Computational Modeling of Thermochemical Evolution of Aluminum Smelter Crust

Abstract

In an aluminum reduction cell, crushed anode cover at room temperature is added onto the exposed bulk electrolyte surface around newly positioned anodes and is heated by high heat flux from this liquid electrolyte. Liquid electrolyte penetrates inside the porous anode cover. Solid cryolite and alumina crystallize from the liquid electrolyte due to the temperature gradient in the anode cover. A solidified crust forms at the bottom part of the anode cover during the heating up period. A thermochemical model which takes into account both the liquid electrolyte penetration and phase transformations has been developed to simulate the temperature evolution, chemical composition development, and liquid front penetration and content in the anode cover. The model is tested against experimental data obtained from industrial cells and laboratory experiments in this paper.

Appendices

Appendix

The equations listed in Section III include the temperature T _L, the concentration Ø _i , and the phase transformation rate m _i . These equations are complicated, especially the conservation equations of energy. They can be more concrete and easy to be solved by finding out the mutual coupling relationships between these variables.

Equations [2], [3], and [4]give

$$ \rho_{\rm{L}} \left( {\frac{{\partial \varepsilon_{\rm{L}} }}{\partial t} + \frac{\partial u}{\partial x}} \right) = m_{1} + m_{2} $$

(30)

Equation [31] is got from energy equations [16] and [17].

Substituting Eqs. [7], [8], and [30] in Eq. [31] gives Eq. [32].

According to definition of specific sensible heat capacity C and latent heat L _i of solid/liquid phase transformation, Eq. [33] is got (ignoring the difference between T _L and T _S).

$$ \varepsilon_{\rm{L}} \rho_{\rm{L}} \frac{{\partial H_{\rm{L}} }}{{\partial T_{\rm{L}} }}\frac{{\partial T_{\rm{L}} }}{\partial t} + u\rho_{\rm{L}} \frac{{\partial H_{\rm{L}} }}{{\partial T_{\rm{L}} }}\frac{{\partial T_{\rm{L}} }}{\partial t} + H_{\rm{L}} \rho_{\rm{L}} \left( {\frac{{\partial \varepsilon_{\rm{L}} }}{\partial t} + \frac{\partial u}{\partial x}} \right) + \left[ {\sum {\left( {\varepsilon_{{\rm{S}},i} \rho_{{\rm{S}},i} \frac{{\partial H_{{\rm{S}},i} }}{{\partial T_{\rm{S}} }}} \right)} } \right]\frac{{\partial T_{\rm{S}} }}{\partial t} + \sum {\left( {\rho_{{\rm{S}},i} H_{{\rm{S}},i} \frac{{\partial \varepsilon_{{_{{\rm{S}},i} }} }}{\partial t}} \right)} = \frac{\partial }{\partial x}\left( {k_{\rm{eff}} \frac{{\partial T_{\rm{L}} }}{\partial x}} \right) + q_{\gamma \alpha } ,\quad x \le h_{\rm{e}} $$

(31)

$$ \varepsilon_{\rm{L}} \rho_{\rm{L}} \frac{{\partial H_{\rm{L}} }}{{\partial T_{\rm{L}} }}\frac{{\partial T_{\rm{L}} }}{\partial t} + u\rho_{\rm{L}} \frac{{\partial H_{\rm{L}} }}{{\partial T_{\rm{L}} }}\frac{{\partial T_{\rm{L}} }}{\partial x} + \left[ {\sum {\left( {\varepsilon_{{\rm{S}},i} \rho_{{\rm{S}},i} \frac{{\partial H_{{\rm{S}},i} }}{{\partial T_{\rm{S}} }}} \right)} } \right]\frac{{\partial T_{\rm{S}} }}{\partial t} + H_{\rm{L}} m - \sum {(H_{{\rm{S}},i} m_{i} } ) = \frac{\partial }{\partial x}\left( {k_{\rm{eff}} \frac{{\partial T_{\rm{L}} }}{\partial x}} \right),\quad x \le h_{\rm{e}} $$

(32)

$$ \varepsilon_{\rm{L}} \rho_{\rm{L}} C_{\rm{L}} \frac{{\partial T_{\rm{L}} }}{\partial t} + u\rho_{\rm{L}} C_{\rm{L}} \frac{{\partial T_{\rm{L}} }}{\partial x} + \sum {(\varepsilon_{{\rm{S}},i} \rho_{{\rm{S}},i} C_{{\rm{S}},i} )} \frac{{\partial T_{\rm{L}} }}{\partial t} + \sum {(L_{i} m_{i} } ) = \frac{\partial }{\partial x}\left( {k_{\rm{eff}} \frac{{\partial T_{\rm{L}} }}{\partial x}} \right) + q_{\gamma \alpha } ,\quad x \le h_{\rm{e}} $$

(33)

According to the mechanism of mass transfer between solid/liquid phases[39]

$$ \phi_{i} = \phi_{i}^{*} - K_{i} m_{i}, $$

(34)

where ϕ ^* _i is saturation concentration, K _i is a coefficient between concentration difference and phase transformation rate, and K _i  < 10⁻⁶ in the case of high surface area S. Equations [35] and [36] are got by substituting Eqs. [30] and [34] in the Eqs. [2] and [3] (In the case of the porous crust with fine pore size, the effect of mass diffusion D in the liquid bath on the model result is likely to be insignificant. Thus, the mass diffusion might be neglected in the application of the model in this case)

$$ (1 - \phi_{1}^{*} + m_{1} K_{1} )m_{1} - (\phi_{1}^{*} - m_{1} K_{1} )m_{2} + K_{1} \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial m_{1} }}{\partial t} + u\frac{{\partial m_{1} }}{\partial x}} \right) = \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial \phi_{1}^{*} }}{\partial t} + u\frac{{\partial \phi_{1}^{*} }}{\partial x}} \right) $$

(35)

$$ (1 - \phi_{2}^{*} + m_{2} K_{2} )m_{2} - (\phi_{2}^{*} - m_{2} K_{2} )m_{1} + K_{2} \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial m_{2} }}{\partial t} + u\frac{{\partial m_{2} }}{\partial x}} \right) = \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial \phi_{2}^{*} }}{\partial t} + u\frac{{\partial \phi_{2}^{*} }}{\partial x}} \right) $$

(36)

The items with K _i are negligible in Eqs. [35] and [36], thus

$$ (1 - \phi_{1}^{*} )m_{1} - \phi_{1}^{*} m_{2} = \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}\frac{{\partial T_{\rm{M}} }}{\partial t} + u\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}\frac{{\partial T_{\rm{M}} }}{\partial x}} \right) $$

(37)

$$ - \phi_{2}^{*} m_{1} + (1 - \phi_{2}^{*} )m_{2} = \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}\frac{{\partial T_{\rm{M}} }}{\partial t} + u\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}\frac{{\partial T_{\rm{M}} }}{\partial x}} \right) $$

(38)

According to the phase diagram, the saturation concentration ϕ ^* _i is function of the equilibrium temperature T _M. For a certain temperature T _M, the saturation concentration ϕ ^* _i is determined.

The solution for above Eqs. [37] and [38] is

$$ \left[ {\begin{array}{*{20}c} {m_{1} } \\ {m_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {1 - \phi_{1}^{*} } & { - \phi_{1}^{*} } \\ { - \phi_{2}^{*} } & {1 - \phi_{2}^{*} } \\ \end{array} } \right]^{ - 1} \left( {\left[ {\begin{array}{*{20}c} {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} \\ {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\rho_{\rm{L}} \varepsilon_{\rm{L}} \frac{{\partial T_{\rm{M}} }}{\partial t}} \\ {\rho_{\rm{L}} u\frac{{\partial T_{\rm{M}} }}{\partial x}} \\ \end{array} } \right]} \right) = [F]\left[ {\begin{array}{*{20}c} {\rho_{\rm{L}} \varepsilon_{\rm{L}} \frac{{\partial T_{\rm{M}} }}{\partial t}} \\ {\rho_{\rm{L}} u\frac{{\partial T_{\rm{M}} }}{\partial x}} \\ \end{array} } \right] $$

(39)

$$ [F] = \left[ {\begin{array}{*{20}c} {1 - \phi_{1}^{*} } & { - \phi_{1}^{*} } \\ { - \phi_{2}^{*} } & {1 - \phi_{2}^{*} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} \\ {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} \\ \end{array} } \right] = \frac{{\left[ {\begin{array}{*{20}c} {1 - \phi_{2}^{*} } & {\phi_{1}^{*} } \\ {\phi_{2}^{*} } & {1 - \phi_{1}^{*} } \\ \end{array} } \right]}}{{1 - \phi_{1}^{*} - \phi_{2}^{*} }}\left[ {\begin{array}{*{20}c} {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{1}^{*} }}{{\partial T_{\rm{M}} }}} \\ {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} & {\frac{{\partial \phi_{2}^{*} }}{{\partial T_{\rm{M}} }}} \\ \end{array} } \right] $$

(40)

Define F _L as

$$ F_{\rm{L}} = [F]_{11} L_{1} + [F]_{21} L_{2} = [F]_{12} L_{1} + [F]_{22} L_{2} $$

(41)

By assuming $T_{\rm{L}} \approx T_{\rm{M}}$, giving

$$ L_{1} m_{1} + L_{2} m_{2} = F_{\rm{L}} \rho_{\rm{L}} \left( {\varepsilon_{\rm{L}} \frac{{\partial T_{\rm{L}} }}{\partial t} + u\frac{{\partial T_{\rm{L}} }}{\partial x}} \right) $$

(42)

Define effective heat capacity C _E as

$$ C_{\rm{E}} = C_{\rm{L}} + F_{\rm{L}} $$

(43)

Substituting C _E, the energy equations are integrated as

$$ C_{\rm{E}} \rho_{\rm{L}} \varepsilon_{\rm{L}} \frac{{\partial T_{\rm{L}} }}{\partial t} + C_{\rm{E}} \rho_{\rm{L}} u\frac{{\partial T_{\rm{L}} }}{\partial x} + \sum {(\varepsilon_{{\rm{S}},i} \rho_{{\rm{S}},i} C_{{\rm{S}},i} )} \frac{{\partial T_{\rm{L}} }}{\partial t} = \frac{\partial }{\partial x}\left( {k_{\rm{eff}} \frac{{\partial T_{\rm{L}} }}{\partial x}} \right) + q_{\gamma \alpha } $$

(44)

Because the coefficient C _E is determined by the temperature and the eutectic curve, the temperature and the concentration are strongly coupled through Eq. [44].

Nomenclature

Symbol	Definition	Value
Variables, properties, and physical constants used in Model
C	Specific heat capacity (J/kg K)
	Liquid electrolyte	1880
	Solid cryolite	1100 to 1660
	Solid alumina	900 to 1260
C E	Effective specific heat capacity (J/kg K)	2 × 103 to 104
D	Diffusion coefficient (m2/s)	10−8
e	Emissivity (m/s2)	0.4
g	Gravity acceleration (m/s2)	9.8
h C	Convective heat transfer coefficient (W/m2 K)	5
h f	Convective heat transfer coefficient (W/m2 K)	500
R	Effective capillary radius (m)	R = 2εL/S
k eff	Effective thermal conductivity (W/mK)
	Alumina-based crust	0.4 to 0.8
	Alumina loose cover	0.1 to 0.3
	Crushed bath crust	1.0 to 1.5
	Crushed bath loose cover	0.2 to 0.8
K R	Reaction rate constant (L/min)	0.05
L	Latent heat (J/kg)
	Eutectic reaction	7.56 × 105
	Chiolite melting	5.12 × 105
S	Surface area per unit volume (m2/m3)
	Sandy alumina	7.7 × 107
ρ	True density (kg/m3)
	Liquid electrolyte	2100
	Solid cryolite	2970
	Solid alumina	3950
ε	Volume fraction
	Initial porosity of sandy alumina	0.7
µ	Viscosity of the liquid electrolyte (mPa s)	2.3
σ	Stefan-Boltzmann constant (kg/s3 k4)	5.67 × 10−8
γL	Surface tension of the liquid electrolyte (N m−1)	0.1
θ	Contact angle between the liquid electrolyte and the solid	cosθ ≈ 1

Symbol	Definition
Other nomenclature
h	Height position in the anode cover/crust (m)
h R	Effective radiative heat transfer coefficient (W/m2K)
h V	Volumetric convective heat transfer coefficient (W/m3K)
H	Enthalpy (J)
J	Diffusion flux (kg/m2s)
K h	Hydraulic conductivity(m/s)
m	Phase transformation rate (kg/m3s)
t	Time (s)
T	Temperature (°C)
p	Pressure (Pa)
Q	Heat flux (w/m2)
q	Volumetric heat generation rate (w/m3)
q γα	Volumetric heat generation rate (w/m3)
u	Superficial fluid velocity (m/s)
x	Coordinate (m)
Ø	Weight fraction in the liquid electrolyte

Subscript	Definition
1,2,3,…8	Alumina, cryolite, AlF3, chiolite, LiF, KF, MgF2, and CaF2
A	Air
B	Bulk electrolyte
C	Loose cover
e	Liquid electrolyte front (eutactic reaction)
L	Liquid electrolyte
M	Interface between the liquid electrolyte and solid phase
p	Chiolite melting
S	Solid phase
γ	Transition alumina