# 4.21: Untitled Page 76

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## 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:

134

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)

135