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Homework_4_Solution

Homework_4_Solution - 3-114 Circular cooling fins of a...

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3-114 Circular cooling fins of a specified dimension are used to enhance heat transfer from a surface. Given that the ends of the fins are connected to another surface, determine (a) - = T x T x ) ( ) ( θ along a fin, and calculate the temperature at 2 / L x = ; (b) the rate of heat transferred from the hot surface through each fin and the fin effectiveness, as well as whether the use of fins is justified; (c) the total rate of heat transfer given the surface area and number of fins being used. Assumptions Steady operating conditions exist. The thermal conductivity and convection coefficients are constant. The radiation of the fins is taken into account through the conduction and convection coefficients. The temperature along the fins varies axially but not radially. Known = = × = = = = = × × = = = = = = = = = - 1 - 2 7 2 2 2 2 2 1 m 623 . 31 m 10 854 . 7 4 m 00314 . 0 m 0.0095 4 1 . 0 1 . 0 fins 625 K W/m 100 K 15 . 273 K 15 . 273 K 15 . 405 K W/m 400 m 0254 . 0 m 001 . 0 c c S s s kA hP m D A D P D -n A n h T T T k L D π π π Unknown tot fin fin Q Q L T x . . , , ), 2 / ( ), ( ε θ Property Relations and Laws mL mL mL mL mL mL mL mL s s mx mx e e C e C C e C e C e C e C e C T T L C C C C T T e C e C x 2 2 2 2 1 2 2 1 2 1 2 1 2 1 2 2 1 1 2 1 1 ) 0 ( 1 ) 0 ( : (3) and (2) From (3) 0 ) ( (2) ) 0 ( K 132 ) 0 ( (1) ) ( - = - = - = - = = + - = - = = + - = + = - - - θ θ θ θ θ θ
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Plugging into (1) and rearranging yields ] [ )] ( [ ) 0 ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( 1 ) 0 ( ) ( ) ( ) ( 2 2 2 mL Sinh L x m Sinh e e e e x e e e e e e e e x L x m L x m mL mL mL mx mx mL mL mL mL mx mx mL - - = - - = - - = - - = - - - - + - - + - θ θ θ θ θ θ (a) ( ) C 61 Sinh[mL] x)] Sinh[m(L T T o s1 = + = - - = T L L T θ (x) ) 2 / ( ) 2 / ( θ (b) W 1.97 = × = - - = - = - = 89 . 0
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