A Novel Hybrid Scheme for Making Feasible Numerical Investigations

Research Note pubs.acs.org/IECR

A Novel Hybrid Scheme for Making Feasible Numerical Investigations of Industrial Three-Phase Flows with Aggregation Henrik Ström,*,† Mia Bondelind,‡ and Srdjan Sasic† †

Department of Applied Mechanics and ‡Department of Civil and Environmental Engineering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden ABSTRACT: This Research Note proposes an innovative modeling and computational scheme for investigating industrial threephase flows with aggregation. The scheme is applied on an industrially relevant application: dissolved air flotation. Aggregate formation is modeled locally using a sophisticated subgrid model, and a concept involving massless tracer particles is invoked to transport the aggregate properties, along with the flow. The model is able to account for the formation and consumption of all entities involved and to track their respective properties, only at a moderate computational cost.

1. INTRODUCTION Multiphase flows are ubiquitous in nature and of great industrial importance. In dense, dispersed multiphase flows, the physics involved are typically very complex and commonly used experimental procedures available for single-phase flows are either ineffective or invasive. Mathematical modeling therefore represents a key step in the investigations of such flows.1 However, the application of detailed numerical models to problems of industrial scale is often intractable, because of excessive computational requirements. In the present Research Note, we introduce and discuss the performance of a novel modeling and simulation scheme applicable to one very challenging type of multiphase flow: industrial three-phase flow with aggregate formation. Three-phase systems represent a particularly challenging form of multiphase flows. They are often characterized by the presence of a continuous (carrier) phase and two dispersed phases. When interacting, the dispersed phases may form a fourth “phase”: the aggregates. Important industrial examples include flotation equipment (for the treatment of wastewater,2 recycled paper,3 or minerals,4,5) and a range of operations within the petrochemical industry.6,7 Many works that deal with the modeling of various aspects of these flotation processes have been previously reported in the literature.8−13 In this work, we focus on flotationin particular, dissolved air flotation (DAF). In flotation, the overall objective is to efficiently separate solid/liquid, liquid/solid/oil, or oil/water mixtures via the formation of buoyant aggregates.14 In a DAF unit, air bubbles are injected into the continuous water phase, which contains solid particles (termed flocs) to be removed from the water. The flocs attach to the bubbles and form buoyant aggregates (Figure 1). These aggregates rise to the surface of the unit where they are removed. Since the aggregates consist of bubbles and flocs with entrained water between them, they represent a complex composite of the three other phases. As a consequence, the aggregates have significantly different hydrodynamic properties and must be treated as a fourth “phase”. In addition, the turbulence of the water−aggregate flow determines the bubble trajectories and the floc concentration field, and also affects the extent of aggregate formation and the aggregate properties. Hence, there © 2013 American Chemical Society

Figure 1. Generic computational cell in a numerical simulation of a three-phase flow with aggregation.

is a feedback of information from the water−aggregate flow field to the fields that determine the local volume fraction of aggregates (as illustrated in Figure 2). Therefore, a model of such an industrial system must be able to handle all the links in this chain in order to deduce any information about the performance of the equipment. The major challenge in treating these types of multiphase flows is not a lack of detailed models, but a lack of models that are computationally feasible enough to be applied in an industrial unit. The local volume fraction of the aggregates and their effect on the carrier-phase flow field is large enough to render Lagrangian particle tracking of the aggregates intractable. At the same time, in an Euler−Euler approach, there is a need to determine and transport the aggregate properties. Whereas this can possibly be accomplished using population balance modeling, the computational cost and complexity of such an approachgiven the variation of both the aggregate sizes and densitiesis insurmountable. Thus, there is a clear need for an approach in which all the effects are taken into Received: Revised: Accepted: Published: 10022

March 19, 2013 June 20, 2013 July 2, 2013 July 2, 2013 dx.doi.org/10.1021/ie400886u | Ind. Eng. Chem. Res. 2013, 52, 10022−10027

Industrial & Engineering Chemistry Research

Research Note

∂ (αkρk Ui , k) = S1, i , k ∂xi

(1)

∂ ∂ (αkρk Ui , k) + (αkρk Ui , kUj , k) ∂t ∂xj = −αk

∂τji , k ∂P + αkρk gi + Fi , k + S2, i , k + ∂xj ∂xi

(2)

Here, αk is the volume fraction of phase k, ρ the density, U the velocity, P the pressure, and gi the gravitational acceleration in coordinate direction i. The momentum exchange between the water and the aggregates (Fi,k) is a function of the size of the aggregates and must be determined locally by the aggregate transport submodel. Included in the stresses (τj,i,k) is the turbulent contribution, to be modeled using any turbulence model of choice (here, the Standard k−ε model15 is chosen, for the sake of simplicity and transparency). The momentum source term, S2,i,k, accounts for the momentum exchange between the air bubbles and the water phase (and is thus zero for k = a), whereas the source term in the continuity equation, S1,i,k, accounts for the formation of the aggregate phase. The Eulerian−Eulerian description of the water and aggregate flow implies that the model is able to resolve flows involving a high volume fraction of aggregates, as illustrated later in Section 3.2. It also includes the momentum exchange between the water, the aggregates and the bubbles, so that the coupling depicted in Figure 2 is taken into account. 2.2. Framework 2: Flocs. The flocs are small and close to neutrally buoyant, and are thus treated as passive tracers in the carrier phase. However, the local concentration of flocs is monitored in order to predict correctly the availability of flocs for aggregate formation. The flocs’ species transport equation is

Figure 2. Information feedback loop between the aggregate properties and concentration, and the turbulent flow field.

account, but that still can be applied to units of industrial size. Here, we report on a hybrid scheme with which numerical simulations of the formation and flow of water, aggregates, flocs, and bubbles in an industrial unit are made feasible. This proof of concept of the proposed scheme is a first step in a comprehensive research campaign in which a final goal is to predict by numerical simulations the flow inside a complete DAF unit.

2. MODELING The proposed scheme is constructed as depicted in Figure 3. On the level of a computational cell, three parallel frameworks

∂C ⎞ ∂ ∂ ∂ ⎛ (αwρw Cf ) + (αwρw Ui ,wCf ) = ⎜αwDf f ⎟ + S3, i ,f ∂t ∂xi ∂xi ⎝ ∂xi ⎠ (3)

Here, Cf is the local concentration of flocs. The diffusivity of flocs in the water phase (Df) is assumed equal to the local turbulent diffusivity, as predicted by the Standard k−ε model with a turbulent Schmidt number of Sc = 0.7. The source term, S3,i,f, accounts for the consumption of flocs in the creation of aggregates and is determined by the aggregate formation submodel. Since the flocs are treated as passive tracers, they do not exchange momentum with any other phase. 2.3. Framework 3: Air Bubbles. The air bubbles are many and considerably lighter than the carrier phase. Consequently, they rise upward and in this manner induce motion in the carrier phase. The behavior of the bubbles in the water is thus modeled using two-way coupled Lagrangian particle tracking. The Eulerian−Lagrangian approach to simulations of bubble motion in unsteady gas−liquid flows compares well with both experiments and Eulerian−Eulerian descriptions.16 In the present work, the usage of the Lagrangian approach offers an efficient way to treat arbitrary bubble size distributions and allow for full flexibility for future extensions where additional physics are incorporated into the bubble equations of motion. Furthermore, this approach also avoids the handling of three (or more) Eulerian phases, with the associated challenges of deriving the three-phase interphase exchange terms.17

Figure 3. Schematic layout of the modeling approach. Three different frameworks are used together on the cell level, together with two subgrid-scale models, to resolve the flow.

are employed to predict the motion of water, aggregates, flocs (solid particles), and bubbles. Two additional submodels are needed to supply and transport the information needed for the aggregate phase in the Eulerian−Eulerian framework, resulting in a hybrid scheme that takes advantage of the different strengths of the Eulerian and Lagrangian methods, respectively. The frameworks and submodels are presented in what follows, in the same order of appearance as in Figure 3. 2.1. Framework 1: Water and Aggregate Phases. This framework is centered around an Eulerian−Eulerian description of the flow of aggregates in water, where the two phases are treated as interpenetrating continua. The continuity equation and the momentum balance equations for the two phases (k is the phase index; k = “w” for water and k = “a” for aggregates) are given as follows: 10023

dx.doi.org/10.1021/ie400886u | Ind. Eng. Chem. Res. 2013, 52, 10022−10027

Industrial & Engineering Chemistry Research

Research Note

It is assumed that aggregates are formed by flocs and bubbles in a ratio of 1:5, as observed experimentally by Fukushi et al. for the currently employed floc size.25 Furthermore, it is assumed that the time scale of formation is significantly shorter than the time scale of transportation, so that aggregate formation takes place until the limiting reactant (i.e., bubbles or flocs) is fully consumed in the cell. Later, bubbles and flocs can be replenished by transportation into the cell from other cells in unexhausted parts of the domain. Computational bubble parcels where all bubbles participate in aggregation are removed from the calculation, and the variable Nb is rescaled for one of the remaining parcels if needed to fulfill conservation of mass for the air. The aggregate density is not only a function of the number of its constituents (bubbles and flocs), but also of the amount of water that is entrained inside the aggregate structure. It must therefore be determined experimentally. To verify the ability of the current scheme to handle the combined variation of aggregate sizes and densities, the density of an aggregate is here calculated from the densities of the participating bubbles and flocs plus a random contribution that is normally distributed about zero with a standard deviation of 10 kg m−3. As a formed aggregate is transported through the system, it may grow by subsequent addition of more bubbles and flocs in downstream cells. Here, we assume that there is no preference for a bubble or a floc to contribute to forming a new aggregate or to building onto a pre-existing one. Consequently, the ratio of continued aggregation to the total consumption rate becomes proportional to the local volume fraction of aggregates (normalized by the aggregate packing limit, pl): rgrowth/(rgrowth + rformation) = αa/αa,pl. This approximation fulfills the prerequisites that, in the absence of the previously formed aggregates, bubbles and flocs participate only in the formation of new aggregates, and that in the limit of a cell being filled with old aggregates, new formation is prohibited. The maximum aggregate size allowed is deduced from the turbulent length scale of the large-scale turbulence. If an aggregate at any time is larger than the local value of this length scale, the aggregate is broken down into aggregates of this size while conserving mass. It is furthermore assumed that the processes of continued aggregation and aggregate breakup have no influence on the aggregate density. 2.5. Submodel 2: Aggregate Transport Submodel. Information about the local aggregate size, needed for the determination of the momentum exchange terms in the Eulerian−Eulerian description, must be transported with the aggregate phase. We choose here to transport this information using massless tracer particles that are tracked in a Lagrangian frame of reference, hence the hybrid nature of the proposed scheme. Whenever aggregates are created in a cell, a new tracer particle is inserted at the cell center, and these tracers move with the velocity of the aggregate phase. Each tracer carries information about the size, density, and mass of the aggregates created in the same time step as when the tracer was inserted. Consequently, the local cell information on the aggregate sizes and densities can always be determined from the mass-weighted average of the information carried by the tracer particles and the properties of the newly created aggregates. If aggregates are transported into a cell in which no tracer particle yet exists (i.e., by turbulent diffusion), a new tracer particle, inheriting the (mass-averaged) properties of the donor cell(s), is inserted. Therefore, these capabilities of the tracer particle approach offer a greater flexibility than the alternate approach, in which the

A large number of computational parcels, each representing Nb bubbles, are tracked through the computational domain using the following particle equations of motion: dxi ,b dt

dUi ,b dt

= Ui ,b

(4)

πd p 3 C DRe = 3πdbμw (Ui ,w − Ui ,b) + (ρ − ρ b ) g i 24 6 w (5)

Here, xb is the position of a bubble, Ub the bubble velocity, db the bubble diameter, μw the dynamic viscosity, CD the bubble drag coefficient, and Re the Reynolds number of the bubble. For the current DAF properties, the bubbles remain within the Stokes flow regime18 and hence CD = 24/Re. The change in momentum for each computational parcel in a cell is multiplied by Nb and summed to obtain the momentum exchange with the water phase (via S2,i,w in eq 2). The velocity of the water, as experienced by the bubbles (Ui,w), is obtained from the solution to eq 2 with an additional fluctuating contribution to simulate the turbulent dispersion of bubbles using the eddy-interaction concept.19,20 The particle equation of motion adopted here takes the drag force and the buoyancy force into account, as these have been shown to be the two most important forces in buoyancy-driven bubbly flows.21 In specific situations, other forces could potentially be of interest (i.e., the virtual mass force, the history force, and the lift force), and there is nothing that precludes their later addition to the current scheme by adding the appropriate expressions to the right-hand side of eq 5. The sum of the volume fractions of all phases must add up to unity, implying that αw + αa + αf + αb = 1

(6) 22

In the limit of small particles and low volume loadings, it becomes acceptable to neglect the volumes occupied by the flocs and the bubbles. It should be stressed here, however, that the effect of the buoyancy of the bubbles on the flow pattern of the other phases can never be neglected, not even at small volume loadings, and this effect is therefore always accounted for via the momentum source term S2,i,w. 2.4. Submodel 1: Aggregate Formation Submodel. The aggregate formation process is perhaps the most challenging aspect of the current problem. The formation of aggregates in a DAF unit consists of two subprocesses: gravity settling and interception.18 In essence, air bubbles act as collectors of flocs. Bondelind et al. presented a comprehensive model that estimates the sizes of the aggregates formed in this way.23 This model uses a description of the forces acting on flocs and bubbles that lead to the creation of aggregates, thus enabling the size of a newly formed aggregate to be obtained. The following parameters are needed to close the aggregate formation submodel: the relative velocity between the flocs and the bubbles, the ratio of flocs to bubbles within an aggregate, the density of the aggregate upon formation, and the growth rate of the previously formed aggregates. The subgrid-scale relative velocity between the flocs and the bubbles in the aggregate formation process is modeled as the sum of two contributions: the floc settling velocity24 and a local turbulent velocity fluctuation. The latter is related to the Kolmogorov velocity scale, but adjusted to account for the short acceleration period of a floc in relation to the lifetime of a turbulent eddy. 10024

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Industrial & Engineering Chemistry Research

Research Note

The current work is primarily intended as a proof of concept, hence the simplistic test case design. Unfortunately, there is a lack of simultaneous experimental measurements of the floc, bubble, and aggregate sizes available in the literature.23 We are therefore only able to assess the predicted behavior in the test case scenario, in terms of orders-of-magnitude estimates and from a qualitative viewpoint. More-detailed experimental data on the aggregation process will be necessary for comprehensive validation studies to be possible, and it is our objective to investigate the possibilities to acquire such data in the continuation of the current work. 3.1. Aggregate Properties. The predicted aggregate diameters for the test case (shown in Figure 5) are in the

bubble parcels would be converted to tracers. The computational overhead due to the aggregate transport submodel is proportional to the number of aggregate-containing cells and the Courant number, and therefore may be optimized by choosing an adequate time step for a given computational mesh. In this way, it becomes possible to keep the overhead cost at or below 5%−10%.

3. RESULTS AND DISCUSSION The performance of the proposed scheme is investigated for a test case using values of the parameters of interest that are typical for the DAF process (see Table 1). The computational Table 1. Test-Case Simulation Settings parameter

value

bubble diameter floc diameter floc inlet concentration water inlet velocity air mass-flow rate superficial gas velocity turbulent intensity of the carrier phase

63 μm 100 μm 4 pM 0.5 m s−1 10−5 kg s−1 0.33 cm s−1 5%

domain is a rectangular two-dimensional box with a uniform inlet of water and flocs at the bottom and with bubbles injected from a point on the inlet boundary. It is depicted in Figure 4

Figure 5. Contour plots of the instantaneous (left) and the timeaveraged (right) aggregate diameter.

approximate range of 250−450 μm, which is significantly larger than the diameters of the constituents (cf. Table 1) and compares with values for DAF units.15,30 The unsteadiness due to the turbulent motion of the bubble plume is visible in the instantaneous snapshot. The ability to handle the variation in density of the aggregates can be verified from Figure 6. The fact that the hybrid scheme allows the tracking of a varying density with the moving aggregates throughout the domain is one of its most important capabilities.

Figure 4. Schematic of the computational domain; the width is 0.25 m, and the height is 0.4 m. Gravitation acts in the negative vertical direction. The domain is divided into 4100 computational cells (indicating that the mesh requirements for the proposed scheme are the same as for traditional Eulerian−Eulerian and Eulerian− Lagrangian simulations). Figure 6. Contour plots of the instantaneous (left) and the timeaveraged (right) aggregate density.

and can be envisaged as representing the contact zone of a DAF unit.2 It should be noted here that the two-dimensional geometry is chosen only to obtain a simple and transparent geometry for the test case, whereas transient three-dimensional simulations will typically be needed to allow a close comparison between simulations and experiments.26−29 The bubbles have no initial upward velocity but only a horizontal velocity component that ranges from −1 m s−1 to +1 m s−1, to ensure that the bubble plume spreads in the horizontal direction. Contour plots are shown either as representing a snapshot in time or displaying time-averaged results. In the latter case, the instantaneous values have been sampled over five flow-through times through the domain.

3.2. Aggregate Volume Fraction. The motivation for treating the aggregate phase as a second Eulerian phase stems from the fact that the aggregate properties are significantly different from those of the water and that the aggregate volume fraction can be very high in an industrial application. To highlight the ability of the model to resolve the flow involving a high volume fraction of aggregates, the inlet floc concentration is increased to produce more aggregates. This action increases the volume loading of aggregates in the domain from ∼6% to 40% in the region of the bubble plume. 10025

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Research Note

Figure 7 illustrates the change of flow pattern in the domain as the presence of aggregates increases. The much higher

4. CONCLUSIONS A novel hybrid scheme is presented that makes feasible numerical simulations of three-phase flows with aggregate formation in units of industrial size. The current work constitutes a proof of concept, where the proposed scheme is tested in investigations of the formation and flow of aggregates in a dissolved air flotation (DAF) unit. The scheme is able to predict realistic aggregate sizes and track two different aggregate properties (size and density) simultaneously through the domain. The transport of the two other dispersed phases (bubbles and flocs) is also resolved, so that the scheme is able to account for the local availability of bubbles and flocs in the aggregate formation process. Finally, the presented scheme possesses the inherent capability to account for a strong coupling between the aggregation process (on the microscale) and the turbulent transport (on the macroscale) of bubbles, flocs, aggregates, and water through the unit. The verification and implementation of the proposed scheme is a first step in the process of making feasible numerical predictions of the flow inside a complete DAF unit. In addition, the all-containing approach to industrial three-phase flows with aggregation represented by this hybrid scheme has a clear potential to serve as a basis for a continued development of a range of other schemes applicable to related challenging industrial processes.

Figure 7. Instantaneous velocity profiles for the aggregates and the water for a low concentration of aggregates (left) and a high concentration of aggregates (right). The velocity magnitude is plotted as a function of the vertical position on the centerline (in the middle of the bubble plume) of the computational domain. The water inlet velocity at vertical position zero is 0.5 m/s in both cases.

volume fraction of aggregates creates significantly stronger buoyancy effects, which give rise to a higher upward velocity in the middle of the domain (where most of the aggregates move) and, in turn, lower upward velocities of clear water at the side walls. These changes in the flow field also have a significant effect on the predicted turbulence levels: the turbulent intensity at the tip of the bubble plume is doubled in the case of a high volume fraction of aggregates. 3.3. Aggregate Formation. In the previous tests, the number of bubbles was such that the flocs were the limiting constituent in the aggregate formation process. To test the capabilities of the scheme to account for the availability of bubbles and flocs, the air flow rate is decreased so that each computational bubble parcel represents a smaller number of bubbles. As a consequence, all the bubbles are consumed before they reach the domain outlet. As can be seen from Figure 8, the model also handles this situation.

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AUTHOR INFORMATION

Corresponding Author

*Tel.:+46 (0) 31 772 13 60. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Figure 8. Contour plots of the instantaneous concentration fields of (from left to right): bubbles, flocs, and aggregates.

After all of the bubbles have been consumed, flocs are entrained into the core flow by the turbulence. However, there can be no continued aggregation, because of the fact that no more bubbles are available. The maximum volume fraction of aggregates is thus observed at the tip of the bubble plume, and this volume fraction does not increase in the downstream direction. 10026

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dx.doi.org/10.1021/ie400886u | Ind. Eng. Chem. Res. 2013, 52, 10022−10027