Please refer to the guidance at the top of the Jupyter Notebook containing the other
piece of coursework.
Your submission for this part should take the form of a separate Jupyter Notebook.
As well as the code itself, you should include graphs and discussions of the results
where required.
Question 2
A company is designing a reactor to break down pollutants in a liquid stream. The reactor is to consist of two parallel plates with flow between then. In order to enhance mixing the inlet will consist of the outlet will be the bottom half of the right hand side of the reactor, while the inlet will cover the entire left hand side. All other boundaries can be considered to be no slip.
The fluid properties are those of water (𝜌𝜌 = 1000 𝑘𝑘𝑘𝑘/𝑚𝑚
and 𝜇𝜇 = 0.001 𝑃𝑃𝑃𝑃. 𝑠𝑠 – note be careful as
to which viscosity is being used in the code – 𝜈𝜈 = 𝜇𝜇⁄𝜌𝜌). You can use an initial pressure drop 0.5 Pa.
a) Code a simulator to solve this problem. You may modify the code that was developed in class. Show the flow and pressure profiles for the conditions given above. Remember to check if the flow has achieved steady state (and note that for some conditions the flow may remain unsteady). (15)
The flow of the contaminant within the water can be described using an advection-diffusion equation with a first order pollutant reaction rate:
𝜕𝜕𝜕𝜕 =−𝐯𝐯∙𝛻𝛻𝜕𝜕+𝐷𝐷∇2𝜕𝜕−𝑘𝑘𝜕𝜕
Where 𝜕𝜕 is the pollutant concentration, 𝐯𝐯 is the liquid velocity vector, 𝐷𝐷 is the diffusivity and 𝑘𝑘 is rate constant. The pollutant concentration at the inlet is 1 kg/m3, while at the outlet the pollutant flows at the same rate as the fluid, which implies that there is zero gradient in the concentration normal to this boundary. All other boundaries have no flux through them. As there is no flux of liquid through these boundaries, this implies that the concentration gradient in the direction normal to these boundaries is zero as well. You can use a diffusivity of 1e-6 m2/s as an initial value (note that
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this is quite high compared to the real diffusivities of chemicals in water, but results in Peclet numbers that are likely to produce more interesting results). For the first order reaction rate you can use a value of 1.0 s-1 as an initial value.
b) Write out a finite difference approximation for the advection diffusion equation given above. You can use an explicit scheme (spatial derivatives calculated at the current time step and time derivate between the current and next time step). You should using upwind approximations for the advection terms (15)
c) Implement this approximation together with appropriate boundary conditions within the
fluid flow simulator. In addition to the Courant number stability criterion that should already
∆𝜕𝜕≪∆𝑥𝑥 . 2𝐷𝐷
be applied for the fluid calculations, there is an additional criterion based on the diffusivity –
Investigate how the parameters in the models influence the results (hint: doing this in terms of dimensionless groups may help the analysis). (20)
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