You are to compare heat transfer processes from a cylindrical pipe in three separate cases. One case has no fins, one has longitudinal fins, and one has circular $($radial$)$ fins. The three systems are shown on the right.

Tube OD: $1\mathrm{.}5$ in

The fins have a measured thickness of $$ in and extend $1\mathrm{.}25$ in

Tube Wall Temperature:

$F$

Air Temperature:

$F$

Tube and Fin Conduction: $9\mathrm{.}55$ BTU$/$h ft $F$

Air Convection: $32\mathrm{.}4BT\frac{U}{h}f{t}^{2}F$

Calculate the heat flow per tube length without fins $($units BTU$/$h ft$).(4)$

We add $10$ longitudinal fins evenly spaced around the tube.

a$)$ Calculate and report the values of $A_o$ and $A_f$ per tube length $($units $f\frac{t^2}{f}t).(4)$

b$)$ Calculate and report the value of $L\sqrt[\phantom{2}]{\frac{h}{k}t}$ for the abscissa of the efficiency chart. $(4)$

c$)$ Report the value of ${}_f.$ Justify your answer using the chart with high precision. $(6)$

d$)$ Determine the heat flow per tube length with the $10$ longitudinal fins. When any of the above answers are wrong, your answer here will not be graded because it is automatically wrong. $(10)$

We use circular $($radial$)$ fins.

a$)$ How many integer number of fins are needed to have at least the same heat transfer area $A_f$ as determined for the $10$ longitudinal fins? $(3)$

b$)$ What are the resulting values of $A_f$ and $A_o$ for this case? $(4)$

c$)$ Calculate and report the value of $(r_o-r_i)\sqrt[\phantom{2}]{\frac{h}{k}t}$ for the abscissa of the efficiency chart. Also calculate and report the value of $\frac{r_o}{r_i}.(4)$

d$)$ Report the value of ${}_f.$ Justify your answer using the chart with high precision. $(6)$

e$)$ Determine the heat flow per unit length. When any of the above answers are wrong, your answer here will not be graded because it is automatically wrong. $(10)$