Measurement of density and
kinematic viscosity
S. Ghosh, M. Muste, F. Stern
Table of contents

Purpose
 Experimental design
 Experimental process
•
•
•
•
•
Test Setup
Data acquisition
Data reduction
Uncertainty analysis
Data analysis
Purpose

Provide hands-on experience with simple table top
facility and measurement systems.

Demonstrate fluids mechanics and experimental fluid
dynamics concepts.

Implementing rigorous uncertainty analysis.

Compare experimental results with benchmark data.
Experimental design
Fd
Sphere
falling at
terminal
velocity
Fb

Fg

Viscosity is a thermodynamic property
and varies with pressure and temperature.

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).

We will measure the kinematic viscosity
through its effect on a falling object.
V
The facility includes:
• A transparent cylinder containing
glycerin.
• Teflon and steel spheres of different
diameters
• Stopwatch
• Micrometer
• Thermometer
Experimental process
Test set-up





Verify the vertical position for the
cylinder.
Open the cylinder lid.
Prepare 10 teflon and 10 steel
spheres.
Clean the spheres.
Test the functionality of stopwatch,
micrometer and thermometer.
Data Acquisition
Experimental procedure:
1.
Measure room temperature.
2.
Measure λ.
3.
Measure sphere diameter using
micrometer.
4.
Release sphere at fluid surface
and then release gate handle.
5.
Release teflon and steel spheres
one by one.
6.
Measure time for each sphere to
travel λ.
7.
Repeat steps 3-6 for all spheres.
At least 10 measurements are
required for each sphere.
Table 1. Gravity and sphere density constants
Definitions
Gravitational
acceleration
Density of steel
Density of teflon
Symbol
g
Value
9.81 m/s2
rs
rt
7991 kg/m3
2148 kg/m3
Table 2. Typical test results (multiple tests)
Trial
TEFLON
STEEL
Dt
tt
Ds
ts
T= 26.4 C
(m)
(sec)
(m)
(sec)
= 0.61 m
0.00661 31.08 0.00359 12.210
1
0.00646 31.06 0.00358 12.140
2
0.00634 30.71 0.00359 12.070
3
0.00632 30.75 0.00359 12.020
4
0.00634 30.89 0.00359 12.180
5
0.00633 30.82 0.00359 12.060
6
0.00637 30.89 0.00359 12.110
7
0.00634 30.71 0.00359 12.120
8
0.00633 31.2 0.00359 12.030
9
0.00634 31.11 0.00359 12.200
10
0.00637 30.91 0.00358 12.114
Average
Std.Dev. (Si) 9.1710-5 0.18 3.1610-6 0.0687
Ret = 0.18 and Res = 0.26
RESULTS
r
n
(kg/m3)
1382.14
1350.94
1305.50
1304.66
1302.38
1306.70
1317.75
1301.50
1320.75
1307.64
1318.80
26.74
(m2/s)
0.000672
0.000683
0.000712
0.000709
0.000720
0.000710
0.000710
0.000717
0.000700
0.000718
0.000706
1.59710-5
Data reduction
D3
Gravity  m g  r sphere 
g
6
Buoyancy:
D3
g
F b = m f g  r fluid 
6
Drag Force: = 3 r fluid  n V D

Terminal velocity attained by an object in free fall 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.

Resistance of the fluid to the motion of a body is defined as drag force and is given by
Stokes expression (see above) for 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).
Data reduction (contd.)

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 - D2s t s
Data reduction (contd.)
Sheet 1
Sheet 2
Experimental Uncertainty
Assessment
• Uncertainty analysis (UA): rigorous methodology for uncertainty
assessment using statistical and engineering concepts.
• ASME (1998) and AIAA (1999) standards are the most recent updates of
UA methodologies, which are internationally recognized as summarized in
IIHR 1999.
•Error: difference between measured and true value.
• Uncertainties (U): estimate of errors in measurements of individual
variables Xi (Uxi) or results (Ur) obtained by combining Uxi.
• Estimates of U made at 95% confidence level.
Definitions
• Bias error b:
Fixed and systematic
•Precision error e:
± and random
 Total error: db+e
Propagation of errors
Block diagram showing elemental error sources, individual measurement
systems measurement of individual variables, data reduction equations, and
experimental results
ELEMENTAL
ERROR SOURCES
1
2
X
1
B ,P
1
1
INDIVIDUAL
MEASUREMENT
SYSTEMS
J
X
2
B ,P
2
X
J
B,P
2
J
J
MEASUREMENT
OF INDIVIDUAL
VARIABLES
EXPERIMENTAL ERROR SOURCES
SPHERE
DIAMETER
FALL
DISTANCE
FALL
TIME
XD
X
B , P
Xt
BD , PD
B t , Pt
2
r = r (X , X ) =
D
r = r (X , X ,......, X )
1
2
J
r
B, P
r
r
DATA REDUCTION
EQUATION
EXPERIMENTAL
RESULT
t
n s,t
Bn , Pn
s,t
s,t
MEASUREMENT
OF INDIVIDUAL
VARIABLES
2
D s t srs - D t t t r t
2
2
Ds t s - D t t t
2
n = n (X D, X t , X r , X  ) =
INDIVIDUAL
MEASUREMENT
SYSTEMS
D g(rr
-1)t
sphere
r
Br , Pr
DATA REDUCTION
EQUATIONS
18
EXPERIMENTAL
RESULTS
Uncertainty equations for single
and multiple tests
Measurements can be made in several ways:

•
Single test (for complex or expensive experiments): one set of
measurements (X1, X2, …, Xj) for r
• According to the present methodology, a test is considered a single
test if the entire test is performed only once, even if the
measurements of one or more variables are made from many
samples (e.g., LDV velocity measurements)
Multiple tests (ideal situations): many sets of measurements (X1, X2,
…, Xj) for r at a fixed test condition with the same measurement system
Uncertainty equations for single
and multiple tests
• The total uncertainty of the result
U r2 = B2r + P2r
• Br : same estimation procedure for single and multiple
tests
• Pr : determined differently for single and multiple
tests
Uncertainty equations for single
and multiple tests: bias limits
• Br :
J 1
J
J
B   B + 2  i k Bik
2
r
• Sensitivity coefficients
i 1
2
i
2
i
i 1 k i +1
r
X i
i 
• Bi: estimate of calibration, data acquisition, data reduction,
conceptual
bias errors for Xi.. Within each category, there may be several elemental
sources of bias. If for variable Xi there are J significant elemental bias
errors [estimated as (Bi)1, (Bi)2, … (Bi)J], the bias limit for Xi is calculated as
J
2
k 1
k
B   Bi 
2
i
• Bike: estimate of correlated bias limits for Xi
L
Bik   Bi  Bk 
 1
and Xk
Precision limits for single test
• Precision limit of the result (end to end):
Pr  tSr
t: coverage factor (t = 2 for N > 10)
Sr: the standard deviation for the N readings of the result. Sr
must be determined from N readings over an
appropriate/sufficient time interval
• Precision limit of the result (individual variables):
J
Pr   (  i Pi )2
i=1
Pi  ti Si
the precision limits for Xi
Often is the case that the time interval for collecting the data is
inappropriate/insufficient and Pi’s or Pr’s must be estimated based on previous
readings or best available information
Precision limits for multiple test
1
r
M
• The average result:
M
r
k 1
k
• Precision limit of the result (end to end):
 M rk  r 2 
S r  

M

1
k

1


tS r
Pr 
M
1/ 2
t: coverage factor (t = 2 for N > 10)
Sr : standard deviation for M readings of the
•
result
The total uncertainty for the average result:

U  B + P  B + 2 Sr
2
r
2
r
2
r
2
r
M

2
• Alternatively Pr can be determined by RSS of the precision
limits of the individual variables
Uncertainty Analysis - density
• Data reduction equation for density r:
2
2
Dt t t r t - Ds t s r s
r=
Dt2 t t - D2s t s
• Total uncertainty for the average density:
U r   Br2 + Pr2
Bias Limit for Density
Correlated Bias : two variables are measured with the
Bias limit Br
same instrument
Br2   D2t BD2t +  t2t Bt2t +  D2s BD2 s +  t2s Bt2s + 2 Dt  Ds BDt BDs + 2 tt  ts Btt Bts
Bias Limit
BD= BD = BD
t
Bt= Bt = Bt
t
t
S
Magnitude
0.000005 m
Percentage Values
Estimation
0.078 % Dt
½ instrument resolution
0.14 % Ds
0.01 s
0.032% tt
0.083% ts
Last significant digit
Sensitivity coefficients
D
2
r 2 Ds t t t s Dt ( r s - r t )
kg


 296,808 4
2
Dt
m
Dt2 tt - Ds2 t s
Ds2 Dt2 t s ( r s - r t )
r
kg
 tt 

 30.60 3
2
tt
m s
Dt2 tt - Ds2 t s
D
2 Dt2 t t t s D s ( r t - r s )
r
kg




527
,
208
2
Ds
m4
Dt2 tt - Ds2 t s
Ds2 Dt2 t t ( r t - r s )
r
kg
 ts 



78
.
1
2
t s
m3  s
Dt2 tt - Ds2 t s
t
s








Precision limit for density
Precision limit Pr
Pr 
2  Sr
M
 r k  r 
S r  
 k 1 M  1
M
2



1/ 2
Typical Uncertainty results
Term
 D BD
t
 t Bt
t
 D BD
s
 t Bt
s
2 Dt  D s BD2
2 t t  t s Bt2
Br
Pr
Ur
Without correlated bias errors
Magnitude
% Values
3
1.48 kg/m
22.30% B r2
2
0.31 kg/m3 0.95% B r
2
-2.63 kg/m3 70.60% B r
2
-0.78 kg/m3 6.15% B r
3.13 kg/m3 0.24%r
U r2
3.3%
With correlated bias errors
Magnitude
% Values
3
1.48 kg/m
147.16% B r2
2
0.31 kg/m3 4.09% B r
2
-2.63 kg/m3 464.72% B r
2
-0.78 kg/m3 38.89% B r
2
-2.79 kg/m3 -522.98% B r
2
-0.69 kg/m3 -31.88% B r
r
1.22 kg/m3 0.09%
U r2
0.47%
16.91 kg/m3 1.28%
U r2
96.70%
16.91 kg/m3 1.29%
U r2
99.53%
17.20kg/m3 1.30%
16.95kg/m3 1.28%
r
r
r
r
Uncertainty Analysis - Viscosity
Data reduction equation for density n:
gD2t
n
( S  1)
18
Total uncertainty for the average viscosity
(teflon sphere):
Un2  Bn2t + Pn2
t
t
Calculating Bias Limit for
Viscosity
Bias limit Bnt (teflon sphere)
Bn2t   D2t BD2 +  r2 Br2 +  t2t Bt2 +  2 B2
Bias Limit
BD= BD = BD
t
Bt= Bt = Bt
t
S
t
No Correlated Bias errors
contributing to viscosity
Magnitude
0.000005 m
Percentage Values
Estimation
0.078 % Dt
½ instrument resolution
0.14 % Ds
0.01 s
0.032% tt
0.083% ts
Last significant digit
Sensitivity coefficients:


5
Dt2 gr t t
n
6 m
t
 

 1.36  10
r r
18r 2
kg  s
2 Dt g rt r  1 tt
n
m
D 

 0.202
18
s
t Dt
2
n Dt g r t r  1
 

 2.27x10 5
tt
t
18
2
m
s2
2 g r r  1t
D
t
t
n
3
    t


1
.
15
x
10

182
m
s
Precision limit for viscosity
Precision limit P υ (teflon sphere)
Pn t 
2  Sn t
M
 n k  n 
Sn  
 k 1 M  1
M
2



1/ 2
Typical Uncertainty results
Term
B
 D BD
t
rBr
 t Bt
t
B
Bn t
Pn t
Magnitude
7.910-4 m
1.110-6 m2/s
4.2710-6 m2/s
2.2910-7 m2/s
-0.9210-6 m2/s
4.510-6 m2/s
1.0110-5 m2/s
Percentage
Values
0.13% 
Bn2
5.97%
Bn2
90.03%
Bn2
0.26%
Bn2
3.74%
t
t
t
t
0.64%n t
Un2
16.43%
n
t
1.43% t
Un2
83.57%
t
Un t
1.1110-5 m2/s
n
1.57% t
Teflon spheres
Presentation of experimental
results: General Format
•
E = B-A
•
UE2 = UA2+UB2
B ± UB
• Data calibrated at UE
level if:
•
|E|  UE
• Unaccounted for bias
and precision limits if:
•
|E| > UE
2.1
Experimental Result (UA= 3%)
Benchmark data (UB = 1.5% )
2.0
1.9
Result R
• EFD result: A ± UA
• Benchmark data:
1.8
1.7
1.6
1.5
1.4
Data not validated
Validated data
1.3
20
25
30
35
Independent variable X i
40
45
Data analysis
R
e
f
e
r
e
n
c
e
d
a
t
a
1
2
6
4
3)
1
2
6
2
Density(kg/m
1
2
6
0
Compare results with
manufacturer’s data
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
R
e
f
e
r
e
n
c
e
d
a
t
a
1
8 2
0
2
2 2
4
2
6
2
8
3
0
3
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
)
(Proctor & Gamble Co (1995))
UA bands
showing %
uncertainty
Flow Visualization using ePIV
• ePIV-(educational) Particle
Image Velocimetry
• Detects motion of particles
using a camera
• Camera details: digital ,
30 frames/second,
600×480 pixel
resolution
• Flash details: 15mW
green continuous diode
laser
Results of ePIV
• Identical particles are tracked in consecutive images to have
quantitative estimate of fluid flow
• Particles have the follow specifications:
• neutrally buoyant : density of SG ~ 1.0
• small enough to follow nearly all fluid motions:
diameter~11μm
•
Qualitative estimates of fluid flow can also be shown
Flow Visualization
• Visualization-a means of viewing fluid flow as a way of examining the
relative motion of the fluid
• Generally fluid motion is highlighted by smoke, die, tuff, particles,
shadowgraphs, Mach-Zehnder interferometer, and many other methods
• Answer the following
questions:
1. Where is the circular cylinder?
2. In what direction is the fluid
traveling?
3. Where is separation occurring?
4. Can you spot the separation
bubbles?
5. What are the dark regions in the
left half of the image?
Flow Visualization-Flow around a
circular cylinder
•
•
•
•
Flow around a sphere is approximated by a circular cylinder
Flow in laboratory exercise has a Reynolds number less than 1.
Flow with ePIV has a Reynolds number range from ~2 to 90.
Reynolds number = Re = (V∙D)/υ = (ρ ∙ V ∙ D)/μ
Re= <1
Glycerine solution with aluminum
powder, V=1.5 mm/s, dia=10 mm
Re= ~2
ePIV, water and 10μm polymer
particels, V=1.5 mm/s, dia=4 mm
Flow Visualization-Flow around a
circular cylinder con’t
Re=1.54
Re=9.6
• Flow separation occurs at Re ~ 5
• Standing eddies occur between
5 < Re < 9
• Length of separation bubble is
found to grow linearly with
Reynolds number until the flow
becomes unstable about Re = 40
• Sinusoidal wake develops at
about Re = 50
• Kármán vortex street develops
around Re = 100
Flow Visualization-Flow around a
circular cylinder con’t
Re=26
Re=30
Re=55
Re=140
Flow Visualization-Flow around a
circular cylinder con’t
• Typical ePIV images
Re=30
Re=60
Re=90
The End