Measurement of Kinematics Viscosity
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Purpose
Design of the Experiment
Measurement Systems
Measurement Procedures
Uncertainty Analysis
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Density
Viscosity
Data Analysis
Discussions
Purpose
–
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Measurement of Kinematic Viscosity of a fluids
Calculate the Uncertainty of the measurement
Comparison of the calculated viscosity to the Manufacturer’s value
Demonstrate the effects of viscosity by comparison of the fall times
for spheres of different densities
Design of the Experiment
Fd
Sphere
falling at
terminal
velocity
Fb

Fg
V
A fluid deforms continuously under the action of a shear stress. The rate of strain in a fluid is
proportional to the shear stress. The proportionality constant is the dynamic viscosity (m).
Viscosity is a thermodynamic property and varies with pressure and temperature. For a given
state of pressure and temperature, there is a large range of values of viscosity between
common fluids. For instance, there is a variation of three orders of magnitude between water
and glycerin, the fluid which will be used in this experiment.
Since the term m/r, where r is the density of the fluid, frequently appears in the equations of fluid
mechanics, it is given a special name, Kinematic viscosity (n).
The most common methods used to determine viscosity are the rotating-concentric-cylinder
method (Engler viscosimeter) and the capillary-flow method (Saybolt viscosimeter).
Alternatively, we will measure the kinematic viscosity through its effect on a falling object.
Forces acting on the body are :
The maximum velocity attained by an object in free fall (terminal velocity) is strongly affected by the
viscosity of the fluid through which it is falling. When terminal velocity is attained, the body
experiences no acceleration, so the forces acting on the body are in equilibrium
D3
g
Gravitational Force : F g = mg = r sphere 
6
D3
g
Buoyancy : F b = m f g  r fluid 
6
Drag Force : = 3 r fluid  n V D
Resistance of the fluid to the motion of the body, (similar friction). This force is described by the Stokes
expression for the drag force on a sphere (valid for Reynolds numbers, Re = VD/n <<1),
where D is the sphere diameter, rfluid is the density of the fluid, rsphere is the density of the falling
sphere, n is the viscosity of the fluid, Fd, Fb, and Fg, denote the drag, buoyancy, and weight forces,
respectively, V is the velocity of the sphere through the fluid (in this case, the terminal velocity), and g
is the acceleration due to gravity (White 1994).
Viscosity and Density
Once terminal velocity is achieved, a summation of the vertical forces must balance. Equating the forces gives:
D g ( r sphere / r fluid - 1) t
n=
18 
2
where t is the time for the sphere to fall a vertical distance .
Using this equation for two different balls, namely, teflon and steel spheres, the following relationship for the density
of the fluid is obtained, where subscripts s and t refer to the steel and teflon balls, respectively.
r fluid =
2
r
D t t Ds t s r s
2
t t
2
t t
D t - D 2s t s
Measurement Systems
In this experiment, we will allow a sphere to fall through a
long transparent cylinder filled with the glycerin.
After the sphere has fallen a long enough distance
so that it achieves terminal velocity, we will measure
the length of time required for the sphere to fall
through the distance, . The block diagram
illustrates the measurement systems and data
reduction equations for the results
The equipment (measurement systems) used here
includes:

A transparent cylinder (beaker) containing glycerin.
A scale is attached to its side to read the distance
the sphere has fallen.

Teflon and steel spherical balls of different sizes

Stopwatch

Micrometer

Thermometer
EXPERIMENTAL ERROR SOURCES
SPHERE
DIAMETER
FALL
DISTANCE
FALL
TIME
XD
X
B , P
Xt
B ,P
D
D
r = r (X , X ) =
t
2
D s t srs- D t t r t
t
2
2
Ds t s - D t t t
2
n = n (X D, X t , X r , X  ) =
n s,t
Bn , Pn
s,t
s,t
MEASUREMENT
OF INDIVIDUAL
VARIABLES
B t , Pt
2
D
INDIVIDUAL
MEASUREMENT
SYSTEMS
D g(rr
-1)t
sphere
r
Br , Pr
DATA REDUCTION
EQUATIONS
18
EXPERIMENTAL
RESULTS
Measurement Procedures
1.
2.
3.
4.
5.
6.
Density and viscosity are functions of temperature. Measure the temperature of the room. We will use this
temperature to compare our measurement with the manufacturer’s data.
Two horizontal lines are marked on the vertical cylinder. We will measure the time required for the spheres
to fall between these two lines. Measure the distance between the two lines, .
Measure the diameter of each sphere (Teflon and Steel) using the micrometer (10 measurements for each).
Release the sphere at the surface of the fluid in the cylinder. Then, release the gate handle.
Release the spheres, one by one, and measure the time for the sphere to travel the length 
Repeat steps 3- 5 for all spheres.
Since the fall time of the sphere is very short, it is important to measure the time as accurately as possible. Start the
stopwatch as soon as the bottom of the ball hits the first mark on the cylinder and stop it as soon as the
bottom of the ball hits the second mark. Two people should cooperate in this measurement with one
looking at the first mark and handling the stopwatch, and the other looking at the second mark. Both
individuals should agree on the value of the measurement.
Uncertainty Analysis
Bias limit
Bias Limit values
Estimation
BD= BDs = BDt
0.000005 m
½ instrument resolution
Bt = Bts = Btt
0.01 s
Last significant digit
B
0.00079 m
½ instrument resolution
The methodology for estimating uncertainties is according to the AIAA S-071-1995 Standard (AIAA, 1995) as
summarized in IIHR (1999) for multiple tests (M = 10). The block diagram for propagation of errors in the
measured density and viscosity is provided in the block diagram. The data reduction equations for density and
viscosity of glycerin are equation (5) and (4), respectively. First, the elemental errors for each of the
independent variable, Xi, in data reduction equations should be identified using the best available information
(for bias errors) and repeated measurements (for precision errors). Table 1 contains the summary of the
elemental errors assumed for the present experiment.
The bias limit, precision limit, and overall uncertainty for the experimental results, namely the density and viscosity
of glycerin, are then found using Eqs. (14), (23) and (24) in IIHR (1999). Note that the in the present analysis
we will neglect the contribution of the correlated bias errors in equation (14).
Density of Glycerin
U rG  Br2G  Pr2G
The total uncertainty for the density measurement is:
The bias limit , and the precision limit , for the result are given by:
j
BrG   B   B   B   B   B
2
i 1
2
i
2
i
2
Dt
2
Dt
2
tt
2
tt
2
Ds
2
Ds
2
ts
PrG 
2
ts
2  S rG
M
Where the sensitivity coefficients (calculated using mean values from previous data for the independent variables):
D
t
2
r G 2 Ds t t t s D t ( r s - r t )
kg


 296,808.22 4
2
Dt
m
Dt2 t t - D s2 t s
D 
s

r G

D s

2 Dt2 t t t s D s ( r t - r s )
D
2
t
2
s
tt - D t s

2
 527,208.8
kg
m4
Ds2 Dt2 t s ( r s - r t )
r G
kg
 tt 


30
.
6
2
t t
m3  s
Dt2 t t - Ds2 t s


Ds2 Dt2 t t ( r t - r s )
r G
kg
 ts 

 78.1 3
2
t s
m s
Dt2 t t - Ds2 t s


The standard deviation for density of glycerin for the 10 repeated measurements is calculated using the following formula
S rG
 M r k  r 2 
 

 k 1 M  1 
1/ 2
Viscosity of Glycerin
Uncertainty assessment for the glycerin viscosity will be based on the measurements conducted with the teflon
spheres. Selection of the teflon spheres experiment is based on a better agreement with Stokes' law requirements
(Re << 1). The total uncertainty for the viscosity measurement is given by equation (24) in IIHR (1999):
Un  Bn2  Pn2
The bias limit , and the precision limit , for viscosity (neglecting correlated bias errors) is given by equations (14) and
(23) in IIHR (1999), respectively
2  Sn
j
2
2 2
2
2
2 2
2
2
2 2
Pn 
n
i
i
D
D
t
t
rG rG
 
B   B   t B t   t B t   B
 B
M
i 1
The sensitivity coefficients, i, (calculated using mean values from previous data for the independent variables)
The standard deviation for the
 r

 r



2
t


t
2D g
1 t
D g
1
viscosity of glycerin for the 10
t
t r

t r
n
m
n
m2
G


G
5


repeated
measurements
is
 

 0.2202
 

 2.27 x10
D
tt t
18
s
18
s2
t Dt
calculated using the following
formula
 r

2 1/ 2
2 g  t  1t
2
M


D
5


n

n
D
g
r
t
t
t
t
k
n

t r
t  1.36 x106 m
S







n
m

n
 G   1.15 x103
r
 

18rG2
kg  s
2
G rG
 k 1 M  1 

18
s
Data Analysis
The following quantities should be obtained:
1.
Temperature
2.
Acceleration (g),
3.
Teflon Density ,
4.
Steel Density
5.
Length
Data reduction includes the following steps
Trial
T = ??
??
Calculate the fluid density for each
measurement using equation (5).
2.
Calculate the kinematic viscosity for each
measurement using equation (4)
for either sphere type.
Calculate the uncertainties for the experimentally
determined glycerin density and kinematic
viscosity. Compare the measured values to the
manufacturer’s values. Figure 3 contains
benchmark data provided by the glycerin
manufacturer.
Dt
(m)
1
2
1.
Teflon
3
4
5
6
7
8
9
10
Average
Std. Dev. (Si)
Steel
tt
(Sec)
Ds
(m)
Results
ts
(sec)
r
n
(kg/m3)
(m2/sec)
Reference Data
Discussions
Reference data for the Density and Viscosity of 100% aqueous glycerin solutions (Proctor & Gamble Co (1995))
Discussions
R
e
f
e
r
e
n
c
e
d
a
t
a
1
2
6
4

1
2
6
2
3)

Density(kg/m
1
2
6
0
1
2
5
8

1
2
5
6

1
2
5
4
1
21
41
61
82
02
22
42
62
83
03
2
T
e
m
p
e
r
a
t
u
r
e
(
D
e
g
r
e
e
s
C
e
l
s
i
u
s
)

2/s)
1
.
4
e
3
1
.
2
e
3
KinematicViscosity(m
1
.
0
e
3
8
.
0
e
4
6
.
0
e
4
4
.
0
e
4
1
8

R
e
f
e
r
e
n
c
e
d
a
t
a
2
0
2
2
2
4
2
6
2
8
T
e
m
p
e
r
a
t
u
r
e
(
d
e
g
r
e
e
s
C
e
l
s
i
u
s
)
3
0
3
2
How does the size of the sphere affect the viscosity?
How does the drag coefficient CD = FD /(1/2) r V2A
vary with viscosity, where V  / t is the velocity of
the sphere?
How does viscosity change with temperature?
We have taken the contributions to uncertainty from
the correlated bias error terms to be zero. For this
experiment, is this a good assumption?
We have ignored the contributions to uncertainty from
sources other than time and distance. If the sphere had
not achieved terminal velocity, how would this affect
the measurement? Is this a bias or precision error?
Would this error make our value for viscosity larger or
smaller?
The Stokes relation is valid only for Re << 1. The
Reynolds number (a dimensionless number that
characterizes the flow) is given by (l V/ n) where l and
V are characteristic length and velocity scales for the
flow. Calculate the Reynolds number for our
experiment. Are we within the limits?