$1$D steady state conduction

This problem involves calculating the heat loss by conduction

through the support legs of a tank storing liquid hydrogen.

A tank storing liquid hydrogen is supported by truncated con$-$

ical legs, which elevate it off the floor. These legs are well$-$

insulated except at the top and bottom circular surfaces, which

are welded to the tank and in contact with the floor, respec$-$

tively. The temperature at the interface of the support and the

$tank(x=x_b)$ is known to be $T_c=60[K].$ The support is in con$-$

tact with the ground at $(x=x_a)$ and is at constant temperature

$T_h=285[K].$ The support has constant thermal conductivity of

$k=10[\frac{W}{m}*K].$

Your task is to determine the heat loss by conduction through the

footer. The design specifies a top$-$side radius of $b=5[cm]$ and

length parameters $x_a=1[cm]$ and $x_b=4[cm].$ Note that these

length parameters refer to the distance from the projected point

of the cone, which is not actually part of the solid material. The

geometric parameter $=\frac{b}{x_b}$ relates the cross$-$sectional radius

to the $x-$coordinate.

a$.)$ Sketch what you expect the temperature profile as a function of position $x$ to look like for this problem. You

should be careful to get the conceptual details of the sketch right based on the information given in the problem

statement and your knowledge of Fourier's law. Label known temperatures and comment on any important

features in your plot.

b$.)$ Draw a differential control volume and use it to derive the ordinary differential equation $($ODE$)$ that governs

this problem. Your ODE should be in terms of temperature, position, and any parameters given in the problem

statement. Simplify the result as much as possible.

c$.)$ Solve the ODE to find an expression for temperature as a function of position. Your solution should have $2$

unknown constants, $C_1$ and $C_2.$

d$.)$ Write mathematical statements of the boundary conditions needed to solve the problem, and use them to

solve analytically for the unknown constants.

e$.)$ Enter your equation$($s$)$ into EES, and calculate the rate of heat transfer through the support. Use the EES

Key Variables feature to report the value you calculate.

f$.)$ Prepare a plot of the temperature as a function of position $x$ between $x=x_{a}dots{x}_b.$