Droplet motion equation

Let us examine the droplet motion (Equation (12.37)) along with its evaporation transport (Equation (12.32)) and the conservation of mass for a single drop ... 

The droplet medium is precisely governed by equations (1.15) and (1.16) which are linked with each other and are very difficult for investigation. Bearing in mind that only heavy droplets should be accounted for, let us use the appropriate law of vertical droplet motion instead of the second equation for the vertical droplet speed v. Imagine the simplest situation that the droplets are constantly generated at the height of fountains z = h with intensity q, /(m2s). 

Interaction of droplets is totally determined under these conditions by the respective contour integrals. Evaluating the latter requires, however, solving Eq. (48) in the precursor. The precursor evolution, as well as droplet motion, can be considered quasistationary when interaction is weak (an a posteriori estimate will be given in the end of this subsection). Since the curvature of the precursor is negligible as well, Eq. (48) reduces to the diffusion equation with variable effective diffusivity D h) ... 

The simplest case of aerosol aggregation is when the rate of aggregation is determined solely by the frequency of particle or droplet encounters due to Brownian motion. The aggregation coefficient can generally be assumed to be independent of the chemical composition of the aerosol. For aerosols with small particle or droplet sizes, Equation 5.11 simplifies to -dNldt = k N. Integrating and rearranging provides an equation for the time for concentration to... 

To analyze the fundamentals of a droplet motion actuated by ctMitmuous electrowetting principles, for illustration, one may consider two infinite parallel plates separated by a distance H, with an intervening liquid. For a steady, fully developed incompressible flow along the x-direction, the Navier-Stokes equation assumes the following simplified form ... 

From the Newton s law, the motion equation of the droplet can be stated as follows ... 

Droplet trajectories for limiting cases can be calculated by combining the equations of motion with the droplet evaporation rate equation to assess the likelihood that drops exit or hit the wall before evaporating. It is best to consider upper bound droplet sizes in addition to the mean size in these calculations. If desired, an instantaneous value for the evaporation rate constant may also be used based on an instantaneous Reynolds number calculated not from the terminal velocity but at a resultant velocity. In this case, equation 37 is substituted for equation 32 ... 

A (fake) potential energy, yielding the equation of motion (A.l), is (cf. the discussion of surface waves on a spherical liquid droplet in Landau adn Lifshitz [116]) ... 

To compute the motion of two immiscible and incompressible fluids such as a gas liquid bubble column and gas-droplets flow, the fluid-velocity distributions outside and inside the interface can be obtained by solving the incompressible Navier-Stokes equation using level-set methods as given by Sussman et al. (1994) ... 

In this model, two level-set functions (d, p) are defined to represent the droplet interface (d) and the moving particle surface (p), respectively. The free surface of the droplet is taken as the zero in the droplet level-set function 0> and the advection equation (Eq. (3)) of the droplet level-set function (droplet surface. The particle level-set function (4>p) is defined as the signed distance from any given point x in the Eulerian system to the particle surface ... 

The Euler Lagrangian approach is very common in the field of dilute dispersed two-phase flow. Already in the mid 1980s, a particle tracking routine was available in the commercial CFD-code FLUENT. In the Euler-Lagrangian approach, the dispersed phase is conceived as a collection of individual particles (solid particles, droplets, bubbles) for which the equations of motion can be solved individually. The particles are conceived as point particles which move... 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

To study the effects due to droplet heating, one must determine the temperature distribution T(r, t) within the droplet. In the absence of any internal motion, the unsteady heat transfer process within the droplet is simply described by the heat conduction equation and its boundary conditions... 

The smaller drops follow. Both liquids accelerate in the holes, because the sum of the cross section of all the holes is less than half the column cross section. However, this motion is retarded within a short distance, whereby a zone of drop compaction results above the trays. These phenomena are modeled based on a balance of maximum and minimum kinetic energy and the cohesive energy of the droplets [1]. After that, the resulting equation for the maximum stable drop diameter in the field of pulsing is ... 

There are three factors that would tend to cause the drag data of Fig. 8 to deviate from Stokes s law. The first is internal circulation however, based on the equation due to Hadamard (1911) and Rybczynski (1911), one would not expect motion within the droplet. The drag force determined by Hadamard and Rybczynski is... 

The case of combustion of an entire spherical surface with forced convection has not yet been solved. Frossling (4) originally proposed a semi-empirical relation for the low-temperature evaporation of droplets in motion. Spalding (60) has applied the equation to his heterogeneous combustion data with some success by including the term containing the transfer number ... 

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