Simplified 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 

If the local cross-sectional area of the chamber is denoted by A(x then, since the mean flow is in the x direction (parallel to the motor axis) and the velocities are parallel to the chamber wall at the wall, integrating equation (3) over the two spatial coordinates normal to the x axis yields 

Here Vj is the x component of v, and the quantities Rj, Vj, and Gj have been assumed to be essentially independent of the spatial coordinates normal to x (alternatively, these quantities may be interpreted as averages over the cross section). 

From the results of Chapters 3 (see also [24]), it may be shown that for nearly all droplet vaporization mechanisms, the dependence of Rj on droplet size may be expressed approximately by the equation 

It will also be assumed that, although the velocity distributions may differ for particles of different sizes, Vj is independent of r. This will be true, 

It will also be assumed that, although the velocity distributions may differ for particles of different sizes, Vj is independent of r. This will be true, for example, when the droplets are so small that they all travel with the velocity of the gas, or when the effect of the gas is negligible but droplets of all sizes are injected with the same average velocity. 

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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... 

In a slightly different form, Eq. (6) is commonly referred to as the Warren spring equation. Representative yield loci determined utilizing the simplified shear cell are shown in Fig. 7 for spray-dried lactose, bolted lactose, and sucrose. The yield locus for each material relates the shear strength to the applied load. 

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... 

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. 

In this chapter, the equation governing the statistical counting procedure for sprays is first derived (Section 11.1) and is applied (Section 11.2) to a very simplified model of rocket-chamber combustion in order to obtain an estimate of the combustion efficiency. This illustrative example and others... 

The presentation of the subject of spray combustion in Chapter 11 is not greatly different from that in the first edition. An updated outlook on the subject has been provided, and the formulation has been generalized to admit time dependences in the conservation equations. The analysis of spray deflagration has been abbreviated, and qualitative aspects of the results therefrom have been anticipated on the basis of simplified physical reasoning. In addition, brief discussions of the topics of spray penetration and of cloud combustion have been added. 

The coefficients of the model are calculated by linear regression (the logarithm of the particle size was used here) and then plotted as a cumulative distribution of a normal plot (Fig. 2). The important coefficients are those that are strongly positive or negative, for example, the spray rate and the interaction between atomization pressure and inlet temperature 35. Others not identified on the diagram are not considered significant and could well be representative mainly of experimental error. The equation can thus be simplified to include only the important terms. However, if interactions are included, their main... 

With increasing complexity of the models, one gets more accurate predictions but also the computation times become considerably larger. Keeping in mind that many of these evaporation models have been developed for use in CFD spray simulations, where hundreds of thousands of droplets have to be considered, computational costs become a primary issue. Therefore, the models discussed in this chapter are limited to the second and third category. The presentation starts with the general conservation equations for mass, species and energy, from which the simplified models are derived. 

The value of k depends on the droplet spectrum, since it relates to the rate of buildup of critical droplets and their distribution. However, Eq. (14.67) does not take into account the flattening effect of the droplet on impact, which results in reduction of 6 and increase of w above the value predicted by Eq. (14.66). Thus, Eq. (14.67) is only likely to be valid under conditions of small impaction velocity. In this case, retention is governed by the surface tension of the spray liquid, the difference between 6 and Or (i.e. the contact angle hysteresis) and the value of 0a-Equation (14.67) can be further simplified by removing the constant terms and standardizing sin a as equal to 1. A further simplification is to replace the second term between square brackets on the right-hand side of Eq. (14.67) by 0m> the arithmetic mean of 0a and 0r. In this way a retention factor, F, may be defined by the simple expression... 

As noted earlier, the chamber shape depends on the type of atomizer employed because the spray angle determines the trajectory of the droplets and therefore the diameter and height of the drying chamber. Typical spray dryer layouts with wheel and nozzle atomization are shown in Figure 10.14. Correlations are available for calculating the drying chamber sizes without pilot tests. However, these equations require simplifying assumptions that make their application... 

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