A mechanistic model for the prediction of ﬂow pattern transitions during separation of liquid-liquid pipe ﬂows

A one-dimensional mechanistic model that predicts the flow pattern transitions during the separation of dispersed liquid-liquid flows in horizontal pipes was developed. The model is able to capture the evolution along the pipe of the four characteristic layers that develop from initially dispersed flows of either oil-in-water or water-in-oil at a range of mixture velocities: a pure water layer at the bottom, a settling (flotation/sedimentation) layer, a dense-packed zone, and a pure oil layer on the top. Coalescence correlations from literature were included in the model to predict the drop growth due to binary drop coalescence and the coalescence rate of drops with their corresponding interface. The model predictions on the evolution of the heights of the different layers were partly compared against available experimental data obtained in a pilot scale two-phase flow facility using tap water and oil (828 kg m -3 , 5.5 mPa s) as test fluids. It was shown that


A mechanistic model for the prediction of flow pattern transitions during separation of liquid-liquid pipe flows Nikola Evripidou, Carlos Avila, Panagiota Angeli
Highlights  A mechanistic model for the prediction of flow pattern transitions in two-phase, oil-water dispersed pipe flows was developed. Mechanistic model can predict the separation of both oil-in-water and water-inoil dispersions. Dispersion separation is controlled by either the drop settling rate or the coalescence rate.

Introduction
Pipe flows of two immiscible liquids are common in the engineering sector and are often encountered in chemical and nuclear plants, and in the oil and gas industry (Danielson 2012).Dispersions can be formed as part of the process or in equipment such as choke valves and bends.These dispersions may be unstable and separate further downstream.The tendency to separate can be exploited to design in-line separators, which are often favourable to other separators as they are simple, small, and lightweight with low operating cost (Zhong et al., 2013).In the oil and gas industry, they can be employed to increase oil recovery, hence have the potential to extend the operational lifetime of older oil fields by making extraction economically viable (Skjefstad and Stanko, 2019).On the other hand, in cases where dispersions are important for enhancing mass transfer, minimising pipe erosion (Wang and Zhang, 2016) or frictional losses during transportation of crude oil (Pilehvari et al. 1988;), the tendency of liquidliquid mixtures to separate can be detrimental.
Unstable dispersions of two immiscible liquids can undergo gravity-controlled separation while flowing through horizontal pipes (Perez, 2005;Voulgaropoulos and Angeli, 2017).Separation is always observed, unless the dispersions are stabilised by surfactants or by flow-induced mixing at high flow velocities.The separation process is rather complex and depends on several factors including the properties of the fluids, the size distribution of the drops present, the mixture velocity, and the pipe diameter and inclination.
Several authors have studied the separation of dispersions experimentally and in different set-ups, including batch and steady-state settlers, and pipe flows.Ryon et al. (1960) was the first to experimentally investigate the separation of liquid-liquid dispersions, while Barnea and Mizrahi (1975) noted the existence of a dense-packed zone.Later, Hartland and Jeelani (1988) explained the batch separation in terms of the physical processes occurring.According to them, dispersions in batch and steady-state settlers consist of settling and dense-packed layers.The drops grow in size due to dropdrop coalescence in the settling layer, they accumulate into a dense-packed zone near the liquid-liquid interface, and finally coalesce with their homophase.They also noted that the rate of settling depends on the drop size and hold-up of the dispersed phase, while the interfacial coalescence rate is a function of the drop size at the coalescing interface and the thickness of the dense-packed layer.More recent works studied the separation of dispersed flows in horizontal pipes.Pérez (2005) and Voulgaropoulos et al. (2019) used a static mixer to generate dispersions which were then fed to a horizontal pipe and observed significant stratification downstream of the inlet at low mixture velocities (cf. Figure 1).Conan et al. (2007) and Voulgaropoulos et al. (2016) employed different multi-nozzle inlet configurations to generate dispersions at low velocities and reported similar findings to those shown in Figure 1.
From experiments, it was concluded that the separation of dispersions in pipes is driven by three main mechanisms: 1) drop settling (i.e.flotation or sedimentation), 2) dropdrop coalescence, and 3) drop-interface coalescence.Assuming a fully dispersed oil-inwater flow in a horizontal pipe, separation would begin as the lower density oil drops begin to float upwards.This results in the formation of a pure water layer at the bottom of the pipe.As the drops continue to float, the thickness of the pure water layer increases.Eventually, some drops reach the top of the pipe where they accumulate into a dense-packed zone.Within the dense-packed zone, the drops come into contact with each other for long enough for coalescence to occur.This process continues and eventually a thin film of pure oil forms at the top of the pipe.Coalescence of drops with the liquid film results in an increase in the thickness of the pure oil layer.This process continues until complete separation is achieved.(Trallero et al., 1997;Angeli and Hewitt, 2000;Elseth, 2001;Simmons and Azzopardi, 2001;Lovick and Angeli, 2004;Voulgaropoulos and Angeli, 2017;Voulgaropoulos et al., 2019).Computational fluid dynamics (CFD) approaches have also been used as they can provide phase distribution with high resolution.Thus far, few CFD studies have been reported on predicting the flow pattern and phase distribution in horizontal oil-water flows (Walvekar et al, 2009;El-Batsh, 2012;Pouraria et al., 2016, Voulgaropoulos et al. 2019).The prediction of the flow patterns and the phase fraction profiles at low mixture velocities where the effect of gravity becomes significant and transitions take place is even more complex.
Accurate models for the prediction of the flow pattern evolution and the separation length in dispersions can provide information for systems where experimental data is sparse.
Previously a few authors attempted to model the separation of dispersions in batch settlers using simplified mechanistic models (Hartland and Jeelani, 1988;Jeelani and Hartland, 1998;Henschke et al, 2002).Later works attempted to extend the batch models to one-dimensional flows in horizontal pipe separators (Pereyra et al., 2013;Othman et al., 2018) by changing the time scale to a length scale using the average mixture velocity and accounting for the change in geometry.Mechanistic models can be an attractive alternative to complex CFD simulations due to their low computational time.Nevertheless, current mechanistic models on separation of dispersions in pipe flows assume that the rate of drop settling is larger than the coalescence rate, hence accumulation of drops into a dense-packed zone is predicted at the liquid-liquid interface.Although this is always true in batch vessels, this is not always the case in pipe flows (Voulgaropoulos, 2017;Evripidou et al., 2019).In dispersed pipe flows of relatively high mixture velocities and small drop diameters the drop settling rate may be similar or even lower than the rate of coalescence and a dense-packed layer may not form.Consequently, current mechanistic models are only valid for certain flow configurations and flow pattern transitions, and cannot be applied to all pipe flows.
In this work, we develop a universal mechanistic model that can predict the flow pattern development and separation of unstable liquid-liquid dispersed pipe flows.New approaches are proposed to predict all possible flow pattern transitions occurring during flow separation, which are not captured in existing models.The model is based on the horizontal pipe separator approach described in Pereyra et al. (2013), which was valid only for systems where a dense-packed layer is formed.In what follows, we identify all flow pattern transitions during separation of dispersions.We then describe the development of the model, placing emphasis on the pipe locations where only a dilute dispersed layer is present, which is often the case in systems of high mixture velocities or small drops, and which are not predicted by previous mechanistic models.Lastly, we demonstrate the applicability of the model to flows of both oil-in-water and water-in-oil dispersions.The model accounts for the main mechanisms that occur during pipe flow and can be used to provide information on the evolution of dispersed flows in systems where sampling is not feasible.

Mechanistic modelling
The  Alternatively, if the rate of drop settling is smaller than the rate of drop-interface coalescence, the dense-packed zone is depleted.This is the case in Figure 2(d) at  = ̿ .
At lengths greater than ̿ (i.e. for  > ̿ ), the settling layer is in direct contact with the pure oil layer and the rate of drop settling (and availability of drops at the interface) limits the rate of coalescence.If the flow is initially fully dispersed, but the drop settling rate is less than the drop-interface coalescence rate, the rate of drop settling will control the rate of coalescence throughout the pipe, and a dense-packed zone will never form  The mechanistic model developed here predicts the changes in thickness of the various layers and the mean drop diameter along the pipe, giving the complete flow profile up to the point of complete phase separation.The model assumes a constant velocity, equal to the mixture velocity  M , across all layers along the spanwise direction, thus neglecting velocity profiles and exchange of momentum between the layers.This is a reasonable assumption for liquid-liquid systems where experiments showed that the slip is very small, especially in the dispersed regions (Rodriguez and Oliemans, 2004;Lovick and Angeli, 2004).The surface tension is assumed to be constant throughout, while the mixture is assumed to be monodispersed.Drop break-up is not considered.Although in reality there is no apparent interface between the two dispersed layers, settling and dense-packed (cf. Figure 1), for modelling purposes, it was assumed that these are two distinct layers.The dispersed-phase fraction within each layer is taken as constant with height and a step-change in the fraction is assumed at the interface between the two layers.
Below the various parts of the model are described in detail.

Settling curve
The primary separation mechanism acting on a dispersed liquid-liquid mixture is density-driven settling.In the presence of a settling layer, the settling curve,  C , changes solely due to the vertical displacement of drops.In an oil-in-water dispersion, the settling curve corresponds to the height of the pure water layer from the bottom of the pipe and can be predicted in terms of pure water layer thickness by Assuming a continuous phase of viscosity  C and density  C , and a dispersed phase of viscosity  D and density  D , the vertical (sedimentation/flotation) velocity  s of drops of size  p , within a settling layer with dispersed-phase fraction  S can be obtained using Equation ( 2) is based on an empirical model developed by Pilhofer and Mewes (1979) from drop settling experiments in batch vessels, but has been modified by Evripidou et al. (2019) who introduced a fitted hindered settling parameter,  h , to better capture delay in settling due to the flow.This correlation was developed for a monodispersed system taking as the average drop diameter the Sauter mean diameter in the settling layer.
In equation ( 2) the two settling parameters are equal to and Other dimensionless numbers include the Archimedes number, Ar, which is given by where  is the gravitational constant, the Hadamard-Rybczynski factor, K HR , given by and a modified friction coefficient, C W , given by The  ∞ is the Reynolds number of a single drop moving vertically in an infinite medium.According to Ishii and Zuber (1979).
In the absence of a settling layer (i.e. for  > ̅ in Figures 2(a) and 2(c)), the thickness of the continuous phase layer can be obtained by where  denotes the internal diameter of the pipe.

Coalescence curve
Accumulation of drops near the top of the pipe due to drop settling results in coalescence.Eventually a continuous oil layer of thickness ℎ D is formed.The increase in ℎ D with pipe length is captured by the coalescence curve,  D , and is determined by the volume rate of coalescence of drops with the oil interface.Assuming a monodispersed mixture at the interface, where all drops have the same diameter  p,I , Pereyra (2011) showed that the evolution of the oil layer thickness along a horizontal pipe is given by where  Ι is the drop-interface coalescence time.In the presence of a dense-packed layer, we set the oil fraction at the interface,  Ι , to 0.9 which is a reasonable value for maximum packing for a polydispersed mixture (Farr and Groot, 2009;Dorr et al., 2013).In the absence of a dense-packed layer (i.e. Figure 2(b) and Figure 2(d) for  > ̿ ), such that the settling layer is in direct contact with the pure oil layer, we suggest that  I is determined by the relative rates of drop settling and drop-interface coalescence.
Assuming monodispersed layers, the average drop diameter at the interface  p,I is taken to be equal to the Sauter mean diameter in the respective dispersed layer (dense-packed or settling).

Drop size evolution
To estimate the coalescence rate within the more dilute settling layer, we computed the Reynolds numbers for the case studies considered here.We found that the settling layer falls into the transition to turbulent flow regime.Thus, we estimated the coalescence rates, by modifying the correlations suggested in Coulaloglou and Tavlarides (1977) to apply to pipe flows.The drop-drop coalescence rates were found to be insignificant, hence drop-drop coalescence within the settling layer is not considered further in this work.

Coalescence time
Of the few correlations available in literature for the calculation of coalescence rates or times (Jeelani and Hartland, 1994;Henschke et al., 2002), the one suggested by Henschke et al. (2002)  respectively.In equations ( 12) and ( 13),  is the interfacial tension between the two phases.  * is the asymmetry parameter describing the asymmetry of the film between adjacent drops.It can be obtained from experimental settling curves and is characteristic for the system used.The Hamaker coefficient  is set to 10 −20  , as proposed by Henschke et al. (2002) for any system.The drop-drop contact area radius  , is calculated by the drop-interface contact area radius  , can be related to the drop-drop contact area radius using equation and the radius of the channel contour formed when three drops approach,   , is given by ). ( La is a modified Laplace number and is given by accounts for the close packing of drops and represents the ratio between the hydrostatic pressure and the interfacial tension.The hydrostatic pressure is a result of the drop-packing height below the draining film, ℎ ̃P.Consequently, ℎ ̃P equals to the thickness of the dense-packed zone if one is present.In the absence of a dense-packed zone, the settling layer is in direct contact with the pure oil layer.In that case, we suggest that ℎ ̃P is taken to be equal to the drop size at the interface  p,I .

Settling layer dispersed-phase fraction
In initially fully dispersed flows, the dispersed-phase volume fraction of the settling layer  S is equal to the oil volume fraction at the inlet,  0 .However, in cases where the flow is partially separated at the inlet, such as the cases shown in Figures 2(c) and 2(d),  S can differ to  0 .In that instance,  S can be obtained from a mass balance on the pipe cross section  S =  pipe  0 −  D,0 −  P,0  P,0  S,0 , where  is the cross-sectional area, while the subscript 0 denotes quantities at the pipe inlet.

Dense-packed layer thickness and dispersed-phase fraction
Assuming that all four characteristic layers are present at a given axial location from the inlet, we calculate the change in the thickness of the dense-packed layer by performing a mass balance on the pipe cross-sectional area.The cross-sectional area of the dense-packed layer is given by where  P is the average dispersed-phase fraction in the dense-packed layer.In the presence of a settling layer the average hold-up of the dense-packed layer is taken equal to as suggested by Henschke et al. (2002).In equation ( 20)  Ι = 0.9.If depletion of the settling layer occurs, equation ( 19) simplifies to In this case, the average holdup in the dense-packed zone is given by  ̅ P .In the absence of a settling layer, we allowed  ̅ P to increase from its previous value.
According to Henschke et al. (2002), since the settling curve is continuous at  =  ̅ (cf.Figures 2(a) and 2(c)), the holdup  P in the region  ≥  ̅ can be calculated by the following exponentially increasing expression: 1 and  2 are coefficients determined based on continuity ensuring that at  =  ̅,  P =  ̅ P | ̅ .The two coefficients are given by and where, according to Pereyra et al. (2013), If at any axial location, the calculated thickness of the dense-packed layer is smaller than the estimated drop size within the layer (i.e. if ℎ P <  P ), we suggest that depletion of the dense-packed layer occurs and equation ( 19) reduces to This transition is shown at the location  = ̿ in Figure 2(d).At that point, the settling layer comes into contact with the pure oil layer and the dense-packed layer is replaced by a monolayer of drops.

Settling layer/Oil layer interface
The oil fraction at the coalescing interface  I , is very high in the presence of a densepacked layer.In this paper, we set that to 0.9.In the absence of the dense-packed layer, where the settling layer is in direct contact with the pure oil layer,  I is determined by the relative rates of settling and drop-interface coalescence and can be estimate by In this case, if  Ι reaches the average hold-up of the dense-packed layer  P as calculated in equation ( 20), we expect a dense-packed layer to form.
If the dense-packed layer does not form, once the thickness of the settling layer becomes smaller than the drop diameter, the oil fraction at the interface is the same as the oil fraction remaining in the dispersion The cross-sectional area of the pure continuous water phase, as shown in Figure 3, is given by

Geometric equations
where The area of the pure oil layer is where  D = 2ℎ D  − 1, and its partial derivative is The area of the dense-packed zone can be calculated from where  P = 2ℎ P  − 1, and its partial derivative by , as The area of the monolayer of drops along the coalescing interface can be calculated from where Finally, a mass balance on the cross-section gives the area of the settling layer as

Model implementation
The equations presented above were solved numerically using gPROMS ModelBuilder along the pipe length at steps of 0.1 .The input parameters required for the model include the fluid properties (densities, viscosities, and surface tension) and the pipe diameter.The drop size at the inlet and the initial thickness of each layer are also needed to initialize the simulation.
The conditions at which transitions between flow patterns occur can be specified to allow the code to transition between sets of equations.For an initial oil-in-water dispersed flow, the conditions for each transition were specified as follows: Dense-packed layer formation (cf. Figure 2(a)).If a uniform settling layer is present at the pipe inlet, a dense-packed layer is assumed to form if the oil interface becomes concentrated.We assume that this occurs once  I =  P .
Depletion of the settling layer (cf. =  ̅ in Figures 2(a) and 2(c)), i.e. transition from stratified mixed flow with dispersed settling and dense-packed layers to stratified mixed flow with a dispersed dense-packed layer only occurs when the settling curve  C meets the dense-packed zone curve  P (i.e. if  C =  P ).Since the equations in gPROMS were solved at discrete lengths along the pipe, if at any step the settling curve is calculated to be higher than the dense-packed layer interface (i.e. if  C >  P ), depletion of the settling layer is assumed.Transition to fully stratified flow (cf. =   in Figure 2) occurs when the settling curve  C meets the coalescence curve  D .Hence, if at any step the settling curve is found higher than the coalescence curve (i.e. if  C >  D ) stratification has been achieved.

Results
The experimental data used to assess the performance of the mechanistic model were obtained in a two-phase liquid-liquid flow facility discussed in detail in Voulgaropoulos et al. (2016).In the experiments, tap water and oil (828   −3 , 5.5  , 0.029   −1 ) were used as test fluids.The test section was a transparent acrylic pipe with an internal diameter of 37  and overall length of around 8 .
Partial dispersions of oil in water were generated at the inlet of the test section using a multi-nozzle mixer.High-speed imaging was employed at three locations along the spanwise dimension of the pipe to enable the identification of the flow patterns.A dualconductance probe was implemented to measure the local volume fractions and the drop size distributions of the mixture along a vertical pipe diameter.Measurements were taken every 2 , spanning the whole pipe diameter.The equations presented in the paper were solved for six cases with different inlet conditions shown in Table 1.The hindered settling parameter  h was taken to be equal to 0.01 as proposed by Voulgaropoulos (2017), while the asymmetric dimple parameter   * was taken to be equal to 0.007, a value that was obtained experimentally by Pereyra et al. (2013)   Meanwhile, the height of the dense-packed layer  P + decreases until it meets the settling curve.At this pipe length, the settling process is complete and all drops are found within the dense-packed layer.Coalescence between drops within the dense-packed layer results in an increase in the average drop diameter at the interface.Coalescence between the drops and the oil-water interface causes the oil layer to increase in thickness (interface decreases in height).Eventually complete separation occurs as the oil layer curve meets the water layer one.Little difference is observed in the length of complete separation between the two different oil fractions flowing at 0.52  −1 .No correlation between the separation length and the oil fraction can be established, as the total separation length is dependent on several variables, including the mixture velocity, the thickness of the dispersed layer at the inlet, and the drop size.This causes the dense-packed layer to decrease in thickness, until it eventually completely depletes and the settling layer comes in direct contact with the pure oil layer.
Settling and coalescence continue in a similar fashion until complete separation occurs.
Longer separation lengths are predicted at  M = 1.04  −1 as settling is the controlling separation mechanism.
The results show that liquid-liquid mixtures at different flow conditions may approach separation in a different manner.These differences result from the relative rates of drop settling and drop-interface coalescence.A higher rate of settling than that of dropinterface coalescence results in the depletion of the settling layer, as seen in the flow profiles of the lower mixture velocity in Figure 4. On the contrary, higher rates of dropinterface coalescence than those of settling may cause the dense-packed layer to deplete if present at the inlet and the settling layer to persist up to the point of complete separation.This shift in the controlling mechanism is not a result of the change in the mixture velocity itself, but rather a consequence of the smaller drops generated at the inlet as a result of the higher mixture velocity.
The suggested model was also validated against experimental data from Pereyra et al.
(2013) to assess its ability to predict the settling curve.The experimental data were obtained in a horizontal pipe separator of 0.1  inner diameter and 6  length, using tap water and mineral oil (857   −3 , 13.6  , 0.029   −1 ) as test fluids.The asymmetric dimple parameter   * was taken as 0.007 as suggested Pereyra et al (2013).The initial drop diameter was assumed to be 250  and a hindered settling parameter  h of 0.2 produced reasonable results.Oil-in-water dispersions with an oil fraction of 0.40 were studied at three different mixture velocities:  M = 0.06  −1 , 0.09  −1 , and 0.13  −1 .The results are presented in Figure 6. and comparison with experimental data obtained by Pereyra et al. (2013).
experimental data, the dimensionless asymmetry coefficient,   * , was assumed to be the same as in the oil-in-water cases.Although this could be the case and such fluid combinations have been identified in the past, Henschke et al. (2002) argues that it cannot be concluded that this is always so, hence the results need to be treated with caution.The results presented in Figure 7 show the predicted length required for separation to be significantly larger for water-in-oil dispersions than for oil-in-water ones.This is due to the lower coalescence rates predicted for the water-in-oil cases.In the case shown in Figure 7(a), the settling layer depletes at a similar axial length as the corresponding oilin-water case (cf. Figure 4(a)), however the lower coalescence rates cause the densepacked layer to persist for longer before complete separation occurs.

Figure 1 :1
Figure 1: Planar Laser-Induced Fluorescence images acquired for a few typical separation of liquid-liquid dispersions is driven by three main mechanisms: drop settling (flotation or sedimentation), drop-drop coalescence, and drop-interface coalescence.Separation can give rise to four distinct layers: a pure layer of the continuous phase, a settling layer (a dilute dispersion where drop settling occurs), a dense-packed layer (a densely packed dispersion where drop-drop coalescence occurs), and a pure layer of the initially dispersed phase.Throughout this paper, the four layers are denoted by the subscripts C, S, P, and D respectively.

Figure 2
Figure2shows schematics of different flow profiles that can arise from fully or partially Figure 2(b)).The length required to reach complete separation of the two immiscible liquids ( =   ) can be determined from the intersection point of the coalescence curve and the settling curve.

Figure 2 :
Figure 2: Schematics of flow profiles showing the evolution of the characteristic layers Drop-drop coalescence can only occur if the contact time between two drops exceeds the drop-drop coalescence time,  C .The relative motion of the drops with respect to each other is negligible within the dense-packed layer and along the coalescing interface, thus drop-drop coalescence is always considered in these locations.Following the assumption that all layers are monodispersed, at every axial location the drops found along the interface or within a dense-packed layer, have the same size  p,I .Hartland and Jeelani (1998) suggested the following expression for the prediction of drop size evolution as a function of drop-drop coalescence time   :

Figure 3 :
Figure 3: Schematic of a pipe with oil-in-water dispersed flow.
of the dense-packed layer (cf. =  ̿ in Figure 2(d)), i.e. transition from stratified mixed flow with dispersed settling and dense-packed layers to stratified mixed flow with a dispersed settling layer is assumed to occur if the thickness of the densepacked layer becomes smaller than the average drop diameter in the layer (i.e. p > ℎ P ).
for a similar system.The predicted nondimensionalized flow profiles are presented in Figures 4 and 5 together with experimental measurements to allow comparison.Nondimensionalization of the layer heights and the pipe length was performed using the internal diameter of the pipe.Dimensionless quantities are denoted with + .

Figure 4 :
Figure 4: Predictions of the flow profile and the Sauter mean diameter for oil-in-water dispersions flowing at  M = 0.52  −1 .

Figure 5 :
Figure 5: Predictions of the flow profile and the Sauter mean diameter for oil-in-water dispersions flowing at  M = 1.04  −1 .

Figure 6 :
Figure 6: Predictions of the flow profiles for oil-in-water dispersions with  0 = 0.40

Figure 7 :
Figure 7: Prediction of the flow profiles for water-in-oil dispersions.
Figure 7(b) shows the formation of a dense-packed layer from an almost fully dispersed inlet, in contrast to the corresponding oil-in-water case (cf. Figure 5(b)) where the dense-packed layer never forms.Finally, in the case in Figure 7(c) the dense-packed layer initially grows in thickness and persists up to the point of complete separation.This is again contrary to the corresponding oil-in-water case (cf. Figure 5(d)) where the dense-packed layer depletes first and the settling layer remains until separation occurs.4 Conclusions In this work, a methodology for the prediction of the separation of dispersed liquidliquid flows in horizontal pipes was presented, which is based on the physical mechanisms of drop settling, drop-interface coalescence, and drop-drop coalescence and uses two fitted parameters,  h and   * .The proposed model predicted the evolution of the pure water layer, the settling layer, the dense-packed zone and the pure oil layer up to the point of complete separation for both oil-in-water and water-in-oil systems.The predicted characteristic layer thicknesses for initial oil-in-water dispersions demonstrated little deviation from the available experimental measurements.The increase in drop size due to drop-drop coalescence was also well captured.It was shown that depending on the inlet and flow conditions, the separation can be either coalescence-controlled or settling-controlled, and different flow patterns and flow pattern transitions can arise.It was also found that the drop size has a major effect on the rate of settling, hence smaller drop sizes are more likely to lead to settling-controlled separation.Nevertheless, in this work we have shown that accurate drop-size measurements of the initial dispersion are not required.The hindered settling parameter can compensate for any inaccuracies in the drop size and produce reasonable results for thickness of the various layers.A relationship between mixture velocity and the hindered settling parameter is likely, but further studies are required to establish it.The continuous phase affects the rate of coalescence.Lower rates of coalescence were predicted for the oil-continuous systems, hence in those cases the dense-packed layer was more likely to form and persist up to the point of complete separation.The model presented here can be used to swiftly predict the evolution of liquid-liquid dispersions for industrial applications.While the findings are encouraging, further experimental data would help validate the model at different pipe length and specifically, data closer to the point of complete separation.In addition, allowing for different drop sizes could increase the accuracy of the model.
was incorporated in the model, as presented inPereyra et al.

Table 1 :
Inlet conditions of the experiments.