Friday, August 19, 2005

 

The efficiency of fuel cells

Posted by ben @ 3:05 PM | Permanent Link

Fuel cells are electrochemical devices, there are no explosions, nothing is "burned", nothing actually even moves (unless you consider gasses and electrons moving). Work is achieved by forming a potential difference between the anode and the cathode, this potential difference is the driving force required to get electrons flowing through a circuit. So calculating the efficiency is a little different for fuel cells (and batteries) compared with heat engines.

First of all, we have to define the Gibbs free energy. This is defined as the "energy available to do external work, neglecting any work done by changes in pressure and/or volume". In a fuel cell, external work involves moving electrons around an external circuit and any "mechanical work" is not used by the fuel cell directly, it is possible to combine a fuel cell with a turbine in a combined cycle system though.

This post isn't a thermo lesson, there are plenty of resources from plenty of people who know plenty more about thermodynamics than I do. But to calculate the Gibbs free energy of the reactions occuring in a fuel cell, you have to first know what reactions are occuring in the fuel cell. We'll take the PEMFC as an example:


Cathode:
1/2O2+2H++2e- <--> H2O

Anode:
H2 <--> 2H++2e-

Overall:
H2+1/2O2 <--> H2O

Gibbs free energy is defined as: dG = dH - TdS

To calculate the Gibbs free energy (dG), we do the old:
dG = n*gf products - n*gf reactants
trick, where n is the number of moles and gf is the Gibbs free energy of formation.

So for the PEMFC, we'll have dG = 1 * gf, H2O - 1/2 * gf,O2 - 1 * gf,H2

gf changes with temperature and state, which means that in order to calculate it, you'd need to do some integrals and find some heat capacities as a function of temperature and it's not stuff you'd want to do by hand (it's not too hard but it's just long) and probably not stuff that most people who might possibly be reading this would ever want to know about. Lots of thermodynamics text books will have tables of these values as a function of temperature.

So, for convenience sake, here is the Gibbs free energy for the PEMFC reaction:
H2 + 1/2O2 <--> H2O





Phase of waterTemp (C)dGrxn (kJ/mole)
Liquid25-237.2
Liquid80-228.2
Gas80-226.1
Gas100-225.2
Gas200-220.4
Gas400-210.3
Gas600-199.6
Gas800-188.6
Gas1000-177.4

If there are no losses in the fuel cell, or if the processes are reversible, then all of the Gibbs free energy is converted into electrical energy, but this never happens.

The efficiency of the fuel cell is defined as:
Eff = dG/dH
or Eff = 1 - TdS/dH
The maximum energy from any fuel is achieved by burning it, this is the "enthalpy of formation", dH. This is found in a similar way for the Gibbs free energy of reaction, I'd recomend looking it up in tables though because it depends on temperature and you'd have to do some more integration.

So, what I'm really leading towards is a comparison of the maximum theoretical efficiencies between a heat engine (using the carnot cycle) and a fuel cell. Of course, the Carnot efficiency is (T1-T2)/T1 where T1 is the operating temperature of the heat engine and T2 is the temperature of the exhaust (assuming exhaust is at 25C):

So then, at low temperatures, fuel cells have a high "theoretical" efficiency compared to a heat engine, as the temperature increases though, the "theoretical" fuel cell efficiency decreases and the "theoretical" heat engine efficiency increases. At around 750C, the Carnot efficiency and the fuel cell efficiency intersect.

Remember though that these are maximum theoretical efficiencies. They are very hard to reach. It gets even a little more complicated as well. In practice, high temperature fuel cells are actually more efficient than low temperature fuel cells because the losses (explained in another post) are smaller at higher temperatures.

So, there you go, that's all about fuel cell efficiencies. Real world efficiencies for fuel cells are around 35-45%. This can be bumped up to 80% for high temperature SOFC fuel cells when recycling the waste heat.


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