Steepest slope

The plot immediately shows whioh of the parameters the 10% NPV is most sensitive to the one with the steepest slope. Consequently the variables can be ranked in order of their relative impact. 

The steepest descent method is a first order minimizer. It uses the first derivative of the potential energy with respect to the Cartesian coordinates. The method moves down the steepest slope of the interatomic forces on the potential energy surface. The descent is accomplished by adding an increment to the coordinates in the direction of the negative gradient of the potential energy, or the force. 

Instead of the definition in Eq. (7-82), the selectivity is often written as log k,). Another way to consider a selectivity-reactivity relationship is to compare the relative effects of a series of substituents on a pair of reactions. This is what is done when Hammett plots are made for a pair of reactions and their p values are compared. The slope of an LEER is a function of the sensitivity of the process being correlated to structural or solvent changes. Thus, in a family of closely related LFERs, the one with the steepest slope is the most selective, and the one with the smallest slope is the least selective.Moreover, the intercept (or some arbitrarily selected abscissa value, usually log fco for fhe reference substituent) should be a measure of reactivity in each reaction series. Thus, a correlation should exist between the slopes (selectivity) and intercepts (reactivity) of a family of related LFERs. It has been suggested that the slopes and intercepts should be linearly related, but the conditions required for linearity are seldom met, and it is instead common to find only a rough correlation, indicative of normal selectivity-reactivity behavior. The Br nsted slopes, p, for the halogenation of a series of carbonyl compounds catalyzed by carboxylate ions show a smooth but nonlinear correlation with log... 

The highest acid concentration (H+(aq)) / largest surface area of magnesium produces the steepest slope and fastest rate. 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

It is evident from Figure 13.10 that the slopes do not vary over a similar range, as required by Equation (53). As a matter of fact, there are examples for which the steepest slope is associated with the coarsest particles (as required by theory) and others for which it occurs with the smallest particles. The quantitative predictions fail on this particular point, but, as we see below, there are some discrepancies between the theoretical model and the actual experimental system that may account for this apparent insensitivity to particle size. Example 13.4 considers another application of Equation (53) to an experimental system. 

In Figure 2, we compare the coke-conversion selectivity behavior as a function of activity for MAT, FFB and riser reactors. The kc is relatively flat for the FFB test, with correspondingly higher negative slopes for MAT and steady state risers. The riser results show the steepest slope indicating that even though the kg observed in the FFB test is relatively independent of activity, the coke selectivity improves with activity in a riser. 

One consequence of the negative-order kinetics is auto catalysis of the crystallization process. If the initial concentration is higher than that of the rate maximum, crystal growth will accelerate initially as the concentration decreases. This is illustrated in Fig. 13. Once past the concentration of the growth-rate maximum, the rate drops off. Interestingly, the positions of steepest slope in the time dependencies of crystal length and width do not coincide (Fig. 13), as the positions of the maxima in Guo and Gioo differ (Fig. 12). 

In Tables II and III the results of partial extraction of OX and DC1 are summarized. Release delay is defined as the intersection of the steepest slope of the release curve with the time axis. The delayed tsg time, (t5gd) is measured from the onset of release. 

The steepest slopes observed are those for —F and —O indicating the dramatic contribution to impulse by a slight increase in AHf. In general, then, — N compounds have a more positive slope than —Cl compounds, demonstrating the relatively better performance of N as a carrier atom over Cl. 

Sensitivity fading since the slope at the extrema is zero, for Pout/ in values equal to zero or one, A(/ values are not determinable with high sensitivity. The highest sensitivity can be obtained at the points with the steepest slope at A[Pg.42]

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR q>e. In many cases, the product produced by such machines is too small for commercial use therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser crystal prc uct. When this is done, the product sample plots on a graph of In n versus L as shown in hne P, Fig. 18-62. The line of steepest slope, line E, represents the particle-size distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lf, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. 

Identification of the Parameters of a Model by the Steepest Slope Method... 

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