Spray equation

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

In the LHF models, it is assumed that droplets are in dynamic and thermodynamic equilibrium with gas in a spray. This means that the droplets have the same velocity and temperature as those of the gas everywhere in the spray, so that slip between the phases can be neglected. The assumptions in this class of models correspond to the conditions in very thin (dilute) sprays. Under such conditions, the spray equation is not needed and the source terms in the gas equations for the coupling of the two phases can be neglected. The gas equations, however, need to be modified by introducing a mixture density that includes the partial density of species in the liquid and gas phases based on their mass fractions. Details of the LHF models have been discussed by Faeth.l589]... 

Here the subscripts on the gradient operators distinguish derivatives with respect to spatial and velocity coordinates, and M is the total number of different kinds of particles (classified according to their chemical composition). The details of the procedure for deriving equation (1) (considering the volume element dr d dy and counting what enters and leaves) are so familiar in fluid dynamics and in kinetic theory that they need not be repeated here. Equation (1) will be called the spray equation. 

The velocity dependence of the distribution function, which is not of primary interest here, may be eliminated from the spray equation by integrating equation (2) over all velocity space. Sincej. 0 very rapidly (at least exponentially) as [ v oo for all physically reasonable flows, the divergence theorem shows that the integral of the last term in equation (2) is zero, whence... 

For the applications discussed in the following sections, the preceding conservation equations are supplemented by ttje ideal gas equation of state, in which Pg (not the fluid density Pf) enters. Since fj appears in each of the conservation equations, it is apparent that they are coupled to the spray equation, which therefore must also be included to obtain a complete set of integrodifferential equations describing spray combustion. 

By substituting equations (60) and (44) into equation (2) and by integrating over r and v, we find that the spray equation assumes the very simple form... 

Since the product n is, in general, not constant, the previously stated assumption = constant is valid only for dilute sprays, f7r n 1. When attention is restricted to dilute sprays, equation (49) (the continuity equation) provides an expression for the gas velocity u. Since m = 0 at x = 0, m = Hence equation (49) becomes... 

Under our present assumptions, the spray equation simplifies to equation (7) with the additional conditions M == l,i w, and = constant. Since the solution to equation (7) given in Section 11.2 will also be valid in the present problem, equation (15) is applicable here and reduces to... 

Drop breakup enters the spray equation via the source term/bu in (19.45). There are various ways of accounting for drop breakup, most of which are also used for a rudimentary description of the atomization process. Some of these approaches are discussed in more detail in Chap. 9, and include the TAB model of O Rourke and Amsden [37], the Wave Breakup model of Reitz and coworkers [46, 40], the Unified Spray Breakup model of Chryssakis and Assanis [10], and the Cascade Atomization and Drop Breakup model of Tanner [54]. 

The description of a dispersed multiphase flow with chemical reactions leads to a complex system of differential and algebraic equations, which can only be solved by specifying appropriate boundary and initial conditions. For the gas phase equations, the boundary conditions are imposed on the gas velocity u, the temperature T, the turbulent kinetic energy k, and its dissipation e. The spray equations require conditions at the nozzle exit and for the interactions of the droplets with the walls. 

The boundary conditions for the spray equations specify the droplet-wall interactions, as well as the mass flow rate and the droplet distribution function at the nozzle exit. Additional details and references can be found in Ref. [4]. 

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