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

418 Numerical Simulation – From Theory to Industry

causing coalescence on the membrane.

times the pore diameter gave nearly mono dispersed emulsions.

greater than 90o, the wall is called the non-wetting.

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,

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

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

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

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 overcome the attaching force.

These four forces can be approximated as follows:

Drag force:

$$F\_D = \frac{1}{2} \mathcal{C}\_d \rho\_{cp} (v^\* - v\_{dp})^2 (\pi D\_p^2 / 4) \tag{4}$$

Surface tension force:

$$F\_t = \pi \sigma D\_p \tag{5}$$

Numerical Simulation of Droplet Dynamics in Membrane Emulsification Systems 421

(12)

For low value of *We* number, the value of inertial force ( *<sup>i</sup> F* ) is very low. In that situation the

\* 2 ( )( ) *<sup>p</sup> d cp dp*

Droplet formation and deformation have been studied for a long time due to complexity with the problem and practical utilities of the phenomenon. Droplet formation in a twophase flow system possesses a rich dynamics with the involvement of several parameters such as average velocity of the liquids, their viscosities, densities, surface tension, surface chemistry and the flow geometry. Droplets formation results the creation of new surfaces which enhance the heat and mass transfer between the phases. Due to enhanced heat and mass transfer, the process has been used for wide ranges of phase-contact applications. Particularly the droplet formation in micro or nano size has received significant attention during last several years. Due to miniature size, the fabrication of experimental facility is expensive and reliable experimentation of microfluidic is very intricate. Hence the viable alternate is the numerical tools for investigating the problems. With the development of high speed computer and advanced algorithm, numerical modeling and simulation have become an essential part in the design and development of numerous engineering systems. Numerical simulations of droplet dynamics i.e. the investigations of two-phase flow in micro scale have been extensively undertaken during last several years. Various types of numerical techniques have been developed to solve the governing equations of the two-

In numerical simulation of droplet dynamics or as a whole in the simulation of two-phase flow, there are several challenges which need to be carefully tackled to obtain reliable results. The main challenge is capturing the moving interface of the two phases accurately, which is not known priory. The accurate tracking of the interface and investigation of twophase flow topology should be the essentiality of a good numerical method. There are several numerical methods based on interface kinematics to track the interface in free surface flows. Among them are: volume of fluid methods, front tracking methods, level set methods, phase field formulations, continuum advection schemes, boundary integral

Volume of fluid method (VOF), earlier known as the volume tracking method, were originally developed by Nichols and Hirt (1975), Noh and Woodward (1976) and further extended by Hirt and Nichols (1981). Since then, the method has been extensively used and significantly improved over the years (Rudman,1997; Rider and Kothe,1998). The VOF

Thus analytically, droplet diameter increases with the increase in surface tension value.

*C vv* 

*D*

diameter of the drop can be approximated as

**1.2. Numerical simulation of droplet dynamics** 

phase flow.

**1.3. Different numerical methods** 

methods, particle-based methods, and moving mesh methods.

Inertial force of dispersed phase:

$$F\_i = \frac{\pi \rho\_{dp} D\_0^2 v\_{dp}^2}{4} \tag{6}$$

Buoyancy force:

$$F\_{\mathcal{B}} = \rho\_c \mathcal{g} V\_{dr} - \mathcal{p}\_c A\_n \tag{7}$$

In above *dp <sup>v</sup>* is the velocity of dispersed phase ( *dp o v v* ), \* *<sup>v</sup>* is the local continuous phase velocity at the centre of the drop, *Dp* is the diameter of the drop, *D*<sup>0</sup> is the pore diameter and Vdr is the droplet volume. The local continuous phase velocity is given by:

$$\boldsymbol{\upsilon}^\* = 2\boldsymbol{\mu}\_{cp} \left[ \mathbf{1} - \left( \frac{\boldsymbol{D}\_h - \boldsymbol{D}\_p}{\boldsymbol{D}\_h} \right)^2 \right] \tag{8}$$

where *cp u* is the average velocity of the continuous phase in the channel. *Dh* is the hydraulic diameter given by:

$$D\_h = \mathcal{Z}bh \;/\; (b+h) \tag{9}$$

where *b* and *h* are the width and height of the continuous phase channel respectively. The drag coefficient *Cd* depends upon the Reynolds number of the droplet (*Rep*) and the viscosity ratio λ (/) *dp cp* . The drop Reynolds number is defined as

$$Re\_p = \frac{\rho\_{dp} \left| \boldsymbol{v}^\* - \boldsymbol{v}\_{dp} \right| D\_p}{\mu\_{dp}} \tag{10}$$

Out of these forces, the only attaching force is the surface tension force and remaining forces, drag and inertial and buoyancy force are detaching forces. Neglecting the buoyancy or gravity force, the balance of forces at the moment of droplet detachment can be written as

$$\frac{1}{2}\mathcal{C}\_{d}\rho\_{cp}(\boldsymbol{v}^\*-\boldsymbol{v}\_{dp})^2(\pi\mathcal{D}\_p^2/4) + \frac{\pi\rho\_{dp}\mathcal{D}\_0^2\boldsymbol{v}\_{dp}^2}{4} = \pi\sigma\mathcal{D}\_p\tag{11}$$

For low value of *We* number, the value of inertial force ( *<sup>i</sup> F* ) is very low. In that situation the diameter of the drop can be approximated as

$$D\_p \propto \frac{\sigma}{\left(\mathcal{C}\_d \rho\_{cp}\right) \left(v^\* - v\_{dp}\right)^2} \tag{12}$$

Thus analytically, droplet diameter increases with the increase in surface tension value.

### **1.2. Numerical simulation of droplet dynamics**

420 Numerical Simulation – From Theory to Industry

Drag force:

Surface tension force:

Buoyancy force:

diameter given by:

viscosity ratio λ (/) *dp cp*

 Inertial force of dispersed phase:

These four forces can be approximated as follows:

<sup>1</sup> \* 22 ( ) ( / 4) <sup>2</sup> *D d cp dp P F C vv D*

*t p F D* 

*B c dr c n F gV p A* 

In above *dp <sup>v</sup>* is the velocity of dispersed phase ( *dp o v v* ), \* *<sup>v</sup>* is the local continuous phase velocity at the centre of the drop, *Dp* is the diameter of the drop, *D*<sup>0</sup> is the pore diameter

> \* 2 1 *h p cp*

where *cp u* is the average velocity of the continuous phase in the channel. *Dh* is the hydraulic

where *b* and *h* are the width and height of the continuous phase channel respectively. The drag coefficient *Cd* depends upon the Reynolds number of the droplet (*Rep*) and the

. The drop Reynolds number is defined as

*p*

<sup>1</sup> \* 22 ( ) ( / 4) <sup>2</sup> *C vv D d cp dp P*

 

\* *dp dp p*

Out of these forces, the only attaching force is the surface tension force and remaining forces, drag and inertial and buoyancy force are detaching forces. Neglecting the buoyancy or gravity force, the balance of forces at the moment of droplet detachment can be written as

 *v vD* 

*dp*

2 2 0 4 *dp dp*

2

*h*

*D D*

*D* 

*D v*

 

(4)

(5)

(6)

(7)

2 /( ) *D bh b h <sup>h</sup>* (9)

*Re* (10)

*Dp* 

(11)

*dp dp D v* (8)

*i*

*F*

and Vdr is the droplet volume. The local continuous phase velocity is given by:

*v u*

Droplet formation and deformation have been studied for a long time due to complexity with the problem and practical utilities of the phenomenon. Droplet formation in a twophase flow system possesses a rich dynamics with the involvement of several parameters such as average velocity of the liquids, their viscosities, densities, surface tension, surface chemistry and the flow geometry. Droplets formation results the creation of new surfaces which enhance the heat and mass transfer between the phases. Due to enhanced heat and mass transfer, the process has been used for wide ranges of phase-contact applications. Particularly the droplet formation in micro or nano size has received significant attention during last several years. Due to miniature size, the fabrication of experimental facility is expensive and reliable experimentation of microfluidic is very intricate. Hence the viable alternate is the numerical tools for investigating the problems. With the development of high speed computer and advanced algorithm, numerical modeling and simulation have become an essential part in the design and development of numerous engineering systems. Numerical simulations of droplet dynamics i.e. the investigations of two-phase flow in micro scale have been extensively undertaken during last several years. Various types of numerical techniques have been developed to solve the governing equations of the twophase flow.
