Radiation, Convection, Conduction

There's a song by Flanders and Swann called First and Second Law. The third verse is:

Heat is work and work is heat
And work is heat and heat is work
Heat will pass by conduction
And heat will pass by convection
And heat will pass by radiation
And that's a physical law

Indeed. But conduction, convection, and radiation get so intimately mixed together that its sometimes hard to say which is which. Convection, for example, always begins with heat passing into a gas by conduction. Without conduction there would be no convection. The converse is not true. And there are cases where convection and radiation look exactly like conduction. One such case is the flow of heat through the filter gas we introduced in our previous post.

Our calculations showed a net 350 W/m² of heat flowing up through the 1 km Lower Gas as a result of The Body being 6.4°C warmer than the lower surface of the Filter Gas. Of this 350 W/m², 80% passed by convection and 20% was the imbalance in the long-wave radiation passing between The Body and the Filter Gas. Suppose we cover The Body with a thin sheet of a material that reflects long-wave radiation. By radiative symmetry, The Body no longer emits radiation to the Filter Gas. Nor does it absorb radiation from the Filter Gas. Referring to our earlier diagram, we have E_B = E_FL = 0 W/m², so for equilibrium at The Body we must have

Q_LG = k(T_B−T_FL)/D_LG = E_S = 350 W/m².

With k = 44 kW/mK and D_LG = 1 km we get T_B−T_FL = 8.0°C. When we stop radiation between the Filter Gas and The Body, The Body warms up by 1.6°C. Heat flows by convection only.

But how does heat flow through the Filter Gas? This is where our confusion begins. Certainly, we will have heat flow by convection. But there is also radiation between layers of the Filter Gas. Do we consider this radiation within the Filter Gas to be conduction, or is it something separate? Suppose we cover the lower and upper surfaces of the Filter Gas with a film that is reflecting on the side facing the filter gas, but black on the side facing away from the filter gas. These films absorb and emit long-wave radiation on their outer surfaces, but neither absorb nor emit radiation on their inner surfaces. The Filter Gas itself is aware only of the temperature of the Upper and Lower Films. It receives heat by conduction at the Lower Film. This heat passes up through the Filter Gas by convection, conduction, and also radiation, all operating together, until it arrives at the Upper Film. The heat passes by conduction into the Upper Film and is radiated out into space by the outer surface.

With these films in place, we know that no heat is entering or leaving the Filter Gas by radiation. If the temperature difference across the Filter Gas is 80°C, the distance between the films is 10 km, and 350 W/m² flows between the films, it looks as if we have a conducting substance between the films with specific conductivity, k, equal to 44 kW/mK. In theory, we could replace our Filter Gas with some other substance with the same specific conductivity.

In practice, there is no substance with specific conductivity even close to 44 kW/mK. The specific conductivity of aluminum is around 200 W/mK. Diamond is around 2 kW/mK. Air is 0.025 W/mK. Specific conductivity refers to the passage of heat through stationary material. To measure the specific conductivity of the Filter Gas, we could use a body of gas that is filled with goose down, so that the gas cannot move up or down. Stationary air is a fine insulator.

But air allowed to move freely in a large volume is a fine transporter of heat. The coefficient of convective heat transfer for air is between 10 and 100 W/m²K, with no mention of the height of the column of air. If we used 10 W/m²K for our Filter Gas, a 35°C temperature difference across the 10-km thickness of the Filter Gas would be sufficient to cause 350 W/m² to flow. As it is, we assumed an 80°C difference, for similarity with the Earth's atmosphere. We see that convection gives our gas layers the power to transport hundreds of times as much heat as an equivalent thickness of metal.

Does a black gas transport more heat than a transparent gas? Our Filter Gas has some absorption length with respect to long-wave radiation, so consider a layer of Filter Gas that is much thinner than the absorption length. The gas below will radiate upwards and the gas above will radiate downwards. The upward radiation will be greater than the downward radiation because the gas below is warmer than the gas above. A black gas transports more heat than a transparent gas.

How much more heat does a black gas transport? If we consider the contribution of radiation to heat transport from The Body to the Filter Gas for a temperature difference of 6.4°C, we see that radiation adds 20% to the total heat transport. A black gas transports slightly more heat than a transparent gas, but not twice as much.

We can simulate the absorption length of our Filter Gas with a transparent layer of gas on the bottom and the top, which is exactly what we do in our latest model. The Lower Gas represents the absorption length of the warm, dense, lower surface of the Filter Gas, while the Upper Gas represents the absorption length of the cold, sparse, upper surface of the Filter Gas.