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1.55 1.19 136.5 156.5 < x2 = 0.05 1.24 0.725 108.9 128.9 < L = 0.10 1.00 0.00 87.0 107.0 < COMMENTS: If the rod were approximated as infinitely long: T(x1) = 148.7°C, T(x2) = 112.0°C, and T(L) = 67.0°C. The assumption would therefore result in significant underestimates of the rod temperature. PROBLEM 3.121 KNOWN: Thickness, length, thermal conductivity, and base temperature of a rectangular fin. Fluid temperature and convection coefficient. FIND: (a) Heat rate per unit width, efficiency, effectiveness, thermal resistance, and tip temperature for different tip conditions, (b) Effect of convection coefficient and thermal conductivity on the heat rate. SCHEMATIC: ASSUMPTIONS: (1) Steady-state, (2) One-dimensional conduction along fin, (3) Constant properties, (4) Negligible radiation, (5) Uniform convection coefficient, (6) Fin width is much longer than thickness (w >> t). ANALYSIS: (a) The fin heat transfer rate for Cases A, B and D are given by Eqs. (3.72), (3.76) and (3.80), where M \u2248 (2 hw2tk)1/2 (Tb - T\u221e) = (2 × 100 W/m2\u22c5K × 0.001m × 180 W/m\u22c5K)1/2 (75°C) w = 450 w W, m\u2248 (2h/kt)1/2 = (200 W/m2\u22c5K/180 W/m\u22c5K ×0.001m)1/2 = 33.3m-1, mL \u2248 33.3m-1 × 0.010m = 0.333, and (h/mk) \u2248 (100 W/m2\u22c5K/33.3m-1 × 180 W/m\u22c5K) = 0.0167. From Table B-1, it follows that sinh mL \u2248 0.340, cosh mL \u2248 1.057, and tanh mL \u2248 0.321. From knowledge of qf, Eqs. (3.86), (3.81) and (3.83) yield ( ) bf f f f t,f b b f q q , , R h 2L t ht q \u3b8 \u3b7 \u3b5 \u3b8 \u3b8 \u2032 \u2032 \u2032\u2248 = \u2032+ \u2248 Case A: From Eq. (3.72), (3.86), (3.81), (3.83) and (3.70), ( ) ( )f sinh mL h / mk cosh mLM 0.340 0.0167 1.057 q 450 W / m 151W / m w cosh mL h / mk sinh mL 1.057 0.0167 0.340 + + × \u2032 = = = + + × < ( )f 2 151W / m 0.96 100 W / m K 0.021m 75 C \u3b7 = = \u22c5 ° < ( )f t,f2 151W / m 75 C 20.1, R 0.50 m K / W 151W / m100 W / m K 0.001m 75 C \u3b5 ° \u2032= = = = \u22c5 \u22c5 ° < ( ) ( ) ( ) b 75 CT L T 25 C 95.6 C cosh mL h / mk sinh mL 1.057 0.0167 0.340 \u3b8 \u221e ° = + = ° + = ° + + < Case B: From Eqs. (3.76), (3.86), (3.81), (3.83) and (3.75) ( )f Mq tanh mL 450 W / m 0.321 144 W / m w \u2032 = = = < f f t,f0.92, 19.2, R 0.52 m K / W\u3b7 \u3b5 \u2032= = = \u22c5 < ( ) b 75 CT L T 25 C 96.0 C cosh mL 1.057 \u3b8 \u221e ° = + = ° + = ° < Continued \u2026.. PROBLEM 3.121 (Cont.) Case D (L \u2192 \u221e): From Eqs. (3.80), (3.86), (3.81), (3.83) and (3.79) f M q 450 W / m w \u2032 = = < ( )f f t,f0, 60.0, R 0.167 m K / W, T L T 25 C\u3b7 \u3b5 \u221e\u2032= = = \u22c5 = = ° < (b) The effect of h on the heat rate is shown below for the aluminum and stainless steel fins. For both materials, there is little difference between the Case A and B results over the entire range of h. The difference (percentage) increases with decreasing h and increasing k, but even for the worst case condition (h = 10 W/m2\u22c5K, k = 180 W/m\u22c5K), the heat rate for Case A (15.7 W/m) is only slightly larger than that for Case B (14.9 W/m). For aluminum, the heat rate is significantly over-predicted by the infinite fin approximation over the entire range of h. For stainless steel, it is over-predicted for small values of h, but results for all three cases are within 1% for h > 500 W/m2\u22c5K. COMMENTS: From the results of Part (a), we see there is a slight reduction in performance (smaller values of f f fq , and ,\u3b7 \u3b5\u2032 as well as a larger value of t,fR )\u2032 associated with insulating the tip. Although \u3b7f = 0 for the infinite fin, fq\u2032 and \u3b5f are substantially larger than results for L = 10 mm, indicating that performance may be significantly improved by increasing L. Va ria tio n o f q f' w ith h (k=1 8 0 W /m .K ) 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 C o n ve ctio n co e ffic ie n t, h (W /m ^2 .K ) 0 5 0 0 1 0 0 0 1 5 0 0 H e a t r a te , qf '(W /m ) q fA' q fB ' q fD ' Va ria tio n o f q f' w i th h (k= 1 5 W /m .K ) 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 C o n ve c tio n co e ffic ie n t, h (W /m ^2 .K ) 0 1 0 0 2 0 0 3 0 0 4 0 0 H e a t r a te , qf '(W /m ) q fA' q fB ' q fD ' PROBLEM 3.122 KNOWN: Thickness, length, thermal conductivity, and base temperature of a rectangular fin. Fluid temperature and convection coefficient. FIND: (a) Heat rate per unit width, efficiency, effectiveness, thermal resistance, and tip temperature for different tip conditions, (b) Effect of fin length and thermal conductivity on the heat rate. SCHEMATIC: ASSUMPTIONS: (1) Steady-state, (2) One-dimensional conduction along fin, (3) Constant properties, (4) Negligible radiation, (5) Uniform convection coefficient, (6) Fin width is much longer than thickness (w >> t). ANALYSIS: (a) The fin heat transfer rate for Cases A, B and D are given by Eqs. (3.72), (3.76) and (3.80), where M \u2248 (2 hw2tk)1/2 (Tb - T\u221e) = (2 × 100 W/m2\u22c5K × 0.001m × 180 W/m\u22c5K)1/2 (75°C) w = 450 w W, m\u2248 (2h/kt)1/2 = (200 W/m2\u22c5K/180 W/m\u22c5K ×0.001m)1/2 = 33.3m-1, mL \u2248 33.3m-1 × 0.010m = 0.333, and (h/mk) \u2248 (100 W/m2\u22c5K/33.3m-1 × 180 W/m\u22c5K) = 0.0167. From Table B-1, it follows that sinh mL \u2248 0.340, cosh mL \u2248 1.057, and tanh mL \u2248 0.321. From knowledge of qf, Eqs. (3.86), (3.81) and (3.83) yield ( ) bf f f f t,f b b f q q , , R h 2L t ht q \u3b8 \u3b7 \u3b5 \u3b8 \u3b8 \u2032 \u2032 \u2032\u2248 \u2248 = \u2032+ Case A: From Eq. (3.72), (3.86), (3.81), (3.83) and (3.70), ( ) ( )f sinh mL h / mk cosh mLM 0.340 0.0167 1.057 q 450 W / m 151W / m w cosh mL h / mk sinh mL 1.057 0.0167 0.340 + + × \u2032 = = = + + × < ( )f 2 151W / m 0.96 100 W / m K 0.021m 75 C \u3b7 = = \u22c5 ° < ( )f t,f2 151W / m 75 C 20.1, R 0.50 m K / W 151W / m100 W / m K 0.001m 75 C \u3b5 ° \u2032= = = = \u22c5 \u22c5 ° < ( ) ( ) ( ) b 75 CT L T 25 C 95.6 C cosh mL h / mk sinh mL 1.057 0.0167 0.340 \u3b8 \u221e ° = + = ° + = ° + + < Case B: From Eqs. (3.76), (3.86), (3.81), (3.83) and (3.75) ( )f Mq tanh mL 450 W / m 0.321 144 W / m w \u2032 = = = < f f t,f0.92, 19.2, R 0.52 m K / W\u3b7 \u3b5 \u2032= = = \u22c5 < ( ) b 75 CT L T 25 C 96.0 C cosh mL 1.057 \u3b8 \u221e ° = + = ° + = ° < Continued \u2026.. PROBLEM 3.122 (Cont.) Case D (L \u2192 \u221e): From Eqs. (3.80), (3.86), (3.81), (3.83) and (3.79) f M q 450 W / m w \u2032 = = < ( )f f t,f0, 60.0, R 0.167 m K / W, T L T 25 C\u3b7 \u3b5 \u221e\u2032= = = \u22c5 = = ° < (b) The effect of L on the heat rate is shown below for the aluminum and stainless steel fins. For both materials, differences between the Case A and B results diminish with increasing L and are within 1% of each other at L \u2248 27 mm and L \u2248 13 mm for the aluminum and steel, respectively. At L = 3 mm, results differ by 14% and 13% for the aluminum and steel, respectively. The Case A and B results approach those of the infinite fin approximation more quickly for stainless steel due to the larger temperature gradients, |dT/dx|, for the smaller value of k. COMMENTS: From the results of Part (a), we see there is a slight reduction in performance (smaller values of f f fq , and ,\u3b7 \u3b5\u2032 as well as a larger value of t,fR )\u2032 associated with insulating the tip. Although \u3b7f = 0 for the infinite fin, fq\u2032 and \u3b5f are substantially larger than results for L = 10 mm, indicating that performance may be significantly improved by increasing L. Va ria tio n o f q f' w ith L (k=1 8 0 W /m .K ) 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 Fin le n g th , L (m ) 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 H e a t r a te , qf '(W /m ) q fA' q fB ' q fD ' Va ria tio n o f q f' w ith L (k=1 5 W /m .K ) 0 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 Fin le n g th , L (m ) 0 3 0 6 0 9 0 1 2 0 1 5 0 H e a t r a te , qf '(W /m ) q fA' q fB ' q fD ' PROBLEM 3.123 KNOWN: Length, thickness and temperature of straight fins of rectangular, triangular and parabolic profiles. Ambient air temperature and convection coefficient. FIND: