BOOKCOMP, Inc. — John Wiley & Sons / Page 281 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 281 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [281], (21) Lines: 1006 to 1027 ——— -5.3346pt PgVar ——— Normal Page PgEnds: T E X [281], (21) TABLE 4.4 Dimensionless Spreading Resistance of an Isothermal Rectangular Area a/b k √ AR s 1 0.4412 2 0.4282 3 0.4114 4 0.3980 4.4.2 Isothermal Rectangular Area Schneider (1978) presented numerical values and a correlation of those values for the dimensionless spreading resistance of an isothermal rectangle for the aspect ratio range: 1 ≤ a/b ≤ 4. The correlation equation is k √ AR s = a b 0.06588 − 0.00232 a b + 0.6786 a/b + 0.8145 (4.52) The numerical values are given in Table 4.4. A comparison of the values for the isothermal rectangular area and the isothermal elliptical area reveals a very close relationship. The maximum difference of approxi- mately −0.7% is found at a/b = 4. It is expected that the close agreement observed for the four aspect ratios will hold for higher aspect ratios because the dimensionless spreading resistance is a weak function of the shape if the areas are geometrically similar. In fact, the correlation values for the rectangle and the analytical values for the ellipse are within ±1.5% over the wider range, 1 ≤ a/b ≤ 13. 4.4.3 Isoflux Regular Polygonal Area The spreading resistances of isoflux regular polygonal areas has been examined extensively. The regular polygonal areas are characterized by the number of sides N ≥ 3, the side dimension s, and the radius of the inscribed circle denoted as r i . The perimeter is P = Ns; the relationship between the inscribed radius and the side dimension is s/r i = 2 tan(π/N). The area of the regular polygon is A = Nr 2 i tan(π/N ). The temperature rises from the minimum values located on the edges to a maximum value at the centroid. It can be found easily by means of integral methods based on the superposition of point sources. The general relationship for the spreading resistance based on the centroid temperature rise is found to be k √ AR s = 1 π N tan(π/N ) ln 1 + sin(π/N ) cos(π/N) N ≥ 3 (4.53) The expression above gives k √ AR s = 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 282 / 2nd Proofs / Heat Transfer Handbook / Bejan 282 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [282], (22) Lines: 1027 to 1066 ——— -0.93408pt PgVar ——— Normal Page PgEnds: T E X [282], (22) Numerical methods are required to find the dimensionless spreading resistance for regular polygons subjected to a uniform heat flux. The corresponding value for the area-averaged basis was reported by Yovanovich and Burde (1977) to be k √ AR s = 0.4600 for the equilateral triangle, which is approximately 4% smaller than the value for the circle. 4.4.4 Arbitrary Singly Connected Area The spreading resistances for isoflux, singly connected source areas were obtained by means of numerical methods applied to the integral formulation of the spreading resistance. The source areas examined were isosceles triangles having a range of aspect ratios, the semicircle, L-shaped source areas (squares with corners removed), and the hyperellipse area defined by x a n + y b n = 1 (4.54) where a and b are the semiaxes along the x and y axes, respectively. The shape parameter n lies in the range 0 <n<∞. Many interesting geometries can be generated by the parameters a, b, and n. The area of the hyperellipse is given by the relationship A = 4ab n Γ(1 + 1/n)Γ(1/n) Γ(1 + 2/n) (m 2 ) (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 shape of 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 temperature rise denoted as R 0 and the area-averaged temperature rise, denoted R. The dimen- sionless spreading resistance was based on the length scale L = √ A. All numer- ical results were found to lie in narrow ranges: 0.4424 ≤ k √ A R ≤ 0.4733 and 0.5197 ≤ k √ AR 0 ≤ 0.5614. The corresponding values for the equilateral trian- gle are k √ A R = 0.4600 and k √ AR 0 = 0.5616, and for the semicircle they are k √ A R = 0.4610 and k √ AR 0 = 0.5456. TABLE 4.5 Effect of n on Dimensionless Spreading Resistances nk √ A Rk √ AR 0 0.5 0.4440 0.5468 1 0.4728 0.5611 2 0.4787 0.5642 4 0.4770 0.5631 ∞ 0.4732 0.5611 BOOKCOMP, Inc. — John Wiley & Sons / Page 283 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 283 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [283], (23) Lines: 1066 to 1105 ——— -0.12532pt PgVar ——— Normal Page * PgEnds: Eject [283], (23) The following approximations were recommended by Yovanovich and Burde (1977) for quick approximate calculations: k √ AR 0 = 5 9 and k √ A R = 0.84 k √ AR 0 . The ratios of the area-averaged and centroid temperature rises for all geometries examined were found to be closely related such that θ/θ 0 = 0.84 ± 1.7%. 4.4.5 Circular Annular Area Analytical methods have been used to obtain the spreading resistance for the isoflux and isothermal circular annulus of radii a and b in the surface of an isotropic half- space having thermal conductivity k. Isoflux Circular Annulus The temperature rise of points in the annular area a ≤ r ≤ b was reported by Yovanovich and Schneider (1977) to have the distribution θ(r) = 2 π qb k E r b − r b E a r + r b 1 − a r 2 K a r (K) (4.56) where the special functions K(x) and E(x) are the complete elliptic integrals of the first and second kinds, respectively, of arbitrary modulus x (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971). The dimensionless spreading resistance, based on the area-averaged temperature rise, of the isoflux circular annulus was reported by Yovanovich and Schneider (1977) to have the form kbR s = 8 3π 2 1 (1 − 2 ) 2 1 + 3 − (1 + 2 )E() + (1 − 2 )K() (4.57) where the modulus is = a/b < 1. When = 0, the annulus becomes a circle of radius b, and the relationship above gives kbR = 8/(3π 3 ), which is in agreement with the result obtained for the isoflux circular area. Isothermal Circular Annulus The spreading resistance for the isothermal cir- cular annulus cannot be obtained directly by the integral method. Mathematically, this is a mixed boundary value problem that requires special solution methods, which are discussed by Sneddon (1966). Smythe (1951) reported the solution for the capac- itance of a charged annulus. Yovanovich and Schneider (1977) used the two results of Smythe to determine the spreading resistance. Yovanovich and Schneider (1977) reported the following relationships for the spreading resistance of an isothermal cir- cular annular contact area: kbR s = 1 π 2 ln 16 + ln[(1 + )/(1 − )] 1 + (4.58) for 1.000 < 1/ < 1.10, and kbR s = π/8 (cos −1 + √ 1 − 2 tanh −1 )[1 + 0.0143 −1 tan 3 (1.28)] (4.59) BOOKCOMP, Inc. — John Wiley & Sons / Page 284 / 2nd Proofs / Heat Transfer Handbook / Bejan 284 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [284], (24) Lines: 1105 to 1171 ——— 0.6811pt PgVar ——— Normal Page PgEnds: T E X [284], (24) for 1.1 < 1/ < ∞. When = 0, the annulus becomes a circle, and the spreading re- sistance gives R s = 1/(4kb), in agreement with the result obtained for the isothermal circular area. 4.4.6 Other Doubly Connected Areas on a Half-Space The numerical data of spreading resistance from Martin et al. (1984) for three dou- bly connected regular polygons: equilateral triangle, square, and circle were nondi- mensionalized as k √ A c R s . The dimensionless spreading resistance is a function of = √ A i /A o , where A i and A o are the inner and outer projected areas of the polygons. The active area is A c = A o − A i . Accurate correlation equations with a maximum relative error of 0.6% were given. For the range 0 ≤ ≤ 0.995, k A c R s = a 0 1 − a 1 a 2 a 3 (4.60) and for the range 0.995 ≤ ≤ 0.9999, kP o R s = a 5 ln a 4 (1/) − 1 (4.61) where P o is the outer perimeter of the polygons and the correlation coefficients: a 0 through a 5 are given in Table 4.6. The correlation coefficient a 0 represents the dimensionless spreading resistance of the full contact area, in agreement with results presented above. Since the results for the square and the circle are very close for all values of the parameter , the correlation equations for the square or the circle may be used for other doubly connected regular polygons, 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 contact conductance h on the spreading resistance of square and circular contact areas. The dimensionless spreading resistance values were correlated with an accuracy of 0.1% by the relationship TABLE 4.6 Correlation Coefficients for Doubly Connected Polygons Circle Square Triangle a 0 0.4789 0.4732 0.4602 a 1 0.99957 0.99980 1.00010 a 2 1.5056 1.5150 1.5101 a 3 0.35931 0.37302 0.38637 a 4 39.66 68.59 115.91 a 5 0.31604 0.31538 0.31529 BOOKCOMP, Inc. — John Wiley & Sons / Page 285 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT SPREADING RESISTANCE IN AN ISOTROPIC HALF-SPACE 285 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [285], (25) Lines: 1171 to 1195 ——— -6.3249pt PgVar ——— Normal Page * PgEnds: Eject [285], (25) TABLE 4.7 Correlation Coefficients for Squares and Circles Circle Square c 1 0.46159 0.45733 c 2 0.017499 0.016463 c 3 0.43900 0.47035 c 4 1.1624 1.1311 k √ AR s = c 1 − c 2 tanh(c 3 ln Bi − c 4 ) 0 ≤ Bi < ∞ (4.62) with Bi = h √ A/k. The correlation coefficients c 1 through c 4 are given in Table 4.7. When Bi ≤ 0.1, the predicted values approach the values corresponding to the isoflux boundary condition, and when Bi ≥ 100, the predicted values are within 0.1% of the values obtained for the isothermal boundary condition. The transition from the isoflux values to the isothermal values occurs in the range 0.1 ≤ Bi ≤ 100. 4.5 TRANSIENT SPREADING RESISTANCE IN AN ISOTROPIC HALF-SPACE Transient spreading resistance occurs during startup and is important in certain micro- electronic systems. The spreading resistance may be defined with respect to the area-averaged temperature or with respect to a single point temperature such as the centroid temperature. Analytical solutions have been reported for a circular area on an isotropic half-space with isothermal, isoflux, and other heat flux distributions (Beck, 1979; Blackwell, 1972; Dryden et al., 1985; Keltner, 1973; Normington and Blackwell, 1964, 1972; Schneider et al., 1976; Turyk and Yovanovich, 1984; Negus and Yovanovich (1989); Yovanovich et al. (1984). Various analytical and numerical methods were employed to obtain short- and long-time solutions. 4.5.1 Isoflux Circular Area Beck (1979) reported the following integral solution for a circular area of radius a which is subjected to a uniform and constant flux q for t>0: 4kaR s = 8 π ∞ 0 erf ζ √ Fo J 2 1 (ζ) dζ ζ 2 (4.63) where erf is the error function, J 1 (x) is a Bessel function of the first kind of order 1 (Abramowitz and Stegun, 1965), and ζ is a dummy variable. The dimensionless time is defined as Fo = αt/a 2 , where α is the thermal diffusivity of the half-space. The spreading resistance is based on the area-averaged temperature. Steady state is obtained when Fo →∞, and the solution goes to 4kaR s = 32/(3π 2 ). Beck (1979) gave approximate solutions for short and long times. For short times where Fo < 0.6, BOOKCOMP, Inc. — John Wiley & Sons / Page 286 / 2nd Proofs / Heat Transfer Handbook / Bejan 286 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [286], (26) Lines: 1195 to 1237 ——— 1.84224pt PgVar ——— Normal Page PgEnds: T E X [286], (26) 4kaR s = 8 π Fo π − Fo π + Fo 2 8π + Fo 3 32π + 15Fo 4 512π (4.64) and for long times where Fo ≥ 0.6, 4kaR s = 32 3π 2 − 2 π 3/2 √ Fo 1 − 1 3(4Fo) + 1 6(4Fo) 2 − 1 12(4Fo) 3 (4.65) The maximum errors of about 0.18% and 0.07% occur at Fo = 0.6 for the short- and long-time expressions, respectively. 4.5.2 Isoflux Hyperellipse The hyperellipse is defined by (x/a) n + (y/b) n = 1 with b ≤ a, where n is the shape parameter and a and b are the axes along the x and y axes, respectively. The hyperellipse reduces to many special cases by setting the values of n and the aspect ratio parameter γ = b/a, which lies in the range 0 ≤ γ ≤ 1. Therefore, the solution developed for the hyperellipse can be used to obtain solutions for many other geometries, such as ellipse and circle, rectangle and square, diamondlike geometries, and so on. The transient dimensionless centroid constriction resistance k √ AR 0 , where R 0 = T 0 /Q, is given by the double-integral solution (Yovanovich, 1997) k √ AR 0 = 2 π √ A π/2 0 r 0 0 erfc r 2 √ A √ Fo dr dω (4.66) with Fo = αt/A, and the area of the hyperellipse is given by A = (4γ/n)B(1 + 1/n, 1/n) and B(x,y) is the beta function (Abramowitz and Stegun, 1965). The upper limit of the radius is given by r 0 = γ/[(sin ω) n + γ n (cos ω) n ] 1/n and the aspect ratio parameter γ = b/a. The solution above has the following characteristics: (1) for small dimensionless times, Fo ≤ 4 ×10 −2 ,k √ AR 0 = (2/ √ π) √ Fo for all values of n and γ; (2) for long dimensionless times, Fo ≥ 10 3 , the results are within 1% of the steady-state values, which are given by the single integral k √ AR 0 = 2γ π √ A π/2 0 dω [(sin ω) n + γ n (cos ω) n ] 1/n (4.67) which depends on the aspect ratio γ and the shape parameter n. The dimensionless spreading resistance depends on the three parameters Fo, γ, and n in the transition region 4 ×10 −2 ≤ Fo ≤ 10 3 in some complicated manner that can be deduced from the solution for the circular area. For this axisymmetric shape we put γ = 1, n = 2 into the hyperellipse double integral, which yields the following closed-form result valid for all dimensionless time (Yovanovich, 1997): k √ AR 0 = √ Fo 1 √ π − 1 √ π exp − 1 4πFo BOOKCOMP, Inc. — John Wiley & Sons / Page 287 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT SPREADING RESISTANCE IN AN ISOTROPIC HALF-SPACE 287 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [287], (27) Lines: 1237 to 1266 ——— -0.04442pt PgVar ——— Normal Page * PgEnds: Eject [287], (27) + 1 2 √ π √ Fo erfc 1 2 √ π √ Fo (4.68) where the dimensionless time for the circle of radius a is Fo = αt/πa 2 . 4.5.3 Isoflux Regular Polygons For regular polygons having sides N ≥ 3, the area is A = Nr 2 i tan(π/N ), where r i is the radius of the inscribed circle. The dimensionless spreading resistance based on the centroid temperature rise k √ AR 0 is given by the following double integral (Yovanovich, 1997): k √ AR 0 = 2 N tan(π/N ) π/N 0 1/ cos ω 0 erfc r 2 √ N tan(π/N) √ Fo dr dω (4.69) where the polygonal area is expressed in terms of the number of sides N, and for convenience the inscribed radius has been set to unity. This double-integral solution has characteristics identical to those of the double-integral solution given above for the hyperellipse; that is, for small dimensionless time, Fo ≤ 4 × 10 −2 , k √ AR 0 = (2/ √ π) √ F 0 for all polygons N ≥ 3; and for long dimensionless times, Fo ≥ 10 3 , the results are within 1% of the steady-state values, which are given by the following closed-form expression (Yovanovich, 1997): k √ AR 0 = 1 π N tan(π/N ) ln 1 + sin(π/N ) cos(π/N) (4.70) The dimensionless spreading resistance k √ AR 0 depends on the parameters: Fo and N in the transition region: 4 × 10 −2 ≤ Fo ≤ 10 3 in some complex manner which, as described above, may be deduced from the solution for the circular area. The steady-state solution gives the values k √ AR o = 0.5617, 0.5611, and 0.5642 for the equilateral triangle, N = 3, the square, N = 4, and the circle, N →∞. The difference between the values for the triangle and the circle is approximately 2.2%, whereas the difference between the values for the square and the circle is less than 0.6%. The following procedure is proposed for computation of the centroid-based transient spreading resistance for the range 4 × 10 −2 ≤ Fo ≤ 10 6 . The closed-form solution for the circle is the basis of the proposed method. For any planar singly connected contact area subjected to a uniform heat flux, take ψ 0 ψ 0 (Fo →∞) = 2 √ Fo 1 − exp − 1 4π · Fo + 1 2 √ Fo erfc 1 2 √ π √ Fo (4.71) BOOKCOMP, Inc. — John Wiley & Sons / Page 288 / 2nd Proofs / Heat Transfer Handbook / Bejan 288 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [288], (28) Lines: 1266 to 1295 ——— 0.62859pt PgVar ——— Long Page PgEnds: T E X [288], (28) where ψ 0 = k √ AR 0 . The right-hand side of eq. (4.71) can be considered to be a universal dimensionless time function that accounts for the transition from small times to near steady state. The procedure proposed should provide quite accurate results for any planar, singly connected area. A simpler expression that is based on the Greene (1989) approximation of the complementary error function is proposed (Yovanovich, 1997): ψ 0 ψ 0 (Fo →∞) = 1 z √ π 1 − e −z 2 + a 1 √ π ze −a 2 (z+a 3 ) 2 (4.72) where z = 1/(2 √ π √ Fo) and the three correlation coefficients are a 1 = 1.5577,a 2 = 0.7182, and a 3 = 0.7856. This approximation will provide values of ψ 0 with maxi- mum errors of less than 0.5% for Fo ≥ 4 ×10 −2 . 4.6 SPREADING RESISTANCE WITHIN A COMPOUND DISK WITH CONDUCTANCE The spreading, one-dimensional flow and total resistances for steady conduction within compound disks is important in many microlectronic applications. The heat enters a compound disk of radius b through a circular area of radius a located in the top surface of the first layer of thickness t 1 and thermal conductivity k 1 , which is in perfect thermal contact with the second layer (called the substrate) of thickness t 2 and thermal conductivity k 2 . The lateral boundary r = b is adiabatic, the face at z = t = t 1 +t 2 is either cooled by a fluid through the film conductance h or it is in contact with a heat sink through a contact conductance h. In either case, h is assumed to be uniform. A compound disk with uniform heat flux and uniform conductance along the lower surface is shown in Fig. 4.4. Figure 4.4 Compound disk with uniform heat flux and conductance. (From Yovanovich et al., 1980.) BOOKCOMP, Inc. — John Wiley & Sons / Page 289 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE WITHIN A COMPOUND DISK WITH CONDUCTANCE 289 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [289], (29) Lines: 1295 to 1351 ——— 0.11618pt PgVar ——— Long Page PgEnds: T E X [289], (29) The boundary condition over the source area may be modeled as either uniform heat flux or isothermal. The complete solution for these two boundary conditions has been reported by Yovanovich et al. (1980). The spreading resistance R s and one- dimensional system resistance R 1D are related to the total system resistance R total in the following manner: R total = R s + R 1D = R s + t 1 k 1 A + t 2 k 2 A + 1 hA (K/W) (4.73) where A = πb 2 . The general solution for the dimensionless spreading resistance pa- rameter ψ = 4k 1 aR s depends on several dimensionless geometric and thermophys- ical parameters: = a/b, τ = t/b,τ 1 = t 1 /b, τ 2 = t 2 /b, κ = k 1 /k 2 , Bi = hb/k 2 , and µ, a parameter that describes the heat flux distribution over the source area. The general relationship between the total heat flow rate Q and the axisymmetric heat flux distribution q(u) is q(u) = Q πa 2 (1 + µ)(1 − u 2 ) µ 0 ≤ u ≤ 1 (4.74) The heat flux distributions corresponding to three values of the parameter µ are presented in Table 4.8, where Q/πa 2 is the average flux on the area. When µ = 0, the heat flux is uniform, and when µ =− 1 2 , the heat flux distribution is called the equivalent isothermal distribution because it produces an almost isothermal area provided that a/b < 0.6. The independent system parameters have the ranges given in Table 4.9. If the first layer conductivity is lower than the substrate conductivity (i.e., 0 < κ < 1), the layer (coating) is said to be thermally resistive, and if the conductivity ratio parameter lies in the range 1 < κ < ∞, the layer is said to be thermally conductive. The general dimensionless spreading resistance relationship was given as (Yo- vanovich et al., 1980): 4k 1 aR s = 8(µ + 1) π ∞ n=1 A n ()B n (τ, τ 1 , κ, Bi) J 1 (δ n ) δ n (4.75) The first layer thermal conductivity and the radius of the heat source were used to nondimensionalize the spreading resistance. If the substrate thermal conductivity TABLE 4.8 Three Heat Flux Distributions µ q(u) − 1 2 Q 2πa 2 √ 1 − u 2 0 Q πa 2 1 2 3Q 2πa 2 1 − u 2 BOOKCOMP, Inc. — John Wiley & Sons / Page 290 / 2nd Proofs / Heat Transfer Handbook / Bejan 290 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [290], (30) Lines: 1351 to 1397 ——— 1.8284pt PgVar ——— Long Page * PgEnds: Eject [290], (30) TABLE 4.9 System Parameter Ranges 0 < < 1 0 < τ 1 < ∞ 0 < τ 2 < ∞ 0 < κ < ∞ 0 < Bi < ∞ is used, the right-hand side of the relationship must be multiplied by the thermal parameter κ. The coefficients A n are functions of the heat flux parameter µ. They are A n = −2 sin δ n δ 2 n J 2 0 (δ n ) for µ =− 1 2 −2J 1 (δ n ) δ 2 n J 2 0 (δ n ) for µ = 0 −2 sin δ n δ 2 n J 2 0 (δ n ) 1 (δ n ) 2 − 1 (δ n ) tan δ n for µ = 1 2 (4.76) The function B n , which depends on the system parameters (τ 1 , τ 2 , κ, Bi), was defined as B n = φ n tanh(δ n τ 1 ) − ϕ n 1 − φ n (4.77) and the two functions that appear in the relationship above are defined as φ n = κ − 1 κ ( cosh δ n τ 1 − ϕ n sinh δ n τ 1 ) cosh δ n τ 1 (4.78) ϕ n = δ n + Bi tanh δ n τ δ n tanh(δ n τ) + Bi (4.79) The eigenvalues δ n are the positive roots of J 1 (·) = 0 (Abramowitz and Stegun, 1965). For the special case of an isotropic disk where κ = 1, B n =−ϕ n , which depends on τ and Bi only. Since the general solution depends on several independent parameters, it is not possible to present the results in tabular or graphical form. The full solution can, however, be programmed easily into computer algebra systems. Characteristics of ϕ n This function accounts for the effects of the parameters: δ n , τ, and Bi. For extreme values of the parameter Bi, it reduces to ϕ n → tanh δ n τ as Bi →∞ coth δ n τ as Bi → 0 (4.80) . / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 281 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [281],. 282 / 2nd Proofs / Heat Transfer Handbook / Bejan 282 THERMAL SPREADING AND CONTACT RESISTANCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [282],. / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF RECTANGULAR SOURCE AREAS 283 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [283],