Black body heat transfer

A solid spherical black body of density ρ\rho and specific heat capacity SS has a radius of RR. If the sphere is initially heated to a temperature of 400 K400 \text{ K} and suspended inside a chamber whose walls are at almost 0 K0 \text{ K}, then what is the time required for the temperature of the sphere to drop down to 200 K?200 \text{ K}?

If this value can be represented as t=abRρS,t = \dfrac ab R \rho S, where aa and bb are coprime positive integers, evaluate the value of a+ba+b.

Details and Assumptions:

  • The chamber is an insulated isothermal enclosure and thermal equilibrium is maintained during the whole process.
  • Explicitly assume that the Stefan-Boltzmann constant is σ=6×108 Wm2 K4\sigma= 6 \times 10^{-8} \text{ Wm}^{-2} \text{ K}^{-4}.
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