BOOKCOMP, Inc. — John Wiley & Sons / Page 291 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE WITHIN A COMPOUND DISK WITH CONDUCTANCE 291 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 [291], (31) Lines: 1397 to 1424 ——— 0.0pt PgVar ——— Long Page * PgEnds: Eject [291], (31) For all values in the range 0 < Bi < ∞ and for all values τ > 0.72, tanh δ n τ ≈ 1 for all n ≥ 1. Therefore, φ n ≈ 1 for values n ≥ 1. Characteristics of B n When τ 1 > 0.72, tanh δ n τ = 1, φ n = 1 for all 0 < Bi < ∞; therefore, B n = 1forn ≥ 1. 4.6.1 Special Cases of the Compound Disk Solution The general solution for the compound disk may be used to obtain spreading re- sistances for several special cases examined previously by many researchers. These special cases arise when some of the system parameters go to certain limits. The spe- cial cases fall into the following two categories: isotropic half-space, semi-infinite flux tube, and finite disk problems; and layered half-space and semi-infinite flux tube problems. Figures 4.5 and 4.6 show the several special cases that arise from the gen- eral case presented above. Results for several special cases are discussed in more detail in subsequent sections. 4.6.2 Half-Space Problems If the dimensions of the compound disk (b, t) become very large relative to the radius a and the first layer thickness t 1 , the general solution approaches the solution for the case of a single layer in perfect thermal contact with an isotropic half-space. In this case → 0, τ 1 → 0 and the spreading resistance depends on the four system parameters (a, t 1 ,k 1 ,k 2 ) and the heat flux parameter µ. If we set t 1 = 0ork 1 = k 2 , the general solution goes to the special case of a circular heat source in perfect contact with an isotropic half-space. In this case the spreading resistance depends on two system parameters (a, k 2 ) and the heat flux parameter µ. The dimensionless spreading resistance is now defined as ψ = 4k 2 aR s , and it is a constant depending on the heat flux parameter. The total resistance is equal to the spreading resistance in both cases because the one-dimensional resistance is negligible. The half-space problems are shown in Figs. 4.5d and 4.6d. 4.6.3 Semi-infinite Flux Tube Problems The general solution goes to the semi-infinite flux tube solutions when the system parameter τ 2 →∞. In this case the spreading resistance will depend on the system parameters (a,b, t 1 ,k 1 ,k 2 ) and the heat flux parameter µ. The dimensionless spread- ing resistance will be a function of the parameters (, τ 1 , κ) and µ. If one sets t 1 = 0 or k 1 = k 2 = k, the dimensionless spreading resistance ψ = 4kaR s depends on the system parameters (a, b, k) and µ or = a/b and µ only. The semi-infinite flux tube problems are shown in Figs. 4.5c and 4.6c. 4.6.4 Isotropic Finite Disk with Conductance In this case, one puts k 1 = k 2 = k or κ = 1. The dimensionless spreading re- sistance ψ = 4kaR s depends on the system parameters (a, b,t, k, h) and µ or the BOOKCOMP, Inc. — John Wiley & Sons / Page 292 / 2nd Proofs / Heat Transfer Handbook / Bejan 292 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 [292], (32) Lines: 1424 to 1433 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [292], (32) q q q q q a a a a a b b b k 1 k 1 k 1 k 1 k 1 t 1 t 1 t 2 t 1 t 1 t 1 k 2 k 2 k 2 k 2 k 2 ( )0< <a ϱ ( ) —0, — ,0< <c ϱϱ ( ) —0, — ,0< <d ϱϱ( ) Bi— , — ,0< <b ϱ ϱϱ b T =0 T =0 h z =0 Ϫ= 1/2 t r q =/kk 12 Bi = /hb k 1 =/tb 1 =/ab =/tb Figure 4.5 Four problems with a single layer on a substrate. (From Yovanovich et al., 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 293 / 2nd Proofs / Heat Transfer Handbook / Bejan 293 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 [293], (33) Lines: 1433 to 1440 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [293], (33) q q q q q a a a a a b b b k 1 k 1 k 1 k 1 k 1 t 1 t 2 k 2 () =1a () =1, —c ϱ () — 0, = 1, —d ϱ( ) =1,Bi—b ϱ b T =0 T =0 h z =0 Ϫ= 1/2 t r q =/kk 12 Bi = /hb k 1 1 1 =/tb =/ab Figure 4.6 Four problems for isotropic systems. (From Yovanovich et al., 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 294 / 2nd Proofs / Heat Transfer Handbook / Bejan 294 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 [294], (34) Lines: 1440 to 1476 ——— 5.87788pt PgVar ——— Long Page PgEnds: T E X [294], (34) dimensionless system parameters ( = a/b, τ = t/b,Bi = hb/ k) and µ. This problem is shown in Fig. 4.6b. These special cases are presented in the following sections. 4.7 SPREADING RESISTANCE OF ISOTROPIC FINITE DISKS WITH CONDUCTANCE The dimensionless spreading resistance for isotropic (κ = 1) finite disks (τ 1 < 0.72) with negligible thermal resistance at the heat sink interface (Bi =∞) is given by the following solutions (Kennedy, 1960; Mikic and Rohsenow, 1966; Yovanovich et al., 1998): For µ =− 1 2 : 4kaR s = 8 π ∞ n=1 J 1 (δ n ) sin δ n δ 3 n J 2 0 (δ n ) tanh δ n τ (4.81) For µ = 0: 4kaR s = 16 π ∞ n=1 J 2 1 (δ n ) δ 3 n J 2 0 (δ n ) tanh δ n τ (4.82) If the external resistance is negligible Bi →∞, the temperature at the lower face of the disk is isothermal. The solutions for isoflux µ = 0 heat source and isothermal base temperature were given by Kennedy (1960) for the centroid temperature and the area-averaged contact area temperature. 4.7.1 Correlation Equations A circular heat source of radius a is attached to one end of a circular disk of thickness t, radius b, and thermal conductivity k. The opposite boundary is cooled by a fluid at temperature T f through a uniform heat transfer coefficient h. The sides of the disk are adiabatic and the region outside the source area is also adiabatic. The flux over the source area is uniform. The heat transfer through the disk is steady. The external resistance is defined as R ext = 1/hA, where A = πb 2 . The solution for the isoflux boundary condition and with external thermal resis- tance was recently reexamined by Song et al. (1994) and Lee et al. (1995). They nondimensionalized the constriction resistance based on the centroid and area- averaged temperatures using the square root of the source area (as recommended by Yovanovich, 1976b, 1991, 1997; Yovanovich and Burde, 1977; Yovanovich and Schneider, 1977; Chow and Yovanovich, 1982; Yovanovich et al., 1984; Yovanovich and Antonetti, 1988) and compared the analytical results against the numerical results reported by Nelson and Sayers (1992) over the full range of independent parameters: Bi = hb/k, = a/b,τ = t/b. Nelson and Sayers (1992) also chose the square root of the source area to report their numerical results. The agreement between the analytical and numerical results were reported to be in excellent agreement. BOOKCOMP, Inc. — John Wiley & Sons / Page 295 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF ISOTROPIC FINITE DISKS WITH CONDUCTANCE 295 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 [295], (35) Lines: 1476 to 1525 ——— -2.5917pt PgVar ——— Long Page PgEnds: T E X [295], (35) Lee et al. (1995) recommended a simple closed-form expression for the dimen- sionless constriction resistance based on the area-averaged and centroid temperatures. They defined the dimensionless spreading resistance parameter as ψ = √ π kaR c , where R c is the constriction resistance, and they recommended the following approx- imations: For the area-averaged temperature ψ ave = 1 2 (1 − ) 3/2 φ c (4.83) and for the centroid temperature: ψ max = 1 √ π (1 − )ϕ c (4.84) with ϕ c = Bi tanh (δ c τ) + δ c Bi + δ c tanh δ c τ (4.85) δ c = π + 1 √ π (4.86) The approximations above are within ±10% of the analytical results (Song et al., 1994; Lee et al., 1995) and the numerical results of Nelson and Sayers (1992). The locations of the maximum errors were not given. 4.7.2 Circular Area on a Single Layer (Coating) on a Half-Space Integral solutions are available for the spreading resistance for a circular source of radius a in contact with an isotropic layer of thickness t 1 and thermal conductivity k 1 which is in perfect thermal contact with an isotropic half-space of thermal conduc- tivity k 2 . The solutions were obtained for two heat flux distributions corresponding to the flux parameter values µ =− 1 2 and µ = 0. Equivalent Isothermal Circular Contact Dryden (1983) obtained the solution for the equivalent isothermal circular contact flux distribution: q(r) = Q 2πa 2 √ 1 − u 2 0 ≤ u ≤ 1 (W/m 2 ) (4.87) The problem is depicted in Fig. 4.7. The dimensionless spreading resistance, based on the area-averaged temperature, is obtained from the integral (Dryden, 1983): ψ = 4k 2 aR s = 4 π k 2 k 1 ∞ 0 λ 2 exp(ζt 1 /a) + λ 1 exp(−ζt 1 /a) λ 2 exp(ζt 1 /a) − λ 1 exp(−ζt 1 /a) J 1 (ζ) sin ζ ζ 2 dζ (4.88) with λ 1 = (1 −k 2 /k 1 )/2 and λ 2 = (1 +k 2 /k 1 )/2. The parameter ζ is a dummy vari- able of integration. The constriction resistance depends on the thermal conductivity BOOKCOMP, Inc. — John Wiley & Sons / Page 296 / 2nd Proofs / Heat Transfer Handbook / Bejan 296 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 [296], (36) Lines: 1525 to 1552 ——— -2.94893pt PgVar ——— Normal Page * PgEnds: Eject [296], (36) Figure 4.7 Layered half-space with an equivalent isothermal flux. (From Yovanovich et al., 1998.) ratio k 1 /k 2 and the relative layer thickness t 1 /a. Dryden (1983) presented simple asymptotes for thermal spreading in thin layers, t 1 /a ≤ 0.1, and in thick layers, t 1 /a ≥ 10. These asymptotes were also presented as dimensionless spreading resis- tances defined as 4k 2 aR s . They are: Thin-layer asymptote: (4k 2 aR s ) thin = 1 + 4 π t 1 a k 2 k 1 − k 1 k 2 (4.89) Thick-layer asymptote: (4k 2 aR s ) thick = k 2 k 1 − 2 π a t 1 k 2 k 1 ln 2 1 + k 1 /k 2 (4.90) These asymptotes provide results that are within 1% of the full solution for relative layer thickness: t 1 /a < 0.5 and t 1 /a > 2. The dimensionless spreading resistance is based on the substrate thermal conduc- tivity k 2 . The general solution above is valid for conductive layers where k 1 /k 2 > 1 as well as for resistive layers where k 1 /k 2 < 1. The infinite integral can be evaluated numerically by means of computer algebra systems, which provide accurate results. 4.7.3 Isoflux Circular Contact Hui and Tan (1994) presented an integral solution for the isoflux circular source. The dimensionless spreading resistance is 4k 2 aR s = 32 3π 2 k 2 k 1 2 + 8 π 1 − k 2 k 1 2 ∞ 0 J 2 1 (ζ)dζ [1 + (k 1 /k 2 ) tanh (ζt 1 /a)]ζ 2 (4.91) BOOKCOMP, Inc. — John Wiley & Sons / Page 297 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF ISOTROPIC FINITE DISKS WITH CONDUCTANCE 297 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 [297], (37) Lines: 1552 to 1602 ——— -2.6009pt PgVar ——— Normal Page * PgEnds: Eject [297], (37) which depends on the thermal conductivity ratio k 1 /k 2 and the relative layer thick- ness t 1 /a. The dimensionless spreading resistance is based on the substrate thermal conductivity k 2 . The general solution above is valid for conductive layers, where k 1 /k 2 > 1, as well as resistive layers, where k 1 /k 2 < 1. 4.7.4 Isoflux, Equivalent Isothermal, and Isothermal Solutions Negus et al. (1985) obtained solutions by application of the Hankel transform method for flux-specified boundary conditions and with a novel technique of linear superpo- sition for the mixed boundary condition (isothermal contact area and zero flux outside the contact area). They reported results for three flux distributions: isoflux, equivalent isothermal flux, and true isothermal source. There results were presented below. Isoflux Contact Area For the isoflux boundary condition, they reported the result for ψ q = 4k 1 aR s : ψ q = 32 3π 2 + 8 π 2 ∞ n=1 (−1) n α n I q (4.92) The first term is the dimensionless isoflux spreading resistance of an isotropic half- space of thermal conductivity k 1 , and the second term accounts for the effect of the layer relative thickness and relative thermal conductivity. The thermal conductivity parameter α is defined as α = 1 − κ 1 + κ with κ = k 1 /k 2 . The layer thickness–conductivity parameter I q is defined as I q = 1 2π 2 2(γ + 1)E 2/(γ + 1) − π 2 √ 2γ I γ − 2πnτ 1 with I γ = 1 + 0.09375 γ 2 + 0.0341797 γ 4 + 0.00320435 γ 6 The relative layer thickness is τ 1 = t/a and the relative thickness parameter is γ = 2n 2 τ 2 1 + 1 The special function E(·) is the complete elliptic integral of the second kind (Abramowitz and Stegun, 1965). Equivalent Isothermal Contact Area For the equivalent isothermal flux boundary condition, they reported the result for ψ ei = 4k 1 aR s : BOOKCOMP, Inc. — John Wiley & Sons / Page 298 / 2nd Proofs / Heat Transfer Handbook / Bejan 298 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 [298], (38) Lines: 1602 to 1654 ——— -0.16101pt PgVar ——— Normal Page PgEnds: T E X [298], (38) ψ ei = 1 + 8 π ∞ n=1 (−1) n α n I ei (4.93) where as discussed above, the first term represents the dimensionless spreading resis- tance of an isothermal contact area on an isotropic half-space of thermal conductivity k 1 and the second term accounts for the effect of the layer relative thickness and the relative thermal conductivity. The thermal conductivity parameter α is defined above. The relative layer thickness parameter I ei is defined as I ei = 1 − β −2 β − β −1 + 1 2 sin −1 β −1 − 2nτ 1 with τ 1 = t/a and β = nτ 1 + n 2 τ 2 1 + 1 Isothermal Contact Area For the isothermal contact area, Negus et al. (1985) reported a correlation equation for their numerical results. They reported that ψ T = 4k 1 aR s in the form ψ T = F 1 tanh F 2 + F 3 (4.94) where F 1 = 0.49472 − 0.49236κ −0.0034κ 2 F 2 = 2.8479 + 1.3337τ + 0.06864τ 2 with τ = log 10 τ 1 F 3 = 0.49300 + 0.57312κ − 0.06628κ 2 where κ = k 1 /k 2 . The correlation equation was developed for resistive layers: 0.01 ≤ κ ≤ 1 over a wide range of relative thickness 0.01 ≤ τ 1 ≤ 100. The maximum relative error associated with the correlation equation is approximately 2.6% at τ 1 = 0.01 and κ = 0.2. Numerical results for ψ q , ψ ei , and ψ T for a range of values of τ 1 and κ were presented in tabular form for easy comparison. They found that the values for ψ q were greater than those for ψ ei and that ψ ei ≤ ψ T . The maximum difference between ψ q and ψ T was approximately 8%. The values for ψ T > ψ ei for very thin layers, τ 1 ≤ 0.1 and for κ ≤ 0.1; however, the differences were less than approximately 8%. For most applications the equivalent isothermal flux solution and the true isothermal solution are simililar. 4.8 CIRCULAR AREA ON A SEMI-INFINITE FLUX TUBE The problem of finding the spreading resistance in an semi-infinite isotropic cir- cular flux tube has been investigated by many researchers (Roess, 1950; Mikic and BOOKCOMP, Inc. — John Wiley & Sons / Page 299 / 2nd Proofs / Heat Transfer Handbook / Bejan CIRCULAR AREA ON A SEMI-INFINITE FLUX TUBE 299 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 [299], (39) Lines: 1654 to 1670 ——— 0.98595pt PgVar ——— Normal Page PgEnds: T E X [299], (39) q a k 1 b t kk 12 /=1 ϱ=/ —tb = / 0.9ab≤ Figure 4.8 Isotropic flux tube with an isoflux area. (From Yovanovich et al., 1998.) Rohsenow, 1966; Gibson, 1976; Yovanovich, 1976a,b; Negus and Yovanovich, 1984a, b; Negus et al., 1989). The system with uniform heat flux on the circular area is shown in Fig. 4.8. This problem corresponds to the case where κ = 1 and τ →∞, and therefore the spreading resistance depends on the system parameters (a,b,k) and the flux distribution parameter µ. The dimensionless spreading resistance defined as ψ = 4kaR s , where R s is the spreading resistance, depends on and µ. The results of several studies are given below. 4.8.1 General Expression for a Circular Contact Area with Arbitrary Flux on a Circular Flux Tube The general expression for the dimensionless spreading (constriction) resistance 4kaR s for a circular contact subjected to an arbitrary axisymmetric flux distribution f (u) (Yovanovich, 1976b) is obtained from the series 4kaR s = 8/π 1 0 uf (u) du ∞ n=1 J 1 (δ n ) δ 2 n J 2 0 (δ n ) 1 0 uf (u)J 0 (δ n u) du (4.95) where δ n are the positive roots of J 1 (·) = 0 and = a/b is the relative size of the source area. BOOKCOMP, Inc. — John Wiley & Sons / Page 300 / 2nd Proofs / Heat Transfer Handbook / Bejan 300 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 [300], (40) Lines: 1670 to 1727 ——— -4.44421pt PgVar ——— Normal Page PgEnds: T E X [300], (40) Flux Distributions of the Form (1 − u 2 ) µ Yovanovich (1976b) reported the following general solution for axisymmetric flux distributions of the form f (u) = (1−u 2 ) µ , where the parameter µ accounts for the shape of the flux distribution. The general expression above reduces to the following general expression: 4kaR s = 16 π (µ + 1)2 µ Γ(µ + 1) 1 ∞ n=1 J 1 (δ n )J µ+1 (δ n ) δ 3 n J 0 (δ n )(δ n ) µ (4.96) where Γ(·) is the gamma function (Abramowitz and Stegun, 1965) and J ν (·) is the Bessel function of arbitrary order ν (Abramowitz and Stegun, 1965). The general expression above can be used to obtain specific solutions for various values of the flux distribution parameter µ. Three particular solutions are considered next. Equivalent Isothermal Circular Source The isothermal contact area requires solution of a difficult mathematical problem that has received much attention by numerous researchers (Roess, 1950; Kennedy, 1960; Mikic and Rohsenow, 1966; Gibson, 1976; Yovanovich, 1976b; Negus and Yovanovich, 1984a,b). Mikic and Rohsenow (1966) proposed use of the flux distribution corresponding to µ =− 1 2 to approximate an isothermal contact area for small relative contact areas 0 < < 0.5. The general expression becomes 4kaR s = 8 π 1 ∞ n=1 J 1 (δ n ) sin δ n δ 3 n J 2 0 (δ n ) (4.97) An accurate correlation equation of this series solution is given below. Isoflux Circular Source The general solution above with µ = 0 yields the isoflux solution reported by Mikic and Rohsenow (1966): 4kaR s = 16 π 1 ∞ n=1 J 2 1 (δ n ) δ 3 n J 2 0 (δ n ) (4.98) An accurate correlation equation of this series solution is given below. Parabolic Flux Distribution Yovanovich (1976b) reported the solution for the parabolic flux distribution corresponding to µ = 1 2 . 4kaR s = 24 π 1 ∞ n=1 J 1 (δ n ) sin δ n δ 3 n J 2 0 (δ n ) 1 (δ n ) 2 − 1 δ n tan δ n (4.99) An accurate correlation equation of this series solution is given below. Asymptotic Values for Dimensionless Spreading Resistances The three series solutions given above converge very slowly as → 0, which corresponds to the . Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE WITHIN A COMPOUND DISK WITH CONDUCTANCE 291 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 [291],. & Sons / Page 292 / 2nd Proofs / Heat Transfer Handbook / Bejan 292 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 [292],. & Sons / Page 293 / 2nd Proofs / Heat Transfer Handbook / Bejan 293 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 [293],