Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 43
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These dimensions are muchsmaller than the dimensions of the contacting bodies. The circular contact area pro- * PgEnds: Ejectduced when two spheres or a sphere and a flat are in contact are two special casesof the elliptical contact. Also, the rectangular contact area, produced when two ideal[319], (59)circular cylinders are in line contact or an ideal cylinder and a flat are in contact, arespecial cases of the elliptical contact area.Figure 4.17 shows the contact between two elastic bodies having physical properties (Young’s modulus and Poisson’s ratio): E1 , ν1 and E2 , ν2 , respectively.
One bodyis a smooth flat and the other body may be a sphere or a circular cylinder having radius D/2. The contact 2a is the diameter of a circular contact area for the sphere/flatcontact and the width of the contact strip for the cylinder/flat contact. A gap is formedadjacent to the contact area, and its local thickness is characterized by δ.Heat transfer across the joint can take place by conduction by means of the contactarea, conduction through the substance in the gap, and by radiation across the gap ifthe substance is “transparent,” or by radiation if the contact is formed in a vacuum.The thermal joint resistance model presented below was given by Yovanovich (1971,1986).
It was developed for the elastic contact of paraboloids (i.e., the elastic contactformed by a ball and the inner and outer races of an instrument bearing).4.15.1Point Contact ModelSemiaxes of an Elliptical Contact Area The general shape of the contact areais an ellipse with semiaxes a and b and area A = πab. The semiaxes are given bythe relationships (Timoshenko and Goodier, 1970)BOOKCOMP, Inc. — John Wiley & Sons / Page 319 / 2nd Proofs / Heat Transfer Handbook / Bejan320123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES3F∆a=m2(A + B)1/3and3F∆b=n2(A + B)1/3(4.145)where F is the total normal load acting on the contact area, and ∆ is a physicalparameter defined by1 − ν221 1 − ν21(m2/N)+(4.146)∆=2E1E2when dissimilar materials form the contact. The physical parameters are Young’smodulus E1 and E2 and Poisson’s ratio ν1 and ν2 .
The geometric parameters A andB are related to the radii of curvature of the two contacting solids (Timoshenko andGoodier, 1970):1111+ ++ 2(A + B) =ρ1ρ1ρ2ρ2=[320], (60)Lines: 2421 to 24721ρ∗(4.147)———0.9983pt PgVarwhere the local radii of curvature of the contacting solids are denoted as ρ1 , ρ1 , ρ2 ,———and ρ2 .
The second relationship between A and B isNormal Page* PgEnds: Eject 1 211 21− +−2(B − A) =ρ1ρ1ρ2ρ2[320], (60)1/21111+2− − cos 2φ(4.148)ρ1ρ1ρ2ρ2The parameter φ is the angle between the principal planes that pass through thecontacting solids.The dimensionless parameters m and n that appear in the equations for the semiaxes are called the Hertz elastic parameters. They are determined by means of thefollowing Hertz relationships (Timoshenko and Goodier, 1970):m= 1/32 E kπ k2andn=2 kE kπ1/3(4.149) where E k is the complete elliptic integral of the second kind of modulus k (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971), andk =1 − k2withk=bn= ≤1ma(4.150)The additional parameters k and k are solutions of the transcendental equation (Timoshenko and Goodier, 1970):BOOKCOMP, Inc.
— John Wiley & Sons / Page 320 / 2nd Proofs / Heat Transfer Handbook / Bejan321JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445B(1/k 2 )E(k ) − K(k )=AK(k ) − E(k )(4.151)where K(k ) and E(k ) are complete elliptic integrals of the first and second kind ofmodulus k .The Hertz solution requires the calculation of k, the ellipticity, K(k ), and E(k ).This requires a numerical solution of the transcendental equation that relates k, K(k ),and E(k ) to the local geometry of the contacting solids through the geometric parameters A and B.
This is usually accomplished by an iterative numerical procedure.To this end, additional geometric parameters have been defined (Timoshenko andGoodier, 1970):cos τ =B −AB +Aandω=A≤1B(4.152)[321], (61)Computed values of m and n, or m/n and n, are presented with τ or ω as theindependent parameter. Table 4.15 shows how k, m, and n depend on the parameterω over a range of values that should cover most practical contact problems. Theparameter k may be computed accurately and efficiently by means of the Newton–Raphson iteration method applied to the following relationships (Yovanovich, 1986):knewTABLE 4.15N (k )=k +D(k )(4.153)Hertz Contact Parameters and Elastoconstriction Parameterωkmnψ∗0.0010.0020.0040.0060.0080.0100.0200.0400.0600.0800.1000.2000.3000.4000.5000.6000.7000.8000.9001.0000.01470.02180.03230.04080.04830.05500.08280.12590.16150.19320.22230.34600.45040.54410.63060.71170.78850.86180.93221.000014.31611.0368.4837.2626.4995.9614.5443.4522.9352.6152.3911.8131.5471.3861.2761.19391.13011.07871.03611.00000.21090.24030.27430.29660.31370.32770.37650.43450.47400.50510.53130.62730.69690.75440.80450.84970.89110.92960.96581.00000.24920.30080.36160.40200.43290.45810.54380.63970.69940.74260.77610.87570.92610.95570.97410.98570.99300.99720.99941.0000BOOKCOMP, Inc.
— John Wiley & Sons / Page 321 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2472 to 2532———6.71815pt PgVar———Normal PagePgEnds: TEX[321], (61)322123456789101112131415161718192021222324252627282930313233343536373839404142434445whereTHERMAL SPREADING AND CONTACT RESISTANCESAE(k )A241+k−k+K(k )BBAE(k )AD(k ) =k k 2 − 2k + k2 kK(k )BBN (k ) = k 2(4.154)(4.155)If the initial guess for k is based on the following correlation of the results given inTable 4.15, the convergence will occur within two to three iterations: 0.6135 2 1/2Ak = 1 − 0.9446(4.156)B[322], (62)Polynomial approximations of the complete elliptic integrals (Abramowitz and Stegun, 1965) may be used to evaluate them with an absolute error less than 10−7 overthe full range of k .Lines: 2532 to 25844.15.2-0.0958pt PgVar———Local Gap Thickness———The local gap thickness is required for the elastogap resistance model developed byNormal PageYovanovich (1986).
The general relationship for the gap thickness can be determinedby means of the following surface displacements (Johnson, 1985; Timoshenko and * PgEnds: EjectGoodier, 1970):δ(x,y) = δ0 + w(x,y) − w0(m)(4.157)where δ0 (x,y) is the local gap thickness under zero load conditions, w(x,y) is the totallocal displacement of the surfaces of the bodies outside the loaded area, and w0 is theapproach of the contact bodies due to loading.The total local displacement of the two bodies is given by 3F∆ ∞x2y2dt1− 2(4.158)− 22π µa +tb + t [(a 2 + t)(b2 + t)t]1/2where µ is the positive root of the equationx2y2+=1a 2 + µ b2 + µ(4.159)When µ > 0, the point of interest lies outside the elliptical contact area:x2y2+ 2 =12ab(4.160)When µ = 0, the point of interest lies inside the contact area, and when x = y =0, w(0, 0) = w0 , the total approach of the contacting bodies isBOOKCOMP, Inc.
— John Wiley & Sons / Page 322 / 2nd Proofs / Heat Transfer Handbook / Bejan[322], (62)323JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445w0 ==3F∆2π∞[(a 203F∆K(k )πadt+ t)(b2 + t)t]1/2(m)(4.161)The relationships for the semiaxes and the local gap thickness are used in thefollowing subsections to develop the general relationships for the contact and gapresistances.4.15.3Contact Resistance of Isothermal Elliptical Contact AreasThe general spreading–constriction resistance model, as proposed by Yovanovich(1971, 1986), is based on the assumption that both bodies forming an elliptical contactarea can be taken to be a conducting half-space. This approximation of actual bodiesis reasonable because the dimensions of the contact area are very small relative to thecharacteristic dimensions of the contacting bodies.If the free (noncontacting) surfaces of the contacting bodies are adiabatic, the totalellipsoidal spreading–constriction resistance of an isothermal elliptical contact areawith a ≥ b is (Yovanovich, 1971, 1986)Rc =ψ2ks a(K/W)2k1 k2k2 + k 2(W/m · K)(4.163)and ψ is the spreading/constriction parameter of the isothermal elliptical contact areadeveloped in the section for spreading resistance of an isothermal elliptical area onan isotropic half-space:ψ=2K(k )π(4.164)in which K(k ) is the complete elliptic integral of the first kind of modulus k and isrelated to the semiaxes 2 1/2bk = 1 −aThe complete elliptic integral can be computed accurately by means of accuratepolynomial approximations and by computer algebra systems.
This important specialfunction can also be approximated by means of the following simple relationships:BOOKCOMP, Inc. — John Wiley & Sons / Page 323 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2584 to 2627———2.00821pt PgVar———Normal Page(4.162) * PgEnds: Ejectwhere a is the semimajor axis, ks is the harmonic mean thermal conductivity of thejoint,ks =[323], (63)[323], (63)324123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES4a ln bK(k ) =2π√(1 + b/a)20 ≤ k < 0.1736(4.165)0.1736 < k ≤ 1These approximations have a maximum error less than 0.8%, which occurs at k =0.1736. The ellipsoidal spreading–constriction parameter approaches the value of 1when a = b, the circular contact area.When the results of the Hertz elastic deformation analysis are substituted intothe results of the ellipsoidal constriction analysis, one obtains the elastoconstrictionresistance relationship developed by Yovanovich (1971, 1986):ks (24F ∆ρ∗ )1/3 Rc =2 K(k )≡ ψ∗π m(4.166)where the effective radius of the ellipsoidal contact is defined as ρ∗ = [2(A + B)]−1 .The left-hand side is a dimensionless group consisting of the known total mechanicalload F , the effective thermal conductivity ks of the joint, the physical parameter∆, and the isothermal elliptical spreading/constriction resistance Rc .
The right-handside is defined to be ψ∗ , which is called the thermal elastoconstriction parameter(Yovanovich, 1971, 1986). Typical values of ψ∗ for a range of values of ω are givenin Table 4.15. The elastoconstriction parameter ψ∗ → 1 when k = b/a = 1, the case *of the circular contact area.4.15.4The thermal resistance of the gas-filled gap depends on three local quantities: the localgap thickness, thermal conductivity of the gas, and temperature difference betweenthe bounding solid surfaces. The gap model is based on the subdivision of the gapinto elemental heat flow channels (flux tubes) having isothermal upper and lowerboundaries and adiabatic sides (Yovanovich and Kitscha, 1974). The heat flow linesin each channel (tube) are assumed to be straight and perpendicular to the plane ofcontact.If the local effective gas conductivity kg (x,y) in each elemental channel is assumedto be uniform across the local gap thickness δ(x,y), the differential gap heat flowrate iskg (x,y) ∆Tg (x,y)dx dyδ(x,y)(W)(4.167)The total gap heat flow rate is given by the double integral Qg = Ag dQg , wherethe integration is performed over the entire effective gap area Ag .The thermal resistance of the gap, Rg , is defined in terms of the overall jointtemperature drop ∆Tj (Yovanovich and Kitscha, 1974):BOOKCOMP, Inc.
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