Bilinear balances energy

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). 

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 set of component/energy balances and normalization equations after modification of bilinear terms can now be written as... 

Q-R factorization is successful in decomposing linear systems of equations. It is also satisfactory when bilinear systems contain component balances and normalization equations. If energy balances are included in the set of process constraints, the procedure has the drawback that only simple thermodynamic relations for the specific enthalpy of the stream can be considered. 

Orthogonal factorizations may be applied to resolve problem (5.3) if the system of equations cp(x, u) = 0 is made up of linear mass balances and bilinear component and energy balances. After replacing the bilinear terms of the original model by the corresponding mass and energy flows, a linear data reconciliation problem results. 

Remark 5 We can also treat the heat loads of each match as variables since they participate linearly in the energy balances. The penalty that we pay, however, is that the objective function no longer satisfies the property of convexity, and hence we will have two possible sources of nonconvexities the objective function and the bilinear equality constraints. 

In general practical situations the mass and energy balances do not yield always linear expressions like equation (2). For example, one device serves for the measurement of mass flow rate and separate analyzers are used to determine the compositions. Consequently the balance may result in bilinear forms as shown in equation (5). 

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