**2. Dimensionless number**

The physical processes and fluid properties are important in atomization morphology and performance, and the mathematical and numerical analysis of atomization is very challenging, so a number of dimensionless numbers are used. First of all, the most important one is Weber number, which represents the ratio of disruptive hydrodynamic forces to the stabilizing surface tension force,

$$We = \frac{\rho\_{\rm g} \left(u\_{\rm g} - u\_{l}\right)^{2} D}{\sigma},\tag{1}$$

*<sup>M</sup>* <sup>¼</sup> *<sup>ρ</sup>gu*<sup>2</sup> *g ρlu*<sup>2</sup> *l*

*Breakup Morphology and Mechanisms of Liquid Atomization*

*DOI: http://dx.doi.org/10.5772/intechopen.84998*

*MR* <sup>¼</sup> *<sup>ρ</sup>gu*<sup>2</sup>

*GLR* <sup>¼</sup> *<sup>ρ</sup>gugAg ρlulAl*

*t* is the real time, and *Ag* and *Al* are the area of gas exit and liquid exit, respectively. Complex fluid is usually a kind of complicated non-Newtonian fluid, which has more dimensionless groups on atomization. For example, in the Bingham model,

the shear rate. Hedstrom number is the useful nondimensional number, which

*He* <sup>¼</sup> *<sup>τ</sup>*0*D*<sup>2</sup>

The ratio of the yield stress to the stabilizing surface tension force is [17]

*<sup>Y</sup>* <sup>¼</sup> *<sup>ρ</sup>gu*<sup>2</sup>

For viscoelastic fluids, Weissenberg number compares the elastic forces to the

where *N*<sup>1</sup> is the first normal-stress difference, and *tR* is the relaxation time. Another dimensionless number on the ratio of first normal stress difference to

*<sup>Z</sup>* <sup>¼</sup> *<sup>N</sup>*<sup>1</sup>

Nozzle is generally used to produce spray. Fuel injection process plays a major role in many aspects of combustion performance. The influence of nozzle structure is remarkable on the atomization performance [19–22]. With the progress of technology

and the ratio of the aerodynamic force to the yield stress is [17]

surface tension force could be defined as follows [18]

the flow is characterized by the following constitutive equations,

depends only on material properties and geometrical parameters,

Here *μ<sup>g</sup>* is the gas viscosity,*c*is the speed of sound in the medium, *f* is the frequency,

where *τ* is the shear stress, *τ*<sup>0</sup> is the yield stress, *η* is the plastic viscosity, and *γ* is

*ρ*

*τ* ¼ *τ*<sup>0</sup> þ *ηγ* and *μ<sup>l</sup>* ¼ *τ*0*=γ* þ *η,* (11)

*gAg ρlu*<sup>2</sup> *<sup>l</sup> Al*

momentum ratio

and mass ratio

viscous forces

**75**

**3. Primary atomization**

*,* (8)

*:* (10)

*<sup>η</sup>*<sup>2</sup> *:* (12)

*<sup>g</sup>=τ*0*:* (14)

*<sup>σ</sup><sup>D</sup> :* (16)

*X* ¼ *τ*0*D=σ,* (13)

*Wi* ¼ *N*1*=τ* or *Wi* ¼ *tRγ,* (15)

(9)

where *ρ<sup>g</sup>* is the gas density, *ug* is the gas velocity, *ul* is the liquid velocity, *D* is the characteristic size (in general, nozzle diameter in primary atomization and drop diameter in secondary atomization), and *σ* is the surface tension. Liquid viscosity will hinder deformation and dissipates energy supplied by aerodynamic forces. The viscosity effect is highly correlated with the Ohnesorge number,

$$Oh = \frac{\mu\_l}{\sqrt{\rho\_l D \sigma}},\tag{2}$$

where *μ<sup>l</sup>* is the liquid viscosity, and *ρ<sup>l</sup>* is the liquid density. Other important dimensionless groups are gaseous Reynolds number

$$\text{Re}\_{\text{g}} = \frac{\rho\_{\text{g}} \mu\_{\text{g}} D}{\mu\_{\text{g}}},\tag{3}$$

liquid Reynolds number

$$\text{Re}\_l = \frac{\rho\_l u\_l D}{\mu\_l},\tag{4}$$

Mach number

$$\mathbf{M}\mathfrak{a} = \mathfrak{u}\_{\mathfrak{g}}/\mathfrak{c},\tag{5}$$

Strouhal number

$$\text{St} = \frac{f \mathcal{D}}{u\_l},\tag{6}$$

the characteristic time [16]

$$T = \frac{\mathfrak{t}u\_{\mathfrak{g}}}{D} \sqrt{\frac{\rho\_{\mathfrak{g}}}{\rho\_{l}}},\tag{7}$$

gas-liquid momentum flux ratio

*Breakup Morphology and Mechanisms of Liquid Atomization DOI: http://dx.doi.org/10.5772/intechopen.84998*

$$M = \frac{\rho\_{\text{g}} u\_{\text{g}}^2}{\rho\_l u\_l^2},\tag{8}$$

momentum ratio

secondary atomization. Atomization has been quantitatively studied for more than a century. However, liquid atomization is a complicate, multiparameter two-phase flow process, which is not well understood. Many empirical theories and equations have been developed and used in atomization. So, there is still a lot of unknown

The physical processes and fluid properties are important in atomization morphology and performance, and the mathematical and numerical analysis of atomization is very challenging, so a number of dimensionless numbers are used. First of all, the most important one is Weber number, which represents the ratio of disrup-

� �<sup>2</sup>

where *ρ<sup>g</sup>* is the gas density, *ug* is the gas velocity, *ul* is the liquid velocity, *D* is the characteristic size (in general, nozzle diameter in primary atomization and drop diameter in secondary atomization), and *σ* is the surface tension. Liquid viscosity will hinder deformation and dissipates energy supplied by aerodynamic forces. The

*Oh* <sup>¼</sup> *<sup>μ</sup><sup>l</sup>* ffiffiffiffiffiffiffiffiffiffi

where *μ<sup>l</sup>* is the liquid viscosity, and *ρ<sup>l</sup>* is the liquid density. Other important

Re*<sup>g</sup>* <sup>¼</sup> *<sup>ρ</sup>gugD μg*

Re*<sup>l</sup>* <sup>¼</sup> *<sup>ρ</sup>lulD μl*

> *St* <sup>¼</sup> *fD ul*

> > ffiffiffiffi *ρg ρl* r

*<sup>T</sup>* <sup>¼</sup> *tug D*

*D*

*<sup>σ</sup> ,* (1)

*<sup>ρ</sup>lD<sup>σ</sup>* <sup>p</sup> *,* (2)

*,* (3)

*,* (4)

*,* (6)

*,* (7)

*Ma* ¼ *ug=c,* (5)

*We* <sup>¼</sup> *<sup>ρ</sup><sup>g</sup> ug* � *ul*

tive hydrodynamic forces to the stabilizing surface tension force,

*Environmental Impact of Aviation and Sustainable Solutions*

viscosity effect is highly correlated with the Ohnesorge number,

dimensionless groups are gaseous Reynolds number

liquid Reynolds number

Mach number

Strouhal number

**74**

the characteristic time [16]

gas-liquid momentum flux ratio

work to be done [1–15].

**2. Dimensionless number**

$$MR = \frac{\rho\_{\text{g}} u\_{\text{g}}^2 A\_{\text{g}}}{\rho\_{l} u\_{l}^2 A\_{l}} \tag{9}$$

and mass ratio

$$\text{GLR} = \frac{\rho\_{\text{g}} u\_{\text{g}} A\_{\text{g}}}{\rho\_{l} u\_{l} A\_{l}}.\tag{10}$$

Here *μ<sup>g</sup>* is the gas viscosity,*c*is the speed of sound in the medium, *f* is the frequency, *t* is the real time, and *Ag* and *Al* are the area of gas exit and liquid exit, respectively.

Complex fluid is usually a kind of complicated non-Newtonian fluid, which has more dimensionless groups on atomization. For example, in the Bingham model, the flow is characterized by the following constitutive equations,

$$
\tau = \tau\_0 + \eta \gamma \text{ and } \mu\_l = \tau\_0/\gamma + \eta,\tag{11}
$$

where *τ* is the shear stress, *τ*<sup>0</sup> is the yield stress, *η* is the plastic viscosity, and *γ* is the shear rate. Hedstrom number is the useful nondimensional number, which depends only on material properties and geometrical parameters,

$$He = \frac{\pi\_0 D^2 \rho}{\eta^2}.\tag{12}$$

The ratio of the yield stress to the stabilizing surface tension force is [17]

$$X = \tau\_0 D / \sigma,\tag{13}$$

and the ratio of the aerodynamic force to the yield stress is [17]

$$Y = \rho\_{\text{g}} u\_{\text{g}}^2 / \tau\_0. \tag{14}$$

For viscoelastic fluids, Weissenberg number compares the elastic forces to the viscous forces

$$\mathbf{W}\mathbf{i} = \mathbf{N}\_1/\mathbf{r} \text{ or } \mathbf{W}\mathbf{i} = \mathbf{t}\_R\mathbf{y},\tag{15}$$

where *N*<sup>1</sup> is the first normal-stress difference, and *tR* is the relaxation time. Another dimensionless number on the ratio of first normal stress difference to surface tension force could be defined as follows [18]

$$Z = \frac{N\_1}{\sigma D}.\tag{16}$$

#### **3. Primary atomization**

Nozzle is generally used to produce spray. Fuel injection process plays a major role in many aspects of combustion performance. The influence of nozzle structure is remarkable on the atomization performance [19–22]. With the progress of technology

*We*-Re*<sup>l</sup>* map or *M*-Re*<sup>g</sup>* map is common. However, these criteria cannot reflect the impact of gas-liquid nozzle exit size. So, there is the modified map of classification by the gas-liquid nozzle exit area ratio *Ag=Al*. **Figure 1** shows that atomization performance will improve with the increase of *Ag=Al*, but the oversized value of *Ag=Al* will produce very little effect and waste energy [41]. These results are con-

In cylindrical gas jet and annular liquid jet, the common atomization modes are bubble (shell) breakup, Christmas tree (cellular) breakup, fiber breakup, and so on [43–55]. Here, the characteristic size is the liquid film thickness *h*, which has an important impact on liquid film breakup. Then, there is the *h=D*-*We* map of breakup regimes based on the Rayleigh-Taylor instability is proposed, which is in well agreement with the experimental results as shown in **Figure 2**. Note that outlet

wall thickness of nozzle can affect the flow field at nozzle outlet [56–61].

Drop is subjected to aerodynamic forces when there is the relative velocity between drop and gas. This force results in drop deformation and, if sufficiently large, will lead to breakup and fragmentation. Differing gas flow conditions can lead to differing drop breakup modes. Based on the morphology, as *We* increases, the vibrational, bag, bag-stamen, multimode, sheet-thinning, shear, and catastrophic breakup mode appear in turn [62–66]. This classification method would lack the quantitative physical mechanism, which may result in the criterion of mode based

The mechanism of drop breakup is a key and hot research area of secondary atomization. The investigation [67] shows the structure and location of turbulent eddies, which do not appear to correlate with drop breakup morphology. The average gas flow fields show no significant differences of drop morphology between bag breakup and sheet-thinning breakup. The results show that the wake structure of gas is unlikely to be the dominant mechanism of secondary atomization. These results agree well with experimental photos that the morphological transition of drop breakup is a strong function of *We*, and the influence of Re*<sup>g</sup>* is

Interfacial instability is very important in atomization [69, 70]. Rayleigh-Taylor (RT) instability is considered as the main driving mechanism responsible for drop breakup in the general bag breakup or Rayleigh-Taylor piercing (including bag breakup, bag-stamen breakup, dual-bag breakup, bag/plume breakup, multibag breakup, etc.) [65, 71–77]. All of these breakup modes have the same characteristic bag structure. With the increase of Weber number, the thin sheet (or membrane) at the periphery of deforming drop appears and breaks up continuously. These breakup types that have continuing shearing and entraining action are all governed by the Kelvin-Helmholtz (KH) instability mainly. This mode can be named as shear breakup (or sheet-thinning breakup, shear-induced entrainment) [65, 78]. So based on the instability in secondary atomization, there are two main modes: general bag

In order to classify the submode of general bag breakup, a dimensionless num-

where *λRT* is the wavelength of the most R-T unstable wave. *NRT* is the number of R-T wave on the windward side, which can also be considered as the bag number

*NRT* ¼ *D=λRT,* (17)

ducive to the design of nozzles.

*Breakup Morphology and Mechanisms of Liquid Atomization*

*DOI: http://dx.doi.org/10.5772/intechopen.84998*

**4. Secondary atomization**

primarily on subjective experience.

breakup and shear breakup.

ber of RT instability wave number is proposed [71]

little [68].

**77**

**Figure 1.** *Influence of Ag=Al on atomization mode.*

[23], it is found that the properties of breakup morphology and fragment distribution in different regimes are different [24–28]. Coaxial air-blast atomizers have many applications [29–31]. In order to obtain the desired results of atomization in the industrial scale, the suitable range of nozzle size and operating condition could be determined with the help of the regime map. There are two basic types on coaxial gas-liquid jets: (I) a cylindrical liquid jet surrounded by an annular gaseous stream and (II) an annular liquid sheet with an inner round gaseous stream [32–35].

In cylindrical liquid jet and annular gas jet, the common atomization modes are Rayleigh-type breakup (axisymmetric and non-axisymmetric), the membrane-type breakup (bag-type and membrane-fiber), fiber-type breakup, superpulsating breakup, atomization, and so on [36–42]. For the traditional classification, the

**Figure 2.** *Influence of h=D on atomization mode.*

*Breakup Morphology and Mechanisms of Liquid Atomization DOI: http://dx.doi.org/10.5772/intechopen.84998*

*We*-Re*<sup>l</sup>* map or *M*-Re*<sup>g</sup>* map is common. However, these criteria cannot reflect the impact of gas-liquid nozzle exit size. So, there is the modified map of classification by the gas-liquid nozzle exit area ratio *Ag=Al*. **Figure 1** shows that atomization performance will improve with the increase of *Ag=Al*, but the oversized value of *Ag=Al* will produce very little effect and waste energy [41]. These results are conducive to the design of nozzles.

In cylindrical gas jet and annular liquid jet, the common atomization modes are bubble (shell) breakup, Christmas tree (cellular) breakup, fiber breakup, and so on [43–55]. Here, the characteristic size is the liquid film thickness *h*, which has an important impact on liquid film breakup. Then, there is the *h=D*-*We* map of breakup regimes based on the Rayleigh-Taylor instability is proposed, which is in well agreement with the experimental results as shown in **Figure 2**. Note that outlet wall thickness of nozzle can affect the flow field at nozzle outlet [56–61].

#### **4. Secondary atomization**

[23], it is found that the properties of breakup morphology and fragment distribution in different regimes are different [24–28]. Coaxial air-blast atomizers have many applications [29–31]. In order to obtain the desired results of atomization in the industrial scale, the suitable range of nozzle size and operating condition could be determined with the help of the regime map. There are two basic types on coaxial gas-liquid jets: (I) a cylindrical liquid jet surrounded by an annular gaseous stream and (II) an annular liquid sheet with an inner round gaseous stream [32–35].

**Figure 1.**

**Figure 2.**

**76**

*Influence of h=D on atomization mode.*

*Influence of Ag=Al on atomization mode.*

*Environmental Impact of Aviation and Sustainable Solutions*

In cylindrical liquid jet and annular gas jet, the common atomization modes are Rayleigh-type breakup (axisymmetric and non-axisymmetric), the membrane-type breakup (bag-type and membrane-fiber), fiber-type breakup, superpulsating breakup, atomization, and so on [36–42]. For the traditional classification, the

Drop is subjected to aerodynamic forces when there is the relative velocity between drop and gas. This force results in drop deformation and, if sufficiently large, will lead to breakup and fragmentation. Differing gas flow conditions can lead to differing drop breakup modes. Based on the morphology, as *We* increases, the vibrational, bag, bag-stamen, multimode, sheet-thinning, shear, and catastrophic breakup mode appear in turn [62–66]. This classification method would lack the quantitative physical mechanism, which may result in the criterion of mode based primarily on subjective experience.

The mechanism of drop breakup is a key and hot research area of secondary atomization. The investigation [67] shows the structure and location of turbulent eddies, which do not appear to correlate with drop breakup morphology. The average gas flow fields show no significant differences of drop morphology between bag breakup and sheet-thinning breakup. The results show that the wake structure of gas is unlikely to be the dominant mechanism of secondary atomization. These results agree well with experimental photos that the morphological transition of drop breakup is a strong function of *We*, and the influence of Re*<sup>g</sup>* is little [68].

Interfacial instability is very important in atomization [69, 70]. Rayleigh-Taylor (RT) instability is considered as the main driving mechanism responsible for drop breakup in the general bag breakup or Rayleigh-Taylor piercing (including bag breakup, bag-stamen breakup, dual-bag breakup, bag/plume breakup, multibag breakup, etc.) [65, 71–77]. All of these breakup modes have the same characteristic bag structure. With the increase of Weber number, the thin sheet (or membrane) at the periphery of deforming drop appears and breaks up continuously. These breakup types that have continuing shearing and entraining action are all governed by the Kelvin-Helmholtz (KH) instability mainly. This mode can be named as shear breakup (or sheet-thinning breakup, shear-induced entrainment) [65, 78]. So based on the instability in secondary atomization, there are two main modes: general bag breakup and shear breakup.

In order to classify the submode of general bag breakup, a dimensionless number of RT instability wave number is proposed [71]

$$N\_{RT} = D/\lambda\_{RT} \,\text{\AA} \tag{17}$$

where *λRT* is the wavelength of the most R-T unstable wave. *NRT* is the number of R-T wave on the windward side, which can also be considered as the bag number

#### **Figure 3.** *Theoretical criterion NRT for general bag breakup.*

approximately. So the theoretical criterion *NRT* could be the new criterion instead of *We* in the range of general bag breakup as shown in **Figure 3**.

When the viscosity of liquid cannot be neglected, *Oh* will be another key parameter [79–84]. Many researches show that the *We* range of drop breakup mode will increase with the increase of *Oh* nonlinearly. The most important transition *We* is the critical Weber number *Wec* occurring at the start of bag breakup. It can establish the criteria for the onset of secondary atomization. Based on the RT instability, the theoretical formula for predicting *Wec* is [81]

$$\left(\frac{We\_0}{We\_c}\right)^{1/2} + C\left(\frac{Oh^2}{We\_c}\right)^{1/3} = 1,\tag{18}$$

where *We*<sup>0</sup> is the critical Weber number when Ohnesorge number tends to zero,

Drop interaction in the continuous gas jet is the important atomization process between primary atomization and secondary atomization [85]. It is the important link between the jet breakup and final spray. Due to airflow, the interaction of two drops evolves in a highly interactive and variable manner. The drop interaction in the airflow yields even more rich atomization morphologies and mechanisms. Behaviors of drop group and the isolated drop in the airflow are significantly different. Experimental photos in **Figure 4** show that there are four main interaction modes, which are coalescence mode, puncture mode, side by side mode, and no direct contact mode [86].

Drop size in atomization is a key parameter that is needed for a lot of fundamental researches and applications [87]. Due to the complicated nature of atomization, most nozzles cannot produce sprays of uniform droplet size. Instead, the spray can be regarded as a spectrum of drop sizes distributed about some defined mean drop size. Now, the most widely used mean diameter is Sauter mean diameter,

*<sup>D</sup>*<sup>32</sup> or *SMD* <sup>¼</sup> <sup>∑</sup>*nid*<sup>3</sup>

where *ni* is the number of droplets per unit volume in size class *i*, and *di* is the

The liquid in prefilming air-blast nozzle is first spread into a very thin sheet or film, which is then exposed to gas operating at the high velocity causing breakup and atomization. By spreading bulk liquid into film, contact area between liquid and gas increases. Generally speaking, SMD will decrease with the increase of gas velocity. However, under some conditions of prefilming atomization, the droplet size increases with the increase of gas velocity, and then decreases with the increase of gas velocity. So, the classical KH-RT atomization model [92–94] is modified and

Droplet size distribution is a crucial parameter of atomization process besides droplet mean diameter. Atomization and spray presents a wide distribution of fragment sizes. Many empirical relationships have been proposed to characterize the distribution of droplet sizes in atomization, for example, Rosin-Rammler, Nukiyama-Tanasawa, log-normal, root-normal, and log-hyperbolic. Atomization process involves a succession of changes of liquid topology, the last being the elongation and capillary breakup of ligaments torn off from the liquid surface. Breakup of liquid ligament (filament or fiber) is the key in primary atomization and secondary atomization, so ligament-mediated spray formation model is proposed [96, 97]. Drop fragments after ligament breakup is found to be gamma distribution. Then, the broad statistics of atomization shows Marshall-Palmer exponential shape

Complex fluids are mixtures that have a coexistence between two or multiphases, which are common in our society and industry [102–104]. Many complex fluids are non-Newtonian fluid, whose characteristics of breakup and atomization

*i* ∑*nid*<sup>2</sup> *i*

(19)

*We*<sup>0</sup> ¼ 11 � 2, and *C* can be considered as a constant 1.48.

*Breakup Morphology and Mechanisms of Liquid Atomization*

*DOI: http://dx.doi.org/10.5772/intechopen.84998*

**5. Fragment size and distribution**

droplet diameter [7, 88–91].

extended to the prefilming air-blast atomization [95].

of overall distribution in spray [98–101].

**6. Complex fluids**

**79**

are unusual [82, 105–107].

**Figure 4.** *Drop interaction regime map.*

where *We*<sup>0</sup> is the critical Weber number when Ohnesorge number tends to zero, *We*<sup>0</sup> ¼ 11 � 2, and *C* can be considered as a constant 1.48.

Drop interaction in the continuous gas jet is the important atomization process between primary atomization and secondary atomization [85]. It is the important link between the jet breakup and final spray. Due to airflow, the interaction of two drops evolves in a highly interactive and variable manner. The drop interaction in the airflow yields even more rich atomization morphologies and mechanisms. Behaviors of drop group and the isolated drop in the airflow are significantly different. Experimental photos in **Figure 4** show that there are four main interaction modes, which are coalescence mode, puncture mode, side by side mode, and no direct contact mode [86].
