Question

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 (? = 2700 kg/m3, and c = 950J/kg

Answer 1

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  

Answer 2

Answer:

T =272.497degrees

%=34.08%

Explanation:

Please see attachment