Thermal Contact Resistance of Non- Conforming Rough Surfaces, Part 2: Thermal Model

Transcription

1 AIAA Thermal Contact eitance of Non- Conforming ough Surface, Part : Thermal Model M. Bahrami, J.. Culham, M. M. Yovanovich and G. E. Schneider Microelectronic Heat Tranfer aboratory Department of Mechanical Engineering Univerity of Waterloo, Waterloo Ontario, Canada N 3G 36th AIAA Thermophyic Conference Meeting and Ehibit June 3-6, 3/Orlando, Florida For permiion to copy or republih, contact the American Intitute of Aeronautic and Atronautic 8 Aleander Bell Drive, Suite 5, eton, VA 9

2 Thermal Contact eitance of Non-Conforming ough Surface, Part : Thermal Model M. Bahrami,J..Culham,M.M.Yovanovich andg.e.schneider Microelectronic Heat Tranfer aboratory Department of Mechanical Engineering Univerity of Waterloo, Waterloo Ontario, Canada N 3G Thermal contact reitance (TC) of non-conforming rough urface i tudied and a new analytical model i developed. TC i conidered a the uperpoition of macro and micro component accounting for the effect of urface curvature and roughne, repectively. The effect of roughne, load and radiu of curvature on TC are invetigated. It i hown that there i a value of urface roughne that minimize the TC. Simple correlation for determining TC, uing the general preure ditribution introduced in Part of thi tudy, are derived which cover the entire range of TC ranging from conforming rough to mooth pherical contact. The comparion of the preent model with more than 7 eperimental data point how good agreement in the entire range of TC. Nomenclature A area, m a radiu of contact, (m) b flu tube radiu, (m) a relative radiu of macrocontact, a /a Hz b pecimen radiu, (mm) B relative macrocontact radiu, a /b CS Carbon Steel c Vicker microhardne coefficient, (GPa) c Vicker microhardne coefficient, ( ) d v Vicker indentation diagonal, (µm) dr increment in radial direction, (m) E equivalent elatic modulu, (GPa) F eternal force, (N) h contact conductance, W/m K H mic microhardne, (GP a) H c (.6σ /m) c,(gp a) k thermal conductivity, (W/mK) m mean abolute urface lope, ( ) n number of microcontact P preure, (Pa) P relative maimum preure, P /P,Hz Q heat flow rate, (W ) thermal reitance, (K/W ) r, z cylindrical coordinate.95/ (.7c ) Y mean urface plane eparation, (m) Ph.D. Candidate, Department of Mechanical Engineering. Aociate Profeor, Director, Microelectronic Heat Tranfer aboratory. Ditinguihed Profeor Emeritu, Department of Mechanical Engineering. Fellow AIAA. Profeor, Department of Mechanical Engineering. Aociate Fellow AIAA. Copyright c 3 by M. Bahrami, J.. Culham, M. M. Yovanovich and G. E. Schneider. Publihed by the American Intitute of Aeronautic and Atronautic, Inc. with permiion. of Greek α non-dimenional parameter, σρ/a Hz γ general preure ditribution eponent δ maimum urface out-of-flatne, (m) ε flu tube relative radiu, a /b η microcontact denity, m κ H B / H BGM λ dimenonle eparation, Y/ σ ξ dimenionle radial poition, r/a ψ preading reitance factor ρ radiu of curvature, (m) σ MS urface roughne, (µm) τ non-dimenional parameter, ρ/a Hz Ω dimenionle parameter Subcript value at origin, olid, a apparent B Brinell b bulk c critical Hz Hertz j joint macro mac macro mic micro r real macro, olid v Vicker Introduction Heat tranfer acro interface formed by mechanical contact of two non-conforming rough olid occur in a wide range of application, uch a: microelec- American Intitute of Aeronautic and Atronautic Paper 3-498

3 Z Q i T temperature profile DT Q S Q i T T microcontact contriction Q F body a Q body F macro contriction reitance Geometrical Analyi Macro-Geometry Mechanical Analyi Macro-Contact Coupled (Bulk Deformation) Thermal Analyi Macro-Contriction eitance Superpoition Thermal Joint eitance Micro-Geometry Micro-Contact (Aperity Deformation) Micro-Contriction eitance Fig. Contact of two pherical rough urface in a vacuum tronic cooling, pacecraft tructure, atellite bolted joint, nuclear engineering, ball bearing, and heat echanger. Due to roughne of the contacting urface, real contact in the form of microcontact occur only at the top of urface aperitie, which are a mall portion of the nominal contact area, normally le than 5 percent. A a reult of curvature or out-of-flatne of the contacting bodie, a macrocontact area i formed, theareawherethemicrocontactareditributed. Thermal energy can be tranferred between contacting bodie by three different mode, i) conduction, through the microcontact, ii) conduction, through the intertitial fluidinthegapbetweentheolid,and iii) thermal radiation acro the gap if the intertitial ubtance i tranparent to radiation. According to Clauing and Chao radiation heat tranfer acro the interface remain mall a long a the body temperature are not too high, i.e., le than 7 K, and in mot typical application can be neglected. In thi tudy the urrounding environment i a vacuum, thu the only remaining heat tranfer mode i conduction at the microcontact. A illutrated in Fig., heatflow i contrained to pa through the macrocontact, and then, in turn through the microcontact. Thi phenomenon lead to a relatively high temperature drop acro the interface. Two et of reitance in erie can be ued to repreent the thermal contact reitance for a joint in a vacuum: the large-cale or macrocopic contriction reitance,, and the mall-cale or microcopic contriction reitance, 3 j mic mac () Many theoretical model for determining thermal contact reitance (TC) have been developed for two limiting cae, i) conforming rough, where contact- Fig. Thermal contact problem ing urface are aumed to be perfectly flat, and ii) elatocontriction, where the effect of roughne i neglected, i.e., contact of two mooth pherical urface. The above limiting cae are implified cae of real contact ince engineering urface have both out-offlatne and roughne imultaneouly. A hown in Fig., TC problem baically conit of three eparate problem: ) geometrical, ) mechanical, and 3) thermal, each ub-problem alo include a micro and macro cale component. The heart of TC i the mechanical analyi. A mechanical model wa developed and preented in the Part of thi tudy. 4 The mechanical analyi determine the macrocontact radiu and the effective preure ditribution for the large-cale contact problem. While the microcontact analyi give the local eparation between the mean plane of the contacting bodie, the local mean ize and the number of microcontact. The reult of the mechanical analyi are ued in the thermal analyi to calculate the microcopic and macrocopic thermal contriction reitance. A few analytical model for contact of two nonconforming rough urface eit in the literature. Bahrami et al. 5 reviewed eiting analytical nonconforming rough TC model and howed through comparion with eperimental data that none of the eiting model cover the above mentioned limiting cae and the tranition region in which both roughne and out-of-flatne are preent and their effect on TC are of the ame importance. Theoretical Background Thermal preading reitance i defined a the difference between the average temperature of the contact area and the average temperature of the heat ink, of American Intitute of Aeronautic and Atronautic Paper 3-498

4 d a) contact of non-conforming rough urface ρ c) rough phere-flat contact, effective radiu of curvature σ σ ρ ρ b) contact of two rough pherical egment ρ σ b effective radiu and roughne Fig. 3 Geometrical modeling which i located far from the contact area, divided by the total heat flow rate Q, 6 T/Q. Thermal conductance i defined in the ame manner a the film coefficient in convective heat tranfer, h Q/ ( TA a ). Conidering the curvature or out-of-flatne of contacting urface in a comprehenive manner i very comple becaue of it random nature. Certain implification mut be introduced to decribe the macrocopic topography of urface uing a few parameter. Theoretical approache by Clauing and Chao, Mikic and ohenow, 3 Yovanovich, Nihino et al., 7 and ambert and Fletcher 8 aumed that a pherical profile might approimate the hape of the macrocopic nonuniformity. According to ambert 9 thi aumption i jutifiable, becaue nominally flat engineering urface are often pherical, or crowned (conve) with a monotonic curvature in at leat one direction. The relationhip between the radiu of curvature and the maimum out-of-flatne i ρ b () δ where δ i the maimum out-of-flatne of the urface. A dicued in Bahrami et al., 4 the contact between two Gauian rough urface can be approimated by the contact between a ingle Gauian urface, having the effective urface characteritic, placed in contact with a perfectly mooth urface. The contact of two phere can be replaced by a flat in contact with a phere incorporating an effective radiu of curvature, effective urface roughne and urface lope a given in Eq. (3) σ p σ σ and m p m m ρ ρ ρ (3) Figure 3 ummarize the geometrical procedure, which ha been widely ued for modeling the actual contact between non-conforming rough bodie. When two non-conforming random rough urface are placed in mechanical contact, many microcontact are formed within the macrocontact area. Microcontact are mall and located far from each other. Thermal contact model are contructed baed on the premie that inide the macrocontact area a number of parallel cylindrical heat channel eit. The real hape of microcontact can be a wide variety of ingly connected area depending on the local profile of the contacting aperitie. Yovanovich et al. tudied the teady tate thermal contriction reitance of ingly connected planar contact of arbitrary hape. By uing an integral formulation and a emi-numerical integration proce applicable to any hape, they propoed a definition for thermal contriction reitance baed on the quare root of the contact area. A non-dimenional contriction reitance baed on the quare root of area wa propoed, which varied by le than 5% for all hape conidered. Yovanovich et al. concluded that the real hape of the contact wa a econd order effect, and an equivalent circular contact, where urface area i preerved, can be ued to repreent the contact. A the baic element for macro and micro thermal analyi, thermal contriction of the flu tube wa employed by many reearcher. Figure 4 illutrate two flu tube in a erie contact. A flu tube conit of a circular heat ink or ource, which i in perfect thermal contact with a long tube. Heat enter the tube fromtheourceandleavethetubeattheotherend. Cooper et al. 3 propoed a imple accurate correlation for calculating the thermal preading reitance of the iothermal flu tube, (ee Bahrami et al. 5 for more detail): flu tube flu tube ψ (ε) k a ( ε).5 k a (4) where ε a/b, k k k / (k k ), and ψ ( ) ithe preading reitance factor. In Eq. (4), it i aumed that the radii of two contacting bodie are the ame, i.e., b b b. In general cae where b 6 b,thermal preading reitance will be, flu tube ψ (a/b) /4ka. Figure 5 illutrate the reitance network analogy for a thermal joint reitance analyi. The total joint reitance can be written a where j,,,, (5) µ, Ã n X i,i!, (6) where n,,i are the number of microcontact and the reitance of each microcontact, repectively. Subcript, ignify bodie,. The Preent Model In addition to the geometrical and mechanical aumption, which were dicued in Bahrami et al., 4 the remaining aumption of the preent model are: 3of American Intitute of Aeronautic and Atronautic Paper 3-498

5 b F k a Q iothermal or ioflu heat contact area Q r P(r) r vacuum adiabatic k Q b dr a Fig. 6 Geometry of contact Fig. 4 body contact plane body Two flu tube in erie contact macrocontact contriction reitance, microcontact contriction reitance microcontact preading reitance macrocontact preading reitance, S, S, Q j joint reitance Fig. 5 Thermal reitance network for nonconforming rough contact in a vacuum contacting olid are iotropic and thick relative to the roughne or wavine radiation heat tranfer i negligible microcontact are circular and teady-tate heat tranfer at microcontact microcontact are iothermal, Cooper et al. 3 proved that all microcontact mut be at the ame temperature, provided the conductivity in each body i independent of direction, poition and temperature. microcontact are flat, it i jutifiable ince urface aperitie have a very mall lope 3 urface are clean and the contact i tatic dr urface element a microcontact flu tube r (r) b (r) O r n (r) a (r) dr A (r) a d (r) Fig. 7 Microcontact ditribution in contact area and thermal reitance network for a urface element Figure 6 how the geometry of the contact with equivalent radiu of curvature and roughne where a i the radiu of the macrocontact area and b i the radiu of the contacting bodie. The flu tube olution i employed to determine the macrocontact thermal reitance, i.e., ( a /b ) 3/ (7) k a Separation between the mean plane of contacting bodie and preure ditribution are not uniform in the contact area, conequently, the number and the average ize of microcontact decreae a the radial poition r increae. Figure 7 illutrate the modeled geometry of the microcontact ditribution, macrocontact area the circle with radiu a, i divided into urface element, dahed ring with increment dr. Figure 7 illutrate the mean average ize of microcontact a mall filled-circle. Around each microcontact a dahed circle illutrate the flu tube aociated with 4of American Intitute of Aeronautic and Atronautic Paper 3-498

6 the microcontact. While microcontact can vary in both ize and hape, a circular contact of equivalent area can be ued to approimate the actual microcontact, ince the local eparation i uniform in each urface element. ocal preading reitance for microcontact can be calculated by applying the flu tube epreion (r) ψ [ε (r)] k a (r) (8) where ε (r) a (r) /b (r) i the local microcontact relative radiu, a (r), ψ ( ) are the local mean average microcontact radiu and the preading reitance factor given by Eq. (4). ocal microcontact relative radiu and microcontact local denity can be calculated from 4 ε (r) r A r (r) A a (r) erfc λ (r) (9) h λ (r) i n ³ m ep 6 σ erfc λ (r) A a () where λ (r) Y (r) / σ, A r and A a are nondimenional eparation, and real and apparent contact area, repectively. The thermal reitance network for the urface element i hown in Fig. 7. In each element n (r) microcontact eit which provide identical parallel path for tranferring thermal energy. Therefore, microcontact thermal reitance for a urface element d (r) i d (r) (r) () n (r) A hown in Fig. 8, urface element form another et of parallel path for tranferring thermal energy in the macrocontact area. Therefore, the effective micro thermal reitance for the joint i P /d (r) () The joint reitance i the um of the macro and micro thermal reitance, i.e., j. eult A eplained in Bahrami et al., 4 a imulation routine wa developed to calculate the thermal joint reitance. A an eample, contact of a 5 (mm) phere with a flat wa conidered and olved with the routine. The contacting bodie are tainle teel and Table lit the urface parameter. The mechanical reult were preented in Bahrami et al. 4 and Fig. 9 and preent thermal output. A epected, the thermal reitance of the microcontact (reitance of the local mean microcontact) increae a r increae. The microcontact relative radiu ε ha it maimum value at the center urface element body contact plane O r body dr a d (r) effective microcontact thermal reitance Fig. 8 Thermal reitance network for urface element Table Input parameter for a typical contact problem ρ 5(mm) F 5(N) σ.4 (µm) E. (GP a) m.7 ( ) c /c 6.7 (GP a) /.5( ) b 5(mm) k 6(W/mK) b k Fig. 9 r/a Hz Micro thermal contact reitance of the contact and decreae with increaing radial poition r. To invetigate the effect of input parameter on thermal joint reitance, the program wa run for a range of each input parameter, while the remaining parameter in Table were held contant. Additionally, elatocontriction thermal reitance introduced by Yovanovich 4 indicated by Hz,waaloincluded in the tudy. Elatocontriction i a limiting cae in which the urface are aumed to be perfectly mooth, i.e., a a Hz and. The effect of roughne on macro, micro, and joint reitance are hown in Fig.. ecall that the joint reitance i the ummation of the macro and micro contact reitance. With relatively mall roughne, 5of American Intitute of Aeronautic and Atronautic Paper 3-498

7 .3 3 j ε a /b.. b k Hz 3 r/a Hz Fig. Microcontact relative radiu 3 4 F (N) 5 Fig. EffectofloadonTC Hz 3 j Hz b k 5 jmin b k j σ (µm) Fig. Effect of roughne on TC the macro thermal reitance dominate the joint reitance and the micro thermal reitance i negligible, alo the joint reitance i cloe to the elatocontriction thermal reitance. By increaing roughne, a become larger thu, the macro thermal reitance decreae, while the micro thermal reitance increae, at ome point they become comparable in ize, by further increae in the roughne micro thermal reitance control the joint reitance. It alo can be een from Fig. that for a fied geometry and load, there i a roughne that minimize the thermal joint reitance. The effectofloadonmicro,macroandjointthermal reitance i hown in Fig.. At light load, due to the mall number and ize of the microcontact, the micro thermal reitance dominate. A the load increae the joint reitance decreae continuouly, micro and macro thermal reitance become comparable in ize and at larger load the macro thermal reitance become the controlling part. At higher load the joint reitance approache the elatocontriction reitance a if no roughne eit. Figure 3 how the effect of radiu of curvature. At very mall radii, the macro thermal reitance dominate ρ (m) Fig. 3 Effect of radiu of curvature on TC due to the mall ize of macrocontact. A the radiu of curvature increae, approaching flat urface, the micro thermal reitance become more important and the macro reitance become maller and eventually when a b the macro reitance fall to zero. Alternative Approach The goal of thi tudy i to develop imple correlation for determining TC. In thi ection, a general epreion for the micro thermal preading reitance i derived, which in conjunction with the macro thermal reitance, Eq. (7), give a correlation to calculate the thermal joint reitance in a vacuum environment. The amount of heat tranferred in a non-conforming rough contact i Q X dq ZZ contact plane dq (3) where dq i the heat tranferred in a urface element. 6of American Intitute of Aeronautic and Atronautic Paper 3-498

8 The local thermal joint conductance i a function of r ZZ Q h (r) T da a (4) contact plane where da a and T contant are the area of a urface element and the temperature drop, repectively. Since the macrocontact area i approimated a a circle Z a Q π T h (r) rdr (5) The effective thermal micro-conductance for a joint i defined a: h Q/A a T. Therefore, the effective microcontact conductance can be found from h π Z a h (r) rdr (6) A a or in term of thermal reitance where / (ha a ), π a (7) h (r) rdr Yovanovich 5 uggeted an epreion for thermal conductance of conforming rough contact a ³ m µ.95 P h.5k (8) σ H mic where H mic and m are the microhardne of the ofter material in contact and the mean abolute lope of aperitie, repectively. Combining Eq. (7) and (8), a relationhip between thermal micro-reitance and preure ditribution can be found a " Z σ a #.95 P (r) rdr (9).5πmk H mic (r) Microhardne depend on everal parameter: mean urface roughne σ, mean abolute lope of aperitie, m, type of material, method of urface preparation, and applied preure. According to Hegazy, 6 urface microhardne can be introduced into the calculation of relative contact preure in the form of the Vicker microhardne H v c (d v) c () where H v i the Vicker microhardne in (GPa), d v d v /d and d (µm), d v i the Vicker indentation diagonal in µm and c and c are correlation coefficient determined from Vicker microhardne meaurement. Song and Yovanovich 7 developed an eplicit epreion relating microhardne to the applied preure µ P P.7c () H mic H where H c (.6σ /m) c, σ σ/σ and σ µm. Sridhar and Yovanovich 8 developed empirical relation to etimate the Vicker microhardne coefficient, uing the bulk hardne of the material. Two leat-quare-cubic fit epreion were reported: c H BGM κ 4.κ.6κ 3 () c.57. κ.4 κ 6.58 κ3 (3) where κ H B /H BGM, H B i the Brinell hardne of the bulk material, and H BGM 3.78 GP a. The above correlation are valid for the range.3 H B 7.6 GP a with the MS percent difference between data and calculated value were reported; 5.3% and.8% for c, and c, repectively. However, in ituation where an effective value for microhardne H mic,e i known the microhardne coefficient can be calculated from c H mic,e and c. Combining Eq.(9), and () give σh.5πk m a [P (r)] (4) rdr where.95/ (.7c ). Bahrami et al. 4 propoed epreion for the preure ditribution of pherical rough contact which cover all poible contact cae including flat contact F/πb F c P P (ξ) ξ γ F F c P,c ξ γ c F F c πb F F c (5) where ξ r/a, γ.5(p /P,Hz )(a /a Hz ). F c i the critical force where a b and it i given by F c 4E ma, b 3ρ.5σρ ª 3/ (6) where ma{, y} return the maimum value between and y. A criterion for defining the flat urface wa derived. 4 It wa hown that if the out-offlatne/wavine and the roughne of a urface are of the ame order of magnitude, the urface i flat, i.e., δ/σ.. Subtituting the preure ditribution, for F F c into Eq. (4) one can obtain σ (H /P ) Z ξ γ ξ dξ.5π mk a (7) Evaluating the integral, one can obtain µ H (8) σ ( γ).5π mk a P For F F c,theeffective microcontact thermal reitance, after evaluating the integral, become µ σ H µ πh b.5πmk b ( γ P c ),c F F c (9) 7of American Intitute of Aeronautic and Atronautic Paper 3-498

9 If an etimate of microhardne i available, et c Hmic, c m, may be etimated from: m.76σ σ [ µ m].5 Start Input F, ρ, σ, mk, E, b, c, c ( ma{, ( )}).5 3 4E Fc b σρ 3ρ σ σ σ m m m [ ] ( ) ( υ ) ( υ) ρ ρ ρ k kk k k E E E π H b F a b Flat urface F c F F c F > Fc γ a P b H ( σ),where.5πkb m H P, c ( γ ) πb H, a b F Fc c ( c ).95/.7 c (.6σ ) [ ] H c m σ σ σ µ m α σ ρ a τ ρ a Hz Hz, (.75 ρ ) 3 a F E P, Hz.5F πa Hz Hz ( a / b ) k a 3/ j End P P P τ α '.75, Hz.37 ' a ' a.5lnp a Hz.4ln P.ln P ' 3 ' ' ' ( a ) γ.5p Fig. 4 Procedure for utilizing the preent model Table data ange of parameter for the eperimental Parameter 7.5 b 4.8 (mm) 5.64 E 4. (GPa) 7.7 F (N) 6.6 k 7. (W/mK).4 m.34 ( ). σ 3.94 (µm).3 ρ / (m) where P,c and γ c are the value at the critical force. The general relationhip for micro thermal reitance can be ummarized a µ πh b F c F µ b a µ H ( γ) F F c P Table 3 eeacher and pecimen material ued in comparion ef. eearcher Material() ½ Ni A Antonetti 9 Ni-Ag B Burde SPS 45, CS Al4 T4 CC Clauing-Chao Bra Anaconda Mg AZ 3B SS33 F Fiher Ni -Carbon Steel H Hegazy 6 Ni SS34 Zircaloy4 Zr-.5%wt Nb K Kitcha Steel -CS MM McMillan-Mikic 3 SS33 M Mikic-ohenow 3 SS35 M Milanez et al. 4 SS34 where.5πb k (m/σ). Figure 4 ummarize the procedure ued to implement the preent model. µ H P,c µ πh b ( γ c ) F F c F F c (3) ComparionWithEperimentalData During the lat four decade a large number of eperimental data have been collected for a wide variety 8of American Intitute of Aeronautic and Atronautic Paper 3-498

10 of material uch a bra, magneium, nickel, ilver and tainle teel in a vacuum. More than 7 data point were collected from an etenive review of the literature, ummarized and compared with the preent model. A ummarized in Table, the eperimental data form a complete et of the material with a wide range of mechanical, thermal, and urface characteritic ued in application where TC i of concern. The data alo include the contact between diimilar metal uch a Ni-Ag and SS-CS. Generally, TC eperimental procedure include two cylindrical pecimen with the ame diameter b which are preed coaially together by applying an eternal load in a vacuum chamber. After reaching teady tate condition, TC i meaured at each load. Thee eperiment have been conducted by many reearcher including Burde and Clauing and Chao. Table 3 indicate the reearcher, reference publication, pecimen deignation, and the material type ued in the eperiment. The comparion include all three region of TC, i.e., the conforming rough, the elatocontriction and the tranition. Table 4 and 5 lit the eperiment number, i.e., the number which wa originally aigned to a particular eperimental data et by the reearcher and geometrical, mechanical and thermal propertie of the eperimental data, a reported. Clauing and Chao, Fiher, Kitcha, and Mikic and ohenow 3 did not report the urface lope m; however the ambert 9 correlation wa ued to etimate thee value (ee Fig. 4). Additionally, the eact value of radii of curvature for conforming rough urface were not reported. Since, thee urface were prepared to be optically flat, radii of curvature in the order of ρ (m) areconidered for thee urface. Figure 5 illutrate the comparion between the preent model and the eperimental data, with j k b j Ω (σ/m)( γ).5πb B µ H P ( B).5 (3) B where B a /b and j i the non-dimenional thermal joint reitance. From Eq. (7), (3), and (3) it can be een that the parameter Ω i the nondimenional TC predicted by the model, i.e., Ω j or j Ω. Therefore the model i hown by the 45-degree line in Fig. (5). The procedure to implement the model and all required relationhip are ummarized in Fig. 4; equivalently, TC can be determined uing Eq. (3). Bahrami et al. 4 propoed the following epreion for calculating a a a.8 α.3 τ.56 a Hz τ.8 (3) Uing Eq. (3) one can find a relationhip for B a a function of non-dimenional and geometrical parame- Table 4 Summary of geometrical, mechanical and thermophyical propertie, rough phere-flat contact ef. E σ/m ρ c / c k b B,A / / B,A- 4..3/ / B,A / / B,A / / B,A / / B,A / / CC,A /- 4..6/ CC,8A / / CC,B / / CC,B / / CC,3B / /.7.7 CC,4B / / CC,3S 3.7./-. 4.6/ CC,M 5.64./ / 96.7 F,A 3../-.9 4./ F,B 3../ / F,3A 3..6/ / K,T /-.4 4./ K,T 3.8.3/-.4 4./ MM,T 3.7.7/ / MM,T / /.7 M,T /-. 4./ M,T / / ter, i.e., B a b.8 µ ahz b α.3 τ.56 τ.8 (33) Eperimental data are ditributed over four decade of Ω from approimately.3 up to 7. The model how good agreement with the data over the entire range of comparion with the eception of a few point. In mot of the conforming rough data et, uch a Hegazy, 6 eperimental data how a lower reitance at relatively light load in comparion with the model and the data approach the model a the load increae. Thi trend can be oberved in almot all conforming rough data et (ee Fig. 5). Thi phenomenon which i called the truncation effect 4 i important at light load when urface are relatively rough. A poible reaon for thi behavior i the Gauian aumption of the urface aperitie which implie that aperitie with infinite height eit. Milanez et al. 4 eperimentally tudied the truncation effect and propoed correlation for maimum aperitie height a function of urface roughne. Becaue of the above-mentioned approimation to account for unreported data, the accuracy of the model i difficult to ae. However, the MS and the average abolute difference between the model and data for the entire et of data are approimately.4% and.%, repectively. 9of American Intitute of Aeronautic and Atronautic Paper 3-498

11 j A, P3435, Ni A, P67, Ni - - : : - : -: : : Ω - : A, P, Ni A, P89, Ni A, P67, Ni-Ag A, P333, Ni-Ag B, A, SPS45-CS B, A, SPS45-CS B, A3, SPS45-CS B, A4, SPS45-CS B, A5, SPS45-CS B, A6, SPS45-CS CC, A, Al4T4 CC, 8A, Al4T4 CC, B, Bra CC, B, Bra CC, 3B, Bra CC, 4B, Bra CC, M, MgAZ3B CC, 3S, SS33 F, A, Ni-CS F, B, Ni-CS F, 3A, Ni-CS H, PNI, Ni H, PNI34, Ni H, PNI56, Ni H, PNI78, Ni H, PNI9, Ni H, PSS, SS34 H, PSS34, SS34 H, PSS56, SS34 H, PSS78, SS34 H, PZ4-Zircaloy4 H, PZ434-Zircaloy4 H, PZ456-Zircaloy4 H, PZ478-Zircaloy4 H, PZN-Zr.5Nb H, PZN34-Zr.5Nb H, PZN56-Zr.5Nb H, PZN78-Zr.5Nb K, T, S-CS K, T, S-CS MM, T, SS33 MM, T, SS33 M, T, SS35 M, T, SS35 M, T, SS34 The Preent Model Fig. 5 Table 5 Summary of geometrical, mechanical and thermophyical propertie for conforming rough contact ef. E σ m c -c k b A,P A,P A,P A,P A,P A,P H,NI H,NI H,NI H,NI H,NI H,SS H,SS H,SS H,SS H,Z H,Z H,Z H,Z H,ZN H,ZN H,ZN H,ZN M,SS Comparion of preent model with eperimental data of Concluding emark TC of non-conforming rough urface wa conidered a the uperpoition of macro and micro thermal reitance component accounting for the effect of urface curvature and roughne, repectively. TC were categorized into three main region, ) the conforming rough limit; where the contacting urface are flat and the effect of urface curvature can be ignored; thu the micro thermal reitance dominate the joint reitance, ) the elatocontriction limit in which the radii of the contacting bodie are relatively mall and the effect of roughne on the TC i negligible and the macro reitance i the controlling part, and 3) the tranition region where the macro and micro thermal reitance are comparable. The reult of the mechanical model preented in Bahrami et al., 4 i.e., the local mean eparation, the local mean radiu and the number of microcontact, were ued to develop an analytical thermal model for determining TC of non-conforming rough contact in a vacuum. The thermal model wa contructed baed on the premie that the mean eparation between the contacting urface in an infiniteimal urface element can be aumed contant. Therefore, the conforming rough model of Cooper et al. 3 could be implemented to calculate the urface element thermal reitance. The urface element thermal reitance were integrated over the macrocontact area to calculate the effective American Intitute of Aeronautic and Atronautic Paper 3-498

12 micro thermal reitance of the contact. The macrocontact reitance wa calculated uing the flu tube olution. The effect of the major parameter, i.e., roughne, load, and radiu of curvature on TC were invetigated. It wa hown that there i a value of urface roughne that minimize TC. Additionally, at large load the effect of roughne on the TC become negligible. By uing the general preure ditribution introduced in Bahrami et al. 4 and the Yovanovich 5 correlation for thermal conductance of conforming rough contact, imple correlation for determining TC were derived which cover the entire range of TC from conforming rough to mooth pherical contact. The procedure for implementing the preent model wa preented in the form of a imple algorithm. The input parameter to utilize the propoed correlation are: load F, theeffective elaticity modulu E,Vickermicrohardne correlation coefficient c and c,effective urface roughne σ and urface lope m, the effective urface out-of-flatne δ or radiu of curvature ρ, radiu of the contacting urface b,andtheharmonic mean of the thermal conductivitie k. The preent model wa compared with more than 7 eperimental data point and howed good agreement over the entire range of TC. The MS difference between the model and the data wa etimated to be approimately.4%. The lit of material in the comparion formed a complete et of the metal ued in application, where TC i of concern. It wa alo hown that the preent model i applicable to diimilar metal. eference Clauing, A. M. and Chao, B. T., Thermal Contact eitance in a Vacuum Environment, Tech. rep., Univerity of Illinoi, Urbana, Illinoi, eport ME-TN-4-, Augut, 963. Yovanovich, M. M., Overall Contriction eitance Between Contacting ough, Wavy Surface, International Journal of Heat and Ma Tranfer, Vol., 969, pp Mikic, B. B. and ohenow, W. M., Thermal Contact Conductance, Tech. rep., Dept. of Mech. Eng. MIT, Cambridge, Maachuett, NASA Contract No. NG -9-65, September, Bahrami, M., Culham, J.., Yovanovich, M. M., and Schneider, G. E., Thermal Contact eitance of Non- Conforming ough Surface Part : Mechanical Model, AIAA Paper No , 36th. AIAA Thermophyic Conference, June 3-6, Orlando, Florida, 3. 5 Bahrami, M., Culham, J.., Yovanovich, M. M., and Schneider, G. E., eview Of Thermal Joint eitance Model For Non-Conforming ough Surface In A Vacuum, Paper No. HT3-475, ASME Heat Tranfer Conference, July -3, io Hotel, a Vega, Nevada, 3. 6 Carlaw, H. S. and Jaeger, J. C., Conduction of Heat in Solid, nd. Edition, Oford Univerity Pre, ondon, UK, Nihino, K., Yamahita, S., and Torii, K., Thermal Contact Conductance Under ow Applied oad in a Vacuum Environment, Eperimental Thermal and Fluid Science, Elevier, Vol., 995, pp ambert, M. A. and Fletcher,. S., Thermal Contact Conductance of Spherical ough Metal, Tranaction of ASME, November, Vol. 9, 997, pp ambert, M. A., Thermal Contact Conductance of Spherical ough Metal, Ph.D. thei, Tea A & M Univerity, Dept. of Mech. Eng.,Tea, USA, 995. Clauing, A. M. and Chao, B. T., Thermal Contact eitance in a Vacuum Environment, Paper No.64-HT-6, Tranaction of ASME: Journal of Heat Tranfer, Vol. 87, 965, pp Hertz, H., On the Contact of Elatic Bodie, Journal fur die reine und angewandie Mathematic, (in German), Vol. 9, 88, pp Yovanovich, M. M., Burde, S. S., and Thompon, C. C., Thermal Contriction eitance of Arbitrary Planar Contact With Contant Flu, AIAA th Thermophyic Conference, San Diego, California, July 4-6, Paper No , 969, pp Cooper, M. G., Mikic, B. B., and Yovanovich, M. M., Thermal Contact Conductance, International Journal of Heat and Ma Tranfer, Vol., 969, pp Yovanovich, M. M., ecent Development In Thermal Contact, Gap and Joint Conductance Theorie and Eperiment, Eighth International Heat Tranfer Conference, San Francico, CA, Augut 7-, 986, pp Yovanovich, M. M., Thermal Contact Correlation, Progre in Aeronautic and Aerodynamic: Spacecraft adiative Tranfer and Temperature Control, in Horton, T.E. (editor), Vol. 83, 98, pp Hegazy, A. A., Thermal Joint Conductance of Conforming ough Surface: Effect of Surface Micro-Hardne Variation, Ph.D. thei, Univerity of Waterloo, Dept. of Mech. Eng., Waterloo, Canada, Song, S. and Yovanovich, M. M., elative Contact Preure: Dependence on Surface oughne and Vicker Microhardne, AIAA Journal of Thermophyic and Heat Tranfer, Vol., No., 988, pp Sridhar, M.. and Yovanovich, M., Empirical Method to Predict Vicker Microhardne, WEA, Vol. 93, 996, pp Antonetti, V. W., On the Ue of Metallic Coating to Enhance Thermal Conductance, Ph.D. thei, Univerity of Waterloo, Dept. of Mech. Eng., Waterloo, Canada, 983. Burde, S. S., Thermal Contact eitance Between Smooth Sphere and ough Flat, Ph.D.thei,UniverityofWaterloo, Dept. of Mech. Eng., Waterloo, Canada, 977. Fiher, N. F., Thermal Contriction eitance of Sphere/ayered Flat Contact: Theory and Eperiment, Mater thei, Univerity of Waterloo, Dept. of Mech. Eng., Waterloo, Canada, 987. Kitcha, W., Thermal eitance of the Sphere-Flat Contact, Mater thei, Univerity of Waterloo, Dept. of Mech. Eng., Waterloo, Canada, McMillan,. and Mikic, B. B., Thermal Contact eitance With Non-Uniform Interface Preure, Tech. rep., Dept. of Mech. Eng. MIT, Cambridge, Maachuett, NASA Contract No. NG -9-(477), November, Milanez, F. H., Yovanovich, M. M., and Mantelli, M. B. H., Thermal Contact Conductance at ow Contact Preure, AIAA Paper No , 36th. AIAA Thermophyic Conference, June 3-6, Orlando, Florida, 3. of American Intitute of Aeronautic and Atronautic Paper 3-498