Question

The next exercises consider different types of boundary conditions. 4. Suppose the concentration of pollutant is not allowed to leave the dam The diffusion equation (3.1.18) for the concentration in the dam still holds and the concentration at the river entrance is still ρ(0)-c, but the boundary condition at the dam wall must be changed to insist that there is no flux of pollutant leaving the dam. Construct the solution with this new boundary condition and plot the profile of the concentration. Where does the pollutant go? Answer: The profile is shown below for the choice VB L- 2: 0.8 0.6 a 0.4 0.2 0 CO 0 0.2 0.4 0.6 0.8 x/L At the wall, ρ(Lyc-0.266.

Answer 1

The diffusion equation is given as, d^2 p(x)/dr^2 Bp(x) = 0, p(0) = c let the dam wall be at x = L therefore No flur at dam means p'(L) = 0 therefore let p(x) = e^lambda x therefore lambda^2 e^lambda x - B e^lambda x = 0 0 + lambda^2 = B = > lambda = plusminus squareroot B 0 + p(x) = c_1 e^squareroot B x + c_2 e^squareroot b x Now, p(0) = c_1 + c_2 = c p'(x) = c_1 squareroot B e^squareroot B x - c_2 squareroot B e^-squareroot B x p'(L) = c_1 squareroot B e^squareroot B L -c_2 e^-squareroot B L = 0 therefore c_1 e^squareroot B L = c_2 e^squareroot B L c_1 = c_2 e^-2 squareroot B L therefore c_2 (1 + e^-2 squareroot B L) = c therefore c_2 = c/1 + e^-2 squareroot L and c_1 = c e^-2 squareroot B L/1 + e^-2 squareroot B L \r\n therefore The solution for