Supplementary material for
“An overall furnace model for the silicomanganese process”
Manuel Sparta · Vetle Kjær Risinggård · Kristian Etienne Einarsrud · Svenn Anton Halvorsen
1 Granular flow and the weak form
The differential equations which are given in Section 3 in the main manuscript are solved using the finite-element method.
Finite-element solvers consider the corresponding integral equations using the so-called weak formulation [1]. The sink term in the equations of motion for granular material explic- itly refers to the weak form. In this Supplementary Material we write out these equations and provide additional details for the implementation of this sink term.
Neglecting for the moment the position-dependence of the viscosity, the Navier–Stokes equations for incompressible flow can be written on vector form as [2]
𝜌 𝜕𝒗
𝜕 𝑡 + (𝒗· ∇)𝒗
=−∇𝑝+𝜂∇2𝒗, (1a)
∇ ·𝒗=0. (1b)
Consider Equation (1b) — the continuity equation. Any so- lution of the continuity equation is also a solution of the following integral equation,
∫
Ω
dΩ𝜙𝑖∇ ·𝒗=0, (2)
M. Sparta
NORCE Norwegian Research Centre AS, Universitetsveien 19, NO- 4630 Kristiansand S, Norway
E-mail: masp@norceresearch.no V. K. Risinggård
NORCE Norwegian Research Centre AS, Universitetsveien 19, NO- 4630 Kristiansand S, Norway
E-mail: veri@norceresearch.no K. E. Einarsrud
Department of Material Science and Engineering, Norwegian Univer- sity of Science and Technology, NO-7491 Trondheim, Norway E-mail: kristian.e.einarsrud@ntnu.no
S. A. Halvorsen
NORCE Norwegian Research Centre AS, Universitetsveien 19, NO- 4630 Kristiansand S, Norway
E-mail: svha@norceresearch.no
where𝜙𝑖 is an arbitrary function and Ω is the domain in which the solution is valid. In fact, this integral equations puts less requirements on𝒗than does the original differential equation. Accordingly, the differential equation is known as the strong formand the integral equation is known as the weak form.However, if we require that 𝒗fulfils the weak- form equation for a large number𝑁of test functions𝜙𝑖, the solution of the weak-form equation approaches the solution of the strong-form equation. Herein lies the power of the finite-element method [1].
The weak form equation can be simplified and supplied with boundary conditions using the following procedure.
Writing out∇ · (𝜙𝑖𝒗)we get∇ · (𝜙𝑖𝒗)=𝒗· ∇𝜙𝑖+𝜙𝑖∇ ·𝒗. We recognize the last term from the continuity equation, hence:
∫
Ω
dΩ 𝜙𝑖∇ ·𝒗=
∫
Ω
dΩ∇ · (𝜙𝑖𝒗) −
∫
Ω
dΩ𝒗· ∇𝜙𝑖 =0. (3) The first term on the right-hand side can be rewritten using the divergence theorem,
∫
Ω
dΩ∇ ·𝑭=
∫
𝜕Ω
d𝑠 𝒏·𝑭, (4)
where 𝑭 is an arbitrary vector field, 𝜕Ω is the boundary of Ω, and𝒏 is the outward-pointing normal of 𝜕Ω. Using the divergence theorem with𝑭 = 𝜙𝑖𝒗gives the weak-form equation [1]
∫
𝜕Ω
d𝑠 𝒏· (𝜙𝑖𝒗)=
∫
Ω
dΩ𝒗· ∇𝜙𝑖. (5)
Similarly, Equation (1a) — the momentum equation — can be written as
∫
Ω
dΩ𝜓𝑖𝜌 𝜕𝒗
𝜕 𝑡 + (𝒗· ∇)𝒗
=
∫
Ω
dΩ 𝜓𝑖(−∇𝑝+𝜂∇2𝒗) (6) using the test function𝜓𝑖. We can now use the same proce- dure as for the continuity equation for each vector component of the momentum equation. Symbolically, we can write the
2 Manuel Sparta et al.
following identity, ∇ · (𝜓𝑖∇𝒗) = 𝜓𝑖∇2𝒗+ ∇𝜓𝑖· ∇𝒗, which holds for each component of𝒗. Using the divergence theorem we get [1]
∫
𝜕Ω
d𝑠 𝒏· (𝜓𝑖∇𝒗)=
∫
Ω
dΩ𝜓𝑖𝜌 𝜕𝒗
𝜕 𝑡
+ (𝒗· ∇)𝒗
+
∫
Ω
dΩ𝜓𝑖∇𝑝+
∫
Ω
dΩ ∇𝜓𝑖·𝜂∇𝒗. (7) By building the solutions𝒗and𝑝from the test functions 𝜓𝑖and𝜙𝑖,
𝒗=Õ
𝑗
𝒗𝑗𝜓𝑗, (8a)
𝑝=Õ
𝑗
𝑝𝑗𝜙𝑗, (8b)
the integrals are reduced to overlap integrals between test functions of our choice, which greatly simplifies their eval- uation [1]. As a result of the association of 𝜓𝑖 with𝒗 and 𝜙𝑖 with𝑝, we call 𝜙𝑖 the pressure test functions and𝜓𝑖 the velocity test functions.
We are now ready to discuss the source term in the granular-flow equations. The continuity equation∇ ·𝒗= 0 states that matter cannot be created nor disappear within the domain of flow. Adding a weak contribution𝑆 𝜙𝑖 amounts to rewriting the continuity equation as
∫
𝜕Ω
d𝑠 𝒏· (𝜙𝑖𝒗)=
∫
Ω
dΩ 𝒗· ∇𝜙𝑖+
∫
Ω
dΩ𝑆 𝜙𝑖. (9)
The added term creates matter at a rate𝑆. We have set𝑆to 𝑆 =
(−𝑟(𝜒void−𝑇), 𝜒void> 𝑇 ,
0, 𝜒void≤𝑇 ,
(10) where 𝜒void is the void volume fraction, 𝑇 is a threshold value, and𝑟is the rate.
Granular flow is characterized by partial slipping at the container walls [3] which is taken into account using Navier- slip boundary conditions [4, 5].
2 Liquid flow
The flow of liquid species through the bottom of the charge and the coke bed mainly takes place by formation of droplets that trickle through the granular layers [6, 7]. In the spirit of our simplified model, we refrain from resolving the trajec- tory of the individual droplets and the liquid flow is taken into account using a velocity field that is computed once at the start of the simulation (using the Navier–Stokes equa- tions (1)) and reused in the advection–diffusion equations of the liquid species throughout all subsequent time steps.
This is equivalent to pre-computing the average trajectories that a droplet would follow when formed in any point of the furnace. A single free parameter (the superficial velocity
at the bottom of the charge) controls the time used by the droplets to follow the trajectories. The superficial velocity represents the average behavior and takes values between 0 m/s (droplets trapped in the coke bed) and 0.5 m/s (small droplets in free fall). The results in the main manuscript have been obtained with a superficial velocity of 1.6 mm/s.
In the current model, all liquid species share the same superficial velocity. Our plan for future implementations is to differentiate between alloy and slag and to link this parameter to the viscosity, so that the alloy trickles faster than the slag.
Furthermore, the viscosity of the slag will depend on its composition.
3 Absolute concentrations
The concentrations of the different chemical species are plot- ted as volume fractions in the main manuscript. Figure S-1 shows the same plots in terms of absolute concentrations (mol/dm3).
References
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Engineering Experiment Station NP-22770, University of Utah, Salt Lake City, Utah, USA (1964). DOI 10.2172/5240257
4. A. Zugliano, R. Artoni, A. Santomaso, A. Primavera, M. Pavliče- vić, inProceedings of the COMSOL Conference 2008 Hannover (Hannover, Germany, 2008)
5. H. Lamb,Hydrodynamics, sixth edn. Cambridge Mathematical Library (Cambridge University Press, 1993)
6. S. Letout, A.P. Ratvik, M. Tangstad, S.T. Johansen, J.E. Olsen, in Progress in Applied CFD, SINTEF Proceedings (SINTEF Academic Press, 2017), pp. 599–604
7. W.M. Husslage, T. Bakker, A.G.S. Steeghs, M.A. Reuter, R.H.
Heerema, Metallurgical and Materials Transactions B36(6), 765 (2005). DOI 10/b9kvsr
Supplementary material for “An overall furnace model for the silicomanganese process” 3
Fig. S-1 Concentration of MnO in the (a) solid and (b) liquid phase. (c) Concentration of liquid Mn. Concentration of SiO2in the (d) solid and (e) liquid phase. (f) Concentrations of of liquid Si. Temperature isotherms are shown.
