Uncertainty ranges

In the feasibility phase the project is tested as a concept. Is it technically feasible and is it economically viable There may be a number of ways to perform a particular task (such as develop an oil field) and these have to be judged against economic criteria, availability of resources, and risk. At this stage estimates of cost and income (production) profiles will carry a considerable uncertainty range, but are used to filter out unrealistic options. Several options may remain under consideration at the end of a feasibility study. 

So the example case results in an uncertainty range from 5.014 to 5.714 with an uncertainty range of 0.7. Therefore if we have a relatively unbiased analytical method, there is a 95% probability that our true analyte value lies between these upper and lower concentration limits. 

The choice of different targets is not only relevant because it leads to different uncertainty ranges, but also because it leads to different strategies. Stabilisation of one type of target, such as temperature, does not imply stabilisation of other possible targets, such as sea-level rise, radiative forcing, concentrations or emissions. 

Figures 7 and 8 illustrate the behavior of the intercepts and slopes from Figure 6 corresponding to the functional forms of Equation 29. The error bars on Figures 7 and 8 represent one standard deviation as determined from isothermal fits. The intercepts have deviations on the order of 0.5% which is consistent with an apparatus analysis. The slopes, however, have much larger uncertainties ranging up to 15%. Increasing the pressure range would greatly reduce this large and important error.

The experimental data are depicted in Fig. 6.1, where the area between the dashed lines represents the uncertainty range that is obtained by using the results in logiojS and Aa and correcting back io I 0. 

Figure 2,5 Polymorphs of A12S105. Dashed area covers uncertainty range among the various experiments. Marked dots B, B, C, C, and H are those plotted in figure 2.4.

Now the uncertainty is added to the results. For result 1 and 4 the situation is still the same. 1 is below, 4 is above the limit. But for the other results we are not sure any more. The uncertainty range tells us, that for both results there is a certain relevant probability that the valne is above lesp. below the limit. 

Let us assume we have an upper limit for compliance of a product and we have four different values, where one is together with its complete expanded uncertainty range above the limit (i) and one is completely below the limit (iv). The other two values have ranges of expanded uncertainty that include ... 

One possibility is the rule described here. With that rule we accept also such values that itself are above the limit, but not with its complete uncertainty range. 

This is a rule that works the other way around. Compliance is stated only when the value and its complete uncertainty range is below the limit. 

Of course it is also possible to decide that in cases where the hmit is within the uncertainty range, other measures have to be taken, e.g. more measurements (to reduee the uneertainty range) or to decide to sell the product at a different price. 

Why is any of this of interest If it is known that some data are normally distributed and one can estimate p and a, then it is possible to state, for example, the probability of finding any particular result (value and uncertainty range) the probability that future measurements on the same system would give results above a certain value and whether the precision of the measurement is lit for purpose. Data are normally distributed if the only effects that cause variation in the result are random. Random processes are so ubiquitous that they can never be eliminated. However, an analyst might aspire to reducing the standard deviation to a minimum, and by knowing the mean and standard deviation predict their effects on the results. 

Thermochemical measurements have been made in an attempt to establish )(Mo=Mo). The interpretation of the data is dependent upon assumptions of the relevant metal-ligand bond energies and, in view of these uncertainties, ranges of values have been quoted 592 196 kJ mol-1 for [Mo2(NMe2)6] and 310-395 kJ mol-1 for [Mo2(OPri)6].203 MO calculations have indicated that the strength of the Mo—Mo triple bond is affected significantly by the nature of the attached ligands and n donors, such as NH2) stabilize this bond. 

Our result thus demonstrates that C3 has a cyclic structure. The estimated cyclic/linear separation is larger than the QCISD(T) results, mainly as a result of N-particle space effects. It is somewhat smaller than the best CCSD(T) results of Scuseria [91], although our uncertainty range includes his values. While these observations seem to suggest a much larger difference between QCISD(T) and CCSD(T) than might have been expected, it should be kept in mind that the QCI calculations are based on a UHF reference, while the CC calculations are based on RHF. Examination of CISD results suggests that half the difference can be attributed directly to reference treatments. 

In individual sensitivity studies where two or several models participated, it was demonstrated that there are differences in the model results, reflecting differences in model formulations of chemical and dynamical processes. A main conclusion of the study is that the uncertainty range of the model estimates of large-scale ozone perturbations is approximately a factor two. 

Large uncertainties remain due to insufficient knowledge about the ozone distributions in the unperturbed as well as the present atmosphere. The ozone distribution is particularly uncertain in the tropics. The best estimate of WMO (1999) of global-mean radiative forcing since the mid-1800s is 0.35 Wm 2 with an uncertainty range +0.15 Wm 2, bracketing the results from a majority of studies. 

Uncertainty range is the range of uncertain variables in a design problem. The uncertainty range can consist of external uncertainties (e.g., supply temperatures and flow rates) and/or internal uncertainties (e.g., heat transfer coefficients). The uncertainty range is typically specified in terms of finite upper and lower bounds on each of the uncertain variables.1... 

Flexible refers to a process which remains feasible for every value of the uncertain variables in the uncertainty range despite desired changes to the process (e.g., supply temperature and flow rate variations due to feedstock changes). 

In all of the resilience analysis techniques reviewed here, the uncertainty range can be extended to include variable target temperatures. In addition, if any of the uncertainties are correlated, then the uncertainty range should include only the independent uncertainties with all the dependent uncertainties expressed in terms of the independent ones. 

Fig. 3. Feasible region R and uncertainty ranges 8 for resilient and nonresilient HENs (a) Resilient HEN (8 C R). (b) Nonresilient HEN (8 R).

The resilience (flexibility) test determines whether a HEN is resilient (flexible) throughout a specified uncertainty range 0,... 

If X d) > 0, then at least one of the feasibility constraints is violated somewhere in the uncertainty range. 

The solution 8C of max-min problem (8) defines a critical point for feasible operation—it is the point where uncertainty range 0 is closest to feasible region R if (d) 0 (Fig. 3a), or it is the point where maximum constraint violations occur if (d) > 0 (Fig. 3b). In qualitative terms, the critical points in the resilience test are the worst case conditions for feasible operation. 

When the critical point must be a vertex of the uncertainty range, the HEN is resilient ( < 0) if and only if it is feasible (i/r s 0) at every vertex. In mathematical terms, semiinfinite problem (6) reduces to a finite optimization problem... 

Example 3 (adapted from Grossmann and Floudas, 1987). The HEN in Fig. 4 was designed for the heat capacity flow rates, target temperatures, and nominal stream supply temperatures shown. This HEN is to be tested for resilience in an uncertainty range of 10 K in all stream supply temperatures. 

It can be shown that the resilience test for this HEN is linear (see Section III,B,1). Therefore the HEN is resilient if and only if it is feasible at every vertex of the specified uncertainty range. 

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