Indeterminable variables

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 most cases, the structural procedure is able to determine whether the measurements can be corrected and whether they enable the computation of all of the state variables of the process. In some configurations this technique, used alone, fails in the detection of indeterminable variables. This situation arises when the Jacobian matrix used for the resolution is singular. 

Equations that contain unmeasured indeterminable variables (NA3)... 

M. U measured or unmeasured variable d, i determinable or indeterminable variable... 

The constant multipliers of the indeterminate variable in a polynomial. For example, in the polynomial x2+3x+7, the coefficients are 1, 3, and 7. common denominator... 

The second class, indeterminate or random errors, is brought about by the effects of uncontrolled variables. Truly random errors are as likely to cause high as low results, and a small random error is much more probable than a large one. By making the observation coarse enough, random errors would cease to exist. Every observation would give the same result, but the result would be less precise than the average of a number of finer observations with random scatter. 

To evaluate the effect of indeterminate error on the data in Table 4.1, ten replicate determinations of the mass of a single penny were made, with results shown in Table 4.7. The standard deviation for the data in Table 4.1 is 0.051, and it is 0.0024 for the data in Table 4.7. The significantly better precision when determining the mass of a single penny suggests that the precision of this analysis is not limited by the balance used to measure mass, but is due to a significant variability in the masses of individual pennies. 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

Let s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in determining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For convenience, entries in the table are represented by the symbol where i identifies the analyst and j indicates the replicate number thus 3 5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors introduced by the analysts. 

Iseler, G. W. et al., Int. Conf. Indium Phosphide Relat. Mater., 1992, 266 Reaction of beryllium, copper, manganese, thorium or zirconium is incandescent when heated with phosphorus [1] and that of cerium, lanthanum, neodymium and praseodymium is violent above 400°C [2], Osmium incandesces in phosphorus vapour, and platinum bums vividly below red-heat [3], Red phosphorus shows very variable vapour pressure between batches (not surprising, it is an indeterminate material). This leads to explosions when preparing indium phosphide by reactions involving fusion with phosphorus in a sealed tube [4],... 

The additional potential required to cause some electrode reactions to proceed at an appreciable rate. The result of an energy barrier to the electrode reaction concerned, it is substantial for gas evolution and for electrodes made of soft metals, e.g. Hg, Pb, Sn and Zn. It increases with current density and decreases with increasing temperature, but its magnitude is variable and indeterminate. It is negligible for the deposition of metals and for changes in oxidation state. 

The random or indeterminate error associated with a measurement or result. Sometimes called the variability, it can be represented statistically by the standard deviation or relative standard deviation (coefficient of variation) (p. 14). 

Steady-state process variables are related by mass and energy conservation laws. Although, for reasons of cost, convenience, or technical feasibility, not every variable is measured, some of them can be estimated using other measurements through balance calculations. Unmeasured variable estimation depends on the structure of the process flowsheet and on the instrument placement. Typically, there is an incomplete set of instruments thus, unmeasured variables are divided into determinable or estimable and indeterminable or inestimable. An unmeasured variable is determinable, or estimable, if its value can be calculated using measurements. Measurements are classified into redundant and nonredundant. A measurement is redundant if it remains determinable when the observation is deleted. 

The unmeasured process variables can be classified into determinable and indeterminable (Fig. 1). 

An unmeasured variable, belonging to the subset u, is indeterminable if it cannot be evaluated from the available measurements using the balance equations. 

Equation 3 cannot be assigned, because it contains two unmeasured process variables. Since /3 and can be calculated from the available information, they are unmeasured but determinable variables. On the other hand, /6 and /7 cannot be calculated from the available information thus, they are indeterminable. 

When the balance equations are formulated around individual units only, it is possible that the classification by output set assignment may not be satisfactory. Some variables classified as indeterminable may actually be determinable if we consider additional balances around groups of units. An erroneous measurement classification is also possible. The problem is in the system of equations used in the classification rather than in the assignment method. The most common problem arises because of the presence of parallel streams between two units. 

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 subset fu rf corresponds to the indeterminable total flowrates. Regarding the subset fyrf, nothing can be said, since some of these variables can be calculated directly from the measurements and some depend on fund Jf, as is explained in Remark 3. 

In order to estimate the unmeasured variables contained in v, the matrix B22 is divided into two parts by column permutation. The first fe columns correspond to the determinable total flowrates, and the (nd —fe) remaining ones belong to indeterminable total flowrates ... 

Unmeasured temperatures or concentrations that correspond to enthalpy or component flows in Vd are determinable if the total flow rate of the stream is measured. Otherwise, they are indeterminable. Measured total flow rates are nonredundant and unmeasured total flow rates are indeterminable. The analysis of intensive constraints between variables may change previous classification. 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

The rank of A2 is equal to 7 therefore, at least 3 unmeasured process variables are indeterminable. 

As the name suggests, indeterminate errors cannot be pin-pointed to any specific well-defined reasons. They are usually manifested due to the minute variations which take place inadvertently in several successive measurements performed by the same analyst, using utmost care, under almost identical experimental parameters. These errors are mostly random in nature and ultimately give rise to high as well as low results with equal probability. They can neither be corrected nor eliminated, and therefore, form the ultimate limitation on the specific measurements. It has been observed that by performing repeated measurement of the same variable, the subsequent statistical treatment of the results would have a positive impact of reducing their importance to a considerable extent. 

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