BOOKCOMP, Inc. — John Wiley & Sons / Page 201 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTENDED SURFACES 201 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 [201], (41) Lines: 2014 to 2081 ——— -2.41492pt PgVar ——— Normal Page PgEnds: T E X [201], (41) 3.5.5 Radiative–Convective Cooling of Solids with Uniform Energy Generation The solutions obtained in Section 3.4.8 for a plane wall (the thermal symmetry case), a solid cylinder, and a solid sphere are now extended to accommodate surface cooling by simultaneous convection and radiation. The surface energy balance for each geometry gives h(T s − T ∞ ) + σ T 4 s − T 4 ∞ + k dT dx x=L = 0 (plane wall) (3.170) h(T s − T ∞ ) + σ T 4 s − T 4 ∞ + k dT dr r=r 0 = 0 (solid cylinder and sphere) (3.171) where is the surface emissivity, σ the Stefan–Boltzmann constant, and T ∞ represents the surrounding or ambient temperature for both convection and radiation. In eqs. (3.170) and (3.171), the last terms can be evaluated using eqs. (3.105), (3.115), and (3.123), respectively. Because eqs. (3.170) and (3.171) require a numerical approach for their solutions, it is convenient to recast them in dimensionless form as N 1 (θ s − 1) +N 2 θ 4 s − 1 + dθ dX X=1 = 0 (plane wall) (3.173) N 1 (θ s − 1) +N 2 θ 4 s − 1 + dθ dR R=1 = 0 (cylinder and sphere) (3.173) where θ = T s /T ∞ ,N 1 = hL/k for the plane wall, N 1 = hr 0 /k for the cylinder and sphere, N 2 = σT 3 ∞ L/k for the plane wall, and N 2 = σT 3 ∞ r 0 /k for the cylinder and sphere, X = x/L,R = r/r 0 , and θ = T/T ∞ . The numerical values for θ s are given in Table 3.10 for a range of values of N 1 and N 2 and ˙qL 2 /kT ∞ = 1 for the plane wall and ˙qr 2 0 /kT ∞ = 1 for the cylinder and sphere. 3.6 EXTENDED SURFACES The term extended surface is used to describe a system in which the area of a surface is increased by the attachment of fins. A fin accommodates energy transfer by conduc- tion within its boundaries, while its exposed surfaces transfer energy to the surround- ings by convection or radiation or both. Fins are commonly used to augment heat transfer from electronic components, automobile radiators, engine and compressor cylinders, control devices, and a host of other applications. A comprehensive treat- ment of extended surface technology is provided by Kraus et al. (2001). In this section we provide the performance characteristics (temperature distribu- tion, rate of heat transfer, and fin efficiency) for convecting, radiating, and convecting- radiating fins. Configurations considered include longitudinal fins, radial fins, and spines. The section concludes with a discussion of optimum fin designs. BOOKCOMP, Inc. — John Wiley & Sons / Page 202 / 2nd Proofs / Heat Transfer Handbook / Bejan 202 CONDUCTION HEAT TRANSFER 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 [202], (42) Lines: 2081 to 2116 ——— -2.70891pt PgVar ——— Normal Page PgEnds: T E X [202], (42) TABLE 3.10 Dimensionless Surface Temperature in Solids with Uniform Energy Generation and Radiative–Convective Surface Cooling θ s N 1 N 2 Plane Wall Solid Cylinder Solid Sphere 0.25 0.25 1.4597 1.2838 1.2075 0.50 0.25 1.4270 1.2559 1.1840 0.75 0.25 1.3970 1.2320 1.1646 1.00 0.25 1.3698 1.2115 1.1484 0.25 0.50 1.2993 1.1759 1.1254 0.50 0.50 1.2838 1.1640 1.1159 0.75 0.50 1.2693 1.1534 1.1076 1.00 0.50 1.2559 1.1439 1.1004 0.25 0.75 1.2258 1.1288 1.0905 0.50 0.75 1.2164 1.1221 1.0853 0.75 0.75 1.2075 1.1159 1.0807 1.00 0.75 1.1991 1.1103 1.0764 0.25 1.00 1.1824 1.1020 1.0710 0.50 1.00 1.1759 1.0976 1.0677 0.75 1.00 1.1698 1.0935 1.0647 1.00 1.00 1.1640 1.0897 1.0619 3.6.1 Longitudinal Convecting Fins The five common profiles of longitudinal fins shown in Fig. 3.19 are rectangular, trapezoidal, triangular, concave parabolic, and convex parabolic. The analytical ex- pressions obtained are based on several assumptions. 1. The heat conduction in the fin is steady and one-dimensional. 2. The fin material is homogeneous and isotropic. 3. There is no energy generation in the fin. 4. The convective environment is characterized by a uniform and constant heat transfer coefficient and temperature. 5. The fin has a constant thermal conductivity. 6. The contact between the base of the fin and the primary surface is perfect. 7. The fin has a constant base temperature. Rectangular Fin For the rectangular fin (Fig. 3.19a), the temperature distribution, rate of heat transfer, and fin efficiency are given for five cases of thermal boundary conditions. 1. Constant base temperature and convecting tip: θ θ b = coshm(b − x) + H sinhm(b −x) coshmb + H sinhmb (3.174) BOOKCOMP, Inc. — John Wiley & Sons / Page 203 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTENDED SURFACES 203 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 [203], (43) Lines: 2116 to 2127 ——— 0.25815pt PgVar ——— Normal Page PgEnds: T E X [203], (43) q f = kmAθ b sinhmb + H coshmb coshmb + H sinhmb (3.175) q id = (hP b +h t A) θ b (3.176) η = q f q id (3.177) T,h ϱ T,h ϱ T,h ϱ T,h ϱ T,h ϱ T,h ϱ T,h ϱ T,h ϱ T,h ϱ t T b T b T b T b T b k k k k k b b b b b L L L L L x x x x x q f q f q f q f q f x e ␦ ␦ b ␦ b ␦ b ␦ b A ()a ()c ()e ()b ()d P =L = fin perimeter 2( )ϩ δ T,h ϱ T,h ϱ insulated tip Figure 3.19 Longitudinal fins of (a) rectangular, (b) trapezoidal, (c) triangular, (d) concave parabolic, and (e) convex parabolic profiles. BOOKCOMP, Inc. — John Wiley & Sons / Page 204 / 2nd Proofs / Heat Transfer Handbook / Bejan 204 CONDUCTION HEAT TRANSFER 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 [204], (44) Lines: 2127 to 2171 ——— 3.91736pt PgVar ——— Normal Page * PgEnds: Eject [204], (44) where θ = T − T ∞ , θ b = T b − T ∞ ,m 2 = hP /kA = 2h/kδ,H = h t /km, and T b is the fin base temperature, T the fin temperature at location x,T ∞ the convective environmental temperature, b the fin height, A the fin cross-sectional area, P the fin perimeter, k the fin thermal conductivity, h the convective heat transfer coefficient for surfaces other than the fin tip, h t the tip convective heat transfer coefficient, q f the fin heat dissipation, and q id the ideal fin heat dissipation. 2. Constant base temperature and insulated tip (H = 0): θ θ b = coshm(b − x) coshmb (3.178) q f = kmAθ b tanhmb (3.179) η = tanhmb mb (3.180) 3. Constant base and tip temperatures: θ θ b = (θ t /θ b ) sinhmx + sinhm(b −x) sinhmb (3.181) q f = kmAθ b coshmb − (θ t /θ b ) sinhmb (3.182) with q id and η given by eqs. (3.176) and (3.177), respectively, T t taken as the pre- scribed tip temperature, and θ t = T t − T ∞ . 4. Convective heating at the base and insulated tip: θ θ f = Bi cosh(mb − x) Bi coshmb + mb sinhmb (3.183) q f = kmAθ f Bi sinhmb Bi coshmb + mb sinhmb (3.184) where Bi = h f b/k, θ f = T f −T ∞ , and h f and T f characterize the convection process at the fin base. Equations (3.176) and (3.177) can be used to find q id and η,butθ b must be found first from eq. (3.183). 5. Infinitely high fin with constant base temperature: θ θ b = e −mx (3.185) q f = kmAθ b (3.186) Because the fin is infinitely high, q id and η cannot be calculated. Instead, one may calculate the fin effectiveness as the ratio of q f to the rate of heat transfer from the base surface without the fin, hAθ b . Thus BOOKCOMP, Inc. — John Wiley & Sons / Page 205 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTENDED SURFACES 205 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 [205], (45) Lines: 2171 to 2213 ——— 6.58131pt PgVar ——— Normal Page * PgEnds: Eject [205], (45) = q f hAθ b = kP hA 1/2 (3.187) Several important conclusions can be drawn from eq. (3.187). First, the fin effec- tiveness is enhanced by choosing a material with high thermal conductivity. Copper has a high value (k = 401 W/m ·K at 300K), but it is heavy and expensive. Aluminum alloys have lower k (k = 168 to 237 W/m ·K at 300 K) but are lighter, offer lower cost, and in most instances are preferable to copper. Second, the fins are more effective when the convecting fluid is a gas (low h) rather than a liquid (higher h). Moreover, there is a greater incentive to use the fin under natural convection (lower h) than under forced convection (higher h). Third, the greater the perimeter/area (P/A) ratio, the higher the effectiveness. This, in turn, suggests the use of thin, closely spaced fins. However, the gap between adjacent fins must be sufficient to prevent interference of the boundary layers on adjacent surfaces. Trapezoidal Fin For a constant base temperature and insulated tip, the tempera- ture distribution, rate of heat transfer, ideal rate of heat transfer, and fin efficiency for a trapezoidal fin (Fig. 3.19b) are θ θ b = I 0 (2m √ bx)K 1 (2m √ bx e ) + K 0 (2m √ bx)I I (2m √ bx e ) I 0 (2mb)K 1 (2m √ bx e ) + K 0 (2mb)I 1 (2m √ bx e ) (3.188) q f = kmδ b Lθ b I 1 (2mb)K 1 (2m √ bx e ) − K 1 (2mb)I I (2m √ bx e ) I 0 (2mb)K 1 (2m √ bx e ) + K 0 (2mb)I 1 (2m √ bx e ) (3.189) q id = 2Lbhθ b (3.190) and eq. (3.177) gives the fin efficiency. In eqs. (3.188) and (3.189), m = √ 2h/kδ b and x e is the distance to the fin tip. The modified Bessel functions appearing here and in subsequent sections are discussed in Section 3.3.5. Triangular Fin The rectangular fin (Fig. 3.19c) is a special case of the trapezoidal fin with x e = 0 and θ θ b = I 0 (2m √ bx) I 0 (2mb) (3.191) q f = kmδ b Lθ b I 1 (2mb) I 0 (2mb) (3.192) η = I 1 (2mb) mbI 0 (2mb) (3.193) Concave Parabolic Fin For the concave parabolic fin shown in Fig. 2.19d, the temperature distribution, rate of heat transfer, and fin efficiency are BOOKCOMP, Inc. — John Wiley & Sons / Page 206 / 2nd Proofs / Heat Transfer Handbook / Bejan 206 CONDUCTION HEAT TRANSFER 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 [206], (46) Lines: 2213 to 2264 ——— 1.4565pt PgVar ——— Short Page * PgEnds: Eject [206], (46) θ θ b = x b −1/2+1/2(1+4m 2 b 2 ) 1/2 (3.194) q f = kδ b Lθ b 2b −1 + 1 + 4m 2 b 2 1/2 (3.195) η = 2 1 + 1 + 4m 2 b 2 1/2 (3.196) Convex Parabolic Fin For the convex parabolic fin shown in Fig. 3.19e, the temperature distribution, rate of heat transfer, and fin efficiency are θ θ b = x b 1/4 I −1/3 4 3 mb 1/4 x 3/4 I −1/3 4 3 mb (3.197) q f = kmδ b Lθ b I 2/3 4 3 mb I −1/3 4 3 mb (3.198) η = 1 mb I 2/3 4 3 mb I −1/3 4 3 mb (3.199) The efficiency of longitudinal fins of rectangular, triangular, concave parabolic, and convex parabolic fins are plotted as a function of mb in Fig. 3.20. 3.6.2 Radial Convecting Fins The radial fin is also referred to as an annular fin or circumferential fin, and the performance of three radial fin profiles is considered. These are the rectangular, triangular, and hyperbolic profiles. Analytical results are presented for the rectangular profile, and graphical results are provided for all three profiles. Rectangular Fin For the radial fin of rectangular profile shown in the inset of Fig. 3.21, the expressions for the temperature distribution, rate of heat transfer, and fin efficiency are θ θ b = K 1 (mr a )I 0 (mr) +I 1 (mr a )K 0 (mr) I 0 (mr b )K 1 (mr a ) + I 1 (mr a )K 0 (mr b ) (3.200) q f = 2πr b kmδθ b I 1 (mr a )K 1 (mr b ) − K 1 (mr a )I 1 (mr b ) I 0 (mr b )K 1 (mr a ) + I 1 (mr a )K 0 (mr b ) (3.201) η = 2r b m r 2 a − r 2 b I 1 (mr a )K 1 (mr b ) − K 1 (mr a )I 1 (mr b ) I 0 (mr b )K 1 (mr a ) + I 1 (mr a )K 0 (mr b ) (3.202) BOOKCOMP, Inc. — John Wiley & Sons / Page 207 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTENDED SURFACES 207 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 [207], (47) Lines: 2264 to 2278 ——— * 29.178pt PgVar ——— Short Page * PgEnds: Eject [207], (47) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Fin efficiency (dimensionless) 012 mb Rectangular Convex parabolic Triangular Concave parabolic Figure 3.20 Efficiencies of longitudinal convecting fins. 1.0 0.9 0.8 0.7 0.6 0.3 0.5 0.2 0.4 0.1 0 Fin efficiency, 0 1.0 2.0 3.0 4.0 5.0 Fin parameter, = 2 . / . mL hk ͌ ␦ b 1 1.5 2 3 4 5 r b ␦ b r a b rr ab / Figure 3.21 Efficiency of radial (annular) fins of rectangular profile. (Adapted from Ullman and Kalman, 1989.) BOOKCOMP, Inc. — John Wiley & Sons / Page 208 / 2nd Proofs / Heat Transfer Handbook / Bejan 208 CONDUCTION HEAT TRANSFER 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 [208], (48) Lines: 2278 to 2307 ——— 3.25099pt PgVar ——— Normal Page PgEnds: T E X [208], (48) 1.0 0.9 0.8 0.7 0.6 0.3 0.5 0.2 0.4 0.1 0 Fin efficiency, 0 1.0 2.0 3.0 4.0 5.0 Fin parameter, = 2 . / . mb hk ͌ b ␦ 1.5 2 3 5 r b ␦ b r a b rr a b / ␦ bb /r 0.01 0.6 0.01–0.6 Figure 3.22 Efficiency of radial (annular) fins of triangular profile. (Adapted from Ullman and Kalman, 1989.) The efficiency of a radial fin of rectangular profile given by eq. (3.202) is plotted as a function of mb in the main body of Fig. 3.21 for r a /r b = 1 (longitudinal fin), 1.5, 2.0, 3.0, 4.0, and 5.0. Triangular Fin The inset in Fig. 3.22 shows a radial fin of triangular profile. The analysis for this profile is given in Kraus et al. (2001) and involves an infinite series that is omitted in favor of numerical results for the fin efficiency, which are graphed in Fig. 3.22. Note that η is a function of m, r a /r b , and δ b /r b . Once η is known, q f = 2π(r 2 a − r 2 b )hθ b η. Hyperbolic Fin A radial fin of hyperbolic profile appears as an inset in Fig. 3.23. The lengthy analytical results are presented in Kraus et al. (2001) and a graph of the fin efficiency is presented in Fig. 3.23. Note that η is a function of m, r a /r b , and δ b /r b , and once η is known, q f = 2π(r 2 a − r 2 b )hθ b η. 3.6.3 Convecting Spines Four commonly used shapes of spines, shown in Fig. 3.24, are the cylindrical, coni- cal, concave parabolic, and convex parabolic. Analytical results for the temperature distribution, rate of heat transfer, and fin efficiency are furnished. BOOKCOMP, Inc. — John Wiley & Sons / Page 209 / 2nd Proofs / Heat Transfer Handbook / Bejan EXTENDED SURFACES 209 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 [209], (49) Lines: 2307 to 2334 ——— 6.09984pt PgVar ——— Normal Page PgEnds: T E X [209], (49) 1.0 0.9 0.8 0.7 0.6 0.3 0.5 0.2 0.4 0.1 0 Fin efficiency, 0 1.0 2.0 3.0 4.0 5.0 Fin parameter, = 2 . / . mb hk ͌ b ␦ 1.5 2 3 4 5 r b ␦ b r a b ␦ bb /r 0.01 0.6 0.01–0.6 1 Figure 3.23 Efficiency of radial (annular) fins of hyperbolic profile. (Adapted from Ullman and Kalman, 1989.) Cylindrical Spine For the cylindrical spine, the results for the rectangular fins are applicable if m = (4h/kd) 1/2 is used instead of m = (2h/kδ) 1/2 . If the spine tip is insulated, eqs. (3.178)–(3.180) can be used. Conical Spine θ θ b = b x 1/2 I 1 (2M √ x) I 1 (2M √ b) (3.203) q f = πkd 2 b Mθ b 4 √ b I 2 (2M √ b) I 1 (2M √ b) (3.204) η = 2 M √ b) I 2 (2M √ b) I 1 (2M √ b) (3.205) where M = (4hb/kd b ) 1/2 . Concave Parabolic Spine θ θ b = x b −3/2+1/2(9+4M 2 ) 1/2 (3.206) BOOKCOMP, Inc. — John Wiley & Sons / Page 210 / 2nd Proofs / Heat Transfer Handbook / Bejan 210 CONDUCTION HEAT TRANSFER 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 [210], (50) Lines: 2334 to 2375 ——— -0.03146pt PgVar ——— Normal Page PgEnds: T E X [210], (50) q f = πkd 2 b θ b −3 + (9 + 4M 2 ) 1/2 8b (3.207) η = 2 1 + (1 + 8 9 m 2 b 2 ) 1/2 (3.208) where M = (4hb/kd b ) 1/2 and m = (2h/kd b ) 1/2 . Convex Parabolic Spine θ θ b = I 0 4 3 Mx 3/4 I 0 4 3 Mb 3/4 (3.209) q f = πkd 2 b Mθ b 2b 1/4 I 1 4 3 Mb 3/4 I 0 4 3 Mb 3/4 (3.210) η = 3 2 √ 2 I 1 4 3 √ 2mb mbI 0 4 3 √ 2mb (3.211) where M = (4hb 1/2 /kd b ) 1/2 and m = (2h/kd b ) 1/2 . Figure 3.25 is a plot of η as a function of mb for the four spines discussed. Th a , Th a , Th a , Th a , T b T b T b T b k k k k b b b b d b d d b d b ()c ()a ()d ()b Figure 3.24 Spines: (a) cylindrical; (b) conical; (c) concave parabolic; (d) convex parabolic. . distribution, rate of heat transfer, and fin efficiency are BOOKCOMP, Inc. — John Wiley & Sons / Page 206 / 2nd Proofs / Heat Transfer Handbook / Bejan 206 CONDUCTION HEAT TRANSFER 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 [206],. Wiley & Sons / Page 204 / 2nd Proofs / Heat Transfer Handbook / Bejan 204 CONDUCTION HEAT TRANSFER 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 [204],. the convective heat transfer coefficient for surfaces other than the fin tip, h t the tip convective heat transfer coefficient, q f the fin heat dissipation, and q id the ideal fin heat dissipation. 2.