Use of Output Set Assignments for Variable Classification

Generally, a chemical plant is composed of several units with several streams and components. The set of material and energy balances constitutes a set of linear/nonlinear equations, which can be represented by an undirected graph. However, when the number of units and streams is large, the graphical representation becomes cumbersome. An alternative representation of the topological structure of the balances in a chemical process is achieved using the occurrence matrix. 

The rows of the occurrence matrix correspond to the balance equations and the columns to the process variables, both measured and unmeasured. An element of the matrix O, is a Boolean 1 or 0, that is. 

To classify the variables, one must first establish what information each equation is to supply, that is, to obtain an output set assignment for the balance equations. 

The output set assignment assigns to any unmeasured process variable one equation, or to two or more variables the same number of equations. This is equivalent to transforming the original undirected graph to a directed one. 

Consider again the system used in the previous examples. For /i, /a, and /4 measured, the corresponding occurrence matrix is given in Table 1. The following assignments can be done  

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In this chapter, the mathematical formulation of the variable classification problem is stated and some structural properties are discussed in terms of graphical techniques. Different strategies are available for carrying out process-variable classification. Both graph-oriented approaches and matrix-based techniques are briefly analyzed in the context of their usefulness for performing variable categorization. The use of output set assignment procedures for variable classification is described and illustrated. 

The output set assignment is not unique however, this does not affect the result of the classification. As Steward (1962) has shown, if there is no structural singularity, the determinable unmeasured variables are always assigned independently of the obtained output set assignment. The classification of the unmeasured variables allows us to define the sequence of calculation for these variables. That is, expressions are obtained to solve them as functions of the measurements. The expressions are also used in the classification of the measured variables and in the formulation of the reconciliation equations. After the reconciliation procedure is applied to the measurements, these equations are used to find an estimate of the unmeasured determinable variables in terms of the reconciled measurements. 

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