14th European Conference on Mixing Warszawa, 10-13 September 2012

MODELLING OF UNSTEADY INTERPHASE MASS TRANSFER FROM ELONGATED FLUID DROPS AT CAPILLARY NUMBERS

CLOSE TO A CRITICAL LEVEL

Antoni Rożeń

Warsaw University of Technology, Faculty of Chemical and Process Engineering, Waryńskiego 1, 00-645 Warszawa, Poland

a.rozen @ichip.pw.edu.pl

Abstract. Single component mass transfer between a liquid drop elongated in an axi-symmetric hyperbolic flow and a surrounding liquid has been studied numerically. Transient values of Sherwood numbers were calculated for a wide range of Peclet number and a viscosity ratio of the contacted liquid phases. The computations were conducted for the largest stable drops in the axi-symmetric hyperbolic flow. An effect of liquid viscosity on a molecular diffusivity of a tracer substance was also accounted for. Results of the exact CFD simulations of mass transfer were used to find out the limitations of a micromixing model. Keywords: mass transfer, extraction, liquid-liquid systems, deformed drop, laminar flow.

1. INTRODUCTION Mixing on the molecular level in liquid-liquid systems depends on the rate of mass

transfer of species (reactants, catalysts, dyes, fillers etc.) between a dispersed phase and a continuous phase. When the contacted liquid phases are very viscous their mixing is usually carried in laminar flow to avoid excessive power consumption and thermal or mechanical degradation of a mixture. A local fluid flow and consequently convective mixing within and outside dispersed filaments or droplets and depend on: magnitude of viscous and capillary forces, difference in viscosity and density of the mixed liquids [1]. Deformation of the dispersed filaments and drops controls growth of the interfacial surface area and reduction of local segregation scales. It also accelerates diffusive mass transport of mixture components [2]. When the dispersed drops become small and the curvature of the interfacial surface grows capillary forces retard deformation. In these conditions convective mixing is sustained by liquid circulation developing in the droplets. This circulation becomes accelerated when the dispersed phase viscosity is lower than the continuous phase viscosity [3,4]. Viscosity of the liquid phases also affects the coefficients of molecular diffusivity of mixture components and consequently influences the rate of mass transfer.

Mentioned above mixing mechanisms were accounted for during calculations of the rate of mass transfer of an inert tracer from an initially spherical liquid drop to a liquid matrix. The drop containing the diffusive tracer was elongated in an axi-symmetric hyperbolic flow. This type of flow is the most energetically effective in creating the new contact surface area and has not be yet investigated in studies concerned with determining mass transfer coefficients for single drops; bibliography can be found in [4]. A model entirely based on computation fluid dynamics and a multi-dimensional micromixing model were used independently to

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calculate transient Sherwood numbers characterizing the rates of mass transfer for the largest stable drops.

2. MODELLING OF MASS TRANSFER Unsteady isothermal mass transfer of a diluted tracer substance from the single liquid

drop to the surrounding liquid has been studied according to the following assumptions: both liquid phases are incompressible (ρ=const) and Newtonian (μ=const), inertia and buoyancy forces are small compared to viscous and capillary forces, the physical properties of both phases including interfacial tension (σ) and molecular diffusivities (D) are independent of the local liquid composition (c), the tracer is well soluble in both phases, there is no interfacial resistance to mass transfer and no effect of mass transfer on a flow field.

The single liquid drop is deformed in an external axi-symmetric stagnation flow, which far from the drop is characterized be the constant scalar gradient (α): 2, 2,x y zv x v y v zα α α∞ ∞ ∞= − = − = (1) The shear rate in this flow field is proportional to the velocity gradient 3γ α= (2) The fluid velocity in the liquid matrix (continuous phase, k=1) surrounding the drop (dispersed phase, k=2) is governed by the continuity and Stokes equations: ( ) 0kv∇⋅ = (3)

( ) 0kτ∇ ⋅ = (4) The stress tensor is related to hydrostatic pressure and the velocity gradient tensor

( )( ) ( ) ( ) ( )2T

k k k k kp I v vτ μ ⎡ ⎤= − + ∇ + ∇⎢ ⎥⎣ ⎦ (5)

At the liquid-liquid interface the fluid velocity vector and the stress tensor satisfy boundary conditions: (1) (2)v v= (6)

( ) ( )(2) (1) ˆ ˆ ˆn n nτ τ σ− ⋅ = ∇ ⋅ (7)

where n̂ is a unit vector perpendicular to the contact surface.

The unsteady convective and diffusive transport of the tracer substance is governed by the local material balance equation

( )( ) 2( ) ( ) ( ) ( )

kk k k k

cv c D c

t∂

+ ⋅∇ = ∇∂

(8)

with the initial condition (t=0) (1) (1) (2) 00 ,c c c c∞= = = (9)

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and boundary conditions at the liquid-liquid interface: ( )(2) (2) (1) (1) ˆ 0D c D c n∇ − ∇ ⋅ = (10)

(2) (1)c mc= (11) where the tracer distribution coefficient m is set to one.

The moving boundary problem defined by Eqs (3-7) was solved by means of CFD software (FIDAP 8.7) using a finite element method allowing to precisely track the liquid-liquid interface and its curvature in time and space. A computational mesh used for the spatial discretization of the problem consisted of 1702 quadric (9 node) elements. The numerical method applied in this study does not require any heuristic procedure to reconstruct the interfacial surface position as it must be done when using a Volume of Fluid Method.

The advection-diffusion Eq (8) is decoupled from the continuity and Stokes Eqs (3-4) according to the initial problem definition. Consequently, the tracer concentration field was updated at each time step by solving numerically the system of Eqs (8-11) with known velocity field and liquid-liquid interface position. Once the concentration field of the tracer substance was found the instantaneous mass transfer coefficient could be computed from the overall mass conservation balance

(2) (2)

(2) (1)

1V dck

S c mc dt∞=−

(12)

where V(2) and S are the drop volume and the interfacial surface area.

The exact numerical solution of system of Eqs (3-11) requires long CPU times due to convergence and stability problems caused by the presence of the moving internal boundary. To avoid this one can use an approximate analytical solution for the continuity and Stokes equations. In this study a model of Yu and Bousmina [5] was used to find out the deformation of the single liquid drop and fluid velocity inside the drop, whereas Eshelby’s model was adopted to find out the liquid velocity outside the drop [6]. An implementation of the approximate flow model into the advection-diffusion Eq (8) significantly reduces CPU time and allows to run computations even in the case of a very large drop deformation.

3. RESULTS

Simulations of mass transfer of the tracer substance from the initially spherical liquid drop into the surrounding liquid were conducted on condition that capillary forces prevent infinite drop deformation by viscous forces. Relation between these two type of forces acting on the drop can be characterized by the capillary number 0 (1)Ca aγ μ σ= (13) where a0 is the initial drop radius. There is a critical value of this number (Cacrit) for a given type of flow and a ratio of viscosity of liquid phases. Exceeding this value leads to the drop destabilization and break-up [7]. Table 1 presents critical capillary numbers determined from CFD simulations for the axi-symmetric hyperbolic flow defined by Eq (1). Table 1. Critical capillary numbers for the axi-symmetric hyperbolic flow . μ(2)/μ(1) 0.1 0.2 0.5 1 2 5 10 Cacrit 0.30 0.27 0.23 0.21 0.19 0.18 0.175

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Normalized velocity vectors ( 0/( )v aα ) obtained for 1t γ and Ca=Cacrit are plotted in Figs 1 and 2. An intensive liquid circulation can be seen for μ(2)/μ(1)=0.1 inside the stable drop (Fig. 1). On the contrary, there is practically no flow inside the stable drop for μ(2)/μ(1)= 10 (Fig. 2). Changing liquid viscosity may also affect the coefficients of molecular diffusion. In this study the ratio of the tracer diffusivities in both phases was found from Stokes-Einstein equation

(1) (1) (2) (2)

B B

D Dk T k Tμ μ

= (14)

where kB is Boltzman constant. Hence, increasing the drop viscosity may slow down both convective and diffusive mass transport. In general relations between convective, diffusive and overall mass transfer rates are characterized by Peclet and Sherwood numbers: 2

0 (1)(2 )Pe a Dγ= (14) 0 (1)2Sh a k D= (15)

Figures 3 and 4 show contours of dimensionless tracer concentration (c/c0) obtained for two different viscosity ratios at time when the average tracer concentration inside the drop was reduced by half. The main resistance to mass transfer is localized in the continuous phase for μ(2)/μ(1)=0.1 (Fig. 3), while for μ(2)/μ(1)=10 its localized in the dispersed phase (Fig. 4). It also takes a significantly shorter time to half the mean tracer content in the less viscous drop.

Figure 1. Normalized velocity vectors for the stabilized drop: μ(2)/μ(1)=0.1, Ca=0.3.

Figure 2. Normalized velocity vectors for the stabilized drop: μ(2)/μ(1)=10, Ca=0.175.

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Transient Sherwood numbers are presented in Figs 5 and 6 for three different values of Peclet number and two different viscosity ratios. First, in all cases a constant value of Sherwood number is reached after a long time; this is not observed for small Peclet numbers (Pe<1). Secondly, when Peclet number is increased, Sherwood number also increases and the gain is proportional approximately to the square root of Peclet number, which is in good agreement with Levich theory [3]. Thirdly, increasing the drop viscosity and keeping other parameters constant reduces the coefficient of mass transfer because circulation flow and molecular diffusion become less effective. Finally, the results of full CFD simulations and the micromixing model predictions are in good agreement for Pe≤100. At higher Peclet numbers mass transport becomes accelerated by the liquid circulation inside the drop, while the approximate analytical solution of the continuity and Stokes equations, implemented in the micromixing model, does not account for it.

Further comparison of the results of the exact CFD model with the results of the micromixing model is shown in Fig. 7, where the limiting values of Sherwood number (Sh∞), obtained for very long process time, are plotted vs. the viscosity ratio of the liquid phases. As it can be seen, the limiting Sherwood numbers predicted by micromixing model are approximately two times smaller than those obtained from the CFD model. It proves that neglecting the presence of liquid circulation in the drop, stabilized by capillary forces, leads to underestimation of Sherwood number for long mixing times. However, the micromixing model still correctly predicts the tendency of the limiting Sherwood number to decrease with

Figure 6. Effect of Peclet number on transient Sherwood number: μ(2)/μ(1)=10, Ca=0.175.

Figure 5. Effect of Peclet number on transient Sherwood number: μ(2)/μ(1)=0.1, Ca=0.3.

Figure 4. Tracer concentration: μ(2)/μ(1)=10, Ca=0.175, Pe=1000, 63.28tγ = .

Figure 3. Tracer concentration: μ(2)/μ(1)=0.1, Ca=0.3, Pe=1000, 6.84tγ = .

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the increase of the viscosity ratio. One should expect a better conformity between two models in a case when the drop deformation cannot be stopped by capillary forces (Ca>Cacrit).

Figure 7. Effect of the viscosity ratio on the limiting Sherwood number: Ca=Cacrit, Pe=1000, 1tγ .

4. CONCLUSIONS Two methods of numerical simulation of mass transfer of the tracer substance from the

single liquid drop, stabilized by capillary forces in the external axi-symmetric hyperbolic flow, have been presented. The first one was based on the exact numerical solution of the system of the continuity, Stokes and material balance equations (CFD model). The second one was based on the implementation of the approximate analytical solution of the continuity and Stokes equations into the advection-diffusion equation and solving the latter numerically (micromixing model). Both models allow to account for the effects of: interfacial tension, difference in viscosity of liquid phases and difference in the tracer molecular diffusivity across the liquid-liquid interface on the rate of mass transfer. They can be successfully used in cases with resistance to mass transfer on both sides of the interface. However, the micromixing model cannot correctly predict the mass transfer coefficient for Peclet numbers exceeding 100, because of the inaccuracies in the flow field approximation. Further work will focus on overcoming this limitation.

5. REFERENCES

[1] Rallison J.M., Acrivos A., 1978. “A numerical study of the deformation and burst of a viscous drop in an extensional flow”, J. Fluid Mech., 89, 191-200. [2] Rożeń A., 2010. “Effect of interfacial tension on micromixing in laminar flow”, Proc. of 19th Int. Congress of Chem. and Proc. Eng., CHISA (Prague, 28.08-01.09), Czech Republic. [3] Levich V.G., 1962. Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, New York. [4] Ubal S., Harrison C.H., Grassia P., Korchinski W.J., 2010. “Numerical simulation of mass transfer in circulating drops”, Chem. Eng. Sci., 65, 2934-2956. [5] Yu W., Bousmina M., 2003. “Ellipsoidal model for droplet deformation in emulsions”, J. Rheol., 47, 1011-1039. [6] Eshelby J.D., 1959. “The elastic field outside an ellipsoidal inclusion”, Proc. R. Soc. Lond. A, 252, 561-569. [7] Grace H.P., 1982. “Dispersion phenomena in highly viscous immiscible fluid systems and application of static mixers as dispersion devices”, Chem. Eng. Commun., 14, 225-277.

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