Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 22
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(3.183).5. Infinitely high fin with constant base temperature:θ= e−mxθb(3.185)qf = kmAθb(3.186)Because the fin is infinitely high, qid and η cannot be calculated. Instead, one maycalculate the fin effectiveness as the ratio of qf to the rate of heat transfer from thebase surface without the fin, hAθb . ThusBOOKCOMP, Inc.
— John Wiley & Sons / Page 204 / 2nd Proofs / Heat Transfer Handbook / Bejan———3.91736pt PgVar[204], (44)EXTENDED SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445qf==hAθbkPhA2051/2(3.187)Several important conclusions can be drawn from eq. (3.187). First, the fin effectiveness is enhanced by choosing a material with high thermal conductivity. Copperhas a high value (k = 401 W/m · K at 300K), but it is heavy and expensive. Aluminumalloys have lower k (k = 168 to 237 W/m · K at 300 K) but are lighter, offer lowercost, and in most instances are preferable to copper. Second, the fins are more effectivewhen 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 underforced convection (higher h). Third, the greater the perimeter/area (P /A) ratio, thehigher 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 ofthe boundary layers on adjacent surfaces.Trapezoidal Fin For a constant base temperature and insulated tip, the temperature distribution, rate of heat transfer, ideal rate of heat transfer, and fin efficiency fora trapezoidal fin (Fig.
3.19b) are√√√θI0 (2m bx)K1 (2m bxe ) + K0 (2m bx)II (2m bxe )=√√θbI0 (2mb)K1 (2m bxe ) + K0 (2mb)I1 (2m bxe )√√I1 (2mb)K1 (2m bxe ) − K1 (2mb)II (2m bxe )qf = kmδb Lθb√√I0 (2mb)K1 (2m bxe ) + K0 (2mb)I1 (2m bxe )(3.188)Triangular Fin The rectangular fin (Fig. 3.19c) is a special case of the trapezoidalfin with xe = 0 and√θI0 (2m bx)=(3.191)θbI0 (2mb)η=I1 (2mb)mbI0 (2mb)(3.192)(3.193)Concave Parabolic Fin For the concave parabolic fin shown in Fig. 2.19d, thetemperature distribution, rate of heat transfer, and fin efficiency areBOOKCOMP, Inc.
— John Wiley & Sons / Page 205 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal Page* PgEnds: Eject[205], (45)√and eq. (3.177) gives the fin efficiency. In eqs. (3.188) and (3.189), m = 2h/kδband xe is the distance to the fin tip. The modified Bessel functions appearing here andin subsequent sections are discussed in Section 3.3.5.I1 (2mb)I0 (2mb)———(3.189)(3.190)qf = kmδb LθbLines: 2171 to 22136.58131pt PgVar√qid = 2Lbhθb[205], (45)206123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFER x −1/2+1/2(1+4m2 b2 )1/2θ=θbb1/2 kδb Lθb qf =−1 + 1 + 4m2 b22b2η=1/21 + 1 + 4m2 b2(3.194)(3.195)(3.196)Convex Parabolic Fin For the convex parabolic fin shown in Fig. 3.19e, thetemperature distribution, rate of heat transfer, and fin efficiency are x 1/4 I−1/3 4 mb1/4 x 3/4 θ3=θbbI−1/3 43 mbqf =I2/3mb[206], (46)(3.198)Lines: 2213 to 22643kmδb LθbI−1/3 43 mb4η=4(3.197)———1 I2/3 3 mbmb I−1/3 43 mb1.4565pt PgVar(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.———Short Page* PgEnds: Eject[206], (46)3.6.2Radial Convecting FinsThe radial fin is also referred to as an annular fin or circumferential fin, and theperformance of three radial fin profiles is considered. These are the rectangular,triangular, and hyperbolic profiles. Analytical results are presented for the rectangularprofile, and graphical results are provided for all three profiles.Rectangular Fin For the radial fin of rectangular profile shown in the inset ofFig. 3.21, the expressions for the temperature distribution, rate of heat transfer, andfin efficiency areθK1 (mra )I0 (mr) + I1 (mra )K0 (mr)=θbI0 (mrb )K1 (mra ) + I1 (mra )K0 (mrb )qf = 2πrb kmδθbη=I1 (mra )K1 (mrb ) − K1 (mra )I1 (mrb )I0 (mrb )K1 (mra ) + I1 (mra )K0 (mrb )2rbI1 (mra )K1 (mrb ) − K1 (mra )I1 (mrb )2 I (mr )K (mr ) + I (mr )K (mr )2m ra − rb 0b1a1a0bBOOKCOMP, Inc.
— John Wiley & Sons / Page 206 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.200)(3.201)(3.202)EXTENDED SURFACES1.0Fin efficiency (dimensionless)0.9RectangularConvex parabolic0.8Triangular0.7Concaveparabolic0.6[207], (47)0.5Lines: 2264 to 22780.401mb———2*1.0[207], (47)0.90.8ra /rb0.711.523450.60.50.4␦brbrab0.30.20.1029.178pt PgVar———Short Page* PgEnds: EjectFigure 3.20 Efficiencies of longitudinal convecting fins.Fin efficiency, 12345678910111213141516171819202122232425262728293031323334353637383940414243444520701.02.03.0Fin parameter, m = L 公2 . h/k . ␦b4.05.0Figure 3.21 Efficiency of radial (annular) fins of rectangular profile. (Adapted from Ullmanand Kalman, 1989.)BOOKCOMP, Inc. — John Wiley & Sons / Page 207 / 2nd Proofs / Heat Transfer Handbook / Bejan2081.0␦b /rb0.90.010.8Fin efficiency, 123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERra / rb0.70.01–0.60.60.5␦b0.6rbrab1.52350.40.30.2[208], (48)0.1001.02.03.0Fin parameter, m = b公2 .
h/k . ␦b4.05.0Figure 3.22 Efficiency of radial (annular) fins of triangular profile. (Adapted from Ullmanand Kalman, 1989.)The efficiency of a radial fin of rectangular profile given by eq. (3.202) is plotted asa function of mb in the main body of Fig. 3.21 for ra /rb = 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. Theanalysis for this profile is given in Kraus et al. (2001) and involves an infinite seriesthat is omitted in favor of numerical results for the fin efficiency, which are graphedin Fig.
3.22. Note that η is a function of m, ra /rb , and δb /rb . Once η is known,qf = 2π(ra2 − rb2 )hθb 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 thefin efficiency is presented in Fig. 3.23.
Note that η is a function of m, ra /rb , and δb /rb ,and once η is known, qf = 2π(ra2 − rb2 )hθb η.3.6.3Convecting SpinesFour commonly used shapes of spines, shown in Fig. 3.24, are the cylindrical, conical, concave parabolic, and convex parabolic. Analytical results for the temperaturedistribution, rate of heat transfer, and fin efficiency are furnished.BOOKCOMP, Inc. — John Wiley & Sons / Page 208 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2278 to 2307———3.25099pt PgVar———Normal PagePgEnds: TEX[208], (48)209EXTENDED SURFACES1.0␦b /rb0.90.010.8Fin efficiency, 123456789101112131415161718192021222324252627282930313233343536373839404142434445␦b0.60.70.611.520.50.40.01–0.6rbbra3450.30.2[209], (49)0.1001.02.03.0Fin parameter, m = b公2 .
h/k . ␦b4.05.0———Figure 3.23 Efficiency of radial (annular) fins of hyperbolic profile. (Adapted from Ullmanand Kalman, 1989.)Cylindrical Spine For the cylindrical spine, the results for the rectangular fins areapplicable if m = (4h/kd)1/2 is used instead of m = (2h/kδ)1/2 . If the spine tip isinsulated, eqs. (3.178)–(3.180) can be used.Conical Spine√ 1/2I1 (2M x)b√xI1 (2M b)√πkdb2 Mθb I2 (2M b)qf =√√4 b I1 (2M b)√2 I2 (2M b)η= √√M b) I1 (2M b)θ=θb(3.203)(3.204)(3.205)where M = (4hb/kdb )1/2 .Concave Parabolic Spine x −3/2+1/2(9+4M 2 )1/2θ=θbbBOOKCOMP, Inc.
— John Wiley & Sons / Page 209 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2307 to 2334(3.206)6.09984pt PgVar———Normal PagePgEnds: TEX[209], (49)210123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERπkdb2 θb −3 + (9 + 4M 2 )1/2qf =8bη=(3.207)21 + (1 + 89 m2 b2 )1/2(3.208)where M = (4hb/kdb )1/2 and m = (2h/kdb )1/2 .Convex Parabolic SpineI0 43 Mx 3/4θ= 4θbI0 3 Mb3/4(3.209)πkdb2 Mθb I1 43 Mb3/4qf =2b1/4 I0 43 Mb3/4 √I1 43 2mb3η= √ √2 2 mbI0 43 2mb[210], (50)(3.210)Lines: 2334 to 2375(3.211)———-0.03146pt PgVarwhere M = (4hb1/2 /kdb )1/2 and m = (2h/kdb )1/2 .Figure 3.25 is a plot of η as a function of mb for the four spines discussed.———Normal PagePgEnds: TEX[210], (50)Ta , hTbTbdkTa , hkdbbb(a)(b)Ta , hTbdbdbkb(c)Ta , hTbkb(d )Figure 3.24 Spines: (a) cylindrical; (b) conical; (c) concave parabolic; (d) convex parabolic.BOOKCOMP, Inc.
— John Wiley & Sons / Page 210 / 2nd Proofs / Heat Transfer Handbook / BejanEXTENDED SURFACES1.0Fin efficiency (dimensionless)123456789101112131415161718192021222324252627282930313233343536373839404142434445211Concave parabolic0.9ConicalConstant crosssectionConvexparabolic0.80.70.60.5[211], (51)01mbLines: 2375 to 24012———0.70709pt PgVarFigure 3.25 Efficiencies of convecting spines.3.6.4———Normal PagePgEnds: TEXLongitudinal Radiating FinsUnlike convecting fins, for which exact analytical solutions abound, few such solutions are available for radiating fins.
Consider the longitudinal fin of rectangularprofile shown in Fig. 3.19a and let the fin radiate to free space at 0 K. The differentialequation governing the temperature in the fin isd 2T2σ 4T=dx 2kδ(3.212)with the boundary conditionsT (x = 0) = TbanddT =0dx x=b(3.213)where is the emissivity of the fin surface and σ is the Stefan–Boltzmann constant(σ = 5.667 × 10−8 W/m 2 · K 4).The solution for the temperature distribution, rate of heat transfer, and fin efficiency are1/220σTt3B(0.3, 0.5) − Bu (0.3, 0.5) = b(3.214)kδqf = 2kδLBOOKCOMP, Inc.
— John Wiley & Sons / Page 211 / 2nd Proofs / Heat Transfer Handbook / Bejan σ 1/2 1/2Tb5 − Tt55kδ(3.215)[211], (51)212123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERη=1/22kδL(σ/5kδ)1/2 Tb5 − Tb52σbLTb4(3.216)where B and Bu are complete and incomplete beta functions discussed in Section3.3.3, u = (Tt /T )5 and Tt is the unknown tip temperature. Because Tt is not known,the solution involves a trial-and-error procedure.Sen and Trinh (1986) reported the solution of eqs.
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