Effects of thermal conductivity on the thermal contact resistance between non-conforming rough surfaces: An experimental and modeling study

Journal Pre-proofs Effects of Thermal Conductivity on the Thermal Contact Resistance Between Non-conforming Rough Surfaces: An Experimental and Modeling Study Dongmei Bi, Mei Jiang, Huanxin Chen, Shanjian Liu, Yaya Liu PII: DOI: Reference:

S1359-4311(19)32906-0 https://doi.org/10.1016/j.applthermaleng.2020.115037 ATE 115037

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Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

29 April 2019 26 January 2020 2 February 2020

Please cite this article as: D. Bi, M. Jiang, H. Chen, S. Liu, Y. Liu, Effects of Thermal Conductivity on the Thermal Contact Resistance Between Non-conforming Rough Surfaces: An Experimental and Modeling Study, Applied Thermal Engineering (2020), doi: https://doi.org/10.1016/j.applthermaleng.2020.115037

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Effects of Thermal Conductivity on the Thermal Contact Resistance Between Non-conforming Rough Surfaces: An Experimental and Modeling Study Dongmei Bi a, Mei Jiang a, Huanxin Chen b, Shanjian Liu a, *, Yaya Liu a a School b School

ARTICLE

of Agricultural Engineering and Food Science, Shandong University of Technology, Zibo, China

of Energy and Power Engineering，Huazhong University of Science and Technology，Wuhan，China

INFO

Article history:

ABSTRACT An experimental investigation of the thermal contact resistance (TCR) between non-conforming rough surfaces was conducted. Experiments demonstrated that

Received

the TCR of OFC-OFC (Oxygen-Free Copper) and SS304-SS304 (Stainless Steel Received in revised form

304) decreased as the thermal conductivity increased. Similarly, the TCR of AlN-

Accepted

AlN ceramics decreased when the thermal conductivity increased from 300 to

Keywords:

347 W·m-1·K-1 at temperatures ranging from 90 to 140 K. The variation between the TCR and the thermal conductivity first exhibited an increasing trend and then

Thermal conductivity Thermal contact resistance

a decreasing trend. Experiments revealed that the effect of the material with the lower thermal conductivity was dominant, such as OFC-SS304 and OFC-AlN

Non-conforming rough surfaces

ceramics. The TCR of the dissimilar contact materials was influenced

Laser photothermal method

significantly by the material that demonstrated a lower thermal conductivity at 0.35 MPa. The TCR of OFC-OFC and the thermal conductivity exhibited an

Modeling

exponential relationship under low pressure conditions and a polynomial relationship when the external pressure ranged from 0.53 to 0.71 MPa. The relationship between the TCR of OFC-SS304 and the harmonic mean thermal conductivity exhibited an exponential relationship from 0.23 to 0.68 MPa. The results presented in this study provided not only a better understanding of the TCR but also a database for engineering applications.

1. Introduction Thermal Contact Resistance (TCR) can be defined as the ratio between the temperature drop and the average heat from across an interface when two objects are in contact. The TCR is an important parameter in the field of heat conduction because it limits the rate of heat removal from the contact regions. This phenomenon can be observed in numerous studies and applications, including aerospace[1,2], superconductivity[3, 4], high speed semiconductor devices[5, 6],

cryogenic interfaces[7, 8], precision machining[9, 10], etc. Change of TCR at a cryogenic temperature and a high heat

* Corresponding author. Tel.: 86 05332780092. E-mail address:[email protected]

flux is influenced by the contact temperature, surface characteristics, material properties, contact pressure and the interstitial medium present at the interface. The effects of these factors on the TCR at an ambient temperature have been extensively investigated in previous studies. The contact pressure acting on the contact interfaces and the contact temperature are the two external factors that can influence the TCR across the joint. Attempts have been made to discern the influence of contact pressure and surface roughness on the TCR. When the contact interfaces undergo thermal, load or combined thermal and load cycles, changes in the TCR and heat flow across the contact interfaces can occur. The influences of the contact temperature and contact pressure on the TCR have also been investigated in previous studies. For instance, the TCR between phenolic resins and carbon-carbon composites, and the TCR between cuprum and aluminum were previously measured. The results revealed that the TCR between materials with a high thermal conductivity (e. g. cuprum) was more easily affected by the contact temperature and pressure than the TCR between materials with a low thermal conductivity (e. g. phenolic resins)[11]. Tang et al.[12] measured the Thermal Contact Conductance (TCC, which is the reciprocal of TCR) between TC4/30CrMnSi from 200 to 350 ℃ and from 0 to 150 MPa. Their results demonstrated that temperature only had a limited impact on the TCC. The TCC reached its peak value at approximately 120 MPa in the processes of loading and unloading, but the variation trends were found to be different for the two processes. The TCR between copper and aluminum nitride was measured based on one-dimensional steadystate heat conduction from 90 to 210 K and from 0.3 to 1.0 MPa. The TCR between aluminum nitride and copper decreased as the contact pressure and contact temperature both increased[13]. The TCR of (Stainless Steel 304) SS304– AlN, SS304–OFC (Oxygen-Free Copper) and SS304–SS304 was measured using the LPM (Laser Photothermal Method) from 70 to 290 K and at a pressure range of 0.2 to 0.7 MPa. It was observed that the TCR varied intensely when the temperature was lower than 150 K[14]. The TCR has also been reported to be greatly influenced by a material’s surface characteristics, including the flatness, waviness and roughness of the material. A previous study demonstrated that the TCR increased linearly with an increase in the average height of the surface roughness[15]. The TCR between the cryocooler and the SC magnet was measured using the plate method. The results revealed that a low surface roughness led to a reduced temperature difference between the cryocooler and the SC magnet. For example, when the surface roughness decreased from 0.369 to 0.210 µm (by 43.0%), the temperature of the SC magnet decreased by 24.2% from 6.12 to 4.64 K[6]. The TCC of TC4-30CrMnSi was also investigated. This study found that the influence of the surface roughness on the TCC significantly increased as the pressure increased from 85 to 115 MPa[12]. Some studies focused on the effect of the interstitial materials on the TCR. The TCR between Ti-6Al-4V and Ti-6Al4V, and TCR between HTC and C/C-SiC exposed to air was much lower and less sensitive to the variation of the contact pressure than that under vacuum[16]. The TCR between two geometrically identical all-aluminum specimens was examined with and without a nanocoating at the interface. It was found that the TCR with implementation of a nanocoating decreased by about 38%[17]. The TCR between a thick copper and an aluminum specimen with different types of thermal media was investigated. The results showed a phase changed material coated on aluminum foils could enhance the thermal transfer between contact interfaces. The addition of the phase change material benefited the considered application in the preceding study compared with greases, foils, mineral materials (graphite, metal) and silicon foils. This was because the addition of a phase change material coated on aluminum foil combined several advantages of mineral foils and greases and still had a positive effect compared to the grease materials[18, 19]. At a cryogenic temperature and high heat flux the TCR between contact interfaces is influenced by contact temperature, surface characteristics, contact pressure, the material properties and the interstitial medium present at the interface. Thermal conductivity is a material property that can characterize the ability of a material to conduct heat. When materials are in contact, the thermal conductivities of the contacting materials greatly influence the heat transport. Thus, the thermal

conductivity of the contacting materials greatly influences the TCR. This study focuses on the effects of the thermal conductivity on the TCR between the non-conforming rough surfaces. This paper was organized as follows: In Section 2 identified the influence of the thermal conductivity on the TCR between similar contact materials. Section 3 presented the effects of the thermal conductivity on the TCR between dissimilar contact materials. Finally, Section 4 provided conclusions and the main takeaways from this study.

2. Experimental methods and materials TCR values were measured by the LPM. The principle of the LPM and a detailed description of this technique has been reported in a previous study[14]. The schematic diagram of the LPM experimental rig was illustrated in Fig.1. A heating laser (MW-NI-GF-808, Changchun Laser Optical Technology Co., Ltd, Changchun, China) was used to generate a thermal wave that passed through the contact materials. A probing laser (He-Ne laser, 632.8 nm) was then implemented to record the thermal wave that passed through the contact materials. A radiation shield was placed inside the vacuum cover. Then, the specimen (contact materials) were mounted on the cold head which could rapidly transfer heat. The specimens were placed in a sample holder with a cylindrical cavity. A smaller cylindrical cavity was located to the right of the sample holder through which the probing laser irradiated on the specimen. The area to the left of the sample holder was threaded. The stroke of the bolt was the compression length, S. The contact force, F, between the specimens was loaded using a spring and the force value could be obtained according to the equation F=μ×S, where μ is the elastic coefficient. A Ni-Cr/Cu+0.13at% Fe thermocouple was welded onto the top of the sample holder and another NiCr/Cu+0.13at% Fe thermocouple was welded on the side of the sample holder to measure the temperature of the specimen. The contact pressure could be obtained according to Eq. (1). The TCR of OFC-OFC, SS304-SS304 and AlN-AlN ceramics was measured by the LPM. The surfaces of the contact materials were polished, and their surface roughness was on the order of 0.1 µm. These surfaces were regarded as the non-conforming rough surfaces. Then the influence of the thermal conductivity on the TCR was analyzed for these materials. The contact materials were shown in Fig. 2, and their parameters were listed in Tab.1.

P

4  S d 2

(1)

Where μ was the elastic coefficient, MPa. S was the compression length, m. d was the diameter of the specimen, 3 mm.

Fig. 1 Schematic diagram of LPM experimental rig

(a)

(b)

(c)

Fig. 2 Materials, (a) OFC, (b) -AlN, (c) SS304 Tab. 1 Parameters of materials Thickness

Roughness(Contour arithmetic

/mm

mean deviation/Ra)

OFC

0.26

0.086 m / 0.03 m

99.95% Cu

SS304

0.20

0.043 m / 0.161 m

OCr18Ni9

AlN

0.30

0.58 m / 0.42 m

N>32.5-33; O<1.0-1.5; C<0.1

Material

Composition

3. Results and discussion 3.1 Similar contact materials 3.1.1 OFC-OFC The thermal conductivity of OFC was depicted in Fig. 3[20]. The variations of the TCR with respect to the thermal conductivity were shown in Fig. 4. The contact pressure was 0.35, 0.53 and 0.71 MPa, respectively. The temperature varied from 85 to 280 K. The TCR of OFC-OFC decreased as the thermal conductivity increased. When the heat transfer increased at the contact interface, the TCR decreased. When the thermal conductivity was lower than 450 W·K-1·m-1, the TCR greatly decreased. In comparison, the TCR decreased more slowly when the thermal conductivity was higher than 450 W·K-1·m-1. At a temperature of 100 K, the thermal conductivity of OFC was 450 W·K-1·m-1. The variations of the TCR of OFC with respect to the thermal conductivity were fitted at different contact pressures. The fitting formulas were shown in Eqs. (2), (3) and (4). At 0.35 MPa, the TCR and the thermal conductivity exhibited an exponential relationship. When the contact pressure ranged from 0.53 to 0.71 MPa, the TCR and the thermal conductivity exhibited a polynomial relationship. The fitting coefficients for all regression equations were all greater than 0.98. Therefore, the fitting formulas could accurately describe the experimental data.

650

Discrete points Fitting curve

600

-1

k/W·m ·K

-1

550

500

450

400 50

100

150

200

250

300

Temperature/K

Fig.3 Thermal conductivity of OFC

7.5E-7

Experimental data Fitting curve

1.25E-6

Experimental data Fitting curve

7E-7

-1

TCR/m ·K·W

1.15E-6

6.5E-7 6E-7

2

2

TCR/m ·K·W

-1

1.2E-6

1.1E-6 1.05E-6 1E-6

5.5E-7

5E-7

400 410 420 430 440 450 460 470 480 -1

400 410 420 430 440 450 460 470 480 -1

-1

(a) 0.35 MPa

(b) 0.5 MPa Experimental data Fitting curve

-1

4E-7

2

TCR/m ·K·W

-1

k/W·m ·K

k/W·m ·K

3.5E-7

3E-7

2.5E-7

400 410 420 430 440 450 460 470 480 -1

-1

k/W·m ·K

(c) 0.71 MPa Fig. 4 0.35 MPa:

Variation of the TCR of OFC-OFC with respect to thermal conductivity

R  1.0197 10 6  9.8411108 exp(0.0891k ) Z  0.9813

(2)

0.53 MPa:

R  5.5686 10 6 - 2.1302 10-8 k  2.1190 10-11 k 2 Z  0.9835

0.71 MPa:

R  8.8960 10 5 - 5.7312 10-7 k  1.2402 10-9 k 2 - 8.9608 10-13 k 3

(3)

Z  0.9959

(4)

Where R was the TCR, k was the thermal conductivity, and z was the correlation coefficient. The heat transfer in OFC depended on electrons. The higher the temperature, the more intense the electron interaction. The intense electron interactions could impede the heat transport of the electrons. Accordingly, when the thermal conductivity of OFC decreased, the TCR increased. Moreover, when the temperature was lower than 100 K, the TCR of OFC-OFC decreased abruptly. From 100 to 300 K, the variation of the TCR with respect to the thermal conductivity was insignificant. This was due to the fact that the thermal conduction in OFC depended on the lattice and electron thermal conductivities, as shown in Eq. (5). The lattice thermal conductivity was proportional to the temperature in Kelvin. The electronic thermal conductivity was proportional to the nth power of the temperature in Kelvin, n was the number of free electrons in each copper atom, and n was equal to 2 for copper. When the temperature was higher than 100 K, the lattice thermal conductivity of OFC was insignificant. When the temperature was lower than 100 K, the influence of the lattice on the thermal conductivity cannot be neglected. Experiments demonstrated that the lower the temperature, the higher the value of the lattice thermal conductivity. Eventually, the TCR of OFC-OFC was reduced more intensely with an increase in the thermal conductivity when the temperature was lower than 100 K.

k =ke  k g

(5)

Where ke was the electronic thermal conductivity and kg was the lattice thermal conductivity. 3.1.2 SS304- SS304 The thermal conductivity of SS304[21] from 25 to 300 K was shown in Fig. 5. The thermal conductivity of SS304 increased with an increase of the contact temperature. The thermal carriers included electrons and phonons. At cryogenic the thermal conductivity mainly dependent on the electron. With the increase of the temperature, the impact of phonon scattering would become more significant. When the contact temperature was lower than 150 K, the change in the thermal conductivity with respect to the temperature was more intense. When the temperature was larger than 150 K, the relationship between the thermal conductivity and contact temperature exhibited a nearly linear relationship.

16

14

12 -1

k/W·m ·K

-1

Discrete points Fitting curve

10

8

6 50

100

150

200

250

300

Temperature/K

Fig. 5 Thermal conductivity of SS304 The variation of the TCR of SS304-SS304 at contact pressures of 0.80, 0.92 and 1.02 MPa with respect to changes in thermal conductivity is shown in Fig. 6. The investigated temperature range varied from 75 to 275 K. The TCR of SS304 decreased with an increase of the thermal conductivity. The TCR exhibited a linear relationship with the thermal conductivity. The change of the TCR with respect to the thermal conductivity was more intense from 120 to 220 K. The

variations of the TCR of SS304 with respect to the thermal conductivity were fitted at different contact pressures. The fitting formulas were shown in Eqs. (6), (7) and (8). The TCR and the thermal conductivity exhibited exponential relationships at 0.80, 0.92 and 1.02 MPa. All of the fitting coefficients were greater than 0.96, so the fitting formulas were suitable to describe the experimental data.

R  1.0058 10 4  0.2863 exp(0.7299k ) ，

z=0.9805

(6)

0.92 MPa：

R  6.0036 10 5  0.0162 exp(0.4634k ) ，

z=0.9929

(7)

1.02 MPa：

R  6.5744 10 5  0.1924 exp(0.7581k ) ，

0.80 MPa：

z=0.9939

(8)

5E-4

Experimental data 0.80MPa Experimental data 0.92MPa Experimental data 1.02MPa Fitting curve 0.80MPa Fitting curve 0.92MPa Fitting curve 1.02MPa

4E-4

2E-4

2

TCR/m ·K·W

-1

3E-4

1E-4

7

8

9

10

11 -1

12

13

14

15

-1

k/W·m ·K

Fig.6 Variation of TCR of SS304-SS304 with the thermal conductivity 3.1.3 AlN- AlN ceramics The thermal conductivity of the AlN ceramics[22] from 30 to 300 K was shown in Fig. 7. The peak of the thermal conductivity was 347 W·K-1·m-1 (140 K). The phonon scattering was the key mechanism of affecting the thermal conductivity of AlN ceramics. At low temperature the phonon scattering from phonon was a significant factor. At high temperature the phonon scattering from grain boundary was a significant factor. So the thermal conductivity increased firstly with respect to the temperature and then decreased with respect to the temperature. The variation of the TCR of AlN-AlN ceramics as the thermal conductivity varied was depicted in Figs. 8, 9. The investigated temperature ranged from 63 K to 257 K and the contact pressure varied from 0.51 and 0.76 MPa. The arrows in Figs.7, 8 indicated the variation direction of the thermal conductivity with respect to the contact temperature. When the thermal conductivity increased from 300 W·m-1·K-1 (90 K) to 347 W·m-1·K-1 (140 K), the TCR decreased by 13.7%. When the thermal conductivity decreased from 347 W·m-1·K-1 (140 K) to 230 W·m-1·K-1 (257 K), the TCR increased. The variation of the TCR with respect to the thermal conductivity exhibited a rotating phenomenon. At 0.51 and 0.76 MPa, the temperature that the rotating phenomenon displayed was 158 and 155 K, respectively. As the contact surfaces came into closer contact at higher contact pressures, the effect of the contact temperature on the surface microscopic deformation reduced. Meanwhile, at the same thermal conductivity, the variation of the TCR at 0.53 MPa was greater than the variation of the TCR at 1.76 MPa. Thus, the temperature not only influenced the thermal conductivity, but it also impacted the surface deformation and the real contact area. At lower temperatures, the surface of the AlN ceramics was harder[23], so the gap between the contact surfaces was larger. At higher temperatures, the surface of the AlN ceramics was softer[23], and microscopic deformation occurred at the contact surfaces, which caused the real contact area to enlarge. Accordingly, the variation of the TCR that was caused by the variation of the thermal conductivity was larger at higher temperatures. The variations of the TCR of AlN-AlN with respect to the thermal conductivity were fitted at 0.53 and 0.76 MPa. The fitting formulas were shown in Eqs. (9), (10), (11) and (12). At 0.53 MPa, the TCR and thermal

conductivity exhibited an exponential relationship from 90 to 158 K, and the TCR and thermal conductivity exhibited a power relationship from 158 to 257 K. At 0.76 MPa, the TCR and thermal conductivity exhibited an exponential relationship from 63 K to 155 K and that exhibited a power relationship from 155 to 248 K. 350

Discrete points Fitting curve

-1

k/W·m ·K

-1

300

250

200

150 50

100

150

200

250

300

Temperature/K

Fig. 7 Thermal conductivity of AlN ceramics

Fig. 8 Variation of TCR of AlN-AlN ceramics with the thermal conductivity (0.53 MPa)

Fig. 9 Variation of TCR of AlN-AlN ceramics with the thermal conductivity (0.76 MPa) 0.53 MPa：

90 K-158 K 158 K-257 K

R  exp(3.140  0.08377 k  1.099 10 4 k 2 ) ， z=0.9706 R  2.233 10 6  5.547 10 4  0.9784 k ， z=0.9963

(9) (10

) 0.76 MPa：

63 K-155 K

 k  6 R  1.816 10 8 exp   6.074 10 ， z=0.9893  63.53 

(1 1)

155 K-248 K

R  1.159 10 6  6.108  0.9862 k ， z=0.9928

(12 )

3.2 Dissimilar contact materials The influence of the thermal conductivity on the TCR of OFC-OFC, SS304-SS304 and AlN-AlN ceramics was also explored. The contact materials were dissimilar and their contact thermal conductivities were different. The extent to which the thermal conductivity influenced the contact materials was unknown. The formula for the harmonic mean thermal conductivity was shown in Eq. (13). The reasons for introducing the harmonic mean thermal conductivity were: (1) The harmonic mean averages should be introduced when the averages are step-by-step. (2) The introduction of the harmonic mean average should be included when the impact of each individual data point cannot be considered [24, 25].

(3) The harmonic mean average will be less affected by the extremum compared with the arithmetric average and the geometric average.

khm =

2k1k2 k1 +k2

(13)

Where khm, k1 and k2 were the harmonic mean thermal conductivity and the thermal conductivity of each of the contact materials, respectively. 3.2.1 OFC-SS304 The variation of the harmonic mean thermal conductivity of OFC-SS304 from 70 to 300 K was depicted in Fig. 10. The harmonic mean thermal conductivity of OFC-SS304 increased with an increase of the contact temperature. The cause of this phenomenon was that the thermal conductivity of OFC was larger than that of SS304. The thermal conductivity of OFC decreased and that of SS304 increased with an increase of the contact temperature, and the materials with the lower harmonic mean average was more pronounced than that of the materials with the higher harmonic mean averages. 30 28 26

Discrete points Fitting curve

22

-1

khm/W·m ·K

-1

24

20 18 16 14 12 50

100

150

200

250

300

Temperature/K Fig. 10 Harmonic mean thermal conductivity of OFC-SS304 The variation of the TCR of OFC-SS304 with respect to the harmonic mean thermal conductivity was depicted in Fig. 11. The investigated temperature range was 70-300 K, while the investigated contact pressure values were 0.23, 0.34, 0.57, 0.62 and 0.68 MPa, respectively. The discrete points in Fig. 11 were the experimental data and the solid line represented the fitted curve. The TCR of OFC-SS304 decreased as the harmonic mean thermal conductivity increased. From 70 to 150 K, the variation of the OFC-SS304 TCR with respect to the harmonic mean thermal conductivity was sporadic. From 70 to 300 K, the thermal conductivity of OFCdecreased and that of SS304

increased with an increase

of the contact temperature. Thus, the influence of the thermal conductivity of SS304 on the OFC-SS304 TCR was

dominant. The variations of the TCR of OFC-SS304 with respect to the harmonic mean thermal conductivity were subsequently fitted. The fitting formulas were shown in Eqs.(14) to (18). The TCR and the harmonic thermal conductivity exhibited exponential relationships from 0.23 to 0.68 MPa. The fitting coefficients were all greater than 0.97, so the fitting formulas could adequately describe the experimental data. 0.23 MPa:

R  1.8250 10 6  1.6990 10 4 exp(0.2218k hm ), Z  0.9967

(14)

0.34 MPa:

R  1.9059 10 6  0.001070 exp(0.3473k hm ),

Z  0.9829

(15)

0.57 MPa:

R  1.3925 10 6  5.7467 10 4 exp(0.3130k hm ), Z  0.9882

(16)

0.62 MPa:

R  9.4017 10 7  3.7338 10 4 exp(0.2916k hm ), Z  0.9812

(17)

0.68 MPa:

R  7.4115 10 7  3.3172 10 4 exp(0.2878k hm ), Z  0.9718

(18)

Fig. 11 Variation of TCR of OFC-SS304 with the harmonic mean thermal conductivity 3.2.2 OFC--AlN ceramics The variation of the harmonic mean thermal conductivity of OFC-AlN ceramics from 70 to 300 K was depicted in Fig. 12. From 70 to 140 K, the harmonic mean thermal conductivity increased with an increase of the contact temperature. From 140 to 300 K, the harmonic mean thermal conductivity decreased with an increase of the contact temperature and the maximum value of the harmonic mean thermal conductivity was 385 W·m-1·K-1. The change in the trend of the harmonic mean thermal conductivity with respect to the temperature change was similar to that of the thermal conductivity for AlN ceramics, but the extreme temperature was 140 and 150 K, respectively. The variation of the TCR of OFC-AlN ceramics with respect to the thermal conductivity was depicted in Fig. 13. The

investigated temperature range was 70-300 K, and the investigated contact pressure values were 0.20 and 0.34 MPa, respectively. The discrete points represented the experimental data and the dashed line represented the fitting curve. When the harmonic mean thermal conductivity of OFC-AlN ceramics was lower than 385 W·m-1·K-1, the TCR of the OFC-AlN ceramics increased with an increase of the harmonic mean thermal conductivity. When the harmonic mean thermal conductivity of the OFC-AlN ceramics was larger than 384 W·m-1·K-1, the TCR of the OFC-AlN ceramics increased with the decrease of the harmonic mean thermal conductivity. The variations of the TCR of OFC-AlN with respect to the harmonic mean thermal conductivity were fitted at 0.20 and 0.34 MPa. The fitting formulas are shown in Eqs.(19) and (20). The harmonic mean thermal conductivity and the TCR exhibited a third order polynomial relationship. The fitting coefficients of the aforementioned equations was 0.9839 and 0.9712, respectively. 400

Discrete points Fitting curve

380

340

-1

khm/W·m ·K

-1

360

320 300 280 260 50

100

150

200

250

300

Temperature/K

Fig. 12 Harmonic mean thermal conductivity of TCR of OFC-AlN ceramics

400 380 360

-1

k/W·m ·K

-1

340 320 300

Experimental data 0.20MPa Experimental data 0.34MPa Fitting curve 020MPa Fitting curve 0.34MPa

280 260 3E-6

4E-6

5E-6

6E-6 2

7E-6

8E-6

9E-6 1E-5 1.1E-5

-1

TCR/m ·K·W

Fig. 13 Variation of TCR of OFC-AlN ceramics with the harmonic mean thermal conductivity

0.20 MPa:

k hm  712.4520  3.9666 108 R  4.5947 1013 R 2  1.6928 1018 R 3

Z  0.9839

(19)

0.34 MPa:

k hm  367.2124  2.5978 108 R  2.7409 1013 R 2  8.2245 1018 R 3

Z  0.9712

(20)

4. Conclusion In this paper, the influence of the thermal conductivity on the TCR between the non-conforming rough surfaces was analyzed. For similar contact materials, OFC-OFC and SS304-SS304, the TCR decreased with an increase of the thermal conductivity. The TCR of the AlN-AlN ceramics decreased when the thermal conductivity varied from 300 (90 K) to 347 (140 K) W·m-1·K-1 . The TCR of the AlN-AlN ceramics increased when the thermal conductivity changed from 347 (140 K) to 230 (257 K) W·m-1·K-1 . The variation of the AlN-AlN ceramics TCR with respect to the thermal conductivity exhibited a rotating phenomenon. A relationship between the TCR and thermal conductivity was established. At 0.35 MPa, the relationship between the TCR of OFC-OFC exhibited an exponential relationship respect to the thermal conductivity, while from 0.53 to 0.71 MPa, the TCR of OFC-OFC exhibited a polynomial relationship respect to the thermal conductivity. The relationship between the TCR of SS304-SS304 exhibited an exponential relationship from 0.80 to 1.02 MPa. At 0.53 and 0.76 MPa, the TCR of AlN-AlN ceramics firstly exhibited an exponential relationship firstly and then exhibited a power relationship with respect to the thermal conductivity. For dissimilar contact materials, i.e. OFC-SS304 and OFC-AlN ceramics, the effect of the material with the lower thermal conductivity was dominant. From 0.23 to 0.68 MPa, the relationship between the TCR of OFC-SS304 and the harmonic mean thermal conductivity exhibited an exponential relationship. At 0.20 and 0.34 MPa, the TCR of OFC-AlN ceramics exhibited a third order polynomials relationship with respect to the harmonic mean thermal conductivity.

Acknowledgements The research was sponsored by National Natural Science Foundation of China (No.51606113, 51406108 and 51376068), Key Research and Development Program of Shandong Province（No. 2017GGX40108）and Distinguished expert of the Taishan scholars of the Shandong Province.

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Highlights 1. Thermal contact resistance (TCR) between thin films was measured by the LPM. 2. The effect of thermal conductivity on TCR was studied. 3. Harmonic mean thermal conductivity of dissimilar contact materials was introduced 4. The relationship between the TCR and thermal conductivity was established.