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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce60b6bf-6db7-4edb-ab85-8ba7df4c46aa%2Fee72d53b-f651-407d-9c89-5797ceea3845%2Fnuuc73u_processed.png&w=3840&q=75)
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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