liquid$-$fuel$-$based portable fuel cells tend to make more

sense than batteries for long$-$operating$-$lifetime applications. Consider a $20-$W lap$-$

top system based on a direct methanol fuel cell. In order to produce $20W,$ this system

requires a $400-c{m}^3$ DMFC$.$ The DMFC is supplied by a $50\%-50\%$ methanol$-$water

fuel reservoir with an energy density of $3400W\frac{h}{L}$ and can convert $20\%$ of this

fuel energy into electricity. In contrast, a lithium$-$ion battery system alternative pro$-$

vides an energy density of $200W\frac{h}{L},100\%$ of which can be converted to electricity.

Based on these specifications, calculate the minimum operating lifetime for which

the fuel cell system will deliver greater volumetric energy density than the lithium

ion battery system. Draw a graph, similar to the one shown in Figure $10\mathrm{.}22,$ which

quantitatively compares the size versus operating lifetime characteristics of the fuel

cell and battery systems.

Figure $10\mathrm{.}22.$ System size versus operating lifetime comparison of a liquid$-$fueled portable power

fuel cell system versus a battery system. The large upfront size cost of the portable fuel cell system

is recouped for long operating missions by the higher energy density of the fuel cell's liquid fuel.

Solve it in detail and show calculation steps