In Part One we looked at a thought experiment designed to make it easier to understand a basic principle – that inner surfaces of systems can have higher temperatures and therefore higher values of radiation than the outside of a system.
We looked at a PVC hollow sphere out in the vastness of space:
The only reason for creating that experiment in its unreal environment was to make the basic physics and the corresponding maths easier to follow.
In that example, with the heat source of 30,000 W, the final equilibrium values (with emissivity = 0.8) were:
T1 = 423 K and T2 = 133 K
In this case the outer wall radiates 30,000 W out into space and so the system is in equilibrium.
Yet the inner wall is radiating 1,824,900 W.
You can see the maths in Part One. From a few comments on this blog, and some comments I saw elsewhere, clearly many people still have problems with it.
That’s a good thing. It’s good because it means that the simple problem is doing its job. It is simple enough that almost everyone realizes the calculated temperatures are correct. It is simple enough that the maths can be followed. But it exposes an idea that many can’t accept – total radiation from a surface inside the system is much higher than the energy source.
The solution is quite simple.
One of the commenters, John N-G, pointed out:
Your point might be driven home more emphatically by noting that the inner surface absorbs only 23.8 W/m2 from the super-light-bulb, yet in the 3m-thick example it emits the afore-mentioned 1452 W/m2 (and also absorbs 1452 W/m2 of wall-emitted radiation).
The inner surface is also in energy balance.
We can consider the values as total energy per second (W), or as energy per second per unit area, (W/m2). We will stay with the total energy because these are the values we have already been working with.
- Energy in = 30,000 W (from the energy source) + 1,824,900 W (from the inner surface surface)
- Energy out = 30,000 W (conducted through the PVC wall to the outer surface) + 1,824,900 W (radiated from the inner surface)
No energy is created or destroyed in the system encompassed by the inner surface.
From some comments related to this and other articles, explaining how equilibrium is reached might help. Once again, the 1st law of thermodynamics is used to calculate the dynamic situation. If there is net energy added to an element of the system then its temperature increases.
The approach to the calculation is a simple numerical model, which divides the sphere wall radially (into 50 equal “slices”):
The inner wall receives 30,000 W. It radiates an amount dependent on temperature but also receives that same amount from the inner wall surface (therefore the net heat received is always = 30,000 W).
The outer wall radiates according to its temperature.
And for all of the “slices” of the wall in between the heat flow is according to the temperature differential (equation in Part One), and the heat gained is according to the simple equation:
dQ = mc.dT – where dQ = change in energy, m = mass, c= specific heat capacity and dT = change in temperature
Spherical symmetry makes the calculation much easier than if it was a box.
If the first law of thermodynamics meant that no inner surface could radiate at a higher value than the outer surface of the system then everything would be at the same temperature.
In the PVC hollow sphere example the inner wall would have to be at 133K – the same as the outer wall. This would mean that no heat could flow from the inner wall to the outer wall – or that Fourier’s law was wrong. And lagging hot water pipes was also “a big con”.
We all know that is not the case – well, maybe not everyone has heard of Fourier, but everyone knows about insulation.
The reason why so many people think that Trenberth and Kiehl’s diagram is flawed is because of an incorrect understanding of the first law of thermodynamics.
If Trenberth and Kiehl’s diagram violates the first law of thermodynamics then so does my PVC sphere. There’s just the small matter of trying to explain how the inner surface would stay at 133K.
At least there is one Get Out of Jail Free card. As a commenter on another blog put it:
The atmosphere cannot both behave like a PVC blackbody and an ideal gas
Analogies prove nothing. But for those brave enough to consider that they might be wrong, I hope the PVC hollow sphere provides some illumination.