Reduced dimensionless

All variables in the system can be expressed in reduced form. Velocity can be expressed as tt = where Uq is a fixed reference velocity and j/ is the dimensionless reduced velocity. Because time, /q, is the quotient of length, Lq, and velocity, Uq, the equation can be manipulated to yield... 

Note that the dependence of the correction tenn is on the dimensionless reduced gradient, not the absolute gradient. 

The results for total backscattering yields and backscattered energy of hydrogen ions on different materials are shown in Fig. 1063 6S> 70 7 . The values are plotted as a function of a dimensionless reduced energy e given by7 ... 

Here, T = eelectrostatic potential, with representing the electrostatic potential, kg the Boltzmann s constant, T the temperature, and e the elementary charge co is the permeability of free space, while Ed is the dielectric constant within the volume element dv. We take Ed as 2.0 inside the membrane and the protein and as 80.0 in the aqueous solution. The integration in Eq. (3) is performed over the volume V of the entire space. 

Equations-of-state are often expressed in an alternative form (Equation 3.21), in terras of the density rather than the volume, and in terms of dimensionless reduced variables instead of the actual temperature, density and pressure. For a given temperature and pressure, Equation 3.21 is solved for the density (or, equivalently, for the volume). The reduced temperature (T), reduced density (p ) and reduced pressure (p) are defined by equations 3.22, 3.23 and 3.24,... 

It is customary to discuss gradient expansions for exchange in terms of dimensionless reduced density gradients... 

It has been common to display the results of computer simulation studies of systems approaching the glassy state as simple linear plots of D versus T, usually in dimensionless reduced units natural to the simulation For example, for soft potentials. 

Dimensionless reduced bonding R(D) Cumulative number of pellets... 

The harmonic oscillator has V = and the harmonic-oscillator Schrodinger equation contains the three constants k, m, and h. We seek to find a dimensionless reduced energy , and a dimensionless reduced x coordinate x, that are defined by... 

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). 

Kenndler, E. Effect of electroosmotic flow on selectivity, efficiency, and resolution in capillary zone electrophoresis expressed by the dimensionless reduced mobility. J. Capillary Electrophoresis 1996, 3 (4), 191. 

Since there is no cut-off in l k uation 189, integral 226 cx)nvcrges only when d < 2, so regularization (bringing the expression with rf < 2 to the dimension d < 4) is required. After this procedure (Freed, 1983), the following expression results for dimensionless reduced osmotic pressure... 

Fig. 1.7 Thermal blob model describing the conformational change of a polymer chain with the temperature. The vertical axis is the dimensionless reduced temperature t with the theta temperature as the reference temperature.

For a better comparison between the behavior of different gases, it is necessary to choose a proper reference state. Gases depart from ideality when the temperature approaches the critical temperature. It is therefore more logical to compare gases at a temperature which is relative to their T. A set of three dimensionless reduced variables was introduced for this purpose this is the principle of corresponding states ... 

Problems 4.30-4.38 apply the Numerov method to several other one-dimensional problems. In solving these problems, you need to (a) find combinations of the constants in the problem that will give a dimensionless reduced energy and length [Eq. (4.68)] ... 

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