# Parsing Elon Musk's solar energy calculation

## Introduction:

Back in 2015, Elon Musk gave an engaging presentation on Tesla’s new home battery products as part of his vision of a solar energy future. A key part of his presentation hinged on a ‘blue square’, a \(10^4 km^2\) section of Texas that is supposedly sufficient to cover USA’s electricity consumption. To give the reader some idea of the size of \(10^4 km^2\), you can fit around ten New York cities in that ‘blue square’.

Thinking about Elon’s solar energy calculation led me to consider the following:

- Can Elon’s argument be justified by a back-of-the-envelope calculation?
- Is solar energy a serious option if we take into account the expected rate of globalisation and technological progress?

Although I’m not American, the USA is a very important case as it represents approximately 20% of the Globe’s total energy consumption while its population represents barely 5% of the world population. If we take into account the rapid Americanisation of all countries then the second question emerges naturally.

Disclaimer: I take it for granted that using fossil fuels isn’t reasonable by any measure.

## Elon Musk’s blue square:

Let’s first note that according to the EIA US electricity consumption was about 3758 TeraWatt Hours in 2015 which we may then convert to watts as follows:

\begin{equation} \frac{3758 \quad \text{TeraWatt hours}}{365 \cdot 24} \approx 429 \quad \text{GigaWatts} \end{equation}

Now, it’s useful to note that according to the US Energy Information Administration total energy consumption in 2015 was around \(9.5 \cdot 10^{16}\) British Thermal Units, or five times the total electricity consumption in the USA. So even if Elon is right that \(10^4 km^2\) may be sufficient for *electricity consumption* it’s a good idea to note that at least five blue squares will be needed for *total USA energy consumption*. That’s more than fifty times the surface area of New York city!

But, is Elon right? Let’s use the following formula for calculating solar energy yield:

\begin{equation} Energy = A \cdot r \cdot H \cdot PR \end{equation}

where \(A\) is the total solar panel area in square meters, \(r\) is the solar panel efficiency, \(H\) is the annual average solar radiation(which varies between regions) and \(PR\) is the performance ratio(usually between 0.5 and 0.9). If we assume Saudi Arabian solar radiation levels,that the performance ratio is nearly 1.0 and about 21% efficiency we have:

\begin{equation} Energy = 10^4 \cdot .21 \cdot 2600 \cdot 1.0 \approx 5.460 \quad \text{PetaWatt Hours} \end{equation}

which I can convert to Watts as follows:

\begin{equation} Wattage = \frac{Energy}{365 \cdot 24} = \frac{5460 \quad \text{TeraWatt Hours}}{365 \cdot 24} \approx 623 \quad \text{GigaWatts} \end{equation}

which more than satisfies the first equation so Elon is right or at least he isn’t wrong in a manner that is obvious.

## Globalisation and Technological progress:

### Upper-bound on potential solar power on Earth due to the sun:

In order to estimate the potential solar power on Earth due to the sun we may use the Stefan-Boltzmann equation for luminosity:

\begin{equation} L_o= \sigma A T^4 \end{equation}

which depends on \(T\) the effective temperature of the sun, \(A\) its surface area and \(\sigma\) the Stefan-Boltzmann constant.

Now if we define 1 \(AU\) to be the average distance between the Earth and the sun, the maximum solar power available to Earth is given by:

\begin{equation} P= L_o \frac{A_{Earth}}{A_{1AU}} = \sigma T^4 \big(\frac{R_s}{1 AU}\big)^2 4 \pi R_{Earth}^2 \approx 174 \quad \text{PetaWatts} \end{equation}

and if we take into account that about 70% of sunlight is lost to outerspace, only about a third of the Globe is terrestrial and the solar energy conversion efficiency is around 20% we have:

\begin{equation} \bar{P}= 0.7 \cdot .3 \cdot .2 \cdot P \approx 7300 \quad \text{TeraWatts} \end{equation}

which is about five hundred times current use. This might sound like a large margin until you realise that energy consumption in developed countries has been growing exponentially during the last two hundred years.

### The rate of globalisation:

Given that new energy infrastructure takes time to build it’s essential to realise that you build it for the foreseeable future and by that I mean at least the next couple decades. Is it possible that within a few decades we might start getting dangerously close to \(\bar{P}\)? I would argue that we are already in trouble because we are rapidly becoming American in our energy consumption patterns.

As I mentioned earlier the USA is responsible for approximately 20% of Earth’s total energy consumption while its population represents barely 5% of the world population. So convergence in American-style economic development implies that eventually the entire world will be consuming around four times more energy per annum. Let’s denote this global energy consumption pattern by \(\hat{P}\) and suppose that this event happens within a decade, then:

\begin{equation} \frac{\bar{P}}{\hat{P}} \approx 100 \end{equation}

so we’re only a factor of a hundred away from doomsday and we haven’t even started building solar panels seriously yet. Can things get worse?

### The rate of technological progress:

As pointed out by Tom Murphy, a professor of Physics at UC San Diego, the rate of energy consumption in the USA has been increasing at an exponential rate. Around 2.3% per year which might not sound like much until you think about the effect of compounding. Let’s suppose that by 2030 all countries have similar energy consumption patterns. In order to figure out how much time human civilisation has left on Earth we must calculate:

\begin{equation} x = \frac{\ln 100}{\ln 1.023} \approx 200 \quad \text{years} \end{equation}

but that’s the result of a simple extrapolation. I’m actually much less optimistic.

At a time when we have two emerging superpowers, China & India, and the decline of the USA that is willing to do everything it can to maintain its hegemony; I believe that we’ll witness an acceleration.

## Discussion:

If we do go all the way with solar energy another important factor consider is the volume of lithium ion battery production that would be necessary in order to supply electricity at night around the Globe. I have yet to do detailed calculations on this but this makes me even less confident that the future can be 100% solar. Yet, if not solar energy then what? I think we can actually make a case for nuclear fusion especially if we start thinking about multi-planetary civilisations. But, I will save this analysis for another day.

# References:

- T. Murphy. Galactic-Scale Energy. 2011.
- D. McKay. Sustainable Energy-without the hot air. 2008.
- EIA. Annual Energy Outlook. 2019.