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# MEASUREMENTS IN FLUID MECHANICS Notes | EduRev

## : MEASUREMENTS IN FLUID MECHANICS Notes | EduRev

``` Page 1

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 56
Module 7 : Lecture 1
MEASUREMENTS IN FLUID MECHANICS
(Incompressible Flow – Part I)

Overview
Accurate measurement in a flowing medium is always desired in many applications.
The basic approach of the given measurement technique depends on the flowing
the medium. Accordingly, the fluid flow diagnostics are classified as measurement of
local properties (velocity, pressure, temperature, density, viscosity, turbulent intensity
etc.), integrated properties (mass and volume flow rate) and global properties (flow
visualization). Also, these properties can be measured directly using certain devices or
can be inferred from few basic measurements. For instance, if one wishes to measure
the flow rate, then a direct measurement of volume/mass flow can be done during a
fixed time interval. However, the secondary approach is to measure some other
quantity such as pressure difference and/or fluid velocity at a point in the flow and
then calculate the flow rate using suitable expressions. In addition, flow-visualization
techniques are sometimes employed to obtain an image of the overall flow field. The
parameters of interest for incompressible flow are the fluid viscosity,
pressure/temperature, fluid velocity and its flow rate.
Measurement of Viscosity
The device used for measurement of viscosity is known as viscometer and it uses the
basic laws of laminar flow. The principles of measurement of some commonly used
viscometers are discussed here;
Rotating Cylinder Viscometer: It consists of two co-axial cylinders suspended co-
axially as shown in the Fig. 7.1.1. The narrow annular space between the cylinders is
filled with a liquid for which the viscosity needs to be measured. The outer cylinder
has the provision to rotate while the inner cylinder is a fixed one and has the provision
to measure the torque and angular rotation. When the outer cylinder rotates, the torque
is transmitted to the inner stationary member through the thin liquid film formed
between the cylinders. Let
12
and rr be the radii of inner and outer cylinders, h be the
depth of immersion in the inner cylinder in the liquid and ( )
21
t rr = - is the annular
Page 2

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 56
Module 7 : Lecture 1
MEASUREMENTS IN FLUID MECHANICS
(Incompressible Flow – Part I)

Overview
Accurate measurement in a flowing medium is always desired in many applications.
The basic approach of the given measurement technique depends on the flowing
the medium. Accordingly, the fluid flow diagnostics are classified as measurement of
local properties (velocity, pressure, temperature, density, viscosity, turbulent intensity
etc.), integrated properties (mass and volume flow rate) and global properties (flow
visualization). Also, these properties can be measured directly using certain devices or
can be inferred from few basic measurements. For instance, if one wishes to measure
the flow rate, then a direct measurement of volume/mass flow can be done during a
fixed time interval. However, the secondary approach is to measure some other
quantity such as pressure difference and/or fluid velocity at a point in the flow and
then calculate the flow rate using suitable expressions. In addition, flow-visualization
techniques are sometimes employed to obtain an image of the overall flow field. The
parameters of interest for incompressible flow are the fluid viscosity,
pressure/temperature, fluid velocity and its flow rate.
Measurement of Viscosity
The device used for measurement of viscosity is known as viscometer and it uses the
basic laws of laminar flow. The principles of measurement of some commonly used
viscometers are discussed here;
Rotating Cylinder Viscometer: It consists of two co-axial cylinders suspended co-
axially as shown in the Fig. 7.1.1. The narrow annular space between the cylinders is
filled with a liquid for which the viscosity needs to be measured. The outer cylinder
has the provision to rotate while the inner cylinder is a fixed one and has the provision
to measure the torque and angular rotation. When the outer cylinder rotates, the torque
is transmitted to the inner stationary member through the thin liquid film formed
between the cylinders. Let
12
and rr be the radii of inner and outer cylinders, h be the
depth of immersion in the inner cylinder in the liquid and ( )
21
t rr = - is the annular
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 56
gap between the cylinders. Considering N as the speed of rotation of the cylinder in
rpm, one can write the expression of shear stress ( ) t from the definition of viscosity
( ) µ , as given below;
2
2
60
rN du
dy t
p
tµ µ
??
= =
??
??
(7.1.1)
This shear stress induces viscous drag in the liquid that can be calculated by
measuring the toque through the mechanism provided in the inner cylinder.
( )
2
11
22
12
2
60
15
or,
rN
T rh r
t
tT T
r r hN CN
p
µp
µ
p
??
= =
??
??
= =
(7.1.2)
Here, C is a constant quantity for a given viscometer.

Fig. 7.1.1: Schematic nomenclature of a rotating cylinder viscometer.

Page 3

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 56
Module 7 : Lecture 1
MEASUREMENTS IN FLUID MECHANICS
(Incompressible Flow – Part I)

Overview
Accurate measurement in a flowing medium is always desired in many applications.
The basic approach of the given measurement technique depends on the flowing
the medium. Accordingly, the fluid flow diagnostics are classified as measurement of
local properties (velocity, pressure, temperature, density, viscosity, turbulent intensity
etc.), integrated properties (mass and volume flow rate) and global properties (flow
visualization). Also, these properties can be measured directly using certain devices or
can be inferred from few basic measurements. For instance, if one wishes to measure
the flow rate, then a direct measurement of volume/mass flow can be done during a
fixed time interval. However, the secondary approach is to measure some other
quantity such as pressure difference and/or fluid velocity at a point in the flow and
then calculate the flow rate using suitable expressions. In addition, flow-visualization
techniques are sometimes employed to obtain an image of the overall flow field. The
parameters of interest for incompressible flow are the fluid viscosity,
pressure/temperature, fluid velocity and its flow rate.
Measurement of Viscosity
The device used for measurement of viscosity is known as viscometer and it uses the
basic laws of laminar flow. The principles of measurement of some commonly used
viscometers are discussed here;
Rotating Cylinder Viscometer: It consists of two co-axial cylinders suspended co-
axially as shown in the Fig. 7.1.1. The narrow annular space between the cylinders is
filled with a liquid for which the viscosity needs to be measured. The outer cylinder
has the provision to rotate while the inner cylinder is a fixed one and has the provision
to measure the torque and angular rotation. When the outer cylinder rotates, the torque
is transmitted to the inner stationary member through the thin liquid film formed
between the cylinders. Let
12
and rr be the radii of inner and outer cylinders, h be the
depth of immersion in the inner cylinder in the liquid and ( )
21
t rr = - is the annular
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 56
gap between the cylinders. Considering N as the speed of rotation of the cylinder in
rpm, one can write the expression of shear stress ( ) t from the definition of viscosity
( ) µ , as given below;
2
2
60
rN du
dy t
p
tµ µ
??
= =
??
??
(7.1.1)
This shear stress induces viscous drag in the liquid that can be calculated by
measuring the toque through the mechanism provided in the inner cylinder.
( )
2
11
22
12
2
60
15
or,
rN
T rh r
t
tT T
r r hN CN
p
µp
µ
p
??
= =
??
??
= =
(7.1.2)
Here, C is a constant quantity for a given viscometer.

Fig. 7.1.1: Schematic nomenclature of a rotating cylinder viscometer.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 56
Falling Sphere Viscometer: It consists of a long container of constant area filled with
a liquid whose viscosity has to be measured. Since the viscosity depends strongly with
the temperature, so this container is kept in a constant temperature bath as shown in
Fig. 7.1.2.

Fig. 7.1.2: Schematic diagram of a falling sphere viscometer.

A perfectly smooth spherical ball is allowed to fall vertically through the liquid
by virtue of its own weight ( ) W . The ball will accelerate inside the liquid, until the
net downward force is zero i.e. the submerged weight of the ball ( )
B
F is equal to the
resisting force ( )
R
F given by Stokes’ law. After this point, the ball will move at
steady velocity which is known as terminal velocity. The equation of motion may be
written as below;
33
66
BR l R s
F F W Dw F Dw
pp
+= ? +=                           (7.1.3)
where, and
ls
ww are the specific weights of the liquid and the ball, respectively. If
the spherical ball has the diameter D that moves at constant fall velocity V in a fluid
having viscosity µ , then using Stokes’ law, one can write the expression for resisting
force ( )
R
F .
3
B
F VD pµ =                                                   (7.1.4)

Page 4

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 56
Module 7 : Lecture 1
MEASUREMENTS IN FLUID MECHANICS
(Incompressible Flow – Part I)

Overview
Accurate measurement in a flowing medium is always desired in many applications.
The basic approach of the given measurement technique depends on the flowing
the medium. Accordingly, the fluid flow diagnostics are classified as measurement of
local properties (velocity, pressure, temperature, density, viscosity, turbulent intensity
etc.), integrated properties (mass and volume flow rate) and global properties (flow
visualization). Also, these properties can be measured directly using certain devices or
can be inferred from few basic measurements. For instance, if one wishes to measure
the flow rate, then a direct measurement of volume/mass flow can be done during a
fixed time interval. However, the secondary approach is to measure some other
quantity such as pressure difference and/or fluid velocity at a point in the flow and
then calculate the flow rate using suitable expressions. In addition, flow-visualization
techniques are sometimes employed to obtain an image of the overall flow field. The
parameters of interest for incompressible flow are the fluid viscosity,
pressure/temperature, fluid velocity and its flow rate.
Measurement of Viscosity
The device used for measurement of viscosity is known as viscometer and it uses the
basic laws of laminar flow. The principles of measurement of some commonly used
viscometers are discussed here;
Rotating Cylinder Viscometer: It consists of two co-axial cylinders suspended co-
axially as shown in the Fig. 7.1.1. The narrow annular space between the cylinders is
filled with a liquid for which the viscosity needs to be measured. The outer cylinder
has the provision to rotate while the inner cylinder is a fixed one and has the provision
to measure the torque and angular rotation. When the outer cylinder rotates, the torque
is transmitted to the inner stationary member through the thin liquid film formed
between the cylinders. Let
12
and rr be the radii of inner and outer cylinders, h be the
depth of immersion in the inner cylinder in the liquid and ( )
21
t rr = - is the annular
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 56
gap between the cylinders. Considering N as the speed of rotation of the cylinder in
rpm, one can write the expression of shear stress ( ) t from the definition of viscosity
( ) µ , as given below;
2
2
60
rN du
dy t
p
tµ µ
??
= =
??
??
(7.1.1)
This shear stress induces viscous drag in the liquid that can be calculated by
measuring the toque through the mechanism provided in the inner cylinder.
( )
2
11
22
12
2
60
15
or,
rN
T rh r
t
tT T
r r hN CN
p
µp
µ
p
??
= =
??
??
= =
(7.1.2)
Here, C is a constant quantity for a given viscometer.

Fig. 7.1.1: Schematic nomenclature of a rotating cylinder viscometer.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 56
Falling Sphere Viscometer: It consists of a long container of constant area filled with
a liquid whose viscosity has to be measured. Since the viscosity depends strongly with
the temperature, so this container is kept in a constant temperature bath as shown in
Fig. 7.1.2.

Fig. 7.1.2: Schematic diagram of a falling sphere viscometer.

A perfectly smooth spherical ball is allowed to fall vertically through the liquid
by virtue of its own weight ( ) W . The ball will accelerate inside the liquid, until the
net downward force is zero i.e. the submerged weight of the ball ( )
B
F is equal to the
resisting force ( )
R
F given by Stokes’ law. After this point, the ball will move at
steady velocity which is known as terminal velocity. The equation of motion may be
written as below;
33
66
BR l R s
F F W Dw F Dw
pp
+= ? +=                           (7.1.3)
where, and
ls
ww are the specific weights of the liquid and the ball, respectively. If
the spherical ball has the diameter D that moves at constant fall velocity V in a fluid
having viscosity µ , then using Stokes’ law, one can write the expression for resisting
force ( )
R
F .
3
B
F VD pµ =                                                   (7.1.4)

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 56
Substituting Eq. (7.1.4) in Eq. (7.1.3) and solving for µ ,
( )
2
where
18
sl
DL
ww V
V t
µ= - =                                    (7.1.5)
The constant fall velocity can be calculated by measuring the time ( ) t taken by the
ball to fall through a distance ( ) L . It should be noted here that the falling sphere
viscometer is applicable for the Reynolds number below 0.1 so that wall will not have
any effect on the fall velocity.

Capillary Tube Viscometer: This type of viscometer is based on laminar flow through
a circular pipe. It has a circular tube attached horizontally to a vessel filled with a
liquid whose viscosity has to be measured. Suitable head
( )
f
h is provided to the
liquid so that it can flow freely through the capillary tube of certain length ( ) L into a
collection tank as shown in Fig. 7.1.3. The flow rate ( ) Q of the liquid having specific
weight
l
w can be measured through the volume flow rate in the tank. The Hagen-
Poiseuille equation for laminar flow can be applied to calculate the viscosity ( ) µ of
the liquid.
4
128
lf
wh d
QL
p
µ
??
=
??
??
(7.1.6)

Fig. 7.1.3: Schematic diagram of a capillary tube viscometer.

Page 5

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 56
Module 7 : Lecture 1
MEASUREMENTS IN FLUID MECHANICS
(Incompressible Flow – Part I)

Overview
Accurate measurement in a flowing medium is always desired in many applications.
The basic approach of the given measurement technique depends on the flowing
the medium. Accordingly, the fluid flow diagnostics are classified as measurement of
local properties (velocity, pressure, temperature, density, viscosity, turbulent intensity
etc.), integrated properties (mass and volume flow rate) and global properties (flow
visualization). Also, these properties can be measured directly using certain devices or
can be inferred from few basic measurements. For instance, if one wishes to measure
the flow rate, then a direct measurement of volume/mass flow can be done during a
fixed time interval. However, the secondary approach is to measure some other
quantity such as pressure difference and/or fluid velocity at a point in the flow and
then calculate the flow rate using suitable expressions. In addition, flow-visualization
techniques are sometimes employed to obtain an image of the overall flow field. The
parameters of interest for incompressible flow are the fluid viscosity,
pressure/temperature, fluid velocity and its flow rate.
Measurement of Viscosity
The device used for measurement of viscosity is known as viscometer and it uses the
basic laws of laminar flow. The principles of measurement of some commonly used
viscometers are discussed here;
Rotating Cylinder Viscometer: It consists of two co-axial cylinders suspended co-
axially as shown in the Fig. 7.1.1. The narrow annular space between the cylinders is
filled with a liquid for which the viscosity needs to be measured. The outer cylinder
has the provision to rotate while the inner cylinder is a fixed one and has the provision
to measure the torque and angular rotation. When the outer cylinder rotates, the torque
is transmitted to the inner stationary member through the thin liquid film formed
between the cylinders. Let
12
and rr be the radii of inner and outer cylinders, h be the
depth of immersion in the inner cylinder in the liquid and ( )
21
t rr = - is the annular
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 56
gap between the cylinders. Considering N as the speed of rotation of the cylinder in
rpm, one can write the expression of shear stress ( ) t from the definition of viscosity
( ) µ , as given below;
2
2
60
rN du
dy t
p
tµ µ
??
= =
??
??
(7.1.1)
This shear stress induces viscous drag in the liquid that can be calculated by
measuring the toque through the mechanism provided in the inner cylinder.
( )
2
11
22
12
2
60
15
or,
rN
T rh r
t
tT T
r r hN CN
p
µp
µ
p
??
= =
??
??
= =
(7.1.2)
Here, C is a constant quantity for a given viscometer.

Fig. 7.1.1: Schematic nomenclature of a rotating cylinder viscometer.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 56
Falling Sphere Viscometer: It consists of a long container of constant area filled with
a liquid whose viscosity has to be measured. Since the viscosity depends strongly with
the temperature, so this container is kept in a constant temperature bath as shown in
Fig. 7.1.2.

Fig. 7.1.2: Schematic diagram of a falling sphere viscometer.

A perfectly smooth spherical ball is allowed to fall vertically through the liquid
by virtue of its own weight ( ) W . The ball will accelerate inside the liquid, until the
net downward force is zero i.e. the submerged weight of the ball ( )
B
F is equal to the
resisting force ( )
R
F given by Stokes’ law. After this point, the ball will move at
steady velocity which is known as terminal velocity. The equation of motion may be
written as below;
33
66
BR l R s
F F W Dw F Dw
pp
+= ? +=                           (7.1.3)
where, and
ls
ww are the specific weights of the liquid and the ball, respectively. If
the spherical ball has the diameter D that moves at constant fall velocity V in a fluid
having viscosity µ , then using Stokes’ law, one can write the expression for resisting
force ( )
R
F .
3
B
F VD pµ =                                                   (7.1.4)

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 56
Substituting Eq. (7.1.4) in Eq. (7.1.3) and solving for µ ,
( )
2
where
18
sl
DL
ww V
V t
µ= - =                                    (7.1.5)
The constant fall velocity can be calculated by measuring the time ( ) t taken by the
ball to fall through a distance ( ) L . It should be noted here that the falling sphere
viscometer is applicable for the Reynolds number below 0.1 so that wall will not have
any effect on the fall velocity.

Capillary Tube Viscometer: This type of viscometer is based on laminar flow through
a circular pipe. It has a circular tube attached horizontally to a vessel filled with a
liquid whose viscosity has to be measured. Suitable head
( )
f
h is provided to the
liquid so that it can flow freely through the capillary tube of certain length ( ) L into a
collection tank as shown in Fig. 7.1.3. The flow rate ( ) Q of the liquid having specific
weight
l
w can be measured through the volume flow rate in the tank. The Hagen-
Poiseuille equation for laminar flow can be applied to calculate the viscosity ( ) µ of
the liquid.
4
128
lf
wh d
QL
p
µ
??
=
??
??
(7.1.6)

Fig. 7.1.3: Schematic diagram of a capillary tube viscometer.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 5 of 56
Saybolt and Redwood Viscometer: The main disadvantage of the capillary tube
viscometer is the errors that arise due to the variation in the head loss and other
parameters. However, the Hagen-Poiseuille formula can be still applied by designing
a efflux type viscometer that works on the principle of vertical gravity flow of a
viscous liquid through a capillary tube. The Saybolt viscometer has a vertical
cylindrical chamber filled with liquid whose viscosity is to be measured (Fig. 7.1.4-a).
It is surrounded by a constant temperature bath and a capillary tube (length 12mm and
diameter 1.75mm) is attached vertically at the bottom of the chamber. For
measurement of viscosity, the stopper at the bottom of the tube is removed and time
for 60ml of liquid to flow is noted which is named as Saybolt seconds. So, Eq. (7.1.6)
can be used for the flow rate ( ) Q is calculated by recording the time (Saybolt
seconds) for collection of 60ml of liquid in the measuring flask. For calculation
purpose of kinematic viscosity ( ) ? , the simplified expression is obtained as below;
1.8
0.002 ; where, in Stokes and in seconds t t
t
µ
??
?
= = -          (7.1.6)
A Redwood viscometer is another efflux type viscometer (Fig. 7.1.4-b) that
works on the same principle of Saybolt viscometer. Here, the stopper is replaced with
an orifice and Redwood seconds is defined for collection of 50ml of liquid to flow out
of orifice. Similar expressions can be written for Redwood viscometer. In general,
both the viscometers are used to compare the viscosities of different liquid. So, the
value of viscosity of the liquid may be obtained by comparison with value of time for
the liquid of known viscosity.

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