Question $3$

Figure $3\mathrm{.}1$ shows a cross sectional view of an apple. The apple has an empty core, inner and outer cortex and skin. Braeburn apples have low porosities $(10\%)$ which makes it harder for gas to move through the tissue structure. Experiments have found that the diffusion coefficient in the inner cortex is low as tissue closest to the center of the apple is more tightly packed. The skin which is exposed to the atmosphere also has high density tissue $($and low diffusivity$)$ which allows gases to accumulate in the tissue structure.

This can potentially become a problem for ethylene gas $($a ripening hormone$)$ which is produced at a constant rate throughout the tissue. Fruit soften faster or can get tissue browning in areas where the concentrations of ethylene are higher. If the diffusion properties of the tissue structure are known, then a position dependent model can used to determine the potential internal ethylene concentration gradients in a fruit.

Figure $3\mathrm{.}1$ Cross section through an apple

a$)$ Starting with a simplified $1-$dimensional geometry, formulate a model of the system that would allow the ethylene concentration to be predicted as a function of radius in the apple. In your formulation discuss the impact of your assumptions on the likely accuracy of predictions. Assume ethylene production follows zero order kinetics and that each tissue section has a different diffusion coefficient. To simplify the geometry, it can be assumed that the core is an empty void space which allows ethylene to accumulate inside. The ethylene concentration in the surrounding air can be considered as zero.

$[10$ marks$]$

b$)$ Define a finite difference grid and derive the finite difference approximations needed to solve the model. $($NB do no more than $5$ cases$).$

$[10$ marks$]$

c$)$ Explain how you would go about calculating the average ethylene concentration in the apple at any time.

$[5$ marks$]$

$[$Total $25$ marks$]$

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