1. Measurement of density and kinematic viscosity
S. Ghosh, M. Muste, F. Stern

2. Table of contents Purpose Experimental design Experimental process
Test Setup
Data acquisition
Data reduction
Uncertainty analysis
Data analysis

3. 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.

4. Experimental design diameters Stopwatch
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.
The facility includes:
A transparent cylinder containing
glycerin.
Teflon and steel spheres of different
diameters
Stopwatch
Micrometer
Thermometer

5. Experimental process

6. 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.

7. Data Acquisition Experimental procedure: Measure room temperature.
Measure sphere diameter using micrometer.
Release sphere at fluid surface and then release gate handle.
Release teflon and steel spheres one by one.
Measure time for each sphere to travel λ.
Repeat steps 3-6 for all spheres. At least 10 measurements are required for each sphere.

8. Data reduction
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).

9. Data reduction (contd.)
Once terminal velocity is achieved, a summation of the vertical forces must balance. Equating the forces gives:
where t is the time for the sphere to fall a vertical distance l.
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.

10. Data reduction (contd.)
Sheet 1
Sheet 2

11. 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.

12. Definitions Bias error b: fixed, systematic
Bias limit B: estimate of b
Precision error e: random
Precision limit P: estimate of e
Total error: d = b + e

13. Propagation of errors
Block diagram showing elemental error sources, individual measurement systems measurement of individual variables, data reduction equations, and experimental results

14. 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

15. Uncertainty equations for single and multiple tests
The total uncertainty of the result
Br : same estimation procedure for single and multiple tests
Pr : determined differently for single and multiple tests

16. Uncertainty equations for single and multiple tests: bias limits
Br :
Sensitivity coefficients
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
Bike: estimate of correlated bias limits for Xi and Xk
Bi for each variable is an estimate of elemental bias errors form different categories: calibration, data acquistionm data reduction, and conceptual bias; eqn. (17)
Bik are the correlated bias limits in Xi and Xk with L the number of correlated bias error sources that are common for measurment of variables Xi and XK

17. Precision limits for single test
Precision limit of the result (end to end):
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):
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

18. Precision limits for multiple test
The average result:
Precision limit of the result (end to end):
t: coverage factor (t = 2 for N > 10)
: standard deviation for M readings of the result
The total uncertainty for the average result:
Alternatively can be determined by RSS of the precision limits of the individual variables

19. Uncertainty Analysis - density
Data reduction equation for density r :
Total uncertainty for the average density:

20. Bias Limit for Density Bias limit Br Sensitivity coefficients
Correlated Bias : two variables are measured with the
same instrument
Bias limit Br
Sensitivity coefficients

21. Precision limit for density

22. Typical Uncertainty results

23. Uncertainty Analysis - Viscosity
Data reduction equation for density n :
Total uncertainty for the average viscosity (teflon sphere):

24. Calculating Bias Limit for Viscosity
Bias limit Bnt (teflon sphere)
Sensitivity coefficients:
No Correlated Bias errors
contributing to viscosity

25. Precision limit for viscosity
Precision limit (teflon sphere)

26. Typical Uncertainty results
Teflon spheres

27. Presentation of experimental results: General Format
EFD result: A ± UA
Benchmark data: B ± UB
E = B-A
UE2 = UA2+UB2
Data calibrated at UE level if:
|E|  UE
Unaccounted for bias and precision limits if:
|E| > UE

28. Data analysis Compare results with manufacturer’s data UA bands
(Proctor & Gamble Co (1995))
UA bands
showing %
uncertainty
Compare results with manufacturer’s data