Assuming sufficient electrical power, is an electrolysed oxyhydrogen rocket practical?

Question

Assume we have access to light electrical power e.g. a (hypothetical) small fusion reactor.

Is it practical to operate a water-fuelled rocket, in which water is converted to hydrogen and oxygen (or their mix, oxyhydrogen) via electrolysis, which are then burnt as propellants in a rocket engine?

1. How much electrical power would be required to hydrolyse sufficient water?

2. Does burning the oxyhydrogen (or oxygen and hydrogen separately) in a rocket engine, with fuel storage in the form of water, offer a higher specific impulse than burning standard rocket fuels?

3. If not, does the practicality of this solution (no cryofuels) make it a good tradeoff?

4. Would this approach offer advantages in the deep space environment, using water as fuel, over simply expelling the water as propellant using fusion heating?

5. If this approach works (saving space and avoiding cryocooling by storing fuel inertly and electrolysing it for burning), are there any other fuel combinations which would work better than water?

Burning liquid hydrogen and oxygen apparently gives one of the best specific impulses available: https://space.stackexchange.com/questions/25541/could-oxyhydrogen-21-h%E2%82%82-o%E2%82%82-mix-be-used-as-a-rocket-fuel

Since I believe water is a lot denser than liquid hydrogen, this could offer substantial space savings. It wouldn't save any weight since the oxygen and hydrogen molecules weigh the same whether they are water or not.

The other big advantage could be the ability to usefully burn water wherever you can find it - if this approach would be more efficient than propulsion via heating water with the fusion plant.

Edit: if water contains hydrogen and oxygen in the wrong proportions for burning, this approach will not be efficient. However I think the proportions are right since they burn back to water.

Another issue is that we would have to burn the hydrogen and oxygen as gas, as they come out of electrolysis. I don't know if this would even produce good thrust or whether they have to be liquid.

Answer 1

The same energy must be put into electrolysis to separate hydrogen and oxygen from water as you can get out of it via combustion. It is a simple energy balance.

Answer 2

First, let's see what type of power will be required.

The Saturn V rocket was able to launch the Apollo missions that that would be a good model for any potential electric rockets.

For a rocket, the power produced can be described by

$\text{Power} (P) = \text{Thrust} (F) \times \text{Velocity} (v)$

Applying this equation to each of the Saturn V stages, we can calculate how much power and energy were produced:

Stage	Thrust Mega-newton	Exhaust m/s	Giga-Watts	Burn sec	Tera-Joules
1	33.4	2580	83	150	12.9
2	5.15	4500	22	360	8.34
3	1.03	4500	4.5	500	2.32
Sum					23.6

If each Mr. Fusion produces "1.21 JigaWatts" (aka, Gigawatts) that would be about 70 units. (We're going to need a lot of old banana peels and and stale beer. The good news is my calculations (which I can provide upon request) show that we will need to fuse about 37 grams of hydrogen for that much energy.)

Moving ahead.

How do we convert electricity to thrust? The OP had the idea of separating the hydrogen and oxygen by electrolysis and piping the result into a combustion chamber. If we burn one kg of hydrogen, we will need 16 kg of oxygen and produce 286 kJ of energy. For 286 kJ, we need 17 kg of water.

For 22.6 TJ of energy, we are going to need:

$ \frac{23.6 \times 10^{12} J}{286\times 10^3 J/kg} \times 17 kg = 1.4 \times 10^9 kg H_2O $

Wup, That puts us over with weight since I was assuming something similar to the original Apollo. ("You're going to need a bigger [rocket]")

Let's say that instead, we ionize the water (maybe using a Mr. Ion unit) and use a linear magnetic field motor to propel it out the exhaust. Because we are working with advanced technologies (aka, science fiction), we can say we will accelerate the ions to 20 km/second. If we push 1kg of water per second out the exhaust at 20km/sec, we will get 20,000 N of thrust. The bad news is kinetic energy goes up the the square of velocity so:

$E=\frac{1}{2} \times M \times V^2 $

for 1 kg at 20,000 m/s that would be:

$E=\frac{1}{2} \times 1 kg \times {20,000}^2 = 200 \times 10^6$ joules and that's every second so we'd need $200\times 10^6$ watts for 20,000 N of thrust. We were talking about needing a max of $33.4 \times 10^6 $ newtons so some quick arithmetic shows

$33.4\times 10^6 N \times \frac{200\times 10^6 W}{20,000N} = 3.34\times 10^9 $ watts

we will need 3.34 gigawatts.

I forgot to mention, that would require 1.67E3 kg of water every second. For the 150 second initial burn, we'd need $150 sec \times 1.67\times 10^3 = 243 \times 10^3 kg $ of water, 1/4 million tonnes. Compare that to the 2.8 thousand tonnes of the Saturn V.

I typed this up and now I'm thinking about it, I must have missed something. The energy is effectively free with the electric rocket, all we need to lift off is the water we need for reaction mass. With the Saturn V, the energy to propel the reaction mass came from combusting the mass itself.

There it is, if I accidentally turned grams to kilograms without noticing, sorry. Please feel free to tell me how wrong I am. I have been wrong before. See here. I was corrected by @Pyrotechnical but I never corrected my answer.

Answer 3

To add just a bit to user1683793's excellent analysis:

Real-world hydrolysis of water to obtain for example combustible rocket fuel is a notoriously inefficient (lossy) process, which requires a lot of extra power input to do the molecule splitting work than the process of combusting the resulting gases will give back.

This so-called overvoltage problem places severe limits on the viability of electrolytically producing hydrogen as a carbon-free replacement for fossil fuels.