Chapter 5, Problem 5.12P

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 $\left(\rho =2700{\text{kg/m}}^{\text{3}},c=950\text{J/kg}\cdot \text{K,}k=240\text{W/m}\cdot \text{K}\right)$ and a charging process for which gas enters the storage unit at a temperature of $T_{g,i}=300°\text{C}\text{.}$ . If the initial temperature of the spheres is $T_i=25°\text{C}$ and the convection coefficient is $h=75{\text{W/m}}^{\text{2}}\cdot \text{K,}$ 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? Is there any advantage to using copper instead of aluminum?