# Over 100% efficiency of H2O separation, help please!

I'm trying to calculate the maximum efficiency of separating water into hydrogen and oxygen using thermal decomposition. I assume I have a system of constant pressure (namely one atmosphere) since higher pressures requires higher temperatures. I have read that at around 3000 Celsius more than half of the water decomposes. So, I'll list the steps I've used to try and solve this:

1. Calculate energy needed to heat 1 mole of water from ambient temperature (20 C) to 3000 C.
2. Calculate energy recovered with a heat exchanger.
3. Divide energy of the separated hydrogen and oxygen gas by the energy used (energy needed - energy recovered)

An assumption I have made here is that the system is already running, so we have hot H2O at 3000 C, then we extract one mole of separated H2O and recover the heat from that by running it in the heat exchanger with the new H2O entering the system which replaces the mass of the separated H2O. Thus the actually percentage of H2O which is separated in the hot tank is only relevant for practical separation efficiencies which I will ignore here (might be a reason to my over 100% efficiency, but I don't think so).

This works well, I integrate the heat capacity (Cp) over temperature for each substance from 20 to 3000 C and get the joules/mole required for heating and how much I can get back. I also add the vaporization energy and formation energy as part of the heating energy and I get an efficiency of around 90%.

However, when I try to add my heat engine to drive the heating itself I run into problems. First of all, I had a very hard time deriving/calculating the heat put into the system for a certain work when the hot side of the heat pump did not stay constant. I tried setting up a differential equation using the Carnot cycles efficiency, but the result said the temperature went down as I put in work, so something went wrong, if anyone could help me there it would be much appreciated.

But then I came to the realization that the vaporization heat will be added to the system at a constant temperature, so I calculated the maximum COP for a heat pump operating between 20 C and 100 C, came out to be around 4.6. Thus I can divide the vaporization heat by 4.6. This is when I run into problems, if I keep the hot temperature at 3000 Celsius I get an efficiency of 99.8%, but if I increase the operating temperature above 3800 Celsius I get efficiencies above 100% (up to 101.5% at 5726 Celsius, above which I don't have any Cp data).

The problem gets even worse because a heat engine could be used to add the heat to the rest of the system too, which would drive up the efficiency even more. What's going on here? Have I completely overlooked something or is it just that the separation energy would make up for this? Also, does anyone have a way to estimate the separation energy?

Thank you for reading this and putting up with me misusing thermodynamics.

• Efficiency for this case should be the total heat content of the fuel that produced the flame that cracked the water (Htot), minus the total heat (and work) that is recovered from heat exchange and from burning the hydrogen that was produced, all divided by Htot. And I assure you, that number will be substantially lower than 100%. Question: why don't you want to use electrolysis? Aug 29, 2019 at 23:13
• @DavidWhite what do you mean by "the flame that cracked the water". The theoretical maximum shouldn't care about how you heat the water, right? Just how much heat you need. I wasn't envisioning a flame to produce the high temperatures, thinking joule heating which would be 100% efficient. I mostly just want to see if it could be better than electrolysis as far as efficiency goes. Aug 29, 2019 at 23:40
• It takes fuel to generate the temperature that cracked the water into hydrogen and oxygen. The heat content of that fuel is the heat input to your process. And regarding doing better than electrolysis, there will be a lot of low temperature heat that cannot be turned into work, so that's probably a losing proposition. Aug 30, 2019 at 0:03
• @David White why do we have to burn a fuel? Why can’t we use electricity to generate the heat :s when talking about any other device you wouldn’t take into account the efficiency of the electricity produced, you would just look at the device and say ”1 Joule of electricity = 0.83 Joule of work, thus 83% efficient” Aug 30, 2019 at 0:11
• Fine. Measure the electricity that was used to crack the water, and consider that to be your energy input into the process. Aug 30, 2019 at 1:46