Modelling of Coupled Mass and Thermal Balances in Hall-Heroult Cells During Anode Change

A spatially-discretised mass balance model, thermal balance model, and cell voltage model must be integrated together to capture the strong coupling effects between process variables, which are mostly left out of previous spatial models. Some important considerations include:

• Alumina consumed during the reaction is replenished by feeders in the cell. Each dump of alumina feed absorbs energy from the electrolyte for preheating and dissolution.
• The freeze (anode freeze and side ledge) can be assumed to be pure cryolite. Therefore, as they form from or dissolve back into the electrolyte, the composition of electrolyte changes.
• Due to the formation or dissolution of freeze, the height of electrolyte changes, which changes the area for heat transfer and therefore affects the thermal balance. The change in electrolyte height depends on the difference between cryolite and electrolyte densities, as newly-formed freeze displaces the electrolyte and vice versa.
• Change in electrolyte composition affects the liquidus temperature, which helps drive the freeze dynamics. The side ledge insulates the cell from further heat loss, so changes in ledge thickness further affects the thermal balances.
• Presence of freeze underneath a new anode acts as additional path resistance which impedes current flow, which has to be redistributed to the other anodes. As the freeze dissolves, more anode area will be exposed and path resistance will decrease and restore to previous value.
• The dissolution of freeze underneath a new anode reduces the local electrolyte temperature, and hence the heat transfer out the sidewall in that zone. This causes a growth in the local side ledge thickness.
• The redistribution of anode current causes the local alumina and carbon anode to be consumed at different rates as per Faraday's law of electrolysis. This changes the mass composition as well as the anode-to-cathode distance (ACD).
• Changes in electrolyte composition, anode height, or ACD due to material consumption changes the total path impedance, which causes anode current to redistribute and cell voltage to change. Both of these consequently affects the heat generation.
• The temperature of the electrolyte also affects cell voltage, which affect heat generation, freeze dissolution, and ultimately electrolyte composition.

The measurement of individual anode currents in a cell is unconventional, but has shown great benefits for cell monitoring and control. ^24–34 As the heat generation in a cell is primarily ohmic, the anode current distribution readily represents the spatial distribution of energy in a cell. Therefore, similar to the work first presented by Cheung, ²⁶ regions in the cell are discretised according to anode locations as shown in Fig. 2 in order to capture the dynamics of local thermal variables during anode change.

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The vertical layers of electrolyte and aluminum metal are not further discretised in this approach as the magnitude of anode current already reflects the total interelectrode path impedance due to electrolyte resistance, bubble layer resistance, as well as overpotentials in the electrochemical reaction. ^21,26 Therefore, local heat generation can be easily calculated from the cell voltage and anode current. By eliminating empirical Eq.s. and constants for electrical resistances from the thermal model, this increases the versatility of the thermal model as actual measurements of cell voltage and anode currents can also be used as model inputs, while for simulation studies, cell voltage and anode current distribution can still be calculated as shown in a later section.

The governing thermal equations, as well as the domains they are applicable in, are summarised in Table I, and more details are provided in the following sections. The key model parameters used in this paper are provided in Tables II and III, and they are specific to the cell technology, cell design and process conditions in this work. As the temperature of an anode changes drastically after replacement, its thermal conductivity is provided as an equation in Table II.

Table I. Common governing thermal Eq.s. in the model.

Parameter	Domain	Governing equation
Heat transfer
Conduction	Electrolyte, metal, ledge, sidewall, shell, ledge–sidewall, sidewall–shell, anode–crust, electrolyte–crust, freeze–anode, anode–assembly, bottom insulation–shell.	$Q={kA}\tfrac{({T}_{1}-{T}_{2})}{L}$	[3]
Convection	Anode-electrolyte, electrolyte–ledge, metal–ledge, electrolyte–freeze, metal–freeze, shell–ambient, crust–hood, assembly–hood, rod–ambient.	Q = hA(Ts − T∞)	[4]
Advection	Electrolyte flow	Q = ρ Avc(T1 − T2)	[5]
Thermal resistance
Heat flow	All subsystems.	$Q=\tfrac{{T}_{1}-{T}_{2}}{R}$	[6]
Conduction	Same as conduction above.	$R=\tfrac{L}{{kA}}$	[7]
Convection	Same as convection above.	$R=\tfrac{1}{{hA}}$	[8]
Derivative of state
Temperature	All subsystems.	$\tfrac{{\rm{d}}(\rho {VcT})}{{\rm{d}}t}={Q}_{{in}}-{Q}_{{out}}+{Q}_{{gen}}$	[9]
Mass of frozen electrolyte	Ledge and anode freeze.	$\tfrac{{\rm{d}}({m}_{{fr}})}{{\rm{d}}t}=\tfrac{{Q}_{{in}}-{Q}_{{out}}}{{\rm{\Delta }}{H}_{{fus}}}$	[10]
Mass of electrolyte	Electrolyte.	$\tfrac{{\rm{d}}({m}_{{elect}})}{{\rm{d}}t}=-\tfrac{{\rm{d}}({m}_{{fr}})}{{\rm{d}}t}$	[11]
Other
Faraday's law of electrolysis	Anode consumption, alumina consumption, metal production.	$\dot{m}=\tfrac{{I}_{{an}}M\eta }{{Fz}}$	[12]

Heat generation

The local electrical power input into the cell can be calculated from the anode current, I_an , and cell voltage, V_cell . However, the energy lost due to voltage drop across bus bar external to the cell, V_ext , and the electrochemical energy used for process reactions do not contribute toward cell heating. The latter amounts to about 6.60 kWh/kg Al (see Table IV), based on a net and gross anode carbon consumption that of a modern technology of 0.40 kg/kg Al and 0.50 kg/kg Al respectively. ³⁵ As this model computes the local heat requirement for the feed and anode only when they are introduced into the electrolyte, the energy that is not used for heat generation reduces to 5.294 kWh/kg Al. Thus, the local heat generation rate at each anode, Q_gen , is given as

$$\begin{eqnarray}&&{Q}_{{gen}}=({V}_{{cell}}-{V}_{{ext}}){I}_{{an}}-5.294{\dot{m}}_{{Al}}({I}_{{an}}),\end{eqnarray} \tag{ 131 }$$

where ${\dot{m}}_{{Al}}$ is the rate of metal production as given by Faraday's law of electrolysis (Eq. 12). In the equation, η is the current efficiency, M is the atomic mass of aluminum (26.98u), F is the Faraday's constant (96,485 C/mol), and z is the number of charges transferred (3). A portion of this heat energy is generated in the anode, with a rate of Q_gen,an , given by

$$\begin{eqnarray}&&{Q}_{{gen},{an}}={I}_{{an}}^{2}\displaystyle \frac{{H}_{{an}}{\rho }_{{res},{an}}}{{A}_{{an}}},\end{eqnarray} \tag{ 14 }$$

where H_an is the anode height, ${\rho }_{{res},{an}}$ is the electrical resistivity of anode, and A_an is the bottom surface area of the anode.

Table IV. Breakdown of energy that does not contribute toward maintaining cell temperature.

Energy Use	kWh/kg Al	References
Local heat requirement for feed and anode	1.306
Preheating feed (alumina and impurities)	0.575	36
Dissolution of feed	0.551	37
Preheating anode (carbon and sulfur)	0.180	35,36
Other energy not used for heat generation	5.294
Preheating other materials and impurities	0.100	35
Enabling and completion of reactions	5.194	Difference
Total	6.60	35,36

Other Heat Gain by Anode

The area around an anode is where ohmic heat generation occurs, but the anode also acts as a heat sink through which heat energy from the electrolyte escapes to the atmosphere. Due to the huge vertical thermal gradient between the electrolyte region and the cell exterior, the heat flow is largely one-dimensional. Therefore, further discretisation of the anode is not necessary, and vertical heat flows will be dealt with by using thermal resistances. Apart from the internal heat generation, an anode also receives thermal energy from three other sources:

• When a cold anode is immersed, cryolite freezes out of the electrolyte and forms around the anode exothermically. Every kilogram of cryolite releases 0.57 MJ of heat energy. ³⁷ As the ACD is very small compared to the anode dimensions, it can be assumed that all the electrolyte under the anode freezes.
• Heat transfer by convection from the electrolyte to the anode.
• Heat transfer by conduction from the freeze to the anode.

Figure 3 shows the heat transfer from the electrolyte with temperature T_elect and the freeze with temperature T_fr to the anode with temperature T_an at the top. As two different thermal gradients exist, Kirchhoff's circuit laws are used by applying the electrical-resistance analogy, ³⁸ and the equivalent thermal resistance circuit is represented in Fig. 4a. It can be further reformulated as Fig. 4b, so loops can be drawn and analysed. R_elect−an is the thermal resistance due to convection between electrolyte and anode surface, R_fr is due to conduction from freeze to the anode surface, while R_an is due to conduction along the height of the anode. The formula for these resistances can be found in Table I. The solution to the thermal resistance circuit in Fig. 4b yields

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{{elect}\to {an}}\\ \quad =\displaystyle \frac{{R}_{{fr}}{T}_{{elect}}+{R}_{{an}}{T}_{{elect}}-{R}_{{fr}}{T}_{{an}}-{R}_{{an}}{T}_{{fr}}}{{R}_{{elect}-{an}}{R}_{{fr}}+{R}_{{elect}-{an}}{R}_{{an}}+{R}_{{fr}}{R}_{{an}}},\end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\begin{array}{l}{Q}_{{electfr}\to {an}}\\ \quad =\displaystyle \frac{{R}_{{elect}-{an}}{T}_{{fr}}+{R}_{{fr}}{T}_{{elect}}-{R}_{{elect}-{an}}{T}_{{an}}-{R}_{{fr}}{T}_{{an}}}{{R}_{{elect}-{an}}{R}_{{fr}}+{R}_{{elect}-{an}}{R}_{{an}}+{R}_{{fr}}{R}_{{an}}},\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{Q}_{{fr}\to {an}}={Q}_{{electfr}\to {an}}-{Q}_{{elect}\to {an}},\end{eqnarray} \tag{ 17 }$$

where Q_elect→an and Q_fr→an are respectively the heat transferred to the bottom anode surface from the electrolyte and the freeze, while Q_electfr→an is the total heat transferred up the anode.

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Heat loss by anode

As Fig. 5 shows, the thermal energy lost from the top of the anode through the rod and yoke can be separated into two parts, since a portion of the assembly is hooded inside the cell and the rest is exposed to external potroom air. It is estimated that more than 45% of heat is lost through the top crust and anode, ¹⁹ amongst which 25% is lost through the stub and rod alone. Despite this, the stub and the rod were not considered in previously published spatial models.

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The thermal resistance circuit is represented in Fig. 6a, consisting of a parallel conduction network up the yoke, followed by conduction up the rod to midpoint of the assembly in the hooded environment. Some heat is lost by convection from the surface area of the internal yoke and rod to the hooded environment inside of the cell. The remaining heat energy conducts up the rod and is lost by convection to the ambient potroom air. Similarly, this circuit can be reformulated as Fig. 6b, where the R₁ is the thermal resistance associated with conduction through the yoke and rod in the hooded environment, R₂ is associated with heat convection to the hooded environment, and R₃ is associated with heat conduction up along the rod to the portion outside of the cell, followed by heat convection to the potroom ambient air. The heat loss from the anode, Q_{an→hood,amb} , is therefore given by one of the loops in Fig. 6b, which is

$$\begin{eqnarray}&&{Q}_{{an}\to {hood},{amb}}=\displaystyle \frac{{R}_{2}{T}_{{an}}+{R}_{3}{T}_{{an}}-{R}_{2}{T}_{{amb}}-{R}_{3}{T}_{{hood}}}{{R}_{1}{R}_{2}+{R}_{1}{R}_{3}+{R}_{2}{R}_{3}}.\end{eqnarray} \tag{ 18 }$$

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Additionally, an industrial practice to mitigate carbon anode burnoffs involves dressing the top of the anodes with crushed frozen electrolyte materials. This is modelled with thermal conduction through the crust and convection to the hooded environment. ²⁶ However, anodes that are newly-set are only dressed about 8 h into their service life, depending on operational procedures. Thus, dressing materials or top crusts are not found in new anodes, so are not included in the anode change model.

Heat balance around anode freeze

The freeze is modelled as an electrically-insulating cuboid that forms under a new anode in the entire interelectrode region, as the ACD is small relative to the anode dimensions. This cuboid changes dimensions as it dissolves back into the electrolyte, thereby exposing more surface area underneath the anode for electric conduction. The heat balances around an anode freeze are as follows:

• Vertical heat conduction from the centre of the freeze to the anode surface, Q_fr→an .
• Heat convection from the metal pad below to the freeze, followed by vertical heat conduction to the centre of the freeze.
• The surfaces where the freeze contacts the electrolyte are at liquidus temperature. Heat from the electrolyte reaches this surface by convection, and leaves this surface for the centre of the freeze by conduction. The heat imbalance causes dissolution of freeze as shown in Eq. 10. ²⁶

Therefore, the freeze shrinks only in the four sides that it interfaces with the electrolyte. It does not melt nor dissolve in the metal pad below, as the melting point of cryolite is higher than the operating temperature, and cryolite is not very soluble in aluminum. ³⁹

An experiment was conducted to observe the freeze underneath new anodes. Newly-set anodes were pulled up three h into their service life, and the freeze observed was photographed from various angles for analysis. It was found that the four sides of the freeze dissolve at different rates, as shown by Fig. 7:

• The side facing the centre channel of the cell dissolves the quickest, aided by the extra electrolyte volume in the region and turbulence induced by bubbles from the opposing anode.
• The side facing the back wall of the cell dissolves slowly, as it is closest to the side ledge and experiences a large heat loss.
• In modern cell operations, anodes are set in pairs. The side facing the other new anode will also dissolve slowly due to the low temperature in the region.
• The remaining side that faces an older anode dissolves at a medium rate, owing to heat generation and bubble turbulence from the older anode.

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The dissolution rate also depends on the location of the anodes as well as electrolyte flow patterns. It can be modelled by using different HTC values for each side of the anode freeze.

Other heat balances

Alumina

Feeders in the cell consecutively dump alumina at a frequency determined by the feeding controller. Each alumina dump is assumed to have dispersed among the feeder zone and the eight subsystems surrounding it. A large portion of the alumina powder, r, dissolves quickly in the electrolyte, while the rest forms agglomerates that dissolve at a slower rate. ⁴⁰ The dissolution of alumina in each of the subsystems around the feeding zone can therefore be modelled as a first-order model with ⁴⁰

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}({C}_{{un},1})}{{\rm{d}}t}=-{k}_{{diss},1}{C}_{{un},1}+\displaystyle \frac{r\,{\dot{m}}_{{fed}}}{{m}_{{elect}}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}({C}_{{un},2})}{{\rm{d}}t}=-{k}_{{diss},2}{C}_{{un},2}+\displaystyle \frac{(1-r){\dot{m}}_{{fed}}}{{m}_{{elect}}},\end{eqnarray} \tag{ 20 }$$

where C_un,1 and C_un,2 are the concentrations of solid alumina that dissolves at a rate of k_diss,1 (0.099 s⁻¹) and k_diss,2 (0.005 s⁻¹) respectively, ⁴⁰ while ${\dot{m}}_{{fed}}$ is the mass feeding rate of alumina with portion r (0.75) dissolving quickly. It is assumed that the alumina fed by the point feeder is dispersed among the four surrounding subsystems. In general, the mass balance of dissolved alumina in each subsystem depends on four components:

• Dissolution of alumina in the electrolyte (Eqs. 19 and 20).
• Consumption via process reactions (Eqs. 1 and 2). This is modelled with Faraday's law of electrolysis (Eq. 12).
• Advective mass transfer of alumina, ${\dot{m}}_{{flow}}$, due to bulk flow of electrolyte (Eq. 5). The electrolyte velocities used in this study was provided courtesy of our industrial partner who performed computational fluid dynamics studies. Details of similar calculations can also be found in Feng et al. ⁴¹ and Zhu et al. ⁴² .
• Dilution as electrolyte mass increases due to the dissolution of freeze, or vice versa. This rate of change in electrolyte mass is given by Eq. 11.

Therefore, the mass balance equation for dissolved alumina in each subsystem is given by

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\rm{d}}({C}_{{diss}})}{{\rm{d}}t}={k}_{{diss},1}{C}_{{un},1}+{k}_{{diss},2}{C}_{{un},2}\\ \quad -\displaystyle \frac{{\dot{m}}_{{consum}}({I}_{{an}})}{{m}_{{elect}}}+\displaystyle \frac{{\dot{m}}_{{flow}}}{{m}_{{elect}}}-\displaystyle \frac{{C}_{{diss}}\tfrac{{\rm{d}}({m}_{{elect}})}{{\rm{d}}t}}{{m}_{{elect}}}.\end{array}\end{eqnarray} \tag{ 21 }$$

Fluorides

Apart from alumina and cryolite, other electrolyte components include AlF₃, CaF₂, MgF₂, LiF, and KF. Dissolution models similar to that of alumina can be considered for the fluorides, but the rate of fluoride loss from a cell is significantly less compared to alumina. With proper feeding of aluminum fluoride to compensate the loss of fluorides, it can be regarded that the net loss of fluorides is negligible. Therefore, the model can be simplified by considering only the advective mass transfer due to electrolyte flow and the changes in concentration due to evolution in freeze and ledge profile. Therefore, the mass balance of each fluoride species can be represented with an equation like the one below.

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}({C}_{{\mathrm{AlF}}_{3}})}{{\rm{d}}t}=\displaystyle \frac{{\dot{m}}_{{flow}}}{{m}_{{elect}}}-\displaystyle \frac{{C}_{{\mathrm{AlF}}_{3}}\tfrac{{\rm{d}}({m}_{{elect}})}{{\rm{d}}t}}{{m}_{{elect}}}.\end{eqnarray} \tag{ 22 }$$

As aforementioned, the change in electrolyte composition affects the liquidus temperature. It is an important variable as the difference between electrolyte temperature and liquidus temperature, or superheat, represents the driving force for the formation or dissolution of freeze. The local liquidus temperature can be calculated with the empirical equation below: ⁴³

$$\begin{eqnarray}&&\begin{array}{l}{T}_{{liq}}=1011+0.5{{\rm{C}}}_{{{AlF}}_{3}}-0.13{{\rm{C}}}_{{{AlF}}_{3}}^{2.2}-\displaystyle \frac{3.45{{\rm{C}}}_{{{CaF}}_{2}}}{1+0.0173{{\rm{C}}}_{{{CaF}}_{2}}}\\ \quad +0.124{{\rm{C}}}_{{{CaF}}_{2}}{{\rm{C}}}_{{{AlF}}_{3}}-0.005\,42{\left({{\rm{C}}}_{{{CaF}}_{2}}{{\rm{C}}}_{{{AlF}}_{3}}\right)}^{1.5}\\ \quad -\displaystyle \frac{7.93{{\rm{C}}}_{{diss}}}{1+0.0936{{\rm{C}}}_{{diss}}-0.0017{{\rm{C}}}_{{diss}}^{2}-0.0023{{\rm{C}}}_{{{AlF}}_{3}}{{\rm{C}}}_{{diss}}}\\ \quad -\displaystyle \frac{8.9{{\rm{C}}}_{{LiF}}}{1+0.0047{{\rm{C}}}_{{LiF}}+0.001{{\rm{C}}}_{{{AlF}}_{3}}^{2}}-3.95\left({{\rm{C}}}_{{{MgF}}_{2}}+{{\rm{C}}}_{{KF}}\right)\end{array}\end{eqnarray} \tag{ 23 }$$

where T_liq is in °C and all the concentration terms in this equation are in wt%.

The ACD is an important variable as it represents the interelectrode electrolyte resistance, which partially governs the amount of anode current flowing through the path, and therefore, local heat generation. There are four sources for the dynamics of ACD:

• Carbon anodes are consumed continuously during the reduction process, decreasing the anode height and increasing the ACD.
• Metal is produced and accumulated at the bottom of the cell during the reduction process, decreasing the ACD.
• After an anode change, the anodes and their assemblies, which were at ambient temperature, warm up considerably. Since the anodes are clamped and suspended in place at the rod (Fig. 5), any thermal expansion decreases the ACD.
• Beam movements as determined by a cell controller to maintain cell pseudo-resistance. Beam movements during metal tapping and anode raise events are not included, as these movements are done to maintain an overall ACD.

Therefore, the dynamic Eqs. for the anode height, H_an , and ACD, D, are

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}({H}_{{an}})}{{\rm{d}}t}=-\mathrm{consum}({I}_{{an}})\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}D}{{\rm{d}}t} & = & \mathrm{consum}({I}_{{an}})-\mathrm{accum}(I)\\ & & -\mathrm{expand}({T}_{{an}})+\mathrm{BM}\end{array}\end{eqnarray} \tag{ 25 }$$

where consum(I_an ) and accum(I) are the rate of anode height decrease and the rate of metal height accumulation respectively, which can be obtained from Faraday's law of electrolysis (Eq. 12). BM is the distance of beam moved due to cell control action, and expand(T_an ) is the vertical thermal expansion of the anode with its assembly. The typical thermal expansion formula only applies as an approximation for a linear expansion over a small temperature range. However, for a larger range and a varying thermal expansion coefficient expressed as AT + B, the new length of a material with initial temperature, T₀, and initial length, L₀, can be solved by integration, giving

$$\begin{eqnarray}&&L={L}_{0}\exp \left(\displaystyle \frac{A}{2}({T}_{{an}}^{2}-{T}_{0}^{2})+B({T}_{{an}}-{T}_{0})\right),\end{eqnarray} \tag{ 26 }$$

where the thermal expansion coefficients for aluminum anode rods, steel yokes and stubs, and carbon anodes are provided in Table V. Initial lengths are also provided in the table but they are dependent on cell design.

Finally, work practices also affect how the beam will be moved. During anode change, the beam is usually raised to increase the ACD, such that more heat will be generated to replenish the deficit energy caused by the changing of anodes. ⁴⁶ New anodes are set also with ACDs that are a couple of centimetres higher than the old anodes they replace. This is mainly done in anticipation that the new anodes are consumed less as they are not carrying full current capacity.

As previously explained, the mass and thermal balance model can use actual measurements of cell voltage and anode currents, for applications such as simulating the local conditions of the cell. However, for simulation studies of behaviours of the local thermal variables, it is important to be able to compute the cell voltage as well as the anode current distribution.

The detailed and well-established cell voltage Eqs. have been published elsewhere, ^32,47 so will not be repeated here for brevity. In short, cell voltage consists of the external voltage drop V_ext , reversible potential E_E , overpotentials arising from the alumina concentration gradient and the electrode surface reactions η_over , as well as the ohmic voltage drop due to electrolyte resistance R_elect , bubble resistance R_bub , anode resistance R_an , and cathode resistance R_ca . The general equation for cell voltage is given by

$$\begin{eqnarray}&&\begin{array}{l}{V}_{{cell}}(I,D,{T}_{{elect}},C,{H}_{{an}})={V}_{{ext}}+{E}_{E}(C)\\ \quad +{\eta }_{{over}}(I,{\tilde{A}}_{{an}})+\left[{R}_{{elect}}(I,D,{\tilde{A}}_{{an}},C)\right.\\ \quad \left.+{R}_{{bub}}(I,{\tilde{A}}_{{an}},C)+{R}_{{an}}({H}_{{an}})+{R}_{{ca}}\right]I\end{array}\end{eqnarray} \tag{ 27 }$$

where C represents electrolyte composition and ${\tilde{A}}_{{an}}$ is the total effective exposed anode area through which current flows. Virtually all the terms on the right-side of the equation are functions of temperature, and therefore they are not expressed as such for simplicity.

When the line current, I, is used for the computation, ${\tilde{A}}_{{an}}$ is the sum of the bottom surface area of all the anodes. In the context of anode change, Eq. (27) is instead used with individual anode currents, while ${\tilde{A}}_{{an}}$ refers to the effective surface area of anode not covered by freeze. This includes any exposed area on the sides of the anode which allows any flowing current to fan out when the bottom surface is all covered. As the array of anodes in the cell are in an electrical parallel connection, the voltage through each path must be the same and all the anode currents must sum up to line current. Therefore, for N anodes, the anode currents and cell voltage at each time point can be computed by simultaneously solving ¹³

$$\begin{eqnarray}&&\left\{\begin{array}{l}{V}_{{cell},1}({I}_{{an},1})={V}_{{cell},2}({I}_{{an},2})\\ {V}_{{cell},1}({I}_{{an},1})={V}_{{cell},3}({I}_{{an},3})\\ \vdots \\ {V}_{{cell},1}({I}_{{an},1})={V}_{{cell},N}({I}_{{an},N})\\ I=\sum _{n=1}^{N}{I}_{{an},n}\end{array}\right..\end{eqnarray} \tag{ 28 }$$

In this work, while the cell voltage equation considered traditional non-slotted anodes, slotted ones are also used in the industry for benefits such as reducing anode thermal stress and helping the release of gas bubbles from underneath the anode. ⁴⁸ Slotted anodes are not considered in this work because calculations show that the electrolyte resistance dominates, as it is in the order of ten times larger than the bubble resistance. If included, the equation will also be highly empirical as there are many ways of slotting an anode, including different slotting orientations, widths, depths, angle of inclination, and profile. ⁴⁸ This is outside the scope of this work, and we therefore consider that the bubble layer thickness and coverage is already representative for slotted anodes. Arkhipov et al. ⁴⁹ have also found that slotted anodes have negligible impact on the anodic voltage drop.

Due to the coupling effects between process variables in this anode change model, Eqs. 3–28 have to be solved simultaneously. With suitable initial conditions, the numerical solution to the differential Eqs. can be found in Mathworks MATLAB®, giving the time-series value of the various process variables. The simulation starts after the freeze formation underneath both of the recently changed anodes.

Realistic initial conditions are used so the simulation results can be validated with measurements from an industrial cell. For example, the initial anode heights used are shown in Fig. 8; they are based on carbon consumption rate consum (I_an ), total anode service life, and anode replacement rota. This is so the distribution of anode current also reflects the anode resistance, and the thermal capacity and generation of each zone is realistic. The ACD, D, is initialised to 2.75 cm for all the anodes, except the new ones which are set 3 cm higher in this study. In this simulation, the alumina concentration is initialised with a value of 4.8 wt%, which is higher than the normal operating band. This is because firstly, the cell is overfed in preparation for the anode replacement to reduce the risk of anode effects, and secondly, lumps of crusts and other materials can fall into the electrolyte during the removal of spent anodes. ⁵⁰ The initial undissolved alumina concentration is assumed to be zero; this value does not significantly change the results.

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The different sides of the anode freeze that is in contact with the electrolyte will dissolve at varying rates subject to local thermal gradients and local cell conditions as previously explained. The latter can be modelled with different, but constant, HTC values for each face, since experiments have shown that a vertical freeze with gradual or steady dynamics does not significantly change the HTC. ⁵¹ The HTCs used for the cell studied are expressed as multiples of the nominal HTC between the electrolyte and the side ledge in Table VI.

As the dissolved alumina concentration is not continuously measured in an industrial cell, the commonly used "demand feed" algorithm ⁵² controls the alumina concentration within a band based on measurements of cell voltage and line current. In this algorithm, the feeding rate of alumina is cycled between over-feeding, base-feeding, and under-feeding, so the trajectory of cell voltage can be used to gauge the level of alumina concentration dissolved in the electrolyte. Although this feeding control algorithm has been added to the model such that feeding control actions can be completely simulated, the best way to validate the mass balance model is to use the actual feed rates from the industrial cell as model input.

Due to the feeding algorithm, feeder position, and the electrolyte flow, there will be spatial and temporal variations in the electrolyte. Figure 11 shows a heat map of the alumina concentration in the cell before anode change, immediately after, and 3 h after anode change. It can be seen that immediately after anode change, the concentration in the proximity of the new anodes is very high, since the electrolyte mass in those regions is assumed to drop dramatically due to rapid freezing. It is also observed that there is a difference in the distribution of alumina concentration before the anode change and 3 h after. One of the contributing factors is the redistribution of anode currents leading to different rates of alumina consumption at different locations.

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The temporal variation can be validated by comparing the simulated and measured concentration level at the tap-end of the cell, as it changes according to the feed rate recorded from the industrial cell, as shown in Fig. 12.

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The electrolyte temperature also experiences spatial and temporal variations. Similar to the alumina concentration, Fig. 13 shows a heat map of the temperature distribution before, immediately after, and 3 h after the anode change operation. It is observed that while the electrolyte temperature before anode change is relatively even, the distribution changes very drastically after the operation with effects lasting for h. This again demonstrates how the anode change operation causes a massive disturbance to the thermal balance.

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In order to validate the results obtained, the simulated and measured electrolyte temperatures at the tap-end, duct-end, between anodes 29–30, and between anodes 7–8 are compared in Fig. 14. The electrolyte temperature between anodes 29–30 is very low due to the deficiency in thermal energy caused by the cold anodes. The electrolyte temperature at the duct-end is also higher than at the tap-end, because the anodes are slightly older at the duct-end, and therefore are shorter with less resistance, assuming that they share the same ACD. Both the simulation and measurements show that the local electrolyte temperature can vary by as much as 13 °C. across the cell, which highlights the importance of having a distributed cell model, especially as superheat can be less than 10 °C. The oscillations observed in the electrolyte temperature are due to the cycling feeding rate shown in Fig. 12. The discrepancy in the duct-end electrolyte temperature in the first hour may be attributed to the changing electrolyte flow pattern upon the replacement of anodes 29–30, as it is observed that the measured temperature in the region remains high within the first hour, while a rising trend like that of the duct-end is expected. It is possible that the electrolyte flow at the duct-end may have temporarily changed as the freeze underneath anodes 29–30 impedes regular flow.

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Figure 15 shows how the freeze underneath one of the newly-set anodes dissolves away with time. As it dissolves, more of the anode is exposed, and the effective surface area increases. This allows more current to pass through, as shown in Fig. 16. Similar to our previous work, ²³ it is observed that the new anodes do not reach full current carrying capacity when all the freeze has dissolved. Instead, the rate of anode current pickup decreases abruptly at around 6–7 h when all the freeze has disappeared, and from this point onwards, the anode current slowly approaches full capacity, taking a minimum of 4 days. ⁵³ This is because even when the freeze has completely dissolved away, the path resistance is still greater because the new anodes have ACDs larger than the other anodes. This is consistent with Cheung's analysis of anode currents, ²⁶ in which it was also noted that at this stage, these anodes behave like the other old anodes. Differences in ACD will eventually get rectified as it has a self-regulating behavior—as the currents through these anodes are less than the other anodes, less carbon is consumed and ACD will decrease at a greater rate compared to the other anodes.

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The freeze dissolution time will differ depending on cell technology, anode location, and local heat transfer conditions. In Cheung's work, ²⁶ it takes on average around 6 h to reach full dissolution, with a standard deviation of 2.7 h. In order to illustrate this effect, another simulation is carried out where anodes at the cell corners (anodes 17–18) are changed instead. Parameters across the simulations are kept very similar to demonstrate just the effect of local heat conditions on the dissolution rate of the anode freeze. Hence, the same HTC as shown in Table VI are used, despite the likelihood that they will change according to anode location and electrolyte flow pattern. The only other difference is in the anode initial heights, since they should reflect the condition when anodes 17 and 18 are changed. The result is shown in Fig. 17; not only does the rate of freeze dissolution differ from the previous simulation study, but the freeze below the two anodes even dissolves at different rates. The freeze below anode 18 dissolves very slowly since it experiences greater heat loss and superheat is low in the area, as this anode is adjacent to two cell walls and more side ledge.

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The results obtained are indirectly validated by comparing the simulated and measured current passing through anode 29 as it slowly picks up over several h, as shown in Fig. 18. Minor differences in the trajectories include the initial anode current load. This is because this model assumes that at the beginning, pieces of freeze at the sides of the anode have already broken off, and therefore has exposed some surface area for electricity conduction. Meanwhile, the bumps in anode current observed at around 11:30 and 14:30 are due to known and unrelated process disturbances elsewhere in the cell.

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This validation demonstrates that with suitable parameters and initial values, the model can be used to predict and study the overall trends of anode current distribution and local cell dynamic response during and after an anode change procedure. When the model is implemented online with feedback from actual process measurements like anode currents and cell voltage, process parameters and initial values can be recursively identified in real-time using online estimation techniques such as an Extended Kalman Filter ⁵⁴ to achieve higher accuracy in the prediction of future process variables.

However, implementation of feedback-based correction is outside the scope of this paper, which focuses on the process mechanisms and model structure; process parameters vary across different cell technologies.