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and modifying the next trial value of x 2 as follows:
1
1 F (x 1 )
T =
(4.6)
x 2 = ( 1 T)x 1 + TF(x 1 )
(4.7)
The direct iteration and Wegstein's methods treat each of the state elements of a
stream as if it were independent of the others, and hence for a single, n -component tear
stream there are n + 1 equations in either method. Some of these ideas are illustrated
in Figure 4.3, due to Rubin (1964), which has been modified minimally to facilitate
the use of Aspen Plus's blocks. Using the traditional approach, the flowsheet appears
to have five tear streams, 1, 3, 4, 6, and 7, using the calculation sequence Mix, B2, B3,
B4, SP1, B5, and SP2. Selection of these streams as tears is illustrated as follows. The
selection Convergence under the setup drop-down menu opens the object manager, and
when New is selected, Figure 4.4 appears. It is then possible to enter the desired tear
streams. Aspen Plus detects whether the sequence is feasible and issues error messages
if it is not.
Through visual trial and error the reader should be able to discover the sequence B3,
B4, SP1, B5, SP2, Mix, and B2 with tear streams 2 and 5, thus reducing the number
of tear streams from five to two and potentially reducing the calculations considerably.
As part of Rubin's paper he developed an algorithm that can produce the minimum
number of tears required for an arbitrary flowsheet. Such an algorithm is a part of Aspen
Plus, and when this problem was solved with Aspen Plus, the optimal sequence was
identified. This problem is available for study at Chapter Four Examples/Rubin.bkp.
The well-known Newton method is an alternative method for solving process flow-
sheets. Details of the method are given in Section 6.5. A significant difference between
Newton and Wegstein is the formulation of equations that are to be solved. Each
equation still has the form of equation (4.2), except that with Newton's method each
of the 2 n c + 1 equations is a function of the 2 n c + 1 unknowns associated with that tear
stream. For example, if there are n t tear streams, n t ( 2 n c + 1 ) unknowns and equations
are to be solved. The Newton method requires partial derivatives of each equation with
respect to each variable for each iteration. To create these numerically, after Aspen Plus
13
4
3
SP2
3A
SP1
1
2
12
10
8
B4
B3
B5
MIX
B2
11
5
9
6
7
Figure 4.3
Rubin's flowsheet modified.
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