# Team:Sydney Australia/MacroscopicDiffusion

In industrial applications, it is often useful to know the sensitivity limits of a particular device. While it is easy to measure the absolute sensitivity of our biosensor by trialling varying concentrations of ethylene, it can be difficult to correlate detected ethylene concentration to ethylene concentration of the nearby fruit. As seen further below, ethylene distribution is not always uniform which adds a layer of complexity to the biosensor’s output.

This problem can be rectified in two ways:
a) producing a sticker than is placed directly on the fruit, so that detected ethylene equals the fruit ethylene concentration
Or b) predicting the diffusion behaviour of ethylene as it moves from the source to the biosensor.

In this diffusion analysis, we model the movement of ethylene in the common scenario of a fruit filled shipping container containing the biosensor.
From the results, we gather a picture of:
•the biosensor’s sensitivity requirements when used in this context, and
•the behaviour of ethylene gas before encountering our biosensor, and what this may mean for our biosensor’s design considerations on an industrial scale.

## Modelling Methodology

The governing principle behind this model is Fick’s Diffusion Law, which describes the average concentration of a diffusing substance over time. Since our model is macroscopic and in room temperature conditions, molecular fluctuations and molecular hydrodynamics terms are not considered.

In 2D, this becomes

In order to obtain numerical solutions for a variety of scenarios involving ethylene diffusion in a storage container, the method of finite differences is used. In this method, the differential equations governing the system are approximated by transforming the infinitesimally small derivative into discrete “timesteps” and “space steps” This divides our system into an array of small squares, called “nodes”, each with length . Our model calculates the local ethylene concentration in each individual node for every time step. With this arrangement, we can now use Fick’s Law to calculate the ethylene concentration at an individual node, and that of it’s immediate neighbours (4 adjacent squares - diagonals are not included)

This is the same equation as above, but is rewritten into a form which can be solved numerically by MATLAB, given the correct boundary and initial conditions. We use MATLAB to plot this as a surface, where height represents local ethylene concentration.

#### Governing Assumptions

But before we can consider any specifics about what we are modelling, some assumptions must be made about the system.
1. There is no movement of the medium (may be slightly off in turbulent journeys e.g. airplanes). In a moving medium, an advection term must be added to the diffusion equation.
2. Air at 25C, atmospheric pressure (may be slightly off in low pressure environments on airplanes). This correlates to an ethylene diffusion coefficient of 0.1543 (Elliott & Watts, 1971)
3. While ethylene production by fruit varies slightly as a function of temperature and time, we model this as a constant linear source of ethylene.
4. The walls of the container do not experience any changes in ethylene concentration. That is, dc/dt=0 around the edges.

#### Initial and Boundary Conditions

The initial and boundary conditions of a system of differential equations are the constraints that the system must follow. A simple example to illustrate this would be gas moving in a pipe. Intuitively, we know that the gas must be constrained in the pipe. But mathematically, we represent this constraint as a boundary condition:
A limitation of using differential equations is that the state of the system at the very beginning cannot be predicted. In our model, the initial and boundary conditions define the scenario we wish to model.

We wanted to test a variety of scenarios involving fruit storage. The scenarios explored are
1. A point source of constant ethylene production. For example, a single box of fruit in the middle of the shipping container, or two boxes placed within a certain distance of each other. This means dc/dt = constant for a given area.
2. A large pallet of fruit at the far wall. This means dc/dt=constant for the given area,
3. A single 1-time dose of ethylene. For example if a storage facility were to inject a known volume of ethylene into a shipping container This means c=constant at time zero, in a given area.
4. A U-shaped distribution of fruit around the walls of the container. For example, a walk-in storage facility with shelves of fruit lining the 3 walls. This means dc/dt=constant for the given area.

For all of these scenarios, the only differences are the initial and boundary conditions. The governing laws, assumptions and equations are identical.

## Results

For all scenarios involving constant sources of ethylene production, the source region has been set to 100 ppm of ethylene. This corresponds to an average ethylene concentration required for avocado ripening (Avocado Industry Council, 2008).

The x and y limits of the simulation region represent the edges of the shipping container. These have been set to a standard 5.99 by 2.35 metres.

#### Single Source

The most simplest simulation is a single 30x30cm box of fruit in the middle of the shipping container, with the rest of the container empty. This box behaves as a constant source of ethylene, which diffuses radially outwards from the source. From the simulation, it is instantly clear that ethylene does not diffuse far at the concentrations associated with fruit ripening. However, a single box of fruit is a very small source of ethylene, so perhaps these results were to be expected.

#### Double Source

The simulation has also been modelled with two 30x30cm boxes of fruit, placed evenly in the middle of the shipping container. This simulation explores how the gases from two separate sources behave once they meet in the middle.

#### Wall pallet

This simulation reflects a more realistic storage scenario, where a large pallet of fruit is placed in the far wall of the shipping container. This is modelled as a large region of constant ethylene production which diffuses unidirectionally towards the entrance.

#### One time dose

The current method of fruit ripening in industry is injecting a standardized volume of ethylene into a sealed container and allowing it to diffuse evenly. In this simulation, we model this common method with a one time dose of 1000ppm ethylene, over a shorter time period of 1 second. While this simulation yields a slightly more uniform distribution of ethylene, better results may be seen with 5x 200ppm doses distributed around the container.

#### U-shaped storage configuration

This storage configuration involves shelves of fruit lining 3 walls of the container, leaving a corridor for access in the middle. This is perhaps the most realistic and practical layout. The video shows ethylene distribution approaching an almost uniform steady state, which is crucial for biosensor calibration. This scenario makes calibration significantly easier than the previous situations where ethylene concentration may vary wildly. Biosensor placement is not limited to certain locations. Unlike other simulations, it may be placed directly at the entrance and still give an output that can be calibrated to represent local fruit ethylene concentration.

## Conclusions

##### How does this affect our final product?

From these simulations, we can see that the application of our biosensor may not be so simple as keeping it locked in a storage facility. The storage layout, biosensor placement and sensitivity must be carefully calibrated to detect the ethylene concentration at the fruit source.
Specifically, the storage layout should be a uniform distribution of fruit source. A single box, or a stack of fruit at the far wall leads to large variations in ethylene concentration throughout the container. As such our biosensor cannot accurately detect the ethylene at the source unless it is placed directly beside it. The ideal storage configuration would be a U-shape around the walls of the container, as this allows maximum storage capacity while maintaining easy biosensor calibration.
These considerations would be taken into account when developing the output device, such as the phone app. When using the app for a U-shaped configuration, the local ethylene concentration readout can be multiplied by a constant which takes into account the decrease in ethylene concentration as a result of diffusion.

## References

Avocados Industry Council. "AIC Ethylene Ripening Protocol". (2008). 1-2. Web.
Elliott, Robert W. and Harry Watts. "Diffusion Of Some Hydrocarbons In Air: A Regularity In The Diffusion Coefficients Of A Homologous Series". Canadian Journal of Chemistry 50.1 (1972): 31-34. Web.

School of Life and Environmental Sciences
The University of Sydney