An isolated system consists of two blocks made of the same substance (Fig 3 9) The internal energies of blocks 1 and 2 are U 1 C 1 T 1 and U 2 C 2 T 2 where C 1 and C 2 are two positive constants Two sides of the blocks face each other exactly The area of each side is A and they are separated by a ...

To determine the ultimate temperature We apply the idea that in thermal equilibrium there can be no net heat movement between the two blocks to get T ...

An isolated system consists of two blocks made of the same substance (Fig. 3.9). The internal energies

Question:

An isolated system consists of two blocks made of the same substance (Fig. 3.9). The internal energies of blocks 1 and 2 are U₁ = C₁T₁ and U₂ = C₂ T₂ where C₁ and C₂ are two positive constants. Two sides of the blocks face each other exactly. The area of each side is A and they are separated by a fixed air gap. We neglect the heat conductivity of the air in the gap. The radiative thermal power that block i exerts on block j, where i, j = 1, 2, is given by,

Figure 3.9

where σ is a constant coefficient.

(a) Determine the final temperature T_fof the system when it reaches equilibrium.
b) Derive the time evolution equation for T₁ (t) and T₂ (t).
c) Consider the particular case where C₁ = C₂ = C and the limit of small temperature variations, i.e. T₁ (t) = T_f +ΔT₁ (t) and T₂ (t) = T_f +ΔT₂ (t) with ΔT₁ (t) ≪ T_fand ΔT₂ (t) ≪ T_f at all times. Show that the temperature difference ΔT (t) = ΔT₁ (t) − ΔT₂ (t) is exponentially decreasing.