ARTICLE pubs.acs.org/IECR
Misting of Newtonian Liquids in Forward Roll Coating Michael S. Owens,*,‡ Madhu Vinjamur,§ L. E. Scriven,^ and C. W. Macosko† †
Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States Ev3 Endovascular, Inc., Plymouth, Minnesota 55441, United States § Department of Chemical Engineering, IIT Bombay, Powai, Mumbai, India 400 076 ‡
ABSTRACT: Misting of Newtonian liquids, in the film-split region of two counter rotating rigid and deformable cylinders was visualized with standard and high-speed cameras. Flow instabilities begin with ribbing and eventually lead to generation of air-borne droplets called mist. As speed was raised, a uniform film thickness evolved into one with ribs, which evolved into continuous sheets of liquid extending downstream of the gap between the cylinders. The edge of each sheet formed a rim whose two ends were attached to the ribs on the cylinders. Still images via high-speed photography revealed a new mechanism for mist generation: the sheets extended downstream, became unstable and ruptured to produce air-borne droplets. A droplet time-of-flight measurement technique quantified the effect of process settings (speed and speed ratio) and material properties (viscosity and surface tension) on droplet size, count, and mass concentration of mist.
’ INTRODUCTION Ribbing or corduroy or pin striping is a steady three-dimensional flow instability in forward roll coating. As speed is raised the steady two-dimensional flow that divides the liquid carried through the gap between counter-rotating cylinders in forward roll metering of a liquid is supplanted at a certain low speed. At the downstream side of the gap where the flow divides, a free surface or meniscus is generated. Viscous force tends to destabilize the free surface, and surface tension force opposes any instability. Ribbing is the outcome of competition between these two forces whose ratio is called capillary number, which is defined by a mean roll speed, V, times effective liquid viscosity, η, divided by surface tension of the liquid, σ. Ca ¼ Vη=σ
ð1Þ
Before the transition to ribbing, the delivered thicknesses are uniform across the cylinders. After the transition, the thicknesses are nonuniform and wavy. Onset of ribbing in symmetric film splitting of Newtonian liquids has been documented extensively by experiments.1-4 Symmetric is meant to be the case of identical cylinders moving at the same surface speed. Reports on flow regimes beyond the onset of ribbing are few and fragmentary. At speeds beyond ribbing, the rib amplitude rises, the wavelength or the ribbing pitch (defined as distance between a point on a rib and corresponding point on a rib immediately next to it) falls and the ribs wander and merge.1 As speed is raised, ribbing pitch falls until ribs become numerous. The onset and development of this behavior with speed, however, has not been documented. Onset of unsteady state ribbing when one of the cylinders had a deformable elastomeric cover has been documented,5 but not the events leading to wandering, merging, and dividing of the ribs. Most of the theoretical studies have addressed the flow instability at the onset of ribbing in Newtonian liquids.1,3,4,6-8 With computer-aided solution of Navier-Stokes equations for r 2011 American Chemical Society
three-dimensional flow, Garfinkel-Castillo and Patera9 produced evidence that the crests of ribs sharpen so that the pattern is no longer sinusoidal at capillary numbers beyond the onset of ribbing instability. Here we report a comprehensive experimental study of flow instabilities of Newtonian liquids, as speed is raised beyond the onset of ribbing. Commercially, forward-roll coating processes operate at speeds significantly higher than those corresponding to ribbing. At these speeds, misting occurs where air-borne droplets are generated in the region of splitting of the liquid film. Mist droplets are defined here to be less than 50 μm in diameter because droplets smaller than this pose the greatest health risk when inhaled,10 a motivating reason to eliminate mist. It has been reported that addition of branched silicone polymers to low viscosity silicones delays the onset of misting.11-13 Experiments to understand the physics of misting have largely been conducted with rheologically complex liquids, such as high molecular weight polymer solutions14-18 and liquid/solid suspensions containing binders, thickeners, and pigments.19-24 It seems to be widely believed that cavitation causes misting. A mechanism is presented here showing that misting can occur without cavitation. Ascanio et al.25 reported flow instabilities that lead to misting of Newtonian liquids in a deformable forward-roll coating configuration where a rigid cylinder placed over a deformable one compresses it because of its weight. They concluded that the pressure drop in the gap was insufficient to cavitate the liquid, an observation also made by Gaskell et al.26 With 20 and 30 wt % aqueous solutions of poly(ethylene glycol), high-speed camera (462 frames per second) images showed c-shaped liquid sheets.25 These reportedly broke into droplets because of film-splitting Received: March 28, 2010 Revised: January 28, 2011 Accepted: February 8, 2011 Published: February 21, 2011 3212
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Figure 1. Schematic of a roll coating configuration employed here. The bottom cylinder carried liquid from the pan to the gap and the excess was rejected. The film split downstream of the gap and each roll was coated with the liquid, which was not scrapped. A Mylar sheet was draped on the top cylinder at a distance of about three-fourths of the circumference from the gap so that it did not interfere with the flow in the gap.
and air entrainment. They did not, however, show the flow transitions and mechanism leading to misting. We present the transitions, measurements of droplet size, number of drops (drop count), and mass concentration of mist to characterize misting. The forces that aid and suppress misting are proposed, and their relative importance is summarized.
’ MATERIALS AND METHODS Experimental Apparatuses. Three forward-roll configurations without substrate or web were used to follow the progression of flow structures as speed was raised: two rigid roll configurations where mist was not observed and a deformable roll configuration where mist was observed. The first configuration comprised two rigid chrome-plated steel cylinders 0.20 m in diameter and 0.45 m in length. They were mounted in precision bearings such that the total indicated run-out was less than one micrometer. The cylinders were horizontally aligned (called sideby-side). The gap, that is, the minimum clearance between the two cylinders was set at 25 μm or more. The film-split proved so sensitive to speed that a change of 1 rpm (0.63 m/min) changed postribbing behavior perceptibly. Each cylinder was driven by a separate motor, and the speed was controlled to 0.01% of the full speed of the motor. The speeds reported here for rigid cylinders are accurate to 0.04 m/min. Unless stated otherwise, the rotation rate of the cylinders is equal and opposite. In the second configuration the same two steel cylinders were positioned vertically as shown in Figure 1. The bottom one was partially submerged in a pan of liquid which was carried to the upstream side of the gap. When they were placed side-by-side, both cylinders carried liquid to the gap. The submerged depth was such that the dipped layers were thick enough that the upstream side of the gap received excess liquid that was rejected back into the pan; this is called the run-back condition and was always maintained. The liquid carried through the gap, controlled to 25 mm or more, split on the downstream side and coated each cylinder surface; it was not removed from the surface during an experiment. Flow structures beyond ribbing and before misting were visualized with these two configurations.
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The third configuration was operated at conditions that readily produced mist droplets. It consisted of two vertically aligned cylinders in which the top cylinder was rigid and the bottom cylinder had a deformable cover. The top cylinder was chromeplated and was 0.1 m in diameter and 0.19 m in length. The bottom cylinder was 0.1 m in diameter and 0.19 m in length and was covered with a 0.33 m thick rubber of hardness 60 durometer (Short A). The gap between the upper and lower cylinders was negative because the rigid cylinder compressed the rubber because of its weight, ∼4 kg. Loading of the two cylinders was not examined here although it controls wet film thickness and uniformity. The rubber covered cylinder was submerged in a pan of liquid that was carried to the upstream side of the gap where excess liquid was rejected. The liquid was not removed from the cylinders after the film split. Mechanism of misting was visualized and misting parameters, such as drop size, drop count, and mass concentration, were measured with this configuration as a function of process parameters and material properties. Flow Visualization. Ideally, flow instabilities in forward-roll coating are recorded at the film-split meniscus where they originate. Coyle4 focused a sheet of light in the film-split region, which reflected at a glancing angle off the meniscus and onto an opal glass. At low surface speeds, the reflected light formed a curved patch of uniform intensity that turned into alternating dark and light curves as speed was raised. This technique enabled him to detect ribbing at much lower capillary numbers than previous researchers. When the rigid cylinders used in this work were mounted vertically, their surfaces reflected light making illumination and clear visualization of the meniscus difficult; therefore, an alternative approach was taken to characterize ribbing and postribbing instabilities. Flow instabilities were indirectly captured here by allowing a 250-μm thick Mylar sheet to touch the flow on the top cylinder surface downstream of the film-split to reveal flow patterns that originate at the film-split. The sheet was placed at a distance of approximately three-fourths of the circumference from the filmsplit, so that it does not affect flow there. The sheet amplified the flow patterns beneath it to provide a measure of ribbing amplitude at different speeds.5 The Mylar sheet captured the onset of ribbing at slightly higher capillary number than those reported by Coyle.4 When the rigid cylinders were placed sideby-side, lighting of the meniscus was less difficult and flow instabilities were recorded. At low cylinder speeds, flow instabilities and patterns were examined with a standard video camera (Cohu model 64152000, made by Cohu Inc., San Diego, California) at 30-frames/s; at high speeds, they were captured with high-speed cameras featuring a Kodak EktaPro Motion Analysis System (1000 frames/s), which contained a Kodak EktaPro Hi-Spec Processor and Kodak EktaPro Intensified Imager (Kodak, San Diego, California). The images were digitized with Scion Image software, version 2.0.4 (Scion Corporation, Frederick, Maryland), for measurement and presentation. Materials. Glycerol (99.7%) was supplied by Brenntag Great Lakes LLC, St. Paul, Minnesota. Aqueous glycerol solutions were prepared by adding deionized water to glycerol, shaking it by hand for 1 to 2 min, and then allowing the solution to equilibrate for 4 h. Low molecular weight polypropylene glycol (PPG) liquids were purchased from Sigma-Aldrich, St. Louis, Missouri, and low molecular weight polydimethylsiloxane (PDMS) oligomers were obtained from Wacker ChemE., Adrian, Michigan and Dow Corning Corporation, Midland, Michigan. 3213
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Table 1. Materials, Their Viscosity and Surface Tension Values, Process Conditions, and Mist Measurementsa material PDMS
a
η, mPa-s
σ, mN/m
V, m/min
V1/V2
Ddrop, μm
74
22
134
1.0
1.60
122.5
0.4
74
22
153
1.0
1.65
180.2
0.6
74
22
172
1.0
1.64
313.9
1.0
74
22
192
1.0
1.72
907.5
3.4
230
22
134
1.0
2.13
284.6
2.6
230
22
153
1.0
2.13
543.0
4.5
230
22
172
1.0
2.20
1,362.2
12.9
230 282
22 22
192 115
1.0 1.0
2.20 2.10
1,705.0 158.4
13.5 2.1
282
22
134
1.0
2.30
336.3
4.4
282
22
153
1.0
2.40
590.2
8.7
282
22
192
1.0
2.1
1,906.0
17.5
282
22
192
1.2
1.9
1,671.0
10.7
282
22
192
1.5
2.0
1,305.0
10.0
282
22
192
1.9
2.0
1,210.0
9.8
282 460
22 22
192 134
3.0 1.0
1.95 2.24
931.0 671.6
6.3 8.5
460
22
153
1.0
2.34
1,416.8
21.3
460
22
172
1.0
2.45
2,148.6
28.1
460
22
192
1.0
2.65
2,143.4
40.2
1160
22
115
1.0
3.00
1,913.0
15.2
drop count (1000)
mist conc., mg/m3
1160
22
134
1.0
3.10
2,143.0
35.6
PPG
348
32
192
1.0
2.2
1,511.0
16.0
glycerol/water
175 145
32 65
192 192
1.0 1.0
2.0 1.65
1,105.0 334.0
4.0 2.2
220
65
192
1.0
1.85
770.0
3.3
Conditions that did not produce mist are not reported here but are reported in the text.
Shear viscosity was measured with an ARES controlled strain Rheometer (TA Instruments, New Castle, Delaware), and all solutions were observed to be Newtonian between shear rates of 1 to 1000 s-1. Glycerol is hygroscopic and its viscosity falls as it absorbs moisture; therefore, both the viscosity and surface tension were measured before and after experiments and an average value is reported here. The surface tension was measured with Kr€uss tensiometer, Model K10ST, (Kr€uss, Matthews, North Carolina) equipped with a Wilhemy plate. Table 1 shows the measured shear viscosity and surface tension of the liquids. All property and misting measurements were carried out at room temperature. Mist Measurements. Mist droplet size, count, and the mass concentration of mist were measured over a three minute sampling period with an Aerosizer DSP27using the vertically aligned roll configuration in which the bottom cylinder was covered with a deformable cover, the top cylinder was rigid, and both rolls were 0.1 m diameter and 0.19 m wide. Mist droplets were drawn into the Aerosizer DSP with a vacuum pump from the filmsplit region at a rate of 2.5 m3/min through a rubber tube 0.18 m in length (1.25 cm inner diameter) placed 7 cm from the film split. The roll coater was kept inside a laminar air flow hood; external air currents did not influence the mist measurements because the configuration was placed in a polycarbonate enclosure. Without the enclosure, the measured droplet size distribution was biased to large sizes because the currents swept away the small droplets. Once the droplets enter the Aerosizer DSP, they combine with a sheath air-stream to straighten the flow path and reduce sample losses due to sedimentation along the inner walls. The droplets
then travel through a nozzle where they experience a 2:1 pressure drop and accelerate near sonic velocity through the droplet counting and sizing region. In this region, the particles travel between two lasers that act as droplet counter and a stop watch to measure time-of-flight. The Aerosizer DSP converts the time-offlight to a number-average droplet size by a proprietary algorithm. The instrument stores the number of measured droplets into memory bins. Each bin had a defined droplet size range. The total mass of droplets was calculated within each bin with the following assumptions: drop size was equal to the median within a bin; drops were spherical and, their density was 1000 kg/m3. The total mass per the three minute sampling time was calculated by summing the masses within each bin; mass concentration was calculated by dividing the total mass per unit time by the volumetric sampling rate, 2.5 m3/minute. The drop count, size, and mass concentration of mist reported here were averages of three measurements and generally had a standard deviation of about 5%, Table 1.
’ RESULTS AND DISCUSSION Film-Split Behavior and Mechanisms. Ribs. Steady ribbing was studied with the rigid rolls in the vertical position using a 70 wt % glycerol/water solution (η= 20 mPa-s, σ = 57 mN/m). The capillary number (Ca) was raised by raising cylinder surface speed, and the gap was adjusted between 25 and 250 μm to extend the results of Coyle4 to smaller ratios of gap/roll diameter. Although the data sets do not overlap, reasonable agreement was found between the light scattering method of Coyle and the 3214
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Figure 2. Still images of film-split region in a side-by-side rigid roll configuration showing ribs and septa as capillary number (Ca) is raised for 100% glycerol. (a) Ribs at Ca = 2.95. (b) Ribs extended downstream of the gap at Ca = 14.8. (c) Ribs evolved into septa at Ca = 26.6. (d) Septa moved downstream at Ca = 47.3.
Mylar sheet method for detecting the onset of steady ribbing.18 The speed was then raised further with a 250 μm gap to determine the onset of unsteady ribbing. As Ca was raised to 0.19, the downstream meniscus receded into the gap, and the rib at the cylinder end became unstable. It widened until it divided, creating a new rib near the cylinder end. On further increase of Ca to 0.21, the ribbing pitch and amplitude fell and ribs moved toward the edges. When Ca was still further raised to 0.25, the ribbing pitch and amplitude fell further, and those near the cylinder ends began to oscillate upstream and downstream. Initially, the oscillations were mild and only near the ends, but became more vigorous and spread toward the middle. Then, the oscillations initiated merger and division of ribs. Between Ca of 0.29 and 0.57, the ribbing pattern became irregular. The crests of the ribs became blunted and the ribbing pitch became difficult to detect. Ribs oscillated in both the crossweb and downstream directions. The same sequence of events was displayed in the side-by-side configuration with a gap of 25 μm. Illumination of the film-split in this configuration was better at higher Ca; therefore, flow behavior at Ca well above 0.57 was visualized with this configuration (Figure 2). Septa. As Ca was raised further, ribs evolved into continuous sheets of liquid extending downstream of the gap forming a forward rim with two of its ends attached to ribs on the cylinder surfaces. These sheets are called septa. Figure 2a shows ribbing at Ca = 2.95 where they became unsteady in the cross-cylinder direction. Between Ca of 2.95 and 14.8, the downstream freesurface at each rib extended from the gap, presumably because of viscous drag, and formed septa as shown in Figure 2b. Once a septum formed, the ends of its forward rim slid along the crest of rib indicating that viscous shear forces were present along the base of the septum.
The curved diverging flow in the septum and its swollen rim suggest that tensile viscous force and surface tension were present; these forces resisted movement of a septum, by viscous drag, downstream of the gap. Figure 2c shows that at Ca of 26.6 the septa extended further from the gap. At a Ca of 47.3, they extended still further (Figure 2d). They became thin as they extended, but did not break. Destabilization of a septum by viscous drag acting along its ends is supported by the three-dimensional stability analysis of ribbing by Garfinkel-Castillo and Patera.9 As Ca was raised, the downstream meniscus receded toward the gap and the ribbing amplitude monotonically grew in the downstream direction. Their analysis and the images therein are interpreted here as the transition of a rib to a septum. With the rigid cylinders configurations, both vertical and side-by-side, postribbing flow behavior, up to formation of septa and their instabilities could be visualized. The septa, however, could not be made unstable enough to generate misting. The deformable roll configuration with a smaller cylinder (0.1 m diameter) and negative gap produced misting at high cylinder speeds or high Ca. Mist. As mentioned in the introduction, cavitation has been proposed in the literature to cause misting with non-Newtonian liquids. As speed or Ca is raised, liquid pressure falls below its vapor pressure, bubbles are proposed to be nucleated, liquid cavitates, and these cavities grow into filaments.21,23,28-32 Highspeed camera images shown in Figure 3 revealed a new mechanism for misting of Newtonian liquids. Events leading to generation of droplets, at a Ca of 44.6, from a septum located near the cylinder end were visualized. The mechanism depicted here was observed across the entire cylinder width, but because of its large size, the septum at the end lent itself for better visualization. Figure 3a shows a stable septum at a reference time of zero. The 3215
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Figure 3. Sequence of side-view images from high speed videos of a septum downstream of the film-split region of the deformable configuration, generated at a Ca of 44.6, with polydimethylsiloxane of viscosity 460 mPa-s. (a) An unsteady but stable septum at t = 0 s. (b) The septum extended downstream at t = 2 ms. (c) A Rayleigh-like instability developed along the septum rim at t = 5 ms. (d) The septum rim broke-up into mist droplets at t = 7 ms.
reflections behind the large septum are several septa in the crosscylinder direction. The septum slid on top of a rib and oscillated in both the downstream and cross-cylinder directions, perhaps because of run-out of the configuration. Figure 3b shows that at 2 ms (ms) relative to time zero, the downstream rim of the septum became extended downstream of the film-split region. The reason for this sudden downstream advancement of the leading edge is unclear. The curved free surface indicates that surface tension was still acting to pull the rim upstream suggesting that the liquid film behind the leading edge is intact. Figure 3c shows that at 5 ms the extended septum rim suffered a Rayleighlike instability resulting in periodic variation in thickness of its rim. Figure 3d shows droplets generated because of break-up of the rim. Prior to break-up, surface tension force tends to pull the septum toward the gap and viscous shear force away from it. When Rayleigh-like instability created fluctuations in the rim thickness, surface tension force destabilizes the rim toward breakup by squeezing it into drops. Normal and radial viscous forces likely slowed break-up in between droplets where extensional kinematics dominated. The balance of surface tension and viscous forces appear to control the stability of the septum and mist droplet size. To understand the relative importance of these forces, number of drops (drop count), drop size, and mass concentration of mist were measured as function of process parameters and material properties, Table 1. Effect of Viscosity. Figure 4 shows that as viscosity was raised, by using different PDMS oligomers, the droplet count rose linearly between 115 and 172 m/min at a speed ratio (ratio of lower and upper cylinder speeds) of unity. The rising drop count with increasing viscosity suggests that viscous forces destabilize septa toward break-up although viscous forces also
Figure 4. Droplet count as a function of poly(dimethylsiloxane) viscosity at four cylinder speeds with a speed ratio of unity. Lines show that the count rises linearly with viscosity.
will permit more liquid to pass through the nip. Nevertheless, it is proposed here that a septum extends downstream as speed is raised; consequently, it thins down in the lateral direction. A thin septum is susceptible to spontaneous rupture to form mist droplets, the greater the tendency of septa to rupture the greater number of droplets. Vrij and Overbeek33 predicted that a critical thickness has to be reached for spontaneous rupture of a liquid film. With higher viscosity liquids, septa extend further downstream because viscous forces tend to move them further away from the gap. Hence, they became thinner and longer than those with low viscosity liquids increasing their propensity to rupture and to produce more droplets. In the moments proceeding break-up of a septum into droplets, a competition of surface tension and viscosity takes place wherein surface tension drives the formation of droplets and viscosity acts to resist. The data in Table 1 shows that the droplet size within the 3216
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Figure 5. Droplet size (0) and droplet count (O) as a function of poly(dimethylsiloxane) viscosity at a cylinder speed of 134 m/min and a speed ration of unity. Linear fitting shows that the droplet count rises faster over the viscosity ranged studied as compared to the droplet size.
PDMS oligomers described above increased from 1.6 to 3.1 um as viscosity was raised from 74 to 1160 mPa s. The mass concentration of mist is the product of both droplet count and droplet size. Although the mass is proportional to d3drop Figure5 shows that at a constant cylinder speed (134 m/min) and over the entire viscosity range of PDMS studied here the drop diameter increases 2, while the droplet count increases nearly 17-18 over the same viscosity range. Consequently droplet count and mass concentration are analogous. Effect of Surface Tension. Figure 6 shows that as surface tension is raised (through Ca) a septum moves toward the filmsplit region making it less likely to reach the critical thickness to rupture into mist droplets. This was tested further by measuring misting of PDMS, PPG, and glycerol/water solutions at a fixed roll velocity (192 m/min) and a speed ratio of unity. It was not possible to obtain equal viscosities of commercial PDMS and PPG to directly compare the effect of surface tension. Consequently, misting is reported as a function of the ratio of surface tension to viscosity. Figure 6 shows the droplet count fell as surface tension was raised. Analyzing the droplet count data as a function of a ratio of surface tension (σ) to viscosity (η) is validated by overlap in data between PDMS and PEG at σ/η = 0.096 and 0.092 and PDMS and glycerol/water at σ /η = 0.297 and 0.295. It is worth noting that over the full range of surface tension and viscosity, examined at 192 m/min, the drop count follows a power-law (power-law exponent of -0.4); however, over a narrower range where misting was significant (inlayed plot, σ/η = 0.09-0.5 m/s) the drop count fell linearly as σ/η was raised. Effect of Cylinder Speed. The misting data above suggests that over a range of viscosities and surface tensions, where misting is significant, capillary number may be the appropriate nondimensional number to describe misting, as it does ribbing. However, the mass concentration of mist of pigmented dispersions has been reported to rise nonlinearly with speed.23,28,34 Blayo et al.22 reported the mass concentration of mist increased according to V2.8, where V equaled the average cylinder speed. However, in these reports the liquids studied where either known to be non-Newtonian or their rheology was not reported. Figure 7 shows that droplet count increased nonlinearly (V3), for each of the PDMS viscosities shown, as speed was raised between 95 and 192 m/min with a speed ratio of unity.
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Figure 6. Drop count versus ratio of surface tension to viscosity for poly(dimethylsiloxane) (0), poly(propylene glycol) (Δ), and glycerol/ water (O) at a speed of 192 m/min. Drop count fell over the entire range according to a power-law with an exponent of -0.4 (R2 = 0.98); over a narrower range of viscosity and surface tension, however, where misting was significant (inlayed plot) the drop count fell linearly.
Figure 7. Drop count rises according to V3 as cylinder speed increases for the 74 (0), 230 ()), 282 (O), and 460 mPa-s (Δ) poly(dimethylsiloxane) oils reported in Table 1.
Therefore, capillary number is not the appropriate nondimensional number for misting. Cylinder speed was found to have little effect on droplet size (Table 1). It is worth noting that the speed range investigated with PDMS oligomers was constrained. Below 115 m/min mist levels were too low to report (background levels were as much as 30% of total measured drop counts). Above 192 m/min high levels of mist saturated the detector. Effect of Cylinder Speed Ratio. To examine the effect of cylinder speed ratio on misting, a PDMS liquid of viscosity 282 mPa s was used. The mean speed was set to192 m/min but the speed ratios were adjusted between 1.0 and 3.0 (Table 1). The mass concentration of mist fell as speed ratio increased according to a power-law with an exponent of -0.7, Figure 8. The average drops size, however, was effectively constant at the speed ratios investigated. As the speed ratio was increased, the film-split meniscus became visibly asymmetric with more liquid coating the cylinder moving faster. Using a lubrication approximation to the twodimensional Navier-Stokes system of equations, Carvalho5 observed an asymmetry in coating thickness in a deformable-forward roll coating with the film thickness ratio proportional to the speed ratio0.7. Septa were also asymmetric; viscous force pulls the end of 3217
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Figure 8. Droplet count as a function of roll speed ratio at a constant mean speed of 192 m/min. Drop count fell according to a power law with a power-law index of -0.7, with R2 = 0.95.
Figure 10. Droplet count as a function of the combination of Ca, V2, and speed ratio. Linear regression shows R2 = 0.93.
Figure 9. Drop diameter as a function of Ca for three Newtonian liquids, poly(dimethylsiloxane), poly(propylene glycol) and glycerol/ water. Linear regression shows R2 = 0.95.
a septum attached to rib on faster moving cylinder further downstream from the gap than the end on slower moving cylinder. At the same mean speed, septa produced with differential speed are likely to be thicker than those with speed ratio of unity. Therefore, they are less likely to rupture and produce less mist. Scaling of Misting. In the prior sections droplet count, size, and mass concentration have been largely examined as a function of single material properties and process parameters. In this section the three forms of mist characterization are presented as a function of the process conditions across all three material systems. In Figure 5, droplet size is shown to rise linearly with an increasing viscosity at constant cylinder speed and surface tension. Figure 9 shows that when all the data is combined that droplet size scales linearly with Ca (R2 = 0.95). In Figures 4 and 5, the droplet count was shown to be rise and fall linearly with viscosity and surface tension, respectively. In Figure 7 droplet count rose with V3 and in Figure 8 was shown to fall according to (V1/V2)0.7. Combining these observations into a single parameter leads to eq 2. V2 Ca 0:7 V1 V2
ð2Þ
Figure 11. Mass concentration of mist as a function of the combination of Ca, V2, and speed ratio. Linear regression shows R2 = 0.95.
Figure 10 shows droplet count rises linearly as a function of eq 2 (R2 = 0.93). Although mass concentration of mist is proportional with d3drop and droplet count, it was shown in Figure 5 that as viscosity rose droplet count increased significantly faster than droplet size. Consequently mass concentration was found to have a similar scaling to material and process parameters as did droplet count. Figure 11 shows that when all the mass concentration data in Table 1 is plotted versus eq 2 the same linear relationship found for the droplet count was found for mass concentration (R2 = 0.96).
’ CONCLUSIONS High-speed visualization showed that as roll speed was raised ribs became unsteady; they oscillated, merged, and divided. At further higher speeds, ribs evolved into continuous sheets of liquid, called septa, extending downstream of the gap forming a forward rim with two of its ends attached to ribs on the rolls. At still higher speeds, mist was produced when septa extended 3218
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Industrial & Engineering Chemistry Research downstream, thinned cross-web, and the leading edge broke-up into droplets. Experimental measurements showed that raising viscosity, cylinder speed, or lowering surface tension generates more mist, and raising differential roll speed suppresses misting. Consolidating all the misting data presented here showed that droplet size scales linearly with Ca, and droplet count and mist concentration both scaled linearly with Ca(V2)/(V1/V2)0.7.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Notes ^
Deceased.
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