Modeling atomization of a round water jet by a high-speed annular air jet based on the self-similarity of droplet breakup

chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

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Modeling atomization of a round water jet by a high-speed annular air jet based on the self-similarity of droplet breakup De-Jun Jiang, Hai-Feng Liu ∗ , Wei-Feng Li, Jian-Liang Xu, Fu-Chen Wang, Xin Gong Key Laboratory of Coal Gasification, Ministry of Education, East China University of Science and Technology, P.O. Box 272, Shanghai 200237, People’s Republic of China

a b s t r a c t Based on the self-similarity of droplet breakup in the secondary atomization region, the atomization process of coaxial air-blast atomizer has been investigated. The relationship of Sauter mean diameter (SMD) with the effects of gas jet velocity, liquid jet velocity and diameter and liquid/gas mass flux ratio was obtained according to the breakup time and motion characteristic of droplet in air stream. The four parameters in the relationship of SMD were estimated according to the experimental results of nine coaxial two-fluid air-blast atomizers with air and water. The results showed that the obtained relationship of SMD can be used to predict the SMD of coaxial air-blast atomizer when m > 13.99d−0.1285 d−0.9415 . g l © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Atomization; Air-blast; Coaxial; Droplet; Breakup; Self-similarity

1.

Introduction

Gasification of liquid fuels in entrained flow gasifiers, and other industrial applications depends on effective atomization of liquid fuel to achieve high rates of mixing and evaporation. Disintegration of liquid jet injected into high-speed gas stream has been studied by many researchers. It is fundamentally different from that which occurs for the same liquid jet discharging into a stagnant gas. For cases in which the momentum flux of the gas stream is of the same order, or exceeds that of the liquid jet, the breakup and atomization is caused by the kinetic energy transfer from the gas to the liquid. This type of atomization is generally referred to as air-blast atomization (Lefebvre, 1989). In the case of interest here, where a low-speed liquid jet is injected at the central axis of a high-speed coaxial air jet, detailed reviews of earlier work have been published by Lefebvre (1989), and more recently by Lin and Reitz (1998), Lasheras and Hopfinger (2000), Sirignano and Mehring (2000), and Babinsky and Sojka (2002). For combustion applications, many empirical correlations are available for the droplet size as a function of injection parameters (Lefebvre, 1989).

∗

The liquid atomization of coaxial air-blast atomization can be divided into a near-field primary breakup region, and a far-field secondary breakup region (Lasheras et al., 1998). The primary breakup, which is dominant in the first few jet diameters, is essentially related to the non-miscible shear instability, and results in the stripping of the liquid jet by the high shear force at the gas/liquid interface. Further downstream, droplet atomization may also occur from the deformation forces exerted on the droplets by the turbulent motion of surrounding air, a process known as secondary atomization. Recently, Varga et al. (2003), Marmottant and Villermaux (2004) and Villermaux et al. (2004) observed and analyzed the successive steps of atomization of a liquid jet when a fast gas stream blows parallel to its surface. Experiments performed with various liquids in a fast air flow showed that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid and a Rayleigh–Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets. Eroglu et al. (1991) measured the breakup length L of a round liquid jet in an annular coaxial air stream and found

Corresponding author. Tel.: +86 21 64251418. E-mail address: hfl[email protected] (H.-F. Liu). Received 26 January 2011; Accepted 10 July 2011 0263-8762/$ – see front matter © 2011 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2011.07.006

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

that the breakup length decreases with increasing Weber number We and increases with increasing liquid jet Reynolds number Re according to the relationship L/dl ∝ We−0.4 Re0.6 . Engelbert et al. (1995) normalized the breakup length by the annular width of the coaxial atomizer and found that the breakup length is a function of the gas/liquid momentum ratio M or gas/liquid energy ratio E. Mayer and Branam (2004) investigated the undisturbed jet length l of this type atomizer, which is the length from the injector plane until the first visible surface disturbances are noticeable. They suggested that the undisturbed jet length is a function of internal jet turbulence, aerodynamic forces and the expected recirculation zone formed directly behind the end plane of central channel between the liquid core flow and the coaxial gas stream. The relationship for the undisturbed jet length is 0.18 l/dl = 1.7(g /l ) ln(Re/We0.5 ) − 0.16. The first major work of the droplet diameter of coaxial airblast atomization was an experimental study by Nukiyama and Tanasawa (1939) who obtained an expression for the  3  2 di / di ) as a function Sauter mean diameter (SMD = of the injection parameters. In particular, they obtained the dependence of the SMD on the atomizing gas velocity, ug . Since these early experiments, many other investigations have led to a plethora of empirical expressions for the mean droplet diameters in coaxial jet sprays (Loranzetto and Lefebvre, 1977; Rizk and Lefebvre, 1984). A summary of a fair number of these correlations for air-blast atomization has been compiled by Lefebvre (1989). The dependence of SMD on the atomizing gas velocity is most often expressed in the form of a power law, SMD ∝ ung , where 0.7 ≤ n ≤ 1.5. Physical explanations for particular values of the exponent n were generally lacking (Lasheras et al., 1998). Varga et al. (2003) obtained the value of n = 1.25 predicted by a phenomenological Rayleigh–Taylor instability model. There is still inconsistency about the dependence of SMD on the liquid jet diameter dl , Loranzetto and Lefebvre (1977) observed, in agreement with Nukiyama and Tanasawa (1939), that dl appears to have little influence on SMD for liquid of low viscosity. Rizk and Lefebvre (1984) found for liquid of low viscosity. But Varga et al. that SMD ∝ d0.6 l (2003) observed that SMD is slightly larger for the case of the smaller liquid nozzle diameter that was reduced by a factor of approximately three, which is actually opposite to intuition. The experimental results of Liu et al. (2006b) showed that the liquid jet diameter has an obvious effect on SMD for a large liquid/gas mass flux ratio m, but has no effect for small m. For a fixed large m, a decrease of SMD with liquid jet diameter increase was observed first, followed by an increase. Therefore, it is necessary to investigate the fundamental physical mechanisms responsible for the dependence of SMD on ug , ul and dl , which are the foundations of atomizer design for reliable atomization performance. Kolmogorov (1941) wrote a stochastic theory for the breakup of solid particles that describes the cascade of uncorrelated breakage events. This theory presents the breakup of solid particles as a random discrete process where the probability of breaking each mother particle into a given number of parts is independent of the size of the mother particle. Kolmogorov’s scenario means that there is self-similarity of solid particles breakup process, which implies the existence of fractal structure (Mandelbrot, 1982). The fractal structure of atomizing spray was observed by Shavit and Chigier (1995), Zhou et al. (2001) and Zhou and Yu (2001). Apte et al. (2003) developed a stochastic subgrid model for large-eddy simulation of coaxial air-blast atomization based on Kolmogorov’s

scenario. Gorokhovski and Saveliev (2003) investigated the breakup of liquid droplet of coaxial air-blast atomization at large Weber number within the framework of Kolmogorov’s scenario of breakup. Liu et al. (2006a) proposed the finite stochastic breakup model (FSBM) of air-blast atomizers for secondary atomization, according to the self-similarity of droplet breakup. In FSBM, the Kolmogorov’s scenario was applied to the breakup process of a liquid droplet in the range from its initial size down to the size of the maximum stable droplet. In this paper, we investigated the atomization process of a round water jet by a high-speed annular air jet based on the self-similarity of the droplet breakup process. The relationships between SMD and gas jet velocity, liquid jet velocity and diameter, and liquid/gas mass flux ratio have developed theoretically, and the parameters in the relationship were determined experimentally using a Malvern Laser Particle Sizer. This paper is organized as follows. After this introduction, we propose the relationships of SMD based on the self-similarity of droplet breakup process in Section 2. In Section 3 we present the experimental set-up and conditions, and in Section 4 estimate the parameters in the relationship with experimental results. Finally, we give a conclusion in Section 5.

2.

Analysis

The liquid atomization of coaxial air-blast atomization can be divided into a near-field primary breakup region, and a farfield secondary atomization region. The SMD of an atomizer is determined by these two processes together. There are several mechanisms of droplet breakup determined by the initial Weber number (Pilch and Erdman, 1997; Chou et al., 1997; Gelfand, 1996; Stone, 1994; Berthoumieu et al., 1999; Lee and Reitz, 2000; Cao et al., 2007; Zhao et al., 2010), such as vibrational breakup, bag breakup, bag-and-stamen breakup and sheet striping. A new breakup mechanism of liquid droplet identified in a continuous and uniform air jet flow, dual-bag breakup, was observed recently by Cao et al. (2007), which also reflects the self-similarity of the droplet breakup process. Therefore, we assumed that the secondary atomization process of coaxial air-blast atomizer is self-similar. The SMD of jth (j is 1, 2, 3, . . .) and (j + 1)th generation droplets of the secondary atomization process are

Nj 3 d i=1 ji , SMDj = N 2 j

(1)

d i=1 ji

Nj+1 SMDj+1 =

i=1 Nj+1 i=1

d3(j+1)i d2(j+1)i

,

(2)

where Nj and Nj+1 are the numbers of jth and (j + 1)th generation droplets, dji and d(j+1)i are the diameters of every jth and (j + 1)th generation droplets, respectively. Obviously, when SMDj  DC , where DC is the maximum stable droplet size (Lefebvre, 1989), according to the self-similarity hypothesis of the droplet breakup process, there is



SMDj+1 SMDj

 ≈ const,

(3)

chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

where • is the average of •. (3) means that {SMDj } is a geometric progression statistically, and it yields SMDj ≈ SMD0 Cj ,

(4)

where C is a positive constant and smaller than 1.0, and SMD0 is the mean droplet size after the primary breakup process. Breakup of a droplet occurs if the breakup time is reached. A dimensionless time characteristic of droplet breakup is given by T=t

ur ε0.5 , D

(5)

where ε = g /l , g and l are the density of gas and liquid, and ur is the relative velocity. The formulation used for this delay was given by Nigmatulin (1991) (it was also used by Berthoumieu et al. (1999) as −0.37

T = 6(1 + 1.2 Lp

−0.25

)(Log We)

where u g = ug /(1 + m), which is about the mean velocity of air jet and the liquid jet after momentum interchange if the flux of surrounding air entrained by the air jet is neglected, and is the characteristic velocity of coaxial jets (Beer and Chigier, 1974). v = v0 /f (Re), v0 = l D2 /18g , g is the gas viscosity, and f (Re) = 1 +

1 2/3 Re , 6

l and l are where Lp is Laplace number Lp = the liquid viscosity and surface tension respectively. For low-viscosity liquid such as water, when D > 10 ␮m (which is the usual droplet size region in air-blast atomization), 1.2 Lp−0.37 ≤ 0.1. So the effect of Lp in (6) can be neglected for low-viscosity liquid. Usually, We ∈ [10, 1000] for air-blast atom−0.25 ization, then (Log We) ∈ [1, 0.760], which means that the effect of We on T is less than 25% and can be neglected for convenience. Therefore, for water and air-blast atomization, it yields

where the Reynolds number of droplet is Re = Dur g /g . If we use the average of v , v , and let  = t/v , it yields ur = ua − ud ≈ ug exp[−(1 + m)].

(7)

Thus, with (5), the average breakup time of droplet of jth generation is SMDj ur ε0.5

∝

SMD0 j C. ur

(8)

We now proceed to analyze the motion of airflow and liquid droplets. It is assumed that (1) The breakup of a droplet mainly takes place near the atomizer orifice, where the momentum interchange between air and liquid mainly takes place too. Thus the flux of surrounding air entrained by the air jet can be neglected. (2) Among the forces forced by airflow upon the droplets, such as drag force, gravity force, Basset force, Magnus force and Saffman force (Liu, 1993), only drag force is considered (Apte et al., 2003).

ur  = ua − ud  ≈ u g

1 − exp[−(1 + m)] ∝ u g . 

(13)

Substituting Eq. (13) into Eq. (8) yields the average breakup time of droplet of jth generation C1 SMD0 j C, u g

tj ≈

Based on conservation of momentum, because the initial velocity and momentum of the liquid jet are much smaller than that of the air jet, we have

tTN =

N 

C1 SMD0  j C1 C SMD0 1 − CN C =

ug u g 1−C N

tj ≈

i=1

C2 = (SMD0 − SMDN ), ug

(15)

where C1 and C2 are constants. The breakup of droplet mainly takes place in the breakup region, which is near the atomizer orifice. In the breakup region, the velocity and turbulence intensity of air are much higher than in any other region in the whole air-droplet twophase flow field. When the total breakup time tTN is equivalent to the droplet resident time  in the breakup region, the breakup process of droplet generally finishes. In the breakup region, the Reynolds number of droplet is much larger than 1, thus Eq. (11) can be simplified as f (Re) ≈

1 2/3 2/3 Re ∝ (u g ) , 6

(16)

and −2/3

(17)

.

Thus, the average acceleration of droplet can be estimated as

 du  d

(9)

(14)

which means that {ti } is also a geometric progression. Therefore, we obtain the total breakup time from j = 1 to N, tTN , as

v ∝ (u g )

ug ≈ ua + mud ,

(12)

Thus, the average relative velocity with respect to time can be estimated as

j=1

tj ∝ T

(11)

(6)

, l l D/2l ,

T ≈ const.

187

dt

=

u  r

v

∝ (u g )

5/3

.

(18)

The velocity of droplet can be estimated as

where m is liquid/gas mass flux ratio, ua and ud are the velocities of air flow and droplets, respectively. The equation of droplets motion is

ud ≈ ul + C3 (u g )

dud 1+m

ua − ud ≈ (ug − ud ), = dt v v

where C3 is a constant. The relative velocity is too small at the end of the breakup region to complete the breakup of a

(10)

5/3

t,

(19)

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

droplet, which means that the velocity of droplet is close to u g , so we can obtain ul + C3 (u g )

5/3

 ≈ C4 u g ,

(20)

where C4 is a constant and of the order of 1. Thus, C4 u g − ul

≈

C3 (u g )

5/3

(21)

.

The existence of the undisturbed length of the liquid jet means that (Mayer and Branam, 2004), from the separate state to the interactional state of the liquid jet and air jet, some time is required, which is defined as the contact time d . The contact time should be deleted from the residence time because it makes no contribution to the breakup process of a droplet. The contact time may be estimated as d ∝

l , u g

(22)

where l is the undisturbed length of the liquid jet. The study of Mayer and Branam (2004) showed that



 0.18

l = 1.7

g

l

ln

Re

We0.5

Fig. 1 – Coaxial air-blast atomizer configuration.

 − 0.16 dl .

(23)

0.18

Because [1.7(g /l ) ln(Re/We0.5 ) − 0.16] does not vary much, which is about 2–4 (Mayer and Branam, 2004), we have

where a1 , a2 , a3 , and a4 are constants. From the above analysis, we know that a1 is of the order of 1 mm, and a3 is of 1. It is obvious that a4 dl a2 ((ug /(1 + m) − a3 ul )/(ug /(1 + m))

d ∝

dl . u g

(24)

2/3

)

< 1.

There is a relation among tTN ,  and d when the breakup process of droplet completed, which can be expressed as

According to Eq. (28), we can find that the liquid jet velocity and liquid jet diameter have reciprocal effect on SMD. Therefore, there may be a non-monotonic trend between SMD and liquid jet diameter, at the same m and ug .

tTN ≈  − d .

3.

(25)

Substituting (15), (21) and (24) into (25) yields C4 u g − ul C2 d (SMD0 − SMDN ) ≈ − C5 l ,

5/3 ug ug C3 (u g )

(26)

where C5 is constant. Thus SMDN ≈ SMD0 −

C4 u g − ul C2 C3 (u g )

2/3

+

C5 d. C2 l

(27)

Note that the above relationships are obtained when SMDN is much larger than the maximum stable droplet size. It means that (27) is useful only when m > mmin . There still is a variable that should be determined in above equation, which is SMD0 , the mean droplet size after the primary breakup process. Villermaux et al. (2004) found that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid and a Rayleigh–Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets. At breakup, the ligament length does not vary much, and is of the order of 1 mm. Here we used the ligament length as SMD0 , which means that SMD0 is a constant. Therefore SMD = a1 − a2

ug /(1 + m) − a3 ul (ug /(1 + m))

2/3

+ a4 dl ,

(28)

Experimental set-up and conditions

The coaxial air-blast atomizer geometry is shown schematically in Fig. 1, consisting of the geometrically simple case of a round liquid jet surrounded by a co-flowing annular air stream. This fundamentally simple geometry provides the well-known nozzle exit conditions and avoids the complicated internal flows that are common to most practical atomizers. The atomizer used in the current experiments was a modular design, consisting of an upper and lower block structure. The lower block section includes a liquid injector and an air nozzle, which can be exchanged to incorporate several atomizer configurations with different diameters. The experiments were conducted at atmospheric pressure, and water and air were used as working fluids. In this configuration, a central liquid jet of diameter dl , and velocity ul is atomized by a high-speed co-flowing annular gas stream with velocity ug and diameter dg . The experimental apparatus is sketched in Fig. 2. A Malvern Laser Particle Sizer of Type 3600 was utilized to measure the droplet size. The focal length of the selected lens was 300 mm. The measurement was carried out at the horizontal plane at a distance of 680 mm away from the orifice of the atomizers, where the atomization process is almost accomplished. The measurement was repeated 20 times at the every operating condition and the arithmetic mean was used to eliminate the influence of randomness in the atomization process.

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

Atomizer Signal detector

Laser Emission Flowmeter Suction

Air Liquid Seperator

Flowmeter Air Compressor Signal Processor

Liquid Tank

Pump

Fig. 2 – Schematic of the experimental apparatus.

Table 1 – Dimensions of atomizers. dl (mm)

Dl (mm)

dg (mm)

Ag (mm2 )

Agl

1 2 3 4 5 6 7 8 9

16.96 9.02 5.00 3.10 2.00 9.02 9.02 5.00 3.10

19.06 11.06 7.06 5.12 5.12 11.06 11.06 7.06 5.12

26.08 20.93 19.10 18.50 18.50 19.10 16.00 16.00 16.00

248.9 248.0 247.4 248.2 248.2 190.4 105.0 161.9 180.5

1.10 3.88 12.6 32.9 79.0 2.98 1.64 8.25 23.9

200 150

SMD, μm

No.

Atomizer 1 50

(29) 0

0

4

8

12

m 200

SMD, μm

200

Calculative value of SMD / µm

100

150

100

150 100 Atomizer 6 50

(29)

0 50

0

5

10

15

m 0 0

50

100

150

200

200

SMD, μm

Experimental value of SMD / µm Fig. 3 – Correlation accuracy plot. There were nine kinds of atomizers used in the experiments, and their dimensions are indicated in Table 1. Atomizers 1–5 had the same gas nozzle cross-sectional area Ag = (/4)(d2g − d2l0 ) with a maximum relative error 0.6%. The liquid jet diameter was varied from 2.00 mm to 16.96 mm.

150 100 Atomizer 8 50

(29)

0 0

2

4

6

8

m

SMD, μm

Calculative value of mmin

150 1 0.8 0.6

100

Atomizer 9

50

0.4

(29) 0.2

0 0

0 0

0.2

0.4

0.6

0.8

1

mmin Fig. 4 – The value of mmin and its calculative value with (30).

1

2

3

4

m Fig. 5 – Fitting results of atomizer 1, 6, 8, and 9. The air velocity is 170 m/s.

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chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

200

200 m=3.84

150

SMD/µm

SMD/µm

m=5.48

100 Experiment (29) (31)

50

150 100 Experiment (29) (31)

50

0

0 0

5

10

15

20

0

5

dl/mm

m=2.19

100

SMD/µm

SMD/µm

20

150 m=2.74

Experiment (29) (31)

50

100

Experiment (29) (31)

50

0

0 0

5

10

15

0

20

5

dl/mm

10

15

20

dl/mm

120

100

SMD/µm

m=1.64

100

SMD/µm

15

dl/mm

150

80 60 Experiment (29) (31)

40 20

m=1.37

80 60 40

Experiment (29) (31)

20

0

0 0

5

10

15

20

0

5

dl/mm

10

15

20

dl/mm

80

80 m=1.10

m=0.82

60 40

SMD/µm

SMD/µm

10

Experiment (29) (31)

20

60 40 Experiment (29) (31)

20

0

0 0

5

10

15

20

0

5

dl/mm

10

15

20

dl/mm

Fig. 6 – The effect of liquid jet diameter on SMD and the comparison of (29) and (31).

Then the investigation of the effect of liquid jet diameter on SMD was carried out with a fixed area of gas nozzle cross section using atomizers 1–5. The gas/liquid nozzle cross-sectional area ratio Agl = (d2g − d2l0 )/d2l was varied from approximately 1.10 to 79.0. The liquid/gas mass flux ratio was varied from 0.137 to 15.6, which is usually less than 1.0 in the former investigations (Lefebvre, 1989; Varga et al., 2003; Nukiyama and Tanasawa, 1939; Loranzetto and Lefebvre, 1977; Rizk and Lefebvre, 1984). The liquid jet velocity ul was varied from 0.012 m/s to 26.5 m/s, and the air jet velocity was varied from 76 m/s to 192 m/s. The calculated liquid/gas mass flux ratio was in the region 2 of 0.137–15.6, the Weber number We = g dl (ug − ul ) /l was in the region of 170–8080, the dynamic pressure ratio M0 = g u2g /l u2l was in the region of 0.03–36,550, and the liquid jet Reynolds number Re = l dl ul /l was in the region of 520–70,740, respectively. According to the experimental conditions and the breakup regime diagram (Lasheras and Hopfinger, 2000), the atomization regimes should belong to membrane breakup, fiber type atomization, and fiber type and recirculating gas cavity.

4.

Results

A detailed description of experimental data has been elucidated by Liu et al. (2006b). The experimental results showed that liquid jet diameter has a clear effect on SMD for large m, but has no effect for small m. A general non-monotonic trend observed clearly is a decrease first and then an increase of SMD with liquid jet diameter for a fixed large m. A similar nonmonotonic trend is also observed between SMD and liquid jet velocity, We, Re, and M at the same m, respectively.

4.1.

Correlation

The parameters in (28) were estimated based on the experimental data using nonlinear least squares fitting. We obtained

SMD = 233.2 − 44.34

ug /(1 + m) − 0.6054ul (ug /(1 + m))

2/3

+ 1.503dl ,

(29)

chemical engineering research and design 9 0 ( 2 0 1 2 ) 185–192

where ug and ul are in meters per second, dl in millimeter and SMD in micrometers. The best applied ranges of (29) are both m > mmin , where d−0.9415 . mmin = 13.99d−0.1285 g l

191

(2010CB227005), PetroChina Innovation Foundation (2009D5006-04-05), and New Century Excellent Talents in University (NCET-08-0775) by Ministry of Education of China.

(30)

References Eq. (29) is accurate within 4.11% (mean of relative error), and their maximum of relative errors is 16%. The values of SMD and the relative error calculated with Eq. (29) are plotted against the measured values in Fig. 3. The value of mmin and its calculative value with Eq. (30) are shown in Fig. 4. In Eq. (29), it can be found that the orders of a1 and a3 agree well with the predicted results in Section 2. At the same time, according to our experimental results, we found that 1.503dl 44.34((ug /(1 + m) − 0.6054ul )/(ug /(1 + m))

2/3

)

< 0.25,

which is also consistent with the analysis in Section 2. When m < mmin , the fitted results are inaccurate, which agrees with the analysis in Section 2. This is illustrated in Fig. 5, as an example, which shows the results of atomizer 1, 6, 8, and 9, when the air velocity is 170 m/s.

4.2.

Effect of liquid jet diameter on SMD

The liquid jet diameter has a clear effect on SMD, especially for the case of large m. Liu et al. (2006a) found that, for a fixed large m, a decrease of SMD with liquid jet diameter increase was observed first, followed by an increase. They have proposed a correlation with five parameters to describe the effect of the liquid jet diameter on SMD SMD = 685.8(ug − 3.297ul )

−0.4813

m0.3665 + 0.1824dl m.

(31)

Fig. 6 is the comparison of (29) and (31) in the aspect of describing the effect of the liquid jet diameter on SMD (atomizer 1–5), where the air jet velocity is about 170 m/s, and the liquid/gas mass flux ratios are varied from 0.82 to 5.48. In the most cases, it seems that Eq. (29) with four parameters predicts SMD much more accurately than Eq. (31) with five parameters.

5.

Conclusions

The atomization process of a round water jet by a high-speed annular air jet has been modeled based on the self-similarity of droplet breakup process. The relationship of SMD with the effects of air jet velocity, liquid jet velocity and diameter, and liquid/gas mass flux ratio were developed, and the parameters in the relationship were determined experimentally. The obtained correlations with four parameters were SMD = 233.2 − 44.34

ug /(1 + m) − 0.6054ul (ug /(1 + m))

2/3

+ 1.503dl ,

where ug and ul are in meters per second, dl in millimeter and SMD in micrometers. The two correlation is accurate within d−0.9415 . The correlation will be 4.11% when m > 13.99d−0.1285 g l useful for people to predict SMD more accurately.

Acknowledgements This study was supported by the National Natural Science Foundation of China (Grant No. 50776033), National Development Programming of Key Fundamental Researches of China

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