Reconciliation of process flow rates by matrix projection. Part II: The nonlinear case

Abstract

Flow rate and concentration measurements in a steady state process are reconciled by weighted least squares so that the conservation laws and other constraints are obeyed. Two projection matrices are constructed in turn, in order to decompose the problem into three subproblems to be solved in sequence. The first matrix eliminates all unmeasured component flow rates and concentrations from the equations; the second then removes the unmeasured total flow rates. The adjustments to component flow rates are iteratively determined, starting with guessed values of unmeasured total flow rates.Chi-square and normal test statistics are derived by linearizing the equations, to allow detection of gross errors in imbalances and adjustments of measurements. C. M. CROWE Department of Chemical EngineeringMcMaster University Hamilton, Ontario, Canada L8S 4L7 SCOPEThe cornerstone for monitoringplant performance is a set of steady state balances for component and total flow rates. Such flow rates are normally obtained from measurements of total flow rates and of concentrations, which are subject to random and sometimes gross errors, and thus in general violate conservation laws. The measurements should be reconciled, in some "best" sense, to obey those laws and any other constraints that are required to be enforced.The linear case, where it is assumed that the total flow rate is measured in any stream in which a concentration is measured, was discussed by Crowe et al., (1983). In this paper, that assumption is omitted so that the balance equations contain products of unknowns and thus are nonlinear (actually bilinear).Conflict with conservation laws can only arise in sections of the process where measurements have been made in all streams in a balance equation. Thus previous work by Vaclavek (1969), Mah et al. (1976), and Romagnoli and Stephanopoulos (198 1) was directed to devising path-tracing or combinatorial algorithms for efficiently finding a maximal set of sections of the process with redundant measurements -the reduced balance scheme (RBS). Unfortunately, such searches must be done separately for each component, or each element in the case of reactions. Crowe et al. (1983) proposed a constructive method of finding the RBS by a projection matrix that effectively blanked out all unmeasured flows in the linear case.The approach here is to extend the technique of Crowe et al. to the nonlinear case by constructing two successive projection matrices. The first eliminates all unmeasured component flow rates and concentrations; the second then eliminates the unmeasured total flow rates from the balance equations. The original problem is thus divided into three sequentially solved subproblems. Guesses of the unmeasured total flow rates are used to solve for the adjustments to component flow rates. These guesses are iteratively updated in the second subproblem until convergence is achieved. Then those unmeasured component flow rates that are determinable are found in the third subproblem.Gross errors in the measurements can arise fro...

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