## Introduction

If you believe that nuclear power is the solution to the energetical (and possibly environmental) issues of the more modern developed nations, the question you should ask yourself is: for how long still?

A few weeks ago I've started a series discussing the “expiration time” for nuclear power under the assumption of a constant growth in energy consumption (with a rate between 2% and 3% per year). The results were not very encouraging for the long term: even at the lower growth rate of 2%, the energy requirements would grow so much that even the entire mass of the Milky Way, converted entirely into energy according to the famous $E=m{c}^{2}$ equation (100% efficient nuclear energy extraction, giving us around 90PJ/kg), would suffice us for less than 5 millenia: not even the time that would be needed to move from one end to the other of said galaxy.

Such is the power of the exponential function.

While the post was not intended as a prediction, but mostly just as a “cautionary tale” about the need to reduce the speed at which energy consumption is growing, it has been criticized for the timescales it considers —timescales ranging from several centuries to millenia, timescales in which “anything may happen”.

I have tried to address the main objections in the follow-up post, discussing primarily two points: one being the choice of 90PJ/kg as upper bound of energy production, and the other being the assumption of energy consumption growing at the current rate for the foreseeable future, and most likely even beyond.

Despite the validity of these longer-term considerations (again: not predictions, just considerations), I don't doubt that many (if not most) people would find it useless to reason over such time spans, refusing to take them into account for shorter-term decision-making.

In this third installment of the series, we're thus going to focus on a much shorter term (say, within the century or so), within which it's much harder to deny a continuing growth in energy consumption, at the current rate (which we will optimistically round down to 2% per year), and it's plausible that nuclear energy extraction will continue within the current order of magnitude of efficiency, or only slightly more (say, no more than ${10}^{-1}$ PJ/kg from the current approximately $1.2\cdot {10}^{-3}$ PJ/kg).

## Some preliminary numbers

If you've gone over the first two installments of the series, you may have noticed that the summary table in part 2 has an exceptionally low number in the upper-left corner: where all other scenarios offer an EET of two centuries or more, the lowest scenario gives us only 14 years. Surely that's too low? How is that possible?

The EET of 14 years is indeed too low. It corresponds to the EET under the following assumptions:

1. constant growth rate of 2% per year;
2. current tech level, extracting around $1.2\cdot {10}^{-3}$ PJ per kg of uranium;
3. $8\cdot {10}^{9}$ kg of available uranium (the amount estimated to be in current known conventional reserves);
4. the worldwide total primary energy consumption ($6\cdot {10}^{5}$ PJ/year) is entirely satisfied from nuclear.

The first two assumptions are entirely reasonable within the timespan of less than two decades, the third assumption is possibly even too generous (it assumes that within these two decades, we'd be able to even just extract all of the uranium from the estimated known conventional reserves).

The last assumption, on the other hand, is completely unrealistic: nuclear power generation today barely covers a fraction of worldwide total primary energy consumption. Existing civilian power plants produce less than 2600TWh (or less than ${10}^{4}$ PJ) of electricity per year (and the amount is going to decrease, if the current initiatives to transition away from nuclear are any indication of the near future).

(That being said, that uranium wouldn't actually last long at full usage shouldn't even be that big of a piece of news for anyone following the field: even back in 2008 there was awareness about how long uranium would last if production was to ramp up, given the discovered deposits. In fact, we actually get more leeway in our estimates because we're using the much larger amount of estimated known conventional reserves.)

But let's try to get a bit more realistic.

## Ramping up nuclear

The first exercise is to see what would happen if we ramped up nuclear power (instead of transitioning away), to try and cover a larger slice of the total primary energy consumption, at the current tech level ($1.2\cdot {10}^{-3}$ PJ/kg).

For simplicity, let's round things a little bit. Assume we currently produce ${10}^{4}$ PJ/year from nuclear (while this is rounded up, in our calculation the final differences is of at best a couple of years over a whole century), and that the readily available uranium from known conventional reserves is ${10}^{10}$ kg (this is a bit on the generous side, but it's one way account for the discovery of some more uranium deposits).

We have two questions: how long will it take to cover the current global primary energy consumption ($6\cdot {10}^{5}$ PJ/year) and how quickly will we run out of uranium. In particularly, we'd like to at least get to satisfy the current primary energy requirements before running out of uranium

The answers to these questions obviously depend on how fast we can ramp up energy production from nuclear power: the faster we ramp up production, the quicker we match primary energy needs, but at the same time, the faster we ramp up production, the quicker we run out of uranium.

(You can follow the exercise by plugging in the relevant numbers in the form found after the table in part 2 of this series, just consider ‘production’ instead of ‘consumption’ in the first and last field.)

It's interesting to see that with anything less than a 4% growth rate for nuclear power generation, we won't even get to produce one whole year's worth of the current primary energy requirement before running out of uranium: at 4%, we would run out of fuel after slightly less than a century, while producing barely more than $5\cdot {10}^{5}$ PJ/year.

Anything less than a 4% growth rate (18 years doubling time) would allow uranium to last for over a century, but without covering the current worldwide primary energy consumption. Ramping up at a 5% rate (more specifically, around 4.82%, 15 years doubling time) would allow us to match the current worldwide primary energy consumption just as we run out of easily accessible uranium, 85 years down the line.

To get some meaningful (multi-year) amount of coverage, we would have to ramp up production even faster, but this would shorten the time of availability of the fuel: for example, at a 7% growth rate (doubling time: 10 years, still realistic considering the time it takes to actually build or expand nuclear power stations) the known uranium reserves would have an EET of only 64 years.

Actually, if the ramping up limit was the current total primary energy consumption, uranium would last a little bit longer: the EET production rate would be $8.8\cdot {10}^{5}$ PJ/year, which is higher than the current consumption. This would buy us a few years if we stopped ramping up as soon as we reached parity, pushing the EET to around 70 years (not much, but still something).

## Playing catchup

On the other hand, assuming that the global primary energy consumption remains constant in the next century is quite a stretch: we can expect it to keep growing at the current rate of at least 2% per year for the foreseeable future.

Given the ramping-up timeline, this would give us at least another doubling, potentially even two: this means that even getting at $6\cdot {10}^{5}$ PJ/year would cover at best only half of the future primary energy needs. We should strive for more. And yet, even a 7% ramp-up rate wouldn't manage to cover a single doubling ($1.2\cdot {10}^{6}$ PJ/year target) before running out of uranium.

We would need at least a 10% ramp-up rate (doubling time: 7 years, which is about the quickest we can do to bring new reactors online) since that would push production to $1.22\cdot {10}^{6}$ PJ/year —just as uranium runs out, 48 years from now.

We could do “better” of course: knowing in advance the number of reactors needed to match the future energy request, we could build all of them at the same time. But that would only get us much closer to the dreaded 14-years EET for conventional uranium reserves (a quick estimate gives us around 30 years at best).

Ultimately, the conclusion remains the same: at the current technological level, and with the current estimates on the quantity of uranium available in conventional resources, we wouldn't be able to cover more than a few decades of global energy requirements at best, even with conservative estimates on how quickly the latter will grow.

## Breeder reactors and the myth of the “renewable” nuclear power

Given that the short expiration time of uranium at current tech level even just to satisfy the current global energy requirements (let alone its increase over the next decades) has been known for decades, one may wonder where the myth of nuclear power as “renewable” comes from.

We can find the answer in a 1983 paper by Bernard L. Cohen published on the American Journal of Physics, vol 51 and titled “Breeder reactors: A renewable energy source”. The abstract reads:

Based on a cost analysis of uranium extracted from seawater, it is concluded that the world’s energy requirements for the next 5 billion years can be met by breeder reactors with no price increase due to fuel costs.

Hence, nuclear power is considered “renewable” in the sense of being able to last as much as other energy sources traditionally considered renewables (such as wind and solar), whose expiration time is essentially given by the time needed for the Sun to run out. (I think that's an acceptable interpretation of the term, so I'm not going to contest that.)

Cohen's work starts from the well-known (even at the time!) short expiration time for traditional nuclear reactors, and shows how moving to breeder reactors would allow unconventional sources of uranium (particularly, as mentioned in the abstract, uranium extracted from seawater) to become cheap (in the economic sense) enough to be feasible without a significant increase in the price of generated electricity.

The combination of $100×$ more effective energy production, and the much higher amount of fuel, lead him to calculate the 5 billion years expieration time —assuming a constant rate of production equal to the total primary energy consumption in 1983.

It should be clear now why Cohen's number don't match up with our initial analysis: uranium would only last long enough to be considered “renewable” at constant production rates, not at ever-increasing rates. In fact, if you want to know the exponential expiration time for seawater uranium in breeder reactors, you just have to look at the second row, second column of the famous table: if energy consumption keeps growing as it is, all the uranium in the sea water fed to breeder reactors wouldn't last us 500 years.

Of course we don't know how accurate of a forecast my “doubling every 30 years” assumption is for future energy consumption (although it's much less far-fetched than some may think) but at the very least we know that Cohen's assumption of constancy was wrong, since consumption has already doubled once since, and it shows no sign of stopping growing anytime soon.

In fact, as I mentioned in the first post, the biggest risk for nuclear comes specifically from the perception of its “renewability”. In some sense, we can expect this to be the opposite of a self-fulfilling prophecy: the appearance of nearly infinite, cheap energy, combined with our inability to understand the exponential function, will more likely encourage an increase in energy consumption, as wasteful behavior devalues in face of the perceived enormity of available energy, ultimately leading to such a steep growth in energy consumption that the source would be consumed in an extremely short time.

By contrast, higher friction against the adoption of nuclear, combined with the much lower energy cap of all other sources, is likely to drive more efforts into efficiency and energy consumption minimization, thus slowing down the growth of energy consumption, and potentially allowing future nuclear power use to last much longer (even though, most likely, still considerably less than the billions of years prospected by Cohen).

## What does it really mean for an energy source to be renewable?

The truth is that, in face of ever-expanding energy requirements, no energy source can be considered truly renewable: the only difference is whether the production of energy from it can keep up with the requirements, or not.

Traditional renewables (wind, solar, hydro, wave, geothermal) can last “forever” (or at least until the Sun dies out) simply because we cannot extract them faster than they regenerate: as such, they won't “die out” (until the Sun does), but at the same time we'll reach a point (and I posit that most likely we're already there) where even if we were able to extract every millijoule as it gets generated, it still wouldn't be enough to match the requirements.

With non-renewables, the energy is all there from the beginning, just waiting for us to extract it. This means that (provided sufficient technological progress) we can extract it at a nearly arbitrary rate, thus keeping up with the growing requirements, but at the cost of exhausting the resource at a disastrous pace.

The importance of reducing energy consumption growth (and thus to avoid the energy Malthusian trap) is thus dual: maximize the usefulness of traditional renewable sources on one hand, and maximize the duration of non-renewable sources on the other. And yet, it would take extremely low growth factors for non-renewable sources to get anywhere close to billions of years in EET.

As an example, consider the case of Cohen's setup (breeder reactors, seawater uranium) in a slightly different scenarios. Assume for example that energy consumption continues to grow at the current pace for slightly more than a century (due to ongoing population growth and developing countries lifting their standards of living), leading to three more doublings, arriving short of $5\cdot {10}^{6}$ PJ/year. Assume also that only at this point humanity switched to breeding reactors fueled by seawater uranium, covering with it the total primary energy requirements, and that from this moment onwards energy consumption kept growing at a lower pace. Depending on how low the new pace is, the EET for the seawater uranium in breeding reactors grows proportionally larger:

Growth
(per year)
EET
(years)
doublings within the EET
(approx)
2%
(no change)
388
(no change1)
11
1%70710
0.5%12759
0.25%22738
0.125%39947
0.0625%68906
0.03125%116045
0.015625%189394
0.0078125%296523
0.00390625%439672.5
0.001953125%608991.7
0.0009765625%780301
0.00048828125%92549< 1

It should be clear that even at very small energy consumption growth factors (the smallest presented factor corresponds to a doubling over more than 140K years) it's simply impossible to have non-renewable resources last billions of years, although some may consider anything over 10K years to be “acceptable”, or at least “not our problem anymore”.

(Side note: even with a 100% conversion of mass to energy, i.e. 90 PJ/kg, the lowest growth rate considered won't give us billions of years: all the seawater uranium would last barely more than a million years, and all of the uranium and thorium estimated to be in the crust would last less than 4 million years, and our entire galaxy 15 million years; to get to a billion years for the Milky Way, growth would have to be lower than ${10}^{-5}%$ per year, at 90 PJ/kg.)

## Does it make sense to make decisions based on something so far into the future?

While it's true that we can't make predictions that far into the future (especially not in the millenia or hundreds thereof that might be provided by the very low growth case), it's true that at the very least we should avoid closing the paths to that future altogether.

Put in another way, we may not be able to look that far, but we are able to determine if we'll get there at all, possibly without passing through a societal collapse.

A quote frequently attributed to Albert Einstein recites something to the tune of:

I do not know with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.

Regardless of how accurate the quote (and its attribution) is, the sentiment is clear: the enormous destructive power (offered by the atom bomb or whatever even worse weapon comes after it) would be enough to throw civilization back to the Stone Age level.

A similar argument can be made here for energy consumption: we don't know when we'll overtop even the most effective form of energy production, but we do know that when that happens it will inevitably lead to civilization collapse —and it will be sudden, despite being perfectly predictable (or I wouldn't be writing this now).

With the famous bacteria in a bottle example, Bartlett highlights, among other things, how even just a few doublings before expiration most people wouldn't realize how close the expiration was, due to the vastness of the available resources and the lack of awareness on how quickly such vastness is consumed by the exponential function, and how even the successful efforts of farsighted individuals to expand the availability would only buy marginal amounts of time before the collapse.

In this perspective, it's never too early to act in the direction of reducing the exponential growth, and in fact, it's actually more likely to be almost always too late. And even if it wasn't too late already, even with the best intentions, there is actually very little, if anything at all, achievable at the individual level that would actually put a dent in the exponential. Frustrating as it may be, even a collective “doing our best” to avoid being wasteful hardly scratches the energy consumption (10 less watts per 24 hours per day per person in the world is still less than 0.5% of the total energy consumption).

The fundamental issue is much more profound, and much more systematic. And the first step in the right direction is to raise awareness about the true nature of the issue: there's a much more urgent problem to address than how to produce energy, and it's how to reduce consumption —and not by the crumbs for which the end user is responsible, but for the entire chain of production, from raw material extraction down to the point of sale.

As I mentioned, this might be the only upside of the transition away from nuclear, and similar “Green New Deal” fairy tale initiatives: promoting consumption reduction by energy starvation —although one would wish there were better ways. And worse, it really won't be enough anyway, as long as it's set in the same system for which growth is such an essential component.

We need a completely new direction.

## A final(?) remark

I'm hardly the first to make such considerations, and I will surely not be the last. Aside from Bartlett whose famous talk on “Arithmetic, Population and Energy” that consciously or not inspired my initial curiosity into looking at the exponential expiration time for nuclear power, others have now and again discussed the finite limits we're set to meet sooner rather than later in our path of growth, including ones I haven't discussed here, such as the waste heat disposal issue.

And yet, awareness of the issue and of its importance is slow in the uptake. It could easily propagate exponentially, and yet (for once that the exponential could work in our favour!) it seems to encounter such large resistance that it barely trickles out with linear growth, and a slow one at that.

Where does this resistance come from? With all the campaigning on climate and “going green” and sustainability, one would expect this crucial side of the issue to be heard more. The numbers and the math behind it aren't even that hard to grasp. So why?

A possible explanation could be that the timeline is still too long to be able to catch people's attention: we can't get people truly involved with the climatological catastrophes we are bound to experience in the next decades, why would they worry about energy suddenly running out three centuries from now?

But I think there's something more to it. And yes, a sizable part of it is the pathetic realization that climate, sustainability and “going green” can be varnish to commercial exploitation (greenwashing, as they call it); full-chain consumption curbing, on the other hand, cannot, as it's the antithesis of what commercial exploitation thrives on.

But beyond that, there's most likely the realization that we're already at the point where any serious effort at sustainability with current standards of living would be in vain without a drastic reduction not so much of the consumption, but rather of the consumers.

1. note that this is in addition to the time necessary to get from $6\cdot {10}^{5}$ to $5\cdot {10}^{6}$ PJ/year; the difference between starting the consumption at the $6\cdot {10}^{5}$ PJ/year level versus starting at the $5\cdot {10}^{6}$ PJ/year level is marginal. ↩