Example 2-13 Series System

Total length of 10-in. pipe to use in calculating capacity is 19 + 2.58 = 21.58 miles. 

By the principles outlined in the examples, gas pipe line systems may be analyzed, paralleled, cross-tied, etc. 

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The explicit form of those equations that satisfy the preliminary data criteria, must then be tested against a series of data sets that have been obtained from different chromatographic systems. As an example, such systems might involve columns packed with different size particles, employed mobile phases or solutes having different but known physical properties such as diffusivity or capacity ratios (k"). 

We have carried out a series of geometry optimizations on nanotubes with diameters less than 2 nm. We will present some results for a selected subset of the moderate band gap nanotubes, and then focus on results for an example chiral systems the chiral [9,2] nanotube with a diameter of 0.8 nm. This nanotube has been chosen because its diameter corresponds to those found in relatively large amounts by Iijima[7] after the synthesis of single-walled nanotubes. 

Obviously for this method to work the ratio T1IT2 must be appreciably smaller than unity. Provided this condition is met, this method is a simple and reliable way to test for an isokinetic relationship or to detect deviations from such a relationship. Exner shows examples of systems plotted both as log 2 vs. log and as AH vs. A5, demonstrating the inadequacy of the latter plot. Exner has also developed a statistical analysis of the Petersen method this analysis yields p and an uncertainty estimate of p. Exner has applied his statistical methods to 100 reaction series, finding that 78 of them follow approximately valid isokinetic relationships. 

Tlie reliability formulas for series and parallel systems can be used to obtain tlie reliability of a system lliat combines features of a series and a parallel system. Consider, for example, tlie system diagrammed in Fig. 20.2.1. Components A, B, C, and D have for llieir respective reliabilities 0.90, 0.80, 0.80, and 0.90. The system fails to operate if A fails, if B and C both fail, or if D fails. Component B and C constitute a parallel subsystem connected in series to components A and D. The reliability of the parallel subsystem is obtained by applying Eq. (20.2.2), which yields... 

In tlie case of random variables assumed to be normally distributed, Monte Carlo simulation is facilitated by use of a table of the normd distribution (Table 20.5.2). Consider, for example, a series system consisting of two electrical components, A and B. Component A lias a time to failure Ta, assumed to be nomially distributed with mean 100 hours and standard deviation 20 hours. Component B has a time to failure Tb, assumed to be normally distributed witli mean 90 hours and stimdiud deviation 10 hours. Tlie system fails whenever component A or component B fails. Tlierefore, tlie time to failure of the system Ts is tlie minimum of time to failure of components A and B. 

First, there is only a limited number of examples of systems which can be classified as being cyclically delocalized systems and, thus, termed homoaromatic. These include a number of bridged bishomotropenylium ions and a somewhat larger series of bridged or caged bis- and trishomocyclopropenium ions. 

Tlie reliability formulas for series and parallel systems can be used to obtain Uie reliability of a system Uiat combines features of a series and a parallel system. Consider, for example, Uie system diagrammed in Fig. 20.2.1. Components A, B, C, and D have for Uieir respective reliabiliues 0.90, 0.80,... 

The system comprising the resistor Re and capacitor C in series provides an example of a class of systems for which, at the zero-frequency or dc limit, current cannot pass. Such systems are considered to have a blocking or ideally polarizable electrode. Depending on the specific conditions, batteries, liquid mercury electrodes, semiconductor devices, passive electrodes, and electroactive polymers provide examples of systems that exhibit such blocking behavior. 

The differences between a single CSTR and a batch reactor are similar to those between semibatch and batch reactors, except that they are usually more pronounced. The addition of more reactors to a series system tends to reduce some of the observed performance differences. A typical example of different behavior is the heat release profile. An advantage often cited for continuous reactor systems is a constant heat load with fully used reactor volume. Batch reactors are not usually operated full, and the heat load is nonuniform. In addition, portions of the batch reaction cycle are devoted to charging and emptying the reactor and sometimes for heating the reagents to polymerization temperature. Thus, the production rate per unit volume can be higher in a continuous system. 

The successful use of Mossbauer spectroscopy for studies of catalysts is demonstrated in an important technical example (catalyst system). In Figure 15 a series of spectra of the Fe/Ti02 catalyst system [30] is shown. The top spectrum represents the freshly prepared catalyst it is obvious that iron is present in the 3-f oxidation state, possibly in the form of Finely dispersed oxide or oxy-hydroxide on Ti02. Under a reducing atmosphere... 

Let us consider for example a system with one degree of freedom, q. If the system has more than one degree of freedom, its Hessian matrix of second derivatives can be diagonalized, and the problem is reduced to several independent one-dimensional problems. The kinetic energy being C = jm q, the potential energy can be expanded in Taylor series in terms of the deviation u from the equilibrium point qQi... 

ILLUSTRATIVE EXAMPLE 21.9 Many systems consisting of several components can be classified as series, parallel, or a combination of both. A series system is one in which the entire system fails to operate if any one of its components fail to operate. If such a system consists of n components which function independently, then the reliability of the system is the product of the reliabilities of the individual components. If denotes the reliability of a series system and R, denotes the reliability of the /th component / = 1, 2,..., then... 

Example 5.3. Consider the reliabihty assessment problems from (Sorensen 2004, Note 6) (series system) and (Sorensen 2004, Note 7) (parallel system), with X = (X,X2), where Xj and Xj are independent standard normal random variables defined in 2 = R, and the limit state functions of Table 3. 

The equations of motion presented so far are second order ODEs with or without constraints. In some cases there might be also additional first order differential equations. Consider, for example, the system shown in Fig. 1.6. If the mass m2 is taken to be zero the mass mi is suspended by an interconnection consisting of a spring and a damper in series. 

A barrier function is a special safety function designed to prevent, control, or mitigate the propagation of failures or energy into an undesired event (UE) or mishap. A safety barrier can be a series of elements that together implement a barrier function, each element consisting of a technical system or human action. For example, series of interlocks are designed into a missile fire control system to prevent the inadvertent and unauthorized outcome of inadvertent missile launch function. The set of interlocks are safety barriers that implement a barrier function. 

The series representation given by Eq. (14) is expected to Sail as c approadies and exceeds unity if A2 > 0. For example, for systems with A2 > 0, extension of the notion that cF(c) may depend on A2MC alone in systems for which 23 ( 2) / 3 is about equal to its asymptotic limit for strong repulsive... 

The IAEA s Technical Reports Series No. 387 [4] presents an overview of concepts and examples of systems discussed in this Safety Guide and may provide useful background material for some users. 

For illustration of how the equipment reliability methods can be applied to human reliability computation, consider the example of the parallel-series system. The two equipment components A and B are switches of electric motors, both independently operating a valve C within a water coolant system of a nuclear power plant (Fig. 5.5.). 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

The steady-state method is also known as divided bar method (see, for example. Beck, 1957 Chekhonin et al., 2012). A cylindrical rock sample is positioned (sandwiched) between two cylinders composed of a reference material with known thermal conductivity. This is a thermal series system. The end of one reference cylinder is heated. After steady state is reached, the temperature drop within the rock sample is compared with the temperature drop within the reference cylinders. The comparison gives thermal conductivity of the rock sample. 

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