$1$: Shelf Life of PET Bottles for Carbonated Beverage Packaging Applications $[30$ pt$]$ A PET bottle used for packaging carbonated beverages loses CO$2$ primarily by permeation through the walls, sorption in the walls, and leakage through an imperfect cap. Two$-$liter PET bottles have received rapid acceptance from bottlers and consumers for carbonated soft drinks. These bottles can have a shelf life of several months $($Shelf life is generally accepted as the time required for $15\%$ loss of CO$2$ from the bottle$).$ The surface area$-$to$-$volume ratio for a $1$ liter and $1/2$ liter bottle is $26\%$ and $59\%$ higher than a $2$ liter bottle, respectively. Hence, to achieve desired shelf lives for these smaller volume bottles, an exterior barrier coating of a high barrier such as polyvinylidene chloride $($PVDC$)$ is applied to the bottle. The shelf life of a $1/2$ liter PET coke bottle is approximately $10$ weeks.

$($a$)$ Develop an algebraic model to predict the shelf life of a PET bottle with a uniform layer of a high barrier polymer on the outside of the bottle. $[10$ pt$]($b$)$ Calculate the thickness of a Saran$($PVDC copolymer$)$ barrier coating required to increase the shelf life to $25$ weeks. For the purposes of this calculation, please assume that the package operates at pseudo$-$steady state. That is$,$ the time to establish a steady state concentration profile across the bottle wall is very rapid compared to the shelf life. Additionally, to simplify the calculation, please assume that the mass transfer area of the bottle can be modeled as a flat sheet of thickness $16\mathrm{.}5$ mil $(1$ mil$=25\mathrm{.}4$ micrometers$)$ and surface area of $55$ in$2.[20$ pt$]$

The average CO$2$ partial pressure difference across the bottle wall is $4\mathrm{.}7$ atm and, for the purposes of this calculation, assume that this average partial pressure difference changes very little over the course of the package shelf life. The initial amount of CO$2$ inside the bottle is $2100$ cm$3($STP$).$ The shelf life is reached when the amount of CO$2$ in the bottle decreases by $15\%(315$ cm$3($STP$))$ to $1785$ cm$3($STP$).$ The permeability of CO$2$ in PET under typical operating conditions is $14\mathrm{.}8$ cm$3$ mil$/(100$ in$2$ day atm$),$ and the permeability of CO$2$ in Saran$$ is $0\mathrm{.}11$ cm$3$ mil$/(100$ in$2$ day atm$).$

In addition to permeation across the wall of the bottle, there is a steady loss of CO$2$ through the bottle cap, and the amount of this loss is $0\mathrm{.}15$ cm$3($STP$)/$day$.$ Moreover, immediately after pressurizing the bottle with CO$2,$ there is a very rapid loss $($i$.$e$.,$ an essentially instantaneous loss$)$ of CO$2$ due to sorption of CO$2$ in to the bottle wall, and the amount of this loss is $145$ cm$3($STP$).$ Finally, there is an effective instantaneous CO$2$ loss, immediately after pressurization, of $47$ cm$3($STP$)$ due to bottle expansion effects