**2.4. Numerical method**

426 Numerical Simulation – From Theory to Industry

\*

introduced and is defined as:

Froude number (*Fr*). They are defined as:

are:

**2.3. Boundary and initial conditions** 

Various types of boundary conditions have been used in the simulation of the membrane emulsification. A fully developed laminar duct flow has been considered at the inlet to the continuous phase channel and the flow has been assumed to be dominant along the stream direction thus: *cp u u* , *v* = 0, *w* = 0. At the inlet of continuous phase, the value of volume fraction has been set to zero (*F* = 0). At the inlets to the micro-pores, a velocity inlet type boundary condition has been used: *v* = *v0*. The other components of the velocity have been put as *u* = 0, *w* = 0 and the value of volume fraction has been set to *F* = 1. Outflow boundary condition has been put at the outlet of the flow channel. Symmetry boundary conditions have been put at both sides of the computational domain. The top of the channel has been treated as walls where no-slip and impermeability boundary conditions have been set. At the starting of the computation, the whole flow domain has been assumed to be filled up with the continuous phase fluid. The static contact angle between the two phases and the wall influences the droplet growth in the dispersed and continuous phase channel. It has been observed that the effect of contact angle in the droplet formation is negligible when its value is greater than 165o (Sang et al., 2009). The

value of static angle used in the present simulation has been set to 170o.

To reduce the number of dependable variables, the governing equations have been expressed in dimensionless form. The diameter of the pore ( *D*<sup>0</sup> ) has been used as length scale and the average velocity ( <sup>0</sup> *v* ) of the dispersed phase has been used as velocity scale for making non-dimensional form of the equation. The non-dimensional equations

> \* \* \* \* .( ) 0 *V*

1 1 . .( ) . Re

In above starred quantities are non-dimensional parameters. The three non-dimensional numbers appeared in the problem are: Reynolds number (*Re*) , Weber number (*We*) and

> 2 0 0 ; *dpv D*

The Reynolds numbers (*Re*) is defined as the ratio of viscous force to the inertia force, Weber number (*We*) is defined as the ratio of inertia force relative to the surface tension force and Froude number (*Fr*) is defined as the ratio of inertia to gravity force. In the present work to incorporate the effect of the continuous phase fluid, the Capillary number has been

(22)

*<sup>v</sup> Fr gD*

. *<sup>F</sup> F* 

(23)

*t* 

\* \* \*\* \* \* \*

*t Fr We*

*We*

*<sup>V</sup> <sup>T</sup> VV p V V*

0 0 ; *dp dp*

 *v D* 

*Re*

Commercial code Ansys Fluent (V12) based on finite volume method has been used in the simulation. The momentum and volume fraction transport equation have been discretized with 2nd order upwind scheme. The PISO (pressure implicit with splitting of operators) algorithm has been used for pressure correction. The VOF/CSF techniques have been used to track the fluid interface between the two immiscible fluids. A geometry reconstruction scheme has been used in the simulation to avoid the diffusion at the interface. The interface was reconstructed by the piecewise-linear interface calculation (PLIC) technique (Youngs, 1982). The unsteady term was treated with first-order implicit time stepping. Simulations were made with very small time steps (~10-7 s). The solutions have been assumed to be converged and therefore iterations have been terminated when the normalized sum of residual mass was less than 10-4 and variation of other variables in successive iteration was less than 10-2. A non-uniform grid was used in the simulation where grids were clustered near the walls and the injection portion of the dispersed phase. The channel was decomposed into 12 105 numbers of control volumes and the pore with 12 103 after a grid independent study.
