A Fractal Model for the Maximum Droplet Diameter in Gas-Liquid Mist Flow

Abstract

Distribution characteristics of liquid droplet size are described using the fractal theory for liquid droplet size distribution in gas-liquid mist flow. Thereby, the fractal expression of the maximum droplet diameter is derived. The fractal model for maximum droplet diameter is obtained based on the internal relationship between maximum droplet diameter and the droplet fractal dimension, which is obtained by analyzing the balance between total droplet surface energy and total gas turbulent kinetic energy. Fractal model predictions of maximum droplet diameter agree with the experimental data. Maximum droplet diameter and droplet fractal dimension are both found to be related to the superficial velocity of gas and liquid. Maximum droplet diameter decreases with an increase in gas superficial velocity but increases with an increase in liquid superficial velocity. Droplet fractal dimension increases with an increase in gas superficial velocity but decreases with an increase in liquid superficial velocity. These are all consistent with the physical facts.

1. Introduction

Since the 1960s, coalescence and breakup phenomena of droplets in gas-liquid mist flow have obtained extensive attention in many physical and chemical process applications [1–3], such as distillation, gas absorption, and multiphase reactions. In nature, the tendency toward droplet coalescence or breakup depends on droplet size [4]. Numerous studies have shown that the main parameters influencing these phenomena are droplet size, system physical properties, and flow properties [5].

Numerous semiempirical models that predict droplet size in gas-liquid mist flow have been developed. According to Woodmansee and Hanratty [6], removal of microwaves on top of liquid film flow surges leads to atomization. Based on Woodmansee and Hanratty’s [6] droplet generation assumptions, Tatterson et al. [7] derived a prediction model for droplet size. Then, an empirical binomial equation was introduced by Andreussi et al. [8] based on their own data, with the first term proportional to gas superficial velocity and the second term linked to the square root of film thickness. Ueda [9] measured size and droplet removal flow rate in upward annular gas-liquid flow. He derived the rate of droplet movement between the liquid film and the gas core by considered droplet transference rates between the gas core and the liquid film. Lastly, Ueda proposed a mean diameter model for droplets under interface tension in gas flow by assuming a gamma-like size distribution for the gas core droplets. The empirical equation contained a single term, and droplet size was proportional to the gas velocity to the power of −0.34, which contrasted to most other equations having a power of −1.0. Sevik and Park [10] studied the assumption that the natural oscillation frequency of the liquid package is equal to the characteristic frequency of the disturbance. Under these conditions, droplets will break when resonance occurs. According to their results, Azzopardi et al. [11] derived an equation linking droplet size to the rupture of liquid packages in gas flow. Using a similar methodology, Lopes and Dukler [12] and Kocamustafaogullari et al. [13] used their own data to optimize this equation. Azzopardi et al. [14] developed a new laser optical technique to study annular flow where droplet size and axial and radial velocity are simultaneously measured and the flow conditions that droplet size distributions depend on are analyzed. Their equation, based on Tatterson et al. [7] and Azzopardi et al. [11], deals with the influence of coalescence by defining a mean free path between collisions. This processing method is similar to the gas kinetic theory.

Shavit and Chigier [15] were the first to observe the existence of fractal characteristics during droplet separation and jet. This observation provided the basis for using fractal theory to describe droplet size distribution. Weixing et al. [16] studied the fractal characteristics associated with the distribution of droplet sizes in the atomization process. They measured the fractal dimensions using droplet-splitting simulations. The results of this experiment led to the creation of a droplet splitting model, which uses a uniformly distributed probability function. Based on Weixing et al. [16], Zhou and Yu [17] measured droplet size distribution data using a dual particle dynamic analyzer and proposed a multifractal model to describe droplet dissolution characteristics. Kolmogorov described the breakup of solid particles as a discrete random process. Apte et al. [18] adopted Kolmogorov’s concept and established a stochastic subgrid model for large-eddy simulation of atomizing spray. Apte et al. showed that the atomization of droplets at high relative liquid-to-gas velocity was not affected by the initial droplet size in the framework of uncorrelated breakup events. According to the description of the droplet dissolution process proposed by Kolmogorov Gorokhovski and Saveliev [19] developed a droplet distribution model in the phase space of droplet position velocity by studying the breakup of droplets at the large Weber number. This numerical method can be used to stochastically model droplet production. According to the self-similarity of droplet breakup, Liu et al. [20] introduced a finite stochastic breakup model for air-blast atomizers, which is used to simulate the process of secondary atomization. Jiang et al. [21] researched the atomization process of the coaxial air-blast atomizer on the basis of the self-similarity of droplet breakup during secondary atomization. According to the breakup time study and the motion features of droplets in gas flow, they observed the factors that affect the Sauter mean diameter (SMD), which are the liquid/gas mass flux ratio, gas jet velocity, liquid jet velocity, and liquid droplet diameter.

In this paper, the distribution characteristics of mist flow droplet size are studied using a fractal mechanism. The fractal model of maximum droplet diameter is obtained by combining the analysis of the droplets total surface energy with that of gas flow total turbulent kinetic energy balance.

2. Model

2.1. Distribution Characteristics of Liquid Droplet Size in Gas-Liquid Mist Flow

Many researchers study the distribution characteristics of liquid droplet size using statistical methods. Results show that cross-sectional distribution characteristics of liquid droplet size have fractal features [15–17]. According to fractal geometry, distribution of the fractal accumulative droplet number, 𝑁, and droplet diameter submit to the following scale relationship [22, 23]: 𝑁()=𝜆𝐿≥𝜆^max𝜆^𝐷𝑓, (1) where 𝜆_max is the maximum droplet diameter. 𝐷_𝑓 is the droplet fractal dimension, 0<𝐷_𝑓<2 in two-dimensional space, and 0<𝐷_𝑓<3 in three-dimensional space [24].

Solving the differential of (1), the liquid droplet number between 𝜆 to 𝜆+𝑑𝜆 can be obtained [25]: −𝑑𝑁=𝐷_𝑓𝜆^𝐷𝑓_max𝜆^{−(𝐷𝑓+1)}𝑑𝜆. (2)

The negative variable in (2) indicates that liquid droplet number increases along with a decrease in droplet size [26, 27] and −𝑑𝑁>0.

The cross-sectional area of gas-liquid mist flow, 𝐴, is expressed as 𝐴=𝜋𝐷²4, (3) where 𝐷 is the diameter of the cross-section.

In cross-section 𝐴, the area occupied by the droplets is 𝐴_l=−^𝜆max_𝜆min𝜋𝜆²4𝑑𝑁=𝜋𝐷_𝑓𝜆²_max42−𝐷_𝑓𝜆1−^min𝜆_max^2−𝐷𝑓. (4)

Generally, 0<𝐷_𝑓<2 in two-dimensional space. In gas-liquid mist flow real situation, 𝜆_min/𝜆_max<10⁻⁶, and 𝐷_𝑓 is greater than 0.5, so (𝜆_min/𝜆_max)^2−𝐷𝑓<10⁻³; (4) can be simplified as 𝐴_l=𝜋𝐷_𝑓𝜆²_max42−𝐷_𝑓. (5)

In cross-section 𝐴, the area occupied by gas is 𝐴_𝑔=𝐴−𝐴_l=𝜋𝐷²4𝐷1−^𝑓𝜆²_max2−𝐷_𝑓𝐷². (6)

In gas-liquid mist flow, the relationship between liquid velocity, 𝜈_𝑙, and liquid superficial velocity, 𝜈_sl, is 𝑣_sl=𝐴_𝑙𝐴𝑣_𝑙. (7)

Substituting (3) and (5) into (7), liquid superficial velocity can be expressed as 𝑣_sl=𝐷_𝑓𝑣_𝑙𝜆²_max2−𝐷_𝑓𝐷². (8)

It can be observed from (8) that 𝐷_𝑓 is proportional to cross-sectional droplet number. An increase in 𝐷_𝑓 indicates that the liquid carried by gas flow disperses into smaller liquid droplets (i.e., liquid coalesces into smaller liquid droplets). This conforms to the physical fact that 𝜆_max decreases along with an increase in 𝐷_𝑓.

Liquid droplet velocity can be synchronously measured by phase Doppler anemometry (Azzopardi and Teixeira [28]) or Ronchi grating technology (Fore and Dukler [29]). The results of these two methodologies are similar: when gas and liquid mix sufficiently, liquid velocity approximately equals that of gas in gas-liquid mist flow: 𝑣_𝑙=𝑣_𝑔. (9)

According to the definition of velocity and superficial velocity of gas and liquid, the relationship between gas and liquid superficial velocity can be obtained from (9): 𝑣_sg=𝐴_𝑔𝐴_𝑙𝑣_sl. (10)

Substituting (5) and (6) into (10), the maximum droplet diameter on the basis of mist flow characteristics can be solved: 𝜆_max=2−𝐷_𝑓𝐷²𝑣_sl𝐷_𝑓𝑣_sg+𝑣_sl^1/2. (11)

As the maximum droplet diameter, 𝜆_max, and the droplet fractal dimension, 𝐷_𝑓, in (11) are both unknown variables, they cannot be directly solved by gas superficial velocity, 𝑣_sg, and liquid superficial velocity, 𝑣_sl. The internal relationship between the maximum droplet diameter, 𝜆_max, and the droplet fractal dimension, 𝐷_𝑓, can be found by analyzing the balance relationship between the surface energy of the dispersed phase and the turbulent kinetic energy of the continuous phase. Therefore, the subsidiary  condition for solving (11) is obtained.

2.2. Droplet Total Surface Energy and Total Gas Turbulent Kinetic Energy Balance Analysis

The droplet breakup model is applied to describe the cause of mist flow formation. According to the balance relationship between the surface energy of the dispersed phase and the turbulent kinetic energy of the continuous phase, Hinze [30] theoretically derived the maximum stability diameter of the dispersed phase in continuous fluid. Similar phenomenon also exists in gas-liquid mist flow. The maximum stability diameter of a droplet can be derived with the aid of the droplet dispersion model.

According to Adamson [31], free surface energy per unit area equals the interfacial tension per unit area in the gas-liquid phase; that is, 𝑒_𝑆=𝜎. (12)

The total free surface energy of dispersed droplets in continuous gas flow is 𝐸_𝑆=−𝑣_𝑙𝜎^𝜆max_𝜆min𝜋𝜆𝑑𝑁=𝜋𝐷_𝑓𝑣_𝑙𝜎𝜆_max1−𝐷_𝑓. (13)

It can be seen that the value of the droplet fractal dimension is 0<𝐷_𝑓<1; when the total free surface energy of the dispersed droplets is greater than 0,  𝐸_𝑠>0. An increase in 𝐷_𝑓 indicates that cross-sectional droplet distribution, contact area between the gas and liquid phase, and the value of 𝐸_𝑆 are large. In limiting cases, when 𝐷_𝑓 equals 0, cross-sectional droplets aggregate which leads to an extremely small contact area and 𝐸_𝑆 tends towards 0; when 𝐷_𝑓 equals 1, tiny liquid droplets dispersed on the cross-section, 𝐸_𝑆, tend toward infinity. An increase in the maximum droplet diameter, 𝜆_max, indicates an increase in the contact area between the gas and liquid phases. Therefore, the rule that 𝐸_S increases with an increase in 𝜆_max and 𝐷_𝑓 conforms to the physical definition of free surface energy.

By substituting (8) into (13), the relationship between total free surface energy of the dispersed droplets and liquid superficial velocity can be obtained: 𝐸_𝑆=𝜋𝜎2−𝐷_𝑓𝐷²1−𝐷_𝑓𝜆_max𝑣_sl. (14)

According to White [32] and Zhang et al. [33], gas flow’s turbulent kinetic energy per unit volume can be expressed as 𝑒_𝑇=32𝜌_𝑔𝑣_𝑟^′2, (15) where 𝜌_𝑔 is gas density and 𝑣^′_𝑟 is the radial velocity of gas.

Thus, the total turbulent kinetic energy of the continuous gas phase can be obtained: 𝐸_𝑇=32𝜌_𝑔𝑣_𝑟^′2𝐴𝑣_sg. (16)

According to Taitel et al. [34] and Chen et al. [35], gas radial velocity; approximately equals friction velocity; that is, 𝑣_𝑟^′21/2=𝑣_sg𝑓_sg2^1/2. (17)

The friction factor 𝑓_sg caused by gas flow is expressed as 1𝑓_sg𝑒=1.14−2𝑙𝑔𝐷+21.25Re^0.9, (18) where 𝑒 is the absolute roughness, 𝑒=1.6×10⁻⁶m. Re is the Reynolds number.

Substituting (17) into (16), the total turbulent kinetic energy of gas-liquid mist flow can be expressed as 𝐸_𝑇=34𝑓_sg𝜌_𝑔𝐴𝑣³_sg. (19)

In high-speed gas flow, total droplet surface energy is equal to the total turbulent kinetic energy of the gas phase [36]: 𝐸_𝑆=𝐸_𝑇. (20)

Substituting (14) and (19) into (20), maximum droplet diameter based on the droplet dispersion model can be obtained: 𝜆_max=16𝜎2−𝐷_𝑓𝑣_sl3𝑓_sg𝜌_𝑔1−𝐷_𝑓𝑣³_sg. (21)

It can be seen from (21) that the maximum droplet diameter increases with a decrease in gas superficial velocity and an increase in liquid superficial velocity. This rule is consistent with physical reality.

2.3. Fractal Model of Maximum Liquid Droplet Diameter

The droplet size distribution rule describes droplet distribution in gas-liquid mist flow. Droplet distribution is controlled by the balance relationship between total droplet surface energy and total gas turbulent kinetic energy. Therefore, the droplet fractal dimension under gas-liquid mist flow conditions can be calculated by substituting (21) into (11): 𝐴𝐷²_𝑓+𝐵𝐷_𝑓+𝐶=0, (22) where 𝐴=3𝐷𝑓_sg𝜌_𝑔𝑣³_sg²+256𝑣_sg𝑣_sl𝜎²+256𝑣_sl𝜎²,𝐵=−23𝐷𝑓_sg𝜌_𝑔𝑣³_sg²+256𝑣_sg𝑣_sl𝜎²+256𝑣_sl𝜎²,𝐶=3𝐷𝑓_sg𝜌_𝑔𝑣³_sg². (23)

Equation (22) has one solution between 0 and 1. Therefore, the liquid droplet fractal dimension in gas-liquid mist flow can be expressed as 𝐷_𝑓=−𝐵−𝐵²−4𝐴𝐶.2𝐴 (24)

By combining (11) with (24), the fractal expression of maximum droplet diameter under gas-liquid mist flow conditions can be obtained: 𝜆_max=2−𝐷_𝑓𝐷²𝑣_sl𝐷_𝑓𝑣_sg+𝑣_sl^1/2,𝐷_𝑓=−𝐵−𝐵²−4𝐴𝐶.2𝐴 (25)

Equation (25) is the analytical solution of the fractal model of maximum droplet diameter in gas-liquid mist flow. The maximum droplet diameter, 𝜆_max, and the droplet fractal dimension, 𝐷_𝑓, can be simultaneously solved using gas superficial velocity, 𝑣_sg, and liquid superficial velocity, 𝑣_sl. The model proposed in this paper contains no empirical constants, and every parameter has clear physical meaning. However, other parameters are required to identify 𝜆_max and 𝐷_𝑓, such as the cross-sectional diameter of gas-liquid mist flow, 𝐷, gas density, 𝜌_𝑔, and interface tension, 𝜎.

3. Results and Discussion

Figure 1 shows a comparison between the proposed model’s predictions and experimental results measured by Fore and Dukler [29]. Agreement between the fractal model predictions and the experimental results of Fore and Dukler in different liquid viscosity and superficial velocity conditions is obtained. Results suggest that the proposed gas-liquid mist flow maximum liquid droplet diameter model is reliable.

Figure 1 

Comparison between the proposed model predictions and experimental results (𝐷=0.15 m, 𝜎=0.072 N/m, and 𝜇𝑙=1.05×10−3 Pa·s).

Figure 2 shows the relationships between maximum droplet diameter,   𝜆_max, droplet fractal dimension, 𝐷_𝑓, and gas superficial velocity, 𝑣_sg, calculated by the present model. It can be seen that 𝜆_max decreases with an increase in 𝑣_sg. Inversely, 𝐷_𝑓 increases with an increase in 𝑣_sg. As 𝑣_sg increases, the turbulent kinetic energy of the gas in the gas-liquid mist flow increases, which forces the original droplets to disperse into smaller droplets, making 𝜆_max decrease and 𝐷_𝑓 increase. These all conform to the physical facts.

Figure 2 

Maximum droplet diameter and droplet fractal dimension versus 𝑣sl at different 𝑣sg for gas-liquid mist flow.

Figure 3 shows the relationships between maximum droplet diameter, 𝜆_max, droplet fractal dimension, 𝐷_𝑓, and liquid superficial velocity, 𝑣_sl, calculated by the present model. 𝜆_max increases with an increase in 𝑣_sl. Inversely, 𝐷_𝑓 decreases with an increase in 𝑣_sl. As 𝑣_sl increases, surface energy of the liquid in the gas-liquid mist flow increases, which forces the original droplets to coalesce into smaller droplets, making 𝜆_max increase and 𝐷_𝑓 decrease. These all conform to the physical facts.

Figure 3 

Maximum droplet diameter and droplet fractal dimension versus 𝑣sg at different 𝑣sl for gas-liquid flow.

4. Conclusion

(1)Based on liquid droplet fractal size distribution properties in gas-liquid mist flow, we describe the characteristics of droplet distribution under mist flow conditions. The fractal expression of maximum droplet diameter is derived and the internal relationship between maximum droplet diameter and droplet fractal dimension is confirmed by analyzing the balance relationship between total droplet surface energy and total gas turbulent kinetic energy. Therefore, the fractal model of the maximum droplet diameter in gas-liquid mist flow is derived. Fractal expressions of cross-sectional area occupied by gas or liquid and the superficial velocity of gas and liquid are obtained. Every parameter proposed in this paper has clear physical meaning.(2)Agreement between predictions of the proposed fractal model and experimental measurements is obtained. Results verify the reliability of the proposed model.(3)The influence of gas and liquid superficial velocity on maximum liquid droplet diameter,𝜆_max, and liquid droplet fractal dimension, 𝐷_𝑓, is analyzed. 𝜆_max increases with a decrease in 𝑣_sg and with an increase in 𝑣_sl. 𝐷_𝑓  increases with an increase in 𝑣_sg and with a decrease in 𝑣_sl. These coincide with the physical facts.

Nomenclature

𝐴:	Cross-sectional area of a conduit, m2
𝐴g:	Cross-sectional area occupied by gas, m2
𝐴𝑙:	Cross-sectional area occupied by liquid, m2
𝐷:	Conduit diameter, m
𝐷𝑓:	Droplet fractal dimensions
𝐸𝑆:	Total surface free energy, W
𝐸𝑇:	Total turbulent kinetic energy, W
𝑒:	Absolute pipe wall roughness, m
𝑒𝑠:	Surface free energy per unit volume, J/m2
𝑒𝑇:	Turbulent kinetic energy per unit area, J/m3
𝑓sg:	Friction factor at gas superficial velocity
𝑁:	Fractal accumulative droplet number
Re:	Reynolds number
𝑣𝑔:	Gas velocity, m/s
𝑣𝑙:	Liquid velocity, m/s
𝑣sg:	Gas superficial velocity, m/s
𝑣sl:	Liquid superficial velocity, m/s
𝑣′𝑟:	Radial velocity, m/s
𝜆:	Droplet diameter, m
𝜆max:	Maximum droplet diameter, m
𝜆min:	Minimum droplet diameter, m
𝜌𝑔:	Gas density, kg/m3
𝜌𝑙:	Liquid density, kg/m3
𝜎:	Interfacial tension, N/m.

Subscripts

𝑔:	Gas phase
𝑙:	Liquid phase
𝑠:	Superficial
𝑇:	Turbulent.

Acknowledgments

The authors are grateful for financial support from the National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019) and Science and Technology Innovation Fund of Southwest Petroleum University (Grant no. GIFSB0701).