his problem is presented in a format that might be representative of an assignment you would

get as an early career engineer. The response to your supervisor or manager would be a memo

with an explanation of how you arrived at your solution. For this homework submit a detailed

description of your method and analysis along with a plots of the temperature distribution

versus time.

Your first assignment at your new job is on a system that deals with steam. In one section of the

system is a tank with walls that are $10$cm thick of a material with a density of $10$ gm$/$cm$3,$ a

thermal conductivity of $1\mathrm{.}0$ cal$/($cm$-$sec$-$C$)$ and a specific heat of $0\mathrm{.}1$ cal$/($g$-$C$).$ The system is

initially at a uniform room temperature of $25$C$.$ During a structural verification test of the

system, steam is introduced into the tank such that the inside walls are at $100$C$.$ The structural

analyst requests that the test be stopped when the temperature half way through the wall

reaches $40$C$.$ Your supervisor asks you if you could compute the time at which this will occur

and you explain that a Fourier series solution of the heat equation could be used based on your

excellent recollection of what you learned in AME $301.$

Use a Fourier series solution of the one$-$dimensional heat equation to solve for the temperature

distribution through the wall as a function of distance and time $($I was able to get good plots

looking at $10$ increments through wall$).$ Using that solution, find the time at which the

temperature is $40$C halfway through the wall but also plot up the temperature distribution as a

function of time $($distance through the wall on the x$-$axis, temperature on the y$-$axis and

separate lines for different times$).$ The solution will include a Fourier series representation of

the temperature distribution. Use at least $20$ terms in the Fourier series, I used $40$ terms and got

a reasonable representation of the initial temperature, the representation at $20$ terms was not

as good but the solution of the time when the center hit $40$C was the same for $20,30$ or $40$

terms. In class, I will show the spreadsheet I used to solve this problem so you can see what the

plots should look like.