Experimentally Validated Computational Fluid Dynamics Simulations

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EXPERIMENTALLY VALIDATED CFD SIMULATIONS OF MULTICOMPONENT HYDRODYNAMICS AND PHASE DISTRIBUTION IN AGITATED HIGH SOLID FRACTION BINARY SUSPENSIONS Li Liu, and Mostafa Barigou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie3032586 • Publication Date (Web): 26 Jul 2013 Downloaded from http://pubs.acs.org on September 1, 2013

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Industrial & Engineering Chemistry Research

EXPERIMENTALLY VALIDATED CFD SIMULATIONS OF MULTICOMPONENT HYDRODYNAMICS AND PHASE DISTRIBUTION IN AGITATED HIGH SOLID FRACTION BINARY SUSPENSIONS

L. Liu and M. Barigou*

School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

Abstract The mixing of dense (5-40 wt%) binary mixtures of glass particles in water has been studied in a stirred vessel at the ‘just-suspended’ speed and at speeds above it, using an Eulerian-Eulerian CFD model. For each phase component, numerical predictions are compared to 3-D distributions of local velocity components and solid concentration measured by an accurate technique of positron emission particle tracking. For the first time, it has been possible to conduct such a detailed ‘pointwise’ validation of a CFD model within opaque dense multicomponent slurries of this type. Predictions of flow number and mean velocity profiles of all phase components are generally excellent. The spatial solids distribution is well predicted except near the base of the vessel and underneath the agitator where it is largely overestimated; however, predictions improve significantly with increasing solid concentration.

Other phenomena and parameters such as particle slip velocities and

homogeneity of suspension are analysed.

Keywords: CFD; mixing; PEPT; stirred vessel; suspension

_______________________ *Corresponding author: Tel: +44 (0)121 414 5277 Fax: +44 (0)121 414 5324 Email: [email protected]

ACS Paragon Plus Environment


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1. Introduction The mixing of solids in liquids finds a wide range of industrial applications such as in the manufacture of fine chemicals, pharmaceuticals, personal/home care products, paper and pulp, polymers, and food. These systems give rise to immense difficulties and continue to pose a formidable challenge to industry as well as academic research.

There are huge practical difficulties in imaging solid-liquid flows and measuring local phase

velocities and phase concentrations, and problems are confounded by large particles with significant inertia, wide particle size distributions and high solid fractions. Consequently, the methods generally adopted for designing stirred vessels for solid-liquid mixing tend to be based on a global ‘black-box’ approach, and a more detailed description of the internal flow structure has long been missing to enable the development of more rational rules for establishing process parameters and equipment design. The advent of powerful measurement and modelling techniques in recent years has reenergised interest in this field and more effort is increasingly being devoted to fully understand the mechanisms behind particle suspension and distribution1-4.

Among the various flow visualisation techniques, laser Doppler velocimetry (LDV) and particle image velocimetry (PIV) have become the most reliable to examine the complex nature of the flow fields in optically transparent single-phase systems5-7. These techniques, however, fail completely in dense systems which are opaque and their application to multiphase flows has, therefore, been restricted to very dilute suspensions1. One of the important aspects in the local description of multiphase mixtures is the measurement of phase distribution which has remained a challenge. Attempts at local measurements in these systems have been mainly limited to the measurement of mean axial solid-concentration profiles at relatively low concentrations, using a single vertical conductivity or capacitance probe traverse or a withdrawal technique8,9. More recent experimental studies using electrical resistance tomography (ERT) demonstrated, albeit mainly qualitatively, how visualisation of gas, solid or liquid distribution can help improve understanding of mixing processes10. However, none of these methods is suitable to probe concentrated suspensions in detail to acquire quantitative information on the local flow dynamics of the different phase components or their individual 3D distribution. Rammohan et al.11 developed a more sophisticated computer automated radioactive particle tracking (CARPT) technique based on gamma-ray emissions. It has been argued for some time that Lagrangian data obtained from a single particle trajectory, where such data can be accurately determined, may unravel valuable mixing information which is not provided by Eulerian observations12-17. We recently reported in a series of papers on the successful use of the technique of positron emission particle tracking (PEPT) to measure the local hydrodynamics as well as the spatial distribution of each phase component in mechanically agitated solid-liquid suspensions16,18-20.

On the computational front, there have been a number of studies on solid-liquid suspensions ranging from dilute to concentrated systems. Direct or Large Eddy Simulations are still far too expensive computationally to deal with multiphase flows in stirred tanks and can only handle dilute systems (~ 5 vol%). Therefore, solution of the Navier-Stokes equations by means of Computational Fluid Dynamics (CFD) codes with the appropriate twophase models incorporated seems to be the most advanced cost-effective method currently available to model such systems.


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In most CFD studies of dilute solid-liquid suspensions ( ~ < 3 wt%), predictions of the phase velocities have been compared to experimental data obtained from optical techniques such as LDV and DPIV published in the work of Oshinowo and Bakker21 and Fan et al.22. On the other hand, the CARPT technique11 was used to obtain the flow field of the solid phase for CFD validation at a solid concentration of only 2.1 wt%23. CFD predictions of dilute solid distributions (up to ~ 1.33 wt%) have also been compared to experimental data obtained from an optical attenuation technique, though measurements were limited to the axial solid distribution.24-26

A number of numerical studies have also addressed suspensions with higher solid loadings up to ~ 34 wt%. The validation work reported in the literature has been very limited, however, relying mostly on single-traverse probe measurements of axial solid distribution2,21,27 but no detailed hydrodynamic data such as the distribution of phase velocities or slip velocities. It is only recently that the axial velocities of both the liquid and solid phase measured by an ultrasound velocity profiler have been used to validate CFD predictions for medium concentrated solid suspensions (13.6 wt%)28. The effects of different CFD approaches such as MFR (multiframe reference) and SG (Sliding Grid), as well as different drag force models on the prediction of the solid distribution have also been studied.26,27,29 However, the lack of detailed hydrodynamic data for validation has limited the development of reliable approaches and models.

Overall, therefore, the accuracy of CFD in the mixing of dense suspensions has not been sufficiently studied and, in particular, little information exists on flow hydrodynamics. Hence, a detailed validation using detailed ‘pointwise’ measurements of 3-D velocities as well as the distribution of the phases and phase components, especially under conditions of moderate to high solid loadings, is needed in order to assess the capability of CFD in these complex flows. The availability of the PEPT technique has provided an exceptional opportunity for studying such systems and for using the unique measurements obtained to rigorously assess with unprecedented detail and accuracy the capability of the numerical models and codes available.

In this paper, the flow of binary mixtures of coarse glass particles containing up to 40 wt% solids in water, is studied numerically by means of a CFD model in a mechanically agitated vessel under different regimes of particle suspension. Experimental results obtained using PEPT in our previous work19 at N = Njs together with new measurements at N>Njs are exploited for validation of a number of suspension characteristics, including most importantly the detailed ‘pointwise’ 3-D velocity field of each phase component and its spatial phase distribution. The results are also analysed to infer for each phase component the local time-average particlefluid slip velocity, phase flow number and suspension uniformity. Finally, verification of the solids mass balance and mass continuity in the vessel is presented.

2. Experimental 2.1 Positron emission particle tracking The two-phase flow field and spatial phase distribution were determined by the technique of positron emission particle tracking (PEPT). PEPT is a non-intrusive technique which can be applied to opaque systems. It uses a single positron-emitting particle as a flow tracer which is then tracked in 3-D space and time to reveal its full



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Lagrangian trajectory. The measurements can be analysed in two different ways: a Lagrangian-statistical analysis exploiting concepts such as residence time, circulation time and trajectory length distribution; or, a Lagrangian-Eulerian analysis used to extract local Eulerian quantities from the purely Lagrangian information contained in the tracer trajectory such as the three velocity components ( u z ,

u r , uθ ) of each phase component,

its local occupancy or time-average concentration20. PEPT has been validated in water by comparing its measurements with PIV and has been shown to be an accurate and reliable technique30. More details about the technique can be found in our earlier papers14,16,18-20,31,32. 2.2 Apparatus and procedures The PEPT experiments were conducted in a fully-baffled flat-base Perspex vessel of diameter T = 288 mm, agitated by a down-pumping 6-blade 45º pitched-turbine (PBT) of diameter D = 0.5T, as illustrated in Fig. 1. The height of the suspension was set at H = T and the impeller off-bottom clearance was 0.25T. The suspending liquid was an aqueous solution of NaCl of density 1150 kg m-3 − NaCl was added to enable the water density to be matched to that of the resin PEPT tracer used to track the liquid − and the solid particles used were spherical glass beads of density 2485 kg m-3. Two nearly-monomodal particle size fractions, (dp ~ 1 mm; ~ 3 mm) were used to make binary solid-liquid suspensions. The two size fractions were mixed in equal proportions with a total solid mass concentration, X, varying from 0 to 40 wt%, i.e., X1 = X2 = 0.5 X. Experiments were conducted under different regimes of solid suspension corresponding to: (i) N = Njs, the minimum speed for particle suspension19; and (ii) speeds above Njs up to 2Njs. Njs was visually determined in the transparent vessel according to the well-known Zwietering criterion33, i.e., no particle remains stationary on the bottom of the tank for longer than 1-2 s. The conditions of the PEPT experiments and CFD simulations conducted are summarized in Table 1.

Each component of the suspension was individually tracked and its full trajectory separately determined in three successive experiments. A neutrally-buoyant radioactive resin tracer of 600 µm diameter was used to track the liquid phase, and from each particle size fraction a representative glass particle was taken, was directly irradiated via a cyclotron and its motion tracked within the agitated slurry. The PEPT tracking time was 30 min in each case, long enough so that adequate data resolution was obtained in every region of the vessel and ergodicity of flow, the theoretical condition which guarantees that a flow follower is representative of all the solid or liquid phase it represents, could be safely assumed. Confirmation of the ergodicity assumption through experimental verifications has been reported in previous papers18,19,34

3. Numerical modelling Numerical simulations were performed using the commercial CFD code ANSYS-CFX 12.0. In simulating solid-liquid flows, the Eulerian-Lagrangian model would be a more realistic approach because it simulates the solid phase as a discrete phase and so allows particle tracking. However, the number of particles that can be tracked is currently very limited, thus restricting the applicability of the model to dilute mixtures. The multifluid Eulerian-Eulerian model was therefore used instead, whereby the liquid and solid phases are both treated as continua.


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The governing equations which form the basis of the model are the continuity equation:

(

∂ ρ qφq ∂t

) + ∂(ρ quiqφq ) = 0

(1)

∂xi

and the momentum equation

(

∂ ρ q uiqφ q ∂t

) + ∂(ρ q uiq u jqφq ) = −φ ∂xi

q

 ∂uiq ∂u jq 2 ∂u kq  ∂P ∂  + φ q µ qeff  + − δ ij  + Fiq + ρ qφ q Fg + Flift ,q + Fvm,q + F B −∇Ps  ∂x j  3 ∂xk ∂xi ∂xi  ∂xi   

(

)

(2)

where ρ is density, φ is volumetric fraction, u is velocity vector, t is time, P is pressure, µ eff is effective viscosity, δ ij is the Kronecker delta ( δ ij = 1 when i = j; δ ij = 0 when i ≠ j), and i, j, k are the three coordinate directions; note that q ≡ d denotes the dispersed phase, and q ≡ c denotes the continuous phase35. The gravitational force is denoted by Fg and the inter-phase drag force is denoted by Fiq. The lift force, Flift,q, and the virtual mass force, Fvm,q, were initially included but then subsequently omitted as they did not affect the results, a finding also previously reported by others36, 37. The term FB is the body force which includes the Coriolis and centrifugal forces induced by using the Multiple Frames Reference (MRF) approach, as discussed further below.

Inclusion of the particle-particle interaction force for the solid phase in equation (2) was attempted through the solid pressure term ( ∇Ps ). As inter-particle interactions increase with solid concentration, the inclusion of this term has been suggested particularly for highly concentrated suspensions (C > 0.2). This solid pressure term is, therefore, a function of the solid concentration38, thus

Ps = Ps (Cs)

(3)

and hence, ∇Ps = G (C s )∇C s

(4)

The function G (Cs) is referred to as the Elasticity Modulus, and is expressed as follows

G( C s ) = G0 e E ( Cs −Csm )

(5)

where G0 is the reference elasticity modulus, E is the compaction modulus, and Csm is the maximum packing parameter (maximum solid loading). There are no generally accepted values for these parameters; however, the values G0 = 1 Pa, E = 20 – 600, have been suggested by Bouillard et al.39. The maximum packing parameter Csm was determined by Thomas40 as 0.625 for spherical particles.



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The inclusion of particle-particle interactions through the solid pressure model which is available in ANSYSCFX, however, led to bad convergence due to linearisation problems in the software for all solid concentrations investigated, and it was not possible to achieve convergence with residual levels below 10-3. In fact, better predictions were obtained without inclusion of the solid pressure model which was consequently removed from the simulations.

The code uses the finite-volume method to discretise the Navier Stokes equations.

The so-called “High

Resolution Advection Scheme” was used to discretise the governing equations. The simulations were run in steady-state, so the transient terms were also omitted. Note that for simplicity, the equations in this section are written for one component only of the dispersed phase, but for a binary suspension they can be readily extended to include a second component.

The inter-phase drag force, Fiq, was modelled via the drag coefficient, CD, as:

Fiq =

3 φd ρ d C D u d −uc (u d −u c ) 4 dp

(6)

CD was estimated using the Gidaspow drag model for densely distributed solid particles38:

(

)

for φd ≤ 0.2

 24  C D = φc−1.65 max 1 + 0.15 Re'0.687 , 0.44   Re' 

and φd > 0.2

Fiq = 150

(1 − φc )2 µc φc d p2

+

7 (1 − φc )ρ c ud − uc 4 dp

where the particle Reynolds number Re′ is defined as Re' = φc

(7)

(8)

ρ c u d − uc d p , and µc is the viscosity of the µc

continuous phase. The model has been found to be particularly useful for higher solid loadings in pipes41, 42 and in stirred vessels26. The above system of differential equations (1) and (2) was closed by employing the well-known mixture k − ε turbulence model where the two phases are assumed to share the same k and ε27. The two-equation turbulence model consists of the differential transport equation (9) for the turbulent kinetic energy, k, and equation (10) for its dissipation rate, ε, as follows:

(

∂ ρ qφq k ∂t

(

∂ ρ qφq ε ∂t

) + ∂(ρ qφquiq k ) = ∂xi

) + ∂(ρ qφquiqε ) = ∂xi

∂ ∂xi

  µ  φq  µ q + qeff   σk  

 ∂k   ∂x  i

  + ρ qφq (Pk − ε )  

∂ ∂xi

  µ  φq  µ q + qeff   σε  

 ∂ε   ∂x  i

  + ρ qφq ε (C1Pk − C2ε )  k 


(9)

(10)

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where Pk is a generation term of turbulence; C1= 1.44 , C 2 = 1.92 , σ k = 1 , and σ ε = 1.314 .

The MFR method was used to take into account the interaction between the stationary baffles and the rotating impeller blades. Consequently, the computational domain was divided into two regions, as shown in Fig. 1: one domain containing all of the rotating elements (hub, blades and shaft), and the other containing the stationary parts (baffles, tank wall and base) separated by three interfaces, a vertical cylindrical interface at r = 94 mm, and two plane horizontal interfaces at z = 37 mm and 107 mm from the bottom of the tank. The interfaces were set sufficiently remote from the impeller to minimize the effects on the results.

A mesh independence study was conducted. There were no significant improvements in either the velocity profiles or the impeller torque beyond a mesh consisting of 1,065858 cells. However, as the computational cost was not prohibitive, an even finer mesh was used to achieve a good prediction of turbulent quantities; this aspect has been discussed by Deglon and Meyer43and Coroneo et al.44. Thus, the computational grid used consisted of ~ 1,504,928 non-uniformly distributed unstructured tetrahedral cells, as illustrated in Fig. 1. Inflated boundary layers were used on the bottom of the tank, walls, baffles, blades and shaft to cover the high velocity gradients in these regions of no slip.

Numerical convergence was deemed to have occurred when the sum of all

normalized residuals fell below 10-4 for all equations; however, most residuals were in fact usually below 10-6 by the end of the simulation.

4. Results and discussion

A detailed quantitative comparison between the CFD predictions and PEPT measurements was conducted at the positions depicted in Fig. 1. For the liquid phase and both components of the solid phase (i.e., 1 mm and 3 mm particles), the azimuthally-averaged distributions of the local velocity components (

u z , u r , uθ ) were

compared:

(i) axially along two radial positions: r = 0.53R, close to the tip of the impeller; and r = 0.90R, half a baffle away from the tank wall; and

(ii) radially along five axial positions: z = 0.016H, close to the base of the vessel; z = 0.08H, approximately halfway between the base and the lower edge of the impeller; z = 0.17H, just under the lower edge of the impeller; z = 0.33H, just above the upper edge of the impeller; and z = 0.83H, in the upper part of the vessel. For ease of comparison, all the velocity plots presented have been normalized by the impeller tip speed ( utip = πDN ).

The azimuthally-averaged spatial distributions of the local solid concentration were also resolved both by PEPT and CFD, and compared at all these radial and axial locations. The solid concentration profiles presented have been normalized by the overall mean volume solid concentration (C).



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4.1 Velocity distributions 4.1.1

Distribution of local velocity components

Azimuthally-averaged axial and radial distributions of the local velocity components ( u z ,

u r , uθ ) measured at

Njs experimentally by PEPT, and the equivalent distributions predicted by the CFD model are shown in Fig. 2 and Fig. 3, respectively, for the liquid phase and the two solid components (dp = 1mm; 3 mm) at a solid concentration of 5 wt%.

The mixture components exhibit similar velocity distributions where

u z is the

predominant velocity component, which is consistent with the performance of a down-pumping mixed flow impeller. The radial distributions exhibit sharp variations in the impeller region (z = 0.17H, 0.33H), but are fairly flat in the upper parts of the vessel (z = 0.83H). Near the tank base,

u r is predominant as each component

of the mixture is forced by the pumping action of the impeller to move towards the wall and is entrained in the single recirculation flow loop, thus, generated.

For the liquid phase and the 1 mm solids, all three velocity components are very well predicted by CFD throughout the vessel. For the 3 mm particles, however, there are some significant discrepancies between CFD and PEPT concerning the axial component

u z . It is interesting to note that the number density of the 1 mm

particles in the suspension is 27 times that of the 3 mm particles. It is perhaps plausible that as the number density of the solids increases, the assumption of a continuum inherent in the Eulerian-Eulerian model becomes somewhat less unrealistic, which might explain why the accuracy of prediction is better for the smaller 1 mm particles.

The numerically predicted axial velocity distributions at Njs for solid concentrations of 10 wt%, 20 wt% and 40 wt%, are compared to the experimental distributions in Figures 4, 5 and 6, respectively. The distributions are similar to those observed at 5 wt% solids, as discussed in the next section.

With increasing solid concentration, the accuracy of prediction for the liquid phase and the 1 mm solids remains high. It is interesting, however, that for the 3 mm particles the numerical prediction improves significantly with X, becoming very good at X = 20 wt% and X = 40 wt%. Again, as X increases, the solid phase tends to behave increasingly more like a continuum, thus, possibly improving the suitability of the Eulerian-Eulerian model. Furthermore, the Gidaspow drag model employed in this study was previously found to be particularly useful at higher solid loadings, which may have perhaps contributed the most to the improved accuracy of prediction at the higher solid concentrations26, 41, 42. On the other hand, however, as solid concentration increases, particleparticle collisions are expected to increase which tends to complicate the physics of the flow to an unknown extent and, hence, it is difficult at present to suggest a completely satisfactory explanation.

Numerical simulations were also conducted to assess the capability of CFD to predict the solid suspension at N >> Njs. Sample results are shown in Fig. 7 for 10 wt% solids at N = 1.8Njs. Apart from some local discrepancies affecting especially the particles around the impeller discharge stream, the flow field of each mixture component is well predicted. At such a high rotational speed, the intensity of turbulence is extremely high, and it is well known that the standard k- ε model used here tends to significantly underestimate the


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turbulence kinetic energy and dissipation rate, especially near the tip of the impeller45, 46; this may explain the disparities observed near the impeller discharge.

Similar results were obtained for the other solid

concentrations.

4.1.2 Reynolds number similarity

The flow structure at high Reynolds numbers presents similar characteristics, a phenomenon often referred to as Reynolds number similarity. In fully turbulent flow, viscosity effects vanish and dimensionless mean values such as velocities normalised by the impeller tip speed should be independent of Reynolds number. This similarity has been reported in Newtonian and non-Newtonian liquids but does not seem to have been tested in multiphase flows of the type studied here47, 48. Results plotted in Fig. 8 show the azimuthally-averaged radial distributions of the normalised velocity components of the three mixture components at the four solid concentrations investigated. The values of the impeller Reynolds number (Reimp = ND2/ν) are given in Table 1, and cover the range 1.5×105 – 2.4×105 for N = Njs. For the liquid phase, the Reynolds number similarity holds at all solid concentrations. For the solid components, some minor disparities are observed but the principle of similarity seems to hold too.

4.2 Spatial distribution of solid components 4.2.1

Method of evaluation of local phase concentration

In addition to location and velocity information, PEPT allows the particle occupancy distribution within the vessel to be determined. By dividing the vessel volume into a grid of nc cells, occupancy can be obtained by calculating the fraction of the total experimental time, t∞, spent by the tracer in each cell during the experiment. Such a definition establishes a mathematical identity between occupancy and probability of presence of the particle tracer, but undesirably makes occupancy highly dependent on the density of the grid so that as the number of cells increases occupancy tends to zero. If the cells are chosen to have equal volume, however, this problem is circumvented by using the ‘ergodic time’ defined as t E = t ∞ / nc , instead of the total experimental time t∞. The ergodic time represents the time that the tracer would spend in any cell if the flow were ergodic, an asymptotic status in which the flow tracer has exactly an equal probability of presence everywhere within the system. Thus, the local occupancy, Oj, can be defined as:

Oj =

∆t j

(11)

tE

where ∆tj is the time that the tracer of the j-th component (solid or liquid) of the mixture spends inside a given cell. For the binary systems examined in this work, the index j assumes values 1, 2 or 3 where j = 1, 2 denotes the solid components, and j = 3 denotes the liquid phase also designated by the symbol L.

On the basis of such a definition, it can be shown that, in a multiphase system, occupancy, Oj, of component j is directly related to its local phase volume concentration, cj, via the relationship



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Oj =

cj

(12)

Cj

where Cj is the mean phase volume concentration of component j in the vessel. Full details of the mathematical treatment can be found in our previous papers18, 19

Thus, using equation (12) the spatial distribution of the local concentration of each flow component was fully resolved in the binary suspensions investigated.

4.2.2 Comparison of experimental and CFD-predicted phase distributions

The ability to predict the spatial phase distribution is of the utmost importance in modelling suspensions in stirred vessels because it impacts directly on the prediction of other important phenomena such as heat and mass transfer.

The azimuthally-averaged axial and radial distributions of the two solid components at the ‘just-

suspended’ state corresponding to Njs are displayed in Figs. 9 and 10, respectively. Overall, the agreement with experiment is good, except near the base of the vessel and around the impeller. The solid distributions are largely overestimated near the base (z = 0.016H), and likewise they are largely underestimated just beneath and above the impeller (z = 0.17H, 0.33H); the errors are worse for the 3 mm particles. The worst CFD predictions occur at the lowest solid concentration of 5 wt%, but they improve significantly both axially and radially with an increase in solids loading, especially close to the tank base.

Particle lift-off from a bed of particles on the bottom of the vessel occurs as a result of the drag and lift forces exerted by the moving fluid. The flow near the base has been described as boundary layer flow which causes particles to be swept across the base of the vessel49. Once small fillets of particles have been formed, particle lift-off is usually seen to be caused by sudden turbulent bursts originating in the turbulent bulk flow above. Fluid-particle interactions are dependent on the relative size of eddies and the particles. Eddies relatively large compared to the particles will tend to entrain them and are responsible for their suspension. Consequently, different agitator designs and configurations generate different convective flows and, thus, achieve different levels of solids suspension at the same power input. The complex nature of the flow field in a stirred vessel is such that there is no fundamental theory to describe the process of particle suspension. The CFD model does not incorporate all of the complex physics of particle sedimentation, particle lift-off, and particle-particle interactions and, hence, the difficulty in achieving accurate prediction of the local solid concentration near the base of the vessel.

Considering the radial solid distributions at Njs shown in Fig. 10, the solids are nearly uniformly distributed except in the central region underneath the impeller where there is a substantial mound of accumulated solids. The axial solid distributions at Njs depicted in Fig. 9, also indicate a fairly uniform suspension in the upper part of the vessel, with the solid concentration reducing gradually upwards until it reaches zero close to the free surface. However, large concentration gradients are found in the lower part of the vessel (z < 0.2H). As


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discussed in more detail later, the homogeneity of the solid distribution at Njs improves when solids loading increases, and this feature is captured by CFD.

At impeller speeds above Njs, the uniformity of the suspension improves significantly, as shown in Fig. 11. The agreement between CFD predictions and PEPT measurements improves somewhat, but predictions remain rather poor near the base.

4.3 Uniformity of suspension 4.3.1 Binary suspensions

The uniformity of suspension can be estimated by the uniformity index, ξ, introduced in our recent works16,18, thus:

ξ=

1

σ +1 2

1

= 1 nc

nc

(13)

2

 ci − C    +1 C  i =1 

∑

where nc, as defined above, is the total number of cells in the PEPT measurement (or CFD computational) grid and i is the cell number. The uniformity of the suspension is at its minimum when ξ → 0 (condition achieved for a theoretically infinite variance, σ2, of the normalised local concentration); and the solids are uniformly distributed within the vessel volume when ξ = 1, i.e., the local solid concentration is equal to the average concentration everywhere in the vessel (σ2 = 0).

Results for the binary suspensions are plotted in Fig. 12. Generally, the 1 mm particles are better distributed than the 3 mm particles at all speeds except when N >> Njs. Initially ξ rises sharply as N increases above Njs, but a plateau is reached at high speeds and it becomes difficult to exceed ξ ~ 0.9, as it is hard to achieve perfect homogeneity below the impeller and near the free surface. This state of dispersion is reached at speeds significantly lower for the lighter 1 mm particles (~ 1.5Njs) than for the 3 mm particles (~ 2Njs). It can also be inferred from the data in Fig. 12 that, for a given particle size, ξ increases as a function of mean solid concentration when the suspensions are considered at the same hydrodynamic mixing regime, i.e., at their Njs speed or the same multiple of their respective Njs value. Note that the Njs values for the different solid

concentrations investigated are displayed in Table 1. This seems to suggest, perhaps counter-intuitively, that for a given state of suspension more concentrated suspensions will tend towards a homogeneous state simply by virtue of their increased solids loading (C → 1).

4.3.2 Comparison of binary and mono-disperse suspensions

We recently studied experimentally the suspension of the same particles considered here but in a mono-disperse mode18. The spatial distributions of these particles obtained from PEPT measurements in the binary and monodisperse modes are compared in Fig. 13. The distributions of the 1 mm particles do not show any significant difference at any solid concentration. The 3 mm particles, however, appear to be better distributed in the binary



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mixture than in the mono mixture at low to medium concentrations (5 wt% and 10 wt%), but this difference disappears at higher concentrations. Such an observation is at present difficult to explain. Compared to the binary suspension, however, significantly higher local solid concentrations are observed in the lower part of the vessel when the 3 mm particles are in a mono-disperse mode, as shown in Fig. 13. This maldistribution is confirmed by the uniformity index plots in Fig. 12 where the values of ξ for the binary suspension are significantly greater than for the mono suspension at the lower solid concentrations investigated.

4.4 Impeller pumping capacity

Recently, we showed how the usual definition of the flow number used to estimate the pumping effectiveness of impellers in single-phase systems can be extended to a two-phase problem18. For each component j of the mixture, Q(j) is calculated along the lower horizontal edge of the impeller blade corresponding to the impeller discharge plane, by integrating the axial velocity profile of the component weighted by its local volume concentration, i.e. by c1 and c2 for the solid components and by (1-c1-c2) for the liquid phase. Thus, the flow number is computed using:

for component 1 of the solid phase (dp = 1 mm):

Q ( S1) 1 = 3 ND ND 3

Fl (S1) =

(S1) 1 z dS

∫c u

(14)

PBT

for component 2 of the solid phase (dp =3 mm):

Fl (S 2 ) =

Q ( S 2) 1 = ND 3 ND 3

(S 2 )dS 2 z

∫c u

(15)

PBT

and for the liquid phase:

Fl (L ) =

Q (L ) 1 = 3 ND ND 3

∫ (1 − c

1

− c2 )u z(L )dS

(16)

PBT

The sum of Q(L), Q(S1) and Q(S2) represents the total volumetric discharge, Q, and introducing the total two-phase flow number, Fl, it follows that:

Fl =

Q = Fl (L ) + Fl (S1) + Fl ( S 2) ND 3

(17)

The computational CFD results are presented in Table 2 alongside the experimental PEPT values for ease of comparison.


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The solid flow numbers Fl (S1) and Fl (S 2) are only a small fraction of the liquid flow number Fl(L), but they increase proportionally to C. This is of course expected because in the calculation of the flow number, the flowrate of each phase (component) is weighted by its local volume concentration (see Eqs. (14), (15) and (16)). The experimentally measured values of Fl (S1) and Fl (S 2) are approximately equal under all conditions investigated, meaning that both solid components (1 mm and 3 mm) are non-preferentially pumped by the PBT. The magnitude of Fl (L ) reduces with increasing solid concentration, and so does the overall flow number Fl because of the higher proportion of heavy solids which are being pumped.

At low solid concentrations and low agitation speeds, Fl (S1) and Fl (S 2) are largely underpredicted by CFD probably because of their small values which are more prone to numerical error. However, the predictions improve greatly at higher mean solid concentrations or higher impeller speeds. The predictions of Fl (L ) are very good throughout, and because of its dominance, the overall flow number Fl is also well predicted with an average error on the order of ~ 10%, as shown in Table 2 in Supporting Information.

There are no published flow number data in solid-liquid suspensions, but in the absence of particles (X = 0) the single-phase flow number obtained from PEPT ( Fl ( L) = 0.87 ) agrees very well with published data for a 6blade pitched turbine (e.g., Fl ( L ) = 0.86 in Gabriele et al.50; and Fl ( L ) = 0.88 in Hockey et al.51.

4.5 Distributions of particle-fluid slip velocities

Information on particle-fluid slip velocity is of real value to processing applications involving the transfer of heat or mass, such as in chemical reactions or the sterilisation of particulate food mixtures. A crude assumption often used in practice takes the free terminal settling velocity of the particle as a representative measure of its mean slip velocity52. In 3-D the magnitude of the local time-average slip velocity vector of a solid component can be estimated using PEPT (or CFD) data, thus:

us =

(uθ

( L)

− uθ( S )

) + (u 2

( L) r

− u r( S )

) + (u 2

( L) z

− u z( S )

)

2

(18)

where, as in the above, the superscripts (L) and (S) refer to the liquid and solid phase, respectively.

Azimuthally-averaged distributions of the local time-average slip velocity, us, inferred from PEPT measurements at Njs, are presented in Fig. 14 at different heights in the vessel for all conditions investigated. The difference between the slip velocity profiles of the two particle sizes shown in Fig. 14 can be measured by the root mean square value, thus:



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∆ rms =

1 nc

 u s1 − u s2   utip i =1  nc

∑

   

2

(19)

where, as before, subscripts 1 and 2 refer to the 1 mm and 3 mm solids, respectively.

At Njs and above it, the slip velocity varies considerably from point to point throughout the vessel. For example at 20 wt% solids, the slip velocity for both particle sizes is within the range 0.01-0.42 ms-1. The minimum values tend to occur near the wall within a layer of fluid approximately one-baffle thick, whilst the maximum values tend to occur below the lower edge of the impeller, in the vicinity of the position (0.70R, 0.20H) corresponding approximately to the bottom of the circulation loop. These values are generally very different from the measured terminal settling velocities in stagnant fluid, i.e., u∞ = 0.18 ms-1 and 0.36 ms-1 for the 1 mm and 3 mm particles, respectively. Even the global volume-average values of us (0.14 ms-1; 0.18 ms-1) are considerably different from the u∞ values. At Njs and at the lower solid concentrations, the larger particles with more inertia display considerably more slip than the smaller ones. This effect, however, reduces considerably at higher solid concentrations, as shown in the ∆ rms plot in Fig. 15, because at high concentrations particle-particle interactions become important and

differences in velocity reduce through particle collisions. The effect also reduces at speeds above Njs; again, because inter-particle interactions also gain importance at higher speeds.

4.6 Verification of mass balance and mass continuity

The ‘pointwise’ measurements obtained from PEPT and CFD enabled the solids mass balance throughout the vessel to be accurately verified, i.e. the measured local values of the two solid volume concentrations balance the experimental parameter Cj = Vj / VT, so that:

1 nc

nc

∑c

(i ) j

≡ Cj

(20)

i =1

where Vj is the volume of component j in the vessel and VT is the total suspension volume. The identity symbol “≡” appears in Eq. (20) because the local concentration is obtained from Eq. 12 through the measurement of local values of the occupancy Oj whose average value is by definition identically equal to 1. Another important tool for checking the accuracy and reliability of flow data is mass continuity. The net mass flux through a volume bounded by a closed surface S should be zero, thus:

∑

S

u ( j ) ⋅ ∆S ≅ 0

(21)

Calculations can be made by considering a closed cylindrical surface S with the same vertical axis, base and diameter as the tank but of a shorter height. Because the vector u is zero over the external surface of the vessel or is parallel to it, the term u·∆S is zero everywhere except in the horizontal plane across the tank, so that S can be reduced to such a plane, Sh, and Eq. (21) is reduced to a zero average of axial velocities across the horizontal


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plane considered. Alternatively, a similar situation can be envisaged where S is a closed cylindrical envelope with the same vertical axis and height as the tank but with a smaller diameter; Eq. (21) then becomes a zero average of radial velocities across the lateral surface, Sr, of the considered cylindrical volume. These two types of surfaces Sh and Sr were numerically introduced in the vessel and the continuity test was applied to all the sets of data collected in this study. The results are summarised in Fig. 16, showing a very good verification of the mass continuity as the velocity average across the given surface in each case is close to zero, generally less than 0.03utip for PEPT and even less for CFD.

5. Conclusions

The three-component flow fields in mechanically agitated dense binary suspensions have been successfully modelled by CFD. The local velocity components of all the mixture components agree very well with PEPT measurements throughout the stirred vessel at Njs and at speeds above it.

This has also led to accurate

predictions of the flow numbers. Furthermore, the Reynolds number similarity was found to hold for all components of the two-phase flow under all conditions investigated.

The spatial distribution of the solid phase is well predicted except close to the base of the vessel and around the impeller where it is largely overestimated. However, the predictions improve significantly at higher solid concentrations, as the performance of the Gidaspow model improves at higher concentrations. However, some improvement might also be due to the Eulerian assumption of a continuum becoming slightly more realistic. Generally, the smaller particles are better distributed than the larger ones except when N >> Njs. The uniformity of suspension improves sharply as N increases above Njs, but a plateau is reached at high speeds and it becomes difficult to reach 100% homogeneity, as it is hard to distribute particles below the impeller and near the free surface. In addition, uniformity increases as a function of mean solid concentration when suspensions are considered at the same hydrodynamic mixing regime, i.e. at their Njs speed or the same multiple of their respective Njs value. The local slip velocity varies widely within the vessel and is generally very different from the particle terminal settling velocity.

At Njs and at lower solid concentrations, the larger particles with more inertia display

considerably more inter-phase slip than the smaller ones. This effect, however, reduces considerably with an increase in solid concentration and impeller speed as particle-particle interactions become important. Acknowledgement

Li Liu’s PhD research was funded by an Elite scholarship from the University of Birmingham. We are grateful to Prof. D.J. Parker, Dr X. Fang and J.F. Gargiuli of the Positron Imaging Centre at The University of Birmingham for their assistance with the PEPT experiments. Supporting Information Available

Table 2 contains a large amount of data on flow numbers obtained from PEPT measurements and CFD predictions for all conditions investigated, and has been placed in SUPPORTING INFORMATION in accord with the publication policy of Industrial & Engineering Chemistry Research. This information is available free of charge via the Internet at http://pubs.acs.org/.



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Nomenclature

c

local volume concentration of solids

-

C

mean volume concentration of solids

-

dp

particle diameter

m

D

impeller diameter

m

Fl

multiphase-phase flow number (Q/ND3)

-

Fl(L)

liquid flow number (Q(L)/ND3)

-

Fl(S1)

flow number (Q(S1)/ND3) for 1mm solids

-

Fl(S2)

flow number (Q(S2)/ND3) for 3 mm solids

-

H

height of suspension

m

N

impeller rotational speed

s-1

nc

number of PEPT or CFD grid cells

-

Njs

minimum speed for particle suspension

s-1

Q

impeller pumping rate

m3 s-1

r

radial distance

m

R

vessel radius

m

Reimp

impeller Reynolds number (ND2/ν)

-

S

surface area

m2

S1, S2

indices denoting 1 mm and 3 mm particles, respectively

Sh Sr

horizontal planar and vertical cylindrical surface areas

m2

T

vessel diameter

m

us

slip velocity

m s-1

u∞

terminal settling velocity

m s-1

utip

impeller tip speed

m s-1

ur uz uθ

cylindrical velocity components

m s-1

X

mean mass concentration of solids

-

z

vertical distance

m

Greek letters


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ν

kinematic liquid viscosity

m2 s-1

ξ

uniformity index

-

σ

standard deviation of normalised c

-

θ

azimuthal coordinate

rad

∆ rms

root mean square

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(12) Villermaux, J. Trajectory Length Distribution (TLD), a novel concept to characterize mixing in flow systems. Chem. Eng. Sci. 1996, 51, 1939-1946. (13) Wittmer, S.; Falk, L.; Pitiot, P.; Vivier, H. Characterization of stirred vessel hydrodynamics by three dimensional trajectography. Can. J. Chem. Eng. 1998, 76, 600-610. (14) Barigou, M. Particle tracking in opaque mixing systems: An overview of the capabilities of PET and PEPT. Chem. Eng. Res. Des. 2004, 82, 1258-1267.



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Table 1. Range of PEPT measurements and CFD simulations.

X (wt%)

C (vol%) N = Njs

N (rpm) (Reimp × 10-5) N > Njs

Binary 5 10

2.4 4.9

380 (1.51) 450 (1.79)

570 (2.27) 675 (2.68)

760 (3.02) 800 (3.18)

20 40

10.4 23.6

510 (2.03) 610 (2.43)

765 (3.04) 765 (3.04)

-

5.2

2.5

360 (1.43)

450 (1.79)

540 (2.15)

720 (2.86)

Mono dp = 1 mm dp = 3 mm

20

1.4

490 (1.91)

613 (2.44)

-

-

5.2

2.5

360 (1.43)

450 (1.79)

540 (2.15)

720 (2.86)

10.6 20

5.2 10.4

410 (1.61) 490 (1.91)

520 (2.07) 613 (2.44)

-

720 (2.86) 735 (2.92)


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Figure 1. Computational grid section in the 45° plane between two baffles.



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liquid phase

dp = 1 mm

dp = 3 mm

Figure 2. Azimuthally-averaged axial distributions of the normalised velocity components of the liquid and solid components at Njs: X = 5 wt%; Njs = 380 rpm.


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liquid phase

dp = 1 mm

dp = 3 mm

Figure 3. Azimuthally-averaged radial distributions of the normalised velocity components of the liquid and solid components at Njs: X = 5 wt%; Njs = 380 rpm.



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liquid phase

dp = 1 mm

dp = 3 mm



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liquid phase

dp = 1 mm

dp = 3 mm




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liquid phase

dp = 1 mm

dp = 3 mm



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liquid phase

dp = 1 mm

dp = 3 mm

Figure 7. Azimuthally-averaged axial distributions of the normalised velocity components of the liquid and solid components at N = 1.8Njs: X = 10 wt%; N = 800 rpm.



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liquid phase

dp = 1 mm

dp = 3 mm

Figure 8. Demonstration of Reynolds number similarity − Azimuthally-averaged radial distributions of the normalised velocity components of the three mixture components based on PEPT measurements: — X = 5 wt%; --- X = 10 wt%; -·-·-· X = 20 wt%; ··· X = 40 wt%; N = Njs; Reimp = 1.5×105 – 2.4×105 (see Table 1).


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Figure 9. Azimuthally-averaged axial distributions of normalised solid volume concentration in suspension at Njs.



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Figure 10. Azimuthally-averaged radial distributions of normalised solid volume concentration in suspension at Njs.


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Figure 11. Azimuthally-averaged axial distributions of normalised solid volume concentration in suspension at N > Njs.



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Figure 12. Variation of suspension uniformity index as a function of impeller speed for different particle sizes in binary- and mono-disperse suspensions based on PEPT measurements.


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Figure 13. Comparison of azimuthally-averaged axial and radial distributions of normalised solid volume concentration based on PEPT measurements in binary- and mono-disperse suspensions at Njs.



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Figure 14. Azimuthally-averaged radial distributions of normalised time-average slip velocity based on PEPT measurements at different solid concentrations and agitation speeds.


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Figure 15. Measure of global difference in slip velocity between the two solid components in the vessel as a function of mean solid concentration based on PEPT measurements.



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.06

S r/u t i p (-)

.05

S

r

.04 .03 .02 .01 0 .05

S z /u ti p (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

S z z

.04

PEPT 1mm PEPT 3mm CFD 1mm CFD 3 mm

.03 .02 .01 0 0

5

10 15 20 25 30 35 40 Mean solid mass concentration, X (wt%)

45

Figure 16. Normalised radial and axial velocities averaged on surfaces Sr (of diameter 0.5T) and Sh (0.3H off the base), respectively, showing excellent verification of mass continuity by both PEPT and CFD.


Experimentally Validated Computational Fluid Dynamics Simulations

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