Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 46
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Also, if b1 = b2 = b,a 1.5ψ1 = ψ2 = 1 −bThe joint and dimensionless joint resistances for this case becomeBOOKCOMP, Inc. — John Wiley & Sons / Page 334 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.205)[334], (74)PgVarJOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445Rj =ψ2ks aandRj∗ = 2bks Rj =(1 − a/b)1.5a/b335(4.206)where ks = 2k1 k2 /(k1 + k2 ).Sridhar and Yovanovich (1994) compared the dimensionless joint resistanceagainst data obtained for contacts between a carbon steel ball and several flats ofNi 200, carbon steel, and tool steel. The nondimensional data and the dimensionlessjoint resistance model are compared in Fig. 4.21 for a range of values of the reciprocal contact strain b/a.
The agreement between the model and the data over the entirerange 20 < b/a < 120 is very good. The points near b/a ≈ 100 are in the elasticcontact region, and the points near b/a ≈ 20 are close to the plastic contact region.In between the points are in the transition region, called the elastic–plastic contactregion.If the material of the flat work-hardens as the deformation takes place, the modelfor predicting the contact radius is much more complex, as described by Sridhar andYovanovich (1994) and Johnson (1985). This case is not given here.[335], (75)Lines: 3068 to 3091———0.47105pt PgVar———Short PagePgEnds: TEX[335], (75)Figure 4.21 Comparison of the data and model for an elastic–plastic contact between ahemisphere and a flat.
(From Sridhar and Yovanovich, 1994.)BOOKCOMP, Inc. — John Wiley & Sons / Page 335 / 2nd Proofs / Heat Transfer Handbook / Bejan336123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES4.15.9Ball-Bearing ResistanceModels have been presented (Yovanovich, 1967, 1971, 1978) for calculating theoverall thermal resistance of slowly rotating instrument bearings, which consist ofmany very smooth balls contained by very smooth inner and outer races. The thermalresistance models for bearings are based on elastic contact of the balls with the innerand outer races and spreading and constriction resistances in the balls and in the innerand outer races.
For each ball there are two elliptical contact areas, one at the innerrace and one at the outer race. The local thickness of the adjoining gap is very complexto model. There are four spreading–constriction zones associated with each ball. Thefull elastoconstriction resistance model must be used to obtain the overall thermalresistance of the bearing. Since these are complex systems, the contact resistancemodels are also complex; therefore, they are not presented here. The references aboveshould be consulted for the development of the contact resistance models and otherpertinent references.4.15.10[336], (76)Lines: 3091 to 3116Line Contact Models———If a long smooth circular cylinder with radius of curvature ρ1 = D1 /2, length L1 , and9.16412ptelastic properties: E1 , ν1 makes contact with another long smooth circular cylinder———with radius of curvature ρ2 = D2 /2, length L1 , and elastic properties: E2 , ν2 , then inNormal Pagegeneral, if the axes of the cylinders are not aligned (i.e., they are crossed), an elliptical * PgEnds: Ejectcontact area is formed with semiaxes a and b, where it is assumed that a < b.
If thecylinder axes are aligned, the contact area becomes a strip of width 2a, and the largeraxes are equal to the length of the cylinder. The general Hertz model presented may[336], (76)be used to find the semiaxes and the local gap thickness if the axes are not aligned. Foraligned axes, the general equations reduce to simple relationships, which are givenbelow.Contact Strip and Local Gap Thickness If the two cylinder axes are aligned,the contact area is a strip of width 2a and length L1 , where (Timoshenko and Goodier,1970; Walowit and Anno, 1975)2F ρ∆a=2πL11/2(m)(4.207)where the effective curvature is111+=ρρ1ρ2(1/m)(4.208)and the contact parameter is∆=121 − ν211 − ν22+E1E2BOOKCOMP, Inc. — John Wiley & Sons / Page 336 / 2nd Proofs / Heat Transfer Handbook / Bejan(m2/N)(4.209)PgVarJOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445337The contact pressure is maximum along the axis of the contact strip, and it is givenby the relationshipP0 =2 F=π aL1F2πL1 ρ∆1/2(N/m2)(4.210)and the pressure distribution has the form (Timoshenko and Goodier, 1970; Walowitand Anno, 1975) x 2P (x) = P0 1 −for 0 ≤ x ≤ a(N/m2)(4.211)aThe mean contact area pressure isPm =F4P0=2aL1π(N/m2)[337], (77)(4.212)The normal approach of the two aligned cylinders is (Timoshenko and Goodier, 1970;Walowit and Anno, 1975)1 − ν222F 1 − ν214ρ14ρ211α=lnln−+−(m)(4.213)πE1a2E2a2where F = F /L1 is the load per unit cylinder length.
The general local gap thicknessrelationship is (Timoshenko and Goodier, 1970; Walowit and Anno, 1975)2δ=ρ1/2ξ21 1/2− 1− 21− 2LL+ ξ(ξ2 − 1)1/2 − cosh−1 ξ − ξ2 + 1 2Lρ2aξ=xL1≤ξ≤L(4.214)(4.215)If a single circular cylinder of diameter D or (ρ1 = D1 /2 = D/2) is in elasticcontact with a flat (ρ2 = ∞), put ρ = D/2 in the relationships above.Contact Resistance at a Line Contact The thermal contact resistance for thevery narrow contact strip of width 2a formed by the elastic contact of a long smoothcircular cylinder of diameter D and a smooth flat whose width is 2b and whose lengthL1 is identical to the cylinder length is given by the approximate relationship (McGeeet al., 1985)41π21Rc =ln −ln(K/W)(4.216)+πL1 k112πL1 k2 π2BOOKCOMP, Inc.
— John Wiley & Sons / Page 337 / 2nd Proofs / Heat Transfer Handbook / Bejan———1.05823pt PgVar———Normal Page* PgEnds: Eject[337], (77)whereL=Lines: 3116 to 3176338123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere the thermal conductivities of the cylinder and flat are k1 and k2 , respectively.The contact parameters are 1 = 2a/D for the cylinder and 2 = a/b for the flat.
Forelastic contacts, 2a/D 1 and 2a/b 1 for most engineering applications. Theapproximate relationship for Rc becomes more accurate for very narrow strips.The width of the flat relative to the cylinder diameter may be 2b > D, 2b = D,or 2b < D. McGee et al. (1985) proposed the use of the dimensionless form of thecontact resistance:Rc∗ = L1 ks Rc =1 ksπks1 ks1ln−+ln2π k1 F ∗2k12π k2 4πF ∗(4.217)F∆L1 D(4.218)whereF∗ =andks =2k1 k2k1 + k 2Gap Resistance at a Line Contact The general elastogap resistance model forline contacts proposed by Yovanovich (1986) reduces for the circular cylinder–flatcontact to[338], (78)Lines: 3176 to 3245———0.48433pt PgVar4aL11=kg,∞ Ig,lRgD———Normal Pagewhere kg,∞ is the gas thermal conductivity and the line contact elastogap integral is * PgEnds: Ejectdefined as (Yovanovich, 1986)Ig,l =2πL1(W/K)(4.219)[338], (78)cosh−1 (ξ)dξ2δ/D + M/D(4.220)xL(4.221)whereL=D2aξ=1≤ξ≤LThis is the coupled elastogap model.
Numerical integration is required to calculatevalues of Ig,l .The gas rarefaction parameter that appears in the gap integral isM = αβΛ(m)(4.222)where the accommodation parameter and other gas parameters are defined asα=2 − α22 − α1+α1α2β=2γ(γ + 1) Prγ=CpCv(4.223)and the molecular mean free path isΛ = Λg,∞Tg Pg,∞Tg,∞ PgBOOKCOMP, Inc. — John Wiley & Sons / Page 338 / 2nd Proofs / Heat Transfer Handbook / Bejan(m)(4.224)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445339where Λg,∞ is the value of the molecular mean free path at the reference temperatureTg,∞ and gas pressure Pg,∞ .Joint Resistance at a Line Contact The joint resistance at a line contact,neglecting radiation heat transfer across the gap, is111=+RjRcRg(W/K)(4.225)McGee et al. (1985) examined the accuracy of the contact, gap, and joint resistancerelationships for helium and argon for gas pressures between 10−6 torr and 1 atm.The effect of contact load was investigated for mechanical loads between 80 and8000 N on specimens fabricated from Keewatin tool steel, type 304 stainless steel,and Zircaloy 4.
The experimental data were compared with the model predictions,and good agreement was obtained over a limited range of experimental parameters.Discrepancies were observed at the very light mechanical loads due to slight amountsof form error (crowning) along the contacting surfaces.[339], (79)Lines: 3245 to 3271———Joint Resistance of Nonconforming Rough Surfaces There is ample em3.31606ptpirical evidence that surfaces may not be conforming and rough, as shown in Fig. 4.1c———and f. The surfaces may be both nonconforming and rough, as shown in Fig. 4.1b andNormal Pagee, where a smooth hemispherical surface is in contact with a flat, rough surface.* PgEnds: EjectIf surfaces are nonconforming and rough, the joint that is formed is more complexfrom the standpoint of defining the micro- and macrogeometry before load is applied,and the definition of the micro- and macrocontacts that are formed after load is ap[339], (79)plied.
The deformation of the contacting asperities may be elastic, plastic, or elastic–plastic. The deformation of the bulk may also be elastic, plastic, or elastic–plastic.The mode of deformation of the micro- and macrogeometry are closely connectedunder conditions that are not understood today.The thermal joint resistance of such a contact is complex because heat can crossthe joint by conduction through the microcontacts and the associated microgaps andby conduction across the macrogap. If the temperature level of the joint is sufficiently high, there may be significant radiation across the microgaps and macrogap.Clearly, this type of joint represents complex thermal and mechanical problems thatare coupled.Many vacuum data have been reported (Clausing and Chao, 1965; Burde, 1977;Kitscha, 1982) that show that the presence of roughness can alter the joint resistanceof a nonconforming surface under light mechanical loads and have negligible effectsat higher loads.
Also, the presence of out-of-flatness can have significant effects onthe joint resistance of a rough surface under vacuum conditions.It is generally accepted that the joint resistance under vacuum conditions may bemodeled as the superposition of microscopic and macroscopic resistance (Clausingand Chao, 1965; Greenwood and Tripp, 1967; Holm, 1967; Yovanovich, 1969; Burdeand Yovanovich, 1978; Lambert, 1995; Lambert and Fletcher, 1997).
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