University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

Experiment B8 Fluids Procedure

Deliverables: Checked lab notebook, Brief Tech Memo

Part I: Viscosity of Engine Oil Theoretical Background Proper lubrication is critical for the operation of internal combustion engines and many other types of machinery. Adding a thin film of oil can prevent two solid surfaces from making physical contact and allow them to easily “glide” past one another. For this to work properly, it is very important to use the correct oil viscosity and density for the given loads and shear stresses involved. This can be difficult, as the viscosity of most fluids significantly decreases as temperature increases. (Think of cold maple syrup compared to warm maple syrup.) In fact, the viscosity of engine oil changes so dramatically with temperature, that the Society of Automotive Engineers (SAE) has developed a standard naming convention to account for it. If you have spent any time working with cars, you have probably seen it. Under the “SAE J300” standard, engine oils are given names like “5W-30”, where the 5 denotes the cold “Winter” viscosity, and the 30 denotes the viscosity at a typical engine operating temperature. As previously mentioned, the viscosity of a fluid typically decreases as temperatures increases. One theory posits that the kinematic viscosity ν of a fluid as a function of temperature obeys the Arrhenius equation ⎛ E ⎞ A , (1) ν = ν0 exp⎜ ⎟ ⎝ kBT ⎠ -23 where ν0 is the exponential pre-factor, EA is the activation energy, kB = 1.38x10 J/K is the Boltzmann constant., and T is the temperature of the fluid in degrees Kelvin. In this lab, you will measure the kinematic viscosity of 10W-30 engine oil as a function of temperature and compare it to Eq. (1) using an “Arrhenius plot”.

Experimental Procedure You will measure the kinematic viscosity of 10W-30 using a device known as a Zahn cup. (No, not Zahm, but Zahn with an N. Sorry to disappoint some of you.) The Zahn cup is essentially a ladle with a small hold drilled into the bottom. The time tZ it takes the fluid to drain out through the hole in the bottom—or “Zahn time”—is related to the kinematic viscosity by the calibration equation

ν = K (tZ − C) (2) where K and C are calibration constants that can be found in the data sheet on the lab webpage.

B8 – Fluids 1 Last Revision: 3/20/18

University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

On the counter, you will find five different beakers containing 10W-30 engine oil. One is in an ice bucket, one is at room temperature, and the other three are on hot plates set to various temperatures.

CAUTION: Lab coats and nitrile gloves are required for this lab exercise. If you spill any oil, please clean it up immediately with the shop towels provided. If you spill a lot of oil, ask the TA or lab instructor for assistance cleaning it up.

1. Sketch the experimental set up in your lab notebook. 2. Make a table in your lab notebook with three columns: Zahn cup hole diameter, oil temperature, and Zahn time tZ. 3. Beginning with any of the five beakers, measure the diameter of the hole in the bottom of the Zahn cup and record it in the table in your lab notebook. 4. Connect the thermocouple to the red HH806AU data logger. Turn it on and check to make sure it is reading a reasonable temperature. 5. Slowly dip the TC into the engine oil. Wait a few seconds for the temperature to equilibrate and record the temperature in your table. Use a shop towel to wipe off the tip when you are finished. 6. Gently submerge the Zahn cup in the oil. With the cup filled, remove it from the oil. Use the stopwatch to time how long it takes to completely drain. Record this time in the table in your lab notebook. 7. Wipe away the oil from the outside and then the inside of the Zahn cup with a shop towel. 8. Set the Zahn cup and TC down on the absorbent padding next to the beaker. 9. Repeat the procedure for the remaining beakers. 10. Convert your Zahn times to kinematic viscosity in centiStokes using Eq. (2). The calibration constants can be found in the Zahn cup data sheet on the lab webpage. Note that the calibration constants depend on the diameter of the hole in the Zahn cup.

B8 – Fluids 2 Last Revision: 3/20/18

University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

Part II: Terminal Velocity of Spheres in Glycerol Theoretical Background Any object moving through a fluid will be subject to a viscous drag force, which depends on the velocity relative to the fluid. For a falling object, the velocity will increase until the viscous drag force Fd balances the gravitational force Fg. When the two forces are balanced, the object falls at a constant speed known as the terminal velocity. Figure 2 illustrates the streamlines around a small sphere falling through a viscous fluid.

4

2 Fd

0

-2 Fg

-4

Figure 1 – Shown above are the streamlines for “Stokes’ Flow” around a sphere. The sphere reaches -4 -2 0 2 4 terminal velocity when the gravitational force Fg is balanced by the viscous drag force Fd.

For a solid sphere surrounded by fluid, the gravitational force is

4 3 Fg = π R (ρS − ρL )g , (3) 3 where R is the radius, ρS is the density of the solid sphere, ρL is the density of the surrounding fluid, and g = 9.8 m/s2 is the acceleration of gravity. (Note that it is not simply mg, because we must account for the “buoyant” force exerted by the surrounding fluid.) According to Stokes’s Law, the viscous drag force on a very slowly moving sphere is

Fd = 6πηRv (4) where η is the dynamic viscosity, R is the radius of the sphere, and v is the velocity.

B8 – Fluids 3 Last Revision: 3/20/18

University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

The terminal velocity can then be calculated by setting Eq. (3) equal to Eq. (4) and solving for v, which yields

2(ρS − ρL )g 2 v = R . (5) T 9η Note that the terminal velocity actually increases with radius R, while it decreases with viscosity η. Furthermore, it is possible to have negative terminal velocity if ρL > ρS. This is the case for highly buoyant objects like hot air balloons and bubbles. The Reynolds number is a commonly used non-dimensional parameter that compares a fluid’s viscous forces to its inertial forces. The Reynolds number is also important, because it determines when a fluid will transition from steady, predictable behavior to chaotic, turbulent behavior. (Low Reynolds number implies steady, predictable behavior. High Reynolds number means turbulent behavior.) For a sphere such the one shown in Fig. 2, the Reynolds number is ρ vD Re = L , (6) η where D = 2R is the diameter of the sphere. Stokes’ Law discussed above is only valid if the Reynolds number is very small, Re << 1. Another commonly used non-dimensional parameter is the drag force coefficient F C = d , (7) d 1 2 2 ρLv A where A is some characteristic area, either the total surface area, or the cross sectional area. Combining −1 Eqs. (4), (6), and (7), it can be shown thatCd ∝ Re for Stokes’ flow around a sphere. Overall, these non-dimensional parameters are widely used by aerospace engineers performing wind tunnel tests. A full size aircraft can rarely fit in a wind tunnel, so it is often necessary to build a smaller scale model. Expressing experimental results in terms of non-dimensional parameters allows engineers to compare scale models to full size aircraft.

Experimental Procedure You will now examine Stokes’ flow around a sphere by measuring the terminal velocity of plastic spheres into a highly viscous fluid known as glycerol. You will use two different types of plastic— Teflon (PTFE) and Delrin—which both have different densities ρS. CAUTION: Lab coats and nitrile gloves are required for this lab exercise. If you spill the tube full of glycerol, ask the TA or lab instructor for assistance cleaning it up. 1. Sketch a schematic of the experiment in your lab notebook. 2. The spheres are sorted by size in a small, compartmented box. The Teflon (PTFE) spheres are in a clear box, and the Delrin spheres are in a black box. Choose which material you would like to use first.

B8 – Fluids 4 Last Revision: 3/20/18

University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

3. Make a table in your lab notebook with three columns: sphere diameter D, distance Δx, and time Δt. Note whether the table is for Teflon or Delrin. 4. Starting with the smallest size, use the tweezers to remove a sphere from the box. Do NOT mix up the spheres by putting them in the wrong compartments. 5. Using the tweezers, slowly submerge the sphere just beneath the surface of the glycerol and let it go. Try not to let an air bubbles become attached to the sphere. Do not worry about removing them from the tube when you are done; the lab instructor will do that at the end of the week. 6. Using the stopwatch, measure the time Δt it takes the sphere to travel a distance Δx down the glass tube. Record the time Δt and distance Δx between the two rings in the table in your lab notebook. 7. Repeat this for each size sphere.

8. Calculate the terminal velocity vT = Δx/ Δt in units of m/s for all the different size spheres that you measured. 9. Leave the spheres at the bottom of the glycerol tube. The lab instructor will remove them at the end of the week. 10. Close the compartmented box, so the spheres do not spill out. 11. Repeat the experiment for the other type of material. When you are finished, you should have data collected for both Teflon and Delrin.

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University of Notre Dame Aerospace and Mechanical Engineering AME 21217: Lab II Spring 2018

Data Analysis and Deliverables Using LaTeX or MS Word, make the following items and give them concise, intelligent captions. Make sure the axes are clearly labeled with units. Plots with multiple data sets on them should have a legend. Additionally, write 1 – 3 paragraphs separate from the caption describing the plots/tables. Any relevant equations should go in these paragraphs. 1. For Part I, make a plot of the measured kinematic viscosity ν (in units of centiStoke) as a function of the temperature (in degrees Kelvin). 2. For Part I, make an “Arrhenius plot” of your measured data: On the y-axis, put the natural log of the measured kinematic viscosity, ln(ν). On the x-axis, put the inverse temperature, 1/T. Label the y-axis “ln(ν [cSt])” and the x-axis “1/T (K-1)”. Add a linear trend line. 3. For Part II, make a plot of the terminal velocity as a function of sphere diameter for both materials. Plot the theoretical curve given by Eq. (5) on top of your data. You will need to look up the relevant material properties, such as the densities ρL and ρS and the dynamic viscosity η. Be sure to include references for these values. 4. For Part II, use your measured data to calculate the Reynolds number and drag force coefficient for each size of sphere. (Note that the drag force Fd = Fg during terminal velocity, so you can use Eq. (3) to calculate Fd.) Plot the log of the drag force coefficient, log(Cd), as a function of the log of the Reynolds number, log(Re). Add a linear trend line. Include a legend to distinguish the Teflon and Delrin data points.

Talking Points – Discuss these in your paragraphs. • Look up the viscosity of 10W-30 and compare to your measured values. • Does your data obey the Arrhenius equation? (Is the data on your Arrhenius plot linear?) • Do your measured terminal velocities agree with Stokes’ law? Explain why they might diverge from one another. • How does your measured drag force coefficient scale with the Reynolds number? Is it what you would expect for Stokes’ flow?

B8 – Fluids 6 Last Revision: 3/20/18

University of Notre Dame Aerospace and Mechanical Engineering AME 21216: Lab I Fall 2017

Appendix A

Equipment Part I • 10W-30 engine oil • Absorbent padding • Zahn cup (Cole-Parmer EW-08700-61) • Hot plates • 500 mL beakers • Ice bucket • 1/16” thermocouples • HH806AU digital thermocouple readout • Stopwatch • Shop towels • Nitrile gloves • Lab Coats (2 per setup) Part II • Glycerol • Newsprint table covering • Glass tubes with cm markings on side • Compartmented boxes with plastic spheres • Tweezers • Stopwatch • Shop towels • Nitrile gloves • Lab coats (2 per setup)

B8 – Fluids 7 Last Revision: 3/20/18