The impingement zone created by the impingement of two free, thin sheets of liquids was studied using photographic techniques. The experimental parameters studied were the impingement angle, single-sheet velocity, single-sheet thickness at impingement, liquid surface tension, and liquid viscosity. Momentum balances of the impingement are in good agreement with observations that a critical mixed sheet velocity must be achieved in order for the liquid to flow in the backward direction of the mixed sheet. The critical mixed sheet velocity for backflow was found to correlate well with the Taylor–Culick velocity. However, if all of the kinetic energy associated with the single-sheet component velocity that is destroyed upon impingement is dissipated into the mixed sheet, then that critical velocity cannot be achieved, and there will be no backflow from the backward part of the mixed sheet. An unusual result of the study is that the projection of the single sheets onto the mixed sheet at impingement presents as a “skin” that can be clearly seen in photographs or a video. This “skin” feature potentially allows for the use of macro photography to measure the thickness of the single sheet at impingement.
I. INTRODUCTION
Micromixing of liquid reactants undergoing fast reactions is important since it is often the rate-limiting step, especially in applications where high yield is important. Examples include production of dyestuffs that employ fast, multiple-pathway reactions;1,2 reactions that occur in viscous media, such as reaction injection molding; rapid crystallization; injection of hypergolic rocket propellants; and production of pharmaceuticals in high yield.
The impingement of free, thin liquid sheets as proposed by Demyanovich3 was shown by Demyanovich and Bourne4 to yield rapid micromixing of relatively large flow rates of liquids. Micromixing times of 1 ms to several milliseconds were achieved for low-viscosity liquids at flow rates of order liters per minute. For example, two equal, single sheets produced at 1.0 bar, with a flow rate of 2.3 l/min each, were impinged at a 30° angle and at a distance of 2 cm from the origin of each sheet. The impingement of the equal sheets resulted in a mixed sheet that exactly bisected the angle of impingement. At impingement, the single-sheet thickness was about 68 µm and the component velocity (“mixing velocity”) that was destroyed upon impingement was 3.3 m/s. The calculated energy dissipation rate under these conditions was 259 000 W/kg, whose calculation was supported by the results of experiments employing competitive–consecutive reactions whose product distribution is sensitive to micromixing.4 High rates of energy dissipation were achieved not because of a high input of energy (in this example, the pressure drop is only 1 bar) but because the mass of liquid in the impingement zone (calculated as less than 0.002 g) is extremely small relative to the total flow rate of the liquids (4.6 l/min).
Besides the extremely small mass of liquid in the impingement zone, it seemed that the key to achieving rapid micromixing was the destruction of the component velocity under conditions that resulted in an inelastic collision in the impingement zone. Under this assumption, the kinetic energy associated with the destroyed component velocity was dissipated leading to turbulence. The mixing of reactants was then modeled assuming that slabs of half-Kolmogorov thickness were produced, which for the example cited above was calculated to be 0.7 µm. The high energy dissipation rate effectively reduced the scale in which diffusion was important from the scale of the single sheets (68 µm) to the half-Kolmogorov thickness (0.7 µm) on a timescale of the liquid residence time in the impingement zone (21 µs in this example). Once the combined liquids flow away from the impingement zone, turbulence dissipates rapidly since energy is not input anywhere else within the mixed sheet. After several milliseconds, the mixed sheet breaks up into droplets.
For the purposes of estimating the energy dissipation rate for an inelastic collision of impinging sheets, Demyanovich and Bourne5 conducted a very limited study that measured the velocity of the mixed sheet formed by two impinging sheets of milk at 30°, 45°, and 60°. The velocity difference between the mixed sheet and the single sheet provided an estimate of the energy released in the impingement zone.
Other than these three impingement angles, the effects of surface tension, higher impingement angles, distance to impingement, single-sheet velocity, and viscosity on the amount of energy dissipated were not investigated. In particular, no study was made on backflow from the impingement zone.
The purpose of this study is to gain a better understanding of the mechanisms that result in the release of energy when two equal sheets of liquid impinge upon each other. The collision in the impingement zone is investigated with microsecond-flash photography and, to a limited extent, a high-speed video. In particular, collisions that result in backflow were further investigated to determine the critical velocity at which this phenomenon occurs. The critical velocity producing backflow was studied as a function of the impingement angle, single-sheet velocity, single-sheet thickness at impingement, surface tension, and viscosity. Momentum balances of impinging sheets undergoing elastic and inelastic collisions are developed and presented.
II. BACKGROUND
A. Prior research on the impingement of free, thin sheets of liquid
To the best of the author’s knowledge, there is no prior published research on the nature of the collision in the impingement zone of impinging thin sheets of liquids. The prior work of Demyanovich and Bourne5 was limited in scope to estimating the energy released in the impingement zone under inelastic collisions at two impingement angles.
B. Prior research on the impingement of liquid jets
Numerous scientific articles are available on the impingement of free liquid jets. Much of the research, however, is focused toward studying the atomization of impinging free liquid jets with less attention focused on liquid-phase mixing. When two liquid jets collide, a thin sheet in a plane determined by the momentum of the individual jets is formed (the plane is a bisection if the two jets are equal). Researchers in the fluid combustion field have studied the atomization of impinging liquid jets.6–12 Research has also been conducted on the liquid-phase mixing of impinging jets particularly as related to the combustion of hypergolic rocket propellants.13–15
A major difference between the impingement of liquid sheets and liquid jets is the characteristic length scale. The characteristic length scale for impinging sheets is much smaller than that for impinging jets. In the example cited earlier, the thickness of the impinging sheets in the impingement zone was 68 μm. The equivalent jet orifice diameter is 1900 μm. The ratio of these characteristic lengths is approximately a factor of 28.
C. Characteristics of expanding thin liquid sheets
Figure 1 (Multimedia view) is a front-view photograph of two equal single sheets of water impinging at an angle of 45°. Each single sheet is formed in the shape of a fan or arc segment. When the single sheets collide, a well-formed impingement line can be seen that produces a mixed sheet in a plane determined by the momentum of the individual sheets (the plane is a bisection if the two sheets are equal). The associated video includes video clips taken at 1000 frames/s, 10 488 frames/s, and 24 046 frames/s and shows a front view of the sheet impingement at an angle of 45°.
25 μs flash photograph of the front view of impingement of two equal water sheets at an impingement angle of 45° and pressure drop of 0.2 bar. Multimedia view: https://doi.org/10.1063/5.0040336.1
25 μs flash photograph of the front view of impingement of two equal water sheets at an impingement angle of 45° and pressure drop of 0.2 bar. Multimedia view: https://doi.org/10.1063/5.0040336.1
Figure 2 shows a simplified illustration of the impingement of two equal single sheets at an angle of 2β. At impingement, a “mixture” of the two liquids is formed that is also in the form of a thin continuous sheet. Figure 2(a) [which is an illustration of Fig. 1 (Multimedia view)] shows that the impingement zone should theoretically be arc shaped; however, as shown in Fig. 1 (Multimedia view), the edges of the zone are flattened due to the surface tension of the liquid.
(a) Simplistic front view of sheets impinging at the distance Ri from the theoretical sheet origin. The centerline is shown as CL. (b) Simplistic side view of sheets impinging at an angle of 2β. The width of the impingement zone, Δr, was assumed to be set by the projection of the single sheets at impingement onto the half-thickness of the mixed sheet. As drawn, (b) ignores the backward portion of the mixed sheet.
(a) Simplistic front view of sheets impinging at the distance Ri from the theoretical sheet origin. The centerline is shown as CL. (b) Simplistic side view of sheets impinging at an angle of 2β. The width of the impingement zone, Δr, was assumed to be set by the projection of the single sheets at impingement onto the half-thickness of the mixed sheet. As drawn, (b) ignores the backward portion of the mixed sheet.
Figure 2(b) shows a simplistic side view of the impingement of two equal single sheets. An important concept is that the apparent width, Δr, of the impingement zone is actually the projection of the single sheet onto/into the mixed sheet. Demyanovich and Bourne4 assumed that this projection defined the limits of the impingement zone for an inelastic collision and proceeded to calculate energy dissipation rates for inelastic collisions based on this geometry. Furthermore, they ignored any potential backward portion of the mixed sheet, which, at low impingement angles, cannot be readily viewed by the naked eye.
It has previously been shown that for a single sheet, the liquid flows from a theoretical point of origin along radial lines with no crossover of liquid from one radial line to the next.16 The limited photographic study by Demyanovich and Bourne5 seemed to confirm that there is no crossover of liquid from one radial line to the next for the mixed sheet as well. However, this observation should not be assumed to be true in the impingement zone.
The radial velocity of the liquid within the sheet (single or mixed) is also found to remain constant if there is insignificant drag on the sheets from the surrounding atmosphere or gas. Dombrowski and Hooper6 found that single-sheet radial velocities remained constant at ambient gas pressure less than 20 bars. Demyanovich and Bourne5 also found that the velocity of liquid within a mixed sheet was constant (within experimental error) at atmospheric pressure.
The flow within a sheet is given by
where Q is the volumetric flow rate, A is the cylindrical shell area normal to liquid flow, and u is the velocity of the liquid sheet. The area of the cylindrical shell is equal to the spread angle of the sheet, φ, multiplied by the radial distance, R, from the origin of the sheet, and the thickness of the sheet, s, at the radial distance, R. Since the volumetric flow rate and velocity are constant, the area normal to flow must be constant; therefore,
The thickness of an expanding single or mixed sheet is inversely proportional to the distance from the origin of the sheet, or alternatively, as the sheet expands, it also thins. The thinning process continues until the sheet ultimately breaks up into droplets, which is often triggered by disturbances such as waves or the formation of holes in the sheet. Typical residence times of a liquid in the form of a sheet are of order several milliseconds.
III. EXPERIMENTAL
A. Photographic study of the impingement zone
A digital camera with a resolution of 25 megapixels, extension tubes, and a macro lens was used to capture still photographs of the impingement zone. Methods for creating and impinging sheets are detailed elsewhere.3 Table I lists the characteristics of the four different impinging-sheet mixing devices studied.
Characteristics of impinging-sheet devices. do = equivalent orifice diameter; ΔP = pressure drop; Q = single-sheet flow rate; φ = sheet spread angle; Ri = distance from the theoretical single-sheet origin to the impingement zone; si = calculated single-sheet thickness at impingement.
Device . | do (cm) . | ΔP (bar) . | Q (l/min) . | φ (deg) . | Ri (cm) . | si (μm) . |
---|---|---|---|---|---|---|
M1 | 0.12 | 0.30 | 0.5 | 94 | 2.0 | 34 |
M2 | 0.19 | 0.30 | 1.25 | 111 | 2.0 | 87 |
M3 | 0.27 | 0.30 | 2.5 | 123 | 2.0 | 156 |
M4 | 0.38 | 0.30 | 5.0 | 110 | 2.0 | 350 |
Device . | do (cm) . | ΔP (bar) . | Q (l/min) . | φ (deg) . | Ri (cm) . | si (μm) . |
---|---|---|---|---|---|---|
M1 | 0.12 | 0.30 | 0.5 | 94 | 2.0 | 34 |
M2 | 0.19 | 0.30 | 1.25 | 111 | 2.0 | 87 |
M3 | 0.27 | 0.30 | 2.5 | 123 | 2.0 | 156 |
M4 | 0.38 | 0.30 | 5.0 | 110 | 2.0 | 350 |
The only way to adequately study the effect of sheet thickness is to use different devices with different equivalent orifices. Although it is possible to change the single-sheet thickness using a single device, primarily by impinging the sheets at different values of Ri, large variations can only be achieved by using different mixing devices.
Photographs were taken in complete darkness with the shutter speed set to 2 s. Just after the shutter was opened, a microsecond flash (ranging from 2 μs to 15 µs) was discharged to freeze the motion in the impingement zone. Photographs were taken with two types of flash units: overview shots with a xenon flash unit that provided flash times of 15 µs and “freeze-action” shots with a LED flash unit that provided flash times of 2 or 4 µs. Initially, a flash position as close as possible but somewhat in front of the sheets and to the side produced good-quality photographs (Figs. 3–5). However, later, it was found that the optimum location of the flash unit seemed to be as close as possible and directly behind the sheets.
Four-microsecond flash photograph of the front view of impingement of two equal water sheets from the M2 device at an impingement angle of 45° and pressure drop of 0.1 bar [front view illustrated in Fig. 2(a)]. A magnified view of the framed area is shown in Fig. 4.
Magnified front view of the framed area shown in Fig. 3. In this photograph, the projection of the front single sheet is black. The “skin” feature of this sheet projection can be clearly seen in the left-hand portion of this photograph. The grayish-colored liquid above the single-sheet projection is the backward portion of the mixed sheet.
Magnified front view of the framed area shown in Fig. 3. In this photograph, the projection of the front single sheet is black. The “skin” feature of this sheet projection can be clearly seen in the left-hand portion of this photograph. The grayish-colored liquid above the single-sheet projection is the backward portion of the mixed sheet.
Projection of the front single sheet of water from the M2 device viewed as a “skin” in this 15 µs photograph taken at a pressure drop of 0.1 bar and an impingement angle of 45°. Note the holes that have formed in the “skin” revealing the mixed sheet underneath.
Projection of the front single sheet of water from the M2 device viewed as a “skin” in this 15 µs photograph taken at a pressure drop of 0.1 bar and an impingement angle of 45°. Note the holes that have formed in the “skin” revealing the mixed sheet underneath.
The most challenging aspect of photographing the impingement zone was focusing the lens. Waves tend to form on the single sheets, and this can lead to an impingement zone that is not absolutely fixed in one location in space. The depth of field in macro photography is generally very shallow. Although not quantitatively measured in this study, the depth of field was typically only fractions of a millimeter (estimated from the microrail used to move the camera back and forth for focusing). During the course of photographing the impingement zone, it was realized that higher quality photographs could be taken if the sheet velocity was in the lower range. Therefore, most of the photographs were taken at pressure drops less than 0.5 bar.
Single sheets produced from the larger size devices also tended to have more surface waves, which resulted in a loss of sharpness when photographing the backward part of the mixed sheet since the backward part of the mixed sheet was directly behind the front single sheet. At low single-sheet velocity, the distortion did not significantly affect the quality of the photographs; however, at high pressure drops, it became almost impossible to focus the lens on the backward part of the mixed sheet.
A very limited high-speed video at 31 000 frames/s was also taken using the macro lens, albeit at significantly lower resolution than afforded by the digital camera. The issues noted previously with still photography were also present with a high-speed video. In particular, it was difficult to provide sufficient lighting on the impingement zone. Normally, as the frame-rate is increased and the shutter speed for each frame is reduced, the amount of light required to achieve an adequate frame exposure increases significantly. In this investigation, the intent was to keep the exposure time for each frame of video at 25 μs or less. Using a macro lens on the video camera only increased the requirement for lighting. However, adequate lighting was generally achieved from a 300-W equivalent LED bulb placed as close as possible behind the impingement zone.
B. A study of the critical velocity producing liquid backflow from the mixed sheet
During the course of photographic investigation, tests at higher impingement angles showed droplets ejecting from the impingement zone in the opposite direction of the forward flow. This result was expected as it had been observed by Demyanovich and Bourne5 while photographing the impingement zone resulting from the collision of two sheets of milk at a 60° impingement angle. The amount of liquid ejected in the backward direction is typically small compared with the forward flow (Table IV provides calculated ratios of forward to backward flows for a perfectly elastic collision).
A series of tests were then conducted to study the effect of sheet thickness at impingement, impingement angle, liquid surface tension, liquid viscosity, and single-sheet velocity on the critical velocity required to cause ejection of droplets from the backward portion of the mixed sheet. Quantitative criteria for droplet ejection could not be readily established. It was noted that droplets tended to first eject from the impingement zone at the centerline of the mixed sheet. It was decided that tests would be conducted to determine the critical velocity at which droplets are first ejected from the backward portion of the mixed sheet at the centerline. Since the centerline is infinitesimally thin, as a practical matter, the experimental centerline included the area ±25° around the true centerline.
Magnification of the impingement zone was accomplished using macro photography with the digital camera. Digital pressure gauges (accurate to 1% of full scale) were used in order to minimize the error in reading the pressure gauges. The determination of the critical pressure drop (critical velocity) was subject to the author’s ability to determine when this phenomenon occurred, and hence, this will introduce error.
Impingement angles of 45°–120° and distances to impingement (R) of 1.5 cm–2.8 cm were studied. Surfactants were not used to study the effects of surface tension because Dombrowski and Fraser17 found that the very short residence time of liquid in the form of a single sheet was insufficient for the long-chain polymer surfactants to achieve a uniform concentration at the interface between the liquid and air. Surface tension was studied using water, a 10% aqueous solution of isopropyl alcohol (IPA), and, to a limited extent, skim milk. Effects of viscosity were primarily studied using 40% aqueous glycerine and, to a lesser extent, using 10% isopropyl alcohol and skim milk.
Surface tension was measured using a du Noüy tensiometer, viscosity using an Ostwald viscometer, and density using a pycnometer. Measured values were compared with available values in the literature. Table II lists the measured properties of the liquids used in these experiments along with comparisons to available literature values. Standard literature values were used for water.
Physical properties at 20 °C of liquids used in the impinging-sheet experiments. meas.—measured in this study; lit.—literature values.
Liquid . | ρ (meas.) (kg/m3) . | ρ (lit.) (kg/m3) . | σ (meas.) (N/m) . | σ (lit.) (N/m) . | μ (meas.) (mPa s) . | μ (lit.) (mPa s) . |
---|---|---|---|---|---|---|
Skim or non-fat milk | 1027 | 103318 | 0.052 | 0.05119 | 1.65 | 1.4720 |
10% isopropyl alcohol (IPA) | 980 | 98221 | 0.0402 | 0.041222 | 1.46 | 1.6623 |
40% aqueous glycerine | 1110 | 110024 | 0.065 | 0.069625 | 3.3 | 3.7226 |
3.1127 |
Liquid . | ρ (meas.) (kg/m3) . | ρ (lit.) (kg/m3) . | σ (meas.) (N/m) . | σ (lit.) (N/m) . | μ (meas.) (mPa s) . | μ (lit.) (mPa s) . |
---|---|---|---|---|---|---|
Skim or non-fat milk | 1027 | 103318 | 0.052 | 0.05119 | 1.65 | 1.4720 |
10% isopropyl alcohol (IPA) | 980 | 98221 | 0.0402 | 0.041222 | 1.46 | 1.6623 |
40% aqueous glycerine | 1110 | 110024 | 0.065 | 0.069625 | 3.3 | 3.7226 |
3.1127 |
IV. RESULTS OF THE PHOTOGRAPHIC STUDY
A. “Skin” appearance of the projection of the single sheet onto the mixed sheet
Figure 3 is a flash photograph of the impingement zone of two equal water sheets impinging at an angle of 45°. The framed area is shown in Fig. 4.
Although it was initially not clear what the photographs would reveal, Fig. 4 clearly shows that there is liquid above the projection of the front single sheet onto the mixed sheet (impingement zone line), which would be in the direction of mixed sheet backflow. This liquid in the backward portion of the mixed sheet appears to have waves that move from left to right or right to left, which are nominally 90° opposed to the impact waves in the forward part of the mixed sheet, which primarily move in the direction of flow (downward direction on the photograph). For the conditions under which Figs. 3 and 4 were taken, there was no ejection of liquid or backflow from the backward portion of the mixed sheet.
Figure 4 shows that the projection of the forward single sheet onto the mixed sheet has the appearance of a “skin.” This effect is also seen in Fig. 5, where holes within the “skin” are shown and the mixed sheet can be seen inside the holes. As the impingement angle is increased, the impact waves on the mixed sheet become more intense and can cause the single-sheet projection to become distorted, as shown in Fig. 6 (impingement at a 90° angle).
Four-microsecond flash photograph of the front view of the impingement of two equal water sheets from the M2 device at an impingement angle of 90° and at a pressure drop of 0.12 bar. Increased impact waves on the surface of the mixed sheet result in significant distortion in the projection of the front single sheet as it impacts the mixed sheet.
Four-microsecond flash photograph of the front view of the impingement of two equal water sheets from the M2 device at an impingement angle of 90° and at a pressure drop of 0.12 bar. Increased impact waves on the surface of the mixed sheet result in significant distortion in the projection of the front single sheet as it impacts the mixed sheet.
Figure 7 (Multimedia view) is an extract from a high-speed video taken at 31 000 frames/s of the impingement of two equal water sheets from the M2 device at an angle of 45° and pressure drop of 0.06 bar. The video is even more striking in that it shows the “skin” projection of the single water sheet as moving up and down with each wave in the forward part of the mixed sheet. The waves in the forward part of the mixed sheet move in the downward direction and appear to be quite vigorous even at this relatively low impingement angle and sheet velocity. The waves in the backward part of the mixed sheet (above the dark single-sheet projection), however, move in the left to right or right to left direction and are much more quiescent. Again, at this low impingement angle and sheet velocity, there is no backflow or ejection of liquid from the backward part of the mixed sheet.
Extracted frame from the video taken at 31 000 frames/s of the impingement of two equal water sheets at an angle of 45° and pressure drop of 0.06 bar. The “skin” appearance of the projection of the single sheet onto the mixed sheet can be clearly viewed. Multimedia view: https://doi.org/10.1063/5.0040336.2
Extracted frame from the video taken at 31 000 frames/s of the impingement of two equal water sheets at an angle of 45° and pressure drop of 0.06 bar. The “skin” appearance of the projection of the single sheet onto the mixed sheet can be clearly viewed. Multimedia view: https://doi.org/10.1063/5.0040336.2
This result seems unusual because the single sheets are continuous, the single sheets combine to form the mixed sheet, and the liquid for both sheets is the same. Therefore, it is unclear why the projection of the single sheet onto the mixed sheet would appear as a skin. The video shows that underneath the skin, there are dark areas that appear to be the troughs of the impact waves. Although no determination was made in this study, it appears that the “skin” appearance of the single-sheet projection onto the mixed sheet occurs due to the surface tension of liquid at an interface with air trapped in the troughs of the impact waves.
B. Photographs of the ejection of droplets from the backward portion of the mixed sheet
At larger impingement angles and/or single-sheet velocities, droplets were ejected as backflow from the backward part of the mixed sheet. Calculation of the critical velocity for mixed sheet backflow (droplet ejection) is discussed in Sec. V. In general, the higher the impingement angle, the lower the critical velocity. Furthermore, the thinner the sheet at impingement, the larger the impingement angle and/or mixed sheet velocity required to cause droplet ejection.
Figure 8 is a photograph of the impingement zone under conditions that resulted in droplet ejection from the backward portion of the mixed sheet. Droplets can be seen above the impingement zone and backward part of the mixed sheet, and it appears that ligaments and tongues of liquid are the source of these droplets.
Four-microsecond photograph of the front view of the impingement of two equal water sheets at an impingement angle of 90° and at a pressure drop of 0.35 bar for the M2 device. The single-sheet velocity is 7.57 m/s, which is greater than the calculated critical velocity of 5.17 m/s required for mixed sheet backflow.
Four-microsecond photograph of the front view of the impingement of two equal water sheets at an impingement angle of 90° and at a pressure drop of 0.35 bar for the M2 device. The single-sheet velocity is 7.57 m/s, which is greater than the calculated critical velocity of 5.17 m/s required for mixed sheet backflow.
Figure 9 shows ligaments beginning to form for potential droplet ejection at a mixed sheet velocity that is almost equal to the calculated critical velocity. The photograph shows four ligaments that could potentially eject droplets. At mixed sheet velocities close to the critical velocity, not all of the ligaments and tongues formed will eject droplets. It was observed that some are absorbed back into the bulk of the liquid.
Four-microsecond flash photograph of the impingement of two equal water sheets from the M3 device at an impingement angle of 80°. The single-sheet velocity is about 98% of the calculated critical velocity for droplet ejection. Note the four ligaments or tongues that could potentially eject droplets.
Four-microsecond flash photograph of the impingement of two equal water sheets from the M3 device at an impingement angle of 80°. The single-sheet velocity is about 98% of the calculated critical velocity for droplet ejection. Note the four ligaments or tongues that could potentially eject droplets.
Figure 10 is a flash photograph of the impingement of two 10% isopropyl alcohol (IPA) sheets from the M1 device at a pressure drop of 0.42 bar and impingement angle of 68°. The viscosity of 10% IPA is about 50% greater than that of water, resulting in a dampening of waves on the mixed sheet. As will be discussed in Sec. V, while viscosity dampens turbulence and has a strong influence on micromixing,4 it does not appear to impact the critical velocity required for droplet ejection from the backward part of the mixed sheet.
Four-microsecond photograph of the front view of the impingement of two equal sheets of 10% aqueous isopropyl alcohol (IPA) at an impingement angle of 68°, pressure drop of 0.42 bar, and single-sheet distance to impingement of 2.0 cm for the M1 device. The higher viscosity of 10% IPA, relative to water, significantly dampens mixed sheet waves. This photograph was taken at a single-sheet velocity that was 3% greater than the calculated critical velocity for the ejection of droplets from the backward portion of the mixed sheet.
Four-microsecond photograph of the front view of the impingement of two equal sheets of 10% aqueous isopropyl alcohol (IPA) at an impingement angle of 68°, pressure drop of 0.42 bar, and single-sheet distance to impingement of 2.0 cm for the M1 device. The higher viscosity of 10% IPA, relative to water, significantly dampens mixed sheet waves. This photograph was taken at a single-sheet velocity that was 3% greater than the calculated critical velocity for the ejection of droplets from the backward portion of the mixed sheet.
Figures 11 and 12 compare the impingement zones of impinging water sheets (Fig. 11) and impinging 40% aqueous glycerol sheets (Fig. 12) for the M2 device at a pressure drop of 0.2 bar and impingement angle of 110°. Both figures show long, thick ligaments forming prior to droplet ejection. The height of each ligament is about 30–40 times the thickness of the single sheet at impingement. In Fig. 11, it is difficult to see the projection of the single sheet onto the mixed sheet, while, in Fig. 12, the skin projection is seen to be riddled with craters and pock-marks.
Two-microsecond photograph of the front view of the impingement of two equal water sheets at an impingement angle of 110° and at a pressure drop of 0.20 bar for the M2 device. The high impingement angle results in long, relatively thick ligaments that eject droplets from the backward portion of the mixed sheet. The single-sheet velocity is 21% greater than the calculated critical velocity required for mixed sheet backflow. The tallest ligament in the photograph is about 33 times the thickness of the single sheet at impingement.
Two-microsecond photograph of the front view of the impingement of two equal water sheets at an impingement angle of 110° and at a pressure drop of 0.20 bar for the M2 device. The high impingement angle results in long, relatively thick ligaments that eject droplets from the backward portion of the mixed sheet. The single-sheet velocity is 21% greater than the calculated critical velocity required for mixed sheet backflow. The tallest ligament in the photograph is about 33 times the thickness of the single sheet at impingement.
Four-microsecond photograph of the front view of the impingement of two equal sheets of 40% aqueous glycerine at an impingement angle of 110° and at a pressure drop of 0.20 bar for the M2 device. The higher viscosity of the aqueous glycerol solution dampens waves on the mixed sheet. The single-sheet velocity is 26% greater than the calculated critical velocity required for mixed sheet backflow. The tallest ligament in the photograph is almost a factor of 40 greater than the single-sheet thickness at impingement.
Four-microsecond photograph of the front view of the impingement of two equal sheets of 40% aqueous glycerine at an impingement angle of 110° and at a pressure drop of 0.20 bar for the M2 device. The higher viscosity of the aqueous glycerol solution dampens waves on the mixed sheet. The single-sheet velocity is 26% greater than the calculated critical velocity required for mixed sheet backflow. The tallest ligament in the photograph is almost a factor of 40 greater than the single-sheet thickness at impingement.
V. DISCUSSION AND ANALYSIS
A. Momentum balance
When the single sheets collide in the impingement zone, momentum is conserved, but kinetic energy may or may not be conserved. A 100% or perfectly elastic collision would mean that no energy is released into the mixed sheet and, instead, streamline flow is preserved. The term “inelastic collision” as used here is defined as a collision that results in the dissipation of kinetic energy from the single-sheet component velocities that are destroyed upon impact. It does not mean a collision where all kinetic energy for both single sheets is dissipated.
The collision might also be somewhere in between inelastic and perfectly elastic as suggested by the mixed sheet velocity measurements at a 60° impingement angle conducted by Demyanovich and Bourne.5 However, it cannot be assumed that one type of collision prevails at all points along the impingement zone line especially at mixed sheet velocities near the mixed sheet critical velocity for liquid ejection from the backward part of the mixed sheet.
1. Momentum balance for a perfectly elastic collision
Figure 13 shows a cross-sectional illustration of a 100% elastic collision. Although the expanding sheets do thin with distance from the origin, this thinning is ignored in Fig. 13. For the scale shown in Fig. 13, the thinning of the sheets is less than 2%. The collision produces a stagnation point (actually a line for the entire impingement zone, but a point in the cross section of the sheet as shown in Fig. 13) around which stream lines are deflected. The stagnation point, P, in Fig. 13 results in flow in the backward direction of the mixed sheet as well as the forward direction.
Cross-sectional side view of two impinging sheets undergoing a 100% elastic collision at an impingement angle of 90°. These illustrations are simplified in that no mixed sheet surface waves are shown. (a) shows the deflection of streamlines around a stagnation point, P, with no crossover of streamlines. (b) illustrates the geometry and parameters for the momentum balances.
Cross-sectional side view of two impinging sheets undergoing a 100% elastic collision at an impingement angle of 90°. These illustrations are simplified in that no mixed sheet surface waves are shown. (a) shows the deflection of streamlines around a stagnation point, P, with no crossover of streamlines. (b) illustrates the geometry and parameters for the momentum balances.
Momentum is conserved in the x direction and y direction. Referring to Fig. 13(b), the general momentum balance equations are (not assuming equal single sheets)
Here,
m1 and m2 are the mass flow rates of the single sheets
mf is the mass flow rate of the mixed sheet in the forward direction
mb is the mass flow rate of the mixed sheet in the backward direction
u1 and u2 are the velocities of the single sheets
v1y and v2y are the component velocities in the y direction of each single sheet prior to impingement
vfy is the velocity of the forward mixed sheet in the y direction
vby is the velocity of the backward mixed sheet in the y direction
v1x and v2x are the component velocities in the x direction of each single sheet prior to impingement
vfx is the velocity of the mixed sheet in the forward x direction
vbx is the velocity of the mixed sheet in the backward x direction.
For equal impinging single sheets, m1 = m2, u1 = u2, v1y = v2y, and v1x = v2x.
The mixed sheet velocity, vm, is assumed to be constant in all directions, so that
In the case of the y-component momentum of the mixed sheet for the equal single-sheet collision,
For the impingement of equal single sheets, the y-component velocities of the single sheets are destroyed and there is no resulting y-component velocity for the mixed sheet. For inelastic collisions, the kinetic energy associated with v1y and v2y is dissipated into the mixed sheet providing rapid micromixing of low-viscosity liquids.4 However, for a 100% elastic collision, there will be a momentum change in the x direction due to the impulse created by the collision. Since there is theoretically no loss of kinetic energy, the mixed sheet velocity, vm, equals the single-sheet velocity, u. For a perfectly elastic collision of equal single sheets (letting u = u1 = u2),
Letting m = m1 = m2 and vx = v1x = v2x, the momentum balance for the system in the x direction is
However, for the single sheets,
where β is the half angle of impingement. Substituting Eq. (9) into Eq. (8) and dividing both sides by u yield
The mass balance on the system is
The amount of liquid from the single sheets that is deflected into the forward part of the mixed sheet is
Substituting this result into Eq. (11) yields the amount of mass flow from the single sheets that becomes the backward flow of the mixed sheet,
For equal single sheets, the ratio of forward flow in the mixed sheet to backward flow in the mixed sheet for a 100% elastic collision is
This ratio is only dependent upon the half angle of impingement.
From Fig. 13(b),
where si is the thickness of the single sheet at impingement and k is the fraction of the single-sheet mass flow that is deflected as backward flow of the mixed sheet.
or
Defining sB as the thickness of the backward portion of the mixed sheet at (near) the impingement zone yields
The thickness of the backward portion of the mixed sheet in the impingement zone, sB, will be important for analyzing the results of the droplet ejection experiments. Table III provides some calculated values of thicknesses as a function of the impingement angle (2β).
Typical widths of single-sheet projection (Δr) and thickness of the backward portion of the mixed sheet (sB) for the M2 device as a function of the impingement angle. QT is the mixed sheet flow rate, and Δr is the calculated width of the projection of the single sheet onto the mixed sheet (Δr = si/sin β).
ΔP (bar) . | QT (l/min) . | 2β (deg) . | si (µm) at R = 2 cm . | Δr (µm) . | sB (µm) . |
---|---|---|---|---|---|
0.3 | 2.5 | 45 | 76.6 | 200.2 | 5.8 |
0.3 | 2.5 | 60 | 76.6 | 153.2 | 10.3 |
0.3 | 2.5 | 90 | 76.6 | 108.4 | 22.4 |
0.3 | 2.5 | 120 | 76.6 | 88.5 | 38.3 |
ΔP (bar) . | QT (l/min) . | 2β (deg) . | si (µm) at R = 2 cm . | Δr (µm) . | sB (µm) . |
---|---|---|---|---|---|
0.3 | 2.5 | 45 | 76.6 | 200.2 | 5.8 |
0.3 | 2.5 | 60 | 76.6 | 153.2 | 10.3 |
0.3 | 2.5 | 90 | 76.6 | 108.4 | 22.4 |
0.3 | 2.5 | 120 | 76.6 | 88.5 | 38.3 |
Table IV compares the ratio of mf/mb for a perfectly elastic collision of impinging sheets with that of impinging jets using equations developed by Hasson and Peck for impinging jets.8 Interestingly, the ratio for impinging jets is the square of the ratio for impinging sheets, which results from the differing geometries.
2. Momentum balance for an inelastic collision
As noted earlier, inelastic collision for this analysis is defined as a collision where the kinetic energy associated with the y-component velocity of the single sheets is dissipated into the liquid. Momentum is conserved in the x and y directions, but kinetic energy is not conserved.
Equations (3) through (6) for the case of a perfectly elastic collision of equal single sheets remain unchanged for the inelastic collision case. The dissipation of kinetic energy associated with the single-sheet component velocity in the y direction results in
where vx is the component velocity of the single sheet in the x direction. The mixed sheet velocity for an inelastic collision is equal to the x component velocity of the equal single sheets (the mixed sheet velocity would equal the single-sheet velocity, u, if the collision were 100% elastic). If some kinetic energy from the x direction flow of the single sheets is also dissipated in the impingement zone, then the mixed sheet velocity will be lower than vx.
The momentum balance in the x direction now simplifies to
As before, continuity of mass flow yields
and solving for mb results in
For an inelastic collision, there is no backflow of liquid from the mixed sheet. This does not mean that there is not a backward part of the mixed sheet, as shown in Fig. 4, just that the liquid is not ejected from this backward portion of the mixed sheet. If additional kinetic energy is lost (from the single-sheet momentum in the x direction), then
Even if additional kinetic energy is dissipated, mb will still be equal to zero.
B. Measurement of sheet thickness
1. Single-sheet thickness at impingement
Prior work on impinging sheets by the author never directly measured the thickness of the single sheets at impingement, even though this is an important parameter. It has been assumed that Eq. (1) provides a calculated single-sheet thickness that is reasonably accurate. The appearance of the projection of the front single sheet as a “skin” in the photographs of this study leads to the possibility that the thickness of the single sheet at impingement could be determined from the photographs. By measuring the height of the projection on the photographs, the single-sheet thickness could potentially be determined from geometry.
Figure 14 is a flash photograph of two sheets of water from the M1 device impinging at an angle of 40°. The projection of the forward single sheet looks quite uniform in height across the photograph. It can be seen that the height of the backward part of the mixed sheet (as measured from the impingement zone) is quite small and much of the backward portion of the mixed sheet is barely visible. Furthermore, by using the smallest device at a relatively low impingement angle, waves on the single sheet prior to impingement and mixed sheet after impingement are less vigorous, and this facilitates creating an impingement zone that appears to allow for reasonable measurements of the single-sheet projection (as opposed to the single-sheet projection shown in Fig. 6).
Four-microsecond photograph of the front view of the impingement of two equal water sheets from the M1 device at an impingement angle of 40°. Ri = 1.9 cm, φ = 99°, and ΔP = 0.53 bar.
Four-microsecond photograph of the front view of the impingement of two equal water sheets from the M1 device at an impingement angle of 40°. Ri = 1.9 cm, φ = 99°, and ΔP = 0.53 bar.
From Eq. (1), the single-sheet thickness at impingement is calculated to be 41 μm. From Fig. 14, the projection of the single-sheet thickness is measured to be 94 μm, which calculates to a single-sheet thickness of 32 μm. The agreement here is to within approximately 22%. However, it can be noted in Fig. 14 that the single-sheet projection is not vertical in the photograph but appears to be leaning back toward the backward part of the mixed sheet. Figure 13 indicates that the projection would appear to lean back due to the fact that the thickness of the backward part of the mixed sheet is less than that of the forward part of the mixed sheet. A projection that appears to be “leaning back” in the photographs would result in a measurement that underestimates the true projection height.
Figure 15 is a flash photograph of two equal single sheets of 10% isopropyl alcohol from the M1 device impinging at an angle of 52° and a pressure drop of 0.3 bar. The higher viscosity of 10% isopropyl alcohol, relative to water, coupled with a low impingement angle and pressure drop (for this device) has greatly dampened the waves on the mixed sheet. Furthermore, the backward portion of the mixed sheet is not easily visible in this particular photograph.
Four-microsecond photograph of the front view of the impingement of two equal 10% isopropyl alcohol sheets from the M1 device at an impingement angle of 52°. Ri = 1.57 cm, φ = 94°, and ΔP = 0.30 bar.
Four-microsecond photograph of the front view of the impingement of two equal 10% isopropyl alcohol sheets from the M1 device at an impingement angle of 52°. Ri = 1.57 cm, φ = 94°, and ΔP = 0.30 bar.
From Eq. (1), the single-sheet thickness at impingement in Fig. 15 is calculated to be 53 μm. From the right-hand side of Fig. 15, where the single-sheet projection appears more vertical, the projection of the single-sheet thickness is measured to be 110 μm–118 µm, which calculates to an average single-sheet thickness of 50 μm. The agreement now is within 6%.
It appears that measurements of the single-sheet thickness could be made from these photographs if the projection of the single sheet onto the mixed sheet is perpendicular to the camera. That can perhaps be facilitated by dampening mixed sheet surface waves to the greatest extent possible. However, while the single-sheet projection in Fig. 15 appears to be vertical on the right-hand side of the photograph, this is definitely not the case in the middle or left-hand portions of Fig. 15. Nevertheless, there is reasonable agreement between Eq. (1) and these measurements. Furthermore, these photographs show that the single-sheet projection is fairly uniform across the spread angle, φ, of the mixed sheet. Equation (1) assumes no variation in sheet thickness across the spread angle of the sheet, which is supported by these photographs.
2. Sheet thickness of the backward part of the mixed sheet
As noted earlier, the thickness of the backward part of the mixed sheet at the rim is an important parameter required for analyzing critical velocities at which the liquid is ejected from the backward part of the mixed sheet. Figure 16(a) is a photograph of two equal single sheets of water impinging at an angle of 73° and a pressure drop of 0.57 bar for the M2 device.
(a) Two-microsecond photograph of the front view of the impingement of two equal water sheets from the M2 device at an impingement angle of 73°. Ri = 2.3 cm, φ = 118°, and ΔP = 0.57 bar. The framed area may contain the backward direction mixed sheet that is turned on the edge so that the thickness can be estimated. The thickness to be measured is shown in (b), which is a magnified view of the framed area shown in (a).
(a) Two-microsecond photograph of the front view of the impingement of two equal water sheets from the M2 device at an impingement angle of 73°. Ri = 2.3 cm, φ = 118°, and ΔP = 0.57 bar. The framed area may contain the backward direction mixed sheet that is turned on the edge so that the thickness can be estimated. The thickness to be measured is shown in (b), which is a magnified view of the framed area shown in (a).
The framed area in the photograph shows a relatively broad “ligament” that may be representative of the backward mixed sheet allowing for an approximate measurement of the thickness. Figure 16(b) shows a magnified view of the framed area and shows the thickness to be measured. An argument could be made for extending this thickness to the end of the white band; measurements will be provided for both.
The measured thicknesses are 14.5 µm for the thickness shown in Fig. 16(b) and 22 µm if the thickness is extended to the end of the white band (these are approximate measurements since they do not take into account the unknown angle at which the edge is turned). From Eq. (1), the calculated single-sheet thickness at impingement for the conditions shown in Fig. 16 is 63 µm. From Eq. (19), the thickness of the backward portion of the mixed sheet for an impingement angle of 73° (β = 36.5°) and a single-sheet thickness equal to 63 µm at impingement is calculated as 12.4 µm.
While the measurement of the single-sheet thickness at impingement can possibly be made by dampening the mixed sheet surface waves, repeatable, reliable measurement of the thickness of the backward mixed sheet is unlikely using a photographic technique because obtaining photographs such as Fig. 16 is a rare occurrence.
C. Analysis of the critical mixed sheet velocity experiments
The experiments where the single-sheet velocity was increased to the point at which droplets were first ejected from the backward portion of the mixed sheet were initially analyzed by assuming that the collision in the impingement zone was perfectly elastic, which means that the mixed sheet velocity, vm, is equal to the single-sheet velocity, u. Since, at this critical velocity, droplets only begin to be ejected and have little momentum, the analysis further assumes that the rim of the backward portion of the mixed sheet is quasi-stationary.
The position of the quasi-stationary rim of the backward portion of the mixed sheet is given by the equilibrium between the inertial forces and the surface tension forces, which pull the rim back toward the impingement zone. This type of analysis has been used by researchers to study the rim of single sheets formed by impinging a jet on a solid surface28 and single sheets formed by impinging jets29 and by many researchers studying the movement of the rim when a soap film is ruptured.29–35
The position of the quasi-stationary rim of the film or sheet is set by an equivalency of surface tension and inertial forces,28–30,34
where σ is the liquid surface tension, ρ is the liquid density, L is a characteristic length scale at the rim, and v is the velocity at the rim. This equation leads to the contraction velocity (also known as the retraction or recession velocity) of the rim that keeps it in a quasi-stationary position. This retraction velocity is generally called the Taylor–Culick velocity, vTC, which is31,33,35
When the impingement of two single sheets is inelastic, the rim of the backward portion of the mixed sheet is “quasi-stationary,” as shown in Fig. 4. As the mixed sheet velocity is increased, droplets begin to be ejected due to elastic collisions in the impingement zone. Since the experiments performed here attempted to determine the single-sheet velocity (which equals the mixed sheet velocity for a 100% elastic collision) at which droplets just begin to be ejected at the centerline of the mixed sheet, it will be assumed that the Taylor–Culick velocity is relevant to the analysis of these data.
The characteristic length scale for Eq. (27) is the thickness of the mixed sheet at (near) the impingement zone in the backflow direction,
Figure 17 is a plot of the critical mixed sheet velocity required to initiate droplet ejection, vmc, vs the Taylor–Culick velocity. The equation for the best-fit, linear regression is
and the R-squared value of the fit is 0.961.
Plot of the mixed sheet critical velocity vs the Taylor–Culick contraction velocity. Note that since a 100% elastic collision was assumed when calculating the backward mixed sheet thickness, the mixed sheet velocity is equal to the single-sheet velocity.
Plot of the mixed sheet critical velocity vs the Taylor–Culick contraction velocity. Note that since a 100% elastic collision was assumed when calculating the backward mixed sheet thickness, the mixed sheet velocity is equal to the single-sheet velocity.
By only focusing on the contraction velocity as the variable to plot on the x-axis, the possible effect of viscosity is not considered a variable, even though the data include data points at different viscosities. In order to include the potential effect of viscosity, a plot of vmc vs the Ohnesorge number, Oh, was constructed. The Ohnesorge number is given by
where μ is the liquid absolute viscosity. Plots of vmc vs Oh, with L = sB, were constructed; however, these plots produced only scatter and no discernable trend in the data.
From Fig. 17, it appears that the critical mixed sheet velocity is not dependent upon viscosity at least up to the viscosities used in these experiments. Similarly, Pandit and Davidson32 found no dependence of the rim velocity for punctured soap films on liquid viscosity at viscosities up to 130 mPa s. Savva and Bush35 noted that viscous forces within a film do not contribute to the film’s momentum budget but can affect the transient time approach to the Taylor–Culick velocity.
Above the line shown in Fig. 17, there will be elastic collisions at places and locations along the impingement zone line that result in droplets being ejected from the backward portion of the mixed sheet. This is consistent with the elastic collision momentum balance, which requires liquid flow away from the impingement zone in the backward direction. Below the line shown in Fig. 17, conditions are such that the liquid cannot break free from the backward mixed sheet due to surface tension forces exceeding inertial forces.
1. Comments on the equilibrium between surface tension and inertial forces
The results shown in Fig. 17 indicate that the mixed sheet critical velocity required to initiate droplet ejection or backflow is roughly two to three times the Taylor–Culick or contraction velocity. The data were analyzed assuming that the Taylor–Culick velocity was applicable, which inherently implies an equilibrium between surface tension and inertial forces at the rim. Although a relationship between the critical velocity and the Taylor–Culick velocity is clearly shown in Fig. 17, these velocities differ by a factor of two to three. In Taylor’s28 experiments with single liquid sheets, the contraction velocity was determined with droplets having almost no velocity as they disengaged from the rim of the liquid sheet. Similarly, the experiments conducted here attempted to determine the critical sheet velocity at which droplets just begin to be ejected and have little momentum.
A likely explanation for why the analysis of the experimental data yielded a critical velocity that is significantly higher than the Taylor–Culick velocity is that the momentum analysis assumed a 100% or perfectly elastic collision. Since water readily deforms, it seems unlikely that the collision is perfectly elastic. However, the analysis calculated a backward flow mixed sheet thickness, sB, assuming a perfectly elastic collision. A collision that is less than perfectly elastic would yield a value of sB that is smaller, which would increase the Taylor–Culick velocity. Furthermore, if the collision is not perfectly elastic, the mixed sheet velocity would be less than the single-sheet velocity. An increase in the Taylor–Culick velocity coupled with a decrease in the inertial velocity would result in a decrease in the ratio of these two velocities.
Work is currently ongoing to measure the coefficient of restitution for impinging sheets under conditions where the mixed sheet velocity equals or exceeds the critical velocity for droplet ejection. Equation (29), however, is still valid for determining the sheet velocity, which will begin to result in elastic collisions in the impingement zone for a specific liquid density, liquid surface tension, sheet impingement angle, and single-sheet thickness at impingement. However, to use Eq. (29), the thickness of the backward portion of the mixed sheet must be calculated assuming a 100% elastic collision in the impingement zone.
VI. CONCLUSIONS
The photographic study of the impingement zone confirms that the mixed sheet produced by the impingement of two free liquid sheets has both a forward and a backward portion. There is always flow in the forward direction, but flow in the backward direction requires that the mixed sheet velocity exceeds the surface tension contraction velocity (Taylor–Culick velocity). The impingement angle has a major influence on whether or not backflow is achieved. The larger the impingement angle, the more readily the backflow is achieved since the thickness of the backward portion of the mixed sheet increases. Backflow can also be achieved by decreasing the surface tension and/or increasing the sheet velocity to the extent that such changes result in a mixed sheet velocity that is at or above the regression line shown in Fig. 17.
The photographs shed some light on differences between the backward and forward portions of the mixed sheet. Figure 4, for example, which is impingement at a 45° angle, shows that the backward portion of the mixed sheet appears to be relatively more quiescent than the forward mixed sheet and the waves tend to move from left to right or right to left as opposed to the downward moving waves in the forward part of the mixed sheet. The video associated with Fig. 7 (Multimedia view) also seems to confirm this observation.
The photographs and a limited video show that the projection of the single sheets onto the mixed sheet at impingement appears as a “skin” on the mixed sheet. Setting conditions so that the mixed sheet waves are greatly dampened may allow for the measurement of the single-sheet projection and, therefore, the single-sheet thickness at impingement.
The experimental results and assumption that the collision of impinging sheets was perfectly elastic resulted in a linear regression equation indicating that the mixed sheet velocity needs to be about two to three times the Taylor–Culick velocity in order for backflow to occur. Since the Taylor–Culick velocity is based on an equilibrium between inertial and surface tension forces, it was expected that the ratio of the inertial velocity to the surface tension contraction velocity would be closer to 1. Most likely, the reason for this discrepancy is that the analysis assumes a perfectly elastic collision in the impingement zone. A collision that is less than 100% elastic is more realistic and would result in a mixed sheet velocity that is lower in value and a Taylor–Culick velocity that is higher in value.
If all of the kinetic energy associated with the single-sheet component velocity that is destroyed upon impingement is dissipated into the mixed sheet, then the critical velocity cannot be achieved and there will be no backflow from the backward part of the mixed sheet.
ACKNOWLEDGMENTS
The author would like to thank Tyler Gerritsen from Gerritsen Design for construction of the microsecond LED flash unit and for conducting verification tests of its specifications.
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request. These data do not include photographs or a high-speed video.