Scaled sensitivity coefficients

The sensitivity coefficients defined by equation (43) relate the absolute change in a solution variable (species concentration) to an absolute change in a parameter (rate constant), and thus have units that depend on the units of the rate constant, which in turn depend on the overall reaction order. The scaled sensitivity coefficients defined by equation (51) relate fractional changes in a solution variable to fractional changes in a parameter. Thus, for example, if <7, = 1, then a 10% increase in parameter dj will lead to a 10% increase in solution variable Likewise, if [Pg.236]

Fig. 7. Scaled sensitivity coefficients for AlCl under the same conditions as in Figs. 5 and 6. Curves are labeled with reaction numbers from Table 10. Solid curves and positive numbers correspond to sensitivities to the forward rate constants, while dotted curves labeled with negative numbers are sensitivities to the reverse rate constants.

Taken together, the analyses of species concentrations, reaction rates, and scaled sensitivity coefficients for this particular example and conditions provide a clear and consistent picture of which species and reactions are most important in the overall process of AICI3 decomposition and reaction with H2 to produce AlCl and HCl. From the 39 reactions included in Table 10, just 5 seem to be essential. These... 

According to (5.44) the semi-logarithmic sensitivity coefficients show the local change in the solutions when the given parameter is perturbed by unity on the logarithmic scale and are invariant under the scaling of the parameters. 

Figure 2b presents the plots of the flow stress taken at =50% as a function of initial strain rate at 450°C in a double logarithmic scale. The strain rate sensitivity coefficient m>0.3 was observed in all the strain rate range examined both for the ECAE and hot rolled (RD) conditions. The flow stress was lower and the strain rate sensitivity was higher in the ECAE condition as compared to the hot rolled one. 

Continuing with the example based on the reaction mechanism of Table 10, the equations for the sensitivity coefficients (equation (50)) were integrated numerically using Matlab. For a mechanism of this size, it was feasible to solve for the species concentrations (15 of them) and all the sensitivity coefficients (39 x 15x2 = 1170 of them) simultaneously. Rather than examining the sensitivity coefficients themselves, it is often more informative to examine scaled or normalized sensitivity coefficients. These are usually defined as... 

Figure 7a shows the variation of p with respect to the cross-correlation coefficient P = PiT,jrj=Pc,Ga or identical mean values, i.e., r= 1, and g = 00. The reliability is seen to decrease as the properties of the two materials become independent. In Fig. 7b, scaled sensitivity measures of p with respect to the standard deviations are shown. It is observed that the reliability is most sensitive to the uncertainty in the load and that the sensitivity to the uncertainty in the material properties diminishes with increasing correlation between the property variables. This is expected, as with increasing correlation the plate becomes statically determinate and the stress distribution becomes... 

Selected time courses of sensitivity coefficients for the endpoint of blood VX concentration following a single subcutaneous dose of 3.2 pg VX kg" BW with cardiac output and tissue blood flows adjusted for brain AChE activity. Inset in (d) is the same data just on a shorter time scale to see the smaller values of the coefficients. Note that some lines completely overlap (a) lines for fractional lung volume and blood in lung (d) lines for length of injection and subcutaneous/blood partition coefficient until the end of the injection (e) lines for dose volume and multiple of dose volume. Coeff coefficient. 

The direction of vector CC is along the tangent of the manifold at point C and is identical for the perturbation of any parameter x. The direction of this vector is identical to the direction of all sensitivity vectors dYIdxic, and the projections of this vector onto the axes are the sensitivity coefficients (see Fig. 8.4b). If the directions of two vectors are identical, then the ratios of their projections onto the axes are identical, even if the lengths of the vectors are different. This explains the scaling law and also why any sensitivity vector can be obtained by multiplying any other nonzero sensitivity vector belonging to a different parameter by an appropriate scalar. 

Only slightly less accurate ( 0.3—0.5%) and more versatile in scale are other titration techniques. Plutonium maybe oxidized in aqueous solution to PuO " 2 using AgO, and then reduced to Pu" " by a known excess of Fe", which is back-titrated with Ce" ". Pu" " may be titrated complexometricaHy with EDTA and a colorimetric indicator such as Arsenazo(I), even in the presence of a large excess of UO " 2- Solution spectrophotometry (Figs. 4 and 5) can be utilized if the plutonium oxidation state is known or controlled. The spectrophotometric method is very sensitive if a colored complex such as Arsenazo(III) is used. Analytically usehil absorption maxima and molar absorption coefficients ( s) are given in Table 10. Laser photoacoustic spectroscopy has been developed for both elemental analysis and speciation (oxidation state) at concentrations of lO " — 10 M (118). Chemical extraction can also be used to enhance this technique. 

Atmospheric sensitivity renders the preparation of ultrapure samples difficult. Nevertheless, vacuum distillation ", ultra-high-vacuum reactive distillation " and crystal growth purification methods " are described zone-refining methods have been applied on a limited scale only - , presumably because of the high volatility of the metals and the unfavorable distribution coefficients. 

The above experimental results largely relate to spectroscopic techniques, which do not give direct information about the spatial scale of the molecular motions. The size of the spatial heterogeneities is estimated by indirect methods such as sensitivity of the dynamics to the probe size or from the differences between translational and rotational diffusion coefficients (rotation-translation paradox). It might be expected that the additional spatial information provided by neutron scattering could help to discriminate between the two scenarios proposed. 

Figure 6.9 Effect of CITREM concentration on the molecular and thermodynamic parameters of complex protein-surfactant nanoparticles in aqueous medium (phosphate buffer, pH = 7.2, ionic strength = 0.05 M 20 °C) (a) extent of protein association, k = Mwcomplex/Mwprotem (b) structure-sensitive parameter, p (c) second virial coefficient, A2 (rnolal scale) (d) effective charge, ZE (net number n of moles of negative charges per mole of original sodium caseinate nanoparticles existing at pH = 7.2 (Mw = 4xl06 Da)). The indicated cmc value refers to the pure CITREM solution. Reproduced from Semenova et al. (2007) with permission.

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