Bilinear Multicomponent and Energy Balances

In the following sections, the use of the Q-R decomposition approach is discussed within the framework of the general multicomponent and energy (biUnear) reconciliation problem. In this case the classification of the measured and unmeasured process variables involves a sequence of steps. 

Component mass and energy balances and normalization equations are first rewritten using the method proposed by Crowe (1986) for the bilinear terms. Streams are divided into three categories depending on the combination of total flowrates (f), concentration (M), and temperature (t) measurements as shown in Table 1. 

The measured variable d is replaced by a consistent measured value plus the correction term Cd) 

In order to display the unmeasured total flow rates of a stream with specific flowrates of category 2, B4 and Ee matrices are defined as 

In order to group all unmeasured total flowrates, zero columns are added to B4 and Eg if necessary. New matrices B5 and E7 are obtained such that  

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A similar approach was undertaken by Mah et al. (1976) in their attempt to organize the analysis of process data and to systematize the estimation and measurement correction problem. In this work, a simple graph-theoretic procedure for single component flow networks was developed. They then extended their treatment to multicomponent flow networks (Kretsovalis and Mah, 1987), and to generalized process networks, including bilinear energy balances and chemical reactions (Kretsovalis and Mah, 1988a,b). 

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