Figure 1
Schematic diagram of the computational domain (A) boundary conditions and (B) mesh structure.
Figure 2
Axial velocity contours for different drop diameters of liquid-liquid extraction systems.
Figure 3
A) Axial velocity, B) Static pressure plots in three liquid-liquid extraction systems, surface tension force model effect on the axial velocity profiles for different drop sizes C: A) 1 to 2mm, and D) 2.5 to 4 mm.
Figure 4
Droplet shapes obtained from CSS simulations for n-butanol drops with the initial diameters of 2, 2.48, 3, 3.48 and 4 mm.
Figure 5
Droplet shapes obtained from CSS simulations for n-butyl acetate drops with the initial diameters of 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0 mm.
Figure 6
Droplet shapes obtained from CSS simulations for toluene drops with the initial diameters of 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 and 4.4 mm.
Figure 7
Transition process of the drop shape changes for a 5mm n-butanol drop from spherical to breakup at times of: 0, 0.1, 0.2, 0.3, 0.4, 0.5 and 0.54s using the CSS model.
Figure 8
Streamline of the different drop diameters in three liquid-liquid extraction systems using the CSS model.
Figure 9
A) n-butanol drop transient velocities using the CSS model, B) transient velocity of oscillating n-butanol drops with the CSF and CSS models.
Figure 10
A) n-butanol drop terminal velocities obtained from the current CSS and CSF models in comparison with the experimental data of Bertakis et al. (2010)Bertakis, E., Groß, S., Grande, J., Fortmeier, O., Reusken, A., Pfennig, A., Validated simulation of droplet sedimentation with finite-element and level-set methods, Chemical Engineering Science 65, 2037-2051 (2010)., CSF and GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)., Lattice-Boltzmann model of Komrakova et al. (2013)Komrakova, A., Eskin, D., Derksen, J., Lattice Boltzmann simulations of a single n-butanol drop rising in water, Physics of Fluids (1994-present), 25, 042102 (2013)., and dynamic mesh model of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). B) Relative errors of the CSF, CSS models of the present work, and CSF and GFM results of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). with respect to the experimental data of Bertakis et al. (2010)Bertakis, E., Groß, S., Grande, J., Fortmeier, O., Reusken, A., Pfennig, A., Validated simulation of droplet sedimentation with finite-element and level-set methods, Chemical Engineering Science 65, 2037-2051 (2010)..
Figure 11
A) n-butyl acetate drops transient velocities using the CSS model, B) Transient velocity of oscillating n-butyl acetate drops with the CSF and CSS models.
Figure 12
A)n-butyl acetate drop terminal velocities obtained from the current CSS and CSF models in comparison with experimental data of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011)., CSF and GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)., and dynamic mesh model of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). B) Relative errors of CSF, CSS models of the present work, and CSF, GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). with respect to Uchar experimental data of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). and C) relative errors of CSF, CSS models of the present work, and CSF , GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). with respect to Uchar experimental data of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011)..
Figure 13
A) Toluene drop transient velocities using the CSS model, B) Transient velocity of oscillating toluene drops with the CSF and CSS models, C) Transient velocity of the current CSS model in comparison with Wegener et al. (2010)Wegener, M., Kraume, M., Paschedag, A.R., Terminal and transient drop rise velocity of single toluene droplets in water, AIChE Journal, 56, 2-10 (2010). experimental data.
Figure 14
A) Toluene drop terminal velocities obtained from the current CSS and CSF models in comparison with experimental data of Wegener et al. (2010)Wegener, M., Kraume, M., Paschedag, A.R., Terminal and transient drop rise velocity of single toluene droplets in water, AIChE Journal, 56, 2-10 (2010)., CSF and GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)., and dynamic mesh model of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). B) Relative errors of CSF, CSS models of the present work, and CSF, GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). with respect to Uchar experimental data of Wegener et al. (2010)Wegener, M., Kraume, M., Paschedag, A.R., Terminal and transient drop rise velocity of single toluene droplets in water, AIChE Journal, 56, 2-10 (2010)., and C) Relative errors of CSF, CSS models of the present work, and CSF , GFM models of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). with respect to Umax experimental data of Wegener et al. (2010)Wegener, M., Kraume, M., Paschedag, A.R., Terminal and transient drop rise velocity of single toluene droplets in water, AIChE Journal, 56, 2-10 (2010)..
Table 1
Binary material properties for standard liquid-liquid extraction systems: n-butyl acetate/water and toluene /water systems at 25 oC and n-butanol/water system at 20 oC.
Table 2
Mesh independence analysis for 2.5 mm drop size for terminal velocity (mm/s) for three binary liquid-liquid extraction systems using the CSF and CSS models.
Table 3
Simulated aspect ratio values of n-butanol drops using the VOF method (ARVOF-CSF and ARVOF-CSS) under steady state conditions and comparison with the level set simulation results of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)..
Table 4
Simulated aspect ratio values of n-butyl acetate drops using the VOF method (ARVOF-CSF and ARVOF-CSS) under steady state conditions and comparison with the level set simulation results of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)..
Table 5
Simulated aspect ratio values of toluene drops using the VOF method (ARVOF-CSF and ARVOF-CSS) under steady state conditions and comparison with the level set simulation results of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)..
Table 6
The average relative error of the present VOF method (VOF-CSS and VOF-CSF), level set model (level set-CSF and level set-GFM) of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)., dynamic mesh model of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011)., and Lattice-Boltzmann model of Komrakova et al. (2013)Komrakova, A., Eskin, D., Derksen, J., Lattice Boltzmann simulations of a single n-butanol drop rising in water, Physics of Fluids (1994-present), 25, 042102 (2013). with respect to the experimental data of Bertakis et al.(2010)Bertakis, E., Groß, S., Grande, J., Fortmeier, O., Reusken, A., Pfennig, A., Validated simulation of droplet sedimentation with finite-element and level-set methods, Chemical Engineering Science 65, 2037-2051 (2010). for n-butanol drops.
Table 7
The average relative error of the present VOF model (VOF-CSS and VOF-CSF), level set model(level set-CSF and level set -GFM) of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014). and dynamic mesh model of Bäumler et al., 2011Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011).) with respect to the experimental data of Bäumler et al. (2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). for n-butyl acetate drops.
Table 8
Relative error average of VOF model (VOF-CSS and VOF-CSF) , level set model(level set-CSF and level set –GFM) of Engberg and Kenig (2014)Engberg, R.F., Kenig, E.Y., Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and sharp interfacial force models, International Journal of Heat and Fluid Flow, 50, 16-26 (2014)., and dynamic mesh model of Bäumler et al.(2011)Bäumler, K., Wegener, M., Paschedag, A., Bänsch, E., Drop rise velocities and fluid dynamic behavior in standard test systems for liquid/liquid extraction—experimental and numerical investigations, Chemical Engineering Science, 66, 426-439 (2011). with respect to the experimental data of Wegener et al. (2010)Wegener, M., Kraume, M., Paschedag, A.R., Terminal and transient drop rise velocity of single toluene droplets in water, AIChE Journal, 56, 2-10 (2010). for toluene drops.
Table A1
n-butanol drop terminal velocities and dimensionless numbers.
Table A2
n-butyl acetate drop terminal velocities and dimensionless numbers.
Table A3
Toluene drop terminal velocities and dimensionless numbers.