$5.$ A beverage dispensing system consists of a bottle of beverage, a dispensing hose with valve, and a CO$2$ system to keep the beverage pressurized. The CO$2$ system includes a small bottle of high$-$pressure liquefied CO$2,$ and a regulator to regulate the gas pressure delivery to the beverage. A typical beverage system contains a $7\mathrm{.}75-$gal beverage bottle, a $5-$lb bottle of liquefied CO$2,$ and a regulator set at $9$ psig. The regulator is connected directly to the CO$2$ bottle, and a $0\mathrm{.}5-$in $($internal diameter$)$ plastic hose connects the regulator to the beverage bottle. The liquefied CO$2$ saturation vapor pressure is $800$ psia. Assume a temperature of $80\backslash$deg F$.$ A pressure relief system must be designed to protect the beverage bottle from overpressure.

The relief device will be installed in the CO$2$ line where it enters the beverage container.

a$.$ Determine the most likely scenario contributing to overpressure of the beverage container.

b$.$ Must two$-$phase flow be considered?

c$.$ Determine the vent area required, assuming a set pressure of $20$ psig and an overpressure of $10\%.$ Also assume a spring$-$type relief.