Stepping technique

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

The Bergmann variation of the Balz Schiemann reaction is a two-step process featuring copper- or copper halide-catalyzed decomposition of aqueous or acetone solutions of arenediazonium fluoroborates containing alkyl or halogen substituents [30] A recent modification is a one-step technique featuring simultaneous diazoti-zation and decomposition by addition of aqueous sodium nitrite at 25 °C to a mixture of fluorobonc acid, copper powder, and 2 isopropyl 6 methylaniline to give 2-isopropyl-6 methylfluorobenzene in 73% yield [37]... 

However, in the case of a root cause analysis system, a much more comprehensive evaluation of the structure of the accident is required. This is necessary to unravel the often complex chain of events and contributing causes that led to the accident occurring. A number of techniques are available to describe complex accidents. Some of these, such as STEP (Sequential Timed Event Plotting) involve the use of charting methods to track the ways in which process and human events combine to give rise to accidents. CCPS (1992d) describes many of these techniques. A case study involving a hydrocarbon leak is used to illustrate the STEP technique in Chapter 7 of this book. The STEP method and related techniques will be described in Section 6.8.3. 

The first case study describes the application of the sequentially timed event plotting (STEP) technique to the incident investigation of a hydrocarbon leak accident. Following the analysis of the event sequence using STEP, the critical event causes are then analyzed using the root cause tree. 

This case study concerns the events leading up to the hydrocarbon explosion which was the starting point for the Piper Alpha offshore disaster. It describes the investigation of the incident using the sequentially timed events plotting (STEP) technique. Based on the STEP work sheet developed, the critical events involved in the incident are identified and analyzed in order to identify their root causes. 

A number of stages are used when applying the STEP technique. These will be illustrated with respect to the investigation carried out for the above incident. 

The case study has documented the investigation and root cause analysis process applied to the hydrocarbon explosion that initiated the Piper Alpha incident. The case study serves to illustrate the use of the STEP technique, which provides a clear graphical representation of the agents and events involved in the incident process. The case study also demonstrates the identification of the critical events in the sequence which significantly influenced the outcome of the incident. Finally the root causes of these critical events were determined. This allows the analyst to evaluate why they occurred and indicated areas to be addressed in developing effechve error reduchon strategies. 

Application of this two-step technique has led to the synthesis of the novel... 

Essentially, stripping analysis is a two-step technique. The first, or deposition, step involves die electrolytic deposition of a small portion of the metal ions hi solution into die mercury electrode to preconcentrate the metals. This is followed by die shipping step (the measurement step), which involves die dissolution (shipping) of die deposit. Different versions of stripping analysis can be employed, depending upon die nature of the deposition and measurement steps. 

Although the mechanisms discussed above are still topics of debate, it is now firmly established that the electrodeposition of conducting polymers proceeds via some kind of nucleation and phase-growth mechanism, akin to the electrodeposition of metals.56,72-74 Both cyclic voltammetry and potential step techniques have been widely used to investigate these processes, and the electrochemical observations have been supported by various types of spectroscopy62,75-78 and microscopy.78-80... 

Kontturi et al. studied TEA ion transfer across water-1,2-DCE microinterfaces covered by different PCs using short potential step techniques [12]. The enhancement in the forward rate constant was observed for all lipids and increased with the surface coverage (Fig. 6). 

The kinetics of CO oxidation from HClOi, solutions on the (100), (111) and (311) single crystal planes of platinum has been investigated. Electrochemical oxidation of CO involves a surface reaction between adsorbed CO molecules and a surface oxide of Pt. To determine the rate of this reaction the electrode was first covered by a monolayer of CO and subsequently exposed to anodic potentials at which Pt oxide is formed. Under these conditions the rate of CO oxidation is controlled by the rate of nucleation and growth of the oxide islands in the CO monolayer. By combination of the single and double potential step techniques the rates of the nucleation and the island growth have been determined independently. The results show that the rate of the two processes significantly depend on the crystallography of the Pt surfaces. 

Whereas with potential step techniques, 2 = ktm, with alternating-current methods, 2 = k/v, and with RDEV, 2 = kb1 jD, where the symbols are as defined in Chapter E) Full treatment of the diffusion-reaction problem is provided in Section 6.2.1. 

The kinetics of following chemical reactions cannot be studied by the single potential step technique in that the response would simply obey the Cottrell equation. In contrast, the double potential step technique, that measures the response exhibited by either the reagent Ox or the product Red, is sensitive to the chemical fate of Red. The cathodic response before the inversion of the applied potential (t < x) is expressed by the Cottrell equation ... 

The responses of the forward and reverse steps are both affected by the chemical complication in fact, the reduction current will be greater than that predicted by the Cottrell equation, whereas the reoxidation current will be lower than that predicted by the Cottrell equation. This implies that to gain information on the regeneration of the reagent it is sufficient to use the single potential step technique. 

What precedes is true for any other electrochemical technique, using in each case the appropriate experimental parameter for varying the diffusion rate (the frequency in impedance methods, the measurement time in potential-step techniques, and so on). 

Example 12.6. Let us consider a much more complex system where the advantages of frequencynlomain solution will be apparent. Rippin and Lamb showed how a frequency-domain stepping technique could be used to find the frequency response of a binary, equimolal-overflow distillation column. The column has many trays and therefore the system is of very high order. 

A new value of frequency is specified and the calculations repeated. Table 12.3 gives a FORTRAN program that performs alt these calculations, The initial part of the program solves for all the steadystate compositions and flow rates, given feed composition and feed flow rate and the desired bottoms and distillate compositions, by converging on the correct value of vapor boilup Vg. Next the coeflicients for the linearized equations arc calculated. Then the stepping technique is used to calculate the intermediate g s and the final P(j transfer functions in the frequency domain. 

A different approach has been proposed by Muscate et al. (52), who developed a two-step technique. In the first step, all nonspecific solutes are... 

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