Droplet spherically symmetric

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

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). 

COSILAB Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrical and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http //www.softpredict.com/cms/ softpredict-home.html for more information. 

Heat Transfer by Conduction. In the theoretical analysis of steady state, heterogeneous combustion as encountered in the burning of a liquid droplet, a spherically symmetric model is employed with a finite cold boundary as a flame holder corresponding to the liquid vapor interface. To permit a detailed analysis of the combustion process the following assumptions are made in the definition of the mathematical model ... 

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

Figure la. Flow configuration for spherically symmetric droplet vaporization... 

Figure la indicates an idealized, spherically symmetric situation when there is no relative velocity between the droplet and the gas stream. The only spacial dependence is the radial distance r. At the droplet surface the inwardly directed heat flux from the gas phase is used to effect vaporization as well as to heat the droplet interior. The fuel vapor... 

Hence a complete analysis of the phenomena of interest will involve the simultaneous descriptions of the chemical reactions in the gas phase, the phase change processes at the interface, the heat, mass, and momentum transport processes in both the gas and liquid phases, and the coupling between them at the interface. The processes are transient and can be one dimensional (spherically symmetric) or two dimensional (axisymmetric). Although extensive research on this problem has been performed, most of it emphasizes the spherical-symmetric, gas-phase transport processes for the vaporization of single-component droplets. Fuchs book (84) provides a good introduction to droplet vaporization whereas Wise and Agoston (21), and Williams (22), have reviewed the state of art to the mid-flfties and the early seventies, respectively. 

In the next section some of the important time scales are identified and transient droplet heating effects during the spherically symmetric, single-component droplet vaporization are reviewed. Spherically symmetric, multicomponent droplet vaporization and droplet vaporization with nonradial convection are discussed in later sections. 

Transient Droplet Heating during Spherically Symmetric Single-Component Droplet Vaporization... 

The discussions so far are quite general and hence are apphcable to all cases involving the spherically symmetric vaporization of a singlecomponent droplet. Equations 6, 7, 8, 9, and 10 show that for a given fuel oxidizer system and for prescribed ambient conditions Yooo and Too, the solutions are determined to within one unknown, H. Three models with different internal heat transport descriptions are presented below. 

Oince the earliest theoretical models by Spalding (I) and Godsave (2) describing the quasi-steady, spherically symmetric combustion of individual fuel droplets in quiescent atmospheres, numerous more elaborate theories have been proposed to provide a better understanding of droplet spray combustion. These theories are based on the premise that the physical and chemical processes involved dmmg single-droplet combustion are fundamental to complex spray combustion processes. 

Consider now the energy equation of the evaporating droplet in spherical-symmetric coordinates in which Cp and X are taken independent of temperature. 

The spherical-symmetric fuel droplet burning problem is the only quiescent case that is mathematically tractable. However, the equations for mass burning... 

Assume a point disclination located in a nematic droplet of radius R. The point disclination can be classified according to their Poincare characteristic angle a as a knot point (a = 0), focus point (0 < a < 7t/2), center (a = 7t/2), saddle-focus point (tt/2 < a < tt) or saddle point (a = 7t/2). For a knot point, one has a spherically symmetrical radial configuration and then... 

Fig. 3.5.10. Singular points in droplets (a) spherically symmetric radial (hedgehog) configuration with the director normal to the surface (b) bipolar structure with the director tangential to the surface (c) singular points in a capillary.

For a drop in a quiescent atmosphere, that is, a drop under non-forced convection, exact expressions for the Nusselt and Sherwood numbers can be derived tmder the assumption that the problem is spherically symmetric. The initial derivation for a single species drop is due to Spalding [26, 27]. An insightful presentation of this approach can be found in Kuo [15]. Essentially the same methodology can also be apphed to multi-species droplets and the resulting expressions for the Sherwood and Nusselt numbers are formally the same as for a single species drop. For more details see Gradinger [12]. 

Without affecting the physics of the problem, we may assume that an emulsion droplet bums in a spherically symmetric manner, with or without an enveloping diffusion flame. This is a valid assumption as long as the relative motion between the surrounding gas and the droplet is low so that no internal motion is induced within the droplet. This may happen if either the gas velocity or the droplet size is small. It has been shown that in a spray flame of ethanol and oxygen formed by a coaxial air-assist nozzle and used for the flame spray pyrolysis, the relative velocity between the gas and droplets close to the nozzle exit is on the order of 10 m/s and... 

The liquid phase model proposed below considers the mass transfer inside the droplet and the changes in liquid phase properties due to the temperature and composition changes. In derivation of the following equations, it has been assumed that liquid circulation is absent, the droplet surface is at local thermodynamic equilibrium state, momentum, energy and mass transfer are spherically symmetric within the droplet, and the two liquids (fuel and water) are immiscible. With these assumptions, the conservation equations for the total mass, mass of water, and the energy equation are written as follows [14] ... 

While an emulsion droplet is burning, its fuel content and water content of the micro-solution droplets continuously evaporate. As a result of water evaporation, the concentration of the solute within the micro-solution droplets increases. Considering a uniform temperature within the micro-solution droplets, with no internal motion or circulation, the only transport equation that has to be considered within the micro-solution droplets is the spherically symmetric mass conservation of solvent, i.e., water, written as follows ... 

In the next sections, a mathematical model assuming spherically symmetric droplets and including a one-dimensional, transient formulation is provided which covers all the five stages of droplet drying. The model is an extension of the work by Gopireddy and Gutheil [21], and it is extended to account for stages IV and V. 

The one-dimensional energy equation of a spherically symmetric bi-component droplet yields... 

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