# Modeling the Settling Velocity of a Sphere in Newtonian and Non-Newtonian Fluids with Machine-Learning Algorithms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{G}) and drag coefficient (C

_{D}) for the dataset were 10

^{−3}< Re

_{G}(-) < 10

^{4}and 10

^{−1}< C

_{D}(-) < 10

^{5}, respectively. The performances of the models were statistically evaluated using an evaluation metric of the coefficient-of-determination (R

^{2}), root-mean-square-error (RMSE), mean-squared-error (MSE), and mean-absolute-error (MAE). The support vector regression with polynomial kernel demonstrated the optimum performance with R

^{2}= 0.92, RMSE = 0.066, MSE = 0.0044, and MAE = 0.044. Its generalization capability was validated using the ten-fold-cross-validation technique, leave-one-feature-out experiment, and leave-one-data-set-out validation. The outcome of the current investigation was a generalized approach to modeling the settling velocity.

## 1. Introduction

_{s}) is the constant free-falling velocity of a solid particle in a stationary liquid when the opposing gravity and drag forces acting on the particle approximately equals one another. It is a significant parameter in the industries that involve solid–liquid two-phase flows. Probably, the most important examples are the slurry transportation in minerals and coal processing, wastewater treatment, dredging, hydraulic fracturing, and drilling operations. It is also essential to the fluidized bed as applied for catalysis, adsorption, ion exchange, water softening, and food processing. The concept of V

_{s}is also vital in mixing operations. In these applications, the settling velocities of amorphous particles experiencing hindrances from the walls, adjacent particles, and the flow conditions in both Newtonian and non-Newtonian liquids are of practical interest. Interestingly, this kind of non-ideal settling velocity in complex operating conditions can be correlated to the ideal V

_{s}of a spherical particle measured with a lab-scale setup or predicted using a reliable model. That is why both measuring and predicting V

_{s}in all kinds of fluids are active research interests since the 1850s [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

- ρ
_{s}: solid density (kg/m^{3}) - ρ
_{l}: liquid density (kg/m^{3}) - d: particle diameter (m)
- g: gravitational acceleration (m/s
^{2}) - μ
_{l}: viscosity of Newtonian liquid (Pa.s)

_{s}are considered to be sufficiently captured with C

_{D}and Re

_{p}. The idea of correlating C

_{D}to Re

_{p}was pioneered by Stokes with a single-parameter equation: ${C}_{D}=\frac{24}{R{e}_{p}}$ [9]. The limitation of this model is that it is applicable only if Re

_{p}< 0.1. Based on the Stokes law, many empirical and semi-empirical correlations have been developed afterward. For example, the C

_{D}-Re

_{p}correlation proposed by Cheng was found to perform better compared to seven (7) other similar models [4]. All of these models were developed for spherical particles in Newtonian fluids. The Cheng correlation is presented with Equation (3).

_{D}-Re

_{p}correlation can be used to predict V

_{s}in a non-Newtonian fluid by replacing Re

_{p}with a generalized particle Reynolds number (Re

_{G}) [6,7,12,16,19,20]. The equivalent viscosity used to define Re

_{G}is usually measured or estimated at a shear rate ($\dot{\gamma}$) of $\frac{{V}_{s}}{d}$. The Re

_{G}for a Herschel Bulkley (HB) fluid is as follows [7,12]:

- τ
_{o}: yield stress (Pa) - τ: shear stress (Pa)
- K: fluid consistency index (-)
- n: flow behavior index (-)

_{o}, K, and n. It should be mentioned that, even after transforming a C

_{D}-Re

_{p}correlation with Re

_{G}, the modified formula requires validation and further modification before applying to predict the V

_{s}in a non-Newtonian fluid (see, for example, [2,7,19,20]). This is because the uncertainty associated with the values predicted using the original C

_{D}-Re

_{G}correlation can be unacceptable. The prediction process also requires repeated iterations as V

_{s}is an implicit function of both dimensionless groups. The iterative procedure is inconvenient from an engineering perspective. Many direct correlations for V

_{s}have been developed to avoid the difficulties associated with implicit C

_{D}-Re

_{G}models. For example, a total of twenty-six (26) explicit correlations were reported by Agwu et al. [1]. Explicit models are, in general, convenient to apply. However, this kind of correlation has some noteworthy limitations, such as rheology specificity, higher uncertainty, constrained range of Re

_{G}, and limited capability of addressing the effect of τ

_{o}[1,2,6,7,11,16,19,20].

_{s}accurately and conveniently in both Newtonian and non-Newtonian fluids. However, the objective is yet to be achieved. A successful generalized model is not available in the literature. This particular limitation of traditional models opens up the opportunity of applying contemporary artificial intelligence (AI) to predict V

_{s}. Different AI-based methodologies were successfully implemented in solving various scientific problems [28,29,30,31,32,33,34,35,36,37]. However, only a limited number of investigations were conducted earlier to check the efficacy of the predictive AI-modules known as the machine learning algorithms (MLAs) in predicting settling velocity [13]. Rooki et al. [14] and Agwu et al. [1] applied one of the popular MLAs, artificial neural network (ANN) based on a commercial computational platform, MATLAB, to predict V

_{s}using datasets comprised of 88 and 676 samples, respectively. Only 70 data points were used for training, while 18 points were reported to be used for testing in Reference [14]. Two independent datasets consisting of 336 and 340 samples were used for modeling and validation, respectively, in Reference [1]. The 336 modeling data were divided further into training (70%), testing (15%), and validating (15%) subsets. That is, only 235 samples were used for training. Even though the performance of ANN was demonstrated to be significantly better than the traditional models, its application was not justified by comparing with other MLAs. For both of these studies, the AI-model was selected arbitrarily without statistical analysis. Besides, the proposed ANN architectures were complex and prone to overfitting.

_{s}can be summarized as follows:

- The traditional models for predicting V
_{s}are categorized as implicit and explicit models.- The semi-mechanistic C
_{D}-Re_{p}models involve implicit correlations of V_{s}. Predicting V_{s}using such correlations is inconvenient from an engineering perspective as it demands iterations. - Most of the existing implicit correlations were developed for Newtonian fluids. The extension of these models to non-Newtonian fluids involves higher uncertainties.
- Considering the inconvenience of implicit models, many explicit correlations were proposed for V
_{s}. This kind of direct model involves complex empirical correlations, which are usually rheology-specific and involve a high degree of uncertainty. - A generalized traditional model applicable to various fluid rheologies is not available in the literature to date.

- Limited efforts have been undertaken to apply AI-based MLAs to develop a generalized model for predicting V
_{s}of spheres in both Newtonian and non-Newtonian fluids. The previous ML investigations were confined to a limited set of fluid rheology, a specific MLA (ANN), and insufficient data.

_{s}. A diverse set of MLAs including support vector regression (SVR), random forest (RF), stochastic gradient boosting (SGB), Bayesian additive regression tree (BART), K-nearest neighbor (KNN) regression, multilayer perceptron (MLP), and ANN were applied. The ten-fold-cross-validation technique with an evaluation metric of a group of statistical parameters was used in identifying the most optimum AI-module. The statistical parameters were mean squared error (MSE), coefficient of determination (R

^{2}), mean absolute error (MAE), and root mean square error (RMSE). A dataset of 967 data points collected from the literature was used for the current analysis. Both Newtonian and non-Newtonian fluids were used for the wet-experiments. The non-Newtonian rheology of the experimental fluids was explained with Bingham, Power-law, and Herschel Buckley models. The leave-one-feature-out analysis was performed to analyze the feature influences on the prediction. Moreover, a leave-one-dataset-out validation was carried out to examine the generalization capability of the optimal model in the prediction of V

_{s}for different types of fluids. All dry-experiments and statistical analyses were conducted using an open-source computation platform, R. The generalized model validated as part of the current study is expected to lead to the development of a universal model to predicting V

_{s}with a decision support system (DSS) for the industry.

## 2. Materials and Methods

#### 2.1. Regression Algorithms

#### 2.1.1. SVR-Radial Basis Function

_{1}, y

_{1}), (x

_{2}, y

_{2}), …, (x

_{N}, y

_{N}), the SVR attempts to find a hypothesis f(x) such that it will be insensitive to at most є-deviations from the actual responses in the training data and as flat as possible. The output of f(x) or the predicted response, denoted as $\widehat{y,}$ can be described by the following expression:

_{i}coefficients as support vectors and the selection of a suitable kernel function for non-linear feature mapping, k(x

_{i}, x), the equation for $\widehat{y}$ can be described as:

#### 2.1.2. SVR-Polynomial

#### 2.1.3. SVR-Linear

#### 2.1.4. Random Forest Regression

_{i}) is fit for each subset by using a random subset of features for splitting at each node, which results in a forest of M regression trees. After fitting an RF model to the whole training data, the prediction of response ($\widehat{y}$). for an unseen test sample (x′) can be made by averaging all the individual regression tree predictions on x′:

#### 2.1.5. Stochastic Gradient Boosting

_{1}, a

_{2}, …} and ${\beta}_{m}$ are expansion coefficients. Both a and ${\beta}_{m}$ are jointly fit to the training data. The parameters of the base learner define the split points of the regression tree [20]. In SGB, the hyper-parameters for tuning are the number of trees (m) and the number of splits to be performed at each node, i.e., the maximum nodes per tree.

#### 2.1.6. Bayesian Additive Regression Tree

_{1}, x

_{2},…, x

_{n}) could be written according to the sum-of-trees model:

#### 2.1.7. K-Nearest Neighbor Regression

#### 2.1.8. Multilayer Perceptron

#### 2.1.9. Artificial Neural Network

#### 2.2. Evaluation Metrics

^{2}, RMSE, and MAE were used to evaluate the regression performance. Brief descriptions of these statistical parameters along with their formulations are described below.

#### 2.2.1. Mean Squared Error

#### 2.2.2. Coefficient of Determination

^{2}is a statistical measure of how well a regression model fits a set of observations. It is defined as the percentage of the response variable variation that is explained by a regression model. The equation for R

^{2}has the following form:

^{2}value ranges between 0 and 100%. Generally, the higher the R

^{2}the better the model fits the data, i.e., the model can explain most of the variances in the observed responses.

#### 2.2.3. Mean Absolute Error

#### 2.2.4. Root Mean Square Error

#### 2.3. Dataset

_{s}, ρ

_{s}, ρ

_{l}, τ

_{o}, K, and n) and one dependent or output parameter (V

_{s}). A statistical description of these parameters along with the Reynolds numbers and drag coefficients is presented in Table 3.

#### 2.4. ML Modeling

#### 2.4.1. Parameter Optimization and Model Selection

_{s}prediction involved associated parameter optimization on a training set and the final model selection based on the validation using the independent test data points. At first, the datasets extracted from the literature were merged. Next, the merged dataset was split into a training (80%) and a test (20%) subset using random sampling. The training of nine ML models was carried out using the training data points (80%). The test dataset (20%), which can be considered as an independent test set, was withheld for model verification. The statistics of training and test data are presented in Table 4.

_{i}

_{,j}is the root mean square deviation of the predicted response from the actual response (V

_{s}) for j-th fold in i-th iteration of the repeated CV. The algorithm used for model selection in the present study is depicted in Figure 2. Note that, the number of folds in CV (k) and repetition times (N) were considered as 10 in this study. The details of the ML models including R packages, hyper-parameter names, and their optimized values after cross-validated training are reported in Table 5.

#### 2.4.2. Feature Importance Analysis and Validation

#### Leave-One-Feature-Out Experiment

^{2}was calculated. Similar to Reference [1], a relative feature influence (RFI) was introduced in the current study for the convenience of analyzing the feature significance. The formula used to calculate RFI is as follows:

_{s}) and is derived using the R

^{2}metrics. The larger the RFI the more significant the feature is.

#### Leave-one-dataset-out validation

#### 2.5. Computing Framework

## 3. Results

#### 3.1. Evaluation of Traditional Modeling Methodologies

#### 3.2. MLA Model Evaluation on the Independent Test Set

^{2}metrics. The performances are also depicted in Figure 7 by plotting the measured values against the predicted values of V

_{s}.

#### 3.3. Feature Importance Analysis

#### 3.4. Leave-One-Dataset-Out Validation

## 4. Discussion

#### 4.1. Limitations of Existing Analytical Models

_{s}requires using a C

_{D}-Re

_{p}or C

_{D}-Re

_{G}correlation to iteratively predict V

_{s}in a Newtonian or non-Newtonian fluid. Although many C

_{D}-Re

_{p}correlations are available in the literature, all of those were proposed for Newtonian fluids. Originally, Cheng [4] also proposed the correlation for Newtonian rheology. Later, the model was extended by replacing Re

_{p}with Re

_{G}to different non-Newtonian fluids [7,20,23]. The results of applying this correlation to the current dataset are presented in Figure 4 by plotting C

_{D}against Re

_{p}and Re

_{G}for Newtonian and non-Newtonian fluids, respectively. As expected, the performance of the Cheng correlation is better for Newtonian fluids compared to the non-Newtonian fluids. Most of the measured values collapse over the model curve for Newtonian fluids. For non-Newtonian fluids, the uncertainty of prediction is considerably high. It can be less than −60% and more than 200% for the current dataset. The values of the model coefficients determined empirically for Newtonian fluids do not necessarily account for the variation in viscosity due to shear in non-Newtonian fluids. That is why this kind of correlation is usually validated and modified before using for a non-Newtonian fluid. The associated complexities and uncertainties render predicting V

_{s}using a C

_{D}-Re

_{p}correlation an inconvenient option for an engineer.

_{s}directly based on the system properties. The following exemplary models were used for the current study.

#### 4.1.1. Ferguson and Church (FC) Model

_{s}in Newtonian fluids was developed by Ferguson and Church [45]. Agwu et al. [1] found its performance as comparable to ANN. The FC model (Equation (22)) applies to both laminar and turbulent conditions.

#### 4.1.2. Okesanya and Kuru (OK) Model

_{s}with the OK model. The most important equations are presented as follows.

- $\overline{\tau}$: surficial shear stress (Pa)
- ${V}^{*}$: shear velocity (m/s)
- $R{e}^{*}$: shear Reynolds number (-)
- $R{e}_{G}^{*}$: shear generalized Reynolds number (-)
- $R{e}_{T}^{*}$: model-specific shear Reynolds number (-)
- $\epsilon $: relative characteristics shear stress (-)$\alpha $: shape factor (-)

_{s}of a spherical particle in a Newtonian fluid is creditable. The agreement between the predicted and measured values is within an acceptable margin of error (Figure 5 and Table 7). However, the model cannot be applied to non-Newtonian fluids in its current form since it requires a constant value for µ

_{l}. On the other hand, the OK model involves a complex numerical scheme. Although mathematical complexity is not uncommon in modeling a non-Newtonian system, this model is afflicted with a significant limitation. Its application is limited to three rheological models: (i) Herschel Bulkley, (ii) Bingham, and (iii) Casson model. The OK model does not apply to other rheological models, such as power-law and Newton’s law. As shown in Figure 6 and Table 7, the model yields result for the Herschel Bulkley fluids with a practical error limit. However, it fails to do so for Bingham fluids. Thus, similar to other existing models, both of these explicit models are rheology-specific and prone to yielding significant errors.

#### 4.2. Analysis of Current AI Model

_{s}of spherical particles in the present study to develop a generalized model applicable to both Newtonian and non-Newtonian fluids. Parametric regression models, such as SVR, MLP, NN, as well as non-parametric regression models such as KNN, including a comparatively latest approach called Bayesian regression were considered. Among the Bagging and Boosting algorithms, RF and SGB were selected. Table 6 shows the results of ML modeling of the selected regression models through repeated cross-validated training. Among all models, the best model was the SVR-Polynomial kernel (R

^{2}= 0.921) with the optimized values of the parameters, such as degree-d of the kernel as three (3), input scaling parameter as 0.1, and the regularization parameter as 1 (see Table 5). As modeling V

_{s}associates non-linear feature interactions, non-linear ML models were found to have superior performances. In the case of SVR, both RBF and Polynomial kernel map the input space to higher dimensional feature space, and, subsequently, the problem becomes linearly separable into that feature space. The SVR-Polynomial kernel demonstrated better performance than its RBF kernel counterpart in the case of independent test data prediction (see Table 7 and Figure 7). It should be noted that the SVR-Poly yielded comparable values of R

^{2}for both training and test datasets (R

^{2}= 0.931 for training dataset; R

^{2}= 0.921 for test dataset), which demonstrates the robustness of the model. The second-best model was the SGB (R

^{2}= 0.906), which is the stochastic version of the general gradient boosting algorithm. The SVR–RBF along with the RF ranked as the third-best model with an identical R

^{2}value of 0.901 for the test dataset. It is important to observe that both bagging (RF) and boosting (SGB) algorithms with stochastic components were found to perform well either by choosing the best possible random set of predictors (RF) or observations (SGB) for splitting at each node of the regression tree and iteration of minimizing the objective loss function. Both models were able to capture the non-linearity of the data and accurately estimate the actual response.

^{2}= 0.773) among the ML models evaluated for the current study. Given the superior performances of other non-linear kernels in SVR, it is apparent that the original input features are non-linearly separable, and, therefore, SVR-Linear inevitably led to an underfitting of the data. The KNN, as a non-parametric regression method, achieved the second-lowest accuracy with R

^{2}= 0.819. The ANN and MLP achieved the next lower accuracies and exhibited comparable performances with R

^{2}= 0.871 and R

^{2}= 0.851, respectively. This is most likely because both ANN and MLP architectures have similar model complexity with a single hidden layer and different numbers of hidden neurons (20 for ANN and 5 for MLP). The BART could achieve a moderate correlation (R

^{2}= 0.88) with the imposition of regularization on each tree while fitting to a small portion of the training data. It helped to achieve a fairly bias-free prediction when several trees were fitted to the whole set of observations.

^{2}values were obtained from the leave-one-feature-out experiments. Then, the R

^{2}values obtained before and after excluding each feature were compared. A relative feature importance indicator was used by taking into account the differences observed in R

^{2}values in the presence or absence of a specific feature. The RFI indicates the influence of a feature on the model in relation to the R

^{2}metric. A decreased ranking-order of the features can be obtained based on their RFI values. As shown in Figure 8, the Particle Density and the Particle Diameter are identified as the two most significant features having respective RFI values of 39.6 and 24.8, followed by a decreasing rank-order of Yield Stress, Flow Consistency Index, Flow Behavior Index, and Liquid density. Specifically, the ordering of the features from the highest to lowest ranks is as follows: ρ

_{s}> d

_{s}> τ

_{o}> K > n > ρ

_{l}(Figure 8). A similar trend of feature influence was reported in a previous ANN-study [1]. It can be explained with the physics of the settling process. The motion of a particle in a stationary fluid is triggered by the gravitational force (F

_{G}). When it is released in the fluid, an acceleration of g (~9.81 m/s

^{2}) acts on it. Afterward, the motion of the particle in the fluid actuates a drag force (F

_{D}) on itself. Its incremental velocity also intensifies F

_{D}. When F

_{D}nearly matches F

_{G}, an equilibrium condition persists and the particle continues moving at a constant velocity known as the settling velocity. Thus, the opposing forces ensue V

_{s}. The primary force in this process is F

_{G}, which is strongly dependent on a spherical particle’s density and size or diameter. The dependency is conceivably stronger on ρ

_{s}than d

_{s}. On the other hand, F

_{D}is the secondary force stemmed from the particle motion. It is known to be a strong function of Newtonian fluid viscosity and weakly dependent on ρ

_{l}. For the non-Newtonian fluids considered in the current study, the rheology was modeled with yield stress, flow consistency index, and flow behavior index. The most important among these parameters is τ

_{o}as F

_{D}cannot be actuated until F

_{G}overcomes it. The viscosity of the non-Newtonian fluid is denoted with K while its shear-thinning behavior is signified with n. Understandably, K plays a more significant role than n in the settling process. Thus, the feature-importance, as shown in Figure 8, is in sync with the physics of the particle settling phenomenon. It justifies the utility of the MLA in predicting V

_{s}.

_{s}. Although the LOOV experiment performs an objective analysis of the ML-model performance, it is largely absent in the previous ML studies of V

_{s}modeling. Most of the previous studies evaluated their proposed models based on test data that come from the same training distribution. This type of in-distribution evaluation does not inform about the performance of the model when applied to an out-distribution data or a completely new dataset. On the other hand, leave-one-out validation is one of the preferred methods to comprehensively evaluate the generalized performance of ML models [34,35,44]. Agwu et al. [1] attempted to conduct a similar validation. However, their effort was constrained by the number of datasets as they used only two independent datasets. In the current study, the LOOV was used to investigate the generalizability of the MLA that was identified as the optimum choice to model the V

_{s}of a sphere in both Newtonian and non-Newtonian fluids. The dry experiments were conducted by training the SVR-Polynomial kernel model on six datasets. The SVR-Polynomial was selected for the LOOV analysis because of its superior performance over the other ML models in predicting V

_{s}for independent test data. Then, the generalization ability of the model was assessed with four evaluation metrics for each dataset. The SVR-Polynomial kernel model achieves moderate to high performances with the R

^{2}values between 0.65 and 0.91 with an average of 0.78. An analogous analysis was reported in Reference [1]. They applied a total of eighteen (18) explicit models along with their ANN-model to predict the V

_{s}for a specific dataset. A statistical comparison of the predictions produced R

^{2}values in the range of 0.23–0.94 with an average of 0.49. An objective comparison of these independent analyses suggests the current model is capable of producing reliable results for both Newtonian and non-Newtonian fluids within an acceptable error margin.

_{s}for a given set of process conditions. Furthermore, R is easy to program, well-documented, and evolving continually due to the freelance contributions from the scientific community. Therefore, R is considered a more economic, flexible, and fast-evolving scientific and statistical computing framework compared to MATLAB. It is the industrial choice for data analysis.

- (a)
- The SVR-Poly can predict V
_{s}of spherical particles in Newtonian and different varieties of non-Newtonian fluids. Whereas, the FC applies to only Newtonian fluids, and the OK could be applied to two non-Newtonian fluids. - (b)
- Prediction uncertainties associated with SVR-Poly are comparable for the Newtonian and non-Newtonian rheologies. That is, the ML model is insensitive to fluid rheology. Unlike the traditional models, it is capable of predicting V
_{s}without a bias to the fluid properties. - (c)
- Prediction accuracy of SVR-Poly is significantly better than the traditional explicit models such as FC and OK models.

## 5. Conclusions

^{2}= 0.92; RMSE = 0.066; MSE = 0.0044; MAE = 0.044). A ten-fold-cross-validation technique was used to ascertain the most suitable model. A comprehensive feature analysis using leave-one-feature-out experiments assured the strong correlation among the parameters selected for modeling and the experimental observations. The correlations were quantified using a relative feature influence indicator. The order of significance for the input parameters was as follows: particle density > particle diameters > yield stress > flow consistency index > flow behavior index > liquid density. Additionally, leave-one-dataset-out validation experiments were conducted to test the reliability of the model for unseen experimental datasets. The model confirmed a reasonable performance in estimating the settling velocity for completely unseen data. It could predict more than 80% of the test data within an error limit of ±20%. The present study provides strong evidence that ML is very effective and accurate as a general predicting tool for the settling velocity of spheres in both Newtonian and non-Newtonian fluids. It can be used as a reliable method in industrial-scale design and operation due to its economical requirements of time and computational resources.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Algorithmic representation of the repeated k-fold cross-validation (CV) algorithm used for parameter optimization and model selection.

**Figure 3.**Algorithmic representation of the leave-one-data-set-out validation involving six datasets.

**Figure 4.**Predictions of V

_{s}with Cheng [4] model and the associated uncertainty for spherical particles in (

**A**) Newtonian fluids and (

**B**) non-Newtonian fluids.

**Figure 5.**Predictions of V

_{s}with Ferguson and Church [45] model (fluid rheology: Newtonian; particle shape: spherical).

**Figure 6.**Predictions of V

_{s}with Okesanya and Kuru [19] model (fluid rheology: non-Newtonian; particle shape: spherical).

**Figure 7.**Performances of the ML models on independent test data: (

**A**) top four models (R

^{2}> 0.9); (

**B**) five low performing models (R

^{2}< 0.9).

**Figure 9.**Comparative performance of the ML model, SVR-Poly, and the traditional explicit models, Ferguson and Church (FC) and Okesanya and Kuru (OK) on the test data.

Reference | Fluids | Rheological Model | Measuring Method | Experimental Error |
---|---|---|---|---|

[2] | Kaolinite-water mixtures | Bingham | Electrical impedance tomography (EIT) | 2% |

[6] | Water/glycerol Aqueous solutions of Carboxymet-hylcellulose | Newtonian Power law | Electronic stopwatch | 3% |

[7] | Solutions of Flowzan | Newtonian Herschel Bulkley | High-speed cera | 8% |

[11] | Aqueous solutions Floxit 5250 L | Bingham | A specialized lighting system | 3–4% |

[15] | Bentonite/Bicarbonate mud. Bentonite/MMH mud. Xanthan gum/seawater mud | Power law | Electronic stopwatch | - |

[16] | Aqueous solutions of Carboxymeth-ylcellulose | Power law | Electronic stopwatch | 1% |

[17] | Water/glycerol Aqueous solutions of Carboxymet-hylcellulose | Newtonian Power law | Calculated from the bed expansion data | - |

[18] | Aqueous solutions of Carbopol | Power law | Electronic stopwatch | - |

[19,20] | Aqueous solutions of Hydrolyzed Polyacrylamide | Power law Bingham Herschel Bulkley | Particle Image Shadowgraphy (PIS) | 3.5% |

[21] | Water/glycerol | Newtonian | PIS | 5% |

[22] | Water/glycerol | Newtonian | High-speed camera | 5% |

Reference | Number of Data Points | Particle Diameter (mm) | Particle Density (kg/m^{3}) | Fluid Density (kg/m^{3}) | Values of Rheological Parameters |
---|---|---|---|---|---|

[2] | 100 | 12.7–19.1 | 2710–7841 | 1174–1357 | τ_{o}: 1.3–30.0 PaK: 0.9800–0.0074 Pa.s ^{n}n: 1 |

[6] | 23 | 1.22–3.16 | 2314–11,444 | 997.9 | τ_{o}: 0 PaK: 0.0010–0.1350 Pa.s ^{n}n: 0.7449–1 |

[7] | 532 | 2–6 | 2230–7700 | 1000–1030 | τ_{o}: 0–3.82 PaK: 0.0010–0.3600 Pa.s ^{n}n: 0.466–1 |

[11] | 58 | 6–15 | 7638–8876 | 997–1490 | τ_{o}: 2.95–20.00 PaK: 0.0090–0.0474 Pa.s ^{n}n: 1 |

[15] | 25 | 1.2–5 | 7730–7949 | 1000–1044.418 | τ_{o}: 0 PaK: 4.0029–19.7360 Pa.s ^{n}n: 0.0614–0.2867 |

[16] | 15 | 1.5–3.5 | 2260–2727 | 1000 | τ_{o}: 0 PaK: 0.0165–0.2648 Pa.s ^{n}n: 0.7529–0.9198 |

[17] | 17 | 3–7 | 2500 | 997–1000 | τ_{o}: 0 PaK: 0.0166–0.5940 Pa.s ^{n}n: 0.561–0.751 |

[18] | 8 | 0.8–5.9 | 1170–2900 | 1000 | τ_{o}: 0 PaK: 0.0462–0.0521 Pa.s ^{n}n: 0.7300 |

[19] | 50 | 1.09–4 | 2510–770 | 994.0–1004.7 | τ_{o}: 0.048–6.646 PaK: 0.158–2.115 Pa.s ^{n}n: 0.507–0.725 |

[20] | 60 | 0.71–4.0 | 2510–5900 | 996–1005 | τ_{o}: 0 PaK: 0.0010–0.1350 Pa.s ^{n}n: 0.7449–1 |

[21] | 20 | 0.71–2 | 2510 | 1000–1180 | τ_{o}: 0 PaK: 0.0010–0.006844 Pa.s ^{n}n: 1 |

[22] | 60 | 1–10 | 2680–7960 | 1224–1250 | τ_{o}: 0 PaK: 0.1350–0.6685 Pa.s ^{n}n: 1 |

Particle Diameter (d_{s}) | Particle Density (ρ_{s}) | Yield Stress (τ_{o}) | Flow Consistency Index (K) | Flow Behavior Index (n) | Fluid Density (ρ_{l}) | Reynolds Number (Re_{G}) | Drag Coefficient (C_{D}) | |
---|---|---|---|---|---|---|---|---|

Unit | m | kg/m^{3} | Pa | Pa.s^{n} | - | kg/m^{3} | - | - |

Maximum | 0.01910 | 11,444 | 30.000 | 19.7360 | 1.0000 | 1490 | 6.7 × 10^{3} | 8.2 × 10^{4} |

Minimum | 0.00071 | 1170 | 0.000 | 0.0010 | 0.0641 | 994 | 9.7 × 10^{−4} | 3.1 × 10^{−1} |

Mean | 0.00554 | 4534.63 | 2.488 | 0.4283 | 0.7186 | 1063.91 | 6.5 × 10^{2} | 3.1 × 10^{2} |

Standard Deviation | 0.00401 | 2370.94 | 5.269 | 1.7624 | 0.2599 | 112.42 | 1.1 × 10^{3} | 3.0 × 10^{3} |

Dataset | Percentage | No. Data Points |
---|---|---|

Training | 80% | 774 |

Test | 20% | 193 |

Total | 100% | 967 |

**Table 5.**The ML models, hyperparameter names, and corresponding optimized values after cross-validated training (the last column includes R packages used for different ML models).

Model | Hyperparameter Names | Optimized Values | R package |
---|---|---|---|

Random Forest | [mtry] | [4] | randomForest |

SVR—RBF Kernel | [sigma, C] | [0.278, 8] | Kernlab |

SVR—Polynomial Kernel | [degree, scale, C] | [3, 0.1, 1] | Kernlab |

SVR—Linear Kernel | [Cost, Loss function] | [0.25, L1-loss] | Kernlab |

Stochastic Gradient Boosting | [n.trees, interaction.depth] | [150, 3] | gbm |

Multilayer Perceptron | [layer1, decay] | [5, 0.0] | RSNNS |

KNN Regression | [no. nearest neighbors] | [5] | Kknn |

Bayesian Additive Regression | [num_trees] | [150] | bartMachine |

Neural Network | [size, decay] | [20, 0.001] | Neuralnet |

Model | Fluid Rheology | MAE | RMSE | MSE | R^{2} |
---|---|---|---|---|---|

Ferguson and Church [45] | Newtonian | 0.030 | 0.002 | 0.0018 | 0.965 |

Non-Newtonian | × | × | × | × | |

Okesanya and Kuru [19] | Newtonian | × | × | × | × |

Bingham plastic | 0.237 | 0.298 | 0.089 | 0.662 | |

Power-law | × | × | × | × | |

Herschel Bulkley | 0.0570 | 0.077 | 0.006 | 0.942 |

Algorithm | Train | Test | Fluid Rheology | ||||||
---|---|---|---|---|---|---|---|---|---|

MAE | RMSE | MSE | R^{2} | MAE | RMSE | MSE | R^{2} | ||

SVR-Poly | 0.035 | 0.054 | 0.0029 | 0.931 | 0.044 | 0.066 | 0.0044 | 0.921 | Newtonian Bingham plastic Powerlaw Herschel Bulkley |

SGB | 0.028 | 0.043 | 0.0019 | 0.955 | 0.045 | 0.074 | 0.0055 | 0.906 | |

SVR-RBF | 0.021 | 0.038 | 0.0014 | 0.965 | 0.038 | 0.074 | 0.0055 | 0.902 | |

RF | 0.017 | 0.029 | 0.0008 | 0.979 | 0.041 | 0.075 | 0.0056 | 0.901 | |

BART | 0.038 | 0.057 | 0.0033 | 0.930 | 0.053 | 0.082 | 0.0067 | 0.880 | |

ANN | 0.029 | 0.046 | 0.0021 | 0.950 | 0.047 | 0.082 | 0.0067 | 0.875 | |

MLP | 0.072 | 0.103 | 0.0106 | 0.750 | 0.065 | 0.091 | 0.0083 | 0.851 | |

KNN-Regression | 0.027 | 0.049 | 0.0024 | 0.941 | 0.055 | 0.099 | 0.0098 | 0.819 | |

SVR-Linear | 0.097 | 0.125 | 0.0156 | 0.631 | 0.092 | 0.117 | 0.0137 | 0.773 |

Experiment Number | Excluded Feature | Test | |||
---|---|---|---|---|---|

MAE | RMSE | MSE | R^{2} | ||

1 | Particle Density | 0.115 | 0.169 | 0.0286 | 0.525 |

2 | Particle Diameter | 0.094 | 0.135 | 0.0182 | 0.673 |

3 | Yield Stress | 0.064 | 0.104 | 0.0108 | 0.804 |

4 | Flow Consistency Index | 0.057 | 0.087 | 0.0076 | 0.863 |

5 | Flow Behaviour Index | 0.052 | 0.079 | 0.0062 | 0.885 |

6 | Liquid Density | 0.046 | 0.072 | 0.0052 | 0.904 |

7 | None | 0.044 | 0.066 | 0.0044 | 0.921 |

Validation Experiment Number | Dataset Name | Fluid Type | Evaluation Metric | |||
---|---|---|---|---|---|---|

MAE | RMSE | MSE | R^{2} | |||

1 | Arabi and Sanders (2016) [2] | Bingham | 0.264 | 0.333 | 0.1109 | 0.715 |

2 | Kelessidis (2004) [16] | Power Law | 0.132 | 0.112 | 0.0125 | 0.913 |

3 | Okesanya et al. (2020) [20] | Power Law | 0.041 | 0.029 | 0.0008 | 0.655 |

4 | Rooki (2012) [15,16,17,18] | Power Law | 0.051 | 0.067 | 0.0045 | 0.848 |

5 | Shahi (2014) [21] | Newtonian | 0.031 | 0.037 | 0.0014 | 0.664 |

6 | Song et al. (2017) [22] | Newtonian | 0.277 | 0.345 | 0.1190 | 0.889 |

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**MDPI and ACS Style**

Rushd, S.; Hafsa, N.; Al-Faiad, M.; Arifuzzaman, M.
Modeling the Settling Velocity of a Sphere in Newtonian and Non-Newtonian Fluids with Machine-Learning Algorithms. *Symmetry* **2021**, *13*, 71.
https://doi.org/10.3390/sym13010071

**AMA Style**

Rushd S, Hafsa N, Al-Faiad M, Arifuzzaman M.
Modeling the Settling Velocity of a Sphere in Newtonian and Non-Newtonian Fluids with Machine-Learning Algorithms. *Symmetry*. 2021; 13(1):71.
https://doi.org/10.3390/sym13010071

**Chicago/Turabian Style**

Rushd, Sayeed, Noor Hafsa, Majdi Al-Faiad, and Md Arifuzzaman.
2021. "Modeling the Settling Velocity of a Sphere in Newtonian and Non-Newtonian Fluids with Machine-Learning Algorithms" *Symmetry* 13, no. 1: 71.
https://doi.org/10.3390/sym13010071