Radiative Cooling TimeThe rate of radiative energy emission from a hot surface is given by the StefanBoltzmann law . Here P is the power emitted from the area, and E is the energy contained by the object. For very hot objects, the role of the ambient temperature can be neglected. If the hot temperature is more than 3.16 times the ambient, then the contribution of ambient terms is less than 1%. For example, for 300K ambient on the earth, an object of temperature higher than 1000K can be treated like a pure radiator into space. If the heat loss is purely radiative and not limited by heat transfer to the radiating surface, then the cooling time can be modeled for a hot object. If the energy of the object can be characterized by pure translational kinetic energy according to equipartition of energy, then Using the chain rule for differentiation Rearranging gives us and integrating gives the cooling time It must be kept in mind that for macroscopic objects, the calculated cooling time for the object as a whole will always be shorter than the real cooling time, so that it gives a lower bound. The above relationship assumes infinite thermal conductivity so that the temperature of the whole object is equal to the surface temperature. In the real world, the surface will cool faster than the interior. The rate of heat transfer from the interior will be expected to limit the rate of radiative loss from the surface.

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Modeling the Radiative Cooling of a Hot SphereThe modeling of the radiative cooling time of a hot sphere can give some insights into the role of radiation in cooling hot objects. Radiation is definitely not the only mechanism involved; the cooling of real macroscopic objects is a multifaceted topic including heat transfer from the interior of the object to the surface. With this caution about the applicability of these results to the real world, a model for the radiative cooling of a sphere will be developed. The model of the cooling time is given by While perhaps instructive as an exploration, this calculation is unrealistic because of several simplifying assumptions:

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