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Frontiers in Chemical Engineering ·

Volume 8 - 2026 | https://doi.org/10.3389/fceng.2026.1676997

Application of pragmatic modelling to metallurgical processes

  • Kurian J. Vachaparambil1*corresponding author
  • Stein Tore Johansen1
  • Bjørn Tore Løvfall1
  1. Process Technology, SINTEF Industry, Trondheim, Norway
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Abstract

Understanding metallurgical processes has historically relied on measured data (which are often very limited), laboratory experiments, simplified unitary operation models, and more recently, computational models (like Computational Fluid Dynamics). Detailed measurements, especially during industrial operations, are often difficult due to the corrosive environment and high temperature of the process. In some cases there is currently no possibility for direct measurements, or if a possible sensor exists the sensor cost is not acceptable. Computational approaches have severe limitations due to the large computational overhead of running these models for large systems with long-term transient variations. In addition we have complex chemistry and multi-physics with strongly coupled phenomena. In such a context, pragmatic models are extremely useful to gain insight into the operations and optimize the process. The pragmatic models are designed to run real-time, or even faster. In this paper, we will provide an overview of applications of the pragmatic modelling techniques to three metallurgical applications: alumina transport in aluminium reduction cells, ladle erosion, and Søderberg electrodes. The pragmatic modelling approach used to develop the simulation framework for these processes and the predictive ability of the models are summarized. Finally, applicability of these pragmatic modelling as well as data-driven approaches (like machine learning) in building digital twins for metallurgical processes are discussed.

Keywords

1 Introduction

Metallurgy has been an integral part of the human evolution for many millennia (Wertime, 1964). The scientific advances and the knowledge gained in this domain can be broadly divided based on the scale of relevance into physical chemistry (at atomic/molecular level), process metallurgy (at equipment/device level) and macro-dynamic metallurgy (at process level), as described by Yin (2021). The physical chemistry at atomic/molecular scales delves into material development (cf., inert anodes for aluminium electrolysis (He et al., 2021), reaction pathway/rates (Rasouli et al., 2022) and thermodynamics/CALPHAD (Jung et al., 2019; Olson and Liu, 2023; Jung and Van Ende, 2020). The process metallurgy involves reactor design and engineering at the scale of the equipment by considering relevant multiscale and multiphysics phenomena (Johansen, 2003; Ludwig et al., 2016). At scales larger than the process level, the macro-dynamic metallurgy delves into the overall process analysis and integration commonly referred to as flowsheet modelling (Schultmann et al., 2004; Jung and Van Ende, 2020).

In process metallurgy, the knowledge gained historically has been based on experiments and unit operation models. The data and observations obtained from experimental campaigns on both industrial and pilot systems provide invaluable knowledge of the metallurgical process, e.g., Ringdalen and Eilertsen (2001); Ray et al. (2007). In addition, lab-scale experiments also provide detailed insights into the phenomena that occur within the metallurgical process, for example, to study the metallothermic reactions that occur in furnaces (Rasouli et al., 2022). Due to the high temperature and corrosive nature of metallurgical processes, it is often difficult to obtain detailed measurements but some process relevant parameters are monitored for process control in industrial systems, see Bearne (1999). Some of these parameters are continuously monitored and others are manually measured, at different sampling frequencies within the system, see Bearne (1999). The difference in the sampling frequencies of the process parameters are related to the difficulty to set up the related sensor in the system, and sometimes due to the lack of robust and affordable sensors. In comparison to experiments, unit operation models refer to models which are based on overall conservation equations, at system level, to simulate the process without resolving its details, see McCabe et al. (1993). For example, the work by Mehrotra (1989) discusses the development of unit operation type models for three metallurgical applications: copper convertor, interfacial phenomena in metal-ceramic systems, and mixing in a gas-stirred liquid bath. These types of unit operations model are used widely in industry as they provide insight into the processes and are very fast, but they require models to treat multiscale and multiphysics nature of the complex processes involved in the process which limits its applications to new processes/phenomena as well as modeling undesirable scenarios during operations. With the availability of computational resources, computational models, like Computational Fluid Dynamics (CFD) and Discrete Element Method (DEM), has become very a popular approach to investigate metallurgical processes as they are based on fundamental conservation equations and provide detailed insights into the process (c.f., Kadkhodabeigi et al. (2011), Peters et al. (2024)). These computational models have been used successfully in literature to model a range of systems and processes of relevance in metallurgy, like multiphase flows, emissions from furnaces, tapping operations, casting process, and electrorefining, see review by Johansen (2003), Ludwig et al. (2016) for more information. It should also be highlighted that the knowledge gained from such computational models can be used to develop sub-models which can be used with unit operation models. Despite the insights provided by computational modelling, these methods are computationally expensive making them unsuitable for real-time predictive applications in industry. With the advances in machine learning techniques, these have also been used within metallurgical process industry to some limited extent (Laha et al., 2015; Bulgarevich et al., 2018; Viertauer et al., 2019), but they are seldom used in process metallurgy as it is constrained by the lack of quality data that covers all operationally relevant scenarios including the abnormal events (or edge cases), see review by Estay et al. (2023).

Thus process metallurgy requires the development of models that combines concepts from both computational and unit operation modelling approaches to allow for fast and accurate simulation of both regular and edge cases encountered during operations. These models, which are referred to as pragmatic models, are fast and treat the relevant physical phenomena crucial to simulate the process/equipment/reactor. The pragmatic modelling refers to an approach/methodology to combine computational modelling techniques with realism (or reasonable approximations) to develop fast physics-based models specifically to monitor and optimize industrial processes. This term was coined almost 10 years ago by Schwarz and Feng (2014) and Zoric et al. (2014) to develop sufficiently accurate and fast models to simulate complex, multiscale, multiphysics and coupled nature of phenomena relevant in industrial processes. These models are developed by bridging decades of knowledge in computational techniques (used in high fidelity simulations), multiscale modelling, industrial modelling (like unit operation models, computational models, reduced order models and engineering correlations) and process knowledge, see Zoric et al. (2014). As such the pragmatic modelling can be seen as a framework that falls within multiscale modelling paradigms relevant to chemical engineering domain. An example of a pragmatic model is LedaFlow™, a commercial transient multiphase flow simulator, which is used for flow assurance in oil and gas applications, marketed by Kongsberg Digital AS 1 . Using LedaFlow™, the transient three-phase flow in kilometers of pipelines can be simulated much faster than realtime (Danielson et al., 2011). The SINDA/FLUINT framework, used to analyze and design thermal/fluid systems, is another example of pragmatic modelling approach used in a multitude of industries (Cullimore and Lin, 1988) 2 . In this work, we present the development of pragmatic models for three very different metallurgical processes and the predictive ability of these models (compared against industrial data). The three metallurgical processes considered for pragmatic model development are alumina transport in aluminium reduction cells, ladle erosion and Søderberg electrodes. These pragmatic models were developed with the aim to be used in the industrial control applications and operational strategy development (like fast look ahead simulations) so their runtime and computation overhead are quite low.

2 Development of pragmatic modelling for metallurgical applications

Based on the pragmatic model development methodology proposed by Zoric et al. (2014), these models are developed with the aim to simulate reactor/system during both regular operation and abnormal events (or edge cases). The aim of these models are to provide predictions of process relevant parameters during operation such that its evolution can be monitored for various operational strategies. In addition, these pragmatic models are developed with the objective to be computationally inexpensive - typically to run on a single processor with fast runtime (i.e., simulation of many operational hours in fraction of that time). To develop these models, the methodology used by the development team can be described as (see Figure 1):

  • Selection of the metallurgical use case: The qualitative goals of the modelling framework for the relevant reactor/system, based on the need and/or strategic objectives of the industry, is identified by the industrial client (case owner) in discussion with process experts (which in this case are metallurgists/operators), in collaboration with development system architects (or model development team).
  • Understanding the process: To develop a pragmatic model which describes the process, the model development (or system architect) team need to have a good understanding of the process, including the operational procedures, as well as undesirable scenarios during operations. This knowledge is critical to develop a pragmatic model and to represent both the process dynamics and the undesirable scenarios. This step most often involves continuous discussions between the developers and the operators/metallurgists to understand their needs and learn from their experiences about the process.
  • Define modelling objectives: Based on the process understanding, a modelling framework is defined by the system architects (or model development team), with the objective to reliably simulate the process scenarios of interest for the operators/metallurgists. The framework is iterated and agreed with the industrial client.
  • Establish and develop a modular physics-based modelling framework: To simulate the metallurgical process, the multiscale and multiphysics nature of the process is divided into explicitly coupled modules. Each of these modules corresponds to the treatment of a physical phenomenon relevant to the process. This involves deriving/determining the governing equations of the relevant phenomena, which often requires (volume and/or ensemble/time) averaging techniques (cf. Johansen, 2003). The modelling framework uses a grid/mesh which should be sufficiently resolved to capture the spatial evolution of a process relevant parameter (like temperature). In some applications, these complex systems can be cut into a set of compartments, an extension of unit operations in classical chemical engineering (see Haag et al., 2018), which numerically is treated as a very coarse grid. This concept of compartmentalization can be extremely useful for many processes as it reduces the mesh needed for the simulation by splitting a process into multiple smaller reactors with local chemistry and exchange between reactors (that are internally ideally mixed) obtained from detailed CFD.
  • The coupling between the different phenomena is enabled via explicit coupling using relevant source/sink terms in the governing equations. For example, when handling the alternating current (AC) flow though a conductor, the electric current flow would be computed based on the material properties (which is temperature dependent), using Maxwell’s equations. The electric current flow is coupled to the energy equation (or temperature model) via the ohmic heating source term. The model development also uses empirical models and experimental observations reported in literature, or is otherwise available, regarding the process along with detailed Computational Fluid Dynamics (CFD) in the development of the closure models for the individual modules. The discretization and implementation of the governing equations is in this work based on Finite Volume Methods and solutions obtained by using linear algebra (Ferziger et al., 2019). Using these explicitly coupled modules for treating physical phenomena allows for quicker development/adaptation of these individual modules during the various stages of the development based on the feedback from the operators/metallurgists.
  • Refinement of the model based on industrial data: In this step, the model developed is compared against industrial measurements data to assess and optimize its predictive abilities. This step also involves obtaining feedback from the operators/metallurgists about the model predictions to incorporate operational procedures used in the process. Based on the discrepancies and feedback, the relevant individual modules are updated to improve the model’s predictions and applicability to the operations. Model improvements and comparison with industrial data/feedback is continuously performed to ensure the model is consistently and accurately performing for the operational scenarios.
  • Application of model in process control: The application of the model to advanced process control or model predictive control should be done only after the model has been thoroughly verified for the scenarios of interest for the process, including its ability to predict undesirable scenarios. Due to the fast runtime of these models, they can be used as a decision support framework for the operators/metallurgists. In addition, models can be used in fast look-ahead of in order to optimize operation. This is particularly useful when we are dealing with processes that evolve over time and where time dependent inputs are controlled by the operator (which is often the case in metallurgical processes).
Illustration of the methodology used in the development of the pragmatic model along with the steps involved.
Figure 1. Illustration of the methodology used in the development of the pragmatic model along with the steps involved.

With the methodology described above, the pragmatic model for each use case considered in this work requires detailed process understanding, implementation and refinement of relevant modules for the treatment of the relevant coupled multiscale and multiphysics phenomena, and eventual application in advanced process control and operational risk minimization. In Section 3, the development and verification of the pragmatic model for the considered use cases are summarized along with an overview of how to adapt the framework for other applications.

3 Pragmatic model use cases in metallurgical industry

3.1 Alumina transport in aluminium reduction cells

Understanding the process: Industrial aluminium production involves electrochemically converting the particle alumina dissolved in cryolite based bath to produce aluminium at the cathode and carbon dioxide at the anode. The dissolution of alumina is a complex phenomena that involves dispersion of fed alumina as a cloud in the bath and formation of agglomerates that could float at the metal-bath interface (called rafts) or sink to form sludge (Bracamonte et al., 2021). Low levels of dissolved alumina in the bath would result in low/high-voltage anode effect (producing perfluorocarbon (PFC) emissions), and high levels and/or inadequate local bath temperature could cause sludging - both of which are undesirable for stable and efficient operation of the aluminium reduction cells (Solheim and Skybakmoen, 2020; Solheim, 2022).

In a typical industrial cell, see Figure 2, the addition of particle alumina occurs via a few feeders located at specific location in the cells. The dissolution of particles is driven by heat and mass transfer, and its transport within the bath is driven by the dispersion and convection (driven by magnetohydrodynamics, turbulence and bubble evolution) as well as dissolution dynamics (Lavoie and Taylor, 2016). The transport and dissolution of the particle alumina also results in associated variation in the temperature due to the endothermic nature of alumina dissolution (Solheim and Skybakmoen, 2020). In industrial cells, control strategies are used for feeding alumina to ensure an optimal level of alumina in the bath (typically around 3-5 wt pct) and operating bath temperature around 950–970 °C, see Bearne (1999). For alumina feeding, these control strategies often results in cycles of underfeeding and overfeeding of particle alumina during operation. With respect to the control measurements, direct measurement of the level of alumina in the bath is limited, it is often performed daily from certain sampling locations in the cell (Bearne, 1999).

Illustration of an industrial aluminium reduction cell.
Figure 2. Illustration of an industrial aluminium reduction cell.

From an operational perspective, it is important to understand the variation in level of alumina within a cell during regular operation to determine optimal sampling locations in the cells, and to understand feeder hole conditions before each feeding instance, see Lavoie and Taylor (2016). In addition, the ability to predict the alumina transport and dissolution in these cells can be used to investigate operational control/design changes.

Modelling objectives: The objective is to develop a model that can predict the distribution of alumina in an industrial aluminium reduction cell during regular operation and predict the associated evolution in temperature. The model should be able to predict the variation of alumina levels and bath temperature related to various alumina feeding strategies and cell designs.

The modelling framework: The physics-based framework used to describe the process has been developed based on the assumptions: bath flow does not vary during regular operation (not valid during and just after anode change operations), sideledge dynamics is neglected, the formation of rafts is not modelled explicitly as they have been observed to last only for limited duration (in the order of minutes) when compared to operational duration of many hours. Based on these assumptions and modelling objectives, the pragmatic model is developed to solve for particle and dissolved alumina levels, bath temperature, tracer distribution by considering operational inputs like current supplied, bath composition and feeding patterns. The model is developed for a coarse grid (based on the concept of compartmentalization of a reactor, see Haag et al. (2018) so that the grid needed to resolve the entire aluminium reduction cell can be reduced significantly (in the order of a few hundred cells). The transport model employed for alumina, bath temperature and tracer uses a bath flow and dispersion model. The time-averaged bath flow (from a detailed magnetohydrodynamics simulation) is interpolated to the grid which is then corrected to ensure mass conservation. The model for particle alumina also accounts for the particle dissolution and feeding, while dissolved alumina transport model accounts for the particle dissolution and electrochemical consumption. The bath temperature model is coupled to particle dissolution via dissolution enthalpy, apart from other source terms that accounts for ohmic heating, heating of alumina particles, and heat transfer through boundaries. The assumptions used in the model development has been successfully verified using measured levels of alumina and tracer levels at various locations during operation in an industrial cell, see Vachaparambil et al. (2025). Readers interested in further information on the pragmatic model for alumina transport in aluminium reduction cells (including its implementation) should refer to our previous publications (Johansen et al., 2022; Vachaparambil et al., 2023; Vachaparambil et al., 2025).

Model predictions: The predictions from the pragmatic model are compared against industrial data from Aloca’s Mosjøen smelter to assess its predictive capability and identify sub-models that require refinement. Some selected model predictions compared against measurements are shown in Figure 3 for illustration. The alumina distribution and the corresponding bath temperature predicted by the model during operation is shown in Figure 4. The model is seen to be able to reasonably capture the observed variations in the alumina levels in the cell using a sludging coefficient (i.e., % of fed alumina that is temporarily lost to the bath). The model at some location (during some instances) has been seen to under-predict alumina in bath which has been attributed to the lack of treatment of self-feeding 3 , see discussion in Vachaparambil et al. (2025). The model has also been seen to be able to predict a reasonable bath temperature distribution during operation. Additionally, the value of the sludging coefficient also provides an estimate on the amount of fed alumina which is being sludged in the cell. In terms of the computational overhead, the model runs about 25 times faster than simulated operational duration. Readers interested in the detailed assessment of the predictive ability of the model (compared to Aloca’s measurements) and uncertainties associated should refer to Vachaparambil et al. (2025).

Comparison of the predicted alumina level from the pragmatic model (labelled PM) with the equivalent experimental data at three selected locations in the industrial aluminium reduction cell. Due to the confidentiality of the industrial data both alumina levels in bath and the time are scaled, see Vachaparambil et al. (2025) for more information. It should be noted that the particle alumina added via each feeder at each feeding instance is 1.85 kg with an uncertainty of 0.25 kg. The model predictions represent three scenarios: the upper and lower bounds of alumina added via a feeder at each feeding instance ( m ), and different levels of sludging coefficient ( B ). The upper bound of the alumina fed, green line, is seen to be not representative of the alumina levels in the cell. Clearly, the alumina fed to the cell is around the lower bound of 1.85–0.25 kg per feeder at each feeding instance. Interested readers should refer to Vachaparambil et al. (2025) for detailed comparison between model predictions and experiments at twelve locations, and for two measurement campaigns conducted in the same cell. The image is adapted from Vachaparambil et al. (2025).
Figure 3. Comparison of the predicted alumina level from the pragmatic model (labelled PM) with the equivalent experimental data at three selected locations in the industrial aluminium reduction cell. Due to the confidentiality of the industrial data both alumina levels in bath and the time are scaled, see Vachaparambil et al. (2025) for more information. It should be noted that the particle alumina added via each feeder at each feeding instance is 1.85 kg with an uncertainty of 0.25 kg. The model predictions represent three scenarios: the upper and lower bounds of alumina added via a feeder at each feeding instance ( m ), and different levels of sludging coefficient ( B ). The upper bound of the alumina fed, green line, is seen to be not representative of the alumina levels in the cell. Clearly, the alumina fed to the cell is around the lower bound of 1.85–0.25 kg per feeder at each feeding instance. Interested readers should refer to Vachaparambil et al. (2025) for detailed comparison between model predictions and experiments at twelve locations, and for two measurement campaigns conducted in the same cell. The image is adapted from Vachaparambil et al. (2025).
Predicted distribution of the dissolved alumina and the corresponding bath temperature by the pragmatic model at an instant during operation in the industrial aluminium reduction cell. The image is adapted from the supplementary material in Vachaparambil et al. (2025).
Figure 4. Predicted distribution of the dissolved alumina and the corresponding bath temperature by the pragmatic model at an instant during operation in the industrial aluminium reduction cell. The image is adapted from the supplementary material in Vachaparambil et al. (2025).

Potential use in industry: This pragmatic model can be used to investigate the evolution of bath temperature and alumina levels in the cell during operation. The typical computational models used to investigate alumina distribution in literature, like Zhan et al. (2014), has very high computational costs due to which these models are used to simulate one or two feeding cycles. In comparison, the pragmatic model approach can simulate operations of several hours or several cycles of feeding in fraction of the operational time. In addition, the pragmatic model can be run on a laptop or desktop and does not require high performance computing systems which are needed for the traditional computational models. Such a model along with accurate industrial magnetohydrodynamics models can be used to investigate alumina transport and bath temperature for various cell designs. Due to the limited measurements of alumina levels in the bath, these model can be used to investigate the non-uniformity in the alumina levels helping to identify the optimal sampling locations (and scheduling) in the bath. These model can also be used to study thermal conditions and alumina level at the feeder hole before each instance of alumina fed to the bath. Finally, the model can be used to generate predictions for various operational scenarios and identify optimal feeding strategies (fast look ahead type applications).

Discussions on knowledge gaps and future possibilities: The pragmatic model considers the dissolution of dispersed alumina particles in the bath. In reality, the particle alumina can disperse forming a cloud in the bath as well as form floating agglomerates (called rafts). Understanding the formation and evolution of rafts are important as they could sink to the metal (forming sludge) in addition to delaying the delaying alumina dissolution. This has been a topic investigated in literature via detailed experiments and modelling (Gylver et al., 2019; Kovács et al., 2020; Bracamonte et al., 2021; Bojarevics and Dupuis, 2022). Future developments of the pragmatic model will incorporate the knowledge gained from previous works in literature to develop a alumina transport and dissolution module that treats raft dynamics. The reintroduction of the sludged alumina to the bath, called self-feeding, is another aspect of the process that requires further investigation. Developing an understanding of the physical mechanism that drives self-feeding is important as it has not been fully understood to the best knowledge of the authors, see Fini et al. (2020) for more information.

3.2 Ladle erosion

Understanding the process: In metallurgical industries, the metal is transported and treated in ladles, which are basically huge cups, built of steel and with inner linings of ceramic materials (see Figure 5). The very inner layer, contacting the liquid steel, is the wear lining which must be replaced at regular intervals. The main challenge is that the wear may lead to steel penetrating the layers of ceramic insulation, melting a hole in the steel casing and causing outflow of liquid steel (≈ 200 tons) into the area surrounding the ladle. This must be avoided at any cost. During operation, the ladle is covered by a lid with heating electrodes, which are inserted from the top, to supply sufficient heat. The additional heat is necessary to maintain metal temperature during periods with different additions to the melt (support melting) and to compensate for heat losses into the refractory and to the external during the refining. The lid also allows for applying suction and partial vacuum above the melt.

Left: A sketch of the cross section of a steel ladle. Right: A tilted and empty but still hot ladle, which has been checked for refractory damage, and will be moved to a storage waiting for next heat. The image is adapted from Johansen et al. (2024a).
Figure 5. Left: A sketch of the cross section of a steel ladle. Right: A tilted and empty but still hot ladle, which has been checked for refractory damage, and will be moved to a storage waiting for next heat. The image is adapted from Johansen et al. (2024a).

During operation, the temperature distribution in the ladle refractory is governed by contact with hot steel, general heat loss from the ladle and wait time until the next addition of steel. A low refractory temperature in the ladle, when steel is added, leads to large and unfortunate thermal stresses. When different additions are added to the steel, these additions may partially melt, and may also stick to the refractory wall for a period. The additions have different heats of melting and will impact the steel temperature. Inert gas is introduced from the bottom of the ladle to enhance mixing and impacts the local heat and mass transfer inside the ladle. In the final step of additions, more alloying materials and slag former is added to enhance the removal of unwanted chemical species and to arrive at the correct alloy quality. At a later stage, vacuum may be applied to enhance the effects of the bubble stirring and the removal of dissolved gases like hydrogen, nitrogen and oxygen. It should be noted that the refractory may dissolve into the slag where these are in contact and some elements of the refractory may dissolve into the steel. Based on the liquid chemistry and the local mass transfer coefficient the local erosion rate may vary.

Understanding ladle erosion is critical to increase number of times a ladle is used during operation without causing any safety issues. Currently, the operators would do a visual assessment of each ladle after it had been emptied, and based on this, experience, and the number of uses since last re-lining determine whether it was safe to use the ladle once more. As slag results in more wear than the steel, it becomes possible to perform a one-time repair of the lining. During the repair, the upper and most eroded layers of the wear bricks were replaced. Then the ladle was put back in production, and the state of the refractory was followed with visual inspections until the operator decided that the entire wear lining should be replaced. At this point, the most complete measurements of the brick erosion were performed. It must be noted that only at this time we fully know the state of the refractory. So using plant data to understand ladle erosion is a challenge, as the data in the first place was not really collected for digital twin and control system development, see Johansen et al. (2024a), Johansen et al. (2024b).

Modelling objectives: The objective is to develop a modelling framework to enable the operators to safely increase the number of uses of a ladle before re-lining.

The modelling framework: As part of solving this challenge, in Johansen et al. (2024b), we outlined and adapted a framework for development of physics-based models for the metallurgical industry. A very first idea was that the model challenge could be solved by using data only, using machine learning. The results from these early attempts were in hindsight found to have little relevance, as the parameters that were found to have leading influence on erosion in fact were later found to be only partially true measurements. The ladles were really 3-dimensional vessels due to the placement of the plug for gas injection and the tapping nozzle (see Figure 5). In order to obtain the fastest possible model we decided that a cylinder-symmetrical model would be sufficient. The thermal processes in the ladle refractory were basically treated as 1-dimensional, solving for the energy in the different refractory layers, ranging from the contact with slag/metal to the gas atmosphere at the outside. At the bottom, the refractory was seen as a vertical stack of disks. At the side wall, we represented each horizontal layer of physical bricks with separate numerical grid points. Radially, or horizontally, wear brick was represented by three numerical grid points, while other bricks, insulation layer and steel shell was represented by single grid points. Vertical heat conduction between brick layers was suppressed. Vertical temperature variation in the ladle wall then became mainly a result of time in contact with metal and slag, and heat redistribution due to radiation. The metal and slag was handled as well mixed regions, which could dynamically change in size, exchange energy with refractory, heater elements and lid. Solid additions (alloy elements, slag formers and additions for refining) could be added at any rate or amount. Models for the wear of the refractory, depending on steel or slag contact were developed. Local mass transfer is a critical element in such models. The pragmatic approach here is to obtain the local heat and mass transfer coefficients between wear brick and steel or slag by using computational fluid dynamics (CFD). We have developed heat and mass transfer models, where only the wall shear velocity is the unknown input. Next, we calculated the flow in the ladle for different purge gas flow rates and vacuum pressures. Using traditional curve fitting we could obtain algebraic relation for the shear velocity versus vertical position, gas flow rate and vacuum pressure. We may note that at the lowest surface pressure (highest vacuum) there is a substantial gas expansion close to the surface, leading to large surface velocities and waves. The model was set to predict the amount of C and Mg that was transferred from the wear lining to the metal and slag during each heat when a ladle was in use. The accumulated erosion would modify the profile of the wear lining. When the ladle was used next time it could be operating under similar or different conditions (steel mass in ladle, operation time, amount of purge gas, additions, vacuum), which would give a different erosion pattern. The erosion from a next heat would incrementally be added to the erosion profile from the previous heat. In this manner the model could be operated over a number of heats, and erosion patterns could be compared with measurements at the time when the ladle was taken out for repair of the most eroded bricks. At this time the model could be compared with data from this repair. The repair was easily done in the model as the repaired bricks were reset to the repaired state and the model could be run until demolition was decided. Now more complete data for the erosion state was obtained and the model could again be compared with data.

The model was developed as “quasi 2D″ because we assumed no vertical energy exchange between brick layers. However, the code structure had a complete 2D grid for the side wall. It would therefore be fast and easy to introduce vertical heat transfer between bricks. Unfortunately, this would slow down the code execution. At an early state of a pragmatic development, speed is of critical importance. Therefore, relaxing this would be an option for further development. In addition, the methodology we have applied, could as well be used to a full 3D model. In this case we could apply 3D CFD and do a more complex curve fitting to get the variation of the wall shear velocity, but now also including one more physical dimension as input parameter. For more details about model assumptions and approximations the reader is referred to Johansen et al. (2024b).

Model predictions: The industrial partner, the Spanish steel company Sidenor, was in possession of substantial amounts of data, ranging over several years. The input data consisted of measurements of mass of steel transferred to the ladle, steel temperatures, addition of a range of materials, such as slag former, alloying elements and refining ingredients. In addition, time series that included heater power, gas purging flow rate and vacuum pressure, were given. The model would need physical data where some was provided by the literature and some was provided by Sidenor. A challenge here was that in many cases proper physical data was not obtainable, even from the material suppliers. In such cases, using the model to run sensitivity studies can provide valuable insight.

The model was then tuned against the available data, as outlined in Johansen et al. (2024b). Main tuning parameters were heater efficiency, wear brick and insulation layer conductivity, and pore length related to the representative depth of the carbon, binding the MgO particles in the wear lining. A prediction example of the steel temperature evolution is seen in Figure 6. The green dots on the experimental temperature line are the actual measurements. It was found that the first measured temperature was not true, but was a left over in the system from the previous heat. The significant upward tuning of the wear brick and insulation thermal conductivities was a strong indication that steel would penetrate in between the bricks. As a result, the bricks might float in the liquid steel subject to the strong buoyancy force. This understanding will add strongly to the understanding of the lining wear mechanisms. This information could impossibly be reduced from ML and AI approaches based on pure data processing. In Figure 7 we see the result from a prediction of the wear state at the time of demolition. The ladle was asymmetric due to the placement of the porous plug and the teeming nozzle. In the empirical examination it was defined a right and left part (seen from above) where the most eroded bricks were measured. The model can quite well represent the wear except close to the feed surface. Here additional mechanisms are in play due to metal and slag splashing. These phenomena were not included in the model and is not critical to the safety as a steel leak around the slag layer would have hand-able consequences.

Predicted steel temperature evolution, measured temperatures and energy added to the ladle (“Power”), for Ladle 5, heat 206217. The image is adapted from Johansen et al. (2024a).
Figure 6. Predicted steel temperature evolution, measured temperatures and energy added to the ladle (“Power”), for Ladle 5, heat 206217. The image is adapted from Johansen et al. (2024a).
Comparison of predicted versus measured eroded thickness for Ladle 11, campaign 80, 2019. The image is adapted from Johansen et al. (2024a).
Figure 7. Comparison of predicted versus measured eroded thickness for Ladle 11, campaign 80, 2019. The image is adapted from Johansen et al. (2024a).

In Figure 8 we see the overall results for measured versus predicted erosion thickness at the time of demolition. These results were based on predicting 5216 heats, which involved 11 different ladles and 61 campaigns. As the active duration of the heats varied between 1 and 3 h, the computations involved much more than 10000 h of real-time simulations, as we also simulated some wait-time. This is the time from when the metal is tapped into the casting machine until the ladle is ready to be filled again. As the measurements consider the most eroded bricks while the model considers an ensemble average, the model should under-predict the erosion. As the number of tuning factors is limited, the prediction is quite good except for the outlier marked A (Ladle 8, campaign 76). The outlier was not readily explained. One possibility may be that special conditions for mushroom formation above the porous plug have been present during this campaign, leading to reduced flow circulation in the ladle. The complexity of ladle lining erosion is extreme as the erosion may vary a lot between neighboring bricks. Several stochastic factors are involved, such as variability in quality between bricks, mechanical damage to bricks during ladle maintenance, and variability in first splashes of steel into the empty ladle.

Comparison between measured and predicted erosion thickness at time of demolition of the wear lining. Symbols represent different ladle numbers. Outlier A is marked. The image is adapted from Johansen et al. (2024a).
Figure 8. Comparison between measured and predicted erosion thickness at time of demolition of the wear lining. Symbols represent different ladle numbers. Outlier A is marked. The image is adapted from Johansen et al. (2024a).

Potential use in industry: This type of model may support the operator during assessment of the state of the ladle. Ideally, we should be able to predict exactly the time when the first brick will fail. This will most likely never be possible. However, by help of this model we can predict the state of the refractory from heat to heat and use a weighed combination of prediction and operator assessment to conclude if one more heat is safe or not. When we know the input conditions for next heat, the model can do a prediction. Depending on the amount of steel and target steel quality the model can assess the incremental wear. Such information may be invaluable in the push for enabling one more heat before demolition. The source code for the ladle erosion model is generally available from GitHub repository 4 .

Discussions on knowledge gaps and future possibilities: Refractory wear depends on a large number of factors, where some may be characterized as unknown, while others are insufficiently known. All material data such as physical and thermodynamic properties should be known for the actual working conditions. This is a significant challenge. In addition the wear brick properties may have variation between bricks. This is highly undesired and the effect of such variations can only be handled by statistical means. The operational data should be collected, validated and reprocessed to be available for future learning and model development. In our case we have treated the slag and metal flow as 2D. The modeled heat flow in the refractory and ladle wall is primarily treated as 1D, but extending the model with axial heat conduction is a simple and easy task. Furthermore, the model can be extended to 3D, both for the refractory part and the bubble driven flow. The bubble driven flow may then be computed in 3D. The resulting refractory wall shear stresses can still be fitted and interpolated, using more complex methods. However, there is a trade off between model accuracy and speed. This has so far not need fully explored. Combining the model predictions with thermal camera images of the ladle steel casing at given times and positions would be valuable to pick up regions with significant deviations (in the hotter direction) which could help to decide on whether one more heat for the refractory or not, but also to support model extensions and tuning. An interesting possibility is to explore differences between model predictions and process data, using ML methods to look for anomalies and explore if we have deviations that can only be explained by yet unidentified factors.

3.3 Søderberg electrodes

Understanding the process: Developed more than 100 years ago, Søderberg electrodes are the dominant electrode technology used in submerged arc furnaces for the production of non-ferrous metals and ferroalloys, see Figure 9. These electrodes are used to supply energy for the alloy forming reactions and with the advances over the years the electrodes used in industry nowadays has a diameter around 2 m and carry alternating currents (AC) around 100–150 kA (Innvrær, 1989). These electrodes have an outer steel casing with slotted ribs/fins into which paste (a mixture of anthracite and coal tar binder) is added periodically during operation. The paste experiences increasing temperature from the heat added to the system by the fan blowing hot air around the upper section of the casing and induction heating from the current supplied and ohmic heating in addition to the external cooling at the current clamps, see Skjeldestad (2019). This results in the unbaked paste to soften, lose volatiles, and undergo baking around 450–500 °C which yields a strong, solid and conductive electrode. Due to the consumption of Søderberg electrodes within furnaces they are lowered into the furnace via an operation referred to as slipping and there are also operational variation in the current supplied.

An illustration of the Søderberg electrode. The image is adapted from Skjeldestad (2019).
Figure 9. An illustration of the Søderberg electrode. The image is adapted from Skjeldestad (2019).

The quality of the baked electrode is critical as breakage of electrode (via soft and hard breakages) has a large economic consequence for the process. The soft breakages are a result of not positioning the baking zone within the current clamp region or due to damage to casing or due to large current in the casing (Innvrær et al., 1995). In comparison, hard breakages, which is typically observed during furnace shutdown scenarios, occur due to large thermal stresses in the electrode (Innvrær et al., 1995). So preventing the electrode breakages requires understanding the impact of the employed operational procedure on the baking zone and thermal stresses.

With increasing current supplied and larger electrodes used in operation, along with advances in the electrode paste, the risk of electrode breakage during operation needs to be understood. Knowledge of the conditions experienced by the electrode during operation can be used to develop operational procedure to minimize these undesirable operational scenarios (like reduce current load, shutdown and high slipping) when using different pastes. In addition, such knowledge can also be used to estimate release of volatiles (like PAH emission) during the operation due to paste softening and baking, see Larsen et al. (2024).

Modelling objectives: The aim is to develop a model that can predict the baking zone position and thermal stresses in Søderberg electrodes during operation based on the operational inputs and temperature dependent paste properties.

The modelling framework: Based on the understanding the process and modelling objectives, an axisymmetric Søderberg electrode model has been developed. The model assumes that the electrode paste added is in the form of briquettes. Due to the axisymmetric assumption, a volume averaged definition of the fin geometry is employed by the model. The model has various modules for treating the relevant physics in the system: Electromagnetic model (for the AC current distribution), temperature model (within the electrode, casings and heated air flow in the upper section of casing), paste flow and distribution (softening, baking and consumption), and thermal stresses. The thermal stress model uses the temporal and spatial variation of temperature during operation to predict the evolution of axial and radial displacements within the electrode. The softening of the paste causes the unbaked paste bed to collapse resulting in a convective flow of paste into the baking zone. The model also has a module to accounts for casing erosion - which has been observed to occur below the current clamps and above the charge in the furnace. The induction heating from the current carrying conductors to the current clamps, relevant in the region above the current clamps, are also treated in the model (via fitted source terms in the temperature model). The model also incorporates the operational changes in time for current supplied, electrode slip, holder motion, casing addition, and paste addition. The casing addition and holder motion/electrode slip procedures are considered to be discrete operations that are treated via adding new mesh elements to the domain and adjusting the boundary conditions. In addition, the temperature dependent properties of the paste are used as input to the model. For readers interested in the additional details of the model, an initial version of the Søderberg electrode model (without a comprehensive treatment for thermal stresses) is available in Vachaparambil et al. (2024).

Model predictions: The model has been set up based on the available industrial data reported in Skjeldestad (2019) and unavailable data (like the electrode slip, boundary heat transfer rates, erosion related parameters and paste properties) were constructed based on the best available estimate (based on discussion with the industry partner, Eramet). The model predictions are compared against the experimental data in Figure 10. The distribution of temperature within the electrode predicted by the model during operation is shown in Figure 11. The model is seen to predict an increasing temperature in the region above the current clamps as a result of induction heating. The discrepancy in this region is due fact that the model employs a heat transfer model for a briquette based paste where as the measurements were obtained for cylinder form of the paste which has a different heat transfer mode in the upper section of the casing. Within the current clamp region, the model also capture the initial temperature decrease driven by the cooling from the current clamps. The subsequent temperature increase is caused by ohmic heating along with conductive heat transfer from the lower section the electrode. The baking isotherm (which is the 475 °C isotherm in Figure 11) profile also qualitatively agrees with equivalent profiles during regular operation reported in literature (Larsen, 2012). From the predicted temperature distribution, the position of the baking isotherm (which is the 475 °C isotherm in Figure 11) can be continuously monitored during operation. Considering the uncertainties in the modelling inputs, the model provides a reasonable estimate of the temperature within the electrode and with a runtime of a few hours for simulating an operation over several days. Model predictions of the thermal stress evolution in Søderberg electrodes during operations will be explored in future works.

Comparison of the predicted temperature at different radial measurement locations within the Søderberg electrode (of diameter (2*R*) equal to 1.9 m) against select thermocouple measurements (labelled TC) extracted from the work of Skjeldestad (2019). Note the number after TC represents the various thermocouple positions along the azimuthal direction at a given radial position used in a measurement campaign, see Skjeldestad (2019).
Figure 10. Comparison of the predicted temperature at different radial measurement locations within the Søderberg electrode (of diameter (2R) equal to 1.9 m) against select thermocouple measurements (labelled TC) extracted from the work of Skjeldestad (2019). Note the number after TC represents the various thermocouple positions along the azimuthal direction at a given radial position used in a measurement campaign, see Skjeldestad (2019).
Predicted temperature distribution within a 1.9 m diameter Søderberg electrode during operation. The gray blocks on the both sides of the electrodes represent the current clamps.
Figure 11. Predicted temperature distribution within a 1.9 m diameter Søderberg electrode during operation. The gray blocks on the both sides of the electrodes represent the current clamps.

Potential use in industry: The model can be used to generate predictions for the evolution of the baking isotherm (and thermal stress) for various operational procedures. Such a model can be used to provide predictions which can be used to develop strategies for the safe operation of the electrode. The predictions of the conditions within the electrode by models can provide insights which can be used to understand the consequences of the operating procedures. This allows for a better electrode management especially when using different electrode pastes in operation. Such a model can be used by the operators/metallurgists as a decision support framework to efficiently operate the Søderberg electrode.

Discussions on knowledge gaps and future possibilities: In this work, the model has been verified against temperatures observed during a measurement campaign reported in Skjeldestad (2019). Future works would need to verify the model predictions for additional operational scenarios especially in cases where electrode breakage has been observed. Another aspect of Søderberg technology which requires further investigation is the characterizing the electrode paste and developing knowledge of its thermo-physical properties. There is very limited data on Søderberg pastes characterization reported in literature (Larsen et al., 2004; White et al., 2020; Beukes et al., 2013; Skjeldestad, 2019). This is a critical knowledge that could aid both in better model predictions and using the appropriate operational procedures (Larsen et al., 2004).

3.4 Application of pragmatic modelling to new industrial use cases

In Sections 3.1–3.3 we have seen some example applications of pragmatic modelling to metallurgical use cases. For a general case, we may concretize a workflow that would lead to a pragmatic model for any situation. The pragmatic model is the simplest model which can give fast, and sufficiently good answers. There could be a short step from a pragmatic model to online process control and operation support tools. Any pragmatic model starts with the simplest possible model which has value for the user. We now give a concrete advise for how to develop a pragmatic model for a general metallurgical case. The recommendation here is aligned with (Figure 1) but now with concrete steps.

  • Step1: A first definition of the metallurgical use-case. This should be a collaboration between the process owners and the developer system architect team.
  • Step 2: A learning phase where the system architect team and the process owners iterate through discussions, on-site visits, investigating what data is available and data quality. If issues with data quality is spotted actions may be taken to mitigate.
  • Step 3: Definition of model requirements, including speed, inputs, outputs and accuracy. This is iterated between process owners and system architects until agreement.
  • Step 4: Develop a modular physics-based model framework. We will assume that we deal with some dynamic process and where a model backbone or overall system model must be defined, and on which sub models will be connected. At this point we must consider how we can exploit application of both Lagrangian and Eulerian frameworks, both as model backbone and to be used in sub-models. The Lagrangian framework can offer accurate transport of multiple properties (the parcel concept) and may in some cased offer possibilities which are very hard to fulfill using Eulerian methods. In general, the sub-models may have their own sub-models on multiple lower levels. An important point here is that these sub-models should be pluggable. As a result, these models can be easily updated when new closure models become available. Typically, the system model may be a 2D or 3D Eulerian representation of the main geometrical elements, or a connected set of sub-processes (unit-operations, transport pipes). A general rule is that sub-models should, as far as possible be look-up models based on pre-calculated results. Pre-calculations may exploit methods for experimental design. This will ease the interpolation inside the pre-calculated data.
  • Step 5: Detailed specification of the model framework, the sub models at all levels, the use and integration with process data and detailed model output. This step is demanding and requires that the system architect team has the appropriate domain knowledge as well as an oversight over critical modeling tools. At this step cost issues will be considered as there will be many commercial tools that may be tempting to use but due to excessive costs may be ruled out. If found acceptable, also the model framework could be based on a commercial tool. It is however not possible to specify a set of methods, models or tools to be used in general. The tool box may contain a significant number of tools and methods, ranging from parametrization methods, factorial designs, statistical methods like ANOVA (analyses of variance), singular value decomposition, reduced order models, computational fluid dynamics and all way to quantum chemistry, when needed. In most applications we will need physical properties, thermodynamic data and reaction kinetics data or corresponding models. Finding useful data here often belongs to the domain of expert knowledge. We have used Python as our main coding language, as it is quite fast and has a vast amount of models and modules available for data exchange and computations.
  • Step 6: Implementation and testing. As the model springs into life, using industrial in-data and comparing with industrial out-data, data issues may be detected or model flaws may be spotted, starting an iteration that tales us back to Step 2. Possibly, elements in the model may need to be redesigned. Model sensitivity analyses is critical.
  • Step 7: Active use of available data to tune and optimize the model. If model is too slow we may step back to step 4 to see what can be done for speedup.
  • Step 8: Apply the model in process control.

We previously made a test to apply the pragmatic modelling concept to drilling of oil-wells, demonstrating the capability of the concept also in other process industries, see Johansen et al. (2017).

4 Building digital twins for metallurgical processes: relevance of pragmatic models and data-driven methods

Digital twins (DTs) refers to a virtual representation of a physical asset or system which is continuously updated with real-time data (Martin, 2021; IBM, 2021). These DTs are used for design, monitoring and improving operations as well as predictive applications, see Martin (2021); IBM (2021). Creation of a DT involve data-driven methods, simulation and modelling techniques (physics-driven methods), or a combination of two approaches (hybrid methods)), using real-time data from the physical asset or system. In this section, we explore the capability of these methods for building DTs for metallurgical processes. It should be mentioned that the use of DTs in industrial applications for decision-making combines operator expertise, model predictions, along with process/sensor data into integrated frameworks (like Papacharalampopoulos et al. (2025), Stavropoulos (2022)).

The applications of data-driven methods towards metallurgical processes, especially towards reactor/system, have been limited (see review by Estay et al. (2023)). An example of data-driven methods used in this domain is the work by Viertauer et al. (2019) where the authors showed that the brick wear (in ladle erosion) can be estimated reasonably using either linear regression model or a deep learning neural network model. In their work, both the linear regression and deep learning neural network model were trained on process data and manual lining measurements which have been cleaned for anomalies and incomplete data. Despite the reasonable predictive ability of the data-driven models, like the one developed by Viertauer et al. (2019), these models are restricted by the inherent inability to simulate process scenarios/conditions which fall beyond the parameter space of the training data which limits the usability of such models especially for edge cases. In addition, training data used for the creation of such models need sufficient and good quality process data as well as control measurements. In metallurgical systems, obtaining such quality data is difficult and requires significant pretreatment to either remove or provide a best estimate for anomalies and incompleteness in the data (cf., Viertauer et al. (2019)). Detailed measurements in the metallurgical processes are often difficult because of extreme environments (involving fumes, dust, high temperature, large currents, and corrosive materials) either due to the lack of methods/sensors or sometimes due to the very high cost of sensors. In this context the data measured needs to be understood (by domain expert and modelling tools like the pragmatic models) so that pretreatment of the data is done carefully without losing critical information about the process in the training data. Thus the use of data-driven methods to make DT in metallurgical processes is often difficult due to the lack of sufficient quality data. Despite the limited use of data-driven methods in simulating reactors/systems for process metallurgy, these methods have relevance in other application in metallurgy where sufficient amount of quality training data is available like for pattern recognition (cf., characterizing microstructure in steel from optical microscopy images (Bulgarevich et al., 2018) and overall process yield optimizations (Laha et al., 2015)).

Apart from data-driven methods, DTs can also be generated based on simulation and modelling techniques. These DTs rely on data assimilation techniques (which historically have been used in weather forecasting models (Rabier, 2005)) to combine real-time data from processes with predictive computational models to reduce errors caused by modelling uncertainties. One of the most popular data assimilation techniques used in such DT application is based on Kalman Filtering (KF). For example, Donato et al. (2024) reported the development of a DT of a Hydrogen powered combustion furnace to predict temperature and NO emissions using KF. For use in such a DT, the predictive computational model used has to be at least faster than real-time. Such fast models are generated from CFD data typically using Reduced Order Models (ROM) (cf., Donato et al. (2024)) or surrogate models (SM) generated using machine learning techniques (cf., Molinaro et al. (2021)). The pragmatic models, like the models discussed in the two works referred to above, are attractive due to its ability to generate large amounts of synthetic data, which covers relevant operational scenarios (including edge cases), at a significantly lower computational overhead than typical CFD simulations (used to generate ROM/SM). In addition, due to the fast runtime of these pragmatic models, they can be used directly in a DT along with data assimilation techniques to explore real-time operational data from the metallurgical processes.

Finally, the hybrid methods used in developing DTs combine both data-driven and physics-driven methods. An example of such a method reported in literature is the hybrid digital twin for the Si refining process at Elkem described by Linnestad et al. (2022), where a physics-based (first principles) refining model is supplemented using a data-driven model that predicts the amount of tapped slag from real-time thermal imaging data. The pragmatic models, like the ones showcased in this work, can also be used in such application employing data assimilation techniques. In fact the use cases discussed in Section.3 employ industrial data for model tuning/calibration albeit not real-time process data, so these pragmatic models can be categorized as hybrid digital twins. The ladle erosion model, discussed in Section.3.2, was developed by optimizing the model using substantial industrial data with the possibility to continuous update the model using new data in the future. Both the aluminium reduction cell (Section.3.1) and Søderberg electrode (Section.3.3) user cases have been optimized using data available from limited industrial campaigns. In hybrid methods, machine learning is a very helpful tool to detect anomalies and possible phenomena that is not picked up by the physics-based model. In addition the use of pragmatic models in industrial applications, especially to build digital twins, requires accounting for model uncertainties. For example, Vachaparambil et al. (2025) highlighted the need for using a realistic non-uniform initialization of alumina in the bath to provide reasonable model predictions. Such uncertainties relating to model inputs (like initial conditions and thermophysical data), simplifications and settings will impact model predictions. So critical output parameters from the model relevant for advanced process control and understanding should be measured directly or estimated using data-driven methods to provide a feedback to update the pragmatic model in real-time to better represent the state of the physical asset or system.

5 Conclusion

In this work, the pragmatic modelling methodology proposed by Zoric et al. (2014) is adapted to develop fast models for metallurgical processes. The industrial use cases presented in this work are alumina transport in alumininum reduction cells, ladle erosion and Søderberg electrodes. In each use case, the pragmatic modelling methodolgy (see Figure 1) is described to provide an understanding of the various stages of the model development. The methodology allows for developing a comprehensive understanding of the process and its bottlenecks, and development of models that can be used in industrial operations. The pragmatic models have been developed by combining approaches used in computational techniques, process knowledge, unit operations and reduced order models specific to each process. In the use cases, the models has been developed in Python based on Finite Volume Methods to treat the coupled multiphysics and transient nature of phenomena that is relevant for the use case. The predictive ability of the developed pragmatic models is seen to be reasonable and they have a low computational overhead.

Comparing the pragmatic modelling methodology for the three use cases in this work provides the readers a practical overview of its application for very different industrial processes. The methodology used in this work can be adapted to other metallurgical processes to develop models that can provide fast predictions for industrial applications. The modular nature of the models allows for incorporating the computational techniques needed to treat the relevant physics while incorporating industrial knowledge. While the choice of the model simplifications and numerical techniques (like coarse-graining or averaging) are process specific, they are necessary to reduce the computational overhead of these models. With its reliable predictions and fast run time, these pragmatic models can be used directly in industries towards advanced process control, and operational strategy development (like fast look ahead simulations) as well as in building digital twins for metallurgical processes.

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Footnotes

Edited by: Xianglong Luo, Guangdong University of Technology, China

Reviewed by: Alexander William Dowling, University of Notre Dame, United States

Panagis Foteinopoulos, University of Patras, Greece

Footnotes

https://ledaflow.com/about/why-ledaflow/

https://spinoff.nasa.gov/spinoff2002/ct_9.html, https://www.crtech.com/products/sindafluint

Self-feeding is the phenomena by which the sludged alumina (or the alumina sinking into the metal) is reintroduced into the bath. The region where self-feeding occurs can be different from the location where the sludging of alumina occured within the cell. It should be noted that self-feeding is a complex phenomena whose underlying physical mechanism is not well understood to the best knowledge of the authors.

https://github.com/SINTEF/refractorywear

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was received for this work and/or its publication. The authors would like to acknowledge funding which has been used in the development of the pragmatic models: SFI Metal Production (Centre for Research-based Innovation, 237738), H2020 project COGNITWIN (870130), Eramet and Research Council of Norway project NextGenSøderberg (313864).

Data availability

The datasets presented in this article are not readily available because the data obtained from industrial measurement campaigns as well as the models (for alumina feeding and Søderberg electrodes) are not available due to its confidentiality. The Ladle erosion model is available for interested users. Requests to access the datasets should be directed to Ladle erosion model, https://github.com/SINTEF/refractorywear. For other information please contact the corresponding author.

Author contributions

KV: Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review and editing. SJ: Conceptualization, Investigation, Methodology, Software, Writing – original draft, Writing – review and editing. BL: Conceptualization, Investigation, Methodology, Software, Writing – review and editing.

Acknowledgements

The authors would like to thank Alcoa, Sidenor, and Eramet for the discussions in the development of the models and for the permission to share the results, and SINTEF for publication support.

References

  1. Bearne G. P. (1999). The development of aluminum reduction cell process control. JOM 51, 16 – 22. doi:10.1007/s11837-999-0035-5
  2. Beukes J., Roos H., Shoko L., van Zyl P., Neomagus H., Strydom C. (2013). The use of thermomechanical analysis to characterise söderberg electrode paste raw materials. Miner. Eng. 46-47, 167 – 176. doi:10.1016/j.mineng.2013.04.016
  3. Bojarevics V., Dupuis M. (2022). “ Advanced alumina dissolution modelling,” in Light metals 2022. Editor Eskin D. (Cham: Springer International Publishing), 339 – 348.
  4. Bracamonte L., Aulie V., Rosenkilde C., Einarsrud K. E., Sandnes E. (2021). “ Dissolution characteristics and concentration measurements of alumina in cryolite melts,” in Light metals 2021. Editor Perander L. (Cham: Springer International Publishing), 495 – 503.
  5. Bulgarevich D. S., Tsukamoto S., Kasuya T., Demura M., Watanabe M. (2018). Pattern recognition with machine learning on optical microscopy images of typical metallurgical microstructures. Sci. Rep. 8, 2078. doi:10.1038/s41598-018-20438-6
  6. Cullimore B. A., Lin C. H. (1988). Fluint - general fluid system analysis with sinda ’85. J. Spacecr. Rockets 25, 97 – 98. doi:10.2514/3.25955
  7. Danielson T. J., Bansal K. M., Djoric B., Duret E.-D., Johansen S. T., Hellan O. (2011). “ Testing and qualification of a new multiphase flow simulator,” in Offshore technology conference of OTC offshore technology conference. OTC–21417–MS. doi:10.4043/21417-MS
  8. Donato L., Galletti C., Parente A. (2024). Self-updating digital twin of a hydrogen-powered furnace using data assimilation. Appl. Therm. Eng. 236, 121431. doi:10.1016/j.applthermaleng.2023.121431
  9. Estay H., Lois-Morales P., Montes-Atenas G., Ruiz del Solar J. (2023). On the challenges of applying machine learning in mineral processing and extractive metallurgy. Minerals 13, 788. doi:10.3390/min13060788
  10. Ferziger J., Perić M., Street R. (2019). Computational methods for fluid dynamics. Springer International Publishing.
  11. Fini M. F., Landry J.-R., Soucy G., Désilets M., Pelletier P., Rivoaland L. (2020). Sludge formation in hall–héroult cells: drawbacks and significant parameters. Mineral Process. Extr. Metallurgy Rev. 41, 59 – 74. doi:10.1080/08827508.2018.1536658
  12. Gylver S. E., Omdahl N. H., Prytz A. K., Meyer A. J., Lossius L. P., Einarsrud K. E. (2019). “ Alumina feeding and raft formation: raft collection and process parameters,” in Light metals 2019. Editor Chesonis C. (Cham: Springer International Publishing), 659 – 666.
  13. Haag J., Gentric C., Lemaitre C., Leclerc J.-P. (2018). Modelling of chemical reactors: from systemic approach to compartmental modelling. Int. J. Chem. React. Eng. 16. doi:10.1515/ijcre-2017-0172
  14. He Y., Zhou K.-c., Zhang Y., Xiong H.-w., Zhang L. (2021). Recent progress of inert anodes for carbon-free aluminium electrolysis: a review and outlook. J. Mater. Chem. A 9, 25272 – 25285. doi:10.1039/D1TA07198J
  15. IBM (2021). What is a digital twin? Available online at: (Accessed April 29, 2025).
  16. Innvrær R. (1989). “ The søderberg electrode system recent research and development new challanges,” in Proceedings of INFACON V, 216 – 226.
  17. Innvrær R., Fidje K., Kalgraf K. (1995). “ Expert system for soderberg electrodes,” in Proceedings of INFACON 7, 555 – 564.
  18. Johansen S. T. (2003). “ Mathematical modeling of metallurgical processes,” in Proceedings of third international conference on CFD in the minerals and process industries (Melbourne, VIC: CSIRO), 5 – 12.
  19. Johansen S. T., Meese E. A., Zoric J., Islam A., Martins D. (2017). “ On pragmatism in industrial modeling part iii: application to operational drilling,” in Progress in applied CFD – CFD2017 proceedings of the 12th international conference on computational fluid dynamics in the oil and gas, metallurgical and process industries (trondheim, Norway: SINTEF proceedings), 2.
  20. Johansen S. T., Einarsrud K. E., Solheim A., Vachaparambil K. J. (2022). “ A pragmatic model for alumina feeding,” in Light metals 2022. Editor Eskin D. (Cham: Springer International Publishing), 503 – 511.
  21. Johansen S. T., Løvfall B. T., Duran T. R. (2024a). A pragmatical physics-based model for predicting ladle lifetime. J. South. Afr. Inst. Min. Metallurg 124, 93 – 110. doi:10.17159/2411-9717/2680/2024
  22. Johansen S. T., Løvfall B. T., Duran T. R., Zoric J. (2024b). Pragmatism in industrial modelling: an application to ladle lifetime in the steel industry. J. South. Afr. Inst. Min. Metall. 124, 111 – 122. doi:10.17159/2411-9717/2681/2024
  23. Jung I.-H., Van Ende M.-A. (2020). Computational thermodynamic calculations: factsage from calphad thermodynamic database to virtual process simulation. Metallurgical Mater. Trans. B 51, 1851 – 1874. doi:10.1007/s11663-020-01908-7
  24. Jung J.-G., Cho Y.-H., Lee J.-M., Kim H.-W., Euh K. (2019). Designing the composition and processing route of aluminum alloys using calphad: case studies. Calphad 64, 236 – 247. doi:10.1016/j.calphad.2018.12.010
  25. Kadkhodabeigi M., Tveit H., Johansen S. T. (2011). Modelling the tapping process in submerged arc furnaces used in high silicon alloys production. ISIJ Int. 51, 193 – 202. doi:10.2355/isijinternational.51.193
  26. Kovács A., Breward C., Einarsrud K., Halvorsen S., Nordgård-Hansen E., Manger E. (2020). A heat and mass transfer problem for the dissolution of an alumina particle in a cryolite bath. Int. J. Heat Mass Transf. 162, 120232. doi:10.1016/j.ijheatmasstransfer.2020.120232
  27. Laha D., Ren Y., Suganthan P. (2015). Modeling of steelmaking process with effective machine learning techniques. Expert Syst. Appl. 42, 4687 – 4696. doi:10.1016/j.eswa.2015.01.030
  28. Larsen B. (2012). “ Electrode models for søderberg electrodes,” in Proceedings of fifth international platinum conference, 275 – 286.
  29. Larsen B., Amaro J., Nascimento S., Fidje K., Gran H. (2004). Melting and densification of electrode paste briquettes in sØderberg electrodes. Proc. INFACON X, 405 – 417.
  30. Larsen S. Y., Spilde T. E., Øyvind K., Zahlsen K., Brunsvik A., Ratvik A. P. (2024). “ Towards eliminating pah emissions from søderberg electrode paste in mn production,” in Proceedings of the 17 international ferro-alloys congress (INFACON XVII), 771 – 776.
  31. Lavoie P., Taylor M. P. (2016). “ Alumina concentration gradients in aluminium reduction cells,” in Advances in molten slags, fluxes, and salts: proceedings of the 10th international conference on molten slags, fluxes and salts 2016. Editors Reddy R. G., Chaubal P., Pistorius P. C., Pal U. (Cham: Springer International Publishing), 791 – 798.
  32. Linnestad K., Hildal K., Jakobsson L. K., Eidnes S., Tjessem V., Bjørnstad E. L. (2022). “ A hybrid digital twin for optimal si-production,” in Proceedings of the silicon for the chemical and solar industry XVI. doi:10.2139/ssrn.4121131
  33. Ludwig A., Wu M., Kharicha A. (2016). Simulation in metallurgical processing: recent developments and future perspectives. JOM 68, 2191 – 2197. doi:10.1007/s11837-016-1992-0
  34. Martin S. (2021). What is a digital twin? Available online at: (Accessed April 29, 2025).
  35. McCabe W., Smith J., Harriott P. (1993). Unit operations of chemical engineering. McGraw-Hill.
  36. Mehrotra S. P. (1989). Role of mathematical modelling in metallurgy. Bull. Mater. Sci. 12, 353 – 368. doi:10.1007/BF02747142
  37. Molinaro R., Singh J.-S., Catsoulis S., Narayanan C., Lakehal D. (2021). Embedding data analytics and cfd into the digital twin concept. Comput. and Fluids 214, 104759. doi:10.1016/j.compfluid.2020.104759
  38. Olson G., Liu Z. (2023). Genomic materials design: calculation of phase dynamics. Calphad 82, 102590. doi:10.1016/j.calphad.2023.102590
  39. Papacharalampopoulos A., Foteinopoulos P., Karagianni O. M., Stavropoulos P. (2025). A Human–ai collaborative framework for additive manufacturing modeling and decision-making. Processes 13, 3877. doi:10.3390/pr13123877
  40. Peters B., Besseron X., Adhav P. (2024). “ Unprecedented insight into the thermal processing of a blast furnace,” in Selected papers from the 15th international conference on industrial applications of computational fluid dynamics (SINTEF Academic Press).
  41. Rabier F. (2005). Overview of global data assimilation developments in numerical weather-prediction centres. Q. J. R. Meteorological Soc. 131, 3215 – 3233. doi:10.1256/qj.05.129
  42. Rasouli A., Herstad K. E., Safarian J., Tranell G. (2022). Magnesiothermic reduction of natural quartz. Metallurgical Mater. Trans. B 53, 2132 – 2142. doi:10.1007/s11663-022-02513-6
  43. Ray C., Sahoo P., Rao S. (2007). “ Electrode management – investigation into soft breaks at 48mva fecr closed furnace,” in Proceedings of INFACON XI, 741 – 751.
  44. Ringdalen E., Eilertsen J. (2001). “ Excavation of a 54 mya hc-ferrochromium furnace,” in Proceedings of INFACON IX, 166 – 173.
  45. Schultmann F., Engels B., Rentz O. (2004). Flowsheeting-based simulation of recycling concepts in the metal industry. J. Clean. Prod. 12, 737 – 751. doi:10.1016/S0959-6526(03)00050-7
  46. Schwarz M. P., Feng Y. (2014). “ Pragmatic cfd modelling approaches to complex multiphase processes,” in Selected papers from 10th international conference on computational fluid dynamics in the oil and gas, metallurgical and process industries (SINTEF Academic Press), 25 – 37.
  47. Skjeldestad A. (2019). Electrode paste behaviour in søderberg electrodes in ferroalloys production. Thesis: NTNU, 325.
  48. Solheim A. (2022). “ Reflections on the low-voltage anode effect in aluminium electrolysis cells,” in Light metals 2022. Editor Eskin D. (Cham: Springer International Publishing), 971 – 978.
  49. Solheim A., Skybakmoen E. (2020). “ Mass- and heat transfer during dissolution of alumina,” in Light metals 2020. Editor Tomsett A. (Cham: Springer International Publishing), 664 – 671.
  50. Stavropoulos P. (2022). Digitization of manufacturing processes: from sensing to twining. Technologies 10, 98. doi:10.3390/technologies10050098
  51. Vachaparambil K. J., Johansen S. T., Solheim A., Einarsrud K. E. (2023). “ A pragmatic model for bath temperature evolution during alumina feeding,” in Light metals 2023. Editor Broek S. (Cham: Springer Nature Switzerland), 113 – 120.
  52. Vachaparambil K. J., Johansen S. T., Larsen S. Y., Kadkhodabeigi M., Pettersen T. (2024). “ A novel electrode model for søderberg electrodes,” in Materials processing fundamentals 2024. Editors Wagstaff S., Anderson A., Sabau A. S., Iloeje C. (Cham: Springer Nature Switzerland), 21 – 30.
  53. Vachaparambil K. J., Johansen S. T., Soncini R. M., Solheim A., Einarsrud K. E., Elstad K. R. (2025). Modelling alumina feeding and transport in an industrial aluminium reduction cell using a pragmatic computational model. Metallurgical Mater. Trans. B 56, 1272 – 1288. doi:10.1007/s11663-024-03360-3
  54. Viertauer A., Mutsam N., Pernkopf F., Gantner A., Grimm G., Winkler W. (2019). “ Refractory condition monitoring and lifetime prognosis for rh degasser,” in AISTech 2019 — proceedings of the iron and steel technology conference, 1081 – 1089.
  55. Wertime T. A. (1964). Man’s first encounters with metallurgy. Science 146, 1257 – 1267. doi:10.1126/science.146.3649.1257
  56. White J. F., Rigas K., Peter Andersson S., Glaser B. (2020). Thermal properties of söderberg electrode materials. Metallurgical Mater. Trans. B 51, 1928 – 1932. doi:10.1007/s11663-020-01890-0
  57. Yin R.-y. (2021). Review on the study of metallurgical process engineering. Int. J. Minerals, Metallurgy Mater. 28, 1253 – 1263. doi:10.1007/s12613-020-2220-z
  58. Zhan S., Li M., Zhou J., Yang J., Zhou Y. (2014). Cfd simulation of dissolution process of alumina in an aluminum reduction cell with two-particle phase population balance model. Appl. Therm. Eng. 73, 805 – 818. doi:10.1016/j.applthermaleng.2014.08.040
  59. Zoric J., Johansen S. T., Einarsrud K. E., Solheim A. (2014). “ On pragmatism in industrial modeling,” in Selected papers from 10th international conference on computational fluid dynamics in the oil and gas, metallurgical and process industries (SINTEF Academic Press), 9 – 24.

Citation

Vachaparambil, K. J., et al. (2026). Application of pragmatic modelling to metallurgical processes. Frontiers in Chemical Engineering, Volume 8. https://doi.org/
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8 - 2026

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Copyright © 2026 Vachaparambil, Johansen and Løvfall.

License: https://creativecommons.org/licenses/by/4.0/

*Correspondence: Kurian J. Vachaparambil

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