S14ThermalConductivity
Abstract
We report measurements of the relationship between thermal contact conductance and applied torque on a bolt compressing a copper-copper junction at 77K. Our data indicate a linear relationship between thermal contact conductance and applied torque of h_s = (410+-10)N + (3150+-90), for h_s in W m-2 K-1 and N in in-lb. Instrument limit error on diode voltage measurements generated low p values in linear regressions of thermal contact conductance, and a more precise multimeter and more frequent measurements would reduce uncertainty significantly.
Introduction
Thermal contact conductance, hc, is defined as the ability of two objects in contact to conduct heat. More specifically, for a given temperature difference between two points on opposite sides of the junction, hc is the heat flow per unit area across the junction, per Kelvin of temperature gradient on the junction. The SI units for hc are W m-2 K-1. In many experiments, improving thermal conductance increases the time that the experiment can be kept cold. Maximum helium efficiency is critical for projects with a limited supply of helium, such as balloon-borne experiments like the E and B Experiment (EBEX) telescope headquartered at the University of Minnesota. Furthermore, results that improve thermal conductance can benefit any project that uses liquid helium, since researchers predict a global shortage in the next decade (Kaplan, 2007).
Previous work by Berman (1956), Greenwood and Johnson (1965), and others indicates that the thermal conductance of a bare copper-copper interface increases linearly with applied pressure at liquid nitrogen temperatures. We cannot accurately measure applied pressure, but Machinery’s Handbook (2000) asserts that the pressure will vary linearly with the fastener tightening torque applied to the bolts which compress the blocks. In constructing a clamped system, it is more practical to measure the torque applied to a bolt which clamps the system than to measure the clamping pressure exerted by that bolt, so we will use fastener tightening torque as our independent variable.
In general, hc simplifies to the sum of heat transfer by radiation, fluid conduction, and solid conduction. For metallic contacts below 873K, the contribution of radiation to this total is less than 1% (Clausing and Chao, 1965), so at cryogenic temperatures this contribution is negligible. Furthermore, our experiment was conducted with ambient pressure less than 10^-7 atm, so fluid conductance was neglected. We are left to assume, therefore, that all thermal energy introduced into the top copper block will be conducted by solid conduction to the surface below. We will denote the contact conductance due to solid conduction by hs, as opposed to the more general hc.
Theory
There are three basic modes of heat transfer: radiation, fluid convection, and solid conduction [1]. We neglect radiation by running our experiment at liquid nitrogen temperature, and we neglect fluid convection by operating it in a vacuum. Therefore we needed only consider solid conduction when analyzing our data
We define the solid contact conductance as:
Equation 1: h_c = (Q/A)/(delta-T)
We predict that h_c varies linearly as a linear function of the applied force between the copper plates. This was shown experimentally by Berman (1956), and was explained from a conceptual standpoint by Greenwood and Johnson (1965). They posited that a supposedly flat surface is actually made up of many hills and valleys of varying size, and when two of these surfaces are in contact, they only actually touch at discrete contact spots. When pressure is applied, the average area of each of the contact spots remains relatively constant, but the number of contact spots increases linearly with the applied pressure, causing the thermal conductance of the junction to rise linearly as well.
Figure 1: Diagram of two “flat” solid surfaces being held in contact with each other. As the pressure is increased, the number of contact spots (circled in red) increases.
Revisiting equation (1) for a moment, we see that in order to calculate the thermal contact conductance as a function of pressure, hc(P), we must measure and find a relationship between Q and ∆T. Our experiment creates heat flow by supplying current I across a resistor R, which produces thermal energy at the rate Q=I^2 R. (2) Since we are neglecting radiation and convection, as discussed earlier, we can assume that all the thermal energy flows through the experimental system by means of solid contact conductance. Let us now rewrite equation (1), incorporating equation (2) and the fact that we will be measuring the applied torque on the screws rather than the pressure itself to more specifically represent our experimental setup: (I^2∙R)/(A∙ΔT)=h_c (N) (3) where N is the applied torque on each screw of the interface in question. We have strong support that, to first order, the pressure between the plates directly corresponds to the torque applied (Oberg and Jones, 2000). As stated in the introduction, the primary goal of this experiment was to test the hypothesis of linearity in hc vs. P, or now hc vs. N, and the claims leading to equation (4), both of which it verified.
Experimental Setup
Our apparatus for this experiment was contained in an Infrared Labs HDL-8 cryostat. During an experiment, the interior of the cryostat was continuously evacuated by a turbopump and rotary vane pump, producing pressure measurements of 7.5*10^-6 torr on a Pirani gauge. Within the cryostat, two layers of thermal shielding isolated the experiment from from the exterior 300K environment; each layer was cooled by a reservoir of liquid nitrogen when the experiment was active. These layers could be removed to access and alter the apparatus when it was at atmospheric pressure. The experimental apparatus was bolted to a cold plate in contact with the inner reservoir of liquid nitrogen.The apparatus consisted of three spacially separated clamped pairs of copper blocks, numbered 3, 6, and 7, and the electrical equipment needed to test them. We named the block of each pair in contact with the cold plate “B” and the one touching the heater “A” to simplify language; for example, the cold-plate block on pair 6 was referred to as 6B. Each clamped pair consisted of two copper blocks clamped with four #8 stainless steel hex cap screws. Before we took our first data set, the surface of each copper block was polished with 400-grit sandpaper and cleaned with Xylene cleaning agent. The screws were tightened with a torque wrench to a particular tightening torque before the cryostat was sealed. The area of contact is 12.6cm^2.
On each clamped pair, a 150Ω resistor was bolted to the A block, where the resistor produced heat flow Q=IR^2, for applied current I and resistance R=150Ω. The temperature sensors depicted in Figure 2 were silicon diode temperature sensors, which were placed above and below the junction. Lakeshore DT-670 and Cryocon S900 models were used in this experiment. Each diode had a calibrated curve of forward voltage bias as a function of temperature, which was provided by the manufacturer. Further, each diode was placed directly in a bath of liquid nitrogen to calibrate its voltage at nitrogen’s boiling point, 77.36 K. Apiezon N thermal grease was applied at each contact between a block and a resistor, a diode, or the cold plate in an attempt to control thermal conductance between components. Wires from the resistors and diodes inside the cryostat were connected to manganin wires which were heat-sunk to the liquid nitrogen reservoir, then connected to apparatus outside the cryostat. Manganin has a temperature coefficient of resistance of 10^-5, so it conducts electricity significantly better than it conducts heat (Goodfellow). An Agilent E3630A power supply powered the resistor, and the current through the resistor was measured with a Keithley 2000-20 multimeter. Both diodes had their forward bias voltage measured with a Keithley 2000 model multimeter on the “diode” setting.
Experimental Procedure
Our experimental procedure was designed to measure the change in temperature on the A and B blocks at equilibrium as a function of increasing heat flow. We began with the cryostat completely open. First, we used a torque wrench to apply a uniform fastener tightening torque to each of the four screws on a clamped pair. Torques of 5, 15, and 25 in-lb were used on all pairs, to acquire control measurements for all three block pairs. Next, we sealed the cryostat and used a rotary vane pump and turbopump to evacuate the system. These pumps ran continuously during the experiment to combat leaks in the cryostat. With the pumps running, we used a funnel to fill the liquid nitrogen reservoirs. We read both diodes on one clamped block pair to measure the temperature of the plates, and began to collect data from those diodes when their forward voltage bias stabilized, indicating a constant temperature.
We recorded the forward voltage bias across the A and B diodes of a block pair at regular intervals throughout our analysis of the thermal conductance of each plate, producing diode voltages as a function of time as our output. The data were used to create a plot of the change in temperature on the A and B blocks from their value at the start of testing (called ΔT_A and ΔT_B), using the standard temperature curves associated with the diodes. We then calculated the temperature gradient ΔT, which we defined as
Equation 2: ΔT = (T_A,final - T_A, initial) - (T_B,final - T_B, initial)
because this value best informed us how the change in temperature from A to B had been affected by the increase in power (see data analysis for a deeper investigation).
Based on the temperature gradient, we made decisions about whether to increase the heat flow. The decision-making process evolved as we conducted new trials, so our early trials did not match this format exactly. When the trial began, we recorded diode voltages with zero applied heat flow until consistently maintained its value over timescales of two minutes. At this point, we recorded the voltage output from the power supply and the current in the wire. Next, we increased the heat flow Q across the resistor by ~0.18W, and repeated the process of evaluating for consistency and increasing eight times, arriving at a maximum heat flow of across the junction. This procedure was repeated for each of the three clamped block pairs.
Analysis and Results
Data analysis proceeded in two main steps. First, voltage data were converted into absolute temperatures using the temperature curves linked at the bottom of this article. We used an assortment of Lakeshore DT670 and Cryocon S900 diodes on our various diodes. From temperature data, we calculated the change in temperature on each diode as a function of time, and temperature gradient, through Equation 2 above.
From these temperature gradient data, we used Kurt Wick's LSQfit algorithm in Microsoft Excel to plot temperature gradient as a function of heat flow, to obtain the heat conductance according to Equation 1. These plots tended to have large error bars (low p-values) due to the large instrument limit errors on temperature gradient measurements. Next, we took these measurements from our various plates and plotted them as a function of applied torque for the two block pairs that gave us physically valid data (the 3 and the 6). Plotting the resultant data in LSQfit produced this graph, which shows a strong linear trend between thermal conductance and fastener tightening torque.
When this experiment has been attempted in the past (see Pittmann, 2012), it failed to obtain meaningful data because temperature sensors could not read the fine changes in temperature reported in our experiment. However, a cursory glance at the temperature gradient v. time plot shows that even with sensitive diodes, our data still showed discrete levels in h_s, rather than smooth curves. Not only did this decrease the accuracy of the averaged temperature gradient at the level, it also caused instrument limit error to be a significant contributing factor to the overall error. The other most significant contribution to error in our measurements was the frequency with which we recorded data. Typically, one researcher would tell the other voltage information from the diodes on the plate at 15-second intervals, which that researcher would plot in Microsoft Excel. At the time of the experiment, we had access to a Lakeshore box which would have allowed us to automatically record voltage data using a GPIB, at much higher frequencies.
Results from block 7 were odd from the beginning – the temperature gradient took longer to come to equilibrium for this block than for any other, and the temperature gradient decreased as a function of applied power, rather than increasing. Thus, we looked for equipment error that would explain this anomalous result. Finally, after the fourth data run, we decided to swap the diodes from blocks 3 and 7 to determine whether the blocks or the diodes on the 7 were the source of the anomalous results. Results from the 7 diodes revealed a strong drift in the voltage output of diode 7B as a function of time. The temperature indicated by the 7B diode increased from 80.6K 80.20K from t=0s 1200s, while the temperature indicated by the 7A diode fell by 0.03K. Reassessing data from previous trials, diode 7B consistently produced temperatures in excess of 80K when in good thermal contact with the cold plate, indicating some internal flaw in the diode that invalidated our assumptions about its temperature-voltage curve. Because it could not be used to calculate temperature without a better understanding of its curve (a task outside the scope of this experiment), we omitted all data that used diode 7B. This diode is the S900 numbered 1756. Conclusion
Our experiment serves as both a proof-of-concept that our experimental procedure can produce reliable data, and as a quantitative measurement of the relation betwen thermal conductance of copper-copper junctions and torque on the clamping bolts. Our results show a method that produces linear results consistent with Greenwood and Johnson’s predictions, which can be used for future trials at liquid nitrogen temperature, and may be extended to liquid helium temperature with a reasonable expectation of success. The quantitative relationship we obtained, h_s = (410+-10)N + (3150+-90) shows that in the torque regime where we took data, a system like ours will have thermal conductance well-modeled by a linear relationship to applied torque.
Suggestions for future groups
Check all your diodes before your first trial! First, dunk the diode in liquid nitrogen to learn its voltage output at nitrogen's boiling point. Next, do your first data run using as many diodes as possible, with all plates at the same torque. If one diode's temperature is excessively high (80K instead of 77K), then you should exchange it for a properly calibrated diode. Check also for long-term drift in voltage values at constant heat flow.
Measure how thermal conductance varies as a function of area. If yes, then does it vary as more screws are added at the same area? In particular, does the thermal conductance vary as a function of the total surface area, or just the area effectively compressed by screws?
If the cryostat can be repaired, measure thermal conductance dependence at liquid helium temperatures. (The original intent of this lab.)
Attempt to find multimeters to measure cryostat output that have higher precision on the diode setting than a Keithley 2000-20 (10uV minimum precision).
Whatever you do, use a computer connection to take data automatically. Hours were wasted taking mediocre data by hand when more numerous data could have been obtained through automation.
Acknowledgements
Dr. S. Hanany, external advisor, for the use of lab space and experimental apparatus
Dr. C. Pryke, internal advisor, for equipment, advice, and suggestions for data-taking
Kurt Wick, technical advisor, for purchasing parts and technical direction
EBEX group - especially Asad Aboobaker, Kate Raach, Bikram Chandra, Chris Geach, Aaron Smith, for sharing past experience
References
Berman, R. (1956). Some Experiments on Thermal Contact at Low Temperatures. Journal of Applied Physics, 27, 318-323. Retrieved from http://scitation.aip.org.
Clausing, A. M., & Chao, B. T. (1965). Thermal contact resistance in a vacuum environment. Journal of Heat Transfer, 243-250. Retrieved from http://heattransfer.asmedigitalcollection.asme.org.
Ditter, A., & Pittman, J. (2012). Technical Design Report: Thermal Conductance as a Function of Applied Force at Copper-Copper Junctions. Unpublished manuscript, Methods of Experimental Physics, School of Physics and Astronomy, University of Minnesota.
Greenwood, G. W., & Johnson, R. H. (1965). The deformation of metals under small stresses during phase transformations. Proceedings of the Royal Society, 283, 403-422. Retrieved from http://rspa.royalsocietypublishing.org.
Kaplan, K. H. (2007). Helium shortage hampers research and industry. Physics Today, 31-32. Retrieved from http://scitation.aip.org.
Oberg, E., & Jones, F. D. (2000). "Torque and Tension in Fasteners." Machinery's handbook: a reference book for the mechanical engineer, designer, manufacturing engineer, draftsman, toolmaker, and machinist. 26th ed. New York: Industrial Press. Print.
Pittman, J. (2012). On the Efficacy of Thermal Grease Across Copper-Copper Junctions. Unpublished manuscript, Methods of Experimental Physics, School of Physics and Astronomy, University of Minnesota.
[1] Tye, R. P. (1969). Thermal conductivity. (Vol. 2, pp. 253-274). New York, NY: Academic Press, Inc.
External Links
Cryocon S900 Temperature Curve Data: http://www.cryocon.com/S900/s900curve.pdf
LakeShore DT-670 Temperature Curve Data: http://www.lakeshore.com/products/Cryogenic-Temperature-Sensors/Silicon-Diodes/DT-670/Pages/Specifications.aspx#curvedt670
