Time Dependent Diffusion
A film of thickness L is filled with a mixture of chemicals A and B. In this mixture, a first order chemical reaction with the reaction constant k1 can occur. During this reaction, chemicals A are consumed. The chemicals are continuously delivered from one side of the film x=0, so that the concentration at this surface is kept constant, n(0)=N. At another side, x=L, the film is covered with a catalyst that reacts with A at the surface with the reaction rate constant k1’’. At the very first moment no chemicals A were present in the film.
a) Formulate the math model for this case for the time-dependent diffusion with these chemical reactions.
b) Reformulate the problem in dimensionless form. List all dimensionless parameters that the concentration profile will depend on. (10 points) Solve the formulated problem for a steady state case and find the distribution of chemicals A in film
c) Analyze the asymptotic cases when k1 * L2/D << 1, k1L^2/D >> 1: k1”L/D <<1, k1”L?D >> 1. What happens with the concentration profiles in these cases?
d) A tube of length L is filled with a mixture of chemicals A and B. In this mixture, a second order chemical reaction with the reaction constant k2 can occur. During this reaction, chemicals A are consumed. Assuming that the chemicals are continuously delivered to the tube end x=0 so that the concentration at this end is kept constant, n(0)=N, and another end is sealed, find the distribution of chemicals in the tube. Plot the concentration profile in dimensionless form n(x/L)/N for three different ratios k2NL2/D =1/2, 1, 3/2
