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Experimental investigation of thermal contact conductance at low temperature based on fractal description
International Communications in Heat and Mass Transfer 33 (2006) 811 – 818 www.elsevier.com/locate/ichmt
Experimental investigation of thermal contact conductance at low temperature based on fractal description☆ Ruiping Xu a , Haidong Feng a , Lanping Zhao b , Lie Xu a,⁎ a
Institute of Refrigeration and Cryogenics Engineering, Shanghai Jiao Tong University, Shanghai, PR China b Thermal Engineering Department, Tongji University, Shanghai, PR China Available online 4 April 2006
Abstract An experimental investigation of thermal contact conductance was conducted with pressed pairs of aluminum alloy 5052 and stainless steel 304 over the low temperature range from 155 to 210 K, with nominal contact pressure from 1 to 7 MPa. The contact surfaces were prepared through bead blasting and characterized with the fractal dimension D and the parameter G of the Weierstrass–Mandelbrot function. The range of fractal dimension is 1.59–1.86 for aluminum and 1.56–1.92 for stainless steel. And the parameter G is in the magnitude of 10−7 m. From the measured results, thermal contact conductance over this temperature range (155–210 K) is less than that near or above room temperature (T > 300 K). The load sensitivity at low temperature is less than that at room temperature. The smaller fractal dimension D characterizes the rougher surface when G is on the same magnitude and results in the smaller value of the contact conductance and insensitivity to the contact pressure. © 2006 Elsevier Ltd. All rights reserved. Keywords: Thermal contact conductance; Low temperature; Fractal dimension
1. Introduction The knowledge of the thermal contact resistance at low temperatures is a basic requirement for many cryogenic technical constructions, thermal insulation and scientific experimental instruments worked over this temperature range. It is necessary to determine the heat transfer across an interface as correctly as possible in order to accurately evaluate the overall thermal performance of various technical systems. Since the process of heat transfer across an interface is a very complex problem depending on the thermo-physical properties of the contact materials, the surface topography and the eventually chosen intermediate layer such as vacuum, air or other fluid, the thermal contact resistance is often exceeded by a factor of 10 or more of the one calculated according to the existing theories. Thus, the experimental investigation is still an effective method to study the contact resistance for practical application. Much of the published experimental works describe measurements near or above room temperature (T > 300 K) or in or near to the liquid helium range (T < 4 K) [1]. However, in the wide intermediate temperature range of 4 to 300 K, only limited research work has just been reported [1–5] and a comparison of the experimental and theoretical data hardly exists for this sub☆
Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. E-mail address: [email protected] (L. Xu).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.023
812
R. Xu et al. / International Communications in Heat and Mass Transfer 33 (2006) 811–818
Nomenclature D De E E′ G Ge H h k P q T σ σe
Fractal dimension Equivalent fractal dimension Elastic modulus ((1 − ν12) / E1 + (1 − ν22) / E2)− 1 Surface characterization parameter Equivalent surface characterization parameter Hardness Thermal contact conductance Thermal conductivity Contact pressure Heat flux across the interface Temperature R.m.s roughness (σ12 + σ22)0.5
ambient temperature range. In addition, traditional surface topography analysis of thermal contact resistance is based on the statistical roughness parameters such as standard deviation of height, slope and curvature, which are always dependent of length scale and resolution of the instrument. In order to obtain the interesting database of contact conductance over this temperature range for designers of cryogenic mission, the present work investigates the thermal contact resistance of stainless steel 304 and aluminum alloy 5052 with experiment over the range from 155 to 210 K. In this investigation the surface topography of aluminum and stainless steel specimens is characterized with the fractal dimension D and characterization parameter G of Weierstrass–Mandelbrot function and its effect on thermal contact conductance is discussed. 2. Experiment The experiment setup consists of a copper heater block, upper and lower specimens, upper and lower heat flux meters and the heat sink, which are all installed in the vacuum chamber, as shown in Fig. 1. The heat sink is an insulated hollow stainless steel cylinder that is full of the liquid nitrogen. The ring type heater can provide different power by adjusting the voltage. Fiberglass reinforced plastic load shaft is placed on the heater block to avoid axial heat losses. The load can be applied to the load shaft and the contact surfaces by a lever and hanging weight arrangement and the load cell measures its value. Stainless steel bellow is used to facilitate vertical displacement of the shaft and seal the vacuum chamber. The polished heat radiation shield surrounds the specimens and the heat flux meters. The chamber is evacuated by the mechanical pump and diffusion pump in the experiment. The axial temperature gradient measurements are made by 16 calibrated T-type thermocouples and Fluke data acquisition system. The test specimens are made from aluminum alloy 5052 (Al5052) and stainless steel 304 (SS304). These specimens were all cut from the same cylindrical metal rods and machined to a size of 50 mm in length and 10 mm in diameter. The specimens were fine turned on both end faces. Each specimen has four holes of 1 mm in diameter at 10 mm intervals. Each of holes is 5 mm deep for locating thermocouples. The test specimens of Al5052 and SS304 are divided into five groups respectively. Their contacting surfaces are then blasted in different ways in order to get isotropy contact surfaces with different roughness. The two-dimensional STRA-1 surface profile meter measures the surface topography of the specimens. The fractal dimension is derived from the structure function [6]. Table 1 lists the machining conditions such as blasting beads, blasting pressures and the corresponding bead blasted surface roughness and the fractal dimension. The test specimens are all cleared with acetone and daubed with oil in order to avoid oxidation. Before the experiment, the specimens are cleared with acetone thoroughly again and indium foils are filled in the gaps between heat flux meters and specimens, lower heat flux meter and the heat sink so as to decrease thermal contact resistance. Thermocouples are installed in the holes of the specimens and some greases enhancing heat conduction are jammed into holes to ensure good contact between them. Then the experiment apparatus is assembled as
R. Xu et al. / International Communications in Heat and Mass Transfer 33 (2006) 811–818
813
Fig. 1. Experimental setup.
shown in Fig. 1. Open vacuum pump and sustain pressure in the vacuum chamber to be less than 2 × 10−2 Pa. Supply liquid nitrogen to heat sink and adjust the voltage of heater to adjust the heating power and make temperature of contact surfaces stabilize. Add weight to hanging arrangement, and measure the contact pressure. The values of temperatures are logged in computer through Fluke data acquisition system when fluctuating values of thermocouples do not exceed 0.2 K during half an hour. Thermal contact conductance (reciprocal of thermal contact resistance) is defined as h = q / (ΔT) where q is the axial heat flux through the column of the flux meters and specimens. The temperature jump at the interface, ΔT, is obtained by extrapolating the temperatures along the axis to the interface indirectly. In the experiment the total error due to heat losses is 5.16%. According to the law of error propagation total uncertainty in the measurement that considers the instrument precision and measurement error is estimated to be 11.2%. Table 1 Machining conditions and fractal and statistical roughness parameters of contact surfaces of aluminum 5052 and stainless steel 304 specimens Specimen
Bead type
Al-1
Glass
15
Al-2
Glass
30
Al-3
Glass
50
Al-4
50
SS-1
Quartz gravel 5# Quartz gravel 9# Steel
SS-2
Garnet
SS-3
Steel
1000
SS-4
Glass
50
SS-5
Alumina
Al-5
PB (kPa)
50 1800 550
150
D
De
G (10−7 m)
Ge (10−7 m)
σ (10−6 m)
1.7101 1.7075 1.5683 1.7286 1.5836 1.5762 1.8650 1.8649 1.8749 1.8103 1.5581 1.5657 1.6088 1.6266 1.6231 1.6471 1.9205 1.9186 1.8464 1.8583
1.7092
2.4645 2.7080 2.3922 2.5674 6.8853 1.7597 1.7424 1.9182 5.0807 4.5551 7.7775 5.2560 2.1781 2.5131 3.5725 4.2017 1.4025 1.6113 2.4610 2.4230
5.8397
3.2692 3.9460 5.4212 3.2456 6.1379 7.1418 1.5049 1.6121 3.2562 3.3351 12.6550 12.5552 4.6345 4.2031 6.9713 7.3975 1.0690 1.0523 2.1328 2.1803
1.6219 1.5936 1.8620 1.8417 1.5609 1.6182 1.6351 1.9200 1.8567
5.3449 6.5866 3.7270 8.4917 11.980 4.1252 8.5514 2.7522 5.1120
σe (10−6 m) 5.12 6.32 9.42 2.21 4.66 17.6 6.25 10.16 1.5 3.05
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3. Results and discussion The variation of thermal contact conductance with nominal contact pressure at different low temperatures is presented in Fig. 2. The contact conductance at low temperature increases according to a simple power law function of contact pressure, i.e. h ∼ Pn. The experimental values of the exponent n are listed in Table 2 and they are mainly centralized in the range 0.45–0.63. In former theoretical study near or above room temperature (> 300 K) the power exponent nt always varies between 0.85–0.99. And in M-T [6] fractal model the contact conductance-load relation follows as h ∼ PηD/2 where η ranges from 1 to 1.33 depending on the fractal dimension D. According to the measured fractal dimension of the contact surfaces, the Majumdar power exponent ηD / 2 is calculated. From Table 2, the range of the experimental exponent basically smaller than the theoretical ones, but it approaches the values of the correlation of Laming and Fletcher and Gyorog which are both obtained from a compilation of much available experimental data. Since the value of power exponent n reflects the sensitivity of contact conductance h to the contact pressure P, the contact conductance is less sensitive to the same contact pressure at low temperature than at room temperature. From Fig. 2, temperature is the other factor influencing the value of contact conductance and it gives rise to large contact conductance. This is caused partly by thermal conductivity that increases with the temperature. For example of stainless steel specimen SS-5 shown in Fig. 3, when the interface temperature increases from 155 to 180 or 210 K, thermal conductivity increases to 1.09 or 1.17 times of that at 155 K, and contact conductance increases to 1.21 or 1.45 times. On the other hand, the variation of hardness and elastic modulus with temperature causes the variation of real contact area and further influences the contact
Fig. 2. Variation of contact conductance with nominal contact pressure for SS304 and Al5052. (a) SS304 at 155 K, (b) SS304 at 180 K, (c) SS304 at 210 K, (d) A15052 at 155 K.
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Table 2 The power exponent obtained from the experiment at low temperature and the theoretical prediction Specimen
Experimental power exponent 155 K
180 K
210 K
SS-1 SS-2 SS-3 SS-4 SS-5 Al-1 Al-2 Al-3 Al-4 Al-5
0.59 0.46 0.55 0.64 0.62 0.52 0.47 0.45 0.59 0.56
0.50 0.47 0.50 0.56 0.61
0.50 0.46 0.57 0.55 0.58
ne
ηD / 2
Former theoretical power exponent nt
0.85–1.14 0.81–1.08 0.80–1.06 0.93–1.24 0.92–1.22 0.78–1.04 0.81–1.08 0.82–1.09 0.96–1.28 0.93–1.23
Laming [7] Fletcher and Gyorog [7] Mal'kov [7] Tien [6] Thomas and Probert [8] Mikic [9] Yovanovich [7] Antonetti [10] Copper et al. [7]
0.5 0.56 0.66 0.85 0.72 0.94 0.95 0.95 0.99
conductance. Usually, the hardness and modules of the materials such as stainless steel and aluminum alloy applied widely in cryogenics engineering increase with the temperature decreasing [11]. The effect of hardness and elastic modulus can be described with the rate of them (H / E'). Since the rate H / E' decreases with temperature increasing, which means the material becomes softer, and the real contact area will increase under the same load. As a result, the contact conductance will increase with temperature, as shown in Fig. 4. Therefore, low temperatures give rise to the small thermal conductivity and the large rate of H / E′ that result in the bad contact heat transfer. Thermal conductivity, hardness and elastic modulus affect the thermal contact conductance at low temperature altogether. The geometric structure of rough surfaces is an important factor to influence the thermal contact conductance. Here the surface profile is described with the fractal dimension D and surface characterization parameter G of the Weierstrass–Mandelbrot function. As shown in Fig. 5, this function can simulate the real profile well. The fractal dimension D determines the proportion of high frequency on the whole surface profile. When D is small, the low frequency components are dominant in surface profile. And as D is increased, the high frequency components become comparable with the low frequency ones. Besides, the fractal dimension D also can influence the amplitude of the profile, which decreases with D increasing. The parameter G can only influence the amplitude of the profile that increases with G and its effect on the profile is less than that of the dimension D. Although these two parameters influence the surface profile, the effect of the fractal dimension is prominent especially when the values of G is on the same magnitude. The variation of contact conductance under the different nominal loads with the fractal dimension D is shown in Fig. 6. It is noted that the parameter G is on the same magnitude of 10− 7 m but not of the same value. There is a basic tendency for contact conductance to increase with the fractal dimension, however the variation of aluminum specimens is different from stainless steel. The contact conductance of aluminum specimen increases slowly over the range of 1.64–1.84. Except this range, contact conductance increases with fractal dimension rapidly. There is no slow-variation region for stainless steel and its contact conductance increases with the fractal dimension D except of the case of D = 1.6182. The reason for
Fig. 3. Variation of thermal contact conductance with thermal conductivity (contact pressure = 3 MPa).
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Fig. 4. Variation of thermal contact conductance with H / E′ (contact pressure = 1 MPa).
this abnormality is not clear and it may be ascribed to the experimental error and machining process. The variation difference of stainless steel and aluminum should be investigated further and one of the possible reasons is the different elastic modulus and hardness of these two materials. Moreover, it is observed that small fractal dimension results in the insensitivity of the contact conductance to contact pressure. As listed in Table 2, the surface with small fractal dimension has the small power exponent n. For example, the power exponent of SS-4 surface with D = 1.92 is 0.64 while that of SS-1 with D = 1.56 is 0.59 at 155 K. In addition, small fractal dimension impairs the effect of thermal conductivity and H / E' controlling contact conductance. The specimen SS-1 is hardly affected by thermal conductivity and H / E', as shown in Figs. 3 and 4.
Fig. 5. The effect of the fractal dimension D and parameter G on the simulated surface profile of SS-5. (a) Measured profile D = 1.8567 G = 5.112 x 10−7, (b) Simulated profile D = 1.8567 G = 5.112 x 10−7, (c) Simulated profile D= 1.8567 G=5.112 x 10−5, (d) Simulated profile D = 1.3 G = 5.112 x 10−7.
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Fig. 6. Variation of thermal contact conductance with the fractal dimension at 155 K. (a) A15052, (b) SS304.
In Fig. 7, the measured thermal contact conductance of SS-5 is compared with the theoretical predictions listed in Table 2. From this Figure, the models of Yovanovich, Cooper, and the Mikic plastic model all overestimate the value of the experimental results while the Mikic elastic model underestimates the ones. The prediction of the Thomas and Probert model can agree with the measured conductance since this model is determined from the various experimental data. The fractal M-T model can predict well the measured values of the contact conductance under low contact pressure while it overestimates the ones slightly under high contact pressure.
4. Conclusion Thermal contact conductance of stainless steel 304 and aluminum 5052 is investigated experimentally over the range of 155–210 K. The contact conductance varies with the contact pressure in a power law function. However, its value is less than that near or above room temperatures and basically centralized between 0.45–0.63. Under the same contact pressure, thermal contact conductance at low temperature is less than that at normal temperature. The effect of temperature on contact conductance can ascribe to thermal conductivity and the rate of hardness and elastic modulus
Fig. 7. Comparison between experimental and model results of SS-5 specimen.
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H / E′. When G is on the same magnitude, a small fractal dimension results in the bad contact heat transfer and reduces the sensitivity of contact conductance to contact pressure. Acknowledgement This work has been financially supported by the National 863 Project of China (No. 2 002AA306412). References [1] E. Gmelin, M. Asen-Palmer, M. Reuther, R. Villar, Thermal boundary resistance of mechanical contacts between solids at sub-ambient temperatures, Journal of Physics. D, Applied Physics 32 (1999) R19–R43. [2] J. Yu, A.L. Yee, R.E. Schwall, Thermal conductance of Cu/Cu and Cu/Si interfaces from 85 K to 300 K, Cryogenics 32 (1992) 610–615. [3] P. Kittel, L.J. Salerno, A.L. Spivak, Thermal conductance of pressed bimetallic contacts at liquid nitrogen temperatures, in: C. Rizzuto (Ed.), Proceedings of the Fifteenth International Cryogenic Engineering Conference, Butterworth–Heinemann, UK, 1994, pp. 389–392. [4] R.C. Niemann, J.D. Gonczy, P.E. Phelan, T.H. Nicol, Design and performance of low-thermal-resistance, high-electrical-isolation heat intercept connections, Cryogenics 35 (11) (1995) 829–832. [5] L. Xu, S.L. Zhou, J. Yang, J.M. Xu, The study on the solid thermal contact resistance at low temperatures, in: T. Haruyama, T. Mitsui, K. Yamafuji (Eds.), Proceedings of the Sixteenth International Cryogenic Engineering Conference/International Cryogenic Materials Conference, Elsevier, Oxford, 1997, pp. 625–628. [6] A. Majumdar, C.L. Tien, Fractal network model for contact conductance, ASME Journal of Heat Transfer 113 (1991) 516–525. [7] C.V. Madhusudana, Thermal Contact Conductance, Springer, Berlin, 1995, pp. 35–39. [8] T.R. Thomas, S.D. Probert, Correlations for thermal contact conductance in vacuo, ASME Journal of Heat Transfer 94 (1972) 276–281. [9] B.B. Mikic, Thermal contact conductance: theoretical considerations, International Journal of Heat and Mass Transfer 17 (1974) 205–214. [10] V.W. Antonetti, T.D. Whittle, R.E. Simons, An approximate thermal contact conductance correlation, ASME Electronic Packaging 115 (1993) 131–134. [11] R.P. Reed, A.F. Clark, Materials at Low Temperatures, American Society for Metals, Metals Park, Ohio 44073, 1983, pp. 371–411.
Experimental investigation of thermal contact conductance at low temperature based on fractal description☆ Ruiping Xu a , Haidong Feng a , Lanping Zhao b , Lie Xu a,⁎ a
Institute of Refrigeration and Cryogenics Engineering, Shanghai Jiao Tong University, Shanghai, PR China b Thermal Engineering Department, Tongji University, Shanghai, PR China Available online 4 April 2006
Abstract An experimental investigation of thermal contact conductance was conducted with pressed pairs of aluminum alloy 5052 and stainless steel 304 over the low temperature range from 155 to 210 K, with nominal contact pressure from 1 to 7 MPa. The contact surfaces were prepared through bead blasting and characterized with the fractal dimension D and the parameter G of the Weierstrass–Mandelbrot function. The range of fractal dimension is 1.59–1.86 for aluminum and 1.56–1.92 for stainless steel. And the parameter G is in the magnitude of 10−7 m. From the measured results, thermal contact conductance over this temperature range (155–210 K) is less than that near or above room temperature (T > 300 K). The load sensitivity at low temperature is less than that at room temperature. The smaller fractal dimension D characterizes the rougher surface when G is on the same magnitude and results in the smaller value of the contact conductance and insensitivity to the contact pressure. © 2006 Elsevier Ltd. All rights reserved. Keywords: Thermal contact conductance; Low temperature; Fractal dimension
1. Introduction The knowledge of the thermal contact resistance at low temperatures is a basic requirement for many cryogenic technical constructions, thermal insulation and scientific experimental instruments worked over this temperature range. It is necessary to determine the heat transfer across an interface as correctly as possible in order to accurately evaluate the overall thermal performance of various technical systems. Since the process of heat transfer across an interface is a very complex problem depending on the thermo-physical properties of the contact materials, the surface topography and the eventually chosen intermediate layer such as vacuum, air or other fluid, the thermal contact resistance is often exceeded by a factor of 10 or more of the one calculated according to the existing theories. Thus, the experimental investigation is still an effective method to study the contact resistance for practical application. Much of the published experimental works describe measurements near or above room temperature (T > 300 K) or in or near to the liquid helium range (T < 4 K) [1]. However, in the wide intermediate temperature range of 4 to 300 K, only limited research work has just been reported [1–5] and a comparison of the experimental and theoretical data hardly exists for this sub☆
Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. E-mail address: [email protected] (L. Xu).
0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.02.023
812
R. Xu et al. / International Communications in Heat and Mass Transfer 33 (2006) 811–818
Nomenclature D De E E′ G Ge H h k P q T σ σe
Fractal dimension Equivalent fractal dimension Elastic modulus ((1 − ν12) / E1 + (1 − ν22) / E2)− 1 Surface characterization parameter Equivalent surface characterization parameter Hardness Thermal contact conductance Thermal conductivity Contact pressure Heat flux across the interface Temperature R.m.s roughness (σ12 + σ22)0.5
ambient temperature range. In addition, traditional surface topography analysis of thermal contact resistance is based on the statistical roughness parameters such as standard deviation of height, slope and curvature, which are always dependent of length scale and resolution of the instrument. In order to obtain the interesting database of contact conductance over this temperature range for designers of cryogenic mission, the present work investigates the thermal contact resistance of stainless steel 304 and aluminum alloy 5052 with experiment over the range from 155 to 210 K. In this investigation the surface topography of aluminum and stainless steel specimens is characterized with the fractal dimension D and characterization parameter G of Weierstrass–Mandelbrot function and its effect on thermal contact conductance is discussed. 2. Experiment The experiment setup consists of a copper heater block, upper and lower specimens, upper and lower heat flux meters and the heat sink, which are all installed in the vacuum chamber, as shown in Fig. 1. The heat sink is an insulated hollow stainless steel cylinder that is full of the liquid nitrogen. The ring type heater can provide different power by adjusting the voltage. Fiberglass reinforced plastic load shaft is placed on the heater block to avoid axial heat losses. The load can be applied to the load shaft and the contact surfaces by a lever and hanging weight arrangement and the load cell measures its value. Stainless steel bellow is used to facilitate vertical displacement of the shaft and seal the vacuum chamber. The polished heat radiation shield surrounds the specimens and the heat flux meters. The chamber is evacuated by the mechanical pump and diffusion pump in the experiment. The axial temperature gradient measurements are made by 16 calibrated T-type thermocouples and Fluke data acquisition system. The test specimens are made from aluminum alloy 5052 (Al5052) and stainless steel 304 (SS304). These specimens were all cut from the same cylindrical metal rods and machined to a size of 50 mm in length and 10 mm in diameter. The specimens were fine turned on both end faces. Each specimen has four holes of 1 mm in diameter at 10 mm intervals. Each of holes is 5 mm deep for locating thermocouples. The test specimens of Al5052 and SS304 are divided into five groups respectively. Their contacting surfaces are then blasted in different ways in order to get isotropy contact surfaces with different roughness. The two-dimensional STRA-1 surface profile meter measures the surface topography of the specimens. The fractal dimension is derived from the structure function [6]. Table 1 lists the machining conditions such as blasting beads, blasting pressures and the corresponding bead blasted surface roughness and the fractal dimension. The test specimens are all cleared with acetone and daubed with oil in order to avoid oxidation. Before the experiment, the specimens are cleared with acetone thoroughly again and indium foils are filled in the gaps between heat flux meters and specimens, lower heat flux meter and the heat sink so as to decrease thermal contact resistance. Thermocouples are installed in the holes of the specimens and some greases enhancing heat conduction are jammed into holes to ensure good contact between them. Then the experiment apparatus is assembled as
R. Xu et al. / International Communications in Heat and Mass Transfer 33 (2006) 811–818
813
Fig. 1. Experimental setup.
shown in Fig. 1. Open vacuum pump and sustain pressure in the vacuum chamber to be less than 2 × 10−2 Pa. Supply liquid nitrogen to heat sink and adjust the voltage of heater to adjust the heating power and make temperature of contact surfaces stabilize. Add weight to hanging arrangement, and measure the contact pressure. The values of temperatures are logged in computer through Fluke data acquisition system when fluctuating values of thermocouples do not exceed 0.2 K during half an hour. Thermal contact conductance (reciprocal of thermal contact resistance) is defined as h = q / (ΔT) where q is the axial heat flux through the column of the flux meters and specimens. The temperature jump at the interface, ΔT, is obtained by extrapolating the temperatures along the axis to the interface indirectly. In the experiment the total error due to heat losses is 5.16%. According to the law of error propagation total uncertainty in the measurement that considers the instrument precision and measurement error is estimated to be 11.2%. Table 1 Machining conditions and fractal and statistical roughness parameters of contact surfaces of aluminum 5052 and stainless steel 304 specimens Specimen
Bead type
Al-1
Glass
15
Al-2
Glass
30
Al-3
Glass
50
Al-4
50
SS-1
Quartz gravel 5# Quartz gravel 9# Steel
SS-2
Garnet
SS-3
Steel
1000
SS-4
Glass
50
SS-5
Alumina
Al-5
PB (kPa)
50 1800 550
150
D
De
G (10−7 m)
Ge (10−7 m)
σ (10−6 m)
1.7101 1.7075 1.5683 1.7286 1.5836 1.5762 1.8650 1.8649 1.8749 1.8103 1.5581 1.5657 1.6088 1.6266 1.6231 1.6471 1.9205 1.9186 1.8464 1.8583
1.7092
2.4645 2.7080 2.3922 2.5674 6.8853 1.7597 1.7424 1.9182 5.0807 4.5551 7.7775 5.2560 2.1781 2.5131 3.5725 4.2017 1.4025 1.6113 2.4610 2.4230
5.8397
3.2692 3.9460 5.4212 3.2456 6.1379 7.1418 1.5049 1.6121 3.2562 3.3351 12.6550 12.5552 4.6345 4.2031 6.9713 7.3975 1.0690 1.0523 2.1328 2.1803
1.6219 1.5936 1.8620 1.8417 1.5609 1.6182 1.6351 1.9200 1.8567
5.3449 6.5866 3.7270 8.4917 11.980 4.1252 8.5514 2.7522 5.1120
σe (10−6 m) 5.12 6.32 9.42 2.21 4.66 17.6 6.25 10.16 1.5 3.05
814
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3. Results and discussion The variation of thermal contact conductance with nominal contact pressure at different low temperatures is presented in Fig. 2. The contact conductance at low temperature increases according to a simple power law function of contact pressure, i.e. h ∼ Pn. The experimental values of the exponent n are listed in Table 2 and they are mainly centralized in the range 0.45–0.63. In former theoretical study near or above room temperature (> 300 K) the power exponent nt always varies between 0.85–0.99. And in M-T [6] fractal model the contact conductance-load relation follows as h ∼ PηD/2 where η ranges from 1 to 1.33 depending on the fractal dimension D. According to the measured fractal dimension of the contact surfaces, the Majumdar power exponent ηD / 2 is calculated. From Table 2, the range of the experimental exponent basically smaller than the theoretical ones, but it approaches the values of the correlation of Laming and Fletcher and Gyorog which are both obtained from a compilation of much available experimental data. Since the value of power exponent n reflects the sensitivity of contact conductance h to the contact pressure P, the contact conductance is less sensitive to the same contact pressure at low temperature than at room temperature. From Fig. 2, temperature is the other factor influencing the value of contact conductance and it gives rise to large contact conductance. This is caused partly by thermal conductivity that increases with the temperature. For example of stainless steel specimen SS-5 shown in Fig. 3, when the interface temperature increases from 155 to 180 or 210 K, thermal conductivity increases to 1.09 or 1.17 times of that at 155 K, and contact conductance increases to 1.21 or 1.45 times. On the other hand, the variation of hardness and elastic modulus with temperature causes the variation of real contact area and further influences the contact
Fig. 2. Variation of contact conductance with nominal contact pressure for SS304 and Al5052. (a) SS304 at 155 K, (b) SS304 at 180 K, (c) SS304 at 210 K, (d) A15052 at 155 K.
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Table 2 The power exponent obtained from the experiment at low temperature and the theoretical prediction Specimen
Experimental power exponent 155 K
180 K
210 K
SS-1 SS-2 SS-3 SS-4 SS-5 Al-1 Al-2 Al-3 Al-4 Al-5
0.59 0.46 0.55 0.64 0.62 0.52 0.47 0.45 0.59 0.56
0.50 0.47 0.50 0.56 0.61
0.50 0.46 0.57 0.55 0.58
ne
ηD / 2
Former theoretical power exponent nt
0.85–1.14 0.81–1.08 0.80–1.06 0.93–1.24 0.92–1.22 0.78–1.04 0.81–1.08 0.82–1.09 0.96–1.28 0.93–1.23
Laming [7] Fletcher and Gyorog [7] Mal'kov [7] Tien [6] Thomas and Probert [8] Mikic [9] Yovanovich [7] Antonetti [10] Copper et al. [7]
0.5 0.56 0.66 0.85 0.72 0.94 0.95 0.95 0.99
conductance. Usually, the hardness and modules of the materials such as stainless steel and aluminum alloy applied widely in cryogenics engineering increase with the temperature decreasing [11]. The effect of hardness and elastic modulus can be described with the rate of them (H / E'). Since the rate H / E' decreases with temperature increasing, which means the material becomes softer, and the real contact area will increase under the same load. As a result, the contact conductance will increase with temperature, as shown in Fig. 4. Therefore, low temperatures give rise to the small thermal conductivity and the large rate of H / E′ that result in the bad contact heat transfer. Thermal conductivity, hardness and elastic modulus affect the thermal contact conductance at low temperature altogether. The geometric structure of rough surfaces is an important factor to influence the thermal contact conductance. Here the surface profile is described with the fractal dimension D and surface characterization parameter G of the Weierstrass–Mandelbrot function. As shown in Fig. 5, this function can simulate the real profile well. The fractal dimension D determines the proportion of high frequency on the whole surface profile. When D is small, the low frequency components are dominant in surface profile. And as D is increased, the high frequency components become comparable with the low frequency ones. Besides, the fractal dimension D also can influence the amplitude of the profile, which decreases with D increasing. The parameter G can only influence the amplitude of the profile that increases with G and its effect on the profile is less than that of the dimension D. Although these two parameters influence the surface profile, the effect of the fractal dimension is prominent especially when the values of G is on the same magnitude. The variation of contact conductance under the different nominal loads with the fractal dimension D is shown in Fig. 6. It is noted that the parameter G is on the same magnitude of 10− 7 m but not of the same value. There is a basic tendency for contact conductance to increase with the fractal dimension, however the variation of aluminum specimens is different from stainless steel. The contact conductance of aluminum specimen increases slowly over the range of 1.64–1.84. Except this range, contact conductance increases with fractal dimension rapidly. There is no slow-variation region for stainless steel and its contact conductance increases with the fractal dimension D except of the case of D = 1.6182. The reason for
Fig. 3. Variation of thermal contact conductance with thermal conductivity (contact pressure = 3 MPa).
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Fig. 4. Variation of thermal contact conductance with H / E′ (contact pressure = 1 MPa).
this abnormality is not clear and it may be ascribed to the experimental error and machining process. The variation difference of stainless steel and aluminum should be investigated further and one of the possible reasons is the different elastic modulus and hardness of these two materials. Moreover, it is observed that small fractal dimension results in the insensitivity of the contact conductance to contact pressure. As listed in Table 2, the surface with small fractal dimension has the small power exponent n. For example, the power exponent of SS-4 surface with D = 1.92 is 0.64 while that of SS-1 with D = 1.56 is 0.59 at 155 K. In addition, small fractal dimension impairs the effect of thermal conductivity and H / E' controlling contact conductance. The specimen SS-1 is hardly affected by thermal conductivity and H / E', as shown in Figs. 3 and 4.
Fig. 5. The effect of the fractal dimension D and parameter G on the simulated surface profile of SS-5. (a) Measured profile D = 1.8567 G = 5.112 x 10−7, (b) Simulated profile D = 1.8567 G = 5.112 x 10−7, (c) Simulated profile D= 1.8567 G=5.112 x 10−5, (d) Simulated profile D = 1.3 G = 5.112 x 10−7.
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Fig. 6. Variation of thermal contact conductance with the fractal dimension at 155 K. (a) A15052, (b) SS304.
In Fig. 7, the measured thermal contact conductance of SS-5 is compared with the theoretical predictions listed in Table 2. From this Figure, the models of Yovanovich, Cooper, and the Mikic plastic model all overestimate the value of the experimental results while the Mikic elastic model underestimates the ones. The prediction of the Thomas and Probert model can agree with the measured conductance since this model is determined from the various experimental data. The fractal M-T model can predict well the measured values of the contact conductance under low contact pressure while it overestimates the ones slightly under high contact pressure.
4. Conclusion Thermal contact conductance of stainless steel 304 and aluminum 5052 is investigated experimentally over the range of 155–210 K. The contact conductance varies with the contact pressure in a power law function. However, its value is less than that near or above room temperatures and basically centralized between 0.45–0.63. Under the same contact pressure, thermal contact conductance at low temperature is less than that at normal temperature. The effect of temperature on contact conductance can ascribe to thermal conductivity and the rate of hardness and elastic modulus
Fig. 7. Comparison between experimental and model results of SS-5 specimen.
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H / E′. When G is on the same magnitude, a small fractal dimension results in the bad contact heat transfer and reduces the sensitivity of contact conductance to contact pressure. Acknowledgement This work has been financially supported by the National 863 Project of China (No. 2 002AA306412). References [1] E. Gmelin, M. Asen-Palmer, M. Reuther, R. Villar, Thermal boundary resistance of mechanical contacts between solids at sub-ambient temperatures, Journal of Physics. D, Applied Physics 32 (1999) R19–R43. [2] J. Yu, A.L. Yee, R.E. Schwall, Thermal conductance of Cu/Cu and Cu/Si interfaces from 85 K to 300 K, Cryogenics 32 (1992) 610–615. [3] P. Kittel, L.J. Salerno, A.L. Spivak, Thermal conductance of pressed bimetallic contacts at liquid nitrogen temperatures, in: C. Rizzuto (Ed.), Proceedings of the Fifteenth International Cryogenic Engineering Conference, Butterworth–Heinemann, UK, 1994, pp. 389–392. [4] R.C. Niemann, J.D. Gonczy, P.E. Phelan, T.H. Nicol, Design and performance of low-thermal-resistance, high-electrical-isolation heat intercept connections, Cryogenics 35 (11) (1995) 829–832. [5] L. Xu, S.L. Zhou, J. Yang, J.M. Xu, The study on the solid thermal contact resistance at low temperatures, in: T. Haruyama, T. Mitsui, K. Yamafuji (Eds.), Proceedings of the Sixteenth International Cryogenic Engineering Conference/International Cryogenic Materials Conference, Elsevier, Oxford, 1997, pp. 625–628. [6] A. Majumdar, C.L. Tien, Fractal network model for contact conductance, ASME Journal of Heat Transfer 113 (1991) 516–525. [7] C.V. Madhusudana, Thermal Contact Conductance, Springer, Berlin, 1995, pp. 35–39. [8] T.R. Thomas, S.D. Probert, Correlations for thermal contact conductance in vacuo, ASME Journal of Heat Transfer 94 (1972) 276–281. [9] B.B. Mikic, Thermal contact conductance: theoretical considerations, International Journal of Heat and Mass Transfer 17 (1974) 205–214. [10] V.W. Antonetti, T.D. Whittle, R.E. Simons, An approximate thermal contact conductance correlation, ASME Electronic Packaging 115 (1993) 131–134. [11] R.P. Reed, A.F. Clark, Materials at Low Temperatures, American Society for Metals, Metals Park, Ohio 44073, 1983, pp. 371–411.