Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 41
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The general relationship for φ(ζ) reduces to simpler relationships fortwo important special cases: the semi-infinite flux channel where t2 → ∞, shownin Fig. 4.11, and the finite isotropic rectangular flux channel where κ = 1, shown inFig. 4.12. The respective relationships areφ(ζ) =(e2ζt1 − 1)κ + (e2ζt1 + 1)(e2ζt1 + 1)κ + (e2ζt1 − 1)(4.113)where the influence of the contact conductance has vanished, andφ(ζ) =(e2ζt + 1)ζL − (1 − e2ζt )Bi(e2ζt − 1)ζL + (1 + e2ζt )Biwhere the influence of κ has vanished.BOOKCOMP, Inc. — John Wiley & Sons / Page 306 / 2nd Proofs / Heat Transfer Handbook / Bejan(4.114)———Short PagePgEnds: TEX[306], (46)SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS123456789101112131415161718192021222324252627282930313233343536373839404142434445307[307], (47)Lines: 2014 to 2021———*17.39099pt PgVar———Short PagePgEnds: TEX[307], (47)Figure 4.11 Rectangular isoflux area on a layer bonded to a rectangular flux channel.
(FromYovanovich et al., 1999.)The dimensionless spreading resistance ψ depends on six independent dimensionless parameters, such as the relative size of the rectangular source area 1 =a/c, 2 = b/d, the layer conductivity ratio κ = k2 /k1 , the relative layer thicknessesτ1 = t1 /L, τ2 = t2 /L, and the Biot number Bi = hL/k1 .The general relationship reduces to several special cases, such as those describedin Table 4.13. The general solution may also be used to obtain the relationship for anisoflux square area on the end of a square semi-infinite flux tube.BOOKCOMP, Inc.
— John Wiley & Sons / Page 307 / 2nd Proofs / Heat Transfer Handbook / Bejan308123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES[308], (48)Lines: 2021 to 2021———0.14703pt PgVar———Normal PagePgEnds: TEXFigure 4.12 Rectangular isoflux area on an isotropic rectangular channel. (From Yovanovichet al., 1999.)TABLE 4.13Summary of Relationships for Isoflux AreaConfigurationLimiting ValuesRectangular heat sourceFinite compound rectangular flux channelSemi-infinite compound rectangular flux channelFinite isotropic rectangular flux channelSemi-infinite isotropic rectangular flux channela, b, c, d, t1 , t2 , k1 , k2 , ht2 → ∞k1 = k2t1 → ∞Strip heat sourceFinite compound rectangular flux channelSemi-infinite compound rectangular flux channelFinite isotropic rectangular flux channelSemi-infinite isotropic rectangular flux channela, c, b = d, t1 , t2 , k1 , k2 , ht2 → ∞k1 = k2t1 → ∞Rectangular source on a half-spaceIsotropic half-spaceCompound half-spacec → ∞, d → ∞, t1 → ∞c → ∞, d → ∞, t2 → ∞BOOKCOMP, Inc.
— John Wiley & Sons / Page 308 / 2nd Proofs / Heat Transfer Handbook / Bejan[308], (48)SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS1234567891011121314151617181920212223242526272829303132333435363738394041424344454.10.1309Square Area on a Semi-infinite Square Flux TubeFor the special case of a square heat source on a semi-infinite square isotropicflux tube, the general solution reduces to a simpler expression which depends onone parameter only. The dimensionless spreading resistance relationship (Mikic andRohsenow, 1966; Yovanovich et al., 1999) was recast in the form∞∞∞2 sin2 mπ1 sin2 mπ sin2 nπk As Rs = 3(4.115)+ 2 2√π m=1m3π m=1 n=1 m2 n2 m2 + n2√where the characteristic length was√ selected as L = As .
The relative size of theheat source was defined as = As /At , where As and At are the source and fluxtube areas, respectively. A correlation equation was reported for eq. (4.115) (Neguset al., 1989):k As Rs = 0.47320 − 0.62075 + 0.11983(4.116)[309], (49)Lines: 2021 to 2064in the range 0 ≤ ≤ 0.5, with a maximum relative error of approximately 0.3%.The constant on the right-hand side of the correlation equation is the value of thedimensionless spreading resistance of an isoflux square source on an isotropic halfspace when the square root of the source area is chosen as the characteristic length.4.10.2 Spreading Resistance of a Rectangle on a Layeron a Half-SpaceThe solution for the rectangular heat source on a compound half-space is obtainedfrom the general relationship for the finite compound flux channel, provided thatt2 → ∞, c → ∞, d → ∞.
No closed-form solution exists for this problem.4.10.3 Spreading Resistance of a Rectangle on an IsotropicHalf-SpaceThe spreading resistance for an isoflux rectangular source of dimensions 2a×2b on anisotropic half-space whose thermal conductivity is k has the closed-form relationship(Carslaw and Jaeger, 1959)3/2 √ 111111k As Rs =(4.117)sinh−1 + sinh−1 1 +1+ 3 − 1+ 2π1 1311where√ 1 = a/b ≥ 1 is the aspect ratio of the rectangle. If the scale length isAs , the dimensionless spreading resistance becomes a weak function of 1.
Fora√square heat source, the numerical value of the dimensionless spreading resistance isk As Rs = 0.4732, which is very close to the numerical value for the isoflux circularsource on an isotropic half-space and other singly connected heat source geometriessuch as an equilateral triangle and a semicircular heat source.L=BOOKCOMP, Inc. — John Wiley & Sons / Page 309 / 2nd Proofs / Heat Transfer Handbook / Bejan———-2.1598pt PgVar———Normal PagePgEnds: TEX[309], (49)310123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCES2c2a2dqa[310], (50)ckLines: 2064 to 2078———t0.08107pt PgVar———Short PagePgEnds: TEXhFigure 4.13 Strip on a finite rectangular channel with cooling.
(From Yovanovich et al.,1999.)4.11STRIP ON A FINITE CHANNEL WITH COOLINGSpreading resistance due to steady conduction from a strip of width 2a and lengthL = 2d through a finite rectangular flux channel of width 2c and thickness t andthermal conductivity k is considered here. A uniform conductance h is specified onthe bottom surface to account for cooling by a fluid or to represent the cooling by aheat sink. The system is shown in Fig. 4.13.This is a special case of the general relationships presented above for a rectangulararea on a compound rectangular channel. A general flux distribution on the strip isgiven byq(x) =Γ(µ + 3/2)Q(a 2 − x 2 )µ√L πa 1+2µ Γ(µ + 1)0≤x≤a(W/m2)(4.118)where Q is the total heat transfer rate from the strip and Γ(·) is the gamma function(Abramowitz and Stegun, 1965).
The parameter µ defines the heat flux distributionBOOKCOMP, Inc. — John Wiley & Sons / Page 310 / 2nd Proofs / Heat Transfer Handbook / Bejan[310], (50)STRIP ON A FINITE CHANNEL WITH COOLING123456789101112131415161718192021222324252627282930313233343536373839404142434445311on the strip, which may have the following values: (1) µ = − 21 to approximate anisothermal strip provided that a/c 1, (2) µ = 0 for an isoflux distribution, and (3)µ = 21 , which gives a parabolic flux distribution.
The three flux distributions areQ1for µ = − 212 − x2LπaQq(x) =(4.119)for µ = 02La2Q 2a − x2for µ = 21Lπa 2The dimensionless spreading resistance relationship based on the mean sourcetemperature is∞ 2 µ+1/2 sin nπΓ(µ + 3/2) kLRs =Jµ+1/2 (nπ)ϕnπ2 nπn2n=1[311], (51)(4.120)Lines: 2078 to 2128———whereϕn =nπ + Bi tanh nπτnπ tanh nπτ + Bi2.76997pt PgVarn = 1, 2, 3, . . .and the three dimensionless system parameters and their ranges are0<=a<1c0 < Bi =hc<∞k0<τ=t<∞cThe general relationship gives the following three relationships for the three fluxdistributions:∞1 sin nπ 2J0 (nπ)ϕnfor µ = − 21π n=1 n2∞ 1 sin2 nπϕnfor µ = 0kLRs =(4.121) 2 π3 n=1n3∞2 sin nπJ1 (nπ)ϕnfor µ = 21 2 3 π n=1 n3The influence of the cooling along the bottom surface on the spreading resistance isgiven by the function ϕn , which depends on two parameters, Bi and τ.
If the channel isrelatively thick (i.e., τ ≥ 0.85), ϕn → 1 for all values n = 1 · · · ∞, and the influenceof the parameter Bi becomes negligible. When τ ≥ 0.85, the finite channel may bemodeled as though it were infinitely thick. This special case is presented next.BOOKCOMP, Inc. — John Wiley & Sons / Page 311 / 2nd Proofs / Heat Transfer Handbook / Bejan———Short PagePgEnds: TEX[311], (51)3121234567891011121314151617181920212223242526272829303132333435363738394041424344454.12THERMAL SPREADING AND CONTACT RESISTANCESSTRIP ON AN INFINITE FLUX CHANNELIf the relative thickness of the rectangular channel becomes very large (i.e., τ → ∞),the relationships given above approach the relationships appropriate for the infinitelythick flux channel.
The dimensionless spreading resistance for this problem dependson two parameters: the relative size of the strip and the heat flux distributionparameter µ. The three relationships for the dimensionless spreading resistance areobtained from the relationships given above with ϕn = 1.Numerical values for LkRs for three values of µ are given in Table 4.14.
From thetabulated values it can be seen that the spreading resistance values for the isothermalstrip are smaller than the values for the isoflux distribution, which are smaller thanthe values for the parabolic distribution for all values of . The differences are largeas → 1; however, the differences become negligibly small as → 0.[312], (52)4.12.1 True Isothermal Strip on an Infinite Flux ChannelThere is a closed-form relationship for the true isothermal area on an infinitely thickflux channel.
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