Velocity species

Then, knowing v one relation between the individual species velocities m... 

Conceptual Basis Species Velocities and Concentration Velocities... 

In chromatography, the species velocity is determined by the ratio t/i/xi of the absolute concentrations of the species in the two phases, as... 

To be retained, cell population retention ratio (R) (i.e., species velocity to average flow velocity ratio) must be <1. As already described, cells shouldi not be eluted according to a pure steric elution mode whose R, for a monodisperse population of size a, is... 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [77] or the average species velocity is used [96]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. 

V local instantaneous mixture velocity, mass average velocity (m/s) modified mass diffusion velocity for species s (m/s) species velocity of molecules of species s generated by chemical reaction (m/s)... 

For a charged species i carried by elutant flow and under the influence of an electric field, the net species velocity, , is the sum of the convective and electrophoretic migration velocities,... 

For a multicomponent fluid, conservation relations can be written for the individual species. Let u, be the species velocity and p, the species density, where the index is used to represent the ith species rather than the component of a... 

This is a surface-related phenomenon based on the mass flux vector of component i and the surface area across which this flux acts. Relative to a stationary reference frame, p, v, is the mass flux vector of component i with units of mass of species i per area per time. It is extremely important to emphasize that p, v, contains contributions from convective mass transfer and molecular mass transfer. The latter process is due to diffusion. When one considers the mass of component i that crosses the surface of the control volume due to mass flux, the species velocity and the surface velocity must be considered. For example. Pi (Vr — Vsurface) is the mass flux vector of component i with respect to the surface... 

Even though the species velocity vectors v are nonzero for reactants and products that diffuse toward and away from the internal catalytic surface, it is customary to neglect convective mass transfer within the pores. In other words, the Reynolds number is vanishingly small and diffusion dominates convective transport. Under these conditions, the dimensionless mass transfer eqnation for component i reduces to... 

This latter form fails to identify the species velocity as a crucial part of the species transport equation, and this often leads to confusion about the mechanical aspects of... 

In addition to the continuity equation and the jump condition, we need a set of N momentum equations to determine the species velocities, and we need chemical kinetic... 

In addition to equations 1.35 and 1.36, we need N momentum equations (Whitaker, 1986a) that are used to determine the N species velocities represented by A =, 2,..., N. There are certain problems for which the N momentum equations consist of the total, or mass average, momentum equation... 

This form of the species momentum equation is acceptable when molecule-molecule collisions are much more frequent than molecule-wall collisions thus equation 1.39 is inappropriate when Knudsen diffusion must be taken into account. The species velocity in equation 1.39 can be decomposed into an average velocity and a diffusion velocity in more than one way (Taylor and Krishna, 1993 Slattery, 1999 Bird etal., 2002), and arguments are often given to justify a particular choice. In this work we prefer a decomposition in terms of the mass average velocity because governing equations, such as the Navier-Stokes equations, are available to determine this velocity. The mass average velocity in equation 1.38 is defined by... 

Internal energy per unit volume Internal energy per unit mass Internal energy associated with bead Contribution to equilibrium internal energy Averaged bead velocity Mass-average fluid velocity Species velocity Set of phase-space coordinates Contributions to relative velocity vector Tensor in dumbbell distribution function Tensors m Rouse distribution function Contributions to the a tensor Finger tensor... 

For the spectroscopy of atoms or ions in gas discharges, optogalvanic spectroscopy (Sect. 1.5) is a very convenient and experimentally simple alternative to fluorescence detection. In favorable cases it may even reach the sensitivity of excitation spectroscopy. For the distinction between spectra of ions and neutral species velocity-modulation spectroscopy (Sect. 1.6) offers an elegant solution. 

An extended Stefan-Maxwell diffusion equation can be constructed by setting up a homogeneous, linear relationship between the vector of diffusion driving forces and the vector of species velocity differences. 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this chapter, we will discuss one particular technique - the Method of Moments, which has been systematically investigated for species dispersion modeling [9, 10]. The Method of Moments was originally proposed to study Taylor dispersion in a circular tube under hydrodynamic flow. Later it was successfully applied to investigate the analyte band dispersion in microfluidic chips (in particular electrophoresis chip). Essentially, the Method of Moments is employed to reduce the transient convection-diffusion equation that contains non-uniform transverse species velocity into a system of simple PDEs governing the spatial moments of the species concentration. Such moments are capable of describing typical characteristics of the species band (such as transverse mass distribution, skew, and variance). 

When a particular component in a mixture is displaced in a given direction, it moves with a certain velocity. This velocity leads to a flux of the species, which is the molar rate of species movement per unit area in any given frame of reference. The nature of the displacements and the forces that cause the displacements leading to species velocity and flux are considered first in this section. Expressions for species velocities and fluxes are then studied to provide the foundations for a quantitative analysis of separation later. 

Tables 3.1.3A-C summarize three items regarding the various fluxes, species velocities and forces Table 3.1.3A identifies the definitions of species fluxes with respect to different frames of references with respect to the individual species velocities Table 3.1.3B provides the interrelationships between the different flux expressions in Table 3.1.3A Table 3.1.3C Illustrates the relation between the important fluxes, bulk velocities, migration velocities and components of chemical potential gradients.

The definition in (3.1.90b) of the migration velocity O) is independent of any of the frames of reference used above to define the fluxes /, and J, Nj, etc. However, if we define Tf,-with respect to the observed species velocity Vt by... 

The preceding treatments were limited to simple molecular diffusion. If the value of or tr was nonzero, it was directly a result of individual species velocity k,-arising from a concentration gradient or a partial pressure gradient. In practical separation processes, there is fluid convection the flow may be laminar or turbulent. The rate of transfer of a species is enhanced considerably by such fluid motion. The fluid motion may or may not be in the direction of the intended species transport More often than not the desired direction of species transport is perpendicular to the direction of bulk motion of the fluid. 

Identify the expression for a species velocity in such a configuration. Specify the exact expression for each term. Assume that both bulk flows, Poiseuille flow and electroosmotic flow, are additive. 

For crossflow membrane processes (Figure 7.0.1(e)-(g)), the species which encounters the least resistance from the membrane (when subjected to its chemical potential based driving force across tbe membrane) will have the highest velocity of movement through the membrane. Correspondingly, its rate of disappearance from the feed stream will be highest, and its concentration in the bulk flow of the reject stream will be lowest Current practice in membrane separation processes does not exploit the differences in individual species velocities through the membrane for multicomponent separation. 

The species conservation equation for solute i (here, salt) in the feed flow channnel may be written using the species conservation equation (6.2.5b) for steady state, no ercternal force based species velocities (t/, = 0) and no chemical reactions as... 

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