Answer / Explanation
Consider a packed bed of 75-mm-diameter aluminum spheres (ρ=2700kg/m3,c=950J/kg⋅K,k=240W/m⋅K) and a charging process for which gas enters the storage unit at a temperature of Tg,i=300∘C.
If the initial temperature of the spheres is Ti=25∘C and the convection coefficient is h=75W/m2⋅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?
To start answering this question, we first define the parameters:
diameter aluminum spheres = 75 mm
charging process for which gas enters the storage unit at a temperature of
Tg,i=300∘C.
If we now recall the equation for internal energy storage, we have:
Q / ρcVθi = 0.90
Where:
1 - exponential ( -t / Tt )
Where:
Tt = ρVc/hAₓ = ρDc / 6h = 2700kg/m³ x 0.075m x 950joule /kg . k/6 x 75 watt/m² . k = 427 sec.
Hence:
t = - Tt In (0.1)
= 427 s x 2.30
= 984 sec.
Now, considering that the corresponding temperature at any point in the sphere is 984
Therefore:
T(984) = Tg,i + ( Ti - Tg,i ) exponential ( - 6h / ρDc )
Hence,
T( 984s) = 300⁰c - 275⁰c exponential ( -6 x 75 watt/m² . K x 984 s / 2700kg/m³) x (0.075 m x 950 joule/kg .K)
T( 984s) = 272.5⁰c
Therefore, we can conclude that prior to the packed bed becoming fully charged, there is a decrease in the temperature of the gas as it navigates through the bed. Also, there is an increase in the time required for the sphere to reach the delegated state of thermal energy storage with an increasing distance from the bed inlet.
So, in summary,
The time it take a sphere near the inlet of the system to accumulate 90% of the maximum possible thermal energy = 984 seconds and the corresponding center temperature = 272.5⁰c
