Article pubs.acs.org/IECR
Rolling Ball Viscometer Calibration with Gas Over Whole Interest Range of Pressure and Temperature Improves Accuracy of Gas Viscosity Measurement Ehsan Davani,*,† Gioia Falcone,‡ Catalin Teodoriu,‡ and William D. McCain, Jr.† †
Petroleum Engineering Department, TAMU 3116, Texas A&M University, College Station, Texas 77843-3116, United States Institute of Petroleum Engineering, Agricolastrasse 10, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany
‡
ABSTRACT: A practical method for measuring gas viscosity uses gas to calibrate the rolling ball viscometer over the whole interest range of pressure and temperature. It reduces the turbulent effect of gas flow around the ball. The viscometer calibration at only atmospheric pressure yields poor results, because it cannot accommodate the turbulent effect at higher pressures. We used nitrogen to calibrate the viscometer for the pressure up to 8,000 psia and temperature up to 260 °F. Optimal tube inclination angle and also ball/tube diameter ratio ensure the laminar flow around the ball and improve the accuracy of the results. Viscosity measurements of ultrapure CO2 confirmed this technique.
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Hersey and Shore5 used the calibration for the first time to test lubricants up to pressures of 57,000 psia and temperature up to 284 °F. Sage and Lacey9 described the use of the rolling ball viscometer to measure the viscosity of hydrocarbon solutions. Later Sage and Lacey10 used it with a tight-fitting ball size to measure the gas viscosities. They calibrated the viscometer with both gas and liquid at atmospheric pressure to measure the viscosity of methane and two natural gases at higher pressure. Comings et al.3 showed that the data of Sage and Lacey10 show a much greater increase with pressure than data obtained from the other investigators, and this mismatch is probably because of the increase of turbulent effect at higher pressures. Lewis8 proposed an equation to calculate the calibration coefficient based on the geometry of the viscometer. This equation is not accurate enough to be replaced with the calibration procedure using a standard fluid; however, it could be used to evaluate the accuracy of calibration results. Carmichael and Sage2 showed that the earlier n-butane gas data (Sage et al.9) are poor except for data measured at atmospheric pressure. This error is probably because Sage et al.9 extended their calibration result at atmospheric pressure to higher pressures, and it reduces the accuracy of the measured data. Viswanathan12 calibrated the viscometer with a standard fluid at constant pressure and different temperatures and measured the viscosity of nitrogen and methane at higher pressures. He found more than 10% error in his results compared with previous published data. The density of gases increases much more rapidly with pressure than does the viscosity. The Reynolds number thus increases rapidly with increasing pressure, and it is impossible to avoid turbulent conditions over portions of the pressure
INTRODUCTION The system of the inclined tube and rolling ball as a viscometer was invented by Flowers.4 Basically a ball rolls by gravitational force in an inclined circular section tube; as long as the flow regime passing around the ball is firmly laminar, the rolling time of the ball linearly relates to the absolute viscosity of the fluid. The theory of the motion assumes that the ball rolls without sliding throughout the tube and only relative measurement can be made; thus, it should be first calibrated with a known viscosity fluid (Figure 1).
Figure 1. A ball is positioned in the upper end of the tube slightly larger than the ball. As the ball rolls down through the tube at a constant angle, the displaced fluid must flow past it through the space between the ball and tube. As long as the flow passing around the ball is strictly laminar flow, the rolling time of the ball multiplied by the difference between the ball and fluid density (buoyancy factor) is a linear function of absolute viscosity of fluid (from Viswanathan12).
Hersey and Wash6 developed a correlation based on a dimensional analysis of variables involved. Hubbard and Brown7 completed the Hersey and Wash6 achievement and developed the systematic dimensional analysis of the rolling ball viscometer and determined a dimensionless calibration curve used to design a viscometer to measure a wide range of viscosities. Block1 carried out a method to determine whether the ball is moving by pure rolling or by a combination of rolling and sliding. © 2012 American Chemical Society
Received: Revised: Accepted: Published: 15276
July 2, 2012 October 19, 2012 October 23, 2012 October 23, 2012 dx.doi.org/10.1021/ie301751y | Ind. Eng. Chem. Res. 2012, 51, 15276−15281
Industrial & Engineering Chemistry Research
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Table 1. Tubes and Balls Properties diameter, in length, in density, g/mL
tube 1
tube 2
ball 1
ball 2
ball 3
ball 4
ball 5
ball 6
0.2598 0.6598
0.261 0.6629
0.252
0.248
0.258
0.2587
0.259
0.26
7.6323
7.6409
7.7908
7.8085
7.7881
7.7848
calibration procedure from other published procedures is that we calibrate the viscometer over the entire interest range of the measurement; all of the previously published procedures (Sage and Lacey;9 Hubbard and Brown7) just calibrated the viscometer at atmospheric pressure with a limited number of temperatures. Our technique minimized the turbulence effect that was always the weakness of the rolling ball viscometer. Laminar flow around the ball is the criteria for the calibration procedure. Ratio of the ball diameter to the inner diameter of the tube (d/D) and inclination angle of the tube to the horizon are two main factors that have a significant impact on the fluid flow around the ball. Ultrahigh pure (99.9999% purity) N2 gas was prepared by Matheson Tri-Gas Company with primary standard purity and was used to calibrate the viscometer. We used six balls and two tubes with different diameters to find the correct size to calibrate the viscometer (Table 1). Previous investigators (Sage and Lacey;9 Hubbard and Brown7) found that the ratio of d/D should be over 0.9 to reach the laminar flow around the ball. We used different combinations of balls and tubes and found that this ratio could be over 0.9 for some points, but, to achieve the laminar flow for our entire interest range of pressure and temperature, it must be above 0.99. After doing more than 100 tests, the laminar flow was obtained using tube 1 and ball 4. Each test was repeated for 20 to 50 times to make sure that the data are repeatable and consistent. Figure 2 shows the result for temperature of 170 °F at five different pressures of 4,000, 5,000, 6,000, 7,000, and 8,000 psi. The straight line shows the laminar flow around the ball.
range covered, if the rolling ball viscometer is calibrated only at atmospheric pressure. Using the gas instead of liquid and calibrating the viscometer over the whole interest range of pressure and temperature provide much more certainty in accuracy of the results. Optimal tube inclination angle and also ball/tube diameter ratio are two important factors that control the flow regime around the ball. An optimal combination of these two factors provides laminar flow around the ball.
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METHOD The general equation for laminar flow in a rolling ball viscometer (Hubbard and Brown7) is μ = K1(ρb − ρ)t
(1) 7
Hubbard and Brown proposed a correlation for the calibration coefficient, K1: 5π Kg sin θd(D + d) 42l
K1 =
(2)
8
Lewis has shown an approximate treatment of the problem in terms of the hydrodynamic of viscous fluids that K is given by 5/2 ⎡ d⎤ K = 0.891⎢1 − ⎥ ⎣ D⎦
(3)
Generally, the value of K1 should be determined by experiment with fluids of known viscosity and density; however, the value of K1 obtained by use of eq 3 serves as a check on the experimental values. Based on the above considerations, the procedure to calibrate a rolling ball viscometer can be summarized as follows: 1. Clean the ball and tube completely. 2. Check the diameter and the density of the ball. Since the diameter and density of the ball are the critical parameters in measurement of viscosity using a rolling ball viscometer, all care should be taken to ensure that both diameter and density are accurate. 3. Place the ball into the tube and fill the tube with the known calibration fluid. 4. Run the test at fixed d/D ratio, inclination angle, temperature, and different pressures. 5. Measure roll-time enough times to obtain statistically consistent values. 6. Multiply the average roll-time by the density difference of the ball and the fluid. 7. Plot the product of roll-time and density difference against the viscosity on a linear scale for each temperature. 8. Use the slope and the y-intercept of the plot to establish the equation for viscosity.
Figure 2. The straight line shows the laminar flow around the ball.
Different inclination angles were used to achieve the laminar flow around the balls. As inclination angle decreases, the chance to get the laminar flow increases, but at very low angles the ball does not fall down, because the gravity force cannot overcome the high static friction between the ball and tube surface; therefore, finding the minimum angle that ball can fall down is as important as the ratio of d/D. Table 2 shows the accuracy of all the measurements.
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CALIBRATION PROCEDURE The calibration procedure mainly gives an estimate of calibration coefficient (K1). It is mostly calibrated with liquid, but, to avoid any further turbulence effect, we used nitrogen as a standard calibration fluid. The main distinction of our 15277
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We plotted all the measured data (Figure 4) and did the regression to find the slope (K1) and the y-intercept (K2) separately and to define them as a function of temperature (Figure 5, Figure 6). By defining the K1 and K2, the equation to measure the viscosity of the gas with the rolling ball viscometer is
Table 2. Measurement Accuracy ball diameter
tube diameter
pressure
temperature
rolling time
weight
0.0001 in
0.0001 in
1 psi
1 °F
1s
0.01 gr
μ = (0.0000143324T ‐0.2601953328)(ρb − ρ)t ‐0.0000000023
After finding the best ball and tube, we used ten different angles from 6 to 15° to find the optimal angle, and for each angle we measured the roll-time for entire interest range of practical pressure and temperature. Then, we calculated the resistance factor and Reynolds number for all of them. The resistance factor7 and Reynolds number are defined as f=
5πg (D + d)2 ρS − ρ 2 t sin θ ρ 42 L2d
T 2 + 0.0000015333T − 0.0002130017
It should be noticed that the K1 and K2 are viscometer characteristics, and they have to be redefined, if any change happens to the geometry of the viscometer. We reproduced the viscosity data for nitrogen with the new equation to check the quality of the data and accuracy of the viscometer equation. Figure 7 shows the error distribution. The reasonable error to reproduce the viscosity values for the nitrogen shows that the calibration procedure has been done successfully and the viscometer is ready to measure the viscosity of the gases at pressure up to 8,000 psi and temperature up to 260 °F.
(4)
2
Re =
Ld ρ D + d μt
(5)
If we plot resistance factor vs Reynolds number for all the angles, we suppose to get the straight line for laminar flow. Figure 3 clearly indicates that from inclination angle of 6 to 11°, the flow around the ball is laminar, but, above the 11°, the data deviate from the straight line and show the turbulent flow around the ball. As temperature goes up the angle that can provide the laminar flow, decreases. At the highest temperature that we used, 260 °F, this angle became 10°; however, as pressure goes up the angle that can provide the laminar flow increases. At the highest pressure that we used, 8,000 psi, this angle became 13°; therefore, to be in the safe area, we chose the angle of 8° that can provide laminar flow over the entire interest range of pressure and temperature. To simplify the calibration procedure, the calibration coefficient is a constant number for the entire viscosity measurements. This incorrect assumption adds some drastic error to the measured data. In reality, the calibration coefficient varies by temperature change; therefore, it should be defined as a function of temperature. Also the y-intercept in Figure 1 is always ignored in calibration procedure. Including the yintercept as a function of temperature in the viscometer equation will improve the accuracy of results considerably; therefore, we changed the main equation for the rolling ball viscometer to μ = K1(ρb − ρ)t + K2
(7)
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CO2 VISCOSITY MEASUREMENT Ultrahigh pure (99.9999% purity) CO2 gas was prepared by Matheson Tri-Gas Company with primary standard purity and was used to verify the new viscometer calibration function. Initial pressure in the gas tank was 2,000 psig. Twenty-one tests were performed on the CO2 gas to cover a wide range of pressures and temperatures. Viscosity measurements were taken at pressures between 4,000 and 8,000 psi and temperatures of 98, 116, 134, 152, 170, 187, 200, 220 and, 240 °F. Figure 8 shows the entire viscosity measurement system that used in this work. Figure 9 shows the pressure, temperature, and angle gauges for the Ruska rolling ball viscometer. The procedure to measure the rolling time of the ball can be summarized as follows: 1. Clean the entire test assembly. Any contaminant can drastically influence the results. 2. Place the ball inside of the tube from the upper end of the viscometer. 3. Evacuate the test assembly using a vacuum pump. 4. Charge the viscometer with the test sample fluid. 5. Set the pressure of the viscometer and allow 30 min to make sure that pressure is stabilized.
(6)
Figure 3. It clearly indicates that from inclination angle of 6 to 11°, the flow around the ball is laminar, but above the 11° the data deviate from straight line and show the turbulent flow around the ball. 15278
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Figure 4. The straight lines show that all the data measured in laminar zone.
Figure 5. K1 should be a function of temperature and not to be assumed constant.
Figure 6. K2 is a function of temperature and should be included in the viscosity measurement.
9. Release the ball by changing toggle switch from the HOLD position to the FALL position. When the ball makes contact at the end of the roll, the clock stops, the buzzer sounds, and the rolling time can be recorded now. Viscosity is sensitive to any temperature variation so it is important to make sure that temperature is balanced throughout the tube. Since there were two heaters located at two ends of the tube, thermocouples were placed at the middle
6. Set the temperature of the viscometer and allow at least one hour reaching thermal equilibrium. 7. Energize the solenoid with the toggle switch in the ON position. Now the ball can be held in position by adjusting the toggle switch to the HOLD position. 8. Bring the ball into a HOLD position at the upper end of the tube by rotating the receptacle arm handle. 15279
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Figure 7. All the relative errors are less than 1%. It shows that the accuracy of the results is acceptable. Also the results indicate no strict relation between error and temperature.
Table 3. CO2 Viscosity Data, cp temperature, °F pressure, psi 4,000 5,000 6,000 7,000 8,000 pressure, psi 4,000 5,000 6,000 7,000 8,000
98
116
0.09579 0.10043 0.10807 0.12341 0.12514
0.08178 0.09088 0.09913 0.10779 0.11493
134
152
0.07470 0.06930 0.08637 0.07561 0.09084 0.08400 0.10284 0.09006 0.10720 0.10184 temperature, °F
170 0.06070 0.07305 0.08160 0.08473 0.09103
187
200
220
240
0.05815 0.06827 0.07164 0.08259 0.08573
0.05188 0.06153 0.07084 0.07945 0.08289
0.04813 0.05713 0.06661 0.07461 0.07900
0.04450 0.05503 0.06265 0.06607 0.07501
Figure 8. Rolling ball viscometer system includes a diaphragm pump and a rolling ball viscometer.
Figure 10. The average relative error for all the temperatures are less than 3%. This shows that the accuracy of the results is acceptable. Also the results indicate no direct relationship between error and temperature.
to unity, confirming that the measured data are acceptable and the proposed technique worked well.
Figure 9. Ruska rolling ball viscometer used in this research.
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CONCLUSIONS Using the gas instead of liquid for the calibration over the whole interest range of pressure and temperature appears to be satisfactory for testing gases in rolling ball viscometer originally designed for testing liquids. Optimal tube inclination angle and also ball/tube diameter ratio prevent the turbulent effect around the ball and enhance the measurement precision.
of the tube to measure the average temperature of test fluid. It is reported in the manual and experimentally confirmed that temperature becomes stable after one hour. Table 3 lists the measured viscosity of ultrahigh pure CO2 at different temperatures and pressures. Figures 10 and 11 show the accuracy of the measured data compare to the NIST value for ultrapure CO2. The R2 is close 15280
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(4) Flowers, A. E. Viscosity Measurement and a New Viscometer. Proc., Am. Soc. Test. Mater. 1928, 14 (2), 665−616. (5) Hersey, M. D.; Shore, H. Viscosity of Lubricants Under Pressure. Mech. Eng. 1928, 50, 221−228. (6) Hersey, M.d.; Wash, J. Theory of Torsion and Rolling Ball Viscosimeters, and the Effect of Pressure on Viscosity. J. Wash. Acad. Sci. 1916, 6, 525−530. (7) Hubbard, R.; Brown, G. Rolling Ball Viscometer. Ind. Eng. Chem., Anal. Ed. 1943, 15 (3), 212−218, DOI: 10.1021/i560115a018. (8) Lewis, H. W. Calibration of Rolling Ball Viscometer. Anal. Chem. 1953, 25 (3), 507−508. (9) Sage, B.; Yale, W.; Lacey, W. Effect of Pressure on Viscosity of NButane and Iso-Butane. Ind. Eng. Chem. 1939, 31, 223−226. (10) Sage, B. H.; Lacey, W. N. Effect of Pressure Upon Viscosity of Methane and Two Natural Gases. Trans. Am. Inst. Min. Metall. Eng. 1938, 127, 118−134. (11) Sage, B. H.; Lacey, W. N. Viscosity of Hydrocarbon Solutions Viscosity of Liquid and Gaseous Propane. Ind. Eng. Chem. 1938, 30 (7), 829−834. (12) Viswanathan, A. Viscosities of Natural Gases at High Pressures and High Temperatures. MS thesis, Texas A&M University, 2007.
Figure 11. Comparison of the measured data of ultrapure CO2 with NIST value confirmed the accuracy of the proposed technique to calibrate the rolling ball viscometer.
Viscosity measurements of ultrapure CO2 confirmed this technique.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +1-979-595-3828. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors wish to acknowledge the sponsors of the Crisman Petroleum Research Institute at Texas A&M University for supporting this project and also would like to thank Mrs. DarlaJean Weatherford for helping with editing and formatting the manuscript.
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NOMENCLATURE d = diameter of the ball, in D = diameter of the tube, in F = resistance factor g = acceleration of gravity, 980 cm/s K = constant K1 = rolling ball viscometer coefficient K2 = y-intercept L = length of tube, in P = pressure, psi Re = Reynolds number t = roll-time, s T = temperature, °F θ = inclination angle of tube to the horizontal, ° μ = dynamic viscosity of fluid, cP π = 3.1416 ρ = density of fluid, gr/ml ρb = density of the ball, gr/ml
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REFERENCES
(1) Block, R. B. On the Resistance to the Uniform Motion of a Solid through a Viscous Liquid. J. Appl. Phys. 1942, 13 (1), 56−65. (2) Carmichael, L.; Sage, B. Viscosity of Hydrocarbons. N-Butane. J. Chem. Eng. Data 1963, 8 (4), 612−616. (3) Comings, E. W.; Mayland, B. J.; Egly, R. S. The Viscosity of Gases at High Pressures. Bull. - Ill., Univ., Eng. Exp. Stn. 1944, 354. 15281
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