1. Introduction
Froth flotation finds extensive application in coal beneficiation, aiming to augment coal quality and its adaptability [
1,
2,
3]. Within this process, a liquid medium acts as a carrier for solid particles and gas bubbles. The hydrodynamic characteristics encompassing fluid flow patterns, velocities, and dynamic features play a pivotal role in governing the distribution and trajectories of solid particles and bubbles. Efficient fluid dynamics are crucial in ensuring homogeneous dispersion within the flotation chamber, preventing agglomeration and obstructions, thus favoring the effective separation of target minerals from contaminants [
4]. The fundamental procedure involves the injection of bubbles into the coal slurry, facilitating the attachment of coal particles onto bubble surfaces, subsequently buoying them to the upper layer of the flotation chamber, thereby forming froth concentrate [
5,
6]. The surface properties of bubbles exert significant influence on their affinity towards mineral particles, thereby impacting separation efficiency. The trajectories and velocities of solid particles within the flotation chamber dictate their frequency and the duration of interaction with bubbles [
6]. Optimized particle movement aids in elevating collision probabilities with bubbles, consequently enhancing flotation efficiency. Additionally, interparticle interactions such as agglomeration, settling, and convective movements also influence their behavior during the flotation process [
7]. A benchmark for successful froth flotation lies in the ash content within the concentrate, a critical indicator of coal quality [
8,
9]. Nonetheless, traditional ash content measurement methodologies entail multistep procedures and protracted durations, lacking real-time adaptability for process control. Operators, relying on visual cues from foam appearances on flotation chamber surfaces, encounter precision limitations. Consequently, there exists an imperative need to devise rapid, precise, and practical techniques for ash content quantification to support real-time process monitoring in flotation. Researchers have explored rapid ash content measurement instruments, including X-ray fluorescence (XRF) analyzers [
10] and γ-ray transmission [
11], among others. However, the accuracy of these radiation-based instruments is susceptible to the intricate heterogeneity inherent in coal samples. Regional disparities in coal composition, mineralogy, and impurity profiles contribute to potential measurement inaccuracies. Particularly in heterogeneous coal blends, higher resolution and intricate calibration procedures are imperative to ensure measurement accuracy. Employing XRF and γ-ray technologies in ash content measurement instruments necessitates the utilization of radioactive sources, entailing radiation risks. Despite their relative expediency, these measurement instruments incur high costs and pose hazards to both operators and the environment, thereby exhibiting limited potential for widespread adoption within the flotation domain.
Zhiping Wen et al. have proposed a methodology employing visual data from coal froth flotation images to predict flotation ash content [
12]. Machine learning techniques have seen widespread adoption in various domains such as object recognition [
13], object detection [
14], and medical image analysis. For instance, Hassan Nateghi utilized machine learning methodologies to determine the solubility of imatinib mesylate, an anticancer pharmaceutical, in a supercritical carbon monoxide environment [
15]. Remarkable strides have been made in industrial sectors as well. Runda Jia integrated machine learning into mineral flotation, specifically focusing on the recognition of flotation froth images, an area of extensive research [
16]. Bei Sun, Zhiping Wen, and their colleagues have employed flotation image characteristics to predict critical variables including flotation recovery and ash content [
12,
17]. The accurate prediction of flotation outcomes holds paramount importance in controlling and optimizing production processes within coal production. By leveraging machine learning for analyzing foam image features, it becomes possible to forecast the ash content in flotation concentrates, thereby fostering increased recovery rates of clean coal and reducing material wastage. Typically, analyses involve features such as bubble size [
18], bubble morphology [
19], and chromatic attributes [
20], as well as texture, among others [
21]. Diverse intelligent algorithms are utilized to establish models that delineate the correlation between foam characteristics and metallurgical parameters, facilitating the creation of predictive models [
22]. However, the risk of critical feature information loss exists, compounded by biases in feature selection that may lead the system to overlook pivotal information crucial for predicting the requisite parameters. Machine learning systems frequently rely on prior knowledge and experiential insights during the process of feature extraction and model establishment. Inadequacies or biases within prior knowledge may consequently impose limitations and inaccuracies within the resultant model. Consequently, the untapped potential of machine learning in flotation monitoring persists as an area ripe for further exploration and refinement.
In recent times, Zhiping Wen and M.R. Hosseini, along with their peers, have embraced the application of deep learning and neural network methodologies to address predicaments surrounding process performance and prognostication [
23,
24]. M.R. Hosseini et al. engaged neural networks to elucidate and model the intricate relationship between procedural parameters, surface bubble dimensions, and operational efficacy during the batch flotation of copper sulfide ores [
23]. Concurrently, V. K. Kalyani et al. harnessed artificial neural networks to scrutinize laboratory-scale froth flotation operations, undertaking estimations of optimal model parameters to compute diverse process parameters across assorted experimental conditions inherent to the coal flotation processes [
25]. Similarly, Mengcheng Tang et al. delved into the anticipation of flotation concentrate ash content by leveraging foam image processing and BP neural network modeling [
26]. Moreover, Gholamhossein Sodeifian et al. employed a multifaceted approach involving response surface methodology (RSM), genetic algorithms, and artificial neural network (ANN) models to elucidate and predict extraction rates, solubility, and concomitant parameters [
27]. Neural networks inherently necessitate voluminous datasets for proficient training to yield commendable performance. In the realm of mineral processing, however, procuring high-fidelity data could prove challenging due to potential noise, incompleteness, or inaccuracies within the datasets. Despite inherent constraints such as data volume dependencies, interpretational constraints, and susceptibility to parameter sensitivity within foam flotation processes, neural networks persist in offering latent value in predictive analytics and optimization strategies, necessitating a judicious appraisal considering both their merits and constraints for pragmatic deployment. Gonzalo Montes-Atenas et al. adeptly prognosticated bubble size and velocity in water and froth flotation slurries through the adept application of deep neural networks (DNNs) entwined with computational fluid dynamics (CFD) [
28]. Likewise, Zhiping Wen et al. envisaged coal flotation concentrate ash content employing a convolutional neural network (CNN) rooted in deep learning principles [
24]. Additionally, Hu Zhang et al. introduced the Feature Reconstruction–Regression Network (FRRN), a resource-efficient deep neural network tailored for monitoring foam flotation performance [
29]. Despite the relative prowess of deep learning in precise flotation performance prognostication, its efficacy hinges upon hyperparameter calibration and demands sizable annotated datasets for robust training, mandating protracted training processes. Furthermore, its implementation necessitates intricate computational resources and specialized expertise in deep learning, posing techno-economic hurdles for ventures constrained by resources and financial limitations. Additionally, extant features exhibit relative limitations, inadequately encompassing the multifaceted interplay between foam image attributes and coal concentrate ash content. Hence, forthcoming research endeavors should delve deeper into exploring multidimensional and intricate image features for enhanced accuracy in flotation outcome prognostication.
The likelihood function, in statistical parlance, elucidates the intricate relationship between parameter probability distribution and the observed dataset, serving as a fundamental tool in statistical modeling [
30]. It quantifies the probability of observing specific data given a certain set of parameters. Within statistical models, the estimation of various unknown parameters necessitates rigorous attention [
31]. Fundamentally, the likelihood function denotes the plausibility of data occurrences under varied parameter scenarios. Maximum likelihood estimation (MLE) represents an approach aimed at estimating model parameters by optimizing the likelihood function, thus seeking parameter values that optimize the probability of observed data occurrences within the selected statistical model [
32]. Essentially, MLE fine-tunes parameter values to optimize the probability of observed data occurrences, resulting in the computed maximum likelihood estimate. In industrial contexts, the imperative nature of interpreting model outputs and assessing credibility necessitates the application of maximum likelihood estimation to determine parameter values optimizing the probability of observed data occurrences within parameterized models. This facilitates a nuanced comprehension of the alignment between features and observed data characteristics [
33]. In our approach, we endeavor to amalgamate the principles of maximum likelihood estimation with the advancements in deep learning. Specifically, leveraging deep neural networks to autonomously extract feature representations from foam images eliminates the need for manual feature engineering. The optimal parameters derived from this process are integrated as neural network weights, effectively harnessing the feature learning prowess inherent in deep neural networks. This amalgamation aims to exploit the strengths of deep learning’s feature learning capabilities alongside statistical estimation techniques, resulting in a comprehensive encapsulation of intricate foam image attributes, thereby significantly enhancing predictive performance concerning coal ash content. The crux of this methodology lies in employing maximum likelihood estimation to infer parameters associated with features, facilitating a deeper understanding of the relationship between features and coal ash content values. This fusion of likelihood function principles with deep learning stands poised to elevate predictive performance while concurrently reinforcing the interpretability and credibility of model outcomes.
This investigation aims to amalgamate deep learning methodologies with the principles of likelihood functions to formulate a predictive model for coal’s refined ash content. The model utilizes parameters such as bubble velocity and reagent dosages, acquired from coal froth flotation processes. The primary objective is to optimize the stability and efficiency of coal flotation techniques, ultimately enhancing the effective utilization of coal resources. This study aspires to steer the coal industry towards a trajectory characterized by cleaner, more efficient, and sustainable practices, thereby making significant contributions to global endeavors for sustainable energy development.
3. Methodology and Modeling
Through the preceding experiments, a comprehensive dataset was obtained, encompassing essential parameters such as froth velocity, reagent dosage, and their corresponding values of coal ash content. This dataset lays a robust groundwork for subsequent modeling and prediction endeavors. We intend to harness the potential of cutting-edge deep learning techniques to process and extract dynamic features from the bubble images. By incorporating the concept of the likelihood function, we aim to establish a sophisticated probabilistic model that comprehensively captures the intricate relationships among reagent dosage, froth velocity, and coal ash content. This approach will offer a more holistic and accurate representation of the complex interplay between these variables. Furthermore, we will seamlessly integrate the optimized parameters derived from the likelihood function into the Keras deep neural network, thereby elevating the accuracy of predicting the coal ash content value.
3.1. Modeling of Froth Velocity Feature Extraction
The traditional methodologies employed for foam image analysis typically concentrate solely on the static attributes of foam while disregarding its dynamic nature, particularly the pertinent froth velocity information. Froth velocity, often denoted as the foam’s fluidity, embodies the dynamic facet of foam behavior, providing critical insights into the flux of concentrate quality [
35]. Notably, research conducted by P.N. Holtham accentuates froth velocity as the most crucial dynamic determinant in assessing flotation performance [
36]. Investigations by A. Jahedsaravani et al. have substantiated the pivotal significance of froth velocity characteristics in the regulation and efficacy of foam flotation control [
1]. Ming Lu et al., through the application of matching algorithms for extracting flotation foam velocity, validated its practicality and effectiveness in industrial production settings [
9]. Additionally, M. Massinaei et al., focusing on the characteristics of froth velocity, formulated predictive control systems to anticipate process states and performance under varied operational conditions [
20]. The velocity of bubbles plays a pivotal role in determining flotation efficiency, as it is intricately linked to bubble generation, flotation, and rupture processes [
37]. Consequently, accurate measurement and analysis of froth velocity are of paramount importance in gaining deeper insights into the flotation process, predicting bubble behavior, and optimizing operational parameters. To perform a meticulous analysis of foam images, we utilized a fusion of sophisticated deep learning techniques and advanced computer vision methodologies. Specifically, the application of the Scale-Invariant Feature Transform (SIFT) algorithm facilitated the extraction of distinctive keypoints and feature descriptors from the images, as illustrated in
Figure 2a. These keypoints represent salient local features within the images, delineating the image characteristics based on their precise spatial locations, scales, and orientations. Subsequently, leveraging keypoint matching techniques and the mean-shift clustering algorithm, we acquired the motion trajectories of foam across the image sequences (
Figure 2b) along with corresponding velocity information (
Figure 2c). Keypoint matching involves identifying analogous keypoints across different images, enabling precise tracking of foam positional variations across diverse image frames [
38]. Meanwhile, the mean-shift clustering algorithm serves as a robust method for density estimation of data points and identification of cluster centroids, effectively capturing and comprehending patterns and velocity characteristics inherent in foam motion [
39]. The employment of the SIFT algorithm enabled precise extraction of distinctive local features from foam images, facilitating accurate tracking and correlation of these salient points and thus ensuring meticulous monitoring and analysis of foam motion. Simultaneously, the utilization of the mean-shift clustering algorithm enhanced our ability to apprehend the motion patterns and velocity dynamics of foam, thereby enabling a more profound insight into the dynamic behaviors of foam during the flotation process. The SIFT algorithm is instrumental in identifying local feature points in the image, each characterized by a unique descriptor [
40,
41]. For the
-th frame of the image
, the SIFT algorithm computes the scale-space extrema of the keypoints
. Subsequently, the image undergoes Gaussian pyramid construction [
42], yielding images at varying scales.
The scale-space extrema of the image at different levels of the Gaussian pyramid are calculated to detect potential keypoints.
The scale-space extrema points are employed as initial candidate keypoints, and the positions and scales of these keypoints are accurately ascertained through interpolation.
In the context of the
-th frame image, each keypoint
undergoes a search process to find the most suitable matching keypoint
in the subsequent frame image
. The matching of keypoints is achieved by evaluating the Euclidean distance between their respective descriptors, which are distinctive representations of image features [
43,
44].
For each identified keypoint
and its corresponding matched keypoint
, the motion vector is calculated to quantify the spatial displacement between them.
The motion vectors undergo normalization, a process that involves standardizing their magnitudes.
In this context, represents the Gaussian convolution result of image at scale , where denotes the Gaussian kernel function. indicates the scale-space extrema of the image at scale , with signifying the maximum value. refers to the position of the extrema, and represents the precise determination of the keypoint’s position and scale through interpolation. represents the feature descriptor of keypoint , and represents the feature descriptors of all keypoints in the frame image. , and , denote the coordinates of keypoints and its matched keypoint , respectively. represents the normalized velocity vector.
Table 4 illustrates the outcomes achieved through the application of deep learning and computer vision methodologies for the identification and extraction of dynamic features pertaining to froth velocity in froth flotation.
3.2. Likelihood Function Modeling of Froth Velocity and Chemical Additions
In this research, we take into consideration two distinct types of flotation reagent dosages, denoted as and , representing the collector and frother, respectively. Our primary objective is to predict the ash content of the clean coal based on these reagent dosages and the froth velocity . To achieve this, we propose a hybrid model that effectively captures the impact of different reagent dosages and froth velocity on the clean coal’s ash content. Additionally, we employ the maximum likelihood estimation method to accurately estimate the parameters of the model.
We possess a dataset consisting of 7500 observational samples, denoted as
with
, where
represents the bubble velocity of the
-th sample, and
and
stand for the two types of reagent dosages for the
-th sample. Furthermore,
signifies the ash content of the
-th clean coal sample. Our proposition postulates that both the bubble velocity
and the two types of reagent dosages,
and
, exert a substantial influence on the ash content
of clean coal. To effectively model the intricate interdependencies among these variables, we propose the utilization of a mixed-effects linear regression model:
In the presented equation, , , , and denote the model’s coefficients, while represents the residual term associated with the -th sample, signifying the unexplained variability remaining after accounting for the model’s explanatory variables.
To estimate the model parameters, we employ the maximum likelihood estimation method. Each sample point
(as depicted in
Figure 3) is independently obtained by sampling from a random variable that follows a Gaussian distribution. Given the model parameters
,
,
, and
, the likelihood function of observing the sample points can be expressed as follows:
The joint likelihood function can be obtained as the product of the individual likelihood functions for all observed samples:
To simplify the computations, it is conventionally employed to take the natural logarithm of the likelihood function, resulting in the following:
Our objective is to identify the parameter values that maximize the natural logarithm of the likelihood function, represented as follows:
The derivation obtained is as follows:
By solving the above system of equations simultaneously, we can obtain the estimated values for the model parameters , , , , and .
3.3. Deep Neural Network Prediction Model with Multi-Feature Input and Hybrid Data Input Using Keras
Having obtained the optimal parameters for the likelihood function, as described earlier, we applied these parameters as the weights for the Keras deep neural network, as depicted in
Figure 4. In this study, we developed an innovative deep neural network model with multiple feature inputs and a hybrid data input scheme, utilizing the Keras framework [
45,
46,
47]. Each neuron in the input layer represents distinct characteristics of foam flotation bubble velocity, collector dosage, and frother dosage. Simultaneously, the output layer comprises a single neuron, which predicts the coal ash content. The architecture of the deep neural network model is presented as follows:
We adopted the mean squared error as the loss function, denoted by the following:
The gradient calculation for each parameter is as follows:
where
During the training procedure of the Keras deep neural network, we employed a series of critical hyperparameter configurations to optimize the model’s performance and mitigate overfitting phenomena. The Adam optimizer was selected as the initializer, given its proficient adaptive learning rate properties, which enable more effective adaptation to intricate optimization tasks [
48]. The learning rate was set to 0.001 to adequately control the parameter update step during training, avoiding the pitfalls of excessively high or low learning rates that could lead to training instability. Following the completion of each training epoch, the learning rate was subjected to decay by dividing the initial learning rate by 200. This gradual reduction in the learning rate during the later stages of training facilitated a more meticulous search for the optimal solution space, thereby enhancing the model’s convergence speed and stability.
To forestall the model from excessively tailoring itself to the training data, an early stopping strategy was introduced. We diligently monitored the loss function on the validation set and immediately terminated the training process if no improvement was discernible over 200 consecutive training iterations. This judicious approach effectively prevented the model from continuing to train in an overfitting state, thereby bolstering its generalization capacity and reducing the risk of overfitting.
5. Conclusions
This study successfully integrates deep learning with the likelihood function, thereby establishing a high-precision prediction model for accurately detecting the ash content of clean coal in the realm of coal froth flotation. Leveraging the Scale-Invariant Feature Transform (SIFT) algorithm, this research extracts salient feature points and descriptors from the images, and through the amalgamation of keypoint matching with mean-shift clustering, it achieves a comprehensive understanding of the flotation process, encompassing froth trajectory and velocity information. Studies have revealed an inverse relationship between the ash content of refined coal and the velocity of bubbles observed during the froth flotation process. This negative correlation suggests that elevated bubble velocities may decrease the duration of the interaction between bubbles and solid particles, thereby impacting the efficacy of flotation. Conversely, lower bubble velocities appear to be conducive to facilitating the separation of coal from the foam, consequently elevating the quality of coal. These revelations illuminate the pivotal impact of dynamic parameters in bubble kinetics and the interplay between coal and foam on the efficiency and quality of flotation processes.
This study proposed the utilization of the maximum likelihood estimation to optimize parameters within the likelihood function, which were subsequently integrated into the Keras deep neural network for training and predictive purposes. Evaluation of the model performance on both the training and validation sets exhibited R-squared values nearing 1, accompanied by minimal values across other assessment metrics. This signifies the exceptional predictive capability of our model in estimating the ash content of refined coal. The harmonious fusion of deep learning and the likelihood function has showcased robust predictive prowess, presenting novel technological avenues for quality control in coal product manufacturing and productivity enhancement. The optimized predictive model developed in this study lays a robust groundwork for real-time monitoring within practical industrial applications, particularly in the domain of froth flotation. This advancement holds the potential to drive the froth flotation sector toward automation and intelligent operations. It is noteworthy that our findings offer an efficient and reliable method for the coal industry and related sectors, specifically for the accurate prediction of ash content in refined coal. This capability enhances product quality and provides crucial support for pivotal decisions in industrial production processes. In essence, our research applies the amalgamation of deep learning and the likelihood function to the realm of coal froth flotation, presenting a robust predictive model with superior performance. This achievement holds significant implications for driving technological innovation and industrial intelligence in related fields, laying a solid foundation for the efficient monitoring and control of froth flotation processes.