Thermodynamics HW Solutions 132

Thermodynamics HW Solutions 132 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-91 A long homogeneous resistance heater wire with specified surface temperature is used to boil water. The temperature of the wire 2 mm from the center is to be determined in steady operation. Assumptions 1 Heat transfer is steady since there is no indication of any change with time. 2 Heat transfer is one-dimensional since there is thermal symmetry about the center line and no change in the axial direction. 3 Thermal conductivity is constant. 4 Heat generation in the wire is uniform. Properties The thermal conductivity is given to be k = 8 W/m °C. Analysis 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 1 = + k g dr dT r dr d r & and 180 ° C (specified surface temperature) Tr T s () 0 == dT dr 0 0 = (thermal symmetry about the centerline) Multiplying both sides of the differential equation by r and rearranging gives g 180°C Resistance wire r r o r k g dr dT
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