Acceleration disperse phase

The objective of this paper is to illustrate the efficacy of inferring the interdroplet forces in a concentrated protein stabilized oil-in-water emulsion from the knowledge of the equilibrium profile of continuous phase liquid holdup (or, dispersed phase faction) when the emulsion is subjected to a centrifugal force field. This is accomplished by demonstrating the sensitivity of continuous phase liquid holdup profile for concentrated oil-in-water emulsions of different interdroplet forces. A Mef discussion of the structure of concentrated oil-in-water emulsion is presented in the next section. A model for centrifugal stability of concentrated emulsion is presented in the subsequent section. This is followed by the simulation of continuous phase liquid holdup profiles for concentrated oil-in-water emulsions for different centrifugal accelerations, protein concentrations, droplet sizes, pH, ionic strengths and the nature of protein-solvent interactions. 

Another role of the surfactant is to initiate interfacial instability, e.g., by creating turbulence and Raykleigh and Kelvin-Helmholtz instabilities. Turbulence eddies tend to disrupt the interface since they create local pressures. Interfacial instabilities may also occur for cylindrical threads of disperse phase during emulsification. Such cylinders undergo deformation and become unstable under certain conditions. The presence of surfactants will accelerate these instabilities as a result of the interfacial tension gradient. 

The mean droplet size, dp, will be influenced by diameter of the orifice, d velocity of the liquid, u interfacial tension, a viscosity of the dispersed phase, p density of the dispersed phase, pd density of the continuous phase, pc, and acceleration due to gravity, g. It would also be acceptable to use the term (pd — pc)g to take account of gravitational forces and there may be some justification in also taking into account the viscosity of the continuous phase. 

The continuous phase film mass transfer rate can be increased by electrostatic acceleration of charged droplets of the dispersed phase in the continuous phase. 

The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called added or virtual mass force. This force was given by the second term in Eq. (8). The constant Cy is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cy is 0.5. 

The cyclone, or inertial separation method, is a common industrial approach for segregating a dispersed phase from a continuous medium based upon the difference in density between the phases. The concept takes advantage of the velocity lag which occurs for dense particles with respect to a lower density medium when both phases are subject to an accelerating flow field, such as within a rotating vortex. The larger the acceleration, the smaller the particle which fails to follow the continuous phase streamlines and will migrate to the outer wall of the cyclone for collection. 

Intennediary formation of FeO in the hot spot is the main oxygen source for the decarburisation. FeO is dispersed in the slag and the metal bath as well as Fe-C droplets, which are accelerated hy the oxygen jet. The subsequent reaction of FeO with carbon dissolved in iron occurs in both phases. In the slag phase, dissolved FeO reacts with dispersed Fe-C droplets, and in the metal bath FeO droplets form the disperse phase. For the decarburisation, several reaction routes can be formulated ... 

When a dispersed phase particle accelerates relative to the continuous phase, some part of the surrounding continuous phase also is accelerated. This extra acceleration of the continuous phase has the effect of added inertia or added mass (Fig. 4.4). 

Maximum stable drop diameter, m Impeller diameter, m Diffusivity of dissolved component or reactant in liquid, m /s Gravitational acceleration, m/s Height of liquid in vessel, m Mass transfer coefficient, m/s Mass transfer coefficient for a single spherical droplet immersed in a liquid flowing at constant velocity past the droplet, m/s Mass of liquid, kg Rate of mass transfer of solute or reactant, kg/s Impeller speed, rotations/s Minimum speed to just suspend solid particles in vessel, rotations/s Minimum impeller speed to completely incorporate dispersed phase into continuous phase in liquid-liquid systems, rotations/s Power dissipation, W Time, s... 

The acceleration of the liquid in the wake of the bubbles can be taken into account through the added mass force given by (5.112), whereas the Eulerian lift force acting on the dispersed phase is normally expressed on the form (5.65). 

The oscillatory measurements are perhaps more useful, since to prevent separation the bulk modulus of the system should balance the gravity forces that is given by hpAg, where h is the height of the disperse phase, Ap is the density difference, and g is acceleration due to gravity. 

The disperse-phase Knudsen number can also be infinity for dense, elastic systems due to initial conditions. For example, if one releases a dense assembly of particles that are initially at rest in a vacuum, the particles will accelerate due to gravity such that they all have the same velocity. For this case, the granular temperature is null... 

However, the disperse-phase granular temperature 0p and the mean acceleration (A) are unknown, and must be modeled using moment closures. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

Gravity separations depend essentially on rite density differences of rite gas, solid, or liquids present in the mix. The particle size of the dispersed phase and the properties of die continuous phase are also factors with rite separation motivated by die acceleration of gravity. The simplest representation of this involves rite assumption of a rigid spherical panicle dispersed in a fluid with rite terminal or free-settling velocity represented by... 

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