Extreme Greenhouse, Part Two

According to atmospheric science sites like this one, atmospheric temperature drops by roughly 80°C as we ascent from altitude 0 km to 10 km. The drop is almost linear, meaning it has constant gradient −8°C/km. At 10 km the temperature stops dropping. The region in which the temperature drops linearly with altitude is the troposphere. Above that is the stratosphere. The boundary between the two is the tropopause.

Consider again our diagram of the extreme greenhouse (we invite you to open the diagram in a new window so you can look at it while you read on). In our previous post, we assumed the filter was a perfect conductor. Let's assume now that the filter presents some thermal resistance. We need T_FL > T_FU so that heat will flow up through the filter to the top surface, where it will be radiated into space. We have,

Q_F = K_F(T_FL-T_FU) = E_FU = E_S,

where Q_F is the heat passing up through the filter. The temperature within the filter will have a constant, negative gradient with altitude. In the case of the Earth and the Sun, we have already shown that E_S (the heat from the sun) is 350 W/m², so that T_FU = 280 K. Now suppose K_F = 350 Wm⁻² / 80 K = 4.4 W/m²K, which will give us an 80°C drop in temperature through the filter, the same drop we see going up through the troposphere. Now we see that T_FL = 360 K. By Stefan's Law, E_FL = 960 Wm⁻². For thermal equilibrium at the body surface, we must have E_B = E_FL + E_S = 1.3 kWm⁻², which implies that we have T_B = 390 K = 117 °C.

Let us imagine the filter in our extreme greenhouse is close to the surface of our body. Recall that we assumed a vacuum between the filter and the body, so that heat transfer between the two must take place by radiation alone. With our chosen value of filter conductivity, we see that the presence of the filter warms our body from 280 K to 390 K, a total of 110 K.

Next we will consider what happens if we allow the filter to come into physical contact with the body, so that heat can pass between the two by conduction, convection, and evaporation as well as by radiation. In that case, we will see that the filter is less able to warm the body.