(a) Explanation Given: W = 200 m m = 0.2 m L = 12 m m = 0.012 m D = 100 m m = 0.1 m t = 1 m m = 0.001 m T 0 = 400 K T L = 350 K T ∞ = 300 K h = 150 W / m K 2 Assumptions: The steady state condition for the given heat rate. Heat conduction in fins is one dimensional. There is negligible radiation, uniform heat transfer with negligible variation in T ∞ The thermal conductivity of pure Aluminium at ( T 0 + T L 2 = 400 + 350 2 = 375 K ) is k = 240 W / m K Calculation: The temperature distribution in the fin is expressed by using the following general solution at distance ( x ) : θ ( x ) = T ( x ) − T ∞ θ ( x ) = C 1 e m x − C 2 e − m x … (1) The temperature distribution at boundary conditions is given by: At x = 0 , θ ( 0 ) = θ 0 θ 0 = T 0 − T ∞ θ 0 = C 1 + C 2 C 2 = θ 0 − C 1 … (2) And at x = L , θ ( L ) = θ L θ L = T L − T ∞ θ L = C 1 e m L + C 2 e − m L … (3) Put the value of C 2 by using eq. (2) in eq. (3): θ L = C 1 e m L + C 2 e − m L θ L = C 1 e m L + ( θ 0 − C 1 ) e − m L θ L = C 1 e m L + θ 0 e − m L − C 1 e − m L θ L = θ 0 e − m L + C 1 ( e m L − e − m L ) C 1 = θ L − θ 0 e − m L e m L − e − m L … (4) Now, put the value of ( C 1 = θ L − θ 0 e − m L e m L − e − m L ) in eq. (2): C 2 = θ 0 − θ L − θ 0 e − m L e m L − e − m L C 2 = θ 0 e m L − θ 0 e − m L − θ L + θ 0 e − m L e m L − e − m L C 2 = θ 0 e m L − θ L e m L − e − m L From eq. (1): θ ( x ) = C 1 e m x − C 2 e − m x θ ( x ) = ( θ L − θ 0 e − m L e m L − e − m L ) e m x − ( θ 0 e m L − θ L e m L − e − m L ) e − m x θ ( x ) = θ L e m x − θ 0 e − m L e m x − θ 0 e m L e − m x + θ L e − m x e m L − e − m L θ ( x ) = θ L e m x − θ 0 e m ( x − L ) + θ 0 e m ( L − x ) + θ L < (b) To determine The maximum power dissipation from top and the bottom plate

6th Edition

ISBN: 9780470501962

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3, Problem 3.142P

(a)

To determine

The rate of fin heat transfer qf0 and qfL in terms of T0 and TL

(b)

To determine

The maximum power dissipation from top and the bottom plate

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Chapter 3, Problem 3.141P

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The rate of fin heat transfer q f 0 and q f L in terms of T 0 and T L

Solution Summary: The author explains the steady state condition for the given heat rate.