Redundant equation system

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. 

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... 

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... 

For n > 3, this equation system will be a redundant system. The coefficients (ao, 01,02) will now be determined so that the sum of squared deviations SSD according to eqn. (b) assumes its minimum value. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

Amster and Hooper, 1986). Other formulations exist for eomponents in parallel with equal reliability values, as shown in equation 4.69, and for eombinations of series, parallel and redundant eomponents in a system (Smith, 1997). The eomplexity of the equations to find the system reliability further inereases with redundaney of eomponents in the system and the number of parallel paths (Burns, 1994) ... 

KDC has a cause and effect relationship between as the primary cause leading to secondary failures. Besides its drastic operational effects on redundant systems, the numerical etlects that reduce sy.stem reliability are pronounced Equation 2.4-5 shows that the probability ut failing a redundant. system composed of n components is the component probability raised to the n-th power. If a common clement couples the subsystems. Equation 2.4-5 is not correct and the failure rate is the failure rate of the common element. KDC is very serious because the time from primary failure to secondary failures may be too short to mitigate. The PSA Procedures Guide (NUREG,/CR-2.3(X)) cl.issities this type as "Type 2."... 

A is an m X n matrix whose (/, j) element is the constraint coefficient aij9 and c, b, 1, u are vectors whose components are cjy bjt ljy ujy respectively. If any of the Equations (7.7) were redundant, that is, linear combinations of the others, they could be deleted without changing any solutions of the system. If there is no solution, or if there is only one solution for Equation (7.7), there can be no optimization. Thus the... 

Typically, process data are improved using spatial, or functional, redundancies in the process model. Measurements are spatially redundant if more than enough data exist to completely define the process model at any instant, that is, the system is overdetermined and requires a solution by least squares fitting. Similarly, data improvement can be performed using temporal redundancies. Measurements are temporally redundant if past measurement values are available and can be used for estimation purposes. Dynamic models composed of algebraic and differential equations provide both spatial and temporal redundancy. 

Let us consider the system of g overmeasured (redundant) variables in m balance equations. Assuming that all of the errors are normally distributed with zero mean and variance I>, it has been shown that the least squares estimate of the measurement errors is given by the solution of the following problem ... 

In the same context as the heat of formation, the JANAF tables have tabulated most conveniently the equilibrium constants of formation for practically every substance of concern in combustion systems. The equilibrium constant of formation (KPt[) is based on the equilibrium equation of formation of a species from its elements in their normal states. Thus by algebraic manipulation it is possible to determine the equilibrium constant of any reaction. In flame temperature calculations, by dealing only with equilibrium constants of formation, there is no chance of choosing a redundant set of equilibrium reactions. Of course, the equilibrium constant of formation for elements in their normal state is one. 

Problem Formulation. The conditions of equilibrium require the equivalence in each phase of temperature, pressure, and chemical potential for each component that is transferable between the phases and are subject to constraints of stoichiometry. A statement of the equivalence of chemical potential is identical to equations 8 and 9. An example is the AJB C D quaternary system. This system contains four binary compounds, AC, BC, AD, and BD, and the conditions of equilibrium allow three equations (of the type given by equation 8) to be written. The fourth possible equation is redundant as a result of the stoichiometric constraint (i.e., equal number of atoms on each sublattice). 

Both the Parameter and Reconcile cases determine (calculate) the same set of parameters. However, these cases do not get the same values for each parameter. A Parameter case has an equal number of unknowns and equations, therefore is considered "square" in mathematical jargon. In the Parameter case, there is no objective function that drives or affects the solution. There are typically the same measurements, and typically many redundant measurements in both the Parameter and Reconcile case. In the Parameter case we determine, by engineering analysis beforehand (before commissioning an online system for instance) by looking at numerous data sets, which measurements are most reliable (consistent and accurate). We "believe" these, that is, we force the model and measurements to be exactly the same at the solution. Some of these measurements may have final control elements (valves) associated with them and others do not. The former are of FIC, TIC, PIC, AIC type whereas the latter are of FI, TI, PI, AI type. How is any model value forced to be exactly equal to the measured value The "offset" between plant and model value is forced to be zero. For normally independent variables such as plant feed rate, tower... 

Another application of the analysis of the stoichiometric matrix is flux balance analysis (Edwards et al. 2002). Often the number of fluxes in the system exceeds the number variable metabolites making equation (3) an underdetermined set of linear equations, that is, many different combinations of fluxes are consistent with system steady state. One approach is to measure the fluxes that enter and exit the cell. Because intracellularly there are many redundant pathways, this does not enable one to determine all fluxes. Isotope labelling may help then (Wiechert 2002). Another approach to then find a smaller number of solutions is to postulate that the solution should satisfy an additional objective. This objective is taken to be associated with optimal functioning of the network, for instance maximization of some flux or combination... 

The central concept involved in coupling is the identification of components, which are the things that are conserved in a reaction system. When chemical reactions are studied, atoms of elements are conserved, but some of these conservation equations may not be independent. Redundant conservation equations are not counted as components C. When the pH is specified, the conservation equation for hydrogen atoms is omitted, and so the number of components for a given system is reduced by one C = C - 1. A test of the conservation matrix A is that the equation A v = 0 must yield a suitable basis for the stoichiometric number matrix v. When it is necessary to recognize that oxygen atoms are available from h2o. A must be used, and C " = C - 1. A test of the conservation matrix A" is that the equation A V " = 0 must yield a suitable basis for the stoichiometric number matrix v. ... 

These are the only independent equations of material balance that can be written down for the ternary system. All the terms in the left-hand side of Eq. 5.6.4 are functions of time. Therefore, we have three equations in two unknowns M and M/, and this redundancy provides for a statistical check of the measurements and allows best values to be calculated. 

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