Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 45
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The dimensionless resistances are givenin Table 4.17. It can be seen that the dimensionless radiative resistance was relativelylarge with respect to the dimensionless gap and contact resistances. The dimensionless gap resistance values varied greatly with the gas pressure. The agreement between the joint resistance model and the data is very good for all test points.[329], (69)Lines: 2859 to 29034.15.7Joint Resistance of a Sphere and a Layered Substrate———Figure 4.18 shows three joints: contact between a hemisphere and a substrate, contactbetween a hemisphere and a layer of finite thickness bounded to a substrate, andcontact between a hemisphere and a very thick layer where t/a 1.In the general case, contact is between an elastic hemisphere of radius ρ and elasticproperties: E3 , ν3 and an elastic layer of thickness t and elastic properties: E1 , ν1 ,which is bonded to an elastic substrate of elastic properties: E2 , ν2 .
The axial load isF . It is assumed that E1 < E2 for layers that are less rigid than the substrate.The contact radius a is much smaller than the dimensions of the hemisphere andthe substrate. The solution for arbitrary layer thickness is complex because the contactradius depends on several parameters [i.e., a = f (F, ρ, t, Ei , νi ), i = 1, 2, 3].
Thecontact radius lies in the range aS ≤ a ≤ aL , where aS corresponds to the very thinTABLE 4.17Effect of Gas Pressure on Gap and Joint Resistances for AirTm(K)Pg(mmHg)Rg∗ModelRr∗ModelRj∗ModelRj∗Test309310311316318321322325321400.0100.040.04.41.80.60.50.2vacuum77.087.697.4138.3168.9231.3245.9352.8∞12951282127012111188115511441113115544.547.950.759.764.772.173.480.5104.746.849.652.359.065.773.174.380.3107.0Source: Kitscha (1982).BOOKCOMP, Inc. — John Wiley & Sons / Page 329 / 2nd Proofs / Heat Transfer Handbook / Bejan-0.06282pt PgVar———Normal PagePgEnds: TEX[329], (69)330123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESFigure 4.18 Contact between a hemisphere and a layer on a substrate: (a) hemisphere andsubstrate; (b) hemisphere and layer of finite thickness; (c) hemisphere and very thick layer.(From Stevanović et al., 2001.)[330], (70)Lines: 2903 to 2927layer limit, t/a → 0 (Fig.
4.18a) and aL corresponds to the very thick layer limit,t/a → ∞ (Fig. 4.18c).For the general case, a contact in a vacuum, and if there is negligible radiation heattransfer across the gap, the joint resistance is equal to the contact resistance, whichis equal to the sum of the spreading–constriction resistances in the hemisphere andlayer–substrate, respectively.The joint resistance is given by Fisher (1985), Fisher and Yovanovich (1989), andStevanović et al.
(2001, 2002)Rj = Rc =ψ1+ 124k3 a4k2 a(K/W)(4.188)where a is the contact radius. The first term on the right-hand side represents the constriction resistance in the hemisphere, and ψ12 is the spreading resistance parameterin the layer–substrate. The thermal conductivities of the hemisphere and the substrateappear in the first and second terms, respectively. The layer–substrate spreading resistance parameter depends on two dimensionless parameters: τ = t/a and κ = k1 /k2 .This parameter was presented above under spreading resistance in a layer on a halfspace. To calculate the joint resistance the contact radius must be found.A special case arises when the rigidity of the layer is much smaller than the rigidityof the hemisphere and the layer. This corresponds to “soft” metallic layers such asindium, lead, and tin; or nonmetallic layers such as rubber or elastomers.
In this case,since E1 E2 and E1 E3 , the hemisphere and substrate may be modeled asperfectly rigid while the layer deforms elastically.The dimensionless numerical values for a/aL obtained from the elastic contactmodel of Chen and Engel (1972) according to Stevanović et al. (2001) are plottedin Fig. 4.19 for a wide range of relative layer thickness τ = t/a and for a range ofvalues of the layer Young’s modulus E1 . The contact model, which is represented bythe correlation equation of the numerical values, is (Stevanović et al., 2002)BOOKCOMP, Inc.
— John Wiley & Sons / Page 330 / 2nd Proofs / Heat Transfer Handbook / Bejan———-1.94696pt PgVar———Normal PagePgEnds: TEX[330], (70)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS1.110.9Chen and EngelModelMODELE1 (MPa)0.050.5501005000.80.7a /aL1234567891011121314151617181920212223242526272829303132333435363738394041424344453310.60.5[331], (71)0.40.3Lines: 2927 to 29430.21.03708pt PgVar——————Normal Page* PgEnds: Eject0.1010⫺210⫺1100101[331], (71)Figure 4.19 Comparison of data and model for contact between a rigid hemisphere and anelastic layer on a rigid substrate. (From Stevanović et al., 2001.)a= 1 − c3 exp(c1 τc2 )aL(4.189)with correlation coefficients: c1 = −1.73, c2 = 0.734, and c3 = 1.04. The referencecontact radius is aL , which corresponds to the very thick layer limit given byaL =3 Fρ4 E131/3fort→∞a(4.190)The maximum difference between the correlation equation and the numericalvalues obtained from the model of Chen and Engel (1972) is approximately 1.9%for τ = 0.02.
The following relationship, based on the Newton–Raphson method, isrecommended for calculation of the contact radius (Stevanović et al., 2001):an+1 =an − aL {1 − 1.04 exp[−1.73(t/an )0.734 ]}1 + 1.321(aL /aS )(t/an )0.734 exp[−1.73(t/an )0.734 ]BOOKCOMP, Inc. — John Wiley & Sons / Page 331 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.191)332123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESIf the first guess is a0 = aL , fewer than six iterations are required to give eight-digitaccuracy.In the general case where the hemisphere, layer, and substrate are elastic, thecontact radius lies in the range aS ≤ a ≤ aL for E2 < E1 .
The two limiting valuesof a are, according to Stevanović et al. (2002),3 F ρ 1/3 aS = 4 E23a=3 F ρ 1/3 aL =4 E13fort→0a(4.192)tfor → ∞awhere the effective Young’s modulus for the two limits are defined asE13 =1 − ν231 − ν21+E1E3−1E23 =1 − ν231 − ν22+E2E3−1[332], (72)(4.193)The dimensionless contact radius and dimensionless layer thickness were definedas (Stevanović et al., 2002)a − aSaL − a St √ 1/3∗τ =αaa∗ =where 0 < a ∗ < 1aLwhere α ==aS(4.194)E23E131/3wherea0 = aS + (aL − aS ) 1 − exp −πBOOKCOMP, Inc.
— John Wiley & Sons / Page 332 / 2nd Proofs / Heat Transfer Handbook / Bejan1/4√ π/4 2t αaS + a L———Normal Page* PgEnds: Eject[332], (72)Since the unknown contact radius a appears on both sides, the numerical solutionof the correlation equation requires an iterative method (Newton–Raphson method)to find its root. For all metal combinations, the following solution is recommended(Stevanović et al., 2002): √ π/4 1/4 t α(4.197)a = aS + (aL − aS ) 1 − exp −πa0———5.84923pt PgVar(4.195)The dimensionless numerical values obtained from the full model of Chen andEngel (1972) for values of α in the range 1.136 ≤ α ≤ 2.037 are shown in Fig.
4.20.The correlation equation is (Stevanović et al., 2002) √ π/4 a − aS1/4 t α(4.196)= 1 − exp −πaL − a SaLines: 2943 to 3001(4.198)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445333[333], (73)Lines: 3001 to 3011———-1.573pt PgVar———Normal PagePgEnds: TEX[333], (73)Figure 4.20 Comparison of the data and model for elastic contact between a hemisphere anda layer on a substrate.
(From Stevanović et al., 2002.)4.15.8 Joint Resistance of Elastic–Plastic Contacts of Hemispheresand Flat Surfaces in a VacuumA model is available for calculating the joint resistance of an elastic–plastic contactof a portion of a hemisphere whose radius of curvature is ρ attached to a cylinderwhose radius is b1 and a cylindrical flat whose radius is b2 . The elastic properties ofthe hemisphere are E1 , ν1 , and the elastic properties of the flat are E2 , ν2 . The thermalconductivities are k1 and k2 , respectively.If the contact strain is very small, the contact is elastic and the Hertz model canbe used to predict the elastic contact radius denoted as ae .
On the other hand, ifthe contact strain is very large, plastic deformation may occur in the flat, which isassumed to be fully work hardened, and the plastic contact radius is denoted ap .Between the fully elastic and fully plastic contact regions there is a transition calledthe elastic–plastic contact region, which is very difficult to model.
In the region thecontact radius is denoted as aep , the elastic–plastic contact radius. The relationshipbetween ae , ap , and aep is ae ≤ aep ≤ ap .BOOKCOMP, Inc. — John Wiley & Sons / Page 333 / 2nd Proofs / Heat Transfer Handbook / Bejan334123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESThe elastic–plastic radius is related to the elastic and plastic contact radii by meansof the composite model based on the method of Churchill and Usagi (1972) forcombining asymptotes (Sridhar and Yovanovich, 1994): aep = aen + apn1/n(m)(4.199)where n is the combination parameter, which is found empirically to have the valuen = 5.
The elastic and plastic contact radii may be obtained from the relationships(Sridhar and Yovanovich, 1994)ae =3 Fρ4 E1/3andap =FπHB1/2(m)(4.200)[334], (74)with the effective modulus1 − ν211 − ν221=+EE1E2(m2/N)(4.201)Lines: 3011 to 3068———The plastic parameter is the Brinell hardness HB of the flat. The elastic–plastic0.81224ptdeformation model assumes that the hemispherical solid is harder than the flat. The———static axial load is F .Short PageThe joint resistance for a smooth hemispherical solid in elastic–plastic contact* PgEnds: Ejectwith smooth flat is given by (Sridhar and Yovanovich, 1994)Rj =ψ1ψ2+4k1 aep4k2 aep(K/W)(4.202)The spreading–constriction resistance parameters for the hemisphere and flat areaep 1.5ψ1 = 1 −b1andaep 1.5ψ2 = 1 −b2(4.203)Alternative Constriction Parameter for a Hemisphere The followingspreading–constriction parameter can be derived from the hemisphere solution:ψ1 = 1.0014 − 0.0438 − 4.02642 + 4.9683(4.204)where = a/b1 .If the contact is in a vacuum and the radiation heat transfer across the gap isnegligible, Rj = Rc .
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