Page 57 - Numerical Methods for Chemical Engineering

46 1 Linear algebra

2 1
2 w
1
2
w 2
2
2 w
1 1 w
1
w 2
1 w 2 2
1
1 w 22
w
2
1
w 2

w 2
1
w
2
w
Figure 1.9 Process diagram for separation system.
1. the total mass flow rate
2 4 5 1
F + F + F = F (1.230)
2. the mass flow rate of species 1
2
5
5
1
4
1
2
4
( w 1 )( F ) + ( w 1 )( F ) + ( w 1 )( F ) = ( w 1 )( F ) (1.231)
3. the mass flow rate of species 2
1
1
4
5
5
2
2
4
( w 2 )( F ) + ( w 2 )( F ) + ( w 2 )( F ) = ( w 2 )( F ) (1.232)
This yields the set of three algebraic equations
x 1 + x 2 + x 3 = 10
(0.04)x 1 + (0.54)x 2 + (0.26)x 3 = 2 (1.233)
(0.93)x 1 + (0.24)x 2 + (0.0)x 3 = 6
Gaussian elimination yields
x 1 = 5.8238 x 2 = 2.4330 x 3 = 1.7433 (1.234)
sep system example.m performs this calculation. For further discussion of the formula-
tion of material and energy balances, and algorithms for their solution, consult Reklaitis
(1983).
Sparse and banded matrices

In the example above, the mathematical formulation of the problem was indeed a linear
system, and we could apply Gaussian elimination directly. Most mathematical problems,
however, are not expressed naturally as linear systems. Still, the availability for linear sys-
tems of rigorous existence and uniqueness conditions and an automated solution procedure