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and produce a calculated set of values for stream 6. The values of stream 6 guessed
and calculated are compared, and if they are not equal within a set tolerance, the cal-
culated values of stream 6 are used as the set of values for the next iteration in the
calculations. The method of direct iteration may require many iterations to converge
and does not guarantee a solution. If a particularly difficult process is encountered, it is
sometimes beneficial to select the convergence option, direct, and limit calculations to
one or two iterations. The maximum number of iterations can be selected under Con-
vOptions/Methods under the tab corresponding to the convergence method, and chosen
as Maximum flowsheet evaluations. This frequently helps in analysis and debugging
of models.
The choice of stream 6 as a recycle stream is intuitive, but it is not the only approach.
If one were to choose stream 2 as a “forward” recycle, one could still complete the
calculations by using the sequence Flash, Split, and React to solve the process. Any
stream selected as a “forward or backward recycle” is termed a
tear stream
.Aspen
Plus has a facility to enable the user to select tear streams and to choose the sequence
of calculations. These options can be found under Convergence, Tear, Convergence,
ConvOrder, and Convergence, Sequence. Whether Aspen Plus or the user selects the
tear streams, it is necessary to initiate the calculations with a set of starting values.
Aspen Plus provides zeros, which in many situations will lead to a converged solution.
If it does not, the user must provide reasonable estimates by entering values on the
stream input form. Prior to rerunning the model, the calculations should be reinitialized
by selecting the Run menu and clicking on Reinitialize.
As an illustration
of the method of direct iteration,
consider solution of
the
equation
f(x)
=
x
2
−
c
(4.1)
Add the quantity
x
to both sides of the equation to form the function
F(x)
,
F(x)
=
x
2
−
c
+
x
=
f(x)
+
x
(4.2)
and look for a solution where
F(x)
=
x
(4.3)
which occurs when
f(x)
=
0. Repeated iterations yield the solution desired. The
derivative
F
(x)
can be evaluated, numerically, from two successive iterations. The
process converges under the conditions
0
<F
(x) <
1
(4.4)
−
1
<F
(x) <
0
(4.5)
Aspen Plus provides several other convergence options which may be selected by
opening Data, Convergence, ConvOptions, Defaults, and the tab Default Methods.
The default method (Wegstein, 1958) is usually suitable for most applications. Weg-
stein's method provides substantial robustness by making use of the first derivative
