Problem $2$: Steady$-$State Heat Conduction in Cylindrical Co$-$ordinates $(25$ Points$)$

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

Heat is generated within a cylindrical battery at the rate of $Q_{\text{g}\text{e}\text{n}}\frac{W}{m^3}($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 $T_{\text{f}\text{l}\text{u}\text{i}\text{d}}$ 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 $Q_{\text{g}\text{e}\text{n}}($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: $1\frac{W}{m}-K.$

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 $Q_{\text{s}\text{e}\text{d}}($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$)$