Scenario decomposition

Basically, there are two different ways to decompose a 2S-MILP (see Figure 9.10). The scenario decomposition separates the 2S-MILP by the constraints associated to a scenario, whereas the stage decomposition separates the variables into first-stage and second-stage decisions. For both approaches, the resulting subproblems are MILPs which can be solved by standard optimization software. 

Scenario Decomposition Based Branch-and-Bound Algorithm... 

Fig. 9.10 Decomposition scenario decomposition and stage decomposition of a 2S-MILP.

The performance of the hybrid ES is compared to that of the commercial MILP solver CPLEX and to that of the state-of-the-art scenario decomposition based branch-and-bound algorithm described in Section 9.3.5 (DDSIP). The results are shown in Figure 9.15. As the reference for the performance of hybrid ES, we take the median from the experiment with feasible initialization (Figure 9.14b). 

Clostermann, E. (2005) Empirical analysis of scenario decomposition in chemical batch scheduling. Diplomarbeit, Universitat Duisburg-Essen. 

Drugs sometimes have quite complicated chemical structures and are, by definition, biologically active compounds. It should not, therefore, come as a surprise that these reactive molecules undergo chemical reactions that result in their decomposition and deterioration, and that these processes begin as soon as the drug is synthesised or the medicine is formulated. Decomposition reactions of this type lead to, at best, drugs and medicines that are less active than intended (i.e. of low efficacy)-, in the worst-case scenario, decomposition can lead to drugs that are actually toxic to the patient. This is clearly bad news to all except lawyers, so the processes of decomposition and deterioration must be understood in order to minimise the risk to patients. 

Solution studies of the complexes outlined in Table 3 are in general agreement that they remain aggregated in solution, with the most common form being dimers.325 Solution studies of 300 in arene solvent indicate that at least two independent aggregates coexist.322 The most likely scenario from the data available is that a dimer-hexamer equilibrium is present in solution. Gas-phase studies of the hydrazides have proved to be problematic due to decomposition of the complexes on volatilization. 

Calculations were made, for many fire scenarios, in which the fire hazard model F.A.S.T. was used to simulate hazard to occupants of a standard room following a fire starting in a plenum above it. A total of 400 m of PVC wire coating was assumed to be present in the plenum. Its decomposition was made a function of the plenum... 

The heat release rate necessary for flashover was calculated, from the equation given by Quintiere et al. [31]. The series of equations is then solved, with the assumption that the temperature increase for flashover is 500 K (leading to an upper level temperature of TUL 795 K) and the plenum temperature for decomposition of the PVC products is 573 K. The results in Table III show that a much more intense fire is required, in all cases, to cause the PVC products to undergo dehydrochlorination than to take the room to flashover. Thus, the heat released by this fire at flashover is insufficient to dehydrochlorinate the PVC products in the plenum, for any of the scenarios. Therefore, the occupants of the room will succumb before there is an effect due to the plenum PVC products. 

In this form, the scenario subproblems are tied together only by the non-anticipativity constraints (9.22). This naturally leads to a decomposition based on the relaxation of the non-anticipativity constraints. 

The main idea of stage decomposition (see Figure 9.10) is to remove the ties between the scenario subproblems of the 2S-MILP by fixing the first-stage variables. The 2S-MILP is written in its intensive form [9], where the resulting master problem is... 

Time decomposition, which is another option in PP/DS optimization to decompose the overall planning problem, has not been used in this scenario. 

Credible cases are identified when the probability of decomposition is low. Energy calculations of known or proposed chemical reactions and side reactions are carried out to determine a more likely level of energy release than the worst-case scenario. Therefore, it is necessary to define the most energetic reactions. Enthalpies of reaction are calculated, followed by calculations of the adiabatic temperature rise of the system and the corresponding pressure rise. 

Fig. 3.18 Scenario of cooling failure with thermal runaway. ATa(j, is the adiabatic temperature rise by desired reaction. ATaa2 is the adiabatic temperature increase by the decomposition reaction. The time required for this increase is TMRad-...

It is somewhat confusing that the term critical diameter is also used by those interested in the potential of an energetic material to undergo thermal runaway. Because, by definition, the energetic material releases heat when it decomposes, it has the potential to increase its local environmental temperature. Depending on the decomposition kinetics of the material, at some critical dimension the charge can self-heat to catastrophic reaction. This can be referred to in terms of the critical diameter or, more often, in terms of the initial environmental temperature that allows this scenario, the critical temperature . 

Scheme 2 Scenario of calculated decomposition pathways of the model precursor...

A reactor of volume 3.5 m3 has a design pressure of 14 barg. A worst case relief scenario has been identified in which a gassy decomposition reaction occurs. The mass of reactants in the reactor would be 2500 kg. An open cel test has been performed in a DIERS bench-scale apparatus, in which the volume of the gas space in the apparatus was 3,800 ml, and the mass of the sample was 44.8 g. The peak rate of pressure rise was 2,263 N/m2s at. a temperature of 246°C, and the corresponding rate of temperature rise was 144°C/minute. (These values include corrections for thermal inertia.) The pressure in the containment vessel corresponding to the peak rate was 20.2 bara. 

In the following section, we will discuss the approach proposed by Ciric and Floudas (1991) for the synthesis of heat exchanger networks without decomposition. Note that we will present the approach for the pseudo-pinch case (which is the most general). The approach for the strict-pinch case (which is a constrained scenario of the pseudo-pinch and as such features more structure) can be found in Ciric and Floudas (1991). 

Without driving (a = 0) one has the typical scenario of spinodal decomposition and there is no anisotropy in the behavior of lx and ly (Fig. 23). Thus, small perturbations grow exponentially and at about t 15 (not shown) a nonlinear saturation of the fastest growing mode becomes important and sharp domain boundaries form. At about t 30 the late stage coarsening starts and the well-known scaling lx ly t1/3 is observed. In Fig. 24 snapshots of the phase separation process are presented for a particular run. 

The scenario presented here was developed by R. Gygax [1, 2]. Let us assume that while the reactor is at the reaction temperature (TP), a cooling failure occurs (point 4 in Figure 3.2). The scenario consists of the description of the temperature evolution after the cooling failure. If, at the instant of failure, unconverted material is still present in the reactor, the temperature increases due to the completion of the reaction. This temperature increase depends on the amount of non-reacted material, thus on the process conditions. It reaches a level called the Maximum Temperature of the Synthesis Reaction (MTSR). At this temperature, a secondary decomposition reaction may be initiated. The heat produced by this reaction may... 

Figure 3.2 Cooling Failure Scenario After a cooling failure, the temperature rises from process temperature to the maximum temperature of synthesis reaction. At this temperature, a secondary decomposition reaction may be triggered. The left-hand part of the scheme is devoted to the desired...

After loss of control of the synthesis reaction, the technical limit will be reached (MTSR > MTT) and the decomposition reaction could theoretically be triggered (MTSR > TD24). In this situation, the safety of the process depends on the heat release rate of both the synthesis reaction and decomposition reaction at the MTT. The evaporative cooling or the emergency pressure relief may serve as a safety barrier. This scenario is similar to class 3, with one important difference if the technical measures fail, the secondary reaction will be triggered. 

At 127 °C, the decomposition reaction is critical, that is, the time to maximum rate (Question 6 in the cooling failure scenario) is shorter than 8 hours (see Table 5.4). 

This chapter describes a runaway scenario. The first section presents a general review of the decomposition reaction characteristics. The second section is devoted to the energy release that defines the consequences of a runaway. The third section deals with triggering conditions of undesired reactions, based on the concept of TMRld. The next section reviews some important aspects for the experimental characterization of decomposition reactions. Finally, the last section gives some examples stemming from industrial practice. 

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