What If... Parameter Sensitivity

As a first approximation, one can assume plug flow through the gauze, in which case the design equation is similar to that for monolith reactors. 

Differential form of the wire gauze design equation 

One of the most important skills of an engineer is to be able to predict the effects of changes of system variables on the operation cf a process. The engineer needs lo determine these effects quickly through approximate but reasonably close calculations, which are sometimes referred to as back-of-the-envelope calculations. This type of calculation is used to answer such questions as Whal will happen if I decrease the particle size What if I triple the flow rate through the reactor  

We will now proceed to show how this type of question can be answered using the packed-bed, mass transfer-limited reactors as a model or example system. Mere we want to learn the effect of changes of the various parameters (e.g., temperature, particle size, superficial velocity) on the conversion. We begin with a rearrangement of the mass transfer correlation, Equation (11-49), to yield 

Find out how the mass transfer coefficient vanes with changes in physical properties and system properties 

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What If... (Parameter Sensitivity) 788 The Shrinking Core Model 792... 

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... 

Realistic predichons of study results based on simulations can be made only with realistic simulation models. Three types of models are necessary to mimic real study observations system (drug-disease) models, covariate distribution models, and study execution models. Often, these models can be developed from previous data sets or obtained from literature on compounds with similar indications or mechanisms of action. To closely mimic the case of intended studies for which simulations are performed, the values of the model parameters (both structural and statistical elements) and the design used in the simulation of a proposed trial may be different from those that were originally derived from an analysis of previous data or other literature. Therefore, before using models, their appropriateness as simulation tools must be evaluated to ensure that they capture observed data reasonably well [19-21]. However, in some circumstances, it is not feasible to develop simulation models from prior data or by extrapolation from similar dmgs. In these circumstances, what-if scenarios or sensitivity analyses can be performed to evaluate the impact of the model uncertainty and the study design on the trial outcome [22, 23]. 

There are two possible routes. The first is to estimate the missing parameters by comparison to force field parameters for similar systems. If for example there are no torsional parameters for rotation armind a H-X-Y-O bond in your molecule, but parameters exist for H-X-Y-C. then it is probably a good approximation to use the same values. In other cases it may be less obvious what to chose. What if your system has an O-X-Y-O torsion, and parameters exist for 0-X-Y-C and C-X-Y-O, but they are very different. What to choose then One or the other, or the average After a choice has been made, the results should ideally be evaluated to determine how sensitive they are to the exact value of the guessed parameters. If for example the guessed parameters can be varied by 50% without seriously affecting the final results, the property of interest is insensitive to the guessed parameters, and can he trusted to the... 

More sophisticated approaches to predict the extent of oral absorption of drugs use mathematical models based on the known physiology and utilizing simple physicochemical measurements as input. The GastroPlus [4] program is a commercial tool that utilizes an advanced compartmental and transit model, based on the work of Amidon and Yu [5], and allows what-if questions to be posed to enable pharmaceutical optimization (see Chapter 17). For instance, the impact of morphology, formulation, and/or particle size changes, and sensitivity analysis to include errors in parameters on the prediction, can be considered. 

P4-2 i What if... you were asked to explore the example problems in this chapter to learn the effects of vary ing. the different parameters This. sensitivity analysis can be carried out by either downloading the examples from the web or by loading the programs from the CD-ROM supplied with the text. For each of the example problems you investigate, write a paragraph describing your findings. 

What if..." questions The What if,." questions are particularly effective when used with the Lh ing Example Problems where one varies the parameters to explore the problem and to carry out a sensitivity analysis. For example, whm if someone suggested ilutr you sftould double the catalyst particle diameter, wluil you say ... 

F9-Z. "What if.. " problem. These problems build in the Living Example Problems in this chapter. By this point in the point the studerrt is in a position to carry out in-depth and meaningful parameter sensitivity studies. If there is only time to assign one or two parts of this problem I would choose part (a) and then either (d) or (e>. However, one could rotate the assignments of (a) through (e) from year to year. 

What if.. problem. A number of parts to Ihjs problem require descriptive answers. Part (h) involves a parameter sensitivity of Living Example Problem 10-7. 

P13 l( Prepare a list of safety considerations for designing and operating chemical reactors. See the August ) 985 issue oJ Chemical Engineering Progress, P13-2b Review the example problems in this chapter and use u software package such as Polymath or MATLAB to carry out a parameter sensitivity analysis to answer the following What if..." questions. 

Optimal TMD parameters found by the above formulation are intrinsically more robust than TMD parameters found by any equivalent deterministic formulation. However, it is important to demonstrate that indeed results obtained by the robust optimization are less sensitive to changes in the estimated structural parameters p mass, stiffness, and damping of the structural elements and of the TMD. Hence, a perturbation is introduced in vector p, such as to verify what would be the actual reliability index of the structure if parameter values were not the nominal values used in designing the TMD. Then, the perturbed reliability index can be compared with the original, and a measure of robustness is obtained the smaller the relative difference, the more robust is the design ... 

If this model is selected, one must then decide what variables to use for the ordinate and the abscissa. The parameters must be dose-sensitive, free of confounding variables, easily determined and preferably linear. We have evaluated this approach for estimating the bioavailability of calcium in mechanically deboned meat products (11). Typically, correlations between various bone parameters and dietary calcium are very high (r = 0.943 to 0.999). This is consistent with what others have found for similar parameters (46,47). These correlations are also similar to the those (r = 0.947 to 0.982) between the amount of calcium consumed and calcium retained (11) a good index procedure. 

Raman spectroscopy s sensitivity to the local molecular enviromnent means that it can be correlated to other material properties besides concentration, such as polymorph form, particle size, or polymer crystallinity. This is a powerful advantage, but it can complicate the development and interpretation of calibration models. For example, if a model is built to predict composition, it can appear to fail if the sample particle size distribution does not match what was used in the calibration set. Some models that appear to fail in the field may actually reflect a change in some aspect of the sample that was not sufficiently varied or represented in the calibration set. It is important to identify any differences between laboratory and plant conditions and perform a series of experiments to test the impact of those factors on the spectra and thus the field robustness of any models. This applies not only to physical parameters like flow rate, turbulence, particulates, temperature, crystal size and shape, and pressure, but also to the presence and concentration of minor constituents and expected contaminants. The significance of some of these parameters may be related to the volume of material probed, so factors that are significant in a microspectroscopy mode may not be when using a WAl probe or transmission mode. Regardless, the large calibration data sets required to address these variables can be burdensome. 

It is safe to say that most graduate courses in chemical reaction engineering today suffer from an excess of mathematical sophistication and insufficient contact with reality. Because of the complexity of many reaction engineering models, it is essential that students be given a balanced and realistic view of what can and cannot be achieved. For example, they must learn that if the intrinsic kinetics of a reaction are not known accurately, this deficiency cannot be made up by a more detailed understanding of the fluid mechanics. In this connection, it would be useful pedagogically to take a complex model and illustrate its sensitivity to various aspects, such as the assumptions inherent in the model, the reaction kinetics, and the parameter estimates. 

In critical cases it may well be worthwhile to make a complete analysis of stability. In many cases, however, enough can be learned by studying what Bilous and Amundson (B7) called parametric sensitivity. These authors derived formulas for calculating the amplification or attenuation of disturbances imposed on an unpacked tubular reactor originally in a steady state, with the idea that if the disturbances grow unduly the performance of the reactor is too sensitive to the conditions imposed on it, that is, to the parameters of the system. The effect of feedback from a control system was not considered. As pointed out by the authors, it would be a much more complicated task to apply their procedure to a packed reactor, but it still would entail far less computation than a study of the transient response. 

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