# 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 C1

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