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
