Cumulative Integral Balances

This is a special type of balance involving hme as a variable, which is rarely, if ever, singled out for discussion in textbooks. It consists of considering a 

Mass initially present (f = 0) = Mass consumed or lost over (0, f) 

Cumulative balances invariably lead to algebraic equations. Paradoxically and in spite of the occurrence of time in the statement (Equation 2.7), they may arise in both steady- and imsteady-state processes. We shall have occasion to demonstrate this in a number of illustrations and practice problems. 

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As is usual in systems of some complexity, a number of different balances can be made depending on the choice of balance space. We may choose, for example, to make an unsteady integral balance about the still, or to make a similar balance about the receiver. They can be instantaneous or cumulative in time and can involve total or component mass balances. 

Cumulative balances such as the two simple expressions (Equations 2.15a and Equation 2.15b) are often overlooked in modeling, or else written out without much thought to their origin. It is important to note that they are quite independent of the unsteady integral balances mentioned previously and consequently serve as additional tools that can be used to supplement the model equations. Typically, they are used as adjuncts to instantaneous balances in batch distillation. We can, for example, solve for either still or distillate composition and obtain... 

The (apparently simplest) method using the accumulation differences (11.1.8) as measured variables in the balance constraint (11.2.4) fails in practice. It has been shown also theoretically that even small systematic errors in the measured (integrated) mass flowrates cumulate and in the end, can lead to absurd values of the holdups (states of the inventories) computed (estimated) by (11.2.9). 

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