# Maximum Theoretical Energy Density of Clockwork

So, I'm slowly working on a concept a little like "Victorian Shadowrun", where magic is suddenly introduced to Victorian Society.

One of the side effects of this is a metal, let's call it Orichalcum which has some rather interesting properties, including an ability to wind a spring of it rather tighter than a steel spring.

Actually, a LOT tighter.

My question is though, how much tighter could a metal be wound than a spring steel? What sort of energy densities are possible? Can we get up to something like 45 MJ/kg (gasoline)? Higher?

I'm assuming there's no residual magic in the metal, just a perfect alignment of atoms etc.

• I'm sure I've looked at this before, but I can't seem to find anything on it. I'd be startled if you could manage more than 25Mj/kg for an ideal system though. I'll see if I can find anything in my notes. – Starfish Prime Dec 12 '19 at 16:08
• I suspect that calculations based purely on stretching of atomic bonds would produce quite a high energy, but with the real world, crystal boundaries etc would make it a lot lot less. – Slarty Dec 12 '19 at 16:26
• Since it's essentially a magic metal, i.e. not one available in the rest of the known universe, you can posit any kind of atomic bond strength you want as well as any deformation limit, etc. Not to mention if you're just looking for energy per unit mass, make it a metal with near-zero density. – Carl Witthoft Dec 12 '19 at 16:33
• Let's assume real world elements, but in any theoretical structure – Riddles Dec 12 '19 at 16:38
• You could look at the exotic molecular spring of solid hydrogen in Phase IV (250+ GPa) or is that outside the remit? – Tantalus' touch. Dec 12 '19 at 16:42

For plausible real-world materials such as a spring made from single-walled carbon nanotubes, you'll get specific energies of the order of 2.125MJ/kg (3.4MJ/l). That's pretty good... much better than lithium ion batteries, but its a long way short of chemical energy storage.

You could handwave your springs being much stronger, but be aware that this would imply that they have a stronger tensile strength than carbon nanotubes, and that has a lot of implications with regards to material technology and manufacturing. Your magical metal would make an excellent armour, for example, and would allow for all sorts of interesting and complex engineering to be performed that would otherwise have been impractical with real world materials of the same era.

• Bother. Not quite enough to power a cyber arm (assuming you have a spring of about 1kg). I may need a little more "hand-wavium" than I'd have liked :( – Riddles Dec 12 '19 at 17:09
• @Riddles if you're prepared to go steampunk instead of rocketpunk, have a look at this thing: a peroxide-driven bionic arm. A portable rocket-fuelled steam-driven prosthesis. What's not to like? You can use your magical unobtanium as fuel, or seals, or other complex and fiddly working parts to make it a little more plausible that it would be available, practical and at least slightly safe given the rest of your available technology. – Starfish Prime Dec 12 '19 at 19:34
• That is scary! I wonder why we haven't heard anything about it since 2007... – Riddles Dec 12 '19 at 21:25
• @Riddles at least in part because the user interface is still more or less absent... there's a lot of neuro-electronic magic still needed. I wonder if also battery technology has marched on a bit since then, and the prospect of carrying a bottle of explosive oxidiser around has become less appealing... – Starfish Prime Dec 12 '19 at 21:33
• Possibly the chance of your arm catching fire may not have enthused the users... – Riddles Dec 13 '19 at 14:05

It looks like somebody did the math for you.

• The energy per unit mass in a bit of the spring that is strained with a strain of $$ε$$ is $$0.5 \cdot Y \cdot ( ε^2 ) / ρ$$ where Y is the Young's modulus, and ρ is the density.

• The stress τ is (roughly) related to the strain by $$τ = Y \cdot ε$$ and the maximum stress you can cope with [in a spring that is to be reused many times] is called the Yield strength, which I'll denote by the symbol $$τ_{max}$$.

Putting these facts together, if $$ε_{max}$$ is the maximum strain $$ε_{max}=τ_{max}/Y$$ The maximum energy per unit mass in the bit of the spring that is maximally strained is $$0.5 \cdot Y \cdot ( τ_{max}/Y ) \cdot 2 / ρ = 0.5 \cdot (τ_{max}) \cdot2 / ( Y \cdot ρ )$$

which can be tabulated as follows

Considering that 1 Wh is equal to 3.6 kJ, you are pretty far from what you can get with gasoline, since you get at best below 24 kJ/kg, 3 orders of magnitude lower, by using carbon fiber.

• Not marked your answer as "the answer" but you definitely get a +1 for reminding me about the equation! My only excuse is not having even had to think about that for 30 years! :) – Riddles Dec 16 '19 at 14:31