Liquid ligament formation dynamics on a spinning wheel

Chemical Engineering Science 119 (2014) 187–198

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Liquid ligament formation dynamics on a spinning wheel Benjamin Bizjan a,b,n, Brane Širok a, Marko Hočevar a, Alen Orbanić b a b

Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, 1000 Ljubljana, Slovenia Abelium d.o.o., Kajuhova 90, 1000 Ljubljana, Slovenia

H I G H L I G H T S








Liquid disintegration on a spinning wheel atomizer was investigated experimentally. Velocity slip between liquid film and wheel surface only significant at slow rotation. As a ligament grows, the trajectory of its free end resembles an involute. Maximum ligament length is proportional to the liquid flow rate and viscosity. Significant effect of end pinch-off on ligament strain rate and breakup mechanism.

art ic l e i nf o

a b s t r a c t

Article history: Received 19 May 2014 Received in revised form 4 August 2014 Accepted 12 August 2014 Available online 20 August 2014

A ligament-type disintegration of liquid on a spinning wheel was investigated experimentally using photographs taken by a high-speed camera. Three different Newtonian liquids were used at various flow rates and the wheel rotational speed was varied in a wide range. Velocity slip between the liquid film and the wheel surface was found to depend primarily on wheel rotational speed and angular position, dropping to approximately 1–1.5% for sufficiently fast rotation. As a liquid ligament grows from the film, the relative pathline of its free (head) end resembles an involute. Ligament strain rate on the film was found to increase steadily until the head droplet pinch-off when a short but significant strain rate reduction was observed. At this point, ligament is rapidly decelerated in the lateral direction which may cause significant longitudinal oscillations, possibly destabilizing its growth. Strain rate then increases again until the ligament detachment from the film which is soon followed by capillary breakup into droplets. The mean ligament length at detachment was determined to increase with a rising liquid flow rate and Ohnesorge number. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Disintegration Fluid dynamics Ligament Liquid film Spinning wheel Strain rate

1. Introduction A spinning wheel apparatus where a stream of liquid flows onto the mantle surface of the wheel has important applications in industry, especially in the production of mineral wool and other fibers (Širok et al., 2008) and in atomization of highly viscous liquids. In case of fiber production, such device is known as a spinning machine or a spinner while in case of atomization, it can be referred to as a spinning wheel atomizer. In this paper, we have studied isothermal atomization of Newtonian fluids, therefore the latter term will be used from this point onwards. A spinning wheel atomizer is a rotary-type atomizer employing centrifugal force as the main disintegration mechanism. While the

n Corresponding author at: Abelium d.o.o., Kajuhova 90, 1000 Ljubljana, Slovenia. Tel.: þ 386 1 542 3614. E-mail addresses: [email protected] (B. Bizjan), [email protected] (B. Širok), [email protected] (M. Hočevar), [email protected] (A. Orbanić).

http://dx.doi.org/10.1016/j.ces.2014.08.031 0009-2509/& 2014 Elsevier Ltd. All rights reserved.

fundamental operating principle is similar to the rotary atomizers with central liquid feed such as spinning discs and cups atomizers, the exact liquid film formation and disintegration mechanism is notably different (Bizjan et al., 2014). Also, there are some key advantages of a spinning wheel over other rotary atomizers. Most notably, a much larger flow rate of liquid can atomize in the ligament formation mode which is preferred as it produces droplets with a relatively narrow size distribution (Liu et al., 2012b). This is due to the fact that liquid film can be made very wide, allowing for the ligaments to form from several parallel radial planes. By using additional wheels, the atomization flow rate can be further increased by several times. However, liquid atomization on a spinning wheel is difficult to model mathematically as it occurs as an unsteady, aperiodic and asymmetric process. For this reason, mathematical and numerical modeling of spinning wheel atomizers and spinners has so far been scarce. Westerlund and Hoikka (1989) developed a relatively simple numerical model for dynamics of continuously growing mineral wool fibers on an industrial spinning machine, but the

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hydrodynamics of ligament formation were not considered. The extent of experimental investigation of the process on this particular type of apparatus is also modest. Širok et al. (2008) formed a regression model for mineral wool fiber diameter on an industrial spinning machine where melt ligaments solidify into fibers. Bajcar et al. (2013) studied silicate melt film dynamics on a spinner wheel in the initial phase of ligament formation. Recently, Bizjan et al. (2014) formed regression models for ligament and droplet diameters of Newtonian liquids on a spinning wheel atomizer while also analyzing ligament spatial distribution on the wheel. Also, a qualitative analysis of ligament formation and disintegration was performed based on the high-speed imaging of the process. However, to properly characterize the liquid disintegration process on spinning wheels, it is necessary to perform an indepth study of liquid film and ligament formation and breakup dynamics. This includes phenomena such as the liquid film velocity slip against the wheel, ligament detachment from the film, head droplet pinch-off and final ligament breakup into a chain of droplets. For this purpose, we conducted an experimental study of the process by means of the high-speed camera visualization followed by the image post-processing. In addition to the process dynamics, a new characteristic parameter, namely the mean ligament length was introduced for additional quantification of the ligament formation process. This paper is organized as follows. Section 2 provides known theoretical background to the rotary atomization mechanism and the underlying fluid dynamics. In Section 3, the experimental setup used for our study is introduced. Section 4 presents the results of image analysis, including the description of the ligament formation and breakup process as well as the quantitative properties such as the liquid film velocity slip, mean ligament length and time-dependent kinematic properties (ligament trajectories and strain rates).

a certain cutoff wavelength are fully damped by viscous and surface tension forces while at higher wavelengths where the centrifugal force is greater than the damping forces, the waves grow exponentially and at different growth rates. The fastest growing wave (wavelength λm) becomes predominant and transforms into the circumferential spacing (s) between the emerging liquid ligaments (Fig. 1) (Eisenklam, 1964). In Fig. 1, f0 denotes the wheel rotational speed in [Hz], R the wheel radius, h the liquid film thickness and B the film width. For a film of an inviscid or low viscosity liquid, surface tension (σ) has a predominant damping effect and λm can be estimated by Eq. (1) for h Z λm/π (Eisenklam, 1964). sffiffiffiffiffiffiffiffiffiffiffi 3σ λm ¼ ð1Þ ρRf 20

2. Theoretical background and methodology

N ¼ 0:360We0:433 q0:810

Liquid disintegration on a spinning wheel atomizer can be divided into two main phases. In the first phase, a liquid stream falls onto the mantle surface of a spinning wheel where it is drawn in motion by the viscous and adhesive forces, forming a thin film slightly wider than the liquid stream. As the film rotates with the wheel, it gradually disintegrates to ligaments that start to form in a radial direction. At this point, the second phase begins in which the ligaments grow longer and thinner until they pinch off from the film and disintegrate in a chain of droplets (Bizjan et al., 2014). 2.1. Liquid film disintegration Initial disintegration of liquid film to ligaments is generally agreed to occur due to the hydrodynamic instabilities that develop on the liquid film and are driven by the shear and centrifugal forces. Initial liquid film disturbances required for development of unstable waves are most likely caused by the Kelvin–Helmholtz instability (Westerlund and Hoikka, 1989) induced by velocity slip between the film and the surrounding gas as well as by fluctuations in the liquid flow rate. The main wave formation mechanism however is the Rayleigh–Taylor instability (Westerlund and Hoikka, 1989; Eisenklam, 1964) which, in case of a spinning wheel, occurs when a layer of denser fluid (i.e., the rotating liquid film) is pushed towards the lighter one (i.e., the surrounding air) by the action of centrifugal force. According to Eisenklam (1964), unstable waves forming on the liquid film surface due to the Rayleigh–Taylor instability grow at different rates depending on their wavelength. The waves below

When the liquid viscosity is so large that its damping effect on wave formation can no longer be neglected, formulations for λm ¼ s become significantly more complex. Liu et al. (2012a, 2012b) investigated ligament spacing for atomization of viscous liquids on spinning cups and supported experimental results with detailed theoretical formulations of the underlying hydrodynamic instabilities. Kamiya (1972) investigated liquid disintegration on spinning disks in ligament formation mode and also developed a mathematical model for unstable wave growth. However, as recently pointed out by Bizjan et al. (2014), the formulations for centrally fed rotary atomizers where liquid spills over the apparatus lip cannot be directly applied to spinning wheels. This is primarily due to the fact that ligament circumferential distribution on the film is highly non-uniform and that the liquid film disintegrates in a direction normal rather than parallel to its surface, allowing for ligaments to form in multiple parallel planes when the film is wide enough. For this reason, Bizjan et al. (2014) used the mean number of ligaments (N) attached to the liquid film instead of ligament spacing (s), obtaining a following power law relation: ð2Þ

In Eq. (2), We is the Weber number (Eq. (3)) and q is the dimensionless flow rate (Eqs. (4) and (5)). As Eq. (2) suggests, the number of ligaments is proportional to the Weber number and liquid flow rate. The definitions of dimensionless numbers that have been used for characterization of liquid disintegration process in this paper are as follows. Weber number for a liquid film on a spinning wheel is defined by We ¼

ρv20 R ρð2π f 0 Þ2 R3 ¼ σ σ

ð3Þ

The dimensionless flow rate q for a spinning wheel was first proposed by Širok et al. (2008) for an industrial mineral wool

Fig. 1. Simplified presentation of ligament formation and disintegration on a spinning wheel atomizer.

B. Bizjan et al. / Chemical Engineering Science 119 (2014) 187–198

spinner as follows: rffiffiffiffiffiffi Q ρ q¼ B Rσ

ð4Þ

Bizjan et al. (2014) assumed the effective liquid film width to be equal to the liquid feed nozzle diameter, B ¼dN (Eq. (5)). This definition of q is also used in the present paper. rffiffiffiffiffiffi Q ρ q¼ ð5Þ dN Rσ Another important parameter is the Ohnesorge number which denotes the ratio of viscous to inertial and surface tension forces. For a liquid film on a spinning wheel it can be defined by pffiffiffiffiffiffiffiffi We μ ¼ pffiffiffiffiffiffiffiffiffi ð6Þ Oh ¼ Re ρRσ In Eq. (6), Re is the Reynolds number, defined as the ratio between the inertial and viscous forces Re ¼

v0Rρ

μ

¼

2π f 0 R2 ρ

μ

ð7Þ

Note that in Eqs. (3), (6) and (7), We, Oh and Re are defined for liquid film rather than ligaments or droplets and are given for the nominal rotational speed of the atomizer (slip between the film and apparatus surface is not considered). Such formulation allows these characteristic numbers to be used as independent variables, i.e. a dimensionless form of spinning wheel input parameters.

When the ligaments start to grow from the unstable liquid film waves, the second liquid disintegration phase begins. According to Bizjan et al. (2014), ligament formation regime occurs in a wide range of Weber numbers and typically starts at log(We)E3. At WeE3  104/q, liquid sheet starts to form intermittently just after the liquid impingement point while the ligaments continue to form elsewhere. This regime is known as partial sheet formation and should be avoided when a narrow ligament and droplet size distribution is required. As a ligament extends from the liquid film, it elongates and thins in diameter at the same time. Bizjan et al. (2014) determined the final ligament diameter (dL) before the onset of disintegration to be a function of Weber number while the Ohnesorge and dimensionless flow rate have no significant effect dL =R ¼ 0:134We  0:345

(Liu, 2000) and rises with the ligament Ohnesorge number (OhL): pffiffiffi μ λOPT ¼ 2π dL ð1 þ3OhL Þ0:5 ; OhL ¼ pffiffiffiffiffiffiffiffiffiffiffi ð9Þ ρσ dL For a ligament of known diameter dL, mean diameter of droplets d to which it disintegrates can be predicted by (Liu, 2000) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6 2 d ¼ 1:5λOPT dL ¼ dL 4:5π 2 ð1 þ 3OhL Þ ð10Þ By combining Eqs. (8) and (10), Bizjan et al. (2014) formed the following model for mean droplet diameter: 0:093

d=R ¼ 0:369Oh

We  1=3

ð8Þ

On a spinning wheel, ligament disintegration is preceded by detachment of the full ligament from its base on the liquid film (Bizjan et al., 2014), which we will refer to as the ligament detachment. If the liquid is a melt that solidifies prior to ligament breakup, fibers are formed and dL is as an output process parameter as it is practically equal to the final fiber diameter. In case of non-solidifying liquids or melts that solidify after ligament breakup, dL is an intermediate parameter from which the mean drop diameter can be calculated (unless measured directly) using the jet breakup theory (Bizjan et al., 2014). According to Lefebvre (1989), the Rayleigh-type ligament breakup occurs when Reynolds and Weber number are low enough for the aerodynamic effects to be neglected, which is typically true for spinning disc, cup and wheel atomizers. In this type of breakup, radially symmetric dilatational waves are formed along the length of a ligament by the interaction of primary disturbances in the liquid and surface tension forces acting to minimize the total surface free energy of the liquid–gas interface (Olesen, 1997). The wavelength of the fastest growing dilatational waves that form along the ligament surface (λOPT, see Fig. 1) is given by Eq. (9)

ð11Þ

According to Eq. (11), droplet diameter reduction can be achieved by increasing the Weber number or by reduction of the Ohnesorge number. As noted by Bizjan et al. (2014), a spinning wheel produces smaller droplets than spinning discs and cups with the same radius and rotational speed. Also, droplet diameters decrease more quickly with the Weber number. An important phenomenon in the process of ligament disintegration that precedes the final breakup into the chain of droplets is the head droplet pinch-off, also known as the end pinching mechanism. Head droplets form from the initial bulbous structure (i.e., tip bulb) on the ligament free end and are significantly larger than the main droplets. Bizjan et al. (2014) determined head droplet diameter (dHD) to only depend on the Weber number, with higher We resulting in smaller head droplets dHD =R ¼ 1:95We  0:45

2.2. Ligament growth and disintegration

189

ð12Þ

The ratio between the head and main droplet diameter (dHD/d) drops with rising Weber numbers (Bizjan et al., 2014), meaning that at faster rotational speeds, droplet size distribution is more uniform, provided that there is no partial sheet formation (Weo3  104/q). Quantitative formulations for spinning wheel operation that have been discussed so far characterize the integral properties of observed process, i.e. temporal and in some cases also spatial averages of instantaneous property values. However, to study the process dynamics on smaller length and time scales, relevant dynamic properties must be selected and measured locally with a sufficient spatial and temporal resolution. So far, several experimental and numerical studies of ligament dynamics have been performed. Stone et al. (1986) and Stone and Leal (1989) investigated breakup dynamics of ligaments produced by droplet elongation under the action of centrifugal force which is a formation mechanism fundamentally similar to the one occurring on a spinning wheel. The authors found that a ligament either contracts to a single drop, or a breakup into smaller droplets occurs by the means of the end pinching mechanism and also the capillary wave instability in case of highly elongated ligaments (i.e. the Rayleigh-type breakup that was already presented). Also, the ligament initial shape was determined to have a strong effect on relaxation dynamics, with bulbous-shaped end being the most prone to the end pinching. A detailed study of the end pinching mechanism was performed by Tong and Wang (2007) who developed a numerical model for relaxation dynamics of free elongated ligaments. Ligament head pinch-off was determined to succeed when liquid flow is predominantly extensional (i.e., flowing from the ligament body into its bulbous end), which results in thinning of the ligament neck up to the point when the head droplet is pinched off by the circumferential surface tension. On the contrary, when a recoiling flow prevails (i.e., flowing in the opposite direction and resulting in ligament head contraction), the ligament neck reopens after a certain time, preventing the pinch-off and thus allowing the ligament to fully contract. According to the authors (Tong and

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Wang, 2007), such situation occurs when the lateral surface tension which generally has a stabilizing effect causes the peak pressure location to shift upstream from the neck. It was also determined that Ohnesorge number has a stabilizing effect as no end pinching occurs above a certain Oh value (Tong and Wang, 2007). Ha and Leal (2001) studied the ligament elongation process experimentally and found that ligament length at the moment of relaxation (detachment from liquid film in our case) also significantly affects stability of the relaxation process. End pinching was determined to occur above a critical elongation ratio (i.e., ratio between critical and initial ligament length) which depends on the strain rate. Shinjo and Umemura (2010) who numerically investigated jet breakup mechanism explained the end pinch-off phenomenon as a consequence of two competing mechanisms known as the shortwave and the long-wave mode. The short wave mode is basically the mechanism as described by Tong and Wang for relaxation of free elongated ligaments. Since a jet is a continuous liquid column, the end pinch-off in short wave mode occurs sequentially with a wavelength of approximately 1.81 ligament diameters (Shinjo and Umemura, 2010). The long wave mechanism was found to be driven by capillary waves (wavelength of approximately 4.5 ligament diameters) that are caused by the ligament tip contraction after the droplet pinch-off. When the waves reach the nozzle (analogous to the ligament foot), they are reflected and destabilized due to the Doppler shift in wavelength (Shinjo and Umemura, 2010). The most unstable wave is reproduced and grows until the end pinch-off occurs. According to Eggers and Villermaux (2008) who experimentally investigated dynamics of ligaments forming from a liquid bulk, ligament longitudinal stretching with a strain rate γ has a damping effect on development of capillary instabilities provided that γ is large enough. When the stretching force is removed (e.g. after the ligament is detached from the liquid film), ligament breakup occurs rapidly. The evolution of ligament surface perturbations follows Eq. (13) (Eggers and Villermaux, 2008):   ∂2 ε ∂ε 3 2 3 2 4 þ þ 2 γ γ ε  ððkr Þ  ðkr Þ expð  3 γ tÞÞexp  γ t ε¼0 L0 L0 2 ∂t 4 ∂t 2 ð13Þ In Eq. (13), γ is the ligament strain rate (Eq. (14)), ε is the amplitude of ligament radius (rL) oscillations (Eq. (15)), rL0 is the radius of unperturbed ligament and k ¼2π/λ is the wave number. All of these variables are given at time t that has passed since the start of ligament stretching.     d Lr ðtÞ Lr ðt ¼ 0Þ d Lr ðtÞ ¼ ð14Þ γ¼ dt Lr ðt ¼ 0Þ dt Lr ðt ¼ 0Þ

ε¼

r L;MAX  r L;MIN 2

ð15Þ

In Eq. (14), Lr(t) is the ligament length at time t. The variables from Eq. (13) are shown in Fig. 2. Note that the model in Eq. (13) neglects viscosity which provides an additional stabilizing effect.

Fig. 2. Variables in the ligament capillary instability model.

Amplitude ratio c ¼ ε(t)/ε(t¼ 0) is introduced to determine if the oscillations are amplified (c4 1) or damped (co 1). From Eq. (13), a proportionality given by Eq. (16) can be derived (Eggers and Villermaux, 2008) for early time (γt⪡1):  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð16Þ c p exp  γ n þ γ n2 =4 þ ðkr L0 Þ2  ðkr L0 Þ4 t n In Eq. (16), tn ¼t/τ is the dimensionless time and dimensionless strain rate, with the time constant qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ ¼ 2ρðrL0 ðt ¼ 0ÞÞ3 σ  1

γn ¼ γτ is the ð17Þ

From Eq. (13) it follows that the wave number (k) decays exponentially (Eggers and Villermaux, 2008), i.e. kðtÞ ¼ kðt ¼ 0Þ expð  γ tÞ. This means that after a sufficiently long time (γt⪢1), Eq. (16) transforms to c p expð  γ t=2Þ and even a very low positive strain rate is sufficient to prevent the development of capillary instabilities. One may be interested in the value of a critical strain rate γCRIT above which the capillary instabilities are suppressed from the very beginning of the ligament growth. For this condition to be met, the amplitude ratio c must have a zero rate of change at t¼0, i.e. dc/dt(t¼0) ¼0. This is true when  γn þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 γ n2 =4 þ ðkrL0 Þ  ðkrL0 Þ ¼ 0, or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ nCRIT ¼ 43 ððkrL0 Þ2  ðkrL0 Þ4 Þ ð18Þ pffiffiffi The instabilities have the fastest growth rate at kr L0 ¼ 2=2 (Eggerspffiffiffiand Villermaux, 2008) for which Eq. (18) yields γ nCRIT ¼ 3=3  0:58. The corresponding dimensional strain rate is pffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ 3 γ CRIT ¼ ¼ ð19Þ 3τ 6ρðr L0 ðt ¼ 0ÞÞ3 Unlike the amplitude ratio, the time until ligament detachment from the bulk was found to be independent of the strain rate as the ligament foot is only weakly stretched (Eggers and Villermaux, 2008).

3. Experimental set-up Experiment was performed on a spinning wheel atomizer by means of high-speed camera visualization (Fig. 3). The wheel with a radius of R¼45 mm and 50 mm wide was spun by a DC-powered motor. Its rotational speed f0 was varied in the 5–25 Hz range with 5 Hz increments using a speed controller. The liquid, namely the glycerol–water mixture with glycerol mass fraction denoted by wgl, was gravity-fed from the supply tank onto the wheel through a circular nozzle with dN ¼3 mm diameter. The nozzle was placed vertically so that the liquid stream impingement point was 25 mm from the wheel's vertical centerline. A protective screen was attached to the nozzle casing to shield the liquid stream from being hit by the ligaments and droplets forming on the wheel. Liquid volumetric flow rate was regulated by a valve on the supply tube. Two different flow rates were used: 1.63 mL/s and 3.27 mL/s. Droplets from the atomizer wheel were contained in a droplet collector. After filtration, collected liquid was returned back to the supply tank using a circulating pump. Three different glycerol–water mixtures were used for our experiment (Table 1). Note that by using different mixtures, liquid viscosity was varied in a wide range (1:10) while density and surface tension coefficients were nearly the same (about 5% variation between liquids). Consequently, in our case Oh (Eq. (6)) can also be used as a measure of liquid viscosity. By using two different flow rates, three kinds of liquids and five different rotational speeds, we produced a total of 30 operating

B. Bizjan et al. / Chemical Engineering Science 119 (2014) 187–198

191

Fig. 3. Schematic diagram of experimental set-up for atomizing process visualization.

Table 1 Properties of glycerol–water mixtures used in the experiment (at 20 1C) and the corresponding liquid film Ohnesorge numbers for R¼ 0.045 m. wgl (dimensionless)

ρ (kg/m³)

μ (Pa s)

σ (N/m)

Oh (dimensionless)

0.85 0.75 0.60

1224 1197 1157

0.107 0.0345 0.0106

0.0647 0.0660 0.0677

5.67  10  2 1.83  10  2 5.65  10  3

Table 2 Image acquisition windows used in the experiment. Window no.

Resolution (pixels)

Acq. rate, fs (Hz)

No. of frames

0 1 2

1536  1536 656  448 448  656

652 4418 4359

907 7287 7287

A positive slip (Z40) occurs when v0 4 vF. points. Physical properties of different liquids are reflected by the Ohnesorge number (Oh). Ligament formation from the liquid film on the wheel and the subsequent breakup into droplets was recorded by a high-speed camera (Fastec Hispec 4 mono 2G) with an 85 mm lens set at aperture 2 and the images were stored on a computer for further processing. The distance from the lens was 1.30 m. The ligaments were illuminated from behind using diffuse illumination generated by a ring of light emitting diodes (LED). The diode ring was positioned behind the wheel and covered by opaque glass functioning as a coaxially mounted background illumination light diffuser. To avoid smearing in the images, camera shutter speed was set to an acceptably low value (5 μs) where illumination was still sufficient. For each of the 30 operating points, a series of images was recorded in three different windows (Table 2 and Fig. 4). The spatial resolution (pixel size) was 0.108 mm/pixel for all windows and was found to be sufficient for detection and tracking of ligaments up to the rotational speed of approximately 40 Hz. An example of recorded images is shown in Fig. 4 along with the definition of wheel angular position ϕ that we will refer to in subsequent image analysis.

4. Results and discussions 4.1. Liquid film velocity slip As pointed out by Bizjan et al. (2014), velocity slip between the liquid film and the wheel surface has an important role in ligament formation due to the reduction of actual film velocity to the nominal one and a consequent reduction in the centrifugal force which drives the formation process. To achieve an optimal atomization efficiency (i.e., smallest ligament and droplet diameters) at a given wheel rotational speed, slip should be as low as possible. Velocity slip Z between the liquid film (velocity vF) and the wheel surface (velocity v0 ¼2πf0R) can be defined by Eq. (20).

Z¼

v0  vF v0

ð20Þ

Liquid film velocity (vf) measurements were conducted by measuring the time in which a typical liquid structure on the film (for example, a ligament foot) passed a wheel circumferential segment of known length. From this data, velocity was calculated. Liquid film velocity was measured for all operating points and at two different angular positions on the wheel (ϕ ¼901 and 1301, angle definition in Fig. 4) and was then used to calculate the slip Z using Eq. (20). Results are shown in Fig. 5. From Fig. 5 it can be seen that the wheel rotational speed f0 and angular position ϕ strongly affect the slip. Variation of film Ohnesorge number and liquid flow rate at a constant f0 also influences the slip, causing notable deviations of Z values between operating points with the same f0. However, no particular relation between Oh, Q and Z was observed. For this reason, in Fig. 5 the individual Z values measured at 6 different combinations of Oh and Q for every f0 were replaced by their mean value and an error bar showing their standard deviation. Contrary to Oh and Q, the effect of wheel rotational speed and angular position on velocity slip is clearly expressed. At ϕ ¼901 (bottom point of the wheel), the slip has a negative value (ZE  2%) at slowest rotation (f0 ¼5 Hz), meaning that the film velocity is larger than wheel circumferential velocity. This is due to the combined effect of a thick liquid film and a relatively low centrifugal to gravitational acceleration ratio (4.5 as opposed to 113 at f0 ¼25 Hz) which allows the film to be significantly accelerated by gravity on the downwards-moving section of the wheel (for  901 o ϕ o901). As f0 is increased, the slip at ϕ ¼901 quickly rises to positive values and reaches a maximum of just over 2% at f0 E10 Hz, which corresponds to WeE104. From this point onwards, the slip slowly drops to approximately 1% at f0 ¼ 20 Hz (We E2.5  104) and remains at this level when f0 is further increased. Reduction in velocity slip can be on one hand explained by liquid film thinning which improves adhesion to the wheel surface. On the other hand the ratio of centrifugal to gravitational acceleration ratio becomes so high that the gravity effect is negligibly low. The remaining slip can be attributed to the force of aerodynamic drag acting upon the liquid film and ligaments.

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Fig. 4. An example of recorded liquid disintegration images and the definition of wheel angular position.

slip between ϕ ¼901 and 1301 is most likely a consequence of the aerodynamic force which slightly decelerates the film over time. From presented results we can conclude that, in terms of its effect on centrifugal force reduction, the slip of liquid film against the wheel surface has a relatively low impact on ligament formation for We 4104. 4.2. Ligament formation regimes

Fig. 5. Liquid film velocity slip at ϕ¼ 901 and 1301. At each rotational speed f0, slip was measured in six different operating points. Dots indicate mean value of these measurements while error bars show their standard deviation.

For the second position on the wheel (ϕ ¼ 1301), a very different trend is observed, with Z 40 for all operating points. At slowest rotation (f0 ¼ 5 Hz), film velocity is again strongly affected by the film thickness and gravity, but this time the slip is quite large (Z E10%). This can be explained by the fact that the film is moving upwards (901o ϕ o2701) and is thus decelerated by the gravity by about 12% of the wheel circumferential velocity in only 401 of wheel rotation. As noted by Bizjan et al. (2014), the quickly increasing slip leads to an abrupt suppression of ligament formation due to the reduction of centrifugal force, especially at positions past ϕ ¼ 1801. As the wheel rotational speed is increased, the slip at ϕ ¼1301 drops rapidly to about 3% at f0 ¼ 10 Hz (WeE 5  103) and then more slowly when f0 is further increased. At the fastest rotation rate (f0 ¼25 Hz, which corresponds to WeE4  104), ZE1.5% was measured, which is slightly more than for ϕ ¼ 1301. Since the effect of gravity is negligible, the increase of

Two typical ligament formation (i.e., growth and disintegration) regimes were identified in our experiments, the first one resulting in ligament contraction to a single droplet (Fig. 6 left) and the second one leading to capillary breakup into a chain of droplets (Fig. 6 right). The phase of ligament growth is similar for both regimes, starting from an initial bulge on the liquid film. As the ligament elongates, it becomes thinner in diameter and its head assumes a bulbous shape (first two images from the top in Fig. 6). Ligament neck becomes very thin and soon the head droplet pinches off from the ligament (third image from the top in Fig. 6). After the end pinch-off, the ligament is temporarily decelerated in the lateral direction and a new ligament head starts to form (third image from the top in Fig. 6). Interestingly, it does not pinch off. As the strain rate of the ligament is resumed, its foot becomes thinner and is eventually detached from the liquid film by a combined action of surface tension and centrifugal force. Now, the ligament enters the relaxation phase. From this point onwards, there is a significant difference between the two formation regimes. In case of the first regime, the ligament slowly contracts into a single droplet (last three images from top to bottom in Fig. 6 left). This mechanism was only observed in experiments with the slowest rotation rate (f0 ¼5 Hz, WeE1600) and for relatively short ligaments. However, the single droplet formation mode is not particularly interesting for practical applications due to the large droplet size which also varies greatly with the ligament length. In the second ligament formation regime, unlike the singledroplet one, the ligament sees practically no contraction after separation from the film (fifth image from the top in Fig. 6 right). Soon after the detachment, the capillary instability causes dilatational waves to grow very rapidly, causing the ligament to disintegrate into a chain of droplets along its whole length (bottom image in Fig. 6 right). Large main droplets nearly uniform in diameter form along with the much smaller satellite droplets. Most of the satellite droplets merge with the main droplets as their velocity in the ligament lateral direction is slightly different. The mean spacing between main droplets for the breakup in Fig. 6 right is 1.40 mm, which is close to the value predicted by Eqs. (8) and (9), i.e. λOPT ¼1.31 mm (the droplet spacing is non-uniform, though).

B. Bizjan et al. / Chemical Engineering Science 119 (2014) 187–198

This ligament formation regime is predominant over a wide range of operating parameters and is most desirable for both liquid atomization (smaller droplets with a narrower size distribution) as well as fiber production (thinner and longer fibers, lower mass fraction of unfiberized material). In Table 3, estimated transitional parameters between both of the described mechanisms are shown. Clearly, at higher liquid film Ohnesorge numbers (Oh), the limit length (L) and length ratio (L/dL) below which a ligament contracts into a single droplet and above which capillary breakup occurs is also higher.

4.3. Mean ligament length The mean ligament length (L) before detachment from the liquid film is an important characteristic measure for the quality and stability of the ligament formation process. Optimally, ligaments should be as long as possible since longer ligaments can typically produce a finer and more uniform droplet distribution; this is due to a larger number of smaller main droplets per head droplet. Also, up to a certain degree, the findings of this analysis may be applied to the production of fibers on a spinner. Longer ligaments can be expected to also yield longer fibers, which is usually a desired property. L was determined for each of the 30 operating points using the following calculation procedure. From the image sequence of a particular operating point, n ¼45 ligaments were chosen at random times and positions on the wheel. Then, the maximum length of a particular ligament (Li) prior to detachment from the film was obtained by measuring its relative radial coordinate r (Fig. 1). Actually, the ligament is slightly longer than r due to its lateral curvature and its shape can be approximated with an involute of a circle (Liu et al., 2012a, 2012b) given by the following parametric equation: ðxe ; ye Þ ¼ Rð sin ξ  ξ cos ξ; cos ξ þ ξ sin ξ 1Þ

193

a quadratic polynomial as c E[3.170 m  2]r2 þ[1.215 m  1]r þ1 (valid for R¼ 0.045 m). Finally, the mean ligament length was calculated as an arithmetic average of Li values: L¼

1 n ∑ L ni¼1 i

ð23Þ

Mean ligament length is shown in Fig. 7 as a function of liquid film Weber and Ohnesorge number and of the liquid flow rate. Dimensionless flow rate q was approximately 0.34 for Q¼1.63 mL/s and 0.69 for Q¼3.27 mL/s (note that q rises slightly when Oh is increased). Fig. 7 shows that ligament length is significantly affected by the liquid properties (i.e., Ohnesorge number), Weber number and the liquid flow rate, ranging between 15 mm and 60 mm (L/R¼ 0.33… 1.33). By assuming a power law dependency between L/R, We, Oh

Table 3 Transition between single droplet and chain of droplets breakup; transitional lengths (L), length to diameter ratios (L/dL) and ligament Ohnesorge numbers (OhL) are shown for f0 ¼ 5 Hz. We

wgl 0.85 0.75 0.60

Oh 3

1.70  10 1.63  103 1.54  103

2

5.67  10 1.83  10  2 5.65  10  3

L (mm)

L/dL

OhL

19 12.5 10

29 18 15.5

4.7  10  1 1.5  10  1 4.7  10  2

ð21Þ

In Eq. (21), ξ Z0 is an angle parameter in radians and the involute origin is on the spinning wheel surface at the 6 o'clock position (ϕ ¼901). Ligament length Li was then calculated by Li ¼ cr

ð22Þ

In Eq. (22), the coefficient c 41 defines the ratio between the radial coordinate r of the ligament head (Fig. 1) and the ligament length. Assuming an involute ligament shape (Eq. (22)), relation between c and r was obtained numerically and approximated with

Fig. 7. Mean ligament length before detachment from the liquid film.

Fig. 6. Ligament growth and disintegration images: left: Oh ¼0.0567, Q¼ 1.63 mL/s, f0 ¼5 Hz, ϕ¼ 190–2201, images are 7.7 ms apart; right: Oh¼ 0.0183, Q¼ 1.63 mL/s, f0 ¼15 Hz, ϕ¼ 70–1351, images are 2.5 ms apart.

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and q, the following least-squares regression model was obtained: L 0:358 0:432 ¼ 5:16We  0:0391 Oh q R

ð24Þ

Measured and regression model-predicted L/R values are well correlated (correlation coefficient R2 ¼0.945). From Eq. (24) and Fig. 7 it is evident that mean ligament length increases if either the liquid flow rate or the Ohnesorge number is increased. An increase in Oh can be attained by increasing liquid viscosity, reducing its density or surface tension coefficient or using a wheel with a smaller diameter. Unlike q and Oh, the effect of Weber number on ligament length is less clearly expressed, but certainly important. From Fig. 7 we can see that L/R changes little with We for the most viscous liquid used (85% glycerol, Oh ¼0.0567) while for other two liquids (75% and 60% glycerol), We affects L/R value more significantly. For all liquid types and flow rates, the mean ligament length starts to decrease at higher Weber numbers. This may be due to the fact that at higher rotational speeds, the ligaments become increasingly thin and detach from the liquid film (which is also thinner) more easily. Apart from the mean ligament length, another important parameter is the standard deviation of L, std(L) introduced to measure the dispersion of the Li distribution. Normalized standard deviation of L will be denoted by I sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n stdðLÞ 1 1 ¼ ð25Þ I¼ ∑ ðLi  LÞ2 L L n1 i ¼ 1 Normalized standard deviations for all operating points are given in Fig. 8. We can see the ligament length standard deviation is between 20% and 60% of the mean ligament length. Also, higher liquid film Ohnesorge numbers result in a reduction of I, suggesting a more uniform ligament length distribution. No particular correlation of I to We and q is observed, though. For the operating point at (f0 ¼5 Hz, Q¼ 3.27 mL/s, Oh ¼5.65  10  3), I is notably lower as with other operating points, which is due to a combination of a large mean ligament length and a narrow length distribution. A visual inspection of ligament formation revealed a much more orderly and predictable ligament formation as for other operating points with the same rotational speed, possibly indicating a transition to a more stable formation regime when Oh becomes sufficiently high.

head droplet pinch-off, reference point for head tracking will be the ligament neck (Fig. 1). In the absolute coordinate system (origin in a fixed point at the wheel bottom), ligament head coordinates will be given as (xabs, yabs). The relative coordinate system will be defined with axes x (tangential to the wheel surface) and y (normal to the wheel surface) and an origin on the liquid film where the ligament foot is attached. For convenience, the local film thickness can be neglected, assuming the coordinate system origin to be at a constant distance of R from the wheel centerline. The angle α between the wheel's vertical centerline and the y-axis will be given as

α ¼ t α vF =R

ð26Þ

In Eq. (26), tα is the time since the origin of the relative coordinate system has passed the wheel bottom position (i.e., ϕ ¼901). Ligament head position in the relative coordinate system will be given as (x,y). Transformation between the absolute and the relative coordinate system positions is defined by ( R sin ðαÞ þ x cos ðαÞ  y sin ðαÞ; window 1 xabs ¼ ð27Þ Rð1  R sin ðαÞÞ  x cos ðαÞ þ y sin ðαÞ; window 2 ( yabs ¼

Rð1  cos ðαÞÞ  x sin ðαÞ  y sin ðαÞ;

Rð1 þ sin ðαÞÞ þx cos ðαÞ  y sin ðαÞ;

window 1 window 2

ð28Þ

Ideally, the relative ligament head pathlines would coincide with the shape of the ligaments. Therefore, a relative ligament head pathline with an involute shape (Eq. (21)) will be taken as a reference. To study the effect of spinning wheel atomizer operating parameters on the ligament pathlines, some typical image sequences were selected, each containing a complete formation of a ligament (consider Table 4 for details). To assure that chosen ligament growth examples were as representative as possible, ligaments were chosen so that their length L was close to the mean ligament length in that operating point.

4.4. Ligament head pathlines A typical kinematic property of a ligament is the path traveled by its head during its formation, known as a pathline or a trajectory. The pathlines can be given either in an absolute or relative coordinate system (Fig. 9). To avoid complications at the Fig. 9. Relative and absolute coordinate system for analysis of ligament motion. Table 4 Selected image sequence cases for quantitative analysis of ligament formation.

Fig. 8. Normalized standard deviation of ligament length.

Case no. wgl

Oh

Q f0 We (mL/s) (Hz)

1 2 3 4 5 6 7 8 9

0.0567 0.0183 0.0183 0.0183 0.0183 0.0183 0.0183 0.0183 0.00565

1.63 1.63 1.63 1.63 1.63 1.63 1.63 3.27 1.63

0.85 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.60

15 5 10 15 15 20 25 15 15

1.53  104 1.63  103 6.52  103 1.47  104 1.47  104 2.61  104 4.08  104 1.47  104 1.38  104

L Window no. ϕ (mm) (deg) 39.2 23.7 23.8 23.3 17.8 21.4 21.4 31.5 18.7

1 2 1 1 2 1 1 1 1

90 130 90 90 130 90 90 90 90

B. Bizjan et al. / Chemical Engineering Science 119 (2014) 187–198

The pathlines traveled by the neck of a growing ligament in the relative coordinate system (Fig. 9) are presented in Fig. 10 for certain cases from Table 4. For reference, the involute curve (Eq. (21)) which represents an ideal ligament pathline (Liu et al., 2012a, 2012b) is also plotted with a solid red line (gray in printed journal). From Fig. 10 it is evident that the wheel rotational speed (f0) has a quite significant effect on the ligament formation kinematics. At the slowest rotation (f0 ¼5 Hz), the ligament head pathline follows the involute well until head droplet pinch-off at x ¼12.5 mm. After this point, the ligament head moves at a nearly constant y coordinate (i.e., approximately tangent to the wheel surface), which can be attributed to a significant reduction of the strain rate as well as deceleration under the force of gravity. As the wheel rotational speed increases, the effect of head droplet pinch-off and gravity on ligament pathlines is reduced. At f0 ¼10 Hz, ligament head initially moves from the wheel following a curve steeper from an involute, but after the head droplet pinchoff at x¼ 4.8 mm, the ligament pathline becomes parallel to the involute. With further increases in rotational speed (f0 410 Hz), ligament head pathlines gradually descend below the involute. This may be due to the effect of liquid film thinning which becomes important at earlier angular positions (ϕ) for higher f0 as the onset of ligament formation also occurs further upstream (Bizjan et al., 2014). Another possible reason is the force of 2 aerodynamic drag that grows proportionally to f 0 and may no longer be small compared to the total sum of forces acting on the ligament. All the cases of ligament formation presented in Fig. 10 were analyzed in acquisition window 1 (ϕ E 901, Fig. 4), with the exception of case nos. 2 and 5 where the ligament growth was observed in window 2 (ϕ E1301, Fig. 4). In case no. 2, (f0 ¼ 5 Hz), window 2 was used due to the absence of fully developed ligaments in window 1. In case no. 5, however window 2 was used for the purpose of ligament kinematics comparison with the case no. 4 which represents the same operating point (f0 ¼15 Hz, Oh¼ 0.0183, Q¼1.83 mL/s), but at a different position on the wheel (i.e., in window 1). As it can be seen from Fig. 10, at ϕ E901 (case no. 4), the ligament head pathlines closely resemble an involute, which suggests the ligament formation is almost unaffected by the thinning of the liquid film. However, the pathline of the ligament forming about 401 later (case no. 5, ϕ E1301) significantly deviates from the involute and other pathlines, especially for x4 5 mm. The fact that it lies below the other pathlines (i.e., y coordinate is lower at the same x coordinate, meaning the radial coordinate r is also lower) indicates that the ligament strain rate (Eq. (14)) is also lower. This most likely occurs due to the liquid film depletion by ligament formation as ϕ increases. The liquid film around the

Fig. 10. Effect of spinning wheel rotational speed (f0) and mean position on the wheel (ϕ) on the ligament head pathlines (Oh¼ 0.0183, Q¼ 1.63 mL/s). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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ligament foot becomes so thin that the liquid supply from the film to the ligament is disrupted, resulting in excessive thinning and an early ligament detachment from the liquid film. Ligament head pathlines can also be shown in the absolute coordinate system (Fig. 11). In Fig. 11, all the pathlines start at the location of the initial ligament formation on the wheel and terminate at the location where the ligaments fully disintegrate. From Fig. 11 we can see that initially, the ligament head pathlines are very close to the wheel surface due to the slow ligament growth. As the ligament head velocity relatively to the liquid film increases, pathline curvature is slowly reduced. After the head droplet pinch-off, the pathlines are slightly inclined towards the spinning wheel due to the ligament contraction. This effect is more pronounced at slower rotational speeds (case no. 2) and shorter ligament lengths (case nos. 5 and 9, respectively). After detachment from the film, ligaments initially move in a direction tangent to their pre-detachment trajectories and soon disintegrate to droplets. Subsequent droplet motion is determined by the forces of inertia, gravity and aerodynamic drag.

4.5. Ligament strain rate As already noted, the strain rate at which the ligament elongates has an important effect on the hydrodynamic stability of its surface. Ligament strain rate was calculated by the following procedure: 1. Calculation of ligament head radial coordinate in the relative coordinate system: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ x2 þy2 ð29Þ 2. Approximation of instantaneous ligament length Lr from r by Eq. (22).

Fig. 11. Ligament head pathlines in the absolute coordinate system for case nos. 2, 4, 5, 7 and 9 (refer to Table 4 for details). Ligament end pinch-off and ligament detachment from the film are marked by red circles and green triangles, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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3. Strain rate calculation by Eq. (30) which is a modified form of Eq. (14). Initial ligament length (Lr(t¼ 0)) was defined as the length when the ligament shape changed from convex to concave. This occurred at Lr E2dHD and rL0 EdHD (Fig. 12).   d Lr ðtÞ γ¼ ð30Þ dt 2dHD Dimensionless strain rate γn can be written as  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d Lr ðtÞ 3 γ n ¼ γτ ¼ 2ρdHD σ  1 dt 2dHD

ð31Þ

In our experiments, derivation of Eqs. (30) and (31) was performed numerically using the central difference scheme with a time step of 1/fs   d Lr ðtÞ d Lr ðt þ1=f s Þ Lr ðt  1=f s Þ ð32Þ ¼ 2dHD ðLr ðtÞÞ  dHD dt 2dHD dt 1=f s In Figs. 13 and 14, γn is shown as a function of dimensionless time (tn) for different cases of ligament growth (Table 4) up to the point of disintegration. For all presented cases, a similar kind of strain rate development can be observed. In the initial phase of ligament growth, γn slowly increases until the head droplet pinchoff when peak values of γn ¼ 0.37–0.94 (γ ¼190–2560 s  1) are attained for f0 ¼5–25 Hz (Fig. 13). After this point, a rapid reduction of γn occurs as a result of ligament contraction. At the slowest rotational speed (f0 ¼5 Hz, Fig. 13), dimensionless strain rate drops to γn E0 while for faster rotation, the magnitude of γn reduction is less severe both in absolute terms as well as relatively to the peak strain rate before the pinch-off. After a short time (tn E1–2), ligament growth is resumed and γn gradually exceeds its previous maximum observed value at the head droplet pinch-off (except at f0 ¼5 Hz). The strain rate peaks at γn ¼0.29–1.31 (γ ¼150–3580 s  1) just before the ligament separates from the liquid film for f0 ¼ 5–25 Hz. After detachment, the stretching forces (e.g., the centrifugal force) are no longer present and γn is quickly though not instantaneously reduced (note that the inertial force within the ligament opposes changes of the strain rate). Theoretically, γn would drop to slightly below zero after a sufficiently long time and the ligament would begin to contract. However, this only occurred for case no. 2 where γn stabilizes at about  0.1 (γ E  50 s  1) until breakup at tn E30. In all the other cases where ligaments formed at higher rotational speeds and Weber numbers, disintegration into a chain of droplets occurred while the strain rate was still positive. The end pinch-off and ligament detachment dimensionless time both decrease when the wheel rotational speed is increased, but the delay between these two occurrences remains roughly the same (tn E7). On the other hand, the end pinch-off and ligament detachment time can be seen to increase with the liquid film Ohnesorge number (Fig. 14). The liquid flow rate seems to have

little effect on the pinch-off time, but the ligament detachment time is significantly increased. This is likely due to the thicker liquid film as the ligaments can elongate more easily and reach longer final lengths. As already noted in the theory section, the critical strain rate that would theoretically suppress the growth of ligament surface perturbations at all times is γn E 0.58. Even for γn ¼0.2, perturbation amplitude only grows by about 4 times of its initial magnitude until tn E 10 and is dampened afterwards (Eggers and Villermaux, 2008). As the final γn value at ligament breakup was still between 0.2 and 1.2 (except case no. 2 where it was negative), the breakup mechanism cannot be explained by the strain rate reduction alone.

Fig. 13. Effect of spinning wheel rotational speed (f0) on the ligament dimensionless strain rate (Oh¼ 0.0183, Q ¼ 1.63 mL/s). Ligament end pinch-off and ligament detachment from the film are marked by red circles and green triangles, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. Effect of liquid flow rate (Q) and film Ohnesorge number (Oh) on the ligament dimensionless strain rate (f0 ¼ 15 Hz). Ligament end pinch-off and ligament detachment from the film are marked by red circles and green triangles, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. Transition from convex to concave ligament shape.

B. Bizjan et al. / Chemical Engineering Science 119 (2014) 187–198

As we can see from our results, ligament disintegration was always preceded first by head droplet pinch-off and then the ligament detachment from the film, but both of these occurrences resulted in a drastic reduction of strain rate. By temporal derivation of Eq. (30) and multiplication with initial ligament diameter (2dHD), we can estimate the peak lateral decelerations to be in the range of 103–5  103 m/s2, which is at least by an order of magnitude larger than the centrifugal acceleration of the liquid film. Such large decelerations may generate additional surface perturbations in the form of capillary waves with significantly larger amplitudes than the initial perturbations at the beginning of ligament growth. These waves may propagate upstream until reaching the ligament foot, then reflect and destabilize due to Doppler shift in wavelength in the already presented long wave mechanism by Shinjo and Umemura (2010). As the ligament foot is weakly stretched and yet quite thin, it is likely to be more sensitive to capillary waves than the rest of the ligament. However, from presented results we could not determine the significance of ligament end pinch-off effect on its detachment from the film.

mechanism may allow for improved modeling of spinning wheel applications such as the liquid atomization or fiber formation. Nomenclature B d dHD dL dN f0 fs g I k L N

5. Conclusions Ligament formation mechanism was studied on a spinning wheel with the liquid stream flowing onto the wheel mantle surface. Experimental results show the Weber and Ohnesorge number and the liquid flow rate have an important effect on ligament formation dynamics and maximum length. To produce long ligaments disintegrating into droplets at relatively long distances from the spinning wheel, the liquid should be of sufficiently high viscosity and supplied at a large flow rate. Regression model for ligament length obtained by the least squares fit is applicable for a wide range of operating conditions (i.e., 103 o Weo105). While the Weber number itself does not have a very significant effect on the ligament length, it is by far the most important parameter for determination of ligament and droplet diameters. Another characteristic parameter affected by We is the velocity slip of the liquid film against the wheel which is significant when the wheel rotation is slow but can be neglected above WeE 104. Also influenced by We are the ligament trajectories which have an involute-like shape in the relative coordinate system but are almost tangent to the wheel in the absolute coordinate system. Aside from these integral parameters, the instantaneous kinematic parameters such as the ligament strain rate have proven to be very informative of the process. Based on the results we can conclude that ligament elongation with a given positive strain rate has a certain stabilizing effect on the ligament formation. However, formation stability cannot be fully determined by the strain rate as the ligament formation process on a spinning wheel is much more complex from the experiments upon which the stability theory is based. Also, there are some specific phenomena involved that must be taken into consideration. In all of our experiments, ligament growth was accompanied with the head droplet pinch-off. The pinch-off is certainly an undesired phenomenon as it produces relatively large droplets and is followed by a sharp strain rate reduction which may destabilize the surface of the ligament and thus contribute to an early ligament detachment from the film and disintegration. Further research should determine the necessary modifications to the existing setup in order to eliminate or at least significantly reduce the presence of ligament end pinching. Another important aspect of the ligament formation process which should be further investigated is the ligament detachment from the liquid film and its effect on the final ligament disintegration. A good understanding of the ligament formation

197

n Oh Q q R r rL Re s t tn vf v0 We wgl x, y xabs, yabs

liquid film width on the wheel, m diameter of droplets formed by ligament disintegration, m head droplet diameter, m mean ligament diameter, m liquid nozzle diameter, m atomizer wheel rotational speed, Hz ¼2π rad/s image acquisition rate, Hz ¼1/s gravitational acceleration, m/s2 normalized standard deviation, dimensionless wave number, 1/m maximum liquid ligament length before detachment/ breakup, m number of ligaments on the wheel perimeter, dimensionless number of statistical samples, dimensionless Ohnesorge number, dimensionless liquid volume flow rate, m3/s liquid dimensionless flow rate, dimensionless atomizer wheel radius, m radial coordinate in the polar coordinate system, m local ligament radius, m Reynolds number, dimensionless ligament circumferential spacing on the liquid film, m time, s dimensionless time, dimensionless liquid film velocity, m/s spinning wheel circumferential velocity, m/s Weber number, dimensionless Glycerol mass fraction in a glycerol–water mixture, dimensionless ligament head coordinates in a relative coordinate system, m ligament head coordinates in an absolute coordinate system, m

Greek letters

γ γn ε λm λOPT μ ρ σ τ ϕ

ligament strain rate, 1/s ligament dimensionless strain rate, dimensionless amplitude of ligament radius oscillations, m wavelength of the most unstable (fastest growing) wave on liquid film, m wavelength of the fastest growing dilatational wave on the ligament, liquid viscosity, Pa s liquid density, kg/m3 liquid surface tension coefficient, N/m ligament time constant, s angular position on the wheel perimeter, deg

Acknowledgments This work is in part supported by Slovenian Research Agency (ARRS), Grants P2-0167 and L2-4270. Operation is also in part financed by the European Union, European Social Fund; Ministry of Economic Development and Technology, Republic of Slovenia, project KROP 2011 at Abelium d.o.o.

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