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Mass Transfer from Ensembles of Newtonian Fluid Spheres at Moderate Reynolds and Peclet Numbers
MASS TRANSFER FROM ENSEMBLES OF NEWTONIAN FLUID SPHERES AT MODERATE REYNOLDS AND PECLET NUMBERS N. Kishore1, R. P. Chhabra1, and V. Eswaran2 1
Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India. Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India.
2
Abstract: In this work, the rate of mass transfer from an ensemble of mono-size spherical Newtonian droplets (free from surfactants) to a Newtonian continuous phase has been numerically studied at moderate Reynolds and Peclet numbers. A simple spherical cell model (so-called free surface cell model) has been used to account for inter-drop hydrodynamic interactions. Extensive numerical results have been obtained to elucidate the effects of the Reynolds number (Reo), the ratio of internal to external fluid viscosity (k), the volume fraction of the dispersed phase (1) and the Schmidt number (Sc) on the local and average Sherwood number (Sh) over the ranges of conditions: 1 Reo 200, 0.2 1 0.6, 0.1 k 50 and 1 Sc 10 000. It has been observed that the effects of viscosity ratio on the local and average Sherwood number is less significant for small values of the Peclet number (Pe) for all values of dispersed phase concentration. As the value of the viscosity ratio increases, the average Sherwood number decreases for all values of the droplet concentration and the Reynolds number. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer in a liquid–liquid system in a new application. However, it is also appropriate to add here that at higher concentrations, fluid spheres interact significantly, deform and coalesce. All these effects are neglected in this study. Therefore, the present results are valid only for dilute to moderate concentration of the dispersed phase. Keywords: ensemble; fluid sphere; viscosity ratio; mass transfer; Sherwood number.
Correspondence to: Professor R. P. Chhabra, Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India. E-mail: [email protected]
DOI: 10.1205/cherd06250 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, August 2007 # 2007 Institution of Chemical Engineers
INTRODUCTION
applications. In most applications, often one fluid is dispersed in the form of droplets moving in another immiscible fluid which leads to enhancement of the rate of mass transfer between the two phases. Some of these applications include, liquid–liquid extraction, distillation, gas absorption, enhanced oil recovery in petroleum industries, production of polymeric alloys and emulsions in paint and detergent industries, fermentation broths, wastewater treatment, and so on (Schramm, 2005). Over the years, extensive literature has been reported on the motion of and mass transfer from single bubbles and drops in Newtonian liquids which has been reviewed thoroughly by Clift et al. (1978), Michaelides (2006) and Chhabra (2006). Although, the detailed kinematics of such studies related to single bubbles and drops provides useful information about the basic underlying physical phenomena, often one encounters clusters of bubbles and drops in chemical and processing industries as mentioned above. Therefore, an adequate
The motion of and mass transfer from drops to viscous liquids is ubiquitous in the chemical, biochemical, and polymer processing industries. In most applications, one encounters clusters of drops, the growth or collapse of which are directly influenced by the rate of mass transfer between the two phases. The rate of mass transfer from a single fluid sphere moving in a continuous fluid medium is greatly influenced by the motion inside the fluid sphere. Experiments have shown that the rate of mass transfer from circulating bubbles and drops is much larger than that from non-circulating bubbles and drops due to the effect of internal circulation on the external flow field. It is readily conceded that one often encounters ensembles of droplets rather than a single drop in most industrial applications. The mass transfer from ensembles of fluid spheres to another immiscible medium is an idealization of many industrially important chemical and processing 1203
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understanding of the rate of mass transfer from clusters of drops is a prerequisite to the understanding and rationalizing the overall efficiency of the contacting equipment. This information can be conveniently expressed using dimensionless parameters such as the Sherwood number, Reynolds number, Schmidt number, viscosity ratio and the volume fraction of the dispersed phase. Most of the literature on mass transfer from clusters of drops pertains to the limiting cases of either zero viscosity ratio (bubbles) or infinite viscosity ratio (solid spheres) in the creeping flow or in the potential flow regimes for small and/or large values of Peclet numbers. Indeed, no prior results are available which relate to mass transfer from ensembles of drops at moderate Reynolds and Peclet numbers for intermediate values of the viscosity ratio and the concentration of the dispersed phase. From a theoretical standpoint, a mathematical description of the inter-drop hydrodynamic interactions is also needed, in addition to the usual conservation equations to model convective transport in these systems. In the literature, two distinct approaches are available to describe inter-drop hydrodynamic interactions. In the first approach, the field equations are solved for specific ordered arrangements (such as square, triangular, simple cubic, body centered cubic, face centered cubic, and so on). While this approach is rigorous, the results naturally depend upon the specific arrangement and extrapolation to even a slightly different configuration is not possible (Chhabra, 2006). These results are frequently expressed in the form of a correction factor to be applied to the case of a single droplet (Stokes’ expression), which is a strong function of the type of packing of droplets. For ordered suspensions, this correction is of the order 11/3, whereas for random suspensions, it is of the order 1. While in the dilute limits, both of these are close to each other, but the two begin to deviate from each other with the increasing values of 1. To date this approach, however, has not only been limited to the zero Reynolds number flow, but has also been used more extensively for solid spheres and bubbles and only scantily for droplets. In the second approach (the so-called cell model), somewhat less rigorous, the inter-drop interactions are approximated by postulating the each fluid sphere to be surrounded by a hypothetical concentric envelope of the continuous phase. The size of the hypothetical envelope is chosen such that the volume fraction of the dispersed phase in each cell is equal to the overall mean volume fraction of the dispersed phase. Thus, the radius of the hypothetical envelope is related to the size of the individual droplet via the mean volume fraction of the system. Qualitatively, this approach is tantamount to imposing an equivalent wall effect, akin to the approach of Di Felice (1996). Thus, this approach converts the difficult many body problem into a conceptually simpler problem involving one droplet confined in a spherical cell. This provided the impetus to the development of the two commonly used cell models, namely, the free surface cell model (Happel, 1958) and the zero vorticity cell model (Kuwabara, 1959). The two models differ only in relation to one of the boundary conditions at the cell boundary. Happel (1958) proposed the cell boundary to be frictionless (zero shear stress) thereby emphasizing the non-interacting nature of cells. On the other hand, Kuwabara (1959) suggested the use of the zero vorticity condition at the cell boundary. While it is virtually impossible to offer a sound theoretical justification for either of these boundary conditions, suffice it to say here that
owing to extra dissipation in the zero vorticity cell model, the resulting values of the resistance are larger than those obtained by using the free surface cell model (Chhabra, 2006). The free surface cell model has been used in this work. However, it will be appropriate to start with a short review of the previously available scant literature.
PREVIOUS LITERATURE Since the pioneering work on the viscous flow past assemblages of solid spheres at low Reynolds number by Happel (1958) and its subsequent extension to ensembles of bubbles and drops (Gal-Or and Waslo, 1968), the free surface cell model has been extensively used to solve the flow past clusters of bubbles, drops and particles in Newtonian fluids and in a wide variety of non-Newtonian fluids. Owing to the non-linear viscosity equation for non-Newtonian fluids, the velocity and stress variational principles have been combined with the cell models to obtain lower and upper bounds on drag coefficients for swarms of spherical bubbles rising in generalized Newtonian fluids including power-law fluids, Carreau model fluids (Gummalam and Chhabra, 1987; Gummalam et al., 1988; Jarzebski and Malinowski, 1986, 1987a, b; Manjunath and Chhabra, 1992; Manjunath et al., 1994; Zhu and Deng, 1994; Zhu, 1995, 2001; Sun and Zhu, 2004). On the other hand, in the limit of potential flow, Chhabra (1998) extended the work of Marrucci (1965) to bubble swarms rising in power-law liquids using the free surface cell model. Similarly, there are a few studies reported on the flow of Newtonian and other generalized Newtonian fluids in fixed and fluidized bed of spheres (Mohan and Raghuraman, 1976a, b; Kawase and Ulbrecht, 1981a, b; Chhabra and Raman, 1984; Satish and Zhu, 1992; Zhu and Satish, 1992; Jaiswal et al., 1991a, b, 1992, 1993, 1994; Dhole et al., 2004). Gal-Or and Waslo (1968) were the first to use the free surface cell model to study the creeping motion of an ensemble of mono-size spherical drops in an another immiscible incompressible Newtonian liquid with and without the presence of surfactants. This approach has also been shown to yield satisfactory predictions of drag on ensembles of drops moving slowly in power-law and other generalized Newtonian fluids (Jarzebski and Malinowski, 1986, 1987a, b; Tripathi and Chhabra, 1994; Zhu and Deng, 1994; Zhu, 2001). Recently, Kishore et al. (2006) have extended this approach to the Newtonian flow past ensembles of mono-size spherical Newtonian droplets at moderate Reynolds numbers up to 500. This model has also been used successfully to capture the sedimentation behaviour of two-fluid spheres (Ferreira et al., 2003) and of composite spheres (Prasad et al., 1990). Most of these studies concerning the hydrodynamics of multi-particle systems have been reviewed recently (Chhabra, 2006). In contrast, there is only a scant literature on mass and heat transfer from clusters of bubbles, drops and particles, even in the creeping flow regime at intermediate Peclet numbers. Pfeffer (1964) combined the free surface cell model and the thin boundary layer solution to obtain an expression for the average Sherwood number in the limit of large Peclet number and the creeping flow of Newtonian liquids through beds of spherical particles as a function of the fractional void volume. Kawase and Ulbrecht (1981a, b) combined the thin concentration boundary layer approximation with the free surface cell model to elucidate the role of power-law index
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MASS TRANSFER FROM ENSEMBLES OF NEWTONIAN FLUID SPHERES on the rate of mass transfer for the creeping flow of power-law liquids in fixed and fluidized beds of spherical particles in the limit of large Peclet numbers. Satish and Zhu (1992) employed a finite difference method to contrast the predictions of the two cell models in terms of the average Sherwood number at high Peclet numbers under the creeping flow conditions. Recently, Shukla et al. (2004) have numerically studied convective heat transfer in fixed and fluidized beds to power-law liquids at intermediate Reynolds and Peclet numbers using the free surface cell model. Most of these studies predict a slight enhancement in the value of Sherwood number in shear-thinning fluids which is consistent with the scant experimental results (Chhabra et al., 2001). In contrast to the aforementioned body of information relating to solid particles, the corresponding mass transfer problem from clusters of spherical bubbles and drops at finite Reynolds and Peclet numbers has not yet been studied even in Newtonian liquids. Admittedly, there are a few analytical results available either for very small or very large values of Peclet numbers in the creeping and/or potential flow regimes. For instance, Ruckenstein (1964) obtained expressions for mass transfer coefficients for ensembles of bubbles and drops using the free surface cell model in the creeping and potential flow regimes for large values of Peclet numbers. LeClair and Hamielec (1971) obtained corresponding results on Sherwood numbers of swarms of spherical bubbles rising in Newtonian liquids using the zero-vorticity cell model for finite values of the Reynolds and Peclet numbers. Gal-Or and Yaron (1973) studied the transient heat and mass transfer to/from ensembles of bubbles and drops at low Reynolds numbers using the approach of Chao (1969). The effect of a first order chemical reaction on mass transfer was studied by Dang and Steinberg (1976). Waslo and Gal-Or (1971) obtained a solution for convective mass or heat transfer in concentrated assemblages of spherical drops or bubbles (rapid internal circulation) and solid particles by employing the boundary layer theory for large values of Peclet numbers in the creeping flow conditions. Yaron and Gal-Or (1971) extended this work to intermediate rates of internal circulations. Later, Yaron and Gal-Or (1973) obtained correlations for high Reynolds numbers steady convective heat and mass transfer and fluid motion of real concentrated systems of bubbles and drops using the free surface cell model. The range of applications shows the extremely versatile nature of this simple approach and hence it will be used here. From the aforementioned discussion, it is safe to conclude that only scant literature is available on mass transfer from ensembles of spherical droplets to a continuous phase for finite Reynolds and Peclet numbers. This work sets out to fill this gap in the literature. In particular, the governing equations together with the free surface cell model have been solved numerically for a wide range of values of the Reynolds number, Peclet number, viscosity ratio and the concentration of the dispersed phase. Based on the numerical results obtained here, simple predictive expression has been developed which facilitate the estimation of the rate of mass transfer in a new application.
PROBLEM STATEMENT AND DESCRIPTION Consider the steady, incompressible and axisymmetric flow of a Newtonian fluid with an uniform velocity Uo and
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concentration Co past an ensemble of immiscible fluid spheres of uniform size (radius R) as shown in Figure 1(a). It is assumed that the drops retain their spherical shape due to large surface tension force. Within the framework of the free surface cell model, the inter-drop interactions are approximated by postulating the each fluid sphere to be surrounded by a hypothetical fluid envelope of radius R1 as shown in Figure 1(b) and detailed in a recent study (Kishore et al., 2006). In general, for dilute systems, droplet deformation would occur beyond a critical value of Weber number of approximately 4. This corresponds to large values of Reynolds numbers in the range of 300– 1000 depending on the physical properties of the fluid systems. Thus, the present study is confined for the cases of dilute to moderate values of dispersed phase concentration, moderate values of the Reynolds number and viscosity ratio where these effects are believed to be negligible. A spherical coordinate system (r, u, f) with its origin fixed at the centre of the drop is employed here with the polar axis (u ¼ 0) being directed along the flow direction as shown in Figure 1(b). Due to the symmetry, vf ¼ 0 and no flow variable depends upon the f-coordinate. The flow and mass transfer characteristics in the continuous and dispersed phases are governed by the equations of continuity, momentum and species continuity. In their dimensionless form these equations can be written in their conservative form as follows: . Continuity equation 1 @ 2 1 @ ðvu Þi,o sin u ¼ 0 r ðvr Þi,o þ r 2 @r r sin u @u
(1)
where i, o represents internal and external flow variables respectively. . r-component of momentum equation i @(vr )i,o 1 @ h 2 1 @ r (vr )2i,o þ þ 2 (vr )i,o (vu )i,o sin u r @r r sin u @u @t (vu )2i,o @pi,o 2 1 @2 2 ¼ þ (r (vr )i,o ) Rei,o r 2 @r 2 r @r @(vr )i,o 1 @ sin u þ 2 (2a) r sin u @u @u
. u-component of momentum equation @(vu )i,o 1 @ 2 1 @ þ 2 ½r (vr )i,o (vu )i,o þ ½(vu )2 sin u r @r r sinu @u i,o @t (vr )i,o (vu )i,o 1 @pi,o 2 1 @ 2 @(vu )i,o ¼ þ þ r r @r r @u Rei,o r 2 @r 1 @ 1 @ 2 @(vr )i,o ½(vu )i,o sin u þ 2 þ 2 (2b) r @u sin u @u r @u
In this problem, on the interface and inside the droplet uniform material concentration is assumed i.e., the Biot number (ratio of the convection to the diffusion mass transfer) is assumed to be small. Therefore, the dimensionless mass transfer equation for the continuous phase in its conservative
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Figure 1. (a) Schematic representation of the flow past ensembles of Newtonian fluid spheres and (b) the cell model idealization.
form can be written as follows: @C 1 @ 2 1 @ þ r ðvr Þ0 C þ ðvu Þ0 C sin u @t r 2 @r r sin u @u 2 1 @ 2 @C 1 @ @C r þ sin u ¼ Pe r 2 @r @r r 2 sin u @u @u
(3)
In the above equations (1)–(3), the velocity terms have been scaled using the free stream velocity Uo, radial distance by radius R, pressure with rUo2 , time using (R=Uo ), and the concentration difference (C 2 Co) using (Cs 2 Co). The dimensionless numbers appearing in these equations are defined as follows:
r Uo (2R) Reo ¼ o mo Pe ¼ Reo Sc ¼ Sc ¼
mo ro Do
(4) Uo (2R) Do
(5)
Reo l k
(vr )o ¼ cos u
(9a)
(t r u )o ¼ 0 C¼0
(9b) (9c)
. At the fluid –fluid interface (r ¼ 1): (vr )i ¼ (vr )o ¼ 0 (vu )i ¼ (vu )o (t r u )i ¼ (t r u )o C¼1
(10a) (10b) (10c) (10d)
. Along the axis of symmetry (u ¼ 0, p):
(7)
where, Cs and Co are the concentrations of the fluid on the surface of the drop and at the cell boundary respectively, r is the density of the fluid, m is the dynamic viscosity of the fluid, Do is the molecular diffusivity, k is the ratio of the internal to external fluid viscosity, i.e., k ¼ mi =mo and l is the ratio of the internal to external fluid density, i.e., l ¼ ri =ro . Within the framework of the free surface cell model, the dimensionless radius of the outer spherical envelope, R1, can be related to the mean volume fraction of the ensemble as follows: R1 ¼ 11=3
. At the cell boundary (r ¼ R1):
(6)
In terms of the viscosity ratio (k) and the density ratio (l), the internal flow Reynolds number can be defined as follows: Rei ¼
the value of R1 for known values of 1 and the radius of the drop. The appropriate dimensionless boundary conditions for this flow are as follows:
(8)
where 1 is the volume fraction of the dispersed phase. Therefore, by simply varying the value of R1, one can simulate emulsions of various concentrations of the dispersed phase including the limiting case of a single drop by setting R1 ! 1, i.e., 1 ¼ 0. Conversely, one can readily calculate
@(vr )i ¼0 @u @(vr )o (vu )o ¼ 0; ¼0 @u @C ¼0 @u
(vu )i ¼ 0;
(11a) (11b) (11c)
. At the centre of the fluid sphere (r ¼ 0): (vr )i and (vu )i remain finite
(12)
Equations (1) –(3), subject to the boundary conditions outlined in equations (9) –(12) provide the theoretical framework to describe mass transfer in ensembles of fluid spheres. These have been numerically solved for the unknown velocity and pressure fields of both phases as discussed in Kishore et al. (2006). Then, using the velocity field of external flow field, the continuous phase mass transfer equation (3) has been solved as discussed in the next section. Once the concentration profile of external fluid is known, one can readily evaluate the local and average values of the Sherwood number (Sh) as follows. By equating the
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The scaling of the field equations and the boundary conditions suggests the local and average Sherwood numbers to be functions of the Reynolds number, the Schmidt number, the viscosity ratio, the density ratio and the volume fraction of the dispersed phase. This study endeavours to establish this functional relationship.
The concentration profile in the external field is obtained with a similar implicit algorithm which uses the QUICK (Leonard, 1979) and second order central difference schemes, respectively, for the convective and the diffusive terms of the species mass transfer equation. As the droplet concentration is increased and for the viscosity ratio, k 1, the convergence became increasingly difficult with the increasing values of the Peclet number. Therefore, the present computations are limited to the maximum value of the Peclet number of 50 000 only. It is useful to add here that, the limiting value of Peclet number may also depend on the volume fraction of the dispersed phase. For high concentration of the dispersed phase (1 70%), this limiting value of the Peclet number could be of the order of 104. After obtaining the steady concentration field for the continuous phase, one can readily calculate the values of the local and average Sherwood numbers using equations (13) and (14) respectively as functions of the Reynolds number, the Schmidt number, the viscosity ratio, the density ratio and the volume fraction of the dispersed phase.
NUMERICAL METHODOLOGY
RESULTS AND DISCUSSION
The velocity and pressure fields in both phases have been numerically obtained by solving equations (1) and (2) along with the appropriate boundary conditions. In this study, a finite difference method based SMAC-implicit algorithm has been used, which is a simplified version of MAC (Harlow and Welch, 1965) implemented on a staggered grid arrangement. As a detailed description of the numerical solution procedure for obtaining the velocity and pressure fields is available in our previous work (Kishore et al., 2006), only the salient features of this methodology as applied to the species continuity equation are presented here. Once the fully converged flow fields have been obtained, the concentration field is obtained by using a time-stepping procedure applied to the species mass transfer equation (3).
As indicated earlier, the dimensional considerations suggest the mass transfer from ensembles of fluid spheres to a continuous phase to be governed by the value of the Reynolds number, the Schmidt number, the volume fraction of the dispersed phase, the viscosity ratio and the density ratio. In this work, numerical results are obtained to elucidate role of each of these parameters except for the density ratio which is assumed to be unity as it is known to exert virtually no influence on the velocity field (Juncu, 1999; Feng and Michaelides, 2001a). Extensive numerical computations have been carried out in the following ranges of conditions: Reo ¼ 1, 5, 10, 20, 50, 100, 200; Sc ¼ 1, 20, 50, 100, 200, 500, 1000, 2000, 10 000 (Maximum Pe ¼ 50 000); k ¼ 0.1, 0.5, 1, 2, 5, 10, 20, 50 and 1 ¼ 0.2, 0.3, 0.4, 0.5, 0.6.
rates of mass transfer by diffusion and by convection at the interface of the fluid sphere, it can easily be seen that: kc (2R) @C Shu ¼ ¼ 2 (13) Do @r r ¼1 where kc is the inter-phase mass transfer coefficient in the continuous phase. Naturally, the value of the Sherwood number Shu will vary with position, u, on the surface of the drop (r ¼ 1) and it is customary to define an average Sherwood number, Shavg as follows: ð 1 p Shavg ¼ Shu sin u d u (14) 2 o
Table 1. Comparison of average Sherwood number of a single spherical bubble, drop and solid particle (with an optimum value of R1 ¼ 150) in Newtonian fluids. Present
Feng and Michaelides (2001b)
Present
Pe ¼ 1
Reo
Feng and Michaelides (2001b)
Present
Pe ¼ 100
Feng and Michaelides (2001b) Pe ¼ 1000
k ¼ 0 (bubble) 1 100 500
2.361 2.428 2.437
2.354 2.423 2.426
8.449 11.166 11.865
10 100 500
2.471 2.488 2.482
2.385 2.417 2.424
9.010 10.947 11.759
10 100 500
2.454 2.468 2.486
2.380 2.413 2.422
8.352 10.227 11.566
10 100 500
2.439 2.453 2.468
2.365 2.398 2.399
1 100 500
2.323 2.402 2.407
2.315 2.384 2.386
8.198 11.223 11.887
22.990 31.840 34.427
23.349 32.441 35.010
8.745 10.604 11.594
23.934 30.414 33.590
22.380 29.726 33.503
8.240 10.141 11.329
21.336 27.584 32.489
21.360 25.499 32.138
14.021 17.730 22.301
14.849 18.756 22.119
11.318 15.764 19.559
11.628 15.946 19.277
k ¼ 0.5
k¼1
k ¼ 10 6.807 6.885 8.168 8.360 9.532 9.077 k ¼ 1 (solid particle) 5.813 5.865 7.696 7.695 8.932 8.524
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KISHORE et al. Table 2. Comparison of the present average Sherwood number for the case of slow motion (Reo ¼ 1) of Newtonian liquids in beds of spherical particles. 1 ¼ 0.6
1 ¼ 0.2
Pe
Pfeffer (1964)
Present
Pfeffer (1964)
Present
100 200 500 1000 2000
15.6032 19.6587 26.6807 33.6160 42.3535
15.8060 19.6133 26.6215 33.4651 42.0104
7.9506 10.0171 13.5933 17.1290 21.5812
8.7719 10.8158 14.3594 17.8598 22.2636
Validation of Results The numerical solution procedure developed to solve the continuity and momentum equations segregated from the species mass transfer equation has been thoroughly validated by benchmarking the values of the drag coefficient over wide ranges of conditions elsewhere (Kishore et al., 2006). However, the solution procedure as applied to the species mass transfer equation is validated here (Table 1). In particular, the present results are compared with the single particle results of Feng and Michaelides (2001b) for a range of values of k with an optimum domain of R1 ¼ 150. It is useful to recall here that an exponential
transformation has been used in the radial direction so as to capture the boundary layer dynamics with adequate accuracy. An inspection of Table 1 shows the two values to be within +2–4% of each other. The present solver has also been validated with the analytical solution of Pfeffer (1964) for the case of heat and mass transfer of Newtonian liquids in beds of solid spheres. Since, this analytical solution is valid in the limits of low Reynolds number and high Peclet number, comparisons have been shown in the range of 100 Pe 2000 (Reo ¼ 1) for two extreme values of particle concentration, i.e., 1 ¼ 0.6 and 0.2 (Table 2). Once again, the two values are seen to be in good agreement with each other.
Figure 2. The effect of Sc on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.2 at Reo ¼ 200 for different values of k. Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A8): 1203–1214
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Figure 3. The effect of Sc on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.2 at Reo ¼ 1 for different values of k.
Similar comparisons are obtained for the other values of 1 and hence are not shows here. Based on these comparisons, the present results are believed to be accurate within +2– 3%. It is worthwhile to add here that the differences of this magnitude are common in such numerical studies due to the differences stemming from grid errors, numerical solution procedure, and so on (Roache, 1994).
Effects of Reo, k, Sc and 1 on Local and Average Sherwood Numbers In general, on the surface of a bubble, drop and solid particle, the local value of Sh decreases as one traverses from the front stagnation point to the rear end of the sphere provided no separation occurs. In the case of recirculation, the local value of Sherwood number decreases from the front stagnation point up to the point of separation and then this value starts increasing as one traverses to the rear end. However, as reported in our previous work (Kishore et al., 2006), no recirculation was observed for any value of k and 1 for Reo 200; therefore, in this work, the local variation in the Sherwood number is found to be continually decreasing from the front stagnation point all the way up to the rear
stagnation point for all values of Reo, k, Sc and 1, as seen in Figures 2–5. Figure 2 shows the effect of Schmidt number on the local variation of the Sherwood number for a range of values of Schmidt number for 1 ¼ 0.2 at Reo ¼ 200 for different values of k. For small values of Peclet numbers, i.e., Pe up to 200, there is almost no variation in the local value of the Sherwood number on the drop surface as compared to that seen for large values of Pe for all values of the viscosity ratio. This is essentially due to the fact that at such low values of Peclet numbers, diffusion is the main mode of mass transfer with a little contribution from the convective mass transfer. Qualitatively as the value of the viscosity ratio increases, the value of Sherwood number at the front stagnation point decreases whereas it is almost unaltered at the rear end; therefore, the overall mass transfer rate decreases as the value of k increases for all values of 1. However, the role of k diminishes at small values of Sc, i.e., for Pe 200. Similar observations can be made from Figure 3 where the effect of Schmidt number on the local Sherwood number is shown for 1 ¼ 0.2 at Reo ¼ 1. However, a dramatic effect of Schmidt number can be seen as the Pe varies from 1 to 10 000 due to the thinning of the concentration boundary layer. Here also, for Pe 100,
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Figure 4. Effect of k on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.6, at Reo ¼ 20 for different values of Sc.
the effect of k was found to be rather small as the rate of mass transfer is predominantly due to diffusion. But for Pe 100, the rate of mass transfer increases significantly due to the increased contribution from the convective mass transfer for a fixed value of the viscosity ratio. Figure 4 shows the effect of viscosity ratio on mass transfer for 1 ¼ 0.6 at Reo ¼ 20 for a range of values of Schmidt numbers. For all values of the Schmidt number and the viscosity ratio, as one traverses from the front stagnation point to the rear end, the local value of the Sherwood number decreases. However, for Sc ¼ 1, as the value of k increases, the local value of the Sherwood number at the front end decreases, whereas at the rear end it increases and thus, there is a cross over of the local value on the top of the drop surface [Figure 4(a)]. Thus, nullifying the effect of k on the overall value of the Sherwood number for small values of Sc (or Pe). Qualitatively similar effects was also seen for other values of the droplet concentration. But for large values of Sc, as the value of k increases, the local value of Sherwood number at the front stagnation point decreases and it is almost constant at the rear end, therefore, the rate of mass transfer decreases. Figure 5 shows the effect of the droplet concentration, 1, on the local variation in the Sherwood number when Reo ¼ 20 and Sc ¼ 100 (Pe ¼ 2000) for
different values of the viscosity ratio. For all values of k, as the value of the droplet concentration increases, the local value of the Sherwood number at the front stagnation increases due to the sharpening of the gradients and at the rear end it is almost constant for all values of 1 and k. Thus, the rate of mass transfer increases as the droplet concentration increases. Similarly, the rate of mass transfer decreases as the value of k increases for all values of droplet concentration. Figure 6 shows the representative results on the effect of Peclet number on the average Sherwood number as a function of the viscosity ratio for different values of Reo for 1 ¼ 0.2. As the value of Peclet number increases, the average Sherwood number increases for all values of the Reynolds number and the viscosity ratio. However, for small values of Pe, there is only a small effect on the average Sherwood number for small values of k. But as the value of k increases, the average Sherwood number decreases for a fixed value of Reo and Sc. Qualitatively similar trends are obtained for the other values of Reo, Sc, k and 1 studied herein and thus are not included here. Finally, it is useful to develop a simple predictive correlation based on the present numerical results for average Sherwood number which can be used to estimate the rate
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Figure 5. Effect of droplet concentration, 1, on the local variation of Sh when Reo ¼ 20 and Sc ¼ 100 for different values of k.
of mass transfer in ensembles of fluid spheres of a liquid – liquid system in a new application. As far as known to us, no such prior correlation is available in the literature. The following expression was found to be satisfactory to correlate the present numerical results: Shavg ¼ 0:89Pe0:48 k 0:17 (1 1)0:33 2 þ 3k 0:5 þ Pe0:1 (1 1)1:54 1þk
(15)
The above equation reproduces the present numerical results (1649 data points) with an average error of 11.86% which rises to maximum of 43.07%. Figure 7 shows the parity plot of the present numerical results and those obtained by equation (15) and no discernable trends could be detected. Finally, it is useful to re-iterate that in concentrated systems, the fluid spheres collide with each other, deform and even coalesce which will influence the rate of mass transfer in these systems. All these effects are neglected in this work. Therefore, the results presented herein are useful for dilute to moderately concentrated systems and these should be used only as a first level approximation for concentrated systems. In this context, the results reported herein
should serve as a basis to incorporate some of these effects in future studies.
CONCLUSIONS The values of the convective inter-phase mass transfer coefficients for ensembles of fluid spheres in another immiscible liquid have been obtained for wide ranges of the pertinent dimensionless parameters, namely, Reo, Sc, k and 1. Irrespective of the values of Reo, k, Sc and 1, as one traverses from the front stagnation point along the surface to the rear end, the local value of Sherwood number decreases. This is presumably so due to the fact that no recirculation was observed in these ranges of conditions. As the value of the viscosity ratio increases, the rate of mass transfer decreases for all values of Reo, Sc and 1. However, for small values of Pe, the effect of k on the local and average Sherwood number is very weak and the mass transfer coefficient is almost constant. For Peclet numbers up to Pe 200, the mass transfer is mainly diffusion controlled. As the value of the droplet concentration increases, the mass transfer coefficient also increases for all values of the Reynolds number, the Schmidt number and the viscosity ratio. As the value of the Peclet number increases, the mean value of Sherwood number also increases for all values of Reo, k and 1.
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Figure 6. Effect of k on the average Sh as a function of Pe for Newtonian flow past an ensemble of 1 ¼ 0.2 for different values of Reo.
However, for small values of Peclet numbers, the average value of Sherwood number is almost constant for all values of the viscosity ratio for a fixed value of 1. As the value of
Schmidt number increases, the concentration boundary layer thins thereby enhancing the rate of mass transfer. Based on the present numerical results a simple predictive correlation has been developed which reproduces the present numerical results with an average error of 11.86%.
NOMENCLATURE C Co Cs Do k kc p Pe QUICK
Figure 7. Comparison of the present average Sh with the proposed correlation values.
r R R1 Re Sc Sh Shu Shavg SMAC Uo
molecular concentration molecular concentration at the cell boundary molecular concentration on the surface of the drop diffusivity, m2 s21 internal to external fluid viscosity ratio, mi/mo continuous phase mass transfer coefficient, m s21 pressure Peclet number quadratic upstream interpolation of convective kinematics radial distance drop radius, m cell boundary Reynolds number Schmidt number Sherwood number Sherwood number along the surface of the drop average Sherwood number simplified marker and cell free stream uniform velocity, m s21
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MASS TRANSFER FROM ENSEMBLES OF NEWTONIAN FLUID SPHERES vr vu vf
r-component of velocity u-component of velocity f-component of velocity
Greek symbols 1 volume fraction of the dispersed phase u streamwise direction, degree f azimuthal direction, degree m dynamic viscosity, Pa s l internal to external fluid density ratio, ri/ro r density of fluid, kg m23 t shear stress, Pa Subscripts i o r u f
internal flow variable external flow variable r-component u-component f-component
REFERENCES Chao, B.T., 1969, Transient heat and mass transfer to a translating droplet, J Heat Transf, Trans, ASME, 91: 273–281. Chhabra, R.P., 1998, Rising velocity of a swarm of spherical bubbles in power-law fluids at high Reynolds numbers, Can J Chem Eng, 76: 137–140. Chhabra, R.P., 2006, Bubbles, Drops and Particles in Non-Newtonian Fluids, 2nd edition (CRC Press, Boca Raton, FL, USA). Chhabra, R.P., Comiti, J. and Machac, I., 2001, Flow of nonNewtonian fluids in fixed and fluidized beds: a review, Chem Eng Sci, 56: 1– 27. Chhabra, R.P. and Raman, J.R., 1984, Slow non-Newtonian flow past an assemblages of rigid spheres, Chem Eng Comm, 27: 23– 46. Clift, R., Grace, J.R. and Weber, M.E., 1978, Bubbles, Drops and Particles (Academic Press, NY, USA). Dang, V.D. and Steinberg, M., 1976, Mass transfer from swarms of bubbles or drops with chemical reactions in continuous phase, AIChE J, 22: 925 –927. Dhole, S.D., Chhabra, R.P. and Eswaran, V., 2004, Power-law fluid flow through beds of spheres at intermediate Reynolds numbers —pressure drop in fixed and distended beds, Chem Eng Res Des, 82: 642–652. Di Felice, R., 1996, Particle-in-a-tube analogy for a multiparticle suspension, Int J Multiphase Flow, 22: 515 –525. Feng, Z.G. and Michaelides, E.E., 2001a, Drag coefficients of viscous spheres at intermediate and high Reynolds numbers, Trans ASME J Fluids Eng, 123: 841– 849. Feng, Z.G. and Michaelides, E.E., 2001b, Heat and mass transfer coefficients of viscous spheres, Int J Heat Mass Transf, 44: 4445–4454. Ferreira, J.M., Soares, A.A. and Chhabra, R.P., 2003, Hydrodynamic behaviour of ensemble of encapsulated liquid drops in creeping motion: A fluid mechanic based model for liquid membranes, Fluid Dyn Res, 32: 201 –215. Gal-Or, B. and Waslo, S., 1968, Hydrodynamics of an ensemble of drops (or bubbles) in the presence or absence of surfactants, Chem Eng Sci, 23: 1431– 1446. Gal-Or, B. and Yaron, I., 1973, Transient mass or heat transfer in ensembles of drops or bubbles translating at low Reynolds numbers, AIChE J, 19: 200 –202. Gummalam, S. and Chhabra, R.P., 1987, Rising velocity of a swarm of spherical bubbles in a power-law non-Newtonian liquid, Can J Chem Eng, 65: 1004– 1008. Gummalam, S., Narayan, K.A. and Chhabra, R.P., 1988, Rising velocity of a swarm of spherical bubbles through a non-Newtonian fluid: Effect of zero viscosity, Int J Multiphase Flow, 14: 361–371. Happel, J., 1958, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles, AIChE J, 4: 197– 201. Harlow, F.G. and Welch, J.E., 1965, Numerical calculation of time dependent viscous incompressible flow of fluid with free surfaces, Phys Fluids, 8: 2182–2188. Jaiswal, A.K., Sundararajan, T. and Chhabra, R.P., 1991a, Hydrodynamics of Newtonian fluid flow through assemblages of rigid
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spherical particles in intermediate Reynolds number regime, Int J Eng Sci, 29: 693–708. Jaiswal, A.K., Sundararajan, T. and Chhabra, R.P., 1991b, Flow characteristics of settling suspensions and fluidized beds of spherical particles, Chem Eng Comm, 106: 139 –150. Jaiswal, A.K., Sundararajan, T. and Chhabra, R.P., 1992, Flow of power-law liquids through particle assemblages at intermediate Reynolds numbers, Can J Chem Eng, 69: 1235– 1241. Jaiswal, A.K., Sundararajan, T. and Chhabra, R.P., 1993, Hydrodynamics of creeping flow of power-law liquids through particle assemblages, Int J Eng Sci, 31: 293– 306. Jaiswal, A.K., Sundararajan, T. and Chhabra, R.P., 1994, Pressure drop for the flow of dilatant fluids through a fixed bed of spherical particles, Can J Chem Eng, 72: 352 –353. Jarzebski, A.B. and Malinowski, J.J., 1986, Drag and mass transfer in multiple drop slow motion in power-law fluids, Chem Eng Sci, 41: 2569–2573. Jarzebski, A.B. and Malinowski, J.J., 1987a, Drag and mass transfer in slow non-Newtonian flows over an ensemble of Newtonian spherical drops and bubbles, Chem Eng Comm, 49: 235– 246. Jarzebski, A.B. and Malinowski, J.J., 1987b, Drag and mass transfer in creeping flow of a Carreau fluid over drops and bubbles, Can J Chem Eng, 65: 680 –684. Juncu, G., 1999, A numerical study of steady viscous flow past a fluid sphere, Int J Heat Mass Transf, 20: 414– 421. Kawase, Y. and Ulbrecht, J.J., 1981a, Drag and mass transfer in nonNewtonian flows through multiparticle systems at low Reynolds numbers, Chem Eng Sci, 36: 1193–1202. Kawase, Y. and Ulbrecht, J.J., 1981b, Motion of and mass transfer from an assemblage of solid spheres moving in non-Newtonian fluids at high Reynolds numbers, Chem Eng Comm, 8: 233–249. Kishore, N., Chhabra, R.P. and Eswaran, V., 2006, Sedimentation in emulsions of mono-size droplets at moderate Reynolds numbers, Chem Eng Res Des, 84: 1180–1193. Kuwabara, S., 1959, The forces experienced by randomly distributed parallel cylinders or spheres in a viscous flow at small Reynolds numbers, J Phys Soc Japan, 14: 527–532. LeClair, B.P. and Hamielec, A.E., 1971, Viscous flow through particle assemblages at intermediate Reynolds numbers– A cell model for transport in bubbles swarms, Can J Chem Eng, 49: 713– 720. Leonard, B.P., 1979, A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Comp Methods Appl Mech Engr, 19: 59–98. Manjunath, M. and Chhabra, R.P., 1992, Free rise velocity of a swarm of spherical gas bubbles through a quiescent power-law liquid, Int J Eng Sci, 30: 871–878. Manjunath, M., Tripathi, A., Chhabra, R.P. and Sundararajan, T., 1994, Numerical simulation of the drag on a swarm of bubbles, Int J Eng Sci, 32: 927–933. Marrucci, G., 1965, Rising velocity of swarm of spherical bubbles, Ind & Eng Chem Fundam, 4: 224–225. Michaelides, E.E., 2006, Particles, Bubbles and Drops: Their Motion, Heat and Mass Transfer (World Scientific, Singapore). Mohan, V. and Raghuraman, J., 1976a, A theoretical study of pressure drop for non-Newtonian creeping flow past an assemblage of spheres, AIChE J, 22: 259 –264. Mohan, V. and Raghuraman, J., 1976b, Bounds on the drag for creeping flow of an Ellis fluid past an assemblage of spheres, Int J Multiphase Flow, 2: 581–589. Pfeffer, R., 1964, Heat and mass transport in multiparticle systems, Ind & Eng Chem Fundam, 3: 380 –383. Prasad, D., Narayan, K.A. and Chhabra, R.P., 1990, Creeping fluid flow relative to an assemblage of composite spheres, Int J Eng Sci, 28: 215–230. Roache, P.J., 1994, Perspective: A method for uniform reporting of grid refinement studies, Trans ASME J Fluids Eng, 116: 405– 413. Ruckenstein, E., 1964, On mass transfer in the continuous phase from spherical bubbles or drops, Chem Eng Sci, 19: 131–146. Satish, M.G. and Zhu, J., 1992, Flow resistance and mass transfer in slow non-Newtonian flow through multiparticle systems, J Appl Mech, 59: 431–437. Schramm, L., 2005, Emulsions, Foams and Suspensions (Wiley – VCH, New York, USA). Shukla, R., Dhole, S.D., Chhabra, R.P. and Eswaran, V., 2004, Convective heat transfer for power-law fluids in packed and fluidized beds of spheres, Chem Eng Sci, 59: 645 –659.
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Sun, D. and Zhu, J., 2004, Approximate solutions of non-Newtonian flows over a swarm of bubbles, Int J Multiphase Flow, 30: 1271–1278. Tripathi, A. and Chhabra, R.P., 1994, Hydrodynamics of creeping motion of an ensemble of power-law fluid drops in an immiscible power-law medium, Int J Eng Sci, 32: 791 –803. Waslo, S. and Gal-Or, B., 1971, Boundary layer theory for mass and heat transfer in clouds of moving drops, bubbles or solid particles, Chem Eng Sci, 26: 829–838. Yaron, I. and Gal-Or, B., 1971, Convective mass or heat transfer from size-distributed drops, bubbles or solid particles, Int J Heat Mass Transf, 14: 727– 737. Yaron, I. and Gal-Or, B., 1973, High Reynolds number fluid dynamics and heat and mass transfer in real concentrated particulate two-lphase systems, Int J Heat Mass Transf, 16: 887–895.
Zhu, J., 1995, Drag and mass transfer for flow of a Carreau fluid past a swarm of Newtonian drops, Int J Multiphase Flow, 21: 935– 940. Zhu, J., 2001, A note on slow non-Newtonian flow over an ensemble of spherical bubbles, Chem Eng Sci, 56: 2237–2241. Zhu, J. and Deng, W., 1994, Non-Newtonian flow past a swarm of Newtonian droplets, Chem Eng Sci, 49: 147–150. Zhu, J. and Satish, M.G., 1992, Non-Newtonian effects on the drag of creeping flow through packed beds, Int J Multiphase Flow, 18: 765– 777. The manuscript was received 27 December 2006 and accepted for publication after revision 24 February 2007.
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Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India. Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India.
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Abstract: In this work, the rate of mass transfer from an ensemble of mono-size spherical Newtonian droplets (free from surfactants) to a Newtonian continuous phase has been numerically studied at moderate Reynolds and Peclet numbers. A simple spherical cell model (so-called free surface cell model) has been used to account for inter-drop hydrodynamic interactions. Extensive numerical results have been obtained to elucidate the effects of the Reynolds number (Reo), the ratio of internal to external fluid viscosity (k), the volume fraction of the dispersed phase (1) and the Schmidt number (Sc) on the local and average Sherwood number (Sh) over the ranges of conditions: 1 Reo 200, 0.2 1 0.6, 0.1 k 50 and 1 Sc 10 000. It has been observed that the effects of viscosity ratio on the local and average Sherwood number is less significant for small values of the Peclet number (Pe) for all values of dispersed phase concentration. As the value of the viscosity ratio increases, the average Sherwood number decreases for all values of the droplet concentration and the Reynolds number. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer in a liquid–liquid system in a new application. However, it is also appropriate to add here that at higher concentrations, fluid spheres interact significantly, deform and coalesce. All these effects are neglected in this study. Therefore, the present results are valid only for dilute to moderate concentration of the dispersed phase. Keywords: ensemble; fluid sphere; viscosity ratio; mass transfer; Sherwood number.
Correspondence to: Professor R. P. Chhabra, Department of Chemical Engineering, Indian Institute of Technology, Kanpur, 208016, India. E-mail: [email protected]
DOI: 10.1205/cherd06250 0263–8762/07/ $30.00 þ 0.00 Chemical Engineering Research and Design Trans IChemE, Part A, August 2007 # 2007 Institution of Chemical Engineers
INTRODUCTION
applications. In most applications, often one fluid is dispersed in the form of droplets moving in another immiscible fluid which leads to enhancement of the rate of mass transfer between the two phases. Some of these applications include, liquid–liquid extraction, distillation, gas absorption, enhanced oil recovery in petroleum industries, production of polymeric alloys and emulsions in paint and detergent industries, fermentation broths, wastewater treatment, and so on (Schramm, 2005). Over the years, extensive literature has been reported on the motion of and mass transfer from single bubbles and drops in Newtonian liquids which has been reviewed thoroughly by Clift et al. (1978), Michaelides (2006) and Chhabra (2006). Although, the detailed kinematics of such studies related to single bubbles and drops provides useful information about the basic underlying physical phenomena, often one encounters clusters of bubbles and drops in chemical and processing industries as mentioned above. Therefore, an adequate
The motion of and mass transfer from drops to viscous liquids is ubiquitous in the chemical, biochemical, and polymer processing industries. In most applications, one encounters clusters of drops, the growth or collapse of which are directly influenced by the rate of mass transfer between the two phases. The rate of mass transfer from a single fluid sphere moving in a continuous fluid medium is greatly influenced by the motion inside the fluid sphere. Experiments have shown that the rate of mass transfer from circulating bubbles and drops is much larger than that from non-circulating bubbles and drops due to the effect of internal circulation on the external flow field. It is readily conceded that one often encounters ensembles of droplets rather than a single drop in most industrial applications. The mass transfer from ensembles of fluid spheres to another immiscible medium is an idealization of many industrially important chemical and processing 1203
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understanding of the rate of mass transfer from clusters of drops is a prerequisite to the understanding and rationalizing the overall efficiency of the contacting equipment. This information can be conveniently expressed using dimensionless parameters such as the Sherwood number, Reynolds number, Schmidt number, viscosity ratio and the volume fraction of the dispersed phase. Most of the literature on mass transfer from clusters of drops pertains to the limiting cases of either zero viscosity ratio (bubbles) or infinite viscosity ratio (solid spheres) in the creeping flow or in the potential flow regimes for small and/or large values of Peclet numbers. Indeed, no prior results are available which relate to mass transfer from ensembles of drops at moderate Reynolds and Peclet numbers for intermediate values of the viscosity ratio and the concentration of the dispersed phase. From a theoretical standpoint, a mathematical description of the inter-drop hydrodynamic interactions is also needed, in addition to the usual conservation equations to model convective transport in these systems. In the literature, two distinct approaches are available to describe inter-drop hydrodynamic interactions. In the first approach, the field equations are solved for specific ordered arrangements (such as square, triangular, simple cubic, body centered cubic, face centered cubic, and so on). While this approach is rigorous, the results naturally depend upon the specific arrangement and extrapolation to even a slightly different configuration is not possible (Chhabra, 2006). These results are frequently expressed in the form of a correction factor to be applied to the case of a single droplet (Stokes’ expression), which is a strong function of the type of packing of droplets. For ordered suspensions, this correction is of the order 11/3, whereas for random suspensions, it is of the order 1. While in the dilute limits, both of these are close to each other, but the two begin to deviate from each other with the increasing values of 1. To date this approach, however, has not only been limited to the zero Reynolds number flow, but has also been used more extensively for solid spheres and bubbles and only scantily for droplets. In the second approach (the so-called cell model), somewhat less rigorous, the inter-drop interactions are approximated by postulating the each fluid sphere to be surrounded by a hypothetical concentric envelope of the continuous phase. The size of the hypothetical envelope is chosen such that the volume fraction of the dispersed phase in each cell is equal to the overall mean volume fraction of the dispersed phase. Thus, the radius of the hypothetical envelope is related to the size of the individual droplet via the mean volume fraction of the system. Qualitatively, this approach is tantamount to imposing an equivalent wall effect, akin to the approach of Di Felice (1996). Thus, this approach converts the difficult many body problem into a conceptually simpler problem involving one droplet confined in a spherical cell. This provided the impetus to the development of the two commonly used cell models, namely, the free surface cell model (Happel, 1958) and the zero vorticity cell model (Kuwabara, 1959). The two models differ only in relation to one of the boundary conditions at the cell boundary. Happel (1958) proposed the cell boundary to be frictionless (zero shear stress) thereby emphasizing the non-interacting nature of cells. On the other hand, Kuwabara (1959) suggested the use of the zero vorticity condition at the cell boundary. While it is virtually impossible to offer a sound theoretical justification for either of these boundary conditions, suffice it to say here that
owing to extra dissipation in the zero vorticity cell model, the resulting values of the resistance are larger than those obtained by using the free surface cell model (Chhabra, 2006). The free surface cell model has been used in this work. However, it will be appropriate to start with a short review of the previously available scant literature.
PREVIOUS LITERATURE Since the pioneering work on the viscous flow past assemblages of solid spheres at low Reynolds number by Happel (1958) and its subsequent extension to ensembles of bubbles and drops (Gal-Or and Waslo, 1968), the free surface cell model has been extensively used to solve the flow past clusters of bubbles, drops and particles in Newtonian fluids and in a wide variety of non-Newtonian fluids. Owing to the non-linear viscosity equation for non-Newtonian fluids, the velocity and stress variational principles have been combined with the cell models to obtain lower and upper bounds on drag coefficients for swarms of spherical bubbles rising in generalized Newtonian fluids including power-law fluids, Carreau model fluids (Gummalam and Chhabra, 1987; Gummalam et al., 1988; Jarzebski and Malinowski, 1986, 1987a, b; Manjunath and Chhabra, 1992; Manjunath et al., 1994; Zhu and Deng, 1994; Zhu, 1995, 2001; Sun and Zhu, 2004). On the other hand, in the limit of potential flow, Chhabra (1998) extended the work of Marrucci (1965) to bubble swarms rising in power-law liquids using the free surface cell model. Similarly, there are a few studies reported on the flow of Newtonian and other generalized Newtonian fluids in fixed and fluidized bed of spheres (Mohan and Raghuraman, 1976a, b; Kawase and Ulbrecht, 1981a, b; Chhabra and Raman, 1984; Satish and Zhu, 1992; Zhu and Satish, 1992; Jaiswal et al., 1991a, b, 1992, 1993, 1994; Dhole et al., 2004). Gal-Or and Waslo (1968) were the first to use the free surface cell model to study the creeping motion of an ensemble of mono-size spherical drops in an another immiscible incompressible Newtonian liquid with and without the presence of surfactants. This approach has also been shown to yield satisfactory predictions of drag on ensembles of drops moving slowly in power-law and other generalized Newtonian fluids (Jarzebski and Malinowski, 1986, 1987a, b; Tripathi and Chhabra, 1994; Zhu and Deng, 1994; Zhu, 2001). Recently, Kishore et al. (2006) have extended this approach to the Newtonian flow past ensembles of mono-size spherical Newtonian droplets at moderate Reynolds numbers up to 500. This model has also been used successfully to capture the sedimentation behaviour of two-fluid spheres (Ferreira et al., 2003) and of composite spheres (Prasad et al., 1990). Most of these studies concerning the hydrodynamics of multi-particle systems have been reviewed recently (Chhabra, 2006). In contrast, there is only a scant literature on mass and heat transfer from clusters of bubbles, drops and particles, even in the creeping flow regime at intermediate Peclet numbers. Pfeffer (1964) combined the free surface cell model and the thin boundary layer solution to obtain an expression for the average Sherwood number in the limit of large Peclet number and the creeping flow of Newtonian liquids through beds of spherical particles as a function of the fractional void volume. Kawase and Ulbrecht (1981a, b) combined the thin concentration boundary layer approximation with the free surface cell model to elucidate the role of power-law index
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MASS TRANSFER FROM ENSEMBLES OF NEWTONIAN FLUID SPHERES on the rate of mass transfer for the creeping flow of power-law liquids in fixed and fluidized beds of spherical particles in the limit of large Peclet numbers. Satish and Zhu (1992) employed a finite difference method to contrast the predictions of the two cell models in terms of the average Sherwood number at high Peclet numbers under the creeping flow conditions. Recently, Shukla et al. (2004) have numerically studied convective heat transfer in fixed and fluidized beds to power-law liquids at intermediate Reynolds and Peclet numbers using the free surface cell model. Most of these studies predict a slight enhancement in the value of Sherwood number in shear-thinning fluids which is consistent with the scant experimental results (Chhabra et al., 2001). In contrast to the aforementioned body of information relating to solid particles, the corresponding mass transfer problem from clusters of spherical bubbles and drops at finite Reynolds and Peclet numbers has not yet been studied even in Newtonian liquids. Admittedly, there are a few analytical results available either for very small or very large values of Peclet numbers in the creeping and/or potential flow regimes. For instance, Ruckenstein (1964) obtained expressions for mass transfer coefficients for ensembles of bubbles and drops using the free surface cell model in the creeping and potential flow regimes for large values of Peclet numbers. LeClair and Hamielec (1971) obtained corresponding results on Sherwood numbers of swarms of spherical bubbles rising in Newtonian liquids using the zero-vorticity cell model for finite values of the Reynolds and Peclet numbers. Gal-Or and Yaron (1973) studied the transient heat and mass transfer to/from ensembles of bubbles and drops at low Reynolds numbers using the approach of Chao (1969). The effect of a first order chemical reaction on mass transfer was studied by Dang and Steinberg (1976). Waslo and Gal-Or (1971) obtained a solution for convective mass or heat transfer in concentrated assemblages of spherical drops or bubbles (rapid internal circulation) and solid particles by employing the boundary layer theory for large values of Peclet numbers in the creeping flow conditions. Yaron and Gal-Or (1971) extended this work to intermediate rates of internal circulations. Later, Yaron and Gal-Or (1973) obtained correlations for high Reynolds numbers steady convective heat and mass transfer and fluid motion of real concentrated systems of bubbles and drops using the free surface cell model. The range of applications shows the extremely versatile nature of this simple approach and hence it will be used here. From the aforementioned discussion, it is safe to conclude that only scant literature is available on mass transfer from ensembles of spherical droplets to a continuous phase for finite Reynolds and Peclet numbers. This work sets out to fill this gap in the literature. In particular, the governing equations together with the free surface cell model have been solved numerically for a wide range of values of the Reynolds number, Peclet number, viscosity ratio and the concentration of the dispersed phase. Based on the numerical results obtained here, simple predictive expression has been developed which facilitate the estimation of the rate of mass transfer in a new application.
PROBLEM STATEMENT AND DESCRIPTION Consider the steady, incompressible and axisymmetric flow of a Newtonian fluid with an uniform velocity Uo and
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concentration Co past an ensemble of immiscible fluid spheres of uniform size (radius R) as shown in Figure 1(a). It is assumed that the drops retain their spherical shape due to large surface tension force. Within the framework of the free surface cell model, the inter-drop interactions are approximated by postulating the each fluid sphere to be surrounded by a hypothetical fluid envelope of radius R1 as shown in Figure 1(b) and detailed in a recent study (Kishore et al., 2006). In general, for dilute systems, droplet deformation would occur beyond a critical value of Weber number of approximately 4. This corresponds to large values of Reynolds numbers in the range of 300– 1000 depending on the physical properties of the fluid systems. Thus, the present study is confined for the cases of dilute to moderate values of dispersed phase concentration, moderate values of the Reynolds number and viscosity ratio where these effects are believed to be negligible. A spherical coordinate system (r, u, f) with its origin fixed at the centre of the drop is employed here with the polar axis (u ¼ 0) being directed along the flow direction as shown in Figure 1(b). Due to the symmetry, vf ¼ 0 and no flow variable depends upon the f-coordinate. The flow and mass transfer characteristics in the continuous and dispersed phases are governed by the equations of continuity, momentum and species continuity. In their dimensionless form these equations can be written in their conservative form as follows: . Continuity equation 1 @ 2 1 @ ðvu Þi,o sin u ¼ 0 r ðvr Þi,o þ r 2 @r r sin u @u
(1)
where i, o represents internal and external flow variables respectively. . r-component of momentum equation i @(vr )i,o 1 @ h 2 1 @ r (vr )2i,o þ þ 2 (vr )i,o (vu )i,o sin u r @r r sin u @u @t (vu )2i,o @pi,o 2 1 @2 2 ¼ þ (r (vr )i,o ) Rei,o r 2 @r 2 r @r @(vr )i,o 1 @ sin u þ 2 (2a) r sin u @u @u
. u-component of momentum equation @(vu )i,o 1 @ 2 1 @ þ 2 ½r (vr )i,o (vu )i,o þ ½(vu )2 sin u r @r r sinu @u i,o @t (vr )i,o (vu )i,o 1 @pi,o 2 1 @ 2 @(vu )i,o ¼ þ þ r r @r r @u Rei,o r 2 @r 1 @ 1 @ 2 @(vr )i,o ½(vu )i,o sin u þ 2 þ 2 (2b) r @u sin u @u r @u
In this problem, on the interface and inside the droplet uniform material concentration is assumed i.e., the Biot number (ratio of the convection to the diffusion mass transfer) is assumed to be small. Therefore, the dimensionless mass transfer equation for the continuous phase in its conservative
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Figure 1. (a) Schematic representation of the flow past ensembles of Newtonian fluid spheres and (b) the cell model idealization.
form can be written as follows: @C 1 @ 2 1 @ þ r ðvr Þ0 C þ ðvu Þ0 C sin u @t r 2 @r r sin u @u 2 1 @ 2 @C 1 @ @C r þ sin u ¼ Pe r 2 @r @r r 2 sin u @u @u
(3)
In the above equations (1)–(3), the velocity terms have been scaled using the free stream velocity Uo, radial distance by radius R, pressure with rUo2 , time using (R=Uo ), and the concentration difference (C 2 Co) using (Cs 2 Co). The dimensionless numbers appearing in these equations are defined as follows:
r Uo (2R) Reo ¼ o mo Pe ¼ Reo Sc ¼ Sc ¼
mo ro Do
(4) Uo (2R) Do
(5)
Reo l k
(vr )o ¼ cos u
(9a)
(t r u )o ¼ 0 C¼0
(9b) (9c)
. At the fluid –fluid interface (r ¼ 1): (vr )i ¼ (vr )o ¼ 0 (vu )i ¼ (vu )o (t r u )i ¼ (t r u )o C¼1
(10a) (10b) (10c) (10d)
. Along the axis of symmetry (u ¼ 0, p):
(7)
where, Cs and Co are the concentrations of the fluid on the surface of the drop and at the cell boundary respectively, r is the density of the fluid, m is the dynamic viscosity of the fluid, Do is the molecular diffusivity, k is the ratio of the internal to external fluid viscosity, i.e., k ¼ mi =mo and l is the ratio of the internal to external fluid density, i.e., l ¼ ri =ro . Within the framework of the free surface cell model, the dimensionless radius of the outer spherical envelope, R1, can be related to the mean volume fraction of the ensemble as follows: R1 ¼ 11=3
. At the cell boundary (r ¼ R1):
(6)
In terms of the viscosity ratio (k) and the density ratio (l), the internal flow Reynolds number can be defined as follows: Rei ¼
the value of R1 for known values of 1 and the radius of the drop. The appropriate dimensionless boundary conditions for this flow are as follows:
(8)
where 1 is the volume fraction of the dispersed phase. Therefore, by simply varying the value of R1, one can simulate emulsions of various concentrations of the dispersed phase including the limiting case of a single drop by setting R1 ! 1, i.e., 1 ¼ 0. Conversely, one can readily calculate
@(vr )i ¼0 @u @(vr )o (vu )o ¼ 0; ¼0 @u @C ¼0 @u
(vu )i ¼ 0;
(11a) (11b) (11c)
. At the centre of the fluid sphere (r ¼ 0): (vr )i and (vu )i remain finite
(12)
Equations (1) –(3), subject to the boundary conditions outlined in equations (9) –(12) provide the theoretical framework to describe mass transfer in ensembles of fluid spheres. These have been numerically solved for the unknown velocity and pressure fields of both phases as discussed in Kishore et al. (2006). Then, using the velocity field of external flow field, the continuous phase mass transfer equation (3) has been solved as discussed in the next section. Once the concentration profile of external fluid is known, one can readily evaluate the local and average values of the Sherwood number (Sh) as follows. By equating the
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The scaling of the field equations and the boundary conditions suggests the local and average Sherwood numbers to be functions of the Reynolds number, the Schmidt number, the viscosity ratio, the density ratio and the volume fraction of the dispersed phase. This study endeavours to establish this functional relationship.
The concentration profile in the external field is obtained with a similar implicit algorithm which uses the QUICK (Leonard, 1979) and second order central difference schemes, respectively, for the convective and the diffusive terms of the species mass transfer equation. As the droplet concentration is increased and for the viscosity ratio, k 1, the convergence became increasingly difficult with the increasing values of the Peclet number. Therefore, the present computations are limited to the maximum value of the Peclet number of 50 000 only. It is useful to add here that, the limiting value of Peclet number may also depend on the volume fraction of the dispersed phase. For high concentration of the dispersed phase (1 70%), this limiting value of the Peclet number could be of the order of 104. After obtaining the steady concentration field for the continuous phase, one can readily calculate the values of the local and average Sherwood numbers using equations (13) and (14) respectively as functions of the Reynolds number, the Schmidt number, the viscosity ratio, the density ratio and the volume fraction of the dispersed phase.
NUMERICAL METHODOLOGY
RESULTS AND DISCUSSION
The velocity and pressure fields in both phases have been numerically obtained by solving equations (1) and (2) along with the appropriate boundary conditions. In this study, a finite difference method based SMAC-implicit algorithm has been used, which is a simplified version of MAC (Harlow and Welch, 1965) implemented on a staggered grid arrangement. As a detailed description of the numerical solution procedure for obtaining the velocity and pressure fields is available in our previous work (Kishore et al., 2006), only the salient features of this methodology as applied to the species continuity equation are presented here. Once the fully converged flow fields have been obtained, the concentration field is obtained by using a time-stepping procedure applied to the species mass transfer equation (3).
As indicated earlier, the dimensional considerations suggest the mass transfer from ensembles of fluid spheres to a continuous phase to be governed by the value of the Reynolds number, the Schmidt number, the volume fraction of the dispersed phase, the viscosity ratio and the density ratio. In this work, numerical results are obtained to elucidate role of each of these parameters except for the density ratio which is assumed to be unity as it is known to exert virtually no influence on the velocity field (Juncu, 1999; Feng and Michaelides, 2001a). Extensive numerical computations have been carried out in the following ranges of conditions: Reo ¼ 1, 5, 10, 20, 50, 100, 200; Sc ¼ 1, 20, 50, 100, 200, 500, 1000, 2000, 10 000 (Maximum Pe ¼ 50 000); k ¼ 0.1, 0.5, 1, 2, 5, 10, 20, 50 and 1 ¼ 0.2, 0.3, 0.4, 0.5, 0.6.
rates of mass transfer by diffusion and by convection at the interface of the fluid sphere, it can easily be seen that: kc (2R) @C Shu ¼ ¼ 2 (13) Do @r r ¼1 where kc is the inter-phase mass transfer coefficient in the continuous phase. Naturally, the value of the Sherwood number Shu will vary with position, u, on the surface of the drop (r ¼ 1) and it is customary to define an average Sherwood number, Shavg as follows: ð 1 p Shavg ¼ Shu sin u d u (14) 2 o
Table 1. Comparison of average Sherwood number of a single spherical bubble, drop and solid particle (with an optimum value of R1 ¼ 150) in Newtonian fluids. Present
Feng and Michaelides (2001b)
Present
Pe ¼ 1
Reo
Feng and Michaelides (2001b)
Present
Pe ¼ 100
Feng and Michaelides (2001b) Pe ¼ 1000
k ¼ 0 (bubble) 1 100 500
2.361 2.428 2.437
2.354 2.423 2.426
8.449 11.166 11.865
10 100 500
2.471 2.488 2.482
2.385 2.417 2.424
9.010 10.947 11.759
10 100 500
2.454 2.468 2.486
2.380 2.413 2.422
8.352 10.227 11.566
10 100 500
2.439 2.453 2.468
2.365 2.398 2.399
1 100 500
2.323 2.402 2.407
2.315 2.384 2.386
8.198 11.223 11.887
22.990 31.840 34.427
23.349 32.441 35.010
8.745 10.604 11.594
23.934 30.414 33.590
22.380 29.726 33.503
8.240 10.141 11.329
21.336 27.584 32.489
21.360 25.499 32.138
14.021 17.730 22.301
14.849 18.756 22.119
11.318 15.764 19.559
11.628 15.946 19.277
k ¼ 0.5
k¼1
k ¼ 10 6.807 6.885 8.168 8.360 9.532 9.077 k ¼ 1 (solid particle) 5.813 5.865 7.696 7.695 8.932 8.524
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KISHORE et al. Table 2. Comparison of the present average Sherwood number for the case of slow motion (Reo ¼ 1) of Newtonian liquids in beds of spherical particles. 1 ¼ 0.6
1 ¼ 0.2
Pe
Pfeffer (1964)
Present
Pfeffer (1964)
Present
100 200 500 1000 2000
15.6032 19.6587 26.6807 33.6160 42.3535
15.8060 19.6133 26.6215 33.4651 42.0104
7.9506 10.0171 13.5933 17.1290 21.5812
8.7719 10.8158 14.3594 17.8598 22.2636
Validation of Results The numerical solution procedure developed to solve the continuity and momentum equations segregated from the species mass transfer equation has been thoroughly validated by benchmarking the values of the drag coefficient over wide ranges of conditions elsewhere (Kishore et al., 2006). However, the solution procedure as applied to the species mass transfer equation is validated here (Table 1). In particular, the present results are compared with the single particle results of Feng and Michaelides (2001b) for a range of values of k with an optimum domain of R1 ¼ 150. It is useful to recall here that an exponential
transformation has been used in the radial direction so as to capture the boundary layer dynamics with adequate accuracy. An inspection of Table 1 shows the two values to be within +2–4% of each other. The present solver has also been validated with the analytical solution of Pfeffer (1964) for the case of heat and mass transfer of Newtonian liquids in beds of solid spheres. Since, this analytical solution is valid in the limits of low Reynolds number and high Peclet number, comparisons have been shown in the range of 100 Pe 2000 (Reo ¼ 1) for two extreme values of particle concentration, i.e., 1 ¼ 0.6 and 0.2 (Table 2). Once again, the two values are seen to be in good agreement with each other.
Figure 2. The effect of Sc on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.2 at Reo ¼ 200 for different values of k. Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A8): 1203–1214
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Figure 3. The effect of Sc on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.2 at Reo ¼ 1 for different values of k.
Similar comparisons are obtained for the other values of 1 and hence are not shows here. Based on these comparisons, the present results are believed to be accurate within +2– 3%. It is worthwhile to add here that the differences of this magnitude are common in such numerical studies due to the differences stemming from grid errors, numerical solution procedure, and so on (Roache, 1994).
Effects of Reo, k, Sc and 1 on Local and Average Sherwood Numbers In general, on the surface of a bubble, drop and solid particle, the local value of Sh decreases as one traverses from the front stagnation point to the rear end of the sphere provided no separation occurs. In the case of recirculation, the local value of Sherwood number decreases from the front stagnation point up to the point of separation and then this value starts increasing as one traverses to the rear end. However, as reported in our previous work (Kishore et al., 2006), no recirculation was observed for any value of k and 1 for Reo 200; therefore, in this work, the local variation in the Sherwood number is found to be continually decreasing from the front stagnation point all the way up to the rear
stagnation point for all values of Reo, k, Sc and 1, as seen in Figures 2–5. Figure 2 shows the effect of Schmidt number on the local variation of the Sherwood number for a range of values of Schmidt number for 1 ¼ 0.2 at Reo ¼ 200 for different values of k. For small values of Peclet numbers, i.e., Pe up to 200, there is almost no variation in the local value of the Sherwood number on the drop surface as compared to that seen for large values of Pe for all values of the viscosity ratio. This is essentially due to the fact that at such low values of Peclet numbers, diffusion is the main mode of mass transfer with a little contribution from the convective mass transfer. Qualitatively as the value of the viscosity ratio increases, the value of Sherwood number at the front stagnation point decreases whereas it is almost unaltered at the rear end; therefore, the overall mass transfer rate decreases as the value of k increases for all values of 1. However, the role of k diminishes at small values of Sc, i.e., for Pe 200. Similar observations can be made from Figure 3 where the effect of Schmidt number on the local Sherwood number is shown for 1 ¼ 0.2 at Reo ¼ 1. However, a dramatic effect of Schmidt number can be seen as the Pe varies from 1 to 10 000 due to the thinning of the concentration boundary layer. Here also, for Pe 100,
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Figure 4. Effect of k on the local variation of Sh for Newtonian flow past an ensemble of 1 ¼ 0.6, at Reo ¼ 20 for different values of Sc.
the effect of k was found to be rather small as the rate of mass transfer is predominantly due to diffusion. But for Pe 100, the rate of mass transfer increases significantly due to the increased contribution from the convective mass transfer for a fixed value of the viscosity ratio. Figure 4 shows the effect of viscosity ratio on mass transfer for 1 ¼ 0.6 at Reo ¼ 20 for a range of values of Schmidt numbers. For all values of the Schmidt number and the viscosity ratio, as one traverses from the front stagnation point to the rear end, the local value of the Sherwood number decreases. However, for Sc ¼ 1, as the value of k increases, the local value of the Sherwood number at the front end decreases, whereas at the rear end it increases and thus, there is a cross over of the local value on the top of the drop surface [Figure 4(a)]. Thus, nullifying the effect of k on the overall value of the Sherwood number for small values of Sc (or Pe). Qualitatively similar effects was also seen for other values of the droplet concentration. But for large values of Sc, as the value of k increases, the local value of Sherwood number at the front stagnation point decreases and it is almost constant at the rear end, therefore, the rate of mass transfer decreases. Figure 5 shows the effect of the droplet concentration, 1, on the local variation in the Sherwood number when Reo ¼ 20 and Sc ¼ 100 (Pe ¼ 2000) for
different values of the viscosity ratio. For all values of k, as the value of the droplet concentration increases, the local value of the Sherwood number at the front stagnation increases due to the sharpening of the gradients and at the rear end it is almost constant for all values of 1 and k. Thus, the rate of mass transfer increases as the droplet concentration increases. Similarly, the rate of mass transfer decreases as the value of k increases for all values of droplet concentration. Figure 6 shows the representative results on the effect of Peclet number on the average Sherwood number as a function of the viscosity ratio for different values of Reo for 1 ¼ 0.2. As the value of Peclet number increases, the average Sherwood number increases for all values of the Reynolds number and the viscosity ratio. However, for small values of Pe, there is only a small effect on the average Sherwood number for small values of k. But as the value of k increases, the average Sherwood number decreases for a fixed value of Reo and Sc. Qualitatively similar trends are obtained for the other values of Reo, Sc, k and 1 studied herein and thus are not included here. Finally, it is useful to develop a simple predictive correlation based on the present numerical results for average Sherwood number which can be used to estimate the rate
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Figure 5. Effect of droplet concentration, 1, on the local variation of Sh when Reo ¼ 20 and Sc ¼ 100 for different values of k.
of mass transfer in ensembles of fluid spheres of a liquid – liquid system in a new application. As far as known to us, no such prior correlation is available in the literature. The following expression was found to be satisfactory to correlate the present numerical results: Shavg ¼ 0:89Pe0:48 k 0:17 (1 1)0:33 2 þ 3k 0:5 þ Pe0:1 (1 1)1:54 1þk
(15)
The above equation reproduces the present numerical results (1649 data points) with an average error of 11.86% which rises to maximum of 43.07%. Figure 7 shows the parity plot of the present numerical results and those obtained by equation (15) and no discernable trends could be detected. Finally, it is useful to re-iterate that in concentrated systems, the fluid spheres collide with each other, deform and even coalesce which will influence the rate of mass transfer in these systems. All these effects are neglected in this work. Therefore, the results presented herein are useful for dilute to moderately concentrated systems and these should be used only as a first level approximation for concentrated systems. In this context, the results reported herein
should serve as a basis to incorporate some of these effects in future studies.
CONCLUSIONS The values of the convective inter-phase mass transfer coefficients for ensembles of fluid spheres in another immiscible liquid have been obtained for wide ranges of the pertinent dimensionless parameters, namely, Reo, Sc, k and 1. Irrespective of the values of Reo, k, Sc and 1, as one traverses from the front stagnation point along the surface to the rear end, the local value of Sherwood number decreases. This is presumably so due to the fact that no recirculation was observed in these ranges of conditions. As the value of the viscosity ratio increases, the rate of mass transfer decreases for all values of Reo, Sc and 1. However, for small values of Pe, the effect of k on the local and average Sherwood number is very weak and the mass transfer coefficient is almost constant. For Peclet numbers up to Pe 200, the mass transfer is mainly diffusion controlled. As the value of the droplet concentration increases, the mass transfer coefficient also increases for all values of the Reynolds number, the Schmidt number and the viscosity ratio. As the value of the Peclet number increases, the mean value of Sherwood number also increases for all values of Reo, k and 1.
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Figure 6. Effect of k on the average Sh as a function of Pe for Newtonian flow past an ensemble of 1 ¼ 0.2 for different values of Reo.
However, for small values of Peclet numbers, the average value of Sherwood number is almost constant for all values of the viscosity ratio for a fixed value of 1. As the value of
Schmidt number increases, the concentration boundary layer thins thereby enhancing the rate of mass transfer. Based on the present numerical results a simple predictive correlation has been developed which reproduces the present numerical results with an average error of 11.86%.
NOMENCLATURE C Co Cs Do k kc p Pe QUICK
Figure 7. Comparison of the present average Sh with the proposed correlation values.
r R R1 Re Sc Sh Shu Shavg SMAC Uo
molecular concentration molecular concentration at the cell boundary molecular concentration on the surface of the drop diffusivity, m2 s21 internal to external fluid viscosity ratio, mi/mo continuous phase mass transfer coefficient, m s21 pressure Peclet number quadratic upstream interpolation of convective kinematics radial distance drop radius, m cell boundary Reynolds number Schmidt number Sherwood number Sherwood number along the surface of the drop average Sherwood number simplified marker and cell free stream uniform velocity, m s21
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r-component of velocity u-component of velocity f-component of velocity
Greek symbols 1 volume fraction of the dispersed phase u streamwise direction, degree f azimuthal direction, degree m dynamic viscosity, Pa s l internal to external fluid density ratio, ri/ro r density of fluid, kg m23 t shear stress, Pa Subscripts i o r u f
internal flow variable external flow variable r-component u-component f-component
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