# 7.1: Introduction to Material Balances for Complex Systems

Most recent paradigm shifts in the mathematical analysis of physical systems are due to the use of computers. In Chapter 4 we encountered the application of matrices in the formulation of material balance problems, and for small matrices those problems could be solved easily. For large matrices solutions are difficult to obtain (see Sec. 4.8), and computer-aided calculations are necessary. In this chapter we consider the transition from simple and small systems to complex and large systems. We begin with some moderately complex processes involving reactors, separators and recycle streams. These systems can be analyzed without the use of computers. In Sec. 7.5 we introduce sequential analysis using iterative methods that require some programming. This sequential analysis forms the basis for process simulators that will be studied in a senior-level design course; however, it is absolutely essential to understand the details presented in this chapter prior to using process simulators for computer-aided design.

In Chapters 4 and 5 we studied multicomponent, multiphase systems without chemical reactions, and in Chapter 6 we learned how to analyze multiple, independent stoichiometric reactions in a general manner. We are now ready to study more complex systems with chemical reactions such as the one shown in Figure $$\PageIndex{1}$$. Here we have identified several distinct control volumes, and the choice of the control volumes that provides the most convenient analysis will be examined in this chapter.

In Sec. 4.7.1 we developed a degree-of-freedom analysis for systems with $$N$$ components, $$M$$ streams, and no chemical reactions. Here we extend that analysis to include chemical reactions in systems for which the governing equations are given by

Axiom I: $\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA=\int_{\mathscr{V}}R_{A} dV , \quad A = 1,2, ..., N \label{1}$

Axiom II $\sum_{A = 1}^{A = N}N_{JA} R_{A} =0 , \quad J = 1, 2, ..., T \label{2}$

When Axiom II is applied to control volumes, we will make use of the global rates of production defined by

$\mathscr{R}_{ A} =\int_{\mathscr{V}}R_{A} dV , \quad A = 1, 2, ..., N \label{3}$

and follow the development in Sec. 6.2.2 so that Equation \ref{2} takes the form

Axiom II $\sum_{A = 1}^{A = N}N_{JA} \mathscr{R}_{ A} =0 , \quad J = 1, 2, ..., T \label{4}$

In terms of the net global rate of production, Axiom I takes the form

Axiom I: $\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA= \mathscr{R}_{ A} , \quad A = 1,2, ..., N \label{5}$

These two results are applicable to any fixed control volume and we will use them throughout this chapter to determine molar flow rates, mass flow rates, mole fractions, etc. In addition to solving problems in terms of Equations \ref{4} and \ref{5}, one can use those equations to derive atomic species balances. This is done in Appendix D where we illustrate how to solve problems in terms of the $$T$$ atomic species rather than in terms of the $$N$$ molecular species.