Page 238 - Separation process principles 2

6.3 Graphical Equilibrium-Stage Method for Trayed Towers 203

Note that the operating line can terminate at the equilibrium
(a) (b)
line, as for operating line 4, but cannot cross it because that
Figure 6.10 Vapor-liquid stream relationships: (a) operating line
would be a violation of the second law of thermodynamics.
(passing streams); (b) equilibrium curve (leaving streams).
The value of Lk, corresponds to a value of XN (leaving
the bottom of the tower) in equilibrium with YN+l, the solute
two stages to the solute concentration in the liquid passing
concentration in the feed gas. It takes an infinite number of
downward between the same two stages. Figure 6.10b illus-
stages for this equilibrium to be achieved. An expression for
trates that the equilibrium curve relates the solute concentra-
Lk, of an absorber can be derived from (6-7) as follows.
tion in the vapor leaving an equilibrium stage to the solute
For stage N, (6-1) becomes, for the minimum absorbent
concentration in the liquid leaving the same stage. This
rate,
makes it possible, in the case of an absorber, to start from
the top of the tower (at the bottom of the Y-X diagram) and
move to the bottom of the tower (at the top of the Y-X
k
Solving (6-8) for XN and substituting the result into (6-7) diagram) by constructing a staircase alternating between
gives the operating line and the equilibrium curve, as shown in
Figure 6.11a. The number of equilibrium stages required for
a particular absorbent flow rate corresponding to the slope
of the operating line, which in Figure 6.11a is for
i (Lf/ V') = 1.5(LAn/ V'), is stepped off by moving up the
For dilute-solute conditions, where Y x y and X w x, (6-9)
staircase, starting from the point (Yl, Xo), on the operating
approaches
line and moving horizontally to the right to the point (Y1, XI)
on the equilibrium curve. From there, a vertical move is
made to the point (Yz, X1) on the operating line. Proceeding
in this manner, the staircase is climbed until the terminal
point (YN+~, XN) on the operating line is reached. As shown
Furthermore, if the entering liquid contains no solute, that is,
in Figure 6.11a, the stages are counted at the points of the
Xo w 0, (6- 10) approaches staircase on the equilibrium curve. As the slope (L'l V') is
L',, = VfKN (fraction of solute absorbed) (6-1 1) increased, fewer equilibrium stages are required. As (L'l V')
is decreased, more stages are required until (LA,/ V') is
This equation is reasonable because it would be expected
reached, at which the operating line and equilibrium curve
that LLi, would increase with increasing V', K-value, and intersect at a so-called pinch point, for which an infinite
fraction of solute absorbed. number of stages is required. Operating line 4 in Figure 6.9
The selection of the actual operating absorbent flow rate
has a pinch point at YN+l, XN. If (Lf/ V') is reduced below
is based on some multiple of L;,, typically from 1.1 to 2. A (L',,/V1), the specified extent of absorption of the solute
value of 1.5 corresponds closely to the value of 1.4 for the cannot be achieved.
optimal absorption factor mentioned earlier. In Figure 6.9, The number of equilibrium stages required for stripping a
operating lines 2 and 3 correspond to 2.0 and 1.5 times LA,, solute is determined in a manner similar to that for absorp-
respectively. As the operating line moves from 1 to 4, the
tion. An illustration is shown in Figure 6.11 b, which refers to
number of required equilibrium stages, N, increases from Figure 6.8b. For given specifications of Yo, XN+1, and the ex-
zero to infinity. Thus, a trade-off exists between L' and N, tent of stripping of the solute, which corresponds to a value
and an optimal value of L' exists. of X1, VA, is determined from the slope of the operating line
A similar derivation of VA,, for the stripper of Figure 6.8, that passes through the points (Yo, XI), and (YN, XN+i) on the
results in an expression analogous to (6-1 1): equilibrium curve. The operating line in Figure 6.11b is for

L' V' = 1.5VAn or a slope of (L'l V') = (L'/VA,)/1.5.
Vkin = - (fraction of solute stripped) (6- 12) In Figure 6.11, the number of equilibrium stages for the
KN
absorber and stripper is exactly three each. These integer
results are coincidental. Ordinarily, the result is some frac-
Number of Equilibrium Stages
tion above an integer number of stages, as is the case in the
As shown in Figure 6.10a, the operating line relates the following example. In practice, the result is usually rounded
solute concentration in the vapor passing upward between to the next highest integer.