Page 60 - Chemical engineering design

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FUNDAMENTALS OF MATERIAL BALANCES
Algebraic symbols are assigned to all the unknown flows and compositions. Balance
equations are then written around each sub-system for the independent components
(chemical species or elements).
Material-balance problems are particular examples of the general design problem
discussed in Chapter 1. The unknowns are compositions or flows, and the relating
equations arise from the conservation law and the stoichiometry of the reactions. For
any problem to have a unique solution it must be possible to write the same number of
independent equations as there are unknowns.
Consider the general material balance problem where there are N s streams each
containing N c independent components. Then the number of variables, N v , is given by:
N v D N c ð N s 2.3
If N e independent balance equations can be written, then the number of variables, N d ,
that must be specified for a unique solution, is given by:
N d D N s ð N c N e 2.4
Consider a simple mixing problem

1
2 Mixer 4
3

Let F n be the total flow in stream n,and x n,m the concentration of component m in
stream n. Then the general balance equation can be written

F 1 x 1,m C F 2 x 2,m C F 3 x 3,m D F 4 x 4,m 2.5
A balance equation can also be written for the total of each stream:
F 1 C F 2 C F 3 D F 4 2.6
but this could be obtained by adding the individual component equations, and so is not
an additional independent equation. There are m independent equations, the number of
independent components.
Consider a separation unit, such as a distillation column, which divides a process stream
into two product streams. Let the feed rate be 10,000 kg/h; composition benzene 60 per
cent, toluene 30 per cent, xylene 10 per cent.

Overhead
product

Feed

System
boundary
Bottom
product