It has been proposed to use large banks of rechargeable lithium ion batteries to power hybrid electric vehicles. The cylindrical batteries, each of which is of radius r; = 9 mm and length L = 65 mm, undergo exothermic electrochemical reactions while being discharged. Since excessively high temperatures damage the batteries, it is proposed to encase them in a phase change material that melts when the batteries discharge (and re-solidifies when the batteries are charged; charging is associated with an endothermic electrochemical reaction). Consider the paraffin of Problems 8.47 and 9.57.

(a) At an instant in time during the discharge of a battery liquid paraffin occupies an annular region of outer radius ro = 19 mm around the battery, which is generating Eg = 1 W of thermal energy. Determine the surface temperature of the battery.

(b) At the time of interest in part (a), what is the rate at which the liquid annulus radius is increasing?

(c) Graph the battery surface temperature versus the outer radius of the liquid-filled annulus. Explain the relative insensitivity of the battery surface temperature to the size of the annulus for 15 mm < ro < 30 mm.