Write a MATLAB function dimensionless f $13$ B $11$ F $1($ x $\_$tilde, mL $,$ mdL $),$ where

$m=\sqrt[\phantom{2}]{\frac{h\text{p}\text{e}\text{r}}{k{A}_c}}=\sqrt[\phantom{2}]{\frac{h}{kL}}$

$B{i}_L=\frac{hL}{k}=m^2{L}^2,($ note $\frac{h}{mk}=mL)$

This function can be used to compute the insulated tip case when $mL=0.$

Exercise this function by considering a fin of rectangular profile $(wt)$ using these

parameters: $$

$k=100\frac{W}{m}-K,L=0\mathrm{.}10m,t=1\mathrm{.}5mm$

and investigate the "insulated tip approximation" by comparing the temperature

distributions along the fin for different convection coefficients as $L$ is varied. Recall

that $L$ for the rectangular profile is

$L=\frac{tw}{2(t+w)}$

and investigate the behavior for $tw10t($maybe $w=[t$:$t$:${10}^{*}t]),$ and consider three

different heat transfer coefficients $h=\left[\begin{array}{ccc}1& 10& 100\end{array}\right]\frac{W}{m^2}-K.$

Also compare the fin efficiency parameters for the convection tip and insulated

approximation. You may wish to write a function etaX$13$B$11$F$1($ mL $,$ mdL $)$ which can

give the solution for the insulated tip case with $mdL=0.$

a$.$ Plot the temperature distributions versus $\frac{x}{L}$ for each $L$ and include both exact and

approximate solutions on the same graph. The graph might look something like

Figure $1.$ Do this for each value of $h$ considered $($one plot for each $h).$

b$.$ Compute the "goodness" of the approximation by comparing how close the two

functions are to each other. One metric that can be used to quantify this is the

average distance between the curves, sometimes called the root$-$mean$-$squared $($RMS$)$

error between them:

$E_{RMS}=\sqrt[\phantom{2}]{\frac{{\displaystyle \underset{?}{\overset{?}{}}}(T_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}}-T_{\text{e}\text{x}\text{a}\text{c}\text{t}}{)}^2}{N}}$

where $N$ is the number of points in the curve.

Create a table showing how $L,E_{RMS},{}_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}},$ and ${}_{\text{e}\text{x}\text{a}\text{c}\text{t}}$ vary with $w.$

c$.$ Some questions to consider:

i$.$ How good is the insulated tip approximation for computing the temperatures

along the fin?

ii$.$ How good is the insulated tip approximation for computing the fin efficiency?

iii. As the fin width increases, what is the limiting value for $L?$ what ratio of $\frac{w}{t}$

approximates this limit well?

Figure $1$ Sample plot showing temperature variation along fin