one question
marylyn50
ass.pdf
Question 3 “Diffusion of Benzene” [75 marks]
The distribution and movement of a substance (solute) within another substance (solvent) is
governed by advection and diffusion. For an example, consider a small amount of food dye
(solute) added to a large tank of water (solvent). This dye will eventually spread evenly
through the entire tank of water via a combination of these two mechanisms.
Advection describes the bulk movement of the solute which will in turn result in dispersion
of the solute. Stirring the water would result in bulk movement of the water and rapid
spread of the dye through the entire tank.
Diffusion instead refers to the slow spread of the solute through the substance without bulk
movement of the solvent. This mechanism would dominate if the water in the tank was kept
completely still and the dye left to slowly diffuse.
The Problem A ship is docked in the Chicago Ship Canal, Figure 1. Suddenly the ship has started to leak
benzene in the canal from a side pipeline. The leak maintains a concentration of benzene at
0.020 mg/L at the point of the leak. Due to the diffusion and advection process the benzene
starts to spread in the canal in the direction of the water flow in the canal.
The equation that governs the diffusion of the benzene in the moving canal water can be
described as follows:
where:
c is the benzene concentration at a particular point in the canal water (mg/L)
u is the velocity of the water in the canal (m/s)
D is the diffusion coefficient (m 2 /s)
x is the distance measured from the side pipeline (m)
t is the time (sec)
The velocity of water in the canal is 0.5 m/s and the water diffusion coefficient is 3.0 m 2 /s.
Assume that at x= 100 m far from the side pipeline (the source of the leak), the concentration
c of benzene in the canal is zero.
(Figure 1)
Side Pipeline
u
x
The Task Your task is to develop a numerical model using the finite difference approach to
approximate the solution of the diffusion equation.
The specific tasks are as follows:
3.1 - Formulate a solution to the diffusion equation using an explicit method
i) Develop a Matlab script file implementing this explicit method
3.2 - Formulate a solution to the diffusion equation using an implicit method
i) Develop a Matlab script file implementing this implicit method
3.3 - Investigate the stability of both the explicit and implicit solution schemes
3.4 - Use both of the models above to:
i) Plot the concentration of the benzene after 5, 10 and 20 sec within the domain
x = 0 ─100 m.
ii) Compare the variation of the benzene concentration at x = 10, 20 and 30
metres form the source of the leak after 20 sec.
iii) Comment on the validity of the assumed boundary condition of c = 0 at x =
100 m.
3.5 - Discussion and review,
Some points to cover in addition to discussing your results:
i) Highlight the assumptions and limitations of your model.
Submission Requirements Give full details of your explicit and implicit approaches and formulations (explain these
equations in your report, not just in the code);
Provide all plots or tables as described above within the report;
Marking Scheme Overall, 70 marks are allocated to this task. The breakdown of marks is:
Details Marks
Explicit Scheme - Application of finite difference approach
- Development of code, including documentation and ease of use
15
Implicit Scheme - Application of finite difference approach
- Development of code, including documentation and ease of use
15
Investigation
i. How robust are the two methods, what are the conditions of stability ii. Comment on the validity of boundary condition of c = 0 at x = 100m
10
Results
i. Plot the concentration of the benzene after 5, 10 and 20 sec within the domain x = 0 ─100 m.
ii. Compare the variation of the benzene concentration at x = 10, 20 and 30 metres after 20 sec
20
Discussion
- Interpretation of results. 10
Presentation - structure
- table of contents
- use of references
(will lose marks you try to pass of others work as your own)
5
Total 75
