Computational Fluid Dynamics Modeling of Corrugated Static Mixers

Article pubs.acs.org/IECR

Computational Fluid Dynamics Modeling of Corrugated Static Mixers for Turbulent Applications Mirella Coroneo,* Giuseppina Montante, and Alessandro Paglianti Department of Chemical, Mining and Environmental Engineering, University of Bologna, Via Terracini 28, I-40131 Bologna, Italy ABSTRACT: This work is aimed at investigating the capability of a fully predictive computational fluid dynamics (CFD) approach to catch the main features of the liquid flow in pipelines equipped with corrugated static mixer inserts. The simulations are based on the numerical solution of the Reynolds averaged Navier−Stokes equations on the three-dimensional domain closely representing the static mixer geometry. As a benchmark, literature experiments on a laboratory scale SMV mixer were considered and the simulation results obtained at different locations were compared with relevant velocity and tracer concentration data. The effect of the turbulence modeling, the near-wall treatment, and the numerical approximations on the accuracy of the results is discussed. The results analysis demonstrated that the selected CFD model can be reliably adopted to evaluate the velocity field and the mixing performances of turbulent flows in static mixers for design and optimization purposes. On the other hand, the application of the same model to an industrial scale corrugated plate mixer highlighted the effects of the distributor geometry and of the distance between consecutive elements on the mixing of two miscible liquids.

1. INTRODUCTION Mixing is a necessary and important operation in several chemical and process plants, and it is often accomplished through mechanical agitation. It can and often does take place however in pipes connecting process equipment; in this case pipelines themselves, usually equipped with static mixing internals, serve as process vessels.1 Static mixing is used mainly when fast blending is required and when long holdups, which are associated with tanks, are not desired;1 they are also adopted as an alternative to mechanical agitation in continuous processes. Compared to stirred tanks, they allow achieving high levels of mixing with short residence times by using the energy of the flow to produce radial redistribution of the passing fluids. In addition, they require lower maintenance than other mixers due to the lack of moving parts,2 lower energy consumption, since the power requirement is limited to the pressure drop overcoming, lower investment costs, and little back mixing.3 An empty pipe exhibiting turbulent flow can be considered as the simplest static mixer where approximately 100 pipe diameters may be needed for complete mixing.4 In order to reduce the mixing length, however, a variety of static mixer geometries have been proposed so far and a careful selection of the proper static mixer for each specific process is needed. Guidelines can be found in the review by Thakur et al.2 and in Etchells and Meyer,1 where the most commonly adopted static mixers are described and compared and the key parameters in the selection of the best one for each specific application are discussed. For what concerns turbulent flow applications, the key design criteria are based on velocity and energy dissipation, in addition to pressure drops, which are highly dependent on the geometry of the mixing elements and are often the limiting factor for in-line mixer application in industrial processes.4 Overall, information on static mixers for turbulent regime is relatively rare;5 comparatively, more extensive information can be found in the open literature for laminar flow applications on both the experimental side and the computational side, as briefly addressed by Regner et al.6 Improvement of the static © XXXX American Chemical Society

mixing performances through geometrical optimization based on the local mixing feature determination might be achieved by computational fluid dynamics (CFD) methods, thus contributing to energy saving in industrial operations. CFD is a powerful and effective tool which permits prediction of the behavior of chemical reactors through the numerical determination of local fluid dynamics and associated transport phenomena in reacting systems, and therefore it is well-suited for static mixer analysis and design.7,8 For turbulent flows, however, a limited number of successful applications of CFD to static mixers, mainly of Kenics9−12 or HEV13 types, have been already reported, while less attention has been devoted to corrugated plate static mixer inserts either on the experimental side14−18 or on the computational side.14,19,20 Corrugated plate static mixers, among which the SMV type is probably the most widely adopted, are highly recommended for single and multiphase turbulent flow applications,1 which include gas/liquid and liquid/liquid contacting, homogenization, dispersion, emulsification, cocurrent mass transfer, heat transfer, and chemical reactions.1,3,4 To the best of our knowledge, the fully predictive three-dimensional simulation of a pipeline equipped with these mixer inserts coupled with a strict evaluation of the fluid dynamic and mixing characteristics has never been performed. In this work, the liquid velocity and the tracer concentration predicted by a CFD simulation strategy based on the solution of the Reynolds averaged Navier−Stokes (RANS) equations, which is nowadays the industrial standard, are compared with literature experimental data and critically discussed for a laboratory scale application. After assessment of the capability of the computational method, the modeling was extended to an Received: February 14, 2012 Revised: September 3, 2012 Accepted: November 20, 2012

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dx.doi.org/10.1021/ie300398z | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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industrial scale apparatus, with the purpose of studying the effects of selected geometrical parameters on the mixing of two miscible liquids.

2. GEOMETRICAL AND WORKING CONDITIONS 2.1. Laboratory Scale Apparatus. The experimental mixer investigated by Karoui et al. by laser Doppler anemometry14 and laser induced fluorescence15 has been considered as benchmark geometry for the simulations. The computational domain consisted of an horizontal pipeline of diameter D equal to 50 mm and length equal to 10D containing one pipe diameter long SMV static mixer insert. The mixer insert, which is depicted in Figure 1, was composed of five Figure 2. Geometrical model of the corrugated plate static mixer inserts in the large diameter pipeline. (a) Single element; (b) two elements.

inclined 45° with respect to the pipeline main axis. The plates were modeled as thin walls since to properly account for the real thickness effect, which was 2 orders of magnitude smaller than the pipe diameter, the overall grid density would have required a huge computational power. For the simulations including two elements, the relative angular position of the mixer inserts was always 90°, as shown in Figure 2b, while their relative distance was varied from 0 to 2D. For this configuration, the mixing process of a solution of sodium hypochlorite at 11 vol % in water was considered, thus reproducing typical water disinfection conditions. Upstream the first mixer, an empty pipe length of 5D was always considered. A single working condition was tested, corresponding to the flow rates of 6021 and 0.69 m3/h for water and the aqueous solution of sodium hypochlorite, respectively, and resulting in a fully turbulent Reynolds number referred to the empty pipe diameter of 2.5 × 106. In order to compare the effects of the distributor geometrical characteristics on the mixing performances, two different configurations have been modeled. The former distributor, which is schematically shown in Figure 3, was a simple

Figure 1. Geometrical model of the laboratory scale SMV static mixer insert.

corrugated plates 2 mm thick; the corrugations formed channels inclined 45° with respect to the pipeline main axis. Further details on the geometrical configuration of the mixer can be found in Karoui et al.14 In the simulations, the pipe lengths upstream and downstream the mixer were equal to 3D and 6D, respectively. As in the experiments of Karoui et al.,14 a constant flow rate of 4.75 m3/h of water at room condition was considered, corresponding to a Reynolds number of 3.4 × 104, as calculated with respect to the empty pipe internal diameter. Beside the single phase flow calculations, the mixing of two miscible liquids was modeled, thus reproducing the experimental conditions of Karoui et al.15 In particular, after the attainment of a fully developed water flow field, the injection of an aqueous solution of rhodamine through a tube of internal diameter equal to 10 mm was considered. The injection tube was positioned coaxially with respect to the main pipeline and the injection point at a distance of 20 mm from the mixer inlet. As in the reference experiments, the simulations were repeated at a constant water flow rate of 4.75 m3/h and for three different rhodamine solution flow rates, corresponding to ratios of 12, 24, and 48 between the inlet rhodamine solution velocity, Vrhod, and the water superficial velocity, Vwater. 2.2. Large Scale Apparatus. An additional set of simulations concerned an industrial scale corrugated plate static mixer. The computational domain consisted of a pipeline of diameter D equal to 920 mm and length equal to 26D, equipped with one or two static mixer inserts, which are depicted in Figure 2. The geometrical features of the inserts were similar to the laboratory scale SMV; the major differences were in the magnitude of the ratio of the element length and of the plate thickness with respect to the pipe diameter. In particular, each insert was 0.25D long and was composed of five plates of 3 mm thickness; the corrugations form channels

Figure 3. Sketch of the large scale pipeline equipped with two static mixer inserts and the single tube distributor.

horizontal tube of diameter equal to 30 mm placed coaxially with respect to the main pipeline. On the other hand, the latter distributor, which is depicted in Figure 4, was designed following the suggestions of Streiff at al.4 with the purpose of reducing the mixing length. It consisted of 30 mm diameter vertical and horizontal tubes, where 13 holes, of 4 mm diameter each, were placed. The additive was injected in the pipeline through the holes cocurrently with respect to the fluid flow and distributed in every single channel of the static mixer. Both B



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Due to the specific geometrical features of the investigated apparatus, the turbulent flows might be significantly affected by the presence of walls. Typically, semiempirical formulas, called “wall functions”, are used to bridge the viscosity-affected region between the walls and the fully turbulent region, obviating the need to resolve the viscosity-affected near-wall region and to modify the turbulence models to account for the presence of walls. In most simulations, the standard wall functions proposed by Launder and Spalding,21 which are typically adopted for turbulent flow simulations, were selected. In order to highlight possible influences on the velocity field, two additional, more complex approaches were selected: the nonequilibrium functions and the enhanced wall treatment, as implemented in FLUENT.24 The nonequilibrium functions are a natural extension of Launder and Spalding’s standard wall functions25 and employ the two-layer concept in computing the budget of turbulence kinetic energy at the wall-adjacent cells and the standard log-law sensitized to pressure-gradient effects. The enhanced wall treatment is a combination of a two-layer model with so-called enhanced wall functions; this approach is designed to resolve the boundary layer down through the viscous sublayer and is based on a function suggested by Kader.26 3.2. Numerical Determination of Miscible Fluids Mixing. After the attainment of the fully developed steadystate flow field of the main liquid stream, the feed of rhodamine or of NaClO solutions in the laboratory and in the large scale apparatus respectively were modeled. For this purpose, the steady-state RANS equations coupled with the k−ε turbulent model equations were solved again adding a velocity inlet boundary condition in order to permit the additive injection. The initial conditions for the calculation were provided by the numerical solution obtained by the pure liquid, single inlet calculations. The mixing of the second liquid stream in water was evaluated by coupling the RANS equations with the Reynolds averaged convection−diffusion equation for the additives:

Figure 4. Geometrical model of the multiple-tube distributor, superimposed on the inlet section of the static mixer.

distributors were positioned approximately at a distance of 0.5D upstream the inlet section of the first mixer insert.

3. MODEL EQUATIONS AND NUMERICAL DETAILS 3.1. Numerical Determination of the Fluid Flow Field. The velocity and the pressure fields were obtained from the numerical solution of the mass and momentum conservation equations under incompressible, isothermal, and steady-state conditions, in the realm of the finite volume CFD code FLUENT 6.3. Since in all cases the flow regime was turbulent, the simulations were based on the solution of the Reynolds averaged Navier−Stokes (RANS) equations, which are obtained from the instantaneous conservation equations splitting the velocity u into its mean U and fluctuating part u′: ∇·(ρ U) = 0

(1) 2

∇·(ρ UU) = −∇p + μ∇ U − ∇·(ρ u′u′)

⎞ ⎛ μ ∇(ρ UΦ) = ∇·⎜ρDm∇Φ + t ∇Φ⎟ σt ⎠ ⎝

(2)

In most cases, the Reynolds stress tensor, ρu′u′, was modeled using the eddy viscosity hypothesis, and in particular the standard k−ε model21 was selected. In order to assess the influence of the turbulence model on the simulations results for the laboratory scale mixer, additional calculations have been also carried out adopting either the “realizable” formulation of the k−ε model22 or the more general Reynolds stress model (RSM).23 The resulting equations were solved in the three-dimensional computational domains described in section 2. For the laboratory scale domain, three unstructured computational grids were adopted consisting of 500 000 (grid A), 2 200 000 (grid B), and 4 400 000 (grid C) cells, respectively, while for the large scale mixers, an unstructured grid of 2 350 000 cells was considered. The conservation equations were integrated in space using a second order upwind discretization scheme for the convective terms, and the SIMPLEC algorithm was used to couple pressure and velocity. No-slip boundary conditions at the walls of the static mixer insert and of the pipelines were imposed, while a velocity inlet boundary condition was selected at the fluid flow entrance and a pressure of 1.01 × 105 Pa at the domain outlet boundary was set.

(3)

where Φ is the additive volumetric fraction, U is the mean velocity vector, ρ is the fluid density, Dm is the molecular diffusivity, μt is the turbulent viscosity, and σt is the turbulent Schmidt number. The molecular diffusivity was fixed to the value of 10−9 m2/s, though it was not a critical value, since the contribution of molecular diffusion to the overall tracer dispersion process is expected to be small with respect to the turbulent diffusion, which is the ratio between the turbulent viscosity and the turbulent Schmidt number. For this last parameter, the commonly suggested value of 0.7 has been adopted.27,28 It is taken smaller than 1 because the scalar spectrum contains higher frequencies than the dynamic spectrum, and as a result scalar eddy diffusion is stronger than momentum eddy diffusion.27,29 As for the physical properties of the secondary streams, the same density and viscosity of water were assumed, since in any case the buoyancy effects on axial mixing are expected to be negligible for small differences in liquid densities.29 Equation 3 was solved adopting a second order upwind discretization scheme, and the solution convergence was carefully checked by monitoring the residuals of all the variables and the mass balance. At the end of the calculations C



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most of the variable residuals were dropped to the order of 10−7. Additional simulations were also carried out with different discretization schemes for eq 3 for assessing the amount of numerical approximations. In the following, a cylindrical coordinate system will be adopted: z, r, and θ are the axial, radial, and tangential axes, respectively, and the origin is placed at the center of the downstream surface of the last static mixer insert; the axial and the radial velocities are positive in the flow direction and from the center of the tube toward the walls, respectively.

4. RESULTS AND DISCUSSION In section 4.1, the calculated results on the laboratory scale SMV static mixer will be compared with the available

Figure 6. Comparison of predictions of different modeling approaches with experimental data by Karoui et al.14 at z = 16 mm. Radial profiles of the mean (a) axial and (b) radial velocities.

velocity field inside the pipeline equipped with one SMV mixer insert. In this configuration, different simulations were carried out, in order to check the influence of the main modeling and numerical settings: the turbulence model, the wall region treatment, the grid size, and the discretization order. The axial and radial velocities along a vertical line placed at 16 mm from the mixer downstream surface, as obtained from different grid densities, are reported in Figure 5, together with the relevant experimental data. As can be observed in Figure 5a, a noticeable difference is obtained comparing the denser B and the coarser A grid axial velocity results, while almost coincident values are obtained with the B and C discretizations. The axial velocity profiles obtained with the two denser grids are closer to the experimental data than that of the coarser grid (grid A). The radial velocity profiles shown in Figure 5b lead to similar conclusions. Also in this case, the effect of the grid refinement is appreciable between the A and B cases, while it is definitely slight between the B and C grids. Overall, the predicted axial and radial velocities catch the main features of the experimental profiles although a closer agreement is obtained in some locations than in others along the radius. Focusing on the mean velocity values, it can be said that for the present computational domain the grid independency is achieved with 2 200 000 cells.

Figure 5. Radial profiles of the mean (a) axial and (b) radial velocity components at z = 16 mm computed by different spatial discretizations. Comparison with the experimental data by Karoui et al.14

experimental data for a quantitative analysis and evaluation of the simulation strategy. Afterward, the geometrical effects on the mixing performances of the large scale apparatus results will be discussed in section 4.2. 4.1. Laboratory Scale SMV Static Mixer. The verification of the present model was performed with the analysis of the D



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In Figure 6, the same experimental data and the finer grid results are compared with the available computational results obtained by Karoui et al.14 in 1997, by adopting a simplified geometrical model and experimental velocity data as boundary conditions, as imposed by the limitations of the CFD codes about one decade ago. As can be observed, the profiles computed with the two CFD approaches are significantly different. Clearly, the description and the discretization of the real equipment geometrical configuration, which are nowadays viable, have allowed achieving more accurate results. Moreover, the fully predictive simulation approach adopted in this work allows overcoming the need for experimental information as boundary conditions of the simplified domain. The present model is able to catch the equipment behavior also very close to the static mixer, as shown in Figure 7, where the radial profiles of the axial and radial velocity components at 2 mm from the mixer outlet section are reported. The grid density effect already discussed is confirmed at this location. In any case, possible differences in the details of the experimental and computational geometries might be the reason for the deviation from the experimental data, since the flow field close to the mixing elements is strictly dependent on geometrical features of the plates. To check the influence of the turbulence model on the velocity flow field, as an alternative to the standard k−ε model, the realizable k−ε model and the Reynolds stress model have also been adopted, while maintaining the same numerical settings of the standard cases. The resulting flow fields inside the static mixer and in the downstream pipeline are reported in Figures 8 and 9, respectively. The velocity vector plots shown in Figure 8 show that at the entrance of the channels the fluid is locally recirculated close to the plate tip and then moves toward the exit following the plate inclination. The three turbulence model predictions are close to each other, as can be also appreciated in Figure 9, where the velocity magnitude maps on a cross section downstream the mixing insert are shown. The results shown in Figures 8 and 9 attest that the adoption of the two different k−ε model versions leads to almost coincident results, while slight variations of the flow field predictions are obtained with the RSM. A wider range of experimental data would be required for the identification of

Figure 7. Radial profiles of the mean (a) axial and (b) radial velocity components at z = 2 mm computed by different spatial discretizations. Comparison with the experimental data by Karoui et al.14

Figure 8. Velocity vector plots inside the SMV element on the horizontal cross section through the pipeline axis (grid C). E



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Figure 9. Maps of velocity magnitude on the vertical cross section z = 30 mm (grid C). The color scale is in meters per second.

Figure 10. Maps of velocity magnitude at different horizontal planes (axial coordinate limited to 10 mm upstream and 50 mm downstream the element). The color scale is in meters per second.

the more accurate model. Since in all cases the flow field features are similarly caught either with the eddy viscosity turbulence models or with the more advanced RSM, the standard k−ε model21 can be adopted for the mean flow field prediction. A closer view of the velocity field predicted by the standard k−ε model on different axial cross sections of the static mixer is reported in Figure 10. The velocity magnitude in the element channels is not uniform, and it exhibits significant differences moving from one channel to the other. Fluid bypass can be observed particularly in the upper and lower sections, due to the geometrical characteristics of the plates. The stagnant zones at the inlet section already observed at the central section are confirmed at the other elevations. Zones of lower velocity take place also at the outlet, but at different positions depending on the cross section elevation. Regarding the treatment of the near-wall region, the results obtained with the three different models show minor differences in the flow field prediction and therefore models more complex than the standard wall functions are not adopted in the following investigation.

As for the turbulence variables predictions, a specific analysis on the grid dependency of the results and on the impact of turbulence models has been performed. The importance of assessing the grid independency of the numerical results by the evaluation of variables related to the turbulent characteristics of the flow, rather than to the mean variables only, has been recently pointed out by Coroneo at al.30 for the turbulent flow predictions in stirred tanks based on a similar computational approach. As an example of the dependency of the turbulent kinetic energy on the computational grid density, the color maps of k on the cross section at z = 16 mm are shown in Figure 11. Major variations are obtained between the grid A and B calculations, but as a difference from the calculated mean velocity, a further increase of the k velocity maximum value is obtained with the finest grid C. Therefore, care must be taken in assessing the grid dependency of the results adding to the mean velocity also the turbulent variables analysis. As for the turbulence model results, the comparison of the predicted k values with the three turbulence models on the horizontal plane along the pipeline axis is shown in Figure 12. The overall features of the maps after the mixer are similar, and F



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Figure 11. Maps of turbulent kinetic energy at z = 16 mm. The color scale is in meters squared per second squared.

Figure 12. Maps of turbulent kinetic energy on the plane at y = 0 (grid C). The color scale is in meters squared per second squared. Figure 14. Effect of eq 3 discretization scheme on the dimensionless additive concentrations along the axial coordinate (Vrhod/Vwater = 24).

as expected, differences in the k values can be appreciated between the k−ε models and the RSM, particularly in the inlet and outlet zones of the mixing element. For what concerns turbulent variables a rigorous comparison with experimental data is not possible, since on the experimental side only the mean values of the three rootmean-square velocity components at selected locations along the pipe diameter at z = 16 mm are available.14 From the three velocity fluctuation mean data, the resulting experimental approximate value of the turbulent kinetic energy is equal to 0.15 m2/s2. The mean k values computed on the same diameter from the standard k−ε, the realizable k−ε, and the RSM simulation results are 0.09, 0.1, and 0.07, respectively. The typical underestimation of the turbulent kinetic energy obtained by the RANS simulation of stirred tanks30 is found also in this application. The comparison of the results leads to the conclusion that on a qualitative basis the two versions of the k−ε model predictions are closer to the experimental data than the RSM.

Figure 13. Experimental15 and calculated dimensionless additive concentrations along the axial coordinate (Vrhod/Vwater = 24).

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From the results discussed above, it is possible to conclude that the solution of the RANS equations coupled with the standard k−ε model and the standard wall functions is not improved by the application of more complex modeling choices. After the turbulent flow field analysis, the mixing behavior of the system has been considered following the dimensionless rhodamine concentrations at r = 0 along the pipe axis. Also in this case, the influence of the grid density was checked at first and the results are shown in Figure 13, where the computed results are compared with the experimental data of Karoui et al.15 As already observed for mean flow field, acceptable results are obtained by the selected CFD modeling approach with both grid B and grid C, while with the coarsest grid a clear underprediction of the concentration along the axial coordinate is obtained. It is worth observing that, as already discussed for the turbulent variables, the mean velocity and the tracer concentration exhibit a different dependency on the grid density. Indeed, the tracer concentration distribution is mainly determined by the local turbulent viscosity and therefore by the turbulent kinetic energy, k, and the dissipation of turbulent kinetic energy, ε, fields.30 The relative error of the finest grid calculations with experimental results 2D after the outlet section of the mixer is always lower than 4%. The influence of the discretization order of the convection− diffusion equation on the simulation results was also analyzed for grid B, comparing the concentration profiles obtained with a first order, a second order, a QUICK, and a MUSCL discretization schemes. As can be observed in Figure 14, except for the first order scheme, which is well-known to be overly diffusive, the adoption of all the other discretization schemes did not produce any difference in the results. For this reason, the second order discretization scheme was selected in the rest of this study. It is also important to underline that the predicted tracer concentration distribution may be affected by numerical diffusion. Although this false diffusion is generally more important in the case of laminar flow, since false diffusion easily masks molecular diffusion, it has to be controlled also in the case of turbulent conditions. A quantitative comprehensive evaluation of the false diffusion magnitude is available for simple first order schemes,31−33 while it is more complex for general flows.33,34 It is well-known that false diffusion decreases with the mesh size, using high order discretization schemes and reducing the angle between the velocity vector and the grid line. In our specific case, the geometry of the mixer and the complex flow field do not permit the use of a grid aligned with the flow direction, but the optimization of the cell size, that leads to a local grid Péclet number lower than 5 in almost all the computational domain, together with the assessment of the accuracy of the discretization scheme, allow excluding a significant influence of the false diffusion on the present computational results. The capability of the model to catch the influence of different rhodamine inlet velocities can be observed in Figure 15, where the simulation results obtained for three different working conditions are compared with the relevant experimental data. In all cases, good predictions are obtained with the finer discretization. The mean relative error between the experimental data and the computational results depends on the value of the rhodamine flow rates: it is equal to 18, 3, and 8% for the velocity ratios of 12, 24, and 48, respectively. It is worth

Figure 15. Experimental15 and calculated dimensionless additive concentrations along the axial coordinate for different velocity ratios and grid C.

Figure 16. Comparison of the CoV values along the pipeline axis. Effect of the distributor geometry.

Figure 17. Effect of LM on the CoV at z = 3D from the outlet section of the first mixer. Solid line: reference value obtained with one mixer.

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Figure 18. Additive volumetric fraction maps (a) on the axial section of the pipeline and (b) at different cross sections.

observing that the local values of concentration exhibit higher gradients closer to the element exit; therefore, for small z/D coordinate, slight differences between the experimental and the computational positions can result in big errors. The relative error at z/D = 4.05, which is typically a sufficient distance to achieve good homogenization, is 10, 1, and 3% for the velocity ratio equal to 12, 24, and 48, respectively. Overall, both the mean flow field and the mixing performances of the laboratory scale static mixer are fairly predicted by the present computational model, thus confirming the usefulness of fully predictive CFD methods for the analysis and optimization of these mixing devices. 4.2. Large Scale Static Mixers. Once the model was shown to be able to predict the main behavior of the mixing process, it was adopted for the analysis of an industrial scale application. First of all, the influence of the distributor design was studied and the results are reported in Figure 16, where the performances of the static mixer with the two distributors are

compared using the coefficient of variation (CoV), which is defined as N

CoV =

∑i = 1 (ci − cmean)2

1

N−1

cmean

(4)

where ci is the local concentration at the ith evaluation point, cmean is the calculated mean concentration of the additive on the cross section, and N is the number of evaluation positions at the actual cross section, which for the presented results correspond to each grid cell in the considered section. The improvement of the mixing performances achieved by the purposely designed distributor with respect to the simple tube is apparent, as reported by the remarkable difference between the two CoV curves along the pipeline axis. The pipeline length for achieving the well-mixed condition, which is often identified by a CoV value of 0.05,1 decreases from 20D to 3D by optimizing the design of the distributor. I



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The same computational strategy was also applied to an industrial scale apparatus for investigating the effect of geometrical configuration on the mixing performances, and the impacts of the distributor design and of the distance between two consecutive inserts were found to be significant. The results clearly show that the mixing performances are worsened when the inserts are not placed one right after the other, while a purposely designed distributor produces an important improvement in the miscible liquids mixing. This work highlights the strong influence of the geometrical features of the apparatus on the performance of in-line mixing, demonstrating that optimized design can be identified by the adoption of CFD modeling with an important reduction of the operating costs and enhancement of the process efficiency.

From the design point of view, besides the influence of the distributor, it is also important to identify what are the effects of the mixing element number and their relative distance, LM, on the apparatus performances. In Figure 17, the CoV values calculated at a distance of 3D from the downstream surface of the first mixer in the case of the optimized distributor is shown for different configurations. In particular, the effect of LM is shown, and as a reference, a continuous line is also added representing the single static element case. It is clearly observable that the introduction of the second mixing element allows significant decrease of the CoV value, thus improving the mixing performances of the process. With increasing LM, however, the computed values of CoV increase up to about a double value with respect to LM = 0. Therefore, for getting better mixing performances with the minimum pipeline length, the distance LM has to be chosen as small as possible. Moreover, it is important to highlight that the mixing element distance does not influence the pressure drop. An almost constant value of approximately 8800 Pa was obtained in all cases for the pressure drops between the upstream surface of the first static mixer insert and a surface at 3D from the downstream section of the first mixer insert. Therefore, in this case the geometrical optimization allows achievement of a better mixing performance without any significant variations of the energetic costs. Finally, the additive concentration maps on different cross sections of the apparatus for the optimized geometrical configuration are shown in Figure 18, in order to highlight the local features of the mixing of the two fluids. The results depicted in Figure 18a show that the local concentration differences along the vertical cross section are importantly dampened downstream the second mixer; also a distance of 3D is required for achieving a perfect mixing, as already identified by the CoV analysis. For a detailed analysis, the additive distribution on the cross sections normal to the pipeline axis are also reported in Figure 18b, where the presence of local gradients is highlighted. The local three-dimensional maps of the concentration obtained by CFD can provide important information in a number of industrial applications, for instance, for reactive inline mixing systems, since they allow performance of advanced analysis on the mixing processes and eventually evaluation of further key process variables, overcoming the limitations of single parameters such as the CoV.35

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AUTHOR INFORMATION

Corresponding Author

*Tel.:+39 051 2090403. Fax: +39 051 6347788. E-mail: mirella. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to thank Dr. Silvio Castagnone for useful discussions. The technical support of Pittaluga s.r.l. is gratefully acknowledged.

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NOMENCLATURE ci = local concentration, kg m−3 cmean = calculated mean concentration, kg m−3 c0 = rhodamine concentration at the additive inlet, kg m−3 CoV = coefficient of variation defined in eq 4, dimensionless D = pipeline diameter, m Dm = molecular diffusivity, m2 s−1 LM = mixer distance, m N = number of evaluation positions, dimensionless p = pressure, Pa U = mean velocity vector, m s−1 u′ = fluctuating velocity vector, m s−1 Vrhod = rhodamine inlet velocity, m s−1 Vwater = water inlet velocity, m s−1 r = radial coordinate, mm z = axial coordinate, mm

Greek Symbols

5. CONCLUSIONS In this work, the fluid dynamic behavior of both a laboratory scale pipeline and an industrial scale pipeline equipped with corrugated plate static mixer inserts was investigated by CFD simulations based on the solution of the RANS equations. For the purpose of verifying this fully predictive modeling strategy, a literature configuration, for which experimental data were available, was considered as a benchmark. The most widespread simulation RANS method for industrial flows has been selected, namely that based on the k−ε model closure and the standard wall function for the treatment of the near-wall region; additionally, different turbulence models and numerical options have been also adopted in order to identify their impact on the turbulent flow field predictions. The model was found reliable enough in predicting the main features of both the velocity field and the mixing behavior, and therefore the reliability of fully predictive RANS models in the design and the rating of the apparatus is confirmed.

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Φ = tracer volumetric fraction, dimensionless μt = turbulent viscosity, Pa s μ = viscosity, kg m−1 s−1 ρ = density, kg m−3 σt = turbulent Schmidt number, dimensionless

REFERENCES

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