Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 35
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The temperature distribution throughout the half-space z ≥ 0 is given bythe infinite integral (Carslaw and Jaeger, 1959)dλq0 a ∞ −λze J0 (λr)J1 (λa)(K)(4.32)θ=k 0λwhere J1 (x) is the Bessel function of the first kind of order 1 (Abramowitz and Stegun,1965), and λ is a dummy variable. The temperature rise in the source area 0 ≤ r ≤ ais axisymmetric and is given by (Carslaw and Jaeger, 1959):dλq0 a ∞J0 (λr)J1 (λa)(K)(4.33)θ(r) =k 0λ0≤r≤a(K)(4.34)where E(r/a) is the complete elliptic integral of the second kind of modulus r/a(Byrd and Friedman, 1971) which is tabulated, and it can be calculated by means ofcomputer algebra systems.
The temperatures at the centroid r = 0 and the edge r = aof the source area are, respectively,θ(0) =q0 akandθ(a) =2 q0 aπ k(K)(4.35)The centroid temperature rise relative to the temperature rise at the edge is greaterby approximately 57%. The values of the dimensionless temperature rise defined askθ(r/a)/(q0 a) are presented in Table 4.1.TABLE 4.1Dimensionless Source Temperaturer/akθ(r/a)/q0 ar/akθ(r/a)/q0 a0.00.10.20.30.40.51.0000.99750.98990.97710.95870.93420.60.70.80.91.00.90280.86300.81260.74590.6366BOOKCOMP, Inc. — John Wiley & Sons / Page 276 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 720 to 777———0.16327pt PgVarThe alternative form of the solution according to Yovanovich (1976c) is2 q0 a r Eθ(r) =π ka[276], (16)———Normal PagePgEnds: TEX[276], (16)SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE123456789101112131415161718192021222324252627282930313233343536373839404142434445The area-averaged source temperature is ∞1 q0 a adλ2πr drθ=J0 (λr)J1 (λa)πa 2 k 0λ0(K)277(4.36)The integrals can be interchanged, giving the result (Carslaw and Jaeger, 1959):2q0 ∞ 2dλ8 q0 a θ=J1 (λa) 2 =(K)(4.37)k 03π kλAccording to the definition of spreading resistance, one obtains for the isoflux circularsource the relation (Carslaw and Jaeger, 1959)Rs =8θ=Q3π21ka(K/W)(4.38)The spreading resistance for the isoflux source area based on the area-averaged temperature rise is greater than the value for the isothermal source by the factor-0.27078pt PgVar4.3.3 Spreading Resistance of an Isothermal Elliptical Source Areaon a Half-SpaceThe spreading resistance for an isothermal elliptical source area with semiaxes a ≥ bis available in closed form.
The results are obtained from a solution that follows theclassical solution presented for finding the capacitance of a charged elliptical diskplaced in free space as given by Jeans (1963), Smythe (1968), and Stratton (1941).Holm (1967) gave the solution for the electrical resistance for current flow froman isopotential elliptical disk. The thermal solution presented next will follow theanalysis of Yovanovich (1971).The elliptical contact area x 2 /a 2 + y 2 /b2 = 1 produces a three-dimensionaltemperature field where the isotherms are ellipsoids described by the relationshipx2y2z2+ 2+=1+ζ b +ζζ(4.39)The three-dimensional Laplace equation in Cartesian coordinates can be transformedinto the one-dimensional Laplace equation in ellipsoidal coordinates:∂ ∂θ2∇ θ=f (ζ)=0(4.40)∂ζ∂ζwhere ζ is the ellipsoidal coordinate for the ellipsoidal temperature rise θ(ζ) and wheref (ζ) = (a 2 + ζ)(b2 + ζ)ζ(4.41)BOOKCOMP, Inc.
— John Wiley & Sons / Page 277 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 777 to 834———(Rs )isoflux32== 1.08076(Rs )isothermal3π2a2[277], (17)———Normal PagePgEnds: TEX[277], (17)278123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESThe solution of the differential equation according to Yovanovich (1971) is ∞dζθ = C2 − C1(K)√f (ζ)ζ(4.42)The boundary conditions are specified in the contact plane (z = 0) whereθ = θ0∂θ=0∂zwithinx2y2+=1a2b2outsidex2y2+=1a2b2(4.43)The regular condition at points remote to the elliptical area is θ → 0 as ζ → ∞. Thiscondition is satisfied by C2 = 0, and the condition in the contact plane is satisfiedby C1 = −Q/4πk, where Q is the total heat flow rate from the isothermal ellipticalarea.
The solution is, therefore, according to Yovanovich (1971), ∞dζQ(K)(4.44)θ=24πk ζ(a + ζ)(b2 + ζ)ζWhen ζ = 0, θ = θ0 , constant for all points within the elliptical area, and whenζ → ∞, θ → 0 for all points far from the elliptical area. According to the definitionof spreading resistance for an isothermal contact area, we find that ∞θ01dζ=Rs =(K/W)(4.45)2Q4πk 0(a + ζ)(b2 + ζ)ζThe last equation can be transformed into a standard form by setting sin t = a/a 2 + ζ. The alternative form for the spreading resistance is1Rs =2πka0π/2dt{1 −[(a 2− b2 )/a 2 ] sin2 t}1/2(K/W)(4.46)The spreading resistance depends on the thermal conductivity of the half-space, thesemimajor axis a, and the aspect ratio of the elliptical area b/a ≤ 1.
It is clear thatwhen the axes are equal (i.e., b = a), the elliptical area becomes a circular area and thespreading resistance is Rs = 1/(4ka). The integral is the complete elliptic integralof the first kind K(κ) of modulus κ = (a 2 − b2 )/a 2 (Byrd and Friedman, 1971;Gradshteyn and Ryzhik, 1965). The spreading resistance for the isothermal ellipticalsource area can be written asRs =1K(κ)2πka(K/W)(4.47)The complete elliptic integral is tabulated (Abramowitz and Stegun, 1965; Magnuset al.
(1966); Byrd and Friedman, 1971). It can also be computed efficiently and veryaccurately by computer algebra systems.BOOKCOMP, Inc. — John Wiley & Sons / Page 278 / 2nd Proofs / Heat Transfer Handbook / Bejan[278], (18)Lines: 834 to 903———2.35738pt PgVar———Long PagePgEnds: TEX[278], (18)SPREADING AND CONSTRICTION RESISTANCES IN AN ISOTROPIC HALF-SPACE123456789101112131415161718192021222324252627282930313233343536373839404142434445279TABLE 4.2 Dimensionless SpreadingResistance of an Isothermal Ellipse√√a/bk A Rsa/bk A Rs123450.44310.43020.41180.39510.38056789100.36780.35660.34660.33770.32974.3.4 Dimensionless Spreading Resistance of an IsothermalElliptical AreaTo compare the spreading resistances of the elliptical area and the circular area, it is[279], (19)necessary to nondimensionalize the two results. For the circle, the radius appears asthe length scale, and for the ellipse, the semimajor axis appears as the length scale.For proper comparison of the two geometries it is important to select a length scaleLines: 903 to 931that best characterizes the two geometries.
The proposed lengthscale is based on the√———square root of the active area of each geometry (i.e., L = A) (Yovanovich, 1976c;0.90663ptPgVarYovanovich and Burde, 1977; Yovanovich et al., 1977). Therefore, the dimensionless———spreading resistances for the circle and ellipse areLong Page√√* PgEnds: Ejectπ(k A Rs )circle =4√a1[279], (19)(k A Rs )ellipse = √K(κ)2 π bwhere κ = 1 − (b/a)2 . The dimensionless spreading resistance values for an isothermal elliptical area are presented in Table 4.2 for a range of the semiaxes ratio a/b.The tabulated values of the dimensionless spreading resistance reveal an interesting trend beginning with the first entry, which corresponds to the circle. The dimensionless resistance values decrease with increasing values of a/b. Ellipses withlarger values of a/b have smaller spreading resistances than the circle; however, thedecrease has a relatively weak dependence on a/b.
For the same area the spreading resistance of the ellipse with a/b = 10 is approximately 74% of the spreadingresistance for the circle.4.3.5Approximations for Dimensionless Spreading ResistanceTwo approximations are presented for quick calculator estimations of the dimensionless spreading resistance for isothermal elliptical areas: √παfor 1 ≤ α ≤ 5 √√( α + 1)2k A Rs =(4.48)1 √ln 4αfor 5 ≤ α < ∞2 παBOOKCOMP, Inc. — John Wiley & Sons / Page 279 / 2nd Proofs / Heat Transfer Handbook / Bejan280123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESwhere α = a/b ≥ 1.
Although both approximations can be used at α = 5, the secondapproximation is slightly more accurate, and therefore it is recommended.4.3.6Flux Distribution over an Isothermal Elliptical AreaThe heat flux distribution over the elliptical area is given by (Yovanovich, 1971) x 2 y 2 −1/2Qq(x,y) =−(W/m2)(4.49)1−2πababThe heat flux is minimum at the centroid, where its magnitude is q0 = Q/2πab, andit is “unbounded” on the perimeter of the ellipse.4.44.4.1SPREADING RESISTANCE OF RECTANGULAR SOURCE AREASIsoflux Rectangular AreaLines: 931 to 1006The dimensionless spreading resistances of the rectangular source area −a ≤ x ≤a, −b ≤ y ≤ b with aspect ratio a/b ≥ 1 are found by means of the integral method(Yovanovich, 1971).
Employing the definition of the spreading resistance based onthe area-averaged temperature rise with Q = 4qab gives the following dimensionlessrelationship (Yovanovich, 1976c; Carslaw and Jaeger, 1959):√ √111 3/2−1 1−1k A Rs =(4.50)sinh+ sinh +1+ 3 − 1+ 2π3where = a/b ≥ 1. Employing the definition based on the centroid temperature rise,the dimensionless spreading resistance is obtained from the relationship (Carslaw andJaeger, 1959)√ √ 11k A Rs =sinh−1 + sinh−1(4.51)π Typical values of the dimensionless spreading resistance for the isoflux rectanglebased on the area-average temperature rise for 1 ≤ a/b ≤ 10 are given in Table 4.3.Table 4.3 Dimensionless SpreadingResistance of an Isoflux Rectangular Area√√a/bk A Rsa/bk A Rs123450.47320.45980.44070.42340.4082BOOKCOMP, Inc. — John Wiley & Sons / Page 280 / 2nd Proofs / Heat Transfer Handbook / Bejan[280], (20)6789100.39500.38330.37290.36360.3552———0.74892pt PgVar———Normal PagePgEnds: TEX[280], (20)SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS123456789101112131415161718192021222324252627282930313233343536373839404142434445281TABLE 4.4 Dimensionless SpreadingResistance of an Isothermal Rectangular Area√a/bk A Rs12344.4.20.44120.42820.41140.3980Isothermal Rectangular AreaSchneider (1978) presented numerical values and a correlation of those values forthe dimensionless spreading resistance of an isothermal rectangle for the aspect ratiorange: 1 ≤ a/b ≤ 4.
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