**1. Introduction**

An emulsion is a two-phase liquid system of two immiscible liquids, where the liquid with lower mass fraction is dispersed in form of small droplets in other surrounding liquid of higher mass fraction. Emulsions are widely used to produce sol–gel, drugs, synthetic materials, and food products. Based on the size of droplet, emulsions can be classified as micro and macro emulsion. Karbstein and Schubert, (1995) have made a limiting droplet size of 0.1 µm, below which the emulsion is termed as micro emulsion and above that size the emulsion is termed as macro emulsion. Size and size distribution of droplets play important roles in the stability of emulsion. There are also other factors such as sedimentation, skimming, droplet aggregation and coalescence, which may affect the stability of the droplets. Thus for making a stable emulsion it is necessary to convert the dispersed phase into tiny droplets and stabilize them against coalescence. Some amount of energy is required in the process to break the dispersed phase into droplets. The amount of energy put in the dispersing phase also controls the resulting droplet size. The stability of newly formed droplets depends on how fast the used emulsifiers are able to occupy the newly created interfaces and how well they stabilize them. The common devices used to produce emulsions are rotor-stator-systems, stirrers and high-pressure homogenizers. During last two decades, new technologies of making emulsion have been developed. Compared to conventional method of emulsification such as rotor-stator method, these new techniques of emulsification have several advantages such as low energy consumption, controllable droplet size with proper distribution and easy scalability. These new methods are based on the microdroplet formation in micrometer sized channels. Three such new methods are T-junction emulsification, flow focusing emulsification, and membrane emulsification. In all these methods, controllable droplet formations are achieved by properly maintaining the combination of continuous and dispersed phase flow rate.

In membrane emulsification process, micro or macro porous membranes are used to generate droplets by pressing the dispersed phase through the porous matrix of the

© 2012 Pathak, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

membrane towards the continuous phase. At the interface, the dispersed phase forms droplets near the region of pores openings and detached by the cross-flowing continuous phase. Sometimes surfactants are used to stabilize the droplets. Compared to the conventional method, membrane emulsification process requires a lower energy input (105 – 106 J/m3) to generate micro-sized droplets (Schubert and Behrend, 2003). Since small droplets are directly formed at the micro-pores of a membrane, rather than by disruption in zones of high energy density, smaller amount of stress is required in the process compared to the conventional method. The main disadvantage of the process is the requirement of longer production time compared to the conventional processes because of the slow rate at which the dispersed phase flows through the membrane (Joscelyne and Trägårdh, 1991, 2000). The longer production time can be reduced by increasing the flow rate of dispersed phase fluid. With the increase in dispersed phase flow rate, the droplet diameter increases first than decreases and the process shifts towards jetting phenomenon (Pathak, 2011). Thus there should be an optimum dispersed phase flux for optimum production time and droplet size in a membrane emulsification system.

Numerical Simulation of Droplet Dynamics in Membrane Emulsification Systems 417

In membrane emulsification process, the size distribution of pores and their relative spatial distribution in the membrane surface control the production of mono-disperse emulsions. The growth and detachment of the droplet i.e. droplet dynamics in membrane emulsification depends on several parameters. Luca et al., (2004) has classified them into

i. **Operating parameters:** cross-flow velocity, transmembrane pressure and disperse

ii. **Membrane parameters:** pore size, active pores, distance between the pores, membrane

Based on these parameters, the size of the droplet in membrane emulsification system depends upon the pore diameter and the dependence of the droplet diameter on pore

 *D*p = *x D*0 (1)

where *D*p is the droplet diameter and *D*0 is the pore diameter. Katoh et al., (1996) experimentally observed the value of *x* in the range of 2 to 12. The major factors affecting the value of *x* are: (i) the shear rates of continuous cross-flow fluid (ii) the dynamic interfacial tension, *γ*; and (iii) the disperse phase flux (*J*d). Other parameters those implicitly control the value of *x* are: the average velocity of the continuous phase flow 〈��〉*,* the viscosities of the disperse phase and the continuous phases (*μ*<sup>d</sup> and *μ*c) the density of the continuous phase (*ρ*c) the thermodynamic temperature, (*T*), the transmembrane pressure (*P*m). The viscosity and dynamic surface tension depend upon temperature and that way temperature can

Cross-flow velocity of continuous phase imparts drag force on the growing droplet for which the droplet detaches at the pore. With the increase in cross-flow velocity, the droplet diameter decreases. The dispersed phase flow rate or velocity influences the droplet dynamics via the inertial force competing with other forces such as drag and surface tension force. The difference between the pressure of dispersed phase in the dispersed phase channel and the average pressure of the continuous phase in the main channel is termed as

��� � �� � ����

where ����� and ������ are the pressure of continuous phase at the inlet and outlet of the main channel. Total transmembrane pressure consists of two parts: one is the capillary pressure (*p*γ) and other is the effective transmembrane pressure (*p*eff) or drag pressure inside the pore. Due to curvature of the droplet and dynamic interfacial tension, a small amount of pressure is required to inflate the droplet. This pressure is the capillary pressure. The capillary

The average pressure of the continuous phase is defined as:

����

� (2)

� � ������ � ������)/2 (3)

iii. **Phase parameters:** interfacial tension, viscosity and density of the processed phases.

three broad categories.

phase flux;

hydrophobicity/hydrophilicity;

diameter can be expressed as:

influence the emulsification process.

transmembrane pressure.

The schematic diagram of a cross-flow membrane emulsification has been shown in Fig. 1. Some commonly used membranes are tubular micro-porous glass (MPG) and shirasu porous glass (SPG) membrane. Some metallic oxides such as ceramic α-Al2O3 or α-Al2O3 coated with titainia oxide or zirconia oxide are also used as membrane. These membranes contain cylindrical, interconnected, uniform micro-pores having pore sizes, typically ranging about 0.05–14 µm. In membrane emulsification system, the time of droplets formation, size and stability of droplets are three important parameters which control the emulsification system. Thus understanding the droplet dynamics in detail may enable to explore the possibilities and limits of membrane emulsification for various applications.

**Figure 1.** Schematic diagram of a cross-flow membrane emulsification process

In membrane emulsification process, the size distribution of pores and their relative spatial distribution in the membrane surface control the production of mono-disperse emulsions. The growth and detachment of the droplet i.e. droplet dynamics in membrane emulsification depends on several parameters. Luca et al., (2004) has classified them into three broad categories.

416 Numerical Simulation – From Theory to Industry

in a membrane emulsification system.

membrane towards the continuous phase. At the interface, the dispersed phase forms droplets near the region of pores openings and detached by the cross-flowing continuous phase. Sometimes surfactants are used to stabilize the droplets. Compared to the conventional method, membrane emulsification process requires a lower energy input (105 – 106 J/m3) to generate micro-sized droplets (Schubert and Behrend, 2003). Since small droplets are directly formed at the micro-pores of a membrane, rather than by disruption in zones of high energy density, smaller amount of stress is required in the process compared to the conventional method. The main disadvantage of the process is the requirement of longer production time compared to the conventional processes because of the slow rate at which the dispersed phase flows through the membrane (Joscelyne and Trägårdh, 1991, 2000). The longer production time can be reduced by increasing the flow rate of dispersed phase fluid. With the increase in dispersed phase flow rate, the droplet diameter increases first than decreases and the process shifts towards jetting phenomenon (Pathak, 2011). Thus there should be an optimum dispersed phase flux for optimum production time and droplet size

The schematic diagram of a cross-flow membrane emulsification has been shown in Fig. 1. Some commonly used membranes are tubular micro-porous glass (MPG) and shirasu porous glass (SPG) membrane. Some metallic oxides such as ceramic α-Al2O3 or α-Al2O3 coated with titainia oxide or zirconia oxide are also used as membrane. These membranes contain cylindrical, interconnected, uniform micro-pores having pore sizes, typically ranging about 0.05–14 µm. In membrane emulsification system, the time of droplets formation, size and stability of droplets are three important parameters which control the emulsification system. Thus understanding the droplet dynamics in detail may enable to explore the possibilities and limits of membrane emulsification for various applications.

**Figure 1.** Schematic diagram of a cross-flow membrane emulsification process


Based on these parameters, the size of the droplet in membrane emulsification system depends upon the pore diameter and the dependence of the droplet diameter on pore diameter can be expressed as:

$$D\_{\mathcal{P}} = \propto D\mathbf{\hat{o}} \tag{1}$$

where *D*p is the droplet diameter and *D*0 is the pore diameter. Katoh et al., (1996) experimentally observed the value of *x* in the range of 2 to 12. The major factors affecting the value of *x* are: (i) the shear rates of continuous cross-flow fluid (ii) the dynamic interfacial tension, *γ*; and (iii) the disperse phase flux (*J*d). Other parameters those implicitly control the value of *x* are: the average velocity of the continuous phase flow 〈��〉*,* the viscosities of the disperse phase and the continuous phases (*μ*<sup>d</sup> and *μ*c) the density of the continuous phase (*ρ*c) the thermodynamic temperature, (*T*), the transmembrane pressure (*P*m). The viscosity and dynamic surface tension depend upon temperature and that way temperature can influence the emulsification process.

Cross-flow velocity of continuous phase imparts drag force on the growing droplet for which the droplet detaches at the pore. With the increase in cross-flow velocity, the droplet diameter decreases. The dispersed phase flow rate or velocity influences the droplet dynamics via the inertial force competing with other forces such as drag and surface tension force. The difference between the pressure of dispersed phase in the dispersed phase channel and the average pressure of the continuous phase in the main channel is termed as transmembrane pressure.

$$
\Delta p\_m = p\_d - \overline{p\_c} \tag{2}
$$

The average pressure of the continuous phase is defined as:

$$
\overrightarrow{p\_c} = (p\_{c,in} + p\_{c,out})/2\tag{3}
$$

where ����� and ������ are the pressure of continuous phase at the inlet and outlet of the main channel. Total transmembrane pressure consists of two parts: one is the capillary pressure (*p*γ) and other is the effective transmembrane pressure (*p*eff) or drag pressure inside the pore. Due to curvature of the droplet and dynamic interfacial tension, a small amount of pressure is required to inflate the droplet. This pressure is the capillary pressure. The capillary

pressure is maximum at the starting of formation of the droplet at the pore and it decrease as the droplet grows. The capillary pressure becomes the minimum at the detachment time of the droplet. The effective or drag pressure part is responsible for the flow rate of dispersed phase. The effective pressure determines the throughput and thus the productivity of the membrane emulsification system. The transmembrane pressure controls the size of the droplet formed in membrane emulsification process. With the increase in transmembrane pressure, several researchers (Katoh et al, 1996; Peng and Williams, 1998; Schröder and Schubert, 1999) have observed the increase in droplet diameter while others (Abrahamse et al., 2002; Vladisavljevic and Schubert, 2003; Vladisavljevic et al., 2004) have observed the decrease in droplet diameter. The variation in droplet diameter with transmembrane pressure results the wide range of droplet size distribution (Abrahamse et al., 2002; Vladisavljevic et al., 2004). The wide range of droplet size also is affected by the steric hindrance between droplets or by the membrane being wetted by the dispersed phase, causing coalescence on the membrane.

Numerical Simulation of Droplet Dynamics in Membrane Emulsification Systems 419

Among different properties of the phase, surface tension controls the droplet dynamics in a greater way than any other properties. Surface tension force holds the droplet and offers the resistance against any deformation. The viscosities of both the phases have also effect on the droplet deformation. The drag force imparted by the continuous phase on the droplet depends upon the viscosity ratio of dispersed phase and continuous phase. For fixed flow rate of continuous phase the drag force increases with the increase in viscosity ratio up to some extent. After that value, the drag force becomes independent of the viscosity ratio. The densities of both phases enter into the droplet dynamics through the buoyancy or gravity force. In micro- or nano-fluidics flow the value of gravity force is very small and it can be

All the parameters discussed in above control the droplet dynamics in membrane emulsification process with different magnitudes and output of the process can be analyzed on the basis of these operating parameters. Besides the individual effects, many of these parameters exhibit coupling effects. Different types of hydrodynamic forces act in the emulsification process. The droplet growth and deformation in membrane emulsification can be explained from the action of

The major forces that act in the process are: drag force imparted by the flowing continuous phase, the interfacial tension force, the inertial force of the dispersed phase and the buoyancy or gravitational force. Different forces acting in the droplet formation process are shown in Fig. 2. Among these forces, interfacial tension force is the attaching force and other are detaching force. The droplet is detached from the pore when the detaching forces

neglected without loss of much accuracy.

**1.1. Different forces acting in membrane emulsification.** 

these and the final droplet size is a result of the interaction of these forces.

**Figure 2.** Different forces acting on the emulsification system

overcome the attaching force.

The design and pore distribution of the membrane are important factors controlling the droplet dynamics in membrane emulsification. Due to presence of multi-pore and multidroplet formation, there is a change in hydrodynamic effects caused by neighboring droplet and interactions between the droplets. The separation distance between the pores controls those hydrodynamic effects. If the separation distance of pores in the flow direction is small, the continuous phase velocity decreases and the boundary layer thickness increases as the flow approaches consecutive rows after crossing the first row. These would lead to an increase in the size of the droplets. With the increase in droplet size, there would be a caution of stability loss and coalescence of the droplets. For high efficiency of the emulsification process, narrow droplet size distribution and higher dispersed phase velocity is required. However, with the increase in dispersed phase flow rate, the droplet formation phenomenon shifts towards jetting (Pathak, 2011) and this requires a greater distance between the pores in the direction of the cross-flowing continuous phase in order to prevent drops from colliding and coalescing. In several experimental studies (Sugiura et al., 2002; Kobayashi et al, 2003, 2006) the droplet size distribution has been observed narrow up to a specific velocity of the dispersed phase, above which the diameter of the droplet distribution has been increased. Timgren et al. (2009) have investigated the effects of pore size distribution on hydrodynamic effects of droplet size and distribution. They observed that for small pore separation distance and with a low dispersed phase velocity the drop formation process was uniform, resulting an emulsion with a narrow drop size distribution. For shortest pore separation distance, with the increase in dispersed phase velocity, they observed the formation of poly dispersed emulsion, whereas pore separations of 15 and 20 times the pore diameter gave nearly mono dispersed emulsions.

The wetting behavior of membrane surface also controls the droplet growth. The wetting behavior of a membrane is represented by the static contact angle between the two liquid phases and the solid boundary. The static contact angle between the two phases and walls controls the evolution of the dispersed phase inside the micro-pore and in the continuous phase flow channel. If the angle is less than 90o the wall is said to be wetting and if it is greater than 90o, the wall is called the non-wetting.

Among different properties of the phase, surface tension controls the droplet dynamics in a greater way than any other properties. Surface tension force holds the droplet and offers the resistance against any deformation. The viscosities of both the phases have also effect on the droplet deformation. The drag force imparted by the continuous phase on the droplet depends upon the viscosity ratio of dispersed phase and continuous phase. For fixed flow rate of continuous phase the drag force increases with the increase in viscosity ratio up to some extent. After that value, the drag force becomes independent of the viscosity ratio. The densities of both phases enter into the droplet dynamics through the buoyancy or gravity force. In micro- or nano-fluidics flow the value of gravity force is very small and it can be neglected without loss of much accuracy.
