(PDF) A Numerical Study of Mixing Performance

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A Numerical Study of Mixing Performance of High-Viscosity Fluid in Novel Static Mixers with Multitwisted Leaves Huibo Meng, Feng Wang, Yanfang Yu, Mingyuan Song, and Jianhua Wu* Engineering and Technology Research Center of Liaoning Province for Chemical Static-Mixing Reaction, School of Energy and Power Engineering, Shenyang University of Chemical Technology, Shenyang 110142, Liaoning, P. R. China ABSTRACT: The flow performance of a high-viscosity fluid in novel static mixers with multitwisted leaves was investigated numerically in the range of Re = 0.1−150. The effects of mixing-segment construction, Reynolds number, and aspect ratio on the chaotic mixing characteristics of different static mixers were evaluated based on the Lagrangian tracking method. The tracer particle distributions, G values, extensional efficiency characteristics, and stretching fields were used to evaluate the dispersion and distribution mixing performances in the new static mixers. Compared with the Kenics static mixer (KSM), the static mixers with three twisted leaves (TKSM) and four twisted leaves (FKSM) achieved chaotic mixing status much earlier and could also maintain this status by successive mixing-element groups. In contrast, there were large unmixed zones in the static mixer with double twisted leaves (DKSM). Stretching rates calculated from pathlines were found to be in good agreement with results reported in the literature. The particle trajectories revealed that the logarithm of the stretching rate increased linearly with the dimensionless axial length. For a given length of static mixer, a decrease in aspect ratio benefited an increasing stretching rate. When the number of multitwisted leaves in the cross section was greater than 2, the range of the probability density curve became larger than that of the KSM. All of the static mixers were found to have small groups of material points experiencing very high stretching. The TKSM and FKSM were found to have higher mixing efficiencies than the KSM, whereas the DKSM exhibited a worse micromixing ability. ratio.12,13 Mixing performances of a KSM at various Reynolds numbers were compared using dynamical systems analysis and variation coefficients.14 Comparing the KSM with a Lightnin Series 45 mixer, Regner et al.15 reported that the intensity of the vortices in the Lightnin mixer was higher than that in the KSM and the formation of vortices was affected by both the pressure drop factor and the rate of striation thinning. For higher Re values close to a point when secondary flow started to have significance for the rate of striation thinning, a lower viscosity of the added liquid resulted in an increase in mixing performance.16,17 The unsteady characteristics of turbulent flow in KSMs have been of interest and attracted more attention during the past 5 years. To evaluate the influence of the number of mixing elements on the velocity distribution and turbulence, the flow field inside a KSM with a diameter of 0.04 m and an aspect ratio of 1.25 was measured using laser Doppler velocimetry.18,19 The self-correlation function, mutual-correlation function, correlation dimension, power spectrum, maximum Lyapunov exponent, and Kolmogorov entropy have been utilized to study the linear and nonlinear characteristics of velocity fluctuation signals in KSMs.20−23 In addition, the chaotic characteristics of instantaneous pressure fluctuation signals in KSMs have been evaluated based on multifractal and recursive analysis.24−28 Tajima et al.29 proposed a new method for ocean disposal of anthropogenic CO2 using a KSM. They conducted experiments

1. INTRODUCTION Fluid mixing is ubiquitous and essential in the chemical process industries. Mixing processes range from simple blending1 to complex chemical reactions2 for which the reaction yield and selectivity are highly dependent on the mixing performance. Improper mixing can result in nonreproducible processing and lowered product quality, with the associated need for more elaborate downstream purification processes and increased waste disposal costs.3 Whether a mixer can yield superior mixing efficiency using less energy has gradually become a key standard for characterizing a specific mixer’s performance.4 Static mixers are employed in-line in a once-through process or in a recycle loop where they supplement or even replace a conventional agitator.5 They offer a more controlled and scalable rate of dilution in fed batch systems and can provide homogenization of feed streams with a minimum residence time.6 Their use in continuous processes is an attractive alternative to conventional agitation because similar and sometimes better performance can be achieved at lower cost.7 Other potential advantages of static mixers over conventionally agitated vessels were summarized by Thakur et al.8 The Kenics static mixer (KSM), called the standard static mixer, has been on the market for about 60 years. Hobbs et al.9 numerically investigated the chaotic properties of mixing in a Kenics mixer. They found that the average stretch grew exponentially with the number of flow periods by calculating the stretching of material elements.10,11 A static mixer equipped with elements twisted by 120° was found to be more energyefficient than the standard Kenics geometry, whereas a static mixer with elements in the same twist direction was found to have segregated islands in the flow. The mixing in both cases exhibited a dependence on the injection location and flow © 2014 American Chemical Society

Received: Revised: Accepted: Published: 4084

September 9, 2013 January 18, 2014 February 19, 2014 February 19, 2014 dx.doi.org/10.1021/ie402970v | Ind. Eng. Chem. Res. 2014, 53, 4084−4095

Industrial & Engineering Chemistry Research

Article

isothermal. The governing continuity and momentum equations can be written as15,32

to investigate the variations in size and distribution of the CO2 drops in the mixer and analyzed the corresponding data using a statistical method.30 A drop dispersion of a high-viscosity polymer material in low-viscosity hexadecane was made using a static mixer. The spherical single beads had a narrow and normal size distribution. The experimentally determined dimensionless diameter of the swollen hydrogel beads was well correlated with the Weber number, degree of swelling, and viscosity ratio.31 Jaworski et al.32 investigated the liquid−liquid drop breakage of a static mixer with 24 standard Kenics inserts by means of advanced numerical modeling tools based on computational fluid dynamics (CFD), whose results agreed well with data reported in a previous experimental study. Tozzi et al.33 evaluated the mixing of the minor component in the major component in a split-and-recombine mixer qualitatively and quantitatively using a magnetic resonance imaging technique, and they observed molecular diffusion of the paramagnetic tracer by establishing the multilamination pattern. As can be noticed from most of the previous studies, new designs for static mixers are being developed, and new applications are being explored.8,34 Wu35 proposed a novel static mixer with higher efficiency of dispersion mixing than the KSM, which is beneficial to the production of propylene oxide from the propylene chlorohydrination reaction. Unfortunately, only a few studies of the flow characteristics of low-viscosity liquids under turbulent conditions have been conducted. Meng et al.36,37 simplified the relationship among all factors influencing turbulent flow resistance in a new type of static mixers using dimensional analysis and the π theorem and developed a general correlation for the friction coefficient of the flow resistance. Gong et al.38 concluded that, compared with the KSM, a static mixer with four twisted leaves (FKSM) could significantly improve the coordination between the velocity field and the temperature field and increase the heat-transfer efficiency by about 20% for Re = 104−105. Zhang et al.39 found that the effects of the dislocation angle on the Nusselt number and pressure drop were more significant than those of the rotation direction of the elements in static mixers with doubletwisted blades. However, turbulence is precluded when highviscosity fluids are mixed, or it must be avoided to protect delicate shear-sensitive materials suspended in the flow. In such cases, chaotic laminar flows can provide a practical alternative, but they have a significant potential to lead to inhomogeneities and poorly mixed regions within the flow system.10 In the current work, the flow performance of a high-viscosity fluid in novel static mixers with multitwisted leaves was investigated numerically in the range of Re = 0.1−150. The effects of mixing-segment construction, Reynolds number, and aspect ratio on the chaotic mixing characteristics of different static mixers were evaluated based on the Lagrangian tracking method. The accuracy of the velocity field from a high-quality CFD solution was validated by comparing the pressure drop with the empirical correlations. The tracer particle distributions, G values, extensional efficiency characteristics, and stretching fields were used to evaluate the dispersion and distribution mixing performance in the new static mixers. The probability density distributions of the stretching rate in the different static mixers were also investigated.

(1)

∇·v = 0 ρ

∂v + v·∇(ρ v) = −∇p + μ∇2 v + F ∂t

(2)

where ρ is the fluid density, μ is the dynamic viscosity, v is the velocity vector, t is the time, and p is the total pressure. The source term F includes the volumetric forces. The governing equations in generalized curvilinear coordinates were discretized on a nonstaggered grid using a finitevolume approach. A pressure-based solver was employed with a second implicit scheme for incompressible flows in the static mixers. The pressure−velocity coupling was handled using SIMPLEC algorithm. The pressure and momentum calculations were carried out by the standard and second-order upwind schemes, respectively. Gradients were needed not only for constructing values of a scalar at the cell faces, but also for computing secondary diffusion terms and velocity derivatives. The gradient of a given variable was used to discretize the convection and diffusion terms in the flow conservation equations. The gradients were computed using ANSYS FLUENT CFD software according to the Green-Gauss cellbased discretization method. The normalized residuals of the continuity and velocity components had to be less than 10−4 and 10−6, respectively. Accordingly, the physical properties of the high-viscosity fluid, namely, density (ρ) and viscosity (μ), were taken as being constant in this study, with values of ρ = 1200 kg/m3 and μ = 0.5 N·s/m2. 2.2. Physical Model. To improve the understanding of how static mixers work and how they could be better utilized in chemical engineering, both a KSM and three novel static mixers were constructed, as shown in Figure 1. A KSM is usually composed of a number of Kenics blades with a twist angle of 180°, as also described elsewhere,32,40 each rotated by 90° relative to the previous one.8−14 Static mixers with double twisted leaves (DKSM), three twisted leaves (TKSM), and four twisted leaves (FKSM), as presented in panels b−d of Figure 1, contained two, three, and four blades, respectively, that were distributed uniformly in the mixer cross section. The upstream groups of mixing elements were rotated with angles of 90°, 60°, and 45° with respect to the downstream groups, with reverse twist directions in the axial direction. For the same mixing length, the KSM contained 6 elements, whereas the novel static mixers contained 12 element groups in the axial direction. Some other geometrical parameters are listed in Table 1. The three-dimensional models of different static mixers constructed by SolidWorks were imported into Gambit for grid meshing. A three-dimensional unstructured mesh was generated for all types of static mixers, and the grid quality was checked for skewness. 2.3. Boundary Conditions. The commercially available CFD software FLUENT was applied to solve the continuity and momentum equations. The open-tube Reynolds number is defined as Re =

Dvρ μ

(3)

where D is the diameter of the pipe containing the static mixer. The inlet velocity employed in the simulations was in the range of v = 0.0010−1.5625 m/s; that is, Re ranged from 0.1 to 150. Considering the viscosity of the fluid, a certain length of empty

2. NUMERICAL METHODOLOGY 2.1. Basic Equations. The general assumptions for the model used are that the flows are laminar, incompressible, and 4085

dx.doi.org/10.1021/ie402970v | Ind. Eng. Chem. Res. 2014, 53, 4084−4095


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empirical correlations,42 which allowed us to apply the numerical method to the studies in the present work.

Figure 1. Static mixers with different groups of mixing elements: (a) KSM, (b) DKSM, (c) TKSM, (d) FKSM.

pipe would be needed for fully developed laminar flow. The length of the inlet tube has usually been twice the diameter in most previous studies. In the present study, a user-defined function (UDF) was used to generate a parabolic inlet profile, which meant that there was no need to reserve a longer inlet pipe for fully developed flow. Thus, the total length of the mixer was decreased, and the number of grids was reduced. The reference pressure point was set at the geometric center of the outlet section, whose boundary condition was set as outflow. 2.4. Model Validation and Grid Independence Test. In CFD, pressure is more sensitive to perturbations than velocity and concentration.41 Consequently, the precision of the computational results for the velocity field greatly depends on the pressure field’s accuracy. The CFD model in this work was validated by comparing the predicted results against empirical correlations available in previous literature. Most empirical correlations for the pressure drop Δp in static mixers are based on the flow resistance Δp0 in empty pipes, which can be computed as Z = Δp/Δp0. Using this formula, we calculated the pressure drop of a KSM constructed for model validation. It is clear in Figure 2a that the numerical results agreed well with the

Figure 2. Model validation and grid independence test.

To ensure that the simulated results were independent of the grid system, grid independence was checked. The grid size increased from 1 to 3 mm with a step size of 0.5 mm at Re = 0.1. The grid quality was checked for skewness, and the maximum Equisize skew values of all meshes were kept below 0.78. Figure 2b shows that the dimensionless magnitude of the velocity of the outlet first increased and then decreased with increasing grid size. The normalized velocities of the outlet at grid sizes of 1 and 1.5 mm differed numerically by 0.11%. However, the normalized velocity deviations increased when the grid size was not less than 2 mm. To ensure computational accuracy and efficiency, a grid size of 1.5 mm was chosen for grid meshing.

Table 1. Geometrical Parameters of Static Mixers KSM tube diameter, D (mm) blade width, W (mm) aspect ratio, Ar blade thickness, δ (mm) twist angle, θ (deg) entrance length, li (mm) mixing-zone length lm (mm) exit length, lo (mm)

40 40 1 2 180 25 240 25

DKSM

1.5

2

360

480

40 20 1 2 180 25 240 25

TKSM

1.5

2

360

480

4086

40 20 1 2 180 25 240 25

FKSM

1.5

2

360

480

40 20 1 2 180 25 240 25

1.5

2

360

480



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Figure 3. Profiles of three-dimensional velocity versus dimensionless axial distance at Re = 100.

Figure 4. Velocity distributions of different cross-sectional profiles at Re = 100: (z − li)/L = (a) 17/8, (b) 9/4, (c) 19/8, (d) 5/2, (z − li)/l = (e,i,m) 41/8, (f,j,n) 21/4, (g,k,o) 43/8, (h,l,p) 11/2.

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3.2. Qualitative Analysis of Mixing. When a fluid passes successively through spatially periodic mixing groups, the streams in the static mixers are divided and forced to change rotation. The interaction between flow division and flow reversal gives rise to many secondary flow vortexes in the plane perpendicular to the predominant axial flow. The flow in the novel static mixers will be complex and chaotic due to the effects of unfolding, stretching, and folding of striations caused by the secondary flows. To evaluate the chaotic mixing performance qualitatively, the Lagrangian tracking method is a good choice. Using this method, a set of 5000 massless particles was distributed uniformly along the diameter at the inlet section, and their trajectories were calculated individually. The position of each particle was recorded when it passed through every periodic section. The performance of distributive mixing, which is the operation of improving the spatial distribution of agglomerates of particles, can be characterized by particle concentrations. Figure 5 displays the particle distributions after some flow

3. RESULTS AND DISCUSSION 3.1. Velocity Fields. To investigate the complexity and periodicity of the velocity fields in static mixers, threedimensional velocities were studied along a line parallel to the axis with the given radial position (x = R/2, y = R/2). Figure 3 shows the velocities in the x, y, and z directions as functions of the axial position at Ar = 1.5. It is clear in Figure 3 that the z-direction velocities in all static mixers always remained above 0. The fluctuation amplitudes of the zdirection velocity decreased as the number of mixing leaves in a cross section increased, so the uniformity of the velocity was improved, which contributed to mixing in the axial direction. As the number of leaves in a mixing group increased, both the x and y velocities become increasingly complex. This is because more mixing leaves are beneficial to the enhancement of radial secondary flow instabilities. With the improvement of multifield synergy,38,39 the fluid in the cross section was continuously updated, which was good for radial diffusion. Based on this analysis, one can conclude that the flows in static mixers have plug-flow behaviors. Moreover, for spatially periodic structures, the fluctuations of the velocities also appeared to be periodic and to have the same periods as the structures, namely, a period of two mixing groups. More detailed descriptions of velocity fields in different static mixers were obtained through two-dimensional cross-sectional profiles of the velocity field. Figure 4 displys the velocity distributions of cross sections of different static mixers. In the figure, z, li, L, and l denote the z coordinate of the cross section, the entrance length, the length of Kenics leaves, and the length of multitwisted leaves, respectively. Some cross sections in the third mixing element of the KSM and the sixth mixing group of static mixers with multitwisted leaves were chosen as objects for analysis. Vectors represent the secondary flow in the x−y plane, whereas contour maps indicate the axial velocity. Some forced vortices were generated around the mixing blades, and their number was the same as the number of mixing blades in the section. The rotation directions of these vortices agreed with the twist directions of the mixing blades. These forced vortices together with the free vortices in the void section constitute Rankine vortices, which can enhance the secondary flow and pulsation of the fluid. The first two columns of Figure 4 show the velocities in the transition of two adjacent mixing-element groups. The fluid field is divided by the upstream mixing group and is reoriented in the space formed by the pipe wall and mixing-element groups, which can be seen exactly from the velocity contours. It can be observed in the first row of Figure 4 that the velocity profiles at 2 < (z − li)/L < 9/4 were still affected by the upstream mixing element for the KSM. The flow patterns gradually became steady at (z − li)/L = 2.3. This means that the cross-sectional mixing in the laminar flow regime took place up to 30% of L at the element-to-element transition, which agrees well with the numerical results reported by Kumar et al.43 Similar steady patterns can be attained at (z − li)/l = 6.23, 6.17, and 6.1 for the DKSM, TKSM, and FKSM, respectively. This indicates that cross-sectional mixing takes place up to 23% 17%, and 10% of l, respectively. For the lengths of the mixing elements, a value of L = 2l was chosen. Therefore, the regions of flow instabilities at the element-to-element transition were reduced largely by the multitwisted blades, which means that the lengths of distribution mixing increased in the novel static mixers

Figure 5. Poincare maps of particle distributions after several periods at Re = 0.1.

periods. The KSM had an almost uniform particle distribution at (z − li)/l = 8, but some unmixed sections still could be seen. As the pathline extended, a good particle distribution was obtained for the KSM at (z − li)/l = 12. Local uniform distributions of massless particles were obtained near the helical blades in the DKSM, and there were large unmixed areas from start to finish. With an increase in dimensionless axial positions, the unmixed areas decreased and still existed at the end section. For the TKSM and FKSM, uniform particle distributions formed at (z − li)/l = 8 and were maintained until the end. 4088



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Owing to the chaotic nature of the flow in static mixers, the distribution structure obtained after some flow periods was robust. In other words, the main role of the mixing-element groups at the beginning is to enhance dispersive ability, whereas that of the successive groups is mainly to maintain this dispersive mixing performance. It would appear that the TKSM and FKSM are more efficient and that the KSM is slightly weaker. The dispersive mixing ability of the DKSM was the weakest and was not fit for the mixing of two immiscible fluids. 3.3. Quantitative Analysis of Mixing. 3.3.1. Calculating G Values. Originally, the root-mean-square G value was developed to quantify the mixing flocculation basins by analogy with the shear rate in a simple, one-dimensional, laminar shear flow (Couette flow).44 Since then, the G value has become a common measurement of mixing and has been used to characterize mixing intensity in a wider variety of mixers or other applications.45 The G value remains entrenched in the engineering literature and still continues to be used because it can be easily calculated from numerical models.45−47 Therefore, the G value was selected as one criterion in this work for assessing the mixing efficiency. For any in-line mixer, the G value is calculated based on the energy losses that occur in the mixer as45 G=

P = μVm

Q Δp μVm

(4)

where P is the power dissipated in the mixer, Vm is the mixer volume, Q is the volumetric flow rate, and Δp is the pressure drop across the mixer. The volume flow rate in the static mixer can be calculated from the velocity field as

Figure 6. G values for different static mixers.

α=

2

Q=

πD v 4

(5)

1/2

⎛ ReΔp ⎞1/2 ⎜ ⎟ ⎝ ρ ⎠

(7)

where |γ| and |ω| are the norms of the rate of deformation tensor and the vorticity tensor, given by

By substituting eqs 3 and 5 into eq 4, one can obtain the following expression for the G value of a static mixer 1 ⎛ πD ⎞ G= ⎜ ⎟ 2 ⎝ Vm ⎠

|γ | |γ | + |ω|

(6)

γ=

1 [∇v + (∇v)T ] 2

ω=

1 [∇v − (∇v)T ] 2

(8)

(9)

where ∇v is the gradient of velocity. The value α = 1 corresponds to pure extension, α = 0.5 corresponds to simple shear, and α = 0 corresponds to pure rotation. Generally speaking, a higher α value means better mixing. Figure 7 shows the extensional efficiency distribution of different cross sections in the same fashion as Figure 4. It is clear in the figure that the transition part of every static mixer covers larger areas for higher α values. This indicates that the transition parts of two mixing-element groups play an important role in the stretching of material, namely, dispersive mixing. On the different sides of the twisted leaves, there exist two high-stretching-efficiency areas whose central positions change with the axial positions of the cross sections. This might be caused by the coupled effect between axial plug flow and radial secondary flow. The average values of extensional efficiency in different cross sections of static mixers are depicted in Figure 8. It can be concluded that the value of α reaches a maximum at (z − li)/l = 0 and (z − li)/l = 12, which correspond to the entrance of the first mixing group and the exit of the last one, respectively. Furthermore, the maximum values in the different static mixers

Figure 6a illustrates the relationship between Re and G for different static mixers. From these logarithmic diagrams, one can see that all of the G values are proportional to the logarithm of Re, which suggests that the static mixing efficiency is improved with increasing Re. For a given Re value, the FKSM was found to have the highest G value, followed by the TKSM, DKSM, and KSM, which indicates that the more mixing leaves in a cross section, the more energy consumed. Figure 6b shows the relationship between G and axial position at Re = 0.1. At the beginning of the mixing pipe, the G values increased gradually, and the rate of increasing decreased. For (z − li)/l > 4, G was nearly stable, and the value difference between adjacent dimensionless positions was less than 1%. The G distributions reveal the different roles of mixing-element groups located at the beginning and the successive part, namely, enhancing dispersive mixing for the beginning part and keeping distribution mixing for the successive part, which confirms the results obtained with particle distributions in section 3.2. 3.3.2. Extensional Efficiency. The extensional efficiency α is an important characteristic for quantifying the relative strength of the pure elongational flow component. It is defined as48 4089



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Figure 7. Extensional efficiency of different cross-sectional profiles at Re = 0.1: (z − li)/L = (a) 17/8, (b) 9/4, (c) 19/8, (d) 5/2, (z − li)/l = (e,i,m) 41/8, (f,j,n) 21/4, (g,k,o) 43/8, (h,l,p) 11/2.

Figure 8. Average extensional efficiency as a function of dimensionless axial position at Re = 0.1.

at (z − li)/l = 2, 4, 6, 8 and 10, which correspond to subsequent element inlets. Elsewhere, the extensional efficiency is close to

were nearly the same. For the KSM, as shown in the top left plot in Figure 8, five slightly less pronounced peaks are present 4090



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To finish all of the calculations, huge amounts of time and resources were used to track fluid particles as described by Hobbs and co-workers9−11 in previous works. Therefore, a new method should be developed soon for high-efficiency calculating. For pathlines used to visualize the flow of massless particles in the problem domain,50 one can obtain position data and velocity gradient data of massless particles through pathlines. With the help of FLUENT, we easily obtained information on all pathlines. By exporting pathline data to files, the prophase integrating course was simplified. To validate the programs written in Matlab using the method mentioned above, we calculated the stretching rate of the numerical model constructed by Saatdjian et al.42 The stretching rates of the first 10 period sections are shown in Figure 9. Comparing the

0.5, the value for an empty tube. This means that only the helix edges have a crucial role in dispersive mixing. Within the helices, only distributive mixing occurs. It can be observed in Figure 8 that the number of slightly less pronounced peaks increased in the static mixers with multihelical blades. The larger ranges of extensional efficiency indicate that more complex flow patterns exist in the novel static mixers than in the KSM. However, a high extensional efficiency in a particular region of a motionless mixer might not be an indicator of good overall mixing because the region might be segregated41 and perhaps no particles pass through these regions, just as in the DKSM. Therefore, other parameters related to stretching need to be discussed for supplement. 3.3.3. Stretching Rate. Ottino49 performed chaos analysis of mixers and provided some indications of the level of chaos. He stated that mixing involved stretching and folding of material elements. The stretching rate determines the rate of the micromixing process, both by increasing the intermaterial area over which interdiffusion of components can occur and also by decreasing the required diffusional distance. Thus, it is of importance to seek a rational method for quantifying the mixing efficiency of the stretching process of laminar flows in different static mixers. By placing a material point in an arbitrary initial location, one can calculate the stretching by tracking the vector attached to the point when it passes through the static mixer pipe, and the positions where the material point experiences high (low) stretching correspond to regions of good (poor) mixing. The positions of material points and their corresponding velocity gradients can be obtained by integration from the Eulerian velocity field. According to Ottino’s theory, the basic measure of deformation with respect to reference configuration is the velocity gradient. Therefore, the position and stretching experienced by each material point can be monitored by integrating and solving the equations42 dx = v(x), dt

x t=0 = x 0

d(I) = (∇v)T ·I, dt

It = 0 = I0

Figure 9. Comparison of the two calculation methods at Re = 0.15.

stretching histories calculated by two methods, we found that our method coincides well with the numerical results of Saatdjian et al. It should be mentioned that many pathlines are needed to make the calculating course more precise. For the model in this work, a type of triangle grid with a size of 0.7 mm (∼1.75% of D) was employed for inlet face mesh refinement, and 5888 pathlines were finally obtained. For the different structures, qualitative differences in stretching rate with respect to different static mixers are expected. An initial stretch vector of I0=(0, 0, 1) was employed for integrating. The stretching histories of different static mixers with Ar = 1.5 at different Reynolds numbers are plotted in Figure 10. It is clear from the figure that there exists an inflection at (z − li)/l = 2 in the curve related to the KSM. This was explained by Hobbs and Muzzio10 as resulting from the initial orientations of stretch vectors being realigned to correspond to the principal stretching directions of the flow at the beginning of the KSM. After this realignment, the initial vector orientations are no longer important, and the logarithm of the stretching rate grows at a linear rate against the dimensionless axial positions. However, the initial vector orientations have little effect on the stretching related to static mixers with multileaves. The DKSM was found to have the poorest stretching efficiency, and its Λn curve was below the curves of all of the other static mixers, whereas the FKSM had the highest stretching efficiency. This can be explained in terms of structure. The FKSM, TKSM, and KSM all have successive structures with lower void rates. For the DKSM, the special arrangement of mixing elements yields a large area without mixing elements where the flow does not encounter bany disturbance. With increasing Re, the Λn curve of the KSM gradually approaches that of the TKSM and is above that of the

(10)

(11)

where x, v, I, and ∇v denote the position vector, velocity vector, stretching vector related to the material point, and velocity gradient, respectively. We used the accumulated stretching λ to represent the stretching value for each material point when it crossed a period section λ=

|I| | I 0|

(12)

After every spatial period, the position and the stretching of every material point were recorded. In this work , the geometric average value of stretching for N material points was applied to represent the stretching rate of each spatial period section N

λg̅ = (∏ λi)1/ N i=1

(13)

For simplification, we introduce the symbol Λn in place of the logarithm of λ̅g Λ n = log λg̅

(14) 4091



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Figure 10. Stretching rates of different static mixers as a function of (z − li)/l at different Re values.

TKSM at Re = 150. This means better stretching is yielded for the KSM than for the TKSM at Re ≥ 150 in creeping flow. Interestingly, we found that the Λn curves have different trends than the G-value curves. The G value is an indicator developed for macromixing,44 whereas the stretching rate is a measure of the micromixing process.10 When the mixing in static mixers is chaotic, the macromixing state is almost steady with little variation, as revealed by the G-value curves (Figure 6) and the particle position distributions (Figure 5). However, material elements are still stretched at the same time. In other words, micromixing is still occurring. This is why the Λn curves always kept increasing whereas the G-value curves exhibit an upward trend at first and is then almost horizontal with increasing values of (z − li)/l. In Figure 11 are plotted , the stretching rates of different static mixers with different aspect ratios at Re = 0.1. Inflections still can be seen in the Λn curves for the KSM at different values of Ar. The Λn curves for all of the static mixers have linear distributions. For the same axial position in a static mixer, Ar = 1 corresponds to the highest stretching rate, whereas Ar = 2 has the lowest stretching value, and Ar = 1.5 has an intermediate value. This indicates that lower aspect ratios correspond to higher stretching efficiencies. For the same mixing length, more mixing elements with low aspect ratios could be arranged than those with higher aspect ratios. Therefore, the fluid passing

through the static mixers with lower aspect ratios will have more opportunities for being stretched and folded. 3.3.4. Statistical Characteristics of the Stretching Rate. The probability density function of the logarithm of the stretching rate is used to describe the distribution of stretching rates of different static mixers. According to its definition, this function can be written as42 Hn(log λ) =

1 d(N log λ) N d log λ

(15)

where d(N log λ) represents the number of material points with stretching rates between log λ and log λ + d(log λ). Hn(log λ) was calculated at the end cross section of the last mixingelement groups and is plotted versus log λ at Figure 12. The value of Hn(log λ) can provide some information about the intensity of the micromixing process. As one can see, the static mixers with more mixing leaves had larger probability density distributions of Hn(log λ), not including the DKSM. Moreover, as the number of mixing leaves contained in a mixing group increased, the probability density curve moved toward higher log λ values, except for the DKSM. This implies that more mixing leaves contribute to the chaos of the flow. The DKSM is a special case because its structure contains a great deal of void space. The probability density curves of all of the static mixers were found to have more fluctuations in areas of high log λ values than in areas of low log λ values. The curves also 4092



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Figure 11. Effect of aspect ratio on stretching rate at Re = 0.1.

The mixing efficiencies of the static mixers were evaluated in terms of G values, particle distribution maps, stretching rates, and extensional efficiency characteristics. A comparison of the particle distribution maps revealed that the fluid in the TKSM and FKSM reached chaotic flow more quickly and the mixingelement groups arranged in the beginning of the mixers played a main role in improving the dispersive mixing whereas the successive ones mainly maintained the dispersive mixing and improved the distribution performance. With an increase in dimensionless axial positions, the unmixed areas in the DKSM decreased and still existed at the end section of the mixer. The results for the G values indicate that the macromixing performances in the DKSM are better than those in the KSM, which is the reverse for micromixing evaluated in terms of stretching rates and particle distribution maps. A smaller aspect ratio was found to correspond to higher stretching in a given mixing length. The probability density functions of stretching rates moved toward higher log λ values as the number of mixing leaves increased, except for the DKSM. The results determined that the ability of micromixing is in the order FKSM > TKSM > KSM > DKSM. Some small groups of material points experienced very high stretching for the four static mixers.

Figure 12. Probability density distribution of the stretching rate at Re = 0.1.

exhibited long tails on the high-stretching side, indicating that a subset of points experienced very high stretching. The absence of extended tails on the low-stretching side of the curves indicates that no large islands of regular, nonchaotic behavior were present in the flow.

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4. CONCLUSIONS The flow fields of three new types of static mixers were studied using a numerical method. Information on positions and velocity gradients was derived from the pathlines of massless particles. By exporting pathline data to files, the prophase integrating course for tracking fluid particles can be simplified significantly. The stretching rate calculated from pathlines was found to coincide well with reference results. It should be mentioned that many pathlines are needed and refinement of the mesh for the inlet face is necessary to make the calculations more precise.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-24-89381016. Fax: +86-24-89381016. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Nos. 21106086 and 21306115), the Program for Liaoning Excellent Talents in University (No. 4093



Article

(8) Thakur, R. K.; Vial, C. H.; Nigam, K. D. P.; Nauman, E. B.; Jelveh, G. D. Static mixers in the process industriesA review. Trans. Inst. Chem. Eng. 2003, 81 (7), 787−826. (9) Hobbs, D. M.; Alvarez, M. M.; Muzzio, F. J. Mixing in globally chaotic flows: A self-similar process. Fractals 1997, 5 (3), 395−425. (10) Hobbs, D. M.; Muzzio, F. J. The Kenics static mixer: A threedimensional chaotic flow. Chem. Eng. J. 1997, 67 (3), 153−166. (11) Hobbs, D. M.; Swanson, P. D.; Muzzio, F. J. Numerical characterization of low Reynolds number flow in the Kenics static mixer. Chem. Eng. Sci. 1998, 53 (8), 1565−1584. (12) Hobbs, D. M.; Muzzio, F. J. Effects of injection location, flow ratio and geometry on Kenics static mixer performance. AIChE J. 1997, 43 (12), 3121−3132. (13) Hobbs, D. M.; Muzzio, F. J. Optimization of a static mixer using dynamical system techniques. Chem. Eng. Sci. 1998, 53 (18), 3199− 3213. (14) Hobbs, D. M.; Muzzio, F. J. Reynolds number effects on laminar mixing in the Kenics static mixer. Chem. Eng. Sci. 1998, 70 (2), 93− 104. (15) Regner, M.; Ö stergren, K.; Trägårdh, C. Effects of geometry and flow rate on secondary flow and the mixing process in static mixersA numerical study. Chem. Eng. Sci. 2006, 61 (18), 6133−6141. (16) Regner, M.; Ö stergren, K.; Trägårdh, C. Influence of viscosity ratio on the mixing process in a static mixer: Numerical study. Ind. Eng. Chem. Res. 2008, 47 (9), 3030−3036. (17) Regner, M.; Ö stergren, K.; Trägårdh, C. An improved numerical method of calculating the striation thinning in static mixers. Comput. Chem. Eng. 2005, 30 (2), 376−380. (18) Wu, J. H.; Zhang, C. M.; Jin, D.; Sun, D.; Gong, B. Experimental study on turbulence properties in a Kenics static mixer. Chin. J. Process Eng. 2008, 8 (4), 714−718 (in Chinese). (19) Gong, B.; Bao, Z. P.; Zhang, C. M.; Wu, J. H. Effect of number of mixing elements on flow field in Kenics static mixer. CIESC J. 2009, 60 (8), 1974−1980 (in Chinese). (20) Yu, Y. F.; Meng, H. B.; Wu, J. H. Delay time correlation of velocity fluctuation signal in the Kenics static mixer. J. Petrochem. Univ. 2008, 21 (4), 73−78 (in Chinese). (21) Meng, H. B.; Yu, Y. F.; Liu, Z. Q.; Wu, J. H. Correlation integral of turbulent instantaneous velocity time series in a Kenics static mixer. J. Beijing Univ. Chem. Technol. Nat. Sci. Ed. 2009, 36 (5), 97−102 (in Chinese). (22) Meng, H. B.; Wu, J. H.; Yu, Y. F.; Chen, X. Chaotic characteristics of velocity in cutting region of Kenics static mixer. Chem. Eng. 2009, 37 (12), 24−27 (in Chinese). (23) Wu, J. H.; Meng, H. B.; Gao, Y. L.; Liu, Z. Q.; Yu, Y. F. Multiscale symbolic analysis of instantaneous velocity time series in the static mixer with twisted leaves. J. Basic Sci. Eng. 2011, 19 (5), 799− 809 (in Chinese). (24) Wu, J. H.; Yu, Y. F.; Wang, Z. Y.; Liu, Z. Q.; Meng, H. B. Recurrence characteristics of instantaneous pulsating pressure signals in static mixer with twisted leaves. CIESC J. 2011, 62 (10), 2845−2853 (in Chinese). (25) Wu, J. H.; Meng, H. B.; Yu, Y. F.; Gong, B. Multi-scale and multi-fractal characteristics of tube-pressure signals in Kenics static mixer. CIESC J. 2009, 60 (8), 1965−1973 (in Chinese). (26) Meng, H. B.; Liu, Z. Q.; Yu, Y. F.; Xiong, Q.; Wu, J. H. Intrinsic mode entropy analysis of tube-wall pressure fluctuation signals in the Kenics static mixer. Int. J. Chem. React. Eng. 2011, 9 (1), 1−9. (27) Meng, H. B.; Gao, Y. L.; Yu, Y. F.; Wu, J. H. Complexity analysis of instantaneous tube-wall pressure fluctuation symbolic series in static mixer with twisted leaves. J. Exp. Fluid Mech. 2011, 25 (5), 30−34 (in Chinese). (28) Meng, H. B.; Liu, Z. Q.; Yu, Y. F.; Wu, J. H. Time-frequency analysis of Hilbert spectrum of pressure fluctuation time series in Kenics static mixer based on empirical mode decomposition. Braz. J. Chem. Eng. 2012, 29 (1), 167−182. (29) Tajima, H.; Yamasaki, A.; Kiyono, F.; Teng, H. New method for ocean disposal of CO2 by a submerged kenics-type static mixer. AIChE J. 2004, 50 (4), 871−878.

LJQ2012035), the Science Foundation for Doctorate Research of the Liaoning Science and Technology Bureau of China (No. 20131090), the Science and Technology Research Project of Education Department of Liaoning Province (No. L2013164), and the Planning Program of Shenyang Science and Technology Bureau (No. F12-188-9-00).

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NOMENCLATURE Ar = aspect ratio of helical leaves D = tube diameter Hn = probability density function of log λ I = stretching vector I0 = initial condition for I L = length of a single-twisted leaf l = length of multitwisted leaves li = entrance length lo = exit length N = number of material particles R = pipe radius Re = Reynolds number P = power lost in the mixer p = total pressure Δp = pressure drop Q = volumetric flow rate v(x) = velocity of a material point as a function of position ∇v = velocity gradient Vm = volume of the static mixer W = blade width x = vector of a material point

Greek Letters

α = extensional efficiency γ = rate of deformation tensor δ = thickness of helical leaf λ = stretching experienced by vector I λ̅g = geometric average value of λ Λn = logarithm of λ̅g μ = viscosity ρ = density ω = vorticity tensor

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REFERENCES

(1) Li, J.; Karimi, I. A. Scheduling gasoline blending operations from recipe determination to shipping using unit slots. Ind. Eng. Chem. Res. 2011, 50 (15), 9156−9174. (2) Bi, R.; Yang, X.; Tan, X.; Zheng, S. A method to predict phosgenation reaction performance to produce toluene diisocyanate in jet reactors. Ind. Eng. Chem. Res. 2013, 52 (44), 15353−15358. (3) Kroon, M. C.; Hartmann, D.; Berkhout, A. J. Toward a sustainable chemical industry: Cyclic innovation applied to ionic liquid-based technology. Ind. Eng. Chem. Res. 2008, 47 (22), 8517− 8525. (4) Jin, Y.; Cheng, Y. Chemical engineering in China: Past, present and future. AIChE J. 2011, 57 (3), 552−560. (5) Tizaoui, C.; Zhang, Y. The modelling of ozone mass transfer in static mixers using back flow cell model. Chem. Eng. J. 2010, 162 (2), 557−564. (6) Barega, E. W.; Zondervan, E.; Haan, A. B. Influence of physical properties and process conditions on entrainment behavior in a staticmixer settler setup. Ind. Eng. Chem. Res. 2013, 52 (8), 2958−2968. (7) Sungwornpatansakul, P.; Hiroi, J.; Nigahara, Y.; Jayasinghe, T. K.; Yoshikawa, K. Enhancement of biodiesel production reaction employing the static mixing. Fuel Process. Technol. 2013, 116, 1−8. 4094



Article

(30) Tajima, H.; Yamasaki, A.; Kiyono, F.; Teng, H. Size distribution of CO2 drops in a static mixer for ocean disposal. AIChE J. 2006, 52 (8), 2991−2996. (31) Tajima, H.; Yoshida, Y.; Abiko, S.; Yamagiwa, K. Size adjustment of spherical temperature-sensitive hydrogel beads by liquid−liquid dispersion using a Kenics static mixer. Chem. Eng. J. 2010, 156 (2), 479−486. (32) Jaworski, Z.; Pianko-Opyrch, P.; Marchisio, D. L.; Nienow, A. W. CFD modelling of turbulent drop breakage in a kenics static mixer and comparison with experimental data. Chem. Eng. Res. Des. 2007, 85 (5), 753−759. (33) Tozzi, E. J.; Bacca, L. A.; Hartt, W. H.; McCarthy, M. J.; McCarthy, K. L. Study of multi-lamination of a non-Newtonian fluid in a split and recombine static mixer using magnetic resonance imaging. Chem. Eng. Sci. 2013, 93 (19), 140−149. (34) Yu, Y. F.; Wang, F.; Meng, H. B.; Wang, Y. F.; Wang, W.; Wu, J. H. Research progress of the characteristics of instantaneous flow in static mixer with twisted leaves. Chem. Ind. Eng. Prog. 2013, 32 (2), 255−262 (in Chinese). (35) Wu, J. H. Propylene chlorohydrination reaction equipment and methods for producing propane epoxy used static mixing with multileaves in a single tube. Chinese Patent CN 200510045606.8, Jun 2, 2007. (36) Meng, H. B.; Wu, J. H.; Yu, Y. F.; Gong, B. Numerical simulation of the flow resistance characteristic in a new type static mixer. J. Petrochem. Univ. 2007, 20 (4), 59−62 (in Chinese). (37) Meng, H. B.; Wu, J. H.; Yu, Y. F.; Chen, X. Numerical simulation of turbulent flow characteristic in new type static mixer. Chem. Eng. (China) 2008, 36 (5), 20−23 (in Chinese). (38) Gong, B.; Zhang, J.; Zhang, C. M.; Wu, J. H. Effect of incorporating assemblies of twist elements on the flow field in a static mixer. J. Beijing Univ. Chem. Technol. 2008, 35 (3), 84−88 (in Chinese). (39) Zhang, J.; Kang, T. X.; Gong, B.; Wu, J. H. Effect of combination method of double-twisted blades on turbulent convective heat transfer. CIESC J. 2011, 62 (S2), 52−60 (in Chinese). (40) Liu, M. Age distribution in the Kenics static micromixer with convection and diffusion. Ind. Eng. Chem. Res. 2012, 51 (20), 7081− 7094. (41) Rauline, D.; Tanguy, P. A.; LeBlevec, J.; Bousquet, J. Numerical investigation of the performance of several static mixers. Can. J. Chem. Eng. 1998, 76 (3), 527−535. (42) Saatdjian, E.; Rodrigo, A. J. S.; Mota, J. P. B. On chaotic advection in a static mixer. Chem. Eng. J. 2012, 187 (1), 289−298. (43) Kumar, V.; Shirke, V.; Nigam, K. D. P. Performance of Kenics static mixer over a wide range of Reynolds number. Chem. Eng. J. 2008, 139 (2), 284−295. (44) Camp, T. R.; Stein, P. C. Velocity gradients and internal work in fluid motion. J. Boston Soc. Civ. Eng. 1943, 30 (4), 219−237. (45) Jones, S. C.; Sotiropoulos, F.; Amirtharajah, A. Numerical modeling of helical static mixers for water treatment. J. Environ. Eng. 2002, 128 (5), 431−440. (46) Rahmani, R. K.; Keith, T. G.; Ayasoufi, A. Three-dimensional numerical simulation and performance study of an industrial helical static mixer. J. Fluids Eng. 2005, 127 (3), 467−483. (47) Rahmani, R. K.; Keith, T. G.; Ayasoufi, A. Numerical simulation and mixing study of pseudoplastic fluids in an industrial helical static mixer. J. Fluids Eng. 2006, 128 (3), 467−480. (48) Heniche, M.; Tanguy, P. A.; Reeder, M. F.; Fasano, J. B. Numerical investigation of blade shape in static mixing. AIChE J. 2005, 51 (1), 44−58. (49) Ottino, J. M. The Kinematics of Mixing: Stretching, Chaos, And Transport; Cambridge University Press: Cambridge, U.K., 1989. (50) ANSYS FLUENT User’s Guide; ANSYS Inc.: Canonsburg, PA, 2009.

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