. Low Temp. Phys., 1995, 100, 281–307. 11. DuPont Co. Polymer Division, Wilmington, DE 19898. 12. Andeen-Hagerling, 31200 Bainbridge Rd, Cleveland, OH 44139. 13. Roark, R.J. and Young, W.C., Formulas for Stress and Strain 5th edn. McGraw Hill, New York, 1975. 14. MKS Inc., 6 Shattuck Rd, Andover, MA 01810-2449. 15. White, G.K., Experimental Techniques in Low Temperature Physics. Clarendon, Oxford, 1959. 16. Benedict, R.P., Fundamentals of Temperature, Pressure and Flow Measurements, 3rd edn, Chap. 18. John Wiley and Sons, New York, 1984.
Cryogenics 1998 Volume 38, Number 6 677
Cryogenics 38 (1998) 673–677 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/98/$19.00
An accurate differential pressure gauge for use in liquid and gaseous helium Chris J. Swanson*, Kris Johnson and Russell J. Donnelly Department of Physics, University of Oregon, Eugene, OR 97403, U.S.A.
Accepted 21 January 1998 We have developed and calibrated an accurate pressure gauge studying the flow of liquid or gaseous helium. The gauge is designed to make measurements of pressure drops in liquid helium flow experiments but could be used for a broad range of other purposes. The gauge is calibrated over a range of pressures from 0 to 200 Pa and can resolve pressures as small as 0.01 Pa. It is accurate to better than 0.5% of the reading or ± 0.01 Pa, whichever is greater, and maintains this accuracy even after warm up and cool down. 1998 Published by Elsevier Science Ltd. All rights reserved Keywords: B. helium gas; B. helium liquid; D. pressure sensor
Introduction Liquid helium and critical helium gas are becoming widely used in the study of high Reynolds number and high Rayleigh number turbulence due to their low kinematic viscosity1–5. However, the cryogenic environment renders most room temperature measuring devices inoperable. To study turbulence at helium temperatures, the standard devices for measuring quantities of interest in turbulence must be replaced by cryogenic alternatives. Many instruments such as thermometers and hot film velocity probes6 have already been developed for cryogenic application. Further, there is an excellent review of cryogenic pressure transducers for a variety of applications7,1. We have developed a gauge which complements previous gauges and meets the following requirements: a reading accuracy of 0.5% of the pressure reading or ± 0.01 Pa, whichever is greater, capability of spanning a range of pressures from 0 to 200 Pa, and most importantly, repeatability through many thermal cycles to room temperature without recalibration. Our gauge follows in large part the design of a high pressure metal diaphragm gauge8 except that the differential pressure range has been significantly reduced and the calibration remains unchanged upon repeated thermal cycling.
Design Low temperature capacitance pressure gauges generally fall into two categories, those using metal diaphragms and those *To whom all correspondence should be addressed 1 Besides capacitive pressure transducers, Validyne (Northridge, CA 91324) manufactures a variable reluctance pressure transducer which has been used in cryogenic applications.
using plastic diaphragms9,10. Initially we designed a gauge using a 25 m thick plastic diaphragm since the plastic style seemed to offer sensitivity to smaller pressure differences. Of the two plastics widely used, Mylar and Kapton11 we selected Kapton due to an observed anisotropic shrinkage of Mylar at 77 K2. However extensive testing found the Kapton calibration curve to vary by as much as 10% after a single thermal cycle. To overcome this problem we attempted to calibrate the gauge using an electrostatic technique10 which could be quickly performed prior to each usage. Comparison of the electrostatic method with a direct pressure calibration also yielded unsatisfactory results, leading us to a metal diaphragm gauge. A drawing of the pressure transducer is shown in Figure 1. The capacitor is made of two parallel plates, which are approximately 1 cm2 in area separated by roughly 40 m. One of the plates is connected by a thin post to a metal diaphragm 150 m thick, across which the differential pressure is applied. Changes in differential pressure move the diaphragm and hence change the capacitance between the plates, which is measured using a high accuracy AC bridge12. The capacitor has been calibrated using helium gas at 2.1 K as the working fluid. The fluid is admitted to the capacitor through top and bottom ports by means of short, 1/8th inch copper tubes. The short copper tubes each attach to 20–25 m filter paper filters. The filters are included to prevent ice or other particles from getting between the electrodes of the capacitor and causing changes in the dielectric constant. The filter 2 We observed that upon submerging a uniformly stretched circular film of Mylar in liquid nitrogen, a series of wrinkles developed. The wrinkles were in a pattern of regular straight lines, such as would be expected if the membrane were stretched in the direction of the lines.
Cryogenics 1998 Volume 38, Number 6 673
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al.
Figure 1 Drawing of the pressure gauge
elements are installed in custom housings built to allow for indium seals. The electrical connections are made through the sides of the main body, as shown in Figure 1. In order to insure that the wires were completely shielded, miniature connectors were screwed into tapped holes in the sides. A thin metal post, soldered to the backside of the connector, passes through the wall to the inside cavity of the capacitor. Embedding the post in Stycast 2850 GT epoxy3 creates a leak tight seal and provides electrical insulation. The most critical feature of the capacitor design involved the placement of the capacitor electrodes. Both centering and spacing were important factors to consider. The copper electrode was centered by gluing it into a short plastic sleeve that fit snugly into the brass body. Subsequently the remaining gap between the electrode and the brass was filled with epoxy. After curing, the electrode was polished flush with the surrounding brass face. In order to minimize any stray capacitance due to charges bound to the epoxy, we trimmed the epoxy back from the surface of the electrode by approx. 0.5 mm. The other side of the device contained the pedestal shaped electrode. The pedestal electrode and membrane are machined out of a single piece of brass. We chose brass in order to match the thermal expansion coefficient of the body. We found that using metals with mismatched expansion coefficients caused the diaphragm to bow slightly in the center, with the maximum deflection being 5–10 times greater than the differential shrinking in the radius. This epoxy is filled with 苲 70% aluminum oxide grit to better match the linear expansion coefficient of the brass. Further, we found it to adhere much better to the metallic surfaces. It is available from Emerson and Cumming, Woburn, MA 01888.
3
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Cryogenics 1998 Volume 38, Number 6
The machining requires special procedures to obtain a flat thin diaphragm of the desired thickness. First, a brass rod is annealed to relieve any internal stresses in the material. It is then turned down to the desired radius, and the indentation at the top of the piece is cut. The brass is then annealed again on a hot plate at 苲 380°C. Failure to pre-relieve these stresses in the metal creates curvature in the diaphragm. A brass rod is then epoxied into the indentation. Holding the work with the brass rod, the rest of the piece can be machined. The solid support of the epoxied brass allows for the metal to be cut down to a thickness of 0.015 cm without tearing or deforming the diaphragm. The unit is then heated to release the epoxy joint, after which it is cleaned with a solvent. We expect that this technique could be used to fabricate a diaphragm down to about 0.0025 cm. Next the pedestal electrode must be installed into the brass body. The body and the pedestal are placed face down on a flat glass plate. The pedestal is then centered by eye and epoxied to the body. While this technique attempts to align the capacitor plate flush with the outside surface, some level of height differential is inevitable. Additionally, the critical distance is not the distance between one electrode and its surrounding surface but the total distance between the electrodes. The simplest and most accurate way to measure the distance between the electrodes is to measure the capacitance. Our capacitance at room temperature is roughly 25 pF, corresponding to a separation of 40 m, and is slightly larger than expected. The source of the larger than expected separation could be from either of two sources. First, the pedestal shaped electrode could be slightly recessed from the surrounding surface, despite our efforts to keep surfaces flush. Second, when the two halves of the capacitor are assembled, indium is squeezed into an O-ring groove. A very thin layer of indium typically spills over the edge of the groove creating a small gap between the halves. In any case, the pressure gauge has sufficient sensitivity for a wide range of plate separations so that exact control over the plate spacing is unnecessary. After assembly, the capacitor was thermally cycled between 77 K and room temperature a few times in order to relax any residual thermal stresses. We also thoroughly leak checked the capacitor at room temperature and 77 K.
Calibration The sensitivity of the device can be predicted from known properties of the material. Assuming a parallel plate configuration, the capacitance is to first order C=
冉 冊
⑀0A y 1+ , d0 d0
(1)
where is the dielectric constant of the fluid, ⑀0 is the permittivity of free space, A is the area of the plates, d0 is the gap between the plates with no pressure applied and y is deflection of one of the plates. Note that a positive value of y decreases the gap. The pressure dependence of the deflection y is given by y=
a4 P, Et3
(2)
where a is the radius of the flexible diaphragm, E is the
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. elastic modulus of the metal, t is the thickness of the diaphragm and  is a coefficient which depends on the boundary condition of the membrane at the wall13. Values of  have been calculated for various boundary conditions and are tabulated in a book by Roark and Young413. Our situation is closest to the case where the membrane is fixed at the wall and guided in the center with pressure being applied to the membrane. In this case  is found to be 0.090. Our geometry differs slightly from the geometry in the book in that a force is applied not only to the membrane but also to the guiding post. We expect this force to increase . To determine an upper limit to how much this could affect , we examine the case where there is no guiding post but a simple membrane with a fixed outer edge. In this case  is 0.188. Thus, we conclude that  is between 0.090 and 0.188. Substituting Equation (2) into Equation (1) we can write the capacitance as C = C0 +
␣C20 P,
(3)
where C0 is the capacitance with no pressure applied, and ␣ ⬅ a4/⑀0AEt3. All of the factors which contribute to ␣ should remain nearly independent of temperature below 5 K and independent of the thermal cycling history of the device. Thus it is expected that ␣, once determined, should remain constant. The explicit dependence on C0 and are included in Equation (3) so that the calibration can be adjusted for differing initial gaps and differing dielectric constants. This explicit dependence on C0 is critical since it renders the calibration effectively insensitive to a host of effects that change the plate separation with no differential pressure applied5. For our capacitor we have a = 0.632 cm, A = 1.105 cm2, E = 1.1 × 1011 N/m2, and t = 0.0151 cm. Using these values the expected value of ␣ ranges from 4.3 to 8.1 × 10−7 (pF Pa)−1 depending on which value of  is most appropriate. It is worth noting also that although we have calibrated this gauge only up to 200 Pa, the maximum pressure that the gauge can withstand without deformation is much higher. It is determined by the yield stress of brass at 4 K, which is 7.6 × 108 Pa. The relation between radial stress, s, and pressure is13 s = 3a2p/4t2.
ferential across the gauge and measuring the resulting capacitance. To obtain the highest accuracy calibration we have used a system represented schematically in Figure 2. The key features of the system are as follows. The gauge is submerged in a liquid helium bath maintained at T = 2.1 K. The gauge is filled with helium gas which is cold trapped at 77 K and filtered at 2.1 K. The gas is maintained at approximately 1300 Pa (i.e. 10 Torr). The differential pressure is measured with a high accuracy MKS Baratron capacitance manometer14, which is connected to the low temperature pressure gauge by means of two 1/4 inch stainless steel tubes. The capacitance is measured with a high accuracy capacitance bridge. Lastly, the two volumes on either side of the capacitor diaphragm are connected to large thermally insulated reservoirs to minimize P-T drifts. Of the two measurements, differential pressure and capacitance, the differential pressure measurement was the most uncertain due to a wide variety of sources. These sources of uncertainty are the Baratron standard, the dielectric constant, drifts in room temperature, hydrostatic pressure differences, and thermomolecular pressure differences. The MKS Baratron differential gauge measures up to a maximum differential pressure of 1330 Pa, with a resolution of 0.002–0.02 Pa depending on the pressure range. At the lowest differential pressures measured, from 0 to 20 Pa, the accuracy of the Baratron is rated at ± 0.006 Pa, and at the higher differential pressures the accuracy is 0.1%. Another source of uncertainty, changes in the dielectric constant of the gas in the low temperature gauge, turns out to be negligible. The dielectric constant is affected by impurities in the helium gas and by variations in temperature and pressure. The uncertainty due to temperature and pressure dependence is negligible since for helium gas is so close to 1. Even the larger of the two effects, uncertainty in the absolute pressure of the gas between the plates, causes variations in of less than 0.01%. The dielectric
(4)
If we set the maximum stress we can apply to be 5-fold lower than the yield stress of brass, we obtain pmax 苲 1 bar. In fact, the electrodes touch each other at a differential pressure of 0.5 bar, further protecting the gauge from inadvertent over-pressure. The calibration is performed by applying a pressure dif-
4
See also 9 for further details on design criteria for determining an optimum membrane thickness. 5 Prominent among these effects is the variation in the internal stresses in the capacitor. We found the capacitor to be extremely sensitive to how the body was mounted. Mounting the capacitor from two different supports with different expansions, for example, caused stresses that required hours to relax. Other variations in C0 were due to differing protocols for cooling the capacitor and variations in gravitational loading due to orientation.
Figure 2 Schematic diagram of the calibration apparatus
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An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. constant can also be affected if ice or other contaminants become lodged between the plates. To eliminate these problems we have cold trapped the helium gas and filtered it with 20–25m filters at 2.1 K. Consider next the uncertainty due to room temperature variations. Small thermal drifts of the tubes connected to the gauge cause similar drifts in the pressure. We found that simply touching one of the copper tubes for a period of 2 s caused a significant change in the differential pressure between the sides. Since we simultaneously measure capacitance and differential pressure, variations in differential pressure do not necessarily contribute to any error. However, the lack of signal stability makes it impossible to signal average to improve precision and raises the concern that the cold gauge is not in thermodynamic equilibrium with the room temperature equipment. To minimize these temperature drift effects we increased the volume of the system and insulated it. This was accomplished by attaching a thermally insulated, 30-l, helium gas bottle to each side of the pressure manifold. This reduced the pressure drifts by more than an order of magnitude. Although helium gas is light at room temperature, the density increases significantly at 2.1 K, giving rise to a small hydrostatic head. Thus, the absolute pressure in the cold gauge at the bottom of the dewar differs from the absolute pressure at the top. This difference is approx. 1– 2 Pa depending on the absolute pressure in the lines and the level of the bath. If the absolute pressure in both tubes is the same then the hydrostatic head is the same and the pressure differential is unaffected. However, on one of the sides the pressure is necessarily elevated above the other in order to obtain the measured pressure differential. This causes our measurement of the pressure differential to be systematically too low by about 0.1%. Finally, we can calculate uncertainties from thermomolecular pressure. If the tubes to the gauge were identical then there would be no thermomolecular effect. However, the junction from 1/8th inch tubing to 1/4th inch tubing occurs at different heights so some thermomolecular pressure difference could occur. The correction for the thermomolecular pressure varies inversely with the absolute pressure in the tube. Thus, we want to keep the pressure as high as possible. On the other hand, we want to keep the pressure well below the saturated vapor pressure of 4176 Pa to prevent condensation. We chose an intermediate value for the absolute pressure of approximately 1300 Pa. At this absolute pressure, the thermomolecular pressure correction is approx. 0.06 Pa15. Thus, we expect differences in thermomolecular pressure between the two sides to be insignificant, even at the low pressures. The capacitance measurement is made using a high precision capacitance bridge from Andeen-Hagerling12. The bridge is an auto balancing bridge simultaneously measuring both capacitance and loss. The capacitance standards are highly stable fused silica capacitors, giving the bridge a stated accuracy of 0.1 × 10−6 pF. The largest uncertainty involved in the measurement of capacitance came from random vibrations of the gauge. We noticed that fluctuations in capacitance decreased by roughly a factor of ten upon cooling the surrounding bath through the lambda transition. We believe that the boiling of the helium in the surrounding bath during cooldown creates vibrations of the capacitor and consequently capacitance fluctuations. At 2.1 K we find that the RMS capacitance fluctuations are approx. 2 × 10−6 pF.
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In conclusion, the two most significant errors were probably due to variations in the gas pressure head and room temperature fluctuations. Since we did not independently monitor the liquid level in the bath we cannot correct for pressure head differences from day to day. Similarly, we are unable to correct for variations in the absolute pressures due to room temperature fluctuations. To better estimate the size of these uncertainties we examine below the variation in calibration on three consecutive days.
Response time The pressure gauge is designed primarily for measurement of mean differential pressure. However, it can be used to measure pressure fluctuations in the fluid. For such measurements the temporal response is of interest. There are at least three factors that will affect the response time of the gauge; the response time of the bridge electronics, the natural frequency of the membrane, and the delay time for a pressure fluctuation to be transmitted through the tubes to the capacitor. The response time of the electronics is set by the manufacturer of our bridge. The bridge excitation frequency is 1 kHz, and the minimum output time constant is 0.01 sec. The natural frequency of the system can be estimated by calculating the mass of the pedestal and the spring constant of the membrane6. The calculation gives f = 630 Hz, well above the 100 Hz limit of the bridge. Finally, one can determine the delay time in the long thin tubes16. In liquid helium for nearly all tubes commonly used this time constant is the time for sound to propagate in the tube. Hence, the greatest limitation is that of the electronics. It should be noted, however, that because the averaging time is reduced, the minimum resolution of the gauge is correspondingly reduced.
Figure 3 Calibration curve. This particular curve corresponds to day 3-run A
6
The response of the membrane to a central force is given in in table 24, case 1f.
16
An accurate differential pressure gauge for use in liquid and gaseous helium: C.J. Swanson et al. Table 1 Regression coefficients from calibration curves Slope (pF/Pa) Day Day Day Day Day Day
1-run 1-run 2-run 2-run 3-run 3-run
A B A B A B
4.1045 4.1023 4.0615 4.0623 4.0651 4.0649
× × × × × ×
10−4 10−4 10−4 10−4 10−4 10−4
Uncertainty (pF/Pa) 0.0010 0.0009 0.0012 0.0010 0.0010 0.0011
× × × × × ×
10−4 10−4 10−4 10−4 10−4 10−4
Results and conclusions A calibration was performed on three consecutive days, warming the capacitor up to room temperature overnight between each day. On each day two different calibration curves were measured. A typical calibration curve is shown in Figure 3. The slopes and intercepts along with their associated uncertainties are determined with a linear regression analysis and are displayed in Table 1. One can see that on any particular day the variation of the slope was in the order of 0.02%, while variations from day to day were much larger. The primary difference between the slopes from day to day is due to variations in C0, in accordance with Equation (3). We suspect that the reason for the variation of C0 is due to differences in rates of cool down. The differences in C0 seen in Table 1 can be accounted for by changes in the capacitor plate separation of only 0.1 m. It should be emphasized that variations from day to day of C0 are in no way problematic, since the pressure depends on C − C0. To properly compare the different calibrations, we use the coefficient ␣. The average value of ␣ is 8.079 × 10−7 with a standard deviation of 0.022 × 10−7 or ± 0.26%. This number agrees reasonably well with the calculated values from 3.8 × 10−7 to 8.1 × 10−7 (pF Pa)−1. To be conservative we estimate that the uncertainty in our value of ␣ is approx. 0.5% and is primarily due to the abovementioned pressure head and room temperature variations. We conclude then that the calibration coefficient, ␣, of the gauge remains constant over a range of differential pressures from 0 to 200 Pa to an accuracy of 0.5%, even after thermal cycling. It is thus possible that in some cases the accuracy of the instrument in liquid helium and critical helium gas applications will be limited by uncertainty in our knowledge of the dielectric constant , not uncertainty in the gauge itself. In an actual flow application the gauge and measurement ports will all typically be submerged in the helium. They will not be connected to long stainless steel tubes spanning a temperature difference of 300 K. We
Intercept (pF)
Uncertainty (pF)
␣ (pF Pa)−1
22.51218 22.51239 22.44146 22.44166 22.43768 22.43790
0.000014 0.000011 0.000009 0.000015 0.000013 0.000014
8.099 8.094 8.065 8.066 8.074 8.074
× × × × × ×
10−7 10−7 10−7 10−7 10−7 10−7
expect that in this case many of the uncertainties involved in the calibration will disappear. Thus, the gauge should be accurate down to its lowest resolution of 0.01 Pa.
Acknowledgements We gratefully acknowledge support from the National Science Foundation under grant DMR 96-14058.
References 1. Donnelly, R.J., High Reynolds Number Flows Using Liquid and Gaseous Helium. Springer, New York, 1991. 2. Barenghi, C.F., Swanson, C.J. and Donelly, R.J., Emerging issues in helium turbulence. J. Low Temp. Phys., 1995, 100, 385–413. 3. Maurer, J., Tabeling, P. and Zocchi, G., Statistics of turbulence between two counter-rotating disks in low temperature helium gas. Europhys. Lett., 1994, 26, 31–36. 4. Wu, X.Z. and Libchaber, A., Non-Boussinesq effects in free thermal convection. Phys. Rev. A., 1991, 43, 2833. 5. Smith, M.R., Donnelly, R.J., Goldenfeld, N. and Vinen, W.F., Decay of vorticity in homogenous turbulence. Phys. Rev. Lett., 1993, 71, 2583–2586. 6. Castaing, B., Chabaud, B. and Hebral, B., Hot wire anemometer operating at cryogenic temperatures. Rev. Sci. Instr., 1992, 63, 4168. 7. Adams, E.D., High-resolution capacitive pressure gauges. Rev. Sci. Instr., 1993, 64, 601–611. 8. Straty, G.C. and Adams, E.D., Highly Sensitive Capacitive Pressure Gauge. Rev. of Sci. Instr., 1969, 40, 1393–1397. 9. Lorenson, C.P., Dynamical properties of superfluid turbulence. Ph.D. thesis, Ohio State University, 1985. 10. Steinhauer, J., Schwab, K., Mukharsky, Y., Davis, J.C. and Packard, R.E., Determination of the energy barrier for phase slips in superfluid 4He. J. Low Temp. Phys., 1995, 100, 281–307. 11. DuPont Co. Polymer Division, Wilmington, DE 19898. 12. Andeen-Hagerling, 31200 Bainbridge Rd, Cleveland, OH 44139. 13. Roark, R.J. and Young, W.C., Formulas for Stress and Strain 5th edn. McGraw Hill, New York, 1975. 14. MKS Inc., 6 Shattuck Rd, Andover, MA 01810-2449. 15. White, G.K., Experimental Techniques in Low Temperature Physics. Clarendon, Oxford, 1959. 16. Benedict, R.P., Fundamentals of Temperature, Pressure and Flow Measurements, 3rd edn, Chap. 18. John Wiley and Sons, New York, 1984.
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