Parameter Feasible Ranges

For any two variables, it is possible to construct a diagram that represents the feasible area on a plane dehned by the two variables as coordinates. If the two product rates are the coordinates, the area is reduced to a straight line, that is, the feasible area is zero. The feasible area varies for different variable pairs the larger the area, the more independent the variables. The process discussed in Example 7.3 is examined in Examples 7.4 through 7.8, each with a different pair of variables selected as the independent variables. 

EXAMPLE 7.4 INDEPENDENT VARIABLES—REFLUX RATIO AND OVERHEAD RATE 

At a given product rate, the reflux ratio may be varied quite independently by changing the vaporization rate in the reboiler and the condensation rate in the condenser, that is, by changing the condenser and reboiler duties. This represents internal circulation within the column and is thus independent of the product rate. Nevertheless, the two quantities are not entirely independent if one variable is fixed, the other may vary only within a certain feasible range. Outside of this range, the vapor or liquid phase may dry up on certain column trays, or one of the products may vanish. 

EXAMPLE 7.5 INDEPENDENT VARIABLES—NQ COMPOSITION IN THE OVERHEAD AND REFLUX RATIO 

FIGURE 7.3 Reflux ratio—product rate feasible region. 

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In all analyses, there is uncertainty about the accuracy of the results that may be dealt with via sensitivity analyses [1, 2]. In these analyses, one essentially asks the question What if These allow one to vary key values over clinically feasible ranges to determine whether the decision remains the same, that is, if the strategy initially found to be cost-effective remains the dominant strategy. By performing sensitivity analyses, one can increase the level of confidence in the conclusions. Sensitivity analyses also allow one to determine threshold values for these key parameters at which the decision would change. For example, in the previous example of a Bayesian evaluation embedded in a decision-analytic model of pancreatic cancer, a sensitivity analysis (Fig. 24.6) was conducted to evaluate the relationship... 

Fig. 2. a) Required number of incoming x-ray photons to observe time-resolved EXAFS of transition metal compounds in H20 solution with a signal-to-noise ratio S/N = 1. No ligand or counterion contributions were included (see Fig. 1). Input parameters are /= 10%, %= 1 % (relative to the absorption edge jump of the selected element). The maxima of curves 2) in Fig. 1 for Fe and Ru correspond to the data points for these elements, b) Feasibility range for time-resolved x-ray absorption spectroscopy. The shaded region indicates the required x-ray dose per data point as a function of the fraction of activated species for the calculated EXAFS experiments on transition metal compounds shown in a). Curves (1) to (3) are extrapolated from experimental results (see section 3. for details) of time-resolved XANES.

The two variables selected to define the column performance become the independent variables, and the others are calculated to satisfy the mass balances, energy balances, and equilibrium relations. Many of the parameters are interdependent to varying degrees, but the two selected as the independent variables must, at least, be independent of each other over certain ranges. In fact, the higher the degree of independence between the two specified variables, the wider the feasible ranges over which the column can operate and the easier for the specified values of the variables to be met. 

If the variable absorbent or stripping gas feed rate in an absorber or stripper is considered as the independent variable, the dependent variables would include product rates and compositions, component recoveries, product temperatures, and so on. When one of these variables is specified, the absorbent or stripping gas stream rate must be calculated to satisfy the specification. The ability of the stream rate to satisfy a given variable specification depends on the relationship between that variable and the stream rate. Each variable may be specified within a certain feasible range, determined by the dependence of the parameter on the stream rate. 

The active variables can be readily identified by screening sensitivity analysis when the sensitivities are ranked according to their absolute values. Such examination partitions the model parameters into two groups the active variables, whose effects on the response(s) are above the experimental noise level, and those below it. In practice, the selection of active variables may depend on considerations other than just the noise-level comparison, such as the certainty with which the parameters are known (i.e., consideration of sensitivity times the range of uncertainty instead of sensitivity alone) or the number of degrees of freedom available for optimization (i.e., the total number of parameters feasible to determine with the given amount of experimental information). 

The archive or frontal set A is initialised by drawing parameters for the STCA system uniformly from their feasible ranges in addition the current best parameter set from manual tuning Q is added to A. Of course many of these randomly selected parameter vectors are dominated by other parameter vectors and these dominated parameters are deleted from A so that A is a non-dominated set (7). In fact, in the work reported here, we found that of 100 randomly initialised parameters only Q and one other parameter vector remained in A after dominated parameter vectors were removed. 

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