Even time discretization

Needless to mention, the exact capturing of time presents further challenges in the analysis. Fundamentally, a decision has to be made on how the time horizon has to be represented. Early methods relied on even discretization of the time horizon (Kondili et al., 1993), although there are still methods published to date that still employ this concept. The first drawback of even time discretization is that it inherently results in a very large number of binary variables, particularly when the granularity of the problem is too small compared to the time horizon of interest. The second drawback is that accurate representation of time might necessitate even smaller time intervals with more binary variables. Even discretization of time is depicted in Fig. 1.8a. 

In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to ... 

The above solutions are all based on ideal or theoretical stages. Even in discrete stage systems, like mixer-settlers, equilibrium may not be attained because of insufficient time for diffusion of solute across the phase boundary or insufficient time for complete clarification of each stage. 

For discrete time, discrete state Markov models numerical solutions for probability of being in any state can be obtained by simple matrix multiplication. This technique can be used to solve many realistic models, regular or absorbing. The technique may be even used on certain non-homogeneous models to include deterministic events as well as probabilistic events. 

Although the measurement of a single lifetime is trivial, severe problems appear if the system shows several lifetimes. In the worst case, the system does not even have discrete time constants but a continuous distribution of lifetimes. The difficulties do not emerge as the lack of a numerical solution but vice versa, as a continuum of solutions fitting well to the experimental results. This is due to the notoriously ill-posed character of the equations used for estimating the time constants from the experimental data. An obvious way to overcome these difficulties is to increase the precision of the measurements. Another, actually more successful. 

High and low limit checks for transmitters have been in practice for quite some time, even when discrete instmmentations were in use. Now with the DCS it is very easy to monitor out of limits for the transmitter and open-circuit and short-circuit tests for sensors like resistance temperature detectors and thermocouples. Most of the transmitters are monitored for out of span (e.g., <4 or >20 mA). Also since smart transmitters have a diagnostic system, they also can detect faults and isolate them, that is, the output of a faulty transmitter could be inhibited generating an alarm. The transmitter is connected via HART/Profibus/fieldbus, and such detections are more explicit and well reported in the system. Also there exists a facility for the operator to select any transmitter manually. 

The string of analysis output results mathematically in a time series data, usually with even time spacing. The mathematical treatments developed for time series analysis can be applied to predict trends, drifts, etc., which have impact on process control decisions. The analysis results can form the input data for statistical process control, developed earlier for discrete production [9,10]. Figure 37.3 shows one of the basic types of presentation charts. Statistical process control has an extensive literature (see Reference 11). 

Large-scale numerical simulation for samples that are many times os large as the critical wavelength is perhaps the only way to develop a quantitative understanding of the dynamics of solidification systems. Even for shallow cells, such calculations will be costly, because of the fine discretizations needed to be sure the dynamics associated with the small capillary length scales are adequately approximated. Such calculations may be feasible with the next generation of supercomputers. 

In the OE of jawed fish only cellular, and little if any, tissue specialisation is achieved. During metamorphosis from tadpole to adult in amphibia, a developmental parallel of water-to-land transition includes the timing of maturation of the AOS. The system as it appears in living amphibians, is already a more or less discrete entity (Fig. 4.3) with its own sub-set of receptors. A process of regionalisation within the bulb, already underway even at the level of organisation in cartilaginous fishes, shows parallel adjustments (Dryer and Graziadei, 1993). 

Third, processing times may require special modeling in chemical industry. While in discrete manufacturing processing times for a certain lot are usually dependent on the lot size, i.e., the number of units to be produced, this is often not true in the chemical industry. Here, processing times are often constant, irrespective of whether a reactor is filled to 70% or 90% of its capacity. This is often referred to as batch production [5], On the other hand, the quality of the material produced may depend on resource utilization. Certain reactions may not even be feasible, if a minimum bound of the procured material is not exceeded. This implies additional restrictions regarding the resource utilization level on the planning situation. 

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