A long cylinder of radius R1 and emissivity ϵis positioned inside an evacuated enclosure whose w

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A long cylinder of radius R1 and emissivity ϵis positioned...

A long cylinder of radius $R_1$ and emissivity $\epsilon$ is positioned inside an evacuated enclosure whose walls are held at temperature $T_0$ and are black. The cylinder attains a steady-state temperature $T_1$ when power $W_1$ per unit length is dissipated in it. Derive an expression for $T_1^*$ in terms of Stefan's constant and the parameters specificd. An intermediate cylinder of external radius $R_2=1.2 R_1$, whose surfaces are black, is next suspended in the enclosure, surrounding the inner cylinder and centred on the same axis. It is now found that the power dissipation in the inner cylinder needed to maintain it at the same steadystate temperature $T_1$ is reduced to $75 \%$ of its former value. Deduce the value of $\epsilon$.

Key Concepts

Stefan-Boltzmann Law

This law states that the power radiated per unit area by a black body is proportional to the fourth power of its absolute temperature, with the proportionality constant being the Stefan’s constant. It provides the fundamental relationship between temperature and radiative power, which is crucial for understanding how bodies at different temperatures exchange thermal radiation.

Emissivity

Emissivity is a measure of a material's efficiency in emitting thermal radiation relative to that of a perfect black body. It is a dimensionless factor between 0 and 1, and it plays a central role in radiative heat transfer, affecting the net radiation exchanged between surfaces and thus influencing thermal equilibrium.

Radiative Heat Transfer

Radiative heat transfer involves the exchange of energy between bodies through electromagnetic radiation. This concept is essential in scenarios where conduction and convection are negligible, as it encompasses the net emission and absorption of thermal radiation based on temperature, surface properties, and geometry.

Geometrical Effects and View Factors

The geometry of the surfaces involved, including their size, shape, and relative positioning, critically influences the amount of radiation exchanged. View factors quantify the fraction of radiation leaving one surface that directly reaches another, which is important for calculating net radiative exchange in complex systems.

Thermal Equilibrium in Radiative Systems

Thermal equilibrium is achieved when the power dissipated in a system is balanced by the power radiated away, leading to a steady-state temperature. This concept underpins the analysis of systems where changes in geometry or surface properties modify the radiative heat exchange and the conditions required to maintain a given temperature.

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