Thermodynamics HW Solutions 146

Thermodynamics HW Solutions 146 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-125 A steam pipe is subjected to convection on both the inner and outer surfaces. The mathematical formulation of the problem and expressions for the variation of temperature in the pipe and on the outer surface temperature are to be obtained for steady one-dimensional heat transfer. Assumptions 1 Heat conduction is steady and one-dimensional since the pipe is long relative to its thickness, and there is thermal symmetry about the center line. 2 Thermal conductivity is constant. 3 There is no heat generation in the pipe. Analysis ( a ) Noting that heat transfer is steady and one-dimensional in the radial r direction, the mathematical formulation of this problem can be expressed as 0 = dr dT r dr d and −= k dT r dr hT Tr ii () [( 1 1 ) ] −=− k dT r dr hTr T o [() ] 2 2 o ( b ) Integrating the differential equation once with respect to r gives r dT d C = 1 r 2 r r 1 T i h i T o h o r Dividing both sides of the equation above by r to bring it to a readily integrable form and then integrating,
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