When I wrote Do Trenberth and Kiehl understand the First Law of Thermodynamics? I imagined that (almost) no one would have a problem with the model created. Instead, I thought perhaps some might question its relevance to climate.
It was a deliberate choice to use conduction to demonstrate the point – the reason is that radiation is less familiar to most people, while conduction is more straightforward and easier to understand.
Here is the model from that article – a heat source in a hollow PVC sphere, located in the depths of space:
Many people have experienced a lagged hot water pipe. The more lagging (insulation), the higher the temperature rises. It seems straightforward.
However, the conceptual barrier that some people have is so large that anything – literally – will be put forward to make the model fit their conceptual idea. In case the case of one blog, claiming that energy can be destroyed in an effort to get the “right” result. A delicious irony that the first law of thermodynamics is cast aside to protect.. the first law of thermodynamics.
The reason this PVC sphere model appears so wrong to many people is for similar reasons that the famous Kiehl & Trenberth diagram seems wrong – the radiation “internally” (earth surface) is higher than the external radiation to space. (Note that the radiation values in the K&T diagram can be measured).
Explaining How the Result is Calculated
..in simple terms.
Solving the maths for the model above is straightforward (refer to the first article for the actual maths). Here is the solution in simple terms:
For the steady state condition the energy radiated from the outer surface must equal the energy source in the center (30,000 W). Otherwise the system will keep accumulating energy.
Given the surface area and the stated emissivity the outer surface temperature (T2) must be 133K (to radiate 30,000 W).
The only way that heat can be transferred from the inner surface to the outer surface is through conduction. This means 30,000 W is conducted through the PVC.
Given the (low) thermal conductivity of PVC and the dimensions, the temperature difference must be 290K, making T1 = 423K.
If the temperature differential is any lower then less than 30,000W will be conducted through the wall. And if that was the case then heat would be accumulated at the inner surface – increasing its temperature until eventually 30,000W did flow through.
Conversely, if the temperature was higher than 423K then more than 30,000W would be conducted through the sphere. This would start to reduce the temperature until only 30,000W was conducted.
Simple really. However, when the result doesn’t seem right, people begin their mental gyrations to get the “right” result.
This article is not written to convince people who have their minds made up. It’s written to help those who are asking the legitimate question:
Haven’t you just created energy? And can’t I use that to run a small power station?
This article is not about proving what has already been demonstrated, it’s about helping with mental models.
- the argument from incredulity
- 3m of PVC can transmit radiation straight through (no it can’t)
- energy disappears under the right circumstances (that was just the first of many flaws in that person’s argument..)
Of course, if someone comes up with yet another alternative calculation of the heat transfer I will be happy to look at it.
In the meantime, let’s create a mental model..
The Power Station
A few people have jubilantly claimed that the model I created, if correct, can run a power station of 1.8 MW, from a source of only 30,000 W.
That’s what it might seem like on the surface. But strangely, the model results were derived by conserving energy. That is, no energy was created or destroyed..
In the steady state condition:
- 30,000 W is produced from the internal source
- 30,000 W is conducted through the PVC “wall”
- 30,000 W is radiated from the outer surface
Energy is not being created or destroyed. Where is the energy accumulation in this model? Where is the usable energy being stockpiled?
- If you want to understand the subject, this point is the one to focus on and think about
- If you don’t want to understand the subject say “he’s created 1.8 MW of energy from 30,000W – ridiculous”, and move on (it sounds good)
The inner surface of the sphere has an area of 1,257 m² (4πr²). Consider one square meter of internal surface, we’ll call it “A” – these kind of models always have catchy names for different components of the model.
- Each second, A receives 23.9 W/m² from the internal heat source (30,000W / 1,257 m²).
- Each second, A conducts 23.9 W/m² through the wall.
- Each second, A absorbs 1,452 W/m² radiated from the rest of the inner wall.
- Each second, A re-radiates 1,452 W/m².
This is another way of saying that no energy is being created or destroyed. Where is the energy to run this power station?
All that happens if we start drawing power out of this system is the temperature internally reduces very quickly.
How Does the Sphere Heat Up?
In my efforts to understand the conceptual problems people have, I believe that this might help. I can’t be certain – this article is about mental models.
Let’s picture the scene when the PVC sphere is “started up”.
Outside it is 0K. Inside it is 0K. Chilly. Very chilly.
Now the 30,000 W heat source is fired up. 30,000 J every second gets radiated out from this source. Every second, 80% of this 30,000 J gets absorbed by the inner surface (with 20% reflected).
At this stage almost no energy is conducted through the PVC sphere. It can’t – because the temperature differential is not nearly high enough. Conduction requires a heat differential. So instead, the energy goes into heating up the inner surface of the sphere.
As the inner surface heats up it begins to conduct heat through to the outer surface – but most of the energy still goes into heating the inner surface.
A necessary consequence of the inner surface being heated up is that it radiates. All of this radiation is absorbed by the rest of the inner surface AND THEN re-radiated. Energy is not being created. This energy can’t be “tapped off” to do anything useful.
A small supply of energy is simply being “bounced around” (not really “bounced” but it might be a useful way to think about it)
This energy is simply the energy that has been accumulated by the inner surface during the initial heating process. It keeps being accumulated until finally the temperature is high enough to conduct the full 30,000 W through to the outer surface.
Now we have reached equilibrium! On our journey to equilibrium, while the inner surface was heating up, it accumulated heat, and this accumulated heat is now radiated, absorbed, re-radiated, absorbed…
You can connect it to a power station and very quickly you will draw down this accumulation of energy. The maximum you can draw out long term will be 30,000 W.
This article is all about mental models – explaining why the actual results for this model don’t violate the First Law of Thermodynamics. The results were calculated from the very simple and standard heat transfer equations.
Analysis of this model, with the results that I have presented (in part one), demonstrates that energy is conserved.
At first glance it might not seem like it to many people – because the inner surface radiation is so high. But the energy is just re-radiated from the energy absorbed. It’s like a small stockpile of energy that is being “bounced around” from wall to wall.
There is only one (legitimate) way to solve the heat transfer equations for this model. Other approaches invent /destroy physics in an attempt to get a low enough value for the radiation emitted from the inner wall.