Steady-State Lumped Systems

The study of stoichiometry and unit operations concerns itself mainly with the application of steady-state macroscopic balances to chemical process problems. When considering small subsystems of chemical plants, the number of describing relations are small and the development of a computational strategy is not difficult. Usually the relations can be solved directly by partitioning the equations, that is, solving each equation of the equation set for a single unknown variable in a sequential manner. As equation sets become coupled, namely, as each relation involves more of the unknown variables, the probability that an equation set can be partitioned decreases. When an equation set cannot be partitioned, the equations must be solved simultaneously or an iterative scheme devised. 

By a solution to a set of m simultaneous equations in n unknowns we mean those values of the unknowns, a i, X2 , Xn that satisfy 

In general, no solution, a number of solutions, or an infinite number of solutions may exist. We will be concerned with the case where m = n and we seek a single, real, physically meaningful solution. 

Systems of linear equations arise with great regularity in material balance problems. An example of a two-dimensional linear system is 

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