Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 44
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— John Wiley & Sons / Page 324 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2627 to 2667———0.29227pt PgVar———Short PagePgEnds: Eject[324], (64)Elastogap Resistance ModeldQg =[324], (64)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445Qg1==Rg∆TjAgkg (x,y) ∆Tg (x,y)dAgδ(x,y) ∆Tj(W/K)325(4.168)The local gap thickness in the general case of two bodies in elastic contact formingan elliptical contact area is given above.The local effective gas conductivity is based on a model for the effective thermalconductivity of a gaseous layer bounded by two infinite isothermal parallel plates.Therefore, for each heat flow channel (tube) the effective thermal conductivity isapproximated by the relation (Yovanovich and Kitscha, 1974)kg (x,y) =kg,∞1 + αβΛ/δ(x,y)(W/m · K)(4.169)where kg,∞ is the gas conductivity under continuum conditions at STP.
The accommodation parameter α is defined asα=Lines: 2667 to 27132 − α12 − α2+α1α2(4.170)where α1 and α2 are the accommodation coefficients at the solid–gas interfaces (Wiedmann and Trumpler, 1946; Hartnett, 1961; Wachman, 1962; Thomas, 1967; Kitschaand Yovanovich, 1975; Madhusudana, 1975, 1996; Semyonov et al., 1984; Wesleyand Yovanovich, 1986; Song and Yovanovich, 1987; Song, 1988; Song et al., 1992a,1993b). The fluid property parameter β is defined by2γβ=(γ + 1)/Pr(4.171)where γ is the ratio of the specific heats and Pr is the Prandtl number.
The mean freepath Λ of the gas molecules is given in terms of Λg,∞ , the mean free path at STP, asfollows:Λ = Λg,∞Tg Pg,∞Tg,∞ Pg(m)(4.172)Two models for determining the local temperature difference, ∆Tg (x,y), are proposed(Yovanovich and Kitscha, 1974). In the first model it is assumed that the boundingsolid surfaces are isothermal at their respective contact temperatures; hence∆Tg (x,y) = ∆Tj(K)(4.173)This is called the thermally decoupled model (Yovanovich and Kitscha, 1974), sinceit assumes that the surface temperature at the solid–gas interface is independent ofthe temperature field within each solid.BOOKCOMP, Inc.
— John Wiley & Sons / Page 325 / 2nd Proofs / Heat Transfer Handbook / Bejan[325], (65)———6.5355pt PgVar———Short PagePgEnds: TEX[325], (65)326123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESIn the second model (Yovanovich and Kitscha, 1974), it is assumed that the temperature distribution of the solid–gas interface is induced by conduction through thesolid–solid contact, under vacuum conditions. This temperature distribution is approximated by the temperature distribution immediately below the surface of an insulated half-space that receives heat from an isothermal elliptical contact.
Solving forthis temperature distribution, using ellipsoidal coordinates it was found that∆Tg (x,y)F (k , ψ)=1−∆TjK(k )(4.174)where F (k , ψ) is the incomplete elliptic integral of the first kind of modulus k andamplitude angle ψ (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971). Themodulus k is given above and the amplitude angle isψ = sin−1a2a2 + µ[326], (66)1/2(4.175)Lines: 2713 to 2741where the parameter µ is defined above. It ranges between µ = 0, the edge of the———elliptical contact area, to µ = ∞, the distant points within the half-space. Since the-0.96779ptPgVarsolid–gas interface temperature is coupled to the interior temperature distribution, it———is called the coupled half-space model temperature drop.Normal PageThe general elastogap model has not been solved.
Two special cases of the general* PgEnds: Ejectmodel have been examined. They are the sphere-flat contact, studied by Yovanovichand Kitscha (1974) and Kitscha and Yovanovich (1975), and the cylinder-flat contact,studied by McGee et al. (1985). The two special cases are discussed below.[326], (66)4.15.5Joint Radiative ResistanceIf the joint is in a vacuum, or the gap is filled with a transparent substance such as dryair, there is heat transfer across the gap by radiation. It is difficult to develop a generalrelationship that would be applicable for all point contact problems because radiationheat transfer occurs in a complex enclosure that consists of at least three nonisothermal convex surfaces. The two contacting surfaces are usually metallic, and the thirdsurface forming the enclosure is frequently a reradiating surface such as insulation.Yovanovich and Kitscha (1974) and Kitscha and Yovanovich (1975) examined anenclosure that was formed by the contact of a metallic hemisphere and a metalliccircular disk of diameter D.
The third boundary of the enclosure was a nonmetalliccircular cylinder of diameter D and height D/2. The metallic surfaces were assumedto be isothermal at temperatures T1 and T2 with T1 > T2 . These temperatures correspond to the extrapolated temperatures from temperatures measured on both sides ofthe joint. The joint temperature was defined as Tj = (T1 + T2 )/2. The dimensionlessradiation resistance was found to have the relationship1 − 2ks1 − 1Dks Rr =++ 1.103(4.176)221πσDTj3BOOKCOMP, Inc. — John Wiley & Sons / Page 326 / 2nd Proofs / Heat Transfer Handbook / Bejan327JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445where σ = 5.67×10−8 W/m2 ·K 4 is the Stefan–Boltzmann constant, 1 and 2 are theemissivities of the hemisphere and disk, respectively, and ks is the effective thermalconductivity of the joint.4.15.6Joint Resistance of Sphere–Flat ContactThe contact, gap, radiative, and joint resistances of the sphere–flat contact shown inFig.
4.17 are presented here. The contact radius a is much smaller than the spherediameter D. Assuming an isothermal contact area, the general elastoconstrictionresistance model (Yovanovich, 1971, 1986; Yovanovich and Kitscha, 1974), becomesRc =12ks a(K/W)(4.177)where ks = 2k1 k2 /(k1 + k2 ) is the harmonic mean thermal conductivity of thecontact, and the contact radius is obtained from the Hertz elastic model (Timoshenkoand Goodier, 1970):2a3F∆ 1/3=(4.178)DD2[327], (67)where F is the mechanical load at the contact and ∆ is the joint physical parameterdefined above.The general-coupled elastogap resistance model for point contacts reduces, forthe sphere–flat contact, to (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich1975; Yovanovich, 1986):———Normal PagePgEnds: TEX1D=kg,0 Ig,pRgL(W/K)(4.179)where L = D/2a is the relative contact size.
The gap integral for point contactsproposed by Yovanovich and Kitscha (1974) and Yovanovich (1975) is defined as√ L2x tan−1 x 2 − 1Ig,p =dx(4.180)2δ/D + 2M/D1The local gap thickness δ is obtained from the relationship x 2 1/22δ=1− 1−DL11−1 122(2 − x ) sin++ x −1 − 22πLxL(4.181)where x = r/a and 1 ≤ x ≤ L. The gap gas rarefaction parameter is defined asM = αβΛ(m)where the gas parameters α, β, and Γ are as defined above.BOOKCOMP, Inc. — John Wiley & Sons / Page 327 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.182)Lines: 2741 to 2803———-2.31227pt PgVar[327], (67)328123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESContacts in a Vacuum The joint resistance for a sphere–flat contact in a vacuumis (Yovanovich and Kitscha, 1974; Kitscha and Yovanovich, 1975)111=+RjRcRr(W/K)(4.183)The models proposed were verified by experiments conducted by Kitscha (1982).
Thetest conditions were: sphere diameter D = 25.4 mm; vacuum pressure Pg = 10−6torr; mean interface temperature range 316 ≤ Tm ≤ 321 K; harmonic mean thermalconductivity of sphere-flat contact ks = 51.5 W/(m · K); emissivities of very smoothsphere and lapped flat (rms roughness is σ = 0.13 µm) 1 = 0.2 and 2 = 0.8, respectively; elastic properties of sphere and flat E1 = E2 = 206 GPa and ν1 = ν2 = 0.3.The dimensionless joint resistance is given by the relationship[328], (68)111+ ∗∗ =∗RjRcRr(4.184)Lines: 2803 to 2859whereRj∗ = Dks RjRc∗ = Dks Rc = L300Rr∗ = 1415Tm———3(4.185)The model and vacuum data are compared for a load range in Table 4.16. The agreement between the joint resistance model and the data is excellent over the full rangeof tests.Effect of Gas Pressure on Joint Resistance According to the Model ofYovanovich and Kitscha (1974), the dimensionless joint resistance with a gas in thegap is given by1111= ∗+ ∗+ ∗Rj∗RcRrRgTABLE 4.16(4.186)Dimensionless Load, Constriction, Radiative, and Joint ResistancesF(N)LD/2aTm(K)Rr∗ModelRj∗ModelRj∗Test16.022.255.687.2195.7266.9467.0115.1103.276.065.450.045.137.43213213213203193183161155115511551164117711881211104.794.771.361.948.043.436.4107.099.470.961.948.842.635.4Source: Kitscha (1982).BOOKCOMP, Inc.
— John Wiley & Sons / Page 328 / 2nd Proofs / Heat Transfer Handbook / Bejan5.01927pt PgVar———Normal PagePgEnds: TEX[328], (68)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445329whereRg∗ = Dks Rg =k s L2kg,∞ Ig,p(4.187)The joint model for the sphere–flat contact is compared against data obtainedfor the following test conditions: sphere diameter D = 25.4 mm; load is 16 N;dimensionless load L = 115.1; mean interface temperature range 309 ≤ Tm ≤ 321K; harmonic mean thermal conductivity of sphere–flat contact ks = 51.5 W/m · K);emissivities of smooth sphere and lapped flat are 1 = 0.2 and 2 = 0.8, respectively.The load was fixed such that L = 115.1 for all tests, while the air pressure wasvaried from 400 mmHg down to a vacuum.
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