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V = Figure 1 2 (Pp-Pf) 9 fl Figure 1. Schematic of the falling ball viscometer F₁ = 6 μRV Re= m 9 R² Viscous drag (6xrv). pv,D μ Velocityv Gravitational pull (weight) mg Figure 2 http://www.schoolphysics.co.uk/ Figure 2. Schematic showing balance of forces 4 Pf and Pp is the mass density of the fluid and sphere, respectively, V is the velocity of the sphere, R is the radius of the sphere, g is acceleration of gravity, and is the dynamic viscosity. Fg = (Pp - Pf) g Stokes' Law is subject to some limitations. Specifically, this relationship is valid only for "laminar" flow with very small Reynolds number. Laminar flow is defined as a condition where fluid particles move along in smooth paths. in lamina (fluid layers gliding over one another). When the flow is not laminar we call it "turbulent". A dimensionless parameter known as the Reynolds number is used to distinguish between these two flow conditions, with less than 2,000 being the threshold for laminar and more than 4,000 for turbulent. The in between range is called transition flow. This number is a ratio between the inertial and viscous forces within the fluid as Re is Reynolds Number, p or pf is the mass density of the fluid is the mass density of the fluid, V (or Vs) is the velocity of the fluids relative to the sphere, and D is the diameter of the sphere. Stokes' Law is valid for Reynold's Number values less than 1.0. In this experiment you may find out that the conditions that make Stokes' Law valid are not applicable. In such case, a more appropriate approach to calculate the drag force is to use the following formula: 2
1 Fp = PACD-² Where A: sphere cross-sectional area; V: velocity of flow or sphere; p: fluid density; Co: drag coefficient that can be estimated by using the graph shown in Figure 3. Table 1 gives you a good first estimate for Co. CD 400 200 100 60 40 20 10. 6 4 2 1 0.6 0.4 0.2 0.1 0.06 10- Sphere Shape Cone Half-sphere-> Cube Angled Cube Long Cylinder Short Cylinder 24 CD= Re - Streamlined Body 1 Streamlined Half-body 10⁰ ADO Drag Coefficient 0.47 0.42 0.50 1.05 (a) Figure 3. Drag coefficient as a function of Reynolds number for smooth cylinder and sphere. Table 1. Typical drag coefficients for some standard shapes 0.80 0.82 1.15 0.04 0.09 B Measured Drag Coefficients 10¹ Smooth sphere- 10² Smooth cylinder 103 Re PUD μ 104 D 10° 106 10² 3
For small diameter fluid columns there is an interaction between the fluid and the wall of the cylinder. For this reason you must correct for the diameter interaction using the following relationship, attributed to Faxen (1922): 1 3 μ(corrected) = µ(observed) 1-2.104() +2.089/ Fluid D is the sphere diameter and Dc is the diameter of the cylinder. The rotary viscometer we will use is a Brookfield viscometer. Figures 4 and 5 show a schematic and picture of such a viscometer. Side view Figure 4. Schematic of the rotary viscometer Figure 5. Picture of the rotary viscometer used in this lab The theory behind the rotary viscometer and the mathematics behind viscosity calculation with such device are beyond the scope of this exercise (even though you will be able to do this type of derivations later this semester). Figure 6 shows a schematic of the velocity profile developing within the liquid in the rotary viscometer. Rotating Cylinder Fixed Cylinder Fixed Cylinder Rotating Cylinder Fluid 000 Coo eoo View from above 3 Te Figure 6. Schematic of the velocity profile inside the rotary viscometer - 0.948 4
Experimental Procedure: Follow the procedure as indicated below and collect the data using the table below. 1) For the "falling ball" viscometer measure the diameter of the cylinder and spheres. 2) Measure the weight of the spheres. 3) Measure the length between the marks on the cylinder wall. 4) Using a thermometer measure the water temperature. 5) Drop (gently) a sphere from the water surface level. 6) Using a stopwatch measure the time it took for the sphere to traverse the distance from top mark to low mark. 7) The experiment is repeated 3 times for each of the 3 types of spheres (large Teflon, small Teflon, small glass) for a total of 9 runs. 8) For the rotary viscometer the lab instructor will demonstrate and provide you with the water viscosity. 9) This will be repeated with the sampling device placed in a beaker with warm water so you can observe how the viscosity of water changes as it gets warmer. Table 1. Column Experimental Data 10) The lab instructor will also use the rotary viscometer to measure the viscosity of cooking oil and provide you with the value. Glass Marble Small Teflon H₂o Large Teflon H₂0 Distance Time Mass Diameter Density (cm) (sec) (g) (cm) 100 100 100 100 100 100 100 100 Table 2. Rotary Viscometer Experimental Data Liquid Temperature (°C) 100 Liquid soap u (cP) V (g/cm³) (cm/s) μ (g/cm.s) D_cylinder (cm) 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5 H corrected Re 5
Results and Calculations: For each sphere type you can calculate the average velocity for your 3 runs and use this to calculate the viscosity inferred for your specific sphere. Correct it for finite size (if appropriate). You can also calculate Reynolds number to check validity of Stokes' law. Using the results estimate the viscosity of the water in the column. How does this compare to the estimate obtained by the rotary viscometer? If your Reynolds number indicated that Stokes' law is not valid use the correct formula and re-calculate the dynamic viscosity. Check and verify that Reynolds number and Co are consistent. Questions: Answer the following questions: 1) Is the viscosity you calculated using the "falling ball" viscometer within expected range? Explain how this compares to the rotary viscometer measurement. 2) Why are the column experimental values different than the rotary experimental values? 3) Which parameter is this experiment most sensitive to? 4) What suggestions do you have for improving the apparatus used and procedures in general? 5) What did you observe in regards to temperature effects? 6) Is Stokes' law valid for our experiment? 7) How different is the calculated viscosity when it is based on Stokes' law and when it is not? 8) Is the finite size correction applicable in our case? How does it change the measured viscosity? 6
Table 1. Sample of Column Experimental Data Distance Time Mass Diameter (cm) (sec) (g) (cm) 100 1.53 4.9 1.54 1.53 5.7 1.55 1.59 5.1 1.55 2.85 1.4 1.21 3.81 1.21 Glass Marble Small Teflon Large Teflon H₂0 100 H₂0 100 100 100 100 100 100 100 Liquid soap 22 65 22 1.3 2.75 1.3 3.13 4.9 2.54 4.9 3.2 Table 2. Sample of Rotary Viscometer Experimental Data Liquid Temperature (°C) 4.9 1.21 0.96 0.84 1.9 1.9 243.9 1.9 &(cP) V Density (g/cm³) (cm/s) (g/cm.s) D_cylinder (cm) 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5 9.5 H corrected Re 7