Chapter 2 Heat Conduction Equation2-133 A cylindrical shell with variable conductivity is subjected to specified temperatures on both sides. The rate of heat transfer through the shell is to be determined. Assumptions 1Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies quadratically. 3 There is no heat generation. PropertiesThe thermal conductivity is given to be . kTkT()()=+021βAnalysis When the variation of thermal conductivity with temperature k(T) is known, the average value of the thermal conductivity in the temperature range between TTis determined from 12and r2T2rr1T1k(T)⎥⎦⎤⎢⎣⎡+++=−⎥⎦⎤⎢⎣⎡−+−=−⎟⎠⎞⎜⎝⎛+=−+=−=∫∫21212201231321201230122012ave3133)1()(212121TTTTkTTTTTTkTTTTkTTdTTkTTdTTkkTTTTTTThis relation is based on the requirement that the rate of heat transfer through a medium with constant
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