Thermal energy storage systems commonly involve a packed bed of solid spheres through which a hot gas flows when the system is being charged. In the charging process heat is transferred from the hot gas to the spheres and it increases the thermal energy stored in the spheres. Here are some variables that are important - the convection coefficient between the particle surface and the gas is h (W/m^2 K) and the particle thermophysical properties are k (W/m K), c_p (J/kg K) and p (kg/m^3). The radius of the particle is R (m). Initially the particles are at a temperature T_i (degree C). The gas flow is started at t=0, and the gas temperature is at T_infinity (degree C) - can be assumed a constant with time.

a. Let the gas flow through the particles for a time t (s). Develop an expression for the total heat transferred to a particle Q (units of J) in this time t. Develop an expression for Q in terms of the variables given. Assume lumped system analysis holds. What is the maximum possible thermal energy (Q_max) a particle can accumulate?

b. Consider a packed bed of 75 mm diameter aluminum spheres (rho = 2700 kg/m^3, c_p= 950 J/kg K, k = 240 W/m K) and a charging process in which gas enters the storage unit at a temperature of T_infinity = 300 degree C. If the initial temperatures of the sphere is T_i = 25 degree C and the convection coefficient is h = 75 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?.