Q$1-$ A pneumatically operated tool is driven from a gas cylinder. The filling pressure absolute is $205$ bar at a temperature of $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and a cylinder volume of $40$ l $.$ The tool is driven with a working pressure pe of $4$ bar and has an air consumption of $\backslash (200\backslash$mathrm$\{$I$\}/\backslash$mathrm$\{$min$\}\backslash ),$ based on $1$ bar and $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$

$1-$a How long can the tool be operated from the cylinder with a duty cycle of $\backslash (100\backslash \%\backslash )?$

$1-$b What is the operating time with a duty cycle of $\backslash (20\backslash \%\backslash )?$

$1-$c What happens if the cylinder, filled at a temperature of $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is used at an ambient temperature of $\backslash (35^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )?$

$1-$d Suppose the gas cylinder is replaced with a compressor that delivers $\backslash (500\backslash$mathrm$\{$l$\}/\backslash$mathrm$\{$min$\}\backslash ).$ The difference between the gas supply and consumption is stored in a $\backslash (0\mathrm{.}5\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ tank$.$ To maintain stable tool operation, a pressure switch starts the compressor when the tank pressure drops to $4\mathrm{.}5$ bar and stops it when the pressure reaches $8$ bar. Assuming constant temperature, calculate the compressor's run and stop times for a $100\backslash \%$ duty cycle.