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Use the separation of variables method to solve the following problem. Consider a long, narrow tube connecting two large, well-mixed reservoirs containing a small concentration
Use the separation of variables method to solve the following problem. Consider a long, narrow tube connecting two large, well-mixed reservoirs containing a small concentration of N2 in another inert gas. The tube length is L = 100 cm. To establish an initial concentration profile in the tube, each reservoir is held at a fixed concentration: Reservoir 1 contains no N2 and reservoir 2 has 2 x 10-6 mol/cm of N2. (a) At t = 0, the concentrations of the reservoirs are rapidly instantaneously) switched. Solve for the concentration in the tube as a function of position and time. The diffusion coefficient of N2 is D = 0.2 cm/s. (b) Plot the concentration c(2,t) as a function of the position along the tube at four different times (initial, early, intermediate, and long times). Use software to plot enough terms of the series solution to have a reasonable representation of the solution. As always, be sure to clearly label your axes. (c) Repeat part (b), but plot the transient part of the solution, ct(2,t)
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