Reconciliation of process flow rates by matrix projection. Part I: Linear case

Abstract

Flow rate measurements in a steady-state process are reconciled by weighted least squares so that the conservation laws are obeyed. A projection matrix is constructed which can be used to decompose the linear problem into the solution of two subproblems, by first removing each balance around process units with an unmeasured component flow rate. The remaining measured flow rates are reconciled, and the unmeasured flow rates can then be obtained from the solution of the conservation equations. The basic case contains constraints which are linear in the component and the total flow rates. The method is extended to cases with bilinear constraints, involving unknown parameters such as split fractions.Chi-square and normal statistics are used to test for overall gross measurement errors, for gross error in each node imbalance which is fully measured, and for each measurement adjustment. C. M. CROWE SCOPETo monitor the performance of a chemical process, we require balanced component and total flow rates in the process streams. These flows can be calculated from judiciously chosen measurements of concentrations, temperatures and total flow rates; but since these measurements are subject to random error, the conservation laws will in general be violated.The basic case considered here is linear in that it is assumed that whenever a concentration or temperature in a stream is measured, so is the total flow rate. Then the component or enthalpy flow can be calculated and used as the raw measurement data. These data are adjusted (reconciled), and the unmeasured flow rates are estimated so that the weighted sum of squares of the adjustments is a minimum and the conservation laws are obeyed. This restriction is then relaxed to allow the inclusion of bilinear constraints, which contain unknown parameters, multiplied by measured quantities.The computational effort can be minimized if a reduced set of balance equations can be obtained from the original balances, such that the reduced set involves no unmeasured flow rate but a maximum number of measured flow rates. This was accomplished originally by Vaclavek (1969b), and later by Mah et al. (1976) and Romagnoli and Stephanopoulos (1981) by algorithms which iteratively eliminate balances involving unmeasured feed or product flow rates and merge two balances with a common unmeasured flow rate. These workers assumed that a stream was either unmeasured or completely measured, an assumption not made here.The approach here is to define a projection matrix which can be directly constructed and which effectively blanks out the unmeasured quantities in producing this reduced set of balances.

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