A fin is an extended surface used to transfer heat from base material (at x=0) t

Question

A fin is an extended surface used to transfer heat from base material (at x=0) to an ambient. Measurement of the temperature distribution along a pin fin gives the data:

x=0,1,2,3,4,5,6,7,8,9,10
T=473.0,446.3,422.6,401.2,382.0,364.3,348.0,332.7,318.1,304.0,290.1

The fin has a length L=10, constant cross-sectional area of 1.6x10(-5), and thermal conductivity k=240 W/m/K. Heat flux is given by qx=-k(dT/dx)

a) Determine the heat flux at x=0 using the forward difference formula to calculate the derivative.
b) Determine the heat flux at x=10. Use the backward difference formula to calculate the derivative.
c) Determine the amount ofheat lost between x=0 and x=10 (the heat flow per unit time in Watts is the heat flux multiplied by the cross-sectional area of the fin)

Explanation / Answer

L =10

k=240 W/m/K

a) qx = - k dT/dx

by forward difference formula at x = 0

dT/dx = (T1 - T0) / (x1 - x0)

q0 = - 240 * [ (446.3 - 473) / (1-0) ] = 6408 W/m2

b) by backward difference formula at x =10

dT/dx = (T10 - T9) / (x10 - x9)

q10 = - 240 * [ ( 290.1 - 304 ) / (10 - 9) ] = 3336 W/m2

c) cross sectional area, A = 1.6 x 10-5

Amount of heat lost = (q0 - q10)A

= (6408 - 3336) x 1.6 x 10-5 = 4915.2 x 10-5 W

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