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Characterization and application of engineered regular rough surfaces in thermal contact resistance
Applied Thermal Engineering 71 (2014) 400e409
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Characterization and application of engineered regular rough surfaces in thermal contact resistance Tengfei Cui a, Qiang Li a, Yimin Xuan a, b, * a b
School of Energy and Power Engineering, Nanjing University of Science & Technology, China College of Energy and Power Engineering, Nanjing University of Aeronautics & Astronautics, China
h i g h l i g h t s Characterization of engineered surfaces are investigated and modeled. Proper surface models are proposed for estimating TCR. The short wave length of regular surfaces is good for reducing the TCR. The real contact area is also severely affected by the flatness of surfaces. The effects of roughness on TCR is not as dominant as wave length and flatness.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2014 Accepted 8 July 2014 Available online 15 July 2014
Proper characterization of rough surfaces is indispensable for accurate estimation of thermal contact resistance (TCR). This work is to establish a proper algorithm of characterizing surface topographies mechanically processed by lathe turning and end-face milling, which is to provide accurate and convenient methods of modeling surfaces for predicting the TCR. The correlations of surface roughness with wavelength and element height in these two types of mechanically machined surfaces are established. Based on the Fourier transforms, the models of the surface topographies of both lathe turning surfaces and end-face milling surfaces are proposed. The presented surface models are applied to both the macroscopic and the multiscale approaches of simulating the TCR to verify the accuracy of the presented surface models. A higher accuracy is verified by the comparisons between the experimental data of the TCR and the numerical results obtained from the presented surface models. The contact conditions of using lathe turning surfaces and end-face milling surfaces are simulated. The results indicate that short wavelength is able to increase real contact area, which is benefit for reducing the TCR at interface. Meanwhile, large flatness angle of lathe turning surfaces will severely decrease the real contact area and lead to increase in the TCR. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Engineered rough surface Surface topography Thermal contact resistance
1. Introduction Thermal contact resistance (TCR) is an important factor in the thermal engineering applications, which always becomes a dominant factor of controlling the heat transfer between solids. Numerous investigations on TCR with experimental and numerical methods were carried out in the past decades [1-4]. It has been found that the TCR (Rc) is affected by several parameters such as
* Corresponding author. School of Energy and Power Engineering, Nanjing University of Science & Technology, China. Tel.: þ86 025 84890688. E-mail address: [email protected] (Y. Xuan). http://dx.doi.org/10.1016/j.applthermaleng.2014.07.020 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
thermal conductivity k, pressure P, material's hardness H, surface roughness Ra, profile slope ks and etc:
Rc ¼ f ðk; P; H; Ra ; ks ; …Þ
(1)
Among these factors, Ra was considered as a dominant factor of TCR, which is one of the parameters to describe surface topography. In the past, one always simply thought that higher Ra caused larger Rc. However, it is not always true. With the deepening of investigations on the TCR, one finds that it is no longer proper to indicate the influence of surface topography just by using Ra. Bendada et al. [5] and Courbon et al. [6] found that huge deviations were preformed in the TCRs between surfaces with different processes but the same roughness. In addition, Zhang et al. [7] and
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Chen et al. [8] found that surface topography also had a big influence on microscale heat and fluid flow, the fluid flow can be totally different even though the roughness is the same. This is because that the real surface topographies are diverse and only parameter Ra is not enough to describe surface topography completely. Different surface topographies will entirely result in quite different consequences for microscale heat transfer and fluid flow. Therefore, great attentions have been given to the models of characterizing and simulating surface topographies. In the characterization of the surface topographies, several surface parameters were proposed based on the surface profiles [9]. Fig. 1 illustrates the commonly used parameters of a surface profile. The peak height Zp is the vertical distance between peak point and the mean line. The valley height Zv is the vertical distance from valley point to the mean line. The element height Zt is the summation of Zp and Zv. The wavelength lw is the length between two neighboring peak points or valley points. It only performs in the surfaces with regular surface topographies and is strongly related to the processing methods. The slope of the profile element ks can be calculated from the element height and the wavelength. The flatness angle a is the angle between the horizontal line and the mean line, which is used to describe and calculate the flatness of surface. Moreover, other statistical parameters can be calculated from the profile heights, such as the root mean square roughness Rq, the arithmetic mean roughness Ra, kurtosis Rku, and skewness Rsk, etc. However, only few parameters are used to guide the machining of rough surface in engineering processes and the mostly-used parameter is Ra. This is because Ra can be easily controlled in common engineering processes without complex surface manufacturing methods. Thus, it is necessarily to investigate the methods of simulating surface topographies just through the parameter Ra. The mathematical definition of Ra is as follows:
Ra ¼
1 l
Zl jzðxÞjdx
(2)
0
Nevertheless, the investigations of surface roughness explicitly or implicitly assumed that the surface topographies are isotropic phenomenon and Gaussian distributions were taken to describe surfaces, which may not be always true for all the case. In fact, most of rough surfaces machined by common processes have highly directional properties and similar surface topographies. Thomas et al. [10] found that the machined surface topographies could be very regular. Several types of surface topographies have been investigated and classified in their work. The experimental results obtained by Guan et al. [11] showed that certain parameters of the tested surface samples with the same manufacturing method are
Fig. 1. The commonly used parameters of surface profile.
401
basically the same. Therefore, it is feasible to obtain the regular surface topographies through the known value of Ra. For simulating surface topographies, Gaussian distribution is usually applied in simulating rough surfaces [12,13]. As for the Fast Fourier Transform (FFT) method of simulating surfaces, the Gaussian distribution, spectral density, and auto-correlation function (ACF) are all employed in simulation of rough surfaces [14e16]. The surface topographies can be controlled by spectral density or ACF. However, in fact, the surface topographies can be still different with the same spectral density or ACF, because the Gaussian distribution is a one-dimensional function that cannot precisely describe three-dimensional questions. Therefore, Gaussian surfaces and the FFT method are suitable in simulating surface topographies with high calculation speed, but low accuracy. For higher precision simulations of surface topographies, fractal models were proposed to cater the need [7e11,17e19]. Those investigation indicated that the surface topographies simulated by fractal models coincide well with the actual surfaces. Patrikar et al. [20] compared two different methods and they confirmed that the fractal model was better than the FFT method in simulating rough surfaces. However, the fractal model needs that the surface topography of the small sampled surface region can represent the total surface topography. Otherwise, it will cost huge computation time to simulate every small region. Furthermore, such fractal descriptions just outlined surface topographies and they cannot be applied to estimate the TCR in the common Euclidean space for any practical applications because almost all existing heat transfer equations are established in this space. Therefore, it is necessary to investigate proper methods of characterizing surface topographies. In this paper, two types of engineered regular surface topographies, which are respectively processed by lathe turning and endface milling, are studied to establish high-precision surface models in the application of predicting the TCR. The correlations between those surface parameters are proposed based on the experimental data. By using the Fourier transformation, the models of the surface topographies processed through these two methods are constructed. Meanwhile, the influences of two surface topographies on simulating TCR are studied. The purpose of this work is to find accurate approaches for predicting the TCR in engineering applications. 2. Characterization of the surfaces processed by lathe turning Lathe turning is a commonly used process in machining the surfaces of columnar parts. Fig. 2(a) illustrates the process of cutting end surface in lathe machine. The columnar part is rotating around the Z-axis, and the cutting tool is moving from the edge of the surface to the center with a constant speed. Usually, a cutting tool moves from the edge of a surface to the center along a regular helical line, which forms the surface topography that performs the helical surface phenomenon (as shown in Fig. 2(b)). Thomas et al. [21] investigated the relationships of Ra with depth of cut, tool nose radius, and feed rate. They found that the depth of cut has no significant effect on surface roughness; the surface roughness was mainly affected by the feed rate and the tool nose radius. Luo et al. [22] and Lin et al. [23] proposed models of simulating the lathe turning surface topographies with several processing parameters. However, those models were too complicated to be used in engineering applications. Therefore, a fast and simple model is in dire need. Fig. 3 exemplifies the profile heights of the surfaces processed by lathe turning with different roughness, which is measured across the centre of column. It can be found that the surface profiles are very regular and periodic, and these periodic domains all have
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Fig. 2. The illustrations of lathe turning used in machining plane surface (a) the process (b) the micrograph of lathe turning surface.
approximate wavelengths and element heights that are strongly related with Ra. Fig. 4(a) shows the mean wavelength measured from different lathe turning surfaces. It can be found that the mean wavelength nearly has a linear relationship with Ra, which had been also mentioned by Thomas et al. [21]. Moreover, there is no obvious effect of materials on wavelength, for that the wavelength is mainly related with the feed speed of cutting tool. In order to provide a simple correlation between the wavelength and Ra for engineering application, a linear equation is proposed to fit the measured results. By considering the deviations caused from the measurements and the precisions of processes, the linear fit equation is fitted as follows:
lw ¼ 53:5*Ra *ð1 þ dl Þ þ 12
ðRa > 0:5 mmÞ
(3)
where dl is a random number to describe the deviations of the wavelength with same Ra. The value range of dl is [0.1, 0.1]. It is noteworthy that this equation is appropriate for the situation when the Ra of lathe turning surfaces is bigger than 0.5 mm, which is the general case in engineering productions. This is because the rough surfaces with Ra < 0.5 mm are commonly manufactured by several processes, which leads that the profile heights no longer all obey the periodic distribution. Fig. 4(b) indicates the relationship between mean element heights and Ra. Due to the periodic performance of surface profiles, a great linear relationship between Zt and Ra can be obtained from the results. The linearly-fitted equation is given as follows:
Fig. 3. The lathe turning surface profiles with different roughness.
Zt ¼ 4:0*Ra
(4)
However, knowing lw and Zt is not enough to reconstruct the real surface topographies. By analyzing a series of surface profiles, one can find that the profile curves of machined surfaces are mainly constructed by three sections. One is the surface flatness which is
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and real surface profiles is approximate to 5%. It is noteworthy that D can also be calculated from the experimental data if a higher accuracy of modeling surfaces wants to be obtained. The profile fractal dimension, D, can be determined by calculating the structure function [7,8]
Str t ¼ < ½f ðx þ tÞ f ðxÞ2 > tð42DÞ
(6)
where Str(t) represents the mean square of the difference in height expected over any spatial distance t. In the two-dimensional space, the range for D lies between 1 (absolutely smooth) and 2 (infinitely rough). A higher value of D means more crumples on profile. During the lathe turning process, the lathe turning surfaces perform helical topographies which are formed by the helical path of the cutting tool. Therefore, combined the function of surface profile, the function of surface topography of lathe turning surface is given as follows:
q lw z x; y ¼ f r 2p
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 ; q ¼ acrcos
x r
(7)
where the x0 and y0 are the coordinate positions of the center of the column. Fig. 6 illustrates the simulated topographies of surfaces processed by lathe turning. 3. Characterization of the surfaces processed by end-face milling
Fig. 4. The relationships between Ra and (a) lw (b) Zt in lathe turning surfaces.
decided by the feed direction of cutting tools. The second is the basic curve of profile that is determined by the geometrical shape of cutting tools. The last is the micro crumples on profiles, which are generated by vibration. In the lathe turning surfaces, the basic profile curves are found to be very similar to triangle waves that can be seen in Fig. 3. Therefore, by transforming the above three parts of lathe turning profile curves into the Fourier series, the mathematical function of lathe turning surface profiles is proposed as follows:
! f x
¼ x sin a þ þ
∞ X
∞ 4Zt X sinð2ð2n þ 1Þpx=lw Þ ð 1Þn 2 p n¼0 ð2n þ 1Þ2
lðD2Þn sin D 1 ln x
(5)
n¼1
The first part on the right-hand side of the above expression is xsina, which is used to calculate the flatness of surfaces. The second part that contains Zt and lw is the Fourier expressions of triangle waves. The last part is imitated from the fractal model to simulate the detailed micro peaks and valleys at profiles, which is related with the discrete frequency modes of the surface roughness ln (l > 1) and the profile fractal dimension D (1 < D < 2). Fig. 5 shows the simulated surface profiles with Ra ¼ 1.0 mm and Ra ¼ 5.0 mm, the wavelengths and the element heights are calculated from Eqs. (3) and (4), l is set to 1.5, D is set to 1.1. It can be found that the simulated profiles are very similar with the profiles shown in Fig. 3. The mean deviation between simulated profiles
Similar to the lathe turning process, end-face milling is another widely used process in machining plane surfaces with regular surface topographies. Fig. 7(a) illustrates the process of end-face milling on a plane surface. The cutting tool is rotating around its center axis and the plate is moving from one side of to another side with a constant speed, which forms the surface topography performed as Fig. 7(b). The surface topographies are strongly related with the radius of cutting tool. Baek [24] and Franco [25] proposed the models of simulating end-face milling surfaces which partly depends on feed speed, cutting tool geometry and tool errors. Fig. 8 exemplifies the profile heights of end-face milling surfaces with different roughness, which is measured along the path of the cutting tool. It can be found that the surface profiles of endface milling surfaces are as regular as the surface profiles of lathe turning surfaces. However, different from the lathe turning surfaces, the mean wavelength of end-face milling surfaces has no significant relationships with Ra and materials, which can be found in Fig. 9(a) and Ref. [25]. Fig. 9(b) indicates the relationship between mean element height and Ra. It is interesting to find out that the correlation between Zt and Ra obtained from the end-face milling surfaces are the same with that obtained from the lathe turning surfaces. This is because the Ra is a statistic parameter that has its shortage for judging which machining approach (either lathe turning or end-face milling) is used for the given surfaces. Therefore, it is not proper to represent the surface topographies by merely using Ra. The linearly-fitted equation is given as follows:
Zt ¼ 4:0*Ra
(8)
Being different from the lathe turning surfaces, the profile curves of end-face milling surfaces are found to be similar to saw-
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Fig. 5. The lathe turning surface profiles with different roughness. Fig. 6. The simulated topographies of lathe turning surface.
tooth waves. It is noteworthy that this is only suitable for end-face milling and deviations are existent by using different geometrical shape of cutting tools. Therefore, by transforming the curves of the end-face milling surfaces profiles to the Fourier series, the mathematical function of surface profile is proposed based on the basic parameters of surface profiles.
! f x
N Zt X sinð2pnx=lw Þ ¼ x sin a p n¼1 n
þ
N X
lðD2Þn sinððD 1Þln xÞ
(9)
n¼1
Fig. 10 illustrates the simulated surface profiles with Ra ¼ 2.0 mm and Ra ¼ 6.2 mm. It can be found that the simulated profiles are similar to the profiles shown in Fig. 8. Besides, it is a commonly encountered situation that the surfaces are milled several times to reduce the surface roughness, which can be seen from Fig. 11(a). In that situation, it is simple to solve this matter by superposing the function of surface profile several times, which can be seen in Fig. 11(b). The mean deviation between simulated profiles and real surface profiles approximate 9%. In the end-face milling process, the surface topographies are related with the moving path and the radius of cutting tool. Therefore, the function of surface topography is given as follows:
zðx; yÞ ¼ f ðrÞ
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xi Þ2 þ ðy yi Þ2 r0
(10)
where, the r0 is the radius of the cutting tool, yi ¼ axi þ b (i ¼ 1,2 … n) is the path of the cutting tool. Fig. 12 illustrates the simulated topographies of the surfaces processed by end-face milling.
4. The application of surface topography in simulating TCR 4.1. The application of the surface topography in macroscale method In order to verify the precisions provided from the surface models in simulating the TCR, these two surface models are applied in a macroscale method which was proposed by Yovanovich [1]. The correlation of predicting the TCR is as follows:
1 ¼ hc ¼ 1:25k m s ðP=HÞ0:95 Rc
(11)
where k is the thermal conductivity, m is the absolute mean asperity slope, s is root mean square surface roughness, P is
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Fig. 7. The illustrations of end-face milling used in machining plane surface (a) the process (b) the micrograph of end-face milling surface.
pressure, and H is micro-hardness. Fig. 13 shows the comparisons between the experimental data of TCR and the data of TCR calculated by the Yovanovich's correlation with the present surface models. The parameters needed in equations are calculated from the surface profiles generated by the present models. The experimental data of the TCR shown in Figs. 13 and 14 are measured by a high-precision TCR instrument with the uncertainty of 5% in Ref. [26]. The measurement is conducted in a vacuum chamber to reduce the heat loss. All the same-size columnar specimens are made by Al material, which are machined by different processes but all of them have the same roughness Ra ¼ 1.6 mm. The lathe turning surfaces have a small flatness angle (nearly 0.1 ) due to the limit of lathe turning process. The wavelength of end-face milling surfaces is nearly 150 mm by controlling the feed speed in end-face milling process. It can be found that the simulated data coincide well with the experimental data, especially when using the endface milling surfaces model. However, the unflatness of the lathe turning surfaces causes big deviations in the prediction of TCR by Yovanovich's model. 4.2. The application of the surface topography in the multiscale method Since the influence of the surface topography on TCR cannot be thoroughly considered and simulated by Yovanovich's method, it is necessary to simulate the TCR with three-dimensional contact surface topographies. Therefore, a multiscale method was proposed to simulate the TCR in our previous works in Ref. [27], which was
Fig. 8. The end-face milling surface profiles with different roughness.
proved to have a high accuracy by comparing the simulated results with the experimental results. With respect to application of the multiscale method, the three-dimensional surface topographies are first generated from the experimental data of the surfaces, and then the elasticeplastic mechanics referenced is applied to simulate the surface deformations under pressure. Finally, the traditional finitedifference method and the microscopic lattice Boltzmann method (LBM) are coupled to calculate heat transfer within the microscopic
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Fig. 9. The relationships between Ra and (a) lw (b) Zt in end-face milling surfaces. Fig. 10. The simulated end-face milling surface profiles with different roughness.
region and around the micro contact points. In order to compare the numerical results with the experimental data, the gaps between two rough surfaces are considered as vacuum and the heat transfer due to radiation and convection are ignored. Heat is only transferred through the interface along the contact points. The evolution equation of the energy distribution function in the lattice Boltzmann model is [28,29]
ei ðx þ vi Dt; y þ vi Dt; z þ vi Dt; t þ DtÞ ¼ 1 Wi ei x; y; z; t þ Wi e0i x; y; z; t þ Dtqv
Wi ¼
Dt l=v
(12)
(13)
where ei is the discrete energy of electrons or phonons, l is the mean free path of electrons or phonons, and v is the group velocity of electrons or phonons. The heat flux is calculated from the following expression
qi ¼
X
vi ei ðx; y; z; tÞ
(14)
Due to the different scale length of two regions, a coupling region is constructed to transmit the physical information calculated through two methods under the constraint condition that the energy conservation is kept. When the convergence criterion is reached, which is usually that the computational residues of
temperature are less than 106, the temperature drop at the interface DT can be calculated from the averaged surface temperature, and the heat flux q can be directly obtained. Therefore, the TCR yields as
Rc ¼ DT=q
(15)
The detailed description of the multiscale method can be referred to the previous work [27]. Fig. 14 shows the comparisons between the experimental data of TCR and the data of TCR simulated by the multiscale method, in which the experimental surfaces are replaced by different kinds of surface models. The surfaces processed by lathe turning and endface milling are simulated by the present models. The random Gaussian surface is generated by FFT method and the W-M surface is generated from the W-M function [20]. All these simulated surfaces are ensured to have the same surface roughness (Ra ¼ 1.6 mm). It can be found that TCR calculated through different surface models are totally different. Especially when the exerted pressure is relatively lower (for example, below 1 MPa), the deviations between the real value of TCR and the numerical results through improper surface models are unacceptable. This is because the surface topography has a dominant influence on the TCR especially for the cases that the exerted pressure is relatively lower, so that any tiny deviations between the simulated surfaces and real surfaces may cause large errors in predicting the TCR. The bigger the
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Fig. 11. The end-face milling surface profiles obtained from (a) measurement (b) simulation. Fig. 12. The simulated surface topographies of end-face milling surfaces with different roughness and different radius of cutting tool.
deviation exists, the larger the error appears. Such as when the pressure equals to 1 MPa, the deviation between the experimental results with milling surfaces and the results calculated from a random Gaussian surface approaches to 100%. Even the W-M surface is applied in the simulation of TCR, the deviation is up to 52% under 1 MPa. Nevertheless, by using the present surface models, the max deviations between the experimental and simulated TCR of lathe turning surface and end-face milling surface, which are present under 1 MPa, are 10% and 13% respectively. The results indicate that present surface models have higher accuracy in simulating TCR than other models. For most cases of electronic cooling, in fact, the exerted pressure is smaller than 1 MPa, which means bigger deviations may exist in predicting the TCR if improper surface roughness models are used. Therefore, it is significant to choose a suitable surface model in the prediction of TCR. Fig. 15 provides the proportions of real contact area in the total contact area between a rigid smooth plane and the surfaces simulated by present models. The materials of those surfaces are Al and the wavelengths of end-face milling surfaces are referred from the real end-face milling surfaces. The deformations of contact surfaces are calculated from the contact elastoplastic mechanics which is also employed in the multiscale method [1,3]. It can be found that the regular surfaces with shorter wavelength signally have larger real contact area. This is because a shorter wavelength
Fig. 13. The comparisons between the experimental data of TCR and simulated data of TCR from Yovanovich's model.
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T. Cui et al. / Applied Thermal Engineering 71 (2014) 400e409
other existing surface models in the applications of predicting the TCR. The numerical and experimental results indicate that the presented surface models are much more valid and realistic. The main conclusions are as follows.
Fig. 14. The comparisons between the experimental data of TCR and simulated data of TCR from multiscale model.
can provide more peaks and valleys in a certain sampling length, which will obviously increase the number of contact points. Therefore, reducing the wavelength of surfaces is beneficial for reducing the TCR. Meanwhile, the surface flatness is another important factor for lathe turning surface. A large flatness angle can seriously decrease the real contact area and increase the TCR. Because the lathe turning surfaces are approximately axially symmetric, the real contact areas are only presented at the edge of surfaces or the center of the surfaces when the lathe turning surfaces have a flatness angle, which is depended on whether the lathe-turning surface is concave or convex. Moreover, a small roughness leads to increase in the real contact area, but its effect may not be so dominant compared to the wavelength and flatness in engineered regular rough surfaces. 5. Conclusions Proper characterization of surface topographies is crucial for accurately estimating the TCR between two contact surfaces. Surfaces machined by two basic engineering manufacturing processes, lathe turning and end-face milling, have been characterized and modeled. These two surface models have been compared with
Fig. 15. The proportion of real contact area in total contact area with different surfaces.
(1) The surface topographies are complex and various. It is not proper to represent the varied surface topographies just through the parameter Ra. Thus, two surface models have been proposed to characterize and simulate two types of surfaces that are machined by lathe turning and end-face milling, respectively. (2) The numerical and experimental results have indicated that big deviation exists between the real TCR and the calculated TCR under low pressure if an improper surface model is applied, and the proposed surface models are much more valid and realistic. (3) Besides reducing the surface roughness, shortening the surface wavelength and guaranteeing a fine surface flatness during processes are other two effective ways to lower the TCR between two rough surfaces. Although the wavelength, the flatness, and the roughness have significant influences on the TCR, the effect of surface roughness may not be so dominant compared to the wavelength and flatness in engineered regular rough surfaces. Acknowledgements This work is sponsored by the National Natural Science Foundation of China (No. 51225602 and No. 51336003). References [1] M.M. Yovanovich, Four decades of research on thermal contact, gap, and joint resistance in microelectronics, IEEE Trans. Compon. Packag. Technol. 28 (2005) 182e206. [2] T. Loulou, E.A. Artyukhin, J.P. Bardon, Estimation of thermal contact resistance during the first stages of metal solidification process: I-experiment principle and modelisation, Int. J. Heat Mass Transf. 42 (1999) 2119e2127. [3] M. Bahrami, M.M. Yovanovich, J.R. Culham, Thermal contact resistance at low contact pressure: effect of elastic deformation, Int. J. Heat Mass Transf. 48 (2005) 3284e3293. [4] X. Zhang, P.Z. Cong, M. Fujii, A study on thermal contact resistance at the interface of two solids, Int. J. Thermophys. 27 (3) (2006) 880e895. [5] A. Bendada, A. Derdouri, M. Lamontagne, Y. Simard, Analysis of thermal contact resistance between polymer and mold in injection molding, Appl. Therm. Eng. 24 (2004) 2029e2040. [6] C. Courbon, T. Mabrouki, J. Rech, D. Mazuyer, E. D'Eramo, On the existence of a thermal contact resistance at the tool-chip interface in dry cutting of AISI 10 45: formation mechanisms and influence on the cutting process, Appl. Therm. Eng. 50 (2013) 1311e1325. [7] C. Zhang, Y. Chen, Z. Deng, M. Shi, Role of rough surface topography on gas slip flow in microchannels, J. Phys. Rev. E 86 (1) (2012) 016319. [8] Y. Chen, C. Zhang, M. Shi, G.P. Peterson, Optimal surface fractal dimension for heat and fluid flow in microchannels, J. Appl. Phys. Lett. 97 (8) (2010) 084101. [9] E.S. Gadelmawla, M.M. Koura, T.M.A. Maksoud, I.M. Elewa, H.H. Soliman, Roughness parameters, J. Mater. Process. Technol. 123 (2002) 133e145. [10] T.R. Thomas, B.G. Rosen, N. Amini, Fractal characterisation of the anisotropy of rough surfaces, Wear 232 (1999) 41e50. [11] Z.Z. Guan, M.H. Ye, X.C. Yin, X.H. Luo, Recognition of surface roughness based on fractal theory and the microscopic images, Appl. Mech. Mater. 241e244 (2013) 3030e3033. [12] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, in: Proc. Roy. Soc. London, 1966, pp. 300e319. [13] X. Zhang, P. Cong, S. Fujiwara, M. Fujii, A new method for numerical simulation of thermal contact resistance in cylindrical coordinates, Int. J. Heat Mass Transf. 47 (2004) 1091e1098. [14] D.E. Newland, An Introduction to Random Vibration and Spectral Analysis, second ed., Longman, London, 1984. [15] Y.Z. Hu, K. Tonder, Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis, Int. J. Mach. Tool. Manuf. 32 (1e2) (1992) 83e90. [16] J.J. Wu, Simulation of rough surfaces with FFT, Tribol. Int. 33 (2000) 47e58. [17] J. Lopez, G. Hansali, H. Zahouani, J.C. Lebosse, T. Mathia, 3D Fractal-based characterisation for engineered surface morphology, Int. J. Mach. Tool. Manuf. 35 (2) (1995) 211e217.
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Characterization and application of engineered regular rough surfaces in thermal contact resistance Tengfei Cui a, Qiang Li a, Yimin Xuan a, b, * a b
School of Energy and Power Engineering, Nanjing University of Science & Technology, China College of Energy and Power Engineering, Nanjing University of Aeronautics & Astronautics, China
h i g h l i g h t s Characterization of engineered surfaces are investigated and modeled. Proper surface models are proposed for estimating TCR. The short wave length of regular surfaces is good for reducing the TCR. The real contact area is also severely affected by the flatness of surfaces. The effects of roughness on TCR is not as dominant as wave length and flatness.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 March 2014 Accepted 8 July 2014 Available online 15 July 2014
Proper characterization of rough surfaces is indispensable for accurate estimation of thermal contact resistance (TCR). This work is to establish a proper algorithm of characterizing surface topographies mechanically processed by lathe turning and end-face milling, which is to provide accurate and convenient methods of modeling surfaces for predicting the TCR. The correlations of surface roughness with wavelength and element height in these two types of mechanically machined surfaces are established. Based on the Fourier transforms, the models of the surface topographies of both lathe turning surfaces and end-face milling surfaces are proposed. The presented surface models are applied to both the macroscopic and the multiscale approaches of simulating the TCR to verify the accuracy of the presented surface models. A higher accuracy is verified by the comparisons between the experimental data of the TCR and the numerical results obtained from the presented surface models. The contact conditions of using lathe turning surfaces and end-face milling surfaces are simulated. The results indicate that short wavelength is able to increase real contact area, which is benefit for reducing the TCR at interface. Meanwhile, large flatness angle of lathe turning surfaces will severely decrease the real contact area and lead to increase in the TCR. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Engineered rough surface Surface topography Thermal contact resistance
1. Introduction Thermal contact resistance (TCR) is an important factor in the thermal engineering applications, which always becomes a dominant factor of controlling the heat transfer between solids. Numerous investigations on TCR with experimental and numerical methods were carried out in the past decades [1-4]. It has been found that the TCR (Rc) is affected by several parameters such as
* Corresponding author. School of Energy and Power Engineering, Nanjing University of Science & Technology, China. Tel.: þ86 025 84890688. E-mail address: [email protected] (Y. Xuan). http://dx.doi.org/10.1016/j.applthermaleng.2014.07.020 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
thermal conductivity k, pressure P, material's hardness H, surface roughness Ra, profile slope ks and etc:
Rc ¼ f ðk; P; H; Ra ; ks ; …Þ
(1)
Among these factors, Ra was considered as a dominant factor of TCR, which is one of the parameters to describe surface topography. In the past, one always simply thought that higher Ra caused larger Rc. However, it is not always true. With the deepening of investigations on the TCR, one finds that it is no longer proper to indicate the influence of surface topography just by using Ra. Bendada et al. [5] and Courbon et al. [6] found that huge deviations were preformed in the TCRs between surfaces with different processes but the same roughness. In addition, Zhang et al. [7] and
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Chen et al. [8] found that surface topography also had a big influence on microscale heat and fluid flow, the fluid flow can be totally different even though the roughness is the same. This is because that the real surface topographies are diverse and only parameter Ra is not enough to describe surface topography completely. Different surface topographies will entirely result in quite different consequences for microscale heat transfer and fluid flow. Therefore, great attentions have been given to the models of characterizing and simulating surface topographies. In the characterization of the surface topographies, several surface parameters were proposed based on the surface profiles [9]. Fig. 1 illustrates the commonly used parameters of a surface profile. The peak height Zp is the vertical distance between peak point and the mean line. The valley height Zv is the vertical distance from valley point to the mean line. The element height Zt is the summation of Zp and Zv. The wavelength lw is the length between two neighboring peak points or valley points. It only performs in the surfaces with regular surface topographies and is strongly related to the processing methods. The slope of the profile element ks can be calculated from the element height and the wavelength. The flatness angle a is the angle between the horizontal line and the mean line, which is used to describe and calculate the flatness of surface. Moreover, other statistical parameters can be calculated from the profile heights, such as the root mean square roughness Rq, the arithmetic mean roughness Ra, kurtosis Rku, and skewness Rsk, etc. However, only few parameters are used to guide the machining of rough surface in engineering processes and the mostly-used parameter is Ra. This is because Ra can be easily controlled in common engineering processes without complex surface manufacturing methods. Thus, it is necessarily to investigate the methods of simulating surface topographies just through the parameter Ra. The mathematical definition of Ra is as follows:
Ra ¼
1 l
Zl jzðxÞjdx
(2)
0
Nevertheless, the investigations of surface roughness explicitly or implicitly assumed that the surface topographies are isotropic phenomenon and Gaussian distributions were taken to describe surfaces, which may not be always true for all the case. In fact, most of rough surfaces machined by common processes have highly directional properties and similar surface topographies. Thomas et al. [10] found that the machined surface topographies could be very regular. Several types of surface topographies have been investigated and classified in their work. The experimental results obtained by Guan et al. [11] showed that certain parameters of the tested surface samples with the same manufacturing method are
Fig. 1. The commonly used parameters of surface profile.
401
basically the same. Therefore, it is feasible to obtain the regular surface topographies through the known value of Ra. For simulating surface topographies, Gaussian distribution is usually applied in simulating rough surfaces [12,13]. As for the Fast Fourier Transform (FFT) method of simulating surfaces, the Gaussian distribution, spectral density, and auto-correlation function (ACF) are all employed in simulation of rough surfaces [14e16]. The surface topographies can be controlled by spectral density or ACF. However, in fact, the surface topographies can be still different with the same spectral density or ACF, because the Gaussian distribution is a one-dimensional function that cannot precisely describe three-dimensional questions. Therefore, Gaussian surfaces and the FFT method are suitable in simulating surface topographies with high calculation speed, but low accuracy. For higher precision simulations of surface topographies, fractal models were proposed to cater the need [7e11,17e19]. Those investigation indicated that the surface topographies simulated by fractal models coincide well with the actual surfaces. Patrikar et al. [20] compared two different methods and they confirmed that the fractal model was better than the FFT method in simulating rough surfaces. However, the fractal model needs that the surface topography of the small sampled surface region can represent the total surface topography. Otherwise, it will cost huge computation time to simulate every small region. Furthermore, such fractal descriptions just outlined surface topographies and they cannot be applied to estimate the TCR in the common Euclidean space for any practical applications because almost all existing heat transfer equations are established in this space. Therefore, it is necessary to investigate proper methods of characterizing surface topographies. In this paper, two types of engineered regular surface topographies, which are respectively processed by lathe turning and endface milling, are studied to establish high-precision surface models in the application of predicting the TCR. The correlations between those surface parameters are proposed based on the experimental data. By using the Fourier transformation, the models of the surface topographies processed through these two methods are constructed. Meanwhile, the influences of two surface topographies on simulating TCR are studied. The purpose of this work is to find accurate approaches for predicting the TCR in engineering applications. 2. Characterization of the surfaces processed by lathe turning Lathe turning is a commonly used process in machining the surfaces of columnar parts. Fig. 2(a) illustrates the process of cutting end surface in lathe machine. The columnar part is rotating around the Z-axis, and the cutting tool is moving from the edge of the surface to the center with a constant speed. Usually, a cutting tool moves from the edge of a surface to the center along a regular helical line, which forms the surface topography that performs the helical surface phenomenon (as shown in Fig. 2(b)). Thomas et al. [21] investigated the relationships of Ra with depth of cut, tool nose radius, and feed rate. They found that the depth of cut has no significant effect on surface roughness; the surface roughness was mainly affected by the feed rate and the tool nose radius. Luo et al. [22] and Lin et al. [23] proposed models of simulating the lathe turning surface topographies with several processing parameters. However, those models were too complicated to be used in engineering applications. Therefore, a fast and simple model is in dire need. Fig. 3 exemplifies the profile heights of the surfaces processed by lathe turning with different roughness, which is measured across the centre of column. It can be found that the surface profiles are very regular and periodic, and these periodic domains all have
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Fig. 2. The illustrations of lathe turning used in machining plane surface (a) the process (b) the micrograph of lathe turning surface.
approximate wavelengths and element heights that are strongly related with Ra. Fig. 4(a) shows the mean wavelength measured from different lathe turning surfaces. It can be found that the mean wavelength nearly has a linear relationship with Ra, which had been also mentioned by Thomas et al. [21]. Moreover, there is no obvious effect of materials on wavelength, for that the wavelength is mainly related with the feed speed of cutting tool. In order to provide a simple correlation between the wavelength and Ra for engineering application, a linear equation is proposed to fit the measured results. By considering the deviations caused from the measurements and the precisions of processes, the linear fit equation is fitted as follows:
lw ¼ 53:5*Ra *ð1 þ dl Þ þ 12
ðRa > 0:5 mmÞ
(3)
where dl is a random number to describe the deviations of the wavelength with same Ra. The value range of dl is [0.1, 0.1]. It is noteworthy that this equation is appropriate for the situation when the Ra of lathe turning surfaces is bigger than 0.5 mm, which is the general case in engineering productions. This is because the rough surfaces with Ra < 0.5 mm are commonly manufactured by several processes, which leads that the profile heights no longer all obey the periodic distribution. Fig. 4(b) indicates the relationship between mean element heights and Ra. Due to the periodic performance of surface profiles, a great linear relationship between Zt and Ra can be obtained from the results. The linearly-fitted equation is given as follows:
Fig. 3. The lathe turning surface profiles with different roughness.
Zt ¼ 4:0*Ra
(4)
However, knowing lw and Zt is not enough to reconstruct the real surface topographies. By analyzing a series of surface profiles, one can find that the profile curves of machined surfaces are mainly constructed by three sections. One is the surface flatness which is
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and real surface profiles is approximate to 5%. It is noteworthy that D can also be calculated from the experimental data if a higher accuracy of modeling surfaces wants to be obtained. The profile fractal dimension, D, can be determined by calculating the structure function [7,8]
Str t ¼ < ½f ðx þ tÞ f ðxÞ2 > tð42DÞ
(6)
where Str(t) represents the mean square of the difference in height expected over any spatial distance t. In the two-dimensional space, the range for D lies between 1 (absolutely smooth) and 2 (infinitely rough). A higher value of D means more crumples on profile. During the lathe turning process, the lathe turning surfaces perform helical topographies which are formed by the helical path of the cutting tool. Therefore, combined the function of surface profile, the function of surface topography of lathe turning surface is given as follows:
q lw z x; y ¼ f r 2p
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0 Þ2 þ ðy y0 Þ2 ; q ¼ acrcos
x r
(7)
where the x0 and y0 are the coordinate positions of the center of the column. Fig. 6 illustrates the simulated topographies of surfaces processed by lathe turning. 3. Characterization of the surfaces processed by end-face milling
Fig. 4. The relationships between Ra and (a) lw (b) Zt in lathe turning surfaces.
decided by the feed direction of cutting tools. The second is the basic curve of profile that is determined by the geometrical shape of cutting tools. The last is the micro crumples on profiles, which are generated by vibration. In the lathe turning surfaces, the basic profile curves are found to be very similar to triangle waves that can be seen in Fig. 3. Therefore, by transforming the above three parts of lathe turning profile curves into the Fourier series, the mathematical function of lathe turning surface profiles is proposed as follows:
! f x
¼ x sin a þ þ
∞ X
∞ 4Zt X sinð2ð2n þ 1Þpx=lw Þ ð 1Þn 2 p n¼0 ð2n þ 1Þ2
lðD2Þn sin D 1 ln x
(5)
n¼1
The first part on the right-hand side of the above expression is xsina, which is used to calculate the flatness of surfaces. The second part that contains Zt and lw is the Fourier expressions of triangle waves. The last part is imitated from the fractal model to simulate the detailed micro peaks and valleys at profiles, which is related with the discrete frequency modes of the surface roughness ln (l > 1) and the profile fractal dimension D (1 < D < 2). Fig. 5 shows the simulated surface profiles with Ra ¼ 1.0 mm and Ra ¼ 5.0 mm, the wavelengths and the element heights are calculated from Eqs. (3) and (4), l is set to 1.5, D is set to 1.1. It can be found that the simulated profiles are very similar with the profiles shown in Fig. 3. The mean deviation between simulated profiles
Similar to the lathe turning process, end-face milling is another widely used process in machining plane surfaces with regular surface topographies. Fig. 7(a) illustrates the process of end-face milling on a plane surface. The cutting tool is rotating around its center axis and the plate is moving from one side of to another side with a constant speed, which forms the surface topography performed as Fig. 7(b). The surface topographies are strongly related with the radius of cutting tool. Baek [24] and Franco [25] proposed the models of simulating end-face milling surfaces which partly depends on feed speed, cutting tool geometry and tool errors. Fig. 8 exemplifies the profile heights of end-face milling surfaces with different roughness, which is measured along the path of the cutting tool. It can be found that the surface profiles of endface milling surfaces are as regular as the surface profiles of lathe turning surfaces. However, different from the lathe turning surfaces, the mean wavelength of end-face milling surfaces has no significant relationships with Ra and materials, which can be found in Fig. 9(a) and Ref. [25]. Fig. 9(b) indicates the relationship between mean element height and Ra. It is interesting to find out that the correlation between Zt and Ra obtained from the end-face milling surfaces are the same with that obtained from the lathe turning surfaces. This is because the Ra is a statistic parameter that has its shortage for judging which machining approach (either lathe turning or end-face milling) is used for the given surfaces. Therefore, it is not proper to represent the surface topographies by merely using Ra. The linearly-fitted equation is given as follows:
Zt ¼ 4:0*Ra
(8)
Being different from the lathe turning surfaces, the profile curves of end-face milling surfaces are found to be similar to saw-
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Fig. 5. The lathe turning surface profiles with different roughness. Fig. 6. The simulated topographies of lathe turning surface.
tooth waves. It is noteworthy that this is only suitable for end-face milling and deviations are existent by using different geometrical shape of cutting tools. Therefore, by transforming the curves of the end-face milling surfaces profiles to the Fourier series, the mathematical function of surface profile is proposed based on the basic parameters of surface profiles.
! f x
N Zt X sinð2pnx=lw Þ ¼ x sin a p n¼1 n
þ
N X
lðD2Þn sinððD 1Þln xÞ
(9)
n¼1
Fig. 10 illustrates the simulated surface profiles with Ra ¼ 2.0 mm and Ra ¼ 6.2 mm. It can be found that the simulated profiles are similar to the profiles shown in Fig. 8. Besides, it is a commonly encountered situation that the surfaces are milled several times to reduce the surface roughness, which can be seen from Fig. 11(a). In that situation, it is simple to solve this matter by superposing the function of surface profile several times, which can be seen in Fig. 11(b). The mean deviation between simulated profiles and real surface profiles approximate 9%. In the end-face milling process, the surface topographies are related with the moving path and the radius of cutting tool. Therefore, the function of surface topography is given as follows:
zðx; yÞ ¼ f ðrÞ
r¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xi Þ2 þ ðy yi Þ2 r0
(10)
where, the r0 is the radius of the cutting tool, yi ¼ axi þ b (i ¼ 1,2 … n) is the path of the cutting tool. Fig. 12 illustrates the simulated topographies of the surfaces processed by end-face milling.
4. The application of surface topography in simulating TCR 4.1. The application of the surface topography in macroscale method In order to verify the precisions provided from the surface models in simulating the TCR, these two surface models are applied in a macroscale method which was proposed by Yovanovich [1]. The correlation of predicting the TCR is as follows:
1 ¼ hc ¼ 1:25k m s ðP=HÞ0:95 Rc
(11)
where k is the thermal conductivity, m is the absolute mean asperity slope, s is root mean square surface roughness, P is
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Fig. 7. The illustrations of end-face milling used in machining plane surface (a) the process (b) the micrograph of end-face milling surface.
pressure, and H is micro-hardness. Fig. 13 shows the comparisons between the experimental data of TCR and the data of TCR calculated by the Yovanovich's correlation with the present surface models. The parameters needed in equations are calculated from the surface profiles generated by the present models. The experimental data of the TCR shown in Figs. 13 and 14 are measured by a high-precision TCR instrument with the uncertainty of 5% in Ref. [26]. The measurement is conducted in a vacuum chamber to reduce the heat loss. All the same-size columnar specimens are made by Al material, which are machined by different processes but all of them have the same roughness Ra ¼ 1.6 mm. The lathe turning surfaces have a small flatness angle (nearly 0.1 ) due to the limit of lathe turning process. The wavelength of end-face milling surfaces is nearly 150 mm by controlling the feed speed in end-face milling process. It can be found that the simulated data coincide well with the experimental data, especially when using the endface milling surfaces model. However, the unflatness of the lathe turning surfaces causes big deviations in the prediction of TCR by Yovanovich's model. 4.2. The application of the surface topography in the multiscale method Since the influence of the surface topography on TCR cannot be thoroughly considered and simulated by Yovanovich's method, it is necessary to simulate the TCR with three-dimensional contact surface topographies. Therefore, a multiscale method was proposed to simulate the TCR in our previous works in Ref. [27], which was
Fig. 8. The end-face milling surface profiles with different roughness.
proved to have a high accuracy by comparing the simulated results with the experimental results. With respect to application of the multiscale method, the three-dimensional surface topographies are first generated from the experimental data of the surfaces, and then the elasticeplastic mechanics referenced is applied to simulate the surface deformations under pressure. Finally, the traditional finitedifference method and the microscopic lattice Boltzmann method (LBM) are coupled to calculate heat transfer within the microscopic
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Fig. 9. The relationships between Ra and (a) lw (b) Zt in end-face milling surfaces. Fig. 10. The simulated end-face milling surface profiles with different roughness.
region and around the micro contact points. In order to compare the numerical results with the experimental data, the gaps between two rough surfaces are considered as vacuum and the heat transfer due to radiation and convection are ignored. Heat is only transferred through the interface along the contact points. The evolution equation of the energy distribution function in the lattice Boltzmann model is [28,29]
ei ðx þ vi Dt; y þ vi Dt; z þ vi Dt; t þ DtÞ ¼ 1 Wi ei x; y; z; t þ Wi e0i x; y; z; t þ Dtqv
Wi ¼
Dt l=v
(12)
(13)
where ei is the discrete energy of electrons or phonons, l is the mean free path of electrons or phonons, and v is the group velocity of electrons or phonons. The heat flux is calculated from the following expression
qi ¼
X
vi ei ðx; y; z; tÞ
(14)
Due to the different scale length of two regions, a coupling region is constructed to transmit the physical information calculated through two methods under the constraint condition that the energy conservation is kept. When the convergence criterion is reached, which is usually that the computational residues of
temperature are less than 106, the temperature drop at the interface DT can be calculated from the averaged surface temperature, and the heat flux q can be directly obtained. Therefore, the TCR yields as
Rc ¼ DT=q
(15)
The detailed description of the multiscale method can be referred to the previous work [27]. Fig. 14 shows the comparisons between the experimental data of TCR and the data of TCR simulated by the multiscale method, in which the experimental surfaces are replaced by different kinds of surface models. The surfaces processed by lathe turning and endface milling are simulated by the present models. The random Gaussian surface is generated by FFT method and the W-M surface is generated from the W-M function [20]. All these simulated surfaces are ensured to have the same surface roughness (Ra ¼ 1.6 mm). It can be found that TCR calculated through different surface models are totally different. Especially when the exerted pressure is relatively lower (for example, below 1 MPa), the deviations between the real value of TCR and the numerical results through improper surface models are unacceptable. This is because the surface topography has a dominant influence on the TCR especially for the cases that the exerted pressure is relatively lower, so that any tiny deviations between the simulated surfaces and real surfaces may cause large errors in predicting the TCR. The bigger the
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Fig. 11. The end-face milling surface profiles obtained from (a) measurement (b) simulation. Fig. 12. The simulated surface topographies of end-face milling surfaces with different roughness and different radius of cutting tool.
deviation exists, the larger the error appears. Such as when the pressure equals to 1 MPa, the deviation between the experimental results with milling surfaces and the results calculated from a random Gaussian surface approaches to 100%. Even the W-M surface is applied in the simulation of TCR, the deviation is up to 52% under 1 MPa. Nevertheless, by using the present surface models, the max deviations between the experimental and simulated TCR of lathe turning surface and end-face milling surface, which are present under 1 MPa, are 10% and 13% respectively. The results indicate that present surface models have higher accuracy in simulating TCR than other models. For most cases of electronic cooling, in fact, the exerted pressure is smaller than 1 MPa, which means bigger deviations may exist in predicting the TCR if improper surface roughness models are used. Therefore, it is significant to choose a suitable surface model in the prediction of TCR. Fig. 15 provides the proportions of real contact area in the total contact area between a rigid smooth plane and the surfaces simulated by present models. The materials of those surfaces are Al and the wavelengths of end-face milling surfaces are referred from the real end-face milling surfaces. The deformations of contact surfaces are calculated from the contact elastoplastic mechanics which is also employed in the multiscale method [1,3]. It can be found that the regular surfaces with shorter wavelength signally have larger real contact area. This is because a shorter wavelength
Fig. 13. The comparisons between the experimental data of TCR and simulated data of TCR from Yovanovich's model.
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other existing surface models in the applications of predicting the TCR. The numerical and experimental results indicate that the presented surface models are much more valid and realistic. The main conclusions are as follows.
Fig. 14. The comparisons between the experimental data of TCR and simulated data of TCR from multiscale model.
can provide more peaks and valleys in a certain sampling length, which will obviously increase the number of contact points. Therefore, reducing the wavelength of surfaces is beneficial for reducing the TCR. Meanwhile, the surface flatness is another important factor for lathe turning surface. A large flatness angle can seriously decrease the real contact area and increase the TCR. Because the lathe turning surfaces are approximately axially symmetric, the real contact areas are only presented at the edge of surfaces or the center of the surfaces when the lathe turning surfaces have a flatness angle, which is depended on whether the lathe-turning surface is concave or convex. Moreover, a small roughness leads to increase in the real contact area, but its effect may not be so dominant compared to the wavelength and flatness in engineered regular rough surfaces. 5. Conclusions Proper characterization of surface topographies is crucial for accurately estimating the TCR between two contact surfaces. Surfaces machined by two basic engineering manufacturing processes, lathe turning and end-face milling, have been characterized and modeled. These two surface models have been compared with
Fig. 15. The proportion of real contact area in total contact area with different surfaces.
(1) The surface topographies are complex and various. It is not proper to represent the varied surface topographies just through the parameter Ra. Thus, two surface models have been proposed to characterize and simulate two types of surfaces that are machined by lathe turning and end-face milling, respectively. (2) The numerical and experimental results have indicated that big deviation exists between the real TCR and the calculated TCR under low pressure if an improper surface model is applied, and the proposed surface models are much more valid and realistic. (3) Besides reducing the surface roughness, shortening the surface wavelength and guaranteeing a fine surface flatness during processes are other two effective ways to lower the TCR between two rough surfaces. Although the wavelength, the flatness, and the roughness have significant influences on the TCR, the effect of surface roughness may not be so dominant compared to the wavelength and flatness in engineered regular rough surfaces. Acknowledgements This work is sponsored by the National Natural Science Foundation of China (No. 51225602 and No. 51336003). References [1] M.M. Yovanovich, Four decades of research on thermal contact, gap, and joint resistance in microelectronics, IEEE Trans. Compon. Packag. Technol. 28 (2005) 182e206. [2] T. Loulou, E.A. Artyukhin, J.P. Bardon, Estimation of thermal contact resistance during the first stages of metal solidification process: I-experiment principle and modelisation, Int. J. Heat Mass Transf. 42 (1999) 2119e2127. [3] M. Bahrami, M.M. Yovanovich, J.R. Culham, Thermal contact resistance at low contact pressure: effect of elastic deformation, Int. J. Heat Mass Transf. 48 (2005) 3284e3293. [4] X. Zhang, P.Z. Cong, M. Fujii, A study on thermal contact resistance at the interface of two solids, Int. J. Thermophys. 27 (3) (2006) 880e895. [5] A. Bendada, A. Derdouri, M. Lamontagne, Y. Simard, Analysis of thermal contact resistance between polymer and mold in injection molding, Appl. Therm. Eng. 24 (2004) 2029e2040. [6] C. Courbon, T. Mabrouki, J. Rech, D. Mazuyer, E. D'Eramo, On the existence of a thermal contact resistance at the tool-chip interface in dry cutting of AISI 10 45: formation mechanisms and influence on the cutting process, Appl. Therm. Eng. 50 (2013) 1311e1325. [7] C. Zhang, Y. Chen, Z. Deng, M. Shi, Role of rough surface topography on gas slip flow in microchannels, J. Phys. Rev. E 86 (1) (2012) 016319. [8] Y. Chen, C. Zhang, M. Shi, G.P. Peterson, Optimal surface fractal dimension for heat and fluid flow in microchannels, J. Appl. Phys. Lett. 97 (8) (2010) 084101. [9] E.S. Gadelmawla, M.M. Koura, T.M.A. Maksoud, I.M. Elewa, H.H. Soliman, Roughness parameters, J. Mater. Process. Technol. 123 (2002) 133e145. [10] T.R. Thomas, B.G. Rosen, N. Amini, Fractal characterisation of the anisotropy of rough surfaces, Wear 232 (1999) 41e50. [11] Z.Z. Guan, M.H. Ye, X.C. Yin, X.H. Luo, Recognition of surface roughness based on fractal theory and the microscopic images, Appl. Mech. Mater. 241e244 (2013) 3030e3033. [12] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, in: Proc. Roy. Soc. London, 1966, pp. 300e319. [13] X. Zhang, P. Cong, S. Fujiwara, M. Fujii, A new method for numerical simulation of thermal contact resistance in cylindrical coordinates, Int. J. Heat Mass Transf. 47 (2004) 1091e1098. [14] D.E. Newland, An Introduction to Random Vibration and Spectral Analysis, second ed., Longman, London, 1984. [15] Y.Z. Hu, K. Tonder, Simulation of 3-D random rough surface by 2-D digital filter and Fourier analysis, Int. J. Mach. Tool. Manuf. 32 (1e2) (1992) 83e90. [16] J.J. Wu, Simulation of rough surfaces with FFT, Tribol. Int. 33 (2000) 47e58. [17] J. Lopez, G. Hansali, H. Zahouani, J.C. Lebosse, T. Mathia, 3D Fractal-based characterisation for engineered surface morphology, Int. J. Mach. Tool. Manuf. 35 (2) (1995) 211e217.
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