A heat and mass transfer problem for the dissolution of an alumina particle in a cryolite bath

doi:10.1016/j.ijheatmasstransfer.2020.120232

International Journal of Heat and Mass Transfer

Volume 162, December 2020, 120232

https://doi.org/10.1016/j.ijheatmasstransfer.2020.120232 Get rights and content

Highlights

•
Formulation of a mathematical model for an initially cold alumina particle in a bath of molten cryolite.
•
The early-time, small-superheat, and small Stefan number limits of the model are investigated using asymptotic analysis.
•
Predictions for the freezing, melting and dissolution time and the physical extent of the temperature disturbance are presented.

Abstract

We investigate the spherically symmetric dissolution of an initially cold alumina particle in a bath of molten cryolite. The cryolite initially freezes on the particle, forming a shell that must melt before the particle can dissolve. We derive asymptotic solutions valid in the limits of small-superheat and of small Stefan number. In the small-superheat limit, the evolution of the boundary exhibits a two-scale behaviour. In the small Stefan number limit, we find that the behaviour of a particle could be limited by either the dissolution (in the case where the temperature differences are small) or by heat transfer (when the latent heat is large and the temperature gradients are large). Our asymptotic predictions are validated by a front fixing numerical scheme that we initiate using the early-time asymptotics.

Introduction

Since its introduction in 1886, the Hall-Héroult process has made aluminium metal into a commodity product. In this continuous process, alumina is dissolved in a cryolite bath from which aluminium is produced by electrolysis [1]. Key economic factors are the high energy consumption and the competitive price of aluminium – the process must be run continuously and efficiently. To achieve a stable process, the alumina concentration has to be kept in a narrow band, which requires a good alumina feeding strategy and rapid dissolution in the bath. In recent years, the electrolyte volume has decreased (because of reduced anode-cathode distance and larger anodes are being used), and the trend has been to increase the amount of aluminium produced per unit time. A deeper and more fundamental understanding of the feeding and dissolution process is needed if further improvements are to be made to the process efficiency.

The industrial process [2] involves several complex steps including the way the alumina is fed into the bath, and the thermochemical reactions between the alumina and the bath. Feeding is typically carried out by breaking the crust that forms at the top of the cell and then pouring alumina as a granular material into the bath, where ideally it sinks and dissolves. However, experimental research [3], [4], [5], [6], [7] on both the laboratory and industrial scale indicate that, in some circumstances, the alumina particles may form a floating structure of undissolved alumina at the surface of the bath. These rafts may remain afloat for much longer times than the feeding cycle [7], ultimately disrupting the process.

Experimental research [8], [9] on the feeding and dissolution process investigates, for example, how the particle dissolution time depends on the bath concentration, the size of the alumina particles, and the initial heating of the particles. Haverkamp and Welch [10] include the effect of stirring of the cryolite bath which accelerates the dissolution by dispersing the alumina particles; this is confirmed by simulations of a mathematical model for the alumina concentration in the bath, and by small sized experiments. An extensive description of various experimental factors influencing the feeding has been covered in a review [2].

Since experiments using molten cryolite are extremely challenging, advanced numerical simulations have proven to be a viable supplement to experiments. For example, Hofer [11] uses a finite element formulation for a model for the Hall-Héroult cell in which the fluid velocity and pressure, the concentration of dissolved alumina, and the position and sizes of the alumina particles are tracked, in order to determine the influence of the flow on the dispersion of both the alumina particles and the dissolved alumina. The finite element framework has also been used by other authors to investigate various aspects of the Hall-Héroult cell [12], [13], [14]. In order to be computationally tractable, all these simulations rely upon sub-grid models which in turn depend upon closure laws formulated on a rigorous mathematical basis.

In contrast to previous experimental studies, which focus on the dissolution of clumps of alumina particles, we consider the behaviour of a single particle since this will be an important ingredient in the closure laws for bath-level simulations. The low temperature of the feedstock compared to the bath, which is maintained only slightly above the melting point to save energy, leads to three distinct process stages in the single particle case: first, some of the bath freezes into a shell around the particle; second, this shell melts; third, after the shell has vanished, the alumina particle dissolves into the cryolite. The first two stages are heat-transfer-driven phase changes and are observed in experiments in which porous alumina lumps are inserted into cryolite [3].

Moving boundary problems are ubiquitous in many disciplines such as materials science, geosciences, biology, or even finance. There is a considerable amount of mathematical literature on spherically symmetric moving boundary problems dealing with bubble nucleation [15], bubble dissolution [16], or mass-transfer-controlled dissolution of an isolated sphere [17]. Due to its rich mathematical structure, the inward solidification of a suddenly cooled liquid sphere has been extensively studied for example in [18], [19], [20], [21], [22], [23], [24]. The analysis in [18] breaks down as the freezing front approaches the centre of the sphere, but this issue was rectified in [19], and improved in [20] by considering the region close to the centre of the sphere when freezing is nearly complete. Similar issues arise in solidification problems in a cylindrical geometry, which was analysed in [22], and for an N-dimensional sphere in [21]. In contrast, the previously mentioned models, which are all one-phase, [23], [24] included temperature evolution in both phases, leading to a two-phase Stefan problem. Furthermore, [25] investigated the integral forms of the equations to calculate bounds for the evolution of the radius. Additionally, the coupling of heat transfer and fluid flow is investigated in [26] by analysing the behaviour of a hot fluid flowing over a cold solid plane.

The paper most relevant to our analysis is [27] in which the problem of a metal particle immersed in its melt is considered. They formulate a two-phase Stefan problem similar to our model, with the difference that the contribution of the melt is lumped into the boundary condition using Newton’s law of cooling with a constant convective coefficient. We will comment on the differences and similarities as they arise. Our aim in this paper is to formulate and solve a mathematical model to describe the freezing, melting and dissolution process for a single alumina particle, which is an essential building block for understanding alumina feeding and distribution in the Hall-Héroult process.

We will present the mathematical model in Section 2. We then nondimensionalise and analyse the thermal problem in Section 3 and present the analysis for the dissolution problem in Section 4. Finally, we summarise the results and present the conclusions in Section 5.

Section snippets

Mathematical model

We consider a spherically symmetric geometry in which a solid spherical alumina particle is immersed in a stationary cryolite bath. We suppose that the initial temperature, $T_{p}^{*}$ , of the particle is much lower than the liquidus temperature, T_m, of the cryolite. The cryolite is initially at a temperature $T_{c}^{*}$ , which is slightly higher than the liquidus temperature. Thus, a frozen shell of cryolite will form on the surface of the particle, which will then melt away as the temperature equilibrates.

Nondimensionalisation

We nondimensionalise (2.1a), (2.2), (2.3), (2.4a), (2.5a) using the scalings $r = a \tilde{r}, t = \frac{ρ_{p} c_{p} a^{2}}{k_{p}} \tilde{t}, R = a \tilde{R}, T_{i} = T_{p}^{*} + (T_{c}^{*} - T_{p}^{*}) {\tilde{T}}_{i} .$ We note that the temperature scale is chosen so that the dimensionless temperature varies between 0 and 1 initially, while the time scale is chosen to be the diffusive time scale inside the particle. Dropping the tildes, the heat equations in (2.1) become $\frac{\partial T_{p}}{\partial t} = \frac{1}{r} \frac{\partial^{2}}{\partial r^{2}} (r T_{p}) in 0 < r < 1,$ $\frac{\partial T_{s}}{\partial t} = \frac{κ_{2}}{r} \frac{\partial^{2}}{\partial r^{2}} (r T_{s}) in 1 < r < R (t),$ $\frac{\partial T_{c}}{\partial t} = \frac{κ_{3}}{r} \frac{\partial^{2}}{\partial r^{2}} (r T_{c}) in r > R (t) .$

The boundary and interface conditions

Nondimensionalisation

Since we assume that dissolution can only start after complete melting of the frozen shell, we nondimensionalise¹ the system (2.6), (2.10) and (2.14), (2.15a), (2.16) using the scalings $r = a \hat{r}, R = a \hat{R}, t - t_{M} = \frac{a^{2}}{D σ} τ, C = C_{f} +$

Summary of scales

In Fig. 13, we plot t_F, t_M and t_D given by (3.51), (3.52) and (4.14) as a function of $β θ_{m} \sqrt{κ_{3}} / ν (1 - θ_{m}),$ where δ is defined in (3.54). There are two qualitatively different regimes. Below the freeze limit (shown by the vertical dot-dashed line given by (3.11)) the particle simply dissolves, since the heat transfer in the cryolite is fast enough that the cryolite temperature remains above the freezing temperature and no freezing occurs. However, above this limit freezing, melting, and dissolution

CRediT authorship contribution statement

A. Kovács: Formal analysis, Software, Visualization, Validation, Writing - original draft. C.J.W. Breward: Writing - review & editing, Supervision, Conceptualization, Methodology, Funding acquisition. K.E. Einarsrud: Writing - review & editing, Conceptualization, Methodology, Supervision. S.A. Halvorsen: Writing - review & editing, Conceptualization, Supervision, Project administration. E. Nordgård-Hansen: Writing - review & editing, Conceptualization, Supervision, Funding acquisition. E.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This publication is based on work supported by the EPSRC Centre For Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with Hydro Aluminium, NORCE and NTNU. Furthermore, this work is partly funded by SFI Metal Production, Centre for Research-based Innovation, 237738. Financial support from the Research Council of Norway and the partners of SFI Metal Production is gratefully acknowledged.

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A heat and mass transfer problem for the dissolution of an alumina particle in a cryolite bath

Highlights

Abstract

Introduction

Section snippets

Mathematical model

Nondimensionalisation

Nondimensionalisation

Summary of scales

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Chem. Eng. Process.

Appl Math Model

Chem Eng Sci

Int J Heat Mass Transf

Int J Heat Mass Transf

Int J Heat Mass Transf

Aluminium electrolysis. Fundamentals of the Hall-Héroult process

A review of alumina feeding and dissolution factors in aluminum reduction cells

Metallurgical and Materials Transactions B

Alumina in Aluminum Smelting and its Behaviour After Addition to Cryolite-based Electrolytes

Study on the Dissolution of Alumina in Cryolite Electrolyte Using the See-Through Cell

Light Metals

Spreading of alumina and raft formation on the surface of cryolitic bath

Light Metals

Alumina feeding and raft formation: Raft collection and process parameters

Lab scale experiments on alumina raft formation

Dissolution of alumina in molten cryolite

Metallurgical and Materials Transactions B

The dissolution of alumina in cryolite melts

Light Metals

Numerical Simulation and Optimization of the Alumina Distribution in an Aluminium Electrolysis Pot

Numerical simulation of alumina-mixing process with a multicomponent flow model coupled with electromagnetic forces in aluminum reduction cells

Jom

Dynamic Modelling of Alumina Feeding in an Aluminium Electrolysis Cell