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Predicting flow and residence time in alumina digestion vessels
Chemical Engineering Science xxx (2016) xxx–xxx
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Predicting flow and residence time in alumina digestion vessels David S. Whyte a, Gary J. Brown a, David F. Fletcher b,⇑ a b
Alcoa of Australia Limited, Kwinana, Western Australia 6966, Australia School of Chemical and Biomolecular Engineering, University of Sydney, Sydney 2006, New South Wales, Australia
a r t i c l e
i n f o
Article history: Received 28 April 2016 Received in revised form 28 November 2016 Accepted 15 December 2016 Available online xxxx Keywords: Residence time distribution Turbulence modelling Scalar transport Particle tracking Digestion vessel
a b s t r a c t Understanding the flow and residence time behaviour in reaction vessels is of crucial importance to improve product yield or the extraction of mineral species from ores. Digestion vessels, used in the Bayer process, are an example of the latter. Transient Computational Fluid Dynamics (CFD) models were used to investigate the impacts of a variety of turbulence models, along with the effects of mesh refinement on the performance of two different vessel geometries. The main focus of the research work was the efficacy of a variety of turbulence models: two-equation (k-e, SST), Reynolds Stress (SSG RSM) and Scale Adaptive Simulation (SAS-SST). The predicted velocity profiles are compared with high quality Laser Doppler Velocimetry (LDV) data. Residence times are calculated via a convected scalar (tracer) and Lagrangian tracking of particles. The calculated residence times are compared with salt tracer estimates from experiments. Generally, there was good agreement between experimental data and results obtained from tracing a passive scalar or tracking neutrally-buoyant particles, with the three results being closer when the SAS-SST model was used compared with the k-e model. In the case of the strongly swirling flow caused by the tangential entry, the tracer was in closer agreement with the experimental data but the Lagrangian tracking results improved considerably with the use of the SAS-SST model. The importance of small-scale turbulence structures in determining the flow profile and residence times of the vessels is highlighted in this study. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Digestion plays a fundamental part in the Bayer process for alumina production, with the objective of this processing stage being to extract the available alumina from the bauxite by dissolving it in a hot caustic soda solution (Gerard and Stroup, 1963). Digester vessels typically have a height to diameter ratio of close to 3:1. Some vessels use mechanical agitation, but those examined in this study have a single inlet stream at the top of the vessel and a single, centrally-located, outlet at the bottom of the vessel, as in the schematic shown in Fig. 1. The flow behaviour in these vessels is therefore dominated by the design and positioning of the inlet nozzle. An understanding of the flow behaviour in these vessels is important, as adequate residence time and mixing are critical to achieving sufficient extraction of alumina from bauxite particles of varying diameter. Past CFD modelling of digestion vessels within the organisation used the standard k-e model in steady-state simulations (Brown and Fletcher, 2005). While the simulations of top-fed digesters
⇑ Corresponding author. E-mail address: [email protected] (D.F. Fletcher).
were considered to give acceptable agreement with experimental data, the same could not be said for the simulations for tangentially-fed vessels. At the time of that work, scale-resolving turbulence models were at their development stage and computing resources were vastly more expensive than they are now. Woloshyn et al. (2006) used CFD to investigate a range of digester feed configurations. Their study considered full-scale digesters using the k-e model and the Reynolds Stress Model (RSM) for turbulence. Residence Time Distributions (RTD) were determined for individual particles of various sizes. However, no validation of the simulations was conducted. Brown et al. (2014) studied the flow in precipitation vessels using CFD. These vessels have flow rates and velocity scales which are comparable with those for the digesters considered here. The work evaluated a variety of turbulence models and validated the numerical predictions against experimental measurements. The authors found that using a turbulence model that was able to capture the unsteady, large-scale turbulence structures (SAS-SST) gave results that matched the experimental data very well. Thus, given improved modelling techniques and computational resources, it is now possible to examine the performance of more sophisticated turbulence modelling approaches. Below, we give a brief description of the experimental study used for validation
http://dx.doi.org/10.1016/j.ces.2016.12.036 0009-2509/Ó 2016 Elsevier Ltd. All rights reserved.
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the vessel walls. These readings were taken at several vertical elevations – those taken at 415 mm and 1465 mm above the cone are reported here as they are representative of the behaviour observed at all locations. The measurement device was held stationary at each point along the radius for approximately two minutes, with 2000 readings taken in each period. Residence Time Distributions (RTD) were determined using a salt tracer. A pulse of salt solution was injected into the inlet pipe once the flow had settled down and the concentration of salt was measured at the outlet over time.
3. Model description 3.1. Turbulence modelling
Fig. 1. Schematic of modelled domain. Both the top and tangential feed lines can be seen. However, only one feed line is active at any time. Note that when the top feed inlet is used there is a deflector plate suspended below the feed nozzle which stops flow from short-circuiting directly to the outlet.
purposes and then compare simulated flow-fields with the experimental data, first for the top-fed vessel and then for the vessel with a tangential inlet. Then the residence time distributions are presented together so that comparison based on the different methods, geometries and turbulence models can be made. The remainder of the paper is presented as follows: Section 2 describes the experimental data used for model validation; Section 3 describes the CFD model used in this study; Section 4 presents the simulation results and comparison with the experimental data and Section 5 contains the conclusions. 2. Experimental study Alcoa commissioned CSIRO Mineral Resources Flagship to generate the experimental data used in this study. A laboratory-scale (1.5 m tall and 0.4 m diameter) model of a full-scale digester was used in the experiments. Water at ambient temperature was used as the working fluid. The data reported here are for an inlet flow rate of 127 L min1. Superficial residence time is a crude estimate of how long a fluid particle will be present within a vessel, and is defined as the volume of the domain divided by the volumetric flow rate through it. In the cases considered here, the superficial residence time, To, is 88 s. The results from the experiments and simulations will be compared using a normalised time (measured time, t, divided by To). Laser Doppler Velocimetry (LDV) was used to measure the velocity components along a line from the central axis to just off
Different turbulence modelling approaches were tested in an effort to determine the most accurate method for calculating the flow field using the most efficient computational approach. If steady simulations can be performed using a RANS turbulence model the computational requirements, in terms of mesh size and computational run time, are vastly smaller than if a transient scale-resolving approach is needed. This knowledge clearly has a huge impact on the computational resources required to perform such analyses. We note here that although a RANS model may give a steady solution this does not mean the real flow is steady. Rather it says that the unsteadiness can be removed by the Reynoldsaveraging process and represented by a simple eddy diffusion hypothesis and there is no underlying transient flow. As flow separation is not a governing feature of the flows, wall functions are considered to be sufficient to predict flow in the wall region. Previous work (Brown and Fletcher, 2005) had shown that the standard k-e model (Launder and Spalding, 1974) could predict the flow in the top-fed digester but was unable to predict the behaviour in the tangentially-fed vessel. The k-e model yields a steadystate solution, where the experiments showed large-scale unsteadiness of the flow. The two-equation SST (Menter, 1994) and Reynolds Stress Model (SSG RSM) Speziale et al., 1991 were also tested in this work. Finally, given the success in predicting unsteady flow in laboratory-scale precipitators (Brown et al., 2014), the Scale Adaptive Simulation (SAS-SST) model of Menter and Egorov, 2010 was considered. When applying the SAS-SST model if the mesh is sufficiently fine the large-scale turbulence structures can be captured if the flow is globally unsteady (Menter and Egorov, 2010). If the flow is not globally unstable, i.e. there is nothing to trigger the generation of large-scale turbulence structures, the simulation model may remain in RANS mode and no structure is resolved. In order to avoid this, a zonal LES region was introduced in the inlet feed pipe in order to convert modelled turbulence into resolved turbulence which ensures that the SAS-SST model will behave in scale-resolving mode and continue to resolve the large-scale turbulence structures in the downstream region. This initiation method uses a harmonic turbulence generator to convert modelled turbulence kinetic energy into random fluctuations which trigger the scale-resolving mode (Menter et al., 2009). In addition the curvature correction term was enabled in the SAS-SST simulations to avoid artificial decay of vortical structures (Smirnov and Menter, 2008).
3.2. Computational domain and mesh The meshes used in this study were informed by the experience in modelling alumina precipitators (Brown et al., 2014). The tangentially-fed digester is inherently 3-dimensional, so no attempt was made to reduce the geometrical complexity.
Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036
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Experience has shown that for systems such as those discussed here, a good starting point for mesh inflation is 10 layers of inflation on all walls with a 1 mm first layer height and a 1.2 expansion ratio. In subsequent meshes, these controls were changed to 20 layers, with a first layer height of 0.5 mm. This gave y+ values typically in the range of 20–30. General guidelines for an appropriate mesh density for the SAS-SST model in this sort of geometry are not available, with the majority of published cases being related to bluff body or aerodynamic flows. Therefore, the edge length of the mesh in the main vessel was systematically reduced and the results assessed. The meshes used are shown in Fig. 2. The inlet region and lower cone were meshed using tetrahedrons, while in the barrel of the digester the mesh was swept in the axial direction. 3.3. Numerical considerations The commercial CFD code ANSYS CFX 15.0 was used for this study. The code uses a vertex-based control volume approach, the Rhie-Chow procedure to couple pressure and velocity fields, a coupled solver to solve for the velocities and pressure simultaneously and an algebraic multi-grid (AMG) solver to resolve the resulting algebraic equations. Spatial and temporal derivatives were calculated using bounded second order differencing schemes in the non-scale resolving simulations and bounded central differencing schemes in the scale-resolving simulations. All simulations were carried out as transients and run for 300 s (3.4 superficial residence times). Simulations that evolved to a
3
steady-state were terminated prior to reaching the target time. The transient simulations were run with adaptive time-stepping with the rms Courant number set to be below one as required for scale-resolving simulations. A few cells in the exit region had Courant number values of around 50 but otherwise most values satisfied the pre-set criterion. The SAS-SST simulation typically required time steps of 1 ms, and for the two-equation models (k-e and SST) the solver reached the upper time step limit specified of 0.1 s. 3.4. Residence time studies Two different and complementary approaches were used to investigate the residence time distributions in the two vessels. Firstly, an Eulerian approach was applied in which a pulse of tracer was injected at the inlet after 300 s (3.4 superficial residence times) when the flow-field was well developed and the scalar flux at the outlet was recorded over time (Brown et al., 2001). In this approach the laminar diffusivity of the tracer was set to zero and the turbulent diffusivity was calculated as the eddy viscosity divided by a turbulent Schmidt number, which was set to the default value of 0.9. It is noteworthy that in this approach in the k-e simulations the eddy viscosity represents the entire effect of turbulence, whereas in the SAS-SST simulations it is only the effect of the unresolved eddies that is treated in this manner, with the effect of the large-scale, resolved eddies being represented by the usual convective term in the equations. The second approach used was Lagrangian particle tracking. In this case representative particles were injected at the inlet and their path through the vessel was calculated by solving Eq. (1)
dx ¼ v; dt
mp
dv ¼ F D þ V p ðqp qf Þg dt
ð1Þ
where x is the particle location, t is time, v is the particle velocity, mp is the particle mass, F D represents drag force, V p is the particle volume, qp ; qf are the particle and fluid densities, and g is the acceleration due to gravity. The drag coefficient was determined from the Schiller-Naumann correlation. When the k-e turbulence model was employed the random walk turbulence model was applied to the particles (Gosman and Ioannides, 1981) so that the velocity used in Eq. (1) is then the sum of the mean and a fluctuating component. In this model a particle is assumed to always be inside a single turbulent eddy. The turbulent velocity, uf , eddy lengthscale, le , and eddy lifetime, se , are calculated as follows
uf ¼ rð2k=3Þ 3=2
le ¼ C 3=4 l k
0:5
=
se ¼ le =k3=2 ¼ 0:2
Fig. 2. Representations of the computational meshes used in the study for the topfed digester. (a) This mesh has 610,000 nodes, with a first inflation layer height of 1.0 mm; (b) this mesh has 2,700,000 nodes, with a first inflation height 0.5 mm. The yellow lines represent the 415 mm (lower) and 1465 mm (upper) elevations used for comparison with experimental data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ð2Þ ð3Þ k
e
ð4Þ
where r is a Gaussian distributed random variable with a zero mean and standard deviation of unity. C l is a constant in the turbulence model taking a value of 0.09 and is included to relate the characteristic length-scale with the eddy dissipation length-scale. A separate fluctuating velocity is computed for each velocity component and therefore the turbulent dispersion is non-isotropic, despite only isotropic information on the turbulence being available. A given fluctuation is used until either the eddy lifetime is exceeded or the particle displacement exceeds the eddy length-scale. It should however be noted that there is no universal value for the model constants and different turbulent flows require different fits to obtain agreement, with the potential variation being significant (Matida et al., 2000).
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Fig. 3. Effect of the number of particles used to determine the residence time distribution for the top feed inlet case simulated using the k-e turbulence model.
When the SAS-SST model was used no additional turbulence dispersion was used. In this case the effect of the resolved turbulence structures is taken into account directly and we show later that the contribution of the unresolved, dissipative turbulence is negligible. 10,000 particles were released from the inlet once the transient simulation had run for 3.4 superficial residence times. A simulation was repeated for the top feed inlet case in which 50,000 instead of 10,000 particles were used to determine the impact of this parameter. The results, shown in Fig. 3, confirm that 10,000 particles are sufficient to capture the residence time curve in a manner independent of the number of particles used. The time at which each particle reached the exit was recorded and the data were binned, with a bin size of 1 s, so that a particle exit time distribution could be plotted. In total four different Lagrangian particle groups were considered for each case. Particles of 10 and 100 lm were used. Two groups used neutrally buoyant particles (having the same density as that of the water, 997 kg m3) and two having particles with a density typical of bauxite (2493 kg m3). The neutrally buoyant particles follow the flow and behave like fluid path-lines. There is no impact of particle size in this case so the two simulations (10
Fig. 4. Transient monitor point data for the top feed simulations: (a) results for the k-e simulation and (b) for the SAS-SST simulation. These show the very different behaviour of the two models - the k-e model results are essentially steady, while the SAS-SST model results show persistent fluctuations.
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and 100 lm) provide two different realisations for the same flow. When the bauxite particles are tracked they have the potential to deviate from the fluid path-lines and will show the impact of a finite Stokes number on the particle trajectories and residence time. Each case was run for an additional three superficial residence times and the data were normalised such that area under a mass flux versus time curve corresponded to the mass of tracer that exited the vessel.
of the experiments (or offered additional insight) than the k-e or SAS-SST models. It is interesting to note that for both inlet configurations the k-e models resulted in effectively steady velocity fields (fluctuations 0.1% of the mean flow) on both the fine and coarse meshes. Statistics of the transient flow were started at the same time as the scalars/particles were introduced and data were captured over the entire 300 s period. In both the SAS-SST predictions and LDV measurements, the standard deviations of the velocity components have been used to indicate the magnitude of flow fluctuations. An effective approach to detecting vortex structures is to plot the sec-
4. Results
ond invariant of the velocity gradient tensor (or Q-criterion) Hunt, 1987:
In this section we will focus on the comparison of the k-e and SAS-SST turbulence models. While the SST and SSG RSM models were considered in the study, neither provided better predictions
Q¼
1 ðjXj2 jSj2 Þ 2
ð5Þ
Fig. 5. Comparison of the predicted vertical components of velocity for the k-e model with coarse and fine meshes on a plane at the mid-height of the top feed model. The results are relatively insensitive to mesh refinement.
Fig. 6. Contours of the vertical component of fluid velocity (negative values indicate down-flow): (a) k-e model using the coarse mesh; (b) SAS-SST model (transient average) using the fine mesh. In both cases, the flow features show qualitative similarities. There is strong down-flow at the wall below the deflector plate. Directly under the deflector plate is a region of strong recirculation – there is a slight difference between the two predictions. However, beneath this region there is fairly uniform down-flow.
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where X and S are the vorticity and shear strain rate tensors, respectively. Vortical structures can be identified by iso-surfaces with Q > 0. Fig. 4 shows the velocity magnitude at four monitor points distributed across the vessel at approximately mid-height for the top feed case. It is evident that after about 200 s the results for the k-e simulation have reached a steady-state, whereas those from the SAS-SST model continue to oscillate. Fig. 5 shows a profile of vertical velocity on the same plane for the k-e model with both the fine and coarse meshes shown in Fig. 2. It is evident that the k-e results are relatively insensitive to the mesh and hence only results for the coarse mesh are presented in the remainder of the paper for this model. 4.1. Flow field top-fed digester Fig. 6 shows a comparison of the vertical velocity component predictions of the k-e and SAS-SST turbulence models. The results are very similar overall with the key difference being the extent
of the recirculation zone under the deflector plate, with that in the SAS-SST simulation being slightly shorter. Fig. 7 shows that both turbulence models predict very similar vertical velocity profiles but show some disagreement with experimental values near the walls at a height of 1465 mm, and uniform down-flow in the lower half of the vessel. The velocity at the walls is hard to measure in the high gradient region and we attribute this disagreement to measurement error. An additional point of agreement between the measurement from LDV and SAS-SST predictions, is the magnitude of the fluctuations seen in the velocity components (Fig. 7). Fluctuations for the k-e model predictions were obtained from qffiffiffiffiffiffi 2 k by assuming isotropy of the fluctuating velocity components. 3 The values obtained are somewhat smaller than those from the SAS-SST predictions, as would be expected, especially in the region near the wall. Overall, we can conclude that in the case of the top feed digester both turbulence modelling approaches perform well in comparison with the experimental data.
Fig. 7. Comparison of the predicted vertical components of velocity with LDV measurements (a) taken along the radius of the vessel at 1465 mm; (b) at 415 mm. Both models predict the same general behaviour. In (a) there is upflow on the central axis (beneath the deflector plate), with faster down-flow at the walls; (b) shows a relatively slow, uniform down-flow in the lower part of the vessel. It should also be noted that the rms of the fluctuations of the SAS-SST predictions (red error bars) and LDV measurements (black error bars) are similar. The rms for the k-e predictions (green error bars) is smaller than that from the SAS-SST predictions, especially in the near wall region. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Isosurface where the Q-criterion takes a value of 10 s2: (a) results from the k-e model using the coarse mesh coloured by velocity; (b) results from the SAS-SST model using the fine mesh coloured by transient average velocity. In (a) it is clear that this model captures only the quasi-steady vortices located in the upper recirculation caused by the deflector plate; (b) highlights the significant turbulence structures resolved by the SAS-SST model in the upper region of the chamber. The deflector plate generates turbulence in the flow that then dissipates in the lower region of the digester.
Fig. 9. Contours of the vertical component of fluid velocity: (a) k-e model results using the coarse mesh; (b) SAS-SST model using the fine mesh showing the transient average vertical velocity. While there are some qualitative similarities between the two cases, the differences are quite clear – the two-equation model predicts that the down-flow region of the central core extends the full height of the vessel, and the flow is more erratic.
Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036
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Fig. 10. Comparison of predicted vertical components of velocity with LDV measurements: (a) taken along the radius of the vessel at 1465 mm; (b) at 415 mm. In (a) it is evident that all the models are consistent, and both deviate from the experimental data at the wall. Part (b) highlights that the key difference between the models is their ability to predict the upflow on the centre-line in the lower region of the digester – only the SAS-SST model is able to capture this behaviour. The meaning of the error bars is as defined in Fig. 7.
As expected, there is a significant difference in the extent and form of the flow structure predicted by the two models. Qcriterion plots, given in Fig. 8, show that the flow predicted by the k-e model is dominated by the recirculation produced by the deflector plate, while the SAS-SST computed flow has a significant amount of structure in the upper section of the vessel, with decay of this turbulence in the lower half of the vessel. 4.2. Flow field tangentially-fed digester It is the model results for the tangentially-fed digester that are most interesting here, as past efforts have shown that twoequation turbulence models are insufficient to accurately predict the flow field or fluid residence times (Brown and Fletcher, 2005). In this case we are interested in both the vertical and tangential components of the velocity. Fig. 9 shows the predicted vertical velocity fields. While both ke and SAS-SST models show global similarity, they have some significant local differences. In particular, the down-flow seen in the
central axis is more intense in the k-e model results. In Fig. 10 we can see that neither of the models predicts the magnitude of the central up-flow shown by the LDV data although the SAS-SST performs much better in this regard. It is also evident the k-e model significantly under-predicts the turbulence fluctuations and the SAS-SST model does much better in this respect. Figs. 11 and 12 show that both models predict the swirling flow profile, with the SAS-SST model being clearly superior at 415 mm from the bottom. Fig. 13 shows an iso-surface of the Q-criterion, coloured by the velocity in the results from the k-e model and by the transientaveraged velocity in the case of the SAS-SST model. Both plots show a similar global structure, but it is evident that vortical structures exist much further into the vessel in the SAS-SST case.
4.3. Residence time top-fed digester As noted above, the k-e model gave a steady flow field so both the tracer calculation and the Lagrangian tracking were done with
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Fig. 11. Contours of the tangential component of fluid velocity for (a) results from the k-e model using the coarse mesh and (b) for the SAS-SST model (transient average) on the fine mesh. There is very little difference between the predictions of the two models.
a frozen flow field, whereas the flow field was evolved with the tracer and particles in the SAS-SST case. Fig. 14 compares the computed results with the experimental data and it is evident that both turbulence models predict the arrival of the tracer and the neutrally buoyant particles at the outlet at a similar time to that observed experimentally. Agreement between the k-e model results and experimental data is generally good but the tracer arrives a little too early, whereas the SAS-SST model results show the tracer arriving slightly later than in the experiment. However, we should note here that the tracer results are for a single experimental trial, as were the SAS-SST results. Repeats of the experimental test and different realisations of the transient SAS-SST simulations would give slightly different results, so this difference in agreement may not be significant. In both cases the neutrally buoyant particles and the 10 lm bauxite particles behave very similarly. In the k-e case there are some significant differences between the Lagrangian method and the tracer. In the SAS-SST case, where the turbulent structures are better resolved, the results from the neutrally buoyant and 10 lm bauxite particles closely follow the fluid tracer result. This suggests that the turbulent dispersion behaviour is different between the two cases and confirms that the Lagrangian approach can give results equivalent to an Eulerian tracer provided sufficient turbulent structure is resolved in the flow. The 100 lm alumina particles show significant short-circuiting in the k-e simulation with the peak flux at around half the superficial residence time. There is some short-circuiting in the SAS-SST simulation results but this is much reduced compared with that seen in the k-e simulation. Our hypothesis is that this change of behaviour is due to increased dispersion of these particles within the vessel due to the turbulent structures being resolved in the SAS-SST simulation. Such differences are potentially very important in predicting the operating performance of a full-scale vessel.
4.4. Residence time tangentially-fed digester Data for the tangentially-fed digester were obtained in a similar manner to those for the top-fed digester and are shown in Fig. 15. The results from the k-e model show that the tracer, neutrally buoyant and 10 lm bauxite particles arrive later than the experimental measurements, whilst there is much closer agreement in the SAS-SST simulations. Both simulations show very similar shaped distributions. These particles also show much greater bypass than the tracer results using both turbulence models. In the tangentially-fed case there is again considerable bypass by the 100 lm bauxite particles, with this effect again more pronounced in the k-e model results with a very high peak at a dimensionless time of 0.4 compared with those from the SAS-SST simulation which has a much broader peak. These differences are again attributed to increased dispersion of these particles within the vessel due to the turbulent structures being resolved in the SAS-SST simulation. The question arises as to whether the discrepancies seen above could be attributed to the impact of the turbulent dispersion caused by the unresolved turbulence. To address this, an additional simulation was run in which the random walk model was applied to 10 lm neutrally buoyant particles in the SAS-SST simulation for the tangential feed case. Fig. 16 shows that whilst there are differences in the results these are very minor and do not bring the results in line with the experimental data.
4.5. Discussion The results presented above show that whilst there is reasonable agreement between the calculated and experimental data there are some differences, especially in the tangentially-fed case. In order to understand whether these differences were a result of
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Fig. 12. Comparison of the predicted tangential velocity with LDV measurements: (a) taken along the radius of the vessel at 1465 mm and (b) at 415 mm. Both plots show that the SAS-SST model provides the most accurate prediction of the velocity profile. The meaning of the error bars is as defined in Fig. 7.
the physical models used or the code implementation of them a number of supplementary simulations were performed. Firstly, an idealised U-bend test case was developed and used to explore simulations with no recirculation zones present when using the steady k-e modelling approach in conjunction with an Eulerian tracer and neutrally buoyant Lagrangian particles using the random walk model. These showed almost identical agreement between the tracer and particle data. We then introduced internal baffles to represent those present in the top feed geometry to create recirculation zones and again results of the two methods were essentially identical. Finally, we introduced recirculation zones adjacent to the wall with the same result. These convinced us that the two methods can give identical results in simple flows. In the top-fed digester the results with the SAS-SST model (see Fig. 14(b)) show that the two methods can also give close agreement in a complex geometry, suggesting that in this case the turbulent structures responsible for distributing the particles are well captured. We have already shown above that adding the turbulent dispersion model to the particle transport equation in the SAS-SST simulations has a small effect in the tangentially-fed case. However, looking at the Q-criterion plots, especially Fig. 13, we
hypothesize that if an even finer mesh than that used here were used we would get more turbulent dispersion of the particles and even better agreement with the tracer results. 5. Conclusions Understanding the flow of the fluid and particles in an alumina digester is central to predicting the device performance. An accurate model for top-fed digesters has been developed. Within the limit of the cases studied, the simulated flow in the top-fed digesters is independent of the turbulence model used. The agreement with experimental data between RTD predictions of the steadystate (k-e model) and the transient SAS-SST models suggests that in this feed orientation, the fine detail of turbulence is not important in predicting the performance of the vessel. However, when the residence time of 100 lm bauxite particles is considered it is clear that the k-e model predicts much greater bypass than the SAS-SST model. To predict the flow field and residence time of the tangentiallyfed digester, it is important that the model resolves the largerscale, unsteady turbulence structures, as can be achieved with
Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036
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Fig. 13. Isosurface where the Q-criterion takes a value of 10 s2 coloured by velocity. Part (a) shows results from the k-e model using the coarse mesh and part (b) is for the SAS-SST model using the fine mesh, coloured by transient average velocity. In part (a) it is evident that the k-e model captures the bulk swirl structure, while in the SAS-SST case resolved turbulence structure exists on top of this swirl, and does not dissipate as in the top-fed digester (see Fig. 8(b)).
Fig. 14. Residence times for (a) k-e model predictions and (b) SAS-SST simulation results for the top-fed digester.
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Fig. 15. Residence times for (a) k-e model predictions and (b) SAS-SST simulations for the tangentially-fed digester.
Fig. 16. The impact of accounting for the turbulent dispersion due to the unresolved turbulence in the SAS-SST simulation of the tangentially-fed digester. Results are presented for 10 lm neutrally buoyant particles.
the SAS-SST model. All models gave reasonable predictions of the swirl profile. The key difference is seen in the vertical velocity component of the flow – the experimental result shows up-flow on the centre-line in the lower regions of the vessel. The SAS-SST model was the only turbulence model that went some way to replicating this behaviour. The SAS-SST model gives close agreement to the RTD curve in the top feed configuration and does a better job than
the k-e model in the tangential feed case. However, it should also be noted that using SAS-SST for modelling a production scale digester may prove computationally prohibitive. The results of the current simulation for digesters and also those from our previous work on precipitators (Brown et al., 2014) show that use of a SAS-SST modelling approach is beneficial in situations where the large-scale turbulence structures cannot be
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modelled using the RANS approximation (tangentially-fed vessels and precipitators). However, in circumstances where the flow turbulence does not result in highly unstable large-scale turbulence structures the two equation k-e model does a reasonably good job. Regarding methods of calculating residence time data, we have seen that there are some differences between using a scalar tracer and tracking Lagrangian particles in complex geometries. The results suggest that as the flow becomes more complex it is important to resolve the turbulence structures to get good agreement with both the velocity field and residence time measurements. We also note that under such circumstances the experimental data are likely very variable and multiple tracer experiments are needed to understand this. Experimental and computational studies are on-going to investigate these issues further.
Acknowledgements The authors would like to acknowledge the significant contribution to this study made by Dr. Jie Wu and Mr. Bon Nguyen of CSIRO who conducted the experimental study. This paper is an extension of a conference paper that was presented at the Eleventh International Conference on CFD in the Minerals and Process Industries (CFD2015), and was nominated for invitation into the CFD2015 Special Issue of Chemical Engineering Science based on its designation as a high-quality paper of relevance to the modelling of fluids-based systems.
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References Brown, G.J., Fletcher, D.F., 2005. Alcoa internal report, October 2005. Brown, G.J., 2001. CFD prediction of particle and fluid residence times in industrial vessels. In: 6th World Congress of Chemical Engineering, Melbourne, Australia. Brown, G.J., Whyte, D.S., Fletcher, D.F., 2014. Dynamic flow modelling in precipitator vessels – a study of turbulence modelling approaches. Appl. Math. Model. 28, 4163–4174. Gerard, G., Stroup, P.T., 1963. Extractive Metallurgy of Aluminum, vol. I. Interscience Publishers, New York. Gosman, A.D., Ioannides, E., 1981. Aspects of computer simulation of liquid fuelled combustors. AIAA Paper, No. 81-0323, 1981. Hunt, J.C.R., 1987. Vorticity and vortex dynamics in complex turbulent flows. Trans. Can. Soc. Mech. Eng. 11 (1), 21–35. Launder, B.E., Spalding, D.B., 1974. The numerical computation of turbulent flows. Comput. Meth. Appl. Mech. Eng. 3, 269–289. Matida, E.A., Nishino, K., Torii, K., 2000. Statistical simulation of particle deposition on the wall from turbulent dispersed pipe flow. Int. J. Heat Fluid Flow 21, 389– 402. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605. Menter, F.R., Egorov, Y., 2010. The scale-adaptive simulation method for unsteady turbulent flow predictions. Part 1: Theory and model description. Flow Turbulence Combust. 85, 113–138. Menter, F.R., Garbaruk, A., Smirnov, P., Cokljat, D., Mathey, F., 2009. Scale adaptive simulation with artificial forcing. In: Proc. 3rd Symposium on Hybrid RANS-LES Methods, Gdansk, Poland, 10–12 June, 2009. Smirnov, P.E., Menter, F.R., 2008, Sensitization of the SST turbulence model to rotation and curvature by applying the Spalart-Shur correction term. ASME Paper GT 2008–50480, Berlin, Germany, 2008. Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach. J. Fluid Mech. 277, 245–272. Woloshyn, J., Oshinowo, L., Rosten, J., 2006. Digester design using CFD. In: 135th Annual Meeting – Proceedings of the Technical Sessions Presented by the TMS Aluminium Committee, pp. 939–944.
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Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Predicting flow and residence time in alumina digestion vessels David S. Whyte a, Gary J. Brown a, David F. Fletcher b,⇑ a b
Alcoa of Australia Limited, Kwinana, Western Australia 6966, Australia School of Chemical and Biomolecular Engineering, University of Sydney, Sydney 2006, New South Wales, Australia
a r t i c l e
i n f o
Article history: Received 28 April 2016 Received in revised form 28 November 2016 Accepted 15 December 2016 Available online xxxx Keywords: Residence time distribution Turbulence modelling Scalar transport Particle tracking Digestion vessel
a b s t r a c t Understanding the flow and residence time behaviour in reaction vessels is of crucial importance to improve product yield or the extraction of mineral species from ores. Digestion vessels, used in the Bayer process, are an example of the latter. Transient Computational Fluid Dynamics (CFD) models were used to investigate the impacts of a variety of turbulence models, along with the effects of mesh refinement on the performance of two different vessel geometries. The main focus of the research work was the efficacy of a variety of turbulence models: two-equation (k-e, SST), Reynolds Stress (SSG RSM) and Scale Adaptive Simulation (SAS-SST). The predicted velocity profiles are compared with high quality Laser Doppler Velocimetry (LDV) data. Residence times are calculated via a convected scalar (tracer) and Lagrangian tracking of particles. The calculated residence times are compared with salt tracer estimates from experiments. Generally, there was good agreement between experimental data and results obtained from tracing a passive scalar or tracking neutrally-buoyant particles, with the three results being closer when the SAS-SST model was used compared with the k-e model. In the case of the strongly swirling flow caused by the tangential entry, the tracer was in closer agreement with the experimental data but the Lagrangian tracking results improved considerably with the use of the SAS-SST model. The importance of small-scale turbulence structures in determining the flow profile and residence times of the vessels is highlighted in this study. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Digestion plays a fundamental part in the Bayer process for alumina production, with the objective of this processing stage being to extract the available alumina from the bauxite by dissolving it in a hot caustic soda solution (Gerard and Stroup, 1963). Digester vessels typically have a height to diameter ratio of close to 3:1. Some vessels use mechanical agitation, but those examined in this study have a single inlet stream at the top of the vessel and a single, centrally-located, outlet at the bottom of the vessel, as in the schematic shown in Fig. 1. The flow behaviour in these vessels is therefore dominated by the design and positioning of the inlet nozzle. An understanding of the flow behaviour in these vessels is important, as adequate residence time and mixing are critical to achieving sufficient extraction of alumina from bauxite particles of varying diameter. Past CFD modelling of digestion vessels within the organisation used the standard k-e model in steady-state simulations (Brown and Fletcher, 2005). While the simulations of top-fed digesters
⇑ Corresponding author. E-mail address: [email protected] (D.F. Fletcher).
were considered to give acceptable agreement with experimental data, the same could not be said for the simulations for tangentially-fed vessels. At the time of that work, scale-resolving turbulence models were at their development stage and computing resources were vastly more expensive than they are now. Woloshyn et al. (2006) used CFD to investigate a range of digester feed configurations. Their study considered full-scale digesters using the k-e model and the Reynolds Stress Model (RSM) for turbulence. Residence Time Distributions (RTD) were determined for individual particles of various sizes. However, no validation of the simulations was conducted. Brown et al. (2014) studied the flow in precipitation vessels using CFD. These vessels have flow rates and velocity scales which are comparable with those for the digesters considered here. The work evaluated a variety of turbulence models and validated the numerical predictions against experimental measurements. The authors found that using a turbulence model that was able to capture the unsteady, large-scale turbulence structures (SAS-SST) gave results that matched the experimental data very well. Thus, given improved modelling techniques and computational resources, it is now possible to examine the performance of more sophisticated turbulence modelling approaches. Below, we give a brief description of the experimental study used for validation
http://dx.doi.org/10.1016/j.ces.2016.12.036 0009-2509/Ó 2016 Elsevier Ltd. All rights reserved.
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the vessel walls. These readings were taken at several vertical elevations – those taken at 415 mm and 1465 mm above the cone are reported here as they are representative of the behaviour observed at all locations. The measurement device was held stationary at each point along the radius for approximately two minutes, with 2000 readings taken in each period. Residence Time Distributions (RTD) were determined using a salt tracer. A pulse of salt solution was injected into the inlet pipe once the flow had settled down and the concentration of salt was measured at the outlet over time.
3. Model description 3.1. Turbulence modelling
Fig. 1. Schematic of modelled domain. Both the top and tangential feed lines can be seen. However, only one feed line is active at any time. Note that when the top feed inlet is used there is a deflector plate suspended below the feed nozzle which stops flow from short-circuiting directly to the outlet.
purposes and then compare simulated flow-fields with the experimental data, first for the top-fed vessel and then for the vessel with a tangential inlet. Then the residence time distributions are presented together so that comparison based on the different methods, geometries and turbulence models can be made. The remainder of the paper is presented as follows: Section 2 describes the experimental data used for model validation; Section 3 describes the CFD model used in this study; Section 4 presents the simulation results and comparison with the experimental data and Section 5 contains the conclusions. 2. Experimental study Alcoa commissioned CSIRO Mineral Resources Flagship to generate the experimental data used in this study. A laboratory-scale (1.5 m tall and 0.4 m diameter) model of a full-scale digester was used in the experiments. Water at ambient temperature was used as the working fluid. The data reported here are for an inlet flow rate of 127 L min1. Superficial residence time is a crude estimate of how long a fluid particle will be present within a vessel, and is defined as the volume of the domain divided by the volumetric flow rate through it. In the cases considered here, the superficial residence time, To, is 88 s. The results from the experiments and simulations will be compared using a normalised time (measured time, t, divided by To). Laser Doppler Velocimetry (LDV) was used to measure the velocity components along a line from the central axis to just off
Different turbulence modelling approaches were tested in an effort to determine the most accurate method for calculating the flow field using the most efficient computational approach. If steady simulations can be performed using a RANS turbulence model the computational requirements, in terms of mesh size and computational run time, are vastly smaller than if a transient scale-resolving approach is needed. This knowledge clearly has a huge impact on the computational resources required to perform such analyses. We note here that although a RANS model may give a steady solution this does not mean the real flow is steady. Rather it says that the unsteadiness can be removed by the Reynoldsaveraging process and represented by a simple eddy diffusion hypothesis and there is no underlying transient flow. As flow separation is not a governing feature of the flows, wall functions are considered to be sufficient to predict flow in the wall region. Previous work (Brown and Fletcher, 2005) had shown that the standard k-e model (Launder and Spalding, 1974) could predict the flow in the top-fed digester but was unable to predict the behaviour in the tangentially-fed vessel. The k-e model yields a steadystate solution, where the experiments showed large-scale unsteadiness of the flow. The two-equation SST (Menter, 1994) and Reynolds Stress Model (SSG RSM) Speziale et al., 1991 were also tested in this work. Finally, given the success in predicting unsteady flow in laboratory-scale precipitators (Brown et al., 2014), the Scale Adaptive Simulation (SAS-SST) model of Menter and Egorov, 2010 was considered. When applying the SAS-SST model if the mesh is sufficiently fine the large-scale turbulence structures can be captured if the flow is globally unsteady (Menter and Egorov, 2010). If the flow is not globally unstable, i.e. there is nothing to trigger the generation of large-scale turbulence structures, the simulation model may remain in RANS mode and no structure is resolved. In order to avoid this, a zonal LES region was introduced in the inlet feed pipe in order to convert modelled turbulence into resolved turbulence which ensures that the SAS-SST model will behave in scale-resolving mode and continue to resolve the large-scale turbulence structures in the downstream region. This initiation method uses a harmonic turbulence generator to convert modelled turbulence kinetic energy into random fluctuations which trigger the scale-resolving mode (Menter et al., 2009). In addition the curvature correction term was enabled in the SAS-SST simulations to avoid artificial decay of vortical structures (Smirnov and Menter, 2008).
3.2. Computational domain and mesh The meshes used in this study were informed by the experience in modelling alumina precipitators (Brown et al., 2014). The tangentially-fed digester is inherently 3-dimensional, so no attempt was made to reduce the geometrical complexity.
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Experience has shown that for systems such as those discussed here, a good starting point for mesh inflation is 10 layers of inflation on all walls with a 1 mm first layer height and a 1.2 expansion ratio. In subsequent meshes, these controls were changed to 20 layers, with a first layer height of 0.5 mm. This gave y+ values typically in the range of 20–30. General guidelines for an appropriate mesh density for the SAS-SST model in this sort of geometry are not available, with the majority of published cases being related to bluff body or aerodynamic flows. Therefore, the edge length of the mesh in the main vessel was systematically reduced and the results assessed. The meshes used are shown in Fig. 2. The inlet region and lower cone were meshed using tetrahedrons, while in the barrel of the digester the mesh was swept in the axial direction. 3.3. Numerical considerations The commercial CFD code ANSYS CFX 15.0 was used for this study. The code uses a vertex-based control volume approach, the Rhie-Chow procedure to couple pressure and velocity fields, a coupled solver to solve for the velocities and pressure simultaneously and an algebraic multi-grid (AMG) solver to resolve the resulting algebraic equations. Spatial and temporal derivatives were calculated using bounded second order differencing schemes in the non-scale resolving simulations and bounded central differencing schemes in the scale-resolving simulations. All simulations were carried out as transients and run for 300 s (3.4 superficial residence times). Simulations that evolved to a
3
steady-state were terminated prior to reaching the target time. The transient simulations were run with adaptive time-stepping with the rms Courant number set to be below one as required for scale-resolving simulations. A few cells in the exit region had Courant number values of around 50 but otherwise most values satisfied the pre-set criterion. The SAS-SST simulation typically required time steps of 1 ms, and for the two-equation models (k-e and SST) the solver reached the upper time step limit specified of 0.1 s. 3.4. Residence time studies Two different and complementary approaches were used to investigate the residence time distributions in the two vessels. Firstly, an Eulerian approach was applied in which a pulse of tracer was injected at the inlet after 300 s (3.4 superficial residence times) when the flow-field was well developed and the scalar flux at the outlet was recorded over time (Brown et al., 2001). In this approach the laminar diffusivity of the tracer was set to zero and the turbulent diffusivity was calculated as the eddy viscosity divided by a turbulent Schmidt number, which was set to the default value of 0.9. It is noteworthy that in this approach in the k-e simulations the eddy viscosity represents the entire effect of turbulence, whereas in the SAS-SST simulations it is only the effect of the unresolved eddies that is treated in this manner, with the effect of the large-scale, resolved eddies being represented by the usual convective term in the equations. The second approach used was Lagrangian particle tracking. In this case representative particles were injected at the inlet and their path through the vessel was calculated by solving Eq. (1)
dx ¼ v; dt
mp
dv ¼ F D þ V p ðqp qf Þg dt
ð1Þ
where x is the particle location, t is time, v is the particle velocity, mp is the particle mass, F D represents drag force, V p is the particle volume, qp ; qf are the particle and fluid densities, and g is the acceleration due to gravity. The drag coefficient was determined from the Schiller-Naumann correlation. When the k-e turbulence model was employed the random walk turbulence model was applied to the particles (Gosman and Ioannides, 1981) so that the velocity used in Eq. (1) is then the sum of the mean and a fluctuating component. In this model a particle is assumed to always be inside a single turbulent eddy. The turbulent velocity, uf , eddy lengthscale, le , and eddy lifetime, se , are calculated as follows
uf ¼ rð2k=3Þ 3=2
le ¼ C 3=4 l k
0:5
=
se ¼ le =k3=2 ¼ 0:2
Fig. 2. Representations of the computational meshes used in the study for the topfed digester. (a) This mesh has 610,000 nodes, with a first inflation layer height of 1.0 mm; (b) this mesh has 2,700,000 nodes, with a first inflation height 0.5 mm. The yellow lines represent the 415 mm (lower) and 1465 mm (upper) elevations used for comparison with experimental data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ð2Þ ð3Þ k
e
ð4Þ
where r is a Gaussian distributed random variable with a zero mean and standard deviation of unity. C l is a constant in the turbulence model taking a value of 0.09 and is included to relate the characteristic length-scale with the eddy dissipation length-scale. A separate fluctuating velocity is computed for each velocity component and therefore the turbulent dispersion is non-isotropic, despite only isotropic information on the turbulence being available. A given fluctuation is used until either the eddy lifetime is exceeded or the particle displacement exceeds the eddy length-scale. It should however be noted that there is no universal value for the model constants and different turbulent flows require different fits to obtain agreement, with the potential variation being significant (Matida et al., 2000).
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Fig. 3. Effect of the number of particles used to determine the residence time distribution for the top feed inlet case simulated using the k-e turbulence model.
When the SAS-SST model was used no additional turbulence dispersion was used. In this case the effect of the resolved turbulence structures is taken into account directly and we show later that the contribution of the unresolved, dissipative turbulence is negligible. 10,000 particles were released from the inlet once the transient simulation had run for 3.4 superficial residence times. A simulation was repeated for the top feed inlet case in which 50,000 instead of 10,000 particles were used to determine the impact of this parameter. The results, shown in Fig. 3, confirm that 10,000 particles are sufficient to capture the residence time curve in a manner independent of the number of particles used. The time at which each particle reached the exit was recorded and the data were binned, with a bin size of 1 s, so that a particle exit time distribution could be plotted. In total four different Lagrangian particle groups were considered for each case. Particles of 10 and 100 lm were used. Two groups used neutrally buoyant particles (having the same density as that of the water, 997 kg m3) and two having particles with a density typical of bauxite (2493 kg m3). The neutrally buoyant particles follow the flow and behave like fluid path-lines. There is no impact of particle size in this case so the two simulations (10
Fig. 4. Transient monitor point data for the top feed simulations: (a) results for the k-e simulation and (b) for the SAS-SST simulation. These show the very different behaviour of the two models - the k-e model results are essentially steady, while the SAS-SST model results show persistent fluctuations.
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and 100 lm) provide two different realisations for the same flow. When the bauxite particles are tracked they have the potential to deviate from the fluid path-lines and will show the impact of a finite Stokes number on the particle trajectories and residence time. Each case was run for an additional three superficial residence times and the data were normalised such that area under a mass flux versus time curve corresponded to the mass of tracer that exited the vessel.
of the experiments (or offered additional insight) than the k-e or SAS-SST models. It is interesting to note that for both inlet configurations the k-e models resulted in effectively steady velocity fields (fluctuations 0.1% of the mean flow) on both the fine and coarse meshes. Statistics of the transient flow were started at the same time as the scalars/particles were introduced and data were captured over the entire 300 s period. In both the SAS-SST predictions and LDV measurements, the standard deviations of the velocity components have been used to indicate the magnitude of flow fluctuations. An effective approach to detecting vortex structures is to plot the sec-
4. Results
ond invariant of the velocity gradient tensor (or Q-criterion) Hunt, 1987:
In this section we will focus on the comparison of the k-e and SAS-SST turbulence models. While the SST and SSG RSM models were considered in the study, neither provided better predictions
Q¼
1 ðjXj2 jSj2 Þ 2
ð5Þ
Fig. 5. Comparison of the predicted vertical components of velocity for the k-e model with coarse and fine meshes on a plane at the mid-height of the top feed model. The results are relatively insensitive to mesh refinement.
Fig. 6. Contours of the vertical component of fluid velocity (negative values indicate down-flow): (a) k-e model using the coarse mesh; (b) SAS-SST model (transient average) using the fine mesh. In both cases, the flow features show qualitative similarities. There is strong down-flow at the wall below the deflector plate. Directly under the deflector plate is a region of strong recirculation – there is a slight difference between the two predictions. However, beneath this region there is fairly uniform down-flow.
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where X and S are the vorticity and shear strain rate tensors, respectively. Vortical structures can be identified by iso-surfaces with Q > 0. Fig. 4 shows the velocity magnitude at four monitor points distributed across the vessel at approximately mid-height for the top feed case. It is evident that after about 200 s the results for the k-e simulation have reached a steady-state, whereas those from the SAS-SST model continue to oscillate. Fig. 5 shows a profile of vertical velocity on the same plane for the k-e model with both the fine and coarse meshes shown in Fig. 2. It is evident that the k-e results are relatively insensitive to the mesh and hence only results for the coarse mesh are presented in the remainder of the paper for this model. 4.1. Flow field top-fed digester Fig. 6 shows a comparison of the vertical velocity component predictions of the k-e and SAS-SST turbulence models. The results are very similar overall with the key difference being the extent
of the recirculation zone under the deflector plate, with that in the SAS-SST simulation being slightly shorter. Fig. 7 shows that both turbulence models predict very similar vertical velocity profiles but show some disagreement with experimental values near the walls at a height of 1465 mm, and uniform down-flow in the lower half of the vessel. The velocity at the walls is hard to measure in the high gradient region and we attribute this disagreement to measurement error. An additional point of agreement between the measurement from LDV and SAS-SST predictions, is the magnitude of the fluctuations seen in the velocity components (Fig. 7). Fluctuations for the k-e model predictions were obtained from qffiffiffiffiffiffi 2 k by assuming isotropy of the fluctuating velocity components. 3 The values obtained are somewhat smaller than those from the SAS-SST predictions, as would be expected, especially in the region near the wall. Overall, we can conclude that in the case of the top feed digester both turbulence modelling approaches perform well in comparison with the experimental data.
Fig. 7. Comparison of the predicted vertical components of velocity with LDV measurements (a) taken along the radius of the vessel at 1465 mm; (b) at 415 mm. Both models predict the same general behaviour. In (a) there is upflow on the central axis (beneath the deflector plate), with faster down-flow at the walls; (b) shows a relatively slow, uniform down-flow in the lower part of the vessel. It should also be noted that the rms of the fluctuations of the SAS-SST predictions (red error bars) and LDV measurements (black error bars) are similar. The rms for the k-e predictions (green error bars) is smaller than that from the SAS-SST predictions, especially in the near wall region. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 8. Isosurface where the Q-criterion takes a value of 10 s2: (a) results from the k-e model using the coarse mesh coloured by velocity; (b) results from the SAS-SST model using the fine mesh coloured by transient average velocity. In (a) it is clear that this model captures only the quasi-steady vortices located in the upper recirculation caused by the deflector plate; (b) highlights the significant turbulence structures resolved by the SAS-SST model in the upper region of the chamber. The deflector plate generates turbulence in the flow that then dissipates in the lower region of the digester.
Fig. 9. Contours of the vertical component of fluid velocity: (a) k-e model results using the coarse mesh; (b) SAS-SST model using the fine mesh showing the transient average vertical velocity. While there are some qualitative similarities between the two cases, the differences are quite clear – the two-equation model predicts that the down-flow region of the central core extends the full height of the vessel, and the flow is more erratic.
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Fig. 10. Comparison of predicted vertical components of velocity with LDV measurements: (a) taken along the radius of the vessel at 1465 mm; (b) at 415 mm. In (a) it is evident that all the models are consistent, and both deviate from the experimental data at the wall. Part (b) highlights that the key difference between the models is their ability to predict the upflow on the centre-line in the lower region of the digester – only the SAS-SST model is able to capture this behaviour. The meaning of the error bars is as defined in Fig. 7.
As expected, there is a significant difference in the extent and form of the flow structure predicted by the two models. Qcriterion plots, given in Fig. 8, show that the flow predicted by the k-e model is dominated by the recirculation produced by the deflector plate, while the SAS-SST computed flow has a significant amount of structure in the upper section of the vessel, with decay of this turbulence in the lower half of the vessel. 4.2. Flow field tangentially-fed digester It is the model results for the tangentially-fed digester that are most interesting here, as past efforts have shown that twoequation turbulence models are insufficient to accurately predict the flow field or fluid residence times (Brown and Fletcher, 2005). In this case we are interested in both the vertical and tangential components of the velocity. Fig. 9 shows the predicted vertical velocity fields. While both ke and SAS-SST models show global similarity, they have some significant local differences. In particular, the down-flow seen in the
central axis is more intense in the k-e model results. In Fig. 10 we can see that neither of the models predicts the magnitude of the central up-flow shown by the LDV data although the SAS-SST performs much better in this regard. It is also evident the k-e model significantly under-predicts the turbulence fluctuations and the SAS-SST model does much better in this respect. Figs. 11 and 12 show that both models predict the swirling flow profile, with the SAS-SST model being clearly superior at 415 mm from the bottom. Fig. 13 shows an iso-surface of the Q-criterion, coloured by the velocity in the results from the k-e model and by the transientaveraged velocity in the case of the SAS-SST model. Both plots show a similar global structure, but it is evident that vortical structures exist much further into the vessel in the SAS-SST case.
4.3. Residence time top-fed digester As noted above, the k-e model gave a steady flow field so both the tracer calculation and the Lagrangian tracking were done with
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Fig. 11. Contours of the tangential component of fluid velocity for (a) results from the k-e model using the coarse mesh and (b) for the SAS-SST model (transient average) on the fine mesh. There is very little difference between the predictions of the two models.
a frozen flow field, whereas the flow field was evolved with the tracer and particles in the SAS-SST case. Fig. 14 compares the computed results with the experimental data and it is evident that both turbulence models predict the arrival of the tracer and the neutrally buoyant particles at the outlet at a similar time to that observed experimentally. Agreement between the k-e model results and experimental data is generally good but the tracer arrives a little too early, whereas the SAS-SST model results show the tracer arriving slightly later than in the experiment. However, we should note here that the tracer results are for a single experimental trial, as were the SAS-SST results. Repeats of the experimental test and different realisations of the transient SAS-SST simulations would give slightly different results, so this difference in agreement may not be significant. In both cases the neutrally buoyant particles and the 10 lm bauxite particles behave very similarly. In the k-e case there are some significant differences between the Lagrangian method and the tracer. In the SAS-SST case, where the turbulent structures are better resolved, the results from the neutrally buoyant and 10 lm bauxite particles closely follow the fluid tracer result. This suggests that the turbulent dispersion behaviour is different between the two cases and confirms that the Lagrangian approach can give results equivalent to an Eulerian tracer provided sufficient turbulent structure is resolved in the flow. The 100 lm alumina particles show significant short-circuiting in the k-e simulation with the peak flux at around half the superficial residence time. There is some short-circuiting in the SAS-SST simulation results but this is much reduced compared with that seen in the k-e simulation. Our hypothesis is that this change of behaviour is due to increased dispersion of these particles within the vessel due to the turbulent structures being resolved in the SAS-SST simulation. Such differences are potentially very important in predicting the operating performance of a full-scale vessel.
4.4. Residence time tangentially-fed digester Data for the tangentially-fed digester were obtained in a similar manner to those for the top-fed digester and are shown in Fig. 15. The results from the k-e model show that the tracer, neutrally buoyant and 10 lm bauxite particles arrive later than the experimental measurements, whilst there is much closer agreement in the SAS-SST simulations. Both simulations show very similar shaped distributions. These particles also show much greater bypass than the tracer results using both turbulence models. In the tangentially-fed case there is again considerable bypass by the 100 lm bauxite particles, with this effect again more pronounced in the k-e model results with a very high peak at a dimensionless time of 0.4 compared with those from the SAS-SST simulation which has a much broader peak. These differences are again attributed to increased dispersion of these particles within the vessel due to the turbulent structures being resolved in the SAS-SST simulation. The question arises as to whether the discrepancies seen above could be attributed to the impact of the turbulent dispersion caused by the unresolved turbulence. To address this, an additional simulation was run in which the random walk model was applied to 10 lm neutrally buoyant particles in the SAS-SST simulation for the tangential feed case. Fig. 16 shows that whilst there are differences in the results these are very minor and do not bring the results in line with the experimental data.
4.5. Discussion The results presented above show that whilst there is reasonable agreement between the calculated and experimental data there are some differences, especially in the tangentially-fed case. In order to understand whether these differences were a result of
Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036
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Fig. 12. Comparison of the predicted tangential velocity with LDV measurements: (a) taken along the radius of the vessel at 1465 mm and (b) at 415 mm. Both plots show that the SAS-SST model provides the most accurate prediction of the velocity profile. The meaning of the error bars is as defined in Fig. 7.
the physical models used or the code implementation of them a number of supplementary simulations were performed. Firstly, an idealised U-bend test case was developed and used to explore simulations with no recirculation zones present when using the steady k-e modelling approach in conjunction with an Eulerian tracer and neutrally buoyant Lagrangian particles using the random walk model. These showed almost identical agreement between the tracer and particle data. We then introduced internal baffles to represent those present in the top feed geometry to create recirculation zones and again results of the two methods were essentially identical. Finally, we introduced recirculation zones adjacent to the wall with the same result. These convinced us that the two methods can give identical results in simple flows. In the top-fed digester the results with the SAS-SST model (see Fig. 14(b)) show that the two methods can also give close agreement in a complex geometry, suggesting that in this case the turbulent structures responsible for distributing the particles are well captured. We have already shown above that adding the turbulent dispersion model to the particle transport equation in the SAS-SST simulations has a small effect in the tangentially-fed case. However, looking at the Q-criterion plots, especially Fig. 13, we
hypothesize that if an even finer mesh than that used here were used we would get more turbulent dispersion of the particles and even better agreement with the tracer results. 5. Conclusions Understanding the flow of the fluid and particles in an alumina digester is central to predicting the device performance. An accurate model for top-fed digesters has been developed. Within the limit of the cases studied, the simulated flow in the top-fed digesters is independent of the turbulence model used. The agreement with experimental data between RTD predictions of the steadystate (k-e model) and the transient SAS-SST models suggests that in this feed orientation, the fine detail of turbulence is not important in predicting the performance of the vessel. However, when the residence time of 100 lm bauxite particles is considered it is clear that the k-e model predicts much greater bypass than the SAS-SST model. To predict the flow field and residence time of the tangentiallyfed digester, it is important that the model resolves the largerscale, unsteady turbulence structures, as can be achieved with
Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036
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Fig. 13. Isosurface where the Q-criterion takes a value of 10 s2 coloured by velocity. Part (a) shows results from the k-e model using the coarse mesh and part (b) is for the SAS-SST model using the fine mesh, coloured by transient average velocity. In part (a) it is evident that the k-e model captures the bulk swirl structure, while in the SAS-SST case resolved turbulence structure exists on top of this swirl, and does not dissipate as in the top-fed digester (see Fig. 8(b)).
Fig. 14. Residence times for (a) k-e model predictions and (b) SAS-SST simulation results for the top-fed digester.
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Fig. 15. Residence times for (a) k-e model predictions and (b) SAS-SST simulations for the tangentially-fed digester.
Fig. 16. The impact of accounting for the turbulent dispersion due to the unresolved turbulence in the SAS-SST simulation of the tangentially-fed digester. Results are presented for 10 lm neutrally buoyant particles.
the SAS-SST model. All models gave reasonable predictions of the swirl profile. The key difference is seen in the vertical velocity component of the flow – the experimental result shows up-flow on the centre-line in the lower regions of the vessel. The SAS-SST model was the only turbulence model that went some way to replicating this behaviour. The SAS-SST model gives close agreement to the RTD curve in the top feed configuration and does a better job than
the k-e model in the tangential feed case. However, it should also be noted that using SAS-SST for modelling a production scale digester may prove computationally prohibitive. The results of the current simulation for digesters and also those from our previous work on precipitators (Brown et al., 2014) show that use of a SAS-SST modelling approach is beneficial in situations where the large-scale turbulence structures cannot be
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modelled using the RANS approximation (tangentially-fed vessels and precipitators). However, in circumstances where the flow turbulence does not result in highly unstable large-scale turbulence structures the two equation k-e model does a reasonably good job. Regarding methods of calculating residence time data, we have seen that there are some differences between using a scalar tracer and tracking Lagrangian particles in complex geometries. The results suggest that as the flow becomes more complex it is important to resolve the turbulence structures to get good agreement with both the velocity field and residence time measurements. We also note that under such circumstances the experimental data are likely very variable and multiple tracer experiments are needed to understand this. Experimental and computational studies are on-going to investigate these issues further.
Acknowledgements The authors would like to acknowledge the significant contribution to this study made by Dr. Jie Wu and Mr. Bon Nguyen of CSIRO who conducted the experimental study. This paper is an extension of a conference paper that was presented at the Eleventh International Conference on CFD in the Minerals and Process Industries (CFD2015), and was nominated for invitation into the CFD2015 Special Issue of Chemical Engineering Science based on its designation as a high-quality paper of relevance to the modelling of fluids-based systems.
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Please cite this article in press as: Whyte, D.S., et al. Predicting flow and residence time in alumina digestion vessels. Chem. Eng. Sci. (2016), http://dx.doi. org/10.1016/j.ces.2016.12.036