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One may still wish to use lumped system analysis even

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Unformatted text preview: of this approach. One may still wish to use lumped system analysis even when the criterion Bi 0.1 is not satisfied, if high accuracy is not a major concern. Note that the Biot number is the ratio of the convection at the surface to conduction within the body, and this number should be as small as possible for lumped system analysis to be applicable. Therefore, small bodies with high thermal conductivity are good candidates for lumped system analysis, especially when they are in a medium that is a poor conductor of heat (such as air or another gas) and motionless. Thus, the hot small copper ball placed in quiescent air, discussed earlier, is most likely to satisfy the criterion for lumped system analysis (Fig. 4–6). Some Remarks on Heat Transfer in Lumped Systems To understand the heat transfer mechanism during the heating or cooling of a solid by the fluid surrounding it, and the criterion for lumped system analysis, consider this analogy (Fig. 4–7). People from the mainland are to go by boat to an island whose entire shore is a harbor, and from the harbor to their destinations on the island by bus. The overcrowding of people at the harbor depends on the boat traffic to the island and the ground transportation system on the island. If there is an excellent ground transportation system with plenty of buses, there will be no overcrowding at the harbor, especially when the boat traffic is light. But when the opposite is true, there will be a huge overcrowding at the harbor, creating a large difference between the populations at the harbor and inland. The chance of overcrowding is much lower in a small island with plenty of fast buses. In heat transfer, a poor ground transportation system corresponds to poor heat conduction in a body, and overcrowding at the harbor to the accumulation of heat and the subsequent rise in temperature near the surface of the body relative to its inner parts. Lumped system analysis is obviously not applicable when there is overcrowding at the surface. Of course, we have disregarded radiation in this analogy and thus the air traffic to the island. Like passengers at the harbor, heat changes vehicles at the surface from convection to conduction. Noting that a surface has zero thickness and thus cannot store any energy, heat reaching the surface of a body by convection must continue its journey within the body by conduction. Consider heat transfer from a hot body to its cooler surroundings. Heat will be transferred from the body to the surrounding fluid as a result of a temperature difference. But this energy will come from the region near the surface, and thus the temperature of the body near the surface will drop. This creates a temperature gradient between the inner and outer regions of the body and initiates heat flow by conduction from the interior of the body toward the outer surface. When the convection heat transfer coefficient h and thus convection heat transfer from the body are high, the temperature of the body near the surface will drop quickly (Fig. 4–8). This will create a larger temperature difference between the inner and outer regions unless the body is able to transfer heat from the inner to the outer regions just as fast. Thus, the magnitude of the maximum temperature difference within the body depends strongly on the ability of a body to conduct heat toward its surface relative to the ability of h = 15 W/m2 ·°C Spherical copper ball k = 401 W/ m·°C D = 12 cm 1 3 V – πD 1 6 Lc = — = —— = – D = 0.02 m 2 6 As π D hLc 15 × 0.02 Bi = —– = ———— = 0.00075 < 0.1 k 401 FIGURE 4–6 Small bodies with high thermal conductivities and low convection coefficients are most likely to satisfy the criterion for lumped system analysis. Boat Bus ISLAND FIGURE 4–7 Analogy between heat transfer to a solid and passenger traffic to an island. T = 20°C 50°C 70°C 85°C 110°C 130°C Convection h = 2000 W/ m2 ·°C FIGURE 4–8 When the convection coefficient h is high and k is low, large temperature differences occur between the inner and outer regions of a large solid. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 214 214 HEAT TRANSFER the surrounding medium to convect this heat away from the surface. The Biot number is a measure of the relative magnitudes of these two competing effects. Recall that heat conduction in a specified direction n per unit surface area is · expressed as q k T/ n, where T/ n is the temperature gradient and k is the thermal conductivity of the solid. Thus, the temperature distribution in the body will be uniform only when its thermal conductivity is infinite, and no such material is known to exist. Therefore, temperature gradients and thus temperature differences must exist within the body, no matter how small, in order for heat conduction to take place. Of course, the temperature gradient and the thermal conductivity are inversely proportional for a given heat flux. Therefore, the larger the thermal conductivity, the smaller the temperature gradient. EXAMPLE 4–...
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