Do Trenberth and Kiehl understand the First Law of Thermodynamics?
But many people claim that they don’t after reviewing their well-known diagram from their 1997 paper, Earth’s Annual Global Mean Energy Budget:
The “problem” is – how can the absorbed solar energy be 235W/m2 when the radiation from the surface is 390W/m2? Where is this energy coming from?
Clearly they have created energy and don’t understand basic physics!
There have been many comments to that effect on this blog and, of course, on many other blogs.
Instead of pointing out that many of these values can be easily measured and checked, we will turn to a simple experiment which might help the many who believe the answer to the title is “No“.
The Thought Experiment
Picture, for the practical among you, building some kind of simple heat chamber in your garage. A wooden or a plastic box, with a light bulb in the center. You want to test whether some new gizmo really works at the high temperatures claimed. Or you want to find the melting point of gold.
The principle is simple – the thicker the material and the higher the energy from the bulb – the hotter it will get inside the heat chamber.
As a method of simplifying the calculations, my chamber will be spherical (because it makes the maths easier than when it is a box) and we will place it in the vastness of space and assume that the ambient temperature is 0.0K. Again, this is just to make the maths simpler to understand.
The inner radius of the sphere is 10m, and the thickness of the wall is 3m. (In a followup comment or post I will show how the values change with x).
The material used for this experiment is PVC which has a thermal conductivity of 0.19 W/m.K – I’ll explain a little more about this parameter in a moment. Probably down at such low temperatures the thermal conductivity won’t be this value but it doesn’t matter too much. We will also assume that the emissivity = 0.8.
You can see on the diagram that the outer surface temperature is T2, the temperature inside the sphere is T1 and the “ambient” is 0K. We don’t yet know what T1 or T2 is, we want to find that out.
In the center, we have our super-light-bulb, which radiates 30,000W. It is mysteriously powered, perhaps it is a nuclear device, or just electric with such a thin power cord we can’t detect it – we don’t really care.
Now – the first law of thermodynamics – energy cannot be created or destroyed. So for our thought experiment, the system receives 30,000W. The “system” is the entire PVC sphere, and everything it encompasses, right to the outer surface. No other source of energy can be detected.
Start Your Engines
Now that we have turned on the energy source the inside of the sphere will heat up. It has to keep heating up until the energy flow out of the sphere is balanced by the energy being added inside the sphere.
How does heat flow out from the center of the sphere?
- First, by conduction to the outer surface of the sphere
- Second, by radiation from the outer surface of the sphere to the vastness of space
Both of these processes are governed by very simple equations which are shown in the maths section at the end. Here, I will just attempt to explain conceptually how the processes work.
We start with consideration of the complete system and after equilibrium is reached the energy gained will be equal to the energy lost.
Energy gained, q = 30,000W = Energy lost
(Note that we are considering energy per second). For a rigid stationary body in the vastness of space, the only mechanism for losing energy is radiating it. All bodies radiate according to their temperature and a property called emissivity. Using the Stefan-Boltzmann law, we can calculate that:
Outer surface temperature, T2 = 133K
With this temperature, at an emissivity of 0.8, the whole sphere is radiating away 30,000W.
So at this point, surely everyone is in agreement. We have calculated the steady-state temperature of the outer surface of the sphere as 133K. We can see that the mysterious energy source of 30,000W is balanced by the outgoing 30,000W radiated away from the outer surface.
The first law of thermodynamics is still intact and no one has to fight about anything..
But What’s the Story Inside?
We also want to calculate T1, the temperature of the inner surface. This is also very easy to calculate. The only mechanism for transferring heat from the inner surface (where the energy source is located) to the outer surface is by conduction.
The maths is below but effectively heat is transferred through a wall when there is a temperature differential between two surfaces. The higher the differential, the more heat. And the property of the material that affects this process is called the thermal conductivity. When this value is high – like for a metal – heat is transferred very effectively. When this value is low – like for a plastic – heat is transferred much less efficiently.
Once the system is generating 30,000W internally the inner wall temperature will keep rising until 30,000W can flow through the wall and be radiated away from the outer surface.
If we use the simple maths to calculate the temperature differential we find that it is 290K.
That is, to get 30,000W to flow through a hollow sphere with inner radius 10m and outer radius 13m and conductivity of 0.19 W/m.K you need a temperature differential of 290K.
Which means that the inner surface is 423K.
Everyone still ok?
What is the Radiation Emitted from the Inner Surface?
With an emissivity of 0.8 and a surface temperature of 423K, the inner surface will be radiating at 1,452 W/m2.
So the total radiation from the inner surface will be 1,824,900W.
What??? You have created energy!!!
Before bringing out the slogans, find out which step is wrong. If you can’t find an incorrect step then perhaps you should consider the possibility that this system is not violating the first law of thermodynamics.
Well, perhaps everyone is comfortable with the idea that with sufficient insulation you can raise the inner temperature of a box or sphere to a very high value – without having to build a power station.
In any case, the system is not creating energy. Inner surfaces are receiving high amounts of radiation while also emitting high amounts of radiation – they are in balance.
And of course, this has nothing whatever to do with the earth’s climate system so everyone can rest easy..
And new article on the real basics – Heat Transfer Basics – Part Zero
In equilibrium, the outer surface of the sphere has to radiate away all of the heat generated internally. This is the first law of thermodynamics.
The internal energy source, q = energy radiated from the outer surface
q = εσT24.4πr22 – the Stefan-Boltzmann equation for emitted energy per m2 x the surface area
where r2 = radius of the outer surface = 10 + 3 = 13
If q = 30,000W, r2 = 13m and ε = 0.8
T2 = q / (εσ.4πr22)1/4 and so T2 = 133K
If you recalculate back using the Stefan-Boltzmann law you will find that 133K with an emissivity of 0.8 radiates at 14.2 W/m2 (corrected-thanks to John N-G) and if you multiply that by the surface area of 4×3.14×132 = 2,124 m2 you find the emitted energy = 30,000W.
Conduction through the Sphere
The equation of heat conduction is very simple:
q = -kA . ΔT/Δx
This is for a planar wall. For a hollow sphere the equation is quite similar:
qr = -kA . dT/dr = – k (4πr2) . dT/dr and the important point is that qr is a constant, independent of r
After a small amount of maths we find that:
qr = 4πk . (T1 – T2) / (1/r1 – 1/r2)
So for the values of k = 0.19, r1 = 10, r2 = 13:
T1 – T2 = 290, therefore, T1 = 423K
Therefore, the radiated energy from the inner surface will be 1,452 W/m2 or a total of 1,824,900W (= εσT14.4πr12).