BOOKCOMP, Inc. — John Wiley & Sons / Page 351 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 351 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 [351], (91) Lines: 3662 to 3692 ——— 11.61588pt PgVar ——— Short Page PgEnds: T E X [351], (91) λ =−0.5444 −0.6636 ln P H e − 0.03204 ln P H e 2 − 0.000771 ln P H e 3 (4.262) Sridhar and Yovanovich (1994) reviewed the plastic and elastic deformation con- tact conductance correlation equations and compared them against vacuum data (Mi- kic and Rohsenow, 1966; Antonetti, 1983; Hegazy, 1985; Nho, 1989; McWaid and Marschall, 1992a, b) for several metal types, having a range of surface roughnesses, over a wide range of apparent contact pressure. Sridhar and Yovanovich (1996a) showed that the elastic deformation model was in better agreement with the vacuum data obtained for joints formed by conforming rough surfaces of tool steel, which is very hard. The elastic asperity contact and thermal conductance models of Greenwood and Williamson (1966), Greenwood (1967), Greenwood and Tripp (1967, 1970), Bush et al. (1975), and Bush and Gibson (1979) are different from the Mikic (1974) elastic contact model presented in this chapter. However, they predict similar trends of contact conductance as a function apparent contact pressure. 4.16.4 Conforming Rough Surface Model: Elastic–Plastic Deformation Sridhar and Yovanovich (1996c) developed an elastic–plastic contact conductance model which is based on the plastic contact model of Cooper et al. (1969) and the elastic contact model of Mikic (1974). The results are summarized below in terms of the geometric parameters A r /A a , the real-to-apparent area ratio; n, the contact spot density; a, the mean contact spot radius; and λ, the dimensionless mean plane separation: A r A a = f ep 2 erfc λ √ 2 (4.263) n = 1 16 m σ 2 exp(−λ 2 ) erfc(λ/ √ 2) (4.264) a = 8 π f ep σ m exp λ 2 2 erfc λ √ 2 (4.265) na = 1 8 2 π f ep m σ exp − λ 2 2 (4.266) h c σ k s m = 1 2 √ 2π f ep exp(−λ 2 /2) 1 − (f ep /2)erfc(λ/ √ 2) 1.5 (4.267) BOOKCOMP, Inc. — John Wiley & Sons / Page 352 / 2nd Proofs / Heat Transfer Handbook / Bejan 352 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 [352], (92) Lines: 3692 to 3734 ——— -1.41464pt PgVar ——— Short Page * PgEnds: Eject [352], (92) λ = √ 2erfc −1 1 f ep 2P H ep (4.268) The important elastic–plastic parameter f ep is a function of the dimensionless contact strain ∗ c , which depends on the amount of work hardening. This physical parameter lies in the range 0.5 ≤ f ep ≤ 1.0. The smallest and largest values correspond to zero and infinitely large contact strain, respectively. The elastic–plastic parameter is related to the contact strain: f ep = 1 + (6.5/ ∗ c ) 2 1/2 1 + (13.0/ ∗ c ) 1.2 1/1.2 0 < ∗ c < ∞ (4.269) The dimensionless contact strain is defined as ∗ c = 1.67 mE S f (4.270) where S f is the material yield or flow stress (Johnson, 1985), which is a complex physical parameter that must be determined by experiment for each metal. The elastic–plastic microhardness H ep can be determined by means of an iterative procedure which requires the following relationship: H ep = 2.76S f 1 + (6.5/ ∗ c ) 2 1/2 (4.271) The elastoplastic contact conductance model moves smoothly between the elastic contact model of Mikic (1974) and the plastic contact conductance model of Cooper et al. (1969), which was modified by Yovanovich (1982), Yovanovich et al. (1982a), and Song and Yovanovich (1988) to include the effect of work-hardened layers on the deformation of the contacting asperities. The dimensionless contact pressure for elastic–plastic deformation of the contacting asperities is obtained from the following approximate explicit relationship: P H ep = 0.9272P c 1 (1.43 σ/m) c 2 1/(1+0.071c 2 ) (4.272) where the coefficients c 1 and c 2 are obtained from Vickers microhardness tests. The Vickers microhardness coefficients are related to Brinell and Rockwell hardness for a wide range of metals. Correlation Equations for Dimensionless Contact Conductance: Elastic– Plastic Model The complex elastic–plastic contact model proposed by Sridhar and Yovanovich (1996a, b, c, d) may be approximated by the following correlation equations for the dimensionless contact conductance: BOOKCOMP, Inc. — John Wiley & Sons / Page 353 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 353 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 [353], (93) Lines: 3734 to 3767 ——— 7.81639pt PgVar ——— Short Page PgEnds: T E X [353], (93) C c = 1.54 P H ep 0.94 0 < ∗ c < 5 (4.273) 1.245b 1 P H ep b 2 5 < ∗ c < 400 (4.274) 1.25 P H ep 0.95 400 < ∗ c < ∞ (4.275) where the elastic–plastic correlation coefficients b 1 and b 2 depend on the dimension- less contact strain: b 1 = 1 + 46,690.2 ( ∗ c ) 2.49 1/30 (4.276) b 2 = 1 1 + 2086.9/( ∗ c ) 1.842 1/600 (4.277) 4.16.5 Gap Conductance for Large Parallel Isothermal Plates Two infinite isothermal surfaces form a gap of uniform thickness d which is much greater than the roughness of both surfaces: d σ 1 and σ 2 . The gap is filled with a stationary monatomic or diatomic gas. The boundary temperatures are T 1 and T 2 , where T 1 >T 2 . The Knudsen number for the gap is defined as Kn = Λ/d, where Λ is the molecular mean free path of the gas, which depends on the gas temperature and its pressure. The gap can be separated into three zones: two boundary zones, which are associated with the two solid boundaries, and a central zone. The boundary zones have thicknesses that are related to the molecular mean free paths Λ 1 and Λ 2 , where Λ 1 = Λ 0 T 1 T 0 P g,0 P g and Λ 2 = Λ 0 T 2 T 0 P g,0 P g (4.278) and Λ 0 ,T 0 , and P g,0 represent the molecular mean free path and the reference temper- ature and gas pressure. In the boundary zones the heat transfer is due to gas molecules that move back and forth between the solid surface and other gas molecules located at distances Λ 1 and Λ 2 from both solid boundaries. The energy exchange between the gas and solid molecules is imperfect. At the hot solid surface at temperature T 1 , the gas molecules that leave the surface after contact are at some temperature T g,1 <T 1 , and at the cold solid surface at temperature T 2 , the gas molecules that leave the surface after contact are at a temperature T g,2 >T 2 . The two boundary zones are called slip regions. In the central zone whose thickness is modeled as d − Λ 1 − Λ 2 , and whose temperature range is T g,1 ≥ T ≥ T g,2 , heat transfer occurs primarily by molecular BOOKCOMP, Inc. — John Wiley & Sons / Page 354 / 2nd Proofs / Heat Transfer Handbook / Bejan 354 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 [354], (94) Lines: 3767 to 3827 ——— 6.58437pt PgVar ——— Normal Page PgEnds: T E X [354], (94) diffusion. Fourier’s law of conduction can be used to determine heat transfer across the central zone. There are two heat flux asymptotes, corresponding to very small and very large Knudsen numbers. They are: for a continuum, Kn → 0 q → q 0 = k g T 1 − T 2 d (4.279) and for free molecules, Kn →∞ q → q ∞ = k g T 1 − T 2 M (4.280) where M = αβΛ = 2 − α 1 α 1 + 2 − α 2 α 2 2γ (γ + 1)Pr Λ (4.281) and k g is the thermal conductivity, α 1 and α 2 the accommodation coefficients, γ the ratio of specific heats, and Pr the Prandtl number. The gap conductance, defined as h g = q/(T 1 − T 2 ), has two asymptotes: for Kn → 0,h g → k g d for Kn →∞,h g → k g M For the entire range of the Knudsen number, the gap conductance is given by the relationship h g = k g d + M for 0 < Kn < ∞ (W/m 2 · K) (4.282) This relatively simple relationship covers the continuum, 0 < Kn < 0.1, slip, 0.1 < Kn < 10, and free molecule, 10 < Kn < ∞, regimes. Song (1988) introduced the dimensionless parameters G = k g h g d and M ∗ = M d (4.283) and recast the relationship above as G = 1 + M ∗ for 0 <M ∗ < ∞ (4.284) The accuracy of the simple parallel-plate gap model was compared against the data (argon and nitrogen) of Teagan and Springer (1968), and the data (argon and helium) of Braun and Frohn (1976). The excellent agreement between the simple gap model and all data is shown in Fig. 4.26. The simple gap model forms the basis of the gap model for the joint formed by two conforming rough surfaces. BOOKCOMP, Inc. — John Wiley & Sons / Page 355 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 355 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 [355], (95) Lines: 3827 to 3835 ——— 0.3591pt PgVar ——— Normal Page * PgEnds: Eject [355], (95) 10 Ϫ3 10 Ϫ1 10 0 10 1 10 2 10 0 10 Ϫ2 10 2 10 1 10 Ϫ1 M * G He (Braun & Frohn, 1976) (Braun & Frohn, 1976) (Teagan & Springer, 1968) (Teagan & Springer, 1968) Ar Ar N 2 Interpolated Model 1G= Mϩ * Figure 4.26 Gap conductance model and data for two large parallel isothermal plates. (From Song et al., 1992a.) 4.16.6 Gap Conductance for Joints between Conforming Rough Surfaces If the gap between two conforming rough surfaces as shown in Fig. 4.22 is occupied by a gas, conduction heat transfer will occur across the gap. This heat transfer is characterized by the gap conductance, defined as h g = ∆T j Q g (W/m 2 · K) (4.285) BOOKCOMP, Inc. — John Wiley & Sons / Page 356 / 2nd Proofs / Heat Transfer Handbook / Bejan 356 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 [356], (96) Lines: 3835 to 3879 ——— 0.05264pt PgVar ——— Normal Page PgEnds: T E X [356], (96) with ∆T j as the effective temperature drop across the gas gap and Q g the heat transfer rate across the gap. Because the local gap thickness and local temperature drop vary in very complicated ways throughout the gap, it is difficult to develop a simple gap conductance model. Several gap conductance models and correlation equations have been presented by a number of researchers (Cetinkale and Fishenden, 1951; Rapier et al., 1963; Shlykov, 1965; Veziroglu, 1967; Lloyd et al., 1973; Garnier and Begej, 1979; Loyalka, 1982; Yovanovich et al., 1982b); they are given in Table 4.19. The parameters that appear in Table 4.19 are b t = 2(CLA 1 +CLA 2 ), where CLA i is the centerline-average surface TABLE 4.19 Models and Correlation Equations for Gap Conductance for Conforming Rough Surfaces Authors Models and Correlations Cetinkale and h g = k g 0.305b t + M Fishenden (1951) Rapier et al. (1963) h g = k g 1.2 2b t + M + 0.8 2b t ln 1 + 2b t M Shlykov (1965) h g = k g b t 10 3 + 10 X + 4 X 2 − 4 1 X 3 + 3 X 2 + 2 X ln(1 + X) Veziroglu (1967) h g = k g 0.264 b t + M for b t > 15 µm kg 1.78 b t + M for b t < 15 µm Lloyd et al. (1973) h g = k g δ + βΛ/(α 1 + α 2 ) δ not given Garnier and Begej (1979) h g = k g exp(−1/Kn) M + 1 − exp(−1/Kn) δ + M δ not given Loyalka (1982) h g = k g δ + M + 0.162(4 − α 1 − α 2 )βΛ δ not given Yovanovich et al. (1982b) h g = k g /σ √ 2π ∞ 0 exp −(Y/σ − t/σ) 2 /2 t/σ + M/σ d t σ Y σ = √ 2 erfc −1 2P H p P H p = P c 1 (1.62σ/m) c 2 1/(1+0.071c 2 ) Source: Song (1988). BOOKCOMP, Inc. — John Wiley & Sons / Page 357 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 357 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 [357], (97) Lines: 3879 to 3919 ——— 0.47626pt PgVar ——— Normal Page PgEnds: T E X [357], (97) roughness of the two contacting surfaces, M = αβΛ,X = b t /M, σ = σ 2 1 + σ 2 2 , where the units of σ are µm. The Knudsen number Kn that appears in the Garnier and Begej (1979) correlation equation is not defined. Song and Yovanovich (1987), Song (1988), and Song et al. (1993b) reviewed the models and correlation equations given in Table 4.19. They found that for some of the correlation equations the required gap thickness δ was not defined, and for other correlation equations an empirically based average gap thickness was specified that is constant, independent of variations of the apparent contact pressure. The gap conductance model developed by Yovanovich et al. (1982b) is the only one that accounts for the effect of mechanical load and physical properties of the contacting asperities on the gap conductance. This model is presented below. The gap conductance model for conforming rough surfaces was developed, modi- fied, and verified by Yovanovich and co-workers (Yovanovich et al., 1982b; Hegazy, 1985; Song and Yovanovich, 1987; Negus and Yovanovich, 1988; Song et al., 1992a, 1993b). The gap contact model is based on surfaces having Gaussian height distributions and also accounts for mechanical deformation of the contacting surface asperities. Development of the gap conductance model is presented in Yovanovich (1982, 1986), Yovanovich et al. (1982b), and Yovanovich and Antonetti (1988). The gap conductance model is expressed in terms of an integral: h g = k g σ 1 √ 2π ∞ 0 exp − (Y/σ − u) 2 /2 u + M/σ du = k g σ I g (W/m 2 · K) (4.286) where k g is the thermal conductivity of the gas trapped in the gap and σ is the effective surface roughness of the joint, and u = t/σ is the dimensionless local gap thickness. The integral I g depends on two independent dimensionless parameters:Y/σ, the mean plane separation; and M/σ, the relative gas rarefaction parameter. The relative mean planes separation for plastic and elastic contact are given by the relationships Y σ plastic = √ 2 erfc −1 2P H p Y σ elastic = √ 2 erfc −1 4P H e (4.287) The relative contact pressures P/H p for plastic deformation and P/H e for elastic deformation can be determined by means of appropriate relationships. The gas rarefaction parameter is M = αβΛ, where the gas parameters are defined as: α = 2 − α 1 α 1 + 2 − α 2 α 2 (4.288) β = 2γ (γ + 1)Pr (4.289) BOOKCOMP, Inc. — John Wiley & Sons / Page 358 / 2nd Proofs / Heat Transfer Handbook / Bejan 358 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 [358], (98) Lines: 3919 to 3952 ——— 7.26427pt PgVar ——— Short Page * PgEnds: Eject [358], (98) Λ = Λ 0 T g T g,0 P g P g,0 (4.290) where α is the accommodation coefficient, which accounts for the efficiency of gas– surface energy exchange. There is a large body of research dealing with experimental and theoretical aspects of α for various gases in contact with metallic surfaces under various surface conditions and temperatures (Wiedmann and Trumpler, 1946; Hart- nett, 1961; Wachman, 1962; Thomas, 1967; Semyonov et al., 1984; Loyalka, 1982). Song and Yovanovich (1987) and Song et al. (1992a, 1993b) examined the several gap conductance models available in the literature and the experimental data and models for the accommodation coefficients. Song and Yovanovich (1987) developed a correlation for the accommodation for engineering surfaces (i.e., surfaces with absorbed layers of gases and oxides). They proposed a correlation that is based on experimental results of numerous investiga- tors for monatomic gases. The relationship was extended by the introduction of a monatomic equivalent molecular weight to diatomic and polyatomic gases. The final correlation is α = exp(C 0 T) M g C 1 + M g + [1 − exp(C 0 T)] 2.4µ (1 + µ) 2 (4.291) with C 0 =−0.57, T = (T s −T 0 )/T 0 ,M g = M g for monatomic gases (= 1.4M g for diatomic and polyatomic gases), C 1 = 6.8 in units of M g (g/mol), and µ = M g /M s , where T s and T 0 = 273 K are the absolute temperatures of the surface and the gas, and M g and M s are the molecular weights of the gas and the solid, respectively. The agreement between the predictions according to the correlation above and the published data for diatomic and polyatomic gases was within ±25%. The gas parameter β depends on the specific heat ratio γ = C p /C v and the Prandtl number Pr. The molecular mean free path of the gas molecules Λ depends on the type of gas, the gas temperature T g and gas pressure P g , and the reference values of the mean free path Λ 0 , the gas temperature T g,0 , and the gas pressure P g,0 , respectively. Wesley and Yovanovich (1986) compared the predictions of the gap conductance model and experimental measurements of gaseous gap conductance between the fuel and clad of a nuclear fuel rod. The agreement was very good and the model was recommended for fuel pin analysis codes. The gap integral can be computed accurately and easily by means of computer algebra systems. Negus and Yovanovich (1988) developed the following correlation equations for the gap integral: I g = f g Y/σ + M/σ (4.292) In the range 2 ≤ Y/σ ≤ 4: BOOKCOMP, Inc. — John Wiley & Sons / Page 359 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 359 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 [359], (99) Lines: 3952 to 3997 ——— 4.09605pt PgVar ——— Short Page PgEnds: T E X [359], (99) f g = 1.063 + 0.0471 4 − Y σ 1.68 ln σ M 0.84 for 0.01 ≤ M σ ≤ 1 1 + 0.06 σ M 0.8 for 1 ≤ M σ < ∞ The correlation equations have a maximum error of approximately 2%. 4.16.7 Joint Conductance for Conforming Rough Surfaces The joint conductance for a joint between two conforming rough surfaces is h j = h c + h g (W/m 2 · K) (4.293) when radiation heat transfer across the gap is neglected. The relationship is applicable to joints that are formed by elastic, plastic, or elastic–plastic deformation of the contacting asperities. The mode of deformation will influence h c and h g through the relative mean plane separation parameter Y/σ. The gap and joint conductances are compared against data (Song, 1988) obtained for three types of gases, argon, helium, and nitrogen, over a gas pressure range between 1 and 700 torr. The gases occupied gaps formed by conforming rough Ni 200 and stainless steel type 304 metals. In all tests the metals forming the joint were identical, and one surface was flat and lapped while the other surface was flat and glass bead blasted. The gap and joint conductance models were compared against data obtained for relatively light contact pressures where the gap and contact conductances were com- parable. Figure 4.27 shows plots of the joint conductance data and the model predic- tions for very rough stainless steel type 304 surfaces at Y/σ = 1.6×10 −4 . Agreement among the data for argon, helium, and nitrogen is very good for gas pressures be- tween approximately 1 and 700 torr. At the low gas pressure of 1 torr, the measured and predicted joint conductance values for the three gases differ by a few percent because h g h c and h j ≈ h c . As the gas pressure increases there is a large increase in the joint conductances because the gap conductances are increasing rapidly. The joint conductances for argon and nitrogen approach asymptotes for gas pressures ap- proaching 1 atm. The joint conductances for helium are greater than for argon and nitrogen, and the values do not approach an asymptote in the same pressure range. The asymptote for helium is approached at gas pressures greater than 1 atm. Figure 4.28 shows the experimental and theoretical gap conductances as points and curves for nitrogen and helium for gas pressures between approximately 10 and 700 torr. The relative contact pressure is 1.7 × 10 −4 is based on the plastic deformation model. The joint was formed by Ni 200 surfaces (one flat and lapped and the second flat and glass bead blasted). The data were obtained by subtracting the theoretical value of h c from the measured values of h j to get the values of h g that appear on the plots. The agreement between the data and the predicted curves is very good. BOOKCOMP, Inc. — John Wiley & Sons / Page 360 / 2nd Proofs / Heat Transfer Handbook / Bejan 360 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 [360], (100) Lines: 3997 to 4013 ——— 0.36307pt PgVar ——— Normal Page PgEnds: T E X [360], (100) 10 0 10 2 10 3 10 4 10 5 10 2 10 1 10 3 P g (torr) h j (W/m . K) 2 Experiment Theory He He Ar Ar N 2 N 2 Vacuum Vacuum Stainless Steel 304 = 4.83 m R= p 14.7 m Y/ = 3.02 PH/ = 1.6 10 e ϫ Ϫ4 Figure 4.27 Joint conductance model and data for conforming rough stainless steel 304 surfaces. (From Song, 1988.) Figure 4.29 shows the experimental data for argon, nitrogen, and helium and the dimensionless theoretical curve for the gap model recast as (Song et al., 1993b) G = 1 + M ∗ (4.294) where G = k g h g Y and M ∗ = M Y = αβΛ Y (4.295) There is excellent agreement between the model and the data over the entire range of the gas–gap parameter M ∗ . The joint was formed by very rough conforming Ni . whose temperature range is T g,1 ≥ T ≥ T g,2 , heat transfer occurs primarily by molecular BOOKCOMP, Inc. — John Wiley & Sons / Page 354 / 2nd Proofs / Heat Transfer Handbook / Bejan 354 THERMAL SPREADING. 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 [354], (94) Lines: 376 7 to 3827 ——— 6.58437pt PgVar ——— Normal Page PgEnds: T E X [354], (94) diffusion. Fourier’s law of conduction can be used to determine heat transfer. 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 353 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 [353],