Problem 1: Drug Diffusion Using the Heat/Diffusion Equation (This problem reviews: -Separation of variables and Fourier series solution to the heat equation-Application of von Neumann boundary conditions-Solution to the Euler formulae to determine the Fourier coefficient)Consider a fluid compartment of length Lseparated into two equal halves by a non-permeable wall, as in Figure 1. The concentration of a particular drug is initially C0in the left half of the compartment, and zero in the right half. At time t= 0 s the wall is removed and the drug is allowed to diffuse through the entire compartment. Assuming the net diffusion of the drug in the yand zdirections is zero, the concentration of the drug C(x, t) as a function of time tand one space dimension xcan be modelled using Fick’s second law: )1(22xCDtC∂∂=∂∂where Dis the mass diffusivity, analogous to the thermal diffusivity ‘c2’ in the heat equation (see lectures). Solve analytically for C(x, t), the concentration of the drug in space and time. Figure 1.Initial state for our drug diffusion problem: a drug of uniform concentration is inside one half of the compartment and is separated from the other half of the compartment by a wall at x L/2. 0 L/2 x
=
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L