Selected studies on the experimental analysis of physical and chemical processes in stirred tanks.

## Abstract

A short review of the state of the art in experimental and computational fluid dynamics (CFD) characterization of micro-hydrodynamics and physicochemical processes in stirred tanks and agglomeration reactors is presented. Results of experimental and computational studies focusing on classical mixing tanks as well as other innovative reactors with various industrial applications are briefly reviewed. The hydrodynamic characterization techniques as well as the influence of the fluid dynamics on the efficiency of the physicochemical processes have been highlighted including some of the limitations of the reported modeling approach and solution strategy. Finally, the need for specialized CFD codes tailored to the specific needs of fluid-particle reactor design and optimization is advocated to advance research in this field.

### Keywords

- physicochemical
- hydrodynamics
- wet agglomeration
- stirred tanks
- CFD

## 1. Introduction

Hydrodynamic and physicochemical interactions play an important role in many industrial unit processes and hence its importance in many engineering applications of fluid flow. Fluid flow investigations in a wide range of process conditions as well as complex biological, physical and chemical processes have been the subject of many scientific publications over the past two decades. Several studies on bench, pilot and industrial scales have been conducted on a wide variety of hydrodynamic conditions and different reactor geometric designs. In many of these studies, the aim is to provide an insight into the fluid flow and process dynamics in terms of the spatial and temporal evolution within the flow device, and in some cases, performance testing of newly designed flow units and processing techniques with potential applications on industrial scale. Regardless of the focus of these studies, it is quite apparent that valuable information can be obtained from the basic study of fluid flow dynamics in process units especially from design and optimization perspective.

A quick survey of the studies in this field shows that many innovative process reactors have been successfully tested on different scales for a wide variety of technical applications ranging from fine particle separation and water purification to cell culture preparation [1, 2, 3, 4, 5, 6]. Experimental data, which are collected in these studies for numerical validation purposes, are often used to characterize the hydrodynamic behaviour as well as to quantify the fluid parameters of interest such as the flow velocity profile, vorticity, turbulent kinetic energy and its rate of dissipation, turbulent intensity, and so on. While there is a large body of scientific literature focusing on the hydrodynamics and physicochemical processes in stirred tank reactors, the aim of the present communication is to briefly summarize developments in this field especially in the application of the knowledge of the fluid dynamics to fluid-particle reactor design, development and optimization.

## 2. Design and formulation of mixing tank problems

### 2.1. Design parameters and process optimization

In fluid engineering problems, research has shown that it is possible to optimize all influencing process parameters in an evolutionary manner right from the conceptual design to the final performance testing phase. This will entail the integration of the fluid flow investigation with the process reactor conceptual design and system optimization [1]. Nowadays, this multistage process design and optimization work flow shown in Figure 1 can be fully automated through the use of computational platform. In formulating and developing a numerical solution strategy to a particular physical problem involving fluid-particle interactions, a sound theoretical understanding and analysis of the problem is often required. This will assist in the selection of appropriate experimental data collection methods and mathematical models that sufficiently encapsulate the physics of the problem. A number of numerical approaches and solution strategies discussed in the subsequent sections have been developed for a multitude of fluid flow scenarios. Therefore, it is important to evaluate each circumstance individually and form an opinion regarding which model would provide the best fit for a particular fluid engineering problem. It has been suggested that the robustness of any mathematical model is a function of the numerical code being used and the flow scenario being modeled [7].

### 2.2. Fluid dynamics and governing equations

The interactions of different phases in fluid flow occur on different scales of the fluid motion as depicted in Figure 2. Fluid dynamics is primarily focused on the macroscopic phenomena of the fluid flow in which the fluid is treated as a continuum. For instance, a fluid element is composed of many molecules, and the fluid dynamics represent the behaviour of the numerous molecules within the system. This concept with certain assumptions forms the basis of the derivation of fluid conservation equations of mass and momentum also known as the Navier-Stokes equation using a fluid control volume [8, 9]. The general form of the governing equations of mass and momentum conservation in any fluid flow system can be written as follows (Eqs. (1) and (2)):

where

### 2.3. Modeling approach and solution strategies

In modeling complex single and multiphase flows in mixing tanks and process reactors, there exist two common numerical solution strategies, namely Eulerian-Eulerian and Eulerian-Lagrangian modeling approach, depending on the scale of the fluid flow as shown schematically in Figure 3. In the former, the fluid domain is treated as an interpenetrating continuum, while in the latter, the discrete or distinct particles of the dispersed phase are tracked in the Lagrangian reference frame. In addition to the flow field, information on the particle population such as the mean size, mass or volume fraction, and number density can be obtained using either of the two approaches [10]. Several variants of these two classes exist such as the Eulerian granular model based on the kinetic theory of granular flow (KTGF), disperse phase model (DPM), discrete element model (DEM) and the macroscopic particle model (MPM). In the case of Eulerian-Eulerian approach, the species distribution of the discrete phase may be accounted for using the population balance model (PBM), while the Eulerian-Lagrangian models can directly compute the particle size distribution while taking into account different collision and interaction mechanisms using DEM [15, 16, 17, 18].

#### 2.3.1. Treatment of flow domain and turbulent flow conditions

Turbulence modeling forms an integral part of the numerical analysis of complex fluid flows since most engineering fluid flows entail certain form of instability. Several closure models have been developed for resolving turbulence parameters in steady-state Reynolds Averaged Navier-Stokes (RANS) equations. The two equation eddy viscosity models such as k-ε and k-ω have been found to perform reasonably well in the modeling of rotating flows in process reactors with the only drawback being the assumption of local isotropic turbulence. The underlying theoretical assumptions underpinning the use of these models can be found in the following reference texts [12, 13]. Since the reactors encountered in most of the practical physicochemical processes contain moving or rotating parts, it is therefore necessary to take this into consideration in the preparation of the computational grid. The most common strategy for steady-state calculations include the single reference frame (SRF), multiple reference frame (MRF) or frozen rotor approach, mixing plane model (MPM) and snapshot approach, while the sliding or dynamic mesh is frequently used in transient calculations of fluid flow. For detailed information on the practical applications of the above-mentioned methods, readers are referred to the following reference texts [20, 21].

#### 2.3.2. Model coupling for multiphase flow problems

Modeling complex physicochemical processes involving fluid flow sometimes necessitates the integration of the existing mathematical models in order to appropriately describe the physics of the problem. This can be achieved through the use of specially developed or customized in-house numerical codes or a modification of the existing ones with several software package vendors offering a platform for software improvement through the use of Application Programming Interface API or Application Customization Toolkit ACT. Such flexibility allows engineers and researchers to extend the capability and versatility of the existing numerical codes. Many software vendors go a step further in this respect by actively encouraging the development of scalable apps that extend the capability of their core software; an excellent example is the mixing tank template released by ANSYS Inc. for the automation of mixing tank simulation process. However, there exist several other flexible options for numerical code development using the open source platform, and the readers are advised to consider available options for their specific problem.

## 3. Experimental analysis of physicochemical processes

Several analytical and instrumental techniques have been developed for the study of complex hydrodynamic-mediated processes found in particle-laden flow—flocculation, wet agglomeration, sedimentation, floatation, fluidization and crystallization that often occur in a wide range of process conditions. These techniques shown in Figure 4 are used either in the quantification of the hydrodynamics of the carrier and dispersed phase, or in the determination of the spatial and temporal evolution of the discrete phase properties such as the change in the particle size and distribution. In the case of the hydrodynamic interactions of the carrier and dispersed phase, a number of laser-based fluid flow techniques such as particle image velocimetry (PIV), particle tracking velocimetry (PTV), laser Doppler anaemometry (LDA), laser Doppler velocimetry (LDV), and more recently, radioactive tracking techniques such as positron emission particle tracking (PEPT) and computer-aided radioactive particle tracking (CARPT) have gained wider acceptance in the scientific community and in industry due to their ease of use and non-intrusive nature [20, 21, 22]. These techniques provide valuable insight into the salient macroscale fluid flow characteristics such as the instantaneous and time-averaged hydrodynamic behaviour of the continuous phase, as well as the influence of the dispersed phase on the fluid flow. This is achieved by coupling the flow field measurements with the particulate phase properties and motion [21]. The experimental data set is subsequently used in the validation of numerical simulation results [18, 23].

The dominant and widely used macroscale experimental fluid flow characterization techniques are the laser velocimetry and radioactive particle tracking techniques such as the PIV or PTV, LDV or LDA with the PIV reported to be a more efficient technique [24]. These on-line methods facilitate the determination of the properties of multiphase particle-laden flow especially at low concentration. These local methods are quite superior to other similar techniques such as optical fiber probing and light scattering due to their non-intrusive nature with little or no interference on the flow while providing time series and time-averaged fluid flow characteristics with a high spatial resolution [18]. The workings of typical field imaging technique such as PIV consist of the tracer particles, laser source for flow illumination and high capacity cameras—complementary metal-oxide semiconductor (CMOS) or charge-coupled device (CCD) for the fluid flow image recording. The captured images are thereafter post-processed and correlated to obtain the hydrodynamic parameters of interest. Table 1 provides a list of recent publications on the experimental analysis of physicochemical processes in stirred tanks. These studies demonstrated the importance of robust and reliable experimental data for complex fluid flow analysis and numerical model validation. Recent advances in experimental techniques have led to the emergence of radioactive particle tracing measurement techniques which aim to improve the ease of data collection, data accuracy and reliability.

Reactor configuration | Stirrer configuration | Experimental technique | Technical application | Tracer particles |
---|---|---|---|---|

Cylindrical tank | Rotating disc | 2D PIV | Mixing/agglomeration | Silver-coated and hollow glass spheres [1] |

Cylindrical tank | Hydrofoil impellers | PEPT | Mixing | Radioactive particles [26] |

Cylindrical tank | Rushton turbine | PIV | Mixing | Polymeric and glass particles [27] |

Cylindrical tank | Hollow blade semi-elliptic disc turbine | TRPIV, PIV | Mixing | Neutrally buoyant glass beads [24] |

Cylindrical tank | Pitched-blade turbine impeller | FPIV | Mixing | Soda-lime glass beads [28] |

Cylindrical tank | Rushton turbine | CARPT | Mixing | Radioactive particles [29] |

Cylindrical tank | Rushton turbine | LDA | Mixing | Hollow glass spheres [30] |

Cylindrical tank | Kenics static mixer | PEPT | Mixing | Radioactive particles [31] |

Cylindrical tank | Pitched-blade turbine | PEPT | Mixing | Radioactive particles [32] |

Cylindrical tank | Pitched-blade turbine | PIV | Mixing | Silica glass spheres [23] |

Square tank | Hydro foil impeller | PIV, image analysis | Mixing/agglomeration | In situ agglomerated flocs [33] |

Cylindrical tank | Six-blade Rushton turbine | 3V3 | Mixing | Opt image polycrystalline particles [34] |

Cylindrical tank | Rushton turbine, pitched-blade turbine | PEPT | Mixing | Monosized silica gel particles [35] |

Cylindrical tank | Six-blade Rushton turbine | PEPT, LDA | Mixing | Ion-exchange resin particles [36] |

Cylindrical tank | Rotor-stator mixer | PIV | Mixing | Polyamide particles [37] |

In order to correlate the hydrodynamic and process conditions with the suspension or dispersion properties especially the change in the species concentration—spatial and temporal evolution of the particle size distribution, a number of laboratory measurement techniques are widely adopted [25]. The choice will depend to a large extent on the concentration and size distribution of the disperse phase and the nature of the flow. Regardless of the chosen analytical approach, such a correlation will facilitate an assessment of the treatment process and the reactor performance under a particular process condition. For instance, the conventional physicochemical simulation tests such as the cylinder, Imhoff cone and jar tests can be combined with parametric analytical techniques such as the Buchner-funnel or pressure filtration test, capillary suction time (CST) test, electrokinetic charge analysis using colloidal titrations (i.e. zeta and streaming potential), laser light scattering or laser diffraction, microscopy, image analysis, photometric dispersion analysis (PDA), fiber optic sensor and HNMR spectroscopy. These techniques have been successfully employed to characterize the physicochemical process in bench, pilot and full-scale studies [38, 39, 40, 41]. A careful consideration of the limitations of each of these approaches will ensure proper selection of an appropriate method.

In most of the physicochemical processes involving particulate flow either as a colloidal dispersion or granular suspension, the species attributes—mean size, particle concentration and distribution and fractal properties of the resulting agglomerates—are the primary parameters of interest [21]. In this case, an appropriate physicochemical simulation such as a jar or cylinder test is often followed by a parametric analysis to characterize the process performance as a function of species attributes. Several other parameters may be of interest depending on the type of reactor and the required solid-liquid separation method. Such parameters may include aggregate mean size, shape and distribution, aggregate volume concentration, aggregate strength, sludge volume index, silting index, residual supernatant turbidity, absorbance or optical density, electrical conductivity, viscosity, zeta or streaming potential, specific resistance to filtration, capillary suction time, and so on [38, 39]. In the case of chemical optimization, a parametric dose-response curve will give reasonably accurate information on the required chemical dose for a particular process condition [42, 43, 44, 45]. Table 2 and Figure 5 show a typical correlation of the agglomerate test properties with the process condition—shear rate. However, regardless of the choice of parametric test, an examination of the supernatant, sediment, filtrate and residue will yield some valuable information on the reactor performance under specific process conditions. Such assessment is carried out either by direct *in situ* measurements such as in particle counting, *ex situ* analysis in which the samples are extracted for measurements or by other indirect parametric indicators. A detailed discussion on the practical applications of different dispersed phase measurement techniques is available elsewhere [40, 41].

Test parameters | Agitation speed | |
---|---|---|

145 rpm | 165 rpm | |

Mean agglomerate diameter, mm | 3.8330 | 3.9182 |

Mean agglomerate compressive strength, Nm m^{−2} | 0.4298 | 0.4351 |

Mean strain rate, s^{−1} | 0.3639 | 0.4088 |

Mean maximum compressive force, N | 4.9476 | 5.2303 |

Considering the wide range of options available to select from, optimizing a given physicochemical condition for a particular process reactor under laboratory conditions is a daunting task. Therefore, in optimizing the design and process parameters for a particular reactor, a statistical correlation of these parameters from a data set is often required, depending on the available time and complexity of the problem, to obtain accurate information on the optimum design and process conditions. A number of statistical methods such as the design of experiment and response surface methodology can be applied to a large set of experimental data to obtain the desired optimization points. This will facilitate an understanding of the influence of different process conditions on the reactor performance which will assist in the selection of optimized operating conditions.

## 4. Modeling physicochemical processes in stirred tank reactor

The use of computational fluid dynamics (CFD) as a research tool to investigate complex fluid-particle interactions has been growing in popularity both in academia and in the industry [46]. CFD provides a powerful alternative and a more robust platform for engineers in the design of equipment and processes involving fluid flow and heat transfer when compared to the classical experimental approach. Nowadays, numerical simulations complement the experimental and analytical techniques and are increasingly being performed in many fluid engineering applications ranging from chemical and mineral processing to civil and environmental process engineering [46]. However, it is worth pointing out that the continual development of reliable empirical, mathematical and computational models relies on a robust and detailed experimental data.

Tables 3 and 4 provide a list of recent experimentally validated numerical studies focusing on the physicochemical analysis of fluid-particle reactors. The former is focused on the analysis of the mixing phenomena in stirred tanks while the latter deals with the technical application of mixing for several industrial processes. The modeling approach in most of these studies is applicable to mixing tanks and process reactors of various geometric designs. Joshi et al. [47, 48] provide a comprehensive review of CFD applications in a single phase mixing tank hydrodynamic analysis focusing on axial and radial flow impellers in a multitude of flow scenarios. Their two-part study, which is one of the most detailed and comprehensive reviews in this field, summarizes developments in mixing tank modeling by bringing together the results of scientific investigations spanning several decades. Similar reviews focusing on turbulent multiphase flows and multiphase reactor modelling, and which provide a more comprehensive discussion on the subject matter are available elsewhere [19, 49, 50, 51].

Reactor configuration | Fluid agitator/application | Experimental validation method | Numerical code/modeling approach | Turbulence models |
---|---|---|---|---|

Cylindrical tank | Grid disc impeller | LDA | CFX/MRF | k-ε [53] |

Cylindrical tank | Grid disc impeller, solid disc, propeller | LDA | CFX/MRF | k-ε [52] |

Cylindrical tank | Rushton turbine, flotation impeller | 2D PIV | Fluent/MRF | k-ε [54] |

Cylindrical tank | Rushton disc impeller | LDA | Fluent/snapshot | k-ε [55] |

Cylindrical tank | Rushton turbine | LDA | Fluent/MRF | k-ε, DES [56] |

Cylindrical tank | Foil impeller, Rushton turbine | Image analysis | PHOENICS/MRF | k-ε [57] |

Cylindrical Tank | Pitched-blade turbine | PEPT | CFX/MRF | k-ε [58] |

Cylindrical tank | Rushton turbine | LDV | Fluent/MRF | k-ε [59, 60] |

Cylindrical tank | Rushton turbine | PLIF | Fluent/MRF | k-ε [61] |

Square tank | Rushton turbine | Power consumption measurements | Fluent/MRF | RSM [62] |

Cylindrical tank | Pitched-blade turbine | 2D PIV | Fluent/sliding mesh | k-ε [63] |

Cylindrical tank | Rushton turbine | CARPT | Fluent/MRF | k-ε [64] |

Cylindrical tank | Rushton turbine | Solids concentration measurements | CFX/MRF | k-ε [65] |

Cylindrical tank | Pfaudler retreat curve impeller | 2D PIV, laser granulometry, nephelometry | Fluent/sliding mesh | k-ε [66] |

Square tank | Rotating cylinder | LDA | Fluent/MRF | k-ε [67] |

Cylindrical tank | Rushton turbine, pitched blade turbine | RPT, LDA | Fluent/MRF | k-ε [68] |

Cylindrical tank | Rushton turbine | LDV | Fluent/MRF | k-ε, LES [69] |

Cylindrical tank | Flat blade turbine, pitched blade turbine, Rushton turbine | LDV | Fluent/MRF | k-ε [70] |

Cylindrical tank | Rushton turbine | LDV | Fluent/MRF | RSM [71] |

Cylindrical tank | Rushton turbine, disc turbine, elliptical blade disc turbine | SPIV | Fluent/sliding mesh | k-ε, LES [72] |

Cylindrical tank | Rushton turbine | 2D PIV, LDA | Fluent/sliding mesh | DES [73] |

Cylindrical tank | Double Rushton turbine | LDA | CFX/MRF | k-ε [74] |

Cylindrical tank | Rushton turbine | Mixing time, power consumption, solids concentration measurements | CFX/MRF/sliding grid | k-ε [75] |

Cylindrical tank | Rushton turbine | Particle size analysis, conductometry | Fluent/MRF | k-ε [76] |

Cylindrical tank | Pitched-blade turbine, double disc impeller | PIV, critical impeller speed measurements | Fluent/MRF | k-ε [77] |

Cylindrical tank | Rushton turbine | LDV | Fluent/MRF | k-ε [78] |

Cylindrical tank | Pitched-blade turbine | PEPT | Fluent/MRF | k-ω, k-ε, RSM [79] |

Cylindrical tank | Rigid, rigid-flexible and punched rigid-flexible impeller | Solids concentration measurements | Fluent/MRF | k-ε [80] |

Cylindrical tank | Flat blade impeller, angle pitch impeller | DPIV | Fluent/MRF | k-ε [81] |

Cylindrical tank | Rotor-stator mixer | PIV | Fluent/MRF/sliding mesh | k-ε, k-ω [82] |

Cylindrical tank | Rotor-stator mixer | LDA | Fluent/sliding mesh | k-ε [83] |

Reactor configuration | Fluid agitator/application | Experimental validation method | Numerical code/modeling approach | Turbulence models |
---|---|---|---|---|

Cylindrical flocculator | Paddle mixer/flocculation | LDA, 2D PIV | CFX/MRF | k-ε, RSM [84] |

Rectangular flocculator | Axial impeller/water purification | 2D PIV | Fluent/MRF | k-ε [85] |

Cylindrical sedimentation tank | Axial impeller/water purification | Laser diffraction | CFX/MRF | k-ε [86] |

Cylindrical Jar testing device | Paddle stirrer/flocculation | LDA | Fluent/MRF | k-ε, k-ω, RSM [7, 87, 88] |

Cylindrical flocculation reactor | Rushton turbine/bio-flocculation | LDV | Fluent/MRF | k-ε [89] |

Cylindrical stirred tank | Pitched turbine blade/silica particle deagglomeration | Laser diffraction/PIDS | Fluent/MRF | k-ε [90] |

Cylindrical stirred bioreactor | Marine impeller/cell cultivation | Tracer and dynamic method | Fluent/MRF | k-ε [91] |

Cylindrical tank | R1342-type impeller/flocculation | Image analysis | Fluent/MRF | k-ε [92] |

Cylindrical tank | Rushton impeller/cell culture | Optical sensor | CFX/MRF | k-ε [93] |

Cylindrical bioreactor | Rushton, scaba and paddle impellers/cell culture | Optical density | Fluent/MRF | k-ε [94] |

Cylindrical tank | Turbine, anchor and oblique impellers/autoclave | Tracer injection | Fluent/MRF | k-ε [95] |

Cylindrical bioreactor | Marine impeller/recombinant protein synthesis | PIV | Fluent/MRF | k-ε [91, 96] |

Cylindrical tube reactor | Impeller/bacterial inactivation | 2D PIV | CFX/MRF | RSM [2] |

Cylindrical bioreactor | Rushton turbine/anaerobic digestion | Gas chromatography | Fluent/MRF | k-ε [97] |

Cylindrical tank | Turbine impeller/polymerization | Droplet size measurements | Fluent/MRF | k-ε [98] |

Cylindrical tank | Rushton impeller/cell cultivation | Dynamic method | Fluent/MRF | k-ε [99] |

Cylindrical tank | Rushton turbine/cell inactivation | PIV | Fluent/MRF | k-ε [100] |

Cylindrical tank | Rushton turbine and propellers/cell culture | Dynamic method | Fluent/MRF | RSM [101] |

Cylindrical crystallizer | Rushton impeller/precipitation | X-ray/laser diffraction | Fluent/MRF | k-ε [102] |

Cylindrical Photobioreactor | Rotating cylinder/algal culture | Optical density | Fluent/SRF | k-ω [103, 104] |

Regardless of the specific focus of each study, most of the studies differ only in terms of stirrer-vessel configurations, experimental validation methods and the choice of modeling approach. In terms of the stirrer-vessel configuration, there is a wide variety of flow inducers available for fluid flow investigation, each with different power demands and flow patterns. In addition to well-established impeller designs employed in most of the studies—Rushton turbine, pitched-blade turbine, propeller, and so on, a few innovative designs have been used with good results [52]. The turbulence models of choice in most of the investigations are the two equation eddy viscosity models such as k-ε and k-ω, and RSM models which are quite efficient in handling rotating flows in stirred tanks and multiphase reactors. The dominant modeling approaches for rotating flow problems are the MRF and sliding mesh. The former is suitable for steady-state problems while the latter is employed for transient calculations. Despite the technical limitations of some of the experimental flow measurement techniques, reasonable agreement was obtained in most of the studies between the experimental data and numerical simulation. In a few of the studies, the model predictions were not quite robust enough when compared to the experimental data set partly due to the complexity of the flow scenario being modeled.

## 5. Conclusions and future perspectives

A review of recent advances in the experimental analysis and numerical modeling of physicochemical processes in stirred tanks and agglomeration reactors have been presented. This review briefly summarizes important findings and major contributions from numerous publications in this field. This short review of the developments in this field clearly shows that significant progress has been made over the past decade in the understanding of complex physicochemical phenomena that are vital for many industrial and environmental processes, especially from experimental and theoretical perspective. However, there is still a gap in knowledge especially in the suitability of the existing mathematical models to accurately predict the reactor performance in a wide range of existing and emerging processes. This clearly calls for a numerical code programming and development to form an integral part of the engineering training and curriculum in future. The successful design, development and optimization of agglomeration units depend on the robustness of the experimental data, mathematical models and simulation tools. This short review is by no means an exhaustive one, and readers are advised to consult other multitudes of scientific publications on the subject matter. In conclusion, numerical modeling along with robust experimental data will continue to be highly indispensable well into the foreseeable future.

## Acknowledgments

The authors would like to acknowledge the financial support from the National Research Foundation (NRF) and The World Academy of Sciences (TWAS) under the funding instrument number UID: 105553.