Local average particle velocity

Batch, semibatch, or continuous-flow operation can be simulated. The continuous phase is assumed well mixed. Particle movement was either random or followed the flow direction of the sum of the local average fluid velocity and the particle gross terminal velocity. The probability of droplet breakup is assigned based on droplet size. Binary breakage was assumed to form two randomly sized particles whose masses equal the parent drop. The probability of coalescence exists when two drops enter the same grid location. Particles are added and removed to simulate flow. 

From the Taylor-Aris formulation for times t> a lD, where a is the capillary radius and D the Stokes-Einstein diffusion coefficient of the particle, the particle of radius will have had sufficient time to sample the full velocity profile. With the local particle velocity taken to be equal to that of the fluid (Eq. 4.2.14), the average particle velocity over the tube cross-section IJ is given by... 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

Equation 2.101 enables calculation of local average quantities such as moments of the particle size distribution. Baldyaga and Orciuch (2001) review expressions for local instantaneous values of particle velocity and diffusivity of particles, Z)pT, required for its solution and recover the distribution using the method of Pope (1979). 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

The counterparts of dissolving particles are the processes of precipitation and crystallization the description and simulation of which involve several additional aspects however. First of all, the interest in commercial operations often relates to the average particle size and the particle size distribution at the completion of the (batch) operation. In precipitation reactors, particle sizes strongly depend on the (variations in the) local concentrations of the reactants, this dependence being quite complicated because of the nonlinear interactions of fluctuations in velocities, reactant concentrations, and temperature. 

To calculate more precisely the average uptake or the local variation in uptake in each airway, the local variations in velocity and concentration profiles must be taken into account. For example, thin momentum and concentration boundary layers occur at bifurcations and gradually increase in thickness with distance downstream. Bell and Friedlander showed that particle and gas transfer to the airway wall is greatest where the boundary layers are thinnest, e.g., at the carina or apex of bifurcations. 

For the motion of a gas-solid suspension in the riser, both the gas and particle velocities have local averaged and random components. Thus, it is desirable to develop a mechanistic model which incorporates a variety of interactive effects due to both the gas and particle velocity components (see Chapter 5) as given in the following [Sinclair and Jackson, 1989] ... 

This model, Model LR (Local-Radial), describes the dependence of X(r) on gas velocity UJj) and particle velocity Ud(r), which are, however, not manipulatable operating parameters, but should be correlated with the average superficial gas velocity t/g and the average superficial particle velocity Ud (= GJpp) as well as boundary conditions. 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

Linking particle tracking and the two-phase fluidised bed model is not straightforward, because individual particle velocities are subsumed in the local averages and therefore not directly available from the fluidised bed model. Consequently, the initial particle velocities for particle tracking have to be obtained from what the fluidised bed model does give, namely the extent of bed expansion, the position and velocity of the bubble, gas velocities and local averages for particle velocities. 

A Eulerian two phase calculation is performed in two dimensions giving locally averaged velocities and volume fractions for the particle and gas/vapour phases. Furthermore, bed expansion and bubble characteristics are obtained. 

Equation (57) employs the local instantaneous values of velocity and concentration. To describe the effects on the process of turbulent fluctuations of the particle velocity as well as species and particle concentrations, one can use Reynolds-averaged form of the population balance... 

To compare mass transfer in packed beds with transfer to a single particle, Sherwood numbers calculated from Eq. (21.62) are plotted in Fig. 21.5 along with the correlation for isolated spheres. The coefficients for packed beds are two to three times those for a single sphere at the same Reynolds number. Most of this difference is due to the higher actual mass velocity in the packed bed. The Reynolds number is based for convenience on the superficial velocity, but the average mass velocity is Gfe, and the local velocity at some points in the bed is even higher. Note that the dashed lines in Fig. 21.5 are not extended to low values of since it is unlikely that the coefficients for a packed bed would ever be lower than those for single particles. 

The actual coefficient is greater than because frequent acceleration and deceleration of the particle raises the average slip velocity and because small eddies in the turbulent liquid penetrate close to the particle surface and increase the local rate of mass transfer. However, if the particles are fully suspended, the ratio kJk T falls within the relatively narrow range of 1.5 to 5 for a wide range of particle sizes and agitation conditions.The effects of particle size, diflusivity, and viscosity follow the trends predicted for k p, but the density difference has almost no effect until it exceeds 0.3 g/cm . For suspended particle, varies with only the... 

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