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Modelling flotation with sedimentation by a system of
conservation laws with discontinuous fluxRaimund Burger‡, Stefan Diehl§, Marıa del Carmen Martı†, Yolanda Vasquez‡
‡CI2MA and Departamento de Ingenierıa Matematica, Universidad de Concepcion, Chile§Centre for Mathematical Sciences, Lund University, Sweden†Departament de Matematiques, Universitat de Valencia, Spain
[email protected], [email protected], [email protected],[email protected]
1. Three-phase flow of solids, gas (bubbles or aggregates) and fluid
Governing partial differential equations (Burger et al. 2019a,b); φ: volume fraction of gas, ϕ:volume fraction of solids within the fluid, A(z): cross-sectional area, t > 0: time, z: height.
A(z)∂φ
∂t+∂
∂z
(A(z)J(φ, z, t)
)= QF(t)φF(t)δ(z − zF), (1)
A(z)∂
∂t
((1− φ)ϕ
)− ∂
∂z
(A(z)F (ϕ, φ, z, t)
)= QF(t)φs,F(t)δ(z − zF)
air supply
feed
concen-trate
washwater
underflow
z
zE
zW
zF
zU
H
QE
QU
QW
QF
zone 3q = q3A = AE
zone 2q = q2A = AE
zone 1
underflow zone
effluent zone
q = q1A = AU
✻
❄
✻
❄
✻
❄
❄
✻
✻
❄
✻
❄
✻
❄
frothregion
bubblyregion
settlingregion
The fluxes J and F are discontinuous functions of z and incorporate batch drift and solidfluxes. Thus, (1) is a system of conservation laws with discontinuous flux and singular sourceterms.
2. Steady States and Operating Charts
Stationary solutions, which have layers of different concentrations of bubbles and particlesseparated by discontinuities in concentration.
0 0.5 1
Gas conc. [ ]
zU
zF
zW
zE
Heig
ht [z
]
(a)
0 0.5 1
Gas conc. [ ]
zU
zF
zW
zE
(b)
0 0.5 1
Gas conc. [ ]
zU
zF
zW
zE
(c)
0 0.5 1
Gas conc. [ ]
zU
zF
zW
zE
(d)
The different steady states depend on the values of the feed input volume fractions of theaggregates φF and the solids φs,F, and on the volumetric flow rates QF, QU and QW. De-sired steady states have a high concentration of aggregates at the top (foam) and zero atthe bottom. Three cases differ only in zone 2, where the solution can be constant low (SSl),constant high (SSh), or have a discontinuity separating these two values (SSd).
φSSl(z) :=
φE = AEj3(φ3)/QE ≥ φ3M in the effluent zone,φ3 = φ3M ≥ φ2 in zone 3,φ2 ∈ [φ2m, φ
M2 ] in zone 2,
0 in zone 1 and underflow,
Industrially relevant steady states andoperating charts:The white region in the Figures show thepossible values for (QU, QF). In each suchpoint, there is unique value of QW.
0 50 1000
20
40
60
80
100
0 50 1000
20
40
60
80
100
3. Numerical Results
Example 1: We start from a tank filledonly with fluid at time t = 0 s, where westart pumping aggregates, solids, fluid andwash water, with φF = 0.3 and φs,F =0.1. A first steady state arises after aboutt = 100 s with a low concentration of aggre-gates, then we ‘close’ the top of the tank att = 150 s. The aggregates interact with thesolid phase in zone 1 and leave throughthe underflow outlet. At t = 350 s, the top ofthe column is opened and a desired steadystate of type SSl is reached after t = 4500 s.
(c)(d)
0
100
0.5
400050
1
2000
0 0
0
500
4000
0.05
3000 2000 1001000
0.1
0.15
Once the system is in steady state, wechange, at t = 4500 s, the feed volume frac-tion of aggregates from φF = 0.3 to 0.4,and simulate the reaction of the system.In the corresponding operating chart, thepoint is no longer in the white region; andno steady state of type SSl is feasible.
(a)(b)
Once this new steady state is reached, wechange, at t = 5000 s, the volumetric flowsso that the new pointlies inside the whiteregion of the operating chart (right). TheFigure shows that a second steady state oftype SSl is slowly reached after t = 16000 s.
0
100
0.5
50
1
15000100000
5000
0
500
0.05
15000
0.1
10000100
0.15
5000
0.2
Example 2: We let the simulation run un-til t = 4500 s when the feed volume fractionφF made a step increase from 0.3 to 0.4(as in Example 1). Instead of waiting witha control action to t = 5000 s, we now makethe control action directly at t = 4500 s. Asteady state of type SSl is quickly reachedat about t = 6500 s.
The dynamics of the entire simulation for Example 1 and 2 can be found in the figures:
4. References and Acknowledgements
1. Burger, R., Diehl, S., and Martı, M.C., 2019a, “A system of conservation laws with discontinuous flux mod-elling flotation with sedimentation.” Preprint 2019-09, CI2MA, UdeC.
2. Burger, R., Diehl, S., and Martı, M.C., and Vasquez, Y., 2019b, “A model of flotation with sedimentation:steady states and numerical simulation of transient operation”, proceeding of FOMPEM 2019.
3. Dickinson, J.E. and Galvin, K.P., 2014, “Fluidized bed desliming in fine particle flotation, Part I.”, ChemicalEngineering Science, 108, pp. 283–298.
4. Galvin, K.P. and Dickinson, J.E., 2014, “Fluidized bed desliming in fine particle flotation Part II: Flotation of amodel feed.”, Chemical Engineering Science, 108, pp. 299–309.
Fondecyt project 1170473; CONICYT/PIA/AFB170001; CRHIAM, Proyecto Conicyt/Fondap/15130015; INRIA Associated Team“Efficient numerical schemes for non-local transport phenomena” (NOLOCO; 2018–2020). SENACYT (Panama).
