Local average fluid 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. 

G is the ratio of the dimensionless number SD in the disturbed region to its value in the normally-operating part of the bed. SD contains the activation energy, heat of reaction, inlet temperature and bed height, all of which have fixed constant values in all regions of the bed. It also contains the possibly variable quantities C, k and F. C is the average heat capacity of the fluid, and depends on the local phase ratio. kg is the specific rate constant, and depends on the local catalyst density and the phase holdup. F is the local average linear velocity, which can vary from point to point for a variety of reasons. 

Mixing parameters local or average fluid velocity or flow, local or average shear rates, blend time, power input, and so on. 

The original work on the flow of fluids through packed beds was carried out by Darcy [1], who examined the rate of flow of water from the local fountains through beds of sand of various thicknesses. He show that the average fluid velocity (um) was directly proportional to the driving pressure (Ap) and inversely proportional to the thickness of the bed, L i.e. 

Average fluid temperature Average velocity Normalized local velocity Streamwise velocity... 

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. 

Note that we have used the fluid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, / is a highly nonlinear function of c. Thus, in a turbulent flow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less influenced by turbulent mixing. 

Velocity distributions in turbulent flowthrough a straight, round tube vary with the flow rate or the Reynolds number. With increasing flow rates the velocity distribution becomes flatter and the laminar sublayer thinner. Dimensionless empirical equations involving viscosity and density are available that correlate the local fluid velocities in the turbulent core, buffer layer, and the laminar sublayer as functions of the distance from the tube axis. The ratio of the average velocity over the entire tube cross section to the maximum local velocity at the tube axis is approximately 0.7-0.85, and increases with the Reynolds number. 

As before, let P be the local stress tensor, and denote by an overbar the statistical average of any quantity. The definition of the fluid-velocity field may be analytically extended to the solid-particle interiors and the pressure therein assumed to vanish. As such, taking the statistical average of the... 

CCT, critical cracking thickness Boltzmann constant (1.381x10 local permeability [m ] fracture resistance [N m ] average permeability in/of compact [m ] particle shape factor compact thickness [m] initial particle number concentration [m refractive index of particle material refractive index of dispersion material number density of ion i dimensionless number dimensionless number Stokes number Peclet number capillary pressure [N-m ] dynamic pressure [N m ] local liquid pressure in the compact [N-m local solid pressure in the compact [N-m ] superficial fluid velocity [m-s q gas constant [J K ] centre to centre distance [m]... 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

Cross-sectional area, m or ft at station a 5 , at station b Net or time-average local fluid velocity in x direction, m/s or ft/s h, velocity of cylinder of plastic fluid in plug flow Un,axi maximum local velocity Uq, velocity of moving plate [Eq. (5.70)]... 

In this type of meter the sensing element, which is small compared to the size of the flow channel, is inserted into the flow stream. A few insertion meters measure the average flow velocity, but the majority measure the local velocity at one point only. The positioning of the sensing element is therefore important if the total flow rate is to be determined. The local measured velocity must bear a constant and known relationship to the average velocity of the fluid. 

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... 

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