Maximum likelihood adjustment of errors in material balances



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Adjustment of measured flow rates is optimal if error is minimized and material balances are satisfied. Corrected flow rates can be calculated by finding a minimum in the weighted sum of squares of the corrections. This method provides the maximum likelihood estimate of the true flow rates. The objective function, derived from the assumption of normal distribution of measurement errors, can be minimized by the LaGrange method of undetermined multipliers. Over-all and component material balances are the equality constraints. Conclusions of this work are: (1) An estimate of true flow rates can be calculated that is better than the measurements. (2) More constraints improve the estimate. (3) Weighting can improve the estimate, but it is not necessary to know the weights to use the method effectively. A computer simulation measured error reduction. Random errors were applied to hypothetical true data to simulate measured flow rates, which were corrected by the error minimization program. The method reduced the initial error sum of squares in over 95 percent of realistic runs by mean amounts of 10 to 70 percent. Maximum likelihood estimators were derived for the transient case and the general non-linear case of significant composition error. A procedure for using the method and simulation in studies involving material balances is presented.