Drainage from a Fluid-Handling Component with Multiple Orifices due

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Drainage from a Fluid-Handling Component with Multiple Orifices due to Inclination or Rotation C. W. Extrand* CPC, 1001 Westgate Drive, St. Paul, Minnesota 55114, United States ABSTRACT: The onset of drainage of liquids from fluid-handling components with two or more orifices was evaluated. The components were filled with water, ethylene glycol, or ethyl alcohol and oriented vertically with their orifices facing downward. The lower end of the components was slowly raised toward the horizon. No flow occurred until a critical angle of inclination was reached. The resistance to drainage was greatest for small, closely spaced orifices and declined precipitously as the size and spacing of the orifices increased. The onset of drainage was successfully modeled as a balance between the hydrostatic pressure in the bulk liquid and the Laplace pressure of the air−liquid interfaces present within the orifices.

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INTRODUCTION In many industrial processes, it would be desirable to avoid liquid spilling or draining from a fluid-handling component with open orifices. In some cases, spills or drainage may be a minor inconvenience. In others, it may be hazardous. Errant liquids can damage electronics, corrode materials, or harm people. With the aim of minimizing spillage or drainage from fluid handing components, several studies were recently undertaken. In the first, the onset of drainage was examined as vertically oriented tubes with an open bottom end raised from a liquid reservoir.1 In the second, vertically oriented tubes or bottles with a single orifice were raised from a reservoir; then, the open orifice at the bottom of the tube or bottle was inclined toward the horizon to further challenge its resistance to drainage.2 It was observed that tubes and bottles with a relatively large orifice were able to resist drainage through a combination of a stable air-liquid interface and atmospheric pressure counteracting the hydrostatic pressure of liquid in the tube. Both of these studies involved a model component with a single orifice, which is a reasonable approximation of a simple monolithic component, such as a tube or a fitting. However, in practice, more complex fluid-handling components have multiple orifices in close proximity. Multiple orifices may be needed to accommodate moving components found in couplers or valves to meet performance requirements (e.g., strength or flow) or to allow for economical manufacturability. Thus, in this work, model components with two or more orifices were evaluated for their ability to resist spillage and drainage when rotated or inclined from a stable vertical orientation. Experimental results are compared to models that assume that the onset of drainage is determined by an interplay of hydrostatic and Laplace pressures.

cylindrically shaped, as depicted in Figure 1. One end is closed and sealed. The other has two circular orifices of diameter D.

Figure 1. Model fluid-handling component with two orifices of diameter D and end-to-end spacing of L. (a) Sky view of the bottom end of a vertical component that has been rotated to an angle of ϕv. If the orifices are oriented parallel to the plane of inclination, then ϕv = 0°; if perpendicular to the plane of inclination, ϕv = 90°. (b) Side view of the component that has been inclined from its original vertical orientation to an angle of α. If the component is oriented vertically, then α = 0°.

The distance between the far ends of the orifices is L. The material of construction is assumed to be relatively hydrophobic with advancing and receding contact angles of θa and θr. The component is filled with a liquid that has a surface tension of γ and a density of ρ and then oriented vertically with its orifices facing downward. If the orifices are sufficiently small, liquid will not drain.

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THEORY Inclination of a Component with Two Orifices. For simplicity, assume that the fluid-handling component is © XXXX American Chemical Society

Received: December 18, 2017 Revised: March 6, 2018

A

DOI: 10.1021/acs.langmuir.7b04271 Langmuir XXXX, XXX, XXX−XXX

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Langmuir The component can be rotated around its vertical axis prior to inclination. The vertical angle of rotation (ϕv) determines the relative height between the orifices during inclination. If ϕv = 90°, then the orifices maintain the same relative height during inclination. On the other hand, if ϕv < 90°, an inclination creates a difference in vertical height (h) between the two orifices, which in turn generates a hydrostatic pressure (Δph) between them Δph = ρgh

Inclination causes the air intruding into the upper orifice to form a bubble and grow. For a component constructed from a material with θr ≤ 90°, it is anticipated that the growing bubble will remain pinned at the edge of the orifice until its buoyancy force exceeds that of the capillary pinning force. Liquid can only flow out of the component if the bubble at the upper orifice detaches and allows air to flow in. The maximum buoyancy force of the bubble ( f b) at its critical radius of curvature for detachment (Ru) can be approximated as

(1)

where g is the acceleration because of earth’s gravitational field (g = 9.81 m/s2). h is related to the end-to-end distance between orifices (L), the cosine of the vertical rotation angle (ϕv), and the sine of the inclination angle (α) h = L ·cos ϕv ·sin α

fb =

4π ρgR u 3 3

(4)

and the corresponding capillary force ( fc) as

fc = πDγ

(2)

(5)

Equating eqs 4 and 5 allows for estimation of the critical radius of curvature for detachment (Ru) in terms of the upper orifice diameter and the properties of the liquid5−7

The increase in local hydrostatic pressure during inclination causes the liquid in the orifices to distort, as shown in Figure 2.

⎛ 3 Dγ ⎞1/3 Ru = ⎜ ⎟ ⎝ 4 ρg ⎠

(6)

In turn, the Laplace pressure at the upper orifice can be estimated as ΔpL,u =

⎛ 4 ρg ⎞1/3 2γ = 2γ ⎜ ⎟ Ru ⎝ 3 Dγ ⎠

(7)

Protrusion of liquid at the lower orifice also generates a Laplace pressure. The maximum Laplace pressure at the lower orifice (ΔpL,l) can be estimated from the critical radius of curvature (Rl) of the protruding liquid and its surface tension (γ)3,4

ΔpL,l =

2γ Rl

(8)

If it is assumed that the shape of the liquid bulging from the orifice can be approximated as a section of sphere, then its radius of curvature (Rl) and the orifice diameter (D) are related through the sine of the advancing contact angle (θa) between the liquid and the component Figure 2. Side cross-sectional view of the inclination of a liquid-filled component with two circular orifices of diameter D and end-to-end spacing of L. (a) Component is oriented vertically (α = 0°). (b) As the component is inclined, liquid bulges outward from the lower orifice and inward from the upper orifice. (c) Critical inclination angle (αc) beyond which liquid will drain from the component.

sin θa =

(9)

Combining eqs 8 and 9 gives the maximum Laplace pressure of the lower orifice in terms of its diameter, the surface tension of the liquid, and the advancing contact angle of the component

ΔpL,l =

The liquid in the upper orifice intrudes inward, whereas the liquid in the lower orifice protrudes outward. The curvature of the liquid creates a Laplace pressure that counteracts the hydrostatic pressure.3,4 If the hydrostatic pressure because of inclination does not exceed the maximum attainable Laplace pressure of a given liquid−component combination, the liquid will not drain. The critical inclination angle required to initiate flow (αc) can be estimated by equating the counteracting hydrostatic (Δph) and Laplace pressures at the upper and lower orifices (ΔpL,u and ΔpL,l)

Δph = ΔpL,u + ΔpL,l

D/2 Rl

4γ ·sin θa D

(10)

Combining eqs 1−3, 7, and 10 and manipulating them algebraically yield a relatively simple equation that allows for estimation of the critical inclination angle for drainage (αc) of a two-orifice component in terms of orifice diameter, orifice spacing, and the properties of the liquid and solid sin αc = 2

⎡ ⎛ ⎞1/3⎤ γ ⎢ 2 sin θa + ⎜ 4 ρg ⎟ ⎥ ρgL ·cos ϕv ⎢⎣ D ⎝ 3 Dγ ⎠ ⎥⎦

(11)

Equation 11 is valid for materials of construction, where θa < 90°. On the other hand, if θa ≥ 90°, then the maximum Laplace

(3) B

DOI: 10.1021/acs.langmuir.7b04271 Langmuir XXXX, XXX, XXX−XXX

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Langmuir pressure achieved at the lower orifice occurs where Rl = D/2 and subsequently, eq 11 reduces to sin αc = 2

⎡ ⎛ ⎞1/3⎤ γ ⎢ 2 + ⎜ 4 ρg ⎟ ⎥ ρgL ·cos ϕv ⎢⎣ D ⎝ 3 Dγ ⎠ ⎥⎦

Combining eqs 1, 3, 7, 10, and 15 gives the critical inclination for drainage (αc) of a fluid-handling component with three equilateral orifices and θa < 90° as ⎡ ⎛ 4 ρg ⎞1/3⎤ 4 γ ⎢ 2 sin θa sin αc = +⎜ ⎟ ⎥ 3 ρgL ⎢⎣ D l ⎝ 3 Duγ ⎠ ⎥⎦

(12)

Other Types of Components. Several other types of orifices and patterns were considered. They are shown in Figure 3. These included components with orifices of two

(16)

If θa ≥ 90°, then ⎡ ⎛ 4 ρg ⎞1/3⎤ 4 γ ⎢2 sin αc = +⎜ ⎟ ⎥ 3 ρgL ⎢⎣ D l ⎝ 3 Duγ ⎠ ⎥⎦

(17)

Inclination of a Component with a Slot with Circular Ends. Figure 3c shows the bottom end of a component with one slot of width D and length L. A slot differs from two orifices in several ways. The capillary pressure generated by liquid bridging in a slot is half that of a circular orifice of the same width ΔpL,l =

A smaller capillary attachment force translates into a smaller bubble ⎛ 3 Dγ ⎞1/3 Ru = ⎜ ⎟ ⎝ 8 ρg ⎠

ΔpL,u =

sin αc = 2

3 L ·sin α 2

(21)

⎡ ⎛ ⎞1/3⎤ γ ⎢ sin θa + ⎜ 8 ρg ⎟ ⎥ ρgL ·cos ϕv ⎢⎣ D ⎝ 3 Dγ ⎠ ⎥⎦

(22)

If θa ≥ 90°, then (13)

⎡ ⎛ 8 ρg ⎞1/3⎤ γ 1 ⎢ +⎜ sin αc = 2 ⎟ ⎥ ρgL ·cos ϕv ⎢⎣ D ⎝ 3 Dγ ⎠ ⎥⎦

(23)

Inclination of a Component with Two Slots with Circular Ends. Figure 3d shows the bottom end of a component with two slots of width D and length L. The distance between the far sides of the slots is Lb. If a two slot component is inclined with its slots parallel to the horizon, then the bubble that forms at the upper slot is pinned along its perimeter and the resulting capillary force is

(14)

Inclination of a Component with Three Equilateral Orifices. Figure 3b shows the bottom end of a component with three circular orifices of diameter D arranged equilaterally, all equally spaced by an end-to-end distance of L with two of them aligned perpendicular to the plane of inclination. Here, the vertical height between the orifices is h = L ·sin 60°·sin α =

⎛ 8 ρg ⎞1/3 2γ = 2γ ⎜ ⎟ Ru ⎝ 3 Dγ ⎠

Combining eqs 1−3, 18, and 21 produces an expression that predicts the critical inclination for drainage (αc) of a component with one slot of width D and length L, where θa < 90°

where θa < 90°. If θa ≥ 90°, then ⎡ ⎛ 4 ρg ⎞1/3⎤ γ ⎢2 +⎜ ⎟ ⎥ ρgL ⎢⎣ D l ⎝ 3 Duγ ⎠ ⎥⎦

(20)

and a larger capillary pressure at the upper end of the slot

different diameters, components with more than two orifices, as well as parallel-sided orifices or slots. These also can be modeled by assuming that the drainage is controlled by competing hydrostatic and Laplace pressures, eq 3, but vary with the type and number of orifices. Inclination of a Component with Two Orifices of Different D. Figure 3a shows the bottom end of a fluidhandling component that has two circular orifices of different diameters. If the lower orifice during an inclination has a diameter of Dl and the upper one has a diameter of Du, then assuming that ϕv = 0°, eq 11 takes the following form

sin αc = 2

(18)

The bubble that forms at the upper end of the slot during inclination experiences less pinning than a circular orificethe contact line there is a semicircle π fc,u = Dγ (19) 2

Figure 3. Sky view of the bottom end of other types of component patterns that were considered. (a) Component with orifices of two different diameters, where the lower orifice has a diameter of Dl and the upper one has a diameter of Du. (b) Component with three circular orifices of diameter D arranged equilaterally, all equally spaced by an end-to-end distance of L, with two of them aligned perpendicular to the plane of inclination. (c) Component with one slot of width D and length L. (d) Component with two slots of width D and length L. The distance between the far sides of the slots is Lb.

⎡ ⎛ 4 ρg ⎞1/3⎤ γ ⎢ 2 sin θa sin αc = 2 +⎜ ⎟ ⎥ ρgL ⎢⎣ D l ⎝ 3 Duγ ⎠ ⎥⎦

2γ sin θa D

fc,u = γ[2(L − D) + πD]

(24)

A larger perimeter results in a larger bubble than a circular orifice

(15) C

DOI: 10.1021/acs.langmuir.7b04271 Langmuir XXXX, XXX, XXX−XXX

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Langmuir ⎛ 3 γ ⎞1/3 [2(L − D) + πD]⎟ Ru = ⎜ ⎝ 4π ρg ⎠

sin ϕh,c = 2

(25)

and a lower Laplace pressure ⎛ 4π ρg ⎞ 2γ 1 = 2γ ⎜ ⎟ Ru ⎝ 3 γ 2(L − D) + πD ⎠

(26)

sin ϕh,c = 2

Combining eqs 1−3, 18, and 26 yields an expression that predicts the critical inclination for drainage (αc) of a component with two slots of width D and length L and a distance between them of Lb where θa < 90° sin αc = 2

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(27)

EXPERIMENTAL DETAILS

Table 1. Surface Tension (γ) and Density (ρ) of the Liquids along with Their Advancing and Receding Contact Angles (θa and θr) on PE (28)

Rotation of a Horizontal Component with Two Orifices. Figure 4 shows the side views of a model fluid-

liquid

γ (mN/m)

ρ (kg/m3)

θa (deg)

θr (deg)

ethyl alcohol (A) ethylene glycol (G) water (W)

22 47 72

789 1110 998

27 61 95

0 56 79

Solids. The capped bottles used to represent a fluid-handling component were purchased from Corning (CentriStar 50 mL selfstanding centrifuge tube). The bottles were composed of polypropylene and the cap of polyethylene (PE). The inner diameter of the bottle and cap was approximately 25 mm. Circular orifices were machined in the PE caps using an end mill installed in a Bridgeport mill. The diameter of the orifices (D, Dl, or Du) ranged from 1.6 to 6.4 mm (1/16 to 1/4 in. diameter). The distance between the far ends of the orifices (L) spanned from 6.4 to 19.1 mm. Alternatively, one or two slots with circular ends were machined into a cap. The width of the slots (D) ranged from 1.6 to 6.4 mm and their length from 6.4 to 19.1 mm. In the cases where two slots were machined, the far edge distance between them is denoted as Lb. The uncertainty in the diameter of the orifices and slots was ≤± 0.05 mm, and the distance between them, ≤± 0.2 mm. Contact Angles. Advancing and receding contact angles were measured using a digital goniometer (Kyowa DMs-401). A PE cap was placed on the stage of the goniometer. To measure an advancing contact angle (θa), a 10 μL water drop was deposited incrementally on the cap, and an image of the resulting sessile drop was captured. To measure a receding contact angle (θr), water was withdrawn from a 10 μL 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 sessile drop images; then, θa and θr were measured directly. Advancing and receding contact angles (θa and θr) of the various liquid−PE combinations are listed in Table 1. Values of θa ranged from 95° for water to 27° for ethyl alcohol (A). Standard deviation and uncertainty in the contact angle measurements were generally ±2°, but was greater (∼±5°) for near-zero values.9,10 Drainage Experiments. Components were filled with liquid and capped. The component was oriented vertically with the orifices facing downward. Unless stated otherwise, the components were oriented such that their two orifices were parallel to the axis of inclination (ϕv = 0°). Components were slowly inclined until the liquid began to flow.

Figure 4. Side views of a model-fluid handling component with two circular orifices of diameter D with end-to-end spacing of L. It is oriented horizontally with its orifices facing outward. (a) Its orifices are oriented parallel to the horizon, ϕh = 0°. (b) Component has been rotated relative to the horizon through an angle of ϕh.

handling component that is oriented horizontally with its orifices facing outward. It has two circular orifices of diameter D with end-to-end spacing of L. If the orifices are relatively small and the component is oriented with its orifices parallel to the horizon (ϕh = 0°), it will not drain. Rotating the component relative to the horizon through an angle of ϕh will increase the height (h) between the orifices by h = L ·sin ϕh

(31)

Liquids. The liquids used were ethyl alcohol (“A”, Sigma-Aldrich, ACS reagent, ≥99.5%), ethylene glycol (“G”, Fisher Scientific, BP2301), and deionized water (“W”). Values of their surface tension (γ) and density (ρ), listed in Table 1, were taken from the literature.4,8 The uncertainty of γ and ρ was estimated to be ±1 mN/m and ±2 kg/m3.

If θa ≥ 90°, then γ sin αc = 2 ρgL b ·cos ϕv ⎡ ⎞1/3⎤ ⎛ 1 ⎢ 1 + ⎜ 4π ρg ⎟ ⎥ ⎢⎣ D ⎝ 3 γ 2(L − D) + πD ⎠ ⎥⎦

⎡ ⎛ 3 ρg ⎞1/3⎤ γ ⎢2 +⎜ ⎟ ⎥ ρgL ⎢⎣ D ⎝ 4 Dγ ⎠ ⎥⎦

The previous analyses assumed that components were constructed from a material with θr ≤ 90°. In the rarer case where θr > 90°, it is hypothesized that the bubble at the upper orifice behaves differently. In the Appendix, this scenario is analyzed for inclination of a two-orifice component.

⎡ γ ⎢ sin θa ρgL b ·cos ϕv ⎢⎣ D

⎛ 4π ρg ⎞1/3⎤ 1 +⎜ ⎟ ⎥ ⎝ 3 γ 2(L − D) + πD ⎠ ⎥⎦

(30)

Equation 30 is valid for materials of construction where θa < 90°. If θa ≥ 90°, eq 30 reduces to

1/3

ΔpL,u =

⎡ ⎛ 3 ρg ⎞1/3⎤ γ ⎢ 2 sin θa +⎜ ⎟ ⎥ ρgL ⎢⎣ D ⎝ 4 Dγ ⎠ ⎥⎦

(29)

Assume that the mechanism is similar to inclination Laplace pressures at the upper and lower orifices counteract hydrostatic pressure. Here, rotation causes a bubble to form in the upper orifice. The growing bubble remains pinned until its buoyancy force exceeds the capillary pinning force. Liquid flows out when bubbles at the upper orifice detach and air flows in. With these assumptions, eqs 1, 3, 7, 10, and 29 can be combined to yield an equation for estimating the critical rotation angle for drainage (ϕh,c) of a two-orifice component D

DOI: 10.1021/acs.langmuir.7b04271 Langmuir XXXX, XXX, XXX−XXX

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Langmuir At that instant, an image was captured with an Apple iPhone 6s. Critical values of the angle of inclination (αc) were measured from the subsequent images. Five replicates were performed for each liquid− pattern combination. Averages and standard deviations were computed and are shown later in tables and graphs. All measurements were made at (22 ± 1) °C. Uncertainty. Estimates of uncertainty in αc were made using standard error propagation techniques involving partial derivatives.11 The uncertainty in αc is expected to be 90°, it is anticipated that the drainage behaves differently. This scenario is analyzed below. Upon inclination of a two-orifice component with θr > 90°, air intrudes at the upper orifice and forms a bubble. The bubble grows but remains pinned at the edge the orifice until the liquid establishes its receding contact angle on the inner surface of the component. At this critical juncture, the radius of curvature of the upper bubble (Ru) is at its minimum Ru =

Article

(34)

Combining eqs 1−3, 33, and 34 yields an equation for estimating the critical inclination angle for drainage (αc) of a two-orifice component, where both θa and θr > 90° γ sin αc = 4 (1 + sin θr) ρgDL ·cos ϕv (35) Here, drainage is initiated by an adhesive failure between the upper bubble and the solid. Retraction of the contact line at the upper orifice causes a decrease in the Laplace pressure, and the drainage commences. All else being equal, it is expected that a two-orifice component with θr < 90° will exhibit more resistance to drainage than the same component with θr > 90°. For example, consider a water-filled component with D = 4.8 mm, L = 15.9 mm, and ϕv = 0°. If θa ≥ 90° and θr < 90°, then the component would be expected to drain when inclined to αc = 44°. In contrast, for a more hydrophobic component where θr = 150°, less inclination would be required to initiate drainage, αc = 36°.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 1-651999-1859. ORCID

C. W. Extrand: 0000-0002-0330-9236 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author thanks L. Hoelscher for helping with machining of the plastic caps and K. Vangsgard for analyzing the composition of the plastic caps and tubes. Also, thanks to M. Acevedo, L. Castillo, J. Doyon, N. Ly, K. Long, K. Sekeroglu, K. Switalla, K. Vangsgard, and J. Wittmayer for their help and support. G

DOI: 10.1021/acs.langmuir.7b04271 Langmuir XXXX, XXX, XXX−XXX