A falling-cylinder viscometer consists of a long vertical cylindrical tube (of radius R), capped at both ends, and a solid cylindrical slug (of radius KR). The system is sketched in Fig. 1. The slug is equipped with fins so that its axis coincides with that of the tube. One can observe the rate of descent of the slug in the cylindrical container when the latter is filled with fluid. The objective of this exercise is to find an equation that gives the viscosity of the fluid (assumed to be incompressible and Newtonian) in terms of the terminal velocity Vof the slug and of the various geometric quantities shown in Fig. 1. --R KR Cylindrical slug descends with speed V through the liquid Liquid fills the cylindrical cavity Figure 1: Sketch of the falling-cylinder viscometer. al - () d(rT,;) AP with P(z) = p(z) + pgz (1.1) dr L Here, AP is the (positive) dynamic pressure change over a generic length L, g is the magnitude of the gravitational field and p is the fluid density. c) Using the Newtonian constitutive equation that relates the viscous stress tensor to the velocity gradient, integrate Eq. 1.1 and show that: In(1/£)] In(1/k)] In(1/£) +1 APR? (1 — €2?) — (1 — к?) (1.2) %3D V 4μLV In(1/k) where u denotes the fluid viscosity and = r/R. State the boundary conditions used, briefly explaining the physical grounds on which they are based.
A falling-cylinder viscometer consists of a long vertical cylindrical tube (of radius R), capped at both ends, and a solid cylindrical slug (of radius KR). The system is sketched in Fig. 1. The slug is equipped with fins so that its axis coincides with that of the tube. One can observe the rate of descent of the slug in the cylindrical container when the latter is filled with fluid. The objective of this exercise is to find an equation that gives the viscosity of the fluid (assumed to be incompressible and Newtonian) in terms of the terminal velocity Vof the slug and of the various geometric quantities shown in Fig. 1. --R KR Cylindrical slug descends with speed V through the liquid Liquid fills the cylindrical cavity Figure 1: Sketch of the falling-cylinder viscometer. al - () d(rT,;) AP with P(z) = p(z) + pgz (1.1) dr L Here, AP is the (positive) dynamic pressure change over a generic length L, g is the magnitude of the gravitational field and p is the fluid density. c) Using the Newtonian constitutive equation that relates the viscous stress tensor to the velocity gradient, integrate Eq. 1.1 and show that: In(1/£)] In(1/k)] In(1/£) +1 APR? (1 — €2?) — (1 — к?) (1.2) %3D V 4μLV In(1/k) where u denotes the fluid viscosity and = r/R. State the boundary conditions used, briefly explaining the physical grounds on which they are based.
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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![A falling-cylinder viscometer consists of a long vertical cylindrical tube (of radius R), capped at both ends,
and a solid cylindrical slug (of radius KR). The system is sketched in Fig. 1. The slug is equipped with fins so
that its axis coincides with that of the tube. One can observe the rate of descent of the slug in the cylindrical
container when the latter is filled with fluid. The objective of this exercise is to find an equation that gives the
viscosity of the fluid (assumed to be incompressible and Newtonian) in terms of the terminal velocity Vof the
slug and of the various geometric quantities shown in Fig. 1.
-KR
Cylindrical slug
descends with
speed V through
the liquid
Liquid fills the
cylindrical cavity
Figure 1: Sketch of the falling-cylinder viscometer.
d(rT,;)
dir) - ()
AP
r
L
with P(2) = p(2) + pgz
(1.1)
dr
Here, AP is the (positive) dynamic pressure change over a generic length L, g is the magnitude of the
gravitational field and p is the fluid density.
c) Using the Newtonian constitutive equation that relates the viscous stress tensor to the velocity gradient,
integrate Eq. 1.1 and show that:
APR?
|(1- ) - (1— к?)
In(1/£)]
In(1/k).
In(1/5)
+1
Vz
(1.2)
V
4μLV
In(1/k)
where u denotes the fluid viscosity and = r/R. State the boundary conditions used, briefly explaining
the physical grounds on which they are based.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F478d30e8-b026-46e6-a171-e144afc9fded%2F84efcddd-32bd-405f-8533-bc8ac7cb2ab2%2Fpdtzsy5_processed.png&w=3840&q=75)
Transcribed Image Text:A falling-cylinder viscometer consists of a long vertical cylindrical tube (of radius R), capped at both ends,
and a solid cylindrical slug (of radius KR). The system is sketched in Fig. 1. The slug is equipped with fins so
that its axis coincides with that of the tube. One can observe the rate of descent of the slug in the cylindrical
container when the latter is filled with fluid. The objective of this exercise is to find an equation that gives the
viscosity of the fluid (assumed to be incompressible and Newtonian) in terms of the terminal velocity Vof the
slug and of the various geometric quantities shown in Fig. 1.
-KR
Cylindrical slug
descends with
speed V through
the liquid
Liquid fills the
cylindrical cavity
Figure 1: Sketch of the falling-cylinder viscometer.
d(rT,;)
dir) - ()
AP
r
L
with P(2) = p(2) + pgz
(1.1)
dr
Here, AP is the (positive) dynamic pressure change over a generic length L, g is the magnitude of the
gravitational field and p is the fluid density.
c) Using the Newtonian constitutive equation that relates the viscous stress tensor to the velocity gradient,
integrate Eq. 1.1 and show that:
APR?
|(1- ) - (1— к?)
In(1/£)]
In(1/k).
In(1/5)
+1
Vz
(1.2)
V
4μLV
In(1/k)
where u denotes the fluid viscosity and = r/R. State the boundary conditions used, briefly explaining
the physical grounds on which they are based.
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