BOOKCOMP, Inc. — John Wiley & Sons / Page 301 / 2nd Proofs / Heat Transfer Handbook / Bejan CIRCULAR AREA ON A SEMI-INFINITE FLUX TUBE 301 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 [301], (41) Lines: 1727 to 1799 ——— 0.26018pt PgVar ——— Normal Page PgEnds: T E X [301], (41) TABLE 4.10 Correlation Coefficients for Three Flux Distributions C n µ =− 1 2 µ = 0 µ = 1 2 C 0 1.00000 1.08085 1.12517 C 1 −1.40981 −1.41002 −1.41038 C 3 0.303641 0.259714 0.235387 C 5 0.0218272 0.0188631 0.0117527 C 7 0.0644683 0.0420278 0.0343458 case of a circular contact area on a half-space. The corresponding half-space results were reported by Strong et al. (1974): 4kaR s = 1forµ =− 1 2 32 3π 2 for µ = 0 1.1252 for µ = 1 2 (4.100) Correlation Equations for Spreading Resistance Since the three series so- lutions presented above for the three heat flux distributions µ =− 1 2 , 0, 1 2 converge slowly as → 0, correlation equations for the dimensionless spreading resistance ψ = 4kaR s for the three flux distributions were developed having the general form ψ = C 0 + C 1 + C 3 3 + C 5 5 + C 7 7 (4.101) with the correlation coefficients given in Table 4.10. The correlation equations, ap- plicable for the parameter range 0 ≤ ≤ 0.8, provide four-decimal-place accuracy. Simple Correlation Equations Yovanovich (1976b) recommended the follow- ing simple correlations for the three flux distributions: 4kaR s = a 1 (1 − a 2 ) (4.102) in the range 0 < ≤ 0.1 with a maximum error of 0.1%, and 4kaR s = a 1 (1 − ) a 3 (4.103) in the range 0 < ≤ 0.3 with a maximum error of 1%. The correlation coefficients for the three flux distributions are given in Table 4.11. TABLE 4.11 Correlation Coefficients for µ =− 1 2 , 0, 1 2 − 1 2 0 1 2 a 1 1 1.0808 1.1252 a 2 1.4197 1.4111 1.4098 a 3 1.50 1.35 1.30 BOOKCOMP, Inc. — John Wiley & Sons / Page 302 / 2nd Proofs / Heat Transfer Handbook / Bejan 302 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 [302], (42) Lines: 1799 to 1820 ——— 0.32205pt PgVar ——— Normal Page * PgEnds: Eject [302], (42) TABLE 4.12 Coefficients for Correlations of Dimensionless Spreading Resistance 4kaR s Flux Tube Geometry and Contact Boundary Condition C 0 C 1 C 3 C 5 C 7 Circle/circle Uniform flux 1.08076 −1.41042 0.26604 −0.00016 0.058266 True isothermal flux 1.00000 −1.40978 0.34406 0.04305 0.02271 Circle/square Uniform flux 1.08076 −1.24110 0.18210 0.00825 0.038916 Equivalent isothermal flux 1.00000 −1.24142 0.20988 0.02715 0.02768 4.8.2 Accurate Correlation Equations for Various Combinations of Source Areas, Flux Tubes, and Boundary Conditions Solutions are also available for various combinations of source areas and flux tube cross-sectional areas, such as circle/circle and circle/square for the uniform flux, true isothermal, and equivalent isothermal boundary conditions (Negus and Yovanovich, 1984a,b). Numerical results were correlated with the polynomial 4kaR s = C 0 + C 1 + C 3 3 + C 5 5 + C 7 7 (4.104) The dimensionless spreading (constriction) resistance coefficient C 0 is the half- space value, and the correlation coefficients C 1 through C 7 are given in Table 4.12. 4.9 MULTIPLE LAYERS ON A CIRCULAR FLUX TUBE The effect of single and multiple isotropic layers or coatings on the end of a circular flux tube has been determined by Antonetti (1983) and Muzychka et al. (1999). The heat enters the end of the circular flux tube of radius b and thermal conductivity k 3 through a coaxial, circular area that is in perfect thermal contact with an isotropic layer of thermal conductivity k 1 and thickness t 1 . This layer is in perfect contact with a second layer of thermal conductivity k 2 and thickness t 2 , which is in perfect contact contact with the flux tube having thermal conductivity k 3 (Fig. 4.9). The lateral boundary of the flux tube is adiabatic and the contact plane outside the contact area is also adiabatic. The boundary condition over the contact area may be isoflux or isothermal. The system is depicted in Fig. 4.9. The dimensionless constric- tion resistance ψ 2 layers = 4k 3 aR c is defined with respect to the thermal conductivity of the flux tube, which is often referred to as the substrate. This constriction resistance depends on several dimensionless parameters: relative contact size = a/b where 0 < < 1; two conductivity ratios: κ 21 = k 2 /k 1 , κ 32 = k 3 /k 2 ; two relative layer thicknesses: τ 1 = t 1 /a, τ 2 = t 2 /a; and the boundary condition over the contact area. The solution for two layers is given as BOOKCOMP, Inc. — John Wiley & Sons / Page 303 / 2nd Proofs / Heat Transfer Handbook / Bejan MULTIPLE LAYERS ON A CIRCULAR FLUX TUBE 303 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 [303], (43) Lines: 1820 to 1849 ——— 13.65111pt PgVar ——— Normal Page PgEnds: T E X [303], (43) ψ 2 layers = 16 π ∞ n=1 φ n, κ 21 κ 32 ϑ + ϑ − (4.105) where φ n, = J 2 1 (δ n ) δ 3 n J 2 0 (δ n ) ρ n, (4.106) and the boundary condition parameter is according to Muzychka et al. (1999): ρ n, = sin δ n 2J 1 (δ n ) isothermal area 1 isoflux area Figure 4.9 Two layers in a flux tube. (From Muzycha et al., 1999.) BOOKCOMP, Inc. — John Wiley & Sons / Page 304 / 2nd Proofs / Heat Transfer Handbook / Bejan 304 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 [304], (44) Lines: 1849 to 1901 ——— -0.11104pt PgVar ——— Normal Page * PgEnds: Eject [304], (44) The thermal conductivity ratios are defined above. The layer parameters ϑ + and ϑ − come from the following general relationship: ϑ ± = (1 + κ 21 )(1 + κ 32 ) ± (1 − κ 21 )(1 + κ 32 ) exp(−2δ n τ 1 ) + (1 − κ 21 )(1 − κ 32 ) exp(−2δ n τ 2 ) ± (1 + κ 21 )(1 − κ 32 ) exp[−2δ n (τ 1 + τ 2 )] The eigenvalues δ n that appear in the solution are the positive roots of J 1 (·) = 0. The two-layer solution may be used to obtain the relationship for a single layer of thermal conductivity k 1 and thickness t 1 in perfect contact with a flux tube of thermal conductivity k 2 . In this case the dimensionless spreading resistance ψ 1 layer depends on the relative contact size , the conductivity ratio κ 21 , and the relative layer thickness τ 1 : ψ 1 layer = 16 π ∞ n=1 φ n, κ 21 ϑ + ϑ − (4.107) and the general layer relationship becomes ϑ ± = 2[(1 + κ 21 ) ± (1 − κ 21 ) exp(−2δ n τ 1 )] 4.10 SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS Consider the spreading resistance R s and total one-dimensional resistance R 1D for the system shown in Fig. 4.10. The system is a rectangular flux channel −c ≤ x ≤ c, −d ≤ y ≤ d, consisting of two isotropic layers having thermal conductivities k 1 ,k 2 and thicknesses t 1 ,t 2 , respectively. The interface between the layers is assumed to be thermally perfect. All four sides of the flux channel are adiabatic. The planar rectangular heat source area −a ≤ x ≤ a,−b ≤ y ≤ b is subjected to a uniform heat flux q, and the region outside the planar source area is adiabatic. The steady heat transfer rate Q = qA = 4qcd occurs in the system and the heat leaves the system through the lower face z = t 1 +t 2 . The heat is removed by a fluid through a uniform heat transfer coefficient h or by a heat sink characterized by an effective heat transfer coefficient h. The total thermal resistance of the system is given by the relation R total = R s + R 1D (K/W) (4.108) where R s is the thermal spreading resistance of the system and R 1D is the one- dimensional thermal resistance, defined as R 1D = 1 A t 1 k 1 + t 2 k 2 + 1 h where A = 4cd (K/W) (4.109) BOOKCOMP, Inc. — John Wiley & Sons / Page 305 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS 305 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 [305], (45) Lines: 1901 to 1920 ——— -0.20218pt PgVar ——— Normal Page PgEnds: T E X [305], (45) Figure 4.10 Rectangular isoflux area on a compound rectangularchannel. (From Yovanovich et al., 1999.) The spreading resistance is given by the general relationship (Yovanovich et al., 1999) R s = 1 2a 2 cdk 1 ∞ m=1 sin 2 (aδ) δ 3 φ m (δ) + 1 2b 2 cdk 1 ∞ n=1 sin 2 (bλ) λ 3 φ n (λ) + 1 a 2 b 2 cdk 1 ∞ m=1 ∞ n=1 sin 2 (aδ) sin 2 (bλ) δ 2 λ 2 β φ m,n (β) (K/W) (4.110) The general relationship for the spreading resistance consists of three terms. The two single summations account for two-dimensional spreading in the x and y direc- tions, respectively, and the double summation accounts for three-dimensional spread- ing from the rectangular heat source. BOOKCOMP, Inc. — John Wiley & Sons / Page 306 / 2nd Proofs / Heat Transfer Handbook / Bejan 306 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 [306], (46) Lines: 1920 to 2014 ——— 3.08873pt PgVar ——— Short Page PgEnds: T E X [306], (46) The eigenvalues δ m and λ n , corresponding to the two strip solutions, depend on the flux channel dimensions and the indices m and n, respectively. The eigenvalues β m,n for the rectangular solution are functions of the other two eigenvalues and both indices: λ n = nπ d δ m = mπ c β m,n = δ 2 m + λ 2 n (4.111) The contributions of the layer thicknesses t 1 ,t 2 , the layer conductivities k 1 ,k 2 , and the uniform conductance h to the spreading resistance are determined by means of the general expression φ(ζ) = α(κζ L −Bi)e 4ζt 1 + (κζL −Bi)e 2ζt 1 + α(κζL −Bi)e 4ζt 1 − (κζL −Bi)e 2ζt 1 + (κζ L +Bi)e 2ζ(2t 1 +t 2 ) + α(κζL +Bi)e 2ζ(t 1 +t 2 ) (κζL +Bi)e 2ζ(2t 1 +t 2 ) − α(κζL +Bi)e 2ζ(t 1 +t 2 ) where the thermal conductivity ratio parameter is α = 1 − κ 1 + κ with κ = k 2 /k 1 ,Bi= hL/k 1 , and L an arbitrary length scale employed to define the dimensionless spreading resistance: ψ = Lk 1 R s (4.112) which is based on the thermal conductivity of the layer adjacent to the heat source. Various system lengths may be used and the appropriate choice depends onthesystem of interest. In all summations φ(ζ) is evaluated in each series using ζ = δ m , λ n , and β m,n as defined above. The general relationship for φ(ζ) reduces to simpler relationships for two important special cases: the semi-infinite flux channel where t 2 →∞, shown in Fig. 4.11, and the finite isotropic rectangular flux channel where κ = 1, shown in Fig. 4.12. The respective relationships are φ(ζ) = (e 2ζt 1 − 1)κ + (e 2ζt 1 + 1) (e 2ζt 1 + 1)κ + (e 2ζt 1 − 1) (4.113) where the influence of the contact conductance has vanished, and φ(ζ) = (e 2ζt + 1)ζL −(1 − e 2ζt )Bi (e 2ζt − 1)ζL +(1 + e 2ζt )Bi (4.114) where the influence of κ has vanished. BOOKCOMP, Inc. — John Wiley & Sons / Page 307 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS 307 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 [307], (47) Lines: 2014 to 2021 ——— * 17.39099pt PgVar ——— Short Page PgEnds: T E X [307], (47) Figure 4.11 Rectangular isoflux area on a layer bonded to a rectangular flux channel. (From Yovanovich et al., 1999.) The dimensionless spreading resistance ψ depends on six independent dimen- sionless parameters, such as the relative size of the rectangular source area 1 = a/c, 2 = b/d, the layer conductivity ratio κ = k 2 /k 1 , the relative layer thicknesses τ 1 = t 1 /L, τ 2 = t 2 /L, and the Biot number Bi = hL/k 1 . The general relationship reduces to several special cases, such as those described in Table 4.13. The general solution may also be used to obtain the relationship for an isoflux square area on the end of a square semi-infinite flux tube. BOOKCOMP, Inc. — John Wiley & Sons / Page 308 / 2nd Proofs / Heat Transfer Handbook / Bejan 308 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 [308], (48) Lines: 2021 to 2021 ——— 0.14703pt PgVar ——— Normal Page PgEnds: T E X [308], (48) Figure 4.12 Rectangular isoflux area on an isotropic rectangular channel. (From Yovanovich et al., 1999.) TABLE 4.13 Summary of Relationships for Isoflux Area Configuration Limiting Values Rectangular heat source Finite compound rectangular flux channel a, b, c, d, t 1 ,t 2 ,k 1 ,k 2 ,h Semi-infinite compound rectangular flux channel t 2 →∞ Finite isotropic rectangular flux channel k 1 = k 2 Semi-infinite isotropic rectangular flux channel t 1 →∞ Strip heat source Finite compound rectangular flux channel a, c,b = d,t 1 ,t 2 ,k 1 ,k 2 ,h Semi-infinite compound rectangular flux channel t 2 →∞ Finite isotropic rectangular flux channel k 1 = k 2 Semi-infinite isotropic rectangular flux channel t 1 →∞ Rectangular source on a half-space Isotropic half-space c →∞,d →∞,t 1 →∞ Compound half-space c →∞,d →∞,t 2 →∞ BOOKCOMP, Inc. — John Wiley & Sons / Page 309 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE IN COMPOUND RECTANGULAR CHANNELS 309 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 [309], (49) Lines: 2021 to 2064 ——— -2.1598pt PgVar ——— Normal Page PgEnds: T E X [309], (49) 4.10.1 Square Area on a Semi-infinite Square Flux Tube For the special case of a square heat source on a semi-infinite square isotropic flux tube, the general solution reduces to a simpler expression which depends on one parameter only. The dimensionless spreading resistance relationship (Mikic and Rohsenow, 1966; Yovanovich et al., 1999) was recast in the form k A s R s = 2 π 3 ∞ m=1 sin 2 mπ m 3 + 1 π 2 2 ∞ m=1 ∞ n=1 sin 2 mπ sin 2 nπ m 2 n 2 √ m 2 + n 2 (4.115) where the characteristic length was selected as L = √ A s . The relative size of the heat source was defined as = √ A s /A t , where A s and A t are the source and flux tube areas, respectively. A correlation equation was reported for eq. (4.115) (Negus et al., 1989): k A s R s = 0.47320 − 0.62075 + 0.1198 3 (4.116) in 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 the dimensionless spreading resistance of an isoflux square source on an isotropic half- space when the square root of the source area is chosen as the characteristic length. 4.10.2 Spreading Resistance of a Rectangle on a Layer on a Half-Space The solution for the rectangular heat source on a compound half-space is obtained from the general relationship for the finite compound flux channel, provided that t 2 →∞,c→∞,d →∞. No closed-form solution exists for this problem. 4.10.3 Spreading Resistance of a Rectangle on an Isotropic Half-Space The spreading resistance for an isoflux rectangular source of dimensions2a×2b on an isotropic half-space whose thermal conductivity is k has the closed-form relationship (Carslaw and Jaeger, 1959) k A s R s = √ π sinh −1 1 + 1 sinh −1 + 3 1 + 1 3 − 1 + 1 2 3/2 (4.117) where = a/b ≥ 1 is the aspect ratio of the rectangle. If the scale length is L = √ A s , the dimensionless spreading resistance becomes a weak function of .For a square heat source, the numerical value of the dimensionless spreading resistance is k √ A s R s = 0.4732, which is very close to the numerical value for the isoflux circular source on an isotropic half-space and other singly connected heat source geometries such as an equilateral triangle and a semicircular heat source. BOOKCOMP, Inc. — John Wiley & Sons / Page 310 / 2nd Proofs / Heat Transfer Handbook / Bejan 310 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 [310], (50) Lines: 2064 to 2078 ——— 0.08107pt PgVar ——— Short Page PgEnds: T E X [310], (50) 2d 2a 2c h k c q a t Figure 4.13 Strip on a finite rectangular channel with cooling. (From Yovanovich et al., 1999.) 4.11 STRIP ON A FINITE CHANNEL WITH COOLING Spreading resistance due to steady conduction from a strip of width 2a and length L = 2d through a finite rectangular flux channel of width 2c and thickness t and thermal conductivity k is considered here. A uniform conductance h is specified on the bottom surface to account for cooling by a fluid or to represent the cooling by a heat sink. The system is shown in Fig. 4.13. This is a special case of the general relationships presented above for a rectangular area on a compound rectangular channel. A general flux distribution on the strip is given by q(x) = Q L Γ(µ + 3/2) √ πa 1+2µ Γ(µ + 1) (a 2 − x 2 ) µ 0 ≤ x ≤ a (W/m 2 ) (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 distribution . the heat leaves the system through the lower face z = t 1 +t 2 . The heat is removed by a fluid through a uniform heat transfer coefficient h or by a heat sink characterized by an effective heat transfer coefficient. singly connected heat source geometries such as an equilateral triangle and a semicircular heat source. BOOKCOMP, Inc. — John Wiley & Sons / Page 310 / 2nd Proofs / Heat Transfer Handbook / Bejan 310. 303 / 2nd Proofs / Heat Transfer Handbook / Bejan MULTIPLE LAYERS ON A CIRCULAR FLUX TUBE 303 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 [303],