Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 50
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The parameters that appear inTable 4.19 are bt = 2(CLA1 + CLA2 ), where CLAi is the centerline-average surfaceTABLE 4.19 Models and Correlation Equations for Gap Conductancefor Conforming Rough SurfacesAuthorsModels and CorrelationsCetinkale andFishenden (1951)kghg =0.305bt + MRapier et al. (1963)hg = kgkgbtLines: 3835 to 38792bt1.20.8ln 1 ++2bt + M2btM———413210 10++ 2 −4ln(1+X)++3XXX3X2XShlykov (1965)hg =Veziroglu (1967)kg 0.264 b + Mthg =kg1.78 bt + Mkgδ + βΛ/(α1 + α2 )Lloyd et al. (1973)hg =Garnier and Begej (1979)hg = kgLoyalka (1982)hg =Yovanovich et al. (1982b)kg /σhg = √2π[356], (96)[356], (96)for bt < 15 µmδ not givenkgδ + M + 0.162(4 − α1 − α2 )βΛ∞0BOOKCOMP, Inc.
— John Wiley & Sons / Page 356 / 2nd Proofs / Heat Transfer Handbook / Bejanδ not givenδ not given exp −(Y/σ − t/σ)2 /2tdt/σ + M/σσ√Y−1 2P= 2 erfcσHp1/(1+0.071c2 )PP=Hpc1 (1.62σ/m)c2Source: Song (1988).———Normal PagePgEnds: TEXfor bt > 15 µmexp(−1/Kn) 1 − exp(−1/Kn)+Mδ+M0.05264pt PgVarCONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445357roughness of the two contacting surfaces, M = αβΛ, X = bt /M, σ = σ21 + σ22 ,where the units of σ are µm.
The Knudsen number Kn that appears in the Garnierand Begej (1979) correlation equation is not defined.Song and Yovanovich (1987), Song (1988), and Song et al. (1993b) reviewed themodels and correlation equations given in Table 4.19. They found that for someof the correlation equations the required gap thickness δ was not defined, and forother correlation equations an empirically based average gap thickness was specifiedthat is constant, independent of variations of the apparent contact pressure. The gapconductance model developed by Yovanovich et al.
(1982b) is the only one thataccounts for the effect of mechanical load and physical properties of the contactingasperities on the gap conductance. This model is presented below.The gap conductance model for conforming rough surfaces was developed, modified, 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 distributionsand 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: ∞exp − (Y /σ − u)2 /2kg 1kghg =(W/m2 · K) (4.286)du = Ig√σ 2π 0u + M/σσwhere kg is the thermal conductivity of the gas trapped in the gap and σ is the effectivesurface roughness of the joint, and u = t/σ is the dimensionless local gap thickness.The integral Ig depends on two independent dimensionless parameters: Y /σ, the meanplane separation; and M/σ, the relative gas rarefaction parameter.The relative mean planes separation for plastic and elastic contact are given by therelationships √Y−1 2P2 erfcplastic =σHp(4.287) √Y−1 4P2 erfcelastic =σHeThe relative contact pressures P /Hp for plastic deformation and P /He for elasticdeformation can be determined by means of appropriate relationships.The gas rarefaction parameter is M = αβΛ, where the gas parameters are definedas:2 − α12 − α2α=+(4.288)α1α2β=2γ(γ + 1)PrBOOKCOMP, Inc.
— John Wiley & Sons / Page 357 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.289)[357], (97)Lines: 3879 to 3919———0.47626pt PgVar———Normal PagePgEnds: TEX[357], (97)358123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESΛ = Λ0Tg PgTg,0 Pg,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 experimentaland theoretical aspects of α for various gases in contact with metallic surfaces undervarious surface conditions and temperatures (Wiedmann and Trumpler, 1946; Hartnett, 1961; Wachman, 1962; Thomas, 1967; Semyonov et al., 1984; Loyalka, 1982).Song and Yovanovich (1987) and Song et al.
(1992a, 1993b) examined the several gapconductance models available in the literature and the experimental data and modelsfor the accommodation coefficients.Song and Yovanovich (1987) developed a correlation for the accommodation forengineering surfaces (i.e., surfaces with absorbed layers of gases and oxides).
Theyproposed a correlation that is based on experimental results of numerous investigators for monatomic gases. The relationship was extended by the introduction of amonatomic equivalent molecular weight to diatomic and polyatomic gases. The finalcorrelation isMg2.4µα = exp(C0 T )(4.291)+ [1 − exp(C0 T )]C1 + M g(1 + µ)2[358], (98)Lines: 3919 to 3952———7.26427pt PgVar———Short Pagewith C0 = −0.57, T = (Ts − T0 )/T0 , Mg = Mg for monatomic gases (= 1.4Mg for * PgEnds: Ejectdiatomic and polyatomic gases), C1 = 6.8 in units of Mg (g/mol), and µ = Mg /Ms ,where Ts and T0 = 273 K are the absolute temperatures of the surface and the gas,[358], (98)and Mg and Ms are the molecular weights of the gas and the solid, respectively.The agreement between the predictions according to the correlation above and thepublished data for diatomic and polyatomic gases was within ±25%.The gas parameter β depends on the specific heat ratio γ = Cp /Cv and thePrandtl number Pr.
The molecular mean free path of the gas molecules Λ dependson the type of gas, the gas temperature Tg and gas pressure Pg , and the referencevalues of the mean free path Λ0 , the gas temperature Tg,0 , and the gas pressure Pg,0 ,respectively.Wesley and Yovanovich (1986) compared the predictions of the gap conductancemodel and experimental measurements of gaseous gap conductance between the fueland clad of a nuclear fuel rod.
The agreement was very good and the model wasrecommended for fuel pin analysis codes.The gap integral can be computed accurately and easily by means of computeralgebra systems. Negus and Yovanovich (1988) developed the following correlationequations for the gap integral:Ig =fgY /σ + M/σIn the range 2 ≤ Y /σ ≤ 4:BOOKCOMP, Inc. — John Wiley & Sons / Page 358 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.292)359CONFORMING ROUGH SURFACE MODELS123456789101112131415161718192021222324252627282930313233343536373839404142434445Y 1.68 σ 0.84 1.063 + 0.0471 4 −lnσMfg = σ 0.8 1 + 0.06Mfor 0.01 ≤for 1 ≤M≤1σM<∞σThe correlation equations have a maximum error of approximately 2%.4.16.7Joint Conductance for Conforming Rough SurfacesThe joint conductance for a joint between two conforming rough surfaces ishj = hc + hg(W/m2 · K)(4.293)[359], (99)when radiation heat transfer across the gap is neglected.
The relationship is applicableto joints that are formed by elastic, plastic, or elastic–plastic deformation of thecontacting asperities. The mode of deformation will influence hc and hg through therelative mean plane separation parameter Y /σ.The gap and joint conductances are compared against data (Song, 1988) obtainedfor three types of gases, argon, helium, and nitrogen, over a gas pressure rangebetween 1 and 700 torr.
The gases occupied gaps formed by conforming rough Ni200 and stainless steel type 304 metals. In all tests the metals forming the joint wereidentical, and one surface was flat and lapped while the other surface was flat andglass bead blasted.The gap and joint conductance models were compared against data obtained forrelatively light contact pressures where the gap and contact conductances were comparable. Figure 4.27 shows plots of the joint conductance data and the model predictions for very rough stainless steel type 304 surfaces at Y /σ = 1.6×10−4 . Agreementamong the data for argon, helium, and nitrogen is very good for gas pressures between approximately 1 and 700 torr.
At the low gas pressure of 1 torr, the measuredand predicted joint conductance values for the three gases differ by a few percentbecause hg hc and hj ≈ hc . As the gas pressure increases there is a large increasein the joint conductances because the gap conductances are increasing rapidly. Thejoint conductances for argon and nitrogen approach asymptotes for gas pressures approaching 1 atm. The joint conductances for helium are greater than for argon andnitrogen, 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 andcurves for nitrogen and helium for gas pressures between approximately 10 and 700torr. The relative contact pressure is 1.7 × 10−4 is based on the plastic deformationmodel.
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