Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 47
Текст из файла (страница 47)
The joint resistance can be modeled asBOOKCOMP, Inc. — John Wiley & Sons / Page 339 / 2nd Proofs / Heat Transfer Handbook / BejanPgVar340123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESRj = R mic + R mac(K/W)(4.226)The microscopic resistance is given by the relationshipR mic =ψmic2ks N aS(K/W)(4.227)where ψmic is the average spreading–constriction resistance parameter, N is the number of microcontacts that are distribution in some complex manner over the contourarea of radius aL and aS represents some average microcontact spot radius, and theharmonic mean thermal conductivity of the joint is ks = 2k1 k2 /(k1 + k2 ).The macroscopic resistance is given by the relationshipR mac =ψmac2ks aL(K/W)(4.228)where ψmac is the spreading–constriction resistance parameter for the contour area ofradius aL .The mechanical model should be capable of predicting the contact parameters:aS , aL , and N .
These parameters are also required for the determination of the thermalspreading–constriction parameters ψmic and ψmac.At this time there is no simple mechanical model available for prediction of the geometric parameters required in microscopic and macroscopic resistance relationships.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.4.16CONFORMING ROUGH SURFACE MODELSThere are models for predicting contact, gap, and joint conductances between conforming (nominally flat) rough surfaces developed by Greenwood and Williamson(1966), Greenwood (1967), Greenwood and Tripp (1970), Cooper et al.
(1969), Mikic (1974), Sayles and Thomas (1976), Yovanovich (1982), and DeVaal (1988).The three mechanical models—elastic, plastic, or elastic–plastic deformation ofthe contacting asperities—are based on the assumptions that the surface asperitieshave Gaussian height distributions about some mean plane passing through eachsurface and that the surface asperities are distributed randomly over the apparentcontact area Aa . Figure 4.22 shows a very small portion of a typical joint formedbetween 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 theplastic or elastic physical properties of the contacting asperities.A very important surface roughness parameter is the surface roughness: either therms (root-mean-square) roughness or the CLA (centerline-average) roughness, whichare defined as (Whitehouse and Archard, 1970; Onions and Archard, 1973; Thomas,1982)BOOKCOMP, Inc.
— John Wiley & Sons / Page 340 / 2nd Proofs / Heat Transfer Handbook / Bejan[340], (80)Lines: 3271 to 3311———-3.56989pt PgVar———Normal PagePgEnds: Eject[340], (80)341CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445CLA roughness =rms roughness =1LL|y(x)| dx(m)(4.229)01LLy 2 (x) dx(m)(4.230)0where 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. ForGaussian asperity heights with respect to the mean plane, these two measures ofsurface roughness are related (Mikic and Rohsenow, 1966):π· CLAσ=2A second very important surface roughness parameter is the absolute mean asperityslope, which is defined as (Cooper et al., 1969; Mikic and Rohsenow, 1966; andDeVaal et al. 1987).[341], (81)Lines: 3311 to 3334———0.60109pt PgVar1yx2m1 = dy1/dx1Ym2 = dy2/dx2[341], (81)Ym = dy/dxy———Normal Page* PgEnds: Ejectx = 公 12 ⫹ 22m = 公m 12 ⫹ m22Figure 4.22 Typical joint between conforming rough surfaces.
(From Hegazy, 1985.)BOOKCOMP, Inc. — John Wiley & Sons / Page 341 / 2nd Proofs / Heat Transfer Handbook / Bejan342123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES1m=LL0 dy(x) dx dx(rad)(4.231)The effective rms surface roughness and the effective absolute mean asperity slopefor a typical joint formed by two conforming rough surfaces are defined as (Cooperet al., 1969; Mikic, 1974; Yovanovich, 1982)σ = σ21 + σ22 and m = m21 + m22(4.232)Antonetti et al.
(1991) reported approximate relationships for m as a function of σ forseveral metal surfaces that were bead-blasted.The three deformation models (elastic, plastic, or elastic–plastic) give relationships for three important geometric parameters of the joint: the relative real contactarea Ar /Aa , the contact spot density n, and the mean contact spot radius a in termsof the relative mean plane separation defined as λ = Y /σ. The mean plane separationY and the effective surface roughness are illustrated in Fig. 4.22 for the joint formedby the mechanical contact of two nominally flat rough surfaces.The models differ in the mode of deformation of the contacting asperities.
Thethree modes of deformation are plastic deformation of the softer contacting asperities,elastic deformation of all contacting asperities, and elastic–plastic deformation of thesofter contacting asperities. For the three deformation models there is one thermalcontact conductance model, given as (Cooper et al., 1969; Yovanovich, 1982)hc =2naksψ()(W/m2 · K)(4.233)where n is the contact spot density, a is the mean contact spot radius, and the effectivethermal conductivity of the joint isks =2k1 k2k1 + k 2(W/m · K)(4.234)and the spreading/constriction parameter ψ, based on isothermal contact spots, isapproximated byψ() = (1 − )1.5for0 < < 0.3(4.235)√where the relative contact spot size is = Ar /Aa .
The geometric parameters n, aand Ar /Aa are related to the relative mean plane separation λ = Y /σ.4.16.1Plastic Contact ModelThe original plastic deformation model of Cooper et al. (1969) has undergone significant modifications during the past 30 years. First, a new, more accurate correlationequation was developed by Yovanovich (1982). Then Yovanovich et al.
(1982a) andBOOKCOMP, Inc. — John Wiley & Sons / Page 342 / 2nd Proofs / Heat Transfer Handbook / Bejan[342], (82)Lines: 3334 to 3384———3.76213pt PgVar———Normal PagePgEnds: TEX[342], (82)CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445343[343], (83)Lines: 3384 to 3393———3.097pt PgVar———Normal PagePgEnds: TEX[343], (83)Figure 4.23 Vickers microhardness versus indentation diagonal for four metal types. (FromHegazy, 1985.)Hegazy (1985) introduced the microhardness layer which appears in most workedmetals.
Figures 4.23 and 4.24 show plots of measured microhardness and macrohardness versus the penetration depth t or the Vickers diagonal dV . These two measures ofindenter penetration are related: dv /t = 7. Figure 4.23 shows the measured Vickersmicrohardness versus indentation diagonal for four metal types (Ni 200, stainlesssteel 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 theVickers microhardness decreases with increasing diagonal (load). The indentationdiagonal was between 8 and 70 µm.Figure 4.24 shows the Vickers microhardness measurements and the Brinell andRockwell macrohardness measurements versus indentation depth.
The Brinell andRockwell macrohardness values are very close because they correspond to large indentations, and therefore, they are a measure of the bulk hardness, which does notchange with load. According to Fig. 4.24, the penetration depths for the Vickers microhardness measurements are between 1 and 10 µm, whereas the larger penetrationBOOKCOMP, Inc. — John Wiley & Sons / Page 343 / 2nd Proofs / Heat Transfer Handbook / Bejan344123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES[344], (84)Lines: 3393 to 3406———3.73402pt PgVarFigure 4.24 Vickers, Brinell, and Rockwell hardness versus indentation depth for four metaltypes.
(From Hegazy, 1985.)———Normal PagePgEnds: TEX[344], (84)depths for the Brinell and Rockwell macrohardness measurements lie between approximately 100 and 1000 µm.The microhardness layer may be defined by means of the Vickers microhardnessmeasurements, which relate the Vickers microhardness HV to the Vickers averageindentation diagonal dV (Yovanovich et al., 1982a; Hegazy, 1985): c2dVHV = c1(GPa)(4.236)d0where d0 represents some convenient reference value for the average diagonal, andc1 and c2 are the correlation coefficients. It is conventional to set d0 = 1 µm. Hegazy(1985) found that c1 is closely related to the metal bulk hardness, such as the Brinellhardness, denoted as HB .The original mechanical contact model (Yovanovich et al., 1982a; Hegazy, 1985)required an iterative procedure to calculate the appropriate microhardness for a givensurface roughness σ and m, given the apparent contact pressure P and the coefficientsc1 and c2 .Song and Yovanovich (1988) developed an explicit relationship for the microhardness Hp , which is presented below.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
