Viscosity Measurement of Water and SAE 30 Motor Oil

Abstract

This lab consisted of testing water and SAE 30 motor oil samples for viscosity measurement by subjecting them to pressure to measure time differences that would ultimately be used to calculate the viscosity of each fluid. The tests were carried out over a constant temperature in typical modified ostwald viscometers at different temperature data points of interest. A kinematic viscosity bath was used to quantify the relationship between viscosity, time and temperature.

Introduction

Any lab conducted must be representative of practical uses.

Fortunately, the study of fluid mechanics is a practical one as it has real life applications from the straw we use to drink, to the way we get the energy to power our cars and homes. For starters, the definition of a fluid is “a substance that deforms continuously when acted on by a shearing stress of any magnitude” where a shearing stress “is created whenever a tangential force acts on a surface. Fluid mechanics is not concerned with fluids at the molecular level, but rather it is more interested in the characterization of the behavior to later create applications for the study.

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In this particular characterization, viscosity allows to study the “fluidity” of a fluid, meaning that viscosity is a measure of a “fluid’s resistance to flow ''.

This lab serves to illustrate this concept using a capillary tube or a small diameter tube where “a height difference H between two fluid surfaces forces the fluid through the capillary tube under the action of gravity” the steady flow through the small tube is known as Poiseuille Flow.

[3] As a result, a parabolic “velocity profile” is created (known as the no-slip boundary condition as fluid next to walls remains motionless) and the resulting reduction in velocity down the fluid wall indicates the wall us resisting fluid flow.

It will soon become apparent that a higher viscosity means more resistance to flow, while the opposite (lower viscosity is lower resistance to flow) is also true. A measure that causes the resistance is the shear stress that occurs between the fluid and the wall. Shear stress is given below in Equation 1. | τ w = μ dr du r=d/2 Eq.(1) Where is the shear stress on the wall, is the dynamic viscosity or just the viscosity of the fluid τ w μ which is dependent on the fluid which also depends on temperature as we will see later, and is the rate dr du of shearing strain (or the velocity gradient). Figure 1 displays the force relationships that with further derivations, yield equation 2. U is the average speed of the fluid flowing through the tube. U = 32μL ρ g d H* * 2 * Eq.(2) 2 Where is the fluid density, g is the acceleration of gravity, d is the diameter of the capillary tube, H is ρ the height difference that forces the fluid through the capillary tube due to gravity, is the dynamic μ viscosity of the fluid, and L is the length of the capillary tube.

From inspection, Eq.(2) demonstrates that if for an increase in viscosity, the velocity U decreases as the viscosity parameter resides in the denominator. For simplicity, the fluid before and after the capillary tube is assumed to be moving slowly such that pressure difference can be found from hydrostatic inspection. Figure 1 shows a depiction of what was discussed above, from the design of the capillary tube to the internal breakdown of what is occuring to the fluid as it flows through the capillary tube.

Depiction of the Capillary Tube Viscometer and FBD of the Fluid in the Tube [3] While Eq.(2) serves as a way of measuring viscosity using the capillary tube as the measuring device, further derivations yield a relationship with the volume of the fluid ​V ​in Equation 3. U = V /td 4π 4 Eq.(3) Where ​U continues to be the average flow speed, ​V is the volume of the fluid over time ​t​, and ​d is the tube diameter. By combining Eq.(2) and Eq.(3) to solve for the dynamic viscosity equation 4 is generated below.

Experimental Setup

For practical purposes, a Modified Ostwald capillary-tube viscometer was utilized to measure the viscosities of both water and SAE 30 motor oil, depicted in Figure 2. Figure 2. Modified Ostwald Viscometer Using derivations again, simple relationships that are applicable to this lab are displayed below in Equation 5, which will be used to calculate the viscosities from the data collected. v = C * t Eq.(5) Where ​t is the time is defined by how long it takes for the fluid to flow through E to F. C is the calibration constant as given by “the geometry of the viscometer and the volume of the fluid in the viscometer”. [3] Table 1 summarizes the different types of viscometers and their associated dimensions and properties.

Table 1 - Viscometer Data

No. C (mm​2​/s​2​) range (mm​2​/s)μ t​min​(s) d​ (mm)

25 0.002 0.5-2 250 0.30

100 0.015 3-15 200 0.63

200 0.1 20 - 100 200 1.01

300 0.25 50 - 250 200 1.27

To put the viscosity relationships to test, the measurements have to occur over a constant temperature. So to do this, each test species was then put into a Ostwald Viscometer (figure 2) that would be then placed in a KV.

The diagram in figure 3 describes the experimental setup used. In the experiment, the temperature was controlled and trials were conducted over three intervals of different temperatures. In order to conduct this experiment, the bath was filled with enough water to cover the viscometers. The 300 viscometer was used for the SAE 30W motor oil while the 25 viscometer was used for the water.

A rubber manual pump was also used to correct the volume of both by applying pressure to the left side of the viscometer so that the fluid could reach the fill point and eventually utilize that to start the timings at Mark E. Once the bath and viscometers with the test species were ready, the lamp was turned on to see the motion of the fluids to conduct the experiment. The set point was set to 30ºC and almost like an oven, once the temperature in the temperature controls display reached the starting point, there was a small waiting period for the viscometers to reach equilibrium.

The actual bath temperature ended up being 30.1ºC for the first round of trials 1 and 2 for both the water and the SAE30W oil. Before beginning the timings, the volume was 5 adjusted as described earlier, and as soon as the fluid reached Mark E, using the built-in timers, the measurements began. Once the fluid reached Mark F, the timings stopped.

Then, using Table 1, the timings were verified to see if the times met the published standard. These measurements were repeated twice, two trials for one temperature. So each trial generated two data points at the same temperature: one for water and another for the motor oil. In total, 6 data points were generated for each fluid, resulting in 12 total results to be analyzed in the next section. The temperatures analyzed were 30ºC, 45ºC, and 55ºC.

Results and Discussion

For the section below it is important to note that the viscometer containing water is 25 so the C calibration constant that applies is 0.002 mm​2​/s​2​and for the viscometer containing SAE 30W motor oil is 300 so the C calibration constant that applies is 0.25 mm​2​/s​2​. Calculations were carried out using Eq.(5) with the corresponding C values. Calculations are displayed in the Appendix A.1.

Table 2. Water Temperature, Time, Viscosity

# Temperature (ºC) Time (s) Viscosity (mm​2​/s)

1 30.1 414.58 0.82916

2 30.1 414.58 0.82916

1 44.9 316.63 0.63326

2 45 316.62 0.63324

1 54.7 268.86 0.53772

2 55 271.47 0.54294

6

Table 3. Motor Oil Temperature, Time, Viscosity

# Temperature (ºC) Time (s) Viscosity (mm​2​/s)

1 30.1 581.58 145.395

2 30.1 588.12 147.03

1 44.9 288.94 72.235

2 45 288.28 72.07

1 54.7 194.43 48.6075

2 55 198.10 49.525

The purpose of the tables above is to define a relationship between viscosity and temperature of different substances. Main sources of error for the viscometer measurements are the observation factors that deal with the timing hand-eye coordination errors, angular misalignment, and temperature regulation pitfalls which can lead to 10-20% deviations in the accuracy and precisions of the final viscosity calculation (Question 1).

The limits of hand-eye coordination can impact the way that the timings start and end, with even a couple seconds of delay, the viscosity varies by that delay. It would need to be much more than just two trials to generate a more accurate landscape to see where the exact errors were and to determine trends in the data. Another source of error is the angular alignment of the viscometer. Data shows [4] that any capillary viscometer can experience errors if the” tilt is just slightly off by more than 10 degrees”.

Although tilts of less than 10 degrees allows for “a precision of less than 2%” which is generally accepted, the effects of a slight tilt in the viscometer can propagate. There is no clear way of telling whether the viscometer from trial 1 was placed at the same angle as for trial 2, and this applies for all of the measurements. And finally, another major source of error can inherently lie in the temperature of the bath.

The contraption that regulates and measures the temperature uses machine arithmetic to compute such values. The calibrations in the Appendix A.2 indicate that the contraption utilizes calibration calculations to generate the measurement. Since measurements use floating point numbers, it can be 7 inferred that even the measurements for temperature can be off, and this can create an error outside of the accepted normal values. For example, the last trial for motor oil is not accepted, as for the temperature measured by the device does not yield the expected results for that temperature, leading to a percentage loss in accuracy for the viscosities calculated. Immediately, due to the trends in both of the data, it can be concluded that the general relationship between viscosity and temperature is that as temperature increases, viscosity decreases.

This relationship exposes an interesting phenomena. Of the two test specimens, SAE30W motor oil appears to be the most dependent on temperature as there is a dramatic ~50% drop in viscosity between both of the trials at the 30ºC and 45ºC temperatures (Question 2 [3]). Comparatively, in water, the drop is less dramatic with a ~20% drop between both of the trials at the 30ºC and 45ºC temperatures. It appears that since oil is significantly more viscous than water, this is why it is more dependent on temperature.

Since the data for SAE30W motor oil at 55ºC is not as reliable, as the ranges are not in target and as pointed out by Prof. Shide Bakhtiari, this data was not used to define which viscosity of the fluid was the most dependent on temperature, therefore only the 30ºC and 45ºC temperatures were taken into account to draw comparisons between the fluids. For Question 3, the Kinematic Viscosity was plotted as a function of temperature for both Water and Oil. Figures 4 and 5 depict such plots. Figure 4. Kinematic Viscosity (mm2/s) vs Temperature (ºC) for Water 8 Figure 5. Kinematic Viscosity (mm2/s) vs Temperature (ºC) for SAE30W Oil In Appendix A.3 the published values for Kinematic Viscosity indicate a 1.12 mm​2​/s value for water at 15.6ºC and a 420 mm​2​/s value for SAE30 Oil also at 15.6ºC. Appendix A.4 also shows a plot for several temperatures for several fluids.

Although it is a bit difficult to extrapolate the viscosity values, it approximately appears that for 35ºC indicates a 0.8 mm​2​/s value for water and a 140 mm​2​/s value for SAE30 Oil. These results compare relatively well for the same temperature where the average value was 0.82916 for water and 146.2125 for SAE 30W oil. There appears to be a ~3% error for water and a ~4% error for the SAE 30W oil.

Observed differences may be caused by the main sources of error as discussed above and because the published values are approximate as it is a bit hard to read from Appendix A.4, then these errors are also approximations of the actual error. Section Uncertainty Analysis delves a bit more into error percentages, unfortunately, no errors dropped below 2% but if the tolerance is 10% or less, then the data should pass standards for error requirements.9

Conclusion

In this lab, forces were applied to three different materials in order to determine many material properties. The main properties found were viscosities of water and SAE 30W oil at different temperatures. From the data, it can be concluded that SAE 30 Motor Oil has a higher viscosity than water, and for both fluids an increase in temperature yields a resulting decrease in viscosity. Practically, this lab teaches that properties in nature are not absolute and there can be relationships that aren’t inherently obvious, as the effect temperature might have on viscosity. As an engineer, these considerations are important to understand as fluids are an important factor in design purposes.

References

1. Munson, Young and Okiishi’s Fundamentals of Fluid Mechanics​ 8th Edition, Wiley, 2016.
2. “DEFINITION OF VISCOSITY.” ​Princeton University​, The Trustees of Princeton University, www.princeton.edu/~gasdyn/Research/T-C_Research_Folder/Viscosity_def.html.
3. P. Krueger, ME2142 Fluid Mechanics Laboratory, ​Viscosity Measurement​.
4. Cannon Instrument Company, ​Analysis of Errors in Viscosity Measurement​, page 3