Real-Time Rheological Monitoring With the Smart Stirrer

This article presents an approach for wireless real-time measurements of dynamic viscosity in the range of 50–1000 mPa $\cdot \text{s}$ utilizing a laboratory hotplate in conjunction with the Smart Stirrer device. The Smart Stirrer is an innovative tool resembling a conventional magnetic stirrer enhanced with an embedded system-on-chip (SoC) Bluetooth low-energy (BLE) module, an inertial measurement unit (IMU), and magnetometer sensors, serving as a platform for viscosity monitoring. The principle of measurement is based on the interaction between the magnetic field of the hotplate’s magnet and the Smart Stirrer’s internal magnets, with variations in a liquid viscosity impacting the rotational dynamics of the Stirrer and measured by the sensors. The paper details the experimental setup, data acquisition, and analysis procedures, demonstrating the capability of the system to accurately and noninvasively measure the viscosity of various liquid solutions in real-time. The method addresses the practical limitations of existing methods and offers an integrative approach for monitoring in situ rheological properties in laboratory settings. It stands out for its simplicity, ease of integration, and potential for broader applications, making it a valuable tool in various fields requiring fluid property measurements.

and in situ measurements of viscosity in fossil products [4].Microelectromechanical systems (MEMSs) are predominantly used, implying that the measurement principle relies on the resonant frequency or damping characteristics of an electromechanical element submerged in the fluid of interest [5].In many configurations, a miniaturized silicon cantilever or oscillating element is set into motion while the viscosity of the fluid causes a drag force on this moving element resulting in changes in its oscillation amplitude and/or resonant frequency [6], [7], [8].Both the shear horizontal acoustic plate mode (SHAPM) and thickness shear mode (TSM) sensor elements have been demonstrated as effective configurations for MEMS-based viscosity measurement systems [9].Each modality can be tuned to optimize its sensitivity and response across various ranges of viscosities and operational conditions.For vibrating wire viscometers, a common approach involves conducting a single or multiharmonic frequency sweep around the sensor resonant frequency to determine its impedance frequency response [10].
Electromechanical systems, such as piezoelectric transducers, are also used in a typical ultrasonic viscometer employing ultrahigh-frequency sound waves as the principal medium for measuring viscosity [11], [12], [13].As generated waves travel through the fluid, their speed and attenuation are directly correlated to the fluid's viscosity.The technique does not require any moving parts in contact with the fluid, thereby eliminating mechanical wear or contamination.In addition to serving the immediate requirements of petroleum and oil industries, MEMSs and Ultrasonic devices are used in biomedical applications [14], [15], food quality monitoring [16], and environmental science [17], where the precise and real-time measurement of fluid properties is invaluable.
In addition, several alternative techniques for real-time viscosity measurement have been explored.An approach to measuring the real-time viscosity of liquid foods uses a dc brush motor, exploiting its dual function as both an actuator and a sensor [18].The measurement principle is based on the correlation between the rotational speed of a stirrer powered by a motor and the viscosity of the liquid.The rotation speed is measured by the noise (a high short-pulse) arising from the contact of the feeding brush passing through the brush.The all-fiber acousto-optic superlattice modulation structure was employed to measure low viscosities in the range of 1.01-5.5 mPa•s [19].The method uses acoustic waves to modulate a fiber Bragg grating (FBG), allowing the determination of viscosity through an analysis of the FBG's bandwidth.Zhang et al. [20] proposed the method based on the relationship between the viscosity and the harmonic spectra of magnetic nanoparticles (MNPs) to measure low-viscosity liquid such as blood plasma.
Ponjavic et al. [21] developed an optical approach that relies on the quantification of the fluorescence lifetime of Thioflavin T introduced into the liquid under study.This technique was validated by conducting measurements on both Newtonian and non-Newtonian fluids confined in a sphereon-flat contact of submicrometer thickness.Similarly, Zhou et al. [22] demonstrated another optical method in which the viscosity of castor oil, selected for its temperature-dependent viscosity characteristics, can be accurately measured through variations in spectral line intensity.They employed principal component analysis to extract the intrinsic features of the transmission spectrum.This processed data was then used to train and validate a radial basis function neural network, providing a machine-learning-based (ML) approach to viscosity measurement.
The integration of optical techniques with ML algorithms has found that the extreme learning machine algorithm was particularly successful in estimating the kinematic viscosity of Fuel Oil-4 when used in conjunction with optical measurements [23].Wang et al. [24], on the other hand, introduced a comprehensive system that merges soft sensing techniques with ML models for viscosity measurement.Their soft sensor comprises physical sensors for temperature and pressure, a temperature estimator, and simulation analysis software designed to calculate material properties.Data from these multiple sources serve as the dataset for an ensemble ML model consisting of random forests and convolutional neural networks.This model is capable of computing melt viscosity, shear stress, and shear rate.
However, in small chemistry laboratories, the deployment of MEMSs, Ultrasound, or optical viscometers can encounter several challenges limiting their applicability.Another critical issue is the challenge of integrating these viscometers into existing experimental setups, which might already be intricate or space-constrained.In a recent study, Cherkasov et al. [25] presented a methodology for assessing viscosity, deploying a so-called Smart Stirrer, a magnetic stirrer bar endowed with an embedded accelerometer.This technique is based on monitoring the deceleration of the stirrer bar as it revolves within the fluid medium under examination.The method is limited to providing a qualitative assessment of viscosity.Additionally, extracting a quantitative viscosity measurement necessitates halting the rotational motion of the underlying hotplate magnet, thereby interrupting the stirring process.
In contrast, the present research introduces a refined approach aimed at the precise, real-time measurement of dynamic viscosity within a chemical reactor.Importantly, this method leverages the capabilities of the Smart Stirrer while obviating the need to suspend the stirring operation.This allows for uninterrupted data acquisition, thereby facilitating a more accurate and comprehensive understanding of rheological properties in dynamically changing systems.A key aspect of our work is that it represents the first-ever method for wireless real-time viscosity measurements.This feature sets our research apart in the context of the existing literature and underscores its novelty and innovation in the field of rheological evaluation.

II. MATERIALS AND METHODS
In the experimental setup, a standard laboratory hotplate (IKA RCT basic) equipped with adjustable temperature and magnetic stirring capabilities was employed.Two beakers made of borosilicate glass with 250 and 400 ml of volume and approximately 65 and 78 mm correspondingly inner diameters (d) were used as the reactors.For consistency, all the experiments and simulations were made with 150 ml of liquids.Test reference liquids comprised poly(vinyl alcohol) (PVA; Mw = 78 000, VWR International Ltd., U.K.) aqueous solutions of varying concentrations as well as glycerin (Poly-Science, USA) at different dilution levels, serving to span a range of viscosities.200 ml of liquid amount was prepared for each solution from both PVA and glycerin sets.The PVA powder was weighed using a laboratory balance, which offers an accuracy of better than 0.5%.The glycerin volumes were measured using a graduated 50 ml cylinder providing an accuracy range of 1%-5%.Viscosity measurements for these reference liquids were performed using an IKA Rotavisc lo-vi rheometer, equipped with calibrated SP-1 and SP-2 standard spindles for measuring viscosities in the ranges of 30-200 mPa•s and higher viscosities, respectively, with an accuracy of 1%.
The Smart Stirrer [Fig.1(a)] with a core nRF52840 BLE SoC (Nordic Semiconductors) is equipped with a magne-Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.tometer, a gyroscope, and accelerometer sensors that provide information about its orientation and motion.The 3-axial magnetometer (LIS3MDL) and the 3-axial accelerometer with a gyroscope implemented in a single IMU sensor (LSM6DSR) are placed in the center of the Stirrer board, and the board is then placed between two cylindrical N52 neodymium-ironboron alloy magnets with oppositely oriented magnetic fields [Fig.1(b)].In this configuration, the magnetic fields at the center of the Stirrer cancel each other out due to the antisymmetrical positioning of the magnets.Consequently, the Stirrer's inherent magnetic field does not interfere with the sensing capabilities of the magnetometer.
The Smart Stirrer was configured to collect the data from a magnetometer, IMU, and temperature sensor subsequently transmitting the data to a central computing device (PC or a Raspberry Pi).The magnetometer and IMU operated at output data rates (ODRs) of 50 and 208 Hz, respectively.During the experiments, the Stirrer was set to rotate at constant rates (specifically, 200, 300, and 400 rpm).The data utilized for subsequent analysis from the magnetometer, IMU, and temperature sensors were derived from the mean values computed over a 30-60 s measurement window.The measurement error was estimated by calculating the mean and standard deviation of the data collected from the magnetometer and IMU sensors.The magnetometer exhibits an error range within 5%.
For the simulation of the process physical model, Comsol Multiphysics was employed, focusing particularly on the interaction of magnetic fields and rotating machinery fluid flow.

III. PRINCIPLE
The principle of measuring real-time viscosity using the Smart Stirrer relies on the interaction between the magnetic fields of the hotplate's magnet and the magnets in the Smart Stirrer.When the Smart Stirrer is placed in a liquid solution and the hotplate is activated, the magnetic field from the hotplate magnet induces the rotation of the Stirrer.Fig. 2(a) shows the simulation result of the distribution of magnetic field in the horizontal x y-plane of the Smart Stirrer.The hotplate magnet is shown as a truncated circle oriented along the x-axis.
As the Smart Stirrer rotates along the z-axis, it encounters resistance due to the fluid viscosity, which impedes its rotation and creates a lag in the mutual orientation of the Smart Stirrer and the hotplate magnet.This angular lag can be detected by the magnetometer embedded in the Smart Stirrer.Fig. 2(b) shows the simulated axial torque τ of the Stirrer as a function of the rotational angle φ between the axis of the hotplate magnet and the Stirrer.As anticipated, the torque is proportional to the rotational angle.This axial torque has a direct influence on the fluid dynamics within the beaker, affecting factors like volumetric flow rate and shear distribution, as explored in the following simulation results.Fig. 3 shows the simulation results for the liquid flow in the laboratory beaker when the Smart Stirrer is rotating.The model employs Rotating Machinery physics to simulate the periodic rotation of the Stirrer.Specifically, the simulation illustrates the volumetric velocity magnitude [Fig.3(a)] and shear rate log( γ ) distribution in a vertical plane cross section [Fig.3(b)].These observations are based on a rotational speed of 300 rpm for the Smart Stirrer in a 250 ml beaker with a 65 mm diameter, containing 150 ml of a fluid with a viscosity of 50 mPa•s.Note that the fluid-air interface exhibits curvature toward the center, indicative of vortex funnel formation in the liquid during the spinning of the Stirrer.Fig. 3(c) displays the torque induced on the rotating Smart Stirrer by fluids of varying viscosities.The data demonstrates a linear dependency between the generated torque and fluid viscosity, with the slope of this correlation increasing as rotational speed grows.In addition, the analysis indicates that the viscous forces acting upon the Smart Stirrer are modestly influenced by the beaker's diameter d-this is reflected in a slight increment in the τ (η) slope with an increase in beaker size.
These findings offer a robust theoretical framework for further empirical investigations and exemplify the complex interrelationships between the axial torque, fluid dynamics, and external parameters like beaker size, thus enhancing our understanding of real-time viscosity measurement using the Smart Stirrer.It is worth noting that the magnitudes of the torque values obtained from the magnetic field interaction and fluid flow simulations are strikingly similar, differing only slightly.This minor discrepancy can be attributed to the approximate characterization of the hotplate magnet's size and material properties.Accurate measurements of these properties proved challenging, leading the authors to make reasonable yet estimated assumptions.

IV. EXPERIMENT AND RESULTS
To empirically validate the hypothesis about the Smart Stirrer's potential as a real-time viscosity measurement device, experiments were conducted using liquids of varying viscosities.Polyvinyl alcohol (PVA) was dissolved in water at different weight ratios, and a series of glycerin solutions were prepared with varying degrees of dilution.These fluids subsequently served as reference samples in all subsequent experiments involving the Smart Stirrer.
Fig. 4 presents the viscosities of these PVA and glycerin solutions, as measured by a rheometer.In Fig. 4(b), experimental data points are represented by dots, while the blue dashed and solid red curves depict calculated theoretical viscosities (η) for binary mixtures of glycerin (η 1 ) and water (η 2 ).The blue curve is based on the logarithmic mixing rule [26] where f is the fraction of glycerin.The red curve follows the equation proposed by Touloukian et al. [27] log where V is the molar volume and η denotes the interaction energy for of flow.Note that the data density at lower viscosities observed in Fig. 3(c) results from the fact that all simulation data were calculated using viscosity values obtained from experimental measurements with PVA and glycerol solutions.Notably, the viscosity of these solutions generally remains relatively low when the concentration of PVA or glycerol is below 70%.The Smart Stirrer is capable of gathering the data at a high acquisition rate, as shown in Fig. 5, which provides a typical sensor reading.The accelerometer graph demonstrates that when the Smart Stirrer is in operation, it assumes a horizontal orientation, as indicated by a 1 g reading on the accelerometer's z-axis, and rotates around the same axis at an angular velocity of approximately 300 rpm.The magnetic induction angle shown in the lower section of Fig. 5 is calculated using the expression α, β, γ = arcsin where M x,y,z represents the magnetic flux density in Gauss for each axis as measured by the magnetometer.For graphical clarity, the induction angle for the x-axis is represented as π/2−α allowing for a more effective scaling and juxtaposition of measurements along each axis.Building upon the high data acquisition capabilities of the Smart Stirrer, further experiments focused on the angle β (corresponding to the y-axis of the stirrer) as a measure of angular lag between the Smart Stirrer and the hotplate magnet.Fig. 6 presents the angle β showing its dependency on liquid viscosity and rotation speed, depicted in varying colors, and also on beaker diameter d, represented by open and closed symbols.In particular, Fig. 6(a) and (b) represents these angular measurements in PVA and glycerin solutions, respectively.The linear dependency of the angle on viscosity is in excellent agreement with the simulated predictions outlined in Fig. 3(c).Moreover, the incremental increase in the slope with respect to the beaker diameter is also in alignment with the aforementioned simulation findings.
To further validate the accuracy of our simulation model, Fig. 7  Comparative analysis of calculated (torque) and experimental (angle) data as a function of Smart Stirrer rotation speed for two fluids of varying viscosities under analogous conditions of speed, viscosity, and reactor dimensions.
shown in Fig. 7(b), which also displays a similar trend in angle dependency on the stirrer rotation speed, when the same conditions for speed, viscosity, and reactor dimensions are considered.
Hence, an empirical model η = η(β, RPM, d) can be established, making the basis for real-time viscosity assessments using the Smart Stirrer.However, the approach does have boundary constraints, particularly at both low and relatively high viscosity regimes.In relatively low viscosities up to 50 mPa•s, the Smart Stirrer exhibits reduced sensitivity.Measuring the lag angle between the Stirrer and the hotplate magnet becomes challenging due to the minimal magnitude and high standard deviation of the measurements.This is illustrated in the inset of Fig. 6(b), where the lag angle exhibits a nonlinear relationship with viscosity.
Conversely, at relatively high viscosities η ≥ 1 Pa•s, the limitation arises from the insufficiency of the magnetic force between the Stirrer and the hotplate to keep up the stable rotation of the Stirrer in viscous media.However, such limitations are also present in the case of conventional laboratory stirrer bars, indicating a common constraint across using the hotplate.
Finally, Fig. 8 presents the measurements capturing dynamic variations in viscosity as a function of temperature for 90% glycerin and an 8% PVA solutions.These measurements not only validate the capability of the Smart Stirrer to monitor viscosity changes in real time, but also highlight its utility in tracking temperature-dependent fluid properties.The data align well with the established literature on the temperature-dependent viscosity of glycerin and PVA solutions [28], [29], [30], further confirming the reliability and accuracy of our experimental setup.
The results substantiate both the reliability and the precision of our experimental apparatus, lending further credence to the Smart Stirrer's potential as a versatile tool for sophisticated fluid analysis.This opens up avenues for extending its application to thermal fluid analysis, an area critical in various industrial and scientific domains, including materials science and chemical engineering.

V. CONCLUSION
The study has demonstrated the capabilities and advantages of employing the Smart Stirrer as a real-time viscosity measurement device.Utilizing sophisticated simulation models along with a broad range of experimental data, we have established a compelling case for the Smart Stirrer's effectiveness and reliability.Through a simple, yet effective experimental design, we were able to achieve real-time viscosity measurements, allowing us to observe and quantify dynamic changes in fluid properties.This methodology can be adapted and extended for a range of applications in materials science and engineering.The implications of empirical relationships for real-time viscosity measurements provide a tool of significant practical utility that is far more adaptable and responsive than traditional methods.
However, it is critical to acknowledge the limitations of the proposed methodology for relatively low (η ≤ 50 mPa•s) viscosity.The upper limit (η ≥ 1000 mPa•s) is associated with the magnetic forces becoming insufficient to ensure stable and uniform rotation of the Smart Stirrer, although similar constraints are observed with conventional laboratory stirrers.An additional critical aspect of our methodology is the need for system calibration for different hotplates and various sizes and shapes of beakers or containers.Such calibration is essential to effectively account for variations in fluid dynamics and magnetic interactions, which arise due to changing conditions in diverse experimental setups.Particularly, differences in hotplate designs and the presence of surrounding ferromagnetic materials can influence the measurement accuracy.
Despite these limitations, the Smart Stirrer represents a significant advancement in the field.Its potential applications extend far beyond the scope of this study, and it could serve as a versatile tool in both material science and chemical engineering.Traditional approaches typically involve wired connections for data transmission and power, which can limit the versatility and adaptability of the measurement system, especially in dynamic or spatially constrained environments.The wireless capability of the Smart Stirrer not only enhances its ease of use, but also expands its applicability in various settings, from laboratory experiments to industrial processes.Real-time monitoring capabilities could accelerate materials discovery by providing immediate, actionable data on fluid properties that are critical to processes like sol-gel synthesis, polymerization reactions, or nanomaterials assembly.Furthermore, the Smart Stirrer's adaptability could facilitate its integration into automated systems, thereby advancing the progress toward fully autonomous laboratories.

Fig. 2 .
Fig. 2. (a) Simulation model of the magnetic field (shown in color) and flux density (arrow streamlines) in the system comprising the hotplate magnet interaction with the Smart Stirrer.(b) Axial torque as a function of the rotational angle between the hotplate magnet and the Smart Stirrer.

Fig. 3 .
Fig. 3. Computational simulation of fluid flow in a reactor featuring a rotating Smart Stirrer.(a) Spatial distribution of the magnitude of flow velocity.(b) Shear rate in a vertical cross section.(c) Torque generated on the Smart Stirrer is influenced by varying liquid viscosities and rotational speeds.

Fig. 4 .
Fig. 4. Viscosity in (a) PVA solutions at varying concentrations and (b) glycerin with varied dilutions.The blue and red curves show theoretical viscosities according to logarithmic mixing rules in (1) and (2) for binary mixtures of glycerin and water.

Fig. 5 .
Fig. 5. Typical sensor outputs from the Smart Stirrer, illustrating (from top to bottom) liquid temperature, Stirrer angular velocity, Stirrer orientation, and magnetic field detection measurements.

Fig. 6 .
Fig. 6.Angular measurements of the dependency of lag angle β on fluid viscosity and Stirrer rotation speed.(a) Relationship in PVA solutions and panel (b) in glycerin solutions.Data points are color-coded to indicate different rotation speeds and symbol-coded (open and closed) for beakers of different diameters.
(a)  shows the calculated torque as a function of the Smart Stirrer's rotation speed in fluids with different viscosities.These calculations are supported by experimental results,

Fig. 7 .
Fig. 7.Comparative analysis of calculated (torque) and experimental (angle) data as a function of Smart Stirrer rotation speed for two fluids of varying viscosities under analogous conditions of speed, viscosity, and reactor dimensions.

Fig. 8 .
Fig. 8. Real-time viscosity measurements for (a) 90% glycerin and (b) 8% PVA solutions as functions of temperature, highlighting the Smart Stirrer's capability in tracking temperature-dependent fluid properties.