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Oct 14, 2024

Integration of experimental study and neural network modeling for estimating iron recovery in Davis tube tests | Scientific Reports

Scientific Reports volume 14, Article number: 22578 (2024) Cite this article 222 Accesses Metrics details Magnetic separation is a common procedure for the enrichment of magnetic iron ores. Davis tube

Scientific Reports volume 14, Article number: 22578 (2024) Cite this article

222 Accesses

Metrics details

Magnetic separation is a common procedure for the enrichment of magnetic iron ores. Davis tube (DT) test is a standard laboratory technique used to determine the optimum magnetic recovery of iron ore using wet low-intensity magnetic separators. However, the DT test is time-consuming and labour-intensive. In this study, based on the results of DT-tests, generalized regression (GRNN) and radial basis function (RBF) neural networks were developed to predict iron recovery through the Fe and FeO content of the feed. First, the DT tests were performed on 613 iron ore samples with varying Fe and FeO content. Then, neural networks were used to model the iron recovery from the DT test, using the Fe and FeO content of the feed as input data. The modeling results showed that GRNN is a better model for predicting iron recovery. The main statistical metrics indicated that GRNN has AAPRE, RMSE, and R2 values of 3.929%, 2.804, and 0.976 respectively for a total of 613 data points. Moreover, sensitivity analysis demonstrated that iron recovery is directly influenced by both Fe and FeO contents, with FeO content having a more pronounced effect. Finally, Leverage analysis showed that GRNN is highly reliable for predicting iron recovery, with only 2.77% of data points flagged as suspicious based on outlier estimation.

The cornerstone of steelmaking lies in the iron extracted from iron ore. Presently, the world’s steel industry leans heavily on iron ores sourced from rich deposits of high-content hematite, goethite, and magnetite1. Magnetic separation stands as the unequivocally most efficient method for concentrating magnetic ores2. At present, the beneficiation of industrial magnetite is conducted using wet process3. The predominant technology employed for magnetite beneficiation involves wet drum low-intensity magnetic separators (WLIMS)4,5. Numerous researchers have employed the wet low-intensity magnetic separator (Davis tube) as a testing tool for concentrating fine magnetic particles6,7,8,9.

The Davis tube (DT) is a standard laboratory device to evaluate the separability of magnetic ores with wet drum low-intensity magnetic separators10,11,12,13,14,15, the separator consists of an inclined tube through which feed material passes, with a divergent magnetic field holding strongly magnetic material while less magnetic material is washed away16,17. Vibration can help produce a cleaner concentrate12,18. The DT test is primarily used to simulate optimal iron recovery rather than the optimal iron grade19. The DT test is chosen as a proxy method due to its use of a small sample size and wet technique, accounting for drag forces, such as friction and hydrodynamics20. Material properties such as liberation degree21,22, structure and texture10,23, magnetic properties7,24,25 and particle size distribution26,27 affect the performance of DT test. In addition, operational settings are also effective, where magnetic field strength and water flow rate being key factors10,28. The DT is commonly used for quality control in wet low-intensity magnetic separation, with applications in simulating iron recovery and predicting concentrate purity29. In the DT test, to obtain iron recovery, the mass and content of iron for the feed and for the product must be measured first, then the iron recovery can be calculated. Given the time-consuming, labor-intensive, and costly nature of the DT test, identifying a model that can accurately forecast its results with an acceptable level of error is crucial. Literature reviews shows that the Multivariate Adaptive Regression Spline (MARS) technique has proven effective in predicting the optimal weight recovery of iron from an iron ore processing plant using data from the DT test. The MARS model achieved determination coefficient (R²) and root mean square error (RMSE) values of 0.978 and 2.538, respectively30. A feed-forward neural network, optimized with a genetic algorithm (GA-ANN), was developed to predict the efficiency and selectivity of the industrial magnetic separation process based on parameters like iron, iron oxide, and sulfur content. The GA-ANN model showed enhanced performance, achieving a mean square error of 0.276 for the selectivity index and 1.782 for separation efficiency, with R² values of 0.95 and 0.92, respectively31. The potential of hyperspectral imaging (HSI) combined with variational mode decomposition (VMD) and a random forest model was explored for accurately predicting total iron (Fet) content in both high-grade and low-grade iron ore samples. The VMD-RF model demonstrated high accuracy, achieving an R² of 0.94 and RMSE of 0.07. These results support cleaner production methods in iron ore processing32.

Literature reviews highlight that the main objective of the DT test is to achieve iron recovery, a process that is both time-consuming and costly. Therefore, this research aims to predict iron recovery in the DT test, an approach that has not been explored previously. This research aims to predict iron recovery in the DT test using a neural network, with iron (Fe) and iron (II) oxide (FeO) contents of the feed as input data, deliberately excluding mineralogical characteristics due to their time-consuming and costly nature. The procedure consisted of two main parts: Experimental and Modeling, which are summarized in Fig. 1.

Flowchart of the test work.

In this work, 613 samples with different Fe and FeO contents were supplied from the main Iranian magnetite iron ore mines. All samples were ground to under 74 microns followed by wet sieving. The resulting samples were oven dried at 100 °C for 12 h. Samples with low and very high Fe and FeO contents were also used to obtain a wide distribution of the data range to achieve a desirable model.

The experiments were performed by Eriez Model EDT Davis tube tester. All the variables were constant during the DT tests. Each sample was ground with a laboratory ball mill to pass through a 74 micron sieve. It has been assumed that all samples have the same liberation of magnetic minerals. Subsequently, 15 g of each sample were used for each DT test. The operating parameters used in the DT test were as follows: The angle of inclination was fixed at 45°, the oscillation rate was maintained at 45 cycles/min and retention time was 2 min, the magnetic field intensity and rate of wash water adjusted to 800 Gauss and 1 L/min respectively. To achieve maximum Fe content of concentrate, this low magnetic field was chosen. The determination of Fe and FeO content (%) was carried out by titration in accordance with ISO 9035:1989 and ISO 2597-1:2006 guidelines.

The statistical description of all test data is shown in Table 1. The feed parameters were used as input and the iron recovery as output to the desired model. Iron recovery of each test was calculated by Eq. (1)33.

where c is Fe content in concentrate, C is the mass of concentrate, f is Fe content in feed and F is the mass of feed that was equal to 15 g in all tests.

Examining Table 1 shows that both the Fe and FeO content of the feed and iron recovery exhibit a wide range. In contrast, the content of Fe in the concentrate is very limited, probably due to the good liberation below 74 microns and the low magnetic intensity of the DT. Due to the low variability in concentrate grade, modeling it is not practical; therefore, predicting concentrate grade was not addressed in this research.

RBF, a widely used type of ANN, is employed for regression and classification tasks on scattered data. By transforming the data into a multi-dimensional space, this particular feed-forward neural network is able to effectively handle diverse scatter data sets. Consequently, it generates accurate results for the given problem34. Typically, RBF neural networks follow a specific structure comprising three layers, with a single hidden layer positioned among the first layer (input layer) and the last layer (output layer)35. The basic concept behind RBF is to use localized radial basis functions φ(xi) to approximate f(x) through a linear superposition of these basis functions. The computation of f(x) can be expressed by Eq. (2)36:

In this equation, φ(xi) shows the transfer function, wT represents the weight vector of the output transfer layer, and b represents the bias term. In modeling with RBF neural networks, various radial basis transfer functions can be employed. One of the main radial basis functions used in RBF neural networks is the Gaussian function36. Equations (3–7) present some examples of radial basis transfer functions37:

In the present study, the Gaussian function was employed, resulting in the Eq. (8) derived by applying the transfer function to the inputs38:

In the equation, φki(x) represents the Gaussian transfer function, N represents the count of kernels used in models, ci represents the centers achieved from clustering, σ denotes the spread coefficient, and M represents the number of data points. In this model, the output can be calculated by performing linear regression of the transfer function in the Eq. (9)38:

In the equation, yk represents the model output, T represents the number of clusters employed in the model, w stands for connection weight, ci denotes the centers, and S represents the number of output and input sets.

The number of neurons in the hidden layer and the spread coefficient (σ) of the transfer function are crucial parameters in the RBF model that require optimization. In order to find the optimum values for variables, various methods can be employed, including trial and error procedures or the utilization of different meta-heuristic approaches. In this particular study, the trial-and-error procedure has been utilized to optimize the aforementioned variables. Figure 2 provides a structure diagram of the RBF neural network that is utilized in the paper.

A structure diagram of the RBF neural network.

This GRNN algorithm39 is essentially a variant of kernel regression, a well-known statistical technique. It is conceived as a normalized RBF neural network, where a hidden neuron is positioned at the central point of all training data sets40. This unique structure and approach enable the GRNN to perform efficient and accurate estimation tasks. In the GRNN technique, there is no iterative calculation procedure involved during the process. Instead, this approach enables the direct prediction of arbitrary functions in the middle of input vectors and output vectors based on the available training data41. The result of the GRNN model can be determined by applying the Eq. (10)42:

In the given equation, Y represents the output value on the basis of the input X, Di2 = (X − Xi)T (X − Xi) represents the squared Euclidean distance, σ is the spread factor, and T denotes the matrix transpose. A GRNN typically consists of four layers: the input layer, pattern layer, summation layer, and output layer. This architecture is illustrated in Fig. 3, which visually depicts the arrangement of these layers within the GRNN model. Also, σ referred to as the smooth parameter or spread factor, plays a crucial role in the GRNN model. It needs to be adjusted during the training process to minimize errors and improve the accuracy of predictions43. The trial-and-error procedure was employed to optimize the aforementioned parameter.

A schematic illustration of GRNN model.

To evaluate model performance, statistical parameters like coefficient of determination (R2), standard deviation (SD), root mean square error (RMSE), average absolute percent relative error (AAPRE), and average percent relative error (APRE) are employed. The definitions are presented in Eqs. (11–15)44:

where IR is the iron recovery (%), N indicates the number of samples, exp and pred represents experimental and predicted respectively.

Cross plots compare model predictions with experimental values, also known as equality plots. A well-performing model shows data points clustered closely around the diagonal line where X = Y. Error distribution plots examine the trend of errors by analyzing their distribution relative to the zero error line. In this study, Ei is computed using Eq. (16)44 and is presented as a percentage relative error:

The proportion of data points that the models accurately estimate is determined by plotting the cumulative frequency of data versus the absolute relative error. A model predicts with less error when much of the data is represented accurately, indicated by a high cumulative frequency and a curve that closely aligns with the y-axis. In this study, Ea is calculated using Eq. (17), expressed as absolute percent relative error44:

Finally, trend analysis illustrates how the model output changes with respect to a specific input, and upward and downward trends can be distinguished.

To evaluate the validity of models for predicting iron recovery based on Fe and FeO contents in feed of each DT test, statistical parameters including, APRE, AAPRE, RMSE, SD and R2 in both GRNN and RBF models were computed to determine the efficiency of prediction. All mentioned statistical data are shown in Table 2. In compliance with Table 2, R2 of both models are very close to 1, which indicates the high performance of the models in forecasting iron recovery. However, GRNN model exhibited superior performance across all statistical metrics compared to RBF model.

In both models, 80% of the data was assigned for model training and 20% for model testing. For statistical comparison through graphical error analysis, cross plot and relative error distribution plot of two models were represented in Figs. 4 and 5 respectively. According to cross plots in Fig. 4, most of the points for both models in the diagram are very close to the Y = X line, and no noticeable difference between the two models can be seen in this diagram.

Cross-plots of developed models.

Error distribution plots of developed models.

In Figs. 6 and 7 relative error plotted against FeO and Fe content of feed. Through these graphs, it can be concluded that the relative error of the GRNN model decreases as Fe and FeO content of the feed increases. As observed, the relative error significantly decreases in recoveries above 25%. For samples with high Fe and FeO feed grades, factors such as the degree of liberation remain relatively constant, making the results primarily dependent on Fe and FeO content. However, at lower Fe and FeO values, the influence of additional factors on iron recovery becomes more significant, leading to increased modeling errors when only Fe and FeO grades are used as inputs.

Figure 8 illustrates the cumulative frequency of the data plotted versus the absolute relative error (Ea, %) for GRNN and RBF models. As observed, Over 70% of the models’ predictions have absolute relative errors less than 4%. The Ea is less than 11% for approximately 90% of the predicted iron recoveries.

Relative error plot versus FeO content of feed for GRNN model.

Relative error plot versus Fe content of feed for GRNN model.

Cumulative frequency plot for the GRNN and RBF models.

In GRNN model as the best model, Iron recovery prediction depending on Fe and FeO content is presented in Figs. 9 and 10. In both diagrams, the experimental data and the model almost coincide with each other. As seen in the diagrams, there is a nearly upward correlation of Iron recovery with Fe and FeO content. Moreover, the GRNN model had good predations for most of data point across various Fe and FeO contents.

Iron recovery prediction in GRNN model as a function of Fe content in feed of DT test.

Iron recovery prediction in GRNN model as a function of FeO content in feed of DT test.

In order to quantitatively investigate the effect of each input parameter, such as the Fe and FeO content of the feed, on the iron recovery of the DT test, the relevancy factor (RF) was introduced for sensitivity analysis. As a matter of fact, iron recovery is influenced more by an input with a higher value of RF. To measure the relevancy factor, Eq. (18) is utilized44.

where inpave, i and inpi, j are average values of i-th input and the j-th values of the i-th input, respectively. Therefore j is from 1 to 613 and i can be 1 or 2. Here, the RF was calculated as 0.81 for Fe content of feed and 0.89 for FeO content of feed. It can be seen that iron recovery is directly related to Fe and FeO contents, but FeO content has a greater significant impact.

The Leverage approach45,46,47 is a method for detection of possible outliers, suspicious data, and detection of the effectiveness and applicability scope of a model. Figure 11 shows the leverage and suspected limit determined using this technique in William’s plot. The deviation between the model and laboratory data is termed as the standardized residual in the current study. As shown in Fig. 11, most points are within the spam of 0 to leverage limit (H*) and − 3 to + 3. The datasets with lesser H and R values are more trustworthy48. Based on the outcomes, less than 3% of points of data were outside the model’s feasibility range. These results indicate that the proposed GRNN model has a high degree of reliability for the estimation of iron recovery of DT test using Fe and FeO content of feed as input data.

William’s plot of GRNN model.

Statistical description of DT concentrate showed that Fe content in concentrate exhibits a narrow distribution. The main reason is low magnetic field strength in the DT tests (800 Gauss) and probably the maximum liberation of magnetic minerals in particle size less than 74 microns.

Relevancy factor and trend plots show that the correlation between input data (Fe and FeO contents of the feed) and iron recovery (as output) in the model is almost positive and both Fe and FeO contents are directly related to iron recovery, but FeO content is more influential.

The leverage approach showed that fewer than 3% of the data points fell outside the model’s feasible range. In addition, over 90% of the predictions from the GRNN models have absolute relative errors of less than 11%. This suggests that the GRNN model is highly reliable for estimating iron recovery in the DT test, using Fe and FeO content of the feed as input data.

Under identical DT test operating conditions, neural network modeling suggests that iron recovery can be accurately predicted using only two variables: the Fe and FeO content of the iron ore samples used in this study. This finding indicates that in the 613 tests conducted, iron recovery was nearly independent of other influencing factors.

The presented model accurately predicts iron recovery at high levels using two parameters: Fe and FeO content in the feed. However, for low iron recoveries (less than 25%), which correspond to low Fe and FeO content, the model shows increased error. This suggests that factors not accounted for in the model, such as the degree of liberation, have a more significant impact on iron recovery at lower levels.

Finally, it is recommended that future research incorporate additional factors such as particle size distribution, degree of liberation, and magnetic field strength, as this study focused on modeling iron recovery based solely on Fe and FeO content under constant DT test conditions. Including these factors could further improve the model’s accuracy.

The databank utilized during this research is available from the corresponding author on reasonable request.

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Department of Mining Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

Mohammad Tahami & Mohammad Ranjbar

Department of Petroleum Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

Mohammad-Reza Mohammadi & Mahin Schaffie

Mineral Industries Research Center, Shahid Bahonar University of Kerman, Kerman, Iran

Mohammad Ranjbar

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M.T.: Investigation, data curation, formal analysis, visualization, writing-original draft; M.-R.M.: Conceptualization, visualization, validation, modeling, writing-original draft; M.S.: Writing-review and editing, methodology, validation, supervision; M.R.: Writing-review and editing, methodology, validation, supervision, project administration.

Correspondence to Mohammad Tahami or Mohammad Ranjbar.

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Tahami, M., Mohammadi, MR., Schaffie, M. et al. Integration of experimental study and neural network modeling for estimating iron recovery in Davis tube tests. Sci Rep 14, 22578 (2024). https://doi.org/10.1038/s41598-024-72850-w

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Received: 11 July 2024

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DOI: https://doi.org/10.1038/s41598-024-72850-w

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