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Types of Fuel Cell

Due to technical issues most fuel cell technologies use hydrogen as a fuel.  Hydrogen can be derived from any hydro-carbon fuel by reforming, but unfortunately incurs a reduction in practical efficiency below ideal because the reforming process involves losses and because of the less favourable thermodynamic properties of hydrogen compared to hydro-carbons (see 'Thermodynamics' section under 'Research & Analysis').

Fuel cell technologies can be classified in many ways, for example by temperature, fuel type, oxidiser type, or charge carrier.  However the most common classification is by electrolyte type:

Polymer Electrolyte (PEFC)
The electrolyte in this fuel cell is an ion exchange membrane that is an excellent proton conductor, such as  fluorinated sulfonic acid polymer.  Charge carriers are hydrated protons, which means that the membrane must remain hydrated, requiring tight water management.  Temperatures are limited to below 120ºC by the polymer properties; which, combined with the water management requirements means that only H2 rich gas with minimal CO (a catalyst poison at low temperature) can be used as fuel.  The low temperature also means that higher catalyst loading (platinum in most cases) than in other types of cell is required in both the anode and cathode.
PEFCs deliver high power density, which offers low weight, volume and cost (despite the platinum requirement).  They are also exhibit a relatively low sensitivity to orientation and contain no corrosive fluids which makes them ideal for transport applications.  Commercialisation is underway with numerous car and bus demonstration systems in use.
Typical operating conditions are 80ºC, 0.285MPa and H2 supply with less than 100ppm CO.  Efficiencies of up to 60%  are predicted for on board H2 supply or 40% with on board reforming.  Smaller units have also been developed, suitable for powering lap tops and other low power devices.

Alkaline Electrolyte (AFC)
The electrolyte in this fuel cell is concentrated (85 wt.) potassium hydroxide(KOH) in high temperature cells (~250ºC), or less concentrated (35-50 wt.) KOH for lower temperature (<120ºC) operation.  The electrolyte is retained in a matrix (usually asbestos), and a wide range of electrocatalysts can be used (e.g., Ni, Ag, metal oxides and noble metals).  Charge is carried by hydrated protons which flow to the cathode (air electrode) as in PEFCs.  Unfortunately the cells are very prone to poisoning, so require an H2 supply containing no other reactive constituents.  This also applies to the oxidiser flow.
AFCs were the first type of fuel cells to be developed (as early as 1930), being used in the early 1960s for the Apollo space vehicle, where their high power to weight ratio, capacity to operate on pure hydrogen (used in the main engines), and potable water by-product made them a perfect solution.  However, they have had relatively little success in terrestrial applications due to the high cost of producing high purity fuel and oxidiser streams, plus corrosion problems.  Typical efficiency is 60%.
See also Metal-Air Fuel Cells .

Phosphoric Acid Electrolyte (PAFC)
The electrolyte in this fuel cell is 100% concentrated phosphoric acid retained in a matrix which is usually silicon carbide.  Operating temperatures are in the range 150 to 220ºC, the lower limit due to loss of ionic conduction and CO poisoning of Pt catalysts, the higher limit due to loss of stability of the acid. Use of 100% concentrated acid makes water management simpler than in AFCs due to the lower water vapour pressure in the cell.  Like PEFCs and AFCs charge carriers are cathode seeking hydrated protons.  Poisoning is less of a problem in PAFCs, than in lower temperature cells, however CO, CO2 and S concentrations must be below a few hundred PPM due to the Pt catalysts.
PAFCs were the first to reach commercialisation with over 60MW of demonstrators in operation.  At present they offer the lowest cost per kW and are used mainly for plants of 50 to 200kW capacity, though some 1 to  11MW plants have been built.
Development interest is however dropping off, and PAFCs are expected to be superseded in the next few years.
Typical efficiency is low at just 40%.

Molten Carbonate Electrolyte (MCFC)
The electrolyte in this fuel cell is usually a combination of alkali carbonates, such as Na and K, which is retained in a ceramic matrix of LiAlO2.  The fuel cell operates at about 600 to 700ºC where the alkali carbonates form a highly conductive molten salt, with anode (fuel electrode) seeking carbonate ions providing ionic conduction.  Due to the high operating temperatures in MCFCs, Ni (anode) and nickel oxide (cathode) are adequate to promote reaction eliminating the need for expensive and impurity sensitive noble metal catalysts, reducing catalyst costs giving greater fuel flexibility.
MCFCs are considered to be second generation fuel cells because they will reach commercialisation after PAFCs. Their much higher operating temperature makes materials selection and mechanical design constraints very different, plus they have anode seeking charge carriers which result in water formation on the fuel side requiring different water management techniques, all of which have taken time to develop.
MCFCs offer higher electrical efficiencies than PAFCs at around 60% plus the possibility of cogeneration (water heating) which makes overall efficiencies of 80% feasible.  At present there are numerous demonstration plants of up to 3MW capacity, though most are of multi-hundred kW capacity.  Most existing plants do not take advantage of the combined cycle possibility (using hot fuel cell gases to drive a bottoming cycle), due to the low Carnot limit, and practical difficulty of designing specialist plant.

Solid Oxide Electrolyte (SOFC)
The electrolyte in this fuel cell is a solid, nonporous metal oxide, usually Y2O3-stabilised ZrO2.  Cells operate at 650 to 1000ºC where efficient conduction of anode seeking oxygen ions takes place.  Typically, the anode is Co-ZrO2 or Ni-ZrO2 cermet, and the cathode is Sr-dopped LaMnO3.  Like the MCFC operating temperatures are high enough to allow internal reforming and promote rapid kinetics with non precious materials.  Even higher temperatures than the MCFC result in by-product heat of a temperature suitable for use directly in standard steam turbine bottoming cycles, but mean that materials and mechanical design requirements are  more stringent.
SOFCs slightly lower electrical efficiencies, around 45%, than MCFCs due to their higher operating temperature (see 'Thermodynamics' section under 'Research & Analysis'), and less advanced development.  However, cogeneration plants of 80% overall efficiency are already in operation, with combined cycle (fuel cell in place of combustion chamber in a gas turbine) plants of 60% electrical efficiency under development.
For more information on the electrochemical operation of SOFC's see 'Cell Construction' section under 'Research & Analysis'

Reference:
This is taken from my Masters project. For context you should read the whole thing .