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Cite This: Langmuir 2017, 33, 12903-12907
Spontaneous Draining of Liquids from Vertically Oriented Tubes C. W. Extrand* CPC, Inc., 1001 Westgate Drive, St. Paul, Minnesota 55114, United States ABSTRACT: This paper describes experiments where liquid-filled tubes, with the bottom ends open, were pulled vertically from a reservoir. If the open diameter on the bottom of the tubes was sufficiently small, liquid was retained. Otherwise, if sufficiently large, the tubes drained from the bottom up. The critical diameter of the opening at the bottom of the tubes ranged from 10−15 mm for water to >5 mm for a dense, low surface tension, perfluoroether oil. The ability of relatively large diameter tubes to retain liquid is attributed to a combination of materials properties and atmospheric pressure. If surface tension were acting alone, the tubes would have had to be much smaller to prevent drainage or spillage.
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Table 1. Liquids and Their Propertiesa
INTRODUCTION In many laboratory and industrial processes, it is desirable to prevent liquid from draining from tubes or spilling from connections. Consider the following two examples. The first example is for dip tubes. Liquid chemicals are often withdrawn from containers using a dip tube. When a container is emptied, the dip tube is removed and transferred to a full one or is removed to allow refilling. Movement of the tube can cause drainage. When the dip tube is reimmersed into a full container, many processes require that the line and dip tube must be purged to remove air. Purging wastes both time and liquid chemical. If the dip tube does not drain when removed from its container, then purging can be minimized or perhaps completely eliminated. The second example is for connectors, couplers, or small valves of various types. Couplers and related devices are often used to connect liquid flow lines. In many cases it is desirable for couplers to have a valve, such that, when a liquid line is disconnected, liquid does not flow or drain from the line. Valved couplers prevent loss of large quantities of liquid. However, dead space inside traditional designs can still allow spillage of small volumes of liquid when disconnected. This spillage can cause corrosion or create hazards to people and equipment. In both cases described above, a simple technology for preventing drainage or spillage would be desirable. This paper describes a variety of experiments that have been performed to understand drainage so that better means can be devised to control and/or prevent drainage and/or spillage.
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a
γ (mN/m)
ρ (kg/m3)
μ (mPa·s)
water glycerol ethylene glycol isopropanol perfluoroether oil
72 65 48 22 17
998 1260 1110 789 1880
1.0 1000 20 2.0 110
γ is surface tension, ρ is density, and μ is viscosity.
Tubes were purchased from McMaster Carr. They consisted of glass (heat-resistant borosilicate glass) and three types of polymers: clear, plasticized polyvinyl chloride (PVC, Food and Dairy grade Tygon, Formulation B-44-4x), semitransparent polyamide 66 (PA66, wearresistant nylon), and semitransparent polyethylene (PE, crackresistant). Glass tubing was cleaned with oxygen plasma (10 min at 800 mTorr and high radio frequency power in a Harrick Plasma Cleaner PDC-001-HP). The nylon tubing was received in straight five foot lengths. The PVC and PE tubing was received in coils. To straighten the PVC and PE tubing, 10 cm lengths were pushed over stainless steel mandrels and heated under vacuum overnight (at 60 °C for PVC and at 80 °C for PE). Unless stated otherwise, capillary rise and drainage experiments were performed on a tensile test machine (Lloyd-Ametek LS1) at 25 °C with 10 cm sections of tubing where ends were cut normal to their length. Wettability of Tube Materials. Wettability of the various tube materials was assessed by contact angles from capillary rise. For a given type of glass or plastic, a tube of the smallest diameter was clamped vertically in the test machine with both ends open. For advancing contact angles (θa), the tube was pushed downward into the liquid at 10 mm/min, stopping periodically to capture images for measuring the height of the fluid column inside the tube (h). On the other hand, for receding contact angles (θr), immersed tubes were pulled upward at the same speed, pausing occasionally to capture images. The height of the column (h) along with tube diameter (D) and properties of the liquid (γ and ρ) were used to estimate contact angles (θ):5
EXPERIMENTAL DETAILS
The liquids used were deionized water, glycerol (Acros, 15892-0010, 99+%), ethylene glycol (Fisher Scientific, BP230-1), isopropanol (Pharmco-Aaper, 99% reagent ACS grade), and a perfluoroether oil (DuPont Krytox 1506 Vacuum Pump Fluid). The properties of the liquids are listed in Table 1. Values of surface tension (γ), density (ρ), and viscosity (μ) were taken from the scientific and supplier literature.1−4 © 2017 American Chemical Society
liquid
Received: September 15, 2017 Revised: October 17, 2017 Published: October 27, 2017 12903
DOI: 10.1021/acs.langmuir.7b03247 Langmuir 2017, 33, 12903−12907
Article
Langmuir ⎛ ρgDh ⎞ θ ≈ arccos⎜ ⎟ ⎝ 4γ ⎠
What happened when tubes were lifted vertically? A capillary bridge formed and then broke, creating an air liquid interface or meniscus on the bottom of the tube. Tubes of large diameter immediately drained from the bottom up. In contrast, tubes of sufficiently small diameter retained liquid. Figure 2 shows the
(1) 2
where g is the acceleration due to gravity (g = 9.81 m/s ). Table 2 lists advancing and receding contact angles (θa and θr) for the various tube materials. The clean glass was completely wet by
Table 2. Tube Materials and Their Wettabilitya
a
tube material
θa (deg)
θr (deg)
clean glass polyamide 66 (PA66) plasticized polyvinyl chloride (PVC) polyethylene (PE)
6±8 70 ± 1 103 ± 1 109 ± 1
3±8 46 ± 3 70 ± 1 99 ± 1
θa and θr are advancing and receding contact angles.
water (and the other liquids), effectively exhibiting zero contact angles. The polymers were relatively hydrophobic, with advancing contact angles for water ranging between 70° and 109° and receding angles between 46° and 99°. Drainage Experiments. In most of the drainage experiments, a 10 cm length of tubing was filled by immersing it into a liquid container. While still immersed, one end on the tube was plugged with a rubber stopper; the other end was covered to prevent drainage. The tube was transferred from the liquid container to the tensile test machine; the tube was oriented vertically with the stoppered end directed upward and clamped, while the other end was immersed in a liquid reservoir. Figure 1 shows such a tube in the tensile test machine. The vertical
Figure 2. Side view images of glass tubes of various diameters (D) pulled from a water reservoir at 100 mm/min. (a) D = 2.7 mm, (b) D = 4.1, (c) 5.6, (d) 7.9, (e) 9.7, (f) 11.2, (g) 11.2, (h) 12.7m, and (i) 15.7 mm.
end result from experiments with clean glass tubes that were filled with water and then lifted from a reservoir−the larger tubes with D ≥ 15.7 mm consistently drained. In contrast, the smaller tubes with D ≤ 9.7 mm retained water. Tubes of different lengths (5−25 cm) or partially filled tubes with air above the liquid showed the same behavior. It should be noted that the transition between retention and drainage was not abrupt. If liquid laden tubes with diameters slightly less than or equal to the critical diameter were tapped, shaken slightly, or tilted, they drained. Tubes with slightly larger diameters sometimes retained liquid when lifted from the reservoir but were easily disturbed; the slightest perturbation initiated drainage. If liquid surface tension (γ) alone were at work, the maximum hydrostatic pressure (Δpmax) that could be supported inside the tubes by an upwardly directed Laplace pressure can estimated as,6 4γ Δpmax = (2) D For water, Δpmax would range from 107 Pa for the smallest tube (D = 2.7 mm) to a mere 30 Pa for the largest nondrainer (D = 9.7 mm). (These values are overestimates, as eq 2 assumes the liquid meniscus is a hemispherical cap.6 None of the menisci were hemispherical shaped but had greater curvature; subsequently, their Laplace pressures were even lower.) With a vertical length of 10 cm, the water inside these tubes generated a hydrostatic pressure of nearly 1000 Pa. Thus, other factors must play a role in preventing drainage. The shape of the meniscus at the bottom of the tubes that retained liquids provides additional evidence that surface tension is not acting alone. The smallest tubes had convex menisci (bulging outward for the bottom of the tube). As the diameter of tubes was increased, their curvature decreased, giving way to concave interfaces for the largest diameter tubes that retained liquid. With this change in curvature, the Laplace pressure pushing upward against the liquid diminished and
Figure 1. Image of a clean glass tube (D = 15.7 mm) containing water in the tensile test machine. The top end of the tube is plugged. The bottom end is immersed into a water reservoir. The water contains red food coloring to improve contrast. tubes were pulled upward from the reservoir at velocities (v) of 10, 100, 1000, or 2000 mm/min. The ascent of the tube was stopped when its bottom was a short distance, usually 10 mm, above the liquid reservoir. The observed behavior was noted. Images and videos of the resulting phenomena were captured with a Nikon digital SLR camera. To improve contrast, red food coloring (McCormick Red) was added to some of the liquids. It made the liquids easier to see but did not affect the properties of the liquid nor the observed behavior.
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RESULTS AND DISCUSSION Consider the clean glass tube (D = 15.7 mm) shown in Figure 1. It is completely full of colored water. The top end of the tube is plugged. The bottom end is immersed into a water reservoir. The tube does not spontaneously drain. With a proper seal, water remains abutted against the stopper indefinitely. On the other hand, if the stopper were dislodged or improperly seated to allow leakage, then the tube would drain from the top down. This was true for all of the combinations of liquids and tubes types, regardless of their diameter. 12904
DOI: 10.1021/acs.langmuir.7b03247 Langmuir 2017, 33, 12903−12907
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Langmuir
(∼3×) considering that the perfluoroether oil has one-quarter the surface tension of water and is nearly twice as dense. Influence of Tube Material and Wettability. Table 3 also lists results from drainage experiments performed with water and plastic tubes. The various tubes had receding contact angles between ∼0° and 99°. The drainage was only weakly dependent on the wettability of the tubes, where the observed onset of flow occurred between 9.5 and 15.7 mm, in agreement with the predicted values from eq 3 of Dc = 10.2−12.7 mm. Tubes with an Angled Bottom. Nearly all of the tubes tested here had flat ends cut normal to their length. Several experiments were done with PVC tubing that had its bottom cut at a 45°, such that it resembled the sharp end of a hypodermic needle. These angled tubes were tested with water. Their behavior was similar to the flat bottomed tubes, except that drainage occurred for smaller diameter tubes, D > 6.4 mm. The presence of a diagonal cut increases the radius of curvature of the interface and the hydrostatic pressure across the opening. Both of the factors are expected to make the tube more susceptible to drainage. Influence of Viscosity, Lift Velocity, and Height. The rate at which liquid-filled tubes were lifted from the reservoir had a subtle influence on drainage. Tubes with diameters slightly larger than Dc (D/Dc ≈ 1−1.3) more frequently retained liquid when lifted at a faster rate. This is somewhat surprising, as the greater acceleration/deacceleration at the beginning/end of the lift provided a greater jolt to the liquid. It may be that inertia carried more liquid within the capillary bridge between the bottom of the tube and the reservoir, producing a more convex curvature and thereby stabilizing the meniscus. The notion that greater stability from faster lift rates is also supported by the size and shape of the menisci of the nondraining tubes. Figure 3 shows side view images of 9.7 mm
inverted, pulling downward on the liquid inside the largest retaining tubes. Why do these relatively large diameter tubes not drain? It is hypothesized that if the air−liquid interfaces at the bottom of the raised vertical tube is sufficiently stable, then atmospheric pressure pushing upon it prevents the liquid from draining. Otherwise, if the inner diameter of the tube is too large, the air−liquid interface is sufficiently unstable that local deformation can initiate drainage. There is rich literature on the dripping and jetting that occurs as flowing liquids exit tubes or other orifices.7,8 The first papers on this subject date from 19th century work of Savart, Plateau, and Rayleigh.9−12 However, it seems that almost nothing has been published on the stability of stationary liquids in plugged vertical tubes. Thus, a simple expression was derived here for estimating the critical orifice diameter (Dc) for the onset of the drainage ⎡ ⎧ ⎪ ⎢ Dc = ⎢(3 + cos θr)⎨3 + cos θr ⎪ ⎢⎣ ⎩ ⎤1/2 2 ⎤−1/2 ⎫ ⎡ ⎞ ⎛ ⎪ ⎥ γ 1 1 1 ⎬ ⎥ +⎢ −⎜ − ⎟⎥ ⎢⎣ 4 3 + cos θr ⎠ ⎥⎦ ⎝2 ⎪ ρg ⎥ ⎭ ⎦
(3)
where γ and ρ are the surface tension and density of the liquid, θr is the receding contact angle, and g is the acceleration due to gravity. Details of the derivation are given in the Appendix. Various Liquids. For water in clean, wettable glass tubes, eq 3 predicts Dc = 13.6 mm, which agrees reasonably well with the experimentally observed behavior, 9.7 mm ≤ Dc ≤ 15.7 mm. The experiments described above for water were repeated with other liquids in clean glass tubes. Results for the various liquids, including water, are listed in Table 3. The other liquids behaved Table 3. Measured and Predicted Diameters (Dc) where Drainage Occurred for the Various Liquid/Tube Combinations
Figure 3. Side view images of 9.7 mm clean glass tubes pulled from a water reservoir at various velocities. (a) v = 10, (b) 100, and (c) 2000 mm/min.
measured Dc (mm) drains? no
yes
predicted Dc (mm) eq 3
water glycerol ethylene glycol isopropanol perfluoroether oil water
≤9.7 ≤9.7 ≤7.9 ≤5.6 ≤2.7
≥15.7 ≥12.7 ≥9.7 ≥9.7 ≥5.6
13.6 11.5 10.4 8.5 4.8
≤11.1
≥12.7
12.7
water
≤7.9
≥12.7
11.6
water
≤9.5
≥12.7
10.2
tube material clean clean clean clean clean
glass glass glass glass glass
polyamide 66 (PA66) polyvinyl chloride (PVC) polyethylene (PE)
liquid
clean glass tubes pulled from a water reservoir at various velocities. The slowest lift rate yielded a less stable, concave meniscus; whereas, at the fastest lift rate produced a more stable, convex meniscus. While viscosity, lift rate, and lift height had little influence on the onset of drainage, they had a much greater effect on the rate and manner in which liquids drained. Liquids with higher viscosities, such as glycerol and the perfluoroether oil, drained more slowly than lower viscosity liquids. The height that the tube was lifted above the liquid reservoir also affected the rate and mode of drainage. If tubes were lifted very quickly from the reservoir, liquid often drained continuously. On the other hand, if a tube was lifted sufficiently slowly, then drainage occurred in intervals. Flow was momentarily stopped as liquid exiting the tube recreated a capillary bridge with the reservoir. Changing the rise height produced similar behavior: lifting tubes to a height >10 mm assured continuous drainage, whereas a lift height ≤10 mm often led to periodic drainage.
similarly. When tubes were lifted vertically, a capillary bridge formed and then broke, creating a meniscus on the bottom of the tube. Smaller diameter tubes retained their liquids. The transition between drainage and retention was gradual. Tubes with D ≈ Dc were easily perturbed. Tubes with D > Dc drained from the bottom up, ranging from D > 9.7 mm for water to D > 2.7 mm for the perfluoroether oil. The difference in critical tube diameters between these two liquids was surprisingly small 12905
DOI: 10.1021/acs.langmuir.7b03247 Langmuir 2017, 33, 12903−12907
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Langmuir Influence of Tube Length and Hydrostatic Pressure. If the stated hypothesis that atmospheric pressure pushing upon a stable liquid meniscus prevents drainage is true, then in the absence of shock or vibration, liquid should be retained in tubes that are much taller than 10 cm. Thus, a few drainage experiments were performed with longer tubes at the historic fire tower in Pequot Lakes, MN. That day, the temperature and atmospheric pressure were 21 °C, 100.5 kPa. Approximately 10 L of tap water were poured into a 40 L bucket. Red food coloring (McCormick Red) was added to the water. Starting with one end, 12 m of PVC with D = 6.4 or 15.9 mm was slowly pushed below the surface of the water in the bucket such that the air inside the tubing was displaced by the water. At this point, the tubing contained no air or bubbles; its entire length was filled with red water. One end of PVC tubing was plugged with a silicone stopper and raised from the bucket to heights as high as 11 m. The open end was kept under water. At several intervals before reaching the maximum height, the top, plugged end was held static. With the open end of the tube submerged, the tubes behaved as a classic water barometer. Atmospheric pressure pushed on the free surface of water in the bucket. That pressure was transmitted through the water, preventing the water from flowing out of the tubes. Below 10 m where atmospheric pressure was greater than hydrostatic pressure, the liquid did not drain. The top of the liquid column inside the tube remained in contact with the stopper. If raised above 10 m, the water separated from the stopper. The greater the height above 10 m, the longer the gap. However, the height of the liquid column did not change. If the top of the tubing was lowered back to 10 m or less, the water closed the gap. (After the initial creation of the gap, the void did not completely close; the vacuum that was initially created was partially “filled” by water vapor.) Periodically during lifting, the open end of the tubes was oriented vertically and gently raised above the surface of the red water in the bucket. For the smaller diameter PVC tubing with D = 6.4 mm, water did not flow from the tube. Thus, it was concluded that in the absence of shock or vibration, sufficiently small diameter tubing could indeed retain a 10 m column of water. On the other hand, when the larger diameter PVC tubing with D = 15.9 mm was lifted from the bucket, it immediately began to drain from the bottom up.
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CONCLUSIONS Vertical tubes pulled from liquid reservoirs retained liquid when their diameters were sufficiently small to inhibit distortion of the air−liquid interface at their open bottom end. Atmospheric pressure pushing on their air−liquid interface prevented drainage from tubes as tall as 10 m. If surface tension were acting alone, the transition between draining and retention would have expected to occur at much smaller tube diameters.
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Figure 4. Depiction of tubes that were filled with liquid, plugged at the top end, oriented vertically, and then immersed in a liquid reservoir. When lifted from the reservoir, the air−liquid interface at the bottom of smaller diameter tubes is stable, and the tube does not drain. (a) A tube with a stable convex meniscus. (b) A tube with a less stable concave meniscus. (c) If the interface is sufficiently long (i.e., the tube diameter is sufficiently large), a Rayleigh instability forms, and then (d) air enters the tube and the tube drains from the bottom up. (e) Crosssectional view of the critical meniscus shape where the hydrostatic pressure and Laplace pressure are equal, Δph = ΔpL, for θr = 0°. (f) Cross-sectional view of the critical meniscus shape where Δph = ΔpL for θr > 0°.
APPENDIX
Derivation of Model for Drainage from a Tube
Consider the liquid-filled tubes depicted in Figure 4. They are circular with inner diameters of D and exhibit advancing and receding contact angles of θa and θr. Their upper end is plugged. The tubes are oriented vertically and pulled upward from the reservoir to expose their open, bottom end to atmospheric pressure. If their diameter is sufficiently small, then liquid will not drain. For the smallest diameter tubes, the resulting meniscus at the bottom opening is convex, Figure 4a.
As tubes of progressively larger diameters are lifted from their liquid reservoir, the shape of the meniscus changes. As tube diameter approaches the critical limit for drainage, the meniscus takes a sinusoidal shape, where liquid protrudes outward from one side of the tube end and inward from the other, Figure 4c. 12906
DOI: 10.1021/acs.langmuir.7b03247 Langmuir 2017, 33, 12903−12907
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Langmuir ⎡ ⎧ ⎪ ⎢ Dc = ⎢(3 + cos θr)⎨3 + cos θr ⎪ ⎢⎣ ⎩
The minimum critical diameter (Dc) required to prevent drainage can be estimated by equating the opposing Laplace pressure (ΔpL) and hydrostatic pressure (Δph) acting on the bulge protruding from the bottom of the tube Δph = ΔpL
(4)
Assume that the maximum Laplace pressure (ΔpL) occurs where the bulge expands to take a shape that can be approximated as hemi-ellipsoid2,5 ΔpL =
γ γ + R1 R2
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1 D 4
ORCID
C. W. Extrand: 0000-0002-0330-9236 Notes
The author declares no competing financial interest.
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(6)
ACKNOWLEDGMENTS The author wishes to thank M. Acevedo, D. Burdge, L. Castillo, J. Doyon, K. Long and D. Meyer, K. Sekeroglu, K. Switalla, K. Vangsgard, J. Wittmayer, and G. Zeien for their help and comments. Also, thanks to executive leadership at CPC and Dover Corporation their continued support of research at CPC.
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3 D 4
(8)
(The minor radius of curvature (R1) of the protruding bulge lies within the drawing plane of Figure 4e,f, whereas the major radius of curvature (R2) is normal to the drawing.) Combining eqs 4−8 gives an equation for estimating the critical diameter (Dc) required to initiate drainage of liquids from tubes where θr = 0° 1/2 ⎡ γ ⎤ Dc = 4⎢(1 + 3−1/2 ) ⎥ ρg ⎦ ⎣
(9)
For the case where the receding contact angle of θr > 0°, the curvature of the lower bulge is less pronounced than a wettable tube of the same diameter, Figure 4f. Here, the principal radii of curvature are approximated as R1 =
1 D 3 + cos θr
REFERENCES
(1) Wu, S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982. (2) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (3) Weast, R. C. Handbook of Chemistry and Physics, 73rd ed.; CRC: Boca Raton, FL, 1992. (4) DuPont Krytox VPF Vacuum Pump Fluids, Product Information; H58530; E. I. du Pont de Nemours and Company, December, 2012. (5) Laplace, P. S. Mécanique Celeste; Courier: Paris, 1805; Vol. 4, Supplément au Xe Livre. (6) Padday, J. F.; Pitt, A. R. The Stability of Axisymmetric Menisci. Philos. Trans. R. Soc., A 1973, 275 (1253), 489−528. (7) Pomeau, Y.; Villermaux, E. Two Hundred Years of Capillarity Research. Phys. Today 2006, 59 (3), 39−44. (8) Lin, S. P.; Reitz, R. D. Drop and Spray Formation from a Liquid Jet. Annu. Rev. Fluid Mech. 1998, 30 (1), 85−105. (9) Savart, F. Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. (Paris) 1833, 53, 337−386. (10) Plateau, J. Experimental and Theoretical Statics of Liquids; Gauthier-Villars: Paris, 1873; Vol. 1. (11) Rayleigh, L. On The Instability Of Jets. Proc. London Math. Soc. 1878, 1−10 (1), 4−13. (12) Rayleigh, L. On the Capillary Phenomena of Jets. Proc. R. Soc. London 1879, 29 (196), 71−97.
(7)
and R2 =
AUTHOR INFORMATION
*E-mail:
[email protected]. Tel: 1-651-9991859.
(5)
where ρ is the density of the liquid and g is the acceleration due to gravity. The wettability of the tubes affects the shape of the bulge. For tubes with a receding contact angle of θr = 0°, Figure 4e, it is assumed that the principal radii of curvature are R1 =
(3)
Corresponding Author
where γ is the surface tension and R1 and R2 are its principal radii of the hemi-ellipsoidal bulge. Conversely, the critical hydrostatic pressure (Δph) can be estimated as
Δph = ρgR1
⎤1/2 2 ⎤−1/2 ⎫ ⎡ ⎞ ⎛ ⎪ ⎥ γ 1 1 1 ⎬ ⎥ +⎢ −⎜ − ⎟⎥ ⎢⎣ 4 3 + cos θr ⎠ ⎥⎦ ⎝2 ⎪ ρg ⎥ ⎭ ⎦
(10)
and 1/2 ⎡ ⎞2 ⎤ ⎛1 1 1 R2 = ⎢ − ⎜ − ⎟⎥ D ⎢⎣ 4 3 + cos θr ⎠ ⎥⎦ ⎝2
(11)
Combining eqs 4−6, 10, and 11 yields the following equation for estimating the critical diameter (Dc) required to initiate drainage of liquids from tubes where θr ≥ 0° 12907
DOI: 10.1021/acs.langmuir.7b03247 Langmuir 2017, 33, 12903−12907
