# 7.29: Untitled Page 174

## 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