Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 48
Текст из файла (страница 48)
Recently, Sridhar and Yovanovich (1996b)developed correlation equations between the Vickers correlation coefficients c1 andBOOKCOMP, Inc. — John Wiley & Sons / Page 344 / 2nd Proofs / Heat Transfer Handbook / BejanCONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445345c2 and Brinell hardness HB over a wide range of metal types. These relationships arealso presented below.Plastic Contact Geometric Parameters For plastic deformation of the contacting asperities, the contact geometric parameters are obtained from the followingrelationships (Cooper et al., 1969; Yovanovich, 1982):1Arλ= erfc √(4.237)Aa221 m 2 exp(−λ2 )n=(4.238)√16 σ erfc(λ/ 2) 28 σλλa=experfc √(4.239)πm22λ21 mexp −(4.240)na = √24 2π σ[345], (85)Lines: 3406 to 3454———-0.74191pt PgVarCorrelation of Geometric Parameters———Normal Page* PgEnds: EjectAr= exp(−0.8141 − 0.61778λ − 0.42476λ2 − 0.004353λ3 )Aa m 2n=exp(−2.6516 + 0.6178λ − 0.5752λ2 + 0.004353λ3 )σσa = (1.156 − 0.4526λ + 0.08269λ2 − 0.005736λ3 )m[345], (85)and for the relative mean plane separationPP 2P 3λ = 0.2591−0.5446 ln−0.02320 ln−0.0005308 lnHpHpHpThe relative mean plane separation for plastic deformation is given by√−1 2Pλ = 2 erfcHp(4.241)(4.242)where Hp is the microhardness of the softer contacting asperities.Relative Contact Pressure The appropriate microhardness may be obtainedfrom the relative contact pressure P /Hp .
For plastic deformation of the contactingasperities, the explicit relationship is (Song and Yovanovich, 1988)1/(1+0.071c2 )PP=Hpc1 (1.62σ/m)c2BOOKCOMP, Inc. — John Wiley & Sons / Page 345 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.243)346123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere the coefficients c1 and c2 are obtained from Vickers microhardness tests.
TheVickers microhardness coefficients are related to the Brinell hardness for a wide rangeof metal types.Vickers Microhardness Correlation Coefficients The correlation coefficients c1 and c2 are obtained from Vickers microhardness measurements. Sridharand Yovanovich (1996b) developed correlation equations for the Vickers coefficients: c1= 4.0 − 5.77HB∗ + 4.0 HB∗3178HBc2 = −0.370 + 0.442c12 − 0.61 HB∗3(4.244)(4.245)where HB is the Brinell hardness (Johnson, 1985; Tabor, 1951) and HB∗ = HB /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 theRockwell C hardness number:BHN = 43.7 + 10.92 HRC −HRC2HRC3+5.18340.26(4.246)for the range 20 ≤ HRC ≤ 65.Dimensionless Contact Conductance: Plastic Deformationsionless contact conductance Cc isCc ≡hc σ1= √ ks m2 2πexp(−λ2 /2)√ 1.511 − 2 erfc(λ/ 2)The dimen-(4.247)The correlation equation of the dimensionless contact conductance obtained fromtheoretical values for a wide range of λ and P /Hp is (Yovanovich, 1982) 0.95Phc σCc ≡= 1.25ks mHp(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 hc for a range of surface roughness σ/m, a range of metaltypes (e.g., Ni 200, SS 304, Zr alloys, etc.), and a range of the relative contact pressure P /Hp (Antonetti, 1983; Hegazy, 1985; Sridhar, 1994; Sridhar and Yovanovich,1994, 1996a). The very good agreement between the contact conductance models andexperiments is shown in Fig. 4.25.In Fig.
4.25 the dimensionless contact conductance model and the vacuum datafor different metal types and a range of surface roughnesses are compared over twoBOOKCOMP, Inc. — John Wiley & Sons / Page 346 / 2nd Proofs / Heat Transfer Handbook / Bejan[346], (86)Lines: 3454 to 3509———2.71118pt PgVar———Long PagePgEnds: TEX[346], (86)CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445347[347], (87)Lines: 3509 to 3531———2.397pt PgVar———Long PagePgEnds: TEX[347], (87)Figure 4.25 Comparison of a plastic contact conductance model and vacuum data. (FromAntonetti, 1983; Hegazy, 1985.)decades of the relative contact pressure defined as P /He , where He was called theeffective microhardness of the joint. The agreement between the theoretical modeldeveloped for conforming rough surfaces that undergo plastic deformation of the contacting asperities is very good over the entire range of dimensionless contact pressure.Because of the relatively high contact pressures and high thermal conductivity of themetals, the effect of radiation heat transfer across the gaps was found to be negligiblefor all tests.4.16.2 Radiation Resistance and Conductance for ConformingRough SurfacesThe radiation heat transfer across gaps formed by conforming rough solids and filledwith a transparent substance (or its in a vacuum) is complex because the geometry ofBOOKCOMP, Inc.
— John Wiley & Sons / Page 347 / 2nd Proofs / Heat Transfer Handbook / Bejan348123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESTABLE 4.18Radiative Conductances for Black Surfaces∆TjTjhr∆TjTjhr1002003004005003504004505005509.9215.4222.9632.8945.53600700800900100060065070075080061.2480.34103.2130.1161.5the microgaps is very difficult to characterize and the temperatures of the boundingsolids vary in some complex manner because they are coupled to heat transfer byconduction through the microcontacts.The radiative resistance and the conductance can be estimated by modeling theheat transfer across the microgaps as equivalent to radiative heat transfer betweentwo gray infinite isothermal smooth plates. The radiative heat transfer is given by Qr = σAa F12 Tj41 − Tj42[348], (88)Lines: 3531 to 3570(W)(4.249)———3.24652pt PgVarwhere σ = 5.67×10−8 W/(m2 · K4) is the Stefan–Boltzmann constant and Tj 1 and Tj 2———are the absolute joint temperatures of the bounding solid surfaces.
These temperaturesNormal Pageare obtained by extrapolation of the temperature distributions within the bounding * PgEnds: Ejectsolids. The radiative parameter is given by1F12=11+ −112(4.250)where 1 and 2 are the emissivities of the bounding surfaces. The radiative resistanceis given byRr =Tj 1 − Tj 2Tj 1 − Tj 2 =QrσAa F12 Tj41 − Tj42(K/W)(4.251)and the radiative conductance by σF12 Tj41 − Tj42Qrhr ==Aa (Tj 1 − Tj 2 )Tj 1 − Tj 2(W/m2 · K)(4.252)The radiative conductance is seen to be a complex parameter which depends on theemissivities 1 and 2 and the joint temperatures Tj 1 and Tj 2 . For many interface problems the following approximation can be used to calculate the radiative conductance:Tj41 − Tj42Tj 1 − Tj 2≈ 4(T j )3where the mean joint temperature is defined asBOOKCOMP, Inc. — John Wiley & Sons / Page 348 / 2nd Proofs / Heat Transfer Handbook / Bejan[348], (88)349CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445Tj =1(Tj 1 + Tj 2 )2(K)If we assume blackbody radiation across the gap, 1 = 1, 2 = 1 gives F12 = 1.This assumption gives the upper bound on the radiation conductance across gapsformed by conforming rough surfaces.
If one further assumes that Tj 2 = 300 Kand Tj 1 = Tj 2 + ∆Tj , one can calculate the radiation conductance for a range ofvalues of ∆Tj and Tj . The values of hr for black surfaces represent the maximumradiative 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 thejoint temperature is Tj = 800 K and ∆Tj = 1000 K, the maximum radiationconductance is approximately 161.5 W/m2 · K. This value is much smaller than thecontact and gap conductances for most applications where Tj < 600 K and ∆Tj <200 K. The radiation conductance becomes relatively important when the interface isformed by two very rough, very hard low-conductivity solids under very light contactpressures. Therefore, for many practical applications, the radiative conductance canbe neglected, but not forgotten.[349], (89)Lines: 3570 to 3612———4.16.3-0.25409pt PgVarElastic Contact Model———The conforming rough surface model proposed by Mikic (1974) for elastic deformaNormal Pagetion of the contacting asperities is summarized below (Sridhar and Yovanovich, 1994,* PgEnds: Eject1996a).Elastic Contact Geometric Parameters The elastic contact geometric parameters are (Mikic, 1974)Ar1λ= erfc √(4.253)Aa421 m 2 exp(−λ2 )n=(4.254)√16 σ erfc(λ/ 2) 2λλ2 σexperfc √a=√(4.255)2πm2λ21 mexp −(4.256)na = √28 πσThe relative mean plane separation is given by√−1 4Pλ = 2erfcHe(4.257)The equivalent elastic microhardness according to Mikic (1974) is defined asHe = CmE whereBOOKCOMP, Inc.
— John Wiley & Sons / Page 349 / 2nd Proofs / Heat Transfer Handbook / Bejan1C = √ = 0.70712(4.258)[349], (89)350123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere the effective Young’s modulus of the contacting asperities is1 − ν211 − ν221=+EE1E2(m2/N)(4.259)Greenwood and Williamson (1966), Greenwood (1967), and Greenwood and Tripp(1970) developed a more complex elastic contact model that gives a dimensionlesselastic microhardness He /mE that depends on the surface roughness bandwidth αand the separation between the mean planes of the asperity “summits,” denoted asλs .
For a typical range of values of α and λs (McWaid and Marschall, 1992a), thevalue of Mikic (1974) (i.e., He /mE = 0.7071) lies in the range obtained with theGreenwood and Williamson (1966) model. There is, at present, no simple correlationfor the model of Greenwood and Williamson (1966).[350], (90)Dimensionless Contact Conductance The dimensionless contact conductance for conforming rough surfaces whose contacting asperities undergo elastic deformation is (Mikic, 1974; Sridhar and Yovanovich, 1994) exp − λ2 /2√ 1.511 − 4 erfc(λ/ 2)1hc σ= √ ks m4 π———(4.260)The power law correlation equation based on calculated values obtained from thetheoretical relationship is (Sridhar and Yovanovich, 1994)Phc σ= 1.54ks mHe0.94(4.261)has an uncertainty of about ±2% for the relative contact pressure range 10−5 ≤P /He ≤ 0.2.Correlation Equations for Surface Parameters The correlation equationsfor Ar /Aa , n, and a for the relative contact pressure range 10−6 ≤ P /He ≤ 0.2 areAr=Aan=12exp(−0.8141 − 0.61778λ − 0.42476λ2 − 0.004353λ3 ) m 2σexp(−2.6516 + 0.6178λ − 0.5752λ2 + 0.004353λ3 )1 σa=√(1.156 − 0.4526λ + 0.08269λ2 − 0.005736λ3 )2mand the relative mean planes separationBOOKCOMP, Inc.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
