Design Example $3\mathrm{.}1$: A process vessel is to be designed for the maximum operating pressure of $500k\frac{N}{m^2}.$ The vessel has the nominal diameter of $1\mathrm{.}2m$ and tangent to tangent lergth of $2\mathrm{.}4m.$ The vessel is made of IS : $2002-1962$ Grade $2$B quality steel having allowable design stress value of $118M\frac{N}{m^2}$ at working temperature. The corrosion allowance is suggested to be $2nm$ for the life span expected for the vessel. The vessel is to be fabricated according to class $2$ of Indian Standard specifications which stipulate the weld joint efficiency of $0\mathrm{.}85.$

$($a$)$ What will be the standard plate thickness to fabricate this vessel $?$

$($b$)$ If a spherical vessel having the same diameter and thickness is fabricated with the same quality steel, what maximum internal pressure the sphere will withstand safely?