Influence of Viscosity Ratio on the Mixing Process in a Static Mixer

Pahl, M. H.; Muschelknautz, E. Static mixers and their applications. Int. Chem. Eng. 1982, 22 ... The online CFM book; http://www.bakker.org, 2000. Th...
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PROCESS DESIGN AND CONTROL Influence of Viscosity Ratio on the Mixing Process in a Static Mixer: Numerical Study Mårten Regner,† Karin O 2 stergren,†,‡ and Christian Tra1 gårdh*,† Department of Food Technology, Engineering and Nutrition, Lund UniVersity, P. O. Box 124, SE-221 00 Lund, Sweden, and The Swedish Institute for Food and Biotechnology, Ideon, SE-223 70 Lund, Sweden

The mixing process in a Lightnin Series 45 static mixer, 40 mm in diameter, has been investigated using computational fluid dynamics and the volume of fluid (VOF) method, a method developed for immiscible fluids but here used for miscible. The mixing process was investigated for two liquids that had viscosities ranging from 0.003 to 51.2 Pa s, and the volumetric flow rate proportion between the liquids was varied between 1/1 and 4/1. The flow rate was 0.1 m/s, and two Reynolds numbers, 1 and 70, were investigated. The mixer performance was evaluated using the rate of striation thinning, and it was found that the greater the difference in viscosity the worse the mixer performance. This effect is due to the difference in elongation rate between the liquids, which exists until equilibrium in shear stress has been reached at the interface between the liquids. For higher Re numbers close to a point when secondary motions start to have significance for the rate of striation thinning, a lower viscosity of the added liquid results in an increase in mixing performance. It was further found that the VOF method can be used to model the mixing of dissimilar liquids in static mixers, but since the thickness of the striations decreases rapidly with the number of mixer elements, the VOF method is most suitable when investigating mixing processes over a small number of mixer elements. 1. Introduction Mixing of fluids is a common unit operation in a large number of processes. This unit operation is used in many applications where a defined degree of homogeneity of a mixture is required. The fluids that are mixed may have different densities and viscosities and may be miscible or immiscible. Common mixing devices are dynamic mixers in agitated tanks for batch operations and static mixers, also called in-line or motionless mixers, for in-line mixing in continuous operations.1 Some of the advantages of static mixers over dynamic mixers in agitated tanks are that they have no moving parts, they require small spaces, and they have low or no maintenance costs and a short residence time. Static mixers are available for different flow conditions, i.e., laminar, transitional, and turbulent flow regimes. A static mixer designed for laminar flow conditions is usually composed of a number of mixer elements, each rotated 90° relative to the previous one. The mixer elements are designed to split the flow into two or more streams, rotate them, and then recombine them. Previous studies on static mixers have focused on characteristics such as residence time distributions, pressure drop, stretching rates, and degree of mixing. These characteristics have been investigated experimentally,2-11 as well as numerically, using computational fluid dynamics (CFD).3,8,12-19 In most of the experimental studies the properties of the liquids were matched as well as possible, and only in a few studies has the effect of viscosity differences been investigated.6,8,11 The conclusions drawn from these investigations were that mixing * To whom correspondence should be addressed. Fax: + 46-46222 4622. E-mail: [email protected]. † Lund University. ‡ The Swedish Institute for Food and Biotechnology.

is best achieved when the liquids have the same viscosity, and that the effect of viscosity decreases with the number of mixer elements. This kind of investigation, to our knowledge, has not been performed using CFD. In the investigations using CFD presented in the literature, the two liquids are modeled by tracking particles, a method that cannot be used to investigate the effects of viscosity differences. In order to study the process of mixing liquids with different physical properties such as density and viscosity, one has to keep track of the location of the liquids in the computational domain. For this purpose interface tracking methods can be used in basically three different ways: a piecewise polynomial, a level set of a particular function, or a collection of volume fractions (volume of fluid).20 In the volume of fluid (VOF) method the volume of each fluid in control volumes containing a portion of the interface is tracked rather than the interface itself. These volume fractions are then used to reconstruct the interface. In order to obtain an acceptable resolution of the interface, at least three cells normal to the interface are required, which makes the VOF method more computationally demanding than particle tracking. The VOF method was developed for immiscible systems and therefore does not allow mass transfer between the phases. However, for miscible systems with high Schmidt numbers, i.e., low diffusivity, in processes with short residence times, the liquids can be approximated as immiscible. For other miscible systems models such as the species transport model can be used. However, when employing tetrahedral meshes as well as any other mesh, the species transport model results in numerical diffusion, even for a higher order scheme, that by an order of magnitude would exceed the molecular diffusion for any liquid having a viscosity in the range of that considered here. The primary aim of this study was to investigate how the mixing performance is affected by the viscosity ratio and

10.1021/ie0708071 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/28/2008

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Figure 1. Geometry of (a) a standard Lightnin Series 45 mixer element and (b) the Lightnin Series 45 static mixer used in the simulations. Table 1. Dimensions of a Lightnin Mixer Element and the Mixer Itself Mixer Element length (mm) diameter (mm) thickness (mm)

60 40 1.4 Mixer

entrance length (mm) exit length (mm) overall length (mm)

5 15 200

volumetric flow rate proportion between two phases when mixing an added liquid into a base liquid in a static mixer, and the underlying cause of the decreasing mixing performance found experimentally when mixing liquids of different viscosities. The secondary aim was to evaluate the VOF method as a tool for investigating mixer performance. 2. Methods A commercial static mixer was chosen as the base configuration for this study, namely the Lightnin Series 45 static mixer21 with an element aspect ratio (L/D) of 1.5. In the Lightnin mixer every alternate element imparts a clockwise rotation to the flow, while the following element imparts a counterclockwise rotation. This will give the flow an alternating clockwise and counterclockwise rotation. The design of a mixer element is shown in Figure 1a, and the elements are assembled as in Figure 1b. The dimensions of a mixing element and the Lightnin mixer are given in Table 1. The mixers in the computational simulations were composed of an entrance section, three mixer elements, and an exit section. All the model liquids in the simulations had the same density, 1000 kg m-3, in order to isolate the effect of viscosity ratio, without including density effects. Two series of numerical experiments were performed: one for Reynolds number 1.25 (denoted Re 1) and one for Reynolds number 70 (denoted Re 70). The viscosity ratio (µ1/µ2) was varied between 1/16 and 16/1 by keeping the viscosity of the base liquid constant (3.2 Pa s for the Re 1 series and 0.057 Pa s for the Re 70 series), and varying the viscosity of the added liquid. The boundary conditions at the inlet were a flat velocity distribution with an axial velocity of 0.1 m s-1 and a ratio of 1, 2, or 4 between the areas covered by the base liquid and the added liquid expressed as liquids 1 and 2, respectively. A static pressure boundary condition was imposed at the outlet (P ) 0 Pa overpressure), and all solid boundaries were stationary with a no-slip condition. When running particle tracking, a reflection boundary condition was used at all walls. The two-phase flow calculations were performed using the CFD software Fluent 6.122 and the VOF method. Different

Figure 2. Average rate of striation thinning as a function of viscosity ratio and volumetric flow rate proportion. The volumetric flow rate proportions are expressed in terms of the Reynolds number at a viscosity ratio of 1.

methods are available in Fluent 6.1 to reconstruct the interface between the phases. In the present study a piecewise linear interface calculation method was chosen for the reconstruction of the interface. This method represents the interface by a plane with the same normal vector as the true interface, whereas the simple line interface calculation method (also available in Fluent 6.1) represents the interface by a plane parallel to one of the faces of the control volume. In the VOF method a single set of momentum equations is shared by the fluids, and in a two-phase system interface tracking is accomplished by solving the continuity equation for the volume fraction of one of the phases and then calculating the volume fraction of the second phase directly by using the fact that the sum of the two volume fractions is unity.22 The properties of an n-phase system are calculated as the volume-fraction-averaged properties, e.g., the viscosity as expressed in eq 1. n

µ)

biµi ∑ i)1

(1)

The rate of striation thinning was calculated by comparing the striation thickness of the two fluids before and after a mixer element in planes normal to the mixer axis. Fourcade et al.3 have outlined a method where particles are placed on a small circle at the inlet and tracked through the mixer, with the rate of striation thinning as defined by Ottino et al.23 See eq 2,

R(t) ) -

d ln s dt

(2)

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Figure 3. Difference in residence time between the two liquids normalized to the average residence time as a function of viscosity ratio and volumetric flow rate proportion, expressed in terms of the Reynolds number at a viscosity ratio of 1.

Figure 5. (a) Difference in elongation rate between the more viscous liquid (1) and the less viscous liquid (2) as a function of axial distance and viscosity ratio at a volumetric flow rate proportion of 1:1. (b) Open cross-sectional area divided by the tube area in the mixer as a function of axial location.

Figure 4. Average relative striation thickness as a function of viscosity ratio and volumetric flow rate proportion, expressed in terms of the Reynolds number at a viscosity ratio of 1.

where s is the striation thickness, and is here extended to be defined as the area of the shape of the cross sections of the phases as calculated using the VOF method. The phase distributions before and after the mixer element were exported to MatLab as TIFF images. The images were converted to binary images, and analyzed using the image analysis functions “bwarea” and “bwperim” in MatLab in order to evaluate the shape of the striations. Particle tracking calculations were performed to calculate the average residence time (T) for every striation at each of the mixer elements. About 39 000 particles were released from the inlet of the domain and tracked through the mixer. The positions of the particles were registered at each element intersection. From the particle residence times the average residence times of the striations were calculated and then used in the calculation of the rate of striation thinning (R) by integration of eq 2 from time t ) 0 to T, assuming a constant rate of striation thinning. The integration results in eq 3,

R)-

( )

safter 1 ln T sbefore

(3)

where T is the residence time of the striation, and sbefore and safter are the striation thickness, defined as above, before and after the mixer element, respectively.

Figure 6. Cross-sectional area fraction of the less viscous liquid as a function of axial distance and viscosity ratio at a volumetric flow rate proportion of 1:1. The dashed line shows the fraction of the total volumetric flow rate.

A series of tests was performed in the same way as described by Regner et al.24 in order to determine the appropriate node density to ensure accurate predictions of the velocity field and the phase distribution. A test case consisting of an entrance section, two mixer elements, and an exit section was laid out and meshed using Gambit 2.1.6 (Fluent Inc., Lebanon, NH). The geometry was meshed using seven different node densities ranging from 4.4 to 14 nodes cm-1, with an approximate doubling of the number of control volumes between each successive mesh. The meshes were evaluated at a Reynolds number of 100 and a volumetric flow rate proportion of 1/1. When changing from a node density of 7.0 nodes cm-1 to 8.9

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3. Results and Discussion

Figure 7. Cross-sectional area fraction of the liquid in the minor stream as a function of axial distance and viscosity ratio at a volumetric flow rate proportion of 2:1 and a Reynolds number of 1.25. The dashed line shows the fraction of the total volumetric flow rate.

nodes cm-1, the local velocity along a line running parallel to the mixer centerline (z-axis) at the coordinates x ) y ) 10 mm changed by less than 1%. In the case of the phase distribution, the location and area of the interface between the phases were evaluated after the first and second mixer elements. The predicted values varied by more than 1% until the node density was changed from 8.9 to 11 nodes cm-1. The next mesh refinement resulted in a change of less than 1% in both velocity and phase distribution. Thus, a node density of 8.9 nodes cm-1 was chosen for the following calculations. The mesh was unstructured and consisted in total of approximately 52 000 tetrahedrons per mixer element.

The rate of striation thinning was evaluated for the second mixer element, in order to avoid any entrance effects, and was found to vary with the viscosity ratio. The viscosity ratio (µ1/ µ2) was varied between 1/16 and 16/1 by keeping the viscosity of the base liquid constant (3.2 Pa s) and varying the viscosity of the added liquid. At a viscosity ratio of 1 this corresponds to a Reynolds number of 1.25. Although the viscosity ratio was varied only moderately, it was seen that the greater the difference in viscosity between the liquids, the lower the average rate of striation thinning except at the highest Reynolds number; see Figure 2. The effect of viscosity ratio on the rate of striation thinning is due to both a difference in residence time between the liquids and a difference in relative striation thickness. The more viscous liquid has a longer residence time in the mixer than the less viscous one, and the difference in residence time increases as the difference in viscosity increases; see Figure 3. For Reynolds number 70 the lower viscosity of the added liquid (viscosity ratio 4/1) caused a drastic increase in the rate of striation thinning which can be explained by a transition into a flow pattern where the secondary motion becomes a more pronounced factor, which is in agreement previous investigations.24 The relative striation thickness also changes since the cross-sectional area fraction of the liquids changes in order to maintain the mass balance in the system; see Figure 4. At a the viscosity ratio of 1 the relative striation thickness was 0.77, as compared to the value of 0.60 for striations of 1 mm in diameter presented in the literature.24 When mixing equal parts of the two liquids (q1/q2 ) 1/1), the average rate of striation thinning was greatest at a viscosity ratio of 1, whereas when mixing the liquids in the proportions

Figure 8. Striation evolution in a Lightnin mixer at different viscosity ratios between a dark (1) and a light (2) colored liquid with the mixing proportions q1/q2 ) 2/1, Re ) 1 (the Reynolds number is based on the viscosity ratio 1/1), and at the beginning of the first, second, and third elements. Viscosity ratios (µ1/µ2) between the liquids and Re were (a) 1/16, (b) 1/1, and (c) 16/1.

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Figure 9. Striation evolution in a Lightnin mixer at different viscosity ratios between a dark (1) and a light (2) colored liquid with the mixing proportions q1/q2 ) 2/1, Re ) 70 (the Reynolds number is based on the viscosity ratio 1/1), and at the beginning of the first, second, and third elements. Viscosity ratios (µ1/µ2) between the liquids and Re were (a) 1/16, (b) 1/1, and (c) 4/1.

2/1 the average rate of striation thinning was almost unaffected within the range of viscosity ratio 1/4 to 4/1 for the case Re 1. The average rate of striation thinning decreased outside this range (except for the highest Reynolds number, case Re 70). As can be seen in Figure 2, the rate of striation thinning was lower when a more viscous liquid was mixed into a less viscous liquid than vice versa (except for the highest Reynolds number). This means that it is more difficult to mix a small proportion of a highly viscous liquid into a less viscous one than it is to mix a less viscous liquid into a more viscous one. Not only the mixing proportions (q1/q2) 1/1 and 2/1 between the base and the added liquid were simulated but also the mixing proportion 4/1. However, in the latter case the striations were too thin, and as a consequence of the VOF method and the mesh size chosen, each striation broke up into several smaller striations. This can be avoided by using a finer mesh or by using adaptive mesh refinement, although it will increase the computational time substantially and require more powerful computers than those available for this study (CPU 2.66 GHz, 3 GB RAM). Mixing systems with the proportion 2/1 with lower viscosities, corresponding to a Reynolds number of 70 at a viscosity ratio of 1, were also simulated since previously presented studies have shown that vortices occur in the flow at flow rates above a Reynolds number of approximately 10.6,10,13,16,24 Three viscosity ratios were simulated: 1/16, 1, and 16. The simulation with a viscosity ratio of 1 resulted in a lower relative striation thickness, 0.77, and thus a higher rate of striation thinning than the previous simulation for the case Re 1. As reported in the literature, vortices could be seen in the flow causing greater stretching of the striations.24 However, since the rate of striation thinning is

based on the striation perimeter and area, it does not reflect any stretching within the phases, and thus the relative striation thickness in this study is greater than the values reported in the literature for thin striations.24 When mixing a liquid with a higher viscosity into the system (µ1/µ2 ) 1/16), vortices could not be seen in the flow. As in the simulations of the mixing proportion (q1/q2) 4/1, the simulation of a viscosity ratio (µ1/µ2) of 16 resulted in striations as thin as or thinner than the mesh size used, even when using a refined mesh with a node density of 18 nodes cm-1, since the striations were more efficiently stretched to thin filaments than in the case of a viscosity ratio of 1 due to the vortices in the flow. A possible explanation of why the rate of striation thinning is highest when mixing liquids of the same viscosity may be that the liquids strive toward a stationary phase distribution with shear stress equilibrium at the interface between the liquids. When mixing liquids of different viscosities, this results in a negative elongation rate (dw/dz) of the more viscous liquid and a positive elongation rate of the less viscous one until stability is reached. The greater the viscosity ratio, the longer the distance before the system reaches stress equilibrium, and the greater the difference in elongation rate between the liquids; see Figure 5a. However, the geometry of a static mixer such as the Lightnin static mixer is likely to delay the shear stress equilibrium at the interface between the liquids due to the twist of the elements and changing cross section area; see Figure 5b. As can be seen in Figures 6 and 7, the cross-sectional area fraction of the liquids changes with the axial distance in the mixer. As the mixer divides the feed streams into more and more striations, the two liquids will be more and more evenly

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distributed over the cross section of the mixer. As a result of this, the difference in average axial velocity of the liquids will decrease, and the cross-sectional area fraction of the liquids will approach their fraction of the total volumetric flow rate (Figures 6 and 7). Thus, the effect of viscosity ratio will decrease with the number of mixer elements, which agrees with experimental results previously presented in the literature.8 Thus, in summary, the main physical factor that a viscosity ratio deviating from 1 negatively influences the mixing performance based on rate of striation thinning and relative striation thickness is that the fluid with lower viscosity is forced to flow faster relative to the other due to the shear stress acting between them. As continuity is always maintained the result will be that the striation thickness of it decreases and the other increases. The exception is if the Re number is high and the added fluid is less viscous, the condition may arise that the secondary flow mixing becomes dominant and thus improves mixing performance in terms of striation thickness and rate of striation thinning. Figures 8 and 9 give a visual picture of shapes and phase distributions at different viscosity ratios and Reynolds numbers analyzed in Figures 2 and 4. 4. Conclusions The viscosity ratio between the two liquids being mixed influences the mixing process: the greater the difference in viscosity, the lower the rate of striation thinning. This is due to a difference in elongation rate between the liquids, which exists until equilibrium in shear stress has been reached at the interface between the liquids. For Re numbers close to conditions that create secondary motions having significance for the striation stretching, a reduction in the viscosity of the added liquid results in a strong reduction in relative striation thickness/increase in striation thinning rate. The effect of the viscosity ratio decreases with increasing number of mixer elements since the liquids will be more and more evenly distributed over the cross section. Hence, the average axial velocity of each liquid will approach the overall axial velocity. Computational fluid dynamics combined with the VOF method can be used for numerical investigations of the mixing of dissimilar liquids in static mixers. However, the level of detail in the phase distribution using VOF is limited by the size of the control volumes. Since the thickness of the striations decreases rapidly especially at large viscosity ratios, the VOF method is most suitable when analyzing mixing processes over a few mixer elements. However, the conclusions drawn from this study have a general character and going to finer striations than done here will not change them even though it is perhaps not recommended to extrapolate these results and completely trust the precision of the extrapolations. Notation A ) striation area (m2) b ) volume fraction D ) tube diameter (m) L ) length of mixer element (m) n ) number of phases O ) striation perimeter (m) q ) volumetric flow rate (m3 s-1)

s ) striation thickness (m) t ) time (s) T ) average residence time (s) w ) velocity in z-direction (m s-1) z ) axial coordinate (m) Greek Symbols R ) rate of striation thinning (s-1) µ ) dynamic viscosity (Pa s) Subscripts after ) after mixer element before ) before mixer element i ) ith phase 1 ) base liquid 2 ) added liquid Literature Cited (1) Myers, K. J.; Bakker, A.; Ryan, D. Avoid agitation by selecting static mixers. Chem. Eng. Prog. 1997, 93, 28. (2) Chen, S. J. In-line, continuous mixing and processing of cosmetic products. J. Soc. Cosmet. Chem. 1973, 24, 639. (3) Fourcade, E.; Wadley, R.; Hoefsloot, H. C. J.; Green, A.; Iedema, P. D. CFD calculation of laminar striation thinning in static mixer reactors. Chem. Eng. Sci. 2001, 56, 6729. (4) Grace, C. D. Static mixing and heat transfer. Chem. Process Eng. 1971, 52, 57. (5) Heywood, N. I.; Viney, L. J.; Stewart, I. W. Mixing efficiencies and energy requirements of various motionless mixer designs for laminar mixing applications. Inst. Chem. Eng. Symp. Ser. 1984, 89, 147. (6) Jaffer, S. A.; Wood, P. E. Quantification of laminar mixing in the Kenics static mixer: an experimental study. Can. J. Chem. Eng. 1998, 76, 516. (7) Joshi, P.; Nigam, K. D. P.; Nauman, E. B. The Kenics static mixers New data and proposed correlations. Chem. Eng. J. Biochem. Eng. 1995, 59, 265. (8) Kova´cs, T.; Troedson, M.; Wormbs, G.; Tra¨gårdh, C.; O ¨ stergren, K.; Innings, F.; Lingnestrand, A.; Strinning, O. Mixing of Newtonian and non-Newtonian fluids in static mixers under laminar flow conditions: experiments and simulations. Proceedings of the 11th European Conference on Mixing, Bamberg, Germany, 2003; p 421. (9) Pahl, M. H.; Muschelknautz, E. Static mixers and their applications. Int. Chem. Eng. 1982, 22, 197. (10) Ujhidy, A.; Ne´meth, J.; Sze´pvo¨lgyi, J. Fluid flow in tubes with helical elements. Chem. Eng. Process. 2003, 42, 1. (11) Wilkinson, W. L.; Cliff, M. J. An investigation into the performance of a static in-line mixer. Proceedings of the 2nd European Conference on Mixing, Cambridge, U.K., 1977; p 15. (12) Bakker, A.; LaRoche, R. D.; Marhsall, E. M. Laminar flow in static mixers with helical elements. The online CFM book; http://www.bakker.org, 2000. (13) Byrde, O.; Sawley, M. L. Optimization of a Kenics static mixer for non-creeping flow conditions. Chem. Eng. J. 1999, 72, 163. (14) Fradette, L.; Li, H. Z.; Choplin, L.; Tanguy, P. 3D-finite element simulation of fluid flow through a SMX static mixer. Comput. Chem. Eng. 1998, 22, 759. (15) Hobbs, D. M.; Muzzio, F. J. The Kenics static mixer: a threedimensional chaotic flow. Chem. Eng. J. 1997, 67, 153. (16) Hobbs, D. M.; Muzzio, F. J. Reynolds number effects on laminar mixing in the Kenics static mixer. Chem. Eng. J. 1998, 70, 93. (17) Hobbs, D. M.; Muzzio, F. J. Optimization of a static mixer using dynamical system techniques. Chem. Eng. Sci. 1998, 53, 3199. (18) Rauline, D.; Tanguy, P. A.; Le Ble´vec, J.-M.; Bousquet, J. Numerical investigation of the performance of several static mixers. Can. J. Chem. Eng. 1998, 76, 527. (19) Zalc, J. M.; Szalai, S. E.; Muzzio, F. J.; Jaffer, S. Characterization of flow and mixing in an SMX static mixer. AIChE J. 2002, 48, 427.

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(20) Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. J. A high-order projection method for tracking fluid interfaces in variable density incompressible flows. J. comput. Phys. 1997, 130, 269. (21) http://www.lightnin-mixers.com/sites/lightnin/ inliner_mixers45.asp. (22) Fluent Inc. Fluent 6.1 User’s guide; Fluent Inc.: Lebanon, NH, 2003. (23) Ottino, J. M.; Ranz, W. E.; Macosko, C. M. A lamellar model for analysis of liquid-liquid mixing. Chem. Eng. Sci. 1979, 34, 877.

(24) Regner, M.; O ¨ stergren, K.; Tra¨gårdh, C. Effects of geometry and flow rate on the mixing process in a static mixersA numerical study. Chem. Eng. Sci. 2006, 61, 6133.

ReceiVed for reView June 12, 2007 ReVised manuscript receiVed January 22, 2008 Accepted January 31, 2008 IE0708071