Particulate phase considerations

The discrete phase is substantially different from the gaseous phase, which is introduced in this section. Again, readers may refer to established literature for more detail [72, 73]. The second phase consists of discrete, non-connected solid particles, that can physically interact with each other and the gas phase by momentum and energy transfer, but not exchange any mass. While the gas phase is treated as a continuum (indexc), the dispersed phase (index p) generally cannot be treated as such. This depends on the classification of the mixture, in particular on the volume fraction of particles αp. This is the volume

discrete volume element ∆V, which approaches an adequately large size with a sufficient number of particles, such that a mixture (indexm) can be defined.

αp = lim ∆V→∆Vm

∆Vp

∆V (2.13)

The bulk density ρ_p has an analogue definition using the particle mass instead of the volume, and can be formulated likewise for the continuous phase respectively. The particle loading can also be expressed in terms of the mass concentration C and the mass loading Z, as the proportions of the mass or mass flow rate.

C= ρp ρc (2.14) Z = m˙p ˙ mc = ρpvp ρ_cvc (2.15)

A second important notion for the classification of particle laden flows is the particle response time, more precisely, the momentum or velocity response time τV. Derived from

a simplified equation for the motion of a single particle, a characteristic time can be obtained, that represents the time scale of response of the particle velocity to a change in relative velocity of the continuous phase.

τV =

ρpd2p

18µ (2.16)

In order to estimate whether a flow is rather dense or dilute, the response time can be compared to the mean time between particle collisions. Therefore, both the volume fraction and the response time determine the classification of the flow regime. If the volume fraction is very low and the response time short, the flow is called dilute; dense in case of the contrary. There is a broad range in which the flow is neither clearly dense nor dilute. However, for micron sized particles, the transition from dilute to dense can be estimated for volume fractions in the order of αp = 0.01-1%, or even lower for dense

materials [73]. A very dilute flow is dominated by particle-fluid interactions, whereas the particle-particle interactions primarily influence the flow in a highly dense regime.

The phase coupling must be distinguished from the diluteness of the flow, although an extremely dense flow most likely exhibits strong phase coupling and in reverse. The

degree of coupling of the phases is based on the momentum and energy exchange between the phases. Therefore, it is useful to classify the flow additionally according to a strong or weak coupling.

Based on the response time, the momentum Stokes number Sk can be formulated as the proportion of that response time and a characteristic flow time scale τF, which

depends on the phenomena of interest. For example, the turbulence interaction could be characterised by the respective turbulent time scale, such as the integral length scale divided by the mean convection velocity.

Sk = τV τF

(2.17)

The momentum coupling parameter can be used for the estimation of the strength of the momentum exchange. It is defined as the relation of the drag force acting on the particles in a volume and the continuous phase momentum flux through the volume. This can be transformed using the mass concentration C (equation 2.14 ) to the following expression.

Πmom = C Sk 1₋ vp vc = C 1 +Sk (2.18)

Hence, in principle, for large Stokes numbers (Sk _≫ 1), the particles have little time to respond to changes in fluid flow. In case of a sufficiently low concentration, the motion of the gas phase can be then considered decoupled from the disperse phase, called a one- way coupling. If the Stokes number is low (Sk _≪ 1), the particles can be considered in velocity equilibrium with the continuous fluid, but for high concentrations, this implies strong momentum exchange. Additionally the momentum exchange for moderate Stokes numbers and concentrations can be increased by a large difference in velocity. In either case, the phases are expected to be coupled; a two-way coupling. For denser flows, the particle-particle interactions become increasingly significant and extend the coupling to a four-way coupling [74]. The above formulations can also be obtained for the energy coupling, based on thermal response times.

For a dense regime, in which particle motion is driven by strong interactions with their neighbours like in the continuous phase, it is possible to reformulate the conservation equations for the gas-particle mixture as a two-fluid model, making use of the volume

fraction and bulk density. Since these equations are derived in a locally fixed, Eulerian reference frame, this approach is called an Eulerian-Eulerian framework, in which both phases are solved as inter-penetrating continua. It requires the volume fraction to be a continuous function in time and space and only few CS applications were faced in this manner.

A different, more common approach is the Eulerian-Lagrangian framework, in which the gas phase is calculated in an Eulerian reference frame and the elements of the dispersed phase are computed in a Lagrangian reference frame that moves with the particles. It is a particularly useful description if the mixture is sufficiently dilute for individual particles to be only influenced by the surrounding fluid. In this case, the respective trajectories are directly calculated by a simple integration of the drag-driven particle momentum conservation equation, so for a one-dimensional case it can be expressed as the following scalar equation. mp dvp dt = 1 2ρCDAp(vc−vp)|vc −vp| (2.19) In this 1D-expression, the connection to the particle Reynolds number can already be noted through the relative velocity magnitude _|vc −vp|. It is now required to define the

particle Reynolds and Mach numbers, which characterise the relative flow of gas around particles. Rep = ρdp|vc −vp| µ (2.20) M ap = | vc−vp| √ γRT (2.21)

Using these and extending the momentum conservation for a single particle to three spatial dimensions, the expression can be transformed to a differential vector equation per unit mass, that shows the role of the particle relaxation time τr and the involvement of the

Reynolds number in it; here finally given in vector notation for 3D. d~vp dt = 1 τr (~vc−~vp) + ~g(ρp−ρc) ρp +F~ (2.22) 1 τr = 18µ ρpd2p · CDRep 24 (2.23)

additional body forces, such as added mass or lift forces, may arise in the term F~ on the right hand side. The calculation of the drag coefficient CD is a subtle topic in itself,

because it depends strongly on the flow conditions and the particle shape. It represents the micro-flow around the particle, in particular depending onRep andM ap, and therefore

varies with location and flow conditions. Some models to calculate these quantities for CS are discussed in the review of numerical approaches in section 2.4.

Having now set the basic terminology of gas-particle nozzle flows, all concepts are introduced that are required to follow the respective details in published CS research as well as the present work.