Computational Fluid Dynamics Simulations of Single Drops in

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CFD simulations of dynamics of single drops in confined geometries Hrushikesh P. Khadamkar, Ashwin Wasudeo Patwardhan, and CHANNAMALLIKARJUN S MATHPATI Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 20 Jun 2017 Downloaded from http://pubs.acs.org on June 20, 2017

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Industrial & Engineering Chemistry Research

CFD simulations of dynamics of single drops in confined geometries Hrushikesh P. Khadamkar1, Ashwin W. Patwardhan1, Channamallikarjun S. Mathpati1*

1. Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India

*Corresponding author: Tel: 91-22-33612106; Fax: 91-22-33611020 E-mail address: [email protected]

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Abstract Understanding of drop rise in confined domains is important as it influences drop rise velocity and drag coefficient. In this work, the motion of single liquid drop rising in a quiescent channel filled with water in presence of wall effect is studied numerically using combined Volume of Fluid and level set method. The ratio of drop diameter to column diameter is used as a parameter to quantify the presence of wall effect. Simulations were carried out for n-butanol and toluene drops. A reduction in rise velocity is seen in all the cases. It is seen that pressure profile, strain rate and vorticity magnitude near the drop surface have a significant effect on the evolution of a drop shape in confined geometries. Drag coefficients have been calculated from the force balance equation. The correlation for correction factor for drag coefficient has been developed to account for the wall effect. Keywords: Drop, wall effect, terminal velocity, drag coefficient, CFD


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1. Introduction The motion of fluid particles (drops and bubbles) rising/settling in a liquid has been a subject of numerous studies. This subject although appears simple but has shown numerous complexities owing to change in particle shape from spherical to oblate ellipsoidal, variations in rising or settling trajectories ranging from rectilinear trajectories in case of spherical particles to zigzag or helical trajectories in the case of oblate ellipsoidal particles, and shape oscillations. The interest in understanding this phenomenon is justified by its application in variety of engineering applications. The application areas include emulsion formation, where one fluid phase must be dispersed in another fluid, the design of efficient mixing devices, the underground multiphase flows in oil and gas reservoirs, and the liquid-liquid extraction. In most practical situations, the motion of fluid particles is confined by walls. In case of enhanced oil recovery, oil drops must be displaced by water in porous rocks. Liquid-liquid contacting is widely used in chemical process industries for reaction and extraction. In most of the liquid-liquid extraction equipment the contact between the liquids is secured through dispersion of the phase as drops. The interface of a drop is mobile. Hence knowledge of drop dynamics should provide the basic information needed for the design of most liquid-liquid contactors in which the drop size is related to the transfer efficiency and the terminal velocity to the capacity of equipment. The drop dynamics are also governed by the presence of neighboring drops and solid surface. The effect of surrounding walls is of importance to the motion of the drop, when characteristic dimension of the drop is of the same order of magnitude as the size of the flow channel. The deformable drops are influenced by the confining surfaces in two ways: their drag is increased, and their shape becomes more elongated. The confining walls exert extra retardation force. This results in the reduction of terminal velocity of drop. The knowledge of this issue is relevant to 3 ACS Paragon Plus Environment


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the design of liquid-liquid extraction equipments such as sieve tray contactor, spray extraction columns. Thus the study of the motion of liquid drops and their behavior in another liquid medium of infinite and restricted extent (column diameter to drop diameter < 10) is of importance to the liquid-liquid extraction process. The wall effect on motion of fluid particles can be expressed in different ways, viz, drag force ratio, velocity ratio, drag coefficient ratio. The velocity ratio is the most commonly used parameter to quantify the wall effect. The velocity ratio is a function of Reynolds number and drop diameter to column diameter ratio. Figure 1 shows the schematic of drop rising through a fluid in a column. The diameter ratio, λ is defined as

λ=

dp

(1)

D

In the literature, the diameter ratio, λ has been extensively used as a model parameter to correlate the effect of wall on terminal rise velocity in confined geometries to that in unconfined geometries. Clift et al.1 proposed that for drops with Eo < 40, wall effects are insignificant (effect on terminal velocity is below 2%) if the following conditions apply: Re p ≤ 0.1

for λ ≤ 0.06

(2)

0.1 < Re p < 100

for λ ≤ 0.08 + 0.02 log10 Re p

(3)

Re p ≥ 100

for λ ≤ 0.12

(4)

On the other hand, for Rep > 100, the wall effect becomes independent of the Reynolds number and Clift et al.1 correlated the literature data as:


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(

Vt Vt ∞ = 1 K U = 1 − λ 2

)

32

(5)

The equation due to Strom and Kintner2 is similar to Equation 5 except for the value of exponent being 1.43. Equation 5 is valid for Eo < 40, Rep > 200 and λ ≤ 0.6. For the case of 1 ≤ Rep ≤ 200, Clift et al.1 recommended the use of plot of Rep and KU -1 vs Best number (ND) for estimation of wall effect on terminal velocities of fluid particles. In the available literature, the free drop movement within a quiescent medium has been studied extensively both experimentally and numerically.

Number of experimental studies3–11

(for review see 1,12,13) have been conducted on unconfined drops. These studies have calculated drop rise velocity, drop shape, drop rise trajectory as a function of Eo and M number. In addition to experimental studies, plenty of literature9–11,14–16 is available on drop dynamics of single drop rise in quiescent continuous phase in an unconfined domains. Table 1 assembles publications from authors dealing with numerical simulations of single drop movement in quiescent continuous phase. These studies involved the use of Volume of Fluid (VOF) method; level set method, combined level set and VOF method to track the drop interface. In addition to this some authors have used Lattice Boltzmann method to study drop dynamics in unconfined domains. The varied parameters were drop size ranging from 1 mm to 5 mm, physical property of continuous and dispersed phase. These numerical studies have been validated (in terms of terminal rise velocity and drop shape) based on their own experimental data and Grace et al.8 chart. Wegener et al.17 have analyzed the published experimental4,5,9,10 and numerical data9,10 for drop dynamics in a quiescent ambient liquid of various groups for different systems. They have recommended the correlations proposed by Hamielec et al.18 and Feng and Michaelides19 to calculate drag coefficient of spherical drops if the system is not contaminated and interfacial



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tension of system is not too small. For the case of oscillating drops, they have recommended the correlations proposed by Thorsen et al.5 and Harper et al.7 (accuracy: 4-7% deviation). They proposed that Clift diagram should be used with caution to calculate terminal velocity. They found good agreement with Clift diagram for spherical drops and low interfacial tension systems. Compared with number of experimental and numerical studies on drop dynamics in an unconfined domain, literature related to effect of wall on fluid particles is scant. Number of authors (for example see,

20–25

) have studied gas bubble dynamics in confined domains.

However, there has not been much activity to study the effect of wall on drop dynamics since the aforementioned early studies described in Clift et al.1.12 Mao et al.26 determined the velocity of single drop moving through Sulzer and Schott structured packings. They used Equation 5 as a basis for correlation of structured packing data. Chhabra and Bangun27 investigated the extent of the wall effects on the free falling velocity of fluid spheres in quiescent Newtonian and pseudo-plastic non-Newtonian media. They showed that the terminal velocity (hence drag coefficient) of the fluid spheres falling in quiescent Newtonian and pseudo-plastic polymer solutions is strongly influenced by the confining walls. They found that in the creeping flow region (Rep 3 mm). Hayashi and Tomiyama30 developed a drag correlation for fluid particles rising along the axis of a vertical pipe at low and intermediate Rep by making use of available drag correlations and interface tracking simulations. The proposed correlation gives accurate estimation of the drag coefficient for fluid particles rising through stagnant liquids in vertical pipes for 0.079 ≤ Eo ≤ 30, −10 ≤ log M ≤ 2, 0.083 ≤ Rep < 200, 0 ≤ γ ≤ 10 and λ ≤ 0.6. While experimental studies on the drag on drops and bubbles in the literature have been carried out using the vessels or containers of finite size, very few investigators have actually set out to glean data on wall effects by using containers of different diameters.12 These studies are limited to analysis of terminal velocity of drop in presence of wall effect.1,12,13 For the case of air bubbles rising in a quiescent liquid, various authors 22,31,32 have studied effect of confinement on drag coefficient and drop shape using numerical simulations. Numbers of experimental studies 31,33–36

have reported effect of confinement on rise trajectories and shape oscillations of bubble

rising in a liquid column. For the case of liquid drops, the majority of numerical studies done so far were concentrated to understand the drop dynamics in unconfined domains. Only few investigators

9,14

have studied the effect of wall on drop rise velocity in order to decide the

domain size to carry out the simulations in an unconfined domains. There appears a large gap in the literature pertaining to dynamics of the confined fluid particles, especially for the case of liquid drops. Thus a more comprehensive approach is required to investigate the effect of wall on the dynamics of liquid drop. These effects have a strong influence on the drop dynamics. 7 ACS Paragon Plus Environment


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Thus, it is necessary to carry out the numerical simulations in a three dimensional computational domain to understand the drop dynamics in the presence of wall effect. Therefore the aim of the present work is to study the effect of wall on drop dynamics in terms of rise velocity, drop shape and drag coefficient using three dimensional CFD simulations. An attempt has been made to develop a correlation for correction factor for calculating drag coefficient for liquid drops in presence of wall effect. 2. Modeling Strategy The three dimensional CFD simulations were performed for drop of toluene (1 mm, 2 mm, 4 mm, 6 mm) or n-butanol (1 mm, 2 mm) rising in a quiescent water. Simulation geometry consisted of rectangular domain of various sizes. Simulations are carried out for various values of diameter ratio, λ (0.11, 0.5, and 0.8). (CLSVOF) method

37–39

The combined level set and volume of fraction

was used to track the drop interface. Modeling aspects of CLSVOF

method have been explained below. The effect of wall on drop rise velocity, internal circulations inside the drop, drop shape, drag coefficient was studied from CFD simulations. The level-set function φ is defined as a signed distance to the interface. Accordingly, the interface is the zero level-set, φ (x,t) and can be expressed as Γ = {x | ϕ ( x, t ) = 0} in a two-phase flow system: + | d | if x ∈ primary phase  ϕ ( x, t ) = 0 if x ∈ Γ − | d | if x ∈ secondary phase 

(6)

where d is the distance from the interface. The normal and curvature of the interface, which is needed in the computation of the surface tension force, can be estimated as


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n=

∇ϕ |ϕ =0 | ∇ϕ |

κ = ∇⋅

(7)

∇ϕ |ϕ =0 | ∇ϕ |

(8)

The evolution of the level-set function can be given by

( )

∂ϕ +∇⋅ uϕ = 0 ∂t where

u

( )

(9)

is the underlying velocity field. The momentum equation can be written as

(

)

( )

T ∂ρu + ∇ ⋅ ρ u u = −∇p + ∇ ⋅ µ ∇u + ∇u  − σκδ (ϕ ) + ρ g   ∂t

(10)

where

δ (ϕ ) =

1 + cos(πϕ a ) if | ϕ | < a and a = 1.5h otherwise δ (ϕ ) = 0 2a

(11)

where h is the grid spacing and σ is the surface tension coefficient. The surface tension force has been modeled using continuum surface force model39. The re-initialization of the level set function was carried out at each time step by using geometrical interface front construction method. The values of the VOF and the level-set function were both used to reconstruct the interface-front. The VOF model provides the size of the cut in the cell where the likely interface passes through, and the gradient of the level-set function determines the direction of the interface. 2.1 Solution Methodology The schematic of computational domain is shown in Figure 2. Simulations were performed with commercially available CFD software Ansys Fluent 14.

The details of

combinations of drop diameter, dp and λ used in simulations are given in Table 2. The physical 9 ACS Paragon Plus Environment


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properties reported by Bertakis et al.9 and Wegener et al.4 were used in all the simulations and are summarized in Table 3. The domain at time t = 0 s, was assumed to be filled with water. The initial drop shape was assumed to be spherical. The initial position of the drop centre was fixed at 5 mm from the bottom wall in all the cases. Initially simulations were carried out to determine the drop terminal velocity in unconfined domain. This exercise was taken up to validate the CFD model which was later used to study the wall effect on drop dynamics. 2.2 Grid size and Time step criteria Resolution of the computational grid is a key factor in any CFD simulation. Therefore, simulations were performed on three different grid sizes (20, 30 and 40 cells per diameter (CPD)) to study the effect of grid size on predictions of drop rise velocity for drop of toluene (dp = 2 mm and 4 mm) in a domain with λ = 0.5. Hexahedral mesh was used in the present study for all the simulations.

The results of grid sensitivity study are presented below in separate

subsection. To obtain stable solutions for transient free-interface problems, two stability criteria with respect to the selected time step should be fulfilled. The first one is related to the Courant number while the second one is related to the treatment of the surface tension. The Courant number is defined as

Co =

umax ⋅ ∆t ∆x

(12)

where umax is the maximum velocity in the computational domain, ∆x is grid size and ∆t is time step. To obtain a stable solution, the Courant number should always be smaller than one, so that, within a selected time step, the interface should only move from one grid element to a 10 ACS Paragon Plus Environment

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neighboring one. Values of cell Courant number near the interface were monitored for all the simulations. It was ensured that range of interfacial Courant number in all the simulations is always less than 0.5. Magnitudes of Courant number near the interface were found to be in the range of 0.01 to 0.04. The second stability criterion is referred to the explicit treatment of the surface tension. The explicit treatment of surface tension is stable when the time step resolves the propagation of capillary waves.39 It is given as

cΦ ⋅ ∆t 1 < 2 ∆x

(13)

The capillary wave phase velocity (CΦ) is given by 40

cΦ =

σ ⋅k (ρ c + ρ d )

(14)

By substituting Equation 14 in Equation 13, we get,

∆t ≤

(ρ c + ρ d ) ⋅ (∆x )3

(15)

4πσ

Equation 15 gives the second stability criterion related to explicit treatment of surface tension. Both stability criteria should be satisfied, and, in this way, the minimum time step obtained by these criteria is selected. In present work, time step of 1 × 10-6 s was used in all the simulations. 2.3 Boundary conditions and discretization schemes The no slip boundary condition was used at the bottom wall and all side walls. At the outlet, outflow boundary condition was employed. The initial velocities in the domain were



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assumed to be zero. PISO algorithm was used for pressure velocity coupling. Momentum equation was discretized using second order upwind scheme. Pressure term was discretized using PRESTO scheme and volume fraction equation was discretized using Geo-reconstruct scheme. Level set equation was discretized using second order upwind scheme. Convergence criteria (Scaled Residuals) were set at 0.0001 for all the equations. The convergence was achieved at every time step. All the simulations were performed on i7 machine with 16 GB of installed memory.

3 Results and discussion 3.1 Effect of grid size For this purpose, simulations were carried out for the drop size of 2 mm and 4 mm in a domain with λ = 0.5. Simulations were carried out till 1 s with hexahedral grid size with resolution of 20, 30 and 40 CPD. Time step for simulation was decided based on stability criteria given by Equation 12 and Equation 15. Table 4 shows the comparison of rise velocity obtained for three grid resolutions. It can be seen from Table 4 that by increasing the grid resolution Reynolds number based on terminal velocity reduces.

The value of Reynolds number based on terminal velocity

obtained from Clift et al.1 correlation is 138.8 and 514.6 for drop of 2 mm and 4 mm diameter respectively. For grid resolution of 40 CPD, value of Ret is 145.8 and 525 for drop size of 2 mm and 4 mm respectively. By increasing the grid resolution from 30 CPD to 40 CPD, calculation time for performing simulation for 1 s of drop rise is ~ 50 days. Further increase in grid resolution will lead to improvement in prediction but at the expense of large computation time. The variation in aspect ratio for both the drop sizes lies within 3% for grid resolutions of 30 CPD 12 ACS Paragon Plus Environment

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and 40 CPD. The drop rise trajectory for drop of 4 mm is also plotted for various grid sizes to check the effect of grid size on rise trajectory of the drop. Figure 3 shows the drop rise trajectory for 4 mm drop of toluene in a domain with λ = 0.5 for various grid resolutions. It can be seen that for all grid resolutions drop rise trajectory has remained practically the same. Hence all the simulations were carried out using grid resolution of 40 cells per diameter. 3.2 Model Validation In the present work, validation of CFD model has been done in two ways. First CFD model has been validated by comparing drag coefficient values obtained from simulations with available correlations in literature for unconfined domain. The drag coefficient for all the cases in the current work has been calculated from force balance. The force balance on buoyancy driven drop can be written as

πd p3

πd p3 dVr 1 2 ρd + Cd Ap ρcVr = (ρd − ρc )g 6 dt 2 6

(16)

The projected area, Ap and rise velocity of the drops have been calculated from the simulations. For this purpose, simulations were carried out for drop sizes of 1 mm and 2 mm for toluene drop and 1 mm for n-butanol drop. Table 5 shows comparison of Cd obtained from predictions for spherical drops with available correlations. The values of Cd for solid particles as well as spherical bubbles are also included in Table 5. For solid particles, value is reported using following correlation41. 0.5 24 3.73 0.00483 ⋅ Re p Cd = + 0.5 − + 0.49 for Rep < 3 × 105 −6 1.5 Re p Re p 1 + 3 ×10 Re p


(17)


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For the case of spherical bubbles, drag coefficient value in Table 5 is reported using following correlation42

Cd =

 16 14.9  1  + 0.78  − 0 . 6 Re p Re p  1 + 10 Re p 

for Rep > 2

(18)

The correlation derived by Feng and Michaelides19 for high Reynolds number for fluid particles is valid up to Rep = 1000 and is given as follows:

C d (Re p , γ ) =

2−γ

C d (Re p , γ ) =

4 γ −2 C d (Re p ,2) + C (Re p , ∞ ) γ +2 γ +2 d

γ

C d (Re p ,0) +

4γ C d (Re p ,2) 6+γ

(19)

(20)

with the functions, C d (Re p ,0 ) =

48 Re p

  1 + 2.21 − 2.14   Re p Re p  

C d (Re p ,2) = 17.0 Re p 24 C d (Re p , ∞ ) = Re p

(21)

−2 / 3

(22)

 Re p 2 / 3  1 +   6  

(23)

Equation 19 is valid for 0 ≤ γ ≤2 and 5 < Rep ≤ 1000. Equation 20 is valid for 2 ≤ γ ≤∞ and 5 < Rep ≤ 1000. The values of Cd obtained from current predictions lie in between the value of Cd for rigid sphere and spherical gas bubble.

The predicted drag values were also compared with the

correlation developed by Feng and Michaelides19. The predicted Cd values were also compared with Cd values obtained (from Equation 16) using terminal rise velocities reported by Bäumler et al.10 as well as with Cd values obtained using terminal velocity given by Henschke's semi14 ACS Paragon Plus Environment

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empirical correlation (for details see

10

). For toluene as well as n-butanol drop, values of Cd

show good agreement with above correlations. The predictions for drop rising in confined domain have been validated as follows. For Rep < 200, model validation has been done by comparing simulation results with the plot of KU -1 vs ND1/3 and Ret vs ND1/3 as suggested by Clift et al.1. Figure 4 shows the plot of KU - 1 vs ND1/3. The symbols in Figure 4 represent the predicted values. It can be seen from Figure 4 that there is good agreement between predicted values and the correlation given in Clift et al.1. The predicted values are lower as compared to Clift correlation. This is expected as the correlation reported by Clift at al.1 is for solid particles. The solid particles will have lower terminal velocity than liquid drop for the same particle size. Figure 5 shows the plot of Ret vs ND1/3. The symbols represent predicted values obtained from simulations. The predicted values are in good agreement with the correlation reported in Clift et al.1. For Rep > 200, simulation results have been compared with plot of 1/KU vs λ (Equation 5) as suggested by Clift et al.1. Figure 6 shows comparison of numerical simulations in terms of velocity ratio, 1/KU to the correlation given by Equation 5. The error bars in Figure 6 represent the spread of data for which Equation 5 was fitted. The Equation 5 predicts the ratio of velocity within ±5% for λ ranging from 0 to 0.3. The accuracy of equation 5 lies within ±10% for λ ranging from 0.3 to 0.4. The accuracy of equation 5 lies within ±15% for λ ranging from 0.4 to 0.5. For λ ranging from 0.5 to 0.6, the accuracy of equation 5 lies within ±30%. For the values of λ > 0.6, the deviation is more than 30%. For this purpose, simulations were performed on domains with value of λ = 0.11, 0.5, 0.8 for drop sizes of 2 mm, 4 mm and 6 mm.

Terminal velocity in an unconfined channel obtained from

Henschke's correlation is used to calculate velocity ratio 1/KU. It can be seen from Figure 6 that simulation results agree well with correlation given by Equation 5. The predicted values of 15 ACS Paragon Plus Environment


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velocity ratio match with Equation 5 within 5% for λ = 0.11 and within 18% for λ = 0.5. Equation 5 is valid for λ < 0.6. Still predicted value of 1/KU at λ = 0.8 for spherical drops is well reproduced by Equation 5. Thus there is good agreement between the predicted values and literature data for terminal drop velocity in terms of drag coefficient in unconfined channels. In addition to that there is good agreement of predicted data in presence of wall effect with correlations reported in Clift et al.1 as seen in Figures 4 to 6. This good agreement between predicted values with literature data validated our CFD modeling methodology. The validated CFD model was used for making predictions for various drop sizes in various domains. 3.3 Effect of λ on drop rise velocity Figure 7 shows the effect of wall on rise velocity of drop for various drop diameters. The horizontal line in Figures 7(a)-(e) indicates the terminal drop rise velocity in unconfined domains. It can be seen from Figures 7(a) - (e) that rise velocity has been hindered for all drops due to the presence of wall. This confirms that wall effect is present in simulations on domains with λ > 0.11. Table 6 assembles the predicted rise velocity in presence of wall effect for all the cases studied in the present work. The value of terminal drop rise velocity for 1 mm toluene drop in unconfined medium is 41.09 mm/s. The terminal rise velocity obtained from prediction in a domain with λ = 0.5 is 20.5 mm/s. In a domain with λ = 0.8, terminal rise velocity was further reduced to 7.5 mm/s. The value of terminal drop rise velocity for 2 mm drop in unconfined medium is 109.38 m/s. The terminal rise velocity obtained from prediction for 2 mm drop in confined domains is 65.1 and 24.2 mm/s for λ = 0.5 and λ = 0.8 respectively. The value of 16 ACS Paragon Plus Environment

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terminal drop rise velocity for 4 mm drop in unconfined medium is 177.8 mm/s (calculated using Henschke correlation). For 4 mm drop, value of rise velocity obtained from prediction in a domain with λ = 0.5 is 117.2 mm/s whereas for a domain with λ = 0.8 value of rise velocity is 48.4 mm/s. The curves for the 6 mm drop diameter show oscillations around a mean value with increasing amplitude (Figure 7 (d)). The value of rise velocity for 6 mm drop in a domain with λ = 0.5 is 113.7 mm/s. The value of rise velocity for 6 mm drop in a domain with λ = 0.8 is 55 mm/s.

The value of terminal drop rise velocity for 1 mm and 2 mm n-butanol drop in

unconfined medium is 29.6 mm/s and 58.5 mm/s (calculated using Henschke correlation). It can be seen from Table 6 that rise velocity of 1 mm (12.9 mm/s) as well as 2 mm (32.5 mm/s) nbutanol drop has been hindered due to the proximity of wall to drop interface. Thus proximity of wall to the drop surface resulted in lowering of the rise velocity of drops. The rise velocity of nbutanol drops are lower as compared to toluene drops of same diameters for λ = 0.5. This is in accordance with Licht and Narasimhamurty43 who showed that rise velocities decrease with lowering of interfacial tension. 3.4 Internal Circulations To obtain a better view of the velocity field inside the drop, the local velocity field in the computational domain needs to be translated in the reference frame related to the moving drop. This can be achieved by subtracting the drop velocity at that instant of time from the axial component of the local velocity field. The velocity vector plots (in terms of the moving drop reference frame) for different drop diameters have been plotted. The velocity vector plots colored by y-velocity in the frame of reference related to the moving drop at various time help to give more insight into the local flow patterns near the drop interface. Figures 8(a) - (d) illustrate the development of internal circulations for 2 mm drop in a 17 ACS Paragon Plus Environment


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domain with λ = 0.5. At t = 0.1 s, the internal circulations inside the drop just start to develop which can be seen from two small but weak circulation cells inside the drop. The small wake formation can be seen behind the drop from Figure 8(a). At t = 0.3 s, the intensity of internal circulations inside the drop has been increased (Figure 8(b)). The wake region behind the drop in turn has been increased as seen in Figure 8(b) when compared to Figure 8(a). At t = 0.5 s (Figure 8(c)), internal circulations inside the drop have attained the full strength. This can be seen from higher values of velocity inside the drop. The wake region in turn also grew in length and as compared with previous times (Figure 8(a) and Figure 8(b)). The strength of internal circulations at t = 1.0 s has remained similar to that at t = 0.5 s as seen in Figure 8(c) - (d). Figures 9(a) - (d) illustrate the development of internal circulations inside the drop of 4 mm diameter in a domain with λ = 0.5. The upward movement of the drop can be seen by negative velocities (-0.09 m/s to -0.18 m/s) observed near the two sides of the drop. Drop of 4 mm diameter traveled at higher speed compared to drop of 2 mm diameter as seen in Figure 7(b) and Figure 7(c). This can be confirmed from Figure 8 and Figure 9 with higher magnitude of downward velocity (-0.26 m/s for dp = 4 mm; -0.087 m/s for dp = 2 mm) at the sides of the drop surface. The drop of 4 mm diameter has become slightly elongated at t = 0.2 s as seen in Figure 9(b). From Figure 9(c) and Figure 9(d), it can be seen that the drop shape has changed from spherical to oblate spherical. The two vigorous circulation cells inside the drop at t = 1.0 s can be seen from Figure 9(d). Figures 10(a) - (d) show the development of internal circulations inside the drop of 6 mm diameter in a domain with λ = 0.5. The drop of 6 mm diameter has been stretched more in horizontal direction at t = 0.2 s (Figure 10(b)) when compared with drop of 4 mm diameter (Figure 9(b)). Internal circulation cells occupied entire interior region of drop. The significant 18 ACS Paragon Plus Environment

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deformation in drop shape can be seen from Figure 10(c) and Figure 10(d). At t = 0.5 s, 6 mm drop underwent severe shape deformation. The strength of internal circulation is higher at corners on downstream side of the drop. At t = 0.8 s, drop of 6 mm diameter has been further deformed and got slightly tilted and shifted by 0.54 mm from central axis.

The internal

circulation inside the drop of 6 mm diameter grew in strength which can be seen from higher values of velocity near the central axis of drop (Figure 10(c) and Figure 10(d)). The strength of internal circulations is quantified by comparing minimum and maximum y-velocity in the frame of reference of the moving drop. The minimum and maximum axial (Vy) velocities in the droplet frame of reference represent minimum and maximum values in the entire domain. Figure 11 (a) - (e) illustrate the strength of internal circulations for drops of toluene and n-butanol in terms of minimum and maximum values of y-velocity in the frame of reference of moving drop. For drop of toluene of 1 mm diameter (Figure 10(a)), the strength of internal circulation lies in the range of -0.08 m/s to 0.07 m/s in domains with λ = 0.5 and 0.8. For drop of 2 mm diameter, it can be seen from Figure 11(b) that strength of internal circulation is highest (0.096 m/s to 0.067 m/s in terms of minimum and maximum values of y-velocity) in a domain with λ = 0.5. This is due to the free surface of drop is farthest from the domain wall. The strength of internal circulation is lower (-0.078 m/s to 0.045 m/s in terms of minimum and maximum values of y-velocity) in a domain with value of λ = 0.8 due to the proximity of wall to the free surface of drop. Figure 11(c) shows the strength of internal circulation for drop of 4 mm diameter in terms of minimum and maximum values of y-velocity in the frame of reference of moving drop. For 4 mm drop, the strength of internal circulation is highest in a domain with λ = 0.5 (-0.268 m/s to 0.126 m/s in terms of minimum and maximum values of y-velocity). The strength of internal circulation in a domain with λ = 0.8 for 4 mm drop is -0.138 m/s to 0.095 m/s 19 ACS Paragon Plus Environment


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in terms of minimum and maximum values of y-velocity. Thus strength of internal circulations diminished with increase in value of λ. Figure 11(d) shows the strength of internal circulation for drop of 6 mm diameter in terms of minimum and maximum values of y-velocity in the frame of reference of moving drop. The strength of internal circulation is higher in a domain with λ = 0.5 (-0.278 m/s to 0.17 m/s in terms of minimum and maximum values of y-velocity) as compared to that in a domain with λ = 0.8 (-0.195 m/s to 0.111 m/s in terms of minimum and maximum values of y-velocity) for drop of 6 mm diameter. Thus strength of internal circulations for 6 mm drop also diminished with increase in value of λ. Figure 11(e) depicts the strength of internal circulation for drops of n-butanol in a domain with λ = 0.5 in terms of minimum and maximum values of y-velocity in the frame of reference of moving drop. It can be seen from Figure 11(e) that strength of internal circulation for 1 mm n-butanol drop is in the range of 0.022 m/s to 0.007 m/s. For 2 mm n-butanol drop, the strength of internal circulation lies in the range of -0.051 m/s to 0.023 m/s y-velocity in the frame of reference of moving drop. For all the drop diameters, the internal circulation was observed. From Figure 11 (a) (e), it can be seen that magnitude of y velocity in the reference frame related to the moving drop in a domain with λ = 0.5 is highest in case of 6 mm drop. This in turn resulted into strong circulation cells inside the 6 mm drop as compared to drops with dp < 6 mm. This is due to the higher value of buoyancy force experienced by the drops with increase in drop size. Also, strength of internal circulation is lower for drops of n-butanol as compared to that of toluene in a domain with λ = 0.5 for same drop sizes. This can be seen by comparing Figure 11 (a) - (b) with Figure 11 (e). Figure 12 shows the variation of axial velocity in droplet frame of reference on a horizontal line passing through centre of mass of the drop for all the cases at t = 0.7 s. This time 20 ACS Paragon Plus Environment

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was chosen as terminal velocity of the drop was reached in all the cases at t = 0.7 s. The x-axis in Figure 12 represents distance normalized by droplet radius.

At normalized distance of

approximately 1 (near the drop interface), axial velocity in droplet frame of reference becomes negative due to downward directed relative velocities. Figure 12(a) illustrates the profile of axial velocity in droplet frame of reference for toluene drops in a domain with λ = 0.5. For the smallest drop diameter of 1 mm, axial velocity in the droplet frame of the reference at the centre of the drop is 0.025 m/s. Internal circulations become more intensive for 2 mm drop. For 2 mm drop, axial velocity in droplet frame of reference ear the centre of the drop is 0.060 m/s. The value of axial velocity is also higher near the interface for 2 mm drop as compared to that in case of 1 mm drop. The internal circulation pattern starts to change for drop diameter of 4 mm and 6mm due to change in shape of the drop. The region with positive axial velocity increases due to elongation of the drop shape. This can clearly be seen for the case of 6 mm drop. For 4 mm drop, axial velocity near the centerline of the drop increased to 0.121 m/s. The value of axial velocity near the centerline got reduced to 0.08 m/s for 6 mm drop. However, magnitude of axial velocity at x/rp ~ 1.4 is higher for 6 mm drop as compared with rest of the drops. Figure 12(b) depicts the profile of axial velocity in droplet frame of reference for toluene drops in a domain with λ = 0.8. It can be seen from Figure 12(b) that an increase in drop diameter led to increase in axial velocity near the centerline of the drop. This can be due to the lesser shape deformation of the drop in a domain with λ = 0.8 as compared to that in domain with λ = 0.5. The value of axial velocity is 0.015 and 0.045 m/s for drop sizes of 1 mm and 2 mm respectively. For drop of 4 mm diameter, axial velocity near the centerline of the drop is 0.092 m/s. For 6 mm drop, axial velocity (0.102 m/s) near the centre of the drop is higher as compared 21 ACS Paragon Plus Environment


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to the rest of the drop sizes. This is in contrast to the behaviour observed for domain with λ = 0.5. It can also be seen from Figures 12 (a) and 12(b) that the magnitude of axial velocity near the interface are considerably lower in domain with λ = 0.8 as compared to that in domain with λ = 0.5. Figure 12(c) shows the profile of axial velocity in droplet frame of reference for nbutanol drops in a domain with λ = 0.5. It can be seen from Figure 12(c) that magnitude of axial velocity near centerline of the drop is 0.007 m/s and 0.023 m/s for drop sizes of 1 mm and 2 mm respectively. Near the drop interface, magnitude of axial velocity is higher for 2 mm drop as compared to 1 mm drop. It can also be seen from Figures 12 (a) and 12 (c) that magnitude of axial velocity in droplet frame of reference at centerline as well as near the interface is lower for n-butanol drops as compared to toluene drops. 3.5 Drop Aspect ratio In order to quantify the change in drop shape, aspect ratio of the drop has been calculated for various drop diameters. Figure 13 depicts the effect of wall on drop aspect ratio for various drop diameters. Aspect ratio is defined as the ratio the length of minor axis to the length of major axis of the drop. The length of minor axis was calculated by plotting the value of level set function on the vertical line passing through center of mass of the drop. The length of major axis was calculated by plotting the value of level set function on the horizontal line passing through center of mass of the drop. The difference between the coordinates where the level set function changes its sign was calculated which represents the length of major and minor axis of the drop. The drop shape can be termed spherical if the value of aspect ratio is within 10% of unity1.


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For 1 mm drop of toluene (Figure 13(a)), the value of aspect ratio varied in the range of 0.98 to 1.0 for all domains (λ = 0.5, λ = 0.8). Thus value of aspect ratio is within 10% of unity for drop size of 1 mm. Thus, the toluene drop of 1 mm size can be termed to remain spherical. For toluene drop of 2 mm diameter (Figure 13(b)), the value of aspect ratio varied in the range of 0.95 to 1.0 for all domains (λ = 0.5, λ = 0.8). Thus value of aspect ratio is within 10% of unity for drop size of 2 mm. Thus, the toluene drop of 2 mm size can be termed to remain spherical. For drop size of 4 mm (Figure 13(c)), aspect ratio varied in the range of 0.4 to 0.99. The value of aspect ratio lies within 10% of unity for a domain with λ = 0.8 throughout its travel path in a domain. The value of aspect ratio lies within 10% of unity till t = 0.1 s in a domain with λ = 0.5. After t = 0.1 s, value of aspect ratio decreased continuously (0.9 to 0.4) with respect to time. Hence drop of 4 mm diameter changed its shape from spherical to ellipsoidal in a domain with value of λ < 0.8. For drop size of 6 mm (Figure 13(d)), the value of aspect ratio varied in the range of 0.47 to 0.99. The value of aspect ratio lies within 10% of unity till t = 0.1 seconds in all domains (λ = 0.5 and λ = 0.8). In a domain with λ = 0.8, aspect ratio varied from 0.9 to 0.76 after t = 0.1 s. Aspect ratio for drop of 6 mm diameter showed oscillatory behavior after t = 0.3 s in domain with λ = 0.8. In a domain with λ = 0.5, aspect ratio decreased continuously to 0.44 till t = 0.38s. After t = 0.38 s, aspect ratio showed oscillatory behavior. Figure 13(e) shows the temporal variation of aspect ratio for drops of toluene and nbutanol in a domain with λ = 0.5. It can be seen from Figure 13(e) that drop of 1 mm diameter of toluene as well as n-butanol remained spherical as aspect ratio value is greater than 0.9. Toluene 23 ACS Paragon Plus Environment


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drop of 2 mm diameter remained spherical throughout its rise path. However, n-butanol drop of 2 mm diameter deviated from its initial spherical shape at t = 0.05 s. The value of aspect ratio for n-butanol drop decreased continuously to 0.49 till t = 0.2 s. After t = 0.2 s, value of aspect ratio remained constant at 0.48. This behavior was observed due to lower interfacial tension of n-butanol-water system as compared to that of toluene-water system. The effect of drop size on aspect ratio can be seen by comparing Figures 13 (a) - (d). The rate of decrease in aspect ratio was higher in case of 6 mm drop than that in case of 4 mm drop in domain with λ = 0.5. Bäumler et al.10 showed that aspect ratio decreased with increase in Eo number. This can be confirmed from Figure 13(b) and Figure 13(c). The aspect ratio for drop of 4 mm (Eo = 0.6) diameter is lower as compared to that in case of drop of 2 mm (Eo = 0.15) diameter in a domain with λ = 0.5. The normalized surface pressure profiles on drop surface in x-y plane passing through center of mass of drop have been plotted in order to understand the drop shape dynamics (Figure 14). Surface pressure profiles have been normalized by the ratio (σ/rp). Figure 14(a) shows normalized pressure distribution on drop surface for 2 mm drop of toluene in a domain with λ = 0.5. The normalized value of pressure distribution for toluene drop of 2 mm diameter varied in between 0.91 to 1.06. The pressure distribution on drop surface has remained similar from t = 0.05 s till t = 1.5 s. Drop of 2 mm diameter has remained spherical which can be seen from Figure 8 and Figure 13(a). Figure 14(b) shows normalized pressure distribution on drop surface for drop of 4 mm diameter. At t = 0.05 s, normalized value of pressure at θ = 90° is 1.09 and at θ = 180° is 1.02. The normalized value of pressure at θ = 180° is 0.66 and 0.16 at t = 0.2 s and t = 0.5 s


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respectively. The normalized value of pressure at θ = 90° is 1.19 and 1.17 at t = 0.2 s and t = 0.5 s respectively. Thus the lowering of pressure at drop equator resulted in elongation of drop surface in horizontal direction. Figure 14(c) illustrates normalized pressure distribution on drop surface for 6 mm drop of toluene. At t = 0.05 s, normalized value of pressure at θ = 180° is 0.92 and at θ = 90° is 1.12 whereas at t = 0.1 s, normalized value of pressure at θ = 180° is 0.69 and at θ = 90° is 1.13. The normalized value of pressure at θ = 180° is -0.93 at t = 0.5 s. The normalized value of pressure at θ = 90° is 0.86 at t = 0.5 s respectively. Thus pressure values at poles of the drop surface have remained almost constant but pressure values at the equator of drop surface have reduced. Thus drop surface was stretched in horizontal direction which can be seen from Figure 10 and Figure 13 (d). Figure 14(d) shows normalized pressure distribution on drop surface for n-butanol drop of 2 mm diameter in a domain with λ = 0.5. The normalized value of pressure distribution for nbutanol drop of 2 mm diameter at θ = 90° (pole of drop) is 1.35 and at θ = 180° (equator of drop) is 0.89 at t = 0.05 s. At t = 0.1 s and t = 0.5 s, value of pressure at θ = 180° is lower as compared to that at t = 0.05 s. The normalized value of pressure at θ = 180° is 0.73 and 0.48 at t = 0.1 s and t = 0.5 s respectively. The normalized value of pressure at θ = 90° is 1.35 and 1.28 at t = 0.1 s and t =0.5 s respectively. Thus drop surface was stretched in horizontal direction due to lowering of pressure near drop equator and drop shape became ellipsoidal which can be confirmed from Figure 13(e). Thus drops underwent shape deformation for all drop sizes due to changes in pressure profiles on drop surface. The pressure profiles show decreasing trend from θ = 90° to θ = 0°.



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Thus pressure at drop equator is less as compared to that at pole on drop surface which resulted into deformation of drop surface. The drop has remained spherical till the normalized pressure value at drop equator is greater than 0.85 which can be confirmed by comparing Figures 14(a) (d). The drop shape deviated from its initial spherical shape if normalized pressure value decreased below 0.85. 3.6 Effect of strain rate and vorticity magnitude on drop dynamics The value of aspect ratio for 2 mm drop is 0.98 whereas the value of aspect ratio for 4 mm drop is 0.91 in a domain with λ = 0.8. These values remained constant after t = 0.5 s. For drop of 6 mm diameter aspect ratio showed oscillatory behaviour after t = 0.3 s. To check whether vorticity and strain rate fields have a relationship with drop dynamics, we have plotted vorticity and strain rates for these cases on a horizontal line passing through centre of mass of drop. The maximum values of vorticity and strain rate were extracted from profiles of vorticity and strain rate on a horizontal line passing through center of mass of drop for various times. Figure 15 shows profiles of maximum strain rate and vorticity. It can be seen from Figure 15 (a) and (b) that for 2 mm drop size, value of vorticity is 606 s-1and 697 s-1 at t = 0.1 s and t = 0.7 s respectively. Value of strain rate is 710 s-1and 760 s-1 at t = 0.1 s and t = 0.7 s respectively. Value of aspect ratio remained constant at 0.98. For 4 mm drop size, value of vorticity is 499s-1 and 679 s-1 at t = 0.2 s and t = 0.7 s respectively. Value of strain rate is 515 s-1 and 760 s-1 at t = 0.2 s and t = 0.7 s respectively. Value of aspect ratio decreased from 0.95 to 0.91. For these cases (2 mm and 4 mm drop size), the variation of vorticity and strain rate with respect to time after t = 0.5 s is minimal (Figures 15 (a)-(d)).


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For 6 mm drop size, an increase in value of vorticity from 296 s-1 (at t = 0.1 s) to 512 s-1 (at t = 0.3 s) as well as strain rate from 307 s-1 (at t = 0.1 s) to 567 s-1 (at t = 0.3 s) can be seen from Figure 15(e) and Figure 15(f) respectively. Aspect ratio decreased continuously (from 0.98 to 0.79) till t = 0.3 s. Lower value of aspect ratio indicates larger deformation of drop shape. Thus drop got deformed and changed its shape from spherical to ellipsoidal as drop surface was subjected to higher vorticity and strain rate. After t = 0.3 s, aspect ratio showed oscillatory behavior. At t = 0.4 s, value of vorticity and strain rate is 517 s-1 and 572 s-1 respectively and value of aspect ratio is 0.81. At t = 0.46 s, the value of vorticity and strain rate is 553 s-1 and 637 s-1 respectively and value of aspect ratio is 0.77. Thus increase in value of vorticity (517 s-1 to 553 s-1) and strain rate (from 572 s-1 to 637 s-1) resulted in more deformation of drop which is indicated by decrease in value of aspect ratio (from 0.81 to 0.77). Similarly At t = 0.54 s, the value of vorticity and strain rate is 516s-1 and 544 s-1 respectively and value of aspect ratio is 0.83. Thus decrease in value of vorticity (553 s-1 to 516 s-1) and strain rate (from 637 s-1 to 544 s1

) resulted in lesser deformation of drop which is indicated by increase in value of aspect ratio

(from 0.77 to 0.83). At t = 0.6 s, the value of vorticity and strain rate is 578 s-1 and 688 s-1 respectively and value of aspect ratio is 0.78. Thus increase in value of vorticity (516s-1 to 578 s1

) and strain rate (from 544 s-1 to 688 s-1) resulted in more deformation of drop which is indicated

by decrease in value of aspect ratio (from 0.83 to 0.78). Thus for any crest in curve of vorticity and strain rate there is a corresponding trough in curve of aspect ratio. Therefore, higher the value of vorticity and strain rate more is the deformation of drop and lower is the value of aspect ratio. The rise velocity of 6 mm drop showed oscillatory behaviour. These oscillations in the velocity field resulted in temporal variation of strain rate as well as vorticity field around the drop as vorticity and strain rate values were calculated from velocity gradient tensor. It can also



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be seen from Figure 15 that the magnitudes of strain rates are higher as compared to that of vorticity for each drop size.

However, we cannot really see the pattern in which these values

are related to whether the shape oscillations in the drop will be present or absent. 3.7 Drag Coefficient, Cd Figures 16(a) - (e) show variation of drag coefficient with respect to time for various drop diameters in domains with different values of λ. Value of Cd is calculated using Equation 16. For all the drops, value of Cd decreased as the time progressed as rise velocity increased with respect to time (Figure 7). Drag coefficient increased in smaller domains as the proximity of wall to the drop surface increased. This can be seen from Figures 14(a) - (d) that drag coefficient is higher in a domain with λ = 0.8 as compared to λ = 0.5. This is expected as an increase in value of λ results in an increase in proximity of wall to the drop surface. For all drops, value of Cd decreased continuously till drop reached its terminal velocity. Drag coefficient remained constant once drop reached its terminal velocity. We have carried out multi-linear regression analysis of the simulation data (Cd values) to come up with the correlation for drag coefficient in presence of wall effect. Equation 24 gives the correlation for the ratio of drag coefficient in presence of wall effect to that in infinite medium as a function of Eo, M and 1 - λ. Cd −1.97 = 2.73(1 − λ ) Eo −0.25 M 0.07 C d∞

(24)

This correlation is valid for 0.04 ≤ Eo ≤ 1.36, 1.95×10-11 ≤ M ≤ 1.23×10-6, 8.5 ≤ Re ≤ 765, 0.11 ≤ λ ≤ 0.8. The R2 value for the correlation is 0.94. The p-value for the exponents of Eo and M is 0.004 and 0.01 respectively. The p-value for the exponent of 1 - λ is less than 0.001. Figure 17


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shows the parity plot for this correlation. It can be seen that there is good agreement between the correlation values and simulation data.

4. Conclusions Drop dynamics in confined and unconfined domains were studied using CFD simulations. The transient rise velocity behavior was evaluated from the CFD simulations in presence of wall effect for various drop diameters. It was found that drop rise velocity is hindered due to the presence of confining walls. The reduction in velocity in presence of confining walls was found to be a function of diameter ratio, λ. The internal circulations inside the drop have been analyzed. The strength of internal circulation diminished with increase in value of λ. Symmetry of internal circulation was lost at dp = 6 mm as drop underwent shape deformation and moved closer to the walls of domain. Drop shape was analyzed by calculating aspect ratio of the drop. The toluene drop of 1 mm and 2 mm diameter and n-butanol drop of 1 mm diameter remained spherical throughout the simulation. The shapes of toluene drop of 4 mm, 6 mm diameter and n-butanol drop of 2 mm diameter changed from spherical to oblate spheroid. The effect of λ on drop shape was investigated. The pressure profile as well as magnitudes of vorticity and strain rate on the drop surface has significant impact on drop shape. The lowering of pressure at equator of the drop resulted in shape deformation. The effect of wall on drag coefficient for drops of various diameters was evaluated from force balance equation. Predicted values of drag coefficient in unconfined domain for spherical drops were in excellent agreement with Feng and Michaelides19 correlation. A correlation was developed for correction factor for drag coefficient in presence of wall effect.

Acknowledgments



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This invited contribution is part of the I&EC Research special issue for the 2017 Class of Influential Researchers. One of the authors (HPK) would like to acknowledge Department of Atomic Energy, Government of India for providing fellowship.

Nomenclature cΦ

capillary wave phase velocity (m/s)

Ap

projected area of drop (m2)

Cd

drag coefficient

Co

Courant number

d

Distance (m)

D

diameter of domain or column (m)

dp

diameter of drop (m)

Eo

Eötvös number

g

acceleration due to gravity ( g = 9.81 m/s2)

h

Grid spacing (m)

k

wavenumber (mm-1)

KU

Ratio of velocity in unconfined domain to confined domain

M

Morton number (M = ((g ⋅ µ c4 ⋅ ∆ρ ) (ρ c2 ⋅ σ 3 )))

ND

Best number N D = 4 ρ ⋅ ∆ρ ⋅ g ⋅ d 3p 3µ 2

p

pressure (N/m2)

Pn

Normalized pressure on drop surface (Pn = ( p (σ rp )))

Re

Reynolds number (Re = (ρ c ⋅ Vt ⋅ d p ) µ c )

(Eo = (g ⋅ ∆ρ ⋅ d

(k = 2π

2 p

σ ))

∆x)

(

)


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Rep

Particle Reynolds number

Ret

Terminal velocity based Reynolds number

rp

radius of drop (m)

t

time (seconds)

u

Velocity (m/s)

umax

maximum velocity in computational domain (m/s)

Vr

rise velocity (m/s)

Vt

terminal rise velocity in confined domain (m/s)

x

x-coordinate (m)

y

y-coordinate (m)

z

z-coordinate (m)

Greek Symbols γ

ratio of viscosities (γ = (µ d µ c ))

Γ

interface

∆t

time step (s)

∆x

Grid size (mm)

θ

angle (°)

κ

Curvature of interface

λ

ratio of drop diameter to domain diameter (λ = d p D )

µ

viscosity (kg/m.s)

ν

kinematic viscosity (m2/s)

ρ

density (kg/m3)



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σ

interfacial tension (N/m)

φ

Level set function

Subscripts ∞

unconfined/infinite domain

c

continuous phase

d

dispersed phase

sph

Spherical

t

terminal

References (1)

Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Courier Corporation, 1978.

(2)

Strom, J. R.; Kintner, R. C. Wall Effect for the Fall of Single Drops. AIChE J. 1958, 4, 153.

(3)

Wegener, M.; Grünig, J.; Stüber, J.; Paschedag, A. R.; Kraume, M. Transient Rise Velocity and Mass Transfer of a Single Drop with Interfacial Instabilities--Experimental Investigations. Chem. Eng. Sci. 2007, 62, 2967.

(4)

Wegener, M.; Kraume, M.; Paschedag, A. R. Terminal and Transient Drop Rise Velocity of Single Toluene Droplets in Water. AIChE J. 2010, 56, 2.

(5)

Thorsen, G.; Stordalen, R. M.; Terjesen, S. G. On the Terminal Velocity of Circulating and Oscillating Liquid Drops. Chem. Eng. Sci. 1968, 23, 413. 32 ACS Paragon Plus Environment

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(6)

Klee, A. J.; Treybal, R. E. Rate of Rise or Fall of Liquid Drops. AIChE J. 1956, 2, 444.

(7)

Harper, J. F.; Moore, D. W.; Pearson, J. R. A. The Effect of the Variation of Surface Tension with Temperature on the Motion of Bubbles and Drops. J. Fluid Mech. 1967, 27, 361.

(8)

Grace, J. R.; Wairegi, T.; Nguyen, T. H. Shapes and Velocities of Single Drops and Bubbles Moving Freely through Immiscible Liquids. Trans. Inst. Chem. Eng 1976, 54, 167.

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1955, 1, 366. List of Figures Figure 1 Schematic of drop rising through a fluid Figure 2 Schematic of Computational Domain of Numerical Simulations (x = z = 1.25 dp, 2 dp and 9 dp, y = 50 dp)

Figure 3 Effect of grid size on drop trajectory for 4 mm toluene drop Figure 4 Validation of CFD model for confined domains with plot of KU -1 vs ND1/3 Figure 5 Validation of CFD model for confined domains with plot of Ret vs ND1/3 Figure 6 Validation of CFD model for confined domains with plot of 1/KU vs λ Figure 7 Effect of wall on rise velocity of drops of toluene and n-butanol of various diameters for different values of λ

Figure 8 The internal circulations within toluene drop (dp = 2 mm) colored by y-velocity (m/s) vectors rising due to buoyancy at various time steps in the frame of reference of the moving drop for λ = 0.5



Figure 11 Strength of internal circulations (in terms of minimum and maximum y-velocity in frame of reference of moving drop) inside the drops of toluene and n-butanol for different values of λ 37 ACS Paragon Plus Environment


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Figure 12 Profile of axial velocity (m/s) in droplet frame of reference for various drop diameters for different values of λ

Figure 13 Effect of wall on drop aspect ratio for various drop diameters Figure 14 Normalized surface pressure distribution on drop surface for various drop diameters in a domain with λ = 0.5

Figure 15 Profiles of maximum vorticity and strain rate and aspect ratio for various drop diameters in a domain with λ = 0.8

Figure 16 Effect of wall on drag coefficient for various drop diameters with respect to time Figure 17 Parity plot for correlation predictions (Equation 24) and simulation data List of Tables Table 1 Summary of previous literature on numerical simulations on drop rise behavior in unconfined domain

Table 2 Combination of drop diameters and λ used in simulations Table 3 Physical properties of systems used in simulations Table 4 Effect of grid size on prediction of Ret and aspect ratio in a domain with λ =0.5 Table 5 Comparison of drag coefficient for spherical drops in an unconfined domain with available correlations

Table 6 Terminal rise velocity of toluene and n-butanol drops in confined domains


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Table 1 Summary of previous literature on numerical simulations on drop rise behaviour in unconfined domain Deformable Drop dp (mm)

Numerical Model details

Domain width

3

CFD 2D, laminar Level set

λ = 0.15

T-W

2, 3, 4

CFD 3D, laminar

Cylindrical domain λ = 0.25

-

B-W

1, 1.5 2 3, 4

CFD 3D, laminar Level set

0.1 ≤ λ(dp = 2 mm) ≤ 0.2 0.1 ≤ λ(dp = 3 mm) ≤ 0.3 0.067 ≤ λ(dp = 4 mm) ≤ 0.285

5 × 10-4

T-W BA-W B-W

1-4.4 1-4 1-3.1

Eiswirth et al.11

T-W

1-5

Komrakova et al.15

B-W

1-4

Francois and Carlson14

B-W

2

T-W B-W

1 2 4 6

Systema

Author

Deshpande and Zimmerman16

Wegener et al.

11

Bertakis et al.

Bäumler et al.

-

9

10

Present Study a

CFD 2D axisymmetric laminar, Mesh Moving method CFD 2D axisymmetric laminar, Level set lattice Boltzmann technique CFD 3D, laminar VOF CFD 3D, Combined LS and VOF Method

∆tmin (s)

-

Grid resolution near the interface/ interior of drop

Validation source

Yes/no

Aspect ratio

yes

no

-

Clift et al.1

no

no

47 040 cells (60 × 32 cells along interface)

Wegener et al.3

yes

no

40 cells along interface

Experiments and Clift et al.1 Experiments and Clift et al.1; Bertakis et al.6, Wegener et al.4

λ = 0.083

1 × 10

-5

yes

yes

59 edges along drop interface

λ = 0.33

5 × 10-5

yes

yes

80 elements along interface

Experiments

-

-

yes

yes

-

Bertakis et al.6; Wegener et al.4

0.14 ≤ λ ≤ 0.33

-

yes

yes

4190 cells inside drop

Clift et al.1; Bertakis et al.6

40 cells per diameter

Clift et al.1; Bäumler et al.10; Wegener et al.4

λ = 0.11 λ = 0.11, 0.5, 0.8 λ = 0.5, 0.8 λ = 0.5, 0.8

1 × 10-6

yes

yes

W = water, T = Toluene, B = n-butanol, BA = n-butyl Acetate, MIBK = Methyl iso-butyl ketone



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Table 2 Combination of drop diameters and λ used in simulations λ 0.11

0.5

0.8

1

(T,B)a

(T,B)

(T)

2

(T,B)

(T,B)

(T)

4

-

(T)

(T)

6

-

(T)

(T)

dp (mm)

a

T = Toluene, B = n-butanol


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Table 3 Physical properties of systems used in simulations ρ,

σ, µ, (mPa s)

(kg/m3)

Mo (-)

Water(c)

997.02

Toluene(d)

862.3

0.552

0.8903

Water(c)

986.51

1.39

1.95 × 10

-11

0.62 al.4

1.63 845.44

Data source

Wegener et 35

n-butanol(d)

γ, (-)

(mN/m)

1.23 × 10-6

3.28


2.36

Bertakis et al.9


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Table 4 Effect of grid size on prediction of Ret and aspect ratio in a domain with λ =0.5 dp (mm)

2

4

Grid resolution

Terminal drop rise

Ret

Aspect ratio

(cells per diameter)

velocity, Vt (mm/s)

(-)

(-)

20

67.2

150.5

0.98

30

65.7

147.3

0.96

40

65.1

145.8

0.94

20

119.6

535.9

0.47

30

118.4

530.6

0.41

40

117.2

525

0.4


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Table 5 Comparison of drag coefficient for spherical drops in an unconfined domain with available correlations dp

Eo

log M

Rep

Cd

Cd,feng

Cd,hen

Cd,bäumler

Cd,solid

Cd,bubble

1

0.04

-10.7

38.1

1.16

1.48

1.05

1.03

1.57

0.75

2

0.15

-10.7

237.8

0.32

0.33

0.29

0.28

0.76

0.22

1

0.85

-5.9

19.4

2.46

2.38

2.16

-

2.55

1.37



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Table 6 Terminal rise velocity of toluene and n-butanol drops in confined domains Vt (mm/s) System

dp (mm)

Vt∞ (mm/s) λ = 0.5

λ = 0.8

1

41.09

20.5

7.5

2

109.38

65.1

24.2

4

177.8

117.2

48.4

6

156.4

113.7

55

1

29.6

12.9

-

2

58.5

32.5

-

Water-toluene

Water-n-butanol


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Figure 1 Schematic of drop rising through a fluid 254x190mm (300 x 300 DPI)

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Figure 2 Schematic of Computational Domain of Numerical Simulations (x = z = 1.25 dp, 2dp and 9dp, y = 50dp) 254x190mm (300 x 300 DPI)


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Figure 3 Effect of grid size on drop trajectory for 4 mm toluene drop 149x128mm (300 x 300 DPI)



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Figure 4 Validation of CFD model for confined domains with plot of KU -1 vs ND1/3 170x158mm (300 x 300 DPI)


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Figure 5 Validation of CFD model for confined domains with plot of Ret vs ND1/3 175x145mm (300 x 300 DPI)



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Figure 6 Validation of CFD model for confined domains with plot of 1/KU vs λ 193x174mm (300 x 300 DPI)


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Figure 7 Effect of wall on rise velocity of drops of toluene and n-butanol of various diameters for different values of λ 341x439mm (300 x 300 DPI)



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Figure 8 The internal circulations within toluene drop (dp = 2 mm) colored by y-velocity (m/s) vectors rising due to buoyancy at various time steps in the frame of reference of the moving drop for λ = 0.5 210x297mm (300 x 300 DPI)


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Figure 11 Strength of internal circulations (in terms of minimum and maximum y-velocity in frame of reference of moving drop) inside the drops of toluene and n-butanol for different values of λ 352x468mm (300 x 300 DPI)



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Figure 12 Profile of axial velocity (m/s) in droplet frame of reference for various drop diameters for different values of λ 325x278mm (300 x 300 DPI)


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Figure 13 Effect of wall on drop aspect ratio for various drop diameters 361x458mm (300 x 300 DPI)



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Figure 14 Normalized surface pressure distribution on drop surface for various drop diameters in a domain with λ = 0.5 271x251mm (300 x 300 DPI)


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Figure 15 Profiles of maximum vorticity and strain rate and aspect ratio for various drop diameters in a domain with λ = 0.8 312x381mm (300 x 300 DPI)



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Figure 16 Effect of wall on drag coefficient for various drop diameters with respect to time 308x406mm (300 x 300 DPI)


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Figure 17 Parity plot for correlation predictions (Equation 24) and simulation data 152x152mm (300 x 300 DPI)



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Computational Fluid Dynamics Simulations of Single Drops in

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