5.12
KNOWN: PROBLEM Diameter, density, specific heat and thermal conductivity of aluminum spheres used in packed bed thermal energy storage system. Convection coefficient and inlet gas temperature. FIND: Time required for sphere to acquire 90% of maximum possible thermal energy and the corresponding center temperature. Potential advantage of using copper in lieu of aluminum. SCHEMATIC:
ASSUMPTIONS: (1) Negligible heat transfer to or from a sphere by radiation or conduction due to contact with other spheres, (2) Constant properties. ANALYSIS: To determine whether a lumped capacitance analysis can be used, first compute Bi = 2 h(ro/3)/k = 75 W/m K (0.025m)/150 W/mK = 0.013 < 0.1. Hence, the lumped capacitance approximation may be made, and a uniform temperature may be assumed to exist in the sphere at any time. From Eq. 5.8a, achievement of 90% of the maximum possible thermal energy storage corresponds to
Q = 0.90 = 1 exp ( t / t ) cV i
where t = Vc / hA s = Dc / 6h = 2700 kg / m 3 0.075m 950 J / kg K / 6 75 W / m 2 K = 427s. Hence,
t = t ln ( 0.1) = 427s 2.30 = 984s
From Eq. (5.6), the corresponding temperature at any location the in sphere is
<
T ( 984s ) = Tg,i + Ti Tg,i exp ( 6ht / Dc )
T ( 984s ) = 300C 275C exp 6 75 W / m K 984s / 2700 kg / m 0.075m 950 J / kg K
2 3
(
)
(
)
<
3
T ( 984 ) s = 272.5C
Obtaining the density and specific heat of copper from Table A-1, we see that (c)Cu 8900 kg/m 3 6 3 6 400 J/kgK = 3.56 10 J/m K > (c)Al = 2.57 10 J/m K. Hence, for an equivalent sphere diameter, the copper can store approximately 38% more thermal energy than the aluminum.
COMMENTS: Before the packed bed becomes fully charged, the temperature of the gas decreases as it passes through the bed. Hence, the time required for a sphere to reach a prescribed state of thermal energy storage increases with increasing distance from the bed inlet.
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