Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 42
Текст из файла (страница 42)
According to Sexl and Burkhard (1969), Veziroglu and Chandra (1969),and Yovanovich et al. (1999), the relationship iskLRs =11lnπ sin(π/2)(4.122)Numerical values are given in Table 4.14. A comparison of the values correspondingto µ = − 21 and those for the true isothermal strip shows close agreement providedthat < 0.5. For very narrow strips where < 0.1, the differences are less than 1%.4.12.2 Spreading Resistance for an Abrupt Change in the CrossSectionIf steady conduction occurs in a two-dimensional channel whose width decreasesfrom 2a to 2b, there is spreading resistance as heat flows through the commonTABLE 4.14Dimensionless Spreading Resistance kLRs in Flux Channelsµ− 21012IsothermalStripChange0.010.10.20.30.40.50.60.70.81.3210.59020.37290.24940.16580.10530.06070.02830.00661.3580.62630.40830.28360.19840.13570.08820.05210.02551.3750.64300.42470.29950.21340.14960.10070.06280.03381.3220.59050.37380.25140.16910.11030.06750.03670.01601.3430.61100.39360.26990.18600.12490.07940.04560.0214BOOKCOMP, Inc.
— John Wiley & Sons / Page 312 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2128 to 2174———-1.6989pt PgVar———Normal PagePgEnds: TEX[312], (52)TRANSIENT SPREADING RESISTANCE123456789101112131415161718192021222324252627282930313233343536373839404142434445313interface.
The true boundary condition at the common interface is unknown. Thetemperature and the heat flux are both nonuniform. Conformal mapping leads to aclosed-form solution for the spreading resistance.The relationship for the spreading resistance is, according to Smythe (1968),11+1 − 21kLRs =+ln+ 2 ln(4.123)2π1−4where = a/b < 1.
Numerical values are given in Table 4.14. An examination of thevalues reveals that they lie between the values for µ = − 21 and µ = 0. The averagevalue of the first two columns corresponding to µ = − 21 and µ = 0 are in very closeagreement with the values in the last column. The differences are less than 1% for ≤ 0.20, and they become negligible as → 0.[313], (53)4.13 TRANSIENT SPREADING RESISTANCE WITHIN ISOTROPICSEMI-INFINITE FLUX TUBES AND CHANNELSTuryk and Yovanovich (1984) reported the analytical solutions for transient spreadingresistance within semi-infinite circular flux tubes and two-dimensional channels. Thecircular contact and the rectangular strip are subjected to uniform and constant heatflux.4.13.1———-5.94008pt PgVar———Normal Page* PgEnds: EjectIsotropic Flux TubeThe dimensionless transient spreading resistance for an isoflux circular source ofradius a supplying heat to a semi-infinite isotropic flux tube of radius b, constantthermal conductivity k, and thermal diffusivity α is given by the series solution√∞16 J12 (δn ) erf(δn Fo)4kaRs =(4.124)π n=1δ3n J02 (δn )where = a/b < 1, Fo = αt/a 2 > 0, and δn are the positive roots of J1 (·) = 0.The average source temperature rise was used to define the spreading resistance.
Theseries solution approaches the steady-state solution presented in an earlier sectionwhen the dimensionless time satisfies the criterion Fo ≥ 1/2 or when the real timesatisfies the criterion t ≥ a 2 /α2 .4.13.2Lines: 2174 to 2219Isotropic Semi-infinite Two-Dimensional ChannelThe dimensionless transient spreading resistance for an isoflux strip of width 2awithin a two-dimensional channel of width 2b, length L, constant thermal conductivity k, and thermal diffusivity α was reported as (Turyk and Yovanovich, 1984)√ ∞1 sin2 mπ erf mπ FoLkRs = 3(4.125)π m=1m3BOOKCOMP, Inc. — John Wiley & Sons / Page 313 / 2nd Proofs / Heat Transfer Handbook / Bejan[313], (53)314123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere = a/b < 1 is the relative size of the contact strip and the dimensionless timeis defined as Fo = αt/a 2 .
There is no half-space solution for the two-dimensionalchannel. The transient solution is within 1% of the steady-state solution when thedimensionless time satisfies the criterion Fo ≥ 1.46/2 .4.14 SPREADING RESISTANCE OF AN ECCENTRIC RECTANGULARAREA ON A RECTANGULAR PLATE WITH COOLINGA rectangular isoflux area with side lengths c and d lies in the surface z = 0 ofa rectangular plate with side dimensions a and b. The plate thickness is t1 and itsthermal conductivity is k1 . The top surface outside the source area is adiabatic, andall sides are adiabatic. The bottom surface at z = t1 is cooled by a fluid or a heat sinkthat is in contact with the entire surface.
In either case the heat transfer coefficient isdenoted as h and is assumed to be uniform. The origin of the Cartesian coordinatesystem (x,y,z) is located in the lower left corner. The system is shown in Fig. 4.14.[314], (54)Lines: 2219 to 2243———0.25099pt PgVar———Long PagePgEnds: TEX[314], (54)Figure 4.14 Isotropic plate with an eccentric rectangular heat source. (From Muzychka et al.,2000.)BOOKCOMP, Inc. — John Wiley & Sons / Page 314 / 2nd Proofs / Heat Transfer Handbook / BejanSPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA123456789101112131415161718192021222324252627282930313233343536373839404142434445315The temperature rise of points in the plate surface z = 0 is given by the relationshipθ(x,y,z) = A0 + B0 z+∞cos λx(A1 cosh λz + B1 sinh λz)m=1+∞cos δy(A2 cosh δz + B2 sinh δz)n=1+∞ ∞cos λx cos δy(A3 cosh βz + B3 sinh βz)(4.126)m=1 n=1[315], (55)The Fourier coefficients are obtained by means of the following relationships:Q t11Q+and B0 = −(4.127)A0 =Lines: 2243 to 2296ab k1hk1 ab 2Xc +c 2Xc −c———λm − sinλm2Q sin221.92044ptPgVarA1 =(4.128)———abck1 λ2m φ(λm )Long Page 2Yc +d 2Yc −d 2Q sinδn − sinδn* PgEnds: Eject22A2 =(4.129)abck1 δ2n φ(δn )[315], (55) 16Q cos(λm Xc ) sin 21 λm c cos(δn Yc ) sin 21 δn dA3 =(4.130)abcdk1 βm,n λm δn φ(βm,n )The other Fourier coefficients are obtained by the relationshipBi = −φ(ζ)Aii = 1, 2, 3(4.131)where ζ is replaced by λm , δn , or βm,n as required.
The eigenvalues aremπnπδn =βm,n = λ2m + δ2nλm =abThe mean temperature rise of the source area is given by the relationship ∞∞cos(λm Xc ) sin 21 λm ccos(δn Yc ) sin 21 δn dθ = θ1D + 2AmAn+2λm cδn dm=1n=1 ∞ ∞cos(δn Yc ) sin 21 δn d cos(λm Xc ) sin 21 λm c+4Amn(4.132)λm cδn dm=1 n=1where the one-dimensional temperature rise isBOOKCOMP, Inc. — John Wiley & Sons / Page 315 / 2nd Proofs / Heat Transfer Handbook / Bejan316123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESθ1D =Qab1t1+k1h(K)(4.133)for an isotropic plate.
The total resistance is related to the spreading resistance andthe one-dimensional resistance:Rtotal =θ= R1D + RsQ(K/W)(4.134)whereR1D1=abt11+k1h(K/W)(4.135)[316], (56)4.14.1Single Eccentric Area on a Compound Rectangular PlateIf a single source is on the top surface of a compound rectangular plate that consistsof two layers having thicknesses t1 and t2 and thermal conductivities k1 and k2 , asshown in Fig. 4.15, the results are identical except for the system parameter φ, whichnow is given by the relationshipφ(ζ) =(αe4ζt1 − e2ζt1 ) + 1(e2ζ(2t1 +t2 ) − αe2ζ(t1 +t2 ) )(αe4ζt1 + e2ζt1 ) + 1(e2ζ(2t1 +t2 ) + αe2ζ(t1 +t2 ) )Lines: 2296 to 2339———9.33221pt PgVar———Normal Page(4.136) * PgEnds: Eject[316], (56)where1=ζ + h/k2ζ − h/k2andα=1−κ1+κ(4.137)with κ = k2 /k1 and ζ is replaced by λm , δn , or βm,n , accordingly.
The one-dimensionaltemperature rise in this case isFigure 4.15 Compound plate with an eccentric rectangular heat source. (From Muzychkaet al., 2000.)BOOKCOMP, Inc. — John Wiley & Sons / Page 316 / 2nd Proofs / Heat Transfer Handbook / BejanSPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA123456789101112131415161718192021222324252627282930313233343536373839404142434445θ1D =4.14.2Qabt1t21++k1k2h317(K)(4.138)Multiple Rectangular Heat Sources on an Isotropic PlateThe multiple rectangular sources on an isotropic plate are shown in Fig. 4.16.
The surface temperature by superposition is given by the following relationship (Muzychkaet al., 2000):T (x,y,0) − Tf =Nθi (x,y,0)(K)(4.139)i=1where θi is the temperature excess for each heat source by itself and N ≥ 2 is thenumber of discrete heat sources. The temperature rise is given by[317], (57)Lines: 2339 to 2361———11.88896pt PgVar———Normal Page* PgEnds: Eject[317], (57)Figure 4.16 Isotropic plate with two eccentric rectangular heat sources. (From Muzychka etal., 2000.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 317 / 2nd Proofs / Heat Transfer Handbook / Bejan318123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESθi (x,y,0) = Ai0 +∞Aim cos λx +m=1+∞Ain cos δyn=1∞ ∞Aimn cos λx cos δy(4.140)m=1 n=1where φ and Ai0 = θ1D are defined above for the isotropic and compound plates.The mean temperature of an arbitrary rectangular area of dimensions cj and dj ,located at Xc,j and Yc,j , may be obtained by integrating over the region Aj = cj dj :θj =1Ajθi dAj =Aj1Aj NAjθi (x,y,0) dAj(4.141)i=1[318], (58)which may be written asLines: 2361 to 2411NN1θj =θi (x,y,0) dAj =θiAji=1i=1(4.142)Aj———Normal PagePgEnds: TEXThe mean temperature of the jth heat source is given byT j − Tf =Nθi(4.143)i=1whereθi =Aio+2∞Aimm=1+4∞ ∞cos(λm Xc,j ) sinλm cjAimn 12λm cjcos(δn Yc,j ) sin+2∞n=1 1δ d2 n jm=1 n=1Aincos(δn Yc,j ) sinδn djcos(λm Xc,j ) sinλm cj δn dj 12λm cj 1δ d2 n j(4.144)Equation (4.143) represents the sum of the effects of all sources over an arbitrarylocation.
Equation (4.143) is evaluated over the region of interest cj , dj located atXc,j , Yc,j , with the coefficients Ai0 , Aim , Ain , and Aimn evaluated at each of the ithsource parameters.4.15JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDSThe elastoconstriction and elastogap resistance models (Yovanovich, 1986) are basedon the Boussinesq point load model (Timoshenko and Goodier, 1970) and the HertzBOOKCOMP, Inc.
— John Wiley & Sons / Page 318 / 2nd Proofs / Heat Transfer Handbook / Bejan———2.45174pt PgVar[318], (58)JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS123456789101112131415161718192021222324252627282930313233343536373839404142434445319zD2E1 , v1␦02aE2 , v2rFigure 4.17 Joint formed by elastic contact of a sphere or a cylinder with a smooth flatsurface. (From Kitscha, 1982.)[319], (59)Lines: 2411 to 2421distributed-load model (Hertz, 1896; Timoshenko and Goodier, 1970; Walowit and———Anno, 1975; Johnson, 1985). Both models assume that bodies have “smooth” sur0.59099ptPgVarfaces, are perfectly elastic, and that the applied load is static and normal to the plane———of the contact area. In the general case the contact area will be elliptical, havingNormal Pagesemimajor and semiminor axes a and b, respectively.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
