Page 153 - Separation process principles 2

118 Chapter 4 Single Equilibrium Stages and Flash Calculations

Regardless of whether only intensive variables or both
intensive and extensive variables are considered, only a few
of the variables are independent; when these are specified,
all other variables become fixed. The number of independent
variables is referred to as the variance or the number of
degrees of freedom, F, for the system.
L
The phase rule of J. Willard Gibbs, which applies only to
the intensive variables at equilibrium, is used to determine F. lndependent equations: lndependent equations:
Same as for (a) plus
The rule states that
Fz,=Vy,+Lx, i=ItoC
FhF + Q = Vh, + Lh,
where C is the number of components and 9 is the number
of phases at equilibrium. Equation (4-1) is derived by count-
ing, at physical equilibrium, the number of intensive vari-
ables and the number of independent equations that relate
these variables. The number of intensive variables, "V, is
(a) (b)
Figure 4.1 Different treatments of degrees of freedom for vapor-
where the 2 refers to the equilibrium temperature and pres-
liquid phase equilibria: (a) Gibbs phase rule (considers equilibrium
sure, while the term C9 is the total number of composition intensive variables only); (b) general analysis (considers all intensive
variables (e.g., mole fractions) for components distributed and extensive variables).
among 9 equilibrium phases. The number of independent
equations, %, relating the intensive variables is
As an example, consider the vapor-liquid equilibrium
(9 = 2) shown in Figure 4.la, where the equilibrium inten-
where the first term, 9, refers to the requirement that mole
sive variables are labels on the sketch located above the list of
or mass fractions sum to one for each phase and the second independent equations relating these variables. Suppose there
term, C(9 - 1), refers to the number of independent K-value
are C = 3 components. From (4-l), F = 3 - 2 + 2 = 3. The
equations of the general form
equilibrium intensive variables are T, P, XI, x2, x3, yl, y2, and
mole fraction of i in phase (1) y3. If values are specified for T, P, and one of the mole frac-
Ki =
mole fraction of i in phase (2) tions, the remaining five mole fractions are fixed and can be
computed from the five independent equations listed in Fig-
where (1) and (2) refer to equilibrium phases. For two '
ure 4.la. Irrational specifications lead to infeasible results.
phases, there are C independent expressions of this type; for
For example, if the components are H20, Nz, and 02, and
three phases, 2C; for four phases, 3C; and so on. For exam- i
T = 100°F and P = 15 psia are specified, a specification of
ple, for three phases (V, L('), L(')), we can write 3C differ- i
XN~ = 0.90 is not feasible because nitrogen is not nearly this 3
ent K-value equations:
soluble in water. j
(1)
K, =yi/x!') i=ltoC In using the Gibbs phase rule, it should be noted that the
K, = yi/~12) i = 1 to c K-values are not variables, but are thermodynamic functions
(2)
1) (2) i=ltoC
KD, = xi /xi that depend on the intensive variables discussed in Chapter 2.
The Gibbs phase rule is limited because it does not deal
However, only 2C of these equations are independent, with feed streams sent to the equilibrium stage nor with ex-
because tensive variables used when designing or analyzing separa-
(2)
KD, = Ki /Ki (1) tion operations. However, the phase rule can be extended for
process applications, by adding the feed stream and exten-
Thus, the term for the number of independent K-value equa- sive variables, and additional independent equations relating
tions is C(9 - l), not C9. feed variables, extensive variables, and the intensive vari-
ables already considered by the rule.
Degrees-of-Freedom Analysis
Consider the single-stage, vapor-liquid (9 = 2) eciuilib-
The degrees of freedom is the number of intensive variables, rium separation process shown in Figure 4. lb. By compari-
"V, less the number of equations, 5%. Thus, from (4-2) and (4-3), son with Figure 4.la, the additional variables are zi, TF, PF,
F, Q, V, and L, or C + 6 variables, all of which are indicated
in the diagram. In general, for 9 phases, the additional
which completes the derivation of (4-1). When the number, variables number C + 9 + 4. The additional independent
.?, of intensive variables is specified, the remaining equations, listed below the 'diagram, are the C component
9 + C(9 - 1) intensive variables are determined from the material balances and the energy balance, or C + 1 equa-
9 + C(9 - 1) equations. tions. Note that, like K-values, stream enthalpies are not