Mathematical modelling of alumina feeding

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MathematicalInstituteUniversityofOxford

Mathematical modelling of alumina feeding

Attila Kovacs

InFoMM CDT, Mathematical Institute, University of Oxford Chris Breward, James Oliver and Andreas Münch (Oxford)

Svenn Halvorsen and Ellen Nordgård-Hansen (Norce) Eirik Manger (Norsk Hydro)

ICIAM 2019

July 18, 2019

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Hall-Héroult cell

Alumina (300K)

Electrolysis in the hot cryolite (1230K)

Cathode Molten aluminium

Anode Anode

Tapping Crust

Current (200 kA)

Insulation

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Alumina feeding processes

A

time

C

E D B

Main question: How does the molten cryolite inltrate and dissolve a porous alumina structure?

Focus of talk is Stage B, raft problem.

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Alumina feeding: raft problem

Fluid Saturated

Uninfiltrated

Dissolution boundary Infiltration boundary

Front movement Mush?

A

time

C

E D B

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Alumina feeding: raft problem

x y

Uninfiltrated

Saturated

Fluid Mush

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1D Inltration problem

Tc

Tm

y Front movement

φ₀

(top)

Solid Alumina Liquid Cryolite

Solid Cryolite

Air 1

Volume fraction Temperature

y =H y =c(t)

y =b(t)

φ₀+ψ

Saturated Mush Uninfiltrated

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1D Inltration problem

c(t)

b(t) H

y 3 Uninfiltrated

Forc(t)<y <H:

∂T₃

∂t = ∂²T₃

∂y² ,

At y =c(t) : T₃ =1−ε, ψc˙ =−Stkpf

∂T₃

∂y , At y =1: ∂T₃

∂y =Nu(T −θe)

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1D Inltration problem

c(t)

b(t) H

y Mushy 2

Forb(t)<y <c(t):

∂ψ

∂t =0,

∂

∂y ((1−φ−ψ)u₂) =0,

−γ K(ψ+φ) (1−φ−ψ)

∂p₂

∂y +β

=u₂,

At y =b(t) : [pi]⁺₋=0, u₁−1−φ−ψ

1−φ u₂=ψb˙1−ρ 1−φ, At y =c(t) : p₂ =−1, u₂ = ˙c

1+ ψρ

1−φ−ψ

,

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1D Inltration problem

c(t)

b(t) H

y 1

Infiltrated For 0<y <b(t):

∂T₁

∂t + ∂

∂y (qφ_cu₁T₁) =α_ip∂²T

∂y²,

∂

∂y ((1−φ)u₁) =0, γ−K(φ)

(1−ψ) ∂p₁

∂y +β

=u₁,

At y=0: T₁ =1, p₁ =0,

At y =b(t) : T₁ =1−ε, ψb˙=−Stkif

∂T₁

∂y , [pi]⁺₋=0, u₁−1−φ−ψ

1−φ u₂=ψb˙1−ρ 1−φ,

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Dimensionless parameters

Meaning and Symbol Denition Value

Inltrating force (γ) KP/(αpµ_c) 10200 Stefan number (St) α_fp[T]cf/L 2.3

Newton cooling (Nu ) hH/kp 3

Diusivity ratio (α_ip ) α_i/α_p 2

Overheat (ε) Tc/Tm−1 0.015

Environment temperature ratio (θ) TA/Tm 0.86 Conductivity ratio (kif) ki/kf 2.5 Conductivity ratio (kpf) kp/kf 0.5

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Dimensionless parameters

Meaning and Symbol Denition Value

Inltrating force (γ) KP/(αpµ_c) 10200 Stefan number (St) α_fp[T]cf/L 2.3

Newton cooling (Nu ) hH/kp 3

Diusivity ratio (α_ip ) α_i/α_p 2

Overheat (ε) T_c/T_m−1 0.015

Environment temperature ratio (θ) TA/Tm 0.86 Conductivity ratio (kif) ki/kf 2.5 Conductivity ratio (kpf) kp/kf 0.5

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Small overheat ( ε 1) limit

In this limit the inltrated region does not exist, so only mushy region and uninltrated is important. For the uninltrated region we

have ∂T₃

∂t = ∂²T₃

∂y² in c(t)<y<1, with boundary conditions,

T₃=1 at y =c(t), T₃ =0 at y =1

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Small overheat ( ε 1) limit

have ∂T₃

∂t = ∂²T₃

T₃=1 at y =c(t), T₃ =0 at y =1 and for the mushy region we have (with suitable boundary conditions)

∂ψ

∂t =0,

∂

∂y ((1−φ−ψ)u₂) =0,

−γ K(ψ+φ) (1−φ−ψ)

∂p₂

∂y +β

=u₂,

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Small overheat ( ε 1) limit

have ∂T₃

∂t = ∂²T₃

T₃=1 at y =c(t), T₃ =0 at y =1 uid equations can be integrated in space and lumped into the boundary conditions giving

ψ=−∂T₃

∂y I

St, c˙= 1

I, c(0) =0 with introducing

I = Z _c(t)

0

1

γK(φ(x))dx, I˙= 1

γIK(ψ(c)), I(0) =0

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Small time asymptotics

• Expandc =O(√

t)as t →0⁺, so need smallt asymptotics to initiate numerical solution, because 0 is singular.

• Seek similarity solution given T ∼T(η), c ∼2λ_c√

t, I ∼2λ_I√ t withη=y/2√

t =O(1) as t→0⁺ andλc,λ_I to be determined from a system of nonlinear equations.

t

y O(√

t)

c(t)

1

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Similarity solutions for St = 1, γ = 10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4 0.6 0.8 1

Temperature(T3)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 0.2 0.4

Similarity variable (η)

Frozenf.ψ

For a parameter set two dierent solutions are possible: faster

propagating with less freeze ( ) or slower moving with more freeze ( ).

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Numerical results (fast)

Stable propagation agrees with small-time solution, then clogs due to the boundary.

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Numerical results (slow)

Unstable solution switches to the stable one in short time, then clogs due to the boundary.

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Regime diagram for early time solutions

Flow number (γ)

StefannumberSt/(γ)

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁻³

10⁻² 10⁻¹ 10⁰

Region of no solution and two solution

and the operating regime for the real system (dashed).

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Conclusions and Future work

Conclusions:

• Developed a multiphase model for the inltration of molten cryolite into a cold porous alumina

• Investigated the relevant small overheat 1−θ1 limit having a new type of Stefan condition coupling Darcy ow to heat equation

• Similarity solution at small times yields nonuniqueness with one stable solution and nonexistence

• Simulations show either clogging or full inltration depending on the top boundary condition

Future work:

• What is the solution when there is nonexistence? (Dierent model)

• What are the next stages of the evolution?

• Rene physics (top boundary, pore size dependent capillary pressure, dropping LTE, composition eects)