Assignment - Solve using Matlab

A project for a fuel cell company requires the analysis of a two-dimensional mixing chamber depicted in Figure 1. The chamber is a channel of height H = 1cm and length L = 5cm and depth 10cm (needed to calculate flow rates). At the left boundary, two inlets of height 0.5cm each inject fluid into the chamber with fully developed parabolic laminar channel profiles with maximum velocity of 1 m/s. The chamber features 2 injection slits in the bottom wall and 2 injections slits in the top wall with location and equal widths shown in the above figure. Through these slits fluid is injected with a constant inflow velocity of 0.5m/s. The right boundary of the chamber is an outlet. All other boundaries are no-slip walls. The chamber is designed to mix two different 200 cSt silicone oils: oil A that

enters the chamber through inlet 1 and slits 2 and 4, and oil B that enters the chamber through inlet 2 and slits 1 and 3. The oils have a diffusion coefficient of α = 1 × 10-5m2 /s. At the initial time, the chamber is filled with oil B at rest. The flow in the chamber is incompressible and two-dimensional. The mixing of the two oils shall be modeled by a convection diffusion equation for the mass fraction Y of oil A. The mass fraction of oil B is 1 - Y. The performance of the mixing chamber shall be measured by the quantity R(t) defined as

R(t) =1/HL ₀∫^H₀∫^L Y(x, y, t)(1 - Y(x, y, t))dxdy

Your task is to write a report that contains the assignment tasks requested below, documents the PDEs and all methods and meshes used to analyze the chamber (simply writing the PDEs solved and methods used is sufficient), and contains a printout of all code used to solve this assignment in an appendix. Note that all results must be backed up by solution verification, i.e., quantitative data must have error bars.

Assignment:

1. Predict the steady state value of R including error bars for the shown mixing chamber.

2. Document the flow evolution and steady state solution of the flow in the mixing chamber using appropriate snapshots of the velocities, mass fractions, and Y (1 - Y ). Plot the temporal evolution of R(t) to steady state.

3. You are tasked to improve the performance R at steady state of the chamber by at least 10% guaranteed. No changes can be made to the left boundary inflow conditions, however the number of slits, slit positions, choice of oil A or B connected to the slits, and slit inlet velocities can be modified under the following constraints:

• Entering through each slit at any moment in time must be either pure oil A, or pure oil B. No mixtures are allowed to enter through the slits.
• The inlet velocity through each slit must be constant in space.
• Neither the viscosities nor the diffusivities of the two oils can be changed.
• The two pumps for providing oil A respective oil B to the slits are limited to a total flow rate of 0.25 liter/s of oil A and 0.25 liter/s of oil B. The pumps furthermore are limited to providing a maximum flow velocity of 1 m/s through each slit.

Document your modifications and repeat tasks 1 &2 for the modified channel.

4. You are tasked to improve the performance R at steady state of the chamber by at least 75% guaranteed under the same constraints and conditions listed in Task 3. Document your modifications and repeat tasks 1 &2 for the modified channel.

5. Repeat tasks 1 - 4 for the configuration shown in Fig. 1 for 5 cSt silicone oils A and B.

Required Content:

• PDEs solved
• Name of methods used to solve the final project
• Mesh resolutions, CFL numbers, Poisson equation convergence criteria used;
• Value of R at steady state with full solution verification/error bars (results for all meshes analyzed, full GCI analysis incl. observed order, asymptotic regime check, GCI values, Richardson extrapolated values and error bars)
• Clearly annotated figures of the temporal evolution and steady state of the flow in the chamber (velocities, mass fractions, Y(1-Y)) using at least 3 transient times and the final steady state, clearly annotated plot of R(t).
• clearly documented changes made to the channel/slits for tasks 3-5.
• all code used to solve this problem in an appendix;
• all the above for each 4) and 5) Tasks.

SOLVE USING MATLAB

And use only below listed functions which are allowed

abs(), sum(), max(), min(), mod(), linspace(), all trigonometric

functions, sqrt(), log(), log10(), exp(), pi, all plot/graph

functions, tic, toc, flipud and transpose for plotting in

2D, ones(), zeros(), size(), length(), reshape(), sign(), global, persistent,

getframe, open(), close(), writeVideoFile(), single, converToSingle, mean()