BOOKCOMP, Inc. — John Wiley & Sons / Page 311 / 2nd Proofs / Heat Transfer Handbook / Bejan STRIP ON A FINITE CHANNEL WITH COOLING 311 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 [311], (51) Lines: 2078 to 2128 ——— 2.76997pt PgVar ——— Short Page PgEnds: T E X [311], (51) on the strip, which may have the following values: (1) µ =− 1 2 to approximate an isothermal strip provided that a/c 1, (2) µ = 0 for an isoflux distribution, and (3) µ = 1 2 , which gives a parabolic flux distribution. The three flux distributions are q(x) = Q Lπ 1 a 2 − x 2 for µ =− 1 2 Q 2La for µ = 0 2Q Lπa 2 a 2 − x 2 for µ = 1 2 (4.119) The dimensionless spreading resistance relationship based on the mean source temperature is kLR s = Γ(µ +3/2) π 2 ∞ n=1 2 nπ µ+1/2 sin nπ n 2 J µ+1/2 (nπ)ϕ n (4.120) where ϕ n = nπ +Bi tanh nπτ nπ tanh nπτ + Bi n = 1, 2, 3, and the three dimensionless system parameters and their ranges are 0 < = a c < 10< Bi = hc k < ∞ 0 < τ = t c < ∞ The general relationship gives the following three relationships for the three flux distributions: kLR s = 1 π 2 ∞ n=1 sin nπ n 2 J 0 (nπ)ϕ n for µ =− 1 2 1 2 π 3 ∞ n=1 sin 2 nπ n 3 ϕ n for µ = 0 2 2 π 3 ∞ n=1 sin nπ n 3 J 1 (nπ)ϕ n for µ = 1 2 (4.121) The influence of the cooling along the bottom surface on the spreading resistance is given by the function ϕ n , which depends on two parameters, Bi and τ. If the channel is relatively thick (i.e., τ ≥ 0.85), ϕ n → 1 for all values n = 1 ···∞, and the influence of the parameter Bi becomes negligible. When τ ≥ 0.85, the finite channel may be modeled as though it were infinitely thick. This special case is presented next. BOOKCOMP, Inc. — John Wiley & Sons / Page 312 / 2nd Proofs / Heat Transfer Handbook / Bejan 312 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 [312], (52) Lines: 2128 to 2174 ——— -1.6989pt PgVar ——— Normal Page PgEnds: T E X [312], (52) 4.12 STRIP ON AN INFINITE FLUX CHANNEL If the relative thickness of the rectangular channel becomes very large (i.e., τ →∞), the relationships given above approach the relationships appropriate for the infinitely thick flux channel. The dimensionless spreading resistance for this problem depends on two parameters: the relative size of the strip and the heat flux distribution parameter µ. The three relationships for the dimensionless spreading resistance are obtained from the relationships given above with ϕ n = 1. Numerical values for LkR s for three values of µ are given in Table 4.14. From the tabulated values it can be seen that the spreading resistance values for the isothermal strip are smaller than the values for the isoflux distribution, which are smaller than the values for the parabolic distribution for all values of . The differences are large as → 1; however, the differences become negligibly small as → 0. 4.12.1 True Isothermal Strip on an Infinite Flux Channel There is a closed-form relationship for the true isothermal area on an infinitely thick flux channel. According to Sexl and Burkhard (1969), Veziroglu and Chandra (1969), and Yovanovich et al. (1999), the relationship is kLR s = 1 π ln 1 sin(π/2) (4.122) Numerical values are given in Table 4.14. A comparison of the values corresponding to µ =− 1 2 and those for the true isothermal strip shows close agreement provided that < 0.5. For very narrow strips where < 0.1, the differences are less than 1%. 4.12.2 Spreading Resistance for an Abrupt Change in the Cross Section If steady conduction occurs in a two-dimensional channel whose width decreases from 2a to 2b, there is spreading resistance as heat flows through the common TABLE 4.14 Dimensionless Spreading Resistance kLR s in Flux Channels µ Isothermal − 1 2 0 1 2 Strip Change 0.01 1.321 1.358 1.375 1.322 1.343 0.1 0.5902 0.6263 0.6430 0.5905 0.6110 0.2 0.3729 0.4083 0.4247 0.3738 0.3936 0.3 0.2494 0.2836 0.2995 0.2514 0.2699 0.4 0.1658 0.1984 0.2134 0.1691 0.1860 0.5 0.1053 0.1357 0.1496 0.1103 0.1249 0.6 0.0607 0.0882 0.1007 0.0675 0.0794 0.7 0.0283 0.0521 0.0628 0.0367 0.0456 0.8 0.0066 0.0255 0.0338 0.0160 0.0214 BOOKCOMP, Inc. — John Wiley & Sons / Page 313 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT SPREADING RESISTANCE 313 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 [313], (53) Lines: 2174 to 2219 ——— -5.94008pt PgVar ——— Normal Page * PgEnds: Eject [313], (53) interface. The true boundary condition at the common interface is unknown. The temperature and the heat flux are both nonuniform. Conformal mapping leads to a closed-form solution for the spreading resistance. The relationship for the spreading resistance is, according to Smythe (1968), kLR s = 1 2π + 1 ln 1 + 1 − + 2ln 1 − 2 4 (4.123) where = a/b < 1. Numerical values are given in Table 4.14. An examination of the values reveals that they lie between the values for µ =− 1 2 and µ = 0. The average value of the first two columns corresponding to µ =− 1 2 and µ = 0 are in very close agreement with the values in the last column. The differences are less than 1% for ≤ 0.20, and they become negligible as → 0. 4.13 TRANSIENT SPREADING RESISTANCE WITHIN ISOTROPIC SEMI-INFINITE FLUX TUBES AND CHANNELS Turyk and Yovanovich (1984) reported the analytical solutions for transient spreading resistance within semi-infinite circular flux tubes and two-dimensional channels. The circular contact and the rectangular strip are subjected to uniform and constant heat flux. 4.13.1 Isotropic Flux Tube The dimensionless transient spreading resistance for an isoflux circular source of radius a supplying heat to a semi-infinite isotropic flux tube of radius b, constant thermal conductivity k, and thermal diffusivity α is given by the series solution 4kaR s = 16 π ∞ n=1 J 2 1 (δ n ) erf(δ n √ Fo) δ 3 n J 2 0 (δ n ) (4.124) where = a/b < 1, Fo = αt/a 2 > 0, and δ n are the positive roots of J 1 (·) = 0. The average source temperature rise was used to define the spreading resistance. The series solution approaches the steady-state solution presented in an earlier section when the dimensionless time satisfies the criterion Fo ≥ 1/ 2 or when the real time satisfies the criterion t ≥ a 2 /α 2 . 4.13.2 Isotropic Semi-infinite Two-Dimensional Channel The dimensionless transient spreading resistance for an isoflux strip of width 2a within a two-dimensional channel of width 2b, length L, constant thermal conduc- tivity k, and thermal diffusivity α was reported as (Turyk and Yovanovich, 1984) LkR s = 1 π 3 ∞ m=1 sin 2 mπ erf mπ √ Fo m 3 (4.125) BOOKCOMP, Inc. — John Wiley & Sons / Page 314 / 2nd Proofs / Heat Transfer Handbook / Bejan 314 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 [314], (54) Lines: 2219 to 2243 ——— 0.25099pt PgVar ——— Long Page PgEnds: T E X [314], (54) where = a/b < 1 is the relative size of the contact strip and the dimensionless time is defined as Fo = αt/a 2 . There is no half-space solution for the two-dimensional channel. The transient solution is within 1% of the steady-state solution when the dimensionless time satisfies the criterion Fo ≥ 1.46/ 2 . 4.14 SPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA ON A RECTANGULAR PLATE WITH COOLING A rectangular isoflux area with side lengths c and d lies in the surface z = 0of a rectangular plate with side dimensions a and b. The plate thickness is t 1 and its thermal conductivity is k 1 . The top surface outside the source area is adiabatic, and all sides are adiabatic. The bottom surface at z = t 1 is cooled by a fluid or a heat sink that is in contact with the entire surface. In either case the heat transfer coefficient is denoted as h and is assumed to be uniform. The origin of the Cartesian coordinate system (x,y,z) is located in the lower left corner. The system is shown in Fig. 4.14. Figure 4.14 Isotropic plate with an eccentricrectangular heat source. (From Muzychka et al., 2000.) BOOKCOMP, Inc. — John Wiley & Sons / Page 315 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA 315 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 [315], (55) Lines: 2243 to 2296 ——— 1.92044pt PgVar ——— Long Page * PgEnds: Eject [315], (55) The temperature rise of points in the plate surface z = 0 is given by the relationship θ(x,y,z) = A 0 + B 0 z + ∞ m=1 cos λx(A 1 cosh λz + B 1 sinh λz) + ∞ n=1 cos δy(A 2 cosh δz + B 2 sinh δz) + ∞ m=1 ∞ n=1 cos λx cos δy(A 3 cosh βz + B 3 sinh βz) (4.126) The Fourier coefficients are obtained by means of the following relationships: A 0 = Q ab t 1 k 1 + 1 h and B 0 =− Q k 1 ab (4.127) A 1 = 2Q sin 2X c +c 2 λ m − sin 2X c −c 2 λ m abck 1 λ 2 m φ(λ m ) (4.128) A 2 = 2Q sin 2Y c +d 2 δ n − sin 2Y c −d 2 δ n abck 1 δ 2 n φ(δ n ) (4.129) A 3 = 16Q cos(λ m X c ) sin 1 2 λ m c cos(δ n Y c ) sin 1 2 δ n d abcdk 1 β m,n λ m δ n φ(β m,n ) (4.130) The other Fourier coefficients are obtained by the relationship B i =−φ(ζ)A i i = 1, 2, 3 (4.131) where ζ is replaced by λ m , δ n ,orβ m,n as required. The eigenvalues are λ m = mπ a δ n = nπ b β m,n = λ 2 m + δ 2 n The mean temperature rise of the source area is given by the relationship θ = θ 1D + 2 ∞ m=1 A m cos(λ m X c ) sin 1 2 λ m c λ m c + 2 ∞ n=1 A n cos(δ n Y c ) sin 1 2 δ n d δ n d + 4 ∞ m=1 ∞ n=1 A mn cos(δ n Y c ) sin 1 2 δ n d cos(λ m X c ) sin 1 2 λ m c λ m cδ n d (4.132) where the one-dimensional temperature rise is BOOKCOMP, Inc. — John Wiley & Sons / Page 316 / 2nd Proofs / Heat Transfer Handbook / Bejan 316 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 [316], (56) Lines: 2296 to 2339 ——— 9.33221pt PgVar ——— Normal Page * PgEnds: Eject [316], (56) θ 1D = Q ab t 1 k 1 + 1 h (K) (4.133) for an isotropic plate. The total resistance is related to the spreading resistance and the one-dimensional resistance: R total = θ Q = R 1D + R s (K/W) (4.134) where R 1D = 1 ab t 1 k 1 + 1 h (K/W) (4.135) 4.14.1 Single Eccentric Area on a Compound Rectangular Plate If a single source is on the top surface of a compound rectangular plate that consists of two layers having thicknesses t 1 and t 2 and thermal conductivities k 1 and k 2 ,as shown in Fig. 4.15, the results are identical except for the system parameter φ, which now is given by the relationship φ(ζ) = (αe 4ζt 1 − e 2ζt 1 ) +(e 2ζ(2t 1 +t 2 ) − αe 2ζ(t 1 +t 2 ) ) (αe 4ζt 1 + e 2ζt 1 ) +(e 2ζ(2t 1 +t 2 ) + αe 2ζ(t 1 +t 2 ) ) (4.136) where = ζ +h/k 2 ζ −h/k 2 and α = 1 −κ 1 +κ (4.137) with κ = k 2 /k 1 and ζ is replaced by λ m , δ n ,orβ m,n , accordingly. The one-dimensional temperature rise in this case is Figure 4.15 Compound plate with an eccentric rectangular heat source. (From Muzychka et al., 2000.) BOOKCOMP, Inc. — John Wiley & Sons / Page 317 / 2nd Proofs / Heat Transfer Handbook / Bejan SPREADING RESISTANCE OF AN ECCENTRIC RECTANGULAR AREA 317 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 [317], (57) Lines: 2339 to 2361 ——— 11.88896pt PgVar ——— Normal Page * PgEnds: Eject [317], (57) θ 1D = Q ab t 1 k 1 + t 2 k 2 + 1 h (K) (4.138) 4.14.2 Multiple Rectangular Heat Sources on an Isotropic Plate The multiple rectangular sources on an isotropic plate are shown in Fig. 4.16. The sur- face temperature by superposition is given by the following relationship (Muzychka et al., 2000): T (x,y,0) − T f = N i=1 θ i (x,y,0) (K) (4.139) where θ i is the temperature excess for each heat source by itself and N ≥ 2 is the number of discrete heat sources. The temperature rise is given by Figure 4.16 Isotropic plate with two eccentric rectangular heat sources. (From Muzychka et al., 2000.) BOOKCOMP, Inc. — John Wiley & Sons / Page 318 / 2nd Proofs / Heat Transfer Handbook / Bejan 318 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 [318], (58) Lines: 2361 to 2411 ——— 2.45174pt PgVar ——— Normal Page PgEnds: T E X [318], (58) θ i (x,y,0) = A i 0 + ∞ m=1 A i m cos λx + ∞ n=1 A i n cos δy + ∞ m=1 ∞ n=1 A i mn cos λx cos δy (4.140) where φ and A i 0 = θ 1D are defined above for the isotropic and compound plates. The mean temperature of an arbitrary rectangular area of dimensions c j and d j , located at X c,j and Y c,j , may be obtained by integrating over the region A j = c j d j : θ j = 1 A j A j θ i dA j = 1 A j A j N i=1 θ i (x,y,0)dA j (4.141) which may be written as θ j = N i=1 1 A j A j θ i (x,y,0)dA j = N i=1 θ i (4.142) The mean temperature of the jth heat source is given by T j − T f = N i=1 θ i (4.143) where θ i = A i o + 2 ∞ m=1 A i m cos(λ m X c,j ) sin 1 2 λ m c j λ m c j + 2 ∞ n=1 A i n cos(δ n Y c,j ) sin 1 2 δ n d j δ n d j + 4 ∞ m=1 ∞ n=1 A i mn cos(δ n Y c,j ) sin 1 2 δ n d j cos(λ m X c,j ) sin 1 2 λ m c j λ m c j δ n d j (4.144) Equation (4.143) represents the sum of the effects of all sources over an arbitrary location. Equation (4.143) is evaluated over the region of interest c j ,d j located at X c,j , Y c,j , with the coefficients A i 0 ,A i m ,A i n , and A i mn evaluated at each of the ith source parameters. 4.15 JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS The elastoconstriction and elastogap resistance models (Yovanovich, 1986) are based on the Boussinesq point load model (Timoshenko and Goodier, 1970) and the Hertz BOOKCOMP, Inc. — John Wiley & Sons / Page 319 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 319 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 [319], (59) Lines: 2411 to 2421 ——— 0.59099pt PgVar ——— Normal Page * PgEnds: Eject [319], (59) 2a D 2 0 ␦ r E 22 , v E 11 , v z Figure 4.17 Joint formed by elastic contact of a sphere or a cylinder with a smooth flat surface. (From Kitscha, 1982.) distributed-load model (Hertz, 1896; Timoshenko and Goodier, 1970; Walowit and Anno, 1975; Johnson, 1985). Both models assume that bodies have “smooth” sur- faces, are perfectly elastic, and that the applied load is static and normal to the plane of the contact area. In the general case the contact area will be elliptical, having semimajor and semiminor axes a and b, respectively. These dimensions are much smaller than the dimensions of the contacting bodies. The circular contact area pro- duced when two spheres or a sphere and a flat are in contact are two special cases of the elliptical contact. Also, the rectangular contact area, produced when two ideal circular cylinders are in line contact or an ideal cylinder and a flat are in contact, are special cases of the elliptical contact area. Figure 4.17 shows the contact between two elastic bodies having physical proper- ties (Young’s modulus and Poisson’s ratio): E 1 , ν 1 and E 2 , ν 2 , respectively. One body is a smooth flat and the other body may be a sphere or a circular cylinder having ra- dius D/2. The contact 2a is the diameter of a circular contact area for the sphere/flat contact and the width of the contact strip for the cylinder/flat contact. A gap is formed adjacent to the contact area, and its local thickness is characterized by δ. Heat transfer across the joint can take place by conduction by means of the contact area, conduction through the substance in the gap, and by radiation across the gap if the substance is “transparent,” or by radiation if the contact is formed in a vacuum. The thermal joint resistance model presented below was given by Yovanovich (1971, 1986). It was developed for the elastic contact of paraboloids (i.e., the elastic contact formed by a ball and the inner and outer races of an instrument bearing). 4.15.1 Point Contact Model Semiaxes of an Elliptical Contact Area The general shape of the contact area is an ellipse with semiaxes a and b and area A = πab. The semiaxes are given by the relationships (Timoshenko and Goodier, 1970) BOOKCOMP, Inc. — John Wiley & Sons / Page 320 / 2nd Proofs / Heat Transfer Handbook / Bejan 320 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 [320], (60) Lines: 2421 to 2472 ——— 0.9983pt PgVar ——— Normal Page * PgEnds: Eject [320], (60) a = m 3F∆ 2(A + B) 1/3 and b = n 3F∆ 2(A + B) 1/3 (4.145) where F is the total normal load acting on the contact area, and ∆ is a physical parameter defined by ∆ = 1 2 1 −ν 2 1 E 1 + 1 −ν 2 2 E 2 (m 2 /N) (4.146) when dissimilar materials form the contact. The physical parameters are Young’s modulus E 1 and E 2 and Poisson’s ratio ν 1 and ν 2 . The geometric parameters A and B are related to the radii of curvature of the two contacting solids (Timoshenko and Goodier, 1970): 2(A + B) = 1 ρ 1 + 1 ρ 1 + 1 ρ 2 + 1 ρ 2 = 1 ρ ∗ (4.147) where the local radii of curvature of the contacting solids are denoted as ρ 1 , ρ 1 , ρ 2 , and ρ 2 . The second relationship between A and B is 2(B − A) = 1 ρ 1 − 1 ρ 1 2 + 1 ρ 2 − 1 ρ 2 2 + 2 1 ρ 1 − 1 ρ 1 1 ρ 2 − 1 ρ 2 cos 2φ 1/2 (4.148) The parameter φ is the angle between the principal planes that pass through the contacting solids. The dimensionless parameters m and n that appear in the equations for the semi- axes are called the Hertz elastic parameters. They are determined by means of the following Hertz relationships (Timoshenko and Goodier, 1970): m = 2 π E k k 2 1/3 and n = 2 π kE k 1/3 (4.149) where E k is the complete elliptic integral of the second kind of modulus k (Abramowitz and Stegun, 1965; Byrd and Friedman, 1971), and k = 1 −k 2 with k = n m = b a ≤ 1 (4.150) The additional parameters k and k are solutions of the transcendental equation (Tim- oshenko and Goodier, 1970): . / 2nd Proofs / Heat Transfer Handbook / Bejan STRIP ON A FINITE CHANNEL WITH COOLING 311 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 [311],. 2nd Proofs / Heat Transfer Handbook / Bejan 312 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 [312],. 0.0521 0.0628 0.0367 0.0456 0.8 0.0066 0.0255 0. 0338 0.0160 0.0214 BOOKCOMP, Inc. — John Wiley & Sons / Page 313 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT SPREADING RESISTANCE 313 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 [313],