Heat Transfer

Thermal conductivity of Aluminum is 167 W/mK

1000 spherical ball bearings of 10-cm diameter are dumped into a 55-gallon drum of water in order to cool quickly after heat treating. The parts are initially at 800◦ C and are made from 6061-T6 Aluminum. The properties of the aluminum may be considered independent of temperature. The water is initially at 20◦ C. The properties of water are assumed to be constant with temperature. The outside of the container is insulated, so no heat is lost from the water to the surroundings during the process. However, the volume of water is sufficiently small that the water itself changes temperature substantially during the cooling process. The heat transfer coefficient between the surface of the parts and the water is 350 W/m2 -K.

A) Assume both the water and the bearings can be treated as lumped capacitances. Derive two ordinary differential equations that describe the temperature of the bearings, Tb, and the temperature of the water, T∞.

B) Prepare the two equations for further analysis by putting them in the form:

 C) Subtract the two ODEs that you derived above from each other. You should end up with a single ODE that can be expressed in terms of the temperature difference, θ = Tb − T∞

where a is a suitable constant. dT dt = a(T - Tx)