As we have seen before, in air systems, pebble beds are usually employed for energy storage. When solar radiation is available, hot air from the collectors enters the top of the storage unit and heats the rocks. As the air flows downward, heat transfer between the air and the rocks results in a stratified distribution of the pebbles, having a high temperature at the top and a low one at the bottom. This is the charging mode of the storage unit. When there is heating demand, hot air is drawn from the top of the unit and cooler air is returned to the bottom of the unit, causing the bed to release its stored energy. This is the discharge mode of the pebble bed storage unit. From this description, it can be realized that the two modes cannot occur at the same time. Unlike water storage, the temperature stratification in pebble bed storage units can be easily maintained.
In the analysis of rock bed storage, it should be taken into account that both the rocks and air change temperature in the direction of airflow and there are temperature differentials between the rocks and air. Therefore, separate energy balance equations are required for the rocks and air. In this analysis, the following assumptions can be made:
1. Forced airflow is one-dimensional.
2. System properties are constant.
3. Conduction heat transfer along the bed is negligible.
4. Heat loss to the environment does not occur.
Therefore, the thermal behavior of the rocks and air can be described by the following two coupled partial differential equations (Hsieh, 1986):
A = cross-sectional area of storage tank (m2). Tb = temperature of the bed material (°C).
pb = density of bed material (kg/m3).
cb = specific heat of bed material (J/kg-K).
x = position along the bed in the flow direction (m). m = mass flow rate of air (kg/s).
e = void fraction of packing = void volume/total volume of bed (dimensionless). hv = volumetric heat transfer coefficient (W/m3-K).
An empirical equation for the determination of the volumetric heat transfer coefficient (hv) is hv = 650(G/d )07 (5.41)
G = air mass velocity per square meter of bed frontal area (kg/s-m2). d = rock diameter (m).
If the energy storage capacity of the air within the bed is neglected, Eq. (5.40) is reduced to mm Ca = -Ahv (Ta - Tb) (5.42)
Equations (5.39) and (5.42) can also be written in terms of the number of transfer units (NTU) as dT
The dimensionless number of transfer units (NTU) is given by h AT
The parameter 9, which is also dimensionless in Eq. (5.43), is equal to
For the long-term study of solar air storage systems, the two-coupled partial differential equations, Eqs. (5.43) and (5.44), can be solved by a finite difference approximation with the aid of a computer.
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