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COMPUTER-AIDED DESIGN 131

resolved in the same manner as in the spread-sheet example; namely, calculate
iteratively until the change in the calculated values is acceptable.
The identification of recycle loops within a process is termed partitioning.
This is done by the software of most flow-sheeting programs; however, in
FLOWTRAN these loops must be identified by the user. Consequently, a
procedure for performing this task is described here. Recycle loops usually can
be identified fairly easily from a process flow diagram. If one can trace a closed
path from an output of a process unit back to the input of that unit, then there
is a recycle. In processes with many units it can be difficult to identify all of the
independent recycle loops by tracing the flow diagram, but it can be done by
hand in the same manner as it is done by some flow-sheeting software. In this
procedure, start with an external feed stream to a process unit and follow one
outlet from that unit. List each process unit encountered until either (a) an
already listed process unit is encountered again or (b) the only remaining output
stream of a unit is not a feed to any process unit. In the first case, a recycle loop
has been identified; all the units in that loop are combined into a single block
(they must be solved together iteratively) and that block is subsequently treated
as is any other single unit and the procedure is continued. In the second case
the identified unit is transferred from the process list to the beginning of a
calculation list. This procedure is continued until all units have been shifted to
the calculation list; this list, from the top down, indicates the order of calcula-
tion. Each block of units obtained in the first step is a recycle loop. This
procedure will be illustrated in Example 4. The final result is one calculation
order among many possible orders. There are algorithms that provide a calcula-
tion order with particular characteristics, such as the fewest possible tears of
recycle loops. Such algorithms are intended to find an efficient calculation
order, that is, one with fewer calculations or faster convergence. However, none
of them can guarantee to be the most effective. Since FLOWTRAN does not
employ such an algorithm, such algorithms will not be reyiewed here.
The iterative calculations of recycle are approached by “tearing” the
recycle loop, that is, by selecting a stream in the loop to be the trial or assumed
stream. Calculations are initiated by assuming values for the conditions of this
stream, and the calculations then continue around the loop back to this
assumed stream. The most recently calculated values for this stream are
compared with the previous values and this procedure is continued (i.e.,
iterated) until consecutive values agree within a preset margin. This method is
called direct or successive substitution. Various mathematical methods are used
to accelerate the convergence of the calculations by guiding the selection of
successive assumed values for the tear stream. These methods include those of
Wegstein and Broyden and modifications thereof.? Partitioning, tearing, and
solution by direct substitution without acceleration are illustrated in Example 4.

tA. W. Westerberg. H. P. Hutchison, R. L. Motard, and P. Winter, “Process Flowsheeting,”
Chapter 3, Cambridge University Press, Cambridge, England, 1979.