MEMORANDUM. To: Dr. Andrew Kean, American Petroleum Institute (API) From: Josh Baida Abel Estrada

Transcription

1 MEMORANDUM To: Dr. Andrew Kean, American Petroleum Institute (API) From: Josh Baida Abel Estrada Cesia Cazares Date: October 27, 2015 RE: Viscosity Measurement Method Development Chris Rini Abstract: You have requested the design of a quick and easy-to-use method for measuring the viscosity of an unknown liquid in the field, specifically Newtonian liquids with viscosities between that of water and glycerin (between about 1 cp and 1,000 cp at room temperature). In order to mitigate potential litigation, derivations from the empirical relations that the method is based on are included. Additionally, uncertainty analysis was performed for the developed method to yield statistical uncertainty of the final measurement. The viscometer designed uses a graduated cylinder filled with the liquid whose viscosity is to be found, into which a stainless steel sphere is dropped. Video of this drop is used to experimentally determine the terminal velocity, which was then used to determine the dynamic viscosity of the fluid. Strengths and Weaknesses of the Design: Viscosity is innately a very difficult property to measure. The fact that it varies with temperature makes its measurement even more challenging. That being said, this method is good for providing a rough approximation of the dynamic viscosity of an unknown Newtonian fluid. Some advantages of this method are that it is very simple and inexpensive, especially when compared to other methods available on the market. It is very likely that your company already has all the tools required to implement this method. It can be set up quickly and in a wide range of temperature and pressure environments. The tools involved are durable and not likely to break during transport. However, this method does have its drawbacks. One of these drawbacks is that the less viscous the fluid is, the less accurate the results are. This is due to the wait time required for the falling ball to reach terminal velocity. This can take longer in less viscous fluids, meaning that a longer tube would be necessary to allow the ball to reach terminal velocity. Not having the ball reach terminal velocity greatly decreases the accuracy of the results. In addition, with less viscous fluids, the camera may not have enough frames per second to capture the exact time when the ball passes a marker.

2 Viscometer Design Specifications: The method developed uses a 250 ml graduated cylinder filled with the given fluid and a stainless steel ball bearing with a mass of g, a diameter of mm, and a density of 7604 kg/m 3. A stopwatch and iphone 5s or above (or other similarly capable high-speed camera) are also required for recording and reviewing data. Screenshots of the computation tools (#1 and #2 created with the EES program, #3 and #4 created with Excel) are included in the appendix. The actual coded programs will be delivered to your company once full payment is received. These programs implement the equations that are derived in the appendices and are used as part of the process of finding the dynamic viscosity. Viscometer Usage Instructions: The user will obtain the density of the fluid whose viscosity they wish to measure. The density of the fluid is easily obtained by taring the graduated cylinder, putting some of the fluid into the graduated cylinder, and obtaining the weight of the fluid (density = mass/volume). Note: If the user is using a different sphere than the one described in the Viscometer Design Specification section, then they will need to measure its properties as well. The person measuring then enters the density of the fluid, (and the mass of the sphere, and the diameter of the sphere if using a different sphere) into computation tool #1. (Computation tools #1 and #2 are integrated into the same EES file.)this will output the theoretical terminal velocity and the theoretical distance it takes for the ball to reach terminal velocity. However, an experimental value of terminal velocity was desired, so this theoretical distance to reach terminal velocity is merely used to ensure that terminal velocity is experimental measure in a range for which it is actually at a constant velocity. The user will then fill the graduated cylinder with the fluid and set up the camera and timer. Then they will start the timer, turn on the camera, and drop the sphere into the filled graduated cylinder. Using the slow motion feature of the camera, they will determine the some distance that the ball falls and the time it takes to fall that distance. Any distance is fine as long as it occurs after the distance to reach terminal velocity (output of computation tool #1).The user must be sure to only measure the time of the falling ball after it has fallen at least this distance through the fluid. The user will then enter the measured distance and time (which is used to find a measured terminal velocity), the density of the fluid, and the density of the sphere into computation tool #2. The second computation tool outputs the experimentally determined dynamic viscosity of the fluid (see Methodology section for an explanation of how it does this). The third computation tool outputs the total resolution uncertainty, It s required inputs are the individual resolution uncertainties of each measuring tool. These include the tools used to measure the time of the fall, distance of the the fall, and the diameter of the sphere. Computation tool #4 gives the statistical uncertainty with a 99% confidence interval (This indicates a 99% confidence that all such measured values will fall in this range, not 99% confidence that the theoretical value will fall in this range). The required inputs are the viscosities of the fluid from each of two trial runs (which come from the output of the first tool). The overall uncertainty is then calculated using equation (8).

3 Theory and Methodology: From analysis of free body diagrams and mass acceleration diagrams, symbolic equations for the terminal velocity of a sphere in a fluid (1) and the dynamic viscosity of that fluid (2) were determined. The terminal velocity equation (1) was then entered into Engineering Equation Solver (EES) to develop a tool wherein the density of the fluid, the density of the ball, the diameter of the ball, and the mass of the ball could be entered and the tool would output the terminal velocity of the ball in the fluid. A stainless steel ball bearing was used, which can be modeled as a smooth sphere. Knowing this, a graph of drag coefficient versus Reynolds number can be used to find the drag coefficient. However, the viscosity depends on the drag coefficient, and the drag coefficient depends on the Reynolds number. Thus, an initial value for the drag coefficient had to be assumed. This initial value was chosen based on the Reynolds number used in other similar experiments and in tables that listed drag coefficients for a sphere. Using this assumed initial value, the viscosity was calculated, which was then used to find the actual Reynolds number. This actual Reynolds number was then used to determine whether the initially assumed drag coefficient matched with the calculated Reynolds number. Figure 1: The effect of surface roughness on the drag coefficient of a sphere in the Reynolds number range for which the laminar boundary layer becomes turbulent Since the initially assumed drag coefficient was too low, a slightly larger drag coefficient was assumed and the above process was iterated until the Reynolds number matched with the initially assumed drag coefficient, meaning that the correct terminal velocity had been found.

4 This theoretically determined terminal velocity was then used in Newtonian equations of motion to find equations for the time and distance necessary to reach terminal velocity in the given fluid (equations 3 and 4 respectively), and thus give the range of the fall wherein velocity was constant. The viscosity equation (2) was then entered into EES to develop a tool that would output the viscosity given the density of the fluid, the density of the ball, the diameter of the ball, and the terminal velocity. And since the terminal velocity was now known via the above described process, this tool could be used to find the viscosity of the fluid. Calibration Data: Because the preliminary experimental data was recorded at about 21 degrees Celsius, a temperature correction factor was developed in order to convert it to 25 degrees Celsius in centipoise (cp). In order to account for interference from boundary layer development and possible creation of turbulence, a correction factor for the diameter of the cylinder compared to the diameter of the ball was used. The equation for the correction factor is as follows: visc_correct = visc_fluid ( Dia/D (Dia/D) (Dia/D) 5 ) (5) where visc_correct, is the corrected viscosity, dia is the diameter of the sphere, and D is the internal diameter of the cylinder. For small diameter fluid columns, there is an interaction between the fluid and the wall of the cylinder, for this reason a consideration for the interaction with the wall is taken into account. The main error is due to the wall effect, for a sphere falling in a cylinder of finite length, the Faxen correction gives better results. The Faxen correction is derived from Faxen s laws.this equation was used considering that measurements are taken sufficiently, far from the tube extremity, no coefficient relating to the tube extremities will be applied. A possible rotation of the ball will not be taken into consideration. It is assumed that the sphere falls in an infinite medium without inertial effects. Uncertainty Analysis: First, the resolution uncertainties were determined based on the resolution of the measuring instruments. For instruments for which the manufacturer did not specify a resolution uncertainty, half of the most precise decimal place was used. Once all of the resolution uncertainties were determined, the general uncertainty propagation method was used to normalize each resolution uncertainty. This converted them all into dimensionless values that could be added to each other. Having accomplished this, the statistical uncertainty was then considered. The viscometer we have created will be used to measure the viscosity of unknown fluids, which means that nothing will be known (6)

5 about the population of measurements of the viscosity. Therefore, the mean and standard deviation of the sample, and not of the population must be relied upon to get the mean of the whole population. In light of this, we will use the Student-t variable (t) instead of the Normal variable (z) for calculating the statistical uncertainty and use equation (7). We have chosen the confidence interval to be 99% confidence. Since we were not provided with information regarding any calibration uncertainties for the measuring tools, we must assume that it is negligible. Finally, we must combine all of the three types of uncertainties using the root sum squared method shown by equation (8). (The uncertainties are not simply summed because all of the uncertainties are not likely to go worst case at the same time.) (7) Conclusion: This method is simple, fast, and inexpensive. It offers a useful approximation for dynamic viscosity and also gives you the additional information of terminal velocity and the time and distance required to reach terminal velocity. It is less accurate than more complicated, more expensive methods, but for low-cost, quick and easy-to-use approximation method that you requested, this viscometer is an ideal choice. For the fluids of interest, Newtonian fluids with viscosities ranging from that of water to that of glycerine, the range of values predicted by the theory was cp (for water) to 934 cp (for glycerol) at 25 degrees Celsius. Based on this, the best-case score for our measurement technique using the formulas put forth at our initial meeting would be 310 as solved for below. (8)

6 Appendix A: Computation Tools #1 and #2 (They are both integrated into the same EES file) Computation Tool #1 Inputs: density of the fluid If using a different sphere than the one described, also need to change: mass of the sphere diameter of the sphere Outputs: theoretical terminal velocity theoretical time to reach terminal velocity theoretical distance to reach terminal velocity Computation Tool #2 Inputs: measured distance measured time density of the fluid If using a different sphere than the one described, also need to change: density of the sphere Output: dynamic viscosity

7 EES Code Terminal Velocity Calculation D = 1.5E-2 A = pi/4*d^2 rho_fluid = 1260 rho_object = 8000 C_d = 0.5 V = pi/6*d^3 m = rho_object*v g = 9.81 t = 0.2 visc_fluid = u_inf = sqrt(2*m*g/(rho_fluid*c_d*a)) y = u_inf^2/g*ln(cosh(g*t/u_inf)) Re = rho_fluid*u_inf*d/visc_fluid visc_fluid = g*(dia)^2*(rho_object-rho_fluid)/(18*u) u =.21 g = 9.81 Dia = 1.5E-2 rho_object = 7600 rho_fluid = 1325 D = 4E-2 visc_correct = visc_fluid*( *dia/d+2.09*(dia/d)^3-0.95*(dia/d)^5) visc_water = 1 diff = (visc_correct-visc_water)/(visc_correct)*100

8 Glycerin Run "Constants" u =.4 g = 9.81 Dia =4.3E-3 rho_object = 7600 D = 2.5E-2 T = 25 rho_fluid=density(glycerin, T=T) mu=viscosity(glycerin, T=T) "Equations" visc_fluid = g*(dia)^2*(rho_object-rho_fluid)/(18*u) visc_correct = visc_fluid*( *dia/d+2.09*(dia/d)^3-0.95*(dia/d)^5) visc_water = diff = (visc_correct-visc_water)/(visc_correct)*100 diff_nocorrect = (visc_fluid-visc_water)/(visc_water)*100

9 Appendix B: Computation Tool #3 Computation Tool #3 Inputs: the individual resolution uncertainties of each measuring tool the resolution uncertainties of tools used to measure the time of the fall, distance of the the fall, and the diameter of the sphere. Output: total resolution uncertainty

10 Appendix C: Computation Tool #4 Computation Tool #4 Inputs: the viscosities of the fluid from each of two trial runs these are calculated by the output of the first tool t for the chosen confidence interval for 99% confidence, t= Output: the statistical uncertainty with a 99% confidence interval (This indicates a 99% confidence that all such measured values will fall in this range, not 99% confidence that the theoretical value will fall in this range),

11 Appendix D; Trial Runs

12 Appendix E: Measured Times and Distances