Thermodynamics HW Solutions 105

Thermodynamics HW Solutions 105 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-63 A spherical container is subjected to specified temperature on the inner surface and convection on the outer surface. The mathematical formulation, the variation of temperature, and the rate of heat transfer are to be determined for steady one-dimensional heat transfer. Assumptions 1 Heat conduction is steady and one-dimensional since there is no change with time and there is thermal symmetry about the midpoint. 2 Thermal conductivity is constant. 3 There is no heat generation. Properties The thermal conductivity is given to be k = 30 W/m °C. Analysis ( a ) Noting that heat transfer is one-dimensional in the radial r direction, the mathematical formulation of this problem can be expressed as 0 2 = dr dT r dr d and Tr T () 11 0 == ° C r 1 r 2 T 1 k T h −= k dT r dr hTr T [( ) ] 2 2 ( b ) Integrating the differential equation once with respect to r gives r dT dr C 2 1 = Dividing both sides of the equation above by r to bring it to a readily integrable form and then integrating,
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