Transient (unsteady) heat transfer over a sphere

[diegojolin]

I AM DOING A PROBLEM ABOUT TRANSIENT HEAT TRANSFER AND IT SAYS:

"A solid sphere of steel has 1 cm of diameter and a initial temperature of T0 =15 ºC. It is placed into a flow of Tinf = 60 ºC where the convection heat coefficient is h = 2000 W/(m^2K):

(Density of steel rho = 7832 Kg/m^3 ; Specific heat c= 434 J / (Kg K) ; Thermal Conductivity of Steel Ks = 63.9 W / ( m K )

1. First develop an expression for the variation with time of the temperature of the sphere (Lumped Capacitance method).

2. Check that the assumptions made to use that method were right and give the order of magnitude of the temperature gradients inside the sphere

3. Compute the time required for the sphere to get to a value of 50ºC when a plastic layer of 4 mm of thickness is placed over the sphere. The thermal conductivity of this layer is 0.3 W/ (m K) "

I have been able to integrate the transient formula this way :

b = (h*A)/(rho*V*c) = 0.354

where A = 4*pi*r^2 and V = 4/3 * pi * r^3

( T(t) - Tinf ) / (T0 - Tinf) = exp ( -b * t)

This expression seems to be valid since my professor has integrated himself several times at class.

To check the assumptions I just computed the Biot number and I saw that I has a value of 0.15 which more or less acceptable.

But how can I give an estimate of the order of magnitude of the temperature gradients??

and even more, how changes this problem when I add a plastic layer? how can I compute the value required?

 

[LawrenceC]

For part 3, note that the density and specific heat were not supplied for the plastic coating. Therefore the only thing left to do is to modify the supplied heat transfer coefficient to compensate for the added resistance of the coating. It has the effect of lessening the Biot number.