Question Two

$[10$ Marks$]$

A pressure$-$relief valve contains a poppet with a $4\mathrm{.}07c{m}^2$ area on which the system pressure acts. The poppe must move $0\mathrm{.}351$ cm from its fully closed position in order to pass pump flow at the pressure$-$relief valy setting $($full pump flow pressure$).$ The pressure required to overcome the external load is $68\mathrm{.}95$ bar. Assum that the pressure$-$relief valve setting is $60\%$ higher than the pressure required to overcome the external loa If the valve$-$cracking pressure is $10\%$ higher than the pressure required to overcome the external load, find ti following:

$($a$)$ The required spring constant of the compression in the valve.

$($b$)$ The required initial compression of the spring from its free length condition as set by the spring adjustment mechanism of the pressure$-$relief valve.