Back-of-the-envelope calculations to why even nuclear won't save us, without curbing energy consumption

## Introduction

Humanity has been looking for alternatives to fossil fuels for over a century, but the problem has started to become more pressing since the 1960s, when people started to reflect on the fact that the resources would sooner or later be exhausted, it was reinforced during the 1970s energy crisis and has been moved to the foreground of both energy and climate discussions, due to the significant impact that burning fossil fuels has on the environment (something that even the oil companies themselves have known for at least half a century, despite their reliance on —and frequent financial support to— “climate skeptics” to deny the significant effect of anthropogenic effects on climate change —something that has been known (or at least suspected) for decades, and they have finally admitted).

For a brief moment, nuclear energy was seen as the most viable alternative, but the enthusiasm behind it received a collective cold shower after the Chernobyl disaster and with the growing issue of the nuclear waste management, that has brought attention back to “renewables” (extracting energy from the wind, the sun or the water) —with its own sets of issue.

Nuclear power still has its fans, whose arguments mainly focus on two aspects:

- nuclear is actually the “greenest” energy source, even compared to “renewables” (especially in the medium/long term);
- nuclear is the only energy source that can keep up with the requirements of modern, advanced societies, especially if you cut out fossil fuels

I'm not going to debate the first point here, but I'll instead focus on the second one.
And my argument won't be to deny the efficiency of nuclear power (in fact, the opposite),
but to show that *despite its efficiency*, **even nuclear power cannot keep up**,
and that the real issue we need to tackle, as we've known for decades if not centuries now,
is our inability to understand the exponential function.

But let's get into the meat of the discussion.

## Fact #1: nuclear energy production has the highest density

This is an undeniable fact by whichever means you measure the density: it is true when you compare it with any of the renewables in terms of energy produced per square meter of occupied land, and it is true if you compare it with any fossil fuel generator in terms of energy produced per unit of mass consumed.

For example, an actual nuclear power plant at the current technological level occupies around 3km² and produces around 1GW, with an effective (surface) density of about 300W/m². By comparison, geothermal can do at best 15W/m², and solar —that can peak at less than 200W/m² on a good day (literally)— will typically do around 7W/m² (considering the Sun cycles) —and everything else is less than a blip compared to that.

In terms of energy density, gasoline and natural gas with their 45MJ/kg and 55MJ/kg respectively
are clear winners among fossil fuels, but their chemical energy density is completely eclipsed
by the nuclear energy density of uranium:
a 1GW plant consumes less than 30 tons of uranium per year, giving us
an *effective* energy density (at our current technological level) of more than 1000GJ/kg:
5 *orders of magnitude* higher than that of the best fossil fuels. In fact,
even going by the worst possible estimates the uranium *ore*
(from which the actual uranium used as fuel is extracted) has an effective energy density
of slightly less than 80MJ/kg, which is still more than 1.5 the *maximum theoretical*
we can get from fossil.

These data points alone could explain why so many people remain solidly convinced that nuclear power is the only viable alternative to fossil fuels, despite the economical, political and social costs of nuclear waste management.

But there's more! The attentive reader will have noticed that I've insisted on the «current technological level» moniker. There's a reason for that: while fossil fuel as an energy source has a long and well-established history, with an associated enormous progress in the efficiency of its exploitation, the same can't be said neither for most renewables, nor for nuclear.

For example, solar irradiance on the Earth surface is around 1kW/m² —about 5 times what we manage to get from it in ideal conditions, and 3 times higher than the surface energy production density of a modern nuclear power plant. A technology breakthrough in solar energy production that could bring the efficiency from 20% to 80% would make solar competitive in massively irradiated regions (think: the Sahara desert).

But the same is true also for nuclear —and in fact, for nuclear, it's
*considerably* more true: indeed, the upper bound on the amount of energy
that can be produced from matter is given us by the famous $E=m{c}^{2}$
mass–energy equivalence equation. If we could convert
1kg of mass *entirely* into energy, this would produce close to 90 *petajoules*
of energy, 90 *million* GJ: 90 *thousand* times more than what a nuclear power plant
can produce today from the fuel pellets fed to it.

If we managed to improve the efficiency of nuclear energy production by a factor of 1000, we'd have an efficiency of only about 1.3%, and it would still completely eclipse any other energy generation method even if they were 100% efficient.

To say that there's room for improvements would be the understatement of the millenia. And this, too, would be an argument in favor of the adoption of nuclear power, and most importantly in investing massively in research for its improvement (especially considering that more efficient production also means less waste to worry about).

And yet, as we'll be seeing momentarily, **even reaching 100% efficiency in nuclear energy
extraction will not save us**.

## Ballpark figure #1: mass of the Earth crust.

Let's now do a quick computation of the total mass of the Earth crust, the “thin” (on a planetary scale) layer whose surface veil is the land we trod upon.

The surface of the earth is marginally more than $S=510\cdot {10}^{6}$ km². To estimate the total mass of the crust, let's pretend, very generously, that the crust can be assumed to be $H=50$ km deep everywhere (this is actually only true for the thickest parts of the continental crust), and of a constant density equal to that of the most dense igneous rocks ($\rho =3500$ kg/m³). Rounding up, this gives us a mass of the crust equal to $S\cdot H\cdot \rho =9\cdot {10}^{22}$ kg.

(This is quite a large overestimation, since the actual average thickness is less than half of $H$, and the average density is less than $3000$ kg/m³, so we're talking about at best a third of the overestimation; but as we shall see, even the generous overestimation of $9\cdot {10}^{19}$ metric tons will not save us.)

## How much energy could we extract from the crust?

Let's play a little game. Let's pretend that we have a 100% efficient mass–energy conversion: 1kg of mass _of any kind _goes in, 90PJ of energy (and no waste!) comes out.

For comparison, the world's yearly primary energy consumption currently amounts
to more than $170\cdot {10}^{3}$ TWh —let's be generous and round it *down* to $600\cdot {10}^{3}$ PJ.

If we had the amazing 100% mass-to-energy conversion technology,
less than 7 (metric) tons of mass would be sufficient
to satisfy the current energy requirements for the whole world in a year.
(For comparison, a modern 1GW nuclear power plant produces 5 tons of *waste* per year.)

If we had this wonderfully 100% efficient technology, it would take $R=1.3\cdot {10}^{19}$ years, at the current energy consumption rate, to exhaust the $9\cdot {10}^{19}$ (metric) tons of the Earth's crust.

(Try it from the other side: $9\cdot {10}^{22}$ kg of mass producing 90 PJ/kg means $8.1\cdot {10}^{24}$ PJ of energy, which divided by $6\cdot {10}^{5}$ PJ of yearly consumption give us a more accurate $R=1.35\cdot {10}^{19}$.)

Needless to say, we wouldn't need to worry about wasting energy ever again,
considering the sun will run out *long* before that (estimated: $5\cdot {10}^{9}$ years).

Or would we?

## Enter the exponential function

Looking again at the world's energy consumption, we can notice that it has been growing at an almost constant rate (a ballpark estimation from the plot gives us a rate of about 2% or 3% per year, corresponding to a doubling time of about 25 to 35 years) —that is, exponentially.

And a widespread idea among supporters of nuclear energy is that with nuclear there's no need to change that —nuclear energy is the solution, after all, given how much it can give us now, and how much potential it still has, there's no need to limit how much energy we use.

The math, however, says different. Since the energy consumption will grow over time, the previously computed ratio $R=1.3\cdot {10}^{19}$ does not tell us anymore the number of years before the crust is consumed —to determine that, we rather need to check how many doublings will fit in that ratio, which we can approximate by ${log}_{2}\left(R\right)$ —and that's less than 64 doublings: at the current growth rate, that means something between 1500 and 2000 years.

For a more detailed computation, we can apply the “Exponential Expiration Time” formula, found for example in Bartlett's work: the EET in our case $\frac{ln(k\cdot 1.35\cdot {10}^{19}+1)}{k}$, which gives us 1351 years for a 3% growth rate, and 2007 years at a 2% growth rate.

This deserves repeating: **at the current rate at which energy consumption grows,
the entire crust of our plane would run out in at most 2000 years in the best-case scenario
that we manage to find a 100% efficient mass to energy conversion method within the next decade**.

## Be more realistic

The actual timespan we can expect is in fact much lower than that.

For example, we're nowhere close to being 100% efficient in mass to energy conversion:
in fact, you'll recall that *even if we manage to improve our efficiency by a thousandfold*,
we'll only be barely more than 1% efficient
—meaning that even the two-orders-of-magnitude-lower
$R=1.35\cdot {10}^{17}$ is still an extremely generous estimate.

But there's more: the mass of the Earth crust is likely one third of that of our gross overestimation, bringing $R$ down to around $R=4.5\cdot {10}^{16}$. But what's worse, the amount of uranium in the crust is currently estimated to be only about 4 parts in a million, which would bring $R$ further down to about $R=1.8\cdot {10}^{11}$.

To wit, that would give us between 747 and 1100 years before we ran out of fuel, assuming we managed to extract all of the uranium and convert it to energy with a 1% efficiency, which is a thousand times better than what we can do now..

I'll take this opportunity to clarify something important about the exponential function —with an example.

At our current tech level, we would have $R=2.34\cdot {10}^{8}$ —all the uranium would be gone in 525 to 768 years. For thorium, which is around 3 times more abundant, the estimate is 562 to 822 years. Now ask yourself: what if we use both? Surely that means over a thousand years (525+562), possibly closer to 2000 (768+822)?

No.

That's not how the exponential function works.

If energy consumption keeps growing at this steady 2-3% rate, thorium and
uranium *combined* would only last 571 to 837 years: switching to thorium after
depleting all the uranium would only add around 50 to 80 years.

## Can it get worse?

It should be clear from even the most optimistic numbers seen so far that nuclear energy by itself
is *not* sustainable in the long term: even if we switched entirely to nuclear power and found a breakthrough
in the next decade or so that would bring the efficiency up by a thousand times, **we won't last more than a
few centuries before running out of energy**, *unless something is done to stop the exponential growth
in energy consumption*.

But it gets worse. I'm not particularly optimistic about humanity's wisdom. In fact, in my experience, the more a resource is abundant, the faster its consumption grows. And this goes for energy too.

In my mind, the biggest threat posed by nuclear power isn't even the risk posed
by the mismanagement of the plants or of the still-radioactive waste.
The biggest threat posed by nuclear power is the “yahoo, practically infinite energy in our hands!”
attitude of its supporters, which is quite likely to lead to energy consumption growing
*at an even higher rate* than the current one, if we ever switch to nuclear on a more extensive scale.

And with an increased growth rate, we'll run out of energy much, much earlier:
at a 7% growth rate in energy consumption (doubling time: 10 years),
all the estimated uranium in the crust would be gone in 237 years at our current tech level,
or 332 years assuming we get the 1% efficiency breakthrough *now*;
and *the entire crust would be depleted in 591 years assuming 100% efficient mass-to-energy conversion
from any material*.

And no, there is no “we'll find something better in the mean time”, because there's *nothing*
better than 100% efficient mass-to-energy conversion. Even harnessing the mass of other celestial bodies
won't do more than extend the expiration time by another few hundred years, maybe a couple of millennia
at best: at a growth rate of 2% and 100% conversion efficiency, *the entire planet of Mars* would last
us for no more than 2105 years —and remember, that's not *in addition to* the depletion of the
crust of our planet: in fact, adding the overestimated mass of Earth's crust to the mass of Mars
won't even budge the expiration time by a single year.

The entire mass of all the celestial bodies in the solar system would last around 2500 years. If we add in the Sun (which means, essentially, just the mass of Sun, actually), we would still run out in 2852 years, at a 2% growth rate and 100% efficiency.

(Wait a second, I'll hear somebody say: how comes the Sun will last for billions of years still,
but if we converted all of its mass into energy using our 100% efficient mechanism it won't even last
3000 years? And the answer, my friend, is again **the exponential function**: the Sun
produces energy at a (more or less) *constant* rate, but we're talking about how quickly it will
be depleted at a *growing* rate. Does that help put things in perspective? No? How about
**the entire Milky Way would last less than 43 centuries**?)

So yes, there is no “we'll find something better”, not at the current growth rate.

The only *sustainable* option is reducing the growth rate of the total energy consumption.

## Degrowth is the answer

Now, with this title I'm not proposing degrowth as the solution, I'm simply stating a fact:
degrowth *will* happen, regardless of whether humanity *choose voluntarily* to go down that path or not.
The only difference is *how* it will happen. But it *will* happen. Because if we don't wisen up and
curb our own growth, we *will* run out of resources, and at the current growth rate that *will*
happen *at best* in a few centuries, with or without nuclear power:
and when it *does* happen (not if, but when), we will have sudden, drastic,
*forceful* degrowth imposed on us by the lack of resources (most importantly, energy).

We're running towards an unbreakable wall. There is no other option but deceleration, and that's because deceleration will happen, whether we want it or not. Our only choice is between slowing down gracefully, and stopping before we hit the wall, or experiencing the sudden, instantaneous and painful deceleration that will happen the moment we hit that wall.

## And now for the “good” news

Slowing down the growth rate is an extremely effective way to extend the EET. Let's have a look at this from our worst-case scenario: at the current technological level, and 3% growth rate, all of the estimated uranium and thorium in Earth's crust will be depleted in 571 years, but with a 2% growth rate it would last 837 years.

Dropping the growth rate to 1%, they would last 1605 years —which is more or less
the EET for *the entire crust at 100% efficient conversion* with a 2.5% growth rate.

Going even lower, to 0.5% growth rate, they would last over 3000 years —more than it would take to deplete the Sun with 100% efficient conversion and a 2% growth rate.

TL;DR:

Increasing the adoption and efficiency of nuclear power generation can buy us *maybe* a few centuries.

Decreasing the growth rate can buy us *millenia*.

Where would you invest with these odds?

(See also the next article on the topic for additional details and comments on the plausibility of a continuing exponential growth.)