Question

A spherical object (50 cm diameter, c=0.46 kJ/kg*C, k = 42 W/m*C, density = 7800 kg/m3 ) is initially at a uniform temperature of 450 C. It is suddenly placed in a constant temperature environment of 100 C. The convection transfer coefficient is 500 W/m2*C. How long would it take for temperature in the center of the sphere to cool down to 150 C? Note: Bi for a sphere is (hD)/(6k).Suggestion: First calculate Biot number, then determine if T is a function of radius and time or simply time alone.

Answer 1

For unsteady state heat transfer,

Biot Number is given by

d = 50 cm = 0.5m

For sphere,

As Bi < 0.1, the body can be assumed to be a lumped capacity model.

Therefore,

On heat balance,

Heat lost by the copper wire = - heat accumulation

On integrating,

Ts = Surrounding temperature

To = Initial temperature of the body

T(t) = Temperature of the body at time t

.........(Cp changed to J/kg-C for unit consistency)

.................................Answer