$1.)$ A cylindrical shell, has an inside diameter of $6$ ft and is $20$ ft long; it is made of $\frac{1}{2}-$in plate and is subjected to a pressure of $100$ psi $.(1)$ Find the stress in pounds per square inch on the metal in the circumferential section of the shell. $(2)$ What is the stress in pounds per square inch in the cylindrical sides due to the transverse pressure?

$2.)$ A drum will be constructed of SA$-285$ Grade B carbon steel by welding. The internal design pressure is $800$ psi $,$ and there are no unreinforced openings in the drum. The drum inside diameter is $48$ in $.$ What is the minimum required wall thickness of the drum?

$3.)$ A boiler drum $4$ ft in diameter is made of $1-$in plate; the tensile strength of the steel is $55,000$; the efficiency of the joint is $85$ percent. What is the estimated design pressure of the drum?

$4.)$ Find the heating surface of an horizontal return tube $($HRT$)$ boiler $78$ in in diameter, $20$ ft long, containing eighty $4-$in tubes $($inside diameter$)$ bricked in one$-$third of the distance from the top. Find the $(1)$ area of the shell, $(2)$ area of the tubes, and $(3)$ area of the heads.

$5.)$ Find the rated horsepower of the HRT boiler in the previous example.

$6.)$ Steam at $450$ psia, dry and saturated, is produced by a boiler when supplied with feedwater at $180F.$ What is the enthalpy of both the steam and feedwater in $Bt\frac{u}{\text{I}}b,$ and what is the amount of heat that must be generated by the boiler in Btu$/$lb$?$ If the boiler produces $120,000l\frac{b}{h}$ of steam, what is the heat output of the boiler in Btu$/$h$?$