Imbibition with solidification in alumina feeding

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MathematicalInstituteUniversityofOxford

Imbibition with solidification in alumina feeding

Attila Kovacs

InFoMM CDT, Mathematical Institute, University of Oxford Chris Breward, James Oliver and Andreas Münch (Oxford) Svenn Anton Halvorsen and Ellen Nordgård-Hansen (NORCE)

Eirik Manger (Hydro Aluminium) APS-DFD (Seattle) 2019

November 25, 2019

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Alumina feeding: raft problem

Fluid Saturated

Uninfiltrated

Dissolution boundary Infiltration boundary

Front movement Mush?

A

time

C

E D B

Source: [Kaszas2016]

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1D Infiltration 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

Similar models e.g. [Mortensen1989]

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1D Infiltration 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 Infiltration 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 Infiltration problem

c(t)

b(t) H

y 1

Infiltrated For 0<y <b(t):

∂T1

∂t + ∂

∂y (qφcu1T1) =α_ip∂²T1

∂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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Similarity solutions for A = 10, 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.S

For a parameter set two different solutions are possible: faster

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

[Tsypkin2005]

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Reason for multiple solutions

Frozen fraction (S)

Functionvalue

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

S 1/A

e^G^(S)²√πerfc (G(S))G(S)S

Nonlinear system splits into“fluid”( ) and “solidification” ( ) parts.

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Parameter dependence

100⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 0.2

0.4 0.6 0.8 1

Flow Number (γ)

FrozenFraction(S)

A= 2 A= 5 A= 10

10⁰ 10¹ 10²

0 0.2 0.4 0.6 0.8 1

Flow Number (γ)

FrozenFraction(S)

n= 0 n= 2/3

n= 1

10⁰ 10¹ 10² 10³ 10⁴ 0

0.2 0.4 0.6 0.8 1

Flow Number (γ)

FrozenFraction(S)

r= 1/2 r= 1 r= 2

Changing the density slightly changes the necessary pressure needed for infiltration, the form of the capillary pressure function changes the behaviour only at large frozen fractions and A has large effect on the solution.

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Regime diagram for early time solutions

Two infiltrating solutions No infiltrating solution

Flow number (γ)

Stefannumber(A/γ)

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

10⁻³ 10⁻² 10⁻¹ 10⁰

Changing the parameters of the operating regime (dashed) can change the behaviour that we would expect happening.

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Further stages

t t

y Air y

F+S Fluid

Air Fluid F+S

Solid Partial infiltration Complete Infiltration

t1

t2

t1

t2

y_max y_c

Depending on the boundary condition at the top, the raft can either get fully infiltrated or partially.

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Numerical results

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

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Numerical results

Unstable solution switches to the stable one on a short timescale, then clogs due to the boundary.

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Time vs Nusselt number

Complete Infiltration

Partial Infiltration

Nusselt number (Nu) InfiltrationTime(tinf)

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

0.3 0.4 0.5

0.6 θe= 0

θ_e= 1

The front stops later than in the isothermal infiltration case ( ), the behaviour is non-monotonous (balance between clogging — infiltrating).

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Complete vs Partial infiltration

Complete Partial

10⁰ 10¹ 10² 10³

0 0.2 0.4 0.6 0.8 1

Convection Coefficient (Nu) EnvironmentTemperature(θe)

0.88 0.9 0.92 0.94 0.96 0.98 1

Measuring the infiltrated height (colour) can be used to identify the convection coefficient, furthermore the infiltrated height is also correlated with the apparent density of the raft.

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Conclusions and Future work

Conclusions:

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

• Investigated the relevant small overheat 1−θ1 limit having an interesting type of Stefan condition coupling Darcy flow to heat equation.

• Similarity solution at small times yields nonuniqueness with one stable solution and nonexistence in certain regions of

parameters.

• Late time simulations show either clogging or complete infiltration depending on the top boundary condition.

Future work:

• Modelling the disintegration of the raft (next stage of the industrial problem)

• Refine physics (dropping LTE, composition effects)

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Thank you!

This work is supported by the EPSRC Centre For Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with NORCE and Hydro Aluminium. 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.

Csilla Kaszás, László I Kiss, Sándor Poncsák, and Jean-François Bilodeau.

Flotation and Infiltration of Artificial Alumina Rafts on the Surface of Molten Cryolite.

InICSOBA, Quebec, 2016.

A. Mortensen, L. J. Masur, J. A. Cornie, and M. C. Flemings.

Infiltration of fibrous preforms by a pure metal: Part I. Theory.

Metallurgical Transactions A, 20(11):2535–2547, 1989.

G. G. Tsypkin.

Two-Valued Solutions in the Problem of Salt Precipitation during Groundwater Evaporation.

Fluid Dynamics, 40(4):593–599, 2005.