Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 49
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— John Wiley & Sons / Page 350 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 3612 to 36625.38135pt PgVar———Short Page* PgEnds: Eject[350], (90)CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445Pλ = −0.5444 − 0.6636 lnHe351P 3P 2− 0.000771 ln− 0.03204 lnHeHe(4.262)Sridhar and Yovanovich (1994) reviewed the plastic and elastic deformation contact conductance correlation equations and compared them against vacuum data (Mikic and Rohsenow, 1966; Antonetti, 1983; Hegazy, 1985; Nho, 1989; McWaid andMarschall, 1992a, b) for several metal types, having a range of surface roughnesses,over a wide range of apparent contact pressure. Sridhar and Yovanovich (1996a)showed that the elastic deformation model was in better agreement with the vacuumdata obtained for joints formed by conforming rough surfaces of tool steel, which isvery hard.The elastic asperity contact and thermal conductance models of Greenwood andWilliamson (1966), Greenwood (1967), Greenwood and Tripp (1967, 1970), Bush etal.
(1975), and Bush and Gibson (1979) are different from the Mikic (1974) elasticcontact model presented in this chapter. However, they predict similar trends ofcontact conductance as a function apparent contact pressure.[351], (91)Lines: 3662 to 3692———11.61588pt PgVar4.16.4 Conforming Rough Surface Model: Elastic–PlasticDeformationSridhar and Yovanovich (1996c) developed an elastic–plastic contact conductancemodel which is based on the plastic contact model of Cooper et al. (1969) and theelastic contact model of Mikic (1974).
The results are summarized below in termsof the geometric parameters Ar /Aa , the real-to-apparent area ratio; n, the contactspot density; a, the mean contact spot radius; and λ, the dimensionless mean planeseparation:fepArλ=erfc √(4.263)Aa221 m 2 exp(−λ2 )(4.264)n=√16 σ erfc(λ/ 2) 28σλλa=fep experfc √(4.265)πm22λ21 2 mfep exp −(4.266)na =8 πσ2fep exp(−λ2 /2)hc σ1(4.267)= √ ks m√ 1.52 2π1 − (fep /2)erfc(λ/ 2)BOOKCOMP, Inc. — John Wiley & Sons / Page 351 / 2nd Proofs / Heat Transfer Handbook / Bejan———Short PagePgEnds: TEX[351], (91)352123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES√2erfc−1λ=1 2Pfep Hep(4.268)The important elastic–plastic parameter fep is a function of the dimensionlesscontact strain ∗c , which depends on the amount of work hardening. This physicalparameter lies in the range 0.5 ≤ fep ≤ 1.0.
The smallest and largest valuescorrespond to zero and infinitely large contact strain, respectively. The elastic–plasticparameter is related to the contact strain:fep = 1 + (6.5/∗c )21+1/20 < ∗c < ∞1/1.2(13.0/∗c )1.2The dimensionless contact strain is defined asmE ∗c = 1.67Sf(4.269)[352], (92)(4.270)Lines: 3692 to 3734where Sf is the material yield or flow stress (Johnson, 1985), which is a complexphysical parameter that must be determined by experiment for each metal.The elastic–plastic microhardness Hep can be determined by means of an iterativeprocedure which requires the following relationship:2.76SfHep = 1 + (6.5/∗c )21/2[352], (92)(4.272)where the coefficients c1 and c2 are obtained from Vickers microhardness tests. TheVickers microhardness coefficients are related to Brinell and Rockwell hardness fora wide range of metals.Correlation Equations for Dimensionless Contact Conductance: Elastic–Plastic Model The complex elastic–plastic contact model proposed by Sridharand Yovanovich (1996a, b, c, d) may be approximated by the following correlationequations for the dimensionless contact conductance:BOOKCOMP, Inc.
— John Wiley & Sons / Page 352 / 2nd Proofs / Heat Transfer Handbook / Bejan———Short Page* PgEnds: Eject(4.271)The elastoplastic contact conductance model moves smoothly between the elasticcontact model of Mikic (1974) and the plastic contact conductance model of Cooperet al. (1969), which was modified by Yovanovich (1982), Yovanovich et al. (1982a),and Song and Yovanovich (1988) to include the effect of work-hardened layers onthe deformation of the contacting asperities.
The dimensionless contact pressure forelastic–plastic deformation of the contacting asperities is obtained from the followingapproximate explicit relationship:1/(1+0.071c2 )P0.9272P=Hepc1 (1.43 σ/m)c2———-1.41464pt PgVarCONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445P 0.941.54HepP b2Cc = 1.245b1Hep0.95 1.25 PHep3530 < ∗c < 5(4.273)5 < ∗c < 400(4.274)400 < ∗c < ∞(4.275)where the elastic–plastic correlation coefficients b1 and b2 depend on the dimensionless contact strain:46,690.2 1/30b1 = 1 + ∗ 2.49(c )1/6001b2 =1 + 2086.9/(∗c )1.842(4.276)[353], (93)(4.277)Lines: 3734 to 3767———4.16.57.81639pt PgVarGap Conductance for Large Parallel Isothermal PlatesTwo infinite isothermal surfaces form a gap of uniform thickness d which is muchgreater than the roughness of both surfaces: d σ1 and σ2 . The gap is filled witha stationary monatomic or diatomic gas.
The boundary temperatures are T1 and T2 ,where T1 > T2 . The Knudsen number for the gap is defined as Kn = Λ/d, whereΛ is the molecular mean free path of the gas, which depends on the gas temperatureand its pressure. The gap can be separated into three zones: two boundary zones,which are associated with the two solid boundaries, and a central zone. The boundaryzones have thicknesses that are related to the molecular mean free paths Λ1 and Λ2 ,whereΛ1 = Λ0T1 Pg,0T0 PgandΛ2 = Λ0T2 Pg,0T0 Pg(4.278)and Λ0 , T0 , and Pg,0 represent the molecular mean free path and the reference temperature and gas pressure. In the boundary zones the heat transfer is due to gas moleculesthat move back and forth between the solid surface and other gas molecules located atdistances Λ1 and Λ2 from both solid boundaries. The energy exchange between thegas and solid molecules is imperfect.
At the hot solid surface at temperature T1 , thegas molecules that leave the surface after contact are at some temperature Tg,1 < T1 ,and at the cold solid surface at temperature T2 , the gas molecules that leave the surfaceafter contact are at a temperature Tg,2 > T2 . The two boundary zones are called slipregions.In the central zone whose thickness is modeled as d − Λ1 − Λ2 , and whosetemperature range is Tg,1 ≥ T ≥ Tg,2 , heat transfer occurs primarily by molecularBOOKCOMP, Inc. — John Wiley & Sons / Page 353 / 2nd Proofs / Heat Transfer Handbook / Bejan———Short PagePgEnds: TEX[353], (93)354123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESdiffusion. Fourier’s law of conduction can be used to determine heat transfer acrossthe central zone.There are two heat flux asymptotes, corresponding to very small and very largeKnudsen numbers.
They are: for a continuum,T1 − T 2dKn → 0q → q0 = kgKn → ∞q → q∞ = kg(4.279)and for free molecules,T1 − T 2M(4.280)whereM = αβΛ =2 − α12 − α2+α1α2[354], (94)2γΛ(γ + 1)Pr(4.281)and kg is the thermal conductivity, α1 and α2 the accommodation coefficients, γ theratio of specific heats, and Pr the Prandtl number.The gap conductance, defined as hg = q/(T1 − T2 ), has two asymptotes:for Kn → 0,hg →kgdfor Kn → ∞,hg →kgMFor the entire range of the Knudsen number, the gap conductance is given by therelationshiphg =kgd +Mfor 0 < Kn < ∞(W/m2 · K)(4.282)This relatively simple relationship covers the continuum, 0 < Kn < 0.1, slip,0.1 < Kn < 10, and free molecule, 10 < Kn < ∞, regimes.
Song (1988) introducedthe dimensionless parametersG=kghg dandM∗ =Md(4.283)and recast the relationship above asG = 1 + M∗for0 < M∗ < ∞(4.284)The accuracy of the simple parallel-plate gap model was compared against the data(argon and nitrogen) of Teagan and Springer (1968), and the data (argon and helium)of Braun and Frohn (1976). The excellent agreement between the simple gap modeland all data is shown in Fig.
4.26. The simple gap model forms the basis of the gapmodel for the joint formed by two conforming rough surfaces.BOOKCOMP, Inc. — John Wiley & Sons / Page 354 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 3767 to 3827———6.58437pt PgVar———Normal PagePgEnds: TEX[354], (94)CONFORMING ROUGH SURFACE MODELS102HeArArN2(Braun & Frohn, 1976)(Braun & Frohn, 1976)(Teagan & Springer, 1968)(Teagan & Springer, 1968)Interpolated Model G = 1 ⫹ M*101[355], (95)G123456789101112131415161718192021222324252627282930313233343536373839404142434445355Lines: 3827 to 3835———0.3591pt PgVar100———Normal Page* PgEnds: Eject[355], (95)10⫺110⫺310⫺210010⫺1101102M*Figure 4.26 Gap conductance model and data for two large parallel isothermal plates. (FromSong et al., 1992a.)4.16.6 Gap Conductance for Joints between ConformingRough SurfacesIf the gap between two conforming rough surfaces as shown in Fig.
4.22 is occupiedby a gas, conduction heat transfer will occur across the gap. This heat transfer ischaracterized by the gap conductance, defined ashg =∆TjQgBOOKCOMP, Inc. — John Wiley & Sons / Page 355 / 2nd Proofs / Heat Transfer Handbook / Bejan(W/m2 · K)(4.285)356123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwith ∆Tj as the effective temperature drop across the gas gap and Qg the heat transferrate across the gap. Because the local gap thickness and local temperature drop varyin very complicated ways throughout the gap, it is difficult to develop a simple gapconductance model.Several gap conductance models and correlation equations have been presented bya number of researchers (Cetinkale and Fishenden, 1951; Rapier et al., 1963; Shlykov,1965; Veziroglu, 1967; Lloyd et al., 1973; Garnier and Begej, 1979; Loyalka, 1982;Yovanovich et al., 1982b); they are given in Table 4.19.
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