Experimental Investigation of Ligament Formation Dynamics of Thin

Article pubs.acs.org/IECR

Experimental Investigation of Ligament Formation Dynamics of Thin Viscous Liquid Film at Spinning Disk Edge Dongxiang Wang,*,†,‡ Xiang Ling,*,‡ Hao Peng,‡ Zhenwei Cui,† and Xinjun Yang†,‡ †

Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology, School of Mechanical Engineering, Jiangnan University, No. 1800 Li Hu Road, Wuxi 214122, China ‡ Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, Nanjing Tech University, No. 5 Xin Mo Fan Road, Nanjing 210009, China ABSTRACT: The ligament formation and break-up process of thin viscous liquid film at the spinning disk edge were performed experimentally. The regression models for ligament number, tail end diameter of ligament, and mean diameter of droplets were proposed. Despite the velocity slippage of rotary cup atomizers being smaller than that of the disk atomizer, it will not improve the atomizing characteristics. The tail end diameter of ligaments and the size of droplets decrease with increasing rotational speed, while liquid flow rate shows smaller influence on the tail end diameter and droplet sizes. For the break-up of a ligament in the centrifugal field, the connected region between joints presents as dumbbell structures, and the dumbbell structures may contract from both ends to the middle for further integration. Despite the capillary wavelength for the break-up of a ligament being about 3.15 times of the ligament diameters due to ligament stretching, the actual dominant mode is the short-wave mode. On the basis of the dispersion relation by Joseph et al.,14 Wang et al.15 proposed a mathematical relation to calculate the ligament number for molten slag for which the viscous effect is predominant over the surface tension effect. In ligament mode, droplets are generated by the break-up of ligaments like liquid jets.16 Ahmed et al.17,18 have proposed two models to simulate the break-up of viscous liquid jet. Shinjo et al.19,20 considered that the break-up of liquid jets involves two break-up modes in the microgravity environment, the short-wave mode (the most unstable break-up wavelength is 1.81 times of the jet diameter) and the long-wave (Rayleigh) mode (the most unstable breakup wavelength is 4.51 times of the jet diameter). For the liquid jet, it is widely believed that the perturbations induced by the injection nozzle strongly affect the break-up of a ligament.21 However, in spinning disk atomizing, there is no injection nozzle for forming each ligament, and the ligament is in the centrifugal field. Which mode, the short-wave mode or the long-wave mode, is dominant is an interesting topic in understanding the spinning disk atomizing process. Meanwhile, there is little research concerning the tail end diameter of a ligament which is the key parameter to determine the droplet sizes.

1. INTRODUCTION The spinning disk has found wide applications in chemical,1,2 food,3 and metallurgical processes4 recently, and has been developed as a process intensification technology. In organic syntheses5,6 or catalyzed7 reactions, the rapid conversion rate can be obtained in the thin liquid film produced by centrifugal and adhesive forces. Moreover, atomization of liquids into gas by the spinning disk can promote the intimate contact and enhance the heat and mass transfer coefficient between two phases in various chemical processes.8 The biggest advantage for such an atomizer is the capability to atomize highly viscous or nonhomogenous liquids robustly.9 In the spinning disk atomization, the droplets typically are generated around or beyond the disk rim by direct drop mode, ligament mode, or sheet mode.10,11 Ahmed et al.9 explained in detail the transition criteria between each mode. The direct drop and sheet modes are in the unsteady atomizing state; a large leading droplet followed by much smaller satellite droplets gives rise to a bimodal size distribution which may not be an advantage for a chemical process. While in ligament mode, especially in the full ligament mode, the atomization process reaches a quasistable state; a quantity of liquid ligaments are drawn out around the disk rim and finally disintegrate into droplets with a more narrow size range. The droplet size can be mainly determined by the ligament number, diameter, and capillary wavelength generated along the ligament. Eisenklam12 and Kamiya13 developed analytical criteria to predict the ligament number for inviscid and viscous liquid, respectively. © 2016 American Chemical Society

Received: Revised: Accepted: Published: 9267

April 14, 2016 August 6, 2016 August 10, 2016 August 10, 2016 DOI: 10.1021/acs.iecr.6b01428 Ind. Eng. Chem. Res. 2016, 55, 9267−9275

Article

9268

Frost10

Q = 0.556−2.78 × 10−6 m3/s, ω = 838−1677 rad/s, D = 0.04−0.12 m, μ = 0.001 Pa s, ρ = 1000 kg/m3, σ = 0.072 N/m ligament

ρ0.67 ω0.65D0.8

σ 0.02Q 0.05μ0.651Rd 0.58

dm = 27.81 water Ahmed26

Q = 1.0−10 × 10−6 m3/s, ω = 50−1000 rad/s, D = 0.04−0.12 m, μ = 0.001−0.022 Pa s, ρ = 1000−1170 kg/m3, σ = 0.033−0.059 N/m ligament

dm = 1.87

σ 0.15Q 0.44μ0.017 ρ0.16 ω0.75D0.8

Q = 2.0−5.0 × 10−6 m3/s, ω = 260−628 rad/s, D = 0.06−0.12 m, μ = 0.02−0.5 Pa s, ρ = 900−1300 kg/m3, σ = 0.06−0.08 N/m ligament

σ 0.26Q 0.32μ0.65 ρ0.29 ω0.79D0.69 dm = 2.0

millet jelly solvent soybean glycerol/ water solution

oils Boize24

Kayano25

Q = 0.125−4.0 × 10−6 m3/s, ω = 78.5−628 rad/s, D = 0.088 m, μ = 0.0073−0.0603 Pa s, ρ = 834−868 kg/m3, σ = 0.028−0.035 N/m direct drop ligament

ρ

dm = 0.006ω−0.98

Q = 0.12−0.77 × 10−6 m3/s, ω = 1990−7330 rad/s, D = 0.02−0.05 m, μ = 0.001 Pa s, ρ = 1000 kg/m3, σ = 0.073 N/m ligament

σ 1.35Q 0.19 D0.66ω1.41μ1.48 0.06

dm = water Ryley23

direct drop ligament sheet

⎛ σ ⎞ dm = 3.8⎜ ⎟ ⎝ ρDω2 ⎠

Here, the following abbreviations apply: ρ is the density, μ is the viscosity, σ is the surface tension, Q is the volume flow rate, ω is the rotational speed of disk, and D is the disk diameter. 2.2. Ligament Mode Disintegration. The disintegration of liquid film results from the unstable waves developed on the liquid film induced by the Kelvin−Helmholtz (KH) and Rayleigh−Taylor (RT) instabilities.27 The initial disturbances required for the formation of waves are most likely caused by the KH instability induced by the velocity slippage at the air− liquid interface.15,27 Nevertheless, the main source which makes these waves much stronger is the RT instability.28 It occurs when a layer of denser fluid is pushed toward the lighter fluid by the equivalent gravity force.29 The unstable waves caused by the RT instability grow at different rates depending on their wavelength. The waves below a certain wavelength are fully damped due to viscosity and surface tension. On the contrary, once the centrifugal force is greater than the viscosity and surface tension, the waves will grow exponentially at different growth rates. The wave with the fastest growth rate will become predominant and finally cause the break-up of the liquid film.12 In the spinning disk atomizing, due to the RT instability caused by the centrifugal force, the

Walton water methyl and salicylate Prewett22

(2)

0.5

⎛ σρD ⎞0.9⎛ μD ⎞⎛ μ ⎞0.84 Q > 19.8⎜ 2 ⎟ ⎜ ⎟ ⎟⎜ ⎝ μ ⎠ ⎝ ρ ⎠⎝ ωρD2 ⎠

correlations

For the first appearance of sheet mode

liquids

(1)

authors

⎛ σρD ⎞0.9⎛ μD ⎞⎛ μ ⎞0.63 Q > 0.46⎜ 2 ⎟ ⎜ ⎟ ⎟⎜ ⎝ μ ⎠ ⎝ ρ ⎠⎝ ωρD2 ⎠

Table 1. Correlations of Mean Diameter (dm) of Droplet by Spinning Disk Atomization

2. THEORETICAL BASIS 2.1. Critical Transition Criterion. The liquid film may disintegrate into droplets around or beyond the disk rim by three modes. Transition from one mode to another occurs with the increase of liquid flow rate, disk diameter, or rotational speed. For a flat disk atomizer, Frost’s criterion10 is utilized to approximately estimate the critical volume flow rate for fully developed ligament mode and sheet mode in this paper, as shown in eqs 1 and 2. For the fully developed ligament mode

break-up mode

ranges of variables

Some correlations for predicting the mean diameter of droplets are proposed in Table 1. For the effect of physical properties, it is insufficient for Ryley23 and Ahmed26 to develop a relationship as only one liquid (water) was investigated. Boize24 considered that the mean diameter is determined only by the rotational speed of the disk for oils in the range 78.5− 628 rad/s, and dm ∝ ω−0.98, which is nearly the same with the expression proposed by Walton and Prewett22 (dm ∝ ω−0.1). Meanwhile, Walton and Prewett22 indicated that the mean diameter is only related with the surface tension and rotational speed. Despite the above investigative effort, it is essential to further explore ligament formation and break-up dynamics. In this paper, we performed an experimental investigation by means of high-speed camera visualization. The evolution of ligament, the number and tail end diameter of ligament, break-up mechanism, and droplet size were analyzed for a deep understanding of ligament type disintegration, which can contribute to the design and optimization of spinning disk atomizers for various chemical processes.

Q = 0.04−2.8 × 10−6 m3/s, ω = 50−10000 rad/s, D = 0.02−0.08 m, μ = 0.001−1.5 Pa s, ρ = 900−1360 kg/m3, σ = 0.031−0.465 N/m

Industrial & Engineering Chemistry Research

DOI: 10.1021/acs.iecr.6b01428 Ind. Eng. Chem. Res. 2016, 55, 9267−9275

Article


first expansion joint structure or the last capillary wave (Figure 1). Wang et al.15 proved that the ratio of ligament length and the tail end diameter of a ligament is L/dL ∝ WeL0.447, which is astonishingly close with Weber (L/dL ∝ WeL0.5). Now that the evolution of ligaments at the disk rim is similar to liquid jets,15 the relationship between critical length (L) and ligament diameter (dL) can be simply expressed as

liquid film disintegrates into a quantity of ligaments around the disk lip with the spacing λm as indicated in Figure 1.

L = c WeL (1 + 3OhL) dL

(6)

where We L = ρd L ν r /σ is the Weber number and OhL = 3μ/ ρσdL is the Ohnesorge number of the ligament stream, respectively. Equation 6 involves the ligament flow parameters, such as tail diameter and velocity. However, for application in practice, it is difficult to always obtain the parameters from experiments. A more useful way is to correlate L/dL with atomizer diameter, rotational speed, volume flow rate, and physical properties of liquid. First, by only allowing for the effect of surface tension force and neglecting the effects of viscous force, 1 + 3OhL ≈ 1, and eq 6 can be simplified as L /dL = c WeL . The ligament stream observed in the experiment is actually very close to the involute of a circle;10,11,13,15 thus, the path line of ligament stream can be expressed as 2

Figure 1. Schematic diagram of ligament mode break-up on a spinning disk atomizer. 13

Kamiya analyzed the combined effect of surface tension and viscosity on the ligament number by equating the dissipation rate of kinetic energy, surface energy, and viscous deformation energy of liquid film to the motive power by centrifugal force at the edge of a spinning disk. The growth rate of disturbance can be expressed as μ 1 α 1 ε = ln = k m(4k m − 3) 2 ( φ − 1) h ≪ D t α0 2 ρR whereφ = 1 +

⎞⎤ ⎟⎟⎥ ⎠⎥⎦

⎛ 2 ⎞1/7 dL a b ρQ = c″We Oh ⎜ 3 ⎟ R ⎝ σR ⎠

(4)

In the ligament mode, the ligaments are stretched to a limit diameter until the tail end of ligaments begins to expand (Figure 1). Droplets are formed by the break-up of the tail end of ligaments which is induced by the capillary waves developed along the ligaments.15 Therefore, the mean diameter of droplets (dm) can be determined simply by the tail end diameter of ligament (dL) and the wavelength of capillary wave (λ); it can be expressed as 3 ⎛ d ⎞2 4 ⎛ dm ⎞ π⎜ ⎟ = π⎜ L ⎟ λ ⎝2⎠ 3 ⎝ 2 ⎠

(9)

As mentioned above, the ligament number is mainly determined by We and St. Meanwhile, it is well-known that the dimensional number Oh or St (St = Oh2) represents the importance of viscosity compared to surface tension. Therefore, with consideration of the viscous effect, eq 9 can be revised as eq 10, and the droplet sizes can be predicted on the basis of eqs 5 and 10.

Weber number, St = 2 μ /σρD is the Stability number, α is the amplitude of disturbance, and αo is the initial amplitude of disturbance. According to Kamiya,13 the wavenumber or ligament number (km) is decided by We and St; it can be obtained by 1 k mSt

(8)

⎛ ρQ 2 ⎞1/7 dL = c′k m−(2/7)We−(2/7)⎜ 3 ⎟ R ⎝ σR ⎠

2

1+

y = R sin(ωt ) − Rωt cos(ωt )

expressed as vr = 2Rω 2L . For viscous incompressible fluid flow, the tail end diameter of a ligament can be estimated by dL = 4Q /πk mvr according to conservation of mass. Finally, the tail end diameter for inviscid liquid is estimated by eq 9.

− k m ), We = ρω2D3/8σ is the

⎡ ⎛ We = k m 2⎢3 + (8k m − 3)St ⎜⎜1 + ⎢⎣ ⎝

(7)

The relative velocity at the tail end of a ligament t = tb can be

(3) 8 (We k m(4k m − 3)2 St 2

x = R cos(ωt ) + Rωt sin(ωt )

(10)

3. EXPERIMENTAL METHODS 3.1. Experimental Apparatus. Experiments were performed on two disk atomizers by means of high-speed camera visualization (Figure 2). The experimental apparatus mainly consisted of a liquid supply system, an atomizing system, and a controlling system. The liquid supply system consisted of pump, storage tank, valve, flowmeter, collector, and connecting tubes. The working fluid was supplied from the storage tank through a circular nozzle with DN = 8 mm in diameter. The nozzle was placed vertically so that the liquid stream impingement point was 5 mm from the mantle surface of a disk. In the atomizing system, two disks with diameter of 0.05 and 0.1 m, respectively, were utilized in this paper. The

(5)

It must be noted that the ligament diameter is always changing along the stretching direction; the ligament diameter (dL) in this paper refers to the tail end of ligament before the 9269


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Industrial & Engineering Chemistry Research N

MSE =

∑i = 1 (ϕie − ϕi p)2 N

(12)

N

R2 =

N

N

[N ∑i = 1 (ϕieϕi p) − ∑i = 1 ϕie ∑i = 1 ϕi p]2 N

N

N

N

[N ∑i = 1 (ϕie)2 − (∑i = 1 ϕie)2 ][N ∑i = 1 (ϕi p)2 − (∑i = 1 ϕi p)2 ]

(13)

4. RESULTS AND DISCUSSION 4.1. Ligament Formation. 4.1.1. Evolution of Ligament. In regard to the growth and break-up of a ligament, the transient mechanism was observed experimentally in details as illustrated in Figure 3 for an example. At t = 0, a torus structure Figure 2. Schematic diagram of experimental apparatus.

spinning disks are made in AISI 304 with thickness of 4 mm. The surface roughness was up to Ra = 1.60 μm by mechanical polishing, and the measurements of roundness were in the range of 0.025 mm. The disk was driven by an ac powered motor for which the rotational speed was adjusted by a frequency converter. The controlling system consisted of highspeed camera, additional light source, data receiver, and computer. 3.2. Experimental Procedure. In this paper, the mixtures of glycerol and water with two mass ratios were utilized (Table 2). A high-speed camera (Phantom V7.3) with a shooting speed Figure 3. Evolution of the ligament during the spinning disk atomization process with Q = 7.1 mL/s, ρ = 1170 kg/m3, μ = 0.0175 Pa s, σ = 0.073 N/m, D = 0.05 m, and ω = 62.8 rad/s.

Table 2. Physical Properties of the Selected Working Fluid (10°C) working fluid [mass %]

density [kg/m3]

viscosity [Pa s]

kinematic viscosity [10−6 m2 s−1]

surface tension [N/m]

40% glycerol 60% glycerol

1113 1170

0.00528 0.0175

4.7 14.9

0.0743 0.073

which is the beginning of the ligament formation appears first at the edge of the bulk film as a result of surface tension. The ligament grows from the initial liquid torus structures due to centrifugal force and keeps stretching and thinning. The head of a ligament starts to present a bulbous shape by absorbing the upstream liquid of ligament (t = 5−15 ms), and promotion of necking phenomena is evident. The ligaments are stretched to a critical length as t increased up to 20 ms; the head separates from the main ligament due to surface tension forces and forms a droplet with a relatively large diameter (also known as the head droplet27,28). The sizes of the head droplets formed by the first time are almost dozens of times larger than that of the subsequent droplets. It must be emphasized that, with pinch-off of the head droplet, capillary waves will form at the ligament tip, grow rapidly, and propagate along the ligament. As a result of these waves, the ligaments will granulate into an array of small spherical droplets. 4.1.2. Number of Ligament. Figure 4a illustrates the number of ligament in the fully developed ligament mode with the increase in rotational speed. The ligament number increases from 25 to 116 when the rotational speed ranges from 62.8 to 314.2 rad/s. For the lower speed, the ligaments are distributed with regular shape throughout the disk edge. Meanwhile, the droplets are mainly formed due to the break-up of ligament tip. However, as the speed rises up, the shapes of ligaments may vary from each other. The ligament may stretch and thin to be so small in diameter that the interfacial stress (i.e., surface tension force) becomes important. Also, for this reason, the break-up of ligament occurs not only at the ligament tip, but part of the liquid ligament may disintegrate integrally first

of 4000 fps and 800 × 600 resolution was utilized to record the evolution of ligament at the disk rim, the images were stored on a computer for further processing. All the experiments were carried out at room temperature. A series of tests were carried out for the following conditions: (1) inlet temperature of mixtures, ambient temperature; (2) range of mass flow rate of mixtures, 0−30 mL/s; (3) range of rotational speed of disk, 62.8−314.2 rad/s. 3.3. Experimental Uncertainty. The volume flow rates of working fluid were measured by an electromagnetic flowmeter (0−0.2 m3/h), which had the uncertainty of ±0.5%. The ambient temperature was measured by PT100 platinum thermistor with the uncertainty of ±0.2 °C. The rotating speed was adjusted by frequency converter with the uncertainty of ±1%. The experimental relative uncertainty of dm and dL were estimated to be 3.75% and 4.52%, respectively. For the statistical analyses of regression, the relative error (RE), normalized mean squared error (MSR) and correlation coefficient (R2) were defined as eqs 11−13 respectively, ϕ represented the dependent variables for regression which can be dL/R, km, or λ/dL. RE =

ϕie − ϕi p ϕie

× 100% (11) 9270


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Figure 4. Number of ligament for ρ = 1170 kg/m3, μ = 0.0175 Pa s, σ = 0.073 N/m, D = 0.05 m, (a) with increasing rotational speed and (b) with increasing volume flow rate.

measurements. Thus, it seems that, for the atomizing of fluids selected in this paper by spinning disk or rotary cup atomizers, the mechanisms of ligament mode disintegration are basically consistent. Despite the velocity slippage of the cup atomizer being smaller than that of the disk atomizer,9 it will not improve the atomizing characteristics significantly. From this perspective, the regular flat spinning disk is recommended rather than the complex shapes of rotary cups within the operating parameters of this paper. According to Kamiya,13 the number of ligament is mainly determined by We and St. With a brief assumption that the ligament number can be predicted by km = cWeaStb and with least-squares fitting, the number of ligament can be predicted by

followed by a second break-up process. In addition, the induced shear stress of air that resulted from the high rotational speed may give another response to this phenomenon. Figure 4b illustrates the number of ligament with the increase of volume flow rate in the fully developed ligament mode. For the operation conditions shown in Figure 4b, the ligament number km = 35 with Q increasing from 4.76 to 11.84 mL/s, which is irrelevant to the volume flow rate of the working fluid. It must be emphasized that this phenomenon was not observed for the first time; it has been observed also by Frost,10 Liu et al.,11 and Kamiya.13 As mentioned before, the liquid film breakup is due to the wave with the fastest growth rate. While the liquid film thickness h at the disk rim is much smaller than the disk diameter D, the volume flow rate would not affect the growth rate of waves on the liquid film12,13 with the precondition that the mantle surface of a spinning disk must be completely wet. In this situation, the wavenumber that determines the number of ligaments is controlled only by the liquid fluid properties (surface tension and viscosity), disk diameter, and rotational speed. Figure 5 illustrates the measured data of 60% glycerine compared with Liu et al.’s11 and Kamiya’s13 model. The

k m = 0.576We 0.458St −0.0619

(14)

Equation 14 is developed for the ranges of operating variables as We = 0.815 × 103 to 2.47 × 105, and St = 6.76 × 10−6 to 1.43 × 10−4. The mean squared error (MSE) is 0.0053, and the correlation coefficient (R2) is 0.9712. It is observed from Figure 6 that eq 14 fits well with the experimental data with the average relative error (RE) limit of +4.88% and −6.23%.

Figure 6. Relation between number of ligament and Weber number.

Figure 5. Number of ligament with increasing rotary speed for working fluids of 60% glycerine.

4.2. Tail End Diameter of Ligament. Apart from the number of ligament, the tail end diameter of the ligament (dL) is primarily an intermediate parameter for predicting the size of the droplets. In this paper, the tail end diameter was determined by the following procedure. For each image recorded at the particular operating point, three ligaments were chosen at a similar region of disk. The diameters of

ligament number predicted by Kamiya13 is at least 30% smaller than our measurements. This model was derived for spinning disks considering both viscosity and surface tension effects, and it overestimates the effect of viscous forces compared to our measurements. It is interesting that the predicted data by Liu et al.11 for rotary cup atomizers is in close proximity to our 9271


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Figure 7. Typical images illustrate the diameter of ligaments with ρ = 1170 kg/m3, μ = 0.0175 Pa s, σ = 0.073 N/m, and D = 0.05 m, (a) with increasing rotational speed and (b) with increasing liquid flow rate.

Figure 8. Tail end diameter of ligaments (a) with increasing rotational speed and (b) with increasing liquid flow rate.

According to the diameter data together with all the experiment input parameters, a data fit was made according to eq 8, and the following relation was obtained

ligaments were measured on the basis of the digital image processing techniques, and finally the mean diameter was obtained by averaging the measured data as illustrated in Figure 7. It is observed from Figure 8a that the tail end diameters of ligaments decrease with the increase in rotational speed. Meanwhile, for the same speed condition, dL increases when the viscosity of fluid increases. For the disk with D = 0.05 m, as the rotational speed rises from 62.8 to 314.2 rad/s, the diameters for 60% glycerol and 40% glycerol drop from 1.12 to 0.544 mm and from 0.957 to 0.497 mm, respectively. However, for the disk with D = 0.1 m, the diameters for the two fluids drop from 0.711 to 0.253 mm and from 0.578 to 0.219 mm, respectively. As is mentioned above, the film breaks up into more ligaments with higher rotational speeds, and the tail end diameter of ligaments drops consequently. Nevertheless, the tail end diameters of ligaments rise very slowly with the increase of liquid flow rate, or in other words, the liquid flow rate shows little influence on the mean diameters of ligaments (Figure 8b). For the disk with D = 0.05 m, as the liquid flow rate rises from 8.29 to 25.02 mL/s, the tail end diameters for 60% glycerol and 40% glycerol just rise from 1.02 to 1.22 mm and from 0.988 to 1.18 mm, respectively, while for the disk with D = 0.1 m, the diameters rise from 0.758 to 0.921 mm and from 0.736 to 0.898 mm, respectively. In the full ligament mode, the number of ligament remains unchanged with increasing volume flow rate as mentioned above. Consequently, with the increase of liquid flow rate, it can be concluded that the tail end diameter of ligaments almost will not rise, while the ligaments stretch longer.

⎛ 2 ⎞1/7 dL 0.251 ρQ −0.321 Oh ⎜ 3 ⎟ = 1.539We R ⎝ σR ⎠

(15)

Equation 15 is developed in the ranges of operating variables: We = 0.815 × 103 to 2.47 × 105, St = 6.76 × 10−6 to 1.43 × 10−4, and ρQ2/σR3 = 0.005 to 0.255. The mean squared error (MSE) is 0.056 with the correlation coefficient (R2) of 0.848. As observed in Figure 9, eq 15 fits well with the experimental data with the average relative error (RE) limit of +13.42% and −16.71%. 4.3. Droplet Formation. 4.3.1. Droplet Formation Mechanism. Figure 10 illustrates a typical process for the formation of droplets for 60% glycerine. As is mentioned in Figure 3, the ligament occurs from the initial bulge at the edge of the bulk liquid; it becomes thinner in diameter and forms a head at the ligament tip. Soon after, the head droplet is pinched off, and the capillary waves grow rapidly and propagate along the ligament (t = 0). Because of these waves, some regions in the upstream of ligament may dilate by absorbing the surrounding fluid and present a flow structure like the expansion joint (t = 2.5 ms), while the connection regions between joints keep contracting and necking happens (t = 5 ms). Under the action of circumferential surface tension force, the expansion joints break-up from the bulk flow and the connection regions appear as dumbbell structures (t = 7.5 ms). At t = 7.5−12.5 ms, the expansion joints gradually form spherical droplets, while the dumbbell structures keep contracting from both ends to the middle and merge into a 9272


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Figure 9. Comparison between the measurement data and the predicted data.

Figure 11. Diameter of ligaments and wavelength of capillary waves.

development of the most unstable wave with a long wavelength (λ = 4.51dL), but by the stretching of ligament due to centrifugal forces, the actual dominant mode is the short-wave mode driven by the propagative capillary wave from ligament tip even though λ/dL > 1.81. 4.3.3. Prediction of Droplet Diameter. The mean diameter of droplets was determined by the digital image processing techniques based on the numerous images obtained. A selfcompiled program was utilized to extract the diameters of droplets produced. Choose an image which was recorded during the quasisteady granulation state first, and then set a processing area on the image that contains droplets. Meanwhile, convert the chosen area into binary images according to a threshold value. Finally, there was extraction of the droplets edge using the Candy operator, and then postprocessing is implemented on the given binary image to calculate the mean diameter of droplets with reference to the standard length scale. It must be emphasized that the droplet sizes change significantly with the radial distance from the center of the spinning disk.26 However, the authors in this paper have been devoted to the formation and break-up of ligaments for a deep understanding of ligament type disintegration in this paper. Consequently, for each image recorded at the particular operating condition, the measurement location of droplet sample remains unchanged. According to the diameter of ligaments and capillary wavelength, the mean diameter of droplets can be obtained by

Figure 10. Break-up of the ligaments for ρ = 1170 kg/m3, μ = 0.0175 Pa s, σ = 0.073 N/m, D = 0.05 m, Q = 7.1 mL/s, and ω = 62.8 rad/s.

droplet. It must be emphasized that the connection region may go through the above evolution process again if the connection region is long enough. Finally, the ligament granulates into an array of large main droplets as well as some satellite droplets. As observed in the experiment, most of the satellite droplets tend to merge with the main droplets as their velocity in the ligament lateral direction is slightly different. 4.3.2. Capillary Wavelength. The measurement method of the capillary wavelength was the same as the tail end diameter of the ligament. It can be indicated in Figure 11 that the mean capillary wavelength is about 3.15 times that of the ligament diameter, meaning λ/dL ≈ 3.15; it seems that the break-up of the ligament is the long-wave mode. It must be emphasized that, in liquid jet, the long-wave (Rayleigh) mode is found to be driven by the short and fast capillary waves caused from the jet tip.20 These waves must be reflected by the nozzle (if they can reach the nozzle by overcoming viscous damping), and the wavelength becomes longer due to the Doppler shift which results from the difference in relative velocities before and after reflection. However, in spinning disk atomization, there is no injection nozzle for each ligament. Although the ligament length extends by the increase of tapping flow, it shows little influence on the diameter of ligament as mentioned above. Consequently, this phenomenon may not be caused by the

⎛ 2 ⎞1/7 dm 0.251 ρQ −0.321 Oh ⎜ 3 ⎟ = 2.582We R ⎝ σR ⎠

(16)

Equation 16 is developed in the ranges of operating variables: We = 0.815 × 103 to 2.47 × 105, St = 6.76 × 10−6 to 1.43 × 10−4, ρQ2/σR3 = 0.005 to 0.255. Figures 12 and 13 illustrate the mean diameter of droplets measured and predicted by eq 16, where the predicted results derived from the correlation by Walton and Prewett22 and Frost10 were utilized as a comparison. It is observed that the prediction by the promoted correlation shows good agreement with the experimental data. The mean diameter of droplets decreases with the increase in rotational speed, while the liquid flow rate shows smaller influence on it. As mentioned before, for full ligament formation mode, the rotational speeds exert tremendous influence on ligament number and diameter. However, as the liquid flow rates increase, the ligament number is wellmaintained, and the tail end diameter of ligaments also rises 9273


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Figure 12. Measured and predicted values of mean diameter at different rotational speeds (a) for disk diameter of 0.05 m and (b) for disk diameter of 0.1 m.

slippage of rotary cup atomizer being potentially smaller than that of the disk atomizer, it will not improve the atomizing characteristics. The tail end diameter of ligaments decreases with the increase in rotational speed. However, the liquid flow rate shows little influence on the mean diameter of ligaments, while the length of ligaments will stretch. Accordingly, the droplet sizes descend with increasing rotational speed, while the liquid flow rate shows a smaller influence. For the break-up of ligament in the centrifugal field, the connected region between joints presents as dumbbell structures. After break-up, the joints tend to be spheres, while the dumbbell structures may contract from both ends to the middle, and they may be further integrated or break up again. Although the capillary wavelength for the break-up of the ligament is about 3.15 times that of the ligament diameters (λ/ dL = 3.15), this phenomenon is not caused by the development of a long-wave mode (λ/dL = 4.51) but is due to the ligament stretching caused by the centrifugal effect; the actual dominant mode is the short-wave mode (λ/dL = 1.81).

Figure 13. Measured and predicted values of mean diameter at different liquid flow rates.

slowly. At lower rotational speed, the mean diameter predicted by Walton and Prewett22 is well above the measurement. Despite the Walton and Prewett22 model having only considered the effect of surface tension on droplet sizes, it is consistent in values with eq 16 as the speed increases. It seems that, for higher rotational speed, the diameter of droplets is mainly determined by the rotational speed and surface tension rather than the viscosity of working fluid for the operation conditions in this paper. Meanwhile, Figures 12 and 13 illustrate that the predicted values by Frost10 slightly underestimate the mean diameter of droplets. It must be emphasized that the operating conditions, disk sizes, working fluids properties, and measurement method are not the same for each test. The present work is performed as a useful complement for the previous fruitful works.

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AUTHOR INFORMATION

Corresponding Authors

*Tel: 86-25-83587321. Fax: 86-25-83600956. E-mail: [email protected]. *E-mail: [email protected]. Funding

Funding for this work comes from National Natural Science Foundation of China (Grant No. 51406078 and 51606086), Fundamental Research Funds for the Central Universities (Grant No. JUSRP115A11), and Major Collegiate Project of Natural Science Foundation of Jiangsu Province (Grant No. 15KJA480001).

5. CONCLUSION This paper described an experimental investigation to characterize the ligament mode break-up of a liquid film at the disk rim in spinning disk atomizing. The droplet size is determined by the tail end diameter of ligament and the capillary wavelength. For the full ligament mode, the regression models for ligament number, tail end diameter of ligament, and mean diameter of droplets are obtained in a wide range of operating conditions (We = 0.815 × 103 to 2.0 × 105, St = 6.76 × 10−6 to 1.43 × 10−4, ρQ2/σR3 = 0.005 to 0.255). Some findings of this paper follow. The ligament number predicted by Kamiya13 overestimates by at least 30% the effect of viscosity. Despite the velocity

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was performed as part of the National Natural Science Foundation of China and Fundamental Research Funds for the Central Universities. The authors gratefully acknowledge the Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology (Nanjing Tech University) for financial support. 9274


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NOMENCLATURE D disk diameter [m] R disk radius [m] Rd radial distance from center of disk [m] dL diameter of ligament [m] dm mean diameter of droplets [m] h liquid film thickness [m] L ligament length [m] km number of ligaments vr velocity of ligament [m/s] Oh Ohnesorge number of the bulk liquid film OhL Ohnesorge number of the ligament stream Q volume of flow rate [m3/s] t time [ms] We Weber number of the bulk liquid film WeL Weber number of the ligament stream Abbreviations

RT Rayleigh−Taylor KH Kelvin−Helmholtz Greek Symbols

ε ω ρ γ μ λ α αo λm

growth rate of disturbance rotational speed [rad/s] density [kg/m3] surface tension [N/m] dynamic viscosity [Pa s] capillary wavelength [m] amplitude of disturbance [mm] initial amplitude of disturbance [mm] ligament spacing

Subscripts

L ligament r relative m mean Superscripts

e experimental p predicted

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Experimental Investigation of Ligament Formation Dynamics of Thin

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