Magnitude criterion

Value of open-loop gain constant K Applying the magnitude criterion to the above point... 

Closed-loop poles (For K = 11.35) Since the closed-loop system is third-order, there are three closed-loop poles. Two of them are given in equation (5.81). The third lies on the real locus that extends from —5 to —oo. Its value is calculated using the magnitude criterion as shown in Figure 5.15. 

Hence point lies on locus. Applying magnitude criterion... 

As before, we substitute Pt + eAf for 7r, however only the leading term is acceptable on the basis of our order of magnitude criterion so that we have... 

The relative insensitivity of this type of diffusion criterion to particle shape and to assumption of exact kinetics, has been discussed in connection with the macroscopic reactant diffusion problem on catalyst granules (7). The condition (15) is a general order-of-magnitude criterion defining the physical conditions of intimacy between the component systems for no mass-transport inhibition. It defines a requirement for realizing the formal kinetics of polystep reactions. 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

The magnitude of the NHV has economic importance because the consumption and cost of motor fuels are frequently expressed in liters/100 km and in Francs/liter in France. From the technical viewpoint, the NHV, establishes the maximum range for a transport system with a given load. This is a decisive criterion for applications like aviation. 

We now turn specifically to the thermodynamics and kinetics of reactions (5. EE) and (5.FF). The criterion for spontaneity in thermodynamics is AG <0 with AG = AH - T AS for an isothermal process. Thus it is both the sign and magnitude of AH and AS and the magnitude of T that determine whether a reaction is thermodynamically favored or not. As usual in thermodynamics, the A s are taken as products minus reactants, so the conclusions apply to the reactions as written. If a reaction is reversed, products and reactants are interchanged and the sign of the AG is reversed also. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

A more difficult criterion to meet with flow markers is that the polymer samples not contain interferents that coelute with or very near the flow marker and either affect its retention time or the ability of the analyst to reproducibly identify the retention time of the peak. Water is a ubiquitous problem in nonaqueous GPC and, when using a refractive index detector, it can cause a variable magnitude, negative area peak that may coelute with certain choices of totally permeated flow markers. This variable area negative peak may alter the apparent position of the flow marker when the flow rate has actually been invariant, thereby causing the user to falsely adjust data to compensate for the flow error. Similar problems can occur with the elution of positive peaks that are not exactly identical in elution to the totally permeated flow marker. Species that often contribute to these problems are residual monomer, reactants, surfactants, by-products, or buffers from the synthesis of the polymer. 

The standard state chosen for the calculation of controls its magnitude and even its sign. The standard state is established when the concentration scale is selected. For most solution kinetic work the molar concentration scale is used, so A values reported by different workers are usually comparable. Nevertheless, an important chemical question is implied Because the sign of AS may depend upon the concentration scale used for the evaluation of the rate constant, which concentration scale should be used when A is to serve as a mechanistic criterion The same question appears in studies of equilibria. The answer (if there is a single answer) is not known, though some analyses of the problem have been made. Further discussion of this issue is given in Section 6.1. 

Taking the minimum drilling cost as a criterion for determining the optimum magnitude of WOB, RPM and T, the following equations are applicable ... 

As is seen, the results of BFCP and SCS are practically identical when ji,jf ji — jf but for the low-frequency transitions SCS yields a halved relative error. In the high-frequency region, the approximate theory provides only the order of magnitude since the basic criterion of SCS for semiclassical relative motion is violated. 

In determining whether a bicyclic system is large enough to accommodate a bridgehead double bond, the most reliable criterion is the size of the ring in which the double bond is located. Bicyclo[3.3.1]non-l-ene" (112) and bicyclo[4.2.1 ] non-l(8)ene (113) are stable compounds. Both can be looked upon as derivatives of tra 5-cyclooctene, which is of course a known compound. Compound 112 has been shown to have a strain energy of the same order of magnitude as that of... 

A typical test for convergence is Ak°H> < TOL where TOL is a user-specified tolerance. This test is suitable only when the unknown parameters are of the same order of magnitude. A more general convergence criterion is... 

When the parameters differ by several orders of magnitude between them, the joint confidence region will have a long and narrow shape even if the parameter estimation problem is well-posed. To avoid unnecessary use of the shape criterion, instead of investigating the properties of matrix A given by Equation 12.2, it is better to use the normalized form of matrix A given below (Kalogerakis and Luus, 1984) as AR. 

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