Classification of variables

This section deals with classification of variables which has close relation to solvability of the balancing problem. Let s start with some simple examples of single-component balance shown in Fig. 2-7. 

Graph 2-7a represents one balancing equation around one node. There is also one unknown flow which can be calculated from the equation. The system is just solvable and the unmeasured flow is observable. The two measured flows are nonredundant. 

In graph 2-7b all three streams are measured. The system is redundant as one of the streams need not be measured and could be calculated. All three streams are redundant but the degree of redundancy is only one (as only one measurement is more than needed to make the system uniquely solvable). The flows in a system with all streams measured will not probably be consistent with the balancing model (the sum of inputs will not equal the sum of outputs). This problem is solved by reconciliation which means adjustment of flows to meet the overall balance. 

In graph 2-7c there are two unmeasured streams. The problem (one equation and two unknowns) is still solvable, but the solution is not unique. We say that the two unmeasured flows are not observable (they are unobservable). The only solution of this obstacle is to complete the measurement (at least one more stream must be measured to make the system fully observable). 

In practice we can meet with even more complicated situations - see Fig. 2-7d. Streams 1, 2, 4 and 5 are measured and redundant (one stream can be calculated from the others). Stream 6 is measured, but nonredundant. Streams 3 and 7 are unmeasured and observable. Streams 8 and 9 are unmeasured and unobservable. The general classification of balancing variables is presented in Fig. 2-8. Anyway, we can see that even in a relatively simple flowsheet with the 

---

The experimental conditions that can be set up by the experimenter are [H]o and [G]o (see Eq. (2.8) and classification of variables). How should the experimental conditions, [H]o and [G]o, be changed for the titration There are so many possibilities described graphically in Fig. 2.6 containing a dilution experiment, a continuous variation experiment, a constant [H]o with different [G]o experiment and two more complicated examples. The criteria to decide the way of change might be as follows ... 

The definition and classification of variables depends on the specific questions to be answered by the experiments. After specification of relevant variables and their type of relation, an experimental design is set up and simulation runs are performed. Analysing the simulation results allows the analyst to parametrize the mathematical model. This results in a metarmodel of the simulated system. This simple mathematical model is finally used to extract quantitative, concise information about the system s behaviour. Figure 4.11 refines Figure 4.6 by adding details about the experimental phase. 

Fig. 2-7. Balancing graphs - classification of variables ---------measured flow ------------unmeasured flow...

Chapter 7 - SolvMlity and Classification of Variables I - Linear Systems... 

More generally, if > 0 then the choice of x = x is not arbitrary, and if L < y then the solution in y is not unique. The classification of variables enables one to decide which of the variables x-, (components of x) is, perhaps, still arbitrary thus must be determined a priori so as to determine a unique solution (a nonredundant variable ), and which of the variables (components of y) is, having satisfied the solvability condition, perhaps still uniquely determined by the given x (an observable variable ). It will be shown later (in Chapter 8) that generally (for a nonlinear system), such a verbal classification is somewhat vague. For a linear system, it can be precisely formulated and the classification based on the partition C = (B, A) only, not on the particular choice of x = X . We call a measured variable redundant if its value is uniquely determined by the other neasured variables and the solvability condition, else nonredundant. There are H (redundant) variables X at most whose values are simultaneously determined by the other measured variables values the number H is called the degree of redundancy. We further call an unmeasured variable y observable if it is uniquely determined by x obeying the condition of solvability, else unobservable. The classification criteria ate (7.1.17) and (7.1.18), from where also (7.1.19). 

Chapter 8 - Solvability and Classification of Variables II - Nonlinear Systems... 

There is a close analogy with the classification of variables in Chapter 7. If m3 is not measured one would call the variables m, and m3 unobservable if m3 = m3 is measured then and tn can be regarded as observable, and m3 as nonredundant. The values of measured composition and temperature variables have to belong to the set (say) iM determined by the condition (8.1.14). With (8.1.12), the condition can be rewritten... 

---