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Thermal conductivity of subcooled liquid oxygen
Cryogenics 45 (2005) 620–625 www.elsevier.com/locate/cryogenics
Thermal conductivity of subcooled liquid oxygen D. Celik a
a,*
, S.W. Van Sciver
a,b
National High Magnetic Field Laboratory, Florida State University, 1800 E Paul Dirac Drive, Tallahassee, FL 32310, United States b Mechanical Engineering Department, FAMU-FSU College of Engineering, Tallahassee, FL 32310, United States Received 16 November 2004; received in revised form 23 November 2004; accepted 17 May 2005
Abstract The thermal conductivity of liquid oxygen below 80 K and pressures up to 1 MPa has been measured using a horizontal, guarded, flat-plate calorimeter. The working equation of the calorimeter is based on the one-dimensional FourierÕs law. The gap between the calorimeter plates was measured in situ from a capacitance measurement. The cooling power to the calorimeter is provided by a twostage Gifford–McMahan cryocooler. The absolute temperatures are measured using platinum resistance thermometers. The results are compared to existing data and analytical models. 2005 Elsevier Ltd. All rights reserved. Keywords: Oxygen (B); Thermal conductivity (C); Calorimeters (D)
1. Introduction Liquid oxygen is the oxidizer for spacecraft engines. It is also used to provide breathing air to the astronauts. The liquid form is also more suitable for storing large quantities for industrial applications, which includes but is not limited to steel blast furnaces, glass making industry and a number of chemical processes. The transport and thermodynamic properties of liquid oxygen have been studied extensively. However, available thermal conductivity data, which are needed when calculating heat transport processes such as temperature distribution in storage vessels, are very limited below about 77 K and non-existent below the freezing point of nitrogen [1–3]. In this paper, we report the thermal conductivity measurement results on subcooled liquid oxygen between 55 and 81 K at pressures up to 1 MPa. The results are compared with the available
*
Corresponding author. Tel.: +1 850 644 2574; fax: +1 850 644 0867. E-mail address: [email protected] (D. Celik). 0011-2275/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2005.05.004
experimental data in the literature and analytical models developed at NIST [4].
2. Description of the experimental apparatus The apparatus used in these measurements is described in Ref. [5]. However, some modifications have been made since that article was published and, from the point of completeness, we describe the modified apparatus in detail in the following paragraphs. The design of the apparatus is based on the guarded, horizontal flat-plate calorimeter [6]; however, we have made two major modifications from the original apparatus in order to eliminate the adjustments to account for the thermal conduction in the calorimeter material and to eliminate the effect of the pressure on the gap between the calorimeter plates. In order to eliminate the adjustment due to the thermal conduction in plate material, we have placed the thermometers directly under the surfaces that are in contact with the liquid. The distance between the center of the thermometers and the plate surfaces are less than 1.7 mm. Combined with the high
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
621
thermal conductivity of copper used for the calorimeter plates, this short distance keeps the required adjustment to the temperature reading due to the conduction in the plate material below 0.001% in the temperature range of the experiments. The effect of pressure on the gap between the plates is eliminated by inserting a separate plate rather than using the cell body as the cold plate. The addition of this separate plate also allows us to use a capacitance meter to measure the gap accurately. The schematic of the calorimeter is given in Fig. 1. A hot plate is formed by sandwiching a film heater with a 100 X resistance between two 76.6 mm diameter copper discs with a thickness of 6.4 mm. Discs are bolted together using #0–80 stainless steel screws. This construction, however, caused a temperature difference between the upper and the lower parts of the hot plate when the heater is energized, as discussed in the next section. A platinum resistance thermometer [7] is inserted in each disk through 2.0 mm diameter radial holes that extends from the side to about 4 mm beyond the center. The center of the radial hole is 1.7 mm below the surface of the discs that is in contact with the liquid. The hot plate is attached to the first thermal guard, which is attached to the second thermal guard each using three 2-mm-diameter G-10 supports and secured with epoxy. The assembly process is done on a thick glass surface to ensure that the components are on the same plane. Both guards are made of copper and the inner diameter of the first guard is 80.0 mm. In addition to a film heater that is sandwiched between the guard body and a copper disc, each guard has a wire heater wrapped around the body. The first thermal guard has two platinum resistance thermometers: one at the center of the top, inserted similar to the hot plate thermometers and the second one is on the side of the guard, which was utilized to verify a uniform temperature distribution in the guard. The second thermal guard has only one platinum resistance thermometer at the center of the top. This whole assembly is supported by three pieces of silica glass on the copper cold plate. The cold plate is placed at the bottom of the measurement cell and elec-
trically insulated from it by means of a piece of Kapton tape for the capacitance measurement, as explained in the next section. The cold plate has a platinum resistance thermometer installed similar to those in the hot plate. The thermometer and heaters are installed using Apiezon-H [7] grease to ensure good thermal contact. The thermometer lead wires are wrapped around their respective parts to minimize heat conduction along the length. All the lead wires are formed from twisted pairs to minimize the electronic noise. The hot plate assembly and the cold plate are placed inside the measurement cell, made of Glidcop Al-25 [8]. The cooling power is provided by a two-stage model GB-37 Gifford–McMahan cryocooler by Cryomech [9]. Compared to other cryogens as the cooling source, the addition of the cryocooler enables a very broad temperature range for the experiments. The temperature of the cell is controlled with a temperature controller, which utilizes a silicon diode temperature sensor in the feedback loop to control the power input to a film heater installed on the thermal link between the cryocooler stage and the measurement cell, see Fig. 2. Gaseous oxygen is condensed inside the cell and pressurized up to 1 MPa using helium gas during the measurements. The solubility of helium in liquid oxygen is very small, less than 0.04 mol% in the range of the experiments [10]. The purity of the gases used for the measurements is 99.996% and contain the following [11]: water < 1 ppm, methane < 0.5 ppm, carbon monoxide < 1 ppm, carbon dioxide < 1 ppm, argon < 20 ppm, and nitrogen < 10 ppm. A cold trap at 77 K is utilized to minimize the impurities; however, some trace
Fig. 1. The schematic of the guarded, flat-plate calorimeter.
Fig. 2. The schematic of the experimental cell.
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D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
amount of nitrogen may still be present together with oxygen in the experimental cell.
0.4 0.3 0.2
The temperature is measured at six different locations using platinum resistance thermometers. The thermometers were calibrated by the manufacturer against the International Temperature Scale of 1990 (ITS-90). Typical accuracies of these calibrated thermometers are reported to be ±12 mK in the temperature range of measurements [7]. The resistance of each thermometer is monitored using a Lakeshore model 370 A/C resistance bridge operating at 13.7 Hz, and a model 3716L low resistance scanner board. The A/C bridge has an accuracy of ±0.03% and a resolution of 40 lX over the operating range (20 X resistance range and 316 lA current) [7]. Pressure in the measurement cell is measured by an MKS model 221D differential pressure guage, which has a resolution of 200 Pa and an accuracy of 0.5% of the read value [12]. The gauge is calibrated in our laboratory against a calibrated US Gauge series 1400 Solfrunt 6’’ test guage [13,14]. The capacitance measurement to determine the gap between the calorimeter plates is performed using an Andeen–Hagerling model 2550 A capacitance bridge. The bridge has an accuracy of 5 ppm, and a resolution of 0.1 aF (107 pF) at its 1 kHz operating frequency [15]. The error in the cell constant caused by the accuracy of the capacitance bridge is about 5 · 104%, which does not cause any measurable error in the thermal conductivity.
0.1
∆T (mK)
3. Temperature, pressure and capacitance measurements
0 -0.1 -0.2 -0.3
60
70
80
T (K)
Fig. 3. The comparison of thermometer calibrations against the cold plate thermometer.
In order to measure the thermal conductivity, a known amount of heat is supplied to the hot plate. The exact amount of heat is determined by multiplying the current going into the heater and the voltage drop along the heater. Both quantities are monitored constantly. Throughout the experiments the nominal value of the heating power was of the order 0.5 W with a standard deviation of less than 105 W. During the measurements the temperature of the upper section of the hot plate is matched by both thermal guards to make sure that all heat generated by the hot plate heater goes to the cold plate. However, a tem-
4. Experimental details
0.250
0.200
0.150
∆T (K)
Determining the thermal conductivity requires the measurement of very small temperature differences between the hot and cold plates, as well as the hot plate and the thermal guards. Therefore, before measuring the thermal conductivity, we have compared the calibration of the thermometers against each other at several points in the temperature range of interest. All thermometer readings were referred to the cold plate thermometer and the observed differences incorporated in the calibration of the other thermometers. Fig. 3 shows the differences between the various thermometer readings and the cold plate thermometer in mK. The thermometer installed in the second thermal guard displays the maximum deviations from the cold plate thermometer. However, it does not have any impact on the measurement results since the second thermal guard is used only to help stabilize the temperature of the first thermal guard.
Hot Plate #1 Hot Plate #2 Thermal Guard-1 Thermal Guard-2
-0.4
0.100
0.050
0.000 0.0
0.1
0.2
0.3
0.4
0.5
P (W)
Fig. 4. Heat load on the hot plate ( : measured temperature difference between the hot plate and the cold plate, : measured temperature difference between the first thermal guard and the hot plate).
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
Q DT Ad DT d
k¼A
ð1Þ
In the above equation, k is the thermal conductivity of the liquid in W/m-K, Q is the heat applied to the hot plate heater in W, DT is the temperature difference between the hot plate lower part and the cold plate in K, A/d (termed as the cell constant) is the ratio of the hot plate area to the gap between the hot plate and the cold plate in m, A*/d* is the ratio of the hot plate lower part circumferential area to the gap between the hot plate and the first thermal guard in m, which is taken as 0.8911 m during the calculations and DT* is the temperature difference between the hot plate lower part and the first thermal guard. The cell constant (A/d) is determined from a capacitance measurement by using the hot plate, cold plate and the thermal guards assembly as a three-terminal capacitor. However, the capacitance measurement had to be corrected for the fringing effects around the edges of the capacitor plates [16]. Consequently, the cell constant was obtained from the following: pR2 C 8pR ¼ R log ð2Þ e0 5d d where C is the measured capacitance of the cell in pF, e0(=8.85 pF/m) is the permittivity of free space, and R and d are the radius of the hot plate and the gap between the hot and cold plates in m, respectively. The second term on the right hand side is due to fringe effects. Compared to the ideal case, the fringe effects causes about a 0.7% increase in the cell constant at room temperature. However, both the area of the plate and the gap are affected by the temperature change. Therefore, the above equation cannot be evaluated directly at lower temperatures. On the other hand, the change in the area of the plate is well defined since the thermal contraction of copper as a function of temperature is known. Consequently, after solving iteratively for d from Eq. (2), the cell constant at the operating condition can be determined at lower temperatures. The effect of this correc-
tion on the thermal conductivity of the liquid is less than 0.06%.
5. Results and discussion Our measured values of thermal conductivity for compressed liquid oxygen are given in Table 1 and shown in Fig. 5. The measurement uncertainties are also Table 1 Thermal conductivity measurement results Pressure (MPa)
T (K)
k (W/m-K)
dP (MPa)
dT (K)
dk (W/m-K)
1.0002 1.0003 0.9998 0.9993 1.0022 1.0002
55.9486 60.1572 65.5879 71.3274 75.6650 80.6776
0.19990 0.19472 0.18754 0.17966 0.17340 0.16640
0.00015 0.00014 0.00002 0.00051 0.00005 0.00030
0.00062 0.00056 0.00038 0.00050 0.00044 0.00024
0.000090 0.000065 0.000055 0.000035 0.000038 0.000034
0.7500 0.7493 0.7496 0.7503 0.7498
55.9423 58.8036 70.7689 75.3548 80.6889
0.19966 0.19616 0.18018 0.17357 0.16614
0.00010 0.00052 0.00003 0.00008 0.00015
0.00150 0.00078 0.00043 0.00071 0.00077
0.000015 0.000044 0.000022 0.000021 0.000047
0.5002 0.5002 0.5020 0.5003 0.5003
56.5024 60.1102 71.1511 75.1652 80.6849
0.19880 0.19435 0.17942 0.17370 0.16587
0.00013 0.00020 0.00017 0.00018 0.00011
0.00017 0.00130 0.00025 0.00036 0.00050
0.000026 0.000031 0.000030 0.000025 0.000032
0.2503 0.2507 0.2498 0.2506 0.2504
55.4962 60.7846 71.1504 75.1497 80.5935
0.19979 0.19325 0.17920 0.17345 0.16576
0.00011 0.00002 0.00002 0.00016 0.00022
0.00120 0.00045 0.00012 0.00043 0.00035
0.000020 0.000021 0.000023 0.000022 0.000036
0.200
1.000 0.750 0.500 0.250
0.195
MPa MPa MPa MPa
0.190
k (W/m-K)
perature difference between the hot plate upper and lower parts, which changes linearly with the power input to the hot plate heater is observed. Fig. 4 shows the temperature differences between the hot plate lower part and the cold plate as well as the temperature difference between the first thermal guard and the hot plate lower part at 80.525 ± 0.025 K and 0.250 MPa for three different power inputs. The change in the thermal conductivity as calculated from Eq. (1) by decreasing the power input from 0.5 W to 0.3 W is 0.28%, which indicates that the way the temperature difference is handled in the working equation is appropriate. The working equation of the calorimeter is given by the following equation:
623
0.185
0.180
0.175
0.170
0.165 55
60
65
70
75
80
T(K)
Fig. 5. Thermal conductivity measurement results at constant pressures (lines are curve fits to experimental results).
624
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included in the table. The errors shown in the table are the standard error values (standard deviation of the mean), and are calculated by taking the ratio of the standard deviation of the samples to the square root of the number of samples [17]. The thermal conductivity values are calculated from the working equation of the calorimeter, Eq. (1). The effect of the correction due to heat load from the thermal guard to the cold plate is less than 3.5%. Overall, the percentage error in these thermal conductivity measurements is less than 0.24%. Fig. 6 shows the comparison of the thermal conductivity versus density at 1 MPa pressure presented in this
paper to those of references [4] and [18]. It needs to be noted that the values reported in [18] are not the results of just another experimental study; but rather the results of a correlation to several experimental studies available to the authors of that particular study. The densities are calculated from the correlation reported in [19]. The differences between our results and those of [18] at 1 MPa are 0.03% and 0.04% at 70 and 80 K, respectively. On the other hand, the maximum deviation between our data and the calculations of [4] is less than 0.8% in the range of measurements, Fig. 7.
6. Summary We have performed high precision thermal conductivity measurements of subcooled liquid oxygen in the temperature range from 55 K to 81 K and pressures up to 1 MPa. The precision of the measurements is better than 0.3% and there is a good agreement between our results and existing data as well as analytical models.
Acknowledgement This research has been supported by NASA through the Research Initiative for Florida Universities under the grant NAG3-2751.
References Fig. 6. Comparison of the thermal conductivity of oxygen as a function of density at 1 MPa pressure.
0.195
0.190
k (W/m-K)
0.185
0.180 60 K 70 K 80 K Ref. [18], 70 K Ref. [18], 80 K Ref. [4], 60 K Ref. [4], 70 K Ref. [4], 80 K
0.175
0.170
0.165
0.25
0.50
0.75
1.00
P (MPa)
Fig. 7. Comparison of thermal conductivity of oxygen at constant temperatures.
[1] Verkin BI. Handbook of properties of condensed phases of hydrogen and oxygen. New York: Hemisphere Publishing Co.; 1991 [chapters 3 and 4]. [2] Roder HM. The thermal conductivity of oxygen. J Res Natl Bur Stand 1982;87(4):279–305. [3] Tsederberg NV, Timrot DL. An experimental determination of the thermal conductivity of liquid oxygen. Sov Phys, Techn Phys 1956;1(8):1791–7. [4] Lemmon EW, Jacobsen RT. Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air. Int J Thermophys 2004;25(1):21–69. [5] Kucukomeroglu T, Celik D, VanSciver SW. Thermal conductivity measurements of subcooled oxygen below 80 K. In: Waynert J, editor. Advances in cryogenic engineering, 49b. p. 1123–9. [6] Roder HM, Diller DE. Thermal conductivity of gaseous and liquid hydrogen. J Chem Phys 1970;52(11):5928–49. [7] Information from the supplier, Lake Shore Cryotronics, Inc., 575 McCorcle Boulevard, Westerville, OH 43082. [8] Information from the supplier, OMG Americas, 2601 Week Drive Research Triangle Park, NC 27709, USA. [9] Information from the supplier, Cryomech, 113 Falso Dr., Syracuse, New York 13211, USA. [10] Sinor JE, Kurata F. Solubility of helium in liquid argon, oxygen and carbon monoxide. J Chem Eng Data 1966;11(4):537–9. [11] Information from the supplier, Airgas South, 945 Yulee St., Tallahassee, FL 32304. [12] Information from the supplier, MKS Industries Inc., 90 Industrial Way, Wilmington, MA 01887-4610.
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625 [13] Information from the supplier, US Gauge, 820 Pennsylvania Blvd., Feasterville, PA 19053. [14] Information from the supplier, Benchmark Calibration Laboratory, Inc., 750 Wylly Avenue, Suite 1, Sanford, FL 32773. [15] Information from the supplier, Andeen–Hagerling, Inc., 31200 Bainbridge Road, Cleveland, OH 44139-2231. [16] Landau LD, Lifshitz EM. Electrodynamics of continuous media. Second ed. Oxford: Butterworth-Heinemann; 1984. p. 19.
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[17] Taylor JR. An introduction to error analysis. Second ed. Sausalito: University Science Books; 1997. p. 102. [18] Laesecke A, Krauss R, Stephan K, Wagner W. Transport properties of fluid Oxygen. J Phys Chem Ref Data 1990;19(5): 1089–122. [19] Weber LA. P–V–T, Thermodynamic and related properties of oxygen from the triple point to 300 K at pressures to 33 MN/m2. J Res Natl Bur Stand-A. Phys Chem 1970;74A(1):93–129.
Thermal conductivity of subcooled liquid oxygen D. Celik a
a,*
, S.W. Van Sciver
a,b
National High Magnetic Field Laboratory, Florida State University, 1800 E Paul Dirac Drive, Tallahassee, FL 32310, United States b Mechanical Engineering Department, FAMU-FSU College of Engineering, Tallahassee, FL 32310, United States Received 16 November 2004; received in revised form 23 November 2004; accepted 17 May 2005
Abstract The thermal conductivity of liquid oxygen below 80 K and pressures up to 1 MPa has been measured using a horizontal, guarded, flat-plate calorimeter. The working equation of the calorimeter is based on the one-dimensional FourierÕs law. The gap between the calorimeter plates was measured in situ from a capacitance measurement. The cooling power to the calorimeter is provided by a twostage Gifford–McMahan cryocooler. The absolute temperatures are measured using platinum resistance thermometers. The results are compared to existing data and analytical models. 2005 Elsevier Ltd. All rights reserved. Keywords: Oxygen (B); Thermal conductivity (C); Calorimeters (D)
1. Introduction Liquid oxygen is the oxidizer for spacecraft engines. It is also used to provide breathing air to the astronauts. The liquid form is also more suitable for storing large quantities for industrial applications, which includes but is not limited to steel blast furnaces, glass making industry and a number of chemical processes. The transport and thermodynamic properties of liquid oxygen have been studied extensively. However, available thermal conductivity data, which are needed when calculating heat transport processes such as temperature distribution in storage vessels, are very limited below about 77 K and non-existent below the freezing point of nitrogen [1–3]. In this paper, we report the thermal conductivity measurement results on subcooled liquid oxygen between 55 and 81 K at pressures up to 1 MPa. The results are compared with the available
*
Corresponding author. Tel.: +1 850 644 2574; fax: +1 850 644 0867. E-mail address: [email protected] (D. Celik). 0011-2275/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2005.05.004
experimental data in the literature and analytical models developed at NIST [4].
2. Description of the experimental apparatus The apparatus used in these measurements is described in Ref. [5]. However, some modifications have been made since that article was published and, from the point of completeness, we describe the modified apparatus in detail in the following paragraphs. The design of the apparatus is based on the guarded, horizontal flat-plate calorimeter [6]; however, we have made two major modifications from the original apparatus in order to eliminate the adjustments to account for the thermal conduction in the calorimeter material and to eliminate the effect of the pressure on the gap between the calorimeter plates. In order to eliminate the adjustment due to the thermal conduction in plate material, we have placed the thermometers directly under the surfaces that are in contact with the liquid. The distance between the center of the thermometers and the plate surfaces are less than 1.7 mm. Combined with the high
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
621
thermal conductivity of copper used for the calorimeter plates, this short distance keeps the required adjustment to the temperature reading due to the conduction in the plate material below 0.001% in the temperature range of the experiments. The effect of pressure on the gap between the plates is eliminated by inserting a separate plate rather than using the cell body as the cold plate. The addition of this separate plate also allows us to use a capacitance meter to measure the gap accurately. The schematic of the calorimeter is given in Fig. 1. A hot plate is formed by sandwiching a film heater with a 100 X resistance between two 76.6 mm diameter copper discs with a thickness of 6.4 mm. Discs are bolted together using #0–80 stainless steel screws. This construction, however, caused a temperature difference between the upper and the lower parts of the hot plate when the heater is energized, as discussed in the next section. A platinum resistance thermometer [7] is inserted in each disk through 2.0 mm diameter radial holes that extends from the side to about 4 mm beyond the center. The center of the radial hole is 1.7 mm below the surface of the discs that is in contact with the liquid. The hot plate is attached to the first thermal guard, which is attached to the second thermal guard each using three 2-mm-diameter G-10 supports and secured with epoxy. The assembly process is done on a thick glass surface to ensure that the components are on the same plane. Both guards are made of copper and the inner diameter of the first guard is 80.0 mm. In addition to a film heater that is sandwiched between the guard body and a copper disc, each guard has a wire heater wrapped around the body. The first thermal guard has two platinum resistance thermometers: one at the center of the top, inserted similar to the hot plate thermometers and the second one is on the side of the guard, which was utilized to verify a uniform temperature distribution in the guard. The second thermal guard has only one platinum resistance thermometer at the center of the top. This whole assembly is supported by three pieces of silica glass on the copper cold plate. The cold plate is placed at the bottom of the measurement cell and elec-
trically insulated from it by means of a piece of Kapton tape for the capacitance measurement, as explained in the next section. The cold plate has a platinum resistance thermometer installed similar to those in the hot plate. The thermometer and heaters are installed using Apiezon-H [7] grease to ensure good thermal contact. The thermometer lead wires are wrapped around their respective parts to minimize heat conduction along the length. All the lead wires are formed from twisted pairs to minimize the electronic noise. The hot plate assembly and the cold plate are placed inside the measurement cell, made of Glidcop Al-25 [8]. The cooling power is provided by a two-stage model GB-37 Gifford–McMahan cryocooler by Cryomech [9]. Compared to other cryogens as the cooling source, the addition of the cryocooler enables a very broad temperature range for the experiments. The temperature of the cell is controlled with a temperature controller, which utilizes a silicon diode temperature sensor in the feedback loop to control the power input to a film heater installed on the thermal link between the cryocooler stage and the measurement cell, see Fig. 2. Gaseous oxygen is condensed inside the cell and pressurized up to 1 MPa using helium gas during the measurements. The solubility of helium in liquid oxygen is very small, less than 0.04 mol% in the range of the experiments [10]. The purity of the gases used for the measurements is 99.996% and contain the following [11]: water < 1 ppm, methane < 0.5 ppm, carbon monoxide < 1 ppm, carbon dioxide < 1 ppm, argon < 20 ppm, and nitrogen < 10 ppm. A cold trap at 77 K is utilized to minimize the impurities; however, some trace
Fig. 1. The schematic of the guarded, flat-plate calorimeter.
Fig. 2. The schematic of the experimental cell.
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D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
amount of nitrogen may still be present together with oxygen in the experimental cell.
0.4 0.3 0.2
The temperature is measured at six different locations using platinum resistance thermometers. The thermometers were calibrated by the manufacturer against the International Temperature Scale of 1990 (ITS-90). Typical accuracies of these calibrated thermometers are reported to be ±12 mK in the temperature range of measurements [7]. The resistance of each thermometer is monitored using a Lakeshore model 370 A/C resistance bridge operating at 13.7 Hz, and a model 3716L low resistance scanner board. The A/C bridge has an accuracy of ±0.03% and a resolution of 40 lX over the operating range (20 X resistance range and 316 lA current) [7]. Pressure in the measurement cell is measured by an MKS model 221D differential pressure guage, which has a resolution of 200 Pa and an accuracy of 0.5% of the read value [12]. The gauge is calibrated in our laboratory against a calibrated US Gauge series 1400 Solfrunt 6’’ test guage [13,14]. The capacitance measurement to determine the gap between the calorimeter plates is performed using an Andeen–Hagerling model 2550 A capacitance bridge. The bridge has an accuracy of 5 ppm, and a resolution of 0.1 aF (107 pF) at its 1 kHz operating frequency [15]. The error in the cell constant caused by the accuracy of the capacitance bridge is about 5 · 104%, which does not cause any measurable error in the thermal conductivity.
0.1
∆T (mK)
3. Temperature, pressure and capacitance measurements
0 -0.1 -0.2 -0.3
60
70
80
T (K)
Fig. 3. The comparison of thermometer calibrations against the cold plate thermometer.
In order to measure the thermal conductivity, a known amount of heat is supplied to the hot plate. The exact amount of heat is determined by multiplying the current going into the heater and the voltage drop along the heater. Both quantities are monitored constantly. Throughout the experiments the nominal value of the heating power was of the order 0.5 W with a standard deviation of less than 105 W. During the measurements the temperature of the upper section of the hot plate is matched by both thermal guards to make sure that all heat generated by the hot plate heater goes to the cold plate. However, a tem-
4. Experimental details
0.250
0.200
0.150
∆T (K)
Determining the thermal conductivity requires the measurement of very small temperature differences between the hot and cold plates, as well as the hot plate and the thermal guards. Therefore, before measuring the thermal conductivity, we have compared the calibration of the thermometers against each other at several points in the temperature range of interest. All thermometer readings were referred to the cold plate thermometer and the observed differences incorporated in the calibration of the other thermometers. Fig. 3 shows the differences between the various thermometer readings and the cold plate thermometer in mK. The thermometer installed in the second thermal guard displays the maximum deviations from the cold plate thermometer. However, it does not have any impact on the measurement results since the second thermal guard is used only to help stabilize the temperature of the first thermal guard.
Hot Plate #1 Hot Plate #2 Thermal Guard-1 Thermal Guard-2
-0.4
0.100
0.050
0.000 0.0
0.1
0.2
0.3
0.4
0.5
P (W)
Fig. 4. Heat load on the hot plate ( : measured temperature difference between the hot plate and the cold plate, : measured temperature difference between the first thermal guard and the hot plate).
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
Q DT Ad DT d
k¼A
ð1Þ
In the above equation, k is the thermal conductivity of the liquid in W/m-K, Q is the heat applied to the hot plate heater in W, DT is the temperature difference between the hot plate lower part and the cold plate in K, A/d (termed as the cell constant) is the ratio of the hot plate area to the gap between the hot plate and the cold plate in m, A*/d* is the ratio of the hot plate lower part circumferential area to the gap between the hot plate and the first thermal guard in m, which is taken as 0.8911 m during the calculations and DT* is the temperature difference between the hot plate lower part and the first thermal guard. The cell constant (A/d) is determined from a capacitance measurement by using the hot plate, cold plate and the thermal guards assembly as a three-terminal capacitor. However, the capacitance measurement had to be corrected for the fringing effects around the edges of the capacitor plates [16]. Consequently, the cell constant was obtained from the following: pR2 C 8pR ¼ R log ð2Þ e0 5d d where C is the measured capacitance of the cell in pF, e0(=8.85 pF/m) is the permittivity of free space, and R and d are the radius of the hot plate and the gap between the hot and cold plates in m, respectively. The second term on the right hand side is due to fringe effects. Compared to the ideal case, the fringe effects causes about a 0.7% increase in the cell constant at room temperature. However, both the area of the plate and the gap are affected by the temperature change. Therefore, the above equation cannot be evaluated directly at lower temperatures. On the other hand, the change in the area of the plate is well defined since the thermal contraction of copper as a function of temperature is known. Consequently, after solving iteratively for d from Eq. (2), the cell constant at the operating condition can be determined at lower temperatures. The effect of this correc-
tion on the thermal conductivity of the liquid is less than 0.06%.
5. Results and discussion Our measured values of thermal conductivity for compressed liquid oxygen are given in Table 1 and shown in Fig. 5. The measurement uncertainties are also Table 1 Thermal conductivity measurement results Pressure (MPa)
T (K)
k (W/m-K)
dP (MPa)
dT (K)
dk (W/m-K)
1.0002 1.0003 0.9998 0.9993 1.0022 1.0002
55.9486 60.1572 65.5879 71.3274 75.6650 80.6776
0.19990 0.19472 0.18754 0.17966 0.17340 0.16640
0.00015 0.00014 0.00002 0.00051 0.00005 0.00030
0.00062 0.00056 0.00038 0.00050 0.00044 0.00024
0.000090 0.000065 0.000055 0.000035 0.000038 0.000034
0.7500 0.7493 0.7496 0.7503 0.7498
55.9423 58.8036 70.7689 75.3548 80.6889
0.19966 0.19616 0.18018 0.17357 0.16614
0.00010 0.00052 0.00003 0.00008 0.00015
0.00150 0.00078 0.00043 0.00071 0.00077
0.000015 0.000044 0.000022 0.000021 0.000047
0.5002 0.5002 0.5020 0.5003 0.5003
56.5024 60.1102 71.1511 75.1652 80.6849
0.19880 0.19435 0.17942 0.17370 0.16587
0.00013 0.00020 0.00017 0.00018 0.00011
0.00017 0.00130 0.00025 0.00036 0.00050
0.000026 0.000031 0.000030 0.000025 0.000032
0.2503 0.2507 0.2498 0.2506 0.2504
55.4962 60.7846 71.1504 75.1497 80.5935
0.19979 0.19325 0.17920 0.17345 0.16576
0.00011 0.00002 0.00002 0.00016 0.00022
0.00120 0.00045 0.00012 0.00043 0.00035
0.000020 0.000021 0.000023 0.000022 0.000036
0.200
1.000 0.750 0.500 0.250
0.195
MPa MPa MPa MPa
0.190
k (W/m-K)
perature difference between the hot plate upper and lower parts, which changes linearly with the power input to the hot plate heater is observed. Fig. 4 shows the temperature differences between the hot plate lower part and the cold plate as well as the temperature difference between the first thermal guard and the hot plate lower part at 80.525 ± 0.025 K and 0.250 MPa for three different power inputs. The change in the thermal conductivity as calculated from Eq. (1) by decreasing the power input from 0.5 W to 0.3 W is 0.28%, which indicates that the way the temperature difference is handled in the working equation is appropriate. The working equation of the calorimeter is given by the following equation:
623
0.185
0.180
0.175
0.170
0.165 55
60
65
70
75
80
T(K)
Fig. 5. Thermal conductivity measurement results at constant pressures (lines are curve fits to experimental results).
624
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625
included in the table. The errors shown in the table are the standard error values (standard deviation of the mean), and are calculated by taking the ratio of the standard deviation of the samples to the square root of the number of samples [17]. The thermal conductivity values are calculated from the working equation of the calorimeter, Eq. (1). The effect of the correction due to heat load from the thermal guard to the cold plate is less than 3.5%. Overall, the percentage error in these thermal conductivity measurements is less than 0.24%. Fig. 6 shows the comparison of the thermal conductivity versus density at 1 MPa pressure presented in this
paper to those of references [4] and [18]. It needs to be noted that the values reported in [18] are not the results of just another experimental study; but rather the results of a correlation to several experimental studies available to the authors of that particular study. The densities are calculated from the correlation reported in [19]. The differences between our results and those of [18] at 1 MPa are 0.03% and 0.04% at 70 and 80 K, respectively. On the other hand, the maximum deviation between our data and the calculations of [4] is less than 0.8% in the range of measurements, Fig. 7.
6. Summary We have performed high precision thermal conductivity measurements of subcooled liquid oxygen in the temperature range from 55 K to 81 K and pressures up to 1 MPa. The precision of the measurements is better than 0.3% and there is a good agreement between our results and existing data as well as analytical models.
Acknowledgement This research has been supported by NASA through the Research Initiative for Florida Universities under the grant NAG3-2751.
References Fig. 6. Comparison of the thermal conductivity of oxygen as a function of density at 1 MPa pressure.
0.195
0.190
k (W/m-K)
0.185
0.180 60 K 70 K 80 K Ref. [18], 70 K Ref. [18], 80 K Ref. [4], 60 K Ref. [4], 70 K Ref. [4], 80 K
0.175
0.170
0.165
0.25
0.50
0.75
1.00
P (MPa)
Fig. 7. Comparison of thermal conductivity of oxygen at constant temperatures.
[1] Verkin BI. Handbook of properties of condensed phases of hydrogen and oxygen. New York: Hemisphere Publishing Co.; 1991 [chapters 3 and 4]. [2] Roder HM. The thermal conductivity of oxygen. J Res Natl Bur Stand 1982;87(4):279–305. [3] Tsederberg NV, Timrot DL. An experimental determination of the thermal conductivity of liquid oxygen. Sov Phys, Techn Phys 1956;1(8):1791–7. [4] Lemmon EW, Jacobsen RT. Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air. Int J Thermophys 2004;25(1):21–69. [5] Kucukomeroglu T, Celik D, VanSciver SW. Thermal conductivity measurements of subcooled oxygen below 80 K. In: Waynert J, editor. Advances in cryogenic engineering, 49b. p. 1123–9. [6] Roder HM, Diller DE. Thermal conductivity of gaseous and liquid hydrogen. J Chem Phys 1970;52(11):5928–49. [7] Information from the supplier, Lake Shore Cryotronics, Inc., 575 McCorcle Boulevard, Westerville, OH 43082. [8] Information from the supplier, OMG Americas, 2601 Week Drive Research Triangle Park, NC 27709, USA. [9] Information from the supplier, Cryomech, 113 Falso Dr., Syracuse, New York 13211, USA. [10] Sinor JE, Kurata F. Solubility of helium in liquid argon, oxygen and carbon monoxide. J Chem Eng Data 1966;11(4):537–9. [11] Information from the supplier, Airgas South, 945 Yulee St., Tallahassee, FL 32304. [12] Information from the supplier, MKS Industries Inc., 90 Industrial Way, Wilmington, MA 01887-4610.
D. Celik, S.W. Van Sciver / Cryogenics 45 (2005) 620–625 [13] Information from the supplier, US Gauge, 820 Pennsylvania Blvd., Feasterville, PA 19053. [14] Information from the supplier, Benchmark Calibration Laboratory, Inc., 750 Wylly Avenue, Suite 1, Sanford, FL 32773. [15] Information from the supplier, Andeen–Hagerling, Inc., 31200 Bainbridge Road, Cleveland, OH 44139-2231. [16] Landau LD, Lifshitz EM. Electrodynamics of continuous media. Second ed. Oxford: Butterworth-Heinemann; 1984. p. 19.
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[17] Taylor JR. An introduction to error analysis. Second ed. Sausalito: University Science Books; 1997. p. 102. [18] Laesecke A, Krauss R, Stephan K, Wagner W. Transport properties of fluid Oxygen. J Phys Chem Ref Data 1990;19(5): 1089–122. [19] Weber LA. P–V–T, Thermodynamic and related properties of oxygen from the triple point to 300 K at pressures to 33 MN/m2. J Res Natl Bur Stand-A. Phys Chem 1970;74A(1):93–129.