Solution of the spray equation

By substituting equation (8) into equation (7), we find that 

The solution to equation (13) is rjj) = jirij). Letting the subscript 0 identify conditions at the injector (x = 0), we may write this solution as 

The final step in the analysis is to obtain the combustion efficiency for a chamber of length x from the size distribution at position x. Let Qj denote the heat released per unit mass of material evaporated from a droplet of kind j, and let p, j represent the density of the liquid in droplets of kind j. The mass of the spray of kind j per unit volume of space is therefore Jo nr p, jGj dr, and the corresponding mass flow rate (mass per second) is Avj times this. Hence, the total amount of heat released per second by spray j between the injector and position x is 

Since the maximum possible heat-release rate (obtained by burning all the droplets) is Yj= i where 

In order to obtain numerical results for rj, we must know Gj o(r). A functional form for Gj o(r), which agrees well with observed size distributions for real sprays [4], [28-33] and simplifies many theoretical computations, is 

Avj times this. Hence, the total amount of heat released per second by spray j between the injector and position x is 

Actually, two of the parameters in equation (19) are determined by the total number of droplets per unit volume and by the average droplet radius the other two govern the shape of the distribution about the mean (for example, the standard deviation). The total number of droplets of kind j per unit volume (irrespective of their size) will be denoted by 

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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... 

For agiven system of metal/alloy and atomization gas, the 2-D velocity distributions of the gas and droplets in the spray can be then calculated using the above-described models, once the initial droplet sizes and velocities are known from the modeling of the atomization stage, as described in the previous subsection. With the uncoupled solution of the gas velocity field in the spray, the simplified Thomas 2-D nonlinear differential equations for droplet trajectories may be solved simultaneously using a 4th-orderRunge-Kutta algorithm, as detailed in Refs. 154 and 156. 

Tanasawa (111) investigated burning droplets from a swirl spray nozzle in an essentially quiescent atmosphere. Again the results are limited and preclude broad application. At this point one may wonder, therefore, how the equations for vaporization rate can be utilized. The solution is to deal with some particular droplet size which is representative of the point of interest. Usually this will be a size considerably greater than average for the spray, when it has diminished to zero, substantially all of the spray will have vaporized. 

It is apparent that effects of eddy currents and general air turbulence might be approximated by modifying Equations 10 and 14. Modifications of the basic equations could possibly account for solute effects. Extension of the theory to nonaqueous solvents requires the appropriate evaluation of Ap and K. Momentum imparted by vertical spray nozzles affects the time required for droplets to reach terminal fall velocity (II). Helicopter and fixed-wing aircraft impart a downward vector to the bulk of the spray and also produce a turbulent effect on the time of droplet fall and should be accounted for. 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

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... 

Aspiration rate is only a small part of the overall transport process in flame spectrometry. The production of aerosol and its transport through the spray chamber are also of great importance. The size distribution of aerosol produced depends upon the surface tension, density, and viscosity of the sample solution. An empirical equation relating aerosol size distribution to these parameters and to nebulizer gas and solution flow rates was first worked out by Nukiyama and Tanasawa,5 who were interested in the size distributions in fuel sprays for rocket motors. Their equation has been extensively exploited in analytical flame spectrometry.2,6-7 Careful matrix matching is therefore essential not only for matching aspiration rates of samples and standards, but also for matching the size distributions of their respective aerosols. Samples and standards with identical size distributions will be transported to the flame with identical efficiencies, a key requirement in analytical flame spectrometry. 

The resulting set of equations Eqs. 8.2 to 8.2 is equivalent to the La-grangian description of the dispersed phase (without collision terms and for mono-disperse sprays) and leads to the same solutions, as shown in [348]. 

The formulation that has been given here is not the only approach to the description of two-phase flows with nonequilibrium processes. Many different viewpoints have been pursued textbooks are available on the subject [43], [44], and a reasonably thorough review recently has been published [45]. Combustion seldom has been considered in this extensive literature. Most of the work that has addressed combustion problems has not allowed for a continuous droplet distribution function but instead has employed a finite number of different, discrete droplet sizes in seeking computer solution sets of conservation equations [5]. The present formulation admits discrete sizes as special cases (through the introduction of delta functions in fj) but also enables influences of continuous distributions to be investigated. A formulation of the present type recently has been extended to encompass thick sprays [25]. Some other formulations of problems of multiphase reacting flows have been mentioned in Sections 7.6 and 7.7. 

The deterministic population balance equations governing the description of mass transfer with reaction in liquid-liquid dispersions present a framework for analysis. However, signiflcant difficulties exist in obtaining solutions for realistic problems. No analytical solutions are available for even the simplest cases of interest. Extension of the solution to multiple reactants for uniform drops is possible using a method of moments but the solution is limited to rate equations which are polynomials (E3). Solutions to the population balance equations for spatially nonhomogeneous dispersions were only treated for nonreacting dispersions (P4), and only a simple case was solved for a spray column (B19). Treatment of unmixed feeds presents a problem. 

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