A small sphere (emissivity =0.90, radius =r1 ) is located at...

A small sphere (emissivity $=0.90,$ radius $=r_{1}$ ) is located at the center of a spherical asbestos shell (thickness $=1.0 \mathrm{~cm},$ outer radius $=r_{2}$ ). The thickness of the shell is small compared to the inner and outer radii of the shell. The temperature of the small sphere is $800.0^{\circ} \mathrm{C},$ while the temperature of the inner surface of the shell is $600.0^{\circ} \mathrm{C},$ both temperatures remaining constant. Assuming that $r_{2} / r_{1}=10.0$ and ignoring any air inside the shell, find the temperature of the outer surface of the shell.

Key Concepts

Stefan?Boltzmann Law

This law relates the power radiated per unit area of a body to the fourth power of its absolute temperature. It is fundamental in problems of radiative heat transfer because it allows one to calculate the radiant energy flux from surfaces, taking into account their temperature and, after adjustment, the emissivity. In problems involving radiative exchange between objects, like concentric spheres, this law becomes essential in establishing energy balances.

Emissivity

Emissivity is a material property that measures how effectively a real surface emits radiation compared to an ideal black body which has an emissivity of 1. It is a crucial parameter in radiative heat transfer problems because it affects the net radiated energy. Accounting for emissivity ensures that the calculated radiative flux reflects the actual performance of the surface in transferring thermal energy.

Radiative Heat Exchange in Enclosures

When dealing with objects enclosed within another, such as a small sphere inside a spherical shell, the geometry simplifies the radiative heat exchange calculation because the view factors are typically unity. This means that nearly all the radiation leaving one surface reaches the other, and the energy transfer can be directly related to the temperatures and areas of the surfaces. This concept is vital for setting up the balance between emitted and absorbed radiation in a closed enclosure.

Thermal Conduction Through a Thin Shell

In cases where a solid shell separates high and low temperature regions, thermal conduction governs the heat transfer across the material. When the shell thickness is small compared to its other dimensions, the temperature gradient can be approximated as constant, simplifying the conduction analysis. This concept connects the inner radiative energy input to the outer radiative output by determining how the shell material conducts heat, and thus influences the outer surface temperature.

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