Coupled electro-thermal field in a high current electrolysis cell or liquid metal batteries

Coupled electro-thermal field exists widely in chemical batteries and electrolysis industry. In this study, a three-dimensional numerical model, which is based on the finite-element software ANSYS, has been built to simulate the electro-thermal field in a magnesium electrolysis cell. The adjustment of the relative position of the anode and cathode can change the energy consumption of the magnesium electrolysis process significantly. Besides, the current intensity has a nonlinear effect on heat balance, and the effects of heat transfer coefficients, electrolysis and air temperature on the heat balance have been released to maintain the thermal stability in a magnesium electrolysis cell. The relationship between structure as well as process parameters and electro-thermal field has been obtained and the simulation results can provide experience for the scale-up design in liquid metal batteries.


Introduction
With the large-scale deployment of renewable energy sources, massive and cheap electricity storage becomes indispensable because the major part of renewable electricity generation (solar, wind) is inherently fluctuating. Liquid metal batteries (LMBs) proposed by professor Sadoway and co-authors [1] are currently discussed as a means to provide economic grid-scale energy storage. The liquid metal battery is high-temperature (700°C thermal field and electric field. Sun et al. [31][32][33] studied the flow field and the electric field in a magnesium electrolysis cell by simulation, where the effect of the electro-magnetic field of the cell was considered. Liu et al. [34] and his colleagues studied the scale-up design of a 300 kA magnesium electrolysis cell using a three-dimensional thermo-electric mathematical model, and the results can be referenced in the scale-up of magnesium electrolysis cells with a high current intensity. In this paper, the study of magnesium electrolysis cell is mainly focused on the thermal balance and energy saving under normal production circumstances. With the help of finite-element software ANSYS, the effects of cell structure and operation parameters (current intensity, electrolysis temperature, ambient temperature, ACD, cell voltage, etc.) on thermal balance of magnesium electrolysis cell have been simulated in a three-dimensional model. Through analysing the thermal field distribution in the anode, cathode and other parts of the actual 120 kA magnesium electrolysis cell, the results of the study are useful to designing and optimizing magnesium electrolysis cells and benefit to the scale-up design of LMBs to a certain extent.

Electric and thermal coupling analysis model 2.1. Physical model
The magnesium electrolysis cell model should be simplified to meet the basic requirements of simulation because of its complicated structure and big size. The physical model in this paper includes graphite anodes, steel cathodes, electrolyte, firebricks, insulation bricks as well as cell shells. The other parts of less impact are not considered in the simulation. According to the symmetry of the long axis, the model is calculated by taking the one-half size of the full cell (as shown in figure 1).
In order to facilitate the finite simulated computation of the magnesium electrolysis cell, some assumptions are made as follows: (1) In the magnesium electrolysis cell, the electric field and thermal field are assumed to be steady. (2) The temperature of the electrolyte and magnesium are equal to the electrolysis temperature. (3) The influences between the two half parts of the cell are ignored. (4) The heat loss of the bus bar part is ignored. (5) The round chamfer is treated as a right angle.

Mathematical model
The heat of the magnesium electrolysis cell is derived from the Joule heat generated by the current, and the Joule heat is dissipated from the outer surface of the cell to the outer space by convection and radiation. For the complex problem of coupling of cell convection and heat transfer, there are two modelling solutions at present: one is to use partition modelling and calculation, and the other is overall modelling and coupling solution. For the calculation object like magnesium electrolysis cell with complex cell shape, coupled with complex boundaries, if the partition model is adopted, the simulation results will vary greatly with the actual ones. Therefore, this study uses overall modelling and coupling solution. According to heat transfer theory, the heat transfer of the constituent components in the magnesium electrolysis cell is subject to the unstable heat transfer control equation (equation 2.2) with an internal heat source. The equation in the three-dimensional model is

Control equation
where k x , k y and k z represent the coefficient of heat conductivity (W m -1°C-1 ). T is the temperature (K), q s the heat intensity, ρ the material density (kg m -3 ) and c the specific heat (W m -1°C-1 ).

Boundary conditions
The boundary conditions of electricity conduction equation are expressed as follows: (1) Take the cathode plate surface as zero potential.
(2) Voltage degree of freedom is applied to the coupling node on the surface of the anode head, and then load the current of the electrolysis cell at either node.
The boundary conditions of heat conduction equation are expressed as follows: (1) The temperature of the electrolyte is constant and equal to the electrolysis temperature.
(2) The temperature around the magnesium electrolysis cell is constant and equals to the actual temperature of the workshop. (3) The heat dissipation coefficient of the outer surface of the electrolysis cell, the heat transfer coefficient between the electrolyte and the side cell surface, the heat transfer coefficient between the electrolyte and the cathode and anode surfaces as well as the heat dissipation coefficient of the cathode and anode will be obtained through the empirical formula which can be found in [35,36]. (4) When applying heat convection and transfer boundary conditions to the interface of the electrolysis cell and external surface, the heat transfer coefficient shall include the influence of radiation heat transfer.

Finite-element analysis of the electro-thermal field model
For the above electro-thermal field model, the Galerkin method with weighted residual approach is used for the solution of finite-element model. The steady thermal field of the conductive part of the electrolysis cell can be expressed as where λ x , λ y and λ z are the heat conduction coefficient in the three dimensions, x, y and z; q vol is the Joule heat of the control unit.
The trial function can be expressed as T(x, y, z) = T(x, y, z, T 1 , T 2 , KT n ), (2.5) where T 1 , T 2 and KT n are undetermined coefficients. Taking equation (2.5) into equation (2.3), the result obtained through weighted residual method is as follows: According to the definition of weighted functions by the Galerkin method, weighted function W s can be expressed as: Owing to the complexity of analysing the temperature field of the space, it is relatively difficult to find the T(x, y, z) satisfying the boundary conditions. And in order to meet the boundary conditions, the Gaussian formula can be used to connect the volume of the simulation area with the surface integral of the boundary. The following equation can be obtained through transformation: where cosα, cosβ and cosγ represent the cosine in the three dimensions x, y and z, respectively. Equation (2.8) defines the whole solving area. Direct solving is quite difficult, so computation of equation (2.8) in the whole area is not adopted. On the contrary, the calculation of each grid unit in each part is conducted, and then they will be combined together to form whole linear equations to solve the question. If the area is divided into E units and n nodes, then the temperature T(x, y, z) will split into the undetermined temperature of the nodes like T 1 , T 2 and T 3 .
The combined equations can be written into the matrix form: In equation (2.10), k represents the temperature stiffness matrix; [T] = [T 1 , T 2 , . . . ., T n ]; P represents the temperature load matrix, the temperature of each node of the magnesium electrolysis cell can be calculated through the above equation.

Finite-element model analysis
The finite-element model of magnesium electrolysis cell is established according to the above methods, and the cell type includes conductive part and un-conductive part. The conductive part adopts unit SOLID69 to analyse the electro-thermal field, while the un-conductive part adopts unit SOLID70 for thermal analysis. In order to get accurate calculation results, hexahedron grid is used to divide the cell. The size of each grid is 0.04 m. There are nine materials used in the electrolysis cell. The characteristics of these materials vary with temperature, so the nonlinear processing is used to treat these materials. The specific finite-element model is shown in figure 2.   The number of hexahedron grids of the finite-element model adopted in this study is 600 000-2 000 000, and accurate calculation results can be obtained if the hexahedron grids are within the number.

Validation and thermal equilibrium of the magnesium electrolysis cell
In this paper, a 120 kA magnesium electrolysis cell in China is set as an example to illustrate the thermoelectric coupling simulation calculation method. According to the above description, a threedimensional half cell model is used here for the simulation of the electro-thermal field coupling in the magnesium electrolysis cell.

Modelling and solving
For the cathode, anode and electrolyte, coupling unit SOLID69 with two degrees of freedom (temperature and voltage) will be adopted, while the others will use SOLID70 unit with only one degree of freedom (temperature). The outer convection heat transfer part will use SURF152 unit to exert boundary conditions and calculate the amount of dissipated heat. The material properties of the magnesium electrolysis cell are nonlinear in the electrolysis process. Therefore, MPDATA order will be used to define the materials properties so as to reflect the distribution of the internal temperature field of electrolysis cell. Tables 1-3 show the structure parameters and materials properties of different cell types.
The model includes anode, cathode, electrolyte, refractory layer, insulation layer, thermal-protective layer, cell cover and chlorine. The size of the grid unit is 0.04 m, and each part of the model uses hexahedron grid. The total number of grid is 668 889 with a calculating time of about 2200 s. The finite-element cell model is shown in figure 3.
The boundary conditions of the model can be expressed as follows: (1) The environment temperature equals 25°C.
(2) The electrolysis temperature equals 700°C.      Figure 4a shows the temperature isopleths of the entire electrolysis cell, while figure 4b,c is the temperature isopleths of anode and cathode. It can be seen that the highest temperature is in the    electrolysis interval while the lowest temperature of the exterior wall closing to 25°C. The lowest temperature of the anode and cathode is 305°C and 114°C, respectively. The APDL which is developed from ANSYS is applied for average temperature estimation on anode head, cathode head, cell cover surface, front wall, back wall, end wall and cell bottom is 338.5°C, 99.2°C, 164.2°C, 61.0°C, 68.2°C, 72.8°C and 52.7°C, respectively. The simulated data are in good agreement with the actual empirical data. The most obvious feature is that the temperature distribution is uneven in every angle, which means that these parts are prone to thermal expansion and contraction, and it is also one of the most important factors affecting the lifespan of the cell. Figure 5a,b are the temperature isopleths for magnesium electrolysis cell in different directions. Figure 5a shows the heat transfer path in the cross section of the cell. The bottom of the cell has the lowest temperature, resulting from the relatively large heat transfer coefficient of the other parts, especially the anode head and the cathode. Owing to the heat resistance, the temperature of the electrolyte drops from 700°C to about 677°C at the cell boundary. Figure 6a examines the thermal field of the whole cell, and the heat dissipation of the anode head is much better than that of the cell cover. Figure 6b shows the temperature isopleths corresponding to figure 6a. It shows that the effect of heat preservation on the end wall and cell bottom is better than that of the cell cover and anode head. The range of the temperature variation is obviously smaller than other parts, especially when the temperature isopleth of the cell bottom reaches 497°C. Figure 6c,d show the vertical view of the temperature field and temperature isopleths, respectively. In figure 6c, the temperature near the cathode is higher than the rest of the surrounding parts because of the faster dissipation speed and bigger thermal conductivity coefficient of the cathode. Therefore, the temperature isopleth concaves inward. But when the temperature drops to about 317°C, the direction of temperature isopleth changes from concave to convex, because the heat dissipation near the cathode is great. Table 5 shows the amount of heat produced and dissipated in each part of 120 kA magnesium electrolysis cell by using APDL language for post-processing.

Energy equilibrium calculation and analysis
The heat generated by the electrolyte accounts for 59% of the total, 32.75% for the anode and only 8.25% for the cathode. On the other hand, the dissipation heat of the anode head occupies 59.4% of the total, followed by 14.99% for the cell cover and 10.93% for the cathode. In general, the difference percentage of heat balance is less than 1%, which meets the requirement of actual production process. Therefore, the parameters of the anode, cell cover and cathode will be adjusted first if adjustment of heat balance is required. Table 6 is a comparison of the thermal balance calculated by analytic method and ANSYS with the thermal analysis in actual production process, and ANSYS is better than analytic method. The heat loss difference reaches 3.75% by analytic method, and it is within the engineering calculation error of 10%. But it is 10 times of the result calculated by finite-element method. The energy loss of the two methods mainly differentiates in the heat dissipation capacity of the cathode.

The influence of structural parameter on heat equilibrium
The thermal balance of the magnesium electrolysis cell is an important criterion for measuring whether a magnesium electrolysis cell can work properly. The research will focus on the effects of the several factors on the thermal balance, which include current intensity, electrolysis temperature, ambient temperature, heat transfer coefficient and the distance between cathode and anode.

The effect of current intensity on the thermal balance
The effects of current intensity of the different cell types, which contain maximum current value all over the world, have been studied in this paper. At the same time, it can provide a basis for designing a larger energy-saving magnesium electrolysis cell in the future. The paper examines the thermal balance of seven cell types including 120, 180, 240, 300, 360, 420 and 480 kA. The main research goal is whether it can be convenient or not to get a large magnesium electrolysis cell, after simplest amplification. As shown in figure 7, with the increase of the current intensity, the heat dissipating capacity of the anode, cathode, cell cover and cell shell increases linearly. The heat dissipation of anode increases from   As shown in figure 8, the thermal balance difference is increasing with the current intensity increasing, which means the simple parallel amplification can lead to thermal unbalance. As the magnesium electrolysis cell is not energy-saving in practical production, increasing the current intensity is not a suitable way to keep the heat balance. Table 7 shows the temperature of each part of different current intensity calculated by APDL. As the current intensity increases, the temperature of anode head can be considered unchanged essentially, and the temperature increase rate of cathode is the largest, reached 9.12°C.

The effect of the electrolysis temperature on the heat balance
Keeping the electrolysis temperature steady is a basic condition to keep the electrochemical reaction. Besides, keeping the electrolysis temperature steady can prevent the electrolyte from cooling crystallization. Generally speaking, different electrolysis temperature levels have different effects on the heat balance of electrolysis cell. The current intensity of 120-300 kA of the magnesium electrolysis cell is mainly researched here; after that, the simulation research is also in the same range. Figure 9 shows the relationship between electrolysis temperature and heat dissipating capacity of anode and cathode. In figure 9a, the heat dissipating capacity of the anode increases slightly with the increase of electrolysis temperature and current intensity. The electrolysis cell with different current intensity (120/180/240/300 kA) has different increase (1854.9/ 2791.6/ 3721.4/ 4652.2 W, respectively).   In figure 9b, the heat dissipating capacity of the cathode increases with electrolysis temperature increasing. The maximum value of that growth appeared in the current intensity of 300 kA with the value of 1299.1 W.
In figure 10a, the heat dissipating capacity increases with the current intensity increasing. With the increase of electrolysis temperature, change of the heat dissipating capacity is smaller. The increasing rate of electrolysis temperature grows with the current intensity increasing, and the maximum of the increase rate is 777.6 W. In figure 10b, the increasing rate is obvious. The maximum increasing rate is 2531.7 W, and the minimum is 1287.0 W.
As shown in figure 11, in the same type of magnesium electrolysis cell, there is different heat balance temperature because of different current intensity. The balance electrolysis temperature of 120 kA is about 700°C, that of 180 kA is 740°C, and the others are more than 750°C. Figure 11b shows the heat balance deviation rate of different current intensity. The cell of 120 kA already reaches balance at 700°C. Therefore, with the increase of electrolysis temperature, the deviation rate increases in contrast. The line of 180 kA is exceptional, and there is an inflection point at 740°C. Before the inflection point, its trend agrees with the trend of 240 and 300 kA, but after the inflection point, its performance is similar to the trend of 120 kA.

The effect of ambient temperature on the heat balance
The ambient temperature is the temperature of the production plant where the magnesium electrolysis cell is located. Generally speaking, ambient temperature of the electrolysis cell will change dramatically with season. However, in the traditional design, it commonly takes 25°C as ambient temperature, and the effect of the ambient temperature on heat balance has been ignored. Besides, the effects of regional   difference and climate difference on heat balance have not been considered. Therefore, it is necessary and significant to research the ambient temperature of the magnesium electrolysis cell. As the figure 12a-d shows, the heat dissipating capacities of the different parts decrease with the increase of ambient temperature in different degrees. In figure 12a, heat dissipating of the anode increases with the current intensity. The heat loss of electrolysis cell 300 kA is 5011.9 W when the temperature is 15°C, and the maximum heat loss of 120 kA is 2003.9 W. In figure 12b, the heat changes slightly compared with figure 12a, and the maximum heat loss is about 658 W from 15°C to 35°C. Figure 12c shows the distribution of heat dissipation of cell cover in different current intensity, and the maximum point of the heat loss appears at 15°C when the current intensity is 300 kA, and its value is 1309.9 W. Figure 12d shows the heat loss of the cell shell, which is similar to the heat loss of cathode. The maximum heat loss is 592 W from 15°C to 35°C. In a word, the heat loss of the cell is about 7570.9 W in total with the increase of ambient temperature. The ambient temperature is a sensitive parameter to the electrolysis  cell. Therefore, in order to optimize the design of the cell, the effect of ambient temperature must be taken into consideration.

The effect of the anode heat transfer coefficient on heat balance
The anode is an essential part of the magnesium electrolysis cell. The heat dissipation of anode accounts for about 60% of the total. Therefore, the adjustment of the heat dissipation of the anode can effectively prevent the heat loss of the entire cell. In this study, changing the heat transfer coefficient of anode is the main method to improve the heat balance of the cell. Figure 13a shows the heat dissipating capacity of anode head in different heat transfer coefficient under different current intensity. With the increase in heat transfer coefficient, the heat of the anode head shows an increasing trend. The heat transfer coefficient of the anode changing from 23 W·(m°C) -1 to 28 W·(m°C) -1 , for the 120, 180, 240 and 300 kA electrolysis cell, makes the heat dissipating capacity of anode increasing by 6688.6, 10 030.1, 13 374.8 and 16 720.4 W respectively. Corresponding to the heat dissipating capacity of anode, its average growth rate is 9.93% by changing the heat transfer coefficient. Figure 13b shows the heat dissipating capacity of cathode. The heat loss is basically unchanged with the heat transfer coefficient of anode head changing, and the maximum of heat dissipating capacity is only 0.2 W, which can be neglected. Therefore, the change of the heat transfer coefficient of anode head has no impact on the heat dissipation capacity of cathode head.
As figure 14 shows, for electrolysis cells in all kinds of current intensity, the heat balance point changes with the heat transfer coefficient of anode. For 120 kA electrolysis cell, its balance point is at  Above all, adjusting the anode heat transfer coefficient is an effective means to make the thermal field of the cell in balance.

The effect of other heat transfer coefficients on heat balance
In the case of constant heat transfer area, changing the heat transfer coefficient of the anode head to improve the thermal balance of the cell is an effective means. It is necessary to study the effects of the cell cover and cell shell heat transfer coefficient on heat balance because the heat dissipating capacity of cell cover, cell shell accounts for 40% of the total. Figure 15a shows the heat dissipating capacity of the anode head in different current intensity decreases slightly with the heat transfer coefficient of other parts increasing. The minimum drop of the heat dissipating capacity is 420.6 W, and the biggest drop is about 1038.7 W. The reason of this phenomenon is that the heat which is flowing to anode originally has been flowed to other parts with the increase of the heat transfer coefficient of other parts. The heat dissipating capacity of cathode, cell cover and cell shell increases by degrees with the heat transfer coefficient increasing which has been shown in figure 15b-d. Among them, the heat dissipating capacity of cell cover changes greatly. With the heat transfer coefficient of 120 kA magnesium electrolysis cell increasing, the heat dissipation of the cathode, cell cover and cell shell increases by 645. 3 Figure 17. (a-b) The heat relationship between ACD and various parts.
As figure 16 shows, the 120 kA magnesium electrolysis cell increasingly deviates from the heat balance point with the increase of the heat transfer coefficient of other parts, and the other electrolysis cells of different current intensity show the opposite trend. The 180 kA electrolysis cell reaches the heat balance, at about 5 W·(m°C) -1 , when the heat transfer coefficient increases by about 2 W·(m°C) -1 , while that of the 240 and 300 kA electrolysis cells still did not reach 2% of difference rates. Comparing with the method which adjusts the heat transfer coefficient of anode, the methods of adjusting the heat transfer coefficient of other parts can be considered next. Although it can meet heat balance to a certain extent, but this way cannot meet all types of magnesium electrolysis cells.     16 18 20 22 (b) (a) Figure 19. The relationship between areas and heat dissipating capacity.

The effect of anode-cathode distance (ACD) on heat balance
The size of ACD not only affects the size of electric potential, but also the area of dissipation. As a result, it will influence the size of heat dissipating capacity, and ultimately the heat balance difference. It is a very important parameter that must be studied deeply during the process of designing a magnesium electrolysis cell. Figure 17a,b show that the heat dissipating capacity of anode and cathode is almost unchanged when the size of ACD is changing. The most obvious change happens on the cell cover and cell shell by adjusting ACD. In figure 17c,d, the growth rates of the heat dissipating capacity increase when the current intensity gets larger. The cell cover and cell shell of 300 kA will be more sensitive to the adjustment of ACD. For the 120 kA electrolysis cell, as ACD has grown from 0.03 to 0.08 m, the heat dissipating capacity of cell cover has changed by 4155.8 W, and cell shell has changed by 2801.3 W; for 180 kA cell, the value of cell cover has increased by 6232.4 W, and cell shell has increased by 4173.8 W; for 240 kA cell, the value of cell cover has increased by 8353.6 W, and cell shell has increased by 5571.4 W; for 300 kA cell, the value of cell cover has increased by10 434.3 W, and cell shell has increased by 6938.6 W. Above all, adjusting ACD can take the heat balance of the cell into control. In this study, the reduction of heat dissipation capacity, which is caused by adjusting ACD, accounts for 26%.
As figure 18 shows, for electrolysis cells in all kinds of current intensity, the heat balance point changes with ACD. Furthermore, with the increase of the current intensity, the balance point moves towards the direction that the polar distance decreases. When ACD is bigger than 0.08 m, heat dissipation rate is much higher than the Joule heat production rate. The cell of 120, 180, 240 and 300 kA reaches the heat balance point when ACD is 0.07, 0.0655, 0.063 and 0.062 m, respectively. And the difference between heat generation and heat dissipation of these four types of cell is about 102 126, 148 241, 194 585 and 240 864 W, respectively. From figure 18b, it can be observed that the changing trend of heat balance difference of these four types of cell is almost unchanged. In a word, the heat balance in electrolysis cell can be achieved through adjusting ACD which is also an extremely effective way.
In figure 19, the total heat dissipation area and the total heat loss was affected by changing ACD in the cells of different current intensity. In figure 19a, there are four ladder sections, and the slope of every ladder section maintains basically the same, which explains that the heat dissipation capacity is related to the size of heat dissipation area rather than current intensity.

Conclusion
The study focuses on electric and thermal fields in electrolysis industry and the conclusions will be of benefit to batteries and electrochemistry. The paper mainly researched the effects of the structure parameters on the thermal field in magnesium electrolysis cells. The influence of current intensity, electrolysis temperature, ambient temperature, heat transfer coefficient and ACD on the heat balance of electrolysis cell is studied by finite-element simulation of actual 120 kA electrolysis cell. Some conclusions which can be obtained by simulation calculation are as follows: (1) Under the condition of same current intensity, the heat dissipating capacity increases linearly with the 120 kA electrolysis cell amplification to 480 kA in parallel. However, the electrolysis cell deviates from the heat balance increasingly. When it is in 240 kA, heat absorbed will be over 5.6%; when it is in 480 kA, the value will close to 7.9%. Therefore, the simple amplification cannot meet the requirements of heat balance, and the design of the electrolysis cell needs to be further optimized.
(2) With the increase of electrolysis temperature, the growth rate of heat dissipation at anode is the largest, followed by cathode, cell shell and cell cover. The ability of changing electrolysis temperature to keeping the heat balance is limited, and the heat balance difference only reaches about 2%. (3) The ambient temperature needs to be considered in the design process of the magnesium electrolysis cell. For the 300 kA electrolysis cell, the heat dissipating capacity, which accounts for 3% of the total, increases by 7570.9 W when the ambient temperature drops from 35°C to 15°C. (4) Commonly, the heat dissipation capacity of anode is about 60% of the total. The simulation results by changing the heat transfer coefficient of anode show that the heat dissipation increases by nearly 9.9% when the anode heat transfer coefficient increases by 5 W·(m°C) -1 . In a word, adjusting the anode heat transfer coefficient is an effective means to make heat loss in balance of the cell. (5) In addition to the anode head, the heat dissipation coefficient of other parts of the cell also has an important effect on the heat balance. When the coefficient increases by 5 W·(m°C) -1 , the total heat dissipation capacity increases by 2.27%. Which means adjusting the heat dissipation coefficient of other parts, such as cell cover and cell shell, can be secondary means in optimizing the magnesium electrolysis cell. (6) The heat dissipation of cell cover and cell shell is sensitive to the change of ACD. The growth of the heat dissipation of the 300 kA magnesium electrolysis cell is 10 434.5 W when ACD increased by 0.05 m. Thus, changing the ACD is an extremely effective way to achieve the heat balance compared with adjusting the heat transfer coefficient of anode.