Sequential processing constraints

In this chapter the mathematical formulation for the sequential processing of both constraints and measurements is presented. Some interesting features are discussed that make this approach well suited for the analysis of a set of measurements and for gross error identification, the subject of the next chapter. 

Let us apply the sequential processing of the constraints to the system defined previously in Example 5.1. We will process one equation at a time starting from the unconstrained problem. 

This chapter discussed the idea of exploiting the sequential processing of information (both constraints and measurements), to allow computations to be done in a recursive way without solving the full-scale reconciliation problem. 

A more systematic approach was developed by Romagnoli and Stephanopoulos (1981) and Romagnoli (1983) to analyze a set of measurement data in the presence of gross errors. The method is based on the idea of exploiting the sequential processing of the information (constraints and measurements), thus allowing the computations to be done in a recursive way without solving the full-scale reconciliation problem. 

The previous approach for solving the reconciliation problem allows the calculation, in a systematic recursive way, of the residual covariance matrix after a measurement is added or deleted from the original adjustment. A combined procedure can be devised by using the sequential treatment of measurements together with the sequential processing of the constraints. 

In many oil/gas fields the production of oil, gas and water are constrained by the processing capacity and other process constraints such as available flow-line transportation capacity. Wang et al. (2002) point out that the available literature does not provide robust procedures on how to formulate and solve typical optimization problems for such systems. Often, the optimization consider the constraints sequentially, or only subproblems are considered (e.g by not including the transportation system to the processing facility). Dutta-Roy Kattapuram (1997) considered the effect of including process constraints for a two-well case that share a common transportation line to the process. They found that failing to include the process constraints (in this case the transportation line) gave a sub-optimal solution of the problem. 

The second classification is the physical model. Examples are the rigorous modiiles found in chemical-process simulators. In sequential modular simulators, distillation and kinetic reactors are two important examples. Compared to relational models, physical models purport to represent the ac tual material, energy, equilibrium, and rate processes present in the unit. They rarely, however, include any equipment constraints as part of the model. Despite their complexity, adjustable parameters oearing some relation to theoiy (e.g., tray efficiency) are required such that the output is properly related to the input and specifications. These modds provide more accurate predictions of output based on input and specifications. However, the interactions between the model parameters and database parameters compromise the relationships between input and output. The nonlinearities of equipment performance are not included and, consequently, significant extrapolations result in large errors. Despite their greater complexity, they should be considered to be approximate as well. 

In TLC the detection process is static (sepaurations achieved in space rather than time) and free from time constraints, or from interference by the mobile phase, which is removed between the development and detection process. Freedom from time constraints permits the utilization of any variety of techniques to enhance detection sensitivity, which if the methods are nondestructive, nay be applied sequentially. Thus, the detection process in TLC is nore flexible and variable than for HPLC. For optical detection the minimum detectable quantities are similar for both technlqpies with, perhaps, a slight advantage for HPLC. Direct comparisons are difficult because of the differences in detection variables and how these are optimized. Detection in TLC, however, is generally limited to optical detection without the equivalent of refractive... 

Commercial process simulators mainly use a form of SQP. To use LP, you must balance the nonlinearity of the plant model (constraints) and the objective function with the error in approximation of the plant by linear models. Infeasible path, sequential modular SQP has proven particularly effective. 

Biologically mediated redox reactions tend to occur as a series of sequential subreactions, each of which is catalyzed by a specific enzyme and is potentially reversible. But despite favorable thermodynamics, kinetic constraints can slow down or prevent attainment of equilibrium. Since the subreactions generally proceed at unequal rates, the net effect is to make the overall redox reaction function as a imidirectional process that does not reach equilibrium. Since no net energy is produced imder conditions of equilibrium, organisms at equilibrium are by definition dead. Thus, redox disequilibrium is an opportunity to obtain energy as a reaction proceeds toward, but ideally for the sake of the organism does not reach, equilibrium. 

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