Chapter 2 Heat Conduction Equation2-89Heat is generated uniformly in a spherical radioactive material with specified surface temperature. The mathematical formulation, the variation of temperature in the sphere, and the center temperature are to be determined for steady one-dimensional heat transfer. Assumptions 1Heat transfer is steady since there is no indication of any changes with time. 2 Heat transfer is one-dimensional since there is thermal symmetry about the mid point.3 Thermal conductivity is constant.4 Heat generation is uniform. PropertiesThe thermal conductivity is given to be k= 15 W/m⋅°C. 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 ro0Ts=80°Ckgrconstantwith 0122==+⎟⎠⎞⎜⎝⎛gkgdrdTrdrdr&&and 80°C (specified surface temperature) TrTs()0==dTdr00=(thermal symmetry about the mid point) (b) Multiplying both sides of the differential equation by r2and rearranging gives 22rkgdrdTrdrd&−=⎟⎠⎞⎜⎝⎛Integrating with respect to r gives rdT
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This note was uploaded on 01/14/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.