Article Cite This: Langmuir XXXX, XXX, XXX−XXX
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Drainage from a Fluid-Handling Component Due to Centrifugal Acceleration Koray Sekeroglu, C. W. Extrand,* and David A. Burdge CPC, 1001 Westgate Drive, St. Paul, Minnesota 55114, United States S Supporting Information *
ABSTRACT: The onset of drainage of liquids from plastic tubes was evaluated. One end of the tubes was plugged, filled with liquid, oriented horizontally, and attached to a rotor. With their open end facing outward, the filled tubes were spun at progressively higher speeds until they began to drain. Resistance to drainage was independent of the tube length, but depended on the surface tension and density of the liquids as well as the diameter and wettability of the tubing. We found that the onset of drainage could be explained in terms of a critical centrifugal acceleration.
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INTRODUCTION Tubing and connectors are essential components in many fluidhandling systems. They allow for efficient design and construction of systems. They also facilitate quick disassembly for cleaning, maintenance, repair, reconfiguration, or disposal. In many industrial processes, it is desirable that disconnected fluid-handling components do not spill or drain. We previously examined how liquids drain from tubes and model connectors that have been raised from a liquid reservoir and then tilted or rotated.1−3 Somewhat surprisingly, liquids did not drain from components with relatively large orifices. It was postulated that this resistance to drainage arises from a combination of liquid surface tension and atmospheric pressure. Because components in the previous studies were moved slowly, the exposed liquid interface of open connectors experienced relatively low accelerations, on the order of earth’s gravitational field. However, in practice, disconnected fluidhandling components are manipulated through a wide range of rapid motions. They are jolted and shaken. Thus, a method is needed to quantify spillage and drainage of components exposed to the much greater accelerations associated with abrupt movements. Various types of spinning apparatuses have been used to initiate movement of sessile liquid drops4−7 or liquids inside open-ended capillary tubes.8 In this work, larger tubes with one open end, meant to represent a disconnected fluid-handling component, were filled with liquid and attached to a spinning apparatus. The component was rotated at progressively higher rates to initiate spillage and drainage. Experimental results are compared to models that assume that onset of drainage is determined by an interplay of surface and body forces.
Figure 1. (a) Side view of a cylindrical tube, one end is open and other end is sealed. (b,c) represent a front view showing the inner diameter (D) and wall thickness (B) of the tube.
(L) is measured between the open end and the proximal end of the seal. The tube, which exhibits advancing and receding contact angles of θa and θr, is filled with a liquid that has surface tension of γ and density of ρ and then oriented horizontally. If its diameter is sufficiently small, liquid will not drain. Next, the tube is attached to the arm of a spinning apparatus with the open end facing outward, such that the distance from the axis of rotation to the open end of the tube is x, Figure 2. The tube is spun at a progressively higher angular velocity (ω) that creates a centrifugal acceleration (a) at the end of the open tube a = ω 2x
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As the centrifugal acceleration increases, liquid at the open end of the tube distorts as shown in Figure 3. The convex bulge
THEORY For simplicity, assume that the component is a cylindrical tube as depicted in Figure 1. One end of the tube is closed and sealed. The other has a circular opening with an inner diameter D. The tube has a wall thickness of B. The length of the tube © XXXX American Chemical Society
(1)
Received: March 20, 2018 Revised: April 27, 2018
A
DOI: 10.1021/acs.langmuir.8b00915 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir
the first scenario, the advancing contact angle is large (θa ≫ 0°) and the lower bulge is pinned at the inner edge of the tube, Figure 3a. The receding contact angle can be any value θr ≥ 0°. Here, the principal radii of curvature are approximated as 1 D 3 + cos θr
R1 =
(7)
and Figure 2. Schematic diagram of the spinning apparatus that comprises a direct current power supply, a frequency generator, and a stepper motor. The rotor turns in the counterclockwise direction with an angular velocity (ω). The distance between the center of rotation and the open end of the tube is denoted as x.
1/2 ⎡ ⎛1 ⎞2 ⎤ 1 1 ⎢ ⎥ −⎜ − R2 = D ⎟ ⎢⎣ 4 3 + cos θr ⎠ ⎥⎦ ⎝2
(8)
Combining 2−8 yields the following estimate of the critical centrifugal acceleration (ac) required to dislodge liquids from tubes, where θa ≫ 0° and θr ≥ 0° ⎧ ⎪ 3 ac = (3 + cos θr)⎨3 + cos θr 2 ⎪ ⎩ −1/2 ⎫ ⎡ ⎞2 ⎤ ⎛1 ⎪ γ 1 1 ⎢ ⎥ ⎬ 2 + −⎜ − ⎟ ⎢⎣ 4 3 + cos θr ⎠ ⎥⎦ ⎝2 ⎪ ρD ⎭
If θr = 0°, then eq 9 reduces to γ ac = 24(1 + 3−1/2 ) 2 ρD Figure 3. Cross-sectional depiction of tubes containing liquid at the critical acceleration that triggers spillage. (a) Interface of the liquid in a tube where θa ≫ 0° and θr ≥ 0°. (b) Interface of the liquid in a wettable tube where θa = θr = 0°.
R1 =
R2 =
(11)
1/2 1⎡ 2 1 2⎤ ⎢⎣D − (D − B) ⎥⎦ 2 4
(12)
where B is the wall thickness of a tube. Combining eqs 2−6, 11, and 12 yields an equation for estimating the critical centrifugal acceleration (ac) required to dislodge liquids from wettable tubes, where θa = θr = 0°
(2)
ac =
(3)
where V and A are the volume and cross-sectional area of the lower bulge 2π 2 V= R1 R 2 (4) 3 A = πR1R 2
D+B 4
and
If the shape of the lower bulge at the critical acceleration is assumed to be a hemiellipsoid, the critical body pressure (Δpb) can be estimated as Δpb = ρVa /A
(10)
In the second scenario, the tubing is completely wet by the liquid (θa = θr = 0°) and the lower bulge is pinned at the outer edge of the tubing, Figure 3b. Here, the principal radii of curvature can be approximated as
on the bottom portion and concave dimple of the top portion become more pronounced. If a critical centrifugal acceleration (ac) is exceeded, then liquid will drain from the spinning tube. The critical centrifugal acceleration to initiate flow (ac) can be estimated by equating the opposing body and surface pressures acting on the protruding lower bulge Δpb = Δps
(9)
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⎤−1/2 ⎤ ⎡ 12γ ⎡ 2 1 ⎢ ⎥ + ⎢D2 − (D − B)2 ⎥ ⎦ ⎥⎦ ρ(D + B) ⎢⎣ D + B ⎣ 4 (13)
EXPERIMENTAL DETAILS
Liquids. The liquids used were deionized water (W), ethylene glycol (EG, Fisher BioReagents, ≥ 99%, CAS 107-21-1), and ethanol (Sigma-Aldrich, >99.5%, CAS 64-17-5). Density (ρ) of the liquids, taken from the literature, were 998, 1110, and 789 kg/m3.11 Solids. Various types of plastic tubing were purchased from McMaster-Carr: polycarbonate (PC, #8585K), polyoxymethylene (POM, #8627K), polyvinylidene fluoride (PVDF, #51105K), nylon 6,6 (PA, #8628K), high-density polyethylene (HDPE, #50375K), polytetrafluorethylene (PTFE, #8547K), and perfluoroalkoxy (PFA, #52705K), with inner diameters (D) that ranged between 3.2 and 9.6 mm and wall thicknesses (B) between 3.3 and 0.55 mm. Tubes were cut to the desired lengths (L ≈ 0.5−4.5 cm) with a razor or a saw. Uncertainty in D and B was estimated to be ≤ ±0.1 mm.
(5)
and R1 and R2 are its principal radii. The curvature of the bulge along with the surface tension of the liquid creates a surface or Laplace pressure (Δps) that counteracts the body pressure9,10 γ γ Δps = + R1 R2 (6) The wettability of the tubes affects the shape of the bulge. (The influences of angular acceleration and air drag are considered to be negligible.) Two scenarios are considered. In B
DOI: 10.1021/acs.langmuir.8b00915 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir Surface Tension. An automatic surface tensiometer (Kyowa DyneMaster DY-300) was used to measure the surface tension of the liquids (γ). Prior to each measurement, the platinum Wilhelmy plate was cleaned in an alcohol flame. Liquid was poured into the glass dish of the tensiometer after cleaning with ethanol and water. The dish was placed on the tensiometer stage. The stage automatically raised the liquid 2.5 mm to prewet the platinum plate and then lowered 2.5 mm for the measurement. With the liquid still contacting the plate, the tensiometer measured a force and used that force and the dimensions of the plate to compute a surface tension for the liquid. The measured surface tension of water, ethylene glycol, and ethanol were 72.5, 48.1, and 22.2 mN/m, which agreed closely with values reported in the literature.10,11 Uncertainty in the measured value of γ was estimated to be ±1 mN/m. Roughness of Solids. Any influence of roughness was captured within the receding contact angles. Nevertheless, we did quantify the roughness using an optical surface profilometer (Bruker Contour GT). Tubing was cut in half lengthwise. 3D images with base dimensions of 0.5 mm × 0.9 mm were captured at three locations on the inner diameter of each type of tubing. Images were flattened with resident software (Vision 64), and roughness parameters were computed. Values of number average roughness (Ra) are listed in Table 1. They ranged from 0.1 μm for PC to 3.7 μm for PTFE.
rotor was spun using a direct current power supply (BK Precision, 1697) and a frequency generator (Stanford Research Systems, DS345). The angular acceleration was ≤0.7 rad/s2. The frequency of rotation was slowly increased to initiate spillage. The critical frequency was converted to a critical angular velocity (ωc) and then in turn converted to critical centrifugal acceleration (ac) using eq 1. Values of ac were divided by the acceleration of the earth’s gravitational field (g = 9.81 m/s2) and reported as a critical acceleration ratio, (ac/g). Averages and standard deviations were computed. Absolute uncertainty in x was estimated to be ≤0.2 cm. Relative uncertainty in ω, ac, and ac/g was estimated to be ≤10%. Mobile phone fixtures that allowed top and side views were built and attached to the rotor. Slow motion videos of spinning tubes were captured with a Samsung Galaxy S7 mobile phone. All spin experiments were performed in triplicate at 23 ± 2 °C.
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RESULTS AND DISCUSSION All liquid-filled tubes behaved similarly when spun. As the spin rate increased, liquid at the open end of the tube distorted, as shown in Figure 3. The bottom portion of the air−liquid interface bulged outward, while the top portion bulged inward. (This behavior also is shown in videos linked to this paper in the Supporting Information section.) The distortion was due to centrifugal acceleration. Videos shot from the above showed that the lateral shape of the bulging liquid was symmetrical in the horizontal plane, indicating that angular acceleration and air drag were relatively unimportant. When the critical centrifugal acceleration ratio (ac/g) for a particular liquid-tube combination was exceeded, liquid spilled and drained. Although the general mechanism was similar, the magnitude of ac/g varied from one experiment to the next. Therefore, we examined a number of parameters, including tube position, tube length, tube diameter, tube composition, type of liquid, and wall thickness of the tubes. Tube Position. It was observed that the critical angular velocity to trigger drainage varied with position of the tube on the spinning rotor. Tubes placed farther from the axis of rotation drained at a lower angular velocity. In contrast, the critical centrifugal acceleration ratio was independent of the tube position. An example is shown in Figure 4 for water in a
Table 1. Number Average Roughness (Ra) of the Inner Diameter of the Various Types of Tubing Along With Advancing (θa) and Receding (θr) Contact Angles of Water (W), Ethylene Glycol (EG), and Ethanol (E) W tube type
Ra (μm)
θa (deg)
θr (deg)
PC PVDF PA POM HDPE PFA PTFE
0.1 2.9 0.4 2.2 0.4 0.2 3.7
105 104 101 98 108 108 112
29 30 23 52 67 82 80
tube type
Ra (μm)
θa (deg)
PC
0.1
53
EG θr (deg) 10 E tube type
Ra (μm)
θa (deg)
θr (deg)
PC
0.1
13
6
Contact Angles. Advancing and receding contact angles were measured using a digital goniometer (Kyowa DropMaster DMs-401). To measure an advancing contact angle (θa), a sessile drop (∼7 μL) was deposited on the outer wall of a tube, additional liquid was added to the drop to advance the contact line, and then, an image was captured. To measure a receding contact angle (θr), liquid was withdrawn from a sessile drop until the contact line retracted; with the needle of the syringe still contacting the drop, an image was captured. Base and tangent lines were constructed on the various drop images and then θa and θr were measured directly. Advancing and receding contact angles (θa and θr) of the various liquid−solid combinations are listed in Table 1. Values of θa and θr ranged from 6° to 112°. Standard deviation in the contact angle measurements was generally ≤ ±3°. Spin Experiments. The spin apparatus, shown in Figure 2, comprised a 36 cm long aluminum rotor attached to a stepper motor (23HS22-2804S) at its centroid. As a safety precaution, the apparatus was bolted to a table and operated under a PC shield. To perform a spillage experiment, one end of a tube was plugged, filled with liquid, oriented horizontally, and attached to the rotor with its open-end facing outward, Figure 2. The distance between the open end of the tube and the axis of rotation (x) was measured and recorded. The
Figure 4. Critical acceleration ratio (ac/g) for water vs distance (x) between the open end of a 3.2 mm diameter PC tube and the axis of rotation. The points are experimental data. Error bars are standard error.
3.2 mm diameter PC tube, where the critical acceleration ratio (ac/g) was 28.9 ± 2.2, regardless of the distance of the open end from the axis of rotation. The points are experimental data. Error bars are standard error. Tube Length. For a given liquid-tube combination, the critical acceleration ratio (ac/g) to instigate drainage also was independent of the tube length. Figure 5 shows critical C
DOI: 10.1021/acs.langmuir.8b00915 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir
Table 2. Experimental and Predicted Values of the Critical Acceleration Ratio (ac/g) for Water in Tubes of Various Composition (D = 3.2 mm, L = 2.5 cm). Predicted Values Came from Eq 9 ac/g tube type PC PVDF PA POM HDPE PFA PTFE
Figure 5. Critical acceleration ratio (ac/g) for water vs length (L) of a 6.4 mm diameter PC tube, x = 15 cm. The points are experimental data. Error bars are standard error.
experimental
predicted
± ± ± ± ± ± ±
26.3 26.2 26.9 23.3 20.9 18.3 18.6
28.9 30.3 25.8 28.9 19.6 13.8 13.7
1.6 1.7 0.9 1.6 2.0 0.9 0.6
Liquid. Most of our experiments were performed with water. Figure 7 shows the critical centrifugal acceleration ratio
centrifugal acceleration ratios (ac/g) for water in 6.4 mm diameter PC tubes of different lengths. Within experimental error, the ac/g values were constant, 3.8 ± 0.5, indicating that the spillage was controlled by forces or pressures acting near the open end of the tubes. Tube Diameter. Figure 6 shows the critical centrifugal acceleration ratio (ac/g) for water versus the inner diameter
Figure 7. Critical acceleration ratio (ac/g) vs distance (x) between the open end of PC tubes (D = 3.2 mm and L = 2.5 cm) and the axis of rotation for various liquids. The points are experimental data for water (W), ethylene glycol (EG), and ethanol (E). Error bars are standard error. The horizontal lines are predicted values from eq 9. Figure 6. Critical acceleration ratio (ac/g) for water vs the inner diameter (D) of PC tubes, where L = 2.5 cm and x = 15 cm. The points are experimental data. Error bars are standard error. The solid curve represents the predicted values from eq 9.
(ac/g) for water compared to two other liquids. The three liquids have similar densities, but differ significantly in their surface tensions, water having the greater surface tension and ethanol the lesser. The data in Figure 7 demonstrate that greater surface tension retards the onset of drainagewater exhibited the greatest resistance to drainage, followed by ethylene glycol and then ethanol. The horizontal lines are predicted values from eq 9. The predictions for water and EG agreed with the experimental data, but not the prediction for ethanol, which was too large. Equation 9 assumes that θa ≫ 0°. However, ethanol nearly completely wets PC. It was hypothesized that the ethanol flowed past the inner edge of the tube wall and was trapped at the outer edge. If true, then the thickness of the tube wall would reduce resistance to drainage. Wall Thickness. To investigate the influence of the wall thickness, tubes were turned on a lathe to reduce their wall thickness (B) while keeping their inner diameter (D) constant and then tested. Figure 8 shows critical centrifugal acceleration ratios (ac/g) versus tube wall thickness (B) for various liquids in PC tubes (D = 3.2 mm, L = 2.5 cm). Water and EG both have large advancing contact angles and are expected to remain pinned on the edge of the inner diameter of the tubes. Thus, wall thickness should not have been a factor and the ac/g values should have been constant. Indeed, they were. On the other hand, if ethanol was pinned at the outer edge of the PC tubes, then wall thickness should have affected the curvature of the ethanol and its ability to resist drainage. Without a doubt, it did.
(D) of PC tubes, where L = 2.5 cm and x = 15 cm. As the inner diameter of the tubes increased, the values of ac/g declined, from 29.8 for D = 3.2 mm to 0.8 for D = 9.6 mm. Increasing the tube diameter allowed more deformation of the air−liquid interface at lower centrifugal accelerations, which ultimately enabled the body forces to overtake surface forces at lower ω values and initiate drainage. The solid curve, which represents the predicted values from eq 9, demonstrates that the ac/g values scale inversely with the square of the tube diameter. Tube Composition. Experimental values of the critical centrifugal acceleration ratio (ac/g) for water in tubes of various compositions are shown in Table 2. The tubes had the same inner diameter and length (D = 6.4 mm and L = 2.5 cm). All were relatively hydrophobic with water θa > 90°, but their receding values ranged from 23° for PA to 82° for PFA. Tubes with smaller receding contact angles spilled at higher critical accelerations than the tubes with larger receding contact angles. Why? Smaller receding contact angles meant that the liquid within the open end of the tube would experience greater contortion and a smaller radius of curvature in its lower bulge and consequently a larger Laplace pressure to prevent drainage. Table 2 also includes predicted ac/g values from eq 9. The measured and estimated values were in general agreement. D
DOI: 10.1021/acs.langmuir.8b00915 Langmuir XXXX, XXX, XXX−XXX
Langmuir
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ACKNOWLEDGMENTS We wish to thank D. Harper for performing the surface roughness measurements. Also, thanks to M. Acevedo, L. Castillo, J. Doyon, N. Ly, K. Switalla, K. Vangsgard, R. Von Wald, and J. Wittmayer for their support and help with publication of this manuscript.
■ Figure 8. Critical acceleration ratio (ac/g) vs tube wall thickness (B) for various liquids in PC tubes (D = 3.2 mm and L = 2.5 cm). The points are experimental data for water (W), EG, and ethanol (E). Error bars are standard error. The solid red and long dash green lines are predicted values for water and ethylene glycol (EG) from eq 9. The short dash blue curve is the predicted value for ethanol from eq 13.
The measured ac/g values declined with increasing wall thickness of the tubes. The solid red and long dash green lines are predicted values for water and EG from eq 9. The short dash blue curve is the predicted value for ethanol from eq 13. The predicted lines and curve fit the data well.
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CONCLUSIONS We have evaluated the onset of drainage of liquids from plastic tubes with one open end using a spinning apparatus. The ability of liquid-tube combinations to resist drainage depended on the surface tension and density of the liquid as well as the diameter and wettability of the tubes. Increasing the surface tension of the liquid increased the resistance to drainage. Conversely, increasing the density of the liquid, tube diameter, or tube wettability decreased the resistance to drainage.
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REFERENCES
(1) Extrand, C. W. Spontaneous Draining of Liquids from Vertically Oriented Tubes. Langmuir 2017, 33, 12903−12907. (2) Extrand, C. W. Drainage from a Fluid-Handling Component Because of Inclination. Langmuir 2018, 34, 126−130. (3) Extrand, C. W. Drainage from a Fluid Handling Component with Multiple Orifices due to Inclination or Rotation. Langmuir 2018, 34, 4159−4165. (4) Goodwin, R.; Rice, D.; Middleman, S. A Model for the Onset of Motion of a Sessile Liquid Drop on a Rotating Disk. J. Colloid Interface Sci. 1988, 125, 162−169. (5) Extrand, C. W.; Gent, A. N. Retention of Liquid Drops by Solid Surfaces. J. Colloid Interface Sci. 1990, 138, 431−442. (6) Tadmor, R.; Bahadur, P.; Leh, A.; N’guessan, H. E.; Jaini, R.; Dang, L. Measurement of Lateral Adhesion Forces at the Interface between a Liquid Drop and a Substrate. Phys. Rev. Lett. 2009, 103, 266101. (7) Tadmor, R.; Das, R.; Gulec, S.; Liu, J.; N’guessan, H. E.; Shah, M.; Wasnik, P. S.; Yadav, S. B. Solid−Liquid Work of Adhesion. Langmuir 2017, 33, 3594−3600. (8) Boadi, D. K.; Marmur, A. Drop Formation and Detachment from Rotating Capillaries. J. Colloid Interface Sci. 1990, 140, 507−524. (9) Laplace, P. S. Mécanique Celeste; Courier: Paris, 1805; Vol. t. 4, Supplément au Xe Livre. (10) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (11) Weast, R. C. Handbook of Chemistry and Physics, 59th ed.; CRC: Boca Raton, FL, 1978−1979.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b00915. Slow motion videos (8× slower) showing spillage of liquids from tubes (L = 2.5 cm, x = 15 cm): (1) Water in a PC tube (D = 6.4 mm) (AVI) (2) Water in a PFA tube (D = 6.4 mm) (AVI) (3) Ethanol in a PC tube (D = 4.8 mm) (AVI)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: 1-651999-1859. ORCID
C. W. Extrand: 0000-0002-0330-9236 Author Contributions
C.W.E. conceived the work and developed the theory and working equations. D.A.B. built the spin apparatus; K.S. refined it and added video capture capability. K.S. developed the experimental methods, collected, and analyzed the data. K.S. created the images, videos, figures, and graphics. C.W.E. and K.S. wrote the manuscript. Notes
The authors declare no competing financial interest. E
DOI: 10.1021/acs.langmuir.8b00915 Langmuir XXXX, XXX, XXX−XXX
