The lumped capacitance formulation can be used when a system is being cooled by thermal radiation, too. Consider a spherical object of radius \(1 \mathrm{~m}\) in low Earth orbit at an initial temperature of \(T_{i}=600 \mathrm{~K}\) being exposed to the vacuum of space where the temperature is effectively \(T_{\infty}=100 \mathrm{~K}\). The emissivity of the object is \(\varepsilon=0.95\), its thermal conductivity is \(k=350 \mathrm{~W} / \mathrm{m} \mathrm{K}\), its density is \(2000 \mathrm{~kg} / \mathrm{m}^{3}\) and its heat capacity is \(500 \mathrm{~J} / \mathrm{kg} \mathrm{K}\).

a. Derive the differential equation governing the temperature of the object, assuming that the lumped capacitance formulation is valid.

b. Unlike convection, there is no clean Biot number that falls out of the equation. However, we can define one using the radiation heat-transfer coefficient, where \(\bar{T}\) is the average temperature of the sphere over the time period of interest.

\[h_{\text {rad }}=\sigma^{r} \varepsilon^{r}\left(\bar{T}^{2}+T_{\infty}^{2}\right)\left(\bar{T}+T_{\infty}\right)\]

At the initial phase of operation, is the lumped capacitance formulation valid? At what initial object temperature would the lumped capacitance formulation become feasible?

c. Assuming the lumped capacitance formulation is valid over the entire range of cooling, how long would it take to cool the object from \(500 \mathrm{~K}\) to \(300 \mathrm{~K}\) ?