Thermodynamics HW Solutions 137

Thermodynamics HW Solutions 137 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-101 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 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies linearly. 3 There is no heat generation. Properties The thermal conductivity is given to be kT k T () ( ) = + 0 1 β . Solution ( a ) The rate of heat transfer through the shell is expressed as & Qk r r TT rr sphere ave = 4 12 21 π where r 1 is the inner radius, r 2 is the outer radius, and + + = = 2 1 ) ( 1 2 0 ave ave T T k T k k r 1 r 2 T 1 k ( T ) r T 2 is 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 & T A dT dr =− where the rate of conduction heat transfer is constant and the heat conduction area A = 4 π r & Q 2 is variable. Separating the variables in the above equation and integrating from r = r 1 where
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