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Falling Ball Viscometry Lab #1

Chase Hilderbrand Joanna Nicholson Eddwie Perez

September 11, 2015

Professor: Dr. Danvers Johnson CWR3201C

INTRODUCTION The objective of the viscometry lab is to use a falling ball viscometer to determine the viscosity of a fluid. The viscosity of a fluid is a quantification of its resistance to deformation by various stresses. This is accomplished by measuring the velocity of a ball bearing, of known diameter, travelling through the unknown fluid. In order to calculate the results, one must assume that the ball had reached terminal velocity, and was not affected by turbulence until the bottom of the tube. The velocities were calculated using marked points on the tube, before the bottom. As well, there is an assumption made that Stokes’ Law is valid, and that minor temperature changes throughout the fluid make extremely minor differences and can be ignored.

THEORY Once a sphere falling through a fluid has reached terminal velocity, it is in equilibrium. The Force of Gravity (FG) is therefore equal to the Resistive Force (FR). FG = FR

(1)

The Force due to gravity is a function of Volume (V), Density (ρ), and gravity (g): 𝐅G = (Vball)(ρball)(g) =

𝟒 𝟑

(𝛑)(𝐫2)(𝛒ball)(𝐠)

(2)

The Resistive Force is a function of the Viscosity (µ), and our measured velocity v): FR = 6πµrv

(3)

By substituting Equation 2, and 3 into Equation 1, we can derive a function for viscosity: µ=

1/18(ρball – ρfluid)(g)(D)2 v

(4)

To find Kinematic Viscosity, (ν):

ν=

µ

(5)

𝛒

The Reynolds Number, (Re) is used to describe the relationship of the kinematic forces to viscous forces. It is a ratio that describes the way in which the ball reacts to the viscosity of the fluid. A low Reynolds number indicates a laminar flow. Re =

𝛒𝐯𝐃 µ

EXPERIMENTAL PROCEDURES 1. Measure the diameter of the available balls and weigh them. Record the diameter and weight. 2. Determine a distance from the surface of the liquid at which the ball reaches terminal velocity by

doing a dry run; dropping the ball and looking for the instant the ball starts to drop at a continuous rate. 3. Once the distance from the surface where terminal velocity begins is established, determine a distance below the point where terminal velocity is reached that is relatively far from the bottom of the tube. 4. Mark two points, one below the location at which terminal velocity is initially reached and the other above the point chosen in step 3. Record the distance between these two points. 5. Using the balls that were measured in step 1 drop one ball in the liquid. Start the timer at P1 and stop the timer at P2. Record the time. 6. Repeat step 5 three more times with the remaining balls measured in step 1. RESULTS Equation 4 was used with the mean data from the above to get a value for viscosity, 0.694 P and a kinematic viscosity of 5.60E-4 m2/s was also calculated.

Viscosity (µ) in (Ns/m2)

Kinematic Viscosity (v) in (m2 /s)

7995.88

0.7624

0.00061

4.08

8055.11

0.6661

0.00053

9.40

4.06

8015.63

0.6889

0.00055

523.56

9.38

4.07

8035.37

0.6895

0.00055

543.48

10.12

4.06

8015.63

0.6636

0.00053

Ball diameter (D) in (mm)

Drop time (t) in (s)

Drop distance s (mm)

Rate of fall v=s/t (mm/s)

Test 1

9.89

2.12

1000

Test 2

9.88

1.84

Test 3

9.89

Test 4 Test 5

Re

Mass of ball (g)

ρ ball (kg/m3)

471.70

7.65

4.05

1000

543.48

10.08

1.91

1000

523.56

9.88

1.91

1000

9.89

1.84

1000

9.89

1.92

1000

521.16

9.32

4.06

8023.52

0.6941

0.00056

0.001

0.018

0.00

2.94

0.15

0.16

22.500

0.46

0.00003

Mean Standard Deviation

FLUID:

Glycerine

Table 1. Falling Ball Data

𝟏

𝒌𝒈

µ = ( ) × (𝝆ball-𝝆gly ) ×𝟗. 𝟖𝟏𝟎 𝟏𝟖 𝒎𝟑

𝒎 𝒔𝟐

∗(

𝟗.𝟖𝟗

𝟏

) 2𝒎𝟐 × 𝟎.𝟒𝟕𝟏 𝟏𝟎𝟎𝟎

𝒎 𝒔

= 𝟎. 𝟕𝟔𝟐𝑷

DISCUSSION The calculated kinematic viscosity was compared to the values in Figure 1 (Munson et al. 2009). An assumption was made that the temperature of the viscometer was approximately 21.0 °C. In comparing the calculated value of dynamic viscosity,the value was close to that of glycerin and suggests that the liquid in the viscometer was likely glycerin. As well, it could be observed that the sphere fell more slowly than it would in water, and in fact, the dynamic viscosity value was

greater than that of water; 762E-3 vs 1.002E-3 (water). The only notably erroneous measurement was the time needed for the ball to travel 1000 mm on the first trial; there seems to be a delay as compared to the other values. However, the overall standard deviation is only 2.94 and does not severely alter the calculated viscosities. CONCLUSIONS The fluid in the viscometer was determined to be that of glycerin based on an assumption of a consistent temperature of the viscometer and fluid and that the terminal velocity was established far enough from the bottom and wall of the tube to prevent any turbulent interference. The largest variance in the data was a delay in time to fall, although it was not very relevant to the final calculations.

Figure 1

Works Cited Munson, B. R., Young, D. F., Okiishi, T. H., Huebsch, W. W. (2009). Fundamentals of Fluid Mechanics, Wiley, Hoboken, NJ, Appendix B. pg. 714.