Generalized sensitivity analysis

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

Hornberger and Spear s original application of generalized sensitivity analysis (GSA) used a binary acceptance-rejection procedure, i.e., they discarded a Monte Carlo realization if they thought that the prediction was inconsistent with the site-specific data (a nonbehavior ) or kept it if they thought it was consistent (a behavior ). The prior probability on each Monte Carlo realization was the reciprocal of the total number of realizations. After the acceptance-rejection procedure was applied, the updated (posterior) probability on each realization that was classified as a behavior was the reciprocal of the number of behaviors, and the posterior probability on nonbehaviors was zero. 

Equation (4.11) is a useful form of the Bayesian acceptance-rejection procedure for generalized sensitivity analysis, in that it applies whenever one s model is predicting an average of a measured quantity. 

Spear R. C. and Hornberger G. M. (1980) Eutrophication in Peel Inlet II. Identification of critical uncertainties via Generalized Sensitivity Analysis. Water Res. 14, 43-59. 

L. Eno and H. Rabitz, "Generalized sensitivity analysis in quantum collision theory", J. Chem. Phys. 71, 4824 (1979)... 

Ever brighter vacuum-ultraviolet sources are being developed that would further boost SPI sensitivity, which already is typically 10 useful yield general, sensitive elemental analysis would then also be available using SPI, making possible a single laser arrangement for both elemental and molecular SALE... 

Finally, it is generally recommended to use estimation approaches, combined with sensitivity analysis, for additives when data are missing, when performing an LCA case study on additive containing products, such as outlined above. Only when they are included it is possible to draw conclusions on the importance of additives over the life cycle of a product. 

However, as a general observation, this study demonstrated the feasibility of the integrated modeling approach to couple an environmental multimedia and a PBPK models, considering multi-exposure pathways, and thus the potential applicability of the 2-FUN tool for health risk assessment. The global sensitivity analysis effectively discovered which input parameters and exposure pathways were the key drivers of Pb concentrations in the arterial blood of adults and children. This information allows us to focus on predominant input parameters and exposure pathways, and then to improve more efficiently the performance of the modeling tool for the risk assessment. 

Most cells used in infrared spectrometry have sodium chloride windows and the path length is likely to vary with use because of corrosion. For quantitative work, therefore, the same cell should be used for samples and standards. In general, quantitative analysis in the infrared region of the spectrum is not practised as widely as in the ultraviolet and visible regions, partly because of the additional care necessary to obtain reliable results and partly because the technique is generally considered to be less sensitive and less precise a precision of 3-8% can be expected. 

Currently, the EPA considers acute pesticide exposures to represent a reasonable certainty of no harm when exposure at the 99.9th percentile is below the RfD. When exposure at the 99.9th percentile exceeds the RfD, the EPA will generally conduct a sensitivity analysis to determine whether particular factors that drive the exposure, such as high residue or high consumption levels, are unusual and so may represent artifacts that artificially skew the exposure distribution curve. 

Transport of NH4+ to the roots in Kirk and Solivas experiment was mainly by diffusion. The additional transport resulting from mass flow of soil solution in the transpiration stream would have increased the influx across the roots by about QQaVa/0.5bD% where Va is the water flux (Tinker and Nye, 2000, pp. 146-148), or about 4% in Kirk and Solivas experiment. A sensitivity analysis showed that rates of diffusion will generally not limit uptake in well-puddled soils, but they may greatly limit uptake in puddled soils that have been drained and re-flooded and in unpuddled flooded soils. 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Fixed capital investment Working capital requirements Total capital investment Total manufacturing expense Packaging and in-plant expense Total product expense General overhead expense Total operating expense Marketing data Cash flow analysis Project profitability Sensitivity analysis Uncertainty analysis... 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

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