Page 162 - Separation process principles 2

1 4.4 Multicomponent Flash, Bubble-Point, and Dew-Point Calculations 127

Table 4.4 Rachford-Rice Procedure for Isothermal-Flash
-
calculations When K-Values Are Independent of Composition
specified variables: F, TF, PF, ZI, z2, . . . , ZC, TV, PV
Steps
(1) TL = TV
(2) PL = Pv
(3) Solve

for = VJF, where Ki = Ki{Tv, Pv].
(4) V = F'4'

Figure 4.11 Rachford-Rice function for Example 4.1.

(7)L=F-V
(8)Q=hvv+h~L-h.~F
root for iteration k + 1 is computed from the recursive
relation

The computational procedure, referred to as the isothermal- *(kt" = q(k) f f( qck)}
flash calculation, is not straightforward because Eq. (4) in f '{9(k)} (4- 10)
Table 4.3 is a nonlinear equation in the unknowns V, L, yi,
where the superscript is the iteration index, and the deriva-
and xi, M~~ solution strategies have been developed, but
tive of f (91, from Eq. (3) in Table 4.4, with respect to Qf is
the generally preferred procedure, as given in Table 4.4,
is that of Rachford and Rice [5] when K-values are f '(~(~'1 x ~r(l - Ki)'
c
I independent (or nearly independent) of equilibrium-phase = i=l [I + q(k)(Ki - I)]' (4-1 1)
compositions.
Equations containing only a single unknown are solved
The iteration can be initiated by assuming 9(') = 0.5. Suffi-
first. Thus, Eqs. (1) and (2) in Table 4.3 are solved, respec-
cient accuracy will be achieved by terminating the iterations
tively, for PL and TL. The unknown Q appears only in
when (Q(~+') - q(k)l/9fk) < 0.0001.
Eq. (6), so Q is computed only after all other equations have
One should check the existence of a valid root
been solved. This leaves Eqs. (3), (4), (5), and (7) in
(0 5 9 5 I), before employing the procedure of Table 4.4,
Table 4.3 to be solved for V, L, and all values of y and x.
by checking to see if the equilibrium condition corresponds
These equations can be partitioned so as to solve for the un-
to subcooled liquid or superheated vapor rather than partial
knowns in a sequential manner by substituting Eq. (5) into
vaporization or partial condensation. A first estimate of
Eq. (4) to eliminate L and combining the result with Eq. (3)
whether a multicomponent feed gives a two-phase equilib-
to obtain Eqs. (5) and (6) in Table 4.4. Here (5) is in xi, but
rium mixture when flashed at a given temperature and pres-
not yi, and (6) is in yi but not xi. Summing these two equa-
sure can be made by inspecting the K-values. If all K-values
tions and combining them with C yi - C xi = 0 to elimi-
are greater than 1, the exit phase is superheated vapor above
nate yi and xi gives Eq. (3) in Table 4.4; a nonlinear equation
the dew point. If all K-values are less than 1, the single exit
in V (or 9 = V/F) only. Upon solving this equation numer-
phase is a subcooled liquid below the bubble point. If one or
ically in an iterative manner for 9 and then V, from Eq. (4)
more K-values are greater than 1 and one or more K-values
of Table 4.4, one can obtain the remaining unknowns di-
are less than 1, the check is made as follows. First, f {9] is
rectly from Eqs. (5) through (8) in Table 4.4. When TF and/or
computed from Eq. (3) for = 0. If the resulting f (0) > 0,
PF are not specified, Eq. (6) of Table 4.3 is not solved for Q.
the mixture is below its bubble point (subcooled liquid).
By this isothermal-flash procedure, the equilibrium-phase
Alternatively, if f {1} < 0, the mixture is above the dew
condition of a mixture at a known temperature (Tv = TL)
point (superheated vapor).
and pressure ( Pv = PL) is determined.
Equation (3) of Table 4.4 can be solved iteratively by
guessing values of \I, between 0 and 1 until the function
f (9) = 0. A typical form of the function, as will be com-
puted in ~xa$le 4.1, is shown in Figure 4.11. The most A 100-hofi feed consisting of 10, 20, 30, and 40 mol% of I
widely employed numerical method for solving Eq. (3) of propane (3), n-butane (4), n-pentane (3, and n-hexane (6), respec-
Table 4.4 is Newton's method [6]. A predicted value of the \I, tively, enters a distillation column at 100 psia (689.5 kPa) and