I’m in the process of writing a couple more in-depth articles but have been much distracted by “First Life” in recent weeks.. sad and unfortunate, because writing Science of Doom articles is much more interesting..
While writing a new article – What’s the Palaver? – Kiehl and Trenberth 1997 – I thought that I should separately explain a few things which related to the earlier article: Do Trenberth and Kiehl understand the First Law of Thermodynamics? Part Three.
I know that many readers already get the point. But clearly some people find the model – and real life – so controversial that they will find many ways to claim “real life” wrong. Stefan-Boltzmann, who was he? Pyrgeometers- clearly a fake product that should be investigated by the Justice Department? And so on.
One of the problems is that radiant heat transfer is not something in accord with everyday life and so – as we all do – people draw on their own experience. But people also draw on confused ideas about the First Law of Thermodynamics to make their case.
In this article, two ideas.
First, is the Atmosphere Made of PVC?
In the original article – Do Trenberth and Kiehl understand the First Law of Thermodynamics? – I used a simple heat conduction problem to demonstrate that temperatures can be much higher inside a system than outside a system, when the system is heated from within.
One commenter explained the link between this and the atmosphere, although perhaps my attempts at humor had slightly back-fired. I had disclaimed any relationship between PVC spheres and the atmosphere..
Well, I confirm the atmosphere is not made of PVC, and that conduction is not important for heat transfer through the atmosphere.
But there is relevance for the atmosphere. Where is the relevance?
Solar radiation heats the climate system “from within”. The atmosphere is mostly transparent to solar radiation so the solar energy initially heats the surface of the earth. Then the surface of the earth heats the atmosphere. Finally the atmosphere radiates energy back out to space.
If it were true that the first law of thermodynamics – the conservation of energy – was violated by a simple “lagged pipe” model – well, that would be the end of an important branch of thermodynamics.
The model showed that the temperature of an inner surface can be higher than an outer surface – and, therefore, radiation from an inner surfaces can be higher than the radiation to space from the outer surface.
The reason for providing the model of the PVC sphere – much simpler than the atmosphere – was to demonstrate that simple point.
Second, What if the Radiation from an Inner Surface CANNOT be Higher than from an Outer Surface
Many people write entertainingly inaccurate articles about this subject (see Interesting Refutation of Some Basics for one example). Apparently, if the radiation from the atmosphere/surface into space is 239 W/m² then the radiation from the inner surface itself cannot be more than 239 W/m². A confusion about the First Law of Thermodynamics.
To be specific, the actual claim from the believers in the Imaginary First Law of Thermodynamics (IFTL) is that the total radiation from the earth’s surface cannot be higher than the total radiation from the climate system into space.
This, according to the IFLT, is not allowed.
Let’s consider the consequences and calculate the results. All we need to connect the two values – when we have the W/m² for both surfaces – is the ratio of surface areas.
If E1 is the radiation in W/m² from inner surface A1, and E2 is the radiation from the outer surface, of area A2:
E1A1 = E2A2
The area of a sphere is proportional to its radius squared (A = 4πr²), so the above equation becomes:
E1r1² = E2r2²
a) The Earth and Climate System
In the case of the Earth and climate system, the radius of the earth and the radius of the “climate system” are almost identical..
The radius of the earth, r1 = 6,380 km or 6.38 x 106 m.
The radiation to space takes place from an average height of around 6km from the surface, so the radius of “the climate system”, r2 = 6.39 x 106 m (at most).
The total radiation to space, E2 = 239 W/m² (measured by satellite).
If the IFTL believers are correct then E1r1² = E2r2²
Therefore, E1 = 239 x (6.39 x 106)² / (6.38 x 106)² = 240 W/m²
Unsurprisingly, this surface radiation value is almost the same as the radiation into space because the two areas are almost identical.
The Stefan-Boltzmann law says that radiation from a surface, E = εσT4
The “currently believed” average value from the earth’s surface is 396 W/m². This is due to the emissivity of the earth’s surface being very close to 1.
So there are three simple choices for why the “believed value” of 396 W/m² is so much higher than the believers in the IFLT appear to claim:
- The Stefan-Boltzmann law is wrong
- The emissivity of the earth’s surface, for the wavelengths under question, is an average of 0.61
- The surface temperature has been massively over-estimated and the “average” temperature of the earth’s surface is actually around -18°C (see note 1).
The 3rd choice should not be ruled out. Perhaps Antartica is a lot larger than measured, or a lot colder. How many temperature stations are there on Antarctica anyway? Maybe there is some cartographical error in estimating the area of this continent from when planes have flown over Antarctica and satellites have crossed the poles.
Perhaps the Gobi desert is a lot colder than people think. No one really makes an effort to measure this stuff, climate scientists just take it all for granted, sitting in their nice warm comfortable offices looking over the results of supercomputer climate models. No one does any field research.
Quite plausible really. It’s not too hard to make the case that the average temperature of the earth is much much much colder than is generally claimed.
b) The PVC Sphere
Let’s review the very simple hollow PVC sphere model. In the original article, the inner radius was 10m and the outer radius was 13m.
Let’s look at what happens as the inner radius is increased up to 10,000m while the wall thickness stays at 3m.
Instead of keeping the internal energy source of 30,000W constant, we will keep the internal energy source per unit area of inner surface constant. In the original example, this value was 23.9 W/m².
Real First Law
With the equations provided in the maths section of Part One, and an energy source of 23.9W/m², here is the temperature difference from inner to outer surface as the inner radius increases:
Note that the x-axis is a log scale. The initial value, 10¹ (=10) was the value from the original example, and the temperature difference was 290K.
As the sphere becomes much larger (and the wall thickness stays constant) the temperature difference tends towards 377K.
Now that is a very interesting number that we can check.
When the wall thickness becomes very thin in comparison to the sphere it is really approximating a planar wall. The equation for heat conduction (per unit area) through a planar wall is:
q = k . ΔT/Δx
where q = W/m², k = conductivity (0.19 W/m.K), ΔT = temperature difference, Δx = wall thickness, m
So for a 3m thick planar PVC wall conducting 23.9 W/m², let’s re-arrange and plug the numbers into the equation:
ΔT = 23.9 x 3 / 0.19 = 377 K
So this is a very simple test. There is no other way to link heat conduction and temperature difference. The simple equations that anyone can check support the PVC sphere model results.
Imaginary First Law
Let’s find out what happens under the imaginary first law. It will be quite surprising for the supporters of the theory.
I couldn’t check the imaginary first law in any textbooks, because it’s.. anyway, as far as I can determine, here are the steps:
1. The radiation from the inner surface must be 23.9 W/m². This means (for an emissivity, ε = 0.8 that has already been prescribed for this model) that the inner surface temperature, T1 = 151.5K (E = εσT4)
2. The inner and outer surface radiation values are related by the equations provided earlier:
E1r1² = E2r2²
3. Therefore, we can calculate the outer surface temperature and therefore the temperature difference.
Here is the graph of temperature difference as the radius increases:
Note the important point that as the radius increases the temperature difference reduces to almost nothing – this is the inevitable consequence of the (flawed) argument that inner surface radiation has to equal outer surface radiation.
Because when r1=10,000m, r2=10,003m, therefore, the areas are almost identical.
Therefore, the radiation values are almost identical, therefore the temperatures are almost identical.
Ouch. This means that somehow 23.9 W/m² is driven by heat conduction across 3m of PVC with no temperature difference.
How can this happen? Well – it can’t. To get 23.9 W/m² across a planar PVC wall 3m thick requires a temperature difference of 377 K.
When r1 = 10,000 m, ΔT = 0.02 K according to my IFTL calculations – and so the conducted heat per unit area, q = 0.0013 W/m². The heat can’t get out, which means the temperature inside increases.. and keeps increasing until the temperature differential is high enough to drive 23.9 W/m² though the wall.
Hopefully, this makes it clear to anyone who hasn’t already made a total nana of themselves that the imaginary first law of thermodynamics, is .. imaginary.
Note 1 – The concept of an average temperature is not really needed to actually do this calculation. Averaging temperatures across different surface materials like oceans, rocks, deserts clearly has some problems – see for example, Why Global Mean Surface Temperature Should be Relegated, Or Mostly Ignored.
All that is really required is to calculate the average radiation value instead. Just find the temperature at each location and calculate the emitted radiation. Then average up all the numbers (area-weighted).
As a note to the note.. To get 240 W/m² with an emissivity close to 1, the “average temperature” can be at most -18°C. With a wider day/night and seasonal variation than we actually experience on earth the “average temperature” would then be lower than -18°C.