I was wondering about the current state of battery power density, and how batteries compare to things like animal body fat or gravity storage.

Notes:

• Hyd+tank assumes tanks like mirai's that weigh 87.5Kg for 5Kg of storage, or 5.5% effective storage by weight.
• Li-Ion and Hydrogen tank weight is not consumed when used, which means you carry "dead-weight" for the whole journey.
• Gravity storage is shown in energy per weight-distance using weights 1000x the others, which gives a more realistic scale for which it would be used.

This shows just how amazing Diesel, Fat, and even Carbs are compared to current battery tech. No wonder flies carry enough energy to stay airborne all day. It also shows how hydrogen storage problems undermine its energy density to about 1/7th of Diesel.

However this is just the energy density of the storage technology. To be truly fair, you should probably take into account the weight of the supporting engines etc to extract that energy, and the efficiency of the whole energy cycle. In particular, Hydrogen fuel cells are 60% efficient, but is also inefficient to generate and hard to store, so its whole energy cycle is pretty bad. Diesel engines peak at about 30% efficiency, and cars are about 20% efficient. Human metabolism is said to be about 25% efficient. Li-Ion batteries have 80~90% charge efficiency (power-out/power-in), and Gravity storage claims a 90% round-trip power efficiency.

So in practice, you get about 1/4 of the energy out of chemical storage, compared to about 90% for Li-Ion and Gravity storage. This means chemical storage is only about 10~20x, and hydrogen is about 4x, the energy density of Li-Ion.

Li-Ion per Kg is 50~100x more energy than Gravity per tonne-metre. Also note that concrete is 2.3 t/m^3, and lead is 11.3 t/m^3, so large weights can take up large amounts of volume too. So a 10t lead weight of nearly 1m^3 volume with a 10m drop would give you about the same energy as 1Kg of Li-Ion Battery, and the same weight of fat would give you 40x as much as Li-Ion.

A Li-Ion Tesla powerwall has 12.5 kWh of capacity, and weighs 114Kg. An equivalent gravity storage unit with a 10m drop would need 500t of weight, or 44m^3 of lead (a 11m*2m*2m block). This is the same as 1Kg or 1.2 litres of Diesel, but a diesel generator is only about 30% efficient and gets about 3kWh/l, so in practice you need 4l of Diesel for 12kWh.

The Li-Ion Telsa big battery in South Australia has 129MWh of storage, and must be at least 10,000 powerwalls, and be at least 1kt of battery.

Large scale gravity storage plans are looking at using at least 100Kt weight with 100m drop which is 27MWh of storage.

I commented on hydrogen efficiency here;

https://www.solarquotes.com.au/blog/lavo-hydrogen-battery-review/#comment-893426

In summary hydrogen efficiencies are, electrolysis 80% (theoretical max 94%), fuel-cell 60% (theoretical max 83%), for 80% x 60% = 48% (theoretical max 94% x 83% = 78%), not taking into account storage losses. Compressing hydrogen to 70MPa (38kg/m^3) is ~6kWh/kg (isothermal min 1.36kWh/kg) after taking into account adiabatic compression and compressor efficiencies. This means storage efficiency is around 85% (theoretical max 96%), so the total efficiency is 48% x 85% = 41% (theoretical max 78% x 96% = 75%). See also;

• https://arxiv.org/pdf/1702.06015.pdf
• https://www.sciencedirect.com/topics/engineering/compressed-hydrogen

Note https://lavo.com.au/ claims to get >50% round-trip efficiency, which seems to be by using metal-hydride chemical compression/storage and using waste-heat from the fuel cell to liberate the hydrogen from the metal-hydride storage, combined with li-Ion battery to act as a buffer for slow-rampup. I still find the 50% claim questionable. Also note the metal-hydride storage cells are 32Kg per 10kWh of effective storage, or an energy density of 0.31kWh/Kg not including the fuel-cell weight. This makes it around the same energy density as Li-Ion.

For compressed gas storage, the mirai pressure tanks are 87.5Kg for 5Kg of storage or 87.5/5 + 1 = 18.5Kg per 1Kg of H2. Assuming 60% fuel-cell efficiency that works out as `33.3*0.6/18.5 ~= 1.1 kWh/Kg, or only 4x the best Li-Ion. However, you still need to factor in the fuel-cell weight.

The specs for available fuel-cells I found are pretty heavy and would eat into the kWh/Kg pretty badly unless you carry a lot of hydrogen;

• 5kW 40% efficient 32.5Kg
• 1kW 40% efficient 4.4Kg
• 0.5kW 56% efficient 5.8Kg
• Mirai 114kW ??% efficient 56Kg

Note the mirai's fuel cell is a pretty amazing 2kW/Kg. This puts the best possible kWh/Kg of a full hydrogen power supply at::

W = P/2  + 18.5*K
E = 0.6 * 33.3*K ~= 20*K
D = E / W = 20*K / (P/2 + 18.5*K)
  = 40*K / (37*K + P)

Where::

P = power output in kW
K = weight of Hydrogen in Kg
W = full system weight in Kg
E = effective energy stored in kWh
D = effective energy density in kWh/Kg

So the lower the P power output and the higher the K hydrogen stored, the higher the energy density. With 3kW per Kg of hydrogen stored we get D=1.0kWh/Kg.

Comparing this to a Tesla Powerwall2 which has P=5kW, E=13.5kWh, W=114Kg, for 0.12kWh/Kg, a similar power/capacity hydrogen system would have K=0.675Kg, W=14.5Kg, for 0.9kWh/Kg, or 7.5x the energy density. Note however, a Powerwall2's energy density is less than half the energy density you can get with higher end Li-Ion batteries, and hydrogen is still half as energy efficient (per kWh you put in, you get half as much out).

A better example is the E=100kWh, W=625Kg, P=400kW (estimated from "ludicrous mode") Tesla model-S battery for D=0.16kWh/Kg. A hydrogen equivalent would have K=5Kg, W=292.5Kg, D=0.34kWh/Kg, or 2x the energy density. It's the fuel-cell power output per Kg that really hurts hydrogen here. If we reduce the power output requirement to 100kW it becomes W=142.5Kg, D=0.70kWh/Kg, or 4x better energy density. This is why many hydrogen solutions include a Li-Ion battery; to give higher peak power output with a lighter/smaller sustained power output fuel-cell.

The benefits of hydrogen also increase with increased E energy stored. For P=100kW, E=400kWh, we get K=20Kg, W=420Kg, D=0.95. Note however that the power required for hovering scales at W^(3/2) (from https://www.mdpi.com/2226-4310/6/3/26/pdf), so your power requirements increase faster-than-linear with weight, while system weight increases linearly with power, so as you scale up you will eventually have insufficient power to even lift your fuel-cell, let alone the extra hydrogen stored.

Note aircraft piston engines are also about 2kW/Kg (DC-7 see https://en.wikipedia.org/wiki/Power-to-weight_ratio#Heat_engines_and_heat_pumps), but they don't need to pay the extra electric motor weight at 10kW/Kg to convert it into mechanical power.