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7.29: Untitled Page 174

  • Page ID
    18307
  • Chapter 7

    This representation can be generalized to obtain the (

    1) th

    i

    value that is

    given by

    ( i1)

    ( i)

    M

    1 C1

     M

    ,

    C  0 3

    . 0 ,

    i  1 , 2 , 3 ,..... (17)

    5

    5 

    This procedure is referred to as Picard’s method (Bradie, 2006), or as a fixed point iteration, or as the method of successive substitution that is described in Appendix B4. Picard’s method is often represented in the form

    x

    f ( x ) ,

    i

    1 , 2 , 3 , e

    ... tc

    (18)

    i

    1

    i

    To illustrate how this iterative calculation is carried out, we assume that (o)

    M

     0 to produce the values listed in Table 7.7a where we see a 5

    converged value given by M  2 333

    .

    . One can avoid these detailed

    5

    calculations by noting that for arbitrarily large values of i we arrive at the fixed point condition given by

    ( i1)

    ( i)

    M

     M

    ,

    i  

    (19)

    5

    5

    and the converged solution for the dimensionless molar flow rate is ()

    1  C

    M

     2 . 333

    (20)

    5

    C

    This indicates that the molar flow rate of dichloroethane ( C H Cl ) in 2

    4

    2

    the recycle stream is

    ( M

    )

     2 . 333 ( M

    )

    (21)

    C2H4Cl2 5

    C2H4Cl2 1

    In terms of the total molar flow rate in Stream #1, this takes the form ( M

    )

     2 . 2867 M

    (22)

    C2H4Cl2 5

    1

    which is exactly the answer obtained in Example 7.5.

    Material Balances for Complex Systems

    325

    Table 7.7a. Converging Values for Dimensionless Recycle Flow Rate (Picard’s Method)

    i

    ( i)

    ( i+1)

    M5

    M5

    0

    0.000

    0.700

    1

    0.700

    1.190

    2

    1.190

    1.533

    3

    1.533

    1.773

    4

    1.773

    1.941

    ….

    ….

    ….

    ….

    ….

    ….

    22

    2.332

    2.333

    23

    2.333

    2.333

    A variation on Picard’s method is called Wegstein’s method (Wegstein, 1958) and in terms of the nomenclature used in Eq. 18 this iterative procedure takes the form (see Appendix B5)

    x

    (1 q) f ( x )

    q x ,

    i

    1 , 2 , 3 , e

    ... tc

    (24)

    i

    1

    i

    i

    in which q is an adjustable parameter. When this adjustable parameter is equal to zero, q  0 , we obtain the original successive substitution scheme given by Eq. 18. When the adjustable parameter is greater than zero and less than one, 0  q  1, we obtain a damped successive substitution process that improves stability for nonlinear systems. When the adjustable parameter is negative, q  0 , we obtain an accelerated successive substitution that may lead to an unstable procedure. For the problem under consideration in this example, Wegstein’s method can be expressed as

    ( i1)

    ( i)

    ( i)

    M

    1 q1 C

    1  M

    q M

    ,

    C  0 3

    . 0

    (25)

    5

    5 

    5

    When the adjustable parameter is given by q   1 . 30 we obtain the accelerated convergence illustrated in Table 7.7b. When confronted with

    326