# 4.20: Untitled Page 75

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## Chapter 4

Control Volume II

Mass fractions:

( ) *, *( ) *, *( ) *, *

*i * 1 *, * 2 *, * 3 *, * 4

(4‐101)

*A i*

*B i*

*C i*

Mass flow rates:

*m*

*, *

*i * 1 *, * 2 *, * 3 *, * 4

(4‐102)

*i*

and we indicate the number of process variables explicitly as I. Three mole fractions in each of four streams 12

II. Four mass molar flow rates

4

For this process the generic degrees of freedom are given by

**Generic Degrees of Freedom**** (A)**

** 16 **

Moving on to the generic specifications and constraints, we list the *three* molecular species balances given by

**v ** **n ** *dA * 0

**v ** **n ** *dA * 0

*A*

*B*

Species balances:

*A*

*A*

(4‐103)

**v ** **n ** *dA * 0

*C*

*A*

along with the *four* mass fraction constraints that apply at the streams that are cut by Control Volume II.

Constraints:

( ) ( ) ( )

1 *, *

*i * 1 *, * 2 *, * 3 *, * 4

(4‐104)

*A i*

*B i*

*C i*

This leads us to the *second step* in our degree of freedom analysis that we express as

I. Balance equations for three molecular species

3

II. Mole fraction constraints for the four streams

4

**Generic Specifications and Constraints**** (B)**

**7 **

Our *third step* in the degree of freedom analysis requires that we list the *particular* *specifications* *and constraints* according to

132

I. Conditions for Stream #1:

*m*

1000 lb */ * h *, * 0 *. * 5 *, * 0 *. * 3

3

1

m

*A*

*B*

II. Conditions for Stream #2:

0 045

*. *

*, *

0 091

*A*

*B*

*. *

2

III. Conditions for Stream #3:

0 069

*. *

*, *

0 901

*A*

*B*

*. *

2

IV. Conditions for Stream #4:

0 955

*. *

*, *

0 041

*A*

*B*

*. *

2

This leads us to the particular specifications and constraints indicated by

**Particular Specifications and Constraints**** (C)** **9**

We summarize these results in Table 4‐6 which indicates that use of Control Volume II will lead to a well‐posed problem. Thus we can use Eqs. 4‐103 and 4-104 to determine the mass flow rates in streams 2, 3, 4 and this calculation is carried out in the following paragraphs.

Often in problems of this type, it is convenient to work with *N * 1 species mass balances and the total mass balance that is given by Eq. 4‐39. In terms of the use of Eqs. 4‐103 with Control Volume II, this approach leads to Control Volume II:

species *A*:

(4‐105a)

species *B*:

(4‐105b)

Total:

(4‐105c)

Here we are confronted with *three equations* and *three unknowns*, and our problem is quite similar to that encountered in Sec. 4.7 where our study of a single distillation column led to a set of *two equations* and *two unknowns*. That problem was solved by Gaussian elimination as indicated by Eqs. 4‐93 through 4‐97. The same procedure can be used with Eqs. 4‐105, and one begins by dividing Eq. 4‐105a by ( )

*A * 2 to obtain the results listed by Eqs. 4‐106.

133