Droplet continuity equation

Derive the general vector form of the overall mass-continuity equation, recognizing that the droplet evaporation represents a source of mass to the system. 

Example 7.7 Absorption of ammonia vapor by lithium nitrate-ammonia solution The following modeling is from Venegas et al. (2004). For simultaneous heat and mass transfer during the absorption of ammonia vapor by lithium nitrate-ammonia (A) solution droplets, the ammonia concentration profile in the liquid phase can be estimated from the continuity equation without a source term... 

Analysis. In the region between the fuel droplets, when Do Dp, the fuel and oxidizer continuity equations can be combined and expressed in the Schwab-Zeldovich form (1,9) ... 

Ostwald ripening is observed when the substance of the emulsion droplets (we will call it component 1) exhibits at least minimal solubility in the continuous phase, p. As discussed above, the chemical potential of this substance in the larger droplets is lower than in the smaller droplets see Equation 5.115. Then a diffusion transport of component 1 from the smaller toward the larger droplets will take place. Consequently, the size distribution of the droplets in the emulsion will change with time. The kinetic theory of Ostwald ripening was developed by Lifshitz and Slyozov, Wagner, and further extended and applied by other authors. " The basic equations of this theory are the following. 

Liquid-mass continuity equation. This equation describes the evolution of droplet diameter by evaporation. Early versions of the model took into account only of evaporation by heat transfer from the superheated vapor, but later versions took some account of direct evaporation of droplets at the wall. 

Vapor continuity equation. Since the droplets are evaporating by heat transfer, the actual quality of the flow is also changing and the change with length can be estimated from the change in droplet diameter. 

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 ... 

Another application for high-speed confocal X-PIV was given by Kinoshita et al., who presented a microflow diagnostic technique for the internal flow of a droplet passing through a microchannel. The confocal j,-PIV system allows a temporal resolution of 2000 frames s in a 228 x 171 p.m region with a confocal depth of 1.88 pm. Kinoshita et al. proposed a three-dimensional velocity measurement method based on confocal p-PIV and the continuity equation [25]. 

In extraction column design, the model equations are normally expressed in terms of superficial phase velocities, L and G, based on unit cross-sectional area. The volume of any stage in the column is then A H, where A is the cross-sectional area of the column. Thus the volume occupied by the total dispersed phase is h A H, where h is the fractional holdup of dispersed phase, i.e., the droplet volume in the stage, divided by the total volume of the stage and the volume occupied by the continuous phase, in the stage, is (1-h) A H. 

As in Section II,A, a set of steady-state mass and energy balances are formulated so that the parameters that must be evaluated can be identified. The annular flow patterns are included in Regime II, and the general equations formulated in Section II,A,2,a, require a detailed knowledge of the hydrodynamics of both continuous phases and droplet interactions. Three simplified cases were formulated, and the discussion in this section is based on Case I. The steady-state mass balances are... 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

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