Page 197 - Chemical engineering design

174
CHEMICAL ENGINEERING
Consider the streams entering unit 1.
α λ λ 1k
13k 3k
1

g
10k α 12k λ 2k

Figure 4.8.

A material balance gives:
4.1
g 10k C ˛ 13k 3k C ˛ 12k 2k D 1k
A similar material balance can be written at the inlet to each unit:
4.2
unit 2: ˛ 21k 1k D 2k
4.3
unit 3: ˛ 32k 2k C g 30k C ˛ 31k 1k D 3k
Rearranging each equation
4.1a
1k ˛ 12k 2k ˛ 13k 3k D g 10k
˛ 21k 1k C 2k D 0 4.2b

˛ 31k 1k ˛ 32k 2k C 3k D g 30k 4.3c
This is simply a set of three simultaneous equations in the unknown flows 1k , 2k , 3k .
These equations are written in matrix form:
i
1 2 3
     
1 1 ˛ 12k ˛ 13k 1k g 10
j 2  ˛ 21k 1 0  ð  2k  D  0 
3 ˛ 31k ˛ 32k 1 3k g 30

There will be a set of such equations for each component.
This procedure for deriving the set of material balance equations is quite general. For
a process with n units there will be a set of n equations for each component.
The matrix form of the n equations will be as shown in Figure 4.9.

 .. .     
1 ˛ 11k ˛ 12k ˛ 13k ˛ 1nk 1k g 10k
 ˛ 21k 1 ˛ 22k ˛ 23k .. . ˛ 2nk   2k   g 20k 
     
  ð   D  
     
˛ n1k .. .. .. ... .. .. ... . . . 1 ˛ nnk nk g n0k
Figure 4.9. Matrix form of equations for n units