Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 36
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The correlation equation is a √a0.6786k A Rs =0.06588 − 0.00232+(4.52)bba/b + 0.8145The numerical values are given in Table 4.4.A comparison of the values for the isothermal rectangular area and the isothermalelliptical area reveals a very close relationship. The maximum difference of approximately −0.7% is found at a/b = 4. It is expected that the close agreement observedfor the four aspect ratios will hold for higher aspect ratios because the dimensionlessspreading resistance is a weak function of the shape if the areas are geometricallysimilar. In fact, the correlation values for the rectangle and the analytical values forthe ellipse are within ±1.5% over the wider range, 1 ≤ a/b ≤ 13.4.4.3Isoflux Regular Polygonal AreaThe spreading resistances of isoflux regular polygonal areas has been examinedextensively.
The regular polygonal areas are characterized by the number of sidesN ≥ 3, the side dimension s, and the radius of the inscribed circle denoted as ri .The perimeter is P = N s; the relationship between the inscribed radius and theside dimension is s/ri = 2 tan(π/N ). The area of the regular polygon is A =N ri2 tan(π/N ). The temperature rises from the minimum values located on the edgesto a maximum value at the centroid. It can be found easily by means of integralmethods based on the superposition of point sources. The general relationship forthe spreading resistance based on the centroid temperature rise is found to be√N11 + sin(π/N )k A Rs =lnN ≥3(4.53)π tan(π/N )cos(π/N )√The expression above gives k A Rs = 0.5516 for the equilateral triangle N = 3,which is approximately 2.3% smaller than the value for the circle where N → ∞.BOOKCOMP, Inc.
— John Wiley & Sons / Page 281 / 2nd Proofs / Heat Transfer Handbook / Bejan[281], (21)Lines: 1006 to 1027———-5.3346pt PgVar———Normal PagePgEnds: TEX[281], (21)282123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESNumerical methods are required to find the dimensionless spreading resistance forregular polygons subjected to a uniform heat flux. The corresponding value√ for thearea-averaged basis was reported by Yovanovich and Burde (1977) to be k A Rs =0.4600 for the equilateral triangle, which is approximately 4% smaller than the valuefor the circle.4.4.4Arbitrary Singly Connected AreaThe spreading resistances for isoflux, singly connected source areas were obtainedby means of numerical methods applied to the integral formulation of the spreadingresistance.
The source areas examined were isosceles triangles having a range ofaspect ratios, the semicircle, L-shaped source areas (squares with corners removed),and the hyperellipse area defined by x n y n+=1(4.54)abwhere a and b are the semiaxes along the x and y axes, respectively. The shapeparameter n lies in the range 0 < n < ∞. Many interesting geometries can begenerated by the parameters a, b, and n. The area of the hyperellipse is given by therelationshipA=4ab Γ(1 + 1/n)Γ(1/n)nΓ(1 + 2/n)(m2)(4.55)where Γ(x) is the gamma function which is tabulated (Abramowitz and Stegun,1965), and it can be computed accurately by means of computer algebra systems.The dimensionless spreading resistance was found to be a weak function of the shapeof the source area for a wide range of values of n. Typical values are given in Table 4.5.The dimensionless spreading resistances were based on the centroid temperaturerise denoted as R0 and the area-averaged temperature rise, denoted√R.
The dimensionless spreading resistance was based on the length scale L√ = A. All numerA R ≤ 0.4733 andical results werefoundtolieinnarrowranges:0.4424≤k√0.5197 ≤√ k A R0 ≤ 0.5614. Thecorrespondingvaluesfortheequilateral trian√glearekAR=0.4600andkAR=0.5616,andforthesemicirclethey are0√√k A R = 0.4610 and k A R0 = 0.5456.TABLE 4.5 Effect of n on DimensionlessSpreading Resistances√√nk ARk A R00.5124∞0.44400.47280.47870.47700.4732BOOKCOMP, Inc. — John Wiley & Sons / Page 282 / 2nd Proofs / Heat Transfer Handbook / Bejan0.54680.56110.56420.56310.5611[282], (22)Lines: 1027 to 1066———-0.93408pt PgVar———Normal PagePgEnds: TEX[282], (22)SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS123456789101112131415161718192021222324252627282930313233343536373839404142434445283The following approximations were recommendedby Yovanovichand√Burde√√(1977) for quick approximate calculations: k A R0 = 59 and k A R = 0.84 k A R0 .The ratios of the area-averaged and centroid temperature rises for all geometriesexamined were found to be closely related such that θ/θ0 = 0.84 ± 1.7%.4.4.5Circular Annular AreaAnalytical methods have been used to obtain the spreading resistance for the isofluxand isothermal circular annulus of radii a and b in the surface of an isotropic halfspace having thermal conductivity k.Isoflux Circular Annulus The temperature rise of points in the annular areaa ≤ r ≤ b was reported by Yovanovich and Schneider (1977) to have the distribution a 2 a 2 qbrr a rθ(r) =KE− E+1−(K) (4.56)π kbbrbrrwhere the special functions K(x) and E(x) are the complete elliptic integrals of thefirst and second kinds, respectively, of arbitrary modulus x (Abramowitz and Stegun,1965; Byrd and Friedman, 1971).
The dimensionless spreading resistance, based onthe area-averaged temperature rise, of the isoflux circular annulus was reported byYovanovich and Schneider (1977) to have the formkbRs =181 + 3 − (1 + 2 )E() + (1 − 2 )K()2223π (1 − )(4.58)for 1.000 < 1/ < 1.10, andkbRs =(cos−1+√π/81−2−1tanh )[1 + 0.0143−1 tan3 (1.28)]BOOKCOMP, Inc. — John Wiley & Sons / Page 283 / 2nd Proofs / Heat Transfer Handbook / Bejan———-0.12532pt PgVar———Normal Page* PgEnds: Eject[283], (23)Isothermal Circular Annulus The spreading resistance for the isothermal circular annulus cannot be obtained directly by the integral method. Mathematically,this is a mixed boundary value problem that requires special solution methods, whichare discussed by Sneddon (1966).
Smythe (1951) reported the solution for the capacitance of a charged annulus. Yovanovich and Schneider (1977) used the two resultsof Smythe to determine the spreading resistance. Yovanovich and Schneider (1977)reported the following relationships for the spreading resistance of an isothermal circular annular contact area:1 ln 16 + ln[(1 + )/(1 − )]π21+Lines: 1066 to 1105(4.57)where the modulus is = a/b < 1. When = 0, the annulus becomes a circle ofradius b, and the relationship above gives kbR = 8/(3π3 ), which is in agreementwith the result obtained for the isoflux circular area.kbRs =[283], (23)(4.59)284123456789101112131415161718192021222324252627282930313233343536373839404142434445THERMAL SPREADING AND CONTACT RESISTANCESfor 1.1 < 1/ < ∞. When = 0, the annulus becomes a circle, and the spreading resistance gives Rs = 1/(4kb), in agreement with the result obtained for the isothermalcircular area.4.4.6Other Doubly Connected Areas on a Half-SpaceThe numerical data of spreading resistance from Martin et al.
(1984) for three doubly connected regular√ polygons: equilateral triangle, square, and circle were nondiAc Rs . The dimensionless spreading resistance is a functionmensionalizedask√of = Ai /Ao , where Ai and Ao are the inner and outer projected areas of thepolygons. The active area is Ac = Ao − Ai .Accurate correlation equations with a maximum relative error of 0.6% were given.For the range 0 ≤ ≤ 0.995, a2 a3(4.60)k Ac Rs = a0 1 −a1Lines: 1105 to 1171and for the range 0.995 ≤ ≤ 0.9999,———a4kPo Rs = a5 ln(1/) − 1(4.61)where Po is the outer perimeter of the polygons and the correlation coefficients: a0through a5 are given in Table 4.6.The correlation coefficient a0 represents the dimensionless spreading resistance ofthe full contact area, in agreement with results presented above. Since the results forthe square and the circle are very close for all values of the parameter , the correlationequations for the square or the circle may be used for other doubly connected regularpolygons, such as pentagons, hexagons, and so on.Effect of Contact Conductance on Spreading Resistance Martin et al.(1984) used a novel numerical technique to determine the effect of a uniform contactconductance h on the spreading resistance of square and circular contact areas.
Thedimensionless spreading resistance values were correlated with an accuracy of 0.1%by the relationshipTABLE 4.6a0a1a2a3a4a5Correlation Coefficients for Doubly Connected PolygonsCircleSquareTriangle0.47890.999571.50560.3593139.660.316040.47320.999801.51500.3730268.590.315380.46021.000101.51010.38637115.910.31529BOOKCOMP, Inc. — John Wiley & Sons / Page 284 / 2nd Proofs / Heat Transfer Handbook / Bejan[284], (24)0.6811pt PgVar———Normal PagePgEnds: TEX[284], (24)TRANSIENT SPREADING RESISTANCE IN AN ISOTROPIC HALF-SPACE123456789101112131415161718192021222324252627282930313233343536373839404142434445TABLE 4.7285Correlation Coefficients for Squares and Circlesc1c2c3c4CircleSquare0.461590.0174990.439001.16240.457330.0164630.470351.1311√k A Rs = c1 − c2 tanh(c3 ln Bi − c4 )0 ≤ Bi < ∞(4.62)√with Bi = h A/k.
The correlation coefficients c1 through c4 are given in Table 4.7.When Bi ≤ 0.1, the predicted values approach the values corresponding to theisoflux boundary condition, and when Bi ≥ 100, the predicted values are within0.1% of the values obtained for the isothermal boundary condition. The transitionfrom the isoflux values to the isothermal values occurs in the range 0.1 ≤ Bi ≤ 100.[285], (25)Lines: 1171 to 11954.5 TRANSIENT SPREADING RESISTANCE IN AN ISOTROPICHALF-SPACE———-6.3249pt PgVar———Normal PageTransient spreading resistance occurs during startup and is important in certain microelectronic systems. The spreading resistance may be defined with respect to the * PgEnds: Ejectarea-averaged temperature or with respect to a single point temperature such as thecentroid temperature. Analytical solutions have been reported for a circular area[285], (25)on an isotropic half-space with isothermal, isoflux, and other heat flux distributions(Beck, 1979; Blackwell, 1972; Dryden et al., 1985; Keltner, 1973; Normington andBlackwell, 1964, 1972; Schneider et al., 1976; Turyk and Yovanovich, 1984; Negusand Yovanovich (1989); Yovanovich et al.
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