Introduction "Thermal energy storage systems commonly involve a packed bed of solid spheres, through which a hot gas flows if the system is being charged, or a cold gas if it is being discharged. In a charging process, heat transfer from the hot gas increases thermal energy stored within the colder spheres; during discharge, the stored energy decreases as heat is transferred from the warmer spheres to the cooler gas."

Consider a packed bed of $75-mm-$diameter aluminum spheres $.$ K$,k=240\frac{W}{m}.K)$ and a charging process for which gas enters the storage unit at a temperature of $Tg,i=300C.$ If the initial temperature of the spheres is $Ti=25C$ and the convection coefficient is $h=75\frac{W}{m^2}.$ K$,$ how long does it take a sphere near the inlet of the system to accumulate $90\%$ of the maximum possible thermal energy? What is the corresponding temperature at the center of the sphere?