Question

A very long cylindrical copper fin is dissipating 0.9 W of heat. It was found that the length of the fin is too difficult to accommodate due to space limitations. Therefore, the length of the fin was shortened such that new heat transfer rate is 0.81 W. However, the shortened fin tip has heat loss due to convection and the end will not be insulated. 2.1 Determine the length of the fin without doing fin tip correction (without correcting the fin length for convection losses). Following information is available. The diameter of the fin = 25 mm; Thermal conductivity of fin material = 396 W/(m. °C); Convection heat transfer coefficient = 10 W/(m2. °C). Fin base temperature and its surrounding medium temperature are not necessary. 2,2 Provide a complete list of assumptions made in solution for 2.1 2.3 If a fin tip correction is made, so that the heat transfer rate from the fin is 0.81 W, calculate the actual length of the fin.

Answer 1

case- 1 - Infinite long fin. q=0.9= I hPKA to h= heat transfer coefficient Pa perimeter of cross section K- Condructivity of A Area of cross section – Tambient e Dase temp tn or wau temp 0.9 0 0.9 АРКА 1000 VIO ***25 x396 *I to = 2:3°C taking fin tip correction Case-2 - Without me the Le length of fin without fintip correction te 9. Unora oo x tanh (mL) 0.81= SOPRA XOox tanh (mi) 0.81 0.9 x tanh (mi). 0.81 = tanh (ma) mar tant [0.8 ML = 1.4722 = 1.4722 0.9 1.4722 10x ** 25/1000 396 * *(4000) = 0.7324 m So, length of fin without fin tip correction = 0.7324m

2:2> Assumption 17 1D, steady state 27 No heat generation 37 heat transfer coefficient is constant throughout the length of sin. Thermal conductivity of the material is constant 47 . 237 Now, fin tip connection is considered 9- JAPKA do tanh (mLc). here, he is the lengen of sin considering fintip correction d. Lea et As, Actual length of fin. ²0.81= 0.9 tanh (me) tanh10.817 mic. 10.9 mLc = 1.4722 Le = 1.4722 m 1.4722 1.4722 the 10***125/1000) NKA 396*** (25/1000) te - 0.7324 m le= ut A LC-Ace = length of fin 0.7324 - IX (0 : 0252) - Length of fin * 0.025) i. Actual fin length = 0.72615 m Actual