4.21: Untitled Page 76
- Page ID
- 18209
Chapter 4
Table 4‐6. Degrees‐of‐Freedom
Stream Variables
compositions
N x M = 12
flow rates
M = 4
Generic Degrees of Freedom (A)
( N x M) + M = 16
Number of Independent Balance Equations
mass/mole balance equations
N = 3
Number of Constraints for Compositions
M = 4
Generic Specifications and Constraints (B)
N + M = 7
Specified Stream Variables
compositions
8
flow rates
1
Constraints for Compositions
0
Auxiliary Constraints
0
Particular Specifications and Constraints (C)
9
Degrees of Freedom (A ‐ B ‐ C)
0
Control Volume II:
A:
(4‐106a)
B:
(4‐106b)
Total:
(4‐106c)
In order to eliminate
from Eq. 4‐106b, one multiplies Eq. 4‐106a by ( ) and
B 2
one subtracts the result from Eq. 4‐106b. To eliminate
from Eq. 4‐106c, one
need only subtract Eq. 4‐106a from Eq. 4‐106c. These two operations lead to the following balance equations:
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Control Volume II:
A:
(4‐107a)
B:
(4‐107b)
Total:
(4‐107c)
Clearly the algebra is becoming quite complex, and it will become worse when we use Eq. 4‐107b to eliminate
from Eq. 4‐107c. Without providing the
details, we continue the elimination process to obtain the solution to Eq. 4‐107c and this leads to the following expression for
:
(4‐108)
Equally complex expressions can be obtained for
and
, and the numerical
values for the three mass flow rates are given by
(4‐109)
In order to determine
, we must make use of the balance equations for Control
Volume I that are given by Eqs. 4‐103. These can be expressed in terms of two species balances and one total mass balance leading to
Control Volume I:
species A:
(4‐110a)
species B:
(4‐110b)
Total:
(4‐110c)
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