EQUATION SHEET FOR EXAM #1 dT dx FOURIER’S LAW: q kA STEFAN-BOLTZMANN LAW: 4 q net A Ts4 Tsur where 5.67 108 NEWTON’S LAW OF COOLING: q W m K 2 hATs T HEAT DIFFUSION EQUATION (CARTESIAN COORDINATES) T T T T k k k q c p x x y y z z t HEAT DIFFUSION EQUATION (CYLINDRICAL COORDINATES) 1 T 1 T T T kr 2 k k q c p r r r r z z t HEAT DIFFUSION EQUATION (SPHERICAL COORDINATES) T T 1 2T 1 T 1 kr 2 k 2 k sin q c p 2 2 r r sin r sin t r r AREAS Surface of a Sphere A 4 r 2 Curved surface of a Circular Cylinder A 2 r L VOLUMES Volume of a Sphere Volume of a Circular Cylinder 4 r3 3 r2L ME 309 HEAT TRANSFER EXAM # 1 CLOSED BOOK AND NOTES MONDAY 27 SEPTEMBER 1999 Radioactive wastes are packed into a long, thin-walled cylindrical container. The wastes generate thermal energy nonuniformly according to the relation generation per volume, b. q o is a constant, and ro is the radius of the container. Steady state is maintained by submerging the container in a liquid that is at a. 2 q q o 1 r ro , where q is the local rate of energy T and provides a uniform convection coefficient, h. Obtain an expression for the total rate at which energy is generated in a unit length of the container and determine an expression for the temperature of the container wall. Determine the temperature distribution T r of the waste in the container in terms of the conductivity of the waste, k, and the variables named above. ro Note: The volumetric (per unit length) integral of a quantity f( r ) in a cylinder is given as: 2 f r r dr 0

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# EQUATION SHEET FOR EXAM #1