# A hemispherical, unheated metal shell is currently being used as a large-scale dome shelter at…

A hemispherical, unheated metal shell is currently being used as a large-scale dome shelter at the South Pole in Antarctica (see Fig. 2 shown below). It is desired to estimate the steady-state temperature profile in the shell during the summer months in order to determine the extent to which uneven thermal expansion of the shell occurs in this extreme environment.Determine an expression for the temperature profile in the shell in terms of the thermal conductivity of the shell material (k), the heat transfer coefficient from the shell to the outside environment (h) (which is assumed not to vary with position on the shell), the shell thickness (d), the dome diameter (D), and any other physical properties needed. You may assume that the heat absorbed by the shell per unit surface area from solar radiation is given by the relation q cos (?), where q is the intensity of absorbed solar radiation per unit surface area on a horizontal surface and ? is the angle shown in Fig. 2. Neglect radial direction gradients in temperature in the shell (i.e., the temperature is a function of ? only) and assume that heat transfer to the air inside the shell is negligible compared with heat transfer to the outside air. Also assume that the shell temperature where the shell contacts the ground is given by the ground temperature T0, which is the same as the outside air temperature. Your final answer should give T in terms of ? with the various physical parameters given above (such as k, h, etc.) used as parameters in your result.Hint: Assume the temperature in the shell is given by T – To = A * cos(?) where A is a function of the various physical parameters given above (such as k, h, etc.). Confirm that this assumed form of the solution solves the governing differential equation and boundary conditions, and in the process of doing so determine A in terms of k, h, etc.

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