A beverage dispensing system consists of a bottle of beverage, a dispensing

hose with a valve, and a $C{O}_2$ system to keep the beverage pressurized. The $C{O}_2$

system includes a small bottle of high$-$pressure liquefied $C{O}_2$ 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 $5lb.$ bottle of liquefied $C{O}_2,$ and a regulator

set at $9$ psig. The regulator is connected directly to the $C{O}_2$ bottle, and a $0\mathrm{.}5$ in

$($internal diameter$)$ plastic hose connects the regulator to the beverage bottle.

The liquefied $C{O}_2$ saturation vapor pressure is $800$ psia. Assume a temperature

of $80F.$ A pressure relief system must be designed to protect the beverage

bottle from overpressure. The relief device will be installed in the $C{O}_2$ line where

it enters the beverage container. Discuss the most likely scenario contributing to

the overpressure of the beverage container. Analyze the need to consider a two$-$

phase flow. Following this, determine the vent area required for a spring$-$type

relief valve. State all assumptions and list all relevant standards used in the

design.