Performance evaluation of different model mixers by numerical simulation

Journal of Food Engineering 71 (2005) 295 303 www.elsevier.com/locate/jfoodeng Performance evaluation of different model mixers by numerical simulation Chenxu Yu, Sundaram Gunasekaran * Food and Bioprocess Engineering Laboratory, Department of Biological Systems Engineering, University of Wisconsin-Madison, 460Henry Mall, Madison, WI 53706, USA Received 7 September 2004; accepted 12 February 2005 Available online 23 May 2005 Abstract Mixing performance of four types of mixers was investigated for Newtonian and shear-thinning fluids using numerical simulation. The four mixers considered were with the following blade designs: rectangular blade, one Z-blade, two Z-blades, and three rotating pins. Finite element method was used to compute velocity profiles, pressure and stress distributions in the mixing chamber. Mixing effectiveness was evaluated by the and volumetric strain rate. It was demonstrated that shear flow was dominant in rectangular, one Z-blade, and two Z-blade mixers; in the mixer with three rotating pins elongational flow was significant making it the most effective for dispersive mixing. The largest volumetric strain rate was predicted in the mixer with two Z-blades. The rectangular blade mixer was deemed due to a large stagnant region observed. The stagnant regions were larger near the wall for all mixers when shear-thinning fluid was used compared to the Newtonian fluid. Thus, for shear-thinning fluids, to achieve the same mixing effect as with Newtonian fluid, either a smaller gap between the blade and the wall or a higher blade speed is required. Overall, the two Z-blade mixer was considered the best in terms of mixing effectiveness. Ó 2005 Published by Elsevier Ltd. 1. Introduction Mixing is an important unit operation in many processes for food, pharmaceutical, paper, and polymer industries. Mixing reduces non-uniformities or gradients in composition, properties, and temperature of materials in and out of a reactor. Generally, the objective of mixing is homogenization. A satisfactory mixing produces a uniform mixture at an optimal time and cost. Information on physical forces acting on mixing components in a mixer, and on the relative importance of compression and relaxation, shear, temperature increase, dwell time, etc. is not clear. Most relations established between the quality of the mixed products and the operating parameters of the mixer are empirical, and are * Corresponding author. Tel.: +1 608 262 1019; fax: +1 608 262 1228. E-mail address: guna@wisc.edu (S. Gunasekaran). inadequate to understand and control the mixing process precisely. Consequently, in the food industry, design of mixing systems is not based on well-established scientific principles (Lindley, 1991). There is a need for research into the mixing of materials of differing characteristics to gain a level of understanding that will allow performance to be predicted and modeling based on a knowledge of the characteristics of the mixer and the properties of the materials. If we can evaluate and predict the performance of mixers of different geometries and boundary conditions according to their ability to produce a homogenous mixture it would allow us to optimize the mixture design. The performance of a mixer can be evaluated if we know the flow pattern developed in the mixer under specific conditions. Of course, it would be ideal if we can gain exact and accurate solution to motion equations with detailed material characteristics. But it is a very difficult, if not impossible, task. Because the geometry of mixers 0260-8774/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.jfoodeng.2005.02.027

296 C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 Fig. 1. Comparison of analytical and numerical results. (a) Axial velocity; (b) angular velocity. Fig. 2. Velocity distribution for a Newtonian fluid in different mixers. is usually not simple, boundary conditions are very complex and rheological properties of food materials change during processing and are too complicated to be known exactly. All these make it very difficult to obtain exact solutions to the motion equations. However, fortunately, to evaluate the performance of a mixer, the knowledge of overall flow pattern in the mixer chamber is adequate (Gramann, 1995). We do not need to know every details of the flow behavior. Even if the materials being mixed are complex, this provision allows us to make some reasonable simplifications and solve the motion equations approximately and still gain very important and useful results. Using these results, overall performance of different mixer designs can be evaluated. The mixing action and physical phenomena that dominate can be considered as either distributive or dispersive. Distributive mixing or laminar mixing of compatible fluids is usually characterized by the distribution of the secondary phase within the primary matrix. This distribution is achieved by imposing large strains on the system such that the interfacial area between the two or more phases increases and the local dimensions (striation thickness of the secondary phase) decrease (Gramann, 1995). The greater the strain rate in the system the faster the decrease in the striation thickness and the more efficient is the distributive mixing. But, imposing large strains on the system is not always sufficient to achieve a homogeneous mixture. In any type of mixing device, initial orientation and position of the two or more fluid components play a significant role in the quality of the mixture. Homogeneous distribution occurs only within the region contained by the streamline cut across by the initial secondary component. In food processing, it is common to have incompatible fluids or agglomerates of solid particles inside a fluid matrix that need to be mixed. To have a uniform blend

C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 297 Fig. 3. Strain rate distribution for a Newtonian fluid in different mixers. different components have to be evenly distributed in the material matrix. However, before they can be distributed, the secondary component must be broken-up or dispersed. This break-up will happen only when the stress applied to the agglomerate is higher than the forces keeping it together. This operation is dispersive mixing and is proportional to the stress applied to the secondary component, thus, dependent on the applied strain rate. The rates of deforming stress or forces the material undergoes while it travels through a mixer are very important when analyzing or optimizing the device. This means that the type of flow generated in a system determines mixing effectiveness. When analyzing viscous flow three types of flow occur, namely, shear flow, Fig. 4. Flow number distribution for a Newtonian fluid in different mixers. elongational flow, and rotational or transitional flow. In food processing, simple shear flow is the most common. While the particles travel and rotate between the plates, the maximum force applied to them is when they are oriented at 45. This force can be expressed as (Osswald & Menges, 1995): F shear ¼ 3pl_cr 2 ð1þ where, l is fluid viscosity, _c is magnitude of the strain rate tensor and r is particle radius (assuming it a sphere). Elongational flow occurs when the velocity is steadily increased while the particles are moving along the streamline. This type of flow is difficult to generate and is usually observed for a short time. The dispersive force exerted on the particles is given by F shear ¼ 6pl_cr 2 ð2þ

298 C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 Fig. 5. Velocity, shear rate and distribution for a Newtonian fluid in a Type IV mixer. (a) Velocity; (b) shear rate; (c). This is twice the shear force exerted on the particle in simple shear flow (Eq. (1). Another type of flow is one where no net deformation occurs on a fluid particle. For example, pure rotational or transitional type flows. These flows are not favorable for mixing because they do not introduce any stress. Elongational flow creates the highest force to break up particles, making it the most desirable for dispersive mixing. The mixing effectiveness of different mixers can be computed by two parameters: and volumetric strain rate. The (also called mixing index) gives information about the type of flow inside the system and the volumetric strain rate gives an estimate of the overall mixing capability. The (k) is defined as k ¼ _c ð3þ _c þ x where, _c is the magnitude of the strain rate tensor; x is the magnitude of the vorticity tensor. The strain rate and vorticity are easily found by tracking particles throughout the process. The varies between 0.0 and 1.0, where 0.0, is a pure rotational flow, 0.5 is shear flow and 1.0 is elongational flow. To evaluate distributive mixing, strain rate is the most important factor; strain rate is also important for dispersive mixing since high stresses are required for good dispersion. To analyze the overall mixing capability, the mixer throughput should also be considered. Taking these factors into account, the volumetric strain rate is defined as _c v _ol ¼ _c a_ve Mixing region ð4þ where, _c a_ve is the average strain rate inside the mixing region. The mixing region is the area (for two-dimensional (2-D) simulation) or volume (for three-dimensional (3-D) simulation) of the section. The objective of this study was to evaluate performance of mixers with different blade designs based on the flow field produced during mixing by numerical simulation. For many operations that are of interest to food and pharmaceutical industries, the fluid matrix is highly viscous with viscosity normally well beyond 10 3 Pa s. This assures that the ReynoldÕs number (Re = DVq/l) is 1, so laminar flow is dominant. Also, the materials can be considered incompressible because under the operating conditions the densities remain constant. Another consideration is that the simulation is conducted to account for flow field that is already in steady state. This, at first, seems unrealistic since the position of the mixing paddles is changing with time, hence the velocity at any particular position must also be changing with time. However, if observed from a coordinate frame that is fixed with the paddles, then after a short time during which the flow develops fully, the flow field enters a steady state. 2. Numerical simulation To simulate the flow field inside a mixer chamber, a finite element scheme was developed. For the isothermal steady flow considered, the general governing equations are: 1. Momentum equations: op oz ¼ o ox l ov z ox þ o oy l ov z oy þ o oz l ov z oz ð5þ

C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 299 5 3 4 4 2 3 3 2 2 1 2 1 1 1 (a) Type I (b) Type II 4 3 3 2 3 2 2 1 2 1 1 1 (c) Type III (d) Type IV Fig. 6. Flow number distribution for Newtonian material in different mixers. (a) 4 3 3 2 2 1 1 Type I (b) 5 4 4 3 3 2 2 1 1 \ Type II (c) 4 3 3 2 2 1 1 Type III Fig. 7. Flow number distribution for Newtonian material in active mixing zone of mixers.

300 C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 Table 1 Volumetric strain rate for Newtonian fluid Shear rate (s 1 ) Volumetric shear rate (s 1 m 3 ) Type I 12.87 202.06 Type II 13.19 207.08 Type III 12.46 195.62 Type IV 258.80 812.63 (s 1 m 2 ) Fig. 9. Shear rate distribution for a power-law fluid in different mixers. l ¼ l 0 _c n 1 ð8þ For Newtonian fluid we used l = 2000 Pa s; for powerlaw model we used l 0 = 16,000 Pa s n and n = 0.5. 2. Continuity equation Fig. 8. Velocity distribution for a power-law fluid in different mixers. 0 ¼ ov x ox þ ov y oy þ ov z oz ð9þ op oy ¼ o ox l ov y ox þ o oy l ov y oy þ o oz l ov y oz ð6þ op ox ¼ o ox l ov x ox þ o oy l ov x oy þ o oz l ov x ð7þ oz Here, l is viscosity, for Newtonian fluid it is a constant; for shear-thinning material we used power-law model, where Four types of mixers were investigated. They are: Type I, mixer with one rectangular blade; Type II, mixer with one Z shape blade; Type III, mixer with two Z shape blades; Type IV, mixer with three rotating pins. Flow inside the chambers of first three types of mixers was simulated with a 3-D finite element approach. The

C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 301 Fig. 10. Flow number distribution for a power-law fluid in different mixers. three mixers were chosen to have same volume and same blade surface area in order to compare their performance directly. The element used was 27-node brick element, the shape functions for velocities are triquadratic, and the shape functions for pressure are trilinear. The pressure degrees of freedom are located at the eight points of 2*2*2 Gaussian integration and are pressure values at those points. This is a discontinuous approximation for pressure, which means the pressure is not continuous across the element boundaries. The finite element form of governing equations were formulated by treating the momentum equations using the Galerkin weighted-residue method, the continuity equation was combined into the momentum equation using a penalty factor (set to 10 7 ). We assumed that the mixer chamber was filled and there was no wall slip. Thus, the motion of the blades or pins determined the boundary conditions. More specifically, the velocity at chamber walls was zero, and at the blades velocity was determined by the rigid body motion of the blades themselves. Translated into a coordinate frame that is fixed with the blades, this will give a Fig. 11. Velocity, shear rate and distribution for a powerlaw fluid in a Type IV mixer. (a) Velocity; (b) shear rate; (c) flow number. new set of boundary conditions that have zero-velocity at the blades and non-zero velocity at the walls. This new set of boundary conditions was used to solve the numerical equations, then the results were translated back into the coordinate frame with moving blades and static walls. The above numerical scheme was justified by following an analytical verification procedure. The problem chosen was Newtonian flow between two rotating concentric cylinders with a pressure drop along the axial direction. The analytical solution of the internal velocity field is given by Bird, Stewart, & Lightfoot (1960) as "! # V z ¼ ðp 0 P l ÞR 2 1 r 2 1 k 2 ln r 4lL R ln 1 R k kr r V h ¼ X 0 R ð10þ r kr k 1 k where, P 0 is entrance pressure and P l is exit pressure at the, R is outer radius, k is ratio of radius of inner and outer cylinders (R i /R 0 ) and l is viscosity, X 0 is the

302 C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 35% 3 30% 2 (a) 25% 20% 15% 10% 5% 0% Type I (b) 2 1 1 Type II 3 2 2 1 1 (c) Type III (d) 40% 35% 30% 25% 20% 15% 10% 5% 0% Type IV Fig. 12. Flow number distribution for power-law material in different mixers. Table 2 Volumetric strain rate for power-law fluid Shear rate (s 1 ) Volumetric shear rate (s 1 m 3 ) Type I 11.35 178.20 Type II 25.11 394.23 Type III 29.36 460.95 Type IV 203.90 640.25 (s 1 m 2 ) angular velocity of the outer cylinder, V z and V h represent the axial and angular velocity components, respectively, r is radial location, L is cylinder length. For the comparison of numerical results to analytical, we considered R i = 1 cm, R 0 = 5 cm, L = 5 cm, (P 0 P l ) = 20 Pa/cm, X 0 = 2rad/s and l = 1 Pa s. The analytical and numerical results for the axial and angular velocity components matched very well (Fig. 1a and b). 3. Results and discussion The velocity fields for type I, II and III mixers for a Newtonian fluid are shown in Fig. 2a d. It can be seen that at the region far away from the blades the velocities are small. This implies that for better mixing performance, as would be expected, large blade is desirable. Also it is apparent that for same surface area, the rectangular blade occupies the smallest space inside the mixer chamber, thus in type I mixer only small part of the material moves rapidly. Fig. 3a c shows the strain rate distribution for type I, II and III mixers for the same Newtonian fluid. Again, same trends can be observed, the highest strain rate occurs at the blade edges, which will benefit both distributive and dispersive mixing. Thus, a large blade edge is desirable. Also, it is apparent that at regions far away from the blade the strain rates are small. Fig. 4a c shows the distribution of for type I, II and III mixers for the same Newtonian fluid. It should be noted that at region far away from the blades the values are not very meaningful because both strain rate and vorticity are small. Even though the value is in shear flow or elongational flow region, since the material is not moving rapidly, it does not contribute much to the mixing performance. Fig. 5a c shows the velocity field, strain rate distribution and distribution of type IV mixer. In this mixer the three pins, each rotating clockwise at 30 rpm, rotated as a group clockwise around the axis of the cylindrical container at 20 rpm. For a filled mixer, it is reasonable to assume that no axial velocity (z-velocity) exists, thus this problem can be simplified to be 2-D. Because of these differences in geometry and motion pattern, it is difficult to compare the computed strain rate of this type IV mixer directly to the other three mixer types. However, the distribution of can still be compared to evaluate the mixing performance. It can be seen that shear is the largest around three pins. Since these pins are moving around, the region between

C. Yu, S. Gunasekaran / Journal of Food Engineering 71 (2005) 295 303 303 them will experience higher shear than other regions. The around the pins is about 1, implying that the elongational flow is dominant, which bodes well for dispersive mixing. Fig. 6a d shows the overall distribution for all four mixer types. It seems that first three mixer types are poor designs because the dominant flow behavior appears to be rotational flow, which does not contribute to mixing. The reason for this phenomenon is that the blades of these three mixers only occupy small space, in the region far away from blades the strain rate is small, and this determines that the overall is small. Thus, to achieve better mixing performance we need to either use a larger blade or decrease the mixer volume. To analyze the real performance of these designs we should focus on the region which is directly affected by the blades, the results are shown in Fig. 7a c, we can see that very different patterns are observed. We call this region as an active-mixing region. It is apparent that in the active-mixing region of all four mixer types the shear flow is dominant. The pin-type mixer (type IV) also shows significant amount of elongational flow which bodes well for dispersive mixing. Among these three blade type mixers, type III (double Z) is supposed to perform best according to analysis. Table 1 shows the average and volumetric strain rate for all four mixer types. Among the three blade-type mixers which can be compared directly, type II mixer provides the largest volumetric strain rate. However, the difference between these three types is small. For the power-law fluid, velocity fields in the three blade-type mixers are shown in Fig. 8a c; the strain rate distributions are shown in Fig. 9a c. Compared to the Newtonian cases, it is clear that type I mixer has almost same characteristics, but both velocity and strain rate distributions in type II and III mixers appear to be more uniform. This implies that type II and III mixers may have larger active-mixing zones for power-law fluids than for Newtonian fluids. This observation is confirmed by the computed distributions (Fig. 10a c). For type I, the pattern was same as that for the Newtonian case. However, for type II and III mixers, the distributions show that shear flow and elongational flow are dominant. Fig. 11a c shows, respectively the velocity field, strain rate and distributions for type IV mixer. They show same trends as observed in the Newtonian case, it implies that type IV mixer is also a good design for power-law fluids. These observations are further confirmed by analysis of the overall distributions. They are shown in Fig. 12a d. Still, dominant flow in type I mixer is rotational flow, thus this design performs poorly. In type II and III mixers the shear flow appears to be dominant with significant parts of elongational flow, especially in type III mixer, elongational flow appears to be extremely important. These suggest that both type II and III mixers are good designs for mixing powerlaw materials with type III mixer being somewhat better. In type IV mixer, compared to the Newtonian case, shear flow is still dominant, but a larger part of the rotational flow is present. This implies that for power-law fluids, type IV mixer may not perform as well as it does for Newtonian fluids. Average volumetric strain rate for all four mixer types are listed in Table 2. Except for type I mixer, there is a significant increases in volumetric strain rate for power-law fluids compared to Newtonian fluids. 4. Conclusions For blade-type mixers (type I, type II, and type III), given the same blade surface area, the blade shape is the critical factor in determining the mixing performance. Mixers tend to perform differently for Newtonian and power-law materials, it implies that optimization based on the Newtonian fluids is not satisfactory for a non-newtonian material. Although mixer geometries were very different, most particles exhibit shear flow in the active-mixing zone. Thus, changing the geometrical design alone is insufficient to cause elongational flow to be dominant. These results indicate that numerical techniques are useful to analyze complicated mixing processes by calculating critical parameters such as and volumetric strain rate, the can be investigated. References Bird, R. B. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transportation phenomenon. New York: Wiley. Gramann, P. J. (1995). Evaluating mixing of polymers in complex systems using the boundary integral method. Ph.D. Thesis, University of Wisconsin-Madison. Lindley, J. A. (1991). Mixing processes for agricultural and food materials: 1. Foundational of mixing. Journal of Agricultural Engineering Research, 48, 153 170. Osswald, T. A., & Menges, G. (1995). Polymer material science. Munich: Hanser Publishers.