By van Jaarsfeld E.J., de Villiers P.U.

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7) where fil (of components rhj, j ~ J+) is the vector of a priori fixed variables. 2). 8) (having deleted reference node a), and A3:/ 1 2 5 9 10 11 11 14 15 -1 h Chapter 3- Mass (Single-component) Balance 49 (having deleted reference node f). 8). The whole space of unmeasured variables (unknowns) is of dimension 7. 10). We can assign an arbitrary value to any one of the variables m4, m6, mT, m8, then the remaining ones are also uniquely determined by the conditions. 5 JUST DETERMINED SYSTEMS The analysis of solvability applies in particular (and more simply) to systems where the set of a priori given variables' values just determines those of the remaining ones.

1), the j-th component mj is uniquely determined by the measured values; thus if m~ is the j-th component of any other solution, we have mi = m]. ) has a solution, the remaining components (say) m~+ where i ~ j determine uniquely mi, +" thus if rh + is another measured vector such that the system is solvable and if mi - + - mi+ f o r a l l i ~ j , we have also rh+ j - m j+ . , K' is observable if and only if arc j separates subgraph Gk~ or also The j-th unmeasured variable is observablei f and only if arc j does not lie in any circuit of subgraph G ~ Recall that G O is the subgraph of G restricted to unmeasured streams (arcs), G o its k-th connected component, K' the number of components that are 38 Material and Energy Balancing in the Process Industries not isolated nodes.

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.