Question

The shell of a space station is a blackened sphere in which a temperature T = 500 K is maintained due to the operation of appliances of the station. The amount of heat given away from a unit surface area is proportional to the fourth power of thermodynamic temperature.

Determine the temperature Tx of the shell if the station is enveloped by a thin spherical ***** screen of nearly the same radius as the radius of the shell.

Answer 1

The total amount of heat Q emitted in space per unit time *** unchanged since it is determined by the energy liberated during the operation of the appliances of the station. Since only the outer surface of the screen emits into space (this radiation depends only on its temperature), the temperature of the screen must be equal to the initial temperature T = 500 K of the station. However, the screen emits the same amount of heat Q inwards. This radiation reaches the envelope of the station and is absorbed by it. Therefore, the total amount of heat supplied to the station per unit time is the sum of the heat Q liberated during the operation of the appliances and the amount of heat Q absorbed by the inner surface of the screen, i.e. is equal to 2Q. According to the heat balance condition, the same amount of heat must be emitted, and hence

where Tx is the required temperature of the envelope of the station. Finally, we obtain