Mathematical model confidence

Once the designer has developed confidence in the analysis techniques pertaining to the various parts of a design concept (whether derived from mathematical models or from physical models), the designer can begin the process of synthesis. Synthesis is basically the combining of the analyses (and any other pertinent information) to... 

In process design, it is frequently discovered that many of the basic data needed to rmderstand a process are lacking. Because most crrrrent mathematical models are not sufficiently accrrrate to permit direct scale-up of the process from laboratory data to full plant size, a pilot plant must be constracted. As models are improved, it may become possible to evaluate design decisions with more confidence, and bypass the pilot plant stage. 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

Mathematical modelling of the dose-response relationship is an alternative approach to quantify the estimated response within the experimental range. This approach can be used to determine the BMD or benchmark concentration (BMC) for inhalation exposure, which can be used in place of the LOAEL or NOAEL (Crump, 1984). The BMD (used here for either BMD or BMC) is defined as the lower confidence limit on a dose that produces a particular level of response (e.g., 1%, 5%, 10%) and has several advantages over the LOAEL or NOAEL (Kimmel Gaylor, 1988 Kimmel, 1990 USEPA, 1995 IPCS, 1999). For example, (1) the BMD approach uses all of the data in fitting a model instead of only data indicating the LOAEL or NOAEL (2) by fitting all of the data, the BMD approach takes into account the slope of the dose-response curve (3) the BMD takes into account variability in the data and (4) the BMD is not limited to one experimental dose. Calculation and use of the BMD approach are described in a US EPA... 

A mathematical model of a plant or a section of a plant can be judged only by comparison with actual plant data. The model may be considered as good when the simulated variables can predict with some level of confidence the plant parameters which are important in determining the cost and quality of the finished product. Failures of the model are likely to be a result of (1) oversimplification of the equations that constitute the model, (2) inadequacy of the numerical solution of the equations. 

When only one factor is involved in the experiment, the predictive ability is often visualised by confidence bands. The size of these confidence bands depends on the magnitude of the experimental error. The shape , however, depends on the experimental design, and can be obtained from the design matrix (Section 2.2.3) and is influenced by the arrangement of experiments, replication procedure and mathematical model. The concept of leverage is used as a measure of such confidence. The mathematical definition is given by... 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

As previously discussed, the physico-chemical phenomena occurring in a monolith SCR catalyst are relatively well understood by now the mathematical models describing such phenomena can then be applied with confidence to the rational design of SCR catalysts and processes. 

Finally, an important step is the validation of the mathematical model performed by comparing its predictions to the experimental data from literature or in-house. The data used for validation must be different from the data used to build the model, or it will be a self-fulfilling prediction. Validation using animal or human in vivo data, whenever available, will significantly increase confidence in the model. 

Normally, the laboratories use fixed and inflexible criteria for this control. They define a limit to the percentage difference between the expected and the obtained value, and in low precision techniques they are obliged to increase this value. Assuming the uncertainty associated with the control standard preparation to be negligible when compared to the instrumental uncertainty, the case-to-case interpolation uncertainties can be used as a fit for each case. If the observed confidence interval includes the expected value, there is reason to think that the system is not under control. The instrumental deviation from control can be used as a guide for instrumental checking or as a warning of the inadequacy of the chosen mathematical model for the calibration. 

A number of mathematical models have been developed to describe the interplay of solubility and these physiological parameters to model dmg absorption. The most simplistic model is the maximal absorbable dose (MAD) calculation. The MAD calculation combines the amount of dmg that can dissolve to form a saturated solution in water equal in volume to the small intestinal volume, with an estimate of the absorption rate and the small intestinal transit time. The maximal absorbable dose is then related to the dose required to achieve the desired therapeutic effect [2], If the estimated MAD is much greater than the predicted dose to achieve a therapeutic effect, this can give confidence enough to take the dmg toward clinical use. Predictions of aqueous solubility may then be useful in predicting the extent of absorption in man. 

The determination of kinetic parameter values from column experiments is predicated upon the ability of the mathematical model to successfully simulate the experimental data. Confidence in the robustness of the parameter values so determined is attained only with a unique solution (i.e., when one suite of parameter values provides a solution that is significantly better than all others). For cases wherein a system is near equilibrium or under extreme nonequilibrium, attainment of a unique solution may prove difficult. A modified miscible-displacement technique, involving flow interruption, that enhances the potential for achieving unique solutions, and thus increases the robustness of optimized values of kinetic parameters, was presented by Brusseau et al. (1989a). In addition, the method has increased sensitivity to nonequilibrium, making it useful for process-level investigation of sorption kinetics. This method would appear to be especially useful for systems com-... 

An additional experiment was chosen to verify the validity of the models and to check their predictive ability, with the condition ofxi = xa = X3 = X4= 0.5, i.e. T, = 57.5 °C, <, = 9 h, 7d = 162.5 °C and = 9 h. The model-predicted and the experimental values of the responses are shown in Table 5. The observed values are within the 95% prediction interval, and within the 95% confidence interval of the predicted ones. The observed values are in good agreement with the predicted ones therefore the three models are valid and have satisfactory predictive ability. It is also confirmed that RSM is an effective tool for the mathematical modeling of the mechanical properties of solid catalysts. 

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