(145b) Computational Investigation on the Hydrodynamics of Structured Fluids in Unbaffled Stirred Tanks
AIChE Annual Meeting
2024
2024 AIChE Annual Meeting
North American Mixing Forum
Computational Analyses of Mixing Processes II
Monday, October 28, 2024 - 12:55pm to 1:20pm
Structured fluids are characterized by their capability to self-assemble and form different structures at a scale larger than the atomistic one. Many fluids fall into the category of structured fluids, like surfactant solutions, suspensions, emulsions, and solutions of polymers in water. The temperature and the composition determine the type of microstructure, which influences the macroscopic properties of the fluid, such as the rheology. For example, these fluids can exhibit Newtonian or non-Newtonian behavior depending on the microstructural phase, with abrupt variations of the local apparent viscosity. Consequently, the fluid dynamics in systems containing this kind of fluids can be strongly affected by the operating conditions, from temperature to local composition and applied strain rate. Poloxamers are nonionic triblock copolymers formed by a central chain of polypropylene oxide in between two chains of polyethylene oxide. These molecules, also known by the commercial name of Pluronics, are found in different physical states, depending on the molecular weight and the ratio between the two repeating units. The different solubility in water of the blocks constituting these polymers leads to an amphiphilic behavior, which results in the formation of microstructures in aqueous solutions. Consequently, poloxamers have surfactant properties and are widely employed in the production of pharmaceutical preparations and creams or detergents for personal care. Even though structured fluids like these are widespread, the design of the industrial processes that use them is made difficult by their complex rheology.
Nonetheless, many industrial sectors, such as food, cosmetics, and pharmaceutical ones, employ structured fluids in the production processes. These processes often involve a mixing step, in equipment in which composition and applied strain rate present substantial local variations. Stirred tanks are one of the most studied mixing equipment, with a historical focus on baffled ones, which usually assure more efficient mixing. However, a renewed industrial interest in unbaffled vessels has been growing in recent times. While the baffles help to redistribute the flux in the radial and axial direction, their absence can be a desired feature for some applications. In particular, it can be an interesting choice for fluids with high viscosity or in general for low Reynolds numbers, to reduce the formation of dead zones. Moreover, unbaffled stirred tanks are preferred for shear-sensitive fluids, as in bioreactors, and during crystallization processes to depress aggregation.
When dealing with complex models, such as the rheology of structured fluids, simulation tools can be a great aid to the design of processes and equipment, given the continuous improvements in the capabilities of software and hardware. Moreover, computational models are a way to investigate the role of the involved phenomena and to reduce the number of traditional experiments needed. This work is inserted in such context and focuses on the impact of the rheological properties of structured fluids in the previously mentioned unbaffled vessels. The aim is to help develop a clearer picture of the hydrodynamics in these kinds of equipment and to facilitate the design of future mixing processes that involve structured fluids.
Given the extensive use of poloxamers for various industrial products, the fluid chosen for this study is a mixture of Pluronic L64 at different concentrations in water. Different microstructures arise depending on the composition of this blend, as it has been already found in a previous work by Pasquino et al. [1] using experimental rheometry and dissipative particle dynamics. In general, the mixture can be considered as highly viscous fluid as the concentration of L64 goes beyond 50% in weight, which is also the limit above which a non-Newtonian behavior is observed. The choice of studying the hydrodynamics of this fluid in an unbaffled stirred tank derives from this high viscosity, together with the shear sensitivity for some applications of this product. The tank has an elliptical bottom and is equipped with a standard Rushton turbine, with a total volume occupied by the fluid equal to 10 L.
Four different blends are analyzed: pure L64 at 25°C, mixtures of L64 in water at 40% and 50% in weight at 25°C, and a mixture of L64 in water at 50% in weight at 30°C. Starting from rheological experiments, a Newtonian model is chosen for the mixtures of L64 and water at 25°C, while a power law is used for the pure L64 and the mixture at 30°C. As previously stated, the microstructures formed in such fluids are influenced also by the temperature, and from this difference in the rheology at fixed concentrations, it is possible to identify a phase transition between 25 and 30°C. The described rheology models are used in Computational Fluid Dynamics (CFD) simulations, to obtain information about the power consumption and the variation of velocity and strain rate in the system. The operating conditions in industrial equipment usually lead to a laminar fluid dynamic regimen, due to the high viscosity of these fluids. Moreover, every simulation is performed at the steady state and with constant composition of the fluid, using the multiple reference frame for the zone around the impeller. Consequently, in investigating the dependence of the power number with respect to the Reynolds number, most of the simulations performed in this work focus on Re lower than 100. The simulations conducted with the mixture of L64 40 wt.% in water, which has the lowest viscosity among the studied blends, explore higher Reynolds numbers, between the laminar and the early transitional regimen. A grid independence study is carried out for different Reynolds numbers, monitoring the power number calculated both using the torque on the impeller and the laminar dissipation energy rate.
In addition to the constitutive equations for viscosity obtained through interpolation of the experimental data with an a priori model, such as the power law, an alternative viscosity model was tested. Such a model is based on the Gaussian Process Regression (GPR), which is a statistical technique akin to machine learning. The GPR actuates a regression on the data, providing a model which outputs single values for the predicted variable, rather than a functional form for viscosity with respect to the strain rate. The GPR viscosity model takes as input the strain rate for every cell of the computational domain and returns the corresponding viscosity prediction, based on the training performed with the experimental rheometry data. One of the most interesting characteristics of the GPR is the capability of returning as output a measure of the quality of the regression. With such information, it is possible to make an evaluation of the training process and consider an enlargement of the training dataset. The model itself can suggest where the regression has a lower quality and help the user identify the optimal values for the next experimental samples. Moreover, the computational performance of the GPR model is evaluated by comparison with the computational times required by the power law model. While the GPR unnecessarily increases the computational costs when the rheogram can be accurately interpolated with a functional form, this can be used as an initial test for the approach. The next step would be creating a multidimensional GPR model in which the viscosity is predicted depending on more input variables, e.g. the strain rate, the composition, and the temperature. In such a case the increased computational cost would be compensated by the complexity of obtaining a multivariate function that can accurately represent the viscosity variations.
The results obtained from the CFD simulation are validated against the results of the experimental setup for the same unbaffled stirred tank. From a qualitative point of view, the trend of the power number with respect to the Reynolds number from the simulations agrees with the experimental one. A quantification of the difference between the computational results and the experimental one can be done through the average relative error, which is about 25%.
In conclusion, computational studies can be a precious aid to the experiment once the desired level of accuracy of the model is achieved. The CFD simulations can be useful to predict integral variables related to the power consumption, but also to analyze more in detail the features of the flow that are not accessible experimentally. On the other hand, the GPR remains a promising tool, under the conditions of a satisfactory training step and a model complex enough to justify the increase in computational costs.
[1] Pasquino et al., Soft Matter, 2019,15, 1396-1404
Nonetheless, many industrial sectors, such as food, cosmetics, and pharmaceutical ones, employ structured fluids in the production processes. These processes often involve a mixing step, in equipment in which composition and applied strain rate present substantial local variations. Stirred tanks are one of the most studied mixing equipment, with a historical focus on baffled ones, which usually assure more efficient mixing. However, a renewed industrial interest in unbaffled vessels has been growing in recent times. While the baffles help to redistribute the flux in the radial and axial direction, their absence can be a desired feature for some applications. In particular, it can be an interesting choice for fluids with high viscosity or in general for low Reynolds numbers, to reduce the formation of dead zones. Moreover, unbaffled stirred tanks are preferred for shear-sensitive fluids, as in bioreactors, and during crystallization processes to depress aggregation.
When dealing with complex models, such as the rheology of structured fluids, simulation tools can be a great aid to the design of processes and equipment, given the continuous improvements in the capabilities of software and hardware. Moreover, computational models are a way to investigate the role of the involved phenomena and to reduce the number of traditional experiments needed. This work is inserted in such context and focuses on the impact of the rheological properties of structured fluids in the previously mentioned unbaffled vessels. The aim is to help develop a clearer picture of the hydrodynamics in these kinds of equipment and to facilitate the design of future mixing processes that involve structured fluids.
Given the extensive use of poloxamers for various industrial products, the fluid chosen for this study is a mixture of Pluronic L64 at different concentrations in water. Different microstructures arise depending on the composition of this blend, as it has been already found in a previous work by Pasquino et al. [1] using experimental rheometry and dissipative particle dynamics. In general, the mixture can be considered as highly viscous fluid as the concentration of L64 goes beyond 50% in weight, which is also the limit above which a non-Newtonian behavior is observed. The choice of studying the hydrodynamics of this fluid in an unbaffled stirred tank derives from this high viscosity, together with the shear sensitivity for some applications of this product. The tank has an elliptical bottom and is equipped with a standard Rushton turbine, with a total volume occupied by the fluid equal to 10 L.
Four different blends are analyzed: pure L64 at 25°C, mixtures of L64 in water at 40% and 50% in weight at 25°C, and a mixture of L64 in water at 50% in weight at 30°C. Starting from rheological experiments, a Newtonian model is chosen for the mixtures of L64 and water at 25°C, while a power law is used for the pure L64 and the mixture at 30°C. As previously stated, the microstructures formed in such fluids are influenced also by the temperature, and from this difference in the rheology at fixed concentrations, it is possible to identify a phase transition between 25 and 30°C. The described rheology models are used in Computational Fluid Dynamics (CFD) simulations, to obtain information about the power consumption and the variation of velocity and strain rate in the system. The operating conditions in industrial equipment usually lead to a laminar fluid dynamic regimen, due to the high viscosity of these fluids. Moreover, every simulation is performed at the steady state and with constant composition of the fluid, using the multiple reference frame for the zone around the impeller. Consequently, in investigating the dependence of the power number with respect to the Reynolds number, most of the simulations performed in this work focus on Re lower than 100. The simulations conducted with the mixture of L64 40 wt.% in water, which has the lowest viscosity among the studied blends, explore higher Reynolds numbers, between the laminar and the early transitional regimen. A grid independence study is carried out for different Reynolds numbers, monitoring the power number calculated both using the torque on the impeller and the laminar dissipation energy rate.
In addition to the constitutive equations for viscosity obtained through interpolation of the experimental data with an a priori model, such as the power law, an alternative viscosity model was tested. Such a model is based on the Gaussian Process Regression (GPR), which is a statistical technique akin to machine learning. The GPR actuates a regression on the data, providing a model which outputs single values for the predicted variable, rather than a functional form for viscosity with respect to the strain rate. The GPR viscosity model takes as input the strain rate for every cell of the computational domain and returns the corresponding viscosity prediction, based on the training performed with the experimental rheometry data. One of the most interesting characteristics of the GPR is the capability of returning as output a measure of the quality of the regression. With such information, it is possible to make an evaluation of the training process and consider an enlargement of the training dataset. The model itself can suggest where the regression has a lower quality and help the user identify the optimal values for the next experimental samples. Moreover, the computational performance of the GPR model is evaluated by comparison with the computational times required by the power law model. While the GPR unnecessarily increases the computational costs when the rheogram can be accurately interpolated with a functional form, this can be used as an initial test for the approach. The next step would be creating a multidimensional GPR model in which the viscosity is predicted depending on more input variables, e.g. the strain rate, the composition, and the temperature. In such a case the increased computational cost would be compensated by the complexity of obtaining a multivariate function that can accurately represent the viscosity variations.
The results obtained from the CFD simulation are validated against the results of the experimental setup for the same unbaffled stirred tank. From a qualitative point of view, the trend of the power number with respect to the Reynolds number from the simulations agrees with the experimental one. A quantification of the difference between the computational results and the experimental one can be done through the average relative error, which is about 25%.
In conclusion, computational studies can be a precious aid to the experiment once the desired level of accuracy of the model is achieved. The CFD simulations can be useful to predict integral variables related to the power consumption, but also to analyze more in detail the features of the flow that are not accessible experimentally. On the other hand, the GPR remains a promising tool, under the conditions of a satisfactory training step and a model complex enough to justify the increase in computational costs.
[1] Pasquino et al., Soft Matter, 2019,15, 1396-1404