# 7.34: Untitled Page 179

- Page ID
- 18312

## Chapter 7

( *M*

) ( *M*

)

CO2 5

N2 5

(31)

CY C

5

( *M*

)

*(* 1

*)*

Y

C H

3

Y

5

2 2 1

2

4

1 ( *M*

) CY *(* 1 C *)*

Y

Y

C

2H4 5

Substitution of this result into Eq. 30 leads to an equation for the molar flow rate of ethylene ( C H ) in Stream 35. This result can be expressed 2

4

in the compact form

( *M*

)

C H

5

(32)

2

4

0

1

( *M*

C CY

)

C

2H4 5

where the two parameters are given by

3 5 Y 1

3 5 Y 2 1

1

Y

2

2

*, *

(33)

CY

Y

Y

At this point we are ready to use a trial‐and‐error procedure to first solve for ( *M*

) and then solve for the parameter .

C2H4 5

Picard’s method

We begin by defining the dimensionless molar flow rate as

( *M*

)

C H

2

4 5

*x *

(34)

so that the governing equation takes the form

*x*

*H*( *x*)

(35)

C

0

1

CY *x*

In order to use Picard’s method (see Appendix B4), we define a new function according to

Definition:

*f *( *x*) *x * *H*( *x*)

(36)

and for any specific value of the dependent variable, *x *, we can *define* a *i*

new value, *x*

, by

*i*1

Definition:

*x *

*f x , *

*i *

*, , ,... , *

(37)

1

( )

1 2 3

*i*

*i*

*Material Balances for Complex Systems *

335

To be explicit, we note that the new value of the dependent variable is given by

*x*

*i*

*x*

*x*

*, *

*i*

*, , , ... , *

*i*

1

1 2 3

*i*

(38)

1C

CY *ix *

In terms of the parameters for this particular problem

37 *. * 6190 *, *

15 *. * 1678 *, *

C 0 *. * 7 *, *

Y 0 *. * 5

(39)

we have the following iterative scheme

15 *. * 1678 *x *

*x*

*x*

37 6190

*i*

*. *

*, *

*i*

1 *, * 2 *, * 3 *, ... , *

*i*

1

*i*

(40)

0 8571

*. *

*i*

*x *

By inspection, one can see that *x * 0 *. * 8571 thus we choose our first guess to be *x * 0 *. * 62 and this leads to the results shown in Table 7.8a.

o

Table 7.8a. Iterative Values for Dimensionless Flow Rate

(Picard’s Method)

*i *

*xi *

*xi*+1

0

0.6200

– 1.4237

1

– 1.4237

45.6633

2

45.6633

98.7402

3

98.7402

151.6598

4

151.6598

204.5328

5

204.5328

257.3835

6

257.3835

…..

7

…..

…..

Clearly Picard’s method *does not converge* for this case and we move on to Wegstein’s method (see Appendix B5).

Wegstein’s method

In this case we replace Eq. 37 with

Definition:

*i*

*x *

1

1 *q* *f*( *x *) *qx , *

*i * 1 *, * 2 *, * 3 *, ... , *

(41)

*i*

*i*

336