14 th European Conference on Mixing Warszawa, 10-13 September 2012 LATTICE-BOLTZMANN METHOD FOR THE SIMULATION OF LAMINAR MIXERS Felix Muggli a, Laurent Chatagny a, Jonas Lätt b a Sulzer Markets & Technology Ltd., Sulzer Innotec, Sulzer Allee 25, CH-8404 Winterthur, Switzerland; b FlowKit Ltd., Route d'oron 2, CH-1010 Lausanne, Switzerland Felix.Muggli@Sulzer.com Abstract. A lattice Boltzmann method is used to simulate the blending of two fluids in static, laminar mixers. The method uses a mesh-based algorithm to solve for the fluid flow, and a meshless technique to trace the interface between the blended fluids. This hybrid approach is highly accurate, because the position of the interface can be traced beyond the resolution of the grid. The numerical diffusion is negligible in this model, and it is possible to reproduce mixing patterns that contain more than one hundred striations with high fidelity. Compared to other methods, this approach makes no assumptions on the used geometry: the subsequent mixer units are for example not required to be identical. Furthermore, although this article focuses on static fluid mixers, the approach naturally extends to fluid mixing in time-dependent flows. Keywords: Laminar Mixer, CFD, Lattice Boltzmann Method, Particle Method. 1. INTRODUCTION Flow simulations are well established within the design and development processes for static laminar mixers. Continuous development of the numerical models and validation of the computational fluid dynamics (CFD) codes against test data did lead to quite reliable simulation tools for the prediction of the pressure loss. However, the correct prediction of the quantitative mixing quality is still quite a challenge. The standard Navier-Stokes solvers usually employed in the industrial design environment tend to predict too optimistic mixing qualities. The reason for this so-called numerical diffusion is well known: the very thin fluid layers, common in laminar mixing, cannot be adequately resolved by a computational mesh of reasonable dimensions. Hence, these fluid layers are actually thinner than the mesh spacing and the computed mixing is then smeared. An alternative to the Reynolds Averaged Navier-Stokes approach is the Lattice- Boltzmann method (LBM). This relatively new method in the field of CFD is based on the Boltzmann equation which describes statistical properties of a gas by means of a velocity distribution function. It basically expresses a balance between transport and collision of molecules. The Boltzmann equation is discretized on a regular grid where the numerical variables are stored on the nodes. At each time step during the simulation a collision term is evaluated and the results are then propagated to the neighboring nodes (streaming). In this article, the LBM is applied to commercially developed static laminar mixers. The code computes the flow pattern in the mixer with high accuracy, and high efficiency thanks to the excellent scalability of the LBM for parallel computing, during the actual computation as well as during pre- and post-processing. Comparisons of the pressure drop show that the flow pattern is computed with an accuracy comparable to that of commercial solvers. 317
The real strength of the method is however found in its ability to include meshless techniques that are used to predict accurate mixing patterns. This article focuses on mixing patterns predicted for the Sulzer SMX mixer and two types of Sulzer helical mixers. In the first case, the computed mixing patterns are shown to match well with results from laboratory experiment. In the second case, the quality of mixing is assessed quantitatively throughout the mixer and compared between two mixer models. 2. NUMERICAL METHOD The numerical approach adopted in this work assumes that the blended fluids have equal viscosity. The fluid motion is therefore simulated through a single-fluid equation, modelled by means of a lattice Boltzmann method. The mixing pattern is computed by tracking the path of passive scalar particles of two species. A sufficient number of particles (of the order of 10 10 ) is injected in order to achieve a continuum approximation. The particles have a Lagrangian movement and are not tied to the fluid grid. In a given control volume, the number of particles n1 and n2 of species 1 and 2 are counted, and the concentration c of fluid 1 is defined as n1 c = n1 + n2 The coefficient of variation COV indicates the mixing quality, and is defined across a slice perpendicular to the mixer tube as ( c c) 2 1 1 COV = where c is the average of c c A The numerical models used in the simulations are summarized in the following table: Fluid Lattice Boltzmann BGK model with D3Q19 lattice [1]. Boundary Condition (mixer) Particles Guo off-lattice model [2]. Second-order Verlet time integration, with linear interpolation of the velocity field in a cell. 3. COMPUTATION OF MIXING QUALITY IN A 4-ELEMENT SMX MIXER The first simulation presented here has been applied to a 4-element SMX mixer, the geometry of which is presented in the image below: Fig. 1: Representation of the 4-element SMX mixer For better visualization, the mixer has been cut in two, and only the lower half is displayed. All four elements of this mixer are identical, and each element is rotated by a 90 angle with respect to the previous one. Further details on SMX mixer can be found in [3,4]. 318
The obtained mixing pattern is displayed in the image below in four sequences, taken at given positions in each of the four mixer units. The left image corresponds to experimental data, which has been obtained through the injection of two hardening, colored fluids. The right image displays the results of lattice Boltzmann simulations. Fig. 2: Comparison of experimental and numerical mixing patterns in the 4-el. SMX mixer a) 1 st mixer unit b) 2 nd mixer unit c) 3 rd mixer unit 319
d) 4 th mixer unit A particularly striking agreement between the mixing patterns obtained in experiment and simulation is observed, and the agreement appears to improve, rather than decrease, along the downstream direction in the mixer. A clear disagreement appears in the central area of the image of the first mixer unit, in which the mixing pattern in the experimental image is not fully symmetric. This must however be associated to the conditions of the experimental setup, in which the flow pattern is not exactly controlled at the point of injection of the fluids in the mixer. It should be mentioned that results of similar quality are also obtained with a different numerical method, based on a so-called trajectory mapping scheme (see [4]). An important relative advantage of the lattice Boltzmann is however provided by its high speed of computation, obtained through efficient parallelization. The mixing pattern shown in the images above where for example computed within 24 hours on a 64-core, Intel-based parallel computer. This leads to a high productivity and interactivity which is not achieved with the non-parallel codes for the trajectory mapping scheme available to the authors. Furthermore, the trajectory mapping scheme relies on the assumption that all mixer elements are identical. It therefore computes the actual mixing pattern for a single mixer element only which then is reapplied to every subsequent element after an adequate rotation. In a long mixer with non-repetitive elements, like the helical mixers described in the next section, this method would therefore be not possible. 4. COMPUTATION OF MIXING QUALITY IN TWO HELICAL MIXERS In this section, simulations for the mixing pattern in two helical mixers are presented. In the first helical mixer (referred to as the long helical mixer in the following), a spacing is introduced between subsequent mixer elements, a fact by which the total mixer length is artificially enhanced. In the second mixer (which we refer to as the short helical mixer ), these additional spaces are skipped. While the tubes of both mixers are of the same length, the tube of the shorter mixer contains an empty space in the outflow area. The number of elements is the same in the two mixers. In both mixers, the successive elements are close to identical, but contain small differences that break the repetitive pattern. The mixers are presented in Fig. 3. As in the previous case, they have been cut in two halves for visualization purposes, and only the lower part is shown. The mixing process is smoother in a helical mixer than in a SMX mixer. The mixing patterns computed in the long helical mixer are shown in Fig. 4, first on a slice in the middle of the mixer (left image), and then on a slice at the mixer exit (right image). In this case, experimental data for a direct comparison was not available. 320
Fig. 3: Representation of the two helical mixers used in the simulations a) long helical mixer b) short helical mixer Fig. 4: Mixing pattern in the long helical mixer, obtained by numerical simulation. We compared the quality of mixing in these two helical mixers, by computing the COV value according to the definition proposed in Section 2. The results are presented in Fig. 5. The value of the COV drops as expected throughout the mixers, as the fluids are blended. At the end of each mixer unit, the COV increases slightly but noticeably. This can be explained by the fact that the fluids, after leaving a mixer element, need to fill the empty space left behind by the central solid structure of the element. In the process, they are dilated (as seen on the left picture of Fig. 4), and the COV increases locally. It can be seen that due to its condensed structure, the short helix mixer blends the two fluids faster, as the slope of the COV curve is steeper. The long helix mixer on the other hand produces a lower COV value at the mixer exit, and provides therefore all in all a blending mechanism of higher quality. 321
Fig. 5: Quality of mixing, represented through the COV value, throughout the long and the short helical mixers. 5. CONCLUSION AND OUTLOOK With the lattice Boltzmann method, a new numerical tool is available which seems particularly adequate for the simulation of static mixing processes. The method easily incorporates meshless techniques to represent fluid mixing with patterns at sub-grid accuracy, and is particularly fast thanks to massive parallelism. It is therefore concluded that this method is quite advantageous for the performance assessment of laminar mixers during the design process in industry. While the results presented in this paper all apply to static mixers, the presented method can be applied without modifications to dynamic mixing processes, because the mesh-based fluid solver and the meshless passive tracers of fluid mixing are dynamically coupled and executed synchronously. Results in dynamic mixers will be presented in future publications. 6. REFERENCES [1] Chen S., Doolen G. D., 1998. Lattice Boltzmann Method for Fluid Flows, Ann. Rev. Fluid Mech., 30, 329-364. [2] Guo Z., Zheng G., Shi B, 2002. An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14, 2007-2010. [3] Visser J. E., Rozendal P. F., Hoogstraten W., Beenackers A. A. C. M., 1999. Three dimensional numerical simulation of flow and heat transfer in the Sulzer SMX static mixer, Ch. Eng. Sc., 54, 2491-2500. [4] Hirschberg S., Koubek R., Moser F., Schöck J., 2009. An improvement of the Sulzer SMX static mixer significantly reducing the pressure drop, Proceedings of 13 th European Conference on Mixing, 14-17. 322