Introduction

There are two branches of the mixing industry, the rotating mixing portion, and the static mixing portion.  Both portions are important in industry, but the rotating industry has undergone more extensive growth and development throughout the years.  

This is because advancement has centered around experimental results, and rotating mixers are easier, and cheaper, to analyse experimentally.  A test stand for static mixers costs on average an order of magnitude more money than that for rotating equipment, and as such, static mixer testing is less prevalent and progress less progressive.  

This presents two interesting situations for companies who provide mixers.  1) There is plenty of data and correlations available for various rotating mixers, so validation of computational methods is easy, and 2) there is a lack of data and consistency around testing of static mixers, which has resulted in no standard for testing in industry.  Most of the testing of static mixers for industrial companies is done by a single third party.  The costs associated with this third-party testing limit the amount of development that happens in static mixer development.  

Both situations have opened the door for computational fluid dynamics (CFD) to be used in analysing the mixing process.  CFD results done with traditional analytical solvers have been used for over a decade to produce comparative results between mixers being considered for a design.  

The results are qualitatively used to make sure that flow is reaching all the critical places in a stirred vessel.  Additionally, simulation results are not used when determining which static mixer is appropriate for an application because traditional CFD results are steady state, whereas mixing in static mixers are driven by transient turbulent eddies, which are smoothed over in traditional CFD results.  

Utilisation of traditional, commercially available solvers is prevalent among industrial companies, but a new simulation solver has been commercially developed with the aid of some of the largest mixer companies in the United States, to improve the mixer development and design process.  

This solver is based around the use of lattice Boltzmann methodology, which utilises specialised computational methods to allow micro-scale turbulence to be capture accurately and effectively.  

The ability to capture turbulent eddies opens a new world for mixing companies regarding developing new mixing technology and predicting mixer behaviour in customer applications.  

What is Lattice Boltzmann?

A lattice Boltzmann solver uses the Boltzmann equation to model the physics in a system.  This equation describes the same physics as the Navier Stokes equations do, and the Navier Stokes equation can be derived from the Boltzmann equation.  

The efficiency of the Boltzmann solver comes from the ability to solve a probability density function of the fluid behavior and derive the fluid properties from this function.  

The mass, momentum and energy are all moments of the integral equation used in the Boltzmann solver.  When this computational efficiency is coupled with modelling efficiencies, the effects on the simulation are profound.  

The solver first fills the system with a structured soup of “particles”, or lattice points.  These lattice points behave as particles would where instead of moving in any direction, they transmit fluid behaviour information in any direction.  

The lattice constrains the infinite possibilities of motion to the nearest neighbouring lattice points as shown in Figure 1.  The probability density function determines the probability of the fluid information being based to all neighbouring points.  

Through the calculation of the moments of the density function, the mass, momentum and energy thermodynamic properties are simulated through the system.  

This lattice is then coupled with a bounce-back function, which reduces the meshing requirements over traditional solvers.  Rather than having to refine meshes in near-wall regions to improve the accuracy of capturing the properties in those regions, when fluid packets of information hit a wall in the lattice Boltzmann solver, that packet is “bounced back” into the fluid domain as shown in Figure 2.  

This happens in a multi-step process where the lattice point pushes its information to all of the nearest neighbours, even those located outside of the wall, any collision is determined and those properties are bounced back to the fluid within the wall.  

Application in Product Improvement

The use of the lattice Boltzmann solver unlocks two of the more critical, yet more difficult pieces of mixer analysis, the transient nature of rotating mixers and the effect of turbulent eddies on static mixers.  

Transient loading and unloading of rotating impellers can lead to advanced wear and tear on equipment.  On the other hand, the lattice Boltzmann solver has the capability to perform transient mixer analysis on a desktop or laptop in just a matter of hours.  

The computational analysis of two similar impellers shows that the lattice Boltzmann solver can capture minuscule fluctuations in the power draw, as well as the differing power draws between the two impellers, whereas traditional solvers can predict the average power draw well for each impeller, but not the instantaneous fluctuations.  

Figure 3 highlights the ability of the Boltzmann solver to capture the power draw fluctuations for the two similar impellers.  These fluctuations are observable in experimental analysis, which highlight the necessity of use of the solver because it allows changes to the structure of various impellers, or new impellers all together to be analysed immediately rather than waiting for physical prototypes to be constructed and tested.  

Additionally, static mixers rely on the transient propagation of turbulent eddies throughout the pipeline, as well as the momentum ratio between the main pipeline flow and an injected side stream to produce the mixing effect.  

The static mixers help create barriers in the system that trip vorticies and promote turbulent eddy formation.  The difficulty with traditional CFD solvers is that in order to capture the turbulent eddies, extremely fine meshes need to be created in the near wall regions where the eddies are formed.  

Alternatively, the bounce back functionality of the Boltzmann solver on the mesh lattice allows more accurate solutions to be produced in a short timeframe with a desktop or a laptop.  

The lattice Boltzmann solver captures the momentum ratio (MR) effect observed in experiments.  Capturing this effect is important because it can impact the radial distribution in the pipe.  

Figure 4 shows the comparison between the simulated results of a high momentum ratio and a low momentum ratio setup with that of experimental means.  

The results show the ability to produce a solution which captures the driving factors in static mixer performance, and thus provide the ability to develop improved mixers without having to involve an expensive third party or expensive lab equipment.  

The design of static mixers centres around the coefficient of variation (COV) which is a mathematical comparison of the deviation to the average of two different species at a specific location in the system.  

One particular static mixer, the UltraTab®, was developed to provide a COV of 0.05, which is industrially accepted as a well-mixed solution, in a short distance from the mixer outlet.  

Traditional CFD solvers predict a COV of 0.25 at a length equal to 3 diameters down stream from the outlet of the mixer, while the lattice Boltzmann solver produces a COV value of 0.05, according to Figures 5.  

The reasoning behind the inability of the traditional CFD solver to predict the correct COV is depicted in Figure  where the lattice Boltzmann solver predicts an even distribution in the pipe of the injected material depicted in Figure 6a) while the traditional solver predicts the injected material aggregates at the top of the pipe in Figure 6b).    

Application in Process Improvement

In addition to improving the products themselves, the lattice Boltzmann solver also can help improve processes for customers.  Systems that involve rotating mixers to produce specific products typically require a very specific blend time, or the time it takes a volume to become fully integrated with injected material.  

There is some subjectivity around determining these results in an experimental setting due to the reliance on an observer’s eye test as to when the system fully changes colour in an acid-base neutralisation.  

A technique has been developed with the lattice Boltzmann solver which determines the blend time by monitoring equal volumes in a system for when each one reaches a steady state concentration of injectant.  

Figure 7 shows that the lattice Boltzmann solver predicts the blend time within 10% error of the observed experimental results.  This is much more accurate than the 30% error that was produced using the traditional CFD solver.  

Given the fact that there is some ambiguity around experimental observations, 10% error is within what is considered acceptable deviation within the mixing industry.  

When that accuracy is coupled with the fact that results can be obtained computationally within about one day, this provides the ability for simulations to be run on specific customer applications in order to provide a tailored solution for the specific process.  

Conclusions

Computational methods already are a part of the toolbox that mixer companies have in order to develop new mixers and provide solutions for customers.  The development of a more efficient, more accurate solver with the help of some of the largest players in the industry, provides a tailored tool that can improve the effectiveness of engineers.  

The lattice Boltzmann solver requires less direct involvement from an engineer for setup and post processing than traditional solvers by a factor of up to 4.  For simple simulations the results are less profound, but for the complex simulations that are required to significantly impact customer applications, the direct time savings for engineers can be on the order of multiple days.  

Not only does the lattice Boltzmann solver provide solutions that were not previously viable at accuracy levels that were not practical, but it also allows companies to deploy their resources in a more effective manner.

One of the other major advantages of the Boltzmann solver is that it can be run on a desktop or laptop computer.  This is because the solver has the unique ability to be run on GPUs or Graphical Processing Units.  

GPUs are much cheaper of an investment than high performance computing options that would be required to analyse through traditional simulation methods.  The lattice Boltzmann solver can improve performance of mixing companies and allow their customers to be serviced with greater accuracy and with greater efficiency.