1. [50 points] The advection diffusion equation is used to compute the distribution of con- centration...

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1. [50 points] The advection diffusion equation is used to compute the distribution of con- centration along the length of a rectangular chemical reactor: ac = D- t 2 U. kc, (1) where c = concentration (mg/m), t = time (min), D = a diffusion coefficient (m/min), x = distance along the tank's longitudinal axis (m) where x = 0 at the tank's inlet, U = velocity in the x direction (m/min), and k = a reaction rate (min) whereby the chemical decays to another form. Develop an explicit scheme to solve this equation numerically, i.e. central difference in space and forward difference in time. Test it for k = 0.15, D = 100, and U = 1 for a tank of length 10m. Use a Ax = 0.5 m, and a step size At = 0.001 min. Assume that the inflow concentration is 100, at the outlet the concentration is not changing with z position, and that the initial concentration in the tank is zero. Perform the simulation from t = 0 to 100 min and plot the final resulting concentrations versus . 1) Write down the original equation and the initial/boundary conditions. 2) Write down the general form of the finite difference formulas you are going to use to convert the given analytical PDE to its finite-difference form. 3) Convert the given analytical PDE to its finite difference form (every single step must be shown). 1. [50 points] The advection diffusion equation is used to compute the distribution of con- centration along the length of a rectangular chemical reactor: ac = D- t 2 U. kc, (1) where c = concentration (mg/m), t = time (min), D = a diffusion coefficient (m/min), x = distance along the tank's longitudinal axis (m) where x = 0 at the tank's inlet, U = velocity in the x direction (m/min), and k = a reaction rate (min) whereby the chemical decays to another form. Develop an explicit scheme to solve this equation numerically, i.e. central difference in space and forward difference in time. Test it for k = 0.15, D = 100, and U = 1 for a tank of length 10m. Use a Ax = 0.5 m, and a step size At = 0.001 min. Assume that the inflow concentration is 100, at the outlet the concentration is not changing with z position, and that the initial concentration in the tank is zero. Perform the simulation from t = 0 to 100 min and plot the final resulting concentrations versus . 1) Write down the original equation and the initial/boundary conditions. 2) Write down the general form of the finite difference formulas you are going to use to convert the given analytical PDE to its finite-difference form. 3) Convert the given analytical PDE to its finite difference form (every single step must be shown).