Some of the flows under investigation in this work are multiphase flows. The dispersed phase in such flows can principally be solved in both Eulerian and Lagrangian frameworks.
In this work, the dispersed phase is solved in a Lagrangian framework, due to some advantages that the Lagrangian framework offers compared to Eulerian methods [224]:
(a) it captures particle/droplet processes such as drag or devolatilization in a closed form and captures their dependence on size due to evolving the full size distribution (i.e., kinetic non-equilibrium between gas and particles of different size is accounted for), (b) it allows for non-equilibrium non-Maxwellian velocity distributions, (c) it shows low numerical diffusion. As a disadvantage, it should be noted that the Lagrangian method is computationally more expensive than standard Eulerian methods. However, given that particles in pulverized coal furnaces show broad size distributions including large particles that are expected to have significant slip velocities, the Lagrangian framework promises to be worth the additional cost. A good alternative that has not been explored in this work is the direct quadrature method of moments (DQMOM) framework developed by Fox [51] and applied by Pedel [156, 157] in the pulverized coal combustion context.
There are different ways to describe the equations for the Lagrangian particle evolu-tion. On the one hand, from a mechanical point of view (i.e., Newton’s second law), the motion of a particle can be described by balancing the forces acting on the particle with its acceleration. On the other hand, the evolution of the dispersed phase can be defined in terms of a ‘droplet distribution function’ (DDF), such as the well known spray equation by Williams [259]. This equation can then be solved by utilizing the stochastic point process representation. In this case, so-called surrogate particles are solved [224]. These surrogate particles are only equivalent to their physical counterparts in a statistical sense.
An advantage of viewing the Lagrangian particle equations as a solution method of the DDF is that not every physical particle needs to be represented in the simulation, but
so-called ‘parcels’ can be used. This is especially important for the work in Chapters 5 and 6, since a total number of coal particles of the order of several billions would have to be solved for. Instead, physical particles are combined in parcels such that all particles within a parcel have the same properties. Clearly, the instantaneous realizations of a par-cel cannot be expected to correspond to the correct physical realizations of the individual particles within the parcel. However, the expectation of the realizations of all parcels corresponds to the expectation of the realizations of all physical particles. Even if parti-cles are not combined in parcels, such as in Chapter 7, but evolve in the LES framework, they should be viewed as surrogate particles, since the influence of subgrid quantities on the particles is modeled by means of a stochastic process. Only in the situation of the carrier-phase DNS in Chapter 8, computational particles can be viewed as corresponding to the physical particles (it should be noted that surrogate particles could be used in a CP-DNS as well, for example if the total number of particles is computationally in-tractable). Interestingly, the Lagrangian solution of the DDF shows similarities to the Lagrangian solution of the transport of the (gas) composition PDF outlined in Chapter 3 and 9. However, the modeling of the surrogate particles may seem more intuitive since the natural choice of solution method for particles is Lagrangian and the modeling for the surrogate particles is mostly equivalent to that of the physical particles.
One of the main assumptions regarding the multiphase flow in this work is that it is dilute. It should be noted that this assumption is not necessary to obtain Eq. 2.52 (neither in the physical nor DDF framework) [224]. However, it is invoked in this work and allows for (a) neglecting the displaced volume of the particles in the transport equations of the gas phase, (b) the use of correlations for drag and heat transfer to be based on isolated particles, (c) neglecting particle collisions and inter-particle effects. A typical rule of thumb for a volume fraction for which neighbor interactions can be neglected is 0.1 [224]. Additionally, a point particle assumption is invoked (not to be confused with point process), assuming that particles are smaller than the smallest turbulence, i.e., Kolmogorov scales. This simplifies the description of interphase exchange terms, since they can be computed assuming that the particles have an infinitesimal size.
The motion of particles, either for a physical particle or modeled surrogate particle or parcel, is described by the balance of particle acceleration and the forces acting on the particle, following Newton’s second law, Eq. 2.52. In the case of the surrogate par-ticles used in the LES, an additional term that represents the influence of the subgrid fluctuations on the particles needs to be considered (last term of Eq. 2.52).
dup = u˜− up
The three terms on the RHS of Eq. 2.52 stand for drag, gravity/buoyancy and forces due to SGS velocity fluctuations. Details on the evaluation of drag and the particle relaxation time are given in Chapter 5 and the last term is detailed in the following.
The last term in Eq. 2.52 describes the influence of the subgrid velocity fluctua-tions on the particle motion. Its treatment mainly follows the methodology proposed by Bini & Jones [12] for the simulation of spray LES, which is based on filtering William’s
spray equation [259]. The model is based on the assumption that the instantaneous state of a particle can be described by a set of independent macroscopic variables, which can be expressed in state space using the fine-grained density function (similar to the gas phase description in terms of transported probability density function, Sec. 3.3.5).
f (ψψψ; t) =
The sample space ψi corresponds to quantity φi, and M is the number of macroscopic variables that describe the particle state. The subscript n denotes the n-th particle. The probability density function of f (ψψψ; t) can be obtained by taking its expectation, i.e., taking the average in the limit of Np → ∞ over independent realizations. In the next step, this PDF is convoluted with the LES filter to obtain the filtered PDF:
P (ψ¯ ψψ, t) = Z
D
E[f (ψψψ; xxx000, t)]G(xxx, xxx000; ∆)dxxx000 (2.54) By differentiating in time, the transport equation for the filtered PDF is obtained:
∂
∂tP +¯ ∂
∂ψi
[E( ˙ψi|ΨΨΨ = ΦΦΦ) ¯P ] = 0, (2.55) where the expectation describes the average of the change of ψi conditioned on ΨΨΨ = ΦΦΦ, i.e., conditioned on the particle state being found at ΨΨΨ = ΦΦΦ within the filter volume. The LES spray equation can now be deduced by selecting appropriate macroscopic variables.
Similarly, a ‘LES coal equation’ can be deduced. For coal, the macroscopic variables of choice are the velocity (vector), temperature, diameter, mass of volatile matter and mass of char, for which the transport equation can be written as:
∂
where aaa is the conditional acceleration. The sample space of temperature, diameter, mass of volatile matter and mass of char is denoted by T , D, Mdevol and Mchar, respectively.
The conditional averages appearing in Eq. 2.56 are modeled such that they are split into a deterministic and stochastic contribution, i.e., the rate of changes of the macroscopic particle quantities are split into a part that is determined by filtered flow field quantities and a part that represents the subgrid influence on the rate of change of the macroscopic quantities. Following Bini & Jones [12], the stochastic contribution is modeled by means of an isotropic Wiener diffusion process with the diffusion coefficient b depending on the SGS velocity fluctuations, a model constant (with a value of unity) and a time scale char-acterizing the interaction of particle and turbulence: b = (C0kSGS/τt)1/2. This diffusion coefficient is multiplied with the Wiener term dW, which represents a three-dimensional Gaussian-distributed random variable with zero mean and a variance of dt (with the
dimension of s1/2). Bini & Jones [13] developed an expression for τt that leads to an appropriate non-Gaussian PDF:
τt= τp2α (∆/√
kSGS)2α−1, with α = 0.8. (2.57) In the direct numerical simulation, discussed in Chapter 8, all turbulent scales are resolved and hence the last term in Eq. 2.52 disappears. After evaluating the particle velocity from Eq. 2.52, the particle position can be obtained by:
dxp = updt. (2.58)
While the equation for the particle acceleration includes a stochastic contribution to describe the influence of subgrid fluctuations, it has not been attempted to include such a term in the equations for particle temperature, mass, etc., as it is omitted in literature.
However, subgrid temperature fluctuations, for example, might have an influence on the evolution of particles.
Particle and gas phase are fully coupled (i.e., ‘two-way coupling’) such that particles feed back and affect the gas phase. This is facilitated through source terms in the bal-ance/conservation equations of the gas phase (either in the large eddy or direct numerical simulation context). These source terms comprise the acceleration/deceleration of the gas phase due to interaction with the particle, the change in enthalpy due to heat transfer between gas phase and particle, and due to the enthalpy going along with mass emitted by the particle and the change in mass itself.
Formally, the source terms from the particle phase can be written in terms of projec-tion P of a Dirac delta function at the particle location onto the Eulerian location x by (see [21, 82]; shown here for momentum):
Fmomentum(x) =
Np
X
n=1
P(δ(xp,n− x))FD,n. (2.59)
However, such definitions are ultimately tied to the numerical method since the numerical method usually provides the projection. Different approaches to eliminate grid depen-dence exist, e.g., based on explicit filtering [21, 82]. The types of projection used in this work are presented in Chapter 4 and more details on the different exchange terms are mainly given in Chapter 5.
Similar to the particle coupling term for the momentum equation, source terms exist for the species mass fraction, mixture fraction and enthalpy equations. These equations follow the form of Eq. 2.59 and are also given in Chapter 4 and 5.