MEC 516/BME 516/AER 316 Virtual Lab 1: Measurement of Dynamic Viscosity

MEC 516/BME 516/AER 316 Virtual Lab 1: Measurement of Dynamic Viscosity 1. Introduction The objective of this lab is to determine the dynamic viscosity (μ) of a commercial gear oil. The viscosity will be measured at room temperature using two methods: (i) a rotational viscometer and (ii) from the terminal velocity of small spheres falling through the oil. The virtual lab consists of a video titled “ Measurement of Dynamic Viscosity”, which can be found in the Lab Info section of the D2L site . It is recommended that you read this entire write -up prior to watching the video. You may watch the v ideo as many times as you wish. 2. Theory Viscosity is a fluid property that describes how readily a fluid flows. More specifically, viscosity is associated with the resistance to the sliding motion of one fluid layer over another. This resistance takes th e form of a shear stress within the fluid. For most common fluids, the shear stress ( τ) is linearly related to the velocity gradient within the fluid: = (1) where μ is a proportionality constant called the dynamic viscosity (or abs olute viscosity). Fluids for which the dynamic viscosity is independent of the velocity gradient are called Newtonian fluids . Most fluids, including the gear oil used for this lab, are Newtonian. A Newtonian fluid will always flow when a shear stress ( τ) is applied, no matter how small that shear stress is. Aside: How do non -Newtonian fluids behave? One of the most common type s of non-Newtonian fluids is called a “shear thinning” fluid. These fluids require a certain threshold of shear stress before they begin to flow. Examples include ketchup, toothpaste, paints and even blood. There are many types of other non -Newtonian fluids, including shear thickening (corn starch in water) and others. 3. M easurement of V iscosity U sing the F alling S phere M ethod In this experiment, the viscosity of an oil is found by measuring the velocity at which a sphere of known size and density falls through the oil. A force balance on a sphere falling through a quiescent (still) viscous fluid is shown in Figure 1. After the sphere has been falling for long enough, the sphere will stop accelerating. When the sphere has reached the steady state velocity, called the terminal velocity, the sum of the forces on the sphere will be zero: ∑ = = 0 + − = 0 (2) At the terminal velocity ( U), the weight of the sphere ( W S) is balanced by the upward drag force ( F D) and the upward buoyancy force (F B). Figure 1 : Force balance on a sphere falling at constant velocity through a viscous fluid. In very slow fl ows, the viscous forces are large compared to the inertial forces. Such a flow is called Creeping Flow or Stokes Flow (after G.G. Stokes, a nineteenth century mathematician). For very slow flow, Stokes obtained an analytical solution for the drag force on the sphere. The solution to the equations of fluid motion gives that the drag force ( F D) on a sphere moving at constant velocity ( U) through a still fluid is: = 3 (3) where D is the diameter of the sphere and μ is the dynamic viscosity of the fluid. From Archimedes’ principle, the upward buoyancy force ( F B) on the sphere is equal to the weight of the fluid it displaces: = ∀ = 3 6 (4) where ρ f is t he density of the fluid. Note that the volume of a sphere is ∀= = = ∀ = 3 6 (5) where m s is the mass of the sphere and ρ s is the density of the sphere. Substituting Equations (3), (4) and (5) into Equation (2), and solving for the dynamic viscosity of the fluid ( μ) gives: = 2 ( − ) 18 ( 6) Equation 6 can be used to calculate the dynamic viscosity of a fluid by measuring the density, diameter and terminal velocity of the sphere. As mentioned above, Equation (6) is applicable only for slow flows. Experiments have shown that Equation (6) gives reasonable results provided the following condi tion is met: < 1 (7) ≡ (7a) This ratio of variables is a dimensionless parameter called the Reynolds number , Re . The Reynolds number determines the character of the flow. As the Reynolds number decreases, the Stokes Flow approximation improves. Note that Stokes Flow may occur in any fluid, provided the variables combine to make the Reynolds number low. A photograph of flow pattern over a sphere at low Reynolds number is shown in Figure 2. In this flow visualization experiment, the sphere was held stationary and the fluid flowed from bottom to top. Lines of dye have been injected into the fluid to show the streamlines. For Stokes flow, the flow is laminar, and streamli nes are symmetrical. There are no eddies or flow recirculation zones in the wake behind the sphere. Figure 2 : Photograph of very slow flow called “Stokes flow” [1]. 4. Apparatus for the F alling Sphere M ethod The apparatus for this experiment is as follows: • Tall graduated cylinder, filled with gear oil • Solid nylon spheres (three sizes) • Hydrometer for measuring the specific gravity of the oil • Stopwatch and ruler to measure the steady state velocity of each sphere • Weigh scale to determine the weight of each sphere • Micrometer to determine the diameter of each sphere • Thermometer to measure the oil temperature The oil is a commercial gear lubricant: Quaker State SAE 80W-90 GL -5 . 5. Procedure for the F alling S phere Method The procedure for the falling sphere method is illustrated on the video, and is as follows: 1. Record the temperature of the room from a thermometer. (The viscosity of oil is highly dependent upon temperature.) 2. Measure the specific grav ity of the oil (SG oil) using a hydrometer or other means. 3. Use a scale to measure the mass of ten spheres of each size. Calculate the average mass for one sphere of each size. 4. Measure the size of each sphere using the micrometer or similarly acc urate instrument. 5. For each size of sphere, measure the steady state velocity of the sphere in the oil using a stopwatch. Use your judgement to decide when steady state is reached. 6. To check the reproducibility of your results, repeat a few measur ements. You can try this at home if you wish. If you don’t have a scale but know the material your sphere(s) are made from, you can calculate the mass of the spheres by calculating the sphere’s volume ( = 4 3 3) then multiply by the material’s density . The density of nylon is 1.15 g/cm 3. You can use any tall clear vessel with a ruler in it to act as a scale. You may not have a hydrometer, but these are readily available at wine and beer- making stores for less than $20. 6. M easurements with the R otational V iscometer The viscosity of the gear oil will be measured using a BYK -Brookfield DVE rotational viscometer, shown in Figure 3. A rotational viscometer turns a spindle that is submerged in the fluid. The rotating spindle shears the fluid, and th e instrument measures the torque produced by the viscous shear stress at the surface of the spindle. At a constant rotational speed, the shaft torque is linearly related to the dynamic viscosity (μ) of the fluid. This type of commercial instrument is used widely in industries where liquids are manufactured. It is used for quality control in the manufacture of food, pharmaceuticals, chemicals, paints and coatings. (a) (b) Figure 3: (a) BYK -Brookfield DVE rotational viscometer with the “s61” spindle attached (b) the control panel and display. (Note: The value shown here in the display in NOT for the gear oil used in this experiment.) The instrument has been set up to measure the dynamic viscosity (μ) in centipoise (cP), where 1 cP=1x10 - 3 N∙s/m 2. To put this into perspective, liquid water at room temperature has a dynamic viscosity of about 1.0 cP. The oil in the present experiment is a few hundred times more viscous t han water. The rotation viscometer has four spindles, shown in Figure 4. The larger spindles have more surface area and require more torque to spin for a given rotational speed. Hence, the largest spindle (s61) is used for low viscosity fluids. The rotational speed can be adjusted from 0.3 to 100 rpm. Higher rotational speeds require more torque. The speed range, combined with the different spindle sizes, gives this instrument a measurement range of 15cP (spindle s61) to 2,000,000 cP (spindle s64). The instrument accuracy is ±1% of the full -scale reading, corresponding to 100% torque [2]. According to operating manual, the fluid container must have an inside diameter of at least 8.25cm [2]. Also, the spindle must be fully submerged in the fluid, including part of the shaft. Each spindle has a notch in the shaft indicating the required liquid level, shown in Figure 4. While instruments of this type are widely used in industry and research, they cost thousands of dollars, and are not typically found in the average kitchen! Figure 4: Spindles for the BYK -Brookfield DVE rotational viscometer. Spindle s61 is for low viscosity fluids and spindle s64 is for the high viscosity fluids. Lab 1 Questions: 1a) Use Equation (6) to calculate the dynamic viscos ity (in N∙s/m 2) using the data for each falling sphere. For each sphere size, check that the low Reynolds number criterion (Equation (7)) is met. Include one sample calculation for one sphere size in your report. Present your results in tabular form listing sphere size, calculated Re and calculated viscosity. Table 1: Sample experimental data Specific Gravity of the oil from hydrometer: SG=0.89 Sphere Diameter (mm) Mass (g) Time to fall 20 cm at terminal velocity (s) Small 6.35 0.15 17.5 Medium 9.51 0.50 9.1 Large 12. 25 1.07 5.1 1b) The viscometer reading for the oil used above is shown in the picture below. How do your results compare to this reading? What are some possible explanations for any discrepancies (if there are any)? 2) Viscosity measurements were made a two different spindle speeds (12 rpm and 6 rpm) with the rotational viscometer. Within the accuracy of the instrument, these two viscosity measurements were the same. Why? What does this tell you about the gear oil? ( Hint: For what broad classification of fluids would you expect the viscosity to depend on shaft rotational speed?) 3) Compare your falling sphere results to the measurements from the rotational viscometer. Which size of sphere produced the most accurate result? Why? 4) Compare your viscosity measurements with the manufacturer’s specifications from the internet. Is this data consistent with your measurements? If a conversion is needed, show this in the sample calculations. When making this comparison, be sure to discuss the effect of temperature on the oil’s viscosity. (A direct comparison at room temperature may not be available.) 5) Viscosity is a very important property for many products and industrial processes. Can you list three products where prop er viscosity is important and discuss why? Are these products Newtonian? As an example , consider ketchup, a non -Newtonian fluid. Manufacturers try to ensure that the viscosity of their ketchup falls in a narrow range so that consumers get a consistent prod uct that flows at reasonable rates. If one batch has an unusually low viscosity, consumers might flood their burgers with it. Conversely, a batch with unusually high viscosity may not flow out of the bottle, frustrating the consumer.