Introduction

As the supply of fossil fuels decreases, it is possible that future stationary or mobile energy systems will use hydrogen fuel cells. This article outlines some rules of thumb for fuel cells and describes the basic calculations required to size a fuel cell appropriately and determine hydrogen fuel requirements.

Fuel Cell Reactions

Many types of fuel cells combine hydrogen with oxygen to produce DC electricity. The by-products of the reaction are water vapor and heat. The proton exchange membrane fuel cell (PEMFC) reactions are as follows:

Anode:     H2 → 2H+ + 2e-
Cathode:  ½O2 + 2H+ + 2e- → H2O
Overall:    H2 + ½O2 → H2O

where H+ denotes hydrogen ions (protons) and e- denotes electrons. Note that two electrons are produced for each hydrogen molecule.

Determining the Number of Cells

Vehicle applications would require approximately P = 100 kW (132 horsepower [hp]) of peak power to supply an electric motor at a total voltage V = 300 V. Thus, the current I in amperes would be given by the power divided by the voltage. Thus,

I = P/V = 100000 W/300 V = 333 A

A typical rule of thumb for the voltage of a single-cell fuel cell Vc is to operate at about 0.7 V. As cells are stacked together within a larger fuel cell, the voltage is additive. Thus, the number of cells required N is given by the total voltage divided by cell voltage, such that:

N = V/Vc = 300 / 0.7 = 429 cells

Determining the Cell Size

Fuel cell performance is described by a polarization plot, as shown below

This plot relates the cell voltage Vc to current density i. Since for this application the cell voltage is known, the current density can be read from the plot as i = 950 mA/cm². To produce 333 A of current, the cell cross-sectional area A required is equal to the total current I divided by the current density. Therefore,

A = I/i = 333 A / (0.950 A/cm²) = 350 cm²

Determining the Hydrogen Requirements

The required hydrogen molar flow rate (in mol/s) is given by the following relationship: ξH2 = IN/zF, where z is the number of electrons produced per mole of fuel and F is Faraday’s constant, given by 96485 Coulombs of charge per mole of electrons. Note that in this expression, z is 2 since 1 mol of H2 produces 2 mol of H+ ions and 2 mol of electrons. With this in mind, we obtain:

Determining the Heat Production

The fuel cell stack heat production Q in W is given by Q = P(1.25/Vc-1) Note that in this formula, 1.25 is the maximum theoretical voltage if all the thermal energy in the hydrogen is converted into electricity. Thus,

This heat can be removed with the vehicle radiator.

Fuel Economy: A Try-It-Yourself Final Note

Note that at a constant highway speed of 88 km/hour, the power requirement is about 25 kW = 25000 W. A trial-and-error solution (or a plot of stack power P in W as a function of the stack current I in A) shows that the fuel cell stack would operate at I = 62 A and V = 403 V, with a cell voltage Vc = 0.94 V, a current density i = 177 mA/cm², and a hydrogen consumption rate of 0.138 mol/s = 0.276 g/s = 0.992 kg/hour. Thus, the fuel cell vehicle can obtain about 88 km = 60 miles on 1 kg of hydrogen.

Note that in this final example, an electric DC/DC voltage converter would be needed to change the fuel cell output to the 300 V required for the electric motor.