Droplet burning

Hence, the constant Kis termed the "buming-constant of the droplet." Integration of the equation 15 produces the droplet burning law ... 

Nevertheless, a diffusion mechanism generally prevails and because it is the slowest step, it determines the regression rate. In evaporation, the mechanism is the conduction of heat from the surrounding atmosphere to the surface in ablation, it is the conduction of heat through the boundary layer in droplet burning, it is the rates at which the fuel diffuses to approach the oxidizer, etc. 

Since the product Dp is independent of pressure, the evaporation rate is essentially independent of pressure. There is a mild effect of pressure on the transfer number, as will be discussed in more detail when the droplet burning case is considered. In order to find a solution for Eq. (6.87) or, more rightly, to evaluate the transfer number B, mAs must be determined. A reasonable assumption would be that the gas surrounding the droplet surface is saturated at the surface temperature Ts. Since vapor pressure data are available, the problem then is to determine Ts. 

As in the case of burning gaseous fuel jets, the diffusion equations are combined readily by assuming Dp = (Alcp), that is, Le = 1. The same procedure can be followed in combining the boundary conditions for the three droplet burning equations to determine the appropriate b variables to simplify the solution for the mass consumption rate. 

Irrespective of these deviations, it is also possible from this transfer number approach to obtain some idea of the species profiles in the droplet burning case. It is best to establish the conditions for the nitrogen profile for this... 

This transfer number approach for a benzene droplet burning in air reveals species profiles characteristic of most hydrocarbons and gives the results summarized below ... 

FIGURE 6.13 Diameter-time measurements of a benzene droplet burning in quiescent air showing diameter-squared dependence (after Godsave [20]). 

The spherical-symmetric fuel droplet burning problem is the only quiescent case that is mathematically tractable. However, the equations for mass burning may be readily solved in one-dimensional form for what may be considered the stagnant film case. If the stagnant film is of thickness <5, the free-stream conditions are thought to exist at some distance 8 from the fuel surface (Fig. 6.16). 

If, indeed, Eqs. (6.171) and (6.172) adequately predict the burning rate of a droplet in laminar convective flow, the droplet will follow a d3/2 burning rate law for a given relative velocity between the gas and the droplet. In this case (3 will be a function of the relative velocity as well as B and other physical parameters of the system. This result should be compared to the d2 law [Eq. (6.172)] for droplet burning in quiescent atmospheres. In turbulent flow, droplets will appear to follow a burning rate law in which the power of the diameter is close to 1. 

Whereas in liquid droplet burning B was not explicitly known because Ts is an unknown, in the problem of heterogeneous burning with fast surface reaction kinetics, B takes the simple form of itn0, which is known provided the mass stoichiometric coefficient i is known. For small values of imom Eq. (9.29) becomes very similar in form to Eq. (9.22) where for the quiescent case hD = Dlrs = alrs. 

Many hypotheses for initiation of liquid expls have been proposed, of which Bowden et al (Refs 13, 14a 27) suggested adiabatic compression of gas bubbles Johansson et al (Ref 28) - vapor or droplet burning Andreev (Ref 29) - droplet formation or suspension behind a burning front is capable of causing a transition to detonation Bolkhovitinov (Ref 33a) - crystallization of the material under pressure Cook et al (Ref 34b) - initiation occurs with the development of a pressure-generated metallic state accompanied by a plasma that provides the postulated requirement of high heat conductivity... 

Burning of Hydrazine Droplets in Oxygen. As a simple illustration of the use of similarity procedures for studying droplet burning, we consider the burning of a liquid hydrazine droplet in pure oxygen. 

The use of the complete similarity relations for the specified 16 reaction steps appears excessively laborious in view of our lack of real knowledge concerning the dominant reaction paths. For this reason, we content ourselves with discussing the expected differences in estimated droplet burning rates when Reaction 20 is replaced by the following artificially concocted, highly simplified reaction scheme ... 

Thus, we find a remarkable lack of sensitivity for the calculated burning rates of an adiabatic droplet-burning process in which the reactions go to completion. This observed lack of sensitivity to reaction rates may well be related to the known successes (11,12, 22) of simplified diffusion-flame theories in theoretical predictions of droplet burning rates. 

Droplet Burning. An analysis completely analogous to that described in Appendix IIA leads to the conclusion that the terms T/Tj (j = 1, 2) in Equations 29 et seq. should be replaced by X Yi cvA Tj/Alj where /(/ = 1... 

Figure 2. Flame structure surrounding fuel droplet burning in oxygen at various...

Although there are many definitions of the mass transfer number, they are mostly generated for specific conditions such as droplet burning or boundary layer burning. All retain the same physical concept, which is the capability of the flame to self-sustain by generating more fuel. If B > 1, then the flame will produce more fuel than that necessary to sustain burning. 

The technique for coupling the chemical kinetic rate equations to the combustion process taking place in a rocket combustion chamber has not been devised. A detailed solution of the combustion chamber kinetics problem requires combination of the relations governing mixing, droplet burning, chemical reaction rates and combustion chamber flow characteristics. It is neither obvious that the complete solution to the complex combustion kinetics problem is possible nor that the efforts in this direction are wisely undertaken on the basis of present understanding of the more fundamental processes. 

Certain crude approaches are available to predict overall results, that is, nonequilibrium compositions. More refined techniques are available for the analysis of simplified models. Solution of the reaction kinetics of homogeneous gas phase combustion is possible through numerical solution of the rate equations. With the exception of the role of an overall highly exothermic reaction, the procedures are similar to those described in the preceding section on nozzle processes. The solution of the droplet burning problem including the role of chemical reaction rates, while not particularly tractable, is feasible. 

Figure 9. Effect of volatility on evaporation time for fuel droplet burning in oxygen-nitrogen atmosphere.

Even within our restricted definition, so many diffusion-flame problems exist that we do not have enough space to consider them all. Therefore, we shall merely illustrate the analytical procedure by means of three examples (Sections 3.1-3.3). The problem of droplet burning (Section... 

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