Boundary cases

Layer of protection analysis (LOPA) is a simplified form of event tree analysis. Instead of analyzing all accident scenarios, LOPA selects a few specific scenarios as representative, or boundary, cases. LOPA uses order-of-magnitLide estimates, rather than specific data, for the frequency of initiating events and for the probability the various layers of protection will fail on demand. In many cases, the simplified results of a LOPA provide sufficient input for deciding whether additional protection is necessary to reduce the likelihood of a given accident type. LOPAs typically require only a small fraction of the effort required for detailed event tree or fault tree analysis. 

The polymers used as matrices in thermoplastic composites as well as fillers have the most diverse physical and chemical structures, thus a wide variety of interactions may form between the two components. Two boundary cases of inter-... 

Once the critical load is exceeded (P > Pc), the crack starts to grow substantially (normally R > 2a, Fig. 6.2.7). For the boundary case of a true cone (i.e., R- 0), crack mechanics becomes independent of the developments in the contact region (i.e., r and Cf°), which is given by the equilibrium equation derived by Roesler (1956)... 

The program discussed above is therefore easily modified to deal with the specified wall heat flux case. A program with this wall thermal boundary case. 

The electrophoretic velocity of the sphere is expressed in a power series of k up to 0(k6). The results for various boundary cases are given as follows ... 

The plant is not only designed for the usage of carbon dioxide, but also for propane and other combustible gases Carbon dioxide and propane were taken as boundary cases to get the basic design data for the necessary heat exchangers... 

If all species consisted of biparental systems with unproblematlcal cohesion, only the boundary cases would spoil the picture, or better stated, would nake the boundary somewhat fuzzy. But there are much more severe cases i... 

This will show that a quantitative description involving all the complexities in simple microporous membranes is not available (if possible). However a number of boundary cases can be described quantitatively, as in Section 9.42.2, and trends in more complex situations can be predicted in combination with the qualitative pictures based on mechanistic considerations. 

Clearly the film thickness, fluid density, and perhaps the orientation of the surfaces [207] are additional thermodynamic variables that may shift or alter phase boundaries. Cases where a single interface stabilizes a different phase than the bulk are central to the field of wetting. The presence of two interfaces separated by only a few nanometers leads to more pervasive phase changes. 

The selectivity and conversion of a chemical reactor depend not only on the kinetics of the reaction but also on the time for which the reaction partners are available for the reaction, that is, the hydrodynamic behavior. By determining the residence-time behavior of a reactor, one can determine its deviation from ideal hydrodynamic behavior (boundary cases plug flow and complete backmixing) and hence decide with which reactor model the real reactor can best be modeled. 

Given Ig, which results from the boundary cases of minimum solvent quantity L ,j and minimum stage number from economic considerations analogous to rectification (Section 2.3.2.3) or absorption (Section 2.3.3) by plotting an L/N diagram. 

Once financial, environmental and reputational forms of risk are excluded one might expect that what is left is a fairly tight definition of the factors which constitute clinical risk. Experience has shown however that there are many boundary cases that can be subject to debate. Clinical risk arises from the manifestation of one or more hazards. Hazards have an intrinsic relationship with harm but it is often in the definition of what constitutes harm where debate arises. 

For IPN s, Donatelli et al. earlier derived an equation especially for semi-IPN s of the first kind (polymer I crosslinked, polymer II linear) and extended this to full IPN s by assuming that the molecular weight of polymer II is infinite. Michel et al. solved the Donatelli equation considering several boundary cases, and reinterpreted the constants involved. However, because of the semi-empirical nature of the Donatelli equation, its intrinsic shortcomings limit its applicability. 

If an SMB process is discretized by an increasing number of columns in the functional zones, the concentration profile converges to that of the TMB model. Thus, the TMB model represents a boundary case of the simulated moving-bed process. If, additionally, only the solution in the steady state of the system is considered, the balance equations in the formulation of the stage model can be simplified in a way that only one nonlinear system of equations has to be solved. Such... 

For modeling of real SMB-SFC processes the results of the triangle theory are applicable only in a limited way. But the model provides a boundary case for the validation of the dynamic SMB-SFC model. Furthermore, the application of the triangle theory enables the derivation of parameters for the evaluation of the quality of products and the separation process. 

The friction factor is a function of the ratio L/d and the fraction

total cross-section area of the perforated plate. Two dilTeient boundary cases (Z/c o 0 and L/d -> oo) can be distinguished (see also Chap. 5) ... 

An interesting boundary case is the polymerisation of pure liquid styrene. The process then continuously passes through the isoger state (a frame work of polystyrene molecules soaked with the monomer) to the xerogel state, which is reached it the entire amount of styrene is polymerised. 

Figure 5.70. Kinetics of heterogeneous bioprocess Diagram of the specific rate of substrate use, as a function of the substrate concentration, s, varying the thickness of the biocatalytically active mass, d, for floe and film geometries. Monod kinetics appear as the boundary case at low d. Unrealistic cases are indicated with dashed lines. (Adapted from Atkinson, 1974 with permission of Pion, London.)...

It would be desirable to apply analytical expressions for the activity coefficient, which are not only able to describe the concentration dependence, but also the temperature dependence correctly. Presently, there is no approach completely fulfilling this task. But the newer approaches, as for example, the Wilson [13], NRTL (nonrandom two liquid theory) [14], and UNIQUAC (universal quasi-chemical theory) equation [15] allow for an improved description of the real behavior of multicomponent systems from the information of the binary systems. These approaches are based on the concept of local composition, introduced by Wilson [13]. This concept assumes that the local composition is different from the overall composition because of the interacting forces. For this approach, different boundary cases can be distinguished ... 

It may not be possible to collect the data representing the extreme, or boundary, case in biomedical situations. An example of extreme biomedical data is a high-temperature spot in the IR image of the face that is caused by SARS. 

Choose examples for each legal size of X up to some size m. Whenever possible, use different constants across the various examples. Make sure that the possible non-determinism of (given a ground value of X) is apparent in the examples. The value of m may be determined by co-routining with the next step, but making sure that examples are chosen at least for all boundary-cases. 

---