Reduced parameters velocity

Assumed reduced parameters h -linear velocity of the mobile phase. 

The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities then, like the reduced plate height, it also is dimensionless. Employing the reduced parameters, the equation of Knox takes the following form... 

Upon substitution of the reduced parameters given above the separation time for a packed column and an open tubular column would be Identical if d 1.73 dp given the current limitations of open tubular column technology the column diameter cannot be reduced to the point %diere these columns can compete with packed columns for fast separations. This is illustrated by the practical txanple in Figure 6.3 (57). Ihe separation speed cannot be Increased for an open tubular column by increasing the reduced velocity since the reduced plate height is increased... 

Assumed reduced parameters h = 3, v = 4.5. These are optimum values fromagraphofreducedplateheightversus reduced linear velocity of the mobile phase. 

A more vigorous treatment similar to the van Deemter equation but developed specifically for HPLC is the Knox equation, which uses a number of reduced parameters where h is the reduced plate height (h/dp) and vis reduced velocity (V dJD. ... 

The velocity profile, Eq. (7.62), can be plotted as a function of position in the gap along the x direction, now given by the reduced parameter, p, of Eq. (7.64). This is done schematically in Figure 7.65. Notice that ai p = X the velocity profiles are flat that is, the flow is plug flow, because the pressure gradient vanishes at these locations. As promised, the pressure profile can also be solved for, for which the result is... 

The two parameters in the rate equation that have been redefined are the column diffusivity H and the linear velocity u. The reduced parameters, which are dimensionless, are... 

The reduced parameters are also helpful in evaluating column performance. The best columns have a reduced plate height of 2 to 5—a number that can be thought of as representing the number of particles between sorptions—and 2 is a practical minimum. The reduced velocity represents the ratio between the flow velocity and the diffusion rate over one particle diameter typical values should be in the range of 3 to 20. 

To illustrate more clearly the effect of these variables on analysis time, reduced parameters can be used for the plate height and velocity. Reduced parameters effectively normalize the plate height and velocity for the particle diameter and the diffusion coefficient to produce dimensionless parameters that allow comparison of different columns and separation conditions. The reduced plate height and reduced velocity are expressed, respectively, as... 

Fig. 5. Differential cross section (weighted with sin 9) as a function of the deflection angle 9 and the reduced parameter A = krm which is proportional to the velocity. The calculation was performed for a LJ 12-6 potential with B = 5000. In the upper part the total cross section multiplied by v° is plotted versus A. The close connection between the number of supernumerary rainbows and the number of the glory undulations is clearly demonstrated. Note that another rainbow oscillation is buried under the forward diffraction peak and not shown in the figure.

To overcome this problem, it is sometimes preferable to use reduced parameters, the reduced plate height, h, and the reduced fluid velocity, v, respectively, in place of H and y. The dimensionless parameters h and V are respectively defined by the expressions ... 

The reduced plate height, h, is independent of the particle diameter. The reduced fluid velocity (u), a concept conceived by Giddings, is a measure of the rate of flow over a particle relative to the rate of diffusion of solute within the particle. Since both reduced parameters, h and V, are normalized for the particle diameter, when h is plotted against v, the different size fractions of the same packing materials should give similar curves. This has been confirmed in practice, so that use of h vs v curves is preferred over that of plate height curves (H versus mobile phase velocity) when comparison of the efficiency of different columns is to be carried out. 

Here, h is the reduced plate height, v is the reduced mobile-phase velocity, and A, B, and C are constants for a given HPLC system. The reduced parameters are given by... 

If the average distance between the inclusions is much less than the characteristic size of a continuous phase over which the macroscopic parameters (velocity, pressure, temperature etc.) change, then it is possible to describe the macroscopic processes in the mixture by the methods of mechanics of continuous mediums as well. To this end, the averaged or macroscopic parameters are introduced. At this point, the concept of multi-velocity continuum [6], representing a set of N continuums, can also be introduced. The number N is the number of considered phases, each of which corresponds to a certain constituent phase of the mixture and fills one and the same volume. For each of these constituent continuums we define in the usual manner the local density, which is called the reduced density... 

Effective occupant restraints, safety systems, and protective equipment not only spread impact energy over the strongest body structures but also reduce contact velocity between the body and the impacted surface or striking object. The design of protective systems is aided by an understanding of injury mechanisms, quantification of human tolerance levels and development of numerical relationships between measurable engineering parameter, such as force, acceleration or deformation, and human injury. These relationships are called injury criteria. 

Effect of larger cross section/reduced fluid velocities shall also be considered on other process parameters, conversion of reactants, and reduction in coeflicient of heat transfer due to reduced turbulence before taking a final decision. 

As explained elsewhere in this book, resolution in SEC can be expressed in terms of the peak standard deviation and the slope of the calibration curve. As in other HPLC modes, the efficiency of SEC columns can be improved by decreasing particle size. The relationship between column efficiency (or plate number N) and velocity can be expressed i dimensionless (reduced) parameters. The reduced plate height h is equal to the ratio of the height of a theoretical plate and the particle size as shown in Equation (1). The reduced velocity v is equal to the product of the linear velocity and particle size dp divided by the solute diffusion coefficient/), as shown in Equation (2). 

From an engineer s point of view it is highly desirable to describe the chromatographic system by reduced parameters. This helps to understand some very basic principles of chromatography and to compare very different systems. Reduced parameters are dimensionless. Instead of the plate height H and mobile phase linear velocity u, one defines the reduced plate height h and the reduced velocity v by the following two equations ... 

FIGURE 2. Fraction I/Iq of the transmitted beam of O atoms as a function of the reduced parameter B/v, where B is the magnetic induction across the beam and v is the velocity of the atoms. (Black triangles, open dots, black dots, open triangles correspond to 1.33, 1.60, 1.80 and 2.21 km s" ). The Ml line is calculated for the ground 3pj state, the dashed line for the metastable E>2 state. In the lower panel, the relative weights Wj of the ij mj> states of 0(3pj) are shown. 

In addition, Figs. 2-4a, 2-4b and 2-4c indicate that the reduced droplet velocity ut increases with the packing size d. The material of the packing elements also has an influence on the parameter ut. It follows from this that the resistance coefficient i[fo in Eq. (2-22) is a function of the size and the surface properties of the packing. The first influencing factor can be expressed dimensionless by the quotient f2(dh/dT). The second factor is linked to the resistance coefficient lr of the dry packing. These two effects are reflected in the general correlation (2-24) ... 

Fractures, cracks, and other defects of the solid mineralic substance change the elastic properties (and other physical properties like electrical, hydrauhc, thermal) dramatically. Elastic wave velocities decrease and a strong dependence on pressure results. As demonstrated by experiments and the foregoing sections, the porosity as the ratio of the volume of the defects (cracks, etc.) to the total volume alone cannot express these effects—microcracks with mily a small porosity can significantly reduce the velocity (see, for example. Fig. 6.8). Other parameters such as aspect ratio and crack density are necessary to describe the physical effects of these defects. 

Deshpande et al. (1992) studied scaling-down of EBRs reducing the length of reactors and ensuring identical hydrodynamic properties in commercial and laboratory units. In order to maintain a given space velocity and to reduce the reactor length, the liquid velocity has to be reduced. To achieve ebullation at this reduced liquid velocity, the particle size has to be reduced as well. The reduction in superficial liquid velocity and particle size has to be achieved in such a manner that the important hydrodynamic parameters, such as the phase holdups, remain unchanged. 

COW/ERGENCE TOLEPJJJCE PAPIW ETER FOR VELOCITIES CONVERGENCE TOLERANCE PARAMETER FOR TEMPERATURE NUMBER OF INTEGRATION POINTS PER ELEMENT MATERIAL PARAMETERS AT FULL INTEGRATION POINTS MATERIAL PARAMETERS AT REDUCED INTEGPATION POINTS... 

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