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Diffusion-convection problems wherein the species diffusivity is concentration-dependent often provide a differential equation of the following general form: |dC |dz where C is the dimensionless concentration and = is the dimensionless distance. Assume vĽ/D, is a constant. Assume that C= 1 at :=0, and C=0 when z is very large. Solve the above differential equation and provide an exact analytical equation for C-A:).

Diffusion-convection problems wherein the species diffusivity is concentration-dependent often provide a differential equation of the following general form: |dC |dz where C is the dimensionless concentration and = is the dimensionless distance. Assume vĽ/D, is a constant. Assume that C= 1 at :=0, and C=0 when z is very large. Solve the above differential equation and provide an exact analytical equation for C-A:).

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Problem #1: Diffusion-convection problems wherein the species diffusivity is concentration-dependent often provide a differential equation of the following general form: where Cis the dimensionless concentration and z is the dimensionless distance. Assume vL/D, is a constant. Assume that C= 1 at z = 0, and C=0 when z is very large. Solve the above differential equation and provide an exact analytical equation for C=ft:). Problem #2: In a diffusion-reaction system, reactant 'A' diffuses through a porous catalyst pellet and reacts in a 1* order reaction to produce 'B'. Efficacy of the catalyst in converting 'A' to 'B' is a function of 'x' such that the mass balance equation describing the system at steady-state is: d?c Dari -kīC = 0 Cis the concentration of 'A' that varies in the x-direction, k is the 1t order reaction rate constant, and D is the diffusion coefficient of 'A'. L is the thickness of the catalyst layer, and can be treated as a constant. First, non-dimensionalize the equation (e.g., assume x = xL and y = CICAO) and then determine the approximate concentration distribution of 'A' in the catalyst layer using regular perturbation assuming that reaction rate is slow compared to diffusion, i.e., kL?ID (=e) is very small. Assume the dimensionless concentration of 'A' to have the following form: y = yo + gyı +y2. You will need to use the following boundary conditions: В.С. 1: В.С. 2: at x = 0; C= CAo or y = 1 at x =L;C= 0 or y = 0 Problem #3: Many one-dimensional transient diffusion-reaction problems in chemical systems involve solving partial differential equations (PDES). One such PDE has the form: a (ac 1 (ac ac ax \ax at Treat all quantities here as dimensionless. Using combination of variables, i.e., 7 = x/VT, prove that the partial analytical solution to the above PDE involves the following expression: C = A +B el+an"ldn Here A and B are constants of integration. Determine the value of the constant a. Do not apply any boundary conditions and / or determine the constants of integration.
Transcribed Image Text:Problem #1: Diffusion-convection problems wherein the species diffusivity is concentration-dependent often provide a differential equation of the following general form: where Cis the dimensionless concentration and z is the dimensionless distance. Assume vL/D, is a constant. Assume that C= 1 at z = 0, and C=0 when z is very large. Solve the above differential equation and provide an exact analytical equation for C=ft:). Problem #2: In a diffusion-reaction system, reactant 'A' diffuses through a porous catalyst pellet and reacts in a 1* order reaction to produce 'B'. Efficacy of the catalyst in converting 'A' to 'B' is a function of 'x' such that the mass balance equation describing the system at steady-state is: d?c Dari -kīC = 0 Cis the concentration of 'A' that varies in the x-direction, k is the 1t order reaction rate constant, and D is the diffusion coefficient of 'A'. L is the thickness of the catalyst layer, and can be treated as a constant. First, non-dimensionalize the equation (e.g., assume x = xL and y = CICAO) and then determine the approximate concentration distribution of 'A' in the catalyst layer using regular perturbation assuming that reaction rate is slow compared to diffusion, i.e., kL?ID (=e) is very small. Assume the dimensionless concentration of 'A' to have the following form: y = yo + gyı +y2. You will need to use the following boundary conditions: В.С. 1: В.С. 2: at x = 0; C= CAo or y = 1 at x =L;C= 0 or y = 0 Problem #3: Many one-dimensional transient diffusion-reaction problems in chemical systems involve solving partial differential equations (PDES). One such PDE has the form: a (ac 1 (ac ac ax \ax at Treat all quantities here as dimensionless. Using combination of variables, i.e., 7 = x/VT, prove that the partial analytical solution to the above PDE involves the following expression: C = A +B el+an"ldn Here A and B are constants of integration. Determine the value of the constant a. Do not apply any boundary conditions and / or determine the constants of integration.
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