Rigorous approach

Two methods are commonly used to correct for the residual current. One method is to extrapolate the total measured current when the analyte s faradaic current is zero. This is the method shown in the voltammograms included in this chapter. The advantage of this method is that it does not require any additional data. On the other hand, extrapolation assumes that changes in the residual current with potential are predictable, which often is not the case. A second, and more rigorous, approach is to obtain a voltammogram for an appropriate blank. The blank s residual current is then subtracted from the total current obtained with the sample. 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

Approximate design equations apphcable only to the case of pure physical desorption are developed later in this sec tion for both packed and plate stripping towers. A more rigorous approach using distiUation concepts may oe found in Sec. 13. A brief discussion of desorption with chemical reac tion is given in the subsec tion Absorption with Chemical Reaction. ... 

For preliminai y-screeniug worh the simpler methods may be adequate, but for final designs one shoiild seriously consider using a more rigorous approach. 

The cost of performing the hazard identification step depends on the size of the problem and the specific techniques used. Techniques such as brainstorming, what-if analyses, or checklists tend to be less expensive than other more structured methods. Hazard and operability (HAZOP) analyses and failure modes and effects analyses (FMEAs) involve many people and tend to be more expensive. But, you can have greater confidence in the exhaustiveness of HAZOP and FMEA techniques—their rigorous approach helps ensure completeness. However, no technique can guarantee that all hazards or potential accidents have been identified. Figure 8 is an example of the hazards identified in a HAZOP study. Hazard identification can require from 10% to 25% of the total effort in a QRA study. 

This example illustrates the simplified approach to film blowing. Unfortunately in practice the situation is more complex in that the film thickness is influenced by draw-down, relaxation of induced stresses/strains and melt flow phenomena such as die swell. In fact the situation is similar to that described for blow moulding (see below) and the type of analysis outlined in that section could be used to allow for the effects of die swell. However, since the most practical problems in film blowing require iterative type solutions involving melt flow characteristics, volume flow rates, swell ratios, etc the study of these is delayed until Chapter 5 where a more rigorous approach to polymer flow has been adopted. 

In this chapter we will develop more rigorous approaches to the analysis of gas turbine plants using both the first and second laws of thermodynamics. 

Young and Wilcock [7] provided a much more rigorous approach which includes an illuminating discussion of how the los.ses arise in the cooling process. They prefer to address the problem by breaking the overall flow into flows through the component ... 

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

A word of caution before we start. A rigorous approach to many of the derivations requires keeping track of several different indices and validating why certain transformations are possible. The derivations will be performed less rigorously, trying to illustrate the flow of arguments, rather than focus on mathematical details. 

Gesztesy, F Grosse, H., and Thaller, B., 1984, A rigorous approach to relativistic corrections of bound state energies for spin-1/2 particles , Ann. Inst. Poincare 40 159. 

Of course, most of what I have said so far is well known. Nevertheless, I hope to have given these issues a new perspective by adopting an almost perversely rigorous approach in demanding that every aspect of electronic configurations should be strictly deducible from quantum mechanics. Although I am not in a position to propose a better explanation, I do not think that we should be complacent about what the present explanation achieves. As I have tried to argue, in terms of deduction from theoretical principles, the present semi-empirical explanation is not fully adequate. 

A rigorous approach to studies of naturally produced toxic materials would involve isolation of the compounds in pure form and demonstration that the substances isolated were actually responsible for the observed inhibition. Synthesis and confirmation of activity of the synthetic vs. the natural material may also be required. These criteria have been met in relatively few cases. Unfortunately, studies with naturally occurring inhibitors have been made primarily by plant physiologists without sufficient assistance from chemists. 

We leave it to the reader to pursue the rigorous approach to probability theory in appropriate texts. 

A less rigorous approach to modeling than the one described above is an empirical procedure whereby water data for a given geometry may be linked with that of another fluid by numerical scaling factors without actually... 

Complete Example 11.4 for the case where Vg/Qg = 20s. Solve the governing ODEs analytically or numerically as you prefer. How does this more rigorous approach change the 95% response time calculated in Example 11.3 ... 

A rigorous approach to evaluating this dissipated energy has previously been undertaken [7] and later improved to allow for the behavior of practical elastomers [9-12], However, we shall present here a simplified argument in order to arrive at the essential scaling law and will subsequently quote the more exact version. 

While our discussion will mainly focus on sifica, other oxide materials can also be used, and they need to be characterized with the same rigorous approach. For example, in the case of meso- and microporous materials such as zeolites, SBA-15, or MCM materials, the pore size, pore distribution, surface composition, and the inner and outer surface areas need to be measured since they can affect the grafting step (and the chemistry thereafter) [5-7]. Some oxides such as alumina or silica-alumina contain Lewis acid centres/sites, which can also participate in the reactivity of the support and the grafted species. These sites need to be characterized and quantified this is typically carried out by using molecular probes (Lewis bases) such as pyridine [8,9],... 

A rigorous approach to the discretization of equipment sizes involves the use of standard equipment sizes already at the start of the cost minimization problem, i.e. Eqn. 7.4-31 is solved but with Vj chosen for each stage from a set of standard equipment units DSVj of type j. The objective function then is formulated as ... 

In this section a phenomenalogical approach to the description of solvent gradients will be adopted based largely on the linear solvent strength model prt osed by Snyder [552-555]. This is the least complicated of the models available and provides a reasonable approximation for typical experimental conditions. Mathematically more rigorous approaches have been developed by Jandera and Churacek [551,556], schoenmakers et al. [520,534,557] and Tomellini et al. [558]. 

The approaches described above give approximate values for the LoD and LoQ. This is sufficient if the analyte levels in test samples are well above the LoD and LoQ. If the detection limits are critical, they should be evaluated by using a more rigorous approach [1, 2, 14]. In addition, the LoD and LoQ sometimes vary with the type of sample and minor variations in measurement conditions. When these parameters are of importance, it is necessary to assess the expected level of change during method validation and build a protocol for checking the parameters, at appropriate intervals, when the method is in routine use. 

These rules are not foolproof but should serve as a first approximation. A more rigorous approach must be used for more quantitative assessments. One possibility is to compute defect formation energies for the compound in question using atomistic simulations or quantum mechanical theory. These formation energies can be inserted into formulas similar to those described for simple defect populations (Chapter 2) ... 

Worked Example 3.12 is somewhat artificial, because most reactions proceed with differing numbers of bonds breaking and forming. A more rigorous approach quantifies the energy per bond - the bond enthalpy AHEE (also called the bond dissociation energy ). 

---