The formation and breakup of molten oxide jets

Chemical Engineering Science 105 (2014) 143–154

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The formation and breakup of molten oxide jets M. Wegener n, L. Muhmood, S. Sun, A.V. Deev CSIRO Process Science and Engineering, Box 312 Clayton South, Victoria 3169, Australia

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T


High-speed images of natural breakup of slag jets at 1600 1C at different flow rates.
The jet length is analysed as a function of time and is normally distributed.
The jet length is predictable taking the instability wave Reynolds number into account.
Short jets break up in a regular manner and produce narrow drop size distributions.
Drop size decreases for higher flow rates and approaches Rayleigh mode.

art ic l e i nf o

a b s t r a c t

Article history: Received 8 August 2013 Received in revised form 8 October 2013 Accepted 15 October 2013 Available online 25 October 2013

Experimental investigations on the capillary breakup of jets of molten oxides (slags) at high temperatures in an inert atmosphere are presented in this paper. The (in)stability of ligaments, threads, or jets of metallurgical slags is of importance in many heat and mass transfer processes related to high-temperature metal production, and aftertreatment of slags as currently investigated in the dry slag granulation process. In order to quantify the dynamic disintegration process of slag jets into droplets, a three-zone high-temperature furnace with maximum temperatures of 1750 1C and optical access was built. In the present study, a molten synthetic calcia/alumina slag at 1600 1C was used to form jets of 1 mm in diameter at different flow rates. The various phenomena of jet formation, appearance of instability waves, jet disintegration and droplet detachment were captured using a high-speed camera. The jet length distribution was calculated and compared with predictions. A Fast Fourier Transform of the temporal development of the jet length was also performed. The jet length showed good agreement with empirical correlations if the instability wave velocity is used in the definition for Reynolds and Weber number. The size distribution of the formed droplets was investigated and compared to theoretical predictions. For higher flow rates, the main peak agrees very well with theoretical equations. For low flow rates near to transition to the dripping regime, short jets formed in a highly repetitive manner. The drop size distribution was found to be very narrow with a mean diameter according to Tate's law taking into account a Harkins and Brown correction factor. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Slag Capillary breakup Surface tension Droplet formation Jet

1. Introduction The (in)stability of liquid jets is a classical problem in fluid dynamics and has been studied since the 19th century (Amini and n

Corresponding author. Tel.: þ 61 395458930. E-mail addresses: [email protected], [email protected] (M. Wegener). 0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.10.030

Dolatabadi, 2010). Since then, an overwhelming amount of work – both experimental and theoretical – has been done to reveal the fundamental mechanisms of disintegrating jets. The fields of application are equally manifold ranging from practical everyday life applications to industrial processes, such as follows:


Liquid/liquid processes: liquid extraction columns (Grant and Middleman, 1996), spray columns, injection of dispersed phase

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M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

into mixing vessels in mixer-settler apparatuses, fibre spinning operations or liquid/liquid jet reactors (Skelland and Walker, 1989), etc. Gas/liquid processes: fuel injection technology, solder spheres (Shimasaki and Taniguchi, 2011), irrigation systems, spraydrying technology (Eggers and Villermaux, 2008), spray forming and coating (Passow et al., 1993; Srinivasan et al., 2011), drop on demand technology (Basaran and Suryo, 2007), etc. Everyday life environment: showers, sinks, pharmaceutical sprays (Eggers and Villermaux, 2008), garden hoses, hair sprays (Birouk and Lekic, 2009), etc.

The formation and breakup of jets is also important for metal production, for example steelmaking processes such as the Basic Oxygen Furnace (BOF) and the Electric Arc Furnace (EAF) (Alam et al., 2009). Gas jets at supersonic speed (M 4 1) are blown into the bath and generate jets and filaments of hot liquid which eventually break into droplets (so-called splashing). Bubbles of inert gases are used to enhance the distribution of alloys in the ladle. Bubbles may escape towards the gas/liquid interface of the melt and may eject as jets which then break up into droplets. Given a certain kinetic energy, this may also cause a slag/metal dispersion resulting in entrapment (Hahn and Neuschütz, 2002; Han and Holappa, 2003). During the last decade, efforts were increased around the world to recover the heat from slags by rotating atomisers. In the most common configurations, molten slag falls onto a spinning disc (Nexhip et al., 2004; Xie et al., 2010; Yoshinaga et al., 1982) or a cup/cylinder (Kashiwaya et al., 2010a, 2010b; Liu et al., 2011; Pickering et al., 1985). Centrifugal forces transport the liquid to the edge of the rotating device where it forms droplets, ligaments, or liquid sheets. The breakup in the ligament regime produces a narrow drop size distribution which is important to control the heat transfer and to avoid particle agglomeration. A similar approach is pursued for Liquid Droplet Heat Exchangers (LDHX) (Bruckner and Mattick, 1983; Bruckner, 1985; Thayer et al., 1983). In one concept, the LDHX is embedded in a solar thermal power plant process. Instead of molten salts, the use of a synthetic slag (which is more temperature stable and relatively cheap) is proposed to increase the process effectivity. Molten slag enters a vertical column via an array of nozzles. Jets form which later disintegrate into desirably uniform-sized droplets. During their fall, the droplets solidify and transfer heat to a counter-current gas stream. The solid particles can eventually be fed back to the heat source. However, little is known about the behaviour of slag jets at high temperatures, and how to control the drop size distribution from slag jets. Accordingly, no LDHX based on metallurgical slags has been built so far, hence fundamental experimental studies are necessary. 1.1. Problem formulation The breakup of liquid jets at ambient temperatures has widely been studied. Metal jets at moderate temperatures have also been investigated to a certain degree in terms of breakup, for example in high-precision solder printing technology (Liu and Orme, 2001; Orme et al., 2000), surface tension measurements of Sn/Pb jets (Bellizia et al., 2003; Howell et al., 2004), or influence of oxygen on the breakup of metal jets (Artem'ev and Kochetov, 1991; Lai and Chen, 2005). However, little is known about the fluid dynamic stability behaviour of molten slag jets. Metals have a high surface tension but usually a relatively low viscosity, whereas slags are both high surface tension and high viscosity liquids. Moreover, the viscosity strongly depends on temperature. The liquidus of common molten oxides such as calcia/silica/alumina strongly depends on the composition and may easily exceed 1400 1C (Ohno et al., 2011; Slag Atlas, 1995). Experimental investigations at these

temperatures are time consuming, expensive, and demanding, especially in terms of material selection and apparatus design. To the authors’ knowledge, few experimental studies on the breakup of slag jets at high temperatures have been published so far. Benda (1983) performed studies in the framework of the LDHX concept. The focus was on the production of uniformly sized droplets from disintegrating jets of molten slags. Some experimental studies were performed in a furnace with and without external excitation. A few photographs show slag jets and proper and improper jet disintegration, but quantitative analysis was restricted to the evaluation of drop size distributions from quenched slag globules. Overall, the experimental results demonstrated the applicability of known theoretical prediction methods to a certain degree. However, many questions remained unanswered, especially those concerning the dynamics of jet formation and breakup. The problem of jet stability is highly complex due to a large number of parameters, such as – but not limited to – nozzle or capillary design, the thermophysical properties of the fluids involved, the interplay of inertia, viscous, capillary and aerodynamic forces, as well as mass transfer and surfactant adsorption (Amini and Dolatabadi, 2010). In most published experimental studies, a controlled jet destabilisation by external disturbances is favoured, since a periodic state may be reached in which only few measurement points are required to measure the growth rate of the instabilities. Mostly, these controlled breakup experiments have been used to validate the available theoretical models of jet stability by comparing experiments with the theoretical dispersion curve (Blaisot and Adeline, 2000). Non-controlled breakup is also of interest, for example in engine injection or the aforementioned spinning disc/cup techniques. Even in an ambient temperature environment, only few experiments on free falling jets have been reported, see references cited in Blaisot and Adeline (2000). In natural mode (i.e. natural breakup: without any deliberate external excitation applied to the jet), a jet is subject to fluctuations in space and time and shows transient behaviour, as will be shown in the present study on natural breakup of molten slag jets. 1.2. Scope of the present work The aim of the present study is to investigate the formation and breakup of jets of molten oxides, namely a synthetic calcia/ alumina slag. This study focusses on jet formation, appearance of capillary waves, unbroken jet length (and its distribution), and frequency distribution of the droplet sizes formed by jet disintegration. Natural jet breakup was explored in detail for different operating conditions (i.e. different flow rates or Reynolds numbers) at 1600 1C in an inert environment. A high-temperature three-zone furnace with optical access was used and the phenomena were video recorded by means of a high-speed camera. Automated image analysis was used to examine the massive amount of recorded data. The discussion highlights the degree of predictability of the relevant breakup phenomena by theoretical and empirical equations.

2. Material and methods The experimental setup consisted of four major components, see Fig. 1: (1) a 3 zone electrically heated tube furnace with a 700 mm heated zone (Tetlow Kilns & Furnaces Pty Ltd Victoria, Australia); (2) a 99.8% high-purity alumina cross tube assembly for optical access and atmosphere control (McDanel Advanced Ceramic Technologies, Pennsylvania, USA); (3) (a–f) a droplet generating device (Mersen Oceania Pty Ltd.) in combination with a backpressure system; and (4) a Phantom v3.11 high-speed camera

M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

18

19

14

Exhaust G1

20

Exhaust G2,3,4

15

17

16

Qw,out Qg,out

21

Qw,in

e

13 f

2700 mm

a 1

3 d

2a 5a

Qw,out

Qw,out

g

2b

7 Qg,in Qw,in

c

b

4

6

G1

Qg,in

Qw,in

G2

5b G3 Qw,out

8 9 10 11

Qw,in

G4

12

Qg,in

1000mm

Argon(UHP)

Fig. 1. Schematic of the experimental device. 1: Heating chamber of three-zone furnace, 2a: vertical alumina tube, 2b: horizontal alumina tube, 3: graphite crucible, shank section, a–g (zoom): droplet generation device, a: crucible hearth, b: graphite capillary, c: tapered bottom, d: hollow graphite stopper, e: slag, f: B-type thermocouple in alumina sheath, g: slag droplet, 4: high-speed camera and computer, 5a, b: water cooled end caps horizontal alumina tube, 6: quartz window, 7: oxygen probe with R-type thermocouple, 8: stainless steel cup to collect slag droplets, 9: metal bellows, 10: graphite stand, 11: balance chamber, 12: gas rotameter, 13: gas bubbler and exhaust alumina tube unit, 14: precision pressure regulator, 15: differential pressure transducer, 16: graphite crucible lifting assembly, 17: graphite shank water cooled end cap, 18: precision linear actuator for stopper, 19: safety relief valve, 20: needle valve, 21: gas bubbler and exhaust graphite crucible unit.

(Vision Research) with a maximum frame rate of 3250 fps at full resolution (1280  800 pixels). The three-zone furnace (1) was equipped with 18 Kanthal Super 1900 1C elements (molybdenum disilicide, six in each zone) and could reach a maximum temperature of 1750 1C. The furnace had four openings to accommodate the alumina cross tube assembly (2) for optical access. The cross tube assembly consisted of two individual tubes, named vertical (2a) and horizontal tube (2b) in the following. The vertical tube (L ¼ 1500 mm, ID ¼ 90 mm) had two holes at its side through which the horizontal alumina tube with a non-circular cross-section (L ¼ 1000 mm, width  height ¼ 50  100 mm) was centred. This horizontal tube had two 38 mm holes in its center position to allow the slag droplets or jets – (g) in zoom section in Fig. 1 – to fall from the graphite capillary (b) through the holes into a stainless steel cup (8) at the bottom of the furnace. The outer joints of the cross were sealed with a high temperature ceramic adhesive (Ceramabond 671, Aremco Products Inc.). All alumina tube ends were water cooled (Q w;in and Q w;out ). Each flange of the horizontal tube (5a and 5b) had an inlet to facilitate gas purging (gas streams G2 and G3). Flange (5b) had a quartz window (6) for optical access, the other flange (5a) was fitted with a 600 mm long oxygen probe (7) (Australian Oxytrol Systems) with an internal R-type thermocouple to measure the oxygen partial pressure and the temperature near the alumina tube junction.

145

The graphite crucible consisted of three main parts: the shank section (3), the hearth section (a) and the stopper (d). The shank section (L ¼ 850 mm, OD ¼ 80 mm, wall thickness 12 mm) connected the hearth section with a lifting assembly (16) – a Linak LA 28 linear actuator – and also guided the stopper. The upper end of the shank had a water cooled end cap (17) and a gas inlet and outlet for the pressurising gas (gas stream G1). The hearth section (L ¼ 205 mm, V  200 mL) had a tapered bottom (c) to facilitate the flow of molten material (e) into the capillary (b). In this study, a knife-edged capillary with nominally 1 mm inner diameter was used (note: the jet diameter was measured at the capillary exit at experimental temperature by image analysis and was found to be 1.09 mm; consequently, this value was used in all further calculations). The outer capillary diameter was 8.21 mm at experimental temperature. The length to diameter ratio of this capillary was 15 at room temperature. The graphite stopper (d) (L ¼ 920 mm, OD ¼ 20 mm) obstructed the entry to the capillary. It was connected to a linear precision stepper actuator (18) (Physik Instrumente GmbH, Germany) with a minimum incremental motion of 0:1 μm located on top of the furnace and in line with the vertical axis. In addition, the hollow graphite stopper housed a B-type thermocouple sheathed by an alumina closed end tube (f). The graphite crucible slid through a flange sealed with a flexible graphite rope into the vertical alumina tube. The water cooled bottom end of the vertical alumina tube was connected via flexible metal bellows (9) to the balance chamber (11). The chamber housed a Sartorius ED3202S-CW precision balance (readability 10 mg, weighing capacity 3200 g, sample rate 10 Hz) and was purged by gas stream G4. The balance pan was protected by a deflector onto which a graphite stand (10) was kept. The stand held the tapered stainless steel cup (8) which collected the slag. The balance transmitted the weight change as a function of time to a computer. For all experiments, ultra high purity argon (BOC Australia) was used. The gas flow was divided into two separate lines. One was used to flush the alumina cross via the three inlets (balance chamber and side flanges, streams G2  4 , controlled by individual gas rotameters (12)). This purging gas exited at the alumina top flange and passed through a bubbler (13) for optical verification of overpressure. The second line, stream G1, pressurised the graphite crucible up to 2 bar. According to the required flow rate and pressure difference for a given capillary diameter, the pressure was controlled by a precision pressure regulator (14) (Norgren, Australia). A pressure transducer (15) (Sensortechnics, Germany) measured the pressure differential between the pressurised graphite crucible and the alumina cross tube. A pressure relief valve (19) was located at the outlet for safety, as well as a needle valve (20) and a bubbler (21). In the present study, a synthetic calcia/alumina slag ð49:1=50:9 wt%Þ was used. The slag was prepared in a multi-step procedure. First, CaCO3 (Sigma-Aldrich Pty Ltd., purity 99.95 wt%) was calcined to CaO at 1000 1C for 12 h in alumina crucibles. Completion of calcination was confirmed by sample weight. Aluminum oxide (Alfa Aesar, purity 99.95 wt%) was also kept in a furnace to remove moisture. In the next step, the desired composition was prepared and kept in a bottle on a roller mixer for 24 h. The mixture was then premelted in a high-temperature muffle furnace using a 95%platinum/5% rhodium crucible, quenched and then crushed. The slag was ground in a ring mill and melted a second time in the muffle furnace to ensure homogeneous composition. The chemical composition was finally analysed by X-ray fluorescence (XRF) measurements. The physical properties density ρ and viscosity μ were estimated from available literature data for a slag with comparable composition, whereas the surface tension s was determined from own measurements by using the pendant drop method. The

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Table 1 Physical properties of calcia/alumina slag at 1600 1C.

Table 2 List of experiments at 1600 1C (calcia/alumina slag, natural breakup).

CaO=Al2 O3 [wt%]

ρ [kg m  3]

μ [Pa s]

s [mN m  1]

Experiment no.

Δp [bar]

_ [g s  1] M

pO2 [10  9 atm]

Frame rate [fps]

49.1/50.9

2742

0.32

584

1 2 3 4 5 6

1.0 1.1 1.2 1.3 1.4 1.5

0.043 0.76 0.83 1.0 1.5 1.44

3.0 3.1 3.6 4.1 3.1 2.8

1000 1000 4000 4000 4000 4000

density was taken from Mukai and Ishikawa (1981) (48 wt%CaO= 52 wt%Al2 O3 at 1600 1C). Available viscosity data show considerable scatter for a calcia/alumina slag at and around 1600 1C. Kozakevitch (1961) reported μ ¼ 0:27 Pa s for 46/54 wt%. Urbain (1983) measured 0.122 Pa s at 1611 1C (47.3/52.4 wt%) whereas (Rossin et al., 1964) found 0.192 Pa s at 1600 1C (48/52 wt%). For a 50/50 wt% calcia/alumina slag, Kozakevitch (1960) reported 0.23 Pa s, and for 1% more calcia (51/49 wt%), the data of Rossin et al. (1964) yielded 0.128 Pa s, both at 1600 1C. Two common viscosity models, the Riboud (Riboud et al., 1981) and the Urbain model (Urbain, 1987) – evaluated at the given temperature and composition of the slag used in the present study (1600 1C, 49.1/ 50.9 wt%) – yield 0.2 and 0.32 Pa s, respectively. The Urbain model is based on the CaO  Al2 O3  SiO2 system and is one of the most widely used models to estimate the dynamic viscosity of slags (Kekkonen et al., 2012). For the present study, in the absence of own data, the value obtained from the Urbain model was used. The physical properties at 1600 1C are given in Table 1. The experimental procedure was as follows: around 500 g of the pre-made slag was filled into the graphite crucible which was then loaded into the furnace. The stopper closed the entry to the capillary. The furnace was then heated at 5 K min  1 to the desired working temperature under argon purge to remove air and to maintain an inert atmosphere. Once the desired temperature was reached, the furnace was held at least another 30 min to equilibrate the temperature in the melt. The pressure was then built up by closing the gas outlet (needle valve (20)) to a minimum flow rate. The stopper was lifted and the molten slag flowed through the gap between stopper and the tapered bottom. Depending on the flow rate or pressure difference, dripping (below critical velocity) or jetting (above critical velocity) occurred. The molten slag passed the optically accessible 100 mm measurement zone of the horizontal alumina tube, and finally accumulated in the stainless steel cup. A Phantom v3.11 high-speed camera was used for the experiments (12 bit depth, 1280  800 pixels CMOS sensor). The slag at 1600 1C was bright enough to record at up to 10,000 fps without any additional lighting. The internal RAM was 16 GB. After each recording, the captured file was transferred from the internal memory to the Phantom CineMag interface, a non-volatile storage device with a capacity of 128 GB. This process took approx. 30 s. To cover the distance from the optical windows to the molten jet within the furnace (4 500 mm), a long working distance lens with high magnification (InFocus KC/ST Video Microscope lens, Infinity, USA) was used. In this study, the focal length was approx. 1030 mm. The lens was additionally equipped with a MidOpt shortpass SP700 hot mirror block filter to block IR radiation. After completion of experiments the data were transferred from the CineMag to the hard disc of a computer for further data analysis. For image analysis, Image-Pro Plus 6.3 (Media Cybernetics) was used. The performed experiments are summarised in Table 2. For all experiments, the temperature was set to 1600 1C. Six different pressures were investigated, ranging from 1.0 to 1.5 bar. Here, Δp is the pressure difference between crucible and alumina cross tube _ was as measured by the pressure transducer. The mass flow rate M determined from the weight gain per unit time transmitted by the balance. The oxygen partial pressure pO2 was relatively constant

Fig. 2. Sequence of jet formation and breakup. Experiment no. 1 (Δp ¼ 1:0 bar, 1000 fps). Time difference between each image: 5 ms.

around 3  10  9 atm. In the experiments listed in Table 2, no additional external excitation was applied.

3. Results and discussion No deliberate external excitation or vibration was applied in these experiments. Consequently, the jet breakup occurred naturally, or in other words breakup was caused by the randomly present ambient perturbations and/or perturbations caused by the equipment (e.g. heating elements, cooling water circulation, cooling fans in furnace shell, and likewise). 3.1. Jet formation At pressures below 0.8 bar, only dripping was observed. At around 0.8 bar, a transition from dripping to jetting commenced. The liquid bridge which connected the main droplet with the remaining liquid at the capillary was stretched until the droplet detached. As the pressure increased, this liquid bridge got longer and eventually formed a proper jet. Fig. 2 shows a sequence of jet formation and droplet detachment for Δp ¼ 1:0 bar (experiment no. 1 in Table 2). The time difference between each image is 5 ms. The outer capillary diameter can be used as a length scale reference. On the first image, an instability wave can be seen which travels along the short jet. On the fourth image, the connecting thread diameter decreases, driven by an increase in the Laplace pressure. The liquid thread eventually detaches and the main droplet accelerates downwards. The free end of the remaining liquid is accelerated upwards by the unbalanced force of surface tension. A droplet forms at the capillary which oscillates shortly from prolate to oblate shape. At these low flow rates, the droplets which detach from the jet vary only slightly in size (between 4.8 and 4.9 mm). The droplet size is far larger than predicted by known theoretical equations for droplets formed from disintegrating jets. According to Tyler (1933), the droplet size is independent of the physical properties in the inviscid case. The volume of the formed droplets

M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

147

6.35Hz

3.5

18

≈157ms

16

3 2.5

12

Magnitude

Jet length (mm)

14

10 8 6

2 1.5 1

4 0.5

2 0 1000

0 1500

2000

2500

0

20

40

60

80

100

Frequency

Time (ms)

Fig. 3. a: Jet length as a function of time. b: Result of Fast Fourier Transform. The mean jet length is 14.22 mm, and the droplet detachment frequency is 6.35 Hz. The corresponding peak is marked with an arrow in the FFT. Experiment no. 1 (Δp ¼ 1:0 bar, 1000 fps).

should be equal to the volume of the liquid column (which is the cross section multiplied by the jet velocity) within one wavelength. The wavelength is the distance between two consecutive swells and necks. These considerations lead to the following equation: dP ¼ 1:89 djet

ð1Þ

Eq. (1) yields 2.06 mm for a 1.09 mm jet. Teng et al. (1995) argued that the droplet size strongly depends on the jet and ambient fluid properties. They defined a modified Ohnesorge number 3μjet þ μamb n Oh ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρjet sdjet

ð2Þ

and gave an equation for the droplet size applicable to lowvelocity jets in gases or liquids:  1=3 dP 3π n ¼ ð1 þ Oh Þ1=6 pffiffiffi ð3Þ djet 2 n

Here, Oh ¼ 0:75, and consequently Eq. (3) predicts a droplet size of 2.25 mm) which is still lower than the size measured in experiment no. 1. Neither Eq. (1) nor Eq. (3) seems to be applicable for short jets at low flow rates near to the dripping regime. A larger diameter is predicted if only the balance between the two main forces present (gravitation and surface tension) is considered, as in Tate's law (Tate, 1864): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6sDcap dP ¼ ð4Þ gρ which yields 5.22 mm with a deviation of approx. 7%. However, this equation does not take into account the remaining liquid at the capillary tip which is mainly formed by the volume of the connecting liquid thread in Fig. 2. This reduces the volume of the detached droplet, and a correction factor – often called the Harkins and Brown factor – has to be applied, as commonly used in the drop weight method (Garandet et al., 1994; Harkins and Brown, 1919; Lee et al., 2008). Strictly speaking, the drop weight method only applies in the slow dripping regime, but since experiment no. 1 exhibited a regular dripping pattern – albeit with a long liquid thread with a diameter equal to the inner capillary diameter – the predicted droplet diameter using the correction factor shall be explored in the following. Lee et al. (2009) proposed a polynomial equation to calculate the Harkins and Brown correction factor based on the capillary radius to the cube root of the detached droplet volume ratio,

Fig. 4. Sequence of jet formation and breakup. Experiment no. 5 (Δp ¼ 1:4 bar, 4000 fps). Time difference between each image: 5 ms.

χ ¼ Rcap =V 1=3 P , the so-called LCP (Lee–Chan–Pogaku) model: Ψ ðχ Þ ¼ 1:000  0:9121ðχ Þ  2:109ðχ Þ2 þ 13:38ðχ Þ3  27:29ðχ Þ4 þ 27:53ðχ Þ5  13:58ðχ Þ6 þ 2:593ðχ Þ7

ð5Þ

Eq. (5) predicts a correction factor Ψ ¼ 0:86 which results in a droplet diameter dP ¼ Ψ  dP;Tate ¼ 4:48 mm, a deviation of approx. 8.2%. The relatively large density of the slag – resulting in low χ values o0:2 – can cause considerable scatter in the determination of the correction factor, as found for metals (Lee et al., 2008). Lee et al. (2008) give an overview of published equations for the correction factor. For example, Clift et al. (1978) give a polynomial equation for 0 r χ r 0:3:

Ψ ðχ Þ ¼ 1:000  0:66031ðχ Þ þ 0:33936ðχ Þ2

ð6Þ

Eq. (6) predicts a higher correction factor, Ψ ¼ 0:91, and therefore a larger droplet diameter: 4.85 mm, which is in agreement with the experimental value within 2.2%. The diameters predicted by Eqs. (4)– (6) are displayed in Fig. 10a, see also discussion in Section 3.3. After droplet detachment, a new jet formation cycle starts. The jet formation and droplet detachment is highly repetitive at this relatively low flow rate, as can be seen in Fig. 3a. It shows the unbroken jet length as a function of time in a 1500 ms time interval. The figure represents the dynamics of jet elongation and jet retraction after droplet detachment. Each cycle lasts approx. 157 ms and reflects the droplet production rate which is 6.35 droplets per second in this case. A Fast Fourier Transform (FFT) of the jet length has been carried out. The result is shown in Fig. 3b. The predominant peak with the largest magnitude is exactly at 6.35 Hz and is marked with an arrow. Fig. 4 shows the jet formation and droplet breakup for Δp ¼ 1:4 bar (experiment no. 5). The elapsed time between each image is again 5 ms. The jet is relatively stable and cylindrical for

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M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

the preponderant part of the jet. Since the flow rate is increased, the jet is significantly longer. Very rarely, wave instabilities can be detected in the cylindrical part. They occur near the detachment point, but in most cases not for more than one wavelength. The fourth image shows the moment of pinch-off. Again, the column retracts due to the unbalanced force of surface tension. This is why the unbroken jet length is not constant and is normally distributed around a mean value (28 mm), see also discussion in Section 3.2. This seems to be a typical behaviour of jet disintegration by natural disturbances in contrast to the steady state laminar Rayleigh jet regime in which the jet is excited with a frequency which corresponds to the fastest growing instability. In an ideal Rayleigh

jet, one droplet is generated from the volume of liquid present within one wavelength. In this case, the droplet formation frequency equals the frequency of the instability waves. The breakup position and hence the length of the jet do not change (Ashgriz and Yarin, 2011). This is referred to as the Rayleigh mode in the following. In most of the experiments in the present study, the jet disintegration shows one of the following main breakup scenarios (some examples are shown in Fig. 5): occasionally, the jet effectively breaks according to the Rayleigh mode (Fig. 5a). The volume of one wavelength forms one droplet. This is especially true for the jets with higher flow rates, see also Fig. 10d–f where the main peak coincides with Tyler's assumption (see also discussion in Section 3.3). However, from what can be observed in the recorded sequences, this does not happen more than for the duration of 2–3 consecutive droplet detachments. Before and after, instead, the droplet collects the liquid volume of 2–3 wavelengths, growing accordingly in diameter (Fig. 5a, b). During this time, the jet slowly moves downwards and the droplet formation frequency decreases dramatically compared to the Rayleigh mode. Another type of breakup mechanism is the formation of satellite droplets after intermediate droplet breakup, i.e. a longer portion of the jet suddenly detaches from the main jet and forms dumbbell-shaped objects which retract quickly to form spherical droplets of different sizes (Fig. 5c, d). A more detailed analysis of this phenomenon may be subject to future investigations. Fig. 6a shows the jet length for experiment no. 5 as a function of time for the whole sequence. It shows a more irregular detachment process compared to Fig. 3a. Accordingly, the FFT in Fig. 6b does not reveal an instructive pattern. As can be seen in Fig. 6a, the jet length is reduced occasionally to around 10 mm. This retraction also happened in the other experiments where – randomly – the jet breakup caused a sharp decrease of the liquid column length due to the high surface tension of the slag. Afterwards, the jet length recovered. Fresh liquid was pumped downwards and accumulated into the main droplet until the next droplet detached. The droplet size was considerably smaller and distributed over a wider droplet size range, see also Section 3.3.

3.2. Jet length Fig. 5. Examples of different breakup mechanisms of slag jets. a: Series of three consecutive droplets each formed by the liquid volume of one wavelength (exp. no. 4). Before and after, a larger droplet forms, shaped by the volume of multiple wavelengths. b: An instability wave pumps liquid into the forming droplet at the tip of the jet (exp. no. 4). c: Intermediate jet disintegration. Dumbbell-shaped liquid objects of different size evolve and retract due to surface tension (exp. no. 5). d: Formed small satellite from intermediate jet breakup and retracted main droplet (exp. no. 2).

Leroux et al. (1996) measured the length of a water jet under different conditions. Similar to our study, no additional excitation was applied. They found that the jet length is normally distributed around a mean value. Fig. 7 shows that the number frequency distribution of the jet length of slags also behaves in a Gaussian manner. The normalised frequency (the number of events in one size class divided by the total number of values) is plotted as a

45

3

40 2.5

30

Magnitude

Jet length (mm)

35

25 20 15

2 1.5 1

10 0.5

5 0

0 0

1000 2000 3000 4000 5000 6000 7000 Time (ms)

0

20

40

60

80

100

Frequency

Fig. 6. a: Jet length as a function of time. b: Result of Fast Fourier Transform. The mean jet length is 28.4 mm. Experiment no. 5 (Δp ¼ 1:4 bar, 4000 fps).

0.16

0.16

0.14

0.14 Normalised frequency

Normalised frequency

M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

0.12 0.1 0.08 0.06 0.04

0.12 0.1 0.08 0.06 0.04 0.02

0.02

0

0 0

5

10

15

20

25

30

35

40

45

0

50

5

10

15

0.16

0.16

0.14

0.14

Normalised frequency

Normalised frequency

20

25

30

35

40

45

50

35

40

45

50

Jet length (mm)

Jet length (mm)

0.12 0.1 0.08 0.06 0.04

0.12 0.1 0.08 0.06 0.04

0.02

0.02

0

0 0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

Jet length (mm)

0.16

0.16

0.14

0.14

0.12 0.1 0.08 0.06 0.04

20

25

30

Jet length (mm)

Normalised frequency

Normalised frequency

149

2 3 4 6

0.12 5 0.1 0.08 0.06 0.04 0.02

0.02

0

0 0

0

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 60 Jet length (mm)

Jet length (mm)

Fig. 7. Normalised jet length frequency distribution for the different operating conditions (experiments 2–6 in Table 2). The symbols represent the experimental results, the dotted curves the normal (Gaussian) distribution. The peak of the normal distribution was chosen to match the maximum value of the experimental distribution. The last figure, f, shows all normal distributions in one plot.

function of the jet diameter for the experiments 2–6. Experiment no. 1 is not included here due to the dripping/retracting behaviour which cannot be represented by a Gaussian distribution. The main results (median, standard, and relative standard deviation of the normal distribution, total number of values) are summarised in Table 3. The trend in the data is clear: the higher the pressure (or the higher the flow rate), the longer the jet. Note that in Fig. 7f, the maximum frequency value is equal (¼ 0.142) in experiments 2–4. This value decreases for experiments 5 and 6 since the distributions become slightly wider. All distributions have in common a slight shoulder at the left flank which represents the jet lengths during retraction. With regard to the distributions, their contribution is rather insignificant. For laminar Rayleigh jets, the unbroken jet length Lb can be approximated taking into account the breakup time according to the fastest

Table 3 Median, standard deviation (SD), relative standard deviation, and total number of considered values for the jet length frequency distribution in Fig. 7. Experiment no.

Median [mm]

SD [mm]

Rel. SD [%]

No. of values [–]

2 3 4 5 6

18.5 22.3 26.5 28.4 32.0

2.8 2.8 2.8 3.8 3.5

15.14 12.56 10.57 13.38 10.94

4605 3145 3145 3145 3145

growing instability wave and the jet velocity. When the amplitude of the perturbation equals the jet radius, the jet disrupts. In the literature, the length of a jet subject only to surface tension and inertia forces is often predicted by the following

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equation (Grant and Middleman, 1996; Weber, 1931)  qffiffiffiffiffiffiffiffiffiffiffiffi  Wejet Lb R0 ¼ ln Wejet þ 3 djet ε0 Rejet

ð7Þ

with the jet Weber number Wejet ¼ ρv2 djet =s and the jet Reynolds number Rejet ¼ ρvdjet =μ. It is worth noting that the ratio of We=Re is also termed the capillary number, Ca. The term lnðR0 =ε0 Þ has been derived from experiments and reflects the ratio of the initial jet radius to the initial disturbance amplitude. Haenlein (1931) found the value 12, Grant and Middleman (1996) reported an average value of 13.4 and gave the correlation lnðR0 =ε0 Þ ¼ 2:66 lnðOhÞ þ 7:68. In general, the logarithmic term lnðR0 =ε0 Þ should be used with care since it closely depends on nozzle behaviour and related parameters (Yim, 1996). Here, however, the expression by Grant and Middleman (1996) was applied to account for the higher viscosity of the slag. To calculate the Weber and Reynolds number, a velocity is required. Usually, this is the jet velocity at the capillary exit. However, the jet is accelerated by gravity, and therefore the jet velocity at the point of droplet detachment is different from the capillary exit velocity. Additionally, liquid accumulates in the main droplet until it breaks off. During this period, no material reaches the electronic balance although constantly fresh material is delivered, as can be seen from the waves at the jet surface moving with a certain velocity. Consequently, no weight change can be measured on the balance. This reduces the average velocity calculated by the mass flow rate, the liquid density and the capillary crosssection. The consequences are displayed in Table 4 and Fig. 8a. In the _ (see table, the volume flow rate Q derived from the mass flow rate M Table 2) gives the velocity at the capillary exit vcap, and accordingly Weber and Reynolds number. The deviation between the jet lengths calculated by Eq. (7) and the experimental data is given in the last column. With one exception, the agreement is poor, with an average Table 4 Comparison of experimental breakup length with values predicted by Eq. (7).

error of 31%. In most cases, Eq. (7) underpredicts all experimental values. Two possible reasons may explain the poor agreement. One has already mentioned in the last paragraph in which the usefulness of the jet velocity at capillary exit is questioned. Another reason could be that the balance may not have measured properly. This can happen if there is only the slightest contact between the vertical alumina tube and the stainless steel cup which collects the slag which may occur e.g. if slag hits the rim of the cup and sticks between cup and alumina wall. The likelihood of a balance measurement error may be underpinned by the fact that the mass flow rate decreases between experiments 5 and 6, see Table 2, although the pressure increases. However, the usefulness of Eq. (7) increases dramatically if the instantaneous velocity of a travelling instability wave near the jet disintegration point is used. This circumvents the necessity to rely solely on the balance data. Thereto, all sequences were screened for travelling waves which occur only randomly. Once a suitable crest of a wave with a measurable amplitude was detected, the displacement with respect to time was measured manually, see Fig. 9. The measurement was repeated for several (but different) waves, and the values averaged. The corresponding mean velocities are listed in Table 5. One can see immediately that the wave velocities are considerably higher, and so are Reynolds and Weber number. Note also that the velocity increases between experiment nos. 5 and 6, which was not the case when the balance data were used (see Table 4). The improvement in the prediction of the breakup length is striking, as can be seen in the last column in Table 5 and Fig. 8b. The mean error reduces to below 15%. It seems that the wave velocity – although it may be different from the average jet velocity at the same spatial position – is a quite valid and reliable approximation which helps to predict the breakup length of highly viscous slag jets at high temperatures. It is interesting to note that the onset of jetting cannot be described by the critical Weber number as defined by Ranz in Lin and Reitz (1998)

2

The jet velocity at capillary exit was calculated with vcap ¼ 4Q =πdjet . Exp. no.

Δp [bar]

Q vcap [mL s  1] [m s  1]

We [–]

Re [–]

Lb (Eq. (7)) [mm]

Lb (exp.) [mm]

Error [%]

1 2 3 4 5 6

1.0 1.1 1.2 1.3 1.4 1.5

0.02 0.28 0.30 0.36 0.55 0.53

0.001 0.45 0.54 0.78 1.76 1.62

0.16 2.81 3.07 3.70 5.54 5.32

0.82 14.44 15.77 19.01 28.51 27.37

14.22 18.5 22.3 26.4 28.4 32.0

94.25 21.92 29.26 28.01 0.38 14.48

ρdjet v2min s

48

ð8Þ

where vmin is the minimum velocity required to form a jet. The Weber numbers in Table 5 are considerably lower, but jets were formed at least in experiments 2–6. More useful is the critical Weber number which divides the region of absolute instability from the region where jets are convectively unstable. This dividing line is comparable with the transition from the dripping to the jetting regime (Amini et al., 2013), or in other words the ability

35

40

30

35

25

Jet length (mm)

Jet length (mm)

0.02 0.30 0.32 0.39 0.59 0.56

Wecr ¼

20 15 10

30 25 20 15 10 Re based on waves

Re based on balance

5

5

Predicted (Eq.7) Experiments

0 0

2

4

Predicted(Eq.7) Experiments

0 6

0

2

4

6

8

Fig. 8. Experimental and predicted jet length (Eq. (7)) as a function of Reynolds number. a: Reynolds and Weber number calculated with the capillary exit velocity vcap based on the mass flow rate measured by the balance. b: Reynolds and Weber number calculated with the average velocity of an instability wave, vwave, near the jet disintegration point.

M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

Fig. 9. Propagating front of an instability wave. The crest of this wave has been chosen to measure its displacement vs. time (arrows). At least 5 different instability wave events were measured for each sequence, and the calculated velocities averaged. The mean values are listed in Table 5. Time difference between each image: 0.5 ms.

Table 5 Comparison of experimental breakup length with values predicted by Eq. (7). In contrast to Table 4, the Reynolds and Weber number was calculated with the average velocity of an instability wave, vwave, near the jet disintegration point. Exp. no.

Δp [bar]

vwave [m s  1]

We [–]

Re [–]

Lb (Eq. (7)) [mm]

Lb (exp.) [mm]

Error [%]

1 2 3 4 5 6

1.0 1.1 1.2 1.3 1.4 1.5

0.27 0.42 0.59 0.69 0.71 0.73

0.37 0.90 1.78 2.44 2.58 2.73

6.04 9.39 13.19 15.43 15.88 16.32

13.13 20.42 28.69 33.55 34.53 35.50

14.22 18.5 22.3 26.4 28.4 32.0

8.31 9.42 22.27 21.32 17.74 9.85

to form a jet (Lin and Reitz, 1998). This critical Weber number depends on the Reynolds number and the density ratio (Leib and Goldstein, 1986) and decreases sharply for low Reynolds numbers. If the Weber numbers from Table 5 are compared to Fig. 3 presented in the review paper by Lin and Reitz (1998), most of the experiments can be located in the region of the convective instability or jetting regime. Absolute instability is characterised by the fact that part of the unstable disturbances will propagate back to the capillary tip to interrupt the formation of a jet (Lin and Reitz, 1998). As could be seen in Fig. 2, experiment no. 1 represents such a case: a short jet has formed, but a longer jet cannot form due to the action of surface tension which forces the jet to break up and retract back to the capillary. 3.3. Drop size distribution For most applications, the drop size distribution is of major importance. The narrower the distribution, the better controllable is a process (e.g. in terms of heat and mass transfer) and the more reliably the quality of a product can be adjusted. The design (and therewith the dimensions and costs) of a liquid droplet heat exchanger (LDHX) closely depends on the predictability and shape of the drop size distribution. Fig. 10 shows the drop size distribution for all 6 cases listed in Table 2. The frequency of droplets belonging to a size class was divided by the total number of measured droplets which gives the normalised frequency distribution. The total number of droplets counted in each case is also given in each figure, preceded by a #. The vertical dotted lines represent the theoretical predictions, Eqs. (1) and (4), respectively. Their relative position is identical in each figure, since jet diameter and physical properties are assumed constant (the constant jet diameter at capillary exit for all back-pressures was verified by image analysis). For experiment no. 1 (Δp ¼ 1:0 bar), as already discussed, the drop size distribution is quite narrow. No satellite droplets were observed, whereas in all other cases a small fraction of very small satellite droplets at around 0.7 mm was found.

151

The droplet size for experiment no. 1 is slightly larger than predicted by Eq. (6) and considerably larger than the droplets of all other experiments. The Weber number is relatively small, hence the surface tension force is relatively large compared to the inertia force. After the detachment of a droplet, the strong unbalanced surface tension force retracts the jet. During this retraction in which the instabilities are mainly transported upstream, fresh liquid is pumped steadily through the capillary and already forms the next droplet during retraction. The Ohnesorge number for the slag used here is 0.24, hence 65 times higher than for water. This means that in the same time the stabilising viscous forces are more dominant which helps to produce larger droplets. As the flow rate is increased, two things become obvious. First, the distribution widens, and second the main peak shifts to smaller droplet sizes. In experiment no. 2, this main peak is already at around 4 mm and shifts in the later cases towards Eq. (1). Additionally, from experiment no. 2 onwards, a peak which matches exactly with the vertical line described by Eq. (1) appears. This peak is small first, but becomes more and more dominant with increasing flow rate until it becomes the dominant peak in the last two cases (Δp ¼ 1:4 and 1.5 bar). During this peak transition, most of the droplet sizes are located in between the two vertical lines. Higher pressures result in higher flow rates. The Weber and the Reynolds number increase and inertia forces become more dominant. Viscosity now helps to generate a relatively long jet, whereas surface tension is the source of instability. Retraction still takes place, but most likely more instabilities are convected downstream with increasing flow rate. The interplay of the competing forces (capillary, viscous, inertia, and gravity forces) leads now to smaller droplets which detach from the disintegrating jet according to Tyler's assumption.

4. Conclusion The capillary breakup of synthetic calcia/alumina slag jets at high temperatures (1600 1C) was investigated in terms of jet formation, jet length, and droplet formation with an attempt at describing the dynamic phenomena in unstable molten oxide jets in a quantitative manner. The transition from the dripping to the jetting mode was identified. For very small flow rates, the jets were rather short, and the droplet formation was found to be highly repetitive exhibiting a characteristic jet retraction pattern due to the high surface tension of the slag. The critical Weber number as proposed by Ranz in Lin and Reitz (1998) did not apply in the slag system, whereas the critical Weber number proposed by Leib and Goldstein (1986) did indeed reflect the onset of jetting in the present study. No external excitation was applied to the jet, hence natural breakup occurred. Consequently, the jet length was unsteady. Similar to findings published by Leroux et al. (1996) it was found that a normal distribution describes the length of slag jets in an appropriate manner. The peak of each length distribution could be predicted by Eq. (7) (Grant and Middleman, 1996; Weber, 1931) within 15% accuracy if the velocity of the travelling instability wave was used to calculate the Reynolds and Weber numbers. The size of the formed droplets was determined and analysed. At the lowest flow rate, the drop size distribution was astonishingly narrow although only in natural breakup mode. The size was found to be nearly 5 times larger than the jet diameter. The distribution became wider and multimodal, with a main peak shifting towards smaller droplet sizes with increasing flow rate. For higher flow rates, very good agreement with Eq. (1) (Tyler, 1933) was observed.

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Fig. 10. Normalised size frequency distribution of droplets formed after disintegration of slag jets subject to different driving pressure differentials. a: 1.0 bar, b: 1.1 bar, c: 1.2 bar, d: 1.3 bar, e: 1.4 bar, f: 1.5 bar. The vertical dotted lines represent sizes predicted with Eqs. (1) and (4) etc. The number following # gives the total number of droplets counted in each case.

In future works, the impact of an external excitation on the instability of jets of molten oxides at high temperatures will be investigated.

Nomenclature Latin djet dP Dcap g ID L Lb

letters jet diameter, m droplet diameter, m diameter of capillary, m gravity, m2 s  1 inner diameter, m length, m jet breakup length, m

_ mass flow rate, kg s  1 M OD outer diameter, m p pressure, Pa pO2 oxygen partial pressure, Pa Q volume flow rate, m3 s  1 Rcap radius of capillary, m R0 initial jet radius, m SD standard deviation t time, s UHP ultra high purity v velocity, m s  1 V volume, m3 Greek letters Δp pressure difference, Pa ε0 initial disturbance, m

M. Wegener et al. / Chemical Engineering Science 105 (2014) 143–154

μ ρ χ

dynamic viscosity, Pa s density, kg m  3 surface tension, N m  1 ratio of capillary radius to cube root of detached droplet

Ψ

volume, χ ¼ Rcap V P Harkins and Brown correction factor

s

 1=3

Subscripts 0 initial, at t¼0 amb ambient phase b breakup cap capillary cr critical g gas jet jet min minimum P droplet w water Dimensionless numbers Ca Capillary number, Ca ¼ μvs  1 ¼ WeRe  1 1 M Mach number, M ¼ vjet vsound pffiffiffiffiffiffiffi Oh Ohnesorge number, Oh ¼ WeRe  1 ¼ μðρsdjet Þ  1=2 Ohn modified Ohnesorge number, Eq. (2) Re Reynolds number, Re ¼ ρvdjet μ  1 We Weber number, We ¼ ρv2 djet s  1

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