A kinetics-based mathematical model of the Peirce-Smith converter has been developed. The model considers mass transfer, heat transfer, and reactions between each of the phases present in the converter. Model validation is carried out using industrial data obtained from both copper and nickel converters. The model is generally able to predict the temperature and compositional variations of the converters to within the errors of the industrial data. However, the interactions between the white metal and slag during the copper blow are not understood sufficiently to model well.

A kinetic model of the peirce-smith converter: Part I. Model formulation and validation

RESUME : Le convertissage par methode Pierce-Smith (PS) est l'etape cle de la production de cuivre et du nickel. Cette operation se poursuit par des etapes sequentielles et presentent un cas ideal pour la programmation mathematique. Ce travail demontre les complexites thermochimiques et les etapes du convertissage au moyen d'un programme lineaire (PL) en nombres entiers mixtes. Ceci est la premiere fois que le convertissage PS est aborde dans un cadre de programmation mathematique et represente un avancement majeur de l'application de la recherche operationnelle aux etapes de production des fonderies de cuivre et de nickel. Les resultats demontrent que le cadre mathematique est fonctionnel, et peut etre utilise quotidiennement pour la gestion optimale des sequences d'operation de l'elaboration de cuivre et de nickel. Le cadre est flexible quant a la definition des contraintes du systeme et de la fonction objective. Cette flexibilite evoque la formulation de divers modes d'operation des fonderies. Le cadre pourra etre exploite en forme de logiciel industriel que les fonderies pourraient utiliser pour coordonner la production journaliere, et de varier leur mode d'operation selon les conditions de l'usine et du marche.
Le cadre a ete formule suivant une methodologie qui est typique de la programmation mathematique, mais qui n'avait jamais ete adaptee au convertissage PS. Premierement, le probleme se pose en forme generale. En effet, Le probleme de convertissage consiste de la coordination des convertisseurs PS avec d'autres operations dans la fonderie afin de maximiser la production durant une periode fixe, tout en respectant les contraintes chimiques, volumetriques et thermiques. Deuxiemement, les diverses composantes et dimensions du systeme sont representees par des structures algebriques generales; c'est-a-dire, des ensembles, des parametres et des variables. Troisiemement, ces composantes sont liees de telle maniere a ce que la formulation puisse etre supportee par des techniques de resolution. Les techniques de resolution par la programmation lineaire (PL) en nombres entiers mixtes sont bien etablies. Par contre, il a ete necessaire d'introduire des simplifications pour pouvoir resoudre le probleme des convertissages par l'adoption d'un cadre hypothetique de PL en nombres entiers mixtes. En considerant la vaste gamme de problemes qui ont ete deja abordees dans ce type de cadre, il semblait raisonnable que le convertissage PS necessiterait seulement des simplifications mineures. En fait, le nouveau cadre mathematique exige certaines simplifications. En particulier, il impose une rigidite artificielle sur deux temperatures variables : la temperature nominale d'effluent gazeux, et la temperature d'ecremage de la scorie. Ces simplifications peuvent etre considerees mineures, puisqu'il existe des logiciels qui ne traitent meme pas ces quantites comme variables. En outre, il semble maintenant plausible de formuler un cadre non lineaire qui fournirait un traitement plus robuste et realiste qui tient compte de ces temperatures. Les resultats de cette etude ont un attrait considerable pour les industries de cuivre et de nickel, puisque l'operation efficace d'une fonderie depend directement de la coordination quotidienne de la production. Suivant les succes du travail actuel, des efforts superficiels provoqueront des changements majeurs dans les operations journalieres des fonderies de cuivre et de nickel.----------ABSTRACT : Peirce-Smith (PS) converting is central to the production of copper and nickel, and is a lucrative, yet previously undeveloped, context for mathematical programming. The thermochemical complexities of PS converting have now been represented within a mixed-integer linear program (MILP). This is the first time that PS converting has been treated within a mathematical programming framework, hence a major advancement in the operations research of copper and nickel smelters.
The MILP framework is now functional, and can be used to construct optimal daily production schedules. The framework offers flexibility in the definition of system constraints and objective functions. This flexibility can accommodate the formulation of alternative modes of operation for smelters. The MILP framework can now be marketed as industrial software, to produce optimal daily schedules, and allow smelters to change their mode of operation in accordance to plant and market conditions.
The framework has been created using a methodology that is typical of mathematical programming, but which had never been adapted to Peirce-Smith converting. Firstly, the problem has been posed in appropriately general terms; indeed, the PS Converter Problem is to coordinate Peirce-Smith converters with other objects in the smelter, so as to maximize production within a fixed period of time, while respecting chemical, volumetric and thermal constraints. Secondly, the various components and dimensions of the system have been represented using general algebraic structures, such as sets, parameters and variables. Thirdly, these components have been related to each other in a manner that can be supported by solution techniques.
The solution techniques for MILP are well established. However, it was initially unclear what degree of simplification would be required in order to fit the PS Converter Problem into a hypothetical MILP framework. Given the vast scope of problems that have already been treated using MILP, it seemed plausible that the PS Converter Problem would require only minor simplifications.
The new MILP framework has indeed required some simplification. More precisely, the framework imposes an artificial rigidity on two classes of temperature variables: nominal offgas temperatures, and skimming temperatures. These simplifications can be considered minor, since there are existing software tools that do not even treat these quantities as variables. Moreover, it now seems plausible to extend the current formulation into a nonlinear framework that would provide a more intensive and realistic treatment of the offgas and skimming temperatures.
The results of this study have a considerable appeal to the copper and nickel industries, since the efficient operation of a smelter is directly linked to its scheduling practice. Following the successes of the current work, superficial efforts will cause major changes in the daily operations of copper and nickel smelters.

Mathematical Programming of Peirce-Smith Converting

A thermodynamic model has been developed to predict the distribution behavior of Ni, Cu, Co, Fe, S, As, Sb, and Bi in the Outokumpu flash-smelting process, the Outokumpu direct high-grade matte smelting process, and the INCO flash-smelting process. In this model, as many as 16 elements (Ni, Cu, Co, Fe, As, Sb, Bi, S, O, Al, Ca, Mg, Si, N, C, and H) are considered, and two nickel sulfide species are used to allow for modeling of sulfur-deficient mattes. The compositions of the matte, slag, and gaseous phases in equilibrium are calculated using Gibbs free energies of formation and the activity coefficients of the components derived from the experimental data. The model predictions are compared with the known industrial data from the Kalgoorlie Nickel Smelter (Kalgoorlie, Australia), the Outokumpu Harjavalta Nickel Smelter (Harjavalta, Finland), the INCO Metals Company (Sudbury, Canada), and from a number of experimental data. An excellent agreement is obtained. It was found that the distribution behaviors of Ni, Co, Cu, Fe, S, As, Sb, and Bi in the nickel smelting furnace depend on process parameters such as the smelting temperature, matte grade, and partial pressure of oxygen in the process.

A Thermodynamic Model of Nickel Smelting and Direct High-Grade Nickel Matte Smelting Processes: Part I. Model Development and Validation

A sensitivity analysis of a kinetics-based model of the Peirce-Smith converter has been carried out, and the model has then been applied to an analysis of copper converter operation. The results of the sensitivity analysis indicate that only factors relating to the mass-transfer rates have a significant effect on the model predictions. However, even with large changes in diffusivities, the model predictions remain within the error of the plant measurements. The converter analysis indicates that considerable improvements to converter productivity can be made, particularly through changes to gas injection practices.

A kinetic model of the Peirce-Smith converter: Part II. Model application and discussion

In Part I[1] of this article, a mathematical model of the nickel converter was developed based on the assumption that the three phases in the converter vessel are in chemical and thermal equilibrium. In this article, a sensitivity analysis of the model is conducted to assess the uncertainties in model predictions and to delineate the critical process variables. Following from the equilibrium assumption, the rate-controlling process in the operation is the rate at which oxygen can be supplied to the matte. Routes to process improvement then lie in increased blast volumes or oxygen enrichment and improved operating procedures to reduce idle time. Temperature was not seen to exert a significant chemical effect on the process. More important to matte composition were variables such as ladle volume and cold charge composition. These variables directly affect either the matte volume or the overall bath composition and so are more important than those variables which have only an indirect effect through changing the conditions of equilibrium. An analysis of heat distribution in the furnace indicates the overwhelming fraction of heat removed by the off-gas.

A mathematical model of the nickel converter: Part II. Application and analysis of converter operation

A mathematical model of the nickel converter has been developed. The primary assumption of the model is that the three phases in the converter are in thermal and chemical equilibrium. All matte, slag, and gas in the converter is brought to equilibrium at the end of each of a series of short time steps throughout an entire charge. An empirical model of both the matte and slag is used to characterize the activity coefficients in each phase. Two nickel sulfide species were used to allow for the modeling of sulfur-deficient mattes. A heat balance is carried out over each time step, considering the major heat flows in the converter. The model was validated by a detailed comparison with measured data from six industrial charges. The overall predicted mass balance was shown to be close to that seen in actual practice, and the heat balance gave a good fit of converter temperature up to the last two or three blows of a charge. At this point, reactions in the converter begin to deviate strongly from “equilibrium,” probably due to the converter reactions coming under liquid-phase mass-transfer control. While the equilibrium assumption does work, it is not strictly valid, and the majority of the charge is probably under gas-phase mass-transfer control.

A mathematical model of the nickel converter: Part I. Model development and verification

A mathematical model of the growth of accretions on submerged single pipes has been developed based on the heat transfer in the vicinity of the injection point. The model is able to predict the formation of either pipe or hemispherical accretions based only on the input con-ditions. It is assumed that a hemispherical accretion is formed only when a porous plug is formed within the tuyere tip by the freezing of bath material during regular penetration episodes. Gas flow within the bath is calculated from physical modeling results, and allowances are made for the possibility of a gas/bath reaction. A preliminary analysis of the model predictions for copper converter and steel ladle stirring applications has been carried out. The model predicts that, under normal operating conditions, the accretions formed in copper converters are very small but can become appreciable under low superheat conditions. The time required to reach steady state depends upon the final accretion length but can be as much as 5 minutes. In ladle stirring applications, the model predicts that the accretion formed will be approximately 4.5-cm high with a slightly larger basal radius. This accretion requires approximately 10 minutes to reach steady state. The final shape and size of an accretion are dependent upon the balance of heat flows around it, with steady state occurring when heat inputs to the accretion exactly equal heat outputs.

Accretion growth on single-pipe tuyeres: Part I. Model development and preliminary analysis

A mathematical model of accretion growth on single-pipe injectors has been used to analyze the process of accretion growth during copper converting and inert gas stirring of steel. A sen-sitivity analysis of the model developed in Part I of this article revealed that the critical unknown factor controlling the growth process is the local bath velocity. It is likely that the asymmetric forms seen in accretions are due primarily to nonuniformity in bath flow around the accretion. The important factors controlling the size of accretions in copper converting are bath superheat, gas flow rate, and oxygen content. Blockage of the tuyeres in the converter is a function of wall and bath superheat and likely is a result of the freezing of matte in the pipe. The effect of gas-flow rate on accretion size is clearly shown in the case of inert gas injection into steel. The transition between the different accretion types found on single-pipe tuyeres, pipes, porous pipes, and hemispheres, depends on events at the tuyere-pipe tip. Conditions of high back-attack frequency, low superheat, and small tuyere diameters tend to lead to accretions forming over the tip. If this takes place at high injection pressures and with a relatively ductile material, a “porous plug” will form, on which a hemisphere will grow. Under other conditions, it may result in a blockage. The presence of accretions at a tuyere has a strong influence on the thermal profile in the surrounding refractory and tuyere pipe. Steep thermal gradients near the tip in both regions will likely lead to refractory deterioration and pipe distortion.

A. K. Kyllo

Papers

A mathematical model of the nickel converter: Part I. Model development and verification

A mathematical model of the nickel converter: Part II. Application and analysis of converter operation

Accretion growth on single-pipe tuyeres: Part I. Model development and preliminary analysis

Accretion growth on single-pipe tuyeres: Part II. Model results and discussion