Sampling problems

As P is increased, the partial cancellation between the kinetic part and the spring part may worsen the statistics on E. This has led to suggestions of alternative ways of estimating E [198]. As P goes up, the springs become stronger, the interactions in become (individually) weaker, and this leads to sampling problems. In... 

Umbrella sampling attempts to overcome the sampling problem by modifying the potenti function so that the unfavourable states are sampled sufficiently. The method can be use with both Monte Carlo and molecular dynamics simulations. The modification of tf potential function can be written as a perturbation ... 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

At the other extreme of Distefano s sample problems, for the largest initial charge, the maximum-stiffness ratio is of the order of 1500, which is considered to be a relatively large value. In this case, more than 10,000 time steps are required to distih 90 percent of the initial change, and the problem is better handled by a stiff integrator. 

These potential sampling problems must be solved in advance of the unit test. The conclusions drawn from any unit test are strongly affected by the accuracy of the sampling methods and the resultant analyses. Methods should be discussed and practiced before the actual unit test. Analysts should use the trial measurements in prehm-inary plant-performance analysis to ensure that the results will be use-bil during the actual unit test. 

Figure 2 and Table 1 illustrate a sample problem (Z, the compressibility factor, is assumed to be 1.0). 

For a sample problem that will include some of the additional iosst-s that are normally encountered in an actual situation, size a compressor to the following given conditions for a hydrocarbon gas ... 

Chapters 7, 8, and 9 demonstrate the consequence modeling techniques for vapor cloud explosions, BLEVEs, and flash fires, respectively, by presenting sample problems. These problems contain sufficient detail to allow an engineer to use the methods presented to evaluate specific hazards. 

Finally, several blast-prediction methods are described and discussed. These methods are demonstrated in Chapter 7 with sample problems. 

Figure 8.3. Graphical presentation for sample problem of the radiation heat flux as a function of time.

In this chapter, applications of the calculation methods used to predict the hazards of BLEVEs, as described in Chapter 6, are demonstrated in the solution of sample problems. Fire-induced BLEVEs are often accompanied by fireballs hence, problems include calculation of radiation effects. A BLEVE may also produce blast waves and propel vessel fragments for long distances. The problems include calculations for estimating these effects as well. Calculation methods for addressing each of these hazards will be demonstrated separately in the following order radiation, blast effects, and fragmentation effects. 

In this section, three examples of blast calculations of BLEVEs and pressure vessel bursts will be given. The first example is designed to illustrate the use of all three methods described in Section 6.3.2. The second is a continuation of sample problem 9.1.5, the BLEVE of a tank truck. A variation in the calculation method is presented instead of determination of the blast parameters at a given distance from the explosion, the distance is calculated at which a given overpressure is reached. The third example is a case study of a BLEVE in San Juan Ixhuatepec (Mexico City). 

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