3.3.1. Introduction

The transport of particles through a gas-phase flow is of importance to a tremendous range of applications. Applications in the area of combustion and fire science include fuel sprays, suppressant transport and metal particle combustion [94, 95, 96, 97, 98, 99, 100, 101]. These applications typically have a strong coupling between the heat and mass transfer. For example, fuel spray combustion is typically limited by the diffusion of the oxidizer towards the particle. In fire suppressant distribution, the cooling associated with the evaporating suppressant can dramatically slow suppressant evaporation. In metal particle combustion, in order for the metal oxide combustion product to condense out, the enthalpy of condensation must be dissipated; this energy dissipation is a combination of radiative and conductive transport, each of which results in differing heat flux consequences. Also relevant are the transport of contaminants through the atmosphere and the dynamics of clouds [102]. A large number of industrial processes share similar physics including powder manufacturing, painting, coating and ink-jet printing.

This report describes the development of a Lagrangian particle and droplet transport model and its integration with a computational fluid dynamics (CFD) code that solves, on an Eulerian mesh, the continuum phase. Conservation of mass, momentum and energy are considered for the coupled system allowing combustion along with evaporating and condensing particles. Since examples of this type of flow are typically sprays, this model is sometimes referred to as a spray model, but it can handle general classes of particulate flows. This model is developed to be suitable for modeling evaporating, condensing or combusting flows of particles in continuum gas-phase flows. This model is based partly on the initial implementation of a dilute spray model in the Vulcan fire-physics computational modeling code [103, 104] as described in [105].

Two significant limitations are stipulated that lead to the simplified conservation equations employed. First, the spray must be dilute, that is the volume fraction of the particle phase must be small (i.e. less than 10 percent). Second, the physical density of the particle should be orders of magnitude greater than the continuum (gas) phase and the particle Reynolds numbers should not be too large or additional terms will appear in the particle evolution equations [106].

3.3.1.1. The Spray Equation

For given physical properties of the particle (composition, density, etc.), the particle field is characterized by the particle locations, velocities, radii and temperatures. This can be expressed in terms of a particle distribution function, f, so that:

(3.620) $f({\bf x}_p, {\bf u}_p, r_p, T_p; t)d{\bf u}_p dr_p d{T_p}$

is the probable number of droplets per unit volume at location ${\bf x}_p$ in the velocity range $({\bf u}_p, {\bf u}_p + d{\bf u}_p)$ , the size range $(r_p, r_p + dr_p)$ , and the temperature range $(T_p, T_p + dT_p)$ . The evolution of this particle distribution can be described by an equation of the Fokker-Planck form [107]:

(3.621) $\dfrac {df}{dt} + {\nabla \cdot ({\bf u}_p f)} + {{\nabla}_u \cdot \left(\dfrac {d {\bf u}_p} {dt} f \right)} + \dfrac {d} {dr_p} \cdot \left(\dfrac {dr_p} {dt} f \right) + \dfrac {d} {dT_p} \cdot \left(\dfrac {dT_p} {dt} f \right) = \left({\dfrac {df}{dt}}\right)_{coll} + \left({\dfrac {df}{dt}}\right)_{brk}.$

Here, expressions for the particle acceleration, evaporation and heating are required in the third through fifth terms on the left-hand side. Similar models for particle collision and breakup appear on the right-hand side. Such models are available in the literature [99, 108] and are described in the earlier report [105]. Unfortunately, Eq. (3.621) is a differential equation in nine dimensions, a fact which makes direct numerical evolution prohibitive in the general case. The standard alternative is to represent $f$ using a fine-grained distribution and Monte Carlo methods. That is, $f$ is represented by a sufficiently large number of discrete distributions, each representing a number of particles, $N_p$ , with the same particular characteristics $({\bf x}_p, {\bf u}_p, r_p, T_p, t)$ . All of the $N_p$ particles in a fine-grained distribution share the same evolution equation, and $f$ is found by summing over the discrete distributions.

In this manner, the evolution of $f$ can be described using evolution equations for individual particles. Such evolution equations are provided in Particle Transport Model.

Because it is typically prohibitive to track all of the particles in a flow, representative parcels of particles are instead tracked. The particles in a given parcel share a common origin and common material properties. To further simplify the parcel evolution equations, each parcel consists of mono-disperse (single-diameter) particles so that all particles in the parcel are described by the same set of evolution equations. For flows where the particle size is distributed over a range of values, it is still necessary to track a statistically significant number of parcels to reproduce the mean behavior. Typically, a large number of parcels (tens or hundreds of thousands) are tracked to describe the evolution of the particle field.

3.3.1.2. A Combined Eulerian-Lagrangian Approach

The typical approach to CFD is to employ Eulerian descriptions of the flow field. Such an Eulerian formulation is employed to evolve the gas-phase continuum flow in the present case using standard methods [109]. To evolve the fine-grained distribution as indicated above, a Lagrangian approach is necessary [110, 111, 112, 113]. Such an approach has been used in other CFD applications including, for example, the popular internal combustion engineering simulation code, KIVA [99]. The coupling between the Eulerian and Lagrangian fields is key to capturing certain relevant physics, and this coupling is described in detail in Particle Transport Model and in Coupling the Lagrangian and Eulerian Fields.

The present paper presumes that turbulent flow fields will be of interest, and that these turbulent flows cannot be fully resolved. Then, in addition to the continuity, momentum, species, and energy equations, there will be representations for the turbulent fluctuations. It is common to employ two-equation models in Reynolds-averaged Navier-Stokes (RANS) approaches while large-eddy simulation (LES) techniques employ estimates of the subgrid fluctuations based on resolved quantities. For the present purposes, the $k-\epsilon$ turbulence model [114] will be presumed with extensions to other methods being straightforward.

When a particle collides with a solid wall, it is assumed to adhere to the wall if the impact velocity (kinetic energy) is sufficiently high, and bounces otherwise. In general, adherence is the predominant result of collisions for the particles considered here. It is well known that fine powders can be convectively lifted from surfaces and transported elsewhere, but this is beyond the scope of the current study. Models for the particle breakup due to hydrodynamic forces and for particle collisions are also available. For these purposes, models developed elsewhere and available in the literature [99, 108] are employed.