Multiphase model

3.3.1 Overview of multiphase models

The models for simulating multiphase flows can generally be grouped into three classes:

 Direct Numerical Simulation (DNS)

 Eulerian-Lagrangian

33 Direct numerical simulations (DNS) can be used to obtain a full resolution of the behaviour of bubbles or particles in a carrying fluid. Multiphase DNS therefore involves simulations in which one fully resolves the temporal and spatial scales relevant to the fluid dynamics [77]. One of the most widely used methods in multiphase DNS is the Volume of Fluid (VOF) method. In the VOF model the interface between the different phases is tracked. The VOF model is suitable for simulating stratified flows, free surface flows and the movement of large bubbles in a liquid [78]. Since the interface between the phases must be resolved, VOF is not applicable for systems with a large number of bubbles or liquid droplets.

In the Eulerian-Lagrangian modelling method, the continuous phase is modelled as a continuum by solving the Navier-Stokes equations. On the other hand, the dispersed (secondary) phase is simulated in a Lagrangian frame of reference by tracking the motion of the individual particles (or bubbles) using an equation of motion. However, the computational cost associated with computing individual particle or bubble trajectories is high and increases with an increase in the number of simulated particles. The Eulerian-Lagrangian method is therefore limited to dilute systems in which the dispersed phase volume fraction is less than 10% [78, 79].

In the Eulerian-Eulerian modelling approach, the different phases are considered as continuous phases that are interacting with each other. Conservation equations governing the balance of mass, momentum, and energy are in this case solved for each phase separately. However, the interaction between the different phases must be accounted for through additional models.

3.3.2 Choice of multiphase model for the present research

The multiphase CFD model used in this research was selected after a thorough literature review of CFD models of bubble column reactors and flotation columns. The ‘hierarchy of models’ concept that was introduced by Delnoij et al. [80] was particularly useful in the selection of a suitable multiphase model for industrial and pilot scale flotation columns. A range of hydrodynamic phenomena characterised by different length and time scales occur in bubble columns. In addition, a wide range of bubble diameters and shapes can also be encountered depending on the physical properties of the liquid phase and the flow regime in which the bubble column is being operated. Bubble columns also differ in sizes and geometry. It is, therefore, difficult to develop a generalized CFD model that is capable of

34 resolving all the different hydrodynamic features occurring at the various length and time scales.

The ‘hierarchy of models’ concept recommends different CFD models depending on the size of the column, the length scales of the particular hydrodynamic phenomena, and the prevailing flow regime. In other words each model is applied for studying specific hydrodynamic phenomena, occurring at a defined scale [81].

According to Delnoij et al. [80] a two-fluid model is recommended to study flow structures occurring in industrial scale bubble columns due to its relatively lower computational cost. An Eulerian-Eulerian two-fluid model was therefore chosen in the present research considering the size of the simulated flotation columns. The Eulerian-Eulerian approach has been applied in several CFD models of pilot-scale bubble columns [22, 32, 82] of similar dimensions as the flotation columns that are simulated in the present research. A review of previous CFD models involving column flotation also found that the Eulerian-Eulerian model was used in most of these studies[11, 13, 14, 17].

3.3.3 Eulerian-Eulerian multiphase model

In the Eulerian-Eulerian approach, both the continuous (primary) phase and the dispersed (secondary) phase are modelled in an Eulerian frame of reference as interpenetrating continua. The gas, solid, and liquid phases are considered as three distinct phases that are interacting with each other. Conservation equations for mass and momentum are therefore solved for each phase separately. Interaction between the phases is then accounted for by means of momentum exchange terms incorporated in their respective momentum equations. In the present research, the multiphase flow in the flotation columns was modelled considering water as the continuous phase (or primary phase) while air bubbles and solid particles were treated as dispersed phases (or secondary phases). The volume averaged mass and momentum conservation equations are as follows, respectively:

𝜕

𝜕𝑡(𝜀𝑞𝜌𝑞) + 𝛻. (𝜀𝑞𝜌𝑞𝒖⃗⃗⃗⃗ ) = 𝑆𝑞 𝑞 (3.3) 𝜕

𝜕𝑡(𝜀𝑞𝜌𝑞𝒖⃗⃗⃗⃗ ) + 𝛻. (𝜀𝑞 𝑞𝜌𝑞𝒖⃗⃗⃗⃗ ⊗ 𝒖𝑞 ⃗⃗⃗⃗ ) = −𝜀𝑞 𝑞𝛻𝑝 + 𝛻. 𝜀𝑞𝜏̿̿̿ + 𝜀𝑞 𝑞𝜌𝑞𝒈⃗⃗ + 𝑀𝐺,𝐿 (3.4) where q is the phase indicator, q = L for the liquid phase, q = G for the gas phase, εq is the

35 source term. 𝑀_𝐺,𝐿 is the interaction force between the phases and ερg is the gravity force, while τ̿ is the qth phase stress-strain tensor given by: q

𝜏_𝑞

̿̿̿ = 𝜀_𝑞𝜇_𝑞(𝛻𝒖⃗⃗⃗⃗ + 𝛻𝒖_𝑞 ⃗⃗⃗⃗ _𝑞𝑇) + 𝜀_𝑞(𝜆_𝑞−2

3𝜇𝑞) 𝛻. 𝒖⃗⃗⃗⃗ 𝐼̿ 𝑞 (3.5) The liquid phase was modelled as incompressible hence its continuity (mass conservation) equation is simplified as follows:

𝜕𝜀𝐿

𝜕𝑡 + ∇. (𝜀𝐿𝜌𝐿𝒖⃗⃗⃗⃗ ) = 0 𝐿 (3.6) Interaction between the phases is generally accounted for through inclusion of the drag force while non drag forces such as the virtual mass and lift force can be neglected [83-85]. Momentum exchange between the phases was therefore accounted for by means of the drag force only in this research. The different drag models that have been used in this research are described in subsequent chapters.