QUESTIONI/60pts] A fluid of viscosity μ flows in the horizontal cylinder (radius R) shown in the figure under a constant pressure gradient dP/dx. Flow aR Open flow The inner core of the cylinder is filled with a porous material. The flow in this porous region is slow and assumed to be a plug-type flow such that the velocity is constant and everywhere the same inside the porous region. Denote this velocity by Uo. The flow in the open (non-porous) region is steady, Newtonian, incompressible and axisymmetric. It will be assumed that only the axial (x) component of the velocity is non-zero. R Porous media flow N.B. All your answers must be expressed in terms of µ, Uo, a, R and dp/dx. (a) Use the continuity and Navier-Stokes equations to determine the expression of the velocity in the open region. (b) What is the expression of the average velocity in the open region? (c) Is the assumption of a linear velocity profile in the open region acceptable when this region becomes very small. Justify your answer by simplifying the expression of the velocity in the limit of a small open region. (d) Determine a relationship between the total flow rate Q (in both the porous and open areas) and the pressure drop (-dp/dx).
QUESTIONI/60pts] A fluid of viscosity μ flows in the horizontal cylinder (radius R) shown in the figure under a constant pressure gradient dP/dx. Flow aR Open flow The inner core of the cylinder is filled with a porous material. The flow in this porous region is slow and assumed to be a plug-type flow such that the velocity is constant and everywhere the same inside the porous region. Denote this velocity by Uo. The flow in the open (non-porous) region is steady, Newtonian, incompressible and axisymmetric. It will be assumed that only the axial (x) component of the velocity is non-zero. R Porous media flow N.B. All your answers must be expressed in terms of µ, Uo, a, R and dp/dx. (a) Use the continuity and Navier-Stokes equations to determine the expression of the velocity in the open region. (b) What is the expression of the average velocity in the open region? (c) Is the assumption of a linear velocity profile in the open region acceptable when this region becomes very small. Justify your answer by simplifying the expression of the velocity in the limit of a small open region. (d) Determine a relationship between the total flow rate Q (in both the porous and open areas) and the pressure drop (-dp/dx).
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
Related questions
Question
![QUESTIONI/60pts]
A fluid of viscosity μ flows in the horizontal
cylinder (radius R) shown in the figure under
a constant pressure gradient dP/dx.
Flow
aR
Open
flow
The inner core of the cylinder is filled with a porous
material. The flow in this porous region is slow and
assumed to be a plug-type flow such that the velocity
is constant and everywhere the same inside the
porous region. Denote this velocity by Uo.
The flow in the open (non-porous) region is steady, Newtonian, incompressible and axisymmetric.
It will be assumed that only the axial (x) component of the velocity is non-zero.
R
Porous
media flow
N.B. All your answers must be expressed in terms of µ, Uo, a, R and dp/dx.
(a) Use the continuity and Navier-Stokes equations to determine the expression of the velocity in
the open region.
(b) What is the expression of the average velocity in the open region?
(c) Is the assumption of a linear velocity profile in the open region acceptable when this region
becomes very small. Justify your answer by simplifying the expression of the velocity in the limit
of a small open region.
(d) Determine a relationship between the total flow rate Q (in both the porous and open areas) and
the pressure drop (-dp/dx).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0df46eb4-a9a8-454c-9c68-5cfef71d3bc8%2F88984fcc-2157-47df-928d-043615785c68%2Fru5ia8_processed.png&w=3840&q=75)
Transcribed Image Text:QUESTIONI/60pts]
A fluid of viscosity μ flows in the horizontal
cylinder (radius R) shown in the figure under
a constant pressure gradient dP/dx.
Flow
aR
Open
flow
The inner core of the cylinder is filled with a porous
material. The flow in this porous region is slow and
assumed to be a plug-type flow such that the velocity
is constant and everywhere the same inside the
porous region. Denote this velocity by Uo.
The flow in the open (non-porous) region is steady, Newtonian, incompressible and axisymmetric.
It will be assumed that only the axial (x) component of the velocity is non-zero.
R
Porous
media flow
N.B. All your answers must be expressed in terms of µ, Uo, a, R and dp/dx.
(a) Use the continuity and Navier-Stokes equations to determine the expression of the velocity in
the open region.
(b) What is the expression of the average velocity in the open region?
(c) Is the assumption of a linear velocity profile in the open region acceptable when this region
becomes very small. Justify your answer by simplifying the expression of the velocity in the limit
of a small open region.
(d) Determine a relationship between the total flow rate Q (in both the porous and open areas) and
the pressure drop (-dp/dx).
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