Rigorous Methods

Going from the Laplace domain to the frequency domain by substituting s = m gives 

To make a Nyquist plot, we merely substitute for z in the function and make a polar plot of as frequency u goes from 0 to oJ2, where 

The reason why we only have to cover this frequency range will be demonstrated in the example below. 

Example 19.9. Let s consider the first-order lag process with a proportional sampled-data controller. 

Note that this function has no poles outside the unit circle, so P = 0. 

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RIGOROUS METHODS FOR MULTICOMPONENT DISTILLATION-TYPE SEPARATIONS... 

When it is desired to compute, with rigorous methods, actual rather than equilibrium stages, Eqs. (13-69) and (13-94) can be modified to include the Murphree vapor-phase efficiency T ij, defined by Eq. (13-29). This is particularly desirable for multistage operations involving feeds containing components of a wide range ol volatility and/or concentration, in which only a rectification (absorption) or stripping action is provided and all components are not sharply separated. In those cases, the use of a different Murphree efficiency for each component and each tray may be necessary to compute recovery accurately. 

Suppose that the process does not have any process MSAs. How can a lean composite line be developed The following shortcut method can be employed to construct the pinch diagram for external MSAs. A more rigorous method is presented in Chapter Six. 

A more rigorous method of tackling nonconvex equilibrium without discretization has been developed by Srinivas and El-Halwagi (1994) and is beyond the scope of this book. 

The elaborate treatment for the H2O system is only possible beeause of its small size. For larger systems, less rigorous methods must be employed. Let us as a more realistic example consider a determination of the relative stability of the C4H6 isomers shown in Figure 11.15. There are experimental values for the first eight structures, which allows an evaluation of the performance of different methods. This in turn enables an estimate of how much trust should be put on the predicted values for 9, 10 and 11. 

The graphical methods of Bras provide helpful short-cuts to avoid the tedious trial-and-error solutions required of the rigorous methods. Reference 9 is the most recent and perhaps the easier to use. The results agree in general within about 10%. 

When the pressure drop ratio (PI - P2)/P1 is less than 0.42, the steam flow is less than the critical flow. While BS 5793 Part 2 Section 2.2 1981 provides an approximate and a rigorous method of calculating flows, it is found in practice that the best results are not... 

A more sensitive and rigorous method of detecting and quantifying allosteric effects is through observation of the kinetics of binding. 

The structures of rutile and anatase, the two tetragonal forms of titanium dioxide, have been determined by rigorous methods (Figs. 1 and 2). They seem at first sight to have little in common beyond the fact... 

Six parameters must be evaluated to determine the positions of 8 Ti and 160 in the unit in case that both the no-parameter positions 4a and 4 b are occupied, and nine parameters in case that all atoms are in positions 8 c. A structure dependent on such a large number of parameters can not be determined by rigorous methods. 

These four equations are the so-called MESH equations for the stage Material balance, Equilibrium, Summation and Heat (energy) balance, equations. MESH equations can be written for each stage, and for the reboiler and condenser. The solution of this set of equations forms the basis of the rigorous methods that have been developed for the analysis for staged separation processes. 

If the approximation had caused an error of 10% or more, you would not be able to use it. You would have to solve by a more rigorous method, such as the quadratic equation. 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

A more rigorous method for interpreting an alkalinity measurement is to use a reaction model to reproduce the titration. The technique is to calculate the effects of adding acid to the original solution, assuming various carbonate contents. When we produce a model that reaches the endpoint pH after adding the acid, we have found the correct carbonate concentration. 

All team members should be familiar with PrHA objectives, the PrHA method to be used, and their roles in performing the PrHA. A 1- or 2-hour overview at the beginning of the first team review session is generally sufficient for this purpose. However, the more demanding PrHA methods, such as fault tree analysis (FTA), require more training and/or a greater depth of experience than less-rigorous methods, such as what-if and checklist analyses. 

When conducting a LOPA study, several methods can be used to determine the frequency. One of the less rigorous methods includes the following steps ... 

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