Chapter 2 Heat Conduction Equation2-101 A spherical shell with variable conductivity is subjected to specified temperatures on both sides. The variation of temperature and the rate of heat transfer through the shell are to be determined. Assumptions 1Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies linearly. 3 There is no heat generation. PropertiesThe thermal conductivity is given to be kTkT()()=+01β. Solution (a) The rate of heat transfer through the shell is expressed as &QkrrTTrrsphereave=−−41221πwhere r1is the inner radius, r2is the outer radius, and ⎟⎠⎞⎜⎝⎛++==21)(120aveaveTTkTkkr1r2T1k(T) rT2is the average thermal conductivity. (b) To determine the temperature distribution in the shell, we begin with the Fourier’s law of heat conduction expressed as &TAdTdr=−where the rate of conduction heat transfer is constant and the heat conduction area A = 4πr&Q2is variable. Separating the variables in the above equation and integrating from r= r1where
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