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Unformatted text preview: Monday January 13, 2008 Monday January 13, 2008 CHEM ENG 2A04 CHEM ENG 2A04 CONDUCTION CONDUCTION HEAT EQUATION HEAT EQUATION A rate equation that allows determination of the conduction heat flux from knowledge of the temperature distribution in a medium Fouriers Law Its most general (vector) form for multidimensional conduction is: q k T =  Implications Implications : : Heat transfer is in the direction of decreasing temperature (basis for minus sign). Direction of heat transfer is perpendicular to lines of constant temperature ( isotherms ). Heat flux vector may be resolved into orthogonal components. T T T q k i k j k k r r z =    r q q z q Cylindrical Coordinates: ( 29 , , T r z sin T T T q k i k j k k r r r =    r q q q Spherical Coordinates: ( 29 , , T r Cartesian Coordinates: ( 29 , , T x y z T T T q k i k j k k x y z =    x q y q z q In angular coordinates , the temperature gradient is still based on temperature change over a length scale and hence has units of C/m and not C/deg. ( 29 or , Heat rate for onedimensional, radial conduction in a cylinder or sphere: Cylinder 2 r r r r q A q rLq = = or, 2 r r r r q A q rq = = Sphere 2 4 r r r r q A q r q = = The Heat Equation A differential equation whose solution provides the temperature distribution in a stationary medium. Based on applying conservation of energy to a differential control volume through which energy transfer is exclusively by conduction. Cartesian Coordinates : Net transfer of thermal energy into the control volume (inflowoutflow) p T T T T k k k q c x x y y z z t + + + = Thermal energy generation Change in thermal energy storage 2...
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