Developing multiphase models for liquid-liquid ...

Paper No S6_Wed_C_37 6th International Conference on Multiphase Flow, ICMF 2007, Leipzig, Germany, July 9 – 13, 2007

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Developing multiphase models for liquid-liquid hydrocyclone flow

Emilio Paladino1, João Aguirre2 and Erick Quintella3

1Department of Mechanical Engineering, Federal University of Rio Grande do Norte – UFRN

[email protected] 2CFD Division, Engineering Simulation and Scientific Software – ESSS

[email protected] 3Process Systems division, Exploration & Production - PETROBRAS

[email protected]

Keywords: Hydrocyclone, CFD, liquid-liquid, two-fluid model Abstract Computational Fluid Dynamics - CFD has become in the last years a more and more used tool in geometry developing and optimization of hydrocyclones. With the advance o computer power, shortcomings as the need of a full Reynolds Stress Model to adequately represent the non-homogeneous turbulent field, have been solved. On the other side most models presented in literature are single phase flow or uses approaches as Lagrangian Particle Tracking of Algebraic Slip (which are basically single phase models with some additional tracking equations o transport equations). For the case being investigated here, which consist in a high oil concentration (>20%) hydrocyclone, a full Eulerian-Eulerian model is needed in order to adequately represent the two-phase flow field. This paper presents some preliminary results and discussions of CFD model development based on full Eulerian-Eulerian approach for oil-water hydrocyclone design. Furthermore, as it is expected a continuous phase inversion at the central region (oil became the continuous phase), an alternative to the traditional continuous-dispersed morphology treatment for the interface length scale, is proposed. In addition a new post-processing parameter which represents the droplet migration velocity is presented, which could be used for preliminary hydrocyclone geometry design using a less costly single phase model. Obviously, this approach is valid, just for dispersed morphologies and low oil concentrations and can be used for preliminary geometry design.

Introduction The need for more compact and efficient separation devices in off-shore operations for processing heavy oil has lead in last years to strong investments in the development of oil-water hydrocyclone separators, in order to substitute the huge, and less efficient for heavy oils, gravitational ones. In terms of computational modeling, efforts are being done to develop adequate models for hydrocyclone flow, mainly in terms of turbulence modeling (Petty & Parks (2001), Cullivan et al. (2004), among others). The common conclusion in these works is that Reynolds Stress Differential models are essential to correctly capture the anisotropy characteristic of turbulence field in these devices, and so, enabling the correct prediction of efficiency and pressure drop. Furthermore, several works state that second-order RSM should be used, like SSG (Cullivan et al. (2004)). Grotchans (1999) showed that strong differences in azimuthal velocity profiles, are encountered when using LLR (Launder et al. (1975)) and SSG (Speziale et al. (1991)) to compute the flow in mineral processing hydrocyclones. Recent works suggest the use of Large Eddy Simulation as turbulence modeling approach (Delgadillo & Rajamani). Nevertheless, few works were found concerning the multiphase flow modeling in hydrocyclones. Even more, typical models in these works follow Lagrangean Particle Tracking (LPT) approach or simplified Eulerian models, as

Algebraic Slip (Drift flux) which just considers dispersed morphologies (Cullivan et al. (2004), Delgadillo & Rajamani). In this last work the Volume-of-Fluid approach (VOF) was used to predict air core region, but particle separation was again predicted using LPT approach. The only reference found to relating a multiphase modeling of a liquid-liquid hydrocyclone was Huang (2005). Nomenclature

ri Volumetric fraction of phase i Ui Velocity of phase i UI Velocity of the interface ρ Density μ Viscosity

ReM Mixture Reynolds number CD Drag coefficient MiI Interfacial momentum transfer term dij Interfacial length scale Aij Interfacial area density σ Surface tension

We Weber Number Ca Capillarity number

Subscripts i, j, oil, water Phase reference M Relative to the mixture


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Nevertheless, results in that work were compared against Colman´s experiments (Colman et al. (1980)), which were performed using polymeric particles as substitutes for oil droplets. Then, phenomena as coalescence and even continuous phase inversion, which is expected as 30% oil volumetric fraction at inlet stream was used in Huang’s simulations, was not considered. The common characteristic of most of these works is that they are related to minerals processing field, and so the phase to be separated is constituted by solid particles, remaining dispersed within the whole domain. In the case of liquid-liquid (oil-water) hydrocyclone other very complex phenomena can take place, as droplet coalescence and break-up and, for higher oil concentrations, which are the focus of these investigations, an “oil core” region could arise where oil becomes the continuous phase, as illustrated in Figure 1. Prediction of the flow conditions for phase inversion is very complex. Some works present models for phase inversion in pipe flow (Brauner & Ullmann (2002)) but no reference was encountered for hydrocyclone flow. Nevertheless, the model presented here does not intend to capture the phase inversion regions, but try to adequately represent the flow within the whole hydrocyclone, using a correlation for the interfacial area density based on volume fraction of each phase.

Figure 1: Expected flow pattern in a high oil concentration hydrocyclone. Preliminary Analysis In this section, some theoretical foundations regarding the droplet break up in turbulent flows are presented. In addition a method for preliminary analysis using a single phase model result is developed. The method is based on a simple analysis of droplet break up criteria combined with the effects of the centrifugal forces which promote the separation. It is known that the bigger the azimuthal velocity the highest the centrifugal force and so the separation efficiency, but also the tendency to droplet break-up and drag increasing. Hinze (1955) and other references suggests that droplet deformation and break up is controlled by a force balance at the droplet interface in the direction of the main flow streamlines, when it moves through an immiscible fluid medium. Break up will depend on the relative importance between the inertial and viscous forces acting on the drop. The process can be controlled by inertial or viscous forces depending on droplet Reynolds number, defined as,

Re C REL dropd

C

V dρμ

= (1)

where VREL is the relative velocity of the droplet to the continuous phase and ρC and μC are the density and viscosity of the continuous phase. If this parameter is small (lower than unity), viscous forces domain the break up, otherwise, inertial forces domain. In each case, break-up can be determined by capillarity number,

Ca C dropdμ γσ

= (2)

when viscous forces control deformation and break-up and Weber number,

_____

We2

C dropu u dρσ

′ ′= (3)

for inertial dominated break-up, where u´u´ is the velocity fluctuation correlation. Hinze (1955) relates this parameter to the eddy dissipation rate, ε. Nevertheless, in this case, this correlation is available as a Reynolds Stress turbulence model was used. For calculations the max value of the three diagonal terms of Re stress tensor was used. For the cases studied here, even when oil constitutes the viscous phase, Re values are much grater then unit except very near from walls. Then inertial break-up criteria was used in this analysis. Hinze (1955) suggest the existence of a critical Weber number beyond which, individual drops become unstable brake up into smaller drops. Then, maximum “admissible” diameter, based on turbulence intensity at each point can be calculated as,

cr_____

2 Wemin ,MAX INLET

C

d Du u

σ

ρ

⎛ ⎞⎜ ⎟=⎜ ⎟

′ ′⎝ ⎠

(4)

A limiting value DINLET equal to the diameter at inlet region, was defined in order to limit the diameter grow in low turbulence regions. Typical values of Wecr suggested in literature round from 0.58 to 1.3 depending on flow type (see Duewell et al. (1998)). Nevertheless, this parameter depends on the flow pattern and specific experiments should be done to have values more adequate for this case. As analyses here are qualitative and for comparative purposes, a value of 1.0 was adopted for Wecr., as suggested by Hinze (1955). From this analysis, a post-processing parameter called “migration velocity”, which should be maximum for better separation, is deduced. This parameter is equivalent to the “terminal velocity” in gravity driven separation, but also it takes into account the possibility of droplet break-up. The “migration velocity” is defined as,

12 2

ˆMAXMIG

C D

VdC r

θρρ

⎛ ⎞Δ= ⎜ ⎟⎝ ⎠

V r (5)

Where Vθ is the Azimuthal velocity at each point and r is the radius measured from hydrocyclone center. Rigorously, r should be computed as the curvature radius of the streamline, but it is computed in this way for the sake of simplicity. The drag coefficient, CD, was computed using equation (21) but using a Reynolds number based on water properties (for single phase flow) and a relative velocity,

Oil


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which in this case, corresponds to “migration velocity” (eq. (1)). This parameter was used for preliminary analysis using single phase flow results. Mathematical Modeling The multiphase model here presented intends to model flows which continuous phase inversion happens within the domain. The challenge is to develop suitable models for the interfacial area density and interfacial momentum transfer, which adequately capture, at least qualitatively, the phase morphology and the characteristics of the interfacial transfer terms. In addition, these models have to be capable of running combined with second order Reynolds Stress Models for turbulence, as serious convergence problems are reported in literature, when combining these kind of turbulence modeling and Eulerian multiphase models (even VOF, as reported by Cullivan et al. (2004)). The formulation of the two-fluid is based on an averaging process (temporal, spatial or assemble averaging, see for example, Ishii & Mishima (1984), Enwald et al. (1996)) of the local and instantaneous mass and momentum equations for each phase,

( ) ( ) 0i i itρ ρ∂

+ ∇ =∂

U

(6)

( ) ( )i i i i i i itρ ρ ρ∂

+ ∇ ⋅ − =∂

U U U Τ f (7)

Together with the respective interfacial conditions,

( )( ) 0k

i i I ij

ρ − ⋅ =∑ U U n

(8)

( )( )k

i i I i i i i ij

ρ σ κ− ⋅ − ⋅ =∑ U U U n T n n

(9)

The interfacial constraints, (Eqs. (8) e (9)) represents the mass and momentum balances across the interface. For the case of mass balance, there are no sources at the interface (when no phase change is present), and the momentum could be imbalanced by surface tension, σ (which effects were also neglected). After the averaging process, one obtains,

( ) ( ) 0=⋅∇+∂∂ Uiiii rrt

ρρ

(10)

( ) ( )

( )i i i i i i i

Turbi i i i iI

r rt

r p r

ρ ρ∂+∇⋅ =

∂∇⋅ + − ∇ + +

U U U

T T f M

(11)

where ri is the volumetric fraction of phase i, Ti

Turb is the phasic turbulent stress tensor and MiI is the interfacial momentum transfer term. Further details about the averaging process could be encountered, for instance, in Wallis (1969), Drew (1983), Ishii & Mishima (1984), and several other works. The main challenge at this point is to define suitable constitutive equations for the interfacial momentum transfer term, MiI. Most applications of the two fluid model consider continuous-dispersed morphologies, remaining the same continuous phase in the whole domain (for instance liquid-

solid hydrocyclones). In these cases this term is deduced from the well known interactions between a continuous stream and particles, analyzing the forces appearing on a body submerged in a free stream trough particle drag correlations. Other forces as virtual mass or lift can also be considered. For a more general flow pattern, we should examine more carefully the origin of these well known particle drag correlations. The general expression for the interfacial momentum transfer term which arises from the averaging process (when no phase change is present), is

íI i iX= − ⋅∇M T (12) Note that ⟨⟩ denote volumetric average, i.e., the quantities are integrated over the volume. The variable Xi is the phase indicator function defined as,

( )1 if phase at time

,0 otherwisei

i tX t

∈⎧= ⎨⎩

rr

and ∇Xi is different from zero only at the interface. Then, this term represents the integration of the stresses over the whole interfacial area, and is proportional to the interfacial area density, this is, the quantity of interfacial area per unit volume. From this analysis, considering that momentum is transferred between phases as a velocity gradient at the interface is present, the general form for the interface momentum transfer term could be expressed as being proportional to a relative velocity (the square of it, in this case) and interfacial area density. This general correlation is implemented in most commercial codes (particularly in CFX 11.0) as,

( )iI i D ij j i j iC Aρ= − −M U U U U (13) This constitutive relation could be applicable, a priori, to any flow pattern, since the drag coefficient CD and the interfacial area density Aij are known. This last parameter is calculated, in a general way, as,

i jij

ij

r rA

d= (14)

For the particle model (continuous-dispersed patterns), the interfacial area density is given by,

''' 6 jij P p

p

rA S n

d= = (15)

where

'''3

6p pp

p p

r rn

V dπ= = (16)

is the particle number density, SP is the particle area and VP is particle volume. The interfacial momentum transfer term for the particle model is then given by,

( )34

i iiI D j i j i

P

rC

dρ

= − −M U U U U (17)

where the projected particle area was considered for drag calculation. The drag coefficient for this case can be calculated from well known correlations for drag over immersed bodies. For small particles, which trend to remain spherical, Schiller Naumann is a good option. One of the main objectives of the research in course is to establish good correlations for interfacial area density and


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interfacial forces for flow computations in a high oil concentration hydrocyclone, where it cannot be considered that one phase remains continuous and the other dispersed over the whole domain. In this work a model is used for the interfacial area density, which basically consist in an asymptotic approximation in the limits of low water or low oil volume fractions. The interfacial area density is then calculated as

/6 oil water

oil wateroil water water oil

r rA

d r d r=

+ (18)

Then ,

/ /0 0

6 6lim and lim

water oil

oil wateroil water oil waterr r

oil water

r rA A

d d→ →= =

(19)

i.e., when one of the phases has a low volume fraction, the continuous-dispersed interfacial area density (eq. (15)) is recovered. This model seems to be valid in the limit of low water or low oil volume fraction, allowing any phase to be the “dispersed phase”, for this limit situation, but not suitable for intermediate volume fractions. Nevertheless, results show that, at least qualitatively, this approach is useful for the hydrocyclone flow. This model for the interfacial area density will be related in this work as “symmetric” model in the results section. Shiller-Naumann correlation is used for the drag coefficient, CD, limited by the minimum value of 0.44, known for the Newton regime. Although, this is valid for the particle model, i.e., dispersed morphologies, it is maintained, as the model hereby proposed represent an asymptotic approximation to the particle model.

( )0.68724 1 0.15ReReD M

M

C = + (20)

For dispersed morphologies, the Reynolds appearing on the correlation is referred to the droplet and is based on relative velocity, droplet diameter and continuous phase fluid properties. In order to the above correlation adequately represent the drag force, when the water or oil constitutes the continuous phase, the drag coefficient is calculated based on the “mixture” Reynolds number,

Re M W O ijM

M

dρμ−

=U U

(21)

where, M o o W Wr rρ ρ ρ= +

M o o W Wr rμ μ μ= + (22)

and the interfacial length scale, dij is calculated, from equations (14) and (18), as

/ 6oil water water oil

oil waterd r d r

d+

= (23)

Then, for the case of low oil or low water volume fraction, dij represents the droplet diameter in equation (18), for the interfacial area density. However, different “diameters” or length scales can be defined for oil and water droplets.

Turbulence Model As reported by several authors, the use of differential Reynolds Stress model is a need for cyclonic type flows. Nevertheless, as mentioned, more recent works suggest the use SSG (Speziale et al. (1991)) model for hydrocyclone flow. This model was used in this work, considering homogeneous interfacial turbulence, i. e., both phases share the same turbulent field. Mixture properties are used for fluids, as given by equations (22). Then, transport as well as production and dissipation terms are calculated considering these properties. This means, for instance, that when water volume fraction is small, at the hydrocyclone core center, oil density and viscosity will be considered for turbulence calculations, and vice-versa. Some attempts to run an inhomogeneous turbulent flow model failed because o divergence and further investigations are being continued in this field. The same model for turbulence calculation was used for single phase computations. Computational Model Mathematical model was implemented on commercial package CFX11.0. This code uses the Element Based Finite Volume Method (Raw, M. J. (1985) and Maliska (2004)) with a fully implicit coupled solver. This coupling includes mass and momentum equations as well as the phase coupling, i.e., solves the whole system of equations for both phases simultaneously (Burns et al. (2001)). Geometry and boundary conditions are showed in Figure 1. It is important to note high aspect ratio of the device, which is a characteristic of liquid-liquid hydrocyclones and the involute shape inlets (Petrobras Patent). Part of the overflow channel was included, in order to move away the boundary condition from the hydrocyclone body.

Figure 1 – Hydrocyclone Geometry and boundary conditions.

A full hexahedral mesh with approximately 175,000 nodes was used. Some snapshots showing mesh details are showed in Figure 2. At the lower right side of the figure, the O-grid structure at the overflow region is showed. This structure extends axially trough the whole domain until the underflow. This permits to have refined grid at the core region with good mesh quality, avoiding wedge elements.

Overflow

Underflow

Inlet 1

Inlet 2


5

Figure 2 – Computational mesh used in simulations

Boundary conditions are summarized in Table 1. At the inlet region, global mass flow and phase volume fractions were specified. Mass flow values showed in Table 1 correspond to an oil volume fraction of 30% at inlets (note that mass flow values depend also of the densities relation). At outlet regions, overflow and underflow, “bulk mass flow rate” condition were specified. This condition allowed imposing a mass flow value for the multiphase mixture and the quantity of each phase flowing out at that region is calculated through volume fractions computed at boundary volumes faces. Then, mass flow split is enforced (as this is a variable normally controlled in operation) and separation efficiency arises from calculations. It is important to point out here that this work does not intend to study specific geometries or operational conditions, but develop adequate models to accomplish this task in future. Then, all operational conditions simulated here and fluid properties used, are hypothetical (although they are within the operational ranges commonly encountered in field operation).

Table 1 – Boundary conditions

Boundary Condition Type Value

INLET Mass flow

1.0 /Totalm kg s= 0.278 /oilm kg s=

0.722 /waterm kg s= OVERFLOW Bulk mass flow 0.3 /Bulkm kg s=

UNDERFLOW Bulk mass flow 0.7 /Bulkm kg s=

WALL No slip condition

For single phase flow, the same mass total flow and split ratio was considered. Properties used for oil and water are presented in Table 2. Although hypothetical, they represent heavy oil properties.

Table 2 – Fluid properties used in simulations

Fluid Density Viscosity Interfacial. tension Oil 1000 kg/m3 1x10-3 Pa s Water 900 kg/m3 1x10-2 Pa s 0.0091 N/m

The simulations were performed in transient way due to the inherent transient characteristic of the hydrocyclone flow Attempts to run the SSG model as steady state (this means “distorted transient”), failed. Due to the numerical low robustness associated to the SSG model, because of the quadratic pressure-strain relation (which results in very high source terms in resulting liner systems), very small timesteps (~2x10-5) had to be used. This resulted in computation times of order of 100 hours in a computer with two Dual Core AMD Opteron™ Processors 275, 2.2 GHz (this means four CPUs) as residence times are of order of 1s. This represents huge CPU times, even for this not too refined mesh. This high computational cost, resulted from the combination of Eulerian model and RSM/SSG model. In the case of single phase flow, timesteps of order of 1x10-4 s or higher could be used reducing, considerably, CPU time. Results and Discussion This section presents some preliminary results for the oil-water flow in a hydrocyclones, obtained using the described computational model. First single phase model results will be presented, using the “migration velocity” concept for analyses. Then, some preliminary results for the two phase flow using the two phase model described in preceding sections will be explored. Single Phase model results A single phase simulation was performed, considering water flow and SSG Reynolds Stress for turbulence computations. Lines positions where all data were plotted are shown in Figure 3. First, the “maximum allowed” (due to turbulence) droplet diameter distributions are plotted along transversal lines in Figure 4 For values of Wecr about unity, and considering values of interfacial tension, σ about 0.01 for oil-water mixture, maximum admitted droplet diameters are, in the most part of the hydrocyclone, over 1mm which was defined as inlet diameter. Then, droplet break-up happens just at boundary layer near walls. Furthermore, the value of 1mm for the inlet diameter was estimated because o high oil concentration, but lower values are expected for production field stream. This means that, a priori, in regions where water is the continuous phase, no droplet break up would happen.

Figure 3 – Lines where data were plotted in charts

Line 1 Line 2 Line 3

Line 430 mm

60 mm120 mm


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0,0

0,2

0,4

0,6

0,8

1,0

1,2

-40 -30 -20 -10 0 10 20 30 40x [mm]

Dm

ax [m

m]

Doil [ mm ] - Line 1



Figure 4 – Radial distribution of maximum droplet diameter (eq. (4)) at three hydrocyclone heights

Distributions of “migration velocity” in transversal lines are shown in Figure 5. Note the near wall regions affected by the variation of droplet diameter (inflection region highlighted by the square). As already mentioned in most part of the separation region, droplet diameter is constant.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

-40 -30 -20 -10 0 10 20 30 40x [mm]

Vm

ig [m

/s]

Vmig - Line 1

Vmig - Line 2

Vmig - Line 3

Figure 5 – Migration velocity distribution in transversal lines – Dinlet=1mm

Figure 6 shows the migration velocity radial distribution when an inlet droplet diameter of 0.1 mm is considered. Note, from Figure 4, that no break-up beyond this diameter will happen (for values of Wecr and σ considered).

0,0

0,1

0,2

0,3

0,4

0,5

-40 -30 -20 -10 0 10 20 30 40x [mm]

Vmig

[m/s

]

Vmig - Line 1

Vmig - Line 2

Vmig - Line 3

Figure 6 – Migration velocity distribution in transversal lines – Dinlet=0.1mm

Finally, Figure 7 shows the oil droplet hypothetical trajectories for 1mm, 0.5mm and 0.1 mm inlet droplet diameter. These trajectories have been calculated through a velocity field obtained by summing the “migration velocity” to the radial velocity component of the water velocity field, maintaining the original values for the other components (azimuthal an axial). Note that, for inlet diameters over 0.5 mm, all droplets goes to the overflow. This means 100% of separation efficiency.

(a) (b) (c)

Figure 7 – Oil droplet hypothetical trajectories for inlet droplet diameter equal to (a) 1mm, (b) 0.5mm, (c) 0.1mm.

From this approach, separation efficiency could be computed by the number of particles leaving the domain by overflow and underflow. Beyond the scope of this analysis and emphasizing that values are just qualitatively representatives, the “migration velocity” concept arises as a useful parameter for preliminary hydrocyclone design, in cases where dispersed oil morphology can be considered over the whole domain. In addition, it is important to emphasize that, for the actual analysis, some hypotheses have been considered, as a value of unity for Wecr, constant oil droplet diameter distribution


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at inlet, etc. More realistic values for these variables should be used for real case analysis. Following section present the preliminary results for a two phase model development in an Eulerian –Eulerian framework. Two phase model results This section presents some results for the multiphase model described in previous sections. First, some variations of parameter Aij have been evaluated, in order to compare results and, mainly, evaluate the model sensibility. As no experimental data was available comparisons were done based on qualitatively expected physical trends. Further work includes model validation though experimental data obtained from an experimental hydrocyclone facility under construction at CENPES1/PETROBRAS. As a first approximation to the for the, a priori, unknown interfacial area density phase morphology, the interfacial length scale, dij, in equation (14) was set as constant, initially, equal to 1mm. Results were compared with the “symmetric” model for the interfacial area, where both oil and water “diameters” where set equal to 1mm. All other parameters were maintained equal in both cases. A surprising (and unexpected) result which shows the importance and strong influence of the interfacial transfer terms into the results is shown in Figure 8 which shows the azimuthal velocity profiles plotted in a line located 30 mm down the overflow.

-8,0

-6,0

-4,0

-2,0

0,0

2,0

4,0

6,0

8,0

-40 -30 -20 -10 0 10 20 30 40

x (mm)

V (m

/s)

Oil.Velocity w - dij=const Water.Velocity w - dij=const Oil.Velocity w [ m s -̂1 ] Water.Velocity w [ m s -̂1 ]

Figure 8 – Azimuthal velocity profiles for “symmetric” and constant dij models

Clearly, results for a constant interfacial area density, are far from the Rankine-vortex-type velocity profile, expected for the hydrocyclone (Bergstrom & Vomhoff (2007)). Obviously this leads to a very low predicted efficiency. Figure 9 show the calculated interfacial area density for the same both cases. It is clear the tendency to separate, i.e., to have smaller values of Aij when dij is constant. This parameter is calculated based on volumetric fraction fields which depend on velocity fields. An explanation for this behaviour could be based on the fact that use of constant values for dij trends to separate phases (in the sense that try

1 Petrobras Research Center, Rio de Janeiro.

to resolve the interface and no “interpenetrating continua” behavior), but this pattern cannot be adequately resolved and probably this behavior of velocity profiles is due to a numerical failure, although simulation have converged for every timestep. Figure 10 shows oil velocity profiles for lines at three hydrocyclone heights, compared with single phase results, showed in Figure 3. Although Rankine vortex profiles are maintained, peaks are displaced and reduced at central region. This can be explained because of the high oil concentration at central region, where it “becomes”2 the continuous phase, characterizing a more viscous mixture.

0,0

200,0

400,0

600,0

800,0

1000,0

1200,0

1400,0

1600,0

-40 -30 -20 -10 0 10 20 30 40x (mm)

Aij

(1/m

)

dij constant

dij from eq. (23)

Figure 9 – Interfacial area density for “symmetric” and constant dij models.

-8,0

-6,0

-4,0

-2,0

0,0

2,0

4,0

6,0

8,0

-40 -30 -20 -10 0 10 20 30 40

x [mm]

V [m

/s]

Two phase - symm. model

Single phase

Figure 10 – Azimuthal velocity profiles for three lines at different heights for single phase model (SP) and “symmetric” two phase model.

It is important to make clear at this point that the interfacial area density Aij is a function of interfacial length scale, dij which is a function of droplet diameter and phase volume

2 Quotations are used as model does not actually predict this phenomenon, but the interfacial area is calculated as if oil was the continuous phase, when water volume fraction trends to zero.


8

fraction. Then, constant dij, does not imply in constant Aij. Furthermore, dij also vary with droplet diameter, which can be make variable through, for instance, a population balance model. Figures 11, 12 and 13 present a general picture of the flow pattern for the two phase “symmetric” case, showing oil and water velocity azimuthal, radial and axial velocity profiles for the three lines showed in Figure 3. It can be seen that only the radial component has a slip velocity, maintaining almost zero relative velocities for the other components, fact commonly reported in literature (se, for instance, Svarovsky & Thew (2003)).

-8,0

-6,0

-4,0

-2,0

0,0

2,0

4,0

6,0

8,0

-40 -30 -20 -10 0 10 20 30 40

x (mm)

V (m

/s)

Voil - Line 1

Vwater - Line 1

Voil - Line 2

Vwater - Line 2

Voil - Line 3

Vwater - Line 3

Figure 11 – Azimuthal velocity profiles for three lines showed at different heights, for two phase model

-2,0

-1,0

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

-40 -30 -20 -10 0 10 20 30 40

x (mm)

V (m

/s)

Voil - Line 1

Vwater - Line 1

Voil - Line 2

Vwater - Line 2

Voil - Line 3

Vwater - Line 3

Figure 12 – Axial velocity profiles for three lines showed at different heights, for two phase model

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

-40 -30 -20 -10 0 10 20 30 40

x (mm)

V (m

/s)

Voil - Line 1Vwater - Line 1Voil - Line 2Vwater - Line 2

Voil - Line 3Vwater - Line 3

Figure 13 – Radial velocity profiles for three lines showed at different heights, for two phase model

Figure 14 shows the relative radial velocity profiles. These velocities characterize the phase separation and have to be maximized for better efficiency.

-0,03

-0,01

0,02

0,04

0,06

0,08

0,10

0,12

0,14

-40 -30 -20 -10 0 10 20 30 40

x (mm)

V (m

/s)

Vrel - Line 1

Vrel - Line 2

Vrel - Line 3

Figure 14 – Relative radial velocity profiles

It is interesting to note that radial velocities increase at the third line, because of the conical region. Furthermore, grater radial relative velocities are captured at the conical region, outside of the core, indicating higher separation at this region. Great differences with “migration velocities”, computed for single phase flow, can be appreciated from these results, even for small droplet diameter (Figure 6). This corroborates that former analysis, based on single phase results, is only valid for very low oil concentrations. Droplet Break-up in Two Phase Model As a first insight to a more complete approach including a population balance model considering break up and coalescence phenomena, a simulation was performed calculating the phase diameter at each grid node using


9

equation (4), i.e., considering the possibility of droplet break up due to turbulence. Values obtained for diameters for each phase were used in equation (23) to compute the interfacial area density. Figure 15 present radial distribution of “maximum” droplet diameter and resulting interfacial area density, Aij, for different heights. It can be observed that there is a substantial reduction in diameter due to turbulence, differently of what was observed for single phase flow (Figure 4). This is because of higher turbulence intensity for two phase flow, mainly in central region, as can be seen in Figure 16. This leads to higher values of interfacial area density distribution, compared with the case where oil and water droplets diameter are constant (Figure 9). At cone region Reynolds stresses are higher and so interfacial area density.

0,00

0,20

0,40

0,60

0,80

1,00

1,20

-40 -20 0 20 40x (mm)

D (m

m)

0

2000

4000

6000

8000

10000

12000

14000

Aij

(1/m

)

D [mm] - Line 1 D [mm] - Line 3

Aij [1/m] - Line 1 Aij [1/m] - Line 3

Figure 15 – Radial distributions of droplet diameters and resulting interfacial length scale.

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

-40 -30 -20 -10 0 10 20 30 40x (mm)

Rey

nold

s St

ress

Line 1

Line 3

Single phase - Line 1

Figure 16 – Reynolds stress distribution for single and two phase flow. Distribution at cone region is also showed for two phase case.

Figure 18 shows distributions of oil volumetric fraction for

both cases through contours at a central plane. Higher oil concentrations at lower regions can be seen when droplet break up is considered. Figure 17 shows the radial relative velocities along the hydrocyclone radius for these cases. Lines are plotted at only at Lines 1 and 3 (from Figure 3) for the sake of clarity. At the cone central region, where higher droplet diameters are present (see Figure 15) radial relative velocities are next to the case of constant droplet diameter. Nevertheless, as expected, radial relatives velocities are lower than former case, in most part of the device leading to lower separation efficiency. Due to local mass conservation, changes on azimuthal and axial velocity profiles are also expected. Nevertheless, as radial velocities almost an order of magnitude lower than other components. Then velocity profiles are very similar to the case considering constant droplet diameter and so these are not showed here.

-0,03

-0,01

0,02

0,04

0,06

0,08

0,10

0,12

-40 -20 0 20 40

x (mm)

V (m

/s)

Vrel - Line 1 - D=consVrel - Line 3 - D=consVrel - Line 1 - D=varVrel - Line 3 - D=var

Figure 17 – Radial relative velocities for two hydrocyclone positions, for constant and variable droplet diameter

Separation efficiency is normally computed as, overfoil

S Inletoil

mm

η = (24)

Carlos Capela3 (personal communication) suggested to calculate a “reduced” efficiency as,

( )Red 1overfoil

S Inletoil

mSPLITm

η = − (25)

where SPLIT is the relation between overflow and inlet total mass flows. This relation takes in account the total mass flow at overflow for efficiency calculation as, using equation (24) efficiency could be 100% is SPLIT is equal to one, independently of the effective separation potential of the device.

3 Hydrocyclone researcher at CENPES/PETROBRAS


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Table 3 – Separation efficiency

Constant drop diam. Variable drop diam. Efficiency 0,813 0,745 Reduced Efficiency 0,57 0,522

Figure 18 – Oil Volume fraction at central plane for: Constant droplet diameter (left) and variable droplet diameter (right).

Conclusions A two phase model for a high phase concentration liquid-liquid hydrocyclone was developed. Model was implemented using an Eulerian-Eulerian approach and a second order Reynolds Stress model for turbulence computations. Although no experimental data was encountered for comparison, results obtained are very consistent with expected physical trends. Preliminary analysis using single phase flow results seems to be good for preliminary design, but it is only valid for very low oil concentrations where dispersed morphology can be ensured in most par of the device. Velocity distributions obtained from two phase flow, show that, for high oil concentrations, strong differences are present between radial relative velocities and the equivalent “migration velocity” from single phase analysis. On the other side theoretical analysis related to the droplet break up and centrifugal forces (mutually opposite effects) seems to be very useful, at least for preliminary design. This analysis was used for local droplet diameter calculation in a two phase model and results obtained followed again

expected trends. Nevertheless, as it was postulated, several hypotheses were made as, constant inlet diameter (considering an hypothetical value of 1mm), We critical number equal to unit etc., for both single and two phase analysis. Then, more detailed analysis is required at this point in order to get realistic results. Furthermore, the approach used here does not consider the “history” of the droplet, i.e., droplets which pass through a high turbulence region could break-up and not coalesce in less turbulent regions. Then a population balance model is necessary to consider the effects of droplet break up and coalescence on the separation efficiency. On the other side, the combination of the “symmetric” model and population balance (for each phase) seems to be a good approach and some tests will be developed in further work. Once again it is emphasized that this work does not intend to obtain real operational result but just develop adequate models for two phase flow within hydrocyclones. Further work also includes investigations of other models for drag coefficient more suitable for intermediate concentration regions CD, (we recall to the fact that model development was based on asymptotic situations of high oil or high water concentrations) and advance in investigations about We critical number for flow within hydrocyclones. Implementation of a population balance model is being investigated, but this approach should take in account oil as well as water as dispersed phases in order to be consistent with the model hereby presented. Within this context, more complex break up models could be investigated. A deeper insight to turbulence modeling in two phase flow is also necessary, and fluid dependent turbulence models are also being investigated. It is expected that all these models can be validated against experiments developed at CENPES/PETROBRAS. Acknowledgements This work was partially supported by ESSS and Petrobras. References Bergstrom, J. & Vomhoff, H., (2007), Experimental Hydrocyclone Flow Field Studies, Separation and Purification Technology, Vol. 53, pp 8-20. Brauner, N. & Ullmann, A., (2002), Modeling of Phase Inversion Phenomenon in Two-Phase Pipe Flows, International Journal of Multiphase Flow, Vol. 28, pp 1177-1204. Burns, A.; Yin, D. W.; Splawski, A. B.; Lo, S.; Guetari, C., (2001), Modeling of Complex Multiphase Flows: A Coupled Solver Approach, Fourth International Conference on Multiphase Flows, New Orleans, LO, USA. Colman, D. & Thew, M. T., (1980), Hydrocyclone to Give a Highly Concentrated Sample of a Ligter Dispersed Phase, Proceedings of International Conference on Hydrocyclones. Cullivan, J. C.; Williams, R. A.; Dyakowski, T.; Cross, C. R., (2004), New Understanding of a Hydrocyclone Flow Field and Separation Mechanism From Computational


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Fluid Dynamics, Minerals Engineering, Vol. 17, pp 651-660. Delgadillo, J. A. & Rajamani, R. K. Exploration of Hydrocyclone Designs Using Computational Fluid Dynamics, International Journal of Mineral Processing, Vol. In Press, Corrected Proof, pp . Drew, D., (1983), Mathematical Modelling of Two-Phase Flows, Annual Review of Fluid Mechanics, Vol. 15, pp 261-291. Duewell, M. R.; Forrester, S. E.; Galvin, K. P.; Evans, G. M., (1998), Prediction of the Droplet Size in a Liquid-Liquid Jet Mixer Emulsification Unit, Port Douglas, Australia. Enwald, H.; Peirano, E.; Almstedt, A.-E., (1996), Eulerian Two-Phase Flow Theory Applied to Fluidization, International Journal of Multiphase Flow, Vol. 22, pp 21-66. Grotjans, H. (1999) Application of higher order turbulence models to cyclone flows, Zyclonabscheiber in der energie- und verfahrenstechnik, Fachtagung Leverkusen, VDI-Verlag, VDI-Berichte, 1511, p. 174, 1999 Hinze, J. O., (1955), Fundamentals of the Hydrodynamic Mechanism of Splitting in Dispersion Processes, AIChE Journal, Vol. 1, pp 289-295. Huang, C. C., (2005), Numerical Simulation of Oil-Water Hydrocyclone Using Reynolds Stress Model for Eulerinan Multiphase Flows, The Canadian Journal of Chemical Engineering, Vol. 83, pp 829-834. Ishii, M. & Mishima, K., (1984), Two-Fluid Model and Hydrodynamic Constitutive Relations, Nuclear Engineering and Design, Vol. 82, pp 107-126. Launder, B. E.; Reece, G. J.; Rodi, W., (1975), Progress in the Developments of a Reynolds-Stress Turbulence Closure, Journal of Fluid Mechanics, Vol. 68, pp 537-566. Maliska, C. R., (2004), Transferência De Calor e Mecânica Dos Fluidos Computacional (in Portuguese), LTC Editora, 2a Edição. Petty, C. A. & Parks, S. M., (2001), Flow Predictions Within Hydrocyclones, Filtration & Separation, Vol. 38, pp 28-34. Raw, M. J., (1985), A New Control-Volume-Based Finite Element Procedure for Numerical Solution of the Fluid Flow and Scalar Transport Equations , University of Waterloo, Canada. Speziale, C. G.; Sarkar, S.; Gatski, T. B., (1991), Modelling the Pressure-Strain Correlation of Turbulence: an Invariant Dynamical Systems Approach, Journal of Fluid Mechanics, Vol. 277, pp 245-272. Svarovsky, L. & Thew, M. T., (2003), Hydrocyclones:

Analysis and Applications, Springer, Wallis, G. B., (1969), One-Dimensional Two-Phase Flow, McGraw-Hill, New York,

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