In the kiwifruit industry, fruit can be packed into trays $($Figure $1\mathrm{.}1).$ Up to $20$ fruit sit on pocket packs $($moulded pastic trays$)$ that support and control the location of the fruit. These are then wrapped in a polyliner $($a HDPE polymer film$)$ and placed into cardboard boxes and stored in coolstores.

Figure $1\mathrm{.}1$ Trays of kiwi fruit. Note: during storage the polyliner is closed to protect the fruit from losing moisture.

Due to the international pressure to reduce the use of plastic packaging, research is undertaken to replace the moulded plastic trays with moulded fibre $($paper based$).$ There is concern about the uptake of moisture from the fruit and being absorbed into the new paper based trays. A mathematical model is required to investigate the proposed change.

The kiwifruit can be assumed to be at a constant water activity $(0\mathrm{.}98)$ with the moisture permeance of the skin limiting moisture release water.day $($:${?}^{-1}*m^{-2}*kP{a}^{-1}\}.$ The $25m$ polyliner has a permeance of $3\mathrm{.}65{10}^{-5}kg$ water.day ${?}^{-1}*m^{-2}*kP{a}^{-1}.$

The moulded fibre tray is porous, and internal mass transfer resistance can be considered negligible. It will absorb moisture at a rate that depends on a mass transfer coefficient and the vapour pressure difference. The moisture soption isotherm can be used to relate the moisture content $(M_C)$ to the equilibrium relative humidity $($or water activity $aw)$ of the paper material. Equation $1\mathrm{.}1$ is the Oswin equation that can be used to describe the isotherm where $C_o,n$ are constants $.$

$M_C=C_o(\frac{aw}{1-aw}{)}^n$

Eq $1\mathrm{.}1$

Relative humidity and water activity are given by Eq $1\mathrm{.}2$

$RH(oraw)=\frac{P_v}{P_s}$

Eq $1\mathrm{.}2$

Where $P_v=$ vapour pressure and $P_s$ is the saturated vapour pressue at a given temperature. Partial pressure and water vapour concentration $kg.m^{-3}$ can be related using the Ideal Gas Law.

Question $1$ continues next page

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a$)$ Formulate a mathematical model to predict the change in moisture content of the moulded fibre tray and the mass of kiwifruit as a function of time during storage at constant conditions. Discuss the implications of each assumptions you make and define all nomenclature used.

$[20$ Marks$]$

b$)$ The model was solved in Matlab. Describe how you might go about checking your code by comparing the solution for simplified cases with analytical solutions.

$[5$ Marks$]$