As discussed in Section 10.6, liquid-fuel-based portable fuel cells tend to make more sense than batteries for long-operating-lifetime applications. Consider a 20-W laptop system based on a direct methanol fuel cell. In order to produce 20W, this system requires a 400-cm³ DMFC. The DMFC is supplied by a 50%–50% methanol–water fuel reservoir with an energy density of 3400 Wh/L and can convert 20% of this fuel energy into electricity. In contrast, a lithium-ion battery system alternative provides an energy density of 200 Wh∕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 10.22, which quantitatively compares the size versus operating lifetime characteristics of the fuel cell and battery systems.