Dimensionless variables, definition

This is the definition assumed in the work of Shoup and Szabo [507] and Aoki and coworkers [57,61,62] and also by Gavaghan in some recent works [260,261]. The above three formulae (12.9, 12.11 and 12.12), are those for this definition of normalised time. One inconvenient side-effect of the definition is that, when one normalises the diffusion (12.2), using the new dimensionless variables definitions... 

It is convenient to introduce the dimensionless variable by the definition so that the Hamiltonian operator becomes... 

The governing dimensionless partial derivative equations are similar to those derived for cyclic voltammetry in Section 6.2.2 for the various dimerization mechanisms and in Section 6.2.1 for the EC mechanism. They are summarized in Table 6.6. The definition of the dimensionless variables is different, however, the normalizing time now being the time tR at which the potential is reversed. Definitions of the new time and space variables and of the kinetic parameter are thus changed (see Table 6.6). The equation systems are then solved numerically according to a finite difference method after discretization of the time and space variables (see Section 2.2.8). Computation of the... 

As can be seen, these expressions are formally identical to those deduced for a planar electrode (compare Eqs. (2.109) and (2.19)), with the only difference being the definition of the dimensionless variable s for planar electrodes and, vgME for DME. Under these conditions, surface concentrations are also given by Eq. (2.20) and condition (2.22) is also fulfilled. 

The inlet conditions are identical to Equation 28. Equations 28,34 and 35 can be written in dimensionless form similar to Equations 30-32. The definition of the dimensionless variables is identical to those of the MAT model, except for... 

Determine the power number at process and operating conditions. Turbulence in agitation can be quantified with respect to another dimensionless variable, the impeller Reynolds number. Although the Reynolds number used in agitation is analogous to that used in pipe flow, the definition of impeller Reynolds number and the values associated with turbulent and laminar conditions are different from those in pipe flow. Impeller Reynolds number NRe is defined as... 

Setting ua i = 0 corresponds to a liquid feed which is innocent of dissolved gaseous reactant, while setting u i - 1 corresponds to a liquid feed which exists in equilibrium with the gas at the feed temperature, pressure, and composition. Definitions for the dimensionless variables which appear in Equations 2-6 are given in the nomenclature. 

To reduce the number of parameters and to analyze their interdependence it is recommended that the model equations as well as the boundary conditions be converted into a dimensionless form. The following definition of dimensionless variables is used Feed concentrations cfeedi should be selected as a reference for concentrations and loadings ... 

Momentum boundary layer problems in the laminar flow regime, particularly (12-18d) and (12-18e), are revisited after a new set of dimensionless variables is introduced. This strategy will be successfiil if one redefines dimensionless independent spatial coordinates (i.e., x and y ) and the x and y components of the dimensionless velocity vector (i.e., v and up such that u and u do not depend explicitly on the Reynolds number, based on simultaneous solution of (12-18d) and (12-18e). The appropriate definitions are... 

By introducing the dimensionless variables t/r and xjL and by using the definition of the dimensionless Bodenstein ° number Bo (Equation 2.2-35) ... 

These equations can be made dimensionless by first choosing an appropriate scaling of the time variable, say, t = xlk. Whereas dimensionless equations are not necessary for carrying out a stability analysis, they often simplify the associated algebra, and sometimes useful relationships between parameters that would not otherwise be readily apparent are revealed. It is also important to note that the particular choice of dimensionless variables does not affect any conclusions regarding number of steady states, stability, or bifurcations in other words, the dimensionless equations have the same dynamical properties as the original equations. Introducing the definition t = into the above equations we find ... 

It should be noted that the definition and use of the dimensionless variables allows a robust solution of the model equations, easy convergence being obtained for all three discretization methods and for very crude solution estimates (for example, all concentrations set at 0.5). 

Nevertheless, the temperature and pressure variables need further consideration. The pressure scale is usually found indirectly by substituting all the other non-dimensional variable definitions into the momentum equation. Accordingly, after substituting the dimensionless variables into the momentum equation, the non-dimensional pressure is defined as p =. Ifps symbolizes the pressure scale, we... 

The reactivity and physical properties of continuous species are defined as a function of a dimensionless variable x e [0, oo). This variable is usually related to a measurable physical quantity, such as molecular weight or boiling point. The fraction of a continuous species belonging to an interval of variable x can be calculated by integrating the time-dependent probability density function p(x, t) over this interval. According to its definition, the integral of this pdf is unit over the whole domain of definition of x at any time. 

In this is a dimensionless variable of distance and time. The inelusion of in the definition makes the variable dimensionless and the faetor of 2 is ineluded so that the solution for Z)j = ZJj = 0 will be the erfc( tj) function. If a solution of this form exists then the partial differential equation can be converted into an ordinary... 

The analysis is the same as in the preceding section, as long as the relationship between potential and time has not been introduced. The same dimensionless approach may also be followed with the exception of the time variable, which may now be normalized against the inversion time tR t = t/tR, leading to the following definition of the normalized current A = /FSC ) [D. In applying equations (6.1) to the first potential step,... 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

Figure H1.1.5 Empirical models that are used to predict the complete flow curve of non-Newtonian fluids or portions of the complete curve. In the full-curve models, K is a constant with time as its dimension and m is a dimensionless constant. See text for definition of other variables in equations.

The enthalpy 7, besides a pair of variables determining the physical state of the material (e.g., p and S or p and v), also depends on the chemical variable n—the depth of occurrence of the irreversible chemical reaction which, for definiteness, we will equate with the concentration (dimensionless, g/g) of the final reaction product. It should be kept in mind that the reaction occurs, particularly at the beginning, irreversibly. 

We must emphasize that, given a constant a, any electromagnetic field may be written in the form (131) and that, with this definition, the Clebsch variables are dimensionless quantities. 

The model contains 7 equations and 12 variables. Fixing the reactor holdup and reaction temperature fixes the Da number. zA 3 and zB 5 are given by the separation performance. Two additional specifications are needed. The plantwide control that we recommend (Figure 4.5) does not rely on self-regulation. The reactor-inlet flow rate of both reactants fKIA and /Rf E are fixed. Fresh feed rates /0A and /0B are used to control inventories at some locations in the plant. Note that an arbitrary flow rate can be used as reference in definition of the dimensionless quantities. 

The values of the indices a, bi,c, c4l dimensional variables must be known in terms of the basic dimensions. These are as follows, the way in which they are found from the definition of the v ariable being indicated in some cases. For example, since ... 

To normalize the governing equations, we introduce a dimensionless position, z = x/a, and two dimensionless dependent variables,/ =/// and u = ua/DD. Note that the normalized velocity m is equivalent to a local Peclet number, indicating the relative magnitudes of the advective and diffusive fluxes of the reactive species. Applying these definitions to the transport equations yields the dimensionless governing equations... 

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