BOOKCOMP, Inc. — John Wiley & Sons / Page 341 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 341 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 [341], (81) Lines: 3311 to 3334 ——— 0.60109pt PgVar ——— Normal Page * PgEnds: Eject [341], (81) CLA roughness = 1 L L 0 |y(x)| dx (m) (4.229) rms roughness = 1 L L 0 y 2 (x) dx (m) (4.230) where y(x) is the distance of points in the surface from the mean plane (Fig. 4.22) and L is the length of a trace that contains a sufficient number of asperities. For Gaussian asperity heights with respect to the mean plane, these two measures of surface roughness are related (Mikic and Rohsenow, 1966): σ = π 2 · CLA A second very important surface roughness parameter is the absolute mean asperity slope, which is defined as (Cooper et al., 1969; Mikic and Rohsenow, 1966; and DeVaal et al. 1987). y x y x 1 2 mdydx 222 =/ mdydx=/ mdydx 111 =/ Y Y ϩ= ͌ 2 1 2 2 mmm= ͌ 2 1 2 2 ϩ Figure 4.22 Typical joint between conforming rough surfaces. (From Hegazy, 1985.) BOOKCOMP, Inc. — John Wiley & Sons / Page 342 / 2nd Proofs / Heat Transfer Handbook / Bejan 342 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 [342], (82) Lines: 3334 to 3384 ——— 3.76213pt PgVar ——— Normal Page PgEnds: T E X [342], (82) m = 1 L L 0 dy(x) dx dx (rad) (4.231) The effective rms surface roughness and the effective absolute mean asperity slope for a typical joint formed by two conforming rough surfaces are defined as (Cooper et al., 1969; Mikic, 1974; Yovanovich, 1982) σ = σ 2 1 + σ 2 2 and m = m 2 1 + m 2 2 (4.232) Antonetti et al. (1991) reported approximate relationships for m as a function of σ for several metal surfaces that were bead-blasted. The three deformation models (elastic, plastic, or elastic–plastic) give relation- ships for three important geometric parameters of the joint: the relative real contact area A r /A a , the contact spot density n, and the mean contact spot radius a in terms of the relative mean plane separation defined as λ = Y/σ. The mean plane separation Y and the effective surface roughness are illustrated in Fig. 4.22 for the joint formed by the mechanical contact of two nominally flat rough surfaces. The models differ in the mode of deformation of the contacting asperities. The three modes of deformation are plastic deformation of the softer contacting asperities, elastic deformation of all contacting asperities, and elastic–plastic deformation of the softer contacting asperities. For the three deformation models there is one thermal contact conductance model, given as (Cooper et al., 1969; Yovanovich, 1982) h c = 2nak s ψ() (W/m 2 · K) (4.233) where n is the contact spot density, a is the mean contact spot radius, and the effective thermal conductivity of the joint is k s = 2k 1 k 2 k 1 + k 2 (W/m · K) (4.234) and the spreading/constriction parameter ψ, based on isothermal contact spots, is approximated by ψ() = (1 −) 1.5 for 0 < < 0.3 (4.235) where the relative contact spot size is = √ A r /A a . The geometric parameters n, a and A r /A a are related to the relative mean plane separation λ = Y/σ. 4.16.1 Plastic Contact Model The original plastic deformation model of Cooper et al. (1969) has undergone signif- icant modifications during the past 30 years. First, a new, more accurate correlation equation was developed by Yovanovich (1982). Then Yovanovich et al. (1982a) and BOOKCOMP, Inc. — John Wiley & Sons / Page 343 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 343 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 [343], (83) Lines: 3384 to 3393 ——— 3.097pt PgVar ——— Normal Page PgEnds: T E X [343], (83) Figure 4.23 Vickers microhardness versus indentation diagonal for four metal types. (From Hegazy, 1985.) Hegazy (1985) introduced the microhardness layer which appears in most worked metals. Figures 4.23 and 4.24 show plots of measured microhardness and macrohard- ness versus the penetration depth t or the Vickers diagonal d V . These two measures of indenter penetration are related: d v /t = 7. Figure 4.23 shows the measured Vickers microhardness versus indentation diagonal for four metal types (Ni 200, stainless steel 304, Zr-4 and Zr-2.5 wt % Nb). The four sets of data show the same trends: that as the load on the indenter increases, the indentation diagonal increases and the Vickers microhardness decreases with increasing diagonal (load). The indentation diagonal was between 8 and 70 µm. Figure 4.24 shows the Vickers microhardness measurements and the Brinell and Rockwell macrohardness measurements versus indentation depth. The Brinell and Rockwell macrohardness values are very close because they correspond to large in- dentations, and therefore, they are a measure of the bulk hardness, which does not change with load. According to Fig. 4.24, the penetration depths for the Vickers mi- crohardness measurements are between 1 and 10 µm, whereas the larger penetration BOOKCOMP, Inc. — John Wiley & Sons / Page 344 / 2nd Proofs / Heat Transfer Handbook / Bejan 344 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 [344], (84) Lines: 3393 to 3406 ——— 3.73402pt PgVar ——— Normal Page PgEnds: T E X [344], (84) Figure 4.24 Vickers, Brinell, and Rockwell hardness versus indentation depth for four metal types. (From Hegazy, 1985.) depths for the Brinell and Rockwell macrohardness measurements lie between ap- proximately 100 and 1000 µm. The microhardness layer may be defined by means of the Vickers microhardness measurements, which relate the Vickers microhardness H V to the Vickers average indentation diagonal d V (Yovanovich et al., 1982a; Hegazy, 1985): H V = c 1 d V d 0 c 2 (GPa) (4.236) where d 0 represents some convenient reference value for the average diagonal, and c 1 and c 2 are the correlation coefficients. It is conventional to set d 0 = 1 µm. Hegazy (1985) found that c 1 is closely related to the metal bulk hardness, such as the Brinell hardness, denoted as H B . The original mechanical contact model (Yovanovich et al., 1982a; Hegazy, 1985) required an iterative procedure to calculate the appropriate microhardness for a given surface roughness σ and m, given the apparent contact pressure P and the coefficients c 1 and c 2 . Song and Yovanovich (1988) developed an explicit relationship for the micro- hardness H p , which is presented below. Recently, Sridhar and Yovanovich (1996b) developed correlation equations between the Vickers correlation coefficients c 1 and BOOKCOMP, Inc. — John Wiley & Sons / Page 345 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 345 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 [345], (85) Lines: 3406 to 3454 ——— -0.74191pt PgVar ——— Normal Page * PgEnds: Eject [345], (85) c 2 and Brinell hardness H B over a wide range of metal types. These relationships are also presented below. Plastic Contact Geometric Parameters For plastic deformation of the con- tacting asperities, the contact geometric parameters are obtained from the following relationships (Cooper et al., 1969; Yovanovich, 1982): A r A a = 1 2 erfc λ √ 2 (4.237) n = 1 16 m σ 2 exp(−λ 2 ) erfc(λ/ √ 2) (4.238) a = 8 π σ m exp λ 2 2 erfc λ √ 2 (4.239) na = 1 4 √ 2π m σ exp − λ 2 2 (4.240) Correlation of Geometric Parameters A r A a = exp(−0.8141 −0.61778λ −0.42476λ 2 − 0.004353λ 3 ) n = m σ 2 exp(−2.6516 +0.6178λ −0.5752λ 2 + 0.004353λ 3 ) a = σ m (1.156 −0.4526λ +0.08269λ 2 − 0.005736λ 3 ) and for the relative mean plane separation λ = 0.2591−0.5446 ln P H p −0.02320 ln P H p 2 −0.0005308 ln P H p 3 (4.241) The relative mean plane separation for plastic deformation is given by λ = √ 2 erfc −1 2P H p (4.242) where H p is the microhardness of the softer contacting asperities. Relative Contact Pressure The appropriate microhardness may be obtained from the relative contact pressure P/H p . For plastic deformation of the contacting asperities, the explicit relationship is (Song and Yovanovich, 1988) P H p = P c 1 (1.62σ/m) c 2 1/(1+0.071c 2 ) (4.243) BOOKCOMP, Inc. — John Wiley & Sons / Page 346 / 2nd Proofs / Heat Transfer Handbook / Bejan 346 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 [346], (86) Lines: 3454 to 3509 ——— 2.71118pt PgVar ——— Long Page PgEnds: T E X [346], (86) where the coefficients c 1 and c 2 are obtained from Vickers microhardness tests. The Vickers microhardness coefficients are related to the Brinell hardness for a widerange of metal types. Vickers Microhardness Correlation Coefficients The correlation co- efficients c 1 and c 2 are obtained from Vickers microhardness measurements. Sridhar and Yovanovich (1996b) developed correlation equations for the Vickers coefficients: c 1 3178 = 4.0 −5.77H ∗ B + 4.0 H ∗ B 2 − 0.61 H ∗ B 3 (4.244) c 2 =−0.370 +0.442 H B c 1 (4.245) where H B is the Brinell hardness (Johnson, 1985; Tabor, 1951) and H ∗ B = H B /3178. The correlation equations are valid for the Brinell hardness range 1300 to 7600 MPa. The correlation equations above were developed for a range of metal types (e.g., Ni200, SS304, Zr alloys, Ti alloys, and tool steel). Sridhar and Yovanovich (1996b) also reported a correlation equation that relates the Brinell hardness number to the Rockwell C hardness number: BHN = 43.7 +10.92 HRC − HRC 2 5.18 + HRC 3 340.26 (4.246) for the range 20 ≤ HRC ≤ 65. Dimensionless Contact Conductance: Plastic Deformation The dimen- sionless contact conductance C c is C c ≡ h c σ k s m = 1 2 √ 2π exp(−λ 2 /2) 1 − 1 2 erfc(λ/ √ 2) 1.5 (4.247) The correlation equation of the dimensionless contact conductance obtained from theoretical values for a wide range of λ and P/H p is (Yovanovich, 1982) C c ≡ h c k s σ m = 1.25 P H p 0.95 (4.248) which agrees with the theoretical values to within ±1.5% in the range 2 ≤ λ ≤ 4.75. It has been demonstrated that the plastic contact conductance model of eq. (4.248) predicts accurate values of h c for a range of surface roughness σ/m, a range of metal types (e.g., Ni 200, SS 304, Zr alloys, etc.), and a range of the relative contact pres- sure P/H p (Antonetti, 1983; Hegazy, 1985; Sridhar, 1994; Sridhar and Yovanovich, 1994, 1996a). Thevery good agreement between the contact conductance models and experiments is shown in Fig. 4.25. In Fig. 4.25 the dimensionless contact conductance model and the vacuum data for different metal types and a range of surface roughnesses are compared over two BOOKCOMP, Inc. — John Wiley & Sons / Page 347 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 347 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 [347], (87) Lines: 3509 to 3531 ——— 2.397pt PgVar ——— Long Page PgEnds: T E X [347], (87) Figure 4.25 Comparison of a plastic contact conductance model and vacuum data. (From Antonetti, 1983; Hegazy, 1985.) decades of the relative contact pressure defined as P/H e , where H e was called the effective microhardness of the joint. The agreement between the theoretical model developed for conforming rough surfaces that undergo plastic deformation ofthe con- tacting asperities is very good over the entire range of dimensionless contact pressure. Because of the relatively high contact pressures and high thermal conductivity of the metals, the effect of radiation heat transfer across the gaps was found to be negligible for all tests. 4.16.2 Radiation Resistance and Conductance for Conforming Rough Surfaces The radiation heat transfer across gaps formed by conforming rough solids and filled with a transparent substance (or its in a vacuum) is complex because the geometry of BOOKCOMP, Inc. — John Wiley & Sons / Page 348 / 2nd Proofs / Heat Transfer Handbook / Bejan 348 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 [348], (88) Lines: 3531 to 3570 ——— 3.24652pt PgVar ——— Normal Page * PgEnds: Eject [348], (88) TABLE 4.18 Radiative Conductances for Black Surfaces ∆T j T j h r ∆T j T j h r 100 350 9.92 600 600 61.24 200 400 15.42 700 650 80.34 300 450 22.96 800 700 103.2 400 500 32.89 900 750 130.1 500 550 45.53 1000 800 161.5 the microgaps is very difficult to characterize and the temperatures of the bounding solids vary in some complex manner because they are coupled to heat transfer by conduction through the microcontacts. The radiative resistance and the conductance can be estimated by modeling the heat transfer across the microgaps as equivalent to radiative heat transfer between two gray infinite isothermal smooth plates. The radiative heat transfer is given by Q r = σA a F 12 T 4 j1 − T 4 j2 (W) (4.249) where σ = 5.67×10 −8 W/(m 2 ·K 4 ) is the Stefan–Boltzmann constant and T j1 and T j2 are the absolute joint temperatures of theboundingsolid surfaces. These temperatures are obtained by extrapolation of the temperature distributions within the bounding solids. The radiative parameter is given by 1 F 12 = 1 1 + 1 2 − 1 (4.250) where 1 and 2 are the emissivities of the bounding surfaces. The radiative resistance is given by R r = T j1 − T j2 Q r = T j1 − T j2 σA a F 12 T 4 j1 − T 4 j2 (K/W) (4.251) and the radiative conductance by h r = Q r A a (T j1 − T j2 ) = σ F 12 T 4 j1 − T 4 j2 T j1 − T j2 (W/m 2 · K) (4.252) The radiative conductance is seen to be a complex parameter which depends on the emissivities 1 and 2 and the joint temperatures T j1 and T j2 . For many interface prob- lems the following approximation can be used to calculate the radiative conductance: T 4 j1 − T 4 j2 T j1 − T j2 ≈ 4(T j ) 3 where the mean joint temperature is defined as BOOKCOMP, Inc. — John Wiley & Sons / Page 349 / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 349 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 [349], (89) Lines: 3570 to 3612 ——— -0.25409pt PgVar ——— Normal Page * PgEnds: Eject [349], (89) T j = 1 2 (T j1 + T j2 ) (K) If we assume blackbody radiation across the gap, 1 = 1, 2 = 1givesF 12 = 1. This assumption gives the upper bound on the radiation conductance across gaps formed by conforming rough surfaces. If one further assumes that T j2 = 300 K and T j1 = T j2 + ∆T j , one can calculate the radiation conductance for a range of values of ∆T j and T j . The values of h r for black surfaces represent the maximum radiative heat transfer across the microgaps. For microgaps formed by real surfaces, the radiative heat transfer rates may be smaller. Table 4.18 shows that when the joint temperature is T j = 800 K and ∆T j = 1000 K, the maximum radiation conductance is approximately 161.5 W/m 2 · K. This value is much smaller than the contact and gap conductances for most applications where T j < 600 K and ∆T j < 200 K. The radiation conductance becomes relatively important when the interface is formed by two very rough, very hard low-conductivity solids under very light contact pressures. Therefore, for many practical applications, the radiative conductance can be neglected, but not forgotten. 4.16.3 Elastic Contact Model The conforming rough surface model proposed by Mikic (1974) for elastic deforma- tion of the contacting asperities is summarized below (Sridhar and Yovanovich, 1994, 1996a). Elastic Contact Geometric Parameters The elastic contact geometric param- eters are (Mikic, 1974) A r A a = 1 4 erfc λ √ 2 (4.253) n = 1 16 m σ 2 exp(−λ 2 ) erfc(λ/ √ 2) (4.254) a = 2 √ π σ m exp λ 2 2 erfc λ √ 2 (4.255) na = 1 8 √ π m σ exp − λ 2 2 (4.256) The relative mean plane separation is given by λ = √ 2erfc −1 4P H e (4.257) The equivalent elastic microhardness according to Mikic (1974) is defined as H e = CmE where C = 1 √ 2 = 0.7071 (4.258) BOOKCOMP, Inc. — John Wiley & Sons / Page 350 / 2nd Proofs / Heat Transfer Handbook / Bejan 350 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 [350], (90) Lines: 3612 to 3662 ——— 5.38135pt PgVar ——— Short Page * PgEnds: Eject [350], (90) where the effective Young’s modulus of the contacting asperities is 1 E = 1 −ν 2 1 E 1 + 1 −ν 2 2 E 2 (m 2 /N) (4.259) Greenwood and Williamson (1966), Greenwood (1967), and Greenwood and Tripp (1970) developed a more complex elastic contact model that gives a dimensionless elastic microhardness H e /mE that depends on the surface roughness bandwidth α and the separation between the mean planes of the asperity “summits,” denoted as λ s . For a typical range of values of α and λ s (McWaid and Marschall, 1992a), the value of Mikic (1974) (i.e., H e /mE = 0.7071) lies in the range obtained with the Greenwood and Williamson (1966) model. There is, at present, no simple correlation for the model of Greenwood and Williamson (1966). Dimensionless Contact Conductance The dimensionless contact conduc- tance for conforming rough surfaces whose contacting asperities undergo elastic de- formation is (Mikic, 1974; Sridhar and Yovanovich, 1994) h c σ k s m = 1 4 √ π exp − λ 2 /2 1 − 1 4 erfc(λ/ √ 2) 1.5 (4.260) The power law correlation equation based on calculated values obtained from the theoretical relationship is (Sridhar and Yovanovich, 1994) h c σ k s m = 1.54 P H e 0.94 (4.261) has an uncertainty of about ±2% for the relative contact pressure range 10 −5 ≤ P/H e ≤ 0.2. Correlation Equations for Surface Parameters The correlation equations for A r /A a ,n, and a for the relative contact pressure range 10 −6 ≤ P/H e ≤ 0.2 are A r A a = 1 2 exp(−0.8141 −0.61778λ −0.42476λ 2 − 0.004353λ 3 ) n = m σ 2 exp(−2.6516 +0.6178λ −0.5752λ 2 + 0.004353λ 3 ) a = 1 √ 2 σ m (1.156 −0.4526λ +0.08269λ 2 − 0.005736λ 3 ) and the relative mean planes separation . / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 341 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 [341],. Proofs / Heat Transfer Handbook / Bejan 342 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 [342],. / 2nd Proofs / Heat Transfer Handbook / Bejan CONFORMING ROUGH SURFACE MODELS 343 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 [343],