**Author details**

434 Numerical Simulation – From Theory to Industry

take part in the droplet formation.

**3.6. Effect of viscosities** 

present work.

**4. Conclusions** 

of the droplet continuously decreases (Fig. 10 and Fig. 12).

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

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

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

The viscosities of both the phases also effect the droplet growth and deformation as discussed in introduction section. The drag force which detaches the droplet depends upon the viscosity ratio of dispersed phase and continuous phase. At a constant value of continuous phase flow rate, the drag force increases with the increase in viscosity ratio up to some extent. After that, the drag force becomes independent of the viscosity ratio. Pathak (2011) has observed negligible effect of dispersed phase viscosity in the membrane emulsification system. Moreover it has been also observed that the effect viscosity on droplet dynamics is not very significant if the ratio of dispersed phase and continuous phase viscosity is high (van Dijke et al., 2010). Since the viscosity ratio of dispersed phase and continuous phase fluid is above 3, the effects of viscosities have not been investigated in the

Droplet formation in a two-pore membrane emulsification has been numerically investigated in this chapter. The dynamics of droplet formation has been investigated by solving the two-phase governing equations using VOF method. The effects of various parameters, viz., dispersed and continuous phase flow rate, surface tension and viscosities on the droplet dynamics have been investigated. The dynamics of evolution of dispersed phase and droplets formation show the dripping and jetting behavior depending upon the Manabendra Pathak *Department of Mechanical Engineering, Indian Institute of Technology Patna, India* 

### **Nomenclature**



Numerical Simulation of Droplet Dynamics in Membrane Emulsification Systems 437

Abrahamse, A. J., van der Padt, A. & Boom, R. M. (2002). Analysis of droplet formation and interactions during cross-flow membrane emulsification, J. Membr. Sci. 204: 125-137.

Brackbill, J. U., Kith, D. B. & Zemach, C. (1992). A Continuum method for modeling surface

Churn, I. L., Glimm, J., McBtyan, O., Plohr, B. & Yanic, S. (1986). Front tracking for gas

Daly, B. J. (1967). Numerical study of two fluid Rayleigh-Taylor instabilities. Phys. Fluids

Glimm, J. & McBryan, O. A. (1985). A computational model for interfaces. Adv. Appl. Math.

Harlow, F. H. & Welch, J. E. (1966). Numerical study of large-amplitude free-surface motion.

Hirt, C. W. & Nichols, B. D. (1981). Volume of fluid (VOF) method for the dynamics of free

Jacqumin, D. (1996). An energy approach to the continuum surface method. AIAA paper,

Joscelyne, S. M. & Trägårdh, G. (1991). Food emulsions using membrane emulsification:

Joscelyne, S. M. & Trägårdh, G. (2000). Membrane emulsification – a literature review. J.

Karbstein, H., & Schubert, H. (1995). Developments in the continuous mechanical production of oil-in-water macro-emulsions. Chem. Engg. Process. 34: 205-211. Katoh, R., Asano, Y., Furuya, A., Sotoyama, K. & Tomita, M. (1996). Preparation of food emulsions using membrane emulsification system. J. Membr. Sci. 113: 131-135. Kobayashi, I., Nakajima, M. & Mukataka, S. (2003). Preparation characteristics of oil-inwater emulsions using differently charged surfactants in straight-through microchannel

Kobayashi, I., Uemura, K. & Nakajima, M. (2006). CFD study of the effect of a fluid flow in a channel on generation of oil-in-water emulsion droplets in straight-through

Luca, G. D., Sindona, A., Giorno, L. & Drioli, E. (2004). Quantitative analysis of coupling

Nichols, B. D. & Hirt, C. W. (1975). Methods for calculating multi-dimensional, transient free surface flows past bodies. Technical Report LA-UR-75-1932, Los Alamos National

Noh, W. F. & Woodward, P. R. (1976). SLIC (simple line interface methods). In: Voore A I V,

Ohta, M., Kikuchi, D., Yoshida, Y. & Sussman, M. (2007). Direct numerical simulation of the slow formation process of single bubbles in a viscous liquid. J. Chem. Eng. Jpn. 40: 939-943.

effects in cross-flow membrane emulsification. J. Membr. Sci. 229: 199-209.

P. J. Zandbergen P J, editors. Lecture Notes in Physics 59: 330-340.

conditions for producing small droplets. J. Food Eng., 39: 59-64.

microchannel emulsification. J. Chem. Eng. Jpn. 39: 855–863.

Antanovskii, L. K. (1995) A Phase Field Model of Capillarity. Phys. Fluids 1: 747-753.

**5. References** 

10:297-307.

6: 422-435.

96-0858.

Phys. Fluids 9: 842-851.

Membr. Sci.169: 107-117.

Laboratory. NM.

tension. J Comp. Phys 100: 335–354.

dynamics. J. Comp. Phys. 62: 83-110.

boundaries. J. Comp. Phys. 39: 201-225.

emulsification. Colloids Surf. A 229: 33–41.

#### *Greek letters*


#### **5. References**

436 Numerical Simulation – From Theory to Industry

*Cd* Drag coefficient

*f* Surface force (N)

*k1* Constant *k2* Constant

*s*

*cp* 

*dp* 

*cp* 

*dp* 

*Greek letters* 

*D0* Diameter of micro-pore (*μ*m) *Dp* Diameter of the droplet (*μ*m)

*F* Volume fraction of dispersed phase fluid

*h* Height of the continuous phase channel (*μ*m)

*pc* Pressure of the continuous phase fluid flow (Pa) *pd* Pressure of the dispersed phase fluid flow (Pa)

*R* Viscosity ratio of continuous phase to dispersed phase fluid

*ucp* Velocity of continuous phase at the inlet to the channel (m/s)

*v\** Local velocity of continuous phase fluid in the channel (m/s) *v0* Velocity of dispersed phase at the inlet to the micro-pore (m/s)

*Qcp* Flow rate of continuous phase fluid (m3/s) *Qdp* Flow rate of dispersed phase fluid (m3/s)

*<sup>Γ</sup>* Position vector of interface of the two phase

Viscosity of continuous phase fluid (m2/s)

Density of continuous phase fluid (kg/ m3)

Density of dispersed phase fluid (kg/ m3)

Surface tension coefficient (N/m)

Viscosity of dispersed phase fluid (m2/s)

Viscosity ratio of dispersed phase to continuous phase fluid

*g* Acceleration due to gravity (m/s2)

*p* Pressure of the flow (Pa)

*p<sup>γ</sup>* Capillary pressure (Pa)

*Re* Reynolds number of the flow *Rp* Reynolds number of the drop *u* Velocity in *x*- direction (m/s)

*v* Velocity in *y*- direction (m/s)

*V* Velocity vector

*We* Webber number

*<sup>φ</sup>* Level set function

Surface curvature (m-1)

*Vdr* Droplet volume((*μ*m)2) *w* Velocity in z- direction (m/s)

*Dh* Hydraulic diameter of the continuous phase channel (*μ*m)


Osher, S. & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Comp. Phys. 79:12-49.

**Section 4** 

**Industrial Applications** 


**Industrial Applications** 

438 Numerical Simulation – From Theory to Industry

Meth. Fluids. 24: 671-691.

Food Sci & Technol. 14: 9-16.

Mechanics, Belgium.

membranes. Colloids Surf A 152: 103-109.

emulsification. J. Phys. Chem. B 106: 9405–9409.

emulsification, Microfluid Nanofluid. 9**:** 77–85.

621–635.

Osher, S. & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed:

Pathak, M. (2011). Numerical simulation of membrane emulsification: effect of flow properties in

Peng, S., J. & Williams, R. A. (1998). Controlled production of emulsions using a crossflow membrane, Part I: droplet formation from a single pore. Chem. Eng. Res. Des., 76: 894-901. Rider, W. J. & Kothe, D. B. (1998). Reconstructing volume tracking. J. Comp. Phys 141:112-52. Rudman, M. (1997). Volume tracking methods for interfacial flow calculations. Int. J. Num.

Sang, L., Hong, Y. & Wang, F. (2009). Investigation of viscosity effect on droplet formation in T-shaped microchannels by numerical and analytical methods. Microfluid Nanofluid 6:

Schröder, V. & Schubert, H. (1999). Production of emulsions using microporous, ceramic

Schubert, H., Ax K, & Behrend, O. (2003). Product engineering of dispersed systems. Trends

Sugiura, S., Nakajima, M., Kumazawa, N., Iwamoto, S. & Seki, M. (2002). Characterization of spontaneous transformation-based droplet formation during microchannel

Sussman, M., Smith, K. M., Hussaini, M. Y., Ohta, M. & Zhi-Wei, R. (2007). A sharp interface

Timgren, A., Trägårdh, G. & Trägårdh. C. (2009). Effects of pore spacing on drop size during cross-flow membrane emulsification—A numerical study. J. Membr. Sci. 337: 232-239. Tryggvason, G., Bunner, B., Ebrat, O. & Tauber, W. (1998). Computations of multi-phase flow by a finite difference/front tracking method. I. Multi-fluid flows. In Lecture Notes for the 29th Computational Fluid Dynamics Lecture Series, Karman Institute for Fluid

van Dijke, K., Kobayashi, I., Schroe¨n, K., Uemura, K., Nakajima, M. & Boom. R. (2010). Effect of viscosities of dispersed and continuous phases in microchannel oil-in-water

Vladisavljevic, G. T. & Schubert, H. (2003). Preparation of emulsions with a narrow particle size distribution using microporous α-alumina membranes. J. Disp. Sci. Technol. 24: 811-819. Vladisavljevic, G. T., Lambrich, U., Kakajima, M. & Schubert, H. (2004). Production of o/w emulsion using SPG membranes, ceramic α-aluminium oxide membranes, microfluidizer

anda silicon microchannel plate - a comparative study. Colloids Surf. A 232: 199-207. Youngs, D. L. (1982). Time-dependent multi-material flow with large fluid distortion. In: Morton K W, Baines M J, editors, Numerical Methods for Fluid Dynamics, Academic Press 1982.

method for incompressible two-phase flows. J. Comp. Phys. 221: 469-505.

algorithms based on Hamilton-Jacobi formulation. J. Comp. Phys. 79:12-49.

the transition from dripping to jetting. J. Membr. Sci. 382: 166-176.

**Chapter 20** 

© 2012 Toader et al., 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,

**Numerical Methods for Analyzing** 

Dumitru Toader, Stefan Haragus and Constantin Blaj

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/49953

**1. Introduction** 

DeCarlo & Lin, 2001).

Mandache, 2004; Istrate et al., 2009).

stiffness ratio dynamical systems).

conceived, allow a better integration of these subsystems.

**the Transients in Medium Voltage Networks** 

The analysis of transient regimes in electric networks is a complex problem due to great number of elements, some nonlinear, as well as, due to the nonsinusoidal variation of

There are available a lot of simulation programs for transients. One of the first of them, dedicated to electro energetic systems, was initiated by H.W.Dommel. Electromagnetic Transient Program (EMTP) was bought by Bonneville Power Administration (USA) (Dommel, 1995). Later on, from this initial program were developed several versions, such as MicroTrans, EMTP-RV, ATP (Alternative Transients Program), PSCAD (Chuco, 2005;

Comparing some of the main simulation programs is not very conclusive, because each one of them are used in some versions trying to solve a large group of problems (Iordache &

Interface becomes more and more friendly, the library becomes larger and the facilities for creating own models are simpler to use. The speed of calculations and the storage capacity do no more represent a problem. It is possible now to implement some complicated algorithms able to solve problems with high speed in the variation of variables, able to solve all kinds of nonlinearities or to handle eigenvalues highly distanced one from another (large

An important criterion for the selection is represented by the integration of complex dynamical systems like rotating electric machines, relays, power electronic devices, FACTS, controllers etc. in the simulation. Some of the programs, depending on the way that were

and reproduction in any medium, provided the original work is properly cited.

currents and voltages. Numerical simulation can solve properly such a problem

Dumitru Toader, Stefan Haragus and Constantin Blaj

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/49953
