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4.16: Untitled Page 71

  • Page ID
    18204
  • Chapter 4

    along with the three mole fraction constraints that apply at the streams that are cut by the control volume illustrated in Figure 4‐9.

    Mole fraction constraints:

    (4‐90)

    We list these specifications and constraints as

    I. Balance equations for three molecular species

    3

    II. Mole fraction constraints for the three streams 3

    and this leads to the generic specifications and constraints given by

    Generic Specifications and Constraints (B)

    6

    Moving on to the third step in our degree of freedom analysis, we list the particular specifications and constraints according to I. Conditions for Stream #1:

     

    1

    M

    1200 mol/h , x

    0 3

    . , x

    0 2

    A

    B

    .

    3

    II. Conditions for Stream #2:

    2

    III. Conditions for Stream #3:

    1

    This leads us to the particular specifications and constraints indicated by

    Particular Specifications and Constraints (C) 6

    and we can see that there are zero degrees of freedom for this problem. We summarize our degree of freedom analysis in Table 4‐2 that provides a template for subsequent problems in which we have N molecular species and M streams.

    When developing the particular specifications and constraints, it is extremely important to understand that the three mole fractions can be specified only in the following manner:

    I. None of the mole fractions are specified in a particular stream.

    II. One of the mole fractions is specified in a particular stream.

    III. Two of the mole fractions are specified in a particular stream.

    The point here is that one cannot specify all three mole fractions in a particular stream because of the constraint on the mole fractions given by Eq. 4‐90. If one specifies all three mole fractions in a particular stream, Eq. 4‐90 for that stream must be deleted and the generic specifications and constraints are no longer generic.

    Multicomponent systems

    124

    Table 4‐2. Degrees‐of‐Freedom

    Stream Variables

    compositions

    N x M = 9

    flow rates

    M = 3

    Generic Degrees of Freedom (A)

    ( N x M) + M = 12

    Number of Independent Balance Equations

    mass/mole balance equations

    N = 3

    Number of Constraints for Compositions

    M = 3

    Generic Constraints (B)

    N + M = 6

    Specified Stream Variables

    compositions

    5

    flow rates

    1

    Constraints for Compositions

    0

    Auxiliary Constraints

    0

    Particular Specifications and Constraints (C)

    6

    Degrees of Freedom (A ‐ B ‐ C)

    0

    There are two important results associated with this degree of freedom analysis. First, we are certain that a solution exits, and this provides motivation for persevering when we encounter difficulties. Second, we are now familiar with the nature of this problem and this should help us to organize a procedure for the development of a solution.

    4.7.2 Solution of macroscopic balance equations

    Before beginning the solution procedure, we should clearly identify what is known and what is unknown, and we do this with an extended version of Table 4‐1 given here as

    125