BOOKCOMP, Inc. — John Wiley & Sons / Page 331 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 331 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [331], (71) Lines: 2927 to 2943 ——— 1.03708pt PgVar ——— Normal Page * PgEnds: Eject [331], (71) aa/ L 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 Ϫ2 10 Ϫ1 10 0 10 1 Chen and Engel Model E MPa 1 () 0.05 0.5 50 100 500 MODEL Figure 4.19 Comparison of data and model for contact between a rigid hemisphere and an elastic layer on a rigid substrate. (From Stevanovi ´ c et al., 2001.) a a L = 1 −c 3 exp(c 1 τ c 2 ) (4.189) with correlation coefficients: c 1 =−1.73,c 2 = 0.734, and c 3 = 1.04. The reference contact radius is a L , which corresponds to the very thick layer limit given by a L = 3 4 F ρ E 13 1/3 for t a →∞ (4.190) The maximum difference between the correlation equation and the numerical values 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, is recommended for calculation of the contact radius (Stevanovi ´ c et al., 2001): a n+1 = a n − a L {1 −1.04 exp[−1.73(t/a n ) 0.734 ]} 1 +1.321(a L /a S )(t/a n ) 0.734 exp[−1.73(t/a n ) 0.734 ] (4.191) BOOKCOMP, Inc. — John Wiley & Sons / Page 332 / 2nd Proofs / Heat Transfer Handbook / Bejan 332 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [332], (72) Lines: 2943 to 3001 ——— 5.84923pt PgVar ——— Normal Page * PgEnds: Eject [332], (72) If the first guess is a 0 = a L , fewer than six iterations are required to give eight-digit accuracy. In the general case where the hemisphere, layer, and substrate are elastic, the contact radius lies in the range a S ≤ a ≤ a L for E 2 <E 1 . The two limiting values of a are, according to Stevanovi ´ c et al. (2002), a = a S = 3 4 F ρ E 23 1/3 for t a → 0 a L = 3 4 F ρ E 13 1/3 for t a →∞ (4.192) where the effective Young’s modulus for the two limits are defined as E 13 = 1 −ν 2 1 E 1 + 1 −ν 2 3 E 3 −1 E 23 = 1 −ν 2 2 E 2 + 1 −ν 2 3 E 3 −1 (4.193) The dimensionless contact radius and dimensionless layer thickness were defined as (Stevanovi ´ c et al., 2002) a ∗ = a −a S a L − a S where 0 <a ∗ < 1 (4.194) τ ∗ = t a √ α 1/3 where α = a L a S = E 23 E 13 1/3 (4.195) The dimensionless numerical values obtained from the full model of Chen and Engel (1972) for values of α in the range 1.136 ≤ α ≤ 2.037 are shown in Fig. 4.20. The correlation equation is (Stevanovi ´ c et al., 2002) a −a S a L − a S = 1 −exp −π 1/4 t √ α a π/4 (4.196) Since the unknown contact radius a appears on both sides, the numerical solution of the correlation equation requires an iterative method (Newton–Raphson method) to find its root. For all metal combinations, the following solution is recommended (Stevanovi ´ c et al., 2002): a = a S + (a L − a S ) 1 −exp −π 1/4 t √ α a 0 π/4 (4.197) where a 0 = a S + (a L − a S ) 1 −exp −π 1/4 2t √ α a S + a L π/4 (4.198) BOOKCOMP, Inc. — John Wiley & Sons / Page 333 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 333 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [333], (73) Lines: 3001 to 3011 ——— -1.573pt PgVar ——— Normal Page PgEnds: T E X [333], (73) Figure 4.20 Comparison of the data and model for elastic contact between a hemisphere and a layer on a substrate. (From Stevanovi ´ c et al., 2002.) 4.15.8 Joint Resistance of Elastic–Plastic Contacts of Hemispheres and Flat Surfaces in a Vacuum A model is available for calculating the joint resistance of an elastic–plastic contact of a portion of a hemisphere whose radius of curvature is ρ attached to a cylinder whose radius is b 1 and a cylindrical flat whose radius is b 2 . The elastic properties of the hemisphere are E 1 , ν 1 , and the elastic properties of the flat are E 2 , ν 2 . The thermal conductivities are k 1 and k 2 , respectively. If the contact strain is very small, the contact is elastic and the Hertz model can be used to predict the elastic contact radius denoted as a e . On the other hand, if the contact strain is very large, plastic deformation may occur in the flat, which is assumed to be fully work hardened, and the plastic contact radius is denoted a p . Between the fully elastic and fully plastic contact regions there is a transition called the elastic–plastic contact region, which is very difficult to model. In the region the contact radius is denoted as a ep , the elastic–plastic contact radius. The relationship between a e ,a p , and a ep is a e ≤ a ep ≤ a p . BOOKCOMP, Inc. — John Wiley & Sons / Page 334 / 2nd Proofs / Heat Transfer Handbook / Bejan 334 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [334], (74) Lines: 3011 to 3068 ——— 0.81224pt PgVar ——— Short Page * PgEnds: Eject [334], (74) The elastic–plastic radius is related to the elastic and plastic contact radii by means of the composite model based on the method of Churchill and Usagi (1972) for combining asymptotes (Sridhar and Yovanovich, 1994): a ep = a n e + a n p 1/n (m) (4.199) where n is the combination parameter, which is found empirically to have the value n = 5. The elastic and plastic contact radii may be obtained from the relationships (Sridhar and Yovanovich, 1994) a e = 3 4 Fρ E 1/3 and a p = F πH B 1/2 (m) (4.200) with the effective modulus 1 E = 1 −ν 2 1 E 1 + 1 −ν 2 2 E 2 (m 2 /N) (4.201) The plastic parameter is the Brinell hardness H B of the flat. The elastic–plastic deformation model assumes that the hemispherical solid is harder than the flat. The static axial load is F . The joint resistance for a smooth hemispherical solid in elastic–plastic contact with smooth flat is given by (Sridhar and Yovanovich, 1994) R j = ψ 1 4k 1 a ep + ψ 2 4k 2 a ep (K/W) (4.202) The spreading–constriction resistance parameters for the hemisphere and flat are ψ 1 = 1 − a ep b 1 1.5 and ψ 2 = 1 − a ep b 2 1.5 (4.203) Alternative Constriction Parameter for a Hemisphere The following spreading–constriction parameter can be derived from the hemisphere solution: ψ 1 = 1.0014 −0.0438 −4.0264 2 + 4.968 3 (4.204) where = a/b 1 . If the contact is in a vacuum and the radiation heat transfer across the gap is negligible, R j = R c . Also, if b 1 = b 2 = b, ψ 1 = ψ 2 = 1 − a b 1.5 (4.205) The joint and dimensionless joint resistances for this case become BOOKCOMP, Inc. — John Wiley & Sons / Page 335 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 335 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [335], (75) Lines: 3068 to 3091 ——— 0.47105pt PgVar ——— Short Page PgEnds: T E X [335], (75) R j = ψ 2k s a and R ∗ j = 2bk s R j = (1 −a/b) 1.5 a/b (4.206) where k s = 2k 1 k 2 /(k 1 + k 2 ). Sridhar and Yovanovich (1994) compared the dimensionless joint resistance against data obtained for contacts between a carbon steel ball and several flats of Ni 200, carbon steel, and tool steel. The nondimensional data and the dimensionless joint resistance model are compared in Fig. 4.21 for a range of values of the recipro- cal contact strain b/a. The agreement between the model and the data over the entire range 20 < b/a < 120 is very good. The points near b/a ≈ 100 are in the elastic contact 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 contact region. If the material of the flat work-hardens as the deformation takes place, the model for predicting the contact radius is much more complex, as described by Sridhar and Yovanovich (1994) and Johnson (1985). This case is not given here. Figure 4.21 Comparison of the data and model for an elastic–plastic contact between a hemisphere and a flat. (From Sridhar and Yovanovich, 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 336 / 2nd Proofs / Heat Transfer Handbook / Bejan 336 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [336], (76) Lines: 3091 to 3116 ——— 9.16412pt PgVar ——— Normal Page * PgEnds: Eject [336], (76) 4.15.9 Ball-Bearing Resistance Models have been presented (Yovanovich, 1967, 1971, 1978) for calculating the overall thermal resistance of slowly rotating instrument bearings, which consist of many very smooth balls contained by very smooth inner and outer races. The thermal resistance models for bearings are based on elastic contact of the balls with the inner and outer races and spreading and constriction resistances in the balls and in the inner and outer races. For each ball there are two elliptical contact areas, one at the inner race and one at the outer race. The local thickness of the adjoining gap is very complex to model. There are four spreading–constriction zones associated with each ball. The full elastoconstriction resistance model must be used to obtain the overall thermal resistance of the bearing. Since these are complex systems, the contact resistance models are also complex; therefore, they are not presented here. The references above should be consulted for the development of the contact resistance models and other pertinent references. 4.15.10 Line Contact Models If a long smooth circular cylinder with radius of curvature ρ 1 = D 1 /2, length L 1 , and elastic properties: E 1 , ν 1 makes contact with another long smooth circular cylinder with radius of curvature ρ 2 = D 2 /2, length L 1 , and elastic properties: E 2 , ν 2 , then in general, if the axes of the cylinders are not aligned (i.e., they are crossed), an elliptical contact area is formed with semiaxes a and b, where it is assumed that a<b.Ifthe cylinder axes are aligned, the contact area becomes a strip of width 2a, and the larger axes are equal to the length of the cylinder. The general Hertz model presented may be used to find the semiaxes and the local gap thickness if the axes are not aligned. For aligned axes, the general equations reduce to simple relationships, which are given below. Contact Strip and Local Gap Thickness If the two cylinder axes are aligned, the contact area is a strip of width 2a and length L 1 , where (Timoshenko and Goodier, 1970; Walowit and Anno, 1975) a = 2 2F ρ∆ πL 1 1/2 (m) (4.207) where the effective curvature is 1 ρ = 1 ρ 1 + 1 ρ 2 (1/m) (4.208) and the contact parameter is ∆ = 1 2 1 −ν 2 1 E 1 + 1 −ν 2 2 E 2 (m 2 /N) (4.209) BOOKCOMP, Inc. — John Wiley & Sons / Page 337 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 337 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [337], (77) Lines: 3116 to 3176 ——— 1.05823pt PgVar ——— Normal Page * PgEnds: Eject [337], (77) The contact pressure is maximum along the axis of the contact strip, and it is given by the relationship P 0 = 2 π F aL 1 = F 2πL 1 ρ∆ 1/2 (N/m 2 ) (4.210) and the pressure distribution has the form (Timoshenko and Goodier, 1970; Walowit and Anno, 1975) P(x)= P 0 1 − x a 2 for 0 ≤ x ≤ a (N/m 2 ) (4.211) The mean contact area pressure is P m = F 2aL 1 = 4P 0 π (N/m 2 ) (4.212) The normal approach of the two aligned cylinders is (Timoshenko and Goodier, 1970; Walowit and Anno, 1975) α = 2F π 1 −ν 2 1 E 1 ln 4ρ 1 a − 1 2 + 1 −ν 2 2 E 2 ln 4ρ 2 a − 1 2 (m) (4.213) where F = F/L 1 is the load per unit cylinder length. The general local gap thickness relationship is (Timoshenko and Goodier, 1970; Walowit and Anno, 1975) 2δ ρ = 1 − 1 L 2 1/2 − 1 − ξ 2 L 2 1/2 + ξ(ξ 2 − 1) 1/2 − cosh −1 ξ −ξ 2 + 1 2L (4.214) where L = ρ 2a ξ = x L 1 ≤ ξ ≤ L (4.215) If a single circular cylinder of diameter D or (ρ 1 = D 1 /2 = D/2) is in elastic contact with a flat (ρ 2 =∞), put ρ = D/2 in the relationships above. Contact Resistance at a Line Contact The thermal contact resistance for the very narrow contact strip of width 2a formed by the elastic contact of a long smooth circular cylinder of diameter D and a smooth flat whose width is 2b and whose length L 1 is identical to the cylinder length is given by the approximate relationship (McGee et al., 1985) R c = 1 πL 1 k 1 ln 4 1 − π 2 + 1 πL 1 k 2 ln 2 π 2 (K/W) (4.216) BOOKCOMP, Inc. — John Wiley & Sons / Page 338 / 2nd Proofs / Heat Transfer Handbook / Bejan 338 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [338], (78) Lines: 3176 to 3245 ——— 0.48433pt PgVar ——— Normal Page * PgEnds: Eject [338], (78) where the thermal conductivities of the cylinder and flat are k 1 and k 2 , respectively. The contact parameters are 1 = 2a/D for the cylinder and 2 = a/b for the flat. For elastic contacts, 2a/D 1 and 2a/b 1 for most engineering applications. The approximate relationship for R c 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 the contact resistance: R ∗ c = L 1 k s R c = 1 2π k s k 1 ln π F ∗ − k s 2k 1 + 1 2π k s k 2 ln 1 4πF ∗ (4.217) where F ∗ = F∆ L 1 D and k s = 2k 1 k 2 k 1 + k 2 (4.218) Gap Resistance at a Line Contact The general elastogap resistance model for line contacts proposed by Yovanovich (1986) reduces for the circular cylinder–flat contact to 1 R g = 4aL 1 D k g,∞ I g,l (W/K) (4.219) where k g,∞ is the gas thermal conductivity and the line contact elastogap integral is defined as (Yovanovich, 1986) I g,l = 2 π L 1 cosh −1 (ξ)dξ 2δ/D +M/D (4.220) where L = D 2a ξ = x L 1 ≤ ξ ≤ L (4.221) This is the coupled elastogap model. Numerical integration is required to calculate values of I g,l . The gas rarefaction parameter that appears in the gap integral is M = αβΛ (m) (4.222) where the accommodation parameter and other gas parameters are defined as α = 2 −α 1 α 1 + 2 −α 2 α 2 β = 2γ (γ +1) Pr γ = C p C v (4.223) and the molecular mean free path is Λ = Λ g,∞ T g T g,∞ P g,∞ P g (m) (4.224) BOOKCOMP, Inc. — John Wiley & Sons / Page 339 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 339 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [339], (79) Lines: 3245 to 3271 ——— 3.31606pt PgVar ——— Normal Page * PgEnds: Eject [339], (79) where Λ g,∞ is the value of the molecular mean free path at the reference temperature T g,∞ and gas pressure P g,∞ . Joint Resistance at a Line Contact The joint resistance at a line contact, neglecting radiation heat transfer across the gap, is 1 R j = 1 R c + 1 R g (W/K) (4.225) McGee et al. (1985) examined the accuracy of the contact, gap, and joint resistance relationships 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 and 8000 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 amounts of form error (crowning) along the contacting surfaces. Joint Resistance of Nonconforming Rough Surfaces There is ample em- pirical 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 and e, where a smooth hemispherical surface is in contact with a flat, rough surface. If surfaces are nonconforming and rough, the joint that is formed is more complex from 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- 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 connected under conditions that are not understood today. The thermal joint resistance of such a contact is complex because heat can cross the joint by conduction through the microcontacts and the associated microgaps and by conduction across the macrogap. If the temperature level of the joint is suffi- ciently high, there may be significant radiation across the microgaps and macrogap. Clearly, this type of joint represents complex thermal and mechanical problems that are 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 resistance of a nonconforming surface under light mechanical loads and have negligible effects at higher loads. Also, the presence of out-of-flatness can have significant effects on the joint resistance of a rough surface under vacuum conditions. It is generally accepted that the joint resistance under vacuum conditions may be modeled as the superposition of microscopic and macroscopic resistance (Clausing and Chao, 1965; Greenwood and Tripp, 1967; Holm, 1967; Yovanovich, 1969; Burde and Yovanovich, 1978; Lambert, 1995; Lambert and Fletcher, 1997). The joint resis- tance can be modeled as BOOKCOMP, Inc. — John Wiley & Sons / Page 340 / 2nd Proofs / Heat Transfer Handbook / Bejan 340 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [340], (80) Lines: 3271 to 3311 ——— -3.56989pt PgVar ——— Normal Page * PgEnds: Eject [340], (80) R j = R mic + R mac (K/W) (4.226) The microscopic resistance is given by the relationship R mic = ψ mic 2k s Na S (K/W) (4.227) where ψ mic is the average spreading–constriction resistance parameter, N is the num- ber of microcontacts that are distribution in some complex manner over the contour area of radius a L and a S represents some average microcontact spot radius, and the harmonic mean thermal conductivity of the joint is k s = 2k 1 k 2 /(k 1 + k 2 ). The macroscopic resistance is given by the relationship R mac = ψ mac 2k s a L (K/W) (4.228) where ψ mac is the spreading–constriction resistance parameter for the contour area of radius a L . The mechanical model should be capable of predicting the contact parameters: a S ,a L , and N. These parameters are also required for the determination of thethermal spreading–constriction parameters ψ mic and ψ mac . At this time there is no simple mechanical model available for prediction of the ge- ometric parameters required in microscopic and macroscopicresistancerelationships. There are publications (e.g., Greenwood and Tripp, 1967; Holm, 1967; Burde and Yovanovich, 1978; Johnson, 1985; Lambert and Fletcher, 1997; Marotta and Fletcher, 2001) that deal with various aspects of this very complex problem. 4.16 CONFORMING ROUGH SURFACE MODELS There are models for predicting contact, gap, and joint conductances between con- forming (nominally flat) rough surfaces developed by Greenwood and Williamson (1966), Greenwood (1967), Greenwood and Tripp (1970), Cooper et al. (1969), Mi- kic (1974), Sayles and Thomas (1976), Yovanovich (1982), and DeVaal (1988). The three mechanical models—elastic, plastic, or elastic–plastic deformation of the contacting asperities—are based on the assumptions that the surface asperities have Gaussian height distributions about some mean plane passing through each surface and that the surface asperities are distributed randomly over the apparent contact area A a . Figure 4.22 shows a very small portion of a typical joint formed between two nominally flat rough surfaces under a mechanical load. Each surface has a mean plane, and the distance between them, denoted as Y , is related to the effective surface roughness, the apparent contact pressure, and the plastic or elastic physical properties of the contacting asperities. A very important surface roughness parameter is the surface roughness: either the rms (root-mean-square) roughness or the CLA (centerline-average) roughness, which are defined as (Whitehouse and Archard, 1970; Onions and Archard, 1973; Thomas, 1982) . become BOOKCOMP, Inc. — John Wiley & Sons / Page 335 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 335 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [ 335] ,. 2nd Proofs / Heat Transfer Handbook / Bejan 332 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [332],. Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 333 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [333],