Migration velocity, equations

This equation has been used for estimating migration velocities of radionuclides (e.g. 66). Here Pr is the density of the rock (kg/m3), p the density of water, e the fissure porosity, af the specific surface of fissures in the bedrock (m2/m3) and ap the specific surface of particles used in the Kd determinations (m2/m3). The distribution coefficient Kd represents ar. equilibrium value for the particular rock under the pertinent conditions. 

As can be observed in Equation (11) the migration velocity is directly proportional to the ionic charge of a compound and the applied electric field strength. It is inversely proportional to the viscosity of the medium and the hydrodynamic radius of the compound. The electric field strength is determined by the applied voltage difference (V, in V) and the... 

As can be derived from the characteristics of the electrophoretic mobility, the migration of a compound can be influenced by altering the charge, the viscosity of the medium, and the dynamic radius of the compounds. Introducing Equation (13) in Equation (11), the electrophoretic mobility is the proportionality factor in the linear relationship between the migration velocity and the electric field strength ... 

This equation shows that for a given collection efficiency, the precipitator size is inversely proportional to particle drift velocity and directly proportional to gas flow rate. Increasing the gas density (migration velocity is a function of gas viscosity) by reducing its temperature or increasing the pressure will reduce the precipitator size. However, theory does not account for gas velocity. This is a variable that influences particle re-entrainment and the drift velocity. This typically requires an ESP design at lower velocities than predicted in theory. 

The proportional constant/in Equation 6.11 is called the friction factor. When the two forces equal each other, the ion moves with a constant migration velocity, which is expressed by the following equation ... 

Each ion has an apparent migration velocity napp easily obtained from the electro-pherogram. If / designates the effective length of the capillary between the injector and the detector, and migration time, then app can be obtained using the equation app == l/tm. 

By combining the apparent mobility and the electro-osmotic flow, which is responsible for the migration of the bulk electrolyte, it is possible to calculate the migration velocity or the electrophoretic mobility of charged species. Using equation (8.3), equation (8.5) can be written as ... 

The capacity factor k to be discussed shortly, is an alternate measure of retention. While k is used more often than R in chromatography, the use of R is advantageous because (i) it is directly proportional to peak migration velocity and is thus a more direct measure of retention than k (ii) most equations describing chromatography are simpler when expressed in terms of R rather than k and (iii) R is a more universal measure of retention R but not k applies to other perpendicular flow methods such as field-flow fractionation. 

The migration velocity u of the EOF can be described in simplified form by means of the Helmholtz equation ... 

Retention, expressed by the capacity factor k of nonionic analytes, is a function of the partition coefficient and the volume of pseudostationary phase (micelles), the volume of the mobile phase, the retention time of the analyte, the dead time (corresponding to the migration velocity of the EOF), and the retention time of the micelles. If the micelles were immobilized in the capillary (i.e., if the pseudostationary phase were stationary ), the capacity factor would be similar to the standard equation of retention in chromatography. 

As the overall aim of parameter determination is the (simulation-based) prediction of the process behavior the final decision about the suitability of the isotherm equation can only be made by comparing experimental and theoretical elution profiles (Section 6.6). Depending on the desired accuracy, this may involve iteration loops for the selection of isotherm equations or in some cases even the methods (Fig. 6.16). In this context it should be remembered that, according to the model equations (e.g. Eq. 6.47), the migration velocity and thus the position of the profiles is a function of the isotherm slope. Therefore, it is most important for the reliability of process simulation that the slopes of the measured and calculated elution profile fit to each other. If the deviations are unacceptable, another isotherm equation should be tested. 

Equation 7.2 shows that to each mobile phase concentration, C, is associated a migration velocity, Uz, given by... 

The migration velocity of the stable concentration shock is given by the following equation... 

In these equations, Uzfi is the migration velocity of the solute at infinite dilution given by Eq. 10.85, a is the Courant number, and the order of the error made in the three models is the first order, 0 h + r). 

Figure 8.3 Huckel equation and typical electropherogram. Influence of the net charge, electric field, volume of the ion and viscosity of the solution upon the migration velocity in an electrolyte animated by an electro-osmotic flow (cf. Section 8.2.2). Higher charge and smaller size confer greater mobility, whereas lower charge and larger size confer lower mobility. The separation depends approximately upon the charge-to-size ratio of each species. Uncharged species move at the same velocity as the electro osmotic flow (see later). The smaller anions arrive last since they would normally go towards the anode. At low pH, electro-osmotic flow is weak and anions may never reach the detector.

Equations 25 and 26 both predict a radial migration velocity (V) increasing as the square of the cross-flow velocity (U) [The Reynolds Number (Re) also includes U). This can explain why the dependence of UF flux on cross-flow velocity (U) can be higher than 0.33 in laminar flow and 0.75 in turbulent flow (e.g., see Figures 3.50 and 3.51). 

Equations 25 and 26 also predict that the radial migration velocity (V) will increase as the tube radius (R) decreases. Thin channels are more effective in depolarizing the membrane surface via the tubular pinch effect. This may explain the larger discrepancies between experimental and theoretical flux values in 15 mil channels (see Figure 3.46) than in 30 mil channels (see Figure 3.47). 

The relative importance of migration and diffusion can be gauged by comparing Ud with the steady-state migrational velocity, u, for an ion of mobility Wj in an electric field (Section 2.3.3). By definition, v = where % is the electric field strength felt by the ion. From the Einstein-Smoluchowski equation, (4.2.2),... 

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