Series Network

Substituting Equation 3.23 into Equation 3.13 yields the following expression for the series system mean time to failure  

MTTFg is the series system mean time to failure. 

It is to be noted that the right-hand side of Equation 3.25 is independent of time t. Thus, the left-hand side of this equation is simply the series system failure rate. It means that whenever we add up failure rates of units/items. 

Assume that an oil and gas industry system is composed of four identical and independently failing emits, and the constant failure rate of each unit is 0.002 failures/h. All four units must operate normally for the oil and gas industry system to operate successfully. Calculate the following  

For constant failure rate of unit i from Equation (3.11) (i.e., Xft) = Xj), we obtain 

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If each step in the series network is first-order, determine values of the rate constants h and 2 ins-1. 

For other reaction networks, a similar set of equations may be developed, with the kinetics terms adapted to account for each reaction occurring. To determine the conversion and selectivity for a given bed depth, Ljh equations 23.4-11 and -14 are numerically integrated from x = 0 to x = Lfl, with simultaneous solution of the algebraic expressions in 23.4-12, -13, -15, and -16. The following example illustrates the approach for a series network. 

Continuity equations for A and B are written similar to equations 23.4-11 to -16 (in this case we have a series network rather than a parallel network) ... 

Although each stack added to a series network would improve the system s efficiency, the incremental benefit obtained with each additional stack diminishes. A finite number of stacks could adequately, but not exactly, approach a reversible process. In a practical network, the number of stacks would be limited by economic, space, and design constraints. 

In order to assess the effect of Cp on the conductance measurements, the analysis of the parallel RC network of Figure 8.9a may be carried out in a manner analogous to the series network discussed previously. The situation differs from the series circuit in that in this case, the voltage is the same across R and C, and the currents ic and iR in Cp and R are different. The instantaneous currents are given by... 

Again, like the series network shown in Equation (1.5.22), the parallel network of Equation (1.5.33) can represent a variety of important reactions. For example, dehydrogenation of alkanes can adhere to this reaction network where the desired product DP is the alkene and the undesired side-product SP is a hydrogenolysis (C — C bond-breaking reaction) product ... 

DMH is an intermediate in peroxide chemistry and could be used for synthesis in the field of lubricant or polyester. So far research has mainly focused on the development of different catalysts and the comparison of their performance [1]. Despite intensive optimization of catalysts and reaction conditions, selectivity to DMH is still low, because molecular oxygen in the gas-phase causes deep oxidation. Beside the development of catalysts and the optimization of reaction conditions, the mode of gas-solid contact and the reactor configurations are important issues as well. As in any parallel-series network with valuable intermediate products, the design of the reactor has a strong influence on the selectivity towards DMH. In principle, modes of contact which maintain a low oxygen concentration in the reactor favour the desired reaction and thus improve selectivity. Obviously, the requirement of a low oxygen concentration in the gas phase can be met by using a redox-type operation in which a... 

Consider the simple series network of Figure 18.2, containing N conductances, Cj,i = 1,..., /V in series. 

Figure 18.2 Simple series network. Thus the flow may be written...

It will always be worth attempting to reduce the complexity of any extensive flow network by applying the rules for parallel and series networks as outlined in the previous two sections. However, there are many plant arrangements where this approach can only proceed... 

Figure 13.3.6 (a) A view of the differential capacitance in the Gouy-Chapman-Stem (GCS) model as a series network of Helmholtz-layer and diffuse-layer capacitances. (b) Potential profile through the solution side of the double layer, according to GCS theory. Calculated from (13.3.23) for 10 M 1 1 electrolyte in water at 25°C. 

Suppose that the following reaction is a series network with square kinetics for the first reaction and half-order kinetics for the second ... 

We can begin by computing the selectivities and yields for the series network in the CSTR versus the PFR first. Consider the simplest series reaction network ... 

All in all, the series network in a transient PER appears to follow the trends that we would expect in evolving to the steady-state condition. The time to reach the steady state is a function of the rate constants, the inlet concentrations, and the holding time, that is, of the volume flow rate and reactor volume. 

Most reactions occur while other reactions are also taking place simultaneously. Very often the products of one reaction are the reactants for the next. Similarly, a reactant may be involved in more than one reaction. When reactions occur in a sequence, this is referred to as a series network. An example would be ... 

We will concern ourselves with this problem using the series network of reactions. We will explore the differences between the complete back-mixed CSTR and the axially distributed but radially well-mixed PFR. 

In this model, the polyaniline chain is assimilated to a series network of parallel resistances and capacitances (Figure 8.21). The resistances./ (/) and the capacitances C(0 will correspond to the emeraldine salt part (the conductive part) and will depend on the emeraldine salt segments length /. The resistance / (/ ) and the capacitances C (/ ) correspond to the emeraldine base part (the insulating part) and depend on the emeraldine base segments length /. 

This is another type of redundancy in which at least n units out of a total of k active units must operate normally for the successful system/network operation. The block diagram of an n-out-of-fc unit system/network is shown in Figure 3.8. The parallel and series networks are special cases of this network torn = 1 and n = k, respectively. 

The series network is the simplest reliability network or configuration, and its block diagram is shown in Figure 3.2. The diagram denotes an m-unit... 

The sodium/sulfur cells are interconnected in a parallel - series network. Sodium/sulfur cells have the disadvantage of failing in a high-resistance mode. 

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