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
i1
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