The case of two different fluxes!

Today I will be investigating the difference between two different types of fluxes that elude almost every climate scientist. Let’s take a typical simple conduction problem you can find in every high school or college textbook:

Figure 1: Conduction through a concrete block

We need to figure out the temperature and radiative emission on the colder side (Tc and Ec).

The formula for conduction is:

         Q         k * A * (Th-Tc)
Power = --- = q = -----------------
         t               L

We can rearrange this equation to suit our needs:

           q * L
Tc = Th - -------    (Equation I)
           A * k

And now we solve for Tc:

            2  *  8            16
Tc = 75 - ----------- = 75 - ------ = 75 - 25 = 50
           0.8 * 0.8          0.64

Tc is 50°C.

What is the conductive heat flux inside the concrete block? That is given simply by:

                        q      2 W
Conductive Heat Flux = --- = -------- = 2.5 W/m²
                        A     0.8 m²

And how much radiation is emitted from the right hand side (Ec)? To figure that out we have to apply Stefan-Boltzmann’s Law:

Cold-side Radiation = Ec = εσ(Tc)⁴   (Equation II)

First we need to know the emissivity of concrete. According to these experts, it is between 0.85 and 0.95, so we will use 0.9. Now we solve:

Ec = 0.9 * 0.00000005670367 * (273.16+50)^4 = 556.58 W/m²

As you can see we have two different flux (W/m²) figures: 2.5 and 557 W/m². We need appropriate labels for them so we don’t confuse them:

Conductive Heat Flux through the medium (CHF)	2.5 W/m²
Cold-Side Radiation from the medium (CSR)	557 W/m²

We can easily see that despite the small CHF through the medium, the emergent CSR is 557/2.5= 223 times larger.

Let’s now combine Equation I and II to summarize what is going on:

         /       q * L  \ 4
Ec = ε*σ*| Th - ------- |
         \       A * k  /

Remembering that CHF is just q/A, we reduce further:

CSR = εσ(Th-CHF*L/k)⁴

It becomes obvious now that CHF and CSR have an inverse relationship. The higher the CHF, the lower the CSR, and the lower the CHF, the higher the CSR.

Why is the distinction between CHF and CSR important?

Professors Davies and Davies have done a good job in measuring Earth’s heat flux:

We present a revised estimate of Earth’s surface heat flux that is based upon a heat flow data-set with 38 347 measurements, which is 55% more than used in previous estimates.

We conclude by discussing our preferred estimate of 47 TW, (rounded from 46.7 TW given that our error estimate is ± 2 TW)

It’s unfortunate that they call their measurement “surface heat flux”, when in reality they measure heat fluxes at various depth ranges which they don’t disclose, and then average that. They measure CHF. The value they get for averaged CHF is 46.7 TW divided by Earth’s surface area: 47×10¹² / 510.1×10¹² = 91.6 mW/m².

91.6 mW/m² is a very small number when compared to the average insolation we receive at the surface, which according to NASA’s official energy budget is 163.3 W/m². 0.0916/163.3 = 1783 times smaller. Perhaps you’ve heard of the idea that the sun supplies 99.95% of our energy? Well guess what? All of this is nonsense, because CHF is irrelevant!!! What is relevant is CSR!

What Davies & Davies should have done is measure all the averaged parameters (Th, k, L) and not just q or q/A (CHF). We need to know the actual CSR before we can start comparing it to insolation. This is key, because without CSR, climate scientists have a completely erroneous view of the way things really are.

Assuming k = 1 and A=1, we examine all the possible temperatures that produce Davies’ CHF (q/A) of 91.6 mW/m².

Depth = 110 meters	Depth = 10 meters	°C/m
10°C	0.84°C	-0.0916
100°C	90.84°C	-0.0916
1000°C	990.84°C	-0.0916
10000°C	9990.84°C	-0.0916

All result in Conductive Heat Flux of 91.6 mW/m²

Let’s think about this: Does it matter whether it’s 0.84°C or 9990.84°C 10 meters below your feet? Of course it does! But you can’t tell the difference using CHF. Only using CSR can we tell the true radiation emerging out of the earth!

We will now transform the above table into CSR, using emissivity = 0.93643 (why this number?).

Depth = 110 meters	Depth = 0 meters	CSR
10°C	-0.076°C	295.31 W/m²
100°C	89.924°C	922.81 W/m²
1000°C	989.924°C	135,150 W/m²
10000°C	9989.924°C	589,112,205 W/m²

Makes a big difference, right? It would be interesting to know the actual Earth-wide averaged CSR, but we will leave that for another day. (Update: That day came: Measuring Geothermal, CSR = 294 W/m²)

In short summary, most climate scientists are clueless about the difference between CHF and CSR and therefore erroneously greatly underestimate the power of geothermal. It may even be the biggest scientific scandal of our time!

Yours Truly, -Zoe

Update

Go on to read Measuring Geothermal (1) and Measuring Geothermal (2).