Redundant equations

With 12 variables and 9 independent linear equality constraints, 3 degrees of freedom exist that can be used to maximize profits. Note that we could have added an overall material balance, xn + xl2 + 7 = 8 + x9 + 10, but this would be a redundant equation since it can be derived by adding the material balances. 

Define the subset of redundant equations to be used for the adjustment of measurements... 

A set of redundant equations is constructed using unassigned equations without indeterminable variables and specific balance equations around disjoint systems of units. The measurements involved in this set are classified as redundant. 

In this case we assign Equation 2 to fs (determinable), leaving two unassigned equations, 1 and 3. Variables fo and fi are still not determinable, but now Equation 1 is not assigned and contains only measured variables thus, this is a redundant equation and the associated variables are also redundant or overmeasured. 

If A22 0, the system possesses unmeasured variables that cannot be determined from the available information (measurements and equations). In such cases the system is indeterminable and additional information is needed. This can be provided by additional balances that may be overlooked, or by making additional measurements (placing a measurement device to an unmeasured process variable). Also, from the classification strategy we can identify those equations that contain only measured variables, i.e., the redundant equations. Thus, we can define the reduced subsystem of equations... 

Thus, we have identified the subset of redundant equations containing only the redundant process variables fa, fa, fu, fH, and fa5. Furthermore, the rank of Ru is equal to 6, which means that at least one of the unmeasured variables is indeterminable. The remaining ones can be written in terms of it, as indicated by Eq. (4.15). In this case, from the orthogonal transformation, the subsets of u are defined as... 

The outlined strategy has been applied to the subsystem of Example 4.4 in Chapter 4. The flow diagram, shown in Fig. 4 of Chapter 4, consists of 7 units interconnected by 15 streams. There are 8 measured flowrates and 7 unmeasured ones. The flowrate measurements with their variances are given in Table 3. In Chapter 4 we identified the subset of redundant equations. In this case it is constituted by one equation that contains the five redundant process variables. By applying the data reconciliation procedure to this reduced set of balances, we obtain the estimates of the measured variables, which are also presented in Table 3. 

As was discussed in Chapters 3 and 4, variable classification allows us to obtain a reduced subsystem of redundant equations that contain only measured and redundant variables. These are used in the reconciliation procedure. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

Thus, we have identified the subset of redundant equations containing only measured (redundant) process variables. Applying the data reconciliation procedure to this reduced set of balances we obtain for the estimate of the measured variables... 

There are 7 redundant equations containing all of the 25 measured variables. 

A data reconciliation procedure was applied to the subset of redundant equations the results are displayed in Table 2. After the reconciliation, the estimation of the unmeasured variables was accomplished and the results are presented in Table 3. 

A data reconciliation procedure was applied to the subset of redundant equations. The results are displayed in Table 4. A global test for gross error detection was also applied and the x2 value was found to be equal to 17.58, indicating the presence of a gross error in the data set. Using the serial elimination procedure described in Chapter 7, a gross error was identified in the measurement of stream 26. The procedure for estimating the amount of bias was then applied and the amount of bias was found... 

The linear models within each pool were subjected to a diversity analysis procedure in order to discard redundant equations based on roughly similar descriptor choices. 

As a check, we will use the third component balance, the one for H2O, a redundant equation... 

The total balance would be F = W + 350 (in kg). Are the three component balances independent Because of the zero terms in the right-hand sides of Eqs. (b) and (c), no variation or combination of Eqs. (b) and (c) will lead to Eq. (a). Are Eqs. (b) and (c) redundant equations No constant exists that when multiplied into Eq. (b) gives Eq. (c), hence the three mass balances are independent. 

Step 9 We have many redundant equations. Overall, process plus separator, and mixing... 

Decide which equations to use if you have redundant equations. 

Note how the rank of a (the rank of a matrix is given by the size of the largest nonzero determinant that can be formed from the matrix), the matrix composed of the first two columns, can at the most be 2 and that the rank of [a, b] is also 2 because the determinant of [a, b] is zero. To obtain a consistent set of equations, one of the three material balances must be eliminated, leaving two equations in two unknowns, P and W, that have a unique solution m = r — 2 and n = 2). The third equation is a redundant equation. It would probably be best to pick the two equations in which the coefficients were known with the greatest precision. 

It is clear from the analysis above that sloppy modeling of a process may lead to a model that does not include all the relevant equations and variables or includes redundant equations and variables. In either case we have an erroneous determination of the degrees of freedom, which may imply incorrectly that we have an infinite number of solutions or no solution at all. 

In the above equations, Ff, Fp and Fj. are molar flow rates of feed, permeate and retentate streams, respectively y f, y p and y are mole fractions of the component in the feed, permeate and retentate streams, respectively Pf and Pp are feed-side and permeate-side pressures, respectively A , is the membrane area and Q, is the permeance (= permeabil-ity/effective membrane thickness) of the i component through the membrane. Note that balance for component n is not included in Equation 4.2 to avoid redundant equations with Equation 4.1. 

Based on their need for reliability, the customers may request in the product specification that the system has a strictly defined high confidence level (e.g., 98 %). The predetermined level of confidence impacts the choice of product s components and selection between different available component grades, balancing cost with quality. The component grade selection is also dependent on how the system is designed, i.e., how many units of a specific component are used and if there is any component redundancy. Equations that are used to calculate the reliability factors are as follows ... 

Parallel paths in design problems arise from the existence of the degrees of freedom in design variables and redundant equations. Steward (l965) and Himmelblau (1967) studied the assignment of admissible output sets for systems of algebraic equations which contained no design... 

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