EFFECT OF SURFACE ASPERITY TRUNCATION ON THERMAL CONTACT CONDUCTANCE

Transcription

1 EFFECT OF SURFACE ASPERITY TRUNCATION ON THERMAL CONTACT CONDUCTANCE Fernando H. Milanez *, M. M. Yovanovich, J. R. Culha Microelectronics Heat Transfer Laboratory Departent of Mechanical Engineering University of Waterloo Waterloo, Ontario, Canada, NL 3G1 Phone: (519) Fax: (519) E-ail: ABSTRACT This paper presents studies on theral contact conductance at light contact loads. Surface profiloetry easureents are presented which show that actual surface asperity height distributions are not perfectly Gaussian. The highest asperities are truncated, leading the existing theral contact conductance odels to underpredict experiental data. These observations have been incorporated into odifications of existing contact conductance odels. The preliinary odel has been copared against theral contact conductance data presented in the open literature, and good agreeent is observed. The truncation leads to an enhanceent of theral contact conductance at light contact pressures. The truncation is a function of the roughness level: the rougher the surface, the ore truncated the surface height distribution. KEY WORDS: light contact pressures, truncated Gaussian odel, bead blasted surfaces, ean separation gap NOMENCLATURE A contact area, a ean contact spot radius, sei-ajor elliptic contact spot axis, b sei-inor elliptic contact spot axis, diensionless contact conductance (Eq. 17) c 1 Vickers icrohardness correlation coefficient, Pa c Vickers icrohardness correlation coefficient E Young s odulus, Pa E equivalent Young s odulus, Pa, (Eq. 4) H c plastic contact hardness, Pa h c contact conductance, W/²K k s haronic ean theral conductivity, W/K k A k B /(k A +k B ) ean absolute roughness profile slope Nb Niobiu Ni Nickel * Research Assistant, ilanez@htlab.uwaterloo.ca Distinguished Professor Eeritus, yov@htlab.uwaterloo.ca Associate Professor, Director MHTL, rix@htlab.uwaterloo.ca n density of contact spots, - P apparent contact pressure, Pa p probability density function SS Stainless Steel v iniu to axiu slope ratio, ( n / x ) Y ean planes separation, Zr Zirconiu Greek sybols f theral constriction factor l noralized ean planes separation,, (Y/σ) n Poisson s ratio s RMS of surface roughness, Subscripts A,B contacting bodies a apparent r real n iniu x axiu trunc truncation TG Truncated Gaussian odel INTRODUCTION Since actual surfaces present deviations fro their idealized geoetrical for, known as roughness and waviness, when two solids are put into contact they will touch only at their highest asperities. The heat transfer across the interface of real solids is not as effective as if the solids were perfectly sooth and flat. A resistance to heat flow, known as theral contact resistance, appears at the interface between solids. Heat transfer across the interface between two solids has been the subject of study by various researchers over any years. Contact heat transfer has any applications in engineering, such as ball bearings, icroelectronic chips and nuclear fuel eleents. When two solids are pressed together, the contacting asperities will defor and for sall spots of solid-solid contact. In the reaining portion of the apparent contact area the bodies are separated by very thin gaps. Heat transfer between two contacting solids can take place by three different odes: conduction through the contact spots, radiation through the

2 gap in the reaining part of the apparent area and conduction through the gas that fills the gap. These heat transfer odes are treated separately and the su of the conductances associated with each of these heat transfer odes is called joint conductance. This work is focused on the contact conductance, which is due to conduction through the contact spots. A theral contact conductance odel is generally coposed of three odels: theral, geoetrical and echanical deforation odels. The theral odel predicts the contact conductance for a given set of contact paraeters: shape, size and nuber of contact spots. These contact paraeters are obtained fro a particular echanical deforation odel, which can be elastic, plastic or elastoplastic. The deforation odel requires a geoetric odel of the surface in order to be able to predict the contact paraeters. Since it is extreely difficult to predict or to characterize the geoetry of actual surfaces by deterinistic eans, statistical analysis has been generally eployed. It is coonly assued that the surface heights of actual surfaces follow the Gaussian distribution. The Gaussian height distribution odel has been used in several theral contact conductance odels, such as the Cooper et al. [1] and the Greenwood and Willliason [] odels, as well as a nuber of other odels derived fro these two. It has been reported in the literature [3-6] that these theral contact odels tend to underpredict experiental data at light contact pressures, and as the pressure increases the odels and easureents agree. The cause of this behavior was unclear up to now and this subject is addressed here. This work presents evidence that the cause for the odels to underpredict the experiental data at light contact pressures is the truncation of the highest asperities. The Gaussian odel fails to predict accurately the contact paraeters at light contact pressures. A new surface geoetric odel, called Truncated Gaussian, is proposed here. Modifications are incorporated to the well-established theral contact conductance odels in order to take into account the truncation of the height distribution of actual surfaces. The next section provides a review of soe of the theral contact conductance odels available in the literature. After that, the asperity truncation proble is identified and the new odels are presented. The new odels are copared against experiental data available in the literature. REVIEW OF EXISTING MODELS Most of the theral contact conductance odels available in the literature eploy the sae theral odel. Cooper et al. [1] first presented the solution for the theral part of the contact conductance proble. They developed a theral odel for the contact between conforing isotropic rough surfaces, such as those obtained by lapping and bead blasting. The contact between surfaces possessing these features generates approxiate circular contact spots randoly distributed over the apparent contact area. The theral contact conductance between conforing isotropic rough surfaces is given by [1,8]: hc ksn a (1) ( 1- A ) 1. 5 r Aa where n is the density of contact spots per unit apparent area, a is the ean contact spot radius and A r /A a is the real-toapparent contact area ratio. The ter in the denoinator of the expression above is called the theral constriction factor and takes into account for the constriction resistance of the heat flow near the contacting spots. DeVaal [7] extended the Cooper et al. [1] isotropic odel to the contact between anisotropic surfaces, such as those obtained by grinding. The contact between such surfaces present elliptical spots rather than circular. The theral contact conductance between conforing anisotropic rough surfaces is given by: hc ksn p a b () f( v, Ar Aa ) where a and b are respectively the ean sei-ajor and seiinor axis of the elliptic contact spots, φ is the theral constriction factor and v is the ratio between the iniu and the axiu slopes of the surface v n / x. DeVaal [7] presents the expressions to copute the theral constriction factor φ in detail. The contact paraeters a, b, n and A r /A a, appearing in Eqs. (1) and (), are obtained fro the surface geoetry and the deforation odels. By assuing that the surface heights and slopes are independent and follow the Gaussian distribution, as well as assuing that the surfaces undergo plastic deforation, Cooper et al. [1] presented an analysis to derive expressions for the contact paraeters. Yovanovich [8] presented the contact paraeter expressions for the isotropic plastic odel in a ore convenient for. Mikic [9] extended the Cooper et al. [1] plastic odel for the case of elastic deforation by assuing that the asperities are spherical near the tips and using results fro the Hertz elastic contact theory. DeVaal [7] developed the contact paraeter expressions for the contact between anisotropic surfaces under plastic deforation. The expressions for the contact paraeters for all these odels are shown in Table 1. The diensionless plastic contact pressure P/H c appearing in Eq. (8) of Table 1 can be coputed using the odel proposed by Song and Yovanovich [10]: P H Ø Œ c Œº c1 P ( 1.6s ) c ø œ œß c The equivalent Young s odulus E appearing in Eq. (9) of Table 1 is: 1-u A 1 -u B E + Ł E A EB ł (3) (4)

3 Sridhar and Yovanovich [5] ade an extensive review of the theral contact conductance odels available in the literature. Most of the odels showed siilar results as the odels reviewed in this section. The authors also copared the odels against experiental data and concluded that the odels based on the Cooper et al. [1] odel, presented in this section, are very accurate especially at high contact pressures. At light loads, the odels tend to underpredict the experiental data. In the next section, it will be shown that the assuption of Gaussian asperity height distribution leads to underestiation of theral contact conductance at light contact loads. A new odel, called Truncated Gaussian odel, is proposed here as the odified geoetry odel. Surface Isotropic Anisotropic (plastic deforation only) Table1. Contact Paraeter Expressions n 1 16 k a p l Y s Ar k n Aa 1 16 Contact Paraeters exp Ł s ł erfc, 1, (- l ) ( l ) s l l exp erfc Ł ł Ł ł erfc P Hc P E for plastic for elastic exp Ł s ł erfc ( A A ) r (- l ) ( l ) a for plastic for elastic s l l a exp erfc p n Ł ł Ł ł s l l b exp erfc p x Ł ł Ł ł l Y s A r Aa P H c erfc ( A A ) r a (5) (6) (7) (8) (9) (10) (11) (1) (13) (14) (15) ACTUAL SURFACE HEIGHT DISTRIBUTIONS The assuption of Gaussian height distribution was first analyzed in ore detail by Greenwood and Williason []. They easured surface roughness profiles of bead blasted aluinu surfaces and concluded that the Gaussian distribution is a good approxiation at least in the range of surface heights between ±σ, where σ is the RMS of the heights of the profile. Figure 1 shows easured surface height distributions obtained fro three different profiles of a typical bead blasted SS 304 surface. The Gaussian odel is also plotted in this graph and it is in good agreeent with the easureents for surface heights in the range, especially in the range of 1.5 z/σ 3.7. In typical engineering applications, the ean separation between the contacting surfaces lies in this range. If this surface is brought into contact with a flat lapped surface, for instance, under a contact pressure of P/H c 10-6, which is a very light contact pressure, one can use Eqs. (7) and (8) to calculate a ean separation gap of Y 4.7σ, according to the Gaussian geoetry odel. However, the easured profile height distributions do not show asperities higher than 3.7σ. The profile height distributions follow the Gaussian distribution up to z 3.7σ, where they are truncated. This is expected to be the axiu ean plane separation under the lightest contact load. Therefore, the Gaussian odel sees to overpredict the ean plane separation under these circustances. Since the actual ean plane separation is saller than predicted by the Gaussian odel, the actual theral contact conductance will be larger than predicted. p (z/σ) SS 304 Bead Blasted Gaussian odel profile 1 profile profile z/σ Fig. 1: Measured height distributions of three different profiles of a SS 304 bead blasted surface and coparison with the Gaussian odel

4 Several other researchers [6, 7, 11, aong others] also easured profile height distributions of actual achined surfaces and concluded that the Gaussian odel is a good approxiation. They presented actual surface profile height easureents truncated between 3 and 4σ, but they did not observe the truncation. Only Song [11] identified the consequences of the asperity truncation on the contact conductance proble. He studied the gap conductance proble and proposed a odified expression to copute the ean plane separation between the contacting surfaces. This expression was derived assuing that the asperity height distribution follows the Gaussian odel but is truncated at soe height level, called here. The odified expression for the ean plane separation λ TG is written in the following for: Y Ø ø TG Ar ltrunc l TG erfc Œ + erfc œ (16) s Œº Aa Ł łœß where, TG stands for Truncated Gaussian and is the noralized height above which the Gaussian distribution is truncated. Therefore, is the height of the highest (truncated) asperities, while λ TG is the separation between two surfaces in contact according to the Truncated Gaussian odel. When the real-to-apparent area ratio A r /A a is sall, that is, the contact pressure is sall (approxiately less than ), the second ter between the square brackets of Eq. (16) has the sae order of agnitude as A r /A a. That eans that the truncation of the asperities is very iportant below this contact pressure level. As the contact pressure increases A r /A a becoes uch larger than erfc( l trunc ), and Eq. (16) can be approxiated by Eqs. (7) and (14), which represent the fully Gaussian odel. Physically, this eans that as the pressure increases, ore and ore asperities coe into contact, and as a consequence, the effect of the very few truncated asperities becoes negligible. Song [11] used Eq. (16) in his gap conductance odel and when he copared the results against experiental data he observed good agreeent. However, when he tried to use the odified ean plane separation expression to predict contact conductance data, the results of the TG odel were uch worse than the fully Gaussian odel. The present authors now believe that Song [11] was not successful in applying the TG geoetry odel in the contact conductance odel because he used the sae expression for the ean contact spot radius as the fully Gaussian odel (Eq. 6). A new expression for the ean contact spot radius is proposed in this work and is presented in the next section. TRUNCATED GAUSSIAN CONTACT CONDUCTANCE MODEL The asperity height distributions shown in Fig. 1 were obtained fro a bead blasted surface, but the authors also analyzed ground and lapped surfaces and found that the results were very siilar to bead blasted surfaces: the distributions were truncated at soe height level between 3 and 4σ, approxiately. These coonly eployed achining processes do not generate asperities higher than this level. The reason for this is still unclear. In the Truncated Gaussian odel, it is assued that the higher asperities are shorter than predicted by the fully Gaussian odel, but they are not issing. The total nuber of asperities reains the sae, although the highest asperities are truncated. Based on this odel, the expression for the contact spot density n, Eqs. (5) and (11), are still valid. The correct expression to copute the ean separation gap is now Eq. (16), instead of Eqs. (7) and (14). Also, the ean contact spot radius a (Eq. 6) ust be corrected using the following expression: a TG ( ltrunc ) ( ltg ) erfc a 1 - (17) erfc where a TG is the ean contact spot radius according to the Truncated Gaussian odel and a is the ean contact spot radius according to fully Gaussian odel (Eq. 6). The expression above was obtained by solving ( n p atg Ar / Aa ) for a TG, where Ar / Aa is obtained fro Eq. (16). The expressions for the sei-ajor and sei-inor axes for the ean elliptical contact spot of the anisotropic plastic odel, Eqs. (1) and (13), becoe siilar to the above expression. The real-to-apparent contact area ratio, the last required contact paraeter is coputed in the sae way as before (Eqs. 7, 8 and 15) because these expressions are obtained fro force balances and do not depend on the geoetric odel used. The next section presents a coparison between the TG contact conductance odel and experiental data available in the literature. COMPARISON BETWEEN TG MODEL AND EXPERIMENTAL DATA Hegazy [4] collected a large quantity of theral contact conductance data between lapped and bead blasted speciens of SS 304, Ni 00, Zr-4 and Zr-Nb possessing various roughness levels. He copared his data with the Cooper et al. [1] isotropic plastic odel and noticed that at light contact pressures the odel underpredicts the data for all the aterials and roughness levels tested. He proposed an explanation for this unexpected behavior as being a consequence of theral strain and flatness deviations of the test speciens. However, he clearly stated that this explanation was not definitive and further work was needed to clarify this phenoenon. This issue is addressed here and is explained in the light of the new Truncated Gaussian geoetric odel. Figures through 4 show the theral contact conductance experiental data obtained by Hegazy [4] for different etals and different roughness levels. The TG odel is also plotted in these graphs as a set of curves for different truncation levels

5 because Hegazy [4] did not provide inforation about the surface height distribution truncation level of his test speciens. The plots show the diensionless theral contact conductance as a function of diensionless contact pressure P/H c. The diensionless contact pressure was coputed using Eq. (3), and the diensionless contact conductance is defined as: hc s Cc (18) ks The lowest curve of each graph is for +, which is equivalent to the fully Gaussian odel. For practical purposes, a value of 5 is sufficient for the TG odel to coincide with the fully Gaussian odel. The curve for the fully Gaussian odel appears as a straight line in the log-log plots. The curves for the TG odel for <5 are concave: they lie above the fully Gaussian odel at light contact pressures and tend to the fully Gaussian odel as the contact pressure increases. The higher the truncation level (saller ), the larger the departure of the TG odel fro the fully Gaussian odel. The TG odel sees to predict the experiental data trend very well. The experiental data lie between the curve of 3.4 and the curve of the fully Gaussian Model ( + ). LEVELS OF TRUNCATION OF REAL SURFACES Figure 5 shows a graph of the values of that best fit Hegazy s [4] data as a function of σ/. Different are observed for distinct etals possessing the sae σ/, although in general decreases with σ/. The values of for different etals are scattered between 3.5 and 4.5 for sall σ/ and tend to approxiately 3.5 for large σ/. σ h c k s Data for SS Hegazy (1985) Hegazy (1985) Data for SS 304 Sridhar CorrelationsC 1 and C easured for C 1 and C (Eqs.7-8) C 1 corrected C 1 corrected RMS RMS 16% 19% σ/ [µ] Isotropic Isotropic Model Model (Eq. (Eq.1) σ/6.64µ - RMS14% 6.64 σ/6.64µ - RMS13% σ/3.6µ σ/3.6µ - RMS% - RMS7% 3.36 σ/40.7µ σ/40.7µ - RMS14% - RMS17% 40.7 σ/57.63µ σ/57.63µ - RMS13% - RMS17% P/H P/H cc c Fig. : Theral contact conductance data for SS 304 fro Hegazy [4] and coparison against the TG odel σ h c k s Hegazy Hegazy Data for (1985) (1985) Ni 00 Data Data - Hegazy for forni Ni00 (1985) Sridhar Correlations C 1 and C easured for C 1 and C (Eqs.7-8) RMS RMS 18% 15% Prediction Prediction (Eq.1) (Eq.1) 8. σ/8.µ σ/8.µ - RMS17% - RMS15% σ/18.05µ σ/18.05µ - RMS17% - RMS15% σ/.55µ - RMS%.55 σ/.55µ - RMS18% σ/41.8µ σ/41.8µ- RMS16% - RMS14% 41.8 σ/59.83µ σ/59.83µ- RMS18% - RMS13% σ/ [µ] 10 - P/H c c Fig. 3: Theral contact conductance data for Ni 00 fro Hegazy [4] and coparison against the TG odel σ h c k s Hegazy Data Hegazy for(1985) (1985) Zr-alloys Data Data - Hegazy forzr-alloys Zr-alloys (1985) Sridhar Correlations C 1 and C easured for C 1 and C (Eqs.7-8) RMS RMS 0% 50% Zr-Nb Prediction (Eq.1) 1.43 Zr-.5wt%Nb -RMS1% -RMS35% Zr-4 -RMS0% 4.34 Zr-4 -RMS60% σ/ [µ] 10 - P/H c c Fig. 4: Theral contact conductance data for Zr-alloys fro Hegazy [4] and coparison against the TG odel The question now is how to predict for different etals and for different roughness levels? Given the difficulty in odeling analytically the bead blasting process or any other achining process, it sees very difficult to predict theoretically. Another option is to easure using a profiloeter, the sae equipent used to easure σ and. Most of the profiloeters available coercially easure a roughness paraeter that represents the height of the highest peak of the profile, generally known as R p [µ]. Song [11] used the R p collected fro a single profiloeter trace as a easure for the truncation ( R p /σ). However, it looks

6 very unlikely that a single trace is able to pass through the peak of the highest asperity of the surface. On the other hand, if one decides to take several different profiles and one of the traces coes across an asperity uch higher than the others, this single asperity could not represent the truncation level of the entire surface either because one single asperity can not support the entire contact load alone, even a very light contact load. In an effort to better understand the truncation of real surface height distributions, the authors decided to undertake a ore detailed study of the surface generation process. The authors chose the bead blasting process for this study for various reasons. One can start fro a flat lapped surface and by bobarding the surface with glass beads at high speeds, one can grow the asperities on the surface at practically any desired RMS roughness level (R q ). Several bead blasting paraeters, such as bead size, air pressure and exposure tie can be adjusted in order to generate the desired roughness level. Moreover, this process has been applied very successfully by other theral contact resistance researchers [3, 4, 6, 11, aong others] to generate randoly distributed asperities on the surface without affecting its flatness, which is very iportant in order to guarantee that the surface geoetry is in accordance with the geoetric odel. Truncation of bead blasted surfaces The bead blasting study consisted of easuring the roughness paraeters R p (axiu profile height), σ (profile height RMS) and (profile ean absolute slope) as well as the general trend of the asperity height distribution as a function of bead blasting exposure tie between 1 and 16 inutes. Three different blasting pressures (10, 0 and 40 psi) and three different glass bead size ranges (1580 µ, µ and µ) were used. Four profiles were assessed over each generated surface, resulting in a total of 136 profile easureents. The iniu and axiu σ/ ratios easured during the tests were 1 and 44 µ, respectively. The first iportant conclusion fro this study was that the general trend of the surface height distribution was Gaussian independent of the blasting paraeter cobinations analyzed. Both the profile height RMS (σ) and ean absolute slope () as well the ratio σ/ increase with increasing exposure tie and blasting pressure, as expected, especially for the saller glass beads. For the largest bead size range tested, the exposure tie did not significantly affect either σ or. The blasting pressure was found to be the ost iportant paraeter in deterining the roughness level. The ain goal of the bead blasting study was to analyze the truncation levels of the surface height distributions for every cobination of blasting paraeters. It was found that the easured R p /σ (noralized axiu profile height) presented very different values for different profiles collected fro the sae surface. The largest R p /σ difference easured fro different profiles on a single surface was ore than 100%. The variation between R p /σ values easured on the sae surface was uch larger than the variation between the ean values of R p /σ fro different surfaces. Also, the average of the four R p /σ readings on each surface varied randoly aong different surfaces. In other words, the R p /σ ratio seeed not to be controlled by any of the bead blasting paraeters. The authors then decided to verify whether the easured R p /σ values could be related to the roughness level of the surface, as observed fro the coparison between the TG odel and the experiental data [4], independent of the blasting paraeters eployed. Figure 6 shows a plot of all 136 easured R p /σ values as a function of σ/ for all cobinations of blasting paraeters analyzed. The R p /σ values lie in a large band, which sees to becoe narrower as σ/ increases. The ean value of R p /σ also sees to experience a slight decrease with increasing σ/. These observations are in accordance with the previous conclusion fro the coparison between the TG odel and the theral data fro Hegazy [4] that decreases with increasing σ/ (Fig. 5). Also, the observation fro the theral tests that is larger than 3.5 also is consistent with the R p /σ easureents presented in Fig. 6. The question of how to predict fro roughness easureents still reains unanswered. However, it is clear that a single profile easureent is not sufficiently accurate to easure because there are only a few truncated asperities and the probability of a single profile trace capture at least one of the truncated asperities is very sall. As it can be see fro Figs. through 4, the TG odel is very sensitive to the value of, and the easured R p /σ present large variations for the sae surface. The authors believe that the best way to obtain inforation on the correct truncation level is fro theral contact conductance experients Truncation values that best fit Hegazy's (1985) data σ/ SS 304 Ni 00 Zr-Nb Zr-4 Fig. 5: values that best fit TG odel to experiental data fro Hegazy [4] versus σ/

7 8 7 Bead Blasted SS 304 ACKNOWLEDGEMENTS The lead author would like to acknowledge the Brazilian Federal Agency for Post-Graduate Education-CAPES for supporting this project. The second and third authors would also like to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada. R p /σ 6 5 REFERENCES [1] Cooper, M., Mikic, B. and Yovanovich, M. M., Theral Contact Conductance, Journal of Heat and Mass Transfer, Vol.1, pp , σ/ [µ] Fig. 6: R p /σ versus σ/ for bead blasted SS 304 surfaces Siilar to the ethod used to obtain the values presented in Fig. 5, by inputting the value that best fits the TG contact conductance odel to the experiental data one can extract inforation on. Therefore, ore theral contact conductance data need to be generated for this purpose, especially in the light contact pressure range. SUMMARY AND CONCLUSIONS The observation that the existing theral contact conductance odels underpredict experiental data at light contact pressures is reported in several previous theral contact conductance studies. This work presents evidence that the well-accepted Gaussian surface height distribution geoetry odel causes the theral contact conductance odels to underpredict the experiental data at light contact pressures. Surface height distribution easureents show that although the distributions follow the Gaussian odel for surface heights larger than 1.5σ, the distributions are truncated generally between 3 and 4σ. A new theral contact conductance odel is proposed based on the Truncated Gaussian geoetry odel. The preliinary results show that the new odel predicts the data trend very well. The new odel requires another surface paraeter, called, in addition to the paraeters σ and. It is not clear at this point how to obtain this third surface paraeter fro profiloeter traces. The use of theral contact conductance data sees to be the best way to obtain this inforation. The truncation of the surface height distribution and its effects on the theral contact conductance proble is a very iportant finding but also very recent. Additional studies are needed in order to clarify the questions raised here, especially regarding to the prediction and/or the control of the truncation level of actual achined surfaces. [] Greenwood, J. A. and Williason, J. B. P, Contact of Noinally Flat Surfaces, Proceedings of the Royal Society of London, pp , [3] Milanez, F. H., Culha, J. R., Yovanovich, M. M., Experiental study on the Hysteresis Effect of Theral Contact Conductance at Light Loads, 40 th AIAA Aerospace Sciences Meeting & Exhibit, January 147, Reno, NV, AIAA , 00. [4] Hegazy, A. A., Theral Joint Conductance of Conforing Rough Surfaces: Effect of Surface Microhardness Variation, Ph. D. Thesis, University of Waterloo, Ontario, Canada, [5] Sridhar, M. R. and Yovanovich, M. M., Theral Contact Conductance of Tool Steel and Coparison with Model, International Journal of Heat and Mass Transfer, Vol. 39, No. 4, pp , [6] Nho, K. M., Experiental Investigation of Heat Flow Rate and Directional Effect on Contact Conductance of Anisotropic Ground/Lapped Interfaces, Ph. D. Thesis, University of Waterloo, Ontario, Canada, [7] DeVaal, J. W., Theral Joint Conductance of Surfaces Prepared by Grinding, Ph. D. Thesis, University of Waterloo, Ontario, [8] Yovanovich, M. M., Theral Contact Correlations, Spacecraft Radiative Heat Transfer and Teperature Control, Edited by T. E. Horton, Progress in Astronautics and Aeronautics, Vol. 83, NY, [9] Mikic, B. B., Theral Contact Conductance; Theoretical Considerations, Journal of Heat and Mass Transfer, Vol. 17, pp. 054, Pergaon Press, [10] Song, S. and Yovanovich, M. M., Relative Contact Pressure: Dependence on Surface Roughness and Vickers Microhardness, Journal of Therophysics and Heat Transfer, Vol., No. 4, pp , [11] Song, S., Analytical and Experiental Study of Heat Transfer Through Gas Layers of Contact Interfaces, Ph. D. Thesis, University of Waterloo, Ontario, 1988.