Disjoint Subsystems

In formulating a model of a very large process such as a whole chemical plant, the possibility exists that a subset of the system equations does not contain any variables in common with the remaining equations in the system. Such a subset of equations may physically correspond to a process unit or group of process units that are not connected in any way to the remaining units in the process. If this situation occurs, the subset of equations, which is called a disjoint subsystem, can be solved completely independently of the remaining equations in the system. Identification of these disjoint subsystems reduces the dimensionality of the complete system to that of the largest disjoint subsystem. 

In order to clarify what a disjoint subsystem is, consider Eqs. (12) ... 

The functions /j and /4 contain only variables xx and x3, variables that do not appear in the remaining two equations, and therefore,/t and/4 constitute a disjoint subsystem (/j itself is also disjoint). The occurrence matrix ofEq. (12) is shown in Fig. 12. If a column permutation n (23) is performed on the... 

New matrices M(2), M(3), etc. are formed until the matrix M(,) is obtained, which contains exactly one nonzero element in each column. Each row of the matrix M(n> corresponds to a subset of equations in the original system. Since each column of M(n) contains only one nonzero element, each variable of the system appears in only one subset of equations represented by the row in which the nonzero element appears. Therefore, the subset of equations represented by a row of M(n) is a disjoint subsystem. For example, the matrix M(1) illustrated in Fig. 13b has only one nonzero element in each column. 

Since the first row of M(1) is made of rows 1 and 4 of the original occurrence matrix,/ and/4 constitute a disjoint subsystem /2 and/3 comprise the other disjoint subsystem. 

Where each D, i l,2...,p represents the topology of each disjoint subsystem. The agregation of the balance equations to form the chain of disjoint subsystems, eliminate the streams among the units of each disjoint system. This leaves only the streams that will connect a particular disjoint system with similar others. These streams together with the streams connected to one unit are called external and the parameters belonging to these streams are called external parameters. 

If a given Du contain at least one external unmeasured parameter, then this disjoint subsystem does not add new parameters to M. Repeat for all D fs. 

If a given disjoint subsystem has only measured external parameters then they ar e all overmeasured, and they are incorporated in the set M. ... 

Here, are the two-electron eqttihbriitm Green s functiorrs of the disjoint subsystems A, B, respectively, and we have used overall charge neutrality to exactly elimirrate a... 

A system can be divided into disjoint subsystems consisting of atoms, or chemical functional groups, which are within a local region in the physical space. We can define a partition matrix for the subsystem a in the space of atomic orbitals. The partition matrix needs to satisfy the normalization condition... 

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