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IEN/CENG 2310 MODELING FOR CHEMICAL AND BIOLOGICAL 

1. A barge carrying chemical wastes, sitting in the middle of a narrow canal, leaked a toxic chemical to the water due to an accident. We would like to know the concentration profile of the chemical at different distances from the barge, over time. As a first approximation, we will assume that the canal is shallow and narrow enough that the concentration will only vary in the x-direction. We set our space domain of interest to be – ≤ ≤ , assuming that at = ± there will no longer be any flux. We will consider two scenarios: (i) the fixed amount of chemical is leaked in one instant, resulting in an initial concentration of within the region – ≤ ≤ , and no more chemical is leaked afterwards; and (ii) the chemical is leaked slowly and continuously, such that the concentration in – ≤ ≤ remains at indefinitely. (a) Assuming the chemical travels by diffusion (to both sides, along the x-axis) only according to Fick’s Law, write down a PDE to describe the dynamics of this process. Include also the initial conditions and boundary conditions as appropriate for scenario (i). (b) Use the method of lines to solve the PDE in Part (a) in MATLAB, allowing the user to specify , , , the time point to stop the simulation , and the diffusion coefficient . (c) Suppose there is a bulk flow of the water in the canal at velocity , which can be positive (meaning that the water is flowing in the +x direction) or negative. Thus, the chemical is transported by both diffusion and convection. Modify the PDE to account for this, and implement it in the MATLAB program, adding a new function argument for . (d) Add a new option to the program to model scenario (ii). Include a new function argument “indefinite” that takes on true/false value in the function definition, so that the user can opt for this indefinite leak scenario. + − 0 + − DELIVERABLES: Parts (a) and (b) will be done in class as a demo. Modify the provided program toxicLeakCanal.m to add the new options in Parts (c) and (d). Rename it “toxicLeakCanal_

_.m”. Test your program