# Laminar flow in a baffled stirred mixer (COMSOL)

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1 AALTO UNIVERSITY School of Chemical Technology CHEM-E7160 Fluid Flow in Process Units Laminar flow in a baffled stirred mixer (COMSOL) Sanna Hyvönen, Nelli Jämsä,

2 Abstract In this simulation experiment a stirred tank with four baffles was modeled. Then the results were compared to literature. The impeller flow patterns were studied and it was concluded that the impeller worked like an axial flow impeller. The shear rate results were also studied, and it was concluded that the shear rate is highest near the impeller blade tip. This is also in compliance with literature. It was concluded that, even though computationally heavy, the computational fluid dynamic simulations provide good estimations for baffled stirred tanks with laminar flow.

4 1. Introduction Navier-Stokes equations form the basis for calculating the motion of fluids. They could be described as Newton s second law of motion for fluids. The Navier-Stokes equations express the conservation of momentum. For a compressible Newtonian fluid, the equation yields the following [1]: where u is fluid velocity, p is fluid pressure, ρ is fluid density and µ dynamic viscosity of the fluid. Different terms of the equation stand for [1]: 1. Inertial forces 2. Pressure forces 3. Viscous forces 4. External forces applied to fluid These equations must be solved together with the continuity equation which stands for conversation of mass in the system [1]: When simulating fluid dynamics, these equations are crucial. The computer solves these equations with particular set of boundary conditions. These boundary conditions are for example inlets, outlets and walls. As a solution, fluid velocity and pressure in a given geometry is predicted. [1] The flow conditions in this simulation experiment are laminar meaning that the impeller Reynolds number is below ten. [3] The formula for impeller Reynolds number is as follows: 1

5 where D is impeller diameter, N is impeller speed, ρ is fluid density and µ dynamic viscosity. The tank in this experiment has baffles. Wall baffling affects significantly on the flow behavior and therefore the quality of mixing. The baffles transform tangential flows to vertical flows, ensure top-to-bottom mixing and they minimize air entrainment. However they increase drag and power consumption. [3] The impeller used in this experiment is a two-blade axial flow impeller. Axial flow impellers are used for blending, solids suspension, solids incorporation, gas inducement and heat transfer. The most important flow characteristics for an impeller can be divided into: flow patterns, pumping, and shear. [3] This experiment consist of modeling a stirred tank using computational fluid dynamics (CFD). The model itself is represented in the next chapter. CDF modeling requires study of many aspects of the process: Domain of interest, here volume occupied by the fluid inside the tank, which is described by a computation grid, a collection of small sub-domains. In these cells the variables are computed and stored. Motion of the impeller. Construction of the computational grid and the solution method used to numerically obtain the flow field. [4] CDF has many benefits and applications including; augmenting design correlation and experimental data, providing comprehensive data, not easily obtained from experiments, reduces scale-up related problems, evaluating plant problems, complementing physical modeling and what-if analysis. [4] The simulation in this report is carried out using COMSOL, but other newer and faster methods were studied briefly. Cudmore et al (2015) developed a model for rotating impellers that could be used as a diagnostic and operational planning tool for mixing equipment. Its requirements were that it should run faster than real time, that it could 2

6 be applied to different operating conditions economically, and that it provides stability limits and the mean square amplitude of the random impeller orbits directly. The solution was a linear lumped-parameter approach of minimal dimensionality. This works provided that the fluid forces on the orbiting impeller can be accurately modelled using the concept of fluid added mass, damping and stiffness. The model was simulated for turbulent flows, so it won t be compared further here. [5] With increasing ability of computation a lot of effort has been put into producing numerical methods for simulating the flow inside stirred tanks. Most of these focus on turbulent flows, since they are more complex and computationally heavy to simulate. Different methods include Reynolds averaged Navier-Stokes (RANS), large eddy simulation (LES), and a turbulence hybrid model-detached eddy simulation (DES). Chara et al (2016) focused on DES, which transfers LES to a RANS-based simulation in boundary layers. They compared experimental data with a DES mode from the CFD package of ANSYS Fluent and found that the DES is a suitable tool to predict turbulent flow in a tank stirred by a Rushton turbine. Again, as this was a simulation for turbulent flow and with a different impeller, no further comparisons to this report were made. [6] 2. Model The modelling of the mixed tank is carried out according to COMSOL tutorial [2]. This simulating experiment uses the rotating machinery feature of the CFD Module in COMSOL Multiphysics 5.2. The model equations in the following model are divided to two parts: Navier-Stokes equations in a rotating frame in the inner domain, Navier-Stokes equations in fixed coordinates in the outer domain. These separate parts need to be coupled together with an identity pair, where a flux continuity boundary condition is applied [2]. The fluid in the tank is water and impeller rotates at a speed of 10 RPM and counterclockwise. Simulation is carried out for 6 seconds with 0.25 second steps. The modeled tank can be seen in figure 1. [2] 3

7 Figure 1. Stirred tank with four baffles. 3. Results Shear rate, velocity field, velocity magnitude and pressure were plotted. These results can be seen from figures

8 Figure 2. Velocity magnitude. Figure 3. Pressure. 5

9 Figure 4. Shear rate. Figure 5. Velocity field represented with arrow surface. 6

10 4. Discussion As can be seen from figure 5, the flow pattern caused by the impeller seems to be typical for a hydrofoil impellers as the liquid is being pumped down. The seen flow pattern is typical for an axial impeller; the flow pattern is produced throughout the entire tank volume as a single stage. Radial flow impeller would have produced two circulating loops, one below and one above the impeller. [3] In figure 4, the shear rate is represented. Shearing force, or shear tress, is related to flow velocities and carries out the mixing process and is responsible for creating fluid intermixing, dispersing gas bubbles and breakage of liquid drops. Shear stress is a function of shear rate, which is defined by velocity gradients, impeller blade pressure drop, turbulence level and viscosity. [3] Shear rate can be described as a time constant, meaning that if shear rate at highest is 120 1/s, and it means that the events in the flow occur on the order of 8 ms. The highest shear rates occur in the immediate vicinity of the impeller as can be seen from figure 4. This volume however is very small meaning that only small part of the material is exposed to these higher rates. The local shear rate depends from mixing speed and distance from the impeller blade tip. [3] High shear rate can break shear sensitive materials such as crystals and biological materials and it is therefore important to design mixing taking into account the shear forces. [3] One down side of computational fluid dynamics is that is computationally heavy. [4] In this experiment 3D-model was used and as well as fine mesh grid and some animations were created from the acquired data. Simulating the experiment took in total 4 hours of computer computing even though the used computer was high tech. 7

11 5. Conclusion The flow pattern and the shear rates of the carried out simulation were studied. It was concluded that the simulated impeller works similarly as an axial flow impeller, creating one clear pattern around the impeller and pumping water down. The shear rate was concluded to be highest in the close distance of the impeller blade tip. These results are in compliance with the literature. It is clear that good simulation results of stirred tank systems with bafflers can be acquired using computational fluid dynamics, even though the simulations are quite heavy and time consuming on regular PCs. References [1] Anonym, https://www.comsol.com/multiphysics/navier-stokes-equations [2] COMSOL tutorial, Laminar Flow in a Baffled Stirred Mixer [3] E. Paul, V. Atiemo-Obeng, S. Kresta, Handbook of Industrial Mixing: Science and Practice (1), Wiley-Interscience, [4] E. Marshall, A. Bakker, Computational Fluid Mixing, Fluent Inc, USA, 2003, ISBN [5] G. Cudmore, A. Holloway, A. Gerber, A model of impeller whirl for baffled mixing vessels, Journal of Fluids and Structures, Volume 54, April 2015, Pages , ISSN , [6] Z. Chara, B. Kysela, J. Konfrst, I. Fort, Study of fluid flow in baffled vessels stirred by a Rushton standard impeller, Applied Mathematics and Computation, Volume 272, Part 3, 1 January 2016, Pages , ISSN , doi: /j.amc

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