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Image by Marilyn Yetzbacher |
The adiabatic method is generally accepted to give the most accurate results. The adiabatic method has been employed for heat capacity measurements by our research group for more than 35 years. Goodwin [1] developed a low-temperature adiabatic calorimeter, which was later modified by Magee [2] for automated measurement and control. It has been used to measure constant-volume heat capacities for many fluids at temperatures from 20 K to 345 K, and at pressures to 35 MPa. In the planning stages of this work, we recognized a need to augment our existing measurement capabilities by developing a new calorimeter which would extend the upped temperature limit to 700 K, without a compromise in accuracy.
In the adiabatic method, heat exchange between the calorimeter and its environment is eliminated as long as a temperature gradient does not exist. However, in actual practice the maintenance of a zero temperature gradient is an ideal situation which cannot be realized. Thus, even the most carefully conducted calorimetric experiment results in some heat loss. This heat loss can be minimized by automated adjustment of the temperature of the surrounding jacket to follow that of the calorimeter.
Since in practice we are only able to minimize heat losses, the primary problem becomes how to accurately evaluate the correction for heat leakage. Fortunately, the magnitude of the heat-loss term is much less important than the accuracy with which it can be determined. Since an accurate calculation of heat leakage is difficult and often not possible, it would be advantageous to develop a technique which eliminated the need to make such a correction altogether. Early attempts to accomplish this goal employed twin calorimeters. Twin calorimeters were used by Joule [3] in the mid-nineteenth century and later by Pfaundler [4]. Such twin devices have in common two calorimeters as nearly identical in construction as possible, supported in nearly identical surroundings.
In the present work [5], we combine the features of twin calorimeters with the adiabatic method, with the goal of realizing virtually complete elimination of the heat-loss correction. Almost identical spherical bombs of 70 cm3 capacity were designed to have a burst pressure in excess of 100 MPa at 700 K. The bombs were fabricated from Inconel 718 because of its high strength and excellent corrosion resistance. It is a weldable, machinable nickel-chromium alloy having exceptionally high tensile strength at elevated temperatures. Its tensile strength is more than twice that of type-316 stainless steel at 700 K. A pair of these bombs was made from Inconel 718 steel sheet by spinning a hardened steel tool in a lathe to produce hemispheres which were subsequently welded together. Each hemisphere was machined with a face which is flat within close tolerances, and has a 60° beveled edge to permit better access by the welding tool when two halves are fused together. After the hemispheres were fused, a 0.16 cm diameter filler rod of the same composition was melted into the groove. After the excess filler material was removed by grinding, the surface was polished to give a smooth appearance to the sphere. A 0.64 cm OD Iconel 718 tube was then welded into a hole of the same size in the top of the sphere. The assembly was then annealed while it underwent an argon purge at 1200 K to remove any heat-induced stress. Each of the spheres was then radiographed to reveal flaws in the weldments, if any. Radiographs of the spheres were made at a wide angle to the weldment. This angled view would reveal any areas which were only partially fused together. Such flaws would have appeared as darkened areas. No flaws were detected. When it was established that the metal was fused through the entire wall, the cells were tested under high vacuum for any leaks. Then they were hydrostatically pressure-tested to 1.75 times the maximum operating pressure (to 35 MPa) to establish their strength. Wall stress calculations indicate that the upper pressure limit at 700 K, when the yield strength is exceeded, is approximately 110 MPa.
The heater wire chosen for the bombs was a metal-sheathed heater which could be brazed onto the surface. An Iconel-sheathed, magnesium oxide insulated, nickel-chromium heater was brazed to the surface with a hard silver alloy that melted at about 900 K. A light steel jig was designed and built to hold the wires in place during brazing.
As shown in the figure, a light copper case was brazed to the sphere, covering two-thirds of its surface. It has two purposes. Chiefly, it intercepts stray heater radiation and conducts it back to the surface, thereby eliminating heat losses. Also, it serves as an anchor for the thermocouple well and platinum resistance thermometer (PRT) sheath which are brazed to it.
A capillary was attached to each sphere to fill and empty the vessels from outside the heated zone of the apparatus. A 0.05 cm inner diameter capillary was welded into a plug, which was in turn welded to the top of each 0.64 cm OD tube. Silver-alloy brazing could not be used to bond the parts of each sphere because of a known tendency of silver to catalyze decomposition of some hydrochlorofluorocarbon sample fluids at temperatures above 400 K. Silver and other brazing alloy components, notably copper, are also susceptible to corrosion with substances such as ammonia. The integrity of the weld seals was verified with a helium leak detector.
Each completed spherical bomb was jacketed in an adiabatic shield coupled to a guard ring, all constructed of Type-6061 aluminum, which was selected for its machinability and high thermal conductivity. The cylindrical surfaces of the guard ring and the sides and bottom of the shield were machined with spiral grooves to accept heaters which were pressed into them. The entire calorimeter assembly, shown in the figure, has been placed inside a high-temperature forced-air convection furnace, specially designed for this application. A differential thermocouple provides a continuous reading of the temperature difference between the sample and the reference cells. A three-junction differential thermopile provides readings of the temperature difference between each bomb and its associated adiabatic shield. Type-K thermocouples encased in magnesium oxide insulation and sealed in a steel sheath were selected for this duty. The measuring ends of the three thermocouples were brazed to the inside surface of the shield with the active portion inserted in three holes which were separated by a 120° angle from each other and located at an elevation near the center of the bomb. The three reference ends were bound together and placed in the thermocouple well which contained thermally conductive grease to a depth of 1 cm. A differential Type-K thermocouple provides readings of the temperature difference between the guard ring and the bomb. It was attached at each end in the same way as the shield thermocouples. A platinum resistance thermometer (PRT), having a calibration traceable to the National Institute of Standards and Technology (NIST), was coated with thermal grease and then inserted into the well brazed to the bottom of each bomb. These two thermometers provide measurements of the absolute temperature of each sphere with a high precision and accuracy. An oscillating quartz crystal pressure transducer, attached to the charging manifold, provides measurements of pressure. These pressures are nearly as accurate as the NIST-traceable piston gauges used to calibrate them. Each of the eight heaters (bomb, shield side, shield bottom, and guard for each cell) in the calorimeter is driven by independently controlled direct current power supplies. All of the instrumentation is connected to a microcomputer through an IEEE-488 standard interface bus. The computer executes FORTRAN code which has both temperature control and data acquisition functions. Except for sample charging, the new calorimeter is completely automated.
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Image by Marilyn Yetzbacher |
The basic principles of the heat capacity experiment are deceptively simple. For a single bomb, we measure the temperature rise (DT) when a measured quantity of heat energy (Q) is supplied to the calorimeter, which contains a mass (m) of substance. Subtracting the heat energy needed to heat the empty bomb (Q0) also widely known as heat equivalent, we calculate the sample heat capacity,
cv =(Q-Q0)/mDT.
For a twin calorimeter we simply replace values for Q with the differential quantity, DQ. It is defined as the heat energy supplied to the sample bomb minus that to the reference. The value of DQ accounts for the heat absorbed by the sample fluid. When we subtract the energy difference between the empty sample bomb and the reference, DQ0, we calculate the specific heat capacity at constant volume,
cv =(DQ-DQ0)/mDT.
Equation (2) is the apparatus working equation. Experience with a single-cell calorimeter, has shown that the quantity Q0 in Eq. (1) is as much as 95% of the measured heat energy Q. As a result, the relative expanded uncertainty (coverage factor k=2 and thus a two-standard-deviation estimate) may be up to 0.4%. This shortcoming of the single-bomb calorimeter leads to an estimated expanded uncertainty for the heat capacity of 0.5% for liquid and 2% for gas samples [2]. A twin-bomb calorimeter overcomes this problem. The result of this technique is to make the uncertainty propagated from the heat equivalent measurements a negligible effect. Thus, we have improved the accuracy of the measurements, allowing us to achieve our goal of a relative expanded uncertainty of 0.4% for the measured specific heat capacities at constant volume.
Uncertainty in cv arises from several sources. Primarily, the accuracy of this method is limited by the uncertainty involved in the temperature rise measurement and the change-of-volume work adjustment [2]. In the following discussion, we use a definition for the expanded uncertainty which is two times the standard uncertainty (i.e. a coverage factor k=2 and thus a two-standard-deviation estimate).
Different sources of uncertainty, including calibration of the platinum resistance thermometer, radiation to or from the thermometer head, and drift of the ice point resistance, contribute to an expanded uncertainty of 3x10-2K for the absolute temperature measurement. Uncertainty in the temperature rise measurement, however, also depends on the reproducibility of temperature measurements. The tempertures assigned to the beginning (T1) and to the end (T2) of a heating interval are determined from a linear fit of temperature with elapsed time, near the integer degree. The experimental ramp rate is approximately +4 x 10 -2K×min-1. This procedure leads to an uncertainty of 5x10-4 K for the interpolated temperatures T1 and T2, leading to values of 7x10-4 K for the uncertainty of the temperature rise, DT=T2-T1. For a typical experimental value of DT=1K, this corresponds to a relative uncertainty of 0.07.
The uncertainty of the change-of-volume work adjustment influences primarily the single-phase values since two-phase experiments are performed over a small pressure range. For water, the ratio of change-of-volume work to total applied heat is as large as 0.04 for the lowest density isochore. Estimated relative uncertainties of 2% in the change-of-volume work are due to both the deviation of the calculated pressure derivatives and the uncertainty of the volume change. This leads to a relative uncertainty in cv of 0.08%.
The energy applied to the calorimeter is the integral of the product of voltage and current from the initial to the final heating time. Voltage and current are measured 80 times during a heating interval of 1 K. The measurements of the electrical quantities have a relative uncertainty of 0.02%. However, we must account for the effect of radiation heat losses or gains which occur when a spurious lag of the controller leads to a small temperature difference of about 10 -2 K between bomb and radiation shield. Since heat transfer by radiation is proportional to T14-T24 @ 4T 3DT, we would expect radiation losses to substantially increase with the bomb temperature, and the losses may be different from the sample and from the reference bombs. Therefore, the uncertainty in the applied heat DQ is evaluated to be 0.5 J×K-1.
The energy difference DQ0 applied to the empty calorimeter has been measured in repeated experiments and fitted to a function of temperature. Its uncertainty is less than 0.5 J×K-1. Its influence on the uncertainty of the heat capacity is relatively small, because the ratio of the total heat DQ to the heat applied to the empty calorimeter DQ0 ranges from 100 J to 600 J. The mass of each sample was determined within 0.01% by differential weighings before and after trapping the sample. The density calculated from this mass and the bomb volume has a relative uncertainty of approximately 0.2%. For pressures, the uncertainty of the gauge of 7 kPa is added to the cross term for the pressure derivative in the change-of-volume work adjustment. However, neither the uncertainty of p nor r contributes appreciably to the combined uncertainty for the measured heat capacity. The relative uncertainty of cv is determined to be 0.3%, by combining the various sources of experimental uncertainty using a root-sum-of-squares formula.
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