MEC511_Lab1

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Toronto Metropolitan University *

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Mechanical Engineering

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Feb 20, 2024

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Course Title: Fluids & Thermodynamics Course Number: MEC 511 Semester/Year: Fall 2023 Instructor: Dr. Jun Cao Assignment/Lab Number: 1 Assignment/Lab Title: Measurement of Dynamic Viscosity Submission Date: Oct 6, 2023 Due Date: Oct 6, 2023 LAST NAME FIRST NAME Student Number Section Signature Table of Contents Summary: 3

Introduction: 4 Apparatus: 5 Procedure: 5 Results: 6 Discussion: 7 Conclusion: 8 Appendices: 8 References: 8 Summary :

In this lab, we delve into the concept of viscosity through conducting an experiment which involves three different-sized metallic spheres submerged in engine oil. By observing the terminal velocities of these spheres as they move at a consistent speed through the oil, we can determine the dynamic viscosity of the fluid. Next, the following equation was used: Given that we know the density and diameter of each sphere, the fluid's density, and the terminal velocities of the spheres as they move within the oil, we can compute the dynamic viscosity of the fluid for each individual metal sphere. Calculation of dynamic viscosity: small = 0.46829 µ ?? ? ? medium = 0.55320 µ ?? ? ? big = 0.49609 µ ?? ? ? Calculation of Reynolds number R Small = 0.1379 R medium = 0.336 R big = 0.889

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Introduction: The purpose of this lab was to determine the dynamic viscosity of engine oil by measuring the terminal velocity of small spheres falling through the oil. This was accomplished through measuring the steady velocity of spheres of three different sizes dropped in motor oil using a stopwatch. Newtonian Fluids are fluids of which their velocity gradient is directly proportional with the shearing stress. This can be expressed by the following equation: Eq 1 τ = µ ?? ?? Where τ is the shear stress, is the dynamic viscosity and is the velocity gradient. µ ?? ?? The force body diagram of the sphere falling through the oil is: Figure 1: Force body diagram of sphere in oil Since the sphere has reached its terminal velocity the sum of forces is equal to 0. This yields the following equation: ? ? = ? ? + ? 𝐵 Eq 2 Where is the weight of the Sphere is the buoyancy force and is the drag force. ? ? ? 𝐵 ? ? Expanding on Eq 2: Eq 3 ? ? = 3πµ?? Where U is the velocity of the sphere and D is its diameter. Eq 4 ? 𝐵 = ρ ? ?? = ρ ? ? π? 3 6

Where is the density of the fluid, g is the acceleration due to gravity and V is the volume of ρ ? the sphere. Eq 5 ? ? = ?? = ρ ? ?? = ρ ? ? π? 3 6 Where is the density of the sphere. ρ ? Substituting Equation 3, 4, and 5 in 6 and making the subject of the equation yields: µ Eq 6 µ = ? 2 ?( ρ ? − ρ ? ) 18? Reynolds number is used to check for accuracy: Eq 7 ρ ? ?? µ < 1 Apparatus: The apparatus for this experiment is shown below: ● Tall graduated cylinder, filled with oil ● Three spheres of various sizes ● Hydrometer for measuring the specific gravity of the oil ● Stopwatch and metre stick to measure the steady velocity of the sphere ● Weight scale to determine the mass of each sphere ● Micrometre to determine the diameter of each sphere ● Thermometre to measure the oil temperature ● Fluorescent light to make spheres more visible during the experiment Procedure: 1. A tall measuring cylinder was filled with engine oil 2. Room temperature was measured with a thermometer 3. Specific gravity was measured with a hydrometer 4. 10 nylon spheres of same size were weighed using a scale 5. Total mass was divided by 10 to get mass of one sphere 6. Using a micrometre, the diameter of a sphere was measured 7. A sphere was dropped into the centre of the measuring cylinder 8. Steps 4 to 7 were repeated for a total of three different sized spheres Note that the spheres were dropped from rest and without spin to minimise errors.

Results: Measured Temperature: 23°C. Measured Specific Gravity of Oil: 0.89. Table 1) Calculated Parameters Small Sphere Medium Sphere Large Sphere Sphere Density ( ) ?? ? 3 0.152 × 10 −3 ?? 4 3 π(3.175× 10 −3 ?) 3 = 1133.7669 ?? ? 3 0.512× 10 −3 ?? 4 3 π(4.755× 10 −3 ?) 3 = 1136.9197 ?? ? 3 1.180× 10 −3 ?? 4 3 π(6.325× 10 −3 ?) 3 = 1113.2990 ?? ? 3 Oil Density ( ) ?? ? 3 890 ?? ? 3 890 ?? ? 3 890 ?? ? 3 Velocity ) ( ? ? 20 × 10 −2 ? 17.5? =0.011428 ? ? 20× 10 −2 ? 9.1? = 0.021978 ? ? 20× 10 −2 ? 5.1? = 0.039215 ? ? Dynamic Viscosity ( ) ?? ? ? = (9.8 ? ? 2 ) (6.35 × 10 −3 ?) 2 (1133.77 ?? ? 3 − 890 ?? ? 3 (18*0.011428 ? ? ) =0.46829 ?? ? ? = (9.8 ? ? 2 ) (9.51 × 10 −3 ?) 2 (1136.92 ?? ? 3 − 890 ?? ? 3 (18*0.021978 ? ? ) =0.55320 ?? ? ? = (9.8 ? ? 2 ) (12.65 × 10 −3 ?) 2 (1113.299 ?? ? 3 − 890 ?? ? 3 ) (18*0.039215 ? ? ) =0.49609 ?? ? ? Reynolds Number = (0.011428 ? ? )(890 ?? ? 3 )(6.35 × 10 −3 ?) 0.46829 ?? ? ? = 0.1379 = (0.021978 ? ? )(890 ?? ? 3 )(9.51 × 10 −3 ?) 0.55320 ?? ? ? = 0.336 = (0.039215 ? ? )(890 ?? ? 3 )(12.65 × 10 −3 ?) 0.49609 ?? ? ? = 0.889 Re < 1? ✔ ✔ ✔ Table 2) Percent Error of Calculated Dynamic Viscosity Small Sphere Medium Sphere Large Sphere Calculated Value 0.46829 0.55320 0.49609

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Accepted Value 0.41501 0.41501 0.41501 Percent Error (%) 12.84 33.3 19.54 The density of the sphere, p s , was obtained by dividing the spheres’ mass by its volume. ρ ? = 𝑀𝑎?? ?????? To calculate the density of the oil, p f , the specific weight of the oil (0.89) and the density of water (1000 kg/m 3 ) must be known, and can be expressed by the equation below: Where ‘density of liquid’ is , ρ ?,?𝑖? ????𝑖?𝑖? ??𝑎?𝑖?? = ????𝑖?? ?? ?𝑖??𝑖? ????𝑖?? ?? ?𝑎??? Where ‘density of liquid’ is , ρ ?,?𝑖? ρ ?,?𝑖? = 0. 89 * 1000 = 890 ??/? 3 The speed of the ball in the oil was determined by the distance travelled divided by the time with the following formula: ? = ?𝑖??𝑎??? ?𝑖?? = ? ? In addition to the dynamic viscosity, the Reynolds number and the “Slow Flow” criteria were determined for each sphere size. Where if the Reynolds number is less than one the Slow Flow criteria would be met. ?????? ???? = ρ ?,?𝑖? (?)(?) µ The dynamic viscosity of the engine oil was calculated using the equation below. µ = ? 2 (ρ ? −ρ ? )? 18?

It was calculated once using the values obtained through the lab, and then the theoretical value was calculated using the values obtained from a datasheet for Quaker State® Hi-Performance Gearplus 80W-90 GL-5 [2]. The percent error could then be calculated using the equation below: % ????? = ?????𝑖????𝑎? − ?ℎ?????𝑖?𝑎? ?ℎ?????𝑖?𝑎? | | | | × 100% Discussion: Did the data for each sphere yield the same dynamic viscosity? If not, why? The data for each sphere did not yield the same dynamic viscosity. The reason for this is that each sphere had a different density and diameter. The dynamic viscosity is proportional to the density and the square of the diameter of the sphere. However, as shown in the calculations, the largest sphere’s density shows to be less than the medium and small spheres, while still maintaining the highest velocity of the three spheres, leading to its dynamic viscosity not following the pattern of the other two spheres. What size of sphere likely gave the most accurate result? The smallest sphere likely gave the most accurate result as can be seen with it having the lowest percentage error of 12.84%. This is due to the fact that it was used first in the experiment and the oil was in a steady state when it was dropped. The other spheres were dropped quickly after, which meant the oil likely did not have enough time to recover from the shearing stress of the previous spheres, resulting in less accurate results for the two larger spheres. Which is seen in the percentage error of 33.3% for the medium sphere and 19.54% for the large sphere. This discrepancy in values between the medium and large spheres can be explained also with external forces and conditions that are introduced when the sphere properties begin to change. How does your viscosity measurement compare with the value in property tables? Give possible reasons for any differences observed. The results had a progressively larger percentage error as the size of the sphere increased, and this is likely due to the reason stated above. Other factors include human error when timing the spheres as they travel through the cylinder. Also, ideal conditions can never be simulated perfectly in a laboratory setting and in this case, the experiment was performed in a cylinder with a limited amount of liquid rather than the ideal case of having infinite fluid. Additionally, the

large sphere produces results that do not follow the trend with the medium or small sphere because the sphere’s properties (i.e, velocity and density) do not follow the pattern, to the point where other forces now have a more noticeable impact in the system, such as turbulence flow in the liquid. Thus, the expected value of the velocity of the large sphere is larger than what is observed. Conclusion: The dynamic viscosity of engine oil was found using three differently sized spheres and dropping them from the same height. The smallest sphere produced the most accurate results and displayed the least percent error difference, with an accuracy of 12.84%. The small and medium spheres follow the rule which states that dynamic viscosity can be accurately measured when the Reynolds number is less than 1. As the Reynolds number increases from the small to the medium spheres, we notice the error discrepancy increasing likewise. Now, although the large sphere has the highest Reynolds number, it is also less than 1. However, since its dynamic viscosity is lower than that of the medium sphere, it shows to have a smaller percentage error, thus breaking the pattern. Appendices: Table 3) Sphere Properties Sphere Diameter [m] Mass [kg] Velocity, U [m/s] Volume [m 3 ] Density [kg/m 3 ] Viscosity [(N*s)/m 2 ] Small 0.00635 0.00015 0.01142 1.34x10 -7 1133.77 0.46829 Medium 0.00951 0.0005 0.02197 4.50x10 -7 1136.92 0.55320 Large 0.01265 0.00118 0.03921 1.059x10 -6 1113.29 0.49609 Table 4) Reynolds Number for Spheres Sphere Reynolds Number Re < 1 Small 0.1379 ✔ Medium 0.336 ✔ Large 0.889 ✔

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References: 1. D. F. Young, T. H. Okiishi, J. I. Hochstein, A. L. Gerhart, and B. R. Munson, Young, Munson and Okiishi's a brief introduction to Fluid Mechanics . Wiley, 2021. 2. “Quaker State hi-performance gearplus 80W-90 GL-5 - shell-livedocs.com,” Quaker State® Hi-Performance Gearplus 80W-90 GL-5 . [Online]. Available: http://www.shell-livedocs.com/data/published/en/8f3184be-5a64-4ed4-a789-0f4a78dc3ac8.pdf..