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Effect of rheological properties of shear thinning liquids on curtain stability
Accepted Manuscript
Effect of rheological properties of shear thinning liquids on curtain stability Alireza Mohammad Karim , Wieslaw J. Suszynski , William B. Griffith , Saswati Pujari , Lorraine F. Francis , Marcio S. Carvalho PII: DOI: Reference:
S0377-0257(18)30217-9 https://doi.org/10.1016/j.jnnfm.2018.11.009 JNNFM 4070
To appear in:
Journal of Non-Newtonian Fluid Mechanics
Received date: Revised date: Accepted date:
21 June 2018 18 November 2018 21 November 2018
Please cite this article as: Alireza Mohammad Karim , Wieslaw J. Suszynski , William B. Griffith , Saswati Pujari , Lorraine F. Francis , Marcio S. Carvalho , Effect of rheological properties of shear thinning liquids on curtain stability, Journal of Non-Newtonian Fluid Mechanics (2018), doi: https://doi.org/10.1016/j.jnnfm.2018.11.009
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Highlights High speed visualization of breakup process of liquid curtains. Critical Weber number for curtain breakup as a function of Ohnesorge number for shear thinning liquids.
Determination of a characteristic viscosity for the flow during curtain breakup process.
Guidance on how to change the viscosity curve to stabilize a liquid curtain.
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Effect of rheological properties of shear thinning liquids on curtain stability
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Alireza Mohammad Karim1, Wieslaw J. Suszynski1, William B. Griffith2, and Saswati Pujari2, Lorraine F. Francis1*, and Marcio S. Carvalho1,3* 1
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Department of Chemical Engineering & Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, Minnesota, 55455, USA 2 The Dow Chemical Company, 400 Arcola Road Collegeville, PA 19426, USA 3 Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marques de Sâo Vicente 225, Gávea, Rio de Janeiro, RJ 22453-900, Brazil
Abstract
Recent results have shown that Newtonian liquid curtains become more stable as the viscosity rises. However, the effect of shear thinning rheology on the dynamics of curtain breakup is not well understood. The role of shear thinning behavior on liquid
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curtain stability was studied by high-speed visualization. The critical condition at the onset of curtain breakup was determined by identifying the flow rate below which the
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curtain broke. Curtain breakup was seen to begin with a hole initiated within the curtain. The results reveal that the dynamics of curtain breakup is governed by the shear viscosity
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at a characteristic deformation rate and, as in the Newtonian liquid case, the stability
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increases as the characteristic viscosity rises.
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corresponding authors: M. S. Carvalho ([email protected]) and L. F. Francis ([email protected])
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1. Introduction Curtain coating is one of the preferred methods for high-speed precision coating of single-layer and multi-layer films. In curtain coating, a liquid sheet forms as the liquid exits the coating die and flows freely downward before it impinges on the surface of the moving substrate to be coated, as sketched in Fig. 1 [1-5]. Curtain coating is challenging,
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especially as the coating thickness decreases. To achieve a thinner coating, a lower flow rate out of the die and a thinner liquid curtain are needed. However, the liquid curtain breaks when the flow rate falls below a certain critical value.
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Figure 1. (a) Schematic of a side view of a stable curtain and representation of web speed, Uweb, and curtain velocity, U, at the impingement zone. (b) Schematic of front view of curtain breakup. The instability of liquid sheet is a phenomenon that was analyzed first by Taylor
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in 1959 [6]. Following Taylor’s analysis, Brown [7] presented the first experimental investigation of the stability of the Newtonian liquid curtain. He proposed a stability
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criterion based on the comparison of the inertial force in the curtain, which pushes down a hole formed within the curtain, with the capillary force, which causes the hole to move upward. This stability condition is therefore based on the Weber number, We, which
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measures the relative strength of inertial and capillary forces (eq. 1): We
QU
1,
(1)
where Q is the volumetric flow rate per unit width of the liquid curtain, U is the liquid curtain velocity,
is the density of the liquid, and
3
is the surface tension of the liquid.
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Many scientists have investigated the validity of the Brown’s stability criterion [8-15]. Several studies [11-15] have found conditions under which the critical parameters at the onset of curtain breakup are not well described by Brown’s criterion. Marangoni, viscous and viscoelastic forces may delay the onset of curtain breakup to lower values of critical flow rates.
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The mechanisms of liquid curtain breakup are more complex than the simple balance between inertial and capillary forces, proposed by Brown [7]. The viscosity of the liquid may have an important effect on curtain stability. Experimental results reported by Karim et al. [4] and numerical analyses presented by Sünderhauf et al. [16] have shown that the viscosity increases the stability of the liquid curtain. Theoretical and
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numerical analysis presented by Savva and Bush [17] concluded the important effect of viscosity is on the growth rate of a hole formed within a liquid curtain. This behavior was confirmed with the experimental results from Karim et al. [4]. This study [4] also revealed that the liquid viscosity affects Newtonian liquid curtain stability in two ways: (1) it slows down the retraction speed of the rim of a hole in the curtain, stabilizing it and
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(2) it slows down the curtain speed inside the viscous boundary layer that is formed near the edge guides, promoting curtain breakup. With properly designed edge guides the first
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effect dominates.
Typical liquids used in curtain coating, are complex colloidal suspensions or
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emulsions [18], which exhibit shear thinning behavior. Despite the broad industrial application, there is still a lack of fundamental understanding of the importance of rate-
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dependent shear viscosity on curtain breakup mechanism and how the viscosity curve should be adjusted, by the use of additives, in order to stabilize liquid curtains. In this work, we present a comprehensive analysis of the relation between the rheological
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characteristics of dilute xanthan gum solutions (i.e., shear thinning liquids) and their liquid curtain stability. The results indicate how the viscosity curve of coating liquids should be adjusted to be able to operate curtain coating process at lower flow rates.
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2. Materials and Methods 2.1. Curtain Coating Experiments The curtain coating set up is shown in Fig. 2. A peristaltic pump delivers liquid from the tank to the coating die to form a curtain between two teflon-coated edge guides at an angle of two degrees. The dynamics of the curtain breakage was recorded and the
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local curtain velocity, u, at different positions (x,y) along the curtain, where x is the distance from the edge guide and y is the distance from the lip of the slide coating die, was also determined. Details of the experimental procedures are described in a previous study of liquid curtain stability of Newtonian solutions [4].
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Figure 2. (a) Schematics of the curtain coating set-up used in the experiments. Dotted lines with arrows indicate the direction of the liquid flow in the curtain coating setup. (b) Front view of the liquid curtain between two Teflon coated converging (bent) edge guides. 5
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2.2. Materials Aqueous solutions of xanthan gum (XG) were used as the shear thinning liquids. Such solutions have been used as model shear-thinning liquids with negligible elasticity
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to study different flow phenomena [19,20]. Xanthan gum powder was added slowly to either distilled water or a glycerol (80 %) - water solution and stirred for about 15 minutes to prevent formation of clumps. Then, the 0.7 mM sodium dodecyl sulfate (SDS) was added and stirred for about five minutes. Next, the solution was homogenized in 1.5 L batches at a speed of 25000 rpm for about four minutes using a homogenizer (IKA model T 18). Finally, the solutions were left for a day to let the entrained air bubbles
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escape.
The shear viscosity curves were obtained using the AR-G2 Rheometer (TA Instruments) with a Couette cell geometry. All xanthan gum solutions tested exhibit shear thinning behavior (see Fig. 3). In the range of shear rate explored, the viscosity curves
viscosity, ̇ is shear rate,
̇
, where
is shear
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shows a low-shear plateau, a power-law region (
is flow consistency index, and n is flow behavior index) and a
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high-shear plateau value. The high shear plateau value approaches the solvent viscosity, i.e.
for the glycerol-water solution. The high shear plateau value for the
water based solutions was not achieved because of inertial effect on the measurements, as
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it is clear if Fig.3. The low shear viscosity plateau
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viscosity and polymer concentration (see Fig. 3).
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is a function of both the solvent
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Figure 3. Rheological properties of xanthan gum solutions in water and glycerol (80 %) water. The characteristic rate of deformation of the curtain breakup flow can be estimated by assuming a radial flow during the hole opening process in a free liquid
width:
̇
⁄
. It is approximated by the ratio of the rim velocity to the curtain
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sheet, i.e. ̇
⁄ . Experimental and theoretical models [4, 16, 17] have shown that
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the rim velocity approaches the Taylor-Culick velocity [6, 21], which is based on a
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balance between inertial and capillary forces:
√ , where t is the curtain
thickness. For the conditions of the present experiments, the characteristic deformation rate is approximately referred to as
̇
. The liquid viscosity at this characteristic rate is
.
Table 1 shows the physical properties of shear thinning solutions prepared for the experiments. The surface tension was measured using the Wilhelmy plate method in the
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digital tensiometer K10ST (Krüss) and density was determined by Krohne MFC 100 Coriolis type mass flow meter used in curtain coating experiment.
Table 1. The physical properties of xanthan gum solutions. Solvent
Density,
0.05
Water
[kg/m ] 993.0
Surface Tension, [mN/m] 47.5
933
Shear viscosity at characteristic shear rate, [mPa.s] 24
0.15
Water
992.7
57.5
2440
102
0.30
Water
986.2
58.3
16290
0.05 0.10
Glycerol (80 %) Glycerol (80 %)
1197.1 1195.3
59.0 58.5
1010 2430
3
Low-Shear Viscosity, [mPa.s]
High-Shear Viscosity, [mPa.s]
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Xanthan Gum [wt%]
325 215 427
The viscoelastic character of the solutions was probed by using the Capillary
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Break-up Extensional Rheometer (CaBER) [22,23]. In the CaBER instrument, a liquid bridge is formed between two cylindrical fixtures. Applying an axial step-strain on the liquid bridge leads to formation of an elongated filament. The CaBER uses a laser micrometer to measure the evolution of mid-point filament diameter as it thins down and breaks.
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For viscoelastic liquids, described by the FENE-P model, after a fast viscous dominated regime, the flow dynamics is governed by the competition between surface
e(-
)
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tension and elastic forces. In this regime, the filament diameter falls exponentially as , where is the relaxation time of the liquid [22]. For viscous liquid
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filaments, Papageorgiou [24] has shown that the diameter of the neck of the liquid filament falls linearly as:
.
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The variation of the mid-point filament diameter with time for the glycerol-water solutions is shown in Figure 4. The breakup of the filament of all water-based solutions was so fast that it could not be accurately measured. The diameter decay with time is
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well described by a linear fit, indicating the viscous breakup regime, as shown in Fig.4. Xanthan gum solutions are shear-thinning liquids with negligible elastic stress. The viscosity in extension can be estimated by the angular coefficient of the linear fit [23]. For these solutions, the estimated extensional viscosity was glycerol-water solution and Trouton ratio, defined as
for the 0.05%
for the 0.1% glycerol-water solution. The , was
and
, respectively. Similar
values of extensional viscosity of Xanthan gum solutions have been reported in the 8
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literature and have been associated with the alignment of semi-rigid molecules [25]. Our prior analysis of curtain breakup with viscoelastic solutions show that viscoelastic stresses strongly affect the dynamics of curtain breakup for Trouton ratio higher than 2000 [15]. The relatively small value of the Trouton ratio of xanthan gum solutions shows their weak extensional thickening behavior. The dynamics of the curtain breakup
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is mainly governed by the viscous stresses.
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Figure 4. Filament diameter versus time during capillary thinning experiments. (a) 0.05
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wt% XG in glycerol (80 %) - water. (b) 0.10 wt% XG in glycerol (80 %) - water.
3. Results
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3.1. Liquid velocity in the curtain
Since the stability condition for the liquid curtain is highly related to the ratio between the speed at which a disturbance (i.e., hole) expands in the curtain and the speed by which the hole is convected by flow, it is crucial to determine the effect of viscous forces on the liquid velocity in the curtain. The velocity of the liquid curtain at different
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distances from the edge guides was measured by tracking the downward motion of air bubbles, as shown in Figure 5. The free fall velocity is described by eq. 2: u (Uexit
g )
,
(2)
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where y is the distance from the die, Uexit is the velocity on the top of the curtain, defined in a first approximation as
is the slot height, which is 508 µm, and g is gravitational
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width of the die and
, where Q is the volumetric flow rate per unit
acceleration. The free fall velocity is shown in Figure 5 as a dashed line.
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The experiments were done using a slide-fed curtain coating die. Therefore, the velocity on the top of the curtain needs to take into account the viscous forces along the
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flow down the inclined plane of the die and along the edge guides of the curtain. The film thickness hdie and velocity at the end of the incline Udie can be estimated by
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assuming a steady state, laminar, one-dimensional, fully-developed, and incompressible flow of a power-law liquid on an inclined plane under effect of gravity [26-28] (the shear rates of the flow down an inclined plane are inside the power-law region of the viscosity curves):
n
hdie
[(
n
1 n
)(
10
g sin K
)
-1 n
Q]
n
1
,
(3)
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⁄
( where
) (
)
(
)
,
(4)
is the inclination angle of the die surface. The curtain velocity estimation
considering viscous effect along the slide die surface is shown as a continuous line in the
U
√Udie
g y.
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plots of Fig.5 and is given by: (5)
For bubbles at different positions with respect to the edge guides, the measured velocity of the curtain is well described by eq. (5), but is systematically lower than the values
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layer formed along the edge guides [28].
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predicted by eq.(5). The velocity difference can be associated with the viscous boundary
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Figure 5. The velocities of the air bubbles within the liquid curtain for (a) 0.15 wt% xanthan gum solution in water, (b) 0.30 wt% xanthan gum solution in water. The measurement uncertainty in u and y is less than the size of the data points.
3.2. Visualization of the liquid curtain breakup
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High-speed visualization was used to analyze the sequence of events that leads to the liquid curtain breakup. Figure 6 shows the expansion of a hole formed within the
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curtain and the eventual complete breakup of the liquid curtain into individual liquid columns for 0.15 wt% xanthan gum solution in water. The entire process for complete
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liquids [4].
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curtain breakup took 998 ms. The breakup process is similar to the one for Newtonian
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Figure 6. The mechanism of the liquid curtain breakage for 0.15 wt% xanthan gum in water solution. (a) t = 50 ms. (b) t = 54.5 ms. (c) t = 62.5 ms. (d) t = 72.5 ms. (e) t = 145 ms. (f) t = 998 ms.
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3.3. Growth dynamics of hole within the shear thinning liquid film As illustrated in Figure 6, liquid curtain breakup is initiated by the formation of a
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hole within the liquid curtain. The capillary force pulls the rim of the hole, causing expansion. At the same time, the curtain flow convects the hole downward. The dynamics
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of the hole-expansion along the horizontal axis (i.e., along the width of the liquid curtain) and the vertical axis (i.e., along the height of the liquid curtain) was analyzed by
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measuring the evolution of the horizontal positions of right, xR, and left, xL, rims and the vertical positions of top, yT, and bottom, yB, rims during a hole expansion for different solutions, the results are presented in Fig. 7.
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Figure 7. (a) The schematics of bounding rims at different locations in the curtain. The dynamics of bounding rim in horizontal and vertical directions: (b) 0.05 wt% xanthan gum in water. (c) 0.15 wt% xanthan gum in water. (d) 0.30 wt% xanthan gum in water. (e) 0.05 wt% xanthan gum in glycerol (80 %) - water solution. (f) 0.10 wt% xanthan gum in glycerol (80 %) - water solution.
The rim speeds in x-direction (ux,L and ux,R), where ux,L and ux,R are the speeds at left and right side of the hole respectively, and y-direction (uy,B and uy,T), where uy,B and
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uy,T are the speeds at bottom and top part of the hole respectively, were calculated from the data in Fig. 7, for the time interval of the hole-expansion for each experiment before the complete curtain breakup. The measured retraction speeds obtained from different polymeric solutions are presented in Figure 8a. The measured retraction speeds are
UT
√ t.
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normalized by the Taylor-Culick [6, 21] speed, UTC, defined as: (6)
In eq. (6), t is the local film thickness at the position of the hole formation, which is evaluated based on the local curtain velocity and mass conservation: Q UCurtain wCurtain
.
(7)
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t
In eq. (7), wCurtain is the local width of the curtain, Q is the critical volumetric flow rate for curtain breakup, and UCurtain is the liquid curtain velocity at the location of holeinitiation.
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The plot in Figure 8a presents both the horizontal and vertical rim speeds, as indicated in Fig. 8b, as a function of the Ohnesorge number, Oh, defined here in terms of :
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the characteristic shear viscosity
1
√
t
.
(8)
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Oh
Therefore, each of the five solutions investigated has a unique Oh. Table 3 presents the
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initial location of the hole in the curtain and its bounding rim speeds for each solution tested. The curtain thickness at the breakup point, the corresponding Taylor-Culick speed
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and Ohnesorge number are also presented.
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Figure 8. (a) Normalized measured speeds of the bounding rim retraction in x direction (i.e. ux,L and ux,R), and y direction (i.e. uy,B and uy,T) versus the Ohnesorge number. (b) Schematics of the location on the bounding rim where the speeds were measured.
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It is important to note that the horizontal retraction speed is in the order of 1 m/s, the same order of magnitude as the curtain velocity at the location of the hole formation (see Table 3). This agrees well with the general stability criterion stating that the curtain becomes unstable if the retraction speed is equal to the curtain velocity. However, the curtain thickness (flow rate) at which the curtain breaks is a strong function of the
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Ohnesorge number. The measured horizontal retraction speed is close to the TaylorCulick speed and falls slightly with Ohnesorge number defined in terms of the characteristic shear viscosity (see Fig. 8).
Xanthan Gum [wt%] 0.05 0.15 0.30 0.05 0.10
Solvent Water Water Water Glycerol (80 %) Glycerol (80 %)
X* [cm] 1.93 1.79 1.83 1.93 1.71
Y* [cm] 3.60 5.57 7.14 6.33 5.25
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Table 3. Details of the curtain breakup for shear thinning solutions. ux,L [m/s] 0.85 1.17 1.07 0.99 0.87
ux,R [m/s] 0.94 1.21 1.07 1.00 0.84
uy,B [m/s] 2.01 2.40 2.32 2.17 1.99
uy,T [m/s] 0.13 0.19 0.05 0.28 0.16
UTC [m/s] 0.71 1.09 1.17 1.27 1.10
UCurtain [m/s] 0.89 1.06 1.21 1.12 1.04
Oh 0.25 0.97 3.27 2.32 4.03
t [µm] 191 98 87 62 80
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3.4. Liquid curtain stability
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X: hole distance from the edge guide; Y: hole distance from exit of the die.
The critical flow rate below which the curtain breaks was recorded for the different shear thinning solutions. The critical flow rate represented by the Weber number
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(ratio of inertial and capillary forces) versus the Ohnesorge number, defined in terms of the characteristic shear viscosity
, is shown in Figure 9. Several experiments were
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performed for each solution. Because the Ohnesorge number is defined in terms of the curtain thickness at the breakup condition, its value is not constant for each solution. The
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critical flow rate decreases as Ohnesorge number increases. It is important to compare the behavior of the 0.3 wt% in water solution with the 0.1 wt% in glycerol-water solution. These solutions have very different high and low shear viscosity but similar characteristic shear viscosity
. Both solutions have similar critical flow rate and curtain thickness at
the breakup. On the other hand, solutions with 0.05 wt% and 0.15 wt% in water have high shear viscosity in the same range but very different characteristic shear viscosity; their critical flow rate and curtain thickness at breakup conditions are very different. The
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breakup dynamics is governed by the shear viscosity evaluated at the characteristic deformation rate of the curtain flow. In Figure 9, we have included the results of critical Weber number of our previous analyses [4] for Newtonian liquids. The behavior of the shear thinning solutions follow the general trend of the Newtonian liquids with the Oh
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defined in terms of the characteristic shear viscosity.
Figure 9. Critical Weber number, We, at which the liquid curtain breaks as a function of the Ohnesorge number, Oh.
4. Final Remarks 18
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The effect of rheological properties on shear thinning curtain stability was studied experimentally by high-speed visualization of the sequence of events that ultimately lead to the curtain breakup and by determining the critical flow rate below which the curtain becomes unstable. The velocity evolution along the curtain was measured at different distances from the edges. It was found the formed viscous boundary layer along the edge
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guide causes the curtain velocity deviates from the free fall velocity. Previous analyses [6,8] have proposed a simple criterion for curtain stability based on the ratio of the curtain velocity, which moves any hole in the curtain downward, to the rim retraction speed, which opens the hole. If the ratio is above one, the curtain is stable. The high-speed visualization images presented here confirm this hypothesis. At
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larger flow rate; a hole is convected down the curtain before it becomes large enough to cause the curtain breakup. Below a critical flow rate, at which the retraction speed is larger than the curtain speed, the hole grows, leading to the curtain breakup. The results show that the dynamics of curtain breakup is governed by the shear viscosity of the liquid evaluated at a characteristic deformation rate of the curtain breakup
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flow. Higher characteristic viscosity leads to more stable curtain; i.e., lower critical flow rate. The dependence of the critical Weber number below which the curtain breaks as a
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function of the Ohnesorge number, Oh, follows the general behavior previously reported
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for Newtonian liquids if the Oh is defined based on this characteristic viscosity.
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Acknowledgements
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This research is supported by the Dow Chemical Company.
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[26] Weinstein, S. J., [1990] “Wave propagation in the flow of shear-thinning fluids down an incline” AIChE J. 36, 1873–1889. [27] Jiang, W. Y., Helenbrook, B. T., Lin, S. P., Weinstein, S. J., [ 5] “Low-Reynoldsnumber instabilities in three-layer flow down an inclined wall” J. Fluid Mech. 539, 387– 416.
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[28] Marston, J. O., Thoroddsen, S. T., Thompson, J., Blyth, M. G., Henry, D., and Uddin, J., [ 14] “Experimental investigation of hysteresis in the break-up of liquid curtains” Chem. Eng. Sci., 117, 248-263.
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Effect of rheological properties of shear thinning liquids on curtain stability Alireza Mohammad Karim , Wieslaw J. Suszynski , William B. Griffith , Saswati Pujari , Lorraine F. Francis , Marcio S. Carvalho PII: DOI: Reference:
S0377-0257(18)30217-9 https://doi.org/10.1016/j.jnnfm.2018.11.009 JNNFM 4070
To appear in:
Journal of Non-Newtonian Fluid Mechanics
Received date: Revised date: Accepted date:
21 June 2018 18 November 2018 21 November 2018
Please cite this article as: Alireza Mohammad Karim , Wieslaw J. Suszynski , William B. Griffith , Saswati Pujari , Lorraine F. Francis , Marcio S. Carvalho , Effect of rheological properties of shear thinning liquids on curtain stability, Journal of Non-Newtonian Fluid Mechanics (2018), doi: https://doi.org/10.1016/j.jnnfm.2018.11.009
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Highlights High speed visualization of breakup process of liquid curtains. Critical Weber number for curtain breakup as a function of Ohnesorge number for shear thinning liquids.
Determination of a characteristic viscosity for the flow during curtain breakup process.
Guidance on how to change the viscosity curve to stabilize a liquid curtain.
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Effect of rheological properties of shear thinning liquids on curtain stability
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Alireza Mohammad Karim1, Wieslaw J. Suszynski1, William B. Griffith2, and Saswati Pujari2, Lorraine F. Francis1*, and Marcio S. Carvalho1,3* 1
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Department of Chemical Engineering & Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, Minnesota, 55455, USA 2 The Dow Chemical Company, 400 Arcola Road Collegeville, PA 19426, USA 3 Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marques de Sâo Vicente 225, Gávea, Rio de Janeiro, RJ 22453-900, Brazil
Abstract
Recent results have shown that Newtonian liquid curtains become more stable as the viscosity rises. However, the effect of shear thinning rheology on the dynamics of curtain breakup is not well understood. The role of shear thinning behavior on liquid
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curtain stability was studied by high-speed visualization. The critical condition at the onset of curtain breakup was determined by identifying the flow rate below which the
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curtain broke. Curtain breakup was seen to begin with a hole initiated within the curtain. The results reveal that the dynamics of curtain breakup is governed by the shear viscosity
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at a characteristic deformation rate and, as in the Newtonian liquid case, the stability
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increases as the characteristic viscosity rises.
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corresponding authors: M. S. Carvalho ([email protected]) and L. F. Francis ([email protected])
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1. Introduction Curtain coating is one of the preferred methods for high-speed precision coating of single-layer and multi-layer films. In curtain coating, a liquid sheet forms as the liquid exits the coating die and flows freely downward before it impinges on the surface of the moving substrate to be coated, as sketched in Fig. 1 [1-5]. Curtain coating is challenging,
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especially as the coating thickness decreases. To achieve a thinner coating, a lower flow rate out of the die and a thinner liquid curtain are needed. However, the liquid curtain breaks when the flow rate falls below a certain critical value.
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Figure 1. (a) Schematic of a side view of a stable curtain and representation of web speed, Uweb, and curtain velocity, U, at the impingement zone. (b) Schematic of front view of curtain breakup. The instability of liquid sheet is a phenomenon that was analyzed first by Taylor
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in 1959 [6]. Following Taylor’s analysis, Brown [7] presented the first experimental investigation of the stability of the Newtonian liquid curtain. He proposed a stability
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criterion based on the comparison of the inertial force in the curtain, which pushes down a hole formed within the curtain, with the capillary force, which causes the hole to move upward. This stability condition is therefore based on the Weber number, We, which
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measures the relative strength of inertial and capillary forces (eq. 1): We
QU
1,
(1)
where Q is the volumetric flow rate per unit width of the liquid curtain, U is the liquid curtain velocity,
is the density of the liquid, and
3
is the surface tension of the liquid.
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Many scientists have investigated the validity of the Brown’s stability criterion [8-15]. Several studies [11-15] have found conditions under which the critical parameters at the onset of curtain breakup are not well described by Brown’s criterion. Marangoni, viscous and viscoelastic forces may delay the onset of curtain breakup to lower values of critical flow rates.
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The mechanisms of liquid curtain breakup are more complex than the simple balance between inertial and capillary forces, proposed by Brown [7]. The viscosity of the liquid may have an important effect on curtain stability. Experimental results reported by Karim et al. [4] and numerical analyses presented by Sünderhauf et al. [16] have shown that the viscosity increases the stability of the liquid curtain. Theoretical and
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numerical analysis presented by Savva and Bush [17] concluded the important effect of viscosity is on the growth rate of a hole formed within a liquid curtain. This behavior was confirmed with the experimental results from Karim et al. [4]. This study [4] also revealed that the liquid viscosity affects Newtonian liquid curtain stability in two ways: (1) it slows down the retraction speed of the rim of a hole in the curtain, stabilizing it and
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(2) it slows down the curtain speed inside the viscous boundary layer that is formed near the edge guides, promoting curtain breakup. With properly designed edge guides the first
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effect dominates.
Typical liquids used in curtain coating, are complex colloidal suspensions or
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emulsions [18], which exhibit shear thinning behavior. Despite the broad industrial application, there is still a lack of fundamental understanding of the importance of rate-
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dependent shear viscosity on curtain breakup mechanism and how the viscosity curve should be adjusted, by the use of additives, in order to stabilize liquid curtains. In this work, we present a comprehensive analysis of the relation between the rheological
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characteristics of dilute xanthan gum solutions (i.e., shear thinning liquids) and their liquid curtain stability. The results indicate how the viscosity curve of coating liquids should be adjusted to be able to operate curtain coating process at lower flow rates.
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2. Materials and Methods 2.1. Curtain Coating Experiments The curtain coating set up is shown in Fig. 2. A peristaltic pump delivers liquid from the tank to the coating die to form a curtain between two teflon-coated edge guides at an angle of two degrees. The dynamics of the curtain breakage was recorded and the
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local curtain velocity, u, at different positions (x,y) along the curtain, where x is the distance from the edge guide and y is the distance from the lip of the slide coating die, was also determined. Details of the experimental procedures are described in a previous study of liquid curtain stability of Newtonian solutions [4].
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(a)
Figure 2. (a) Schematics of the curtain coating set-up used in the experiments. Dotted lines with arrows indicate the direction of the liquid flow in the curtain coating setup. (b) Front view of the liquid curtain between two Teflon coated converging (bent) edge guides. 5
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2.2. Materials Aqueous solutions of xanthan gum (XG) were used as the shear thinning liquids. Such solutions have been used as model shear-thinning liquids with negligible elasticity
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to study different flow phenomena [19,20]. Xanthan gum powder was added slowly to either distilled water or a glycerol (80 %) - water solution and stirred for about 15 minutes to prevent formation of clumps. Then, the 0.7 mM sodium dodecyl sulfate (SDS) was added and stirred for about five minutes. Next, the solution was homogenized in 1.5 L batches at a speed of 25000 rpm for about four minutes using a homogenizer (IKA model T 18). Finally, the solutions were left for a day to let the entrained air bubbles
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escape.
The shear viscosity curves were obtained using the AR-G2 Rheometer (TA Instruments) with a Couette cell geometry. All xanthan gum solutions tested exhibit shear thinning behavior (see Fig. 3). In the range of shear rate explored, the viscosity curves
viscosity, ̇ is shear rate,
̇
, where
is shear
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shows a low-shear plateau, a power-law region (
is flow consistency index, and n is flow behavior index) and a
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high-shear plateau value. The high shear plateau value approaches the solvent viscosity, i.e.
for the glycerol-water solution. The high shear plateau value for the
water based solutions was not achieved because of inertial effect on the measurements, as
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it is clear if Fig.3. The low shear viscosity plateau
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viscosity and polymer concentration (see Fig. 3).
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is a function of both the solvent
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Figure 3. Rheological properties of xanthan gum solutions in water and glycerol (80 %) water. The characteristic rate of deformation of the curtain breakup flow can be estimated by assuming a radial flow during the hole opening process in a free liquid
width:
̇
⁄
. It is approximated by the ratio of the rim velocity to the curtain
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sheet, i.e. ̇
⁄ . Experimental and theoretical models [4, 16, 17] have shown that
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the rim velocity approaches the Taylor-Culick velocity [6, 21], which is based on a
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balance between inertial and capillary forces:
√ , where t is the curtain
thickness. For the conditions of the present experiments, the characteristic deformation rate is approximately referred to as
̇
. The liquid viscosity at this characteristic rate is
.
Table 1 shows the physical properties of shear thinning solutions prepared for the experiments. The surface tension was measured using the Wilhelmy plate method in the
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digital tensiometer K10ST (Krüss) and density was determined by Krohne MFC 100 Coriolis type mass flow meter used in curtain coating experiment.
Table 1. The physical properties of xanthan gum solutions. Solvent
Density,
0.05
Water
[kg/m ] 993.0
Surface Tension, [mN/m] 47.5
933
Shear viscosity at characteristic shear rate, [mPa.s] 24
0.15
Water
992.7
57.5
2440
102
0.30
Water
986.2
58.3
16290
0.05 0.10
Glycerol (80 %) Glycerol (80 %)
1197.1 1195.3
59.0 58.5
1010 2430
3
Low-Shear Viscosity, [mPa.s]
High-Shear Viscosity, [mPa.s]
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Xanthan Gum [wt%]
325 215 427
The viscoelastic character of the solutions was probed by using the Capillary
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Break-up Extensional Rheometer (CaBER) [22,23]. In the CaBER instrument, a liquid bridge is formed between two cylindrical fixtures. Applying an axial step-strain on the liquid bridge leads to formation of an elongated filament. The CaBER uses a laser micrometer to measure the evolution of mid-point filament diameter as it thins down and breaks.
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For viscoelastic liquids, described by the FENE-P model, after a fast viscous dominated regime, the flow dynamics is governed by the competition between surface
e(-
)
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tension and elastic forces. In this regime, the filament diameter falls exponentially as , where is the relaxation time of the liquid [22]. For viscous liquid
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filaments, Papageorgiou [24] has shown that the diameter of the neck of the liquid filament falls linearly as:
.
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The variation of the mid-point filament diameter with time for the glycerol-water solutions is shown in Figure 4. The breakup of the filament of all water-based solutions was so fast that it could not be accurately measured. The diameter decay with time is
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well described by a linear fit, indicating the viscous breakup regime, as shown in Fig.4. Xanthan gum solutions are shear-thinning liquids with negligible elastic stress. The viscosity in extension can be estimated by the angular coefficient of the linear fit [23]. For these solutions, the estimated extensional viscosity was glycerol-water solution and Trouton ratio, defined as
for the 0.05%
for the 0.1% glycerol-water solution. The , was
and
, respectively. Similar
values of extensional viscosity of Xanthan gum solutions have been reported in the 8
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literature and have been associated with the alignment of semi-rigid molecules [25]. Our prior analysis of curtain breakup with viscoelastic solutions show that viscoelastic stresses strongly affect the dynamics of curtain breakup for Trouton ratio higher than 2000 [15]. The relatively small value of the Trouton ratio of xanthan gum solutions shows their weak extensional thickening behavior. The dynamics of the curtain breakup
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is mainly governed by the viscous stresses.
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Figure 4. Filament diameter versus time during capillary thinning experiments. (a) 0.05
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wt% XG in glycerol (80 %) - water. (b) 0.10 wt% XG in glycerol (80 %) - water.
3. Results
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3.1. Liquid velocity in the curtain
Since the stability condition for the liquid curtain is highly related to the ratio between the speed at which a disturbance (i.e., hole) expands in the curtain and the speed by which the hole is convected by flow, it is crucial to determine the effect of viscous forces on the liquid velocity in the curtain. The velocity of the liquid curtain at different
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distances from the edge guides was measured by tracking the downward motion of air bubbles, as shown in Figure 5. The free fall velocity is described by eq. 2: u (Uexit
g )
,
(2)
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where y is the distance from the die, Uexit is the velocity on the top of the curtain, defined in a first approximation as
is the slot height, which is 508 µm, and g is gravitational
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width of the die and
, where Q is the volumetric flow rate per unit
acceleration. The free fall velocity is shown in Figure 5 as a dashed line.
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The experiments were done using a slide-fed curtain coating die. Therefore, the velocity on the top of the curtain needs to take into account the viscous forces along the
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flow down the inclined plane of the die and along the edge guides of the curtain. The film thickness hdie and velocity at the end of the incline Udie can be estimated by
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assuming a steady state, laminar, one-dimensional, fully-developed, and incompressible flow of a power-law liquid on an inclined plane under effect of gravity [26-28] (the shear rates of the flow down an inclined plane are inside the power-law region of the viscosity curves):
n
hdie
[(
n
1 n
)(
10
g sin K
)
-1 n
Q]
n
1
,
(3)
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⁄
( where
) (
)
(
)
,
(4)
is the inclination angle of the die surface. The curtain velocity estimation
considering viscous effect along the slide die surface is shown as a continuous line in the
U
√Udie
g y.
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plots of Fig.5 and is given by: (5)
For bubbles at different positions with respect to the edge guides, the measured velocity of the curtain is well described by eq. (5), but is systematically lower than the values
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layer formed along the edge guides [28].
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predicted by eq.(5). The velocity difference can be associated with the viscous boundary
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Figure 5. The velocities of the air bubbles within the liquid curtain for (a) 0.15 wt% xanthan gum solution in water, (b) 0.30 wt% xanthan gum solution in water. The measurement uncertainty in u and y is less than the size of the data points.
3.2. Visualization of the liquid curtain breakup
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High-speed visualization was used to analyze the sequence of events that leads to the liquid curtain breakup. Figure 6 shows the expansion of a hole formed within the
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curtain and the eventual complete breakup of the liquid curtain into individual liquid columns for 0.15 wt% xanthan gum solution in water. The entire process for complete
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liquids [4].
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curtain breakup took 998 ms. The breakup process is similar to the one for Newtonian
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Figure 6. The mechanism of the liquid curtain breakage for 0.15 wt% xanthan gum in water solution. (a) t = 50 ms. (b) t = 54.5 ms. (c) t = 62.5 ms. (d) t = 72.5 ms. (e) t = 145 ms. (f) t = 998 ms.
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3.3. Growth dynamics of hole within the shear thinning liquid film As illustrated in Figure 6, liquid curtain breakup is initiated by the formation of a
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hole within the liquid curtain. The capillary force pulls the rim of the hole, causing expansion. At the same time, the curtain flow convects the hole downward. The dynamics
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of the hole-expansion along the horizontal axis (i.e., along the width of the liquid curtain) and the vertical axis (i.e., along the height of the liquid curtain) was analyzed by
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measuring the evolution of the horizontal positions of right, xR, and left, xL, rims and the vertical positions of top, yT, and bottom, yB, rims during a hole expansion for different solutions, the results are presented in Fig. 7.
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Figure 7. (a) The schematics of bounding rims at different locations in the curtain. The dynamics of bounding rim in horizontal and vertical directions: (b) 0.05 wt% xanthan gum in water. (c) 0.15 wt% xanthan gum in water. (d) 0.30 wt% xanthan gum in water. (e) 0.05 wt% xanthan gum in glycerol (80 %) - water solution. (f) 0.10 wt% xanthan gum in glycerol (80 %) - water solution.
The rim speeds in x-direction (ux,L and ux,R), where ux,L and ux,R are the speeds at left and right side of the hole respectively, and y-direction (uy,B and uy,T), where uy,B and
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uy,T are the speeds at bottom and top part of the hole respectively, were calculated from the data in Fig. 7, for the time interval of the hole-expansion for each experiment before the complete curtain breakup. The measured retraction speeds obtained from different polymeric solutions are presented in Figure 8a. The measured retraction speeds are
UT
√ t.
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normalized by the Taylor-Culick [6, 21] speed, UTC, defined as: (6)
In eq. (6), t is the local film thickness at the position of the hole formation, which is evaluated based on the local curtain velocity and mass conservation: Q UCurtain wCurtain
.
(7)
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t
In eq. (7), wCurtain is the local width of the curtain, Q is the critical volumetric flow rate for curtain breakup, and UCurtain is the liquid curtain velocity at the location of holeinitiation.
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The plot in Figure 8a presents both the horizontal and vertical rim speeds, as indicated in Fig. 8b, as a function of the Ohnesorge number, Oh, defined here in terms of :
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the characteristic shear viscosity
1
√
t
.
(8)
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Oh
Therefore, each of the five solutions investigated has a unique Oh. Table 3 presents the
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initial location of the hole in the curtain and its bounding rim speeds for each solution tested. The curtain thickness at the breakup point, the corresponding Taylor-Culick speed
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and Ohnesorge number are also presented.
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(a)
Figure 8. (a) Normalized measured speeds of the bounding rim retraction in x direction (i.e. ux,L and ux,R), and y direction (i.e. uy,B and uy,T) versus the Ohnesorge number. (b) Schematics of the location on the bounding rim where the speeds were measured.
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It is important to note that the horizontal retraction speed is in the order of 1 m/s, the same order of magnitude as the curtain velocity at the location of the hole formation (see Table 3). This agrees well with the general stability criterion stating that the curtain becomes unstable if the retraction speed is equal to the curtain velocity. However, the curtain thickness (flow rate) at which the curtain breaks is a strong function of the
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Ohnesorge number. The measured horizontal retraction speed is close to the TaylorCulick speed and falls slightly with Ohnesorge number defined in terms of the characteristic shear viscosity (see Fig. 8).
Xanthan Gum [wt%] 0.05 0.15 0.30 0.05 0.10
Solvent Water Water Water Glycerol (80 %) Glycerol (80 %)
X* [cm] 1.93 1.79 1.83 1.93 1.71
Y* [cm] 3.60 5.57 7.14 6.33 5.25
*
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Table 3. Details of the curtain breakup for shear thinning solutions. ux,L [m/s] 0.85 1.17 1.07 0.99 0.87
ux,R [m/s] 0.94 1.21 1.07 1.00 0.84
uy,B [m/s] 2.01 2.40 2.32 2.17 1.99
uy,T [m/s] 0.13 0.19 0.05 0.28 0.16
UTC [m/s] 0.71 1.09 1.17 1.27 1.10
UCurtain [m/s] 0.89 1.06 1.21 1.12 1.04
Oh 0.25 0.97 3.27 2.32 4.03
t [µm] 191 98 87 62 80
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3.4. Liquid curtain stability
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X: hole distance from the edge guide; Y: hole distance from exit of the die.
The critical flow rate below which the curtain breaks was recorded for the different shear thinning solutions. The critical flow rate represented by the Weber number
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(ratio of inertial and capillary forces) versus the Ohnesorge number, defined in terms of the characteristic shear viscosity
, is shown in Figure 9. Several experiments were
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performed for each solution. Because the Ohnesorge number is defined in terms of the curtain thickness at the breakup condition, its value is not constant for each solution. The
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critical flow rate decreases as Ohnesorge number increases. It is important to compare the behavior of the 0.3 wt% in water solution with the 0.1 wt% in glycerol-water solution. These solutions have very different high and low shear viscosity but similar characteristic shear viscosity
. Both solutions have similar critical flow rate and curtain thickness at
the breakup. On the other hand, solutions with 0.05 wt% and 0.15 wt% in water have high shear viscosity in the same range but very different characteristic shear viscosity; their critical flow rate and curtain thickness at breakup conditions are very different. The
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breakup dynamics is governed by the shear viscosity evaluated at the characteristic deformation rate of the curtain flow. In Figure 9, we have included the results of critical Weber number of our previous analyses [4] for Newtonian liquids. The behavior of the shear thinning solutions follow the general trend of the Newtonian liquids with the Oh
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defined in terms of the characteristic shear viscosity.
Figure 9. Critical Weber number, We, at which the liquid curtain breaks as a function of the Ohnesorge number, Oh.
4. Final Remarks 18
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The effect of rheological properties on shear thinning curtain stability was studied experimentally by high-speed visualization of the sequence of events that ultimately lead to the curtain breakup and by determining the critical flow rate below which the curtain becomes unstable. The velocity evolution along the curtain was measured at different distances from the edges. It was found the formed viscous boundary layer along the edge
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guide causes the curtain velocity deviates from the free fall velocity. Previous analyses [6,8] have proposed a simple criterion for curtain stability based on the ratio of the curtain velocity, which moves any hole in the curtain downward, to the rim retraction speed, which opens the hole. If the ratio is above one, the curtain is stable. The high-speed visualization images presented here confirm this hypothesis. At
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larger flow rate; a hole is convected down the curtain before it becomes large enough to cause the curtain breakup. Below a critical flow rate, at which the retraction speed is larger than the curtain speed, the hole grows, leading to the curtain breakup. The results show that the dynamics of curtain breakup is governed by the shear viscosity of the liquid evaluated at a characteristic deformation rate of the curtain breakup
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flow. Higher characteristic viscosity leads to more stable curtain; i.e., lower critical flow rate. The dependence of the critical Weber number below which the curtain breaks as a
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function of the Ohnesorge number, Oh, follows the general behavior previously reported
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for Newtonian liquids if the Oh is defined based on this characteristic viscosity.
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Acknowledgements
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This research is supported by the Dow Chemical Company.
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[ 3] McKinley, G.H. and Tripathi, A., [ ], “How to Extract the Newtonian Viscosity from Capillary Breakup Measurements in a Filament Rheometer”, J. Rheol., 44(3), 653671. [24] Papageorgiou, D. T., [1995] “On the breakup of viscous liquid threads”, Physics of Fluids 7, 1529-1544. [25] Khagram. M., Gupta, R. K. and Sridhar, T., [1985] “Extensional Flow of Xanthan Gum Solutions”, Journal of Rheology 29, 191-207.
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[28] Marston, J. O., Thoroddsen, S. T., Thompson, J., Blyth, M. G., Henry, D., and Uddin, J., [ 14] “Experimental investigation of hysteresis in the break-up of liquid curtains” Chem. Eng. Sci., 117, 248-263.
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