Extrapolation outside

Smoothed data presented at rounded temperatures, such as are available in Tables 6.2 and 6.4, plus the C° values at 298 K listed in Table 6.1 and 6.3, are especially suitable for substitution in the foregoing parabolic equations. The use of such a parabolic fit is appropriate for interpolation, but data extrapolated outside the original temperature range should not be sought. 

Instead of radical reactions, models based on molecular reactions have been proposed for the cracking of simple alkanes and Hquid feeds like naphtha and gas oil (40—42). However, the vaUdity of these models is limited, and caimot be extrapolated outside the range with confidence. With sophisticated algorithms and high speed computers available, this molecular reaction approach is not recommended. 

It is inadvisable to extrapolate outside the regions given in Figure 9.7. For high scaled-pressure values (i.e., scaled energy larger than 0.8), method 3 should be used. 

Such plots should produce a straight line with the slope being equal to -AH /R and the intercept equal to n kB/h) + AS" /R. As the available temperature range typically is 100°, the error in AH will typically be 0.1-0.5 kcal/mol. The activation entropy is determined by extrapolating outside the data points to 7 = cxc (1/7 = 0), and is usually somewhat less well defined, a typical error may be 5 e.u. 

A word of caution should be added with regard to the calculation of the burn-out flux for a pressure intermediate to the main pressure groups that have been correlated this calculation must not be done by taking intermediate y values from Table II. The recommended procedure is to estimate the burn-out flux for the required conditions for the main pressure groups above and below the required pressure, and then to interpolate linearly. It must be also emphasized that while the above correlations can be used with confidence within the experimental ranges of the data, extrapolation outside these ranges should not be taken very far without allowing for a possible reduction in the accuracy obtained. 

All predictions must be taken for what they are, namely, generalizations based on current knowledge and understanding. There is a temptation for a user to assume that a computer-generated answer must be correct. To determine whether this is in fact the case, a number of factors concerning the model must be addressed. The statistical evaluation of a model was addressed above. Another very important criterion is to ensure that a prediction is an interpolation within the model space, and not an extrapolation outside of it. To determine this, the concept of the applicability domain of a model has been introduced [106]. 

Autocatalytic decomposition can be determined by 1ST techniques. The sensitivity of 1ST equipment enables measurements to be made at relatively low temperatures, which results in the potential to examine a wide temperature range. This is an advantage because extrapolations outside the temperature range actually examined in tests are reduced. The importance here is that the kinetics of decomposition at high temperatures are not always the same as at the lower temperatures of likely processing. 

The corollary is that we should always keep in mind the experimental range. Extrapolation outside that range is unwise. This will become particularly clear when we discuss the yield phenomenon - an area of great interest in many practical situations. Whatever the origins claimed... 

Equations 18-20 give about the same results and the standard obtained in the procedure that led to Equation 20 is shown in Figure 7. Since these equations contain only steric parameters, the picture obtained from the MTD method can directly be used to compare the biological activity of the compounds. MTD and MTD are in principle the same in this example because no electronic and hydrophobic parameters are involved. However, extrapolation outside the hypermolecule is not permissible. 

However, they can have very exciting properties when the higher terms become large, as they can have multiple maxima and minima. For instance, take y = a + bx + cx, which is dominated by the constant term a at very small values of x. But at sufficient large values of x, say x > ajb, the linear term begins to be more important than the constant term and at even larger values of x, such as x > a/c and x > b/c, the quadratic term takes over. At the value of x = —bflc, the first derivative dy/dx = 0, so that the function is either at a maximum or a minimum. If the function of interest is required by theory to be monotonic, then it may be hazardous to extrapolate outside of a limited range with a polynomial. 

It is common within the industry to characterize chemical processes in terms of one or a few global reaction steps, assigning an Arrhenius rate expression to describe the rate of each reaction. If knowledge of the detailed chemistry is inadequate or the chemical scheme is to be combined with computational fluid dynamics for a complex flow description, a simplified chemistry may be necessary. It is important, however, to realize that such a chemical description can only be used for the narrow range of conditions (temperature, composition, etc.) for which it is developed. Any extrapolation outside these conditions may be erroneous or even disastrous. 

The PLS model for fed-batch fermentation performs well when the fermentation is operating within conditions that are represented in the 20 training batches used in developing the model. As with many model-based systems, model performance is poor when extrapolating outside of the operating conditions of the training set. 

Table 3.2 summarizes the chemistries that lead to the precipitation of solid phases. The actual parameters for these equilibria are in Appendix B. The range of temperatures used in model parameterizations or validation and the maximum molal concentrations are both fundamentally important in properly applying the model. Extrapolations outside these tempera-ture/compositional ranges require careful scrutiny of model calculations. 

You can interpolate linearly for any API oil value between these equations and with extrapolation outside to 90°API. Temperature coverage is good from 50 to 300°F. If outside of this range, use the American Society for Testing and Materials (ASTM) Standard Viscosity-Temperature Charts for Liquid Petroleum Products (ASTM D-341 [6]). The values derived by Eqs. (1.1) to (1.4) are found to be within a small percentage of error by the ASTM D-341 method. 

In many practical applications, the engineer often has only plant performance data to use to backcalculate kinetic parameters. Data of this type are seldom extensive enough to permit precise calculation of all parameters since the plant normally operates in a fairly narrow window of operating conditions. However, useful simplified kinetics and parameters can often be determined that describe the major kinetics inside this region. Extrapolation outside the region from which the data has been obtained is very risky. 

From the perspective of the design engineer, the advantage of this approach is that the expressions for the adsorbed-phase concentrations are simple and explicit. However, the expressions do not reduce to Henry s law in the low-concentration limit, which is a thermodynamic requirement for physical adsorption. They therefore suffer from the disadvantage of any purely empirical equations, and they do not provide a reliable basis for extrapolation outside the range of experimental study. 

When sufficient data are available, use of the benchmark dose (BMD) or benchmark concentration (BMC) approach is preferable to the traditional health-based guidance value approaches (IPCS, 1999a, 2005 USEPA, 2000 Sonich-Mullin et al 2001). The BMDL (or BMCL) is the lower confidence limit on a dose (the BMD) (or concentration, BMC) that produces a particular level of response or change from the control mean (e.g. 10% response rate for quantal responses one standard deviation from the control mean for a continuous response) and can be used in place of the NOAEL. The BMD/BMC approach provides several advantages for dose-response evaluation 1) the model fits all of the available data and takes into account the slope of the dose-response curve 2) it accounts for variability in the data and 3) the BMD/BMC is not limited to one experimental exposure level, and the model can extrapolate outside of the experimental range. 

Both models have the disadvantage that the effect increases with increasing concentrations without an upper limit which is a highly unphysiologic behavior. Therefore, such models should be used with caution or better should not be used for extrapolation outside the observed range. The gold standard of the empirical PD models is the sigmoid Amax model with or without the Hill coefficient ... 

Related Calculations. Graphic representation of liquid-liquid equilibrium is convenient only for binary systems and isothermal ternary systems. Detailed discussion of such diagrams appears in A. W. Francis, Liquid-Liquid Equilibrium, Interscience, New York, 1963. Thermodynamic correlations of liquid-liquid systems using available models for liquid-phase nonideality are not always satisfactory, especially when one is trying to extrapolate outside the range of the data. 

Finally, the method is only applicable in the temperature range of 300-425 K. Extrapolation outside this range is not recommended. The group parameters are not temperature-dependent. Consequently, predicted phase equilibria extrapolate poorly with respect to temperature. 

A dose with a specified low level of excess health risk, generally in the range of 1% to 10%, that can be estimated from data with little or no extrapolation outside the experimental dose range. For laboratory animals, an exposure (usually at low concentrations) of long duration, such as months or years. For human populations, an exposure that lasts at least 7 years and could last as long as a lifetime. 

Mechanistic Modeling. In mechanistic modeling, an intrinsic reaction network is determined, based on the most probable mechanistic description. Rate constants are established individually for these elementary reactions through kinetic measurements. This type of model allows confident extrapolation outside the range of the data base used in its development. 

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