A PEM fuel cell has the following parameters:

Operating temperature: $75\backslash$deg C

Anode: Pt$/$C$,1\mathrm{.}5$ atm H$2($g$)$

Cathode: PtCo$/$C$,2$ atm O$2($g$)$

Bipolar plate: $10$ mm thick graphite

Membrane: $60$m thick nafion, resistivity $0\mathrm{.}015$ cm$2$

Limiting Current Density: $2\mathrm{.}4$ A cm$-2$

i$)$ Using an Excel spreadsheet model, calculate and show separately on the same plot the:

$$ Overpotential due to the anode,

$$ Overpotential due to the cathode, and

$$ Overall total activation overpotential for the operating current density range.

ii$)$ Expanding the model to include all overpotential losses in the fuel cell, plot the fuel cell polarisation curve and corresponding power curve $($on the same plot$).$ The following expression can be used to simplify the concentration overpotential:

$=(/)*$ ln $[/()]$

where j is the current density and jL is the limiting current density.

For both parts of question d$),$ show all working and equations used and justify any assumptions you have made, citing literature sources where relevant.

e$)$ i$)$ Engineers designing a fuel cell stack for an automotive application use the fuel cell in question d$)$ as a basis for their system. The electronics of the drive train require an output voltage of $150$ V and peak power of $114$ kW$.$ Size the fuel cell stack required for this car. State your reasoning for your selection.

ii$)$ Estimate the total cost of metals in the catalysts for this fuel cell stack with reference to typical loadings of the catalysts from literature

iii$)$ The car has a target range of $350$ km$.$ Stating all assumptions and referencing reasonable values from the literature, size the hydrogen tank required.

f$)$ What are the potential causes of performance degradation over the lifetime of the vehicle, and what tests might you use to characterise this in an aged fuel cell?