Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 40
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The general expression becomes∞4kaRs =8 1 J1 (δn ) sin δn π n=1 δ3n J02 (δn )∞(4.98)An accurate correlation equation of this series solution is given below.Parabolic Flux Distribution Yovanovich (1976b) reported the solution for theparabolic flux distribution corresponding to µ = 21 .4kaRs =∞24 1 J1 (δn ) sin δn 11−π n=1 δ3n J02 (δn )(δn )2δn tan δn (4.99)An accurate correlation equation of this series solution is given below.Asymptotic Values for Dimensionless Spreading Resistances The threeseries solutions given above converge very slowly as → 0, which corresponds to theBOOKCOMP, Inc.
— John Wiley & Sons / Page 300 / 2nd Proofs / Heat Transfer Handbook / Bejan———-4.44421pt PgVar———Normal PagePgEnds: TEX[300], (40)Isoflux Circular Source The general solution above with µ = 0 yields theisoflux solution reported by Mikic and Rohsenow (1966):16 1 J12 (δn )π n=1 δ3n J02 (δn )Lines: 1670 to 1727(4.97)An accurate correlation equation of this series solution is given below.4kaRs =[300], (40)301CIRCULAR AREA ON A SEMI-INFINITE FLUX TUBE123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 4.10Correlation Coefficients for Three Flux DistributionsCnµ = − 21µ=0C0C1C3C5C71.00000−1.409810.3036410.02182720.06446831.08085−1.410020.2597140.01886310.0420278µ=121.12517−1.410380.2353870.01175270.0343458case of a circular contact area on a half-space.
The corresponding half-space resultswere reported by Strong et al. (1974):1for µ = − 21 324kaRs =(4.100)for µ = 03π21.1252for µ = 21Correlation Equations for Spreading Resistance Since the three series solutions presented above for the three heat flux distributions µ = − 21 , 0, 21 convergeslowly as → 0, correlation equations for the dimensionless spreading resistanceψ = 4kaRs for the three flux distributions were developed having the general formψ = C0 + C1 + C3 3 + C5 5 + C7 7Simple Correlation Equations Yovanovich (1976b) recommended the following simple correlations for the three flux distributions:4kaRs = a1 (1 − a2 )(4.102)in the range 0 < ≤ 0.1 with a maximum error of 0.1%, and4kaRs = a1 (1 − )a3(4.103)in the range 0 < ≤ 0.3 with a maximum error of 1%.
The correlation coefficientsfor the three flux distributions are given in Table 4.11.a1a2a3Correlation Coefficients for µ = − 21 , 0, 21− 2101211.41971.501.08081.41111.351.12521.40981.30BOOKCOMP, Inc. — John Wiley & Sons / Page 301 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1727 to 1799———0.26018pt PgVar———Normal PagePgEnds: TEX(4.101)with the correlation coefficients given in Table 4.10. The correlation equations, applicable for the parameter range 0 ≤ ≤ 0.8, provide four-decimal-place accuracy.TABLE 4.11[301], (41)[301], (41)302123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESTABLE 4.124kaRsCoefficients for Correlations of Dimensionless Spreading ResistanceFlux Tube Geometry andContact Boundary ConditionCircle/circleUniform fluxTrue isothermal fluxCircle/squareUniform fluxEquivalent isothermal fluxC0C1C3C5C71.080761.00000−1.41042−1.409780.266040.34406−0.000160.043050.0582660.022711.080761.00000−1.24110−1.241420.182100.209880.008250.027150.0389160.027684.8.2 Accurate Correlation Equations for Various Combinationsof Source Areas, Flux Tubes, and Boundary Conditions[302], (42)Solutions are also available for various combinations of source areas and flux tubecross-sectional areas, such as circle/circle and circle/square for the uniform flux, trueisothermal, and equivalent isothermal boundary conditions (Negus and Yovanovich,1984a,b).Numerical results were correlated with the polynomial4kaRs = C0 + C1 + C3 3 + C5 5 + C7 7MULTIPLE LAYERS ON A CIRCULAR FLUX TUBEThe effect of single and multiple isotropic layers or coatings on the end of a circularflux tube has been determined by Antonetti (1983) and Muzychka et al.
(1999). Theheat enters the end of the circular flux tube of radius b and thermal conductivity k3through a coaxial, circular area that is in perfect thermal contact with an isotropiclayer of thermal conductivity k1 and thickness t1 . This layer is in perfect contact witha second layer of thermal conductivity k2 and thickness t2 , which is in perfect contactcontact with the flux tube having thermal conductivity k3 (Fig. 4.9).The lateral boundary of the flux tube is adiabatic and the contact plane outside thecontact area is also adiabatic.
The boundary condition over the contact area may beisoflux or isothermal. The system is depicted in Fig. 4.9. The dimensionless constriction resistance ψ2 layers = 4k3 aRc is defined with respect to the thermal conductivityof the flux tube, which is often referred to as the substrate. This constriction resistancedepends on several dimensionless parameters: relative contact size = a/b where0 < < 1; two conductivity ratios: κ21 = k2 /k1 , κ32 = k3 /k2 ; two relative layerthicknesses: τ1 = t1 /a, τ2 = t2 /a; and the boundary condition over the contact area.The solution for two layers is given asBOOKCOMP, Inc. — John Wiley & Sons / Page 302 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.32205pt PgVar———Normal Page(4.104) * PgEnds: EjectThe dimensionless spreading (constriction) resistance coefficient C0 is the halfspace value, and the correlation coefficients C1 through C7 are given in Table 4.12.4.9Lines: 1799 to 1820[302], (42)303MULTIPLE LAYERS ON A CIRCULAR FLUX TUBE123456789101112131415161718192021222324252627282930313233343536373839404142434445∞ψ2 layers =ϑ+16 φn, κ21 κ32 −π n=1ϑ(4.105)whereφn, =J12 (δn )ρn,δ3n J02 (δn )(4.106)and the boundary condition parameter is according to Muzychka et al.
(1999): sin δn isothermal areaρn, = 2J1 (δn )1isoflux area[303], (43)Lines: 1820 to 1849———13.65111pt PgVar———Normal PagePgEnds: TEX[303], (43)Figure 4.9 Two layers in a flux tube. (From Muzycha et al., 1999.)BOOKCOMP, Inc. — John Wiley & Sons / Page 303 / 2nd Proofs / Heat Transfer Handbook / Bejan304123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESThe thermal conductivity ratios are defined above. The layer parameters ϑ + andϑ come from the following general relationship:−ϑ ± = (1 + κ21 )(1 + κ32 ) ± (1 − κ21 )(1 + κ32 ) exp(−2δn τ1 )+ (1 − κ21 )(1 − κ32 ) exp(−2δn τ2 )± (1 + κ21 )(1 − κ32 ) exp[−2δn (τ1 + τ2 )]The eigenvalues δn that appear in the solution are the positive roots of J1 (·) = 0.The two-layer solution may be used to obtain the relationship for a single layerof thermal conductivity k1 and thickness t1 in perfect contact with a flux tube ofthermal conductivity k2 .
In this case the dimensionless spreading resistance ψ1 layerdepends on the relative contact size , the conductivity ratio κ21 , and the relative layerthickness τ1 :ψ1 layer =16π∞n=1φn, κ21ϑϑ−(4.107)Lines: 1849 to 1901———and the general layer relationship becomes-0.11104pt PgVarϑ ± = 2[(1 + κ21 ) ± (1 − κ21 ) exp(−2δn τ1 )]———Normal Page* PgEnds: Eject4.10 SPREADING RESISTANCE IN COMPOUND RECTANGULARCHANNELSConsider the spreading resistance Rs and total one-dimensional resistance R1D forthe system shown in Fig.
4.10. The system is a rectangular flux channel −c ≤ x ≤c, −d ≤ y ≤ d, consisting of two isotropic layers having thermal conductivitiesk1 , k2 and thicknesses t1 , t2 , respectively. The interface between the layers is assumedto be thermally perfect. All four sides of the flux channel are adiabatic. The planarrectangular heat source area −a ≤ x ≤ a, −b ≤ y ≤ b is subjected to a uniformheat flux q, and the region outside the planar source area is adiabatic. The steady heattransfer rate Q = qA = 4qcd occurs in the system and the heat leaves the systemthrough the lower face z = t1 + t2 . The heat is removed by a fluid through a uniformheat transfer coefficient h or by a heat sink characterized by an effective heat transfercoefficient h.The total thermal resistance of the system is given by the relationR total = Rs + R1D(K/W)(4.108)where Rs is the thermal spreading resistance of the system and R1D is the onedimensional thermal resistance, defined as1 t1t21R1D =where A = 4cd(K/W)(4.109)++A k1k2hBOOKCOMP, Inc.
— John Wiley & Sons / Page 304 / 2nd Proofs / Heat Transfer Handbook / Bejan[304], (44)+[304], (44)SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS123456789101112131415161718192021222324252627282930313233343536373839404142434445305[305], (45)Lines: 1901 to 1920———-0.20218pt PgVar———Normal PagePgEnds: TEXFigure 4.10 Rectangular isoflux area on a compound rectangular channel. (From Yovanovichet al., 1999.)The spreading resistance is given by the general relationship (Yovanovich et al., 1999)Rs =∞sin2 (aδ)1φm (δ)2a 2 cdk1 m=1δ3+∞1sin2 (bλ)φn (λ)2b2 cdk1 n=1λ3+∞∞ 1sin2 (aδ) sin2 (bλ)φm,n (β)22a b cdk1 m=1 n=1δ2 λ2 β(K/W)(4.110)The general relationship for the spreading resistance consists of three terms.
Thetwo single summations account for two-dimensional spreading in the x and y directions, respectively, and the double summation accounts for three-dimensional spreading from the rectangular heat source.BOOKCOMP, Inc. — John Wiley & Sons / Page 305 / 2nd Proofs / Heat Transfer Handbook / Bejan[305], (45)306123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESThe eigenvalues δm and λn , corresponding to the two strip solutions, depend onthe flux channel dimensions and the indices m and n, respectively.
The eigenvaluesβm,n for the rectangular solution are functions of the other two eigenvalues and bothindices:nπmπ(4.111)δm =βm,n = δ2m + λ2nλn =dcThe contributions of the layer thicknesses t1 , t2 , the layer conductivities k1 , k2 , andthe uniform conductance h to the spreading resistance are determined by means of thegeneral expressionφ(ζ) =α(κζL − Bi)e4ζt1 + (κζL − Bi)e2ζt1 +α(κζL − Bi)e4ζt1 − (κζL − Bi)e2ζt1 +[306], (46)(κζL + Bi)e2ζ(2t1 +t2 ) + α(κζL + Bi)e2ζ(t1 +t2 )(κζL + Bi)e2ζ(2t1 +t2 ) − α(κζL + Bi)e2ζ(t1 +t2 )Lines: 1920 to 2014where the thermal conductivity ratio parameter isα=———3.08873pt PgVar1−κ1+κwith κ = k2 /k1 , Bi = hL/k1 , and L an arbitrary length scale employed to define thedimensionless spreading resistance:ψ = L k1 R s(4.112)which is based on the thermal conductivity of the layer adjacent to the heat source.Various system lengths may be used and the appropriate choice depends on the systemof interest.In all summations φ(ζ) is evaluated in each series using ζ = δm , λn , and βm,n asdefined above.
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