**3. Results and discussions**

In the present work, the dynamics of droplet formation in two pores of membrane emulsification has been investigated for different flow rates (velocities) of dispersed phase, continuous phase, surface tension and viscosity of the two phases. It is to be noted that In case of membrane emulsification process, the dispersed phase fluid gets more space to interact with the continuous phase fluid compared to the confined geometries in case of Tjunction emulsification. Due to this the evolution of dispersed phase becomes different than the case of T-junction emulsion and the dependence of the process on different properties of both the phases also changes.

## **3.1. Growth and detachment of the droplets**

In membrane emulsification system, the flow rate of dispersed phase controls the droplet dynamics via its inertial force competing with the drag force imparted by the continuous phase and interfacial tension force. In order to investigate the effect of dispersed phase flow rate, simulations have been made for different values of *We* number by changing the dispersed phase velocity i.e. inertial force and keeping the surface tension force fixed. Before discussing the effects of *We*, the growth and detachment of the droplet for a constant values of *We* (0.0086) and *Ca* (0.028) at different time levels have been shown in Fig. 4. It has been observed that both the droplet grow at their respective micro-pore and detached by the

continuous phase at the same location. As the first droplet is being detached and carried away by continuous phase fluid, the second droplet starts to grow at the pore and the repetition of droplets detachment takes place periodically with constant volume of droplet. Simulation has also revealed that the growth rate of the droplet at the downstream micro pore is different than the upstream micro pore and the droplet at that pore requires more detachment time. The presence of upstream droplet changes the hydrodynamic effects and reduces the viscous drag force of continuous phase on the downstream droplet. Due to reduction in drag force, the surface tension force can hold the droplet for longer time and detachment takes place lately. Due to this the droplet size is greater than the droplet formed at the upstream pore. Moreover due to low drag force of the continuous phase, the inertia force of dispersed phase has some effect in forming the necking of the droplet in the downstream pore. The diameter of the droplet after detachment has been found as 41.3 µm, which is about 4.13 times the pore diameter. This ratio is within the range of 1-12 observed by Katoh et al. (1996).

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droplets decreases. It has been observed that in the transverse (*z*-) direction the droplet remains almost spherical throughout the formation process; therefore, the distance between

**Figure 5.** Growth of droplets at horizontal x-z plane at different time level, *We* = 0.0086, *Ca* = 0.0208

**Figure 6.** Growth of droplets at horizontal *x*-*z* plane for different values of *We*, *Ca* = 0.0208

To check the effect of *We* on the droplet growth along transverse direction, the simulated results of droplet growth at horizontal plane for different values of *We* and at different time levels have been shown in Fig 6. With the increase in *We* the droplet size increases and the distance between the two droplets decreases along the continuous phase fluid flow direction. At high value of *We* the necking phenomenon has been observed which makes the insufficient distance between the pore for avoiding coalescence. On the other hand at high value of *We* the growth of the droplet in transverse direction is same as the case in lower value of *We*. Thus the spacing of pores in transverse direction can be fixed based on droplet

In order to show the effect of dispersed phase flow rate in droplet dynamics, the droplet growth before the detachment has been shown in Fig.7 for different values of *We* (0.0021

With the fixed distance along cross-flow and transverse direction, the maximum porosity of the membrane can be calculated. The porosity is defined as the ratio of the total pore cross-

the pores in that direction can be fixed at seven times the pore diameter.

sectional area and the total membrane surface area.

diameter and irrespective of the dripping or jetting mode.

**3.2. Effect of dispersed phase flow rate** 

**Figure 4.** Growth and detachment of droplet at different time level, *We* = 0.0086, *Ca* = 0.0208

Thus the distance between the two pores should be at least five times the pore diameter for simultaneous growing of two spherical droplets without any hindrance from the neighboring pore. On the other hand the droplet deforms in the direction of flow of the continuous phase as shown in Fig. 4. Considering the deformation and growth of the droplet, it can be concluded that a distance of almost 10 times the pore diameter between the two pores in *x*-direction i.e. in cross-flow direction is needed to avoid contact, and thus avoid coalescence at two neighboring pores for this particular flow rate of dispersed phase and surface tension.

The investigation of droplet growth in transverse direction i.e. *z*-direction is important to design the pore distance in transverse direction. The growth of the droplet in a horizontal plane (*x*-*z*) at at *y/D0* = 25 µm has been shown in Fig. 5 at different time levels for *We* = 0.0086 and *Ca* = 0.0208. As the time progresses, the droplet at the first pore grows and deforms along the cross-flow fluid direction. Thus the distance between the two forming droplets decreases. It has been observed that in the transverse (*z*-) direction the droplet remains almost spherical throughout the formation process; therefore, the distance between the pores in that direction can be fixed at seven times the pore diameter.

**Figure 5.** Growth of droplets at horizontal x-z plane at different time level, *We* = 0.0086, *Ca* = 0.0208

With the fixed distance along cross-flow and transverse direction, the maximum porosity of the membrane can be calculated. The porosity is defined as the ratio of the total pore crosssectional area and the total membrane surface area.

**Figure 6.** Growth of droplets at horizontal *x*-*z* plane for different values of *We*, *Ca* = 0.0208

To check the effect of *We* on the droplet growth along transverse direction, the simulated results of droplet growth at horizontal plane for different values of *We* and at different time levels have been shown in Fig 6. With the increase in *We* the droplet size increases and the distance between the two droplets decreases along the continuous phase fluid flow direction. At high value of *We* the necking phenomenon has been observed which makes the insufficient distance between the pore for avoiding coalescence. On the other hand at high value of *We* the growth of the droplet in transverse direction is same as the case in lower value of *We*. Thus the spacing of pores in transverse direction can be fixed based on droplet diameter and irrespective of the dripping or jetting mode.

## **3.2. Effect of dispersed phase flow rate**

428 Numerical Simulation – From Theory to Industry

by Katoh et al. (1996).

and surface tension.

continuous phase at the same location. As the first droplet is being detached and carried away by continuous phase fluid, the second droplet starts to grow at the pore and the repetition of droplets detachment takes place periodically with constant volume of droplet. Simulation has also revealed that the growth rate of the droplet at the downstream micro pore is different than the upstream micro pore and the droplet at that pore requires more detachment time. The presence of upstream droplet changes the hydrodynamic effects and reduces the viscous drag force of continuous phase on the downstream droplet. Due to reduction in drag force, the surface tension force can hold the droplet for longer time and detachment takes place lately. Due to this the droplet size is greater than the droplet formed at the upstream pore. Moreover due to low drag force of the continuous phase, the inertia force of dispersed phase has some effect in forming the necking of the droplet in the downstream pore. The diameter of the droplet after detachment has been found as 41.3 µm, which is about 4.13 times the pore diameter. This ratio is within the range of 1-12 observed

**Figure 4.** Growth and detachment of droplet at different time level, *We* = 0.0086, *Ca* = 0.0208

Thus the distance between the two pores should be at least five times the pore diameter for simultaneous growing of two spherical droplets without any hindrance from the neighboring pore. On the other hand the droplet deforms in the direction of flow of the continuous phase as shown in Fig. 4. Considering the deformation and growth of the droplet, it can be concluded that a distance of almost 10 times the pore diameter between the two pores in *x*-direction i.e. in cross-flow direction is needed to avoid contact, and thus avoid coalescence at two neighboring pores for this particular flow rate of dispersed phase

The investigation of droplet growth in transverse direction i.e. *z*-direction is important to design the pore distance in transverse direction. The growth of the droplet in a horizontal plane (*x*-*z*) at at *y/D0* = 25 µm has been shown in Fig. 5 at different time levels for *We* = 0.0086 and *Ca* = 0.0208. As the time progresses, the droplet at the first pore grows and deforms along the cross-flow fluid direction. Thus the distance between the two forming

In order to show the effect of dispersed phase flow rate in droplet dynamics, the droplet growth before the detachment has been shown in Fig.7 for different values of *We* (0.0021 to 0.215) number. The qualitative difference in droplet growth for different values of *We* can be seen in the figure. Some key phenomena such as dripping at low *We* number, necking and jetting at higher *We* number have been observed. During the droplet formation in membrane emulsification process, the inertial force of the dispersed flow acts as detaching force, and acts against the attaching force of surface tension. When inertia force of the dispersed is less than the drag force or interfacial tension force, it cannot influence the droplet dynamics and the droplet growth and detachment are controlled by the drag force competing with the surface tension force. At the low value of *We* (*We* =0.0021) number, the droplet forms and breakups at the micro-pore which is termed as the dripping mode.

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*t* = 0.125 have been shown in Fig. 8. Recirculating flows have been observed inside the both

droplets and the center of recirculation is different in both the droplets.

**Figure 8.** Velocity vector at the central plane, *We* = 0.0086, *Ca* = 0.0208

**Figure 9.** Velocity vector at the horizontal plane, *We* = 0.0086, *Ca* = 0.0208

''shade'' of another droplet experiences a different velocity profile.

is visible from the velocity field, where velocity field is weak.

As the continuous liquid phase interacts the dispersed phase, it imparts a viscous drag force on the evolving interface between the two phases. The viscous drag force produces shear stress along the interface that faces the continuous phase fluid. This initiates the recirculation inside the both droplets. The acceleration of the dispersed phase out of the pore also affects the motion inside the forming drop, especially at an early stage of drop formation. The centre point of the rotational flow inside the drop is at the top of the drop, which is controlled by the above two factors. The dispersed phase inside the interface front finally flows along the continuous liquid phase and is accelerated by the viscous drag. From the velocity diagram it can be seen that the upstream droplet has disturbed the approaching velocity field for the downstream growing droplet. Thus the droplet that grows in the

The velocity field inside and outside the dispersed phase in horizontal plane (*x*-*z*) at the time level (0.125) for *W*e = 0.0086 has been shown in Fig. 9. The wake formed by the first droplet

**Figure 7.** Growth of the droplets for different values of *We*, *Ca* = 0.0208

At intermediate values of Weber number (0.0086 and 0.077), the point of droplet detachment has been moved away from the micro-pore, and formation of dispersed phase thread and necking have been observed. With the increase in inertial force of the dispersed phase, the effective pressure overcomes the capillary pressure inside the liquid thread leading to a stretched filament and also distends the droplet neck noticeably. With the increasing in dispersed phase flow rate further (0.215), two nodes form in the liquid filament and extension of the droplet neck occurs. The detachment point of the droplets moves further downstream from the pore. Thus jetting occurs and the droplet forms at the tip of the droplet. A decrease in the resultant droplet size can be observed. Thus at higher value of *We* number, due to formation of the jetting and the droplet formation from the tip, the distance of 10 pore diameter between the pores is not sufficient to avoid the contact and coalescence of two neighboring droplets.

## **3.3. Velocity field during droplet growth**

The droplet dynamics in membrane emulsification process is controlled by the evolving velocity field outside and inside the dispersed phase since the drag force is correlated to velocity. The velocity fields for *We* = 0.0086 in the central vertical plane (*x*-*y*) at the time level *t* = 0.125 have been shown in Fig. 8. Recirculating flows have been observed inside the both droplets and the center of recirculation is different in both the droplets.

**Figure 8.** Velocity vector at the central plane, *We* = 0.0086, *Ca* = 0.0208

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dripping mode.

of two neighboring droplets.

**3.3. Velocity field during droplet growth** 

to 0.215) number. The qualitative difference in droplet growth for different values of *We* can be seen in the figure. Some key phenomena such as dripping at low *We* number, necking and jetting at higher *We* number have been observed. During the droplet formation in membrane emulsification process, the inertial force of the dispersed flow acts as detaching force, and acts against the attaching force of surface tension. When inertia force of the dispersed is less than the drag force or interfacial tension force, it cannot influence the droplet dynamics and the droplet growth and detachment are controlled by the drag force competing with the surface tension force. At the low value of *We* (*We* =0.0021) number, the droplet forms and breakups at the micro-pore which is termed as the

**Figure 7.** Growth of the droplets for different values of *We*, *Ca* = 0.0208

At intermediate values of Weber number (0.0086 and 0.077), the point of droplet detachment has been moved away from the micro-pore, and formation of dispersed phase thread and necking have been observed. With the increase in inertial force of the dispersed phase, the effective pressure overcomes the capillary pressure inside the liquid thread leading to a stretched filament and also distends the droplet neck noticeably. With the increasing in dispersed phase flow rate further (0.215), two nodes form in the liquid filament and extension of the droplet neck occurs. The detachment point of the droplets moves further downstream from the pore. Thus jetting occurs and the droplet forms at the tip of the droplet. A decrease in the resultant droplet size can be observed. Thus at higher value of *We* number, due to formation of the jetting and the droplet formation from the tip, the distance of 10 pore diameter between the pores is not sufficient to avoid the contact and coalescence

The droplet dynamics in membrane emulsification process is controlled by the evolving velocity field outside and inside the dispersed phase since the drag force is correlated to velocity. The velocity fields for *We* = 0.0086 in the central vertical plane (*x*-*y*) at the time level

**Figure 9.** Velocity vector at the horizontal plane, *We* = 0.0086, *Ca* = 0.0208

As the continuous liquid phase interacts the dispersed phase, it imparts a viscous drag force on the evolving interface between the two phases. The viscous drag force produces shear stress along the interface that faces the continuous phase fluid. This initiates the recirculation inside the both droplets. The acceleration of the dispersed phase out of the pore also affects the motion inside the forming drop, especially at an early stage of drop formation. The centre point of the rotational flow inside the drop is at the top of the drop, which is controlled by the above two factors. The dispersed phase inside the interface front finally flows along the continuous liquid phase and is accelerated by the viscous drag. From the velocity diagram it can be seen that the upstream droplet has disturbed the approaching velocity field for the downstream growing droplet. Thus the droplet that grows in the ''shade'' of another droplet experiences a different velocity profile.

The velocity field inside and outside the dispersed phase in horizontal plane (*x*-*z*) at the time level (0.125) for *W*e = 0.0086 has been shown in Fig. 9. The wake formed by the first droplet is visible from the velocity field, where velocity field is weak.

### **3.4. Effect of continuous phase velocity**

To show the effect of continuous phase velocity, the simulation has been made for same range of Weber number (0.00 to 0.215) but with higher continuous phase flow rate (*Qcp* = 0.54 l/h). The comparison of droplet diameter for two values of continuous phase velocities i.e. flow rates has been shown in Fig. 10. It has been observed that for the higher continuous phase flow rate, the diameter of the droplets is smaller compared to lower continuous phase flow rate. At a particular value of continuous phase flow rate, the droplet diameter increases with the increase in droplet diameter, than decreases with the increase in droplet diameter due to jetting phenomenon. It can be seen that with the increase in continuous phase velocity, the droplet diameter decreases. Due to increase in velocity, the continuous phase fluid imparts higher drag force and droplet detaches within short interval. It has been also observed that the reduction of droplet diameter during transition from dripping to jetting is less in case of higher value of continuous phase velocity compared to its lower value.

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**Figure 11.** Detachment time of the droplet for different *We* number

sufficient to avoid the coalescence of the droplets at two neighbor pores.

**Figure 12.** Effect of surface tension on droplet diameter

To show the effect of surface tension on the droplet formation, the simulation has been made for different values of surface tension (*Ca* = 0.0208 to 0.0625) at constant value of inertial force. The droplet diameters for different values of *Ca* and two values of continuous phase flow rate have been shown in Fig. 12. With the decrease in surface tension force i.e. increase of *Ca*, the droplet diameter decreases. At low value of surface tension, the attaching surface tension force cannot hold the droplet for longer time against the other detaching forces leading to formation of droplet with smaller size. At very low value of surface tension (0.0008 N/m) even at low value of *We* (0.0021) the phenomenon shows the jetting behavior as shown in Fig. 13. Due to jetting phenomenon, the spacing between the micro-pore is not

**3.5. Effect of surface tension** 

**Figure 10.** Effect of continuous phase velocity on droplet diameter

The detachment time of droplets for two values of flow rates of continuous phase have been shown in Fig. 11. With lower value of continuous phase flow rate, the detachment time decreases exponentially with the increase in *We* number up to some value of *We* number, after that the decrease rate reduces. At a particular *We* number, the detachment time has been observed less for higher continuous phase flow rate compared to lower continuous phase flow rate. For lower flow rate of continuous phase, the detachment time decreases with the increase of *We* number, but it does not follow the same trend as for the lower flow rate.

**Figure 11.** Detachment time of the droplet for different *We* number

#### **3.5. Effect of surface tension**

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compared to its lower value.

rate.

**Figure 10.** Effect of continuous phase velocity on droplet diameter

The detachment time of droplets for two values of flow rates of continuous phase have been shown in Fig. 11. With lower value of continuous phase flow rate, the detachment time decreases exponentially with the increase in *We* number up to some value of *We* number, after that the decrease rate reduces. At a particular *We* number, the detachment time has been observed less for higher continuous phase flow rate compared to lower continuous phase flow rate. For lower flow rate of continuous phase, the detachment time decreases with the increase of *We* number, but it does not follow the same trend as for the lower flow

**3.4. Effect of continuous phase velocity** 

To show the effect of continuous phase velocity, the simulation has been made for same range of Weber number (0.00 to 0.215) but with higher continuous phase flow rate (*Qcp* = 0.54 l/h). The comparison of droplet diameter for two values of continuous phase velocities i.e. flow rates has been shown in Fig. 10. It has been observed that for the higher continuous phase flow rate, the diameter of the droplets is smaller compared to lower continuous phase flow rate. At a particular value of continuous phase flow rate, the droplet diameter increases with the increase in droplet diameter, than decreases with the increase in droplet diameter due to jetting phenomenon. It can be seen that with the increase in continuous phase velocity, the droplet diameter decreases. Due to increase in velocity, the continuous phase fluid imparts higher drag force and droplet detaches within short interval. It has been also observed that the reduction of droplet diameter during transition from dripping to jetting is less in case of higher value of continuous phase velocity

> To show the effect of surface tension on the droplet formation, the simulation has been made for different values of surface tension (*Ca* = 0.0208 to 0.0625) at constant value of inertial force. The droplet diameters for different values of *Ca* and two values of continuous phase flow rate have been shown in Fig. 12. With the decrease in surface tension force i.e. increase of *Ca*, the droplet diameter decreases. At low value of surface tension, the attaching surface tension force cannot hold the droplet for longer time against the other detaching forces leading to formation of droplet with smaller size. At very low value of surface tension (0.0008 N/m) even at low value of *We* (0.0021) the phenomenon shows the jetting behavior as shown in Fig. 13. Due to jetting phenomenon, the spacing between the micro-pore is not sufficient to avoid the coalescence of the droplets at two neighbor pores.

**Figure 12.** Effect of surface tension on droplet diameter

From the present investigation, it can be seen that dripping to jetting transition can be possible in two ways: one at constant surface tension while varying the inertial force (varying the dispersed phase flow rate) and another at constant inertial force while varying the surface tension force. A qualitative difference in flow pattern in both the transitions has been observed. During dispersed phase controlled transition, the diameter of the drop increases first then decreases rapidly while in surface tension controlled transition, the size of the droplet continuously decreases (Fig. 10 and Fig. 12).

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operating conditions and properties of two-phase liquids in the emulsification system. At constant continuous phase flow rate, the dripping phenomenon occurs at low dispersed phase velocity i.e. at low *We* number and transits towards jetting with the increase in dispersed phase flow rate. At constant continuous phase flow rate, with the increase in dispersed phase flow rate, the droplet size increases initially but decreases as the system transits towards jetting. At constant dispersed phase flow rate, with the increase in continuous phase flow rate, the droplet size decreases and also detachment time. Two ways of dripping to jetting transition have been observed, one with the increasing dispersed phase flow rate at constant continuous phase flow rate and other way is reducing the surface tension at constant dispersed phase flow rate. Both the transitions show different physical structures. The effect of inertia force has been observed the negligible for high value of surface tension and significant for lower surface tension value. The distance between the pore in continuous flow direction depends upon the operating parameters leading to dripping to jetting mode but the pore distance in transverse direction is not affected by the dripping or jetting behavior. Thus at higher value of *We* number, due to formation of the jetting and the droplet formation from the tip, the distance of 10 pore diameter between the pores is not sufficient to avoid the contact and coalescence of two neighboring droplets. The droplet size in the process scales with four main forces: drag forces imparted by the continuous phase, inertia force imparted by dispersed phase, surface tension force and the gravity force. In dripping mode inertial force of dispersed phase has negligible effect as the surface tension and drag force are dominant whereas in jetting mode inertial force of dispersed phase and surface tension force take part in the droplet formation. The evolving vortices are observed in the initial stage of dripping mode but it disappears in later stage. Three important factors must be considered in order to obtain a high production rate in membrane emulsification. (i) A proper combination of continuous phase, dispersed phase flow rate and surface tension so that droplet formation is made just before the starting of surface instability in jetting region. (ii) A proper distribution of pores so that coalescence of droplets does not occur during the droplet growth. (iii) The crossflow velocity must be high enough to provide a sufficient wall shear stress at the membrane surface to transport the drops away from the pore opening and, thus avoid the static hindrance and drop coalescence.

**Author details** 

Manabendra Pathak

**Nomenclature** 

*India* 

*Department of Mechanical Engineering, Indian Institute of Technology Patna,* 

*An* Area of the droplet neck ((*μ*m)2)

*Ca* Capillary number

*b* Width of the continuous phase channel (*μ*m)

Surface tension force is dominant over the inertial force in dripping mode, hence the droplet size increases and in jetting mode inertial force overcomes the surface tension force for which drop size decreases. In surface controlled breakup, the drag force and interfacial force take part in the droplet formation.

**Figure 13.** Jetting behavior at low surface tension value
