The upper operational temperature limit for a lithium ion battery is $343$ K $.$

Heat is generated within a cylindrical battery at the rate of $Q_{\text{e}\text{s}\text{s}}\frac{W}{m^3}($as a result of

electrochemical reactions within the battery$).$ Assume that the end caps of the battery

$($shown as positive and negative terminals in the image above are insulated$).$

The circumferential surface of the battery $($denoted by "metal case" in the image above$)$

is subject to a heat transfer coefficient $($ h $)$ of $20\frac{W}{m^2}-K$ and a surrounding $I_{\text{b}\text{h}\text{u}\text{i}\text{d}}$

temperature of $300$ K during its operation such that steady state temperature conditions

are being maintained inside the battery.

Your goal is to determine the maximum heat generation rate $Q_{ggn}($in $\frac{W}{m^3})$ within the

battery for the conditions shown such that the maximum temperature within the battery

does not exceed $343$ K $.$

The average radial thermal conductivity of the battery is: $1\frac{W}{m}-$K The size of the

battery is $14$ mm diameter and $50$ mm length

a$)$ Simplify the differential equation for energy conservation associated with this

problem $(2$ points$)$

b$)$ Identify the boundary conditions $(2$ points$)$

c$)$ Solve the differential equation to obtain a general solution $(2$ points$)$

d$)$ Solve for the constants to obtain the temperature profile $(4$ points$)$

e$)$ Determine the maximum heat generation rate $Q_{\text{e}\text{a}\text{t}}($in $\frac{W}{m^3})$ for your battery size $(5$

points$)$

f$)$ Show that heat generated within the battery is equal to the heat lost from the

boundaries to ensure that your solution corresponds to steady$-$state conditions $(10$

points$)$