Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

Sloman, Benjamin M.; Please, Colin P.; Gorder, Robert A. Van

doi:10.1017/jfm.2020.727

Cited by 4 publications

(3 citation statements)

References 68 publications

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“…The dripping of material is included in the furnace model of [1], at a 'dripping-rate' dependent on the local temperature. In [48], the melting and flow of the charge material are studied, with the charge modelled as a single fluid, with temperature-dependent viscosity, and the free-boundary is defined as the isotherm where the material reaches the 'dripping temperature'. Rather than modelling the flow of the material, or prescribing the temperature of the surface r = s(t), we choose instead to define the boundary r = s(t) to be the point where enough of the solid material has been reacted away that the structure loses integrity, and any remaining material collapses into the crater.…”

Section: Boundary Conditions For the Mass Flow Problem In The Charge Materialsmentioning

confidence: 99%

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

Luckins¹,

Oliver²,

Please³

et al. 2021

Eur. J. Appl. Math

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Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

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Section: Boundary Conditions For the Mass Flow Problem In The Charge Materialsmentioning

confidence: 99%

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

Luckins¹,

Oliver²,

Please³

et al. 2021

Eur. J. Appl. Math

Self Cite

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show abstract

“…For instance, when designing a kiln for the production of rutile titanium dioxide, the ‘kiln length’ must be determined so that the chemical reactions have completed by the time the reactant material reaches the end of the kiln (Agrawal & Ghoshdastidar 2017), else the process will fail. There is a similar free boundary at the edge of the reacting material bed within a silicon furnace, where the material collapses or drips into the craters which form beneath the electrodes in the furnace (Andresen 1995; Sloman, Please & Van Gorder 2020).…”

Section: Introductionmentioning

confidence: 99%

Modelling and analysis of an endothermic reacting counter-current flow

Luckins

Oliver²,

Please³

et al. 2022

J. Fluid Mech.

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We study the endothermic reaction and flow of a granular solid reactant, where energy for the reaction is provided by a counter-current flow of hot gases through the porous reactant bed. Research into reacting flows typically focusses on exothermic combustion processes. However, endothermic processes are common in the metallurgy industry, including the production of cement, silicon and rutile titanium dioxide. Several common features are observed in experimental and numerical studies of these processes, including critical temperatures of the reactant at which the chemical reaction begins, and regions of the reactor with uniform reactant temperature. Motivated specifically by the processes in a silicon furnace, we analyse a model of endothermic, reacting counter-current flow using the method of matched asymptotic expansions. Assuming the Péclet number in the solid is large, we explore the full range of values for the dimensionless inter-phase heat-transfer rate, finding six distinguished limits. In all limits, we find a diffusive boundary layer in which there is a fast chemical reaction rate due to the high temperatures, analogous to exothermic flame fronts. Outside this region, the counter-current flow is crucial to the chemical processes. For intermediate values of the heat-transfer rate, we find the same qualitative properties as those observed across the metallurgy industry, and we quantify the dependence of these properties on the flow rate and heat-transfer rate. In the limit of large heat-transfer coefficient, we derive the single-temperature limit, in which the solution structure is dependent on the direction of net heat flux through the domain.

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“…On the slower timescale, there are many models in the silicon-production literature for the chemical reactions, heat transfer and motion of the raw materials in the furnace, ranging from large-scale CFD simulations (Andresen, 1995) to mathematical analysis of the continuum equations for the heat and mass transfer (Sloman et al, 2017(Sloman et al, , 2018(Sloman et al, , 2020.…”

Section: Introductionmentioning

confidence: 99%

Modelling alternating current effects in a submerged arc furnace

Luckins

Oliver

Please

et al. 2022

IMA Journal of Applied Mathematics

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Modelling the production of silicon in a submerged arc furnace (SAF) requires accounting for the wide range of timescales of the different physical and chemical processes: the electric current which is used to heat the furnace varies over a timescale of around $10^{-2}\,$ s, whereas the flow and chemical consumption of the raw materials in the furnace occurs over several hours. Models for the silicon furnace generally either include only the fast-timescale, or only the slow-timescale processes. In a prior work, we developed a model incorporating effects on both the fast and slow timescales, and used a multiple-timescales analysis to homogenise the fast variations, deriving an averaged model for the slow evolution of the raw materials. For simplicity, in the previous work we focussed on the electrical behaviour around the base of a single electrode, and prescribed the current in this electrode to be sinusoidal, with given amplitude. In this paper, we extend our previous analysis to include the full electrical system, modelled using an equivalent circuit system. In this way, we demonstrate how the two furnace-modelling approaches (on the fast and slow timescales) may be combined in a computationally efficient way. Our previously derived model for the arc resistance is based on the assumption that the dominant heat loss from the arc is by radiation (we will refer to this as the radiation model). Alternative arc models include the empirical Cassie and Mayr models, which are commonly used in the SAF literature. We compare these various arc models, explore the dependence of the solution of our model on the model parameters and compare our solutions with measurements from an operational silicon furnace. In particular, we show that only the radiation arc model has a rising current-voltage characteristic at high currents. Simulations of the model show that there is an upper limit on the length of the furnace arc, above which all the current bypasses the arc and flows through the surrounding material.

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Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

Abstract: Abstract

Cited by 4 publications

References 68 publications

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

Modelling and analysis of an endothermic reacting counter-current flow

Modelling alternating current effects in a submerged arc furnace

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