Experiment #2 Fluid Properties: Viscosity Jordan Hines Performed on January 31, 2011 Report due February 7, 2011
Lab Group: Elizabeth Hildebrandt & Anthony Freeman
Table of Contents
Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………….....…p. 1 Procedure …………………………………………………………………………........p. 2 Results ………………………………………………………………...…………….pp. 3-4 Discussion and Conclusion …………………………………………………..……...pp.5-6 Appendix ……………………………………………………..…….………………pp. 7-8
Object The purpose of this experiment was to determine the dynamic viscosity of a given fluid using the falling sphere viscometer method.
Theory A fluid’s dynamic viscosity is defined as its resistance to an externally applied shear force. This definition implies that a fluid will resist any change in form. Due to this, if a solid object is placed into a fluid of lesser density, the object will fall through the fluid, exerting a shear force on the fluid surrounding it. The object will displace fluid as it falls which will cause a buoyant reaction force from the fluid onto the object. During this process, there are three forces present: the weight due to gravity, the buoyant force, and the drag force. The force balance equation of these three forces is shown below: Weight – Buoyant Force – Drag Force = 0 (Equation 1) The force of weight in Equation 1 acts in the downward direction and is dependent upon the acceleration due to gravity, the density of the sphere, and the size of the sphere. The density and size of the sphere determine its mass, and this mass is multiplied by the gravitational acceleration to determine the force of weight. The buoyant force in Equation 1 is caused by the pressure gradient exerted by the fluid on the object. The lateral forces of this pressure are equal and opposite and therefore negate one another. The pressure on the submerged object acting in the vertical direction is lesser on the top than on the bottom thus causing a net upward force on the object. The drag force in Equation 1 is caused by the viscous effects of the fluid as it moves over the surface of the submerged sphere. The force attempts to resist the sphere’s motion through the fluid. As such, it acts in the vertical direction, but opposite to the motion of the sphere. ρsg(4/3)πR3 – ρg(4/3)πR3 - 6πμVR = 0 (Equation 2) Equation 2 represents the same force balance equation as Equation 1 but in less general terms. In this equation, the variables ρs, ρ, g, R, μ, and V represent the density of the sphere being placed into the fluid, the density of the fluid itself, the acceleration due to gravity, the radius of the sphere, the dynamic viscosity of the fluid, and the terminal velocity of the sphere respectively. Equation 2 can be rearranged to solve for the dynamic viscosity of a fluid. This is shown below in Equation 3: μ = [ρg(4/3)πR3 - ρsg(4/3)πR3] / 6πVR (Equation 3)
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Procedure Equipment:
Large, transparent cylinder with affixed scale Suave Naturals Fresh Mountain Strawberry shampoo Digital Calipers Stainless Steel ball bearings iPhone with default stopwatch application
Experiment: 1. Take note of the ambient air temperature. This will be used as the temperature of the 2.
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5. 6.
fluid. Measure the diameter of the stainless steel ball bearing using the digital calipers. Record this measurement onto your data sheet. Also, take note of the density of the material out of which the ball bearings are made. Fill the large, transparent cylinder with shampoo. There should be an attached ruler so that velocity measurements can be taken. The cylinder should be covered at the top by a piece of material with a circular hole at its center having a diameter larger than that of the largest ball bearing to be used. Drop the ball bearing into the hole at the center of the cover of the cylinder. Observe the bearing until it reaches terminal velocity and take note of its position with respect to the ruler at that point; start the time measuring device at this point. Assign a second point a fixed distance from the first point; this will be the stopping point for the time measurement. Record the time taken to travel between the two points. Repeat this step five more times, recording the time value each time. Divide each time measurement by the length between the two points to determine the velocity for each trial. These velocities will be used to determine the drag force. Plug all known information into Equation 3 to solve for the dynamic viscosity of the fluid for each trial. Average these values to determine the dynamic viscosity of the fluid.
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Results Table 2.1. Initial conditions for temperature, density, and vertical distance traveled by sphere. Stainless Steel Ball Bearing Density
ρs = 8000 kg/m3
Shampoo Density Displacement on Scale
ρ = 1014 kg/m3 z = 0.3048 m
Ambient Lab Temperature
T = 23 oC
All initial conditions listed in Table 2.1 were measured except for the density of the ball bearings. This value for density was obtained from Fundamentals of Material Science and Engineering: An Integrated Approach by Callister and Rethwish. The density of the shampoo was measured by another group in the laboratory using a hydrometer. The displacement distance was found using the scale attached to the cylinder in which the experiment was performed, and the ambient air temperature was found on the electronic thermostat in the laboratory. After all six trials had been performed, the terminal velocity for each trial was obtained using the following equation: V = z/t (Equation 4) Where z is the vertical distance (in meters) which the ball bearing travelled through the shampoo, and t is the time (in seconds) measured during which the ball bearing travelled between the two points. The calculated values can be found in Table 2.2 below. Table 2.2. Measured results from falling sphere viscometer. Sphere Trial Time (sec.) Diameter (m) Velocity (m/s) 1 2
9.3 9.6
0.00950 0.00948
0.0328 0.0318
3 4 5 6
9.6 13.2 12.9 20.4
0.00948 0.00784 0.00789 0.00633
0.0318 0.0231 0.0236 0.0149
After obtaining the values for the terminal velocity for each trial, these values were used (along with Equation 3) to calculate the value of the dynamic viscosity for each trial. These values of viscosity can be found in Table 2.3.
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Table 2.3. Calculated dynamic viscosity for each trial Trial 1 2 3 4 5 6
Viscosity (N∙s/m2) 10.484 10.777 10.777 10.135 10.031 10.211
Once the viscosity had been calculated for each trial, the Reynolds number of each trial must be calculated in order to determine the validity of the calculations. Equation 1 above is only valid for Reynolds numbers less than 1. The Reynolds number for each trial is shown in Table 2.4 below: Figure 2.4. Reynolds numbers of individual time trials Trial Reynolds Number 1 0.03011 2 0.02832 3 0.02832 4 0.01811 5 0.01884 6 0.00939 As can be seen in the table above, the Reynolds number for the individual trials are all below 1; thus, the equation and resulting calculations are valid. With this, the average of all the viscosity values was taken to find the actual dynamic viscosity of the shampoo. The result is shown below: μAVG = 10.402 N∙s/m2
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Discussion and Conclusion
Terminal Velocity in meters per second
The terminal velocity of two spheres having different diameters would not be identical. Assuming that both spheres are made of the same material, the difference in diameter would cause a change in the geometry such that the weight of the spheres would be significantly different. This change in size and weight would change all three forces listed in Equation 1 above. The weight would increase proportionately to the buoyancy force due to the fact that the buoyancy force is directly related to the mass of fluid which is being displaced. The change in size would also cause a change in drag force due to the increase in surface area of the sphere. This increase in drag force would not be enough to counteract the relatively large increase in the net downward force created by the weight minus the buoyancy force. Due to this, the terminal velocity of the larger sphere would be significantly higher than that of the smaller sphere. This can be seen in Figure 2.1 below. (1&2) 0.0350 Actual Data 0.0300
Poly. (Actual Data)
0.0250 0.0200 0.0150 0.0100 0.00600 0.00650 0.00700 0.00750 0.00800 0.00850 0.00900 0.00950 0.01000 Sphere Diameter in meters
Figure 2.1. Terminal velocity as a function of sphere. The viscosity found for two different size spheres should, theoretically, be the same due to the fact that viscosity is a property of the fluid through which the sphere is moving and not a property of the sphere itself. (3) The largest shortcoming for this experiment is human error. Because the starting and stopping of the timing device is performed by a human, it is expected that some error will be present. Also, the distance across which the sphere is travelling is measured by the human eye, which accounts for more error. Also, in this experiment, the assumption is made that both the fluid through which the sphere is travelling and the sphere itself are perfectly homogenous materials with uniform densities. This is not likely to be the case due to the fact that all of the bearings and all of the shampoo were not likely to have come from one batch of materials which would allow for less variation in material properties. Further, the density of the shampoo was obtained through experiment which allows for further human error. Finally, less error could have been achieved by limiting variables within the experiment. By this, I mean to say that a lower degree of error likely would have been achieved had we used the same ball bearing for 5
each trial. This would have limited variables in the experiment and allowed us to average the time for each trial and use that value of average time to calculate a terminal velocity and then a single viscosity. (4) The temperature of the ambient air should be recorded because the fluid is likely to be at the same temperature given the amount of time that the fluid has been surrounded by air at the same temperature. The temperature of the fluid is the significant measurement because the density of the fluid is directly related to its temperature. If the temperature were greater, the fluid would be less dense; conversely, if the temperature were lower, the opposite would be true. (5) This method could be used in gasses, however, there would likely be a much larger degree of error due to the speed at which the sphere would fall and the ability of a human to react quickly enough to measure such an occurrence. If some form of automation was used to measure the time in which the sphere moved between two given points in a gas, this experiment could likely be performed successfully in such an environment. (6) This method could be successfully completed in opaque fluids, however, it would require methods other that human observation due to a human’s inability to see in order to record the time the sphere takes to pass from the starting point to the stopping point. A method such as thermal imaging could be used to accomplish such an experiment. This method would not work, however, with nonhomogeneous fluids such as those mentioned. (7 & 8)
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Appendix Data Usage The following is the calculation used to obtain the terminal velocity of the falling sphere: V = z/t V = 0.3048 m / 9.6 s V = 3.175 x 10-2 m/s The following calculation is used to obtain the dynamic viscosity of the shampoo: μ = [ρg(4/3)πR3 - ρsg(4/3)πR3] / 6πVR μ = {[(1014 kg/m )(9.81m/s2)(4/3)(0.00475 m)3] – [(8000 kg/m3)(9.81m/s2)(4/3) (0.00475 m)3]}/[6π(0.0328 m/s)(0.00475 m)] μ = 10.484 N∙s/m2 3
The following calculation is used to obtain the Reynolds number for each trial: Re = ρVD/ μ Re = (1014 kg/m3)(0.0328 m/s)(0.00950 m)/(10.484 N∙s/m2) Re = 0.03011
Bibliography
Introduction to Fluid Mechanics, 3rd Edition William S. Janna (1993) A Manual for the Mechanics of Fluids Laboratory William S. Janna (2008) Fundamentals of Material Science and Engineering: An Integrated Approach W.D. Callister, Jr and D.G. Rethwish (2008)
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