ChemE 109 – Numerical and Mathematical Methods in Chemical and Biological Engineering Fall 2019 PROJECT Steady-state and transient analysis of a diffusion-reaction process Assigned: Wednesday, November 20. Due: Monday, December 16. Process description and modeling The objective of this project is to study the steady-state and transient characteristics of the following reaction scheme: A + 2B k1−−→ D A k2−−→ P B k3−−→ U U k4−−→ F which takes place in the reaction zone shown below: The reactants A and B enter the reaction zone through the left membrane. The concentrations of the species A and B outside of the film which is located in the left side of the reaction zone are kept at the fixed values CAF and CBF . In the right side of the reaction zone, all the species deposit with different deposition rates. The boundaries in the reaction zone are at r = 0 and r = L. Under the assumptions that the molecular diffusion inside the reaction zone is one-dimensional and follows Fick’s law, and that the diffusion coefficients DA, DB, DU and DF , of the species A, B, U and F , respectively are constant and equal (i.e. DA = DB = DU = DF = D¯), the following equations can be derived, describing the change in CA, CB, CU and CF with position and time: ∂CA ∂t = D¯ ∂2CA ∂r2 − k1CAC2B − k2CA ∂CB ∂t = D¯ ∂2CB ∂r2 − 2k1CAC2B − k3CB ∂CU ∂t = D¯ ∂2CU ∂r2 + k3CB − k4CU ∂CF ∂t = D¯ ∂2CF ∂r2 + k4CU (1) where: CA: concentration of species A 1 Figure 1: Schematic of reaction zone. 2 CB: concentration of species B CU : concentration of species U CF : concentration of species F k1: rate constant for the first reaction k2: rate constant for the second reaction k3: rate constant for the third reaction k4: rate constant for the fourth reaction D¯: diffusion coefficient r: distance t: time Defining the following dimensionless variables and parameters: yA = CA CAF , yB = CB CAF , yU = CU CAF , yF = CF CAF , β = CBF CAF , τ = k1C 2 AF t, D = D¯ k1C2AFL 2 x = r L , γ = k2 k1C2AF , δ = k3 k1C2AF , ζ = k4 k1C2AF (2) the system of Eq.1 can be written as: ∂yA ∂τ = D ∂2yA ∂x2 − yAy2B − γyA ∂yB ∂τ = D ∂2yB ∂x2 − 2yAy2B − δyB ∂yU ∂τ = D ∂2yU ∂x2 + δyB − ζyU ∂yF ∂τ = D ∂2yF ∂x2 + ζyU (3) The boundary conditions are: at x = 0: ∂yA ∂x (0, t) = −(1− yA), ∂yB ∂x (0, t) = −(β − yB), yU(0, t) = 0, and yF (0, t) = 0, at x = 1: ∂yA ∂x (1, t) = −yA, ∂yB ∂x (1, t) = −ηy2B, ∂yU ∂x (1, t) = −θyU , and ∂yF ∂x (1, t) = 0, Questions 1. (50 pts.) In the first part of the project, the objective is to analyze the steady-state characteristics of the system. Using the O(∆x2) centered finite difference approximation method compute and plot the steady-state profiles yA vs. x, yB vs. x, yU vs. x, and yF vs. x for each of the following three cases: a. (8 pts.) δ = 0, ζ = 0, D = 0.1, β = 1.5,γ = 0.05, = 0.0, η = 0.0, θ = 0.0. b. (10 pts.) δ = 0.05, ζ = 0.0, D = 0.1, β = 1.5, γ = 0.02, = 0.1, η = 0.05, θ = 0.1. c. (12 pts.) δ = 0.05, ζ = 0.03, D = 0.1, β = 1.5, γ = 0.02, = 0.1, η = 0.05, θ = 0.1. Make sure you use enough nodal points to accurately calculate the solutions. Include your rationale in the discussion (5 pts.). Explain the criteria that you employed to judge the 3 accuracy of the computed results (5 pts.). Discuss and explain the nature of the profiles that you obtained for the three cases (10 pts.). 2. (40 pts.) In the second part of the project, the objective is to study the time-dependent behavior of the system. For the case (c) from part 1, compute and plot (separately) the evolution of the spatial profiles of yA, yB, yF and yU in time (25 pts.). Use as initial conditions yA(x, 0) = yB(x, 0) = yU(x, 0) = yF (x, 0) = 0. Include enough profiles (in the same plot) to show clearly the time evolution until a steady-state is reached. Assume that the steady-state is obtained when you reach 99% of the steady-state values found in the previous question. Explain the method that you used to integrate the system in time and criteria that you employed to judge the accuracy of the computed results (5 pts.). Discuss the relationship between the plots that you obtained and your results from the previous question (10 pts.). Requirements and Report Format You are expected to work independently on the project. You may use parts of programs given in class or programs you have written on your own, but you may not share pieces of code among fellow students. The project should be clearly written (deductions, up to 5 points, will be made for messiness, poor organization and poor writing), according to the following format: • Title page. Please include the title of the project and your name. • Table of contents. Indicate page numbers for all report sections and figures. • Introduction. Briefly describe the problem, the development of the dimensionless mathematical model, and the strategy and methods you used to obtain solutions. Specify the class programs that you used, as well as any major modifications made to the class programs. The introduction will be worth 10 points. • Results and Discussion. Address the questions posed in the previous section. Point values are indicated for each item. Make sure to include all required figures and tables, with a brief explanation of what they represent. Interpret your results adequately. • Appendices. Include listings of all programs you used to generate your results. Clearly indicate all sections of the programs which are significant modifications of the original class codes. Please also upload your code to courseweb. If you have any questions regarding the requirements of the project, contact Professor Panagiotis D. Christofides or TAs. Good Luck!!! 4
