Four-Phase Eulerian-Eulerian Model for Prediction of Multiphase Flow ...

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Four-Phase Eulerian-Eulerian Model for Prediction of Multiphase Flow in Cyclones H. F. Meier1, A. A. Vegini1 and M. Mori2 1Chemical

2School

Engineering Department, University of Blumenau, Blumenau-SC, Brazil e-mail: [email protected] and [email protected]

of Chemical Engineering, University of Campinas, P.O. Box 6066, Zip Code 13083-970, Campinas-SP, Brazil, e-mail: [email protected] Received: 9 January 2011, Accepted: 10 March 2011

Abstract The purpose of this work is to present a new code, the CYCLO-EE3, which is based on computational fluid dynamics (CFD) techniques. The model used is based on the Eulerian-Eulerian approach and it is composed of several time differential equations in a 3-D-space domain with a 3-D symmetric cyclone inlet. The solid phase is treated like a hypothetical fluid and drag forces between phases are responsible for the gas-solid interaction. The CYCLO- EE3 code makes possible the use of up to three solid phases, each one with size of particle, density and specific volumetric fraction. The mathematical model is completed using a hybrid turbulence model composed of the combination of the (k-ε) standard model and Prandtl’s longitudinal mixing model to represent the turbulence of the gas phase. The model is solved using the finite volume method with staggered grids in the cylindrical coordinate system. The numerical results allowed prediction of the pressure drop and the collection efficiency as well as the complete fluid dynamic behavior of the gas-solid flow for the cyclone. The collection efficiency and pressure drop from an experimental study reported by literature were used to validate the model. A comparison between the experimental data and the numerical results shows that the CYCLO-EE3 code predicts the results well. Keywords: Cyclone, Computational fluid dynamics, Multiphase flow, EulerianEulerian model, Collection efficiency.

1. INTRODUCTION Cyclone separators are a very useful piece of equipment for removal of particles from air streams. The particle-laden gas enters at the top of the cyclone cylinder tangentially and moves down the cyclone into the conical section in a spiral-shaped path. Under the influence of the centrifugal force developed by the swirling air stream, the particles coat the sides of the chamber and are retained in the boundary layer. Cyclones have been extensively used in industry to separate particles from air or gas streams for both air pollution control and process use. The main reasons for the widespread use of cyclones are that they are inexpensive, have no moving parts and can be constructed to endure severe operating conditions such as high temperature and pressure. Despite the apparently simple structure, the flow behavior in cyclones is very complex, including several phenomena such as vortex breakdown, reversal of flow and high turbulence intensity. Pressure drop and collection efficiency are generally the design parameters of interest determining the performance of a cyclone. An accurate prediction of pressure drop and collection efficiency is very important because they are directly related to operating costs. There are plenty of models developed on the basis of experimental data for calculating the main design parameters of

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Four-Phase Eulerian-Eulerian Model for Prediction of Multiphase Flow in Cyclones

cyclones. Therefore, most of them are simplified models that cannot predict the fluid dynamics phenomena that are very important for the reliable functioning of cyclones. Multiphase flow equations in cyclones have been developed and analyzed by many researchers. More than several hundreds of CFD studies have been carried out using the Eulerian-Lagrangean approach, which takes into account the influence of the particles in the gas flow by means of oneway, two-way and four-way coupling between discrete particles and continuous gas1 ( Zhao et al. 2006, Wan et al. 2008, Yoshida et al. 2009, ). Only a few useful studies have been dealt with twofluid approach (the Eulerian-Eulerian approach). In the Eulerian-Eulerian approach the solid phase is also treated as a continuous phase, exposed to the analogous conservation equations for the fluid phase (Meier et al. 2002ab, Ropelato et al. 2005, Cortés and Gil 2007, Qian at al. 2007). In this work, a new mathematical model based on Eulerian-Eulerian approach and a new numerical CFD code named CYCLO-EE3 were developed and validated for diluted multiphase system in cyclones. The CYCLO-EE3 has capability to consider one, two and three solid phases and numerical results were validated with the experimental data from the literature (Zhao et al. 2004). The code has also a great potential for predicting global and individual collection efficiency, pressure drop, the main parameters used for cyclone design, as well as flow field characteristics in order to better understanding the behavior of the multiphase flow in cyclones. 2. MATHEMATICAL MODELLING In general, the Eulerian-Eulerian model has been used to represent the multiphase flow in cyclones considering a two-phase system, where a real fluid (gas phase) interacts with one phase treated like a hypothetical fluid (solid phase) by means of drag forces and mass and energy fluxes in the interface between phases. The main assumption of this model is that the phases interpenetrate which both phases have fluid dynamic properties at the same point in the time-space domain, i.e., continuous properties, such as density, viscosity, thermal conductivity, etc. Only the molecular behavior of the material is ignored. Details about the two-phase Eulerian-Eulerian model for cyclones modeling and simulation can be obtained in Meier and Mori (1998), Meier and Mori (1999), Meier et al. (2002 a,b) and Vegini et al. (2008). The four-phase flow model is an extension of the two-phase flow model, where three solid phases characterized by different particles diameters are used to represent the solid phase as whole. Next section presents details on the mathematical formulation of the four-phase flow model. 2.1 The four-phase flow model The development of a four-phase flow model for gas-solid flow in cyclones requires some assumptions: (a) three different solid phases can represent the behavior of the solid in the gas phase. The main assumption of this model establishes that the phases can interpenetrate and all phases have fluid dynamics properties at the same point in the time-space domain; (b) conservation equations for mass, momentum and energy are also applied to four phases, increasing substantially the number of partial differential equations of the model and, consequently, the computational efforts increase significantly in order to solve them; (c) turbulence acts only on the gas phase and can be modeled by RANS turbulence approach with first order closure by means of an anisotropic model from a combination of the two equation turbulence model(k-ε) and zero-equation model (mixture-length of Prandtl). This kind of model is named hybrid first order closure in the turbulence theory, and it considers that the space and time turbulent scales are different between the swirl movement (outside of the plane) and the axial-radial movement (inside of the plane) (based on works of Duggins and Fritting (1987) and Dyakowski and Williams (1993)); (d) disturbance in the flow near the inlet region due to asymmetry of the tangential or in the volute gas inlet quickly disappears, which makes it possible to use axial symmetry and apply the 3-D symmetry model. This assumption has been used in several CFD works with success (Malhotra et al. (1994) and Madsen et al. (1994)), and can be used as good alternative to reduce the higher computational time due to the increasing of the phases in the multiphase model; (e) drag forces between solids phase and gas phase are responsible for all gas-solid interaction, and they do not have any important contribution of the solid-solid interaction, for example by collision between particles. This assumption agrees well for diluted operating conditions and three-way coupling; (f) the strain 1

See Sommerfeld 2000, for conceptual details of the several ways coupling between phases.

Journal of Computational Multiphase Flows

H. F. Meier, A. A. Vegini and M. Mori

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tensor acting on the solid phases due to molecular mechanisms similar to the gas phase are neglected due to the diluted conditions of the gas-solid flows in cyclones. This implies that the solid phases obey the Euler momentum equation without pressure and viscous terms. Conservation Equations Mass conservation in the gas phase,

(

)

(

)

∂ f ρ + ∇. f ρ v = 0 g g g ∂t g g -

(1)

Mass conservation in the solid phase, ∂ ( f ρ ) + ∇. ( f ρ v ) = 0 s1 s1 s1 ∂t s1 s1 ∂ ( f ρ ) + ∇. ( f ρ v ) = 0 s2 s2 s2 ∂t s2 s2 ∂ ( f ρ ) + ∇. ( f ρ v ) = 0 s3 s3 s3 ∂t s3 s3

-

(2)

Momentum conservation in the gas phase,

(

)

(

)

( )

3

(

∂ f ρ v + ∇. f ρ v v = − f ∇. T eff + f ρ g − ∇p + ∑ Fdrag g g g g g g g g ∂t g g g i =1 -

)

g,si

(3)

Momentum conservation in the solid phase,

(

)

∂ ( f ρ v ) + ∇. ( f ρ v v ) = f ρ g − F s1 s1 s1 s1 s1 s1 drag s1, g ∂t s1 s1 s1 ∂ ( f ρ v ) + ∇. ( f ρ v v ) = f ρ g − F s2 s2 s2 s2 s2 s2 drag s2, g ∂t s2 s2 s2 ∂ ( f ρ v ) + ∇. ( f ρ v v ) = f ρ g − F s3 s3 s3 s3 s3 s3 drag s3, g ∂t s3 s3 s3

(

(

)

)

(4)

On the left side of the Equations (1) to (4) the variables f, ρ and v are volume fraction, density and velocity, respectively, with subscripts g, s1,s2 and s3 indicating gas phase and solid 1, solid 2 and solid 3 phases. The variables on the right side of the Equations (3) and (4), Tgeff, g, p and Fdrag, represent respectively: the effective strain tensor on the gas phase (molecular + Reynolds Tensor); the gravity acceleration; the thermodynamic pressure; and the drag force acting on the phases. Constitutive Equations The model adopted for the effective shear stress is based on the Boussinesq approximation, where the relation of shear stress to deformation rate is similar to that in the general Newtonian fluid model. In this case, the tensor is directly proportional to the deformation rate: Tgeff = −2µ eff D g

(5)

where the deformation rate is

(

)

T D g = 1 ∇v g + ∇v g   2

(6)

and the effective viscosity is a combination of molecular viscosity and turbulent viscosity:

µ ef = µ g + µ (t)

Volume 3 · Number 2 · 2011

(7)

96


In Eq. (7) the turbulent viscosity (µ(t)) can be obtained from an anisotropic model developed by Meier (1998) from a combination of the (k-e) standard turbulence model and Prandtl’s mixing length theory. In the cylindrical coordinate system, for example, Prandtl’s model can be written as follows: 1

(µ (t))θ,θ = (µ (t))r,θ = (µ (t)) z,θ

 ∂u  2  u  2  ∂w  2  2  g  +  g  +  g    ∂r   r   ∂z   2     v     r ∂  g   +      ∂r  r    (t ) 2 2 = µ + ρg r l  2    ∂u in ∂wg   g + 1   +    2  ∂z  ∂ r      2   ∂vg     +    ∂z         

( )

(8)

where the first term on the right side of Eq. (8), (m (t))in, physically represents an additional turbulence contribution due to inlet effects according to Pericleous (1997) and l is the mixing length by analogy with the kinetic theory of gases. A correlation for (m (t))in suggested by Pericleous is

( µ( ) ) t

in

( )

= ρg kg

1

2

Dh . 10

(9)

Where Dh is the hydraulic diameter of the cyclone inlet. The other component of the anisotropic turbulent viscosity with the application of the (k-ε) standard model is (µ (t))r,z = (µ (t))r,r = (µ (t)) z,z = C µ ρ g

k g2 εg

(10)

where Cµ is a constant, kg is the turbulent kinetic energy and εg is the dissipation rate of turbulent kinetic energy. There are also two additional conservation equations: ef ∂ f ρ k + ∇. f ρ v k = f ∇.  µ ∇k  + f G − ρ ε  g g g g g g g g g g g g g   σk ∂t

(

)

(

)

(

)

(11)

and ef ∂ f ρ ε + ∇. f ρ v ε = f ∇.  µ ∇ε  + f C G − C ρ ε ε g  g g g g g g g 1 g 2 g g  σε  ∂t g g g kg

(

)

(

)

(

)

(12)

where σk, σe, C1 and C2 are constants of the model and Gg is a source of turbulence and can be predicted by the following equation: Gg = −Tg(t) : ∇v g .

(13)

This kind of decomposition of the Reynolds stress components with mixture length and k-e model produces anisotropic behavior due to the time scale of the swirling movement in cyclones which is different as due to the time scale in axial and radial movement. The anisotropy of the Reynolds Stress can be expressed in their components according to the following representation:



97

(14)

Now the drag forces between phases can be modeled by a standard equation:

(F )

drag g,si

(

= Fdrag

)

si, g

(

)

(

)

= β g,si v g − v si = β si,g v g − v si ;

i=1, 2 and 3

(15)

where βg,si is the interface coefficient and can be predicted for concentrated flows (fg 0.8, a model proposed by Wen and Yuu (Gidaspow, 1994) relates the interface coefficient to the drag coefficient, CD, by the following equation: f ρ v − v si f si ; β g,si = 3 C Dg,si g g g 4 d psiϕ psi

i=1, 2 and 3

(17)

In the literature there are a large number of empirical correlations for drag coefficient calculation as a function of the Reynolds number for the particles. A drag coefficient that establishes equations for all flow regimes Stokes, transition and Newton regimes was published by Coelho and Massarani (Massarani, 1997) and can be expressed as 1.18

C Dg,si

0.85    0.85 24  =  − K  2si  K1si Re psi  

;

i=1, 2 and 3

(18)

with  ϕ  K1si = 0.843 log10  psi  ;  0.065  K 2si = 5.31 − 4.88 ϕ psi ;

i=1, 2 and 3

i=1, 2 and 3

(19)

(20)

and Re psi =


ρ g v g − v psi d psi ; µg

i=1, 2 and 3

(21)

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2.2 Boundary and initial conditions The initial conditions used for numerical simulation of the single-phase dynamic flow were startup conditions, where all velocity components were considered to be equal to zero and the pressure field was the atmospheric field. For the gas-solid flow, profiles for the gas were obtained from the steady state of single-phase flow with an abrupt entry of the solid phase. The boundary conditions used were a uniform profile at the inlet, no-slip for gas and slip for solid on the wall, axial symmetry conditions and continuity conditions at the exits. More details can be found in Meier et al. (2002a, 2002b). 2.3 Numerical methods The CYCLO-EE3 code, developed by Meier (1998), was used as base and it was adapted for the new condition. This program is composed of three moduli: the preprocessor modulus, the processor modulus and the postprocessor modulus. The preprocessor modulus is responsible for generating the grid, which was developed in a cylindrical coordinate system with a multiblock scheme and staggered grids. The numerical solutions were obtained through the processor modulus, which uses the finite volume method to yield a system of discretized equations for the partial differential equations by using the SIMPLEC method for pressure-velocity coupling and high order differential schemes for the discretization of the variables on the faces of the control volumes. The system of nonlinear algebraic equations is solved by the traditional tri-diagonal matrix algorithm (TDMA) in a line-byline fashion. The time solution is obtained from an implicit first-order procedure to guarantee stability and high convergence rate. Details about this solution strategy can be found in Patankar (1980), Maliska (1995), Meier (1998), and Meier et al. (2000). Animation techniques generated with the postprocessor modulus are applied to visualize the behavior of the flow in the cyclone during the online analysis processing of virtual sensors located in several parts of the cyclone, and after offline analysis processing of the velocity components (axial, radial and tangential velocities for all phases), pressure, volume fractions and turbulent properties. 3. RESULTS AND DISCUSSION 3.1 Case study In this study, experimental data on collection efficiency and pressure drop obtained by Zhao et al. (2004) are compared with the numerical results predicted by the CYCLO-EE3 code for the model with one solid phase (two-phase model) and three solid phases (four-phase model). The experimental study was performed with a cyclone having a conventional tangential single inlet. The geometry and dimensions of the cyclone are given in Fig. 1. The experiments carried out by Zhao et al. (2004) were performed with air under ambient conditions as a function of flow rate and with a dust load of 5.0 g/m3. The solid particles were talcum powder with a lognormal size distribution with a skeletal density of 2700 kg/m3, a mass median particle diameter of 5.97 µm and a geometric standard deviation of 2.08. The operating conditions that were used in this study are summarized in Table 1. In addition, for the numerical simulation it is necessary to define the particle size and volumetric fraction for the two-phase model (one solid phase) and for the four-phase model (three solid phases). By analyzing the definition of the lognormal size distribution, it is possible to conclude that for the two-phase model, the mass median particle diameter (d50 = 5.97 mm) can be used since it is the particle diameter that divides the frequency distribution in half, i.e., fifty percent of the solid phase has particles with a larger diameter and fifty percent of the solid phase has particles with a smaller diameter.



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Figure 1. Definition of cyclone dimensions.

Table 1. Operating conditions.

For the four-phase model it is necessary to define three sizes of particles and their respective volumetric fractions. With the definition of mass median particle diameter and the geometric standard deviation, d15.78 =

d50 . σg

(22)

and d84.13 = d50σ g .


(23)

100


where σg is the geometric standard deviation; d50 is the mass median particle diameter, which is equal to or less than 50% of the mass of particle present; d15.78 is the particle diameter, which is equal to or less than 15.78% of the mass of particles present; and d84.13 is the particle diameter, which is equal to or less than 84.13% of the mass of particles present, it is possible to define three different particle diameters and their respective volumetric fractions. In Table 2 a summary of the values used in each model for the numerical simulation is presented, and in Figure 2 it can be seen the particle diameter distribution curve represented by EE1 and EE3, two and four phase models, respectively. Table 2. Particle diameter and fraction volumetric.

Figure 2. Particle diameter distribution for EE1 and EE3 models.

3.2 Results The main purpose of the present study is to examine the performance of a new code, the CYCLOEE3, which can use more than one solid phase in the model to predict the design parameters of cyclones. The study reported here was carried out to validate the code by comparing experimental data and numerical solution of the CYCLO-EE3 code. 3.2.1. Mesh Sensitivity Analysis The sensitivity analysis of the numerical solutions with the grid concentration was carried out using the Case 3 of the Table 1 and changing the grid from 802 to 46,590 volumes in an axis symmetry cylindrical coordinate system. Pressure drop between the inlet and the exit of the vortex finder of the cyclone was chosen as a global variable which represents all effects of the numerical approximation due to the discretization errors in the numerical domain. Figure 3 presents the results for five grid concentration and it can be seen that the pressure drop practically has been stabilized at 15.3 mBar with the grid (d) with 18,185 volumes. Therefore, all validation studies presented in the next section were performed with the grid (d).



101

Figure 3. Mesh sensitivity analysis.

3.2.2. Validation of the Model The study was planned using the operating conditions shown in the Table 1. The comparison of the Zhao experiment and the numerical simulation for the two-phase model and the four-phase model is shown in Figures 4 and 5. The maps of volumetric fraction for the operating condition in Case 3 with four-phase model can be observed in Figures 6, 7 and 8. Pressure drop The cyclone pressure drop as a function of the inlet air velocity (Cases 1 to 4) for flow without particles (gas phase) and flow with particles (two-phase model and four-phase model) and the experimental values are shown in Figure 4. A comparison of the results shows that the data predicted by the numerical simulation (gas phase, two-phase model and four-phase model), as expected, display the same trend as the experimental values, i.e., the pressure drop increases with an increase in the inlet velocity. As can be observed, the predicted results are the same in all cases, even for the gas phase. This can be explained by the lower solid particle loading; therefore, the pressure drop in the cyclone is practically due to the gas alone. The CYCLO-EE3 code shows a good prediction of cyclone pressure drop at different operational inlet velocities (Fig. 4) the of within 11-24% of the experimental value. The over prediction of the model for pressure drop in all inlet velocities can be explained due to the axis symmetry assumption that establishes an ideal vortex flow with high vortex preservation. In fact this promotes an increase in the tangential velocity field responsible by the major part of the pressure drop in cyclones. On the other hand, the effect of the particle concentration and phases on the gas phase can practically be neglected for all diluted cases used in this work.

Figure 4. Pressure drop as a function of inlet velocities Volume 3 · Number 2 · 2011

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Collection efficiency The Fig. 5 shows the numerical results and measured overall efficiency of the cyclone as a function of inlet velocity for the same cases above. Good agreement between the numerical calculations of the four-phase model and the experimental data can be observed. The two-phase model overestimates the collection efficiency with a prediction absolute error of about 4%. According to Figure 5, the four-phase model can predict the overall efficiency better than the usual two-phase model, and this suggest that the increment of the number of solid phases can be a trend to improve the Eulerian-Eulerian approach in gas-solid flows in cyclones.

Figure 5. Collection efficiency at different inlet velocities.

Maps of volumetric fraction The CFD technique has great potential for predicting the flow field characteristics inside the cyclone. Figures 6, 7 and 8 represent maps of volumetric fractions for each solid phase of the fourphase model. It can be seen, mainly in the Figures 7 and 8, the smallest solid particle escape from the top through a “vortex-finder” tube, causing loss of material and a reduction in the global efficiency of collection as expected. The largest particles were pushed to the external walls of the cyclone due to the centrifugal force promoted by the swirling flow, increasing the particle volume fraction near the wall from inlet to the underflow direction (see Figure 6). It makes possible to suppose that the particle-particle interaction in this region can be important on the collection efficiency, and consequently suggest the need of more experimental studies in order to predict this effect not considered in this work. Likewise, Figures 7 and 8 show a large dispersion of the solid volume fraction for the smallest particle diameters from the wall to the center which reduces the collection efficiency where the particle-particle interaction acts as a force which drives the particles to the wall. In fact, in neglecting the particle-particle interaction in the four-phase model one can justify the under prediction of the global collection efficiency showed in Figure 5, except for higher inlet velocity where the centrifugal force prevails.



Figure 6. Volumetric fraction map of solid phase 1 (dp=12.42 mm).

Figure 7. Volumetric fraction map of solid phase 2 (dp=5.97 mm).


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104


Figure 8. Volumetric fraction map of solid phase 3 (dp=2.87 mm). 4. CONCLUSIONS In this paper the four-phase model described above was compared with experimental data to validate the CYCLO-EE3 code. It was demonstrated that the four-phase model is more appropriate and accurate than the usual two-phase model for the design of cyclones by means of pressure drop and collection efficiency. Another advantage of the CYCLO-EE3 code is the possibility of visualizing the dynamic flow inside the cyclone. As a consequence, it can be said that the CFD techniques are able to predict the flow field in great detail, thus providing a better understanding of the fluid dynamics of the cyclones. In conclusion, it can be said that the CYCLO-EE3 code based on Eulerian-Eulerian approach to deal with particles of different diameter sizes has great potential for the optimization and design of cyclones, and the increment of the phase number can be an alternative to predict the particulate flows with heterogeneities in the particle diameter distribution. As a future work the authors are extending the model to represent up to five solid phases (model EE5) under diluted and concentrated conditions, and also a possibility for development of new drag correlation for solid-solid interaction in order to improve the model. ACKNOWLEDGEMENTS The authors are very grateful for the assistance received from University of Campinas and University of Blumenau and to CAPES, which made this work possible. REFERENCES Cortés, C., Gil, A., 2007. Modeling the gas and particle flow inside cyclone separators , Progress in Energy and Combustion Science, vol. 33, pp. 409-452. Dyakowski, T. and Williams, R. A., 1993. Modelling Turbulent Flow within a Small-Diameter Hydrocyclone, Chem. Eng. Scie., vol. 48, pp. 1143-1152. Gidaspow, D., 1994. Multiphase Flow and Fluidization – Continuum and Kinetic Theory Descriptions. Ed. Academic Press Inc., London. Madsen, H. J. , Thorstensen, J. H. , Salimi, P. , Hassing, N. H. and Rusaas, J., 1994. Prediction of the Performance of Gas Cyclones, Second CFDS International User Conference, December. Malhotra, A., Branion, R. M. R. and Hauptmann, E. G., 1994. Modelling the Flow in a Hydrocyclone, Canad. J. Chem. Eng., vol. 72, pp. 953-960.



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